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+arXiv:2301.01212v1  [q-fin.RM]  31 Dec 2022
+Assessment of creditworthiness models privacy-preserving
+training with synthetic data
+Ricardo Mu˜noz-Cancino1, Cristi´an Bravo2, Sebasti´an A. R´ıos3, and Manuel Gra˜na4
+1,3Business Intelligence Research Center (CEINE), Industrial Engineering Department, University of
+Chile, Beauchef 851, Santiago 8370456, Chile
+2Department of Statistical and Actuarial Sciences, The University of Western Ontario,1151 Richmond
+Street, London, Ontario, N6A 5B7, Canada.
+4Computational Intelligence Group, University of Basque Country, 20018 San Sebasti´an, Spain.
+Abstract
+Credit scoring models are the primary instrument used by financial institutions to manage
+credit risk. The scarcity of research on behavioral scoring is due to the difficult data access.
+Financial institutions have to maintain the privacy and security of borrowers’ information
+refrain them from collaborating in research initiatives. In this work, we present a methodology
+that allows us to evaluate the performance of models trained with synthetic data when they
+are applied to real-world data. Our results show that synthetic data quality is increasingly
+poor when the number of attributes increases. However, creditworthiness assessment models
+trained with synthetic data show a reduction of 3% of AUC and 6% of KS when compared with
+models trained with real data. These results have a significant impact since they encourage
+credit risk investigation from synthetic data, making it possible to maintain borrowers’ privacy
+and to address problems that until now have been hampered by the availability of information.
+Keywords: credit scoring; synthetic data; generative adversarial networks; variational au-
+toencoders
+1
+Introduction
+For decades financial institutions have used mathematical models to determine borrowers’ credit-
+worthiness and consequently manage credit risk. The main objective of these models is to char-
+acterize each borrower with the probability of not complying with their contractual obligations
+The Basel Committee on Banking Supervision (2000), avoiding to give loans to applicants that
+will not be able to pay them back. Despite all the years of research on credit scoring, there is still
+little done on behavioral scoring models, which are the credit scoring models used on those clients
+who have already been granted a loan, because it requires large volumes of data and a relevant
+∗NOTICE: This is a preprint of a published work. Changes resulting from the publishing process, such as editing, corrections,
+structural formatting, and other quality control mechanisms may not be reflected in this version of the document. Please cite this work
+as follows: Mu˜noz-Cancino, R., Bravo, C., R´ıos, S.A., Gra˜na, M. (2022). Assessment of Creditworthiness Models Privacy-Preserving
+Training with Synthetic Data. In: , et al. Hybrid Artificial Intelligent Systems. HAIS 2022. Lecture Notes in Computer Science(), vol
+13469. Springer, Cham. https://doi.org/10.1007/978-3-031-15471-3 32
+∗E-mail
+addresses:
+rimunoz@uchile.cl
+(Ricardo
+Mu˜noz-Cancino),
+cbravoro@uwo.ca
+(Cristi´an
+Bravo),
+srios@dii.uchile.cl (Sebasti´an A. R´ıos), manuel.grana@ehu.es (Manuel Gra˜na)
+1
+
+historical depth Goh and Lee (2019); Kennedy et al. (2013). In addition, financial institutions are
+often reluctant to collaborate in this type of investigation due to concerns about data security and
+personal privacy. Until now, the use of synthetic data in credit scoring is mainly restricted to bal-
+ancing the minority class in classification problems using the traditional SMOTE Gici´c and Subasi
+(2019), variational autoencoders Wan et al. (2017), and lately generative adversarial networks
+Fiore et al. (2019); Lei et al. (2020); Ngwenduna and Mbuvha (2021). In these studies, synthetic
+records of the minority class are generated, and the original data set is augmented. In this paper,
+we present a framework that allows us to train a model on synthetic data and then apply it to
+real-world data. We also analyze if the model copes with data drift by applying both models to
+real-world data representing the same problem but obtaining the dataset one year later. The main
+findings of our work are:
+• It is possible to train a model on synthetic data that achieves good performance in real
+situations.
+• As the number of features increases, the synthesized data quality gets worse.
+• There is a performance cost for working in a privacy-preserving environment.
+This cost
+corresponds to a loss of predictive power of approximately 3% if measured in AUC and 6%
+in KS.
+2
+Related Work
+2.1
+Credit Scoring
+Credit scoring aims to manage credit risk, defined as the potential for a borrower to default on
+established contractual obligations The Basel Committee on Banking Supervision (2000). These
+models intensively use borrower data, demographic information, payment behavior, and even al-
+ternative data sources such as social networks Mu˜noz-Cancino et al. (2021); ´Oskarsd´ottir et al.
+(2019), psychometrics Djeundje et al. (2021), and geolocation Simumba et al. (2021).
+2.2
+Generative models for synthetic data generation
+Generative models are a subset of machine learning models whose main objective is to learn
+the real-data distribution and then to generate consistent samples from the learned distribution.
+Working with synthetic data allows addressing problems where real-data is expensive to obtain,
+where a large dataset is needed to train a model, or where the real-data is sensitive or cannot
+be shared Torres (2018). For years, statistical methods were the most used ones to estimate the
+real-world data joint distribution. In this group, Gaussian Mixture Models are the most utilized
+for this task when there are fewer continuous variables. At the same time, Bayesian Networks are
+commonly used for discrete variables. The main problem of these methods is dealing with datasets
+containing numerical and categorical variables. They also present problems when the continuous
+variables have more than one mode and the categorical variables present small categories Xu et al.
+(2020). During the last years, deep learning models have gained popularity to generate synthetic
+data due to their performance and because they allow us to deal with the problems mentioned
+above. The generative adversarial networks and the variational autoencoders stand out within
+these models.
+2
+
+2.2.1
+Generative Adversarial Networks
+Generative adversarial networks are a deep learning framework based on a game theory scenario
+where a generator network G(·) must compete with a discriminator network D(·). The generator
+network produces samples of synthetic data that attempt to emulate real data. In contrast, the
+discriminator network aims to differentiate between real examples from the training dataset and
+synthetic samples obtained from the generator Goodfellow et al. (2016).
+Its most basic form,
+vanilla GAN, G(·) maps a vector z from a multivariate Gaussian distribution N (0, I) to a vector
+ˆx in the data domain X. While D(·) outputs a probability that indicates whether ˆx is a real
+training samples or a fake sample drawn from G(·) Xu et al. (2020).
+The generator G(·) and
+the discriminator D(·) are alternatively optimized to train a GAN. Vanilla GANs have two main
+problems, representing unbalanced categorical features and expressing numerical features having
+multiple modes. To solve this, Xu et al. (2019) Xu et al. (2019) present a conditional generator
+(CTGAN) that samples records from each category according to the log-frequency; this way, the
+generator can explore all discrete values. Moreover, the multimodal distributions are handled using
+kernel density estimation to assess the number of modes in each numerical feature.
+2.2.2
+Variational autoencoders
+Autoencoders (AE) are an unsupervised machine learning method that enables two main objec-
+tives: low-dimensional representation and synthetic data generation.
+Variational Autoencoder
+Kingma and Welling (2013) interpret the latent space produced by the encoder as a probability
+distribution modeling the training samples as independent random variables, assuming the poste-
+rior distribution defined by the encoder qθ(z|x) and generative distribution pφ(x|z) defined by the
+decoder. To accomplish that the encoder produces two vectors as output, one of means and the
+other of standard deviations, which are the parameters to be optimized in the model. Xu et al.
+(2019) Xu et al. (2019) present TVAE, a variational autoencoder adaption for tabular data, using
+the same pre-processing as in CTGAN and the evidence lower bound (ELBO) loss.
+3
+Methodology and Experimental Design
+3.1
+Dataset
+In this work, we use a dataset provided by a financial institution already used for research on credit
+scoring Mu˜noz-Cancino et al. (2021,2). This dataset includes each borrower financial information
+and social interactions features over two periods: January 2018 and January 2019; each dataset
+contains 500,000 individuals. Each borrower is labeled based on their payment behavior in the
+following 12-month observation period. Each borrower in the 2018 dataset is labeled as a defaulter
+if it was more than 90 days past due between February 2018 and January 2019 and is labeled as a
+non-defaulter if it was not more than 90 days past due. Borrowers from the Jan-2019 dataset are
+similarly tagged. This dataset contains three feature subsets: XF in corresponds to the borrower’s
+financial information, XDegree corresponds to the number of connections the borrower has in the
+social interaction network, and XSocInt are the features extracted from the social interactions.
+3
+
+3.2
+Synthetic data generation
+A step to privacy-preserving credit scoring model building is to generate a synthetic dataset that
+mimics real-world behavior. In order to accomplish this, we compare the performance of two state-
+of-the-art synthetic data generators, CTGAN and TVAE, defined in Sect. 2. The first experiment
+(S01) only compares these methods using borrowers’ features XF in. The objective of this stage
+is to find a method to generate synthetic data from real data, and it is not part of this study
+to find the best way to generate them. Despite not generating an exhaustive search for the best
+hyper-parameters, we will test two different architectures (Arch) for each synthesizer. Arch A is
+the default configuration for both methods. In the case of CTGAN, Arch B set up the generator
+with two linear residual layers and the discriminator with two linear layers, both of size (64, 64).
+In the case of TVAE Arch B, set hidden layers of (64, 64) for both the encoder and the decoder.
+Then, in experiment S02, we train a new synthesizer using the best architecture from S01. This
+experiment uses the borrowers’ features XF in and exclusively one feature from the network data,
+the node degree XDegree. We only include node degree because its feature enables us to reconstruct
+an entire network using the random graphs generators. Finally, in experiment S03, the borrowers
+and social interaction features (XF in + XDegree + XSocInt) are used to train a synthesizer. This
+experiment corresponds to the traditional approach to generating synthetic data from a dataset
+using social interaction features.
+3.3
+Borrower’s creditworthiness assessment
+The objective of this stage is to have a framework that allows us to estimate the borrower’s cred-
+itworthiness from a feature set.
+This modeling framework is based on previous investigations
+Mu˜noz-Cancino et al. (2021,2). This stage begins by discarding attributes with low or null predic-
+tive power and selecting uncorrelated attributes. The correlation-based selection method begins by
+selecting the attribute with the highest predictive power. It then discards the possible selections
+if the correlation exceeds a threshold ρ. This step is repeated until no attributes are left to select.
+To ensure the model generalization capability, we work under a K-fold cross-validation scheme;
+in this way, the feature selection and the model training use K-1 folds, and the evaluation is car-
+ried out with the remaining fold. Additionally, we use two holdout datasets, one generated with
+information from the same year as the training dataset but not contained. The second contains
+information from one year later. Both the results of the validation fold and the holdout dataset
+are stored to use a t-test later to compare different models (Flach, 2012, Ch. 12).
+3.4
+Evaluation Metrics
+In this section, we describe a set of metrics that will help us to evaluate the performance of the
+synthetic data generators and the classification models used for creditworthiness assessment. The
+area under the curve (AUC) is a performance measure used to evaluate classification models
+Bradley (1997). The AUC is an overall measure of performance that can be interpreted as the
+average of the true positive rate for all possible values of the false positive rate. A higher AUC
+indicates a higher overall performance of the classification model Park Seong Ho (2004). Another
+classification performance measure is the F-measure. This metric is calculated as the harmonic
+mean between precision and recall. It is beneficial for dichotomous outputs and when there is
+no preference between maximizing the model’s precision or recall Hripcsak and Rothschild (2005).
+Kolmogorov-Smirnov (KS) statistic measures the distance separating two cumulative distributions
+4
+
+Hodges (1958). The KS statistic ranges between 0 and 1 and is defined as D = maxx |F1(x)−F2(x)|,
+where F1 and F2 are two cumulative distributions. In the case of creditworthiness assessment,
+we are interested in the difference between the cumulative distributions of defaulters and non-
+defaulters, and a higher D indicates a higher discriminatory power.
+However, in the case of
+synthetic data generation, we are interested in the real data distribution and the synthetic data
+distribution being as similar as possible; in this way, a lower D indicates a better synthetic data
+generation. In order for all the acceptance criteria to be the same, we define the KSTest as 1−D;
+in this way, a higher KSTest indicates a better synthetic data generator. In the synthetic data
+generation problem, the KS is only valid to measure the performance for continuous features; to
+handle categorical features, we will use the chi-square test (CS). The CS is a famous test to assess
+the independence of two events McHugh (2013). We will call CSTest to the resulting p-value for
+this test. Therefore a small value indicates we can reject the null hypothesis that synthetic data
+has the real data distribution. In the synthetic data generation problem, we want to maximize the
+CSTest.
+3.5
+Experimental setup
+The parameters of the univariate selection are set at KSmin = 0.01 and AUCmin = 0.53, i.e., we
+discard feature with a univarite performance lower than KSmin or AUCmin. In the multivariate
+selection process, we set ρ = 0.7 in the process to avoid high correlated features Akoglu (2018). The
+N-Fold Cross-Validation stage is carried out considering N = 10, and in each iteration, the results
+of regularized logistic regression and gradient boosting Friedman (2001) models are displayed.
+4
+Results and Discussion
+In this section, we present the results of our methodology. We start with the implementation
+details. Then, we compare the synthesizers, and finally, we analyze the creditworthiness assessment
+performance of the models trained using synthetic data.
+4.1
+Implementation Details
+In this work, we used the Python implementations of Networkx v2.6.3 Hagberg et al. (2008) and
+Synthetic Data Vault (SDV) v5.0.0 Patki et al. (2016) for networks statistics and synthetic data
+generation, respectively. To conduct the experiments, we used a laptop with 8 CPU cores Intel i7
+and 32 GB of RAM.
+4.2
+Synthetic Data Generation Performance
+The first objective is to analyze the performance of the methods to generate synthetic data pre-
+sented above, CTGAN and TVAE. Table 1 shows the results obtained. The features Synthesizer
+training features corresponds to the training feature set, while Arq indicates the network architec-
+ture defined in Sect. 3.2. The experiment S01 consisted in comparing both synthesizer using two
+different architectures. It is observed that a reduction in the number of layers reduces the execution
+times considerably in both cases, being TVAE, the one that presented the fastest execution times.
+KSTest show us the performance to synthesize continuous features, where TVAE achieves better
+performance than CTGAN. The difference between TVAE architectures is almost negligible when
+5
+
+evaluate continuous features performance. The performance to synthesize categorical features is
+measured using CSTest. In this case, TVAE obtained higher performance again, the differences
+between architectures is slightly higher to architecture A. Another popular approach to measuring
+the synthesizer performance is training a classifier to distinguish between real and synthetic data.
+The column Logistic Detection in Table 1 shows the result after training a logistic regression model;
+the value displayed corresponds to the complementary F-measure. In this way, values closer to 1
+indicate that the classifier cannot distinguish between real and synthetic data, and values closer
+to 0 mean the classifier efficiently detects synthetic data. It can be seen that TVAE achieve the
+best performance, but this performance decreases as we include more features to the synthesizer.
+Table 1: Synthetic data generators performance
+Experiment
+Synthesizer training features
+Synthesizer
+Arch
+Exec Time (m)
+CSTest
+KSTest
+Logistic Detection
+S01
+XF in
+CTGAN
+A
+410
+0.836
+0.864
+0.697
+B
+260
+0.861
+0.846
+0.749
+TVAE
+A
+230
+0.962
+0.868
+0.803
+B
+130
+0.952
+0.861
+0.756
+S02
+XF in + XDegree
+TVAE
+B
+140
+0.935
+0.836
+0.644
+S03
+XF in + XDegree + XSocInt
+TVAE
+A
+400
+0.924
+0.809
+0.539
+S03
+XF in + XDegree + XSocInt
+TVAE
+B
+320
+0.907
+0.825
+0.542
+S03
+XF in + XDegree + XSocInt
+TVAE
+B
+465
+0.930
+0.819
+0.513
+4.3
+Creditworthiness assessment performance on real data
+This section establishes a comparison line for the performance of the models trained with synthetic
+data. In order to establish this comparison, we first trained classifiers using real-world data and
+tested their performance using the holdout datasets previously defined. Table 2 shows the results
+of training models according to the methodology described in 3.3. The performance is measured
+using AUC and KS on the two holdout datasets; the 10-folds mean and its standard deviation are
+shown for each statistic. For each feature set, we trained two classifiers, logistic regression and
+gradient boosting. The results show that gradient boosting obtains better results compared to
+logistic regression. More details of this comparison are shown in Table 3, where we quantify the
+higher predictive power of gradient boosting.
+Table 2: Creditworthiness assessment performance for models trained on real data
+Classifier training
+features
+Classifier
+Holdout 2018
+Holdout 2019
+AUC
+KS
+AUC
+KS
+XF in
+GB
+0.88 ± 0.001
+0.59 ± 0.002
+0.82 ± 0.001
+0.50 ± 0.002
+XF in
+LR
+0.87 ± 0.001
+0.58 ± 0.001
+0.82 ± 0.001
+0.50 ± 0.002
+XF in + XDegree + XSocInt
+GB
+0.88 ± 0.001
+0.59 ± 0.002
+0.82 ± 0.001
+0.50 ± 0.002
+XF in + XDegree + XSocInt
+LR
+0.87 ± 0.001
+0.58 ± 0.002
+0.83 ± 0.001
+0.50 ± 0.002
+XDegree + XSocInt
+GB
+0.61 ± 0.002
+0.17 ± 0.002
+0.62 ± 0.001
+0.18 ± 0.002
+XDegree + XSocInt
+LR
+0.60 ± 0.001
+0.17 ± 0.002
+0.61 ± 0.001
+0.18 ± 0.002
+Based on the results presented above, we will select gradient boosting for the comparisons
+against the models trained on synthetic data that we will present in the next section.
+4.4
+Creditworthiness assessment performance on synthetic data
+This section aims to know how the performance of a creditworthiness assessment model (the
+classifier) behaves when trained on synthetic data and applied to real-world data. Table 4 shows
+6
+
+Table 3: Gradient boosting and logistic regression comparison on real data (holdout 2018)
+Classifier training features
+AUC diff (%)
+KS diff (%)
+AUC diff p-value
+KS diff p-value
+XF in
+0.70%
+1.65%
+0.000
+0.000
+XF in + XDegree + XSocInt
+0.84%
+1.91%
+0.000
+0.000
+XDegree + XSocInt
+1.65%
+2.36%
+0.000
+0.000
+the performance indicators on real-world data. Considering all synthesizers are trained with at
+least the feature set XF in, the results of training the classifier with XF in are also displayed for all
+synthesizers. It is observed that regardless of the synthesizer, training the classifier incorporating
+at least feature set XF in produces similar performances in 2018 except in S02. However, when we
+analyze how much the model degrades, the model trained with synthetic XF in from synthesizer
+S01 is the one that suffers a minor discrimination power loss. It can be explained in part that a
+better synthesizer manages to capture better the proper relationship between the borrower features
+and the default.
+Table 4: Creditworthiness assessment performance on real data for model trained on synthetic data
+Synthesizer
+Experiment
+Classifier training
+features
+holdout 2018
+holdout 2019
+AUC
+KS
+AUC
+KS
+S01
+XF in
+0.85 ± 0.003
+0.53 ± 0.002
+0.82 ± 0.002
+0.48 ± 0.002
+S02
+XF in
+0.82 ± 0.001
+0.51 ± 0.001
+0.80 ± 0.001
+0.46 ± 0.002
+S03
+XF in
+0.85 ± 0.002
+0.55 ± 0.002
+0.80 ± 0.002
+0.46 ± 0.002
+S03
+XF in + XDegree + XSocInt
+0.85 ± 0.002
+0.56 ± 0.003
+0.80 ± 0.002
+0.47 ± 0.003
+S03
+XDegree + XSocInt
+0.60 ± 0.002
+0.16 ± 0.002
+0.61 ± 0.003
+0.18 ± 0.002
+The comparison of performance obtained by the models trained with synthetic data against
+the models trained on real-world data is presented in Table 5. We can understand this comparison
+as the cost of using synthetic data, and it corresponds to the loss of predictive power to preserve
+the borrower’s privacy. We can observe that in the best cases, this decrease in predictive power is
+approximately 3% and 6% when we measure the performance in AUC and KS, respectively.
+Table 5: Comparison between models trained using synthetic data and models trained on real data. ∗∗
+denotes when the difference is statistically significant using 0.05 as the p-value threshold, while ∗ uses 0.1.
+Synthesizer
+Experiment
+Classifier training
+features
+holdout 2018
+holdout 2019
+AUC diff
+KS diff
+AUC diff
+KS diff
+S01
+XF in
+-3.59%∗∗
+-10.09%∗∗
+-0.86%∗∗
+-3.92%∗∗
+S02
+XF in
+-6.24%∗∗
+-13.24%∗∗
+-3.32%∗∗
+-6.48%∗∗
+S03
+XF in
+-2.81%∗∗
+-6.01%∗∗
+-3.21%∗∗
+-6.70%∗∗
+S03
+XF in + XDegree + XSocInt
+-3.12%∗∗
+-5.68%∗∗
+-2.54%∗∗
+-4.73%∗∗
+S03
+XDegree + XSocInt
+-1.85%∗∗
+-4.31%∗∗
+-0.69%∗∗
+1.10%∗
+5
+Conclusions
+This work aimed to use synthetic data to train creditworthiness assessment models. We used a
+massive dataset of 1 million individuals and trained state-of-the-art synthesizer methods to obtain
+synthetic data and achieve this goal. Then, we presented a training framework that allows us to
+analyze trained models with synthetic data and observe their performance on real-world data. In
+addition, we observed their performance one year after being trained to see how susceptible they
+7
+
+are to data drift. Our results show that lower quality synthetic data is obtained as we increase the
+number of attributes in the synthesizer. Despite this, it is possible to use these data to train models
+that obtain good results in real-world scenarios, with only a reduction in the predictive power of
+approximately 3% and 6% when we measure the performance in AUC and KS, respectively. These
+findings are of great relevance since they allow us to train accurate creditworthiness models. At the
+same time, we keep borrowers’ privacy and encourage financial institutions to strengthen ties with
+academia and foster collaboration and research in credit scoring without the privacy and security
+restrictions.
+6
+Future Work
+Our future work will delve into how to synthesize social interactions’ information in the form of
+graphs and not as added attributes to the training dataset since, as we show, this deteriorates the
+quality of the synthetic data.
+Acknowledgements
+This work would not have been accomplished without the financial support of CONICYT-PFCHA
+/ DOCTORADO BECAS CHILE / 2019-21190345. The second author acknowledges the support
+of the Natural Sciences and Engineering Research Council of Canada (NSERC) [Discovery Grant
+RGPIN-2020-07114]. This research was undertaken, in part, thanks to funding from the Canada
+Research Chairs program. The last author thanks the support of MICIN UNDER project PID2020-
+116346GB-I00.
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+

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+Spectral edge-to-edge topological state transfer in diamond photonic lattices
+Gabriel C´aceres-Aravena†,1, 2 Basti´an Real†,1, 2 Diego Guzm´an-Silva,1, 2 Paloma Vildoso,1, 2
+Ignacio Salinas,1, 2 Alberto Amo,3 Tomoki Ozawa,4 and Rodrigo A. Vicencio1, 2, ∗
+1Departamento de F´ısica, Facultad de Ciencias F´ısicas y Matem´aticas, Universidad de Chile, Chile
+2Millenium Institute for Research in Optics - MIRO, Chile
+3Univ. Lille, CNRS, UMR 8523—PhLAM—Physique des Lasers Atomes et Mol´ecules, F-59000 Lille, France
+4Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, Sendai 980-8577, Japan
+(Dated: January 12, 2023)
+Transfer of information between topological edge states is a robust way of spatially manipulating
+quantum states while preserving their coherence in lattice environments. This method is particularly
+efficient when the edge modes are kept within the topological gap of the lattice during the transfer.
+In this work we show experimentally the transfer of photonic modes between topological edge states
+located at opposite ends of a dimerized one-dimensional photonic lattice. We use a diamond lattice
+of coupled waveguides and show that the transfer is insensitive both to the presence of a high density
+of states in the form of a flat band at an energy close to that of the edge states, and to the presence
+of disorder in the hoppings. We explore dynamics in the waveguide lattice using wavelength-scan
+method, where different input wavelength translates into different effective waveguide length. These
+results open the way to the implementation of more efficient protocols based on the active driving
+of the hoppings.
+Topological edge states are a remarkable resource to
+engineer photonic systems with isolated modes protected
+from the presence of disorder. In two-dimensional lat-
+tices, they can be used to fabricate topological edge mode
+lasers with distributed gain and quantized orbital mo-
+mentum [1, 2], to transfer single photons around cor-
+ners in elaborated photonic circuits [3, 4], and to de-
+sign topological frequency combs with enhanced effi-
+ciency [5, 6].
+One dimensional systems such as the
+Su-Schrieffer-Heeger (SSH) lattice are particularly in-
+teresting because topological edge and interface modes
+are hosted deep into the topological gap of the lattice.
+This gap protection has been shown to be beneficial
+to preserve the quantum state of photons in boundary
+modes [7, 8]. Interestingly, the presence of topological
+edge modes on both sides of one-dimensional lattices can
+be used to transfer a state from one edge of the lattice
+to the other with high fidelity with the advantage of be-
+ing protected from certain types of disorder due to the
+topological nature of the system. Such edge state trans-
+fer is a promising route to store and manipulate photonic
+quantum states in on-chip lattice environments.
+Most topological edge transfer protocols rely on the
+adiabatic evolution of the lattice such that an edge mode
+is driven into quasi-bulk modes and again into an edge
+mode at the other side [9–17].
+While these protocols
+present an optimized transfer rate and fidelity, they are
+limited by the adiabaticity condition that requires the
+adiabatic passage to be slow enough to avoid the Zen-
+ner coupling of the edge state information into the bulk
+modes [18, 19]. Furthermore, the presence of disorder in
+the lattice would enhance this coupling. A variation of
+these protocols include counter-adiabatic driving meth-
+ods [20]. Recently, a different route has been proposed
+based on the coherent coupling of edge modes within the
+gap [19, 21, 22]. The great advantage of this approach
+is that edge modes are kept well into the topological gap
+throughout the protocol, ensuring a high fidelity in re-
+duced times. The simplest version of the coherent state
+transfer of topological edge states is via passive evanes-
+cent coupling of the exponential tails of edge modes at op-
+posite sides of the finite size lattice. In this case, coherent
+transfer between edge modes takes place at a frequency
+determined by the tail overlap, which can be controlled
+via the size of the gap. Observation of such coherent os-
+cillations was reported in a short SSH lattice for Rydberg
+atoms [23].
+In this work, we demonstrate coherent edge-to-edge
+transfer of light in a dimerized diamond lattice of coupled
+waveguides employing a spectral tomographic technique.
+More importantly, we show experimentally that the fi-
+delity of the transfer is robust to a number of perturba-
+tions in the system. First, we show that orthogonality of
+eigenmodes in our undriven protocol preserves the trans-
+fer even in the presence of a high density of states in the
+form of a flat band at energies close to that of the edge
+states. Second, we demonstrate that the transfer mech-
+anism is robust to the presence of lattice defects thanks
+to the underlying chiral symmetry of the system. The
+experimental proof of principle we report in this work
+can be signi���cantly sped up by applying a number of
+driven techniques based on the modulating of the hop-
+pings in time and the use of concatenated topological
+lattices [19, 22, 24]. Such techniques are readily imple-
+mentable in lattices of coupled waveguides.
+To demonstrate the topological edge transfer we use
+a diamond lattice of coupled waveguides with different
+intracell (t1) and intercell (t2) hoppings, as sketched in
+Fig. 1(a). The lattice has three sites per unit cell, denoted
+as A, B and C sites. Considering a tight-binding coupled-
+mode approach, the evolution of the optical field at every
+arXiv:2301.04189v1  [physics.optics]  10 Jan 2023
+
+2
+d2
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+(a)
+(b)
+(e1) (e2)
++
++
+Figure 1. (a) Sketch of a dimerized diamond lattice, with A,
+B and C the sites of the unit cell. Thick (thin) line denotes a
+strong (weak) hopping, and t1 (t2) indicates the intra(inter)-
+cell coupling constant. Top (bottom) panel schematizes the
+trivial (topological) case t1 > t2 (t1 < t2).
+(b) Spectrum
+as a function of δ for a finite (lines) and an infinite (shaded
+area) lattice.
+The vertical line denotes δ = 1.
+The color
+indicates the IPR for all the states. Inset: amplitude profiles
+of edge states at δ = 0.4. (c) Sketch of the fs laser writing
+technique. (d) Microscope image of a diamond lattice with
+{d1, d2} = {35, 25} µm (δ = 0.37) and dv = 32 µm. Output
+images for the lattice in (d) and for the excitation at (e1) a
+B left edge site and (e2) an in-phase A-C right edge sites.
+Yellow ellipses indicate the excited sites.
+site of the n-th unit cell is written as:
+−i∂zAn = t1Bn + t2Bn+1 ,
+−i∂zBn = t1(An + Cn) + t2(An−1 + Cn−1) ,
+(1)
+−i∂zCn = t1Bn + t2Bn+1 .
+Here An, Bn and Cn are the amplitudes of the opti-
+cal field at the n-th unit cell.
+z describes the coordi-
+nate along the waveguides and the dynamical variable.
+Moreover, the hopping strengths among nearest-neighbor
+(NN) sites can be varied experimentally by adjusting the
+lattice distances [25]. We then define the control param-
+eter δ ≡ t1/t2 to characterize the different regimes. We
+assume an infinite system and impose a Bloch-like ansatz
+in Eq. (1), obtaining the following bands:
+βz(kx) = 0, ±t2
+�
+2 [δ2 + 2δ cos(kxa) + 1] ,
+(2)
+where βz is the propagation constant (energy), a is
+the lattice constant and kx the quasimomentum.
+The
+spectrum is composed of two dispersive and one flat
+band (FB) [see shaded areas and horizontal light-blue
+line at βz = 0 in Fig. 1(b), respectively].
+The gap
+in between both dispersive bands has a size equal to
+2
+√
+2t2|δ − 1|. For δ = 1, this gap closes and the three
+bands touch each other at the edges of the Brillouin
+zone [26].
+The diamond lattice possesses the smallest
+experimentally reported FB states [26, 27], with an in-
+verse participation ratio (IPR) [28] of 1/2 [represented
+as light-blue color in Fig. 1(b)].
+Specifically, in the
+bases of Wannier functions in the A, B and C sites,
+the FB eigenvector is given by: |vF B⟩ = {1, 0, −1}/
+√
+2,
+and the ones corresponding to the dispersive bands
+are |v±⟩ = {eiφ(kx), ±
+√
+2, eiφ(kx)}/
+√
+2, where φ(kx) =
+arctan(− sin(kxa)/[δ + cos(kxa)]).
+Even though this lattice has three sites per unit
+cell, it exhibits similar topological features to the SSH
+model [29], when varying the parameter δ [30]. Indeed,
+a quantized Zak phase of a value 0 or π can be found
+when δ > 1 (t1 > t2) or δ < 1 (t1 < t2), respectively.
+In this case, the nontrivial phase is protected by inver-
+sion symmetry between An and Cn and by chiral sym-
+metry [30, 31]. Thus, we expect the appearance of two
+edge states at zero propagation constant on a lattice with
+open boundaries when δ < 1. To corroborate this, we
+compute the spectrum as a function of δ for dimerized
+diamond lattices of 9 unit cells [see full lines in Fig. 1(b)].
+It can be clearly seen that two states at zero frequency
+(lighter blue) transform into two dispersive states (darker
+blue) around δ = 1. When increasing δ, the degeneracy
+between them is removed at around δ = 0.7 (splitting
+∆βe
+z ∼ 0.06) [32] due to the finite size of the lattice. The
+flat band remains unchanged at βz = 0, for any value
+of δ.
+The IPR (denoted by color) shows very clearly
+the transition from localized edge states (IPR = 1 or
+1/2, light blue) into extended propagating modes (IPR
+∼ 1/N, black).
+Figs. 1(b)-insets show the two edge states for δ = 0.4.
+They exhibit exponentially localized amplitudes at both
+edges. On the left edge, these states present a null ampli-
+tude at A and C sites, whereas the states have a null am-
+plitude at B sites at the right edge. Moreover, one edge
+state is antisymmetric (bottom inset) and the other one is
+symmetric (top inset), with respect to the opposite edge.
+They decay exponentially into the bulk as (−δ)|n−ne| for
+a semi-infinite system, exhibiting a phase shift of π at
+consecutive B or A,C sites, depending on the specific
+edge (ne). Notice in Fig. 1(b) that the edge states are
+degenerate for δ ≲ 0.7; consequently, the sum of these
+states gives a state fully localized at the left edge with
+amplitude on B sites only and, conversely, the subtrac-
+tion of them gives a state fully localized at the right edge
+with amplitude on A and C sublattices. For δ ≳ 0.7, the
+degeneracy of the edge states is lifted, and their frequency
+deviates from 0 to ±βe
+z. Therefore, the excitation of sites
+at the edges is expected to induce an oscillatory pattern
+in between both surfaces with a frequency βe
+z [33], with
+a long-distance state transfer occurring on a dynamical
+
+(a)
+>1
+Trivial
+A
+Bs<1
+Topological
+A
+B
+t1
+C4
+-1
+1
+2
+βz
+0
+t2
+-2
+-4
+0
+0.5
+1
+1.5
+23
+scale ztransfer = π/βe
+z [32].
+We fabricate several dimerized diamond photonic lat-
+tices, of 9 unit cells each, by using a femtosecond (fs)
+laser writing technique [25, 32, 34], as it is sketched in
+Fig. 1(c).
+For a first set of experiments, the diamond
+geometry is defined by distances d1, d2 and dv = 32 µm,
+as described in Fig. 1(d). For these values, the diagonal
+(NN) distance was swept in the interval {25.6, 43.1} µm,
+as d1 and d2 were varied in the interval {20, 40} µm
+in steps of 1 µm.
+The hopping coefficients (which de-
+cay exponentially on waveguide separation [25]) range in
+the interval ∼ {0.03, 0.21} cm−1 at a wavelength of 640
+nm [32]. Fig. 1(d) shows an output facet of a lattice with
+d1 = 35 and d2 = 25 µm, with t1 = 0.05 and t2 = 0.14
+cm−1 (δ = 0.37). We first test the quality of the lat-
+tices by exciting them at different input positions using
+a 640 nm laser beam (see Ref. [32] for a complete charac-
+terization). For example, topological edge states can be
+efficiently excited by injecting light directly at the lattice
+boundaries [28, 32, 35, 36]. Fig. 1(e1) shows the output
+profile after a B-edge site excitation, with a clear expo-
+nential decaying profile from the edge into the bulk. The
+excitation at the right boundary requires a more com-
+plicated input condition with two in-phase beams. The
+result of this is shown in Fig. 1(e2), with an output profile
+formed by A and C sites mostly.
+We propose a novel technique to characterize the lat-
+tice dynamics. Instead of measuring the output profiles
+at different z values (which also implies the fabrication of
+a larger number of lattices), we implement a wavelength-
+scan method: The dynamics of a wavepacket injected in
+the input facet of a lattice is revealed when varying the
+input wavelength coming from a Supercontinuum (SC)
+laser source. In general, the lattice dynamics depends on
+the excitation wavelength λ: the longer the wavelength
+the wider the mode profile and the larger the coupling
+constants [25, 32]. In this way, by tuning the input wave-
+length, we can study the same lattice at different effective
+lengths.
+We first consider a diamond lattice with d1 = d2 =
+30 µm. We excite a B site at the central 5-th unit cell and
+scan the input wavelength in the interval 600 − 760 nm,
+with a step of 10 nm. Fig. 2(a) shows the output inten-
+sity for three selected λ’s [32]. Fig. 2(b) shows the second
+moment (width), defined as m2 ≡ �
+n(n−nc)2Pn, versus
+wavelength, with Pn ≡ |An|2 +|Bn|2 +|Cn|2 the unit cell
+power and nc ≡ �
+n nPn the center of mass. We observe
+a growing diffraction pattern [27], with a width that in-
+creases almost linearly with the input wavelength [a lin-
+ear fit is included in Fig. 2(b)]. m2 ∼ z corresponds to a
+diffusive regime [37], as expected for discrete diffraction;
+therefore, a λ increment produces an effectively larger
+propagation distance z or a larger coupling constant t1,2.
+A dimerized diamond lattice has two hoppings which
+simultaneously change while λ is modified. Since we ob-
+serve a linear dependence of coupling constants on wave-
+length, we can assume δ as a constant, as a first approx-
+(c)
+600 nm
+680 nm
+650 nm
+700 nm
+(a)
+(b)
+Figure 2. (a) Output intensity profiles for a B-site central
+excitation, for a lattice with d1 = d2 = 30 µm, at the indicated
+wavelength.
+(b) m2 versus λ.
+(c) md versus d1 (bottom)
+and δ (top). Insets in (c) show the profile at 700 nm for the
+indicated case. The bar shows the standard deviation. Yellow
+ellipses indicate the excited sites.
+imation. We use the wavelength-scan method to exper-
+imentally determine nc for all the output profiles, after
+exciting a B site at the central (5-th) unit cell of 17 dimer-
+ized lattices, having different values of δ. For each lattice,
+we average nc over λ and obtain the averaged beam dis-
+placement md, from which the topological invariant can
+be inferred [38, 39]. A topologically trivial lattice has a
+md = 0, as an indication of a zero Zak phase. A topolog-
+ically non-trivial system will shift this value to md ∼ 0.5,
+corresponding to a π Zak phase [38]. Our collected results
+are shown in Fig. 2(c). We observe that for d1 < 30 µm
+(δ > 1) the lattice is topologically trivial and the prop-
+agation shows a md around zero. For 30 ⩽ d1 ⩽ 34 µm
+(0.4 < δ ⩽ 1), a transition region without a well defined
+topological phase is observed. For d1 ⩾ 35 µm (δ ⩽ 0.4),
+the lattices express a clear averaged beam displacement
+around 0.5, implying a nontrivial Zak phase. Therefore,
+the wavelength-scan method gives us valuable informa-
+tion about the dynamics on a specific lattice, and it be-
+comes a key method to determine its topological phase
+on a finite size configuration.
+The number of unit cells in the lattice affects the edge
+
+4
+¥3
+2
+0.5
+0.3
+0.1
+0.8
+0.6
+0.4
+0.2
+0.0
+25
+30
+35
+40
+di [um]10
+8
+6
+2
+4
+2
+0
+600
+650
+700
+750
+Wavelength ^ [nm]4
+(d)
+650 nm
+680 nm
+710 nm
+740 nm
+(a)
+650 nm
+670 nm
+690 nm
+710 nm
+730 nm
+660 nm
+680 nm
+700 nm
+720 nm
+740 nm
+640 nm
+(c)
+(b)
+Figure 3. (a) Output profiles of a non-trivial diamond pho-
+tonic lattice at different λ’s, after a B-edge excitation (see
+yellow ellipse). (b) Fidelity versus wavelength for topological
+(black), trivial (gray), topological + defect (red), and trivial
++ defect (orange) lattices. (c) Microscope image of a topolog-
+ical lattice plus two coupling defects (see dashed rectangle).
+(d) Same than (a) for a topological lattice with a coupling
+defect.
+state properties:
+the fewer unit cells, the shorter the
+range of δ in which the edge states keep degenerate in
+βz [32].
+When the degeneracy is lifted, the two edge
+modes hybridize. Therefore, an input on one edge will
+excite coherently both modes and result in periodic os-
+cillations of the amplitude at the two edges.
+Then, a
+transfer of light from one edge to the other becomes pos-
+sible [23, 33].
+To experimentally demonstrate this we
+fabricate a topological lattice with 9 unit cells and dis-
+tances d1 = 18, d2 = 14, and dv = 14 µm (t1 = 0.30 and
+t2 = 0.42 cm−1 at 640 nm, and δ = 0.71). The trivial
+lattice (δ = 1.40) is obtained by inverting these distances
+to d1 = 14 and d2 = 18 µm. We decreased the distances
+to increase the coupling coefficients and favor a faster
+transport in between the edges, while staying at the non-
+degenerate situation. Again, we use a SC laser source in
+the range 610 − 740 nm and sweep the input wavelength
+in steps of 10 nm. We excite the system by injecting light
+at the B left edge waveguide, as shown in Fig. 3(a). For
+λ ≲ 670 nm, the intensity profiles are well localized at the
+left edge, with most of the light intensity at the B sublat-
+tices, with a profile resembling the edge state [Fig. 1(e1)].
+The edge states splitting manifests for λ ≈ 680 nm, where
+we start observing a smooth population of the opposite
+edge, with a weak excitation of the lattice center (a weak
+background radiation is always observed because of the
+excitation of dispersive modes [32]). The connection in
+between both edge patterns [see Figs. 1(b)-insets and (e)],
+with a B-site exponential decaying profile at the left sur-
+face and an A,C exponential profile at the right edge,
+becomes evident for ∼ 710 nm. The spectral state trans-
+fer phenomenon starts occurring at λ ≳ 720 nm: the
+light injected at one edge is mostly transferred into the
+opposite edge [32]. This shows a very interesting trans-
+port mechanism which does not require that the light
+explores the whole lattice; in this case, the light is sud-
+denly transferred from one edge into the other without
+interacting with the lattice bulk.
+We define the fidelity F for an edge-to-edge light
+transfer by measuring the normalized transferred in-
+tensity at the opposite lattice edge: F ≡ (|Aedge|2 +
+|Cedge|2)/ �
+n Pn. If all the light reaches the two right-
+most sites F = 1, and F = 0 in the fully opposite
+case. We show our results in Fig. 3(b), where we plot
+the fidelity F versus λ, for topological and trivial lat-
+tices. We observe how the topological (black) and the
+trivial (gray) cases have a similar dynamical scale; i.e.,
+both processes occur approximately at the same speed.
+However, the fidelity at the A, C surface is larger for the
+topological lattice (∼ 61%). The trivial lattice presents
+a standard discrete diffraction pattern [27], with the en-
+ergy exploring the whole lattice while it moves from one
+edge into the other [32], as the wavelength increases [sim-
+ilar to Fig. 2(a)]. Therefore, once the light arrives at the
+A, C right edge it is reflected back due to the absence of
+the edge states. The fidelity in this case decreases to a
+∼ 40%.
+A remarkable feature of the state transfer between
+topological edge states is the resilience to certain types of
+disorder. Although the fabrication process can produce
+random on-site or inter-site defects, we fabricate a cou-
+ple of lattices with a symmetric coupling defect, as the
+one shown in Fig. 3(c). We design a different distance in
+between the fourth and the fifth cells and inside the fifth
+cell [see dashed rectangle in Fig. 3(c)]. Specifically, we
+set this distance to 23 µm, implying a coupling defect of
+0.18 cm−1. Fig. 3(d) shows a set of output images at the
+indicated values of λ, for the topological lattice with a
+defect. We notice that this defect produces some reflec-
+tion and trapping of energy at short wavelengths, con-
+sequently, not all the energy is edge-to-edge transferred.
+Despite this, a significant amount of the light excites the
+topological right-edge state composed of A and C sites.
+The fidelity is ∼ 26% for the topological case, whereas it
+drops to ∼ 15% for the trivial lattice.
+These numbers show that a trivial lattice undergoes
+a stronger back reflection caused by the defect, because
+the light explores the whole lattice and interacts strongly
+with it. On the other hand, in the topological case, the
+light does not travel across the lattice and excite effi-
+ciently the edge state without the need of arriving at the
+boundary by standard transportation. The fidelity is not
+perfect in none of the topological cases because a single
+B-site input always excites part of the dispersive spec-
+
+0.6
+0.5
+0.4
+F 0.3
+0.2
+0.1
+0.0
+620
+640
+660
+680
+700
+720
+740
+Wavelength ^ [nm]888888885
+trum, in which the modes extend over the entire lattice.
+Nonetheless, the strong difference between the topologi-
+cal and trivial cases is the key of success for a topological
+state transfer process, which occurs due to the excitation
+of exponentially localized edge states which live at both
+edges simultaneously and deep in the gap of the lattice
+spectrum. This could be a proof of concept for a long
+distance sensor, which detects away from the interaction
+region.
+The wavelength-scan method proposed in this work of-
+fers a tool for investigating the dynamics in lattices of
+coupled waveguides. Using this method, we evidenced
+the nontrivial topology of dimerized diamond lattices by
+measuring the averaged beam displacement and, addi-
+tionally, we demonstrated an edge-to-edge transfer of
+light via the excitation of the topological edge states.
+This transfer is partially robust to defects across the
+lattice bulk and, hence, it has the potential for a pre-
+cise wavelength filter, as well as an efficient information
+transport mechanism between two distant ports or by
+concatenating several topological lattices in the quantum
+domain [12, 14].
+Acknowledgments. This work was supported in part
+by Millennium Science Initiative Program ICN17−012,
+and FONDECYT Grant 1191205.
+A.A. acknowledges
+the support of European Research Council grant Emer-
+genTopo (865151), the French government through the
+Programme Investissement d’Avenir (I-SITE ULNE /
+ANR-16-IDEX-0004 ULNE) managed by the Agence Na-
+tionale de la Recherche, the Labex CEMPI (ANR-11-
+LABX-0007) and the CPER Wavetech. T.O. acknowl-
+edges the support from JSPS KAKENHI Grant No.
+JP20H01845, JST PRESTO Grant No. JPMJPR19L2,
+and JST CREST Grant No.JPMJCR19T1.
+†Both authors contributed equally.
+∗ rvicencio@uchile.cl
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+Controllable tunnelling of single flux-quanta mediated by quantum phase-slip in
+disordered superconducting loops
+J.A. Potter,∗ J.C. Fenton, and P.A. Warburton
+London Centre for Nanotechnology, UCL, 17-19 Gordon Street, London WC1H 0AH, United Kingdom
+(Dated: January 12, 2023)
+Quantum phase-slip (QPS) is the exact dual to the well-known Josephson effect. Although there
+are numerous proposals for applications of QPS devices, experimental work to develop these remains
+in the relatively early stages. Significant barriers to exploiting QPS nanowires for useful technologies
+still exist, such as establishing robust nanowire fabrication methods that allow coupling to low-loss
+circuits, and demonstrating control over the QPS process with an experimenter-controlled external
+bias. Here we report experiments which show that both of these barriers have been overcome. We
+present measurements at 300 mK of NbN coplanar waveguide (CPW) resonators embedded with
+nanowires fabricated using a neon focused ion-beam. The internal quality factor exceeds 2 × 104 —
+significantly higher than previously reported in comparable experiments. The resonator frequency
+tunes periodically with an applied magnetic field, revealing tunnelling of the order parameter that
+always occurs at half-integer values of the applied flux. In contrast to previous studies of single QPS,
+the order-parameter tunnelling is shown to be adiabatic, demonstrating improved control over energy
+dissipation in nanowire QPS circuits. Our results highlight a promising pathway towards realising
+low-loss nanowire-based QPS devices.
+I.
+INTRODUCTION
+Quantum circuits based on superconducting materials
+are currently the leading candidate for the implementa-
+tion of a scalable quantum computer, already beginning
+to tackle relevant computation and simulation problems
+[1, 2] and recently demonstrating ‘quantum advantage’
+[3]. The Josephson junction is near ubiquitous in these
+circuits, providing the necessary nonlinearity. A quantum
+phase-slip nanowire is the flux-charge dual to the Joseph-
+son junction [4], and in theory every Josephson junction-
+based circuit has a QPS dual. As well as being proposed
+as a qubit with favourable properties over traditional
+Josephson-based technology [5, 6], the QPS nanowire’s
+dual property of a nonlinear quantum capacitance enables
+potential applications such as novel qubit-qubit couplers
+[7], parametric amplification for qubit readout, and a
+primary quantum current standard [8]. As yet, the huge
+potential of QPS nanowires remains to be fully exploited,
+and two key reasons for this stand out. Firstly, consistent
+and reliable fabrication of materials and nanowires with
+the requisite properties remains challenging; secondly, full
+control over individual QPS events in a nanowire has not
+yet been demonstrated.
+The QPS phenomenon is most pronounced in quasi-
+one-dimensional superconducting nanowires, by which we
+mean that the cross-sectional dimensions of the nanowire
+are comparable to the coherence length ξ [9]. In these
+nanowires, quantum (or indeed thermal) fluctuations can
+lead to complete suppression of the superconducting order
+parameter over the cross-section of the wire. This in turn
+leads to a sudden change of 2π in the phase difference
+∗ jamie.potter@npl.co.uk
+Present address:
+National Physical Laboratory, Teddington,
+TW11 0LW, United Kingdom
+between the two ends of the wire, accompanied by the
+transfer of a quantised amount of magnetic flux equal to
+the magnetic flux quantum Φ0 = h/2e through the wire.
+This tunnelling of a flux quantum can be coherent [10–12]
+or incoherent [13], depending on the relative scales of
+the phase-slip energy ES and the inductive energy EL
+of the nanowire. When ES/EL < 1, the magnetic flux
+quantum number is well-defined, and incoherent transfer
+of individual flux-quanta is observable. This is in direct
+analogy to small Josephson junctions with large charging
+energy, where tunnelling of single Cooper pairs can be
+observed [14]. It is the incoherent QPS regime that is
+the focus of this paper. Incoherent QPS occur probabilis-
+tically, with a frequency characterised by the phase-slip
+rate ΓS ≡ ES/h. However if one waits for a time τ that
+is much longer than 1/ΓS, then flux-quantum tunnelling
+will occur with extremely high likelihood. A device dis-
+playing deterministic transfer of quantised amounts of
+magnetic flux may find useful applications in tasks such
+as energy-efficient classical digital logic processing [15–19].
+Development of such a device is a key enabling technol-
+ogy for control of superconducting quantum processors
+at technologically-useful scales.
+Historically, phase slips were studied in externally con-
+nected, current-biased nanowires, where the collective
+effect of many phase slips manifests as a resistance below
+Tc [20–25]. However, in order to isolate a single phase-
+slip, it is necessary to incorporate the nanowire into a
+flux-biased superconducting loop. The flux-dependent
+energy states of a continuous superconducting loop are
+described by a set of parabolas (see Fig. 1(a)), where
+each parabola corresponds to a unique value N of the
+phase winding number, or equivalently the number of flux
+quanta in the loop. A single phase-slip corresponds to
+a transition between neighbouring parabolas, and if no
+lower-energy state is available at a particular external flux
+bias, then phase slips are forbidden. When sweeping the
+arXiv:2301.04411v1  [cond-mat.supr-con]  11 Jan 2023
+
+2
+externally applied flux, tunnelling of a single flux-quantum
+becomes allowed at Φapp = (N + 1/2)Φ0 (known as the
+degeneracy point, highlighted in red in Fig. 1(a)) [26, 27].
+Tunnelling will occur when the system passes the degen-
+eracy point if the rate of phase slips is much greater than
+the rate at which the flux is swept. However, if the flux is
+swept slowly with respect to ΓS, tunnelling will cause the
+system to enter a metastable state and it will then un-
+dergo irreversible relaxation to the ground state at some
+later time. A number of recent experiments [28–31] have
+demonstrated relaxation via QPS from a higher-energy
+metastable state, but controlled single-flux-quantum tun-
+nelling when passing through the degeneracy point has
+not previously been demonstrated.
+In this paper, we demonstrate for the first time single-
+flux-quantum tunnelling occurring at the degeneracy point
+in a continuous superconducting loop. The flux quanta
+tunnel through NbN nanowires embedded in the loop and
+this is read out via coupling to a high-quality coplanar
+waveguide (CPW) resonator. An important innovation in
+our fabrication technique is that the nanowires were fabri-
+cated by neon focused-ion-beam (FIB) milling. The FIB
+process enables the repeatable fabrication of nanowires
+with width w ≈ 25 nm ensuring a large phase-slip rate,
+while introducing minimal losses to the host resonator.
+FIB milling has previously been shown to be compati-
+ble with low-dissipation superconducting circuits [32, 33];
+however, this is the first report of the use of FIB to fabri-
+cate a device in which quantum phase-slip behaviour has
+been measured. Our results show flux-periodic tuning of
+the resonant frequency ν0 while maintaining a high intrin-
+sic quality factor Qi at all values of applied flux. We show
+that a single quantum phase-slip always occurs when ad-
+jacent winding-number states become degenerate. This is
+ensured by a phase-slip rate — we estimate ΓS = 35 MHz
+— which is large in comparison to 1/τE, where τE is the
+experimental timescale. However, the phase-slip rate is
+less than the inductive energy (EL/h > 1 THz), and less
+than the thermal energy (kBT/h ≈ 6 GHz), which ex-
+cludes the possibility of avoided crossings associated with
+coherent QPS.
+Our demonstration of the ability to control the tun-
+nelling of single flux-quanta represents important progress
+towards applications of QPS devices in quantum infor-
+mation processing that have been proposed elsewhere.
+The low loss in our device suggests the potential for high
+tunnelling rates in QPS devices without a significant in-
+crease in T1-type decoherence. In addition to this, there is
+potential for a QPS digital logic processing device, based
+on the deterministic transfer of single flux-quanta [15–
+18]. Utilising quantum tunnelling of flux in such a device
+should enable significant reduction in the heat dissipa-
+tion associated with each gate [34], a reduction that will
+be necessary for the scaling up of systems beyond the
+1,000-qubit level.
+FIG. 1. (a) Flux-dependent energy spectrum of a continuous
+superconducting loop, with blue dashed line highlighting the
+ground state. A single flux-quantum may tunnel into the loop
+at the degeneracy point — highlighted in red. N is the winding
+number and Φ0 the flux quantum. (b) Optical (main image;
+blue contrast) and SEM (inset; grey contrast) images of the
+NbN nanowire-embedded loop located at the short-circuited
+termination of the CPW resonator. The nanowires in this
+device are 25-nm wide and 200-nm long. The lower and upper
+leads to the loop are connected to the CPW centre conductor
+and superconducting ground plane respectively.
+II.
+FABRICATION AND EXPERIMENTAL
+DETAILS
+We fabricated nanowire-embedded resonators from 10-
+nm-thick films of superconducting NbN. The NbN was
+deposited on c-axis oriented sapphire substrates by dc
+magnetron sputtering of a 99.99% pure niobium target in
+a 1:1 Ar:N2 atmosphere at a pressure of 5 × 10−3 mbar
+and a power of 150 W. The resulting film was measured to
+have critical temperature Tc = 8.55 K and sheet resistance
+Rsq = 1.2 kΩ/sq.
+
+(a)
+N=0
+N=1
+N=2
+Energy
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+Applied Flux (Φo)
+(b)
+sapphire
+NbN
+μm
+100um3
+FIG. 2. (a) Upper panel: Single-tone spectroscopy of nanowire-embedded resonator measured at T = 305 mK and ⟨n⟩ ≈ 50.
+|S21| is plotted as a function of frequency and applied magnetic field. The top axis shows the applied magnetic flux Φapp seen by
+the nanowire loop, which is inferred from the periodicity of the resonator tuning. Dashed white lines show resonant frequencies
+corresponding to the calculated energy states of the loop. Lower panel: Magnetic-field dependence of the measured intrinsic
+quality factor. (b) Calculated energy states of the loop and values extracted from the measured resonant frequency.
+Quarter-wavelength resonators were patterned by
+electron-beam lithography (EBL) into a 300-nm layer
+of PMMA resist. Multiple resonators on each chip are
+capacitively coupled to a common feedline and are pat-
+terned with narrow loops galvanically coupled at the
+short-circuited end. At this stage, the loops contain ‘pre-
+cursor’ nanowires designed with a width of 400 nm. The
+pattern was transferred into the NbN film by reactive ion
+etching (RIE) using a 2:5 volume ratio of CHF3 and SF6
+at 100 W and 100 mbar.
+Nanowires are then patterned into the loops using a
+neon focused-ion-beam, whereby a beam of Ne ions is
+accelerated from an atomically-defined tip onto the sample
+surface with spot size down to 2 nm and sufficient energy
+to sputter the metal film [35, 36]. The precursor wires
+were milled to a width of 25 nm (see Fig. 1(c)) using a 15-
+keV Ne beam. A Ne dose of 0.5 nC/µm−2 was sufficient
+to clear the 10-nm film.
+The sample was wire-bonded to a copper printed-circuit-
+board (PCB) and enclosed within an ECCOSORB-lined
+brass sample box. This was cooled to a temperature of
+T ∼ 300 mK using a 3He refrigerator. Measurement of the
+rf response of the device was made using a Vector Network
+Analyser (VNA) via an input line with 60 dB of atten-
+uation to reduce thermal noise from room temperature.
+Signals in the output line were amplified with a high-
+electron-mobility-transistor (HEMT) amplifier. Global
+magnetic field was applied perpendicular to the plane of
+the loop using a superconducting solenoid and a precision
+current source. The lines supplying current to the coil
+were filtered at room temperature with an upper cut-off
+frequency of 9.2 kHz.
+III.
+RESULTS AND DISCUSSION
+A. Flux Dependence of Resonant Frequency
+In this paper, we present results on a single NbN
+nanowire-embedded CPW resonator (see Supplemental
+Information for comparison of multiple devices). We mea-
+sured the forward transmission (S21) through the on-chip
+feedline, where the λ/4 resonators appear as a notch-type
+resonance. The upper panel of Fig. 2(a) shows the main
+result of this work: under an applied magnetic field, the
+resonance tuning shows discontinuous changes of gradi-
+ent at periodic values of the applied field. As we will
+demonstrate in the remainder of this paper, these dis-
+continuities occur when two stable states with winding
+number differing by one become degenerate and are due
+to single-flux-quantum tunnelling mediated by quantum
+phase-slips in the nanowire [37].
+The magnetic-field periodicity is 153 µT, which cor-
+responds to a single flux-quantum in our loop assum-
+ing a flux-focusing factor F = 1.7 [38].
+The data in
+Fig. 2(a) corresponds to a single direction of magnetic
+field sweep, but sweeps in the opposite direction were
+found to give the same result. We also observe a non-
+periodic, parabolic decrease of the resonant frequency as
+the magnitude of the applied field is increased. This is the
+expected [39, 40] kinetic-inductance tuning of the NbN
+resonator, and can be parametrised by a phenomenologi-
+cal field-scale B⋆ = 8 mT.
+
+(a)
+Dapp(@o)
+(b)
+-3
+-2
+-1
+0
+2
+3
+S21| (dB)
+45.0
+500
+3.384
+3.383
+45.2
+400
+Frequency (GHz)
+3.382
+45.4
+3.381
+300
+45.6
+3.380
+45.8
+E (GHz)
+3.379
+200
+3.378
+46.0
+100
+3.377
+46.2
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+10×-01
+N=-1 N=0
+N=1
+N=2
+calculated
+-100
+extracted from measured vo
+3
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+4
+-2
+0
+Applied Field (mT)
+Φapp(Φo)4
+Figure 2(a) also shows periodic variation of Qi of the
+resonance as a function of applied field with the same
+field period as the resonant frequency. Quality factors
+were obtained by an analytical fit [41] to the equation
+|S21(ν)| = aeiαe−i2πντ
+�
+1 −
+�
+Ql
+|Qc|
+�
+eiφ
+1 + 2iQl(ν/ν0 − 1)
+�
+,
+(1)
+where ν is the probe frequency and ν0 is the resonance
+frequency.
+Qc and Ql are the coupling and loaded
+quality factors respectively, and obey the relationship
+1/Ql = 1/Qi + 1/Qc.
+φ accounts for the effect of
+impedance mismatches in the circuit, the scale factor
+a represents the change in amplitude due to any attenua-
+tion/amplification in the measurement chain, α describes
+any initial phase offset of the signal, and τ accounts
+for frequency-dependent cable delay. We find that Qi
+decreases approximately quadratically from 4 × 103 at
+δΦ ≡ (Φapp −NΦ0)/Φ0 = 0 to 3.4×103 at δΦ = 1/2. We
+attribute this small change to non-equilibrium quasiparti-
+cles excited by the induced screening current providing an
+extra loss mechanism [42]. We observe no sharp decrease
+in Qi at δΦ = 1/2, which suggests that the heat dissipated
+by the quantum phase-slip itself is not large enough to
+cause extra losses in the resonator. We note that the
+intrinsic quality factor exceeds any currently reported in
+the literature for QPS devices, and discuss this further in
+the Supplemental Information.
+The periodic tuning of the resonance is well fitted by
+a model of an inductive superconducting loop remaining
+in its ground state (see Fig. 2(b)), where the system is
+allowed to move between adjacent parabolas by trans-
+ferring a single flux-quantum through the nanowire at
+Φapp = (N + 1/2)Φ0.
+The loop is made up of a wide section and a narrow
+section (as shown in Fig. 1(b)), and so can be modelled
+as two nonlinear kinetic inductances in series. The flux-
+dependent kinetic inductance of the loop Lk(Φ) is there-
+fore
+Lk(I) = Lk,1(0)
+�
+1 +
+�
+Is
+I⋆,1
+�2�
++ Lk,2(0)
+�
+1 +
+�
+Is
+I⋆,2
+�2�
+,
+(2)
+where I⋆,1 and I⋆,2 are known to be of the order of the
+critical current in the wide and narrow section respectively
+[43]. Since the screening current is Is = Φ/Lk, we can
+insert this into Eq. 2 and solve for Lk(Φ).
+The input impedance Zin of a λ/4 CPW resonator
+terminated by an inductive load, as a function of frequency
+and load impedance, is
+Zin(ν, ZL) = Z0
+ZL + iZ0 tan
+� 2πνl
+c
+�
+Z0 + iZL tan
+� 2πνl
+c
+�,
+(3)
+where Z0 is the characteristic impedance of the resonator,
+ZL(Φ) = i2πνLk(Φ) is the impedance of the inductive
+load, c is the speed of light in the resonator, and l is its
+FIG. 3. |S21| measured at Φapp = 0 and Φapp = Φ0/2 along
+with fit (black line) to a linear resonance model. This shows
+that at Φapp = Φ0/2 the response is linear, so the current in
+the nanowire is well below the critical current.
+length. At resonance, Im{Zin} → ∞, so given Lk(Φ) one
+can numerically calculate ν0(Φ), or given ν0(Φ) one can
+numerically calculate Lk(Φ).
+One can also calculate the flux-dependent free energy
+of the loop E(Φ) from Lk(Φ), using the relation
+L−1
+k
+= d2E
+dΦ2 .
+(4)
+To obtain the free energy, we simply numerically integrate
+the inverse of the inductance twice with respect to flux.
+We calculated Lk(Φ) for our device from Eq. 2 using
+only independently determined parameters. A critical
+current density of Jc = 4.4 × 105 Acm−2 was obtained
+from a dc measurement of a track etched into the NbN
+film, and a sheet kinetic inductance of Lsq = 0.34 nH/sq
+was inferred from the zero-field ν0 of the resonator. The
+geometry of the loop was measured by SEM and this was
+used to calculate Lk(0) and I⋆ (we set I⋆ = Ic). We then
+calculated the white dashed lines in Fig. 2(a) using Eq. 3,
+and the solid black lines in Fig. 2(b) from Eq. 4. The blue
+points in Fig. 2(b) were extracted from the measured ν0
+using Eqs. 3 and 4.
+B. Mechanisms for Flux Quantum Transfer
+The periodic tuning of the resonator and the associated
+fit to the calculated energy states of the loop constitute
+strong evidence that the loop remains in its ground state
+for all values of Φapp, and this is made possible by a
+single flux-quantum transferring into or out of the loop at
+δΦ = 1/2. It is important to establish the mechanism by
+which the flux is able to enter the loop, so we now turn our
+attention to the transitions between flux states. Across
+multiple devices on multiple chips, we found an onset
+
+-45.0
+6Φ= 0
+-45.2
+45.4
+(dB
+45.6
+S21
+45.8
+-46.0
+6Φ = 1/2
+46.2
+3.378
+3.380
+3.382
+3.384
+Frequency(GHz)5
+of flux-periodic tuning in devices containing nanowires
+with w ≲ 35 nm. This dependence of the behaviour on
+nanowire width suggests that the flux tunnelling occurs
+in the nanowires, and not in the wider part of the loop.
+We can now examine some possible physical processes
+that could occur in the nanowires and see how the data
+fits with them.
+-Does the nanowire current exceed its critical current
+Inw
+c ?
+Niobium-nitride resonators commonly exhibit a
+nonlinear S21 response when they are driven with a high
+microwave power [44] as a result of the current-induced
+nonlinear kinetic inductance. As we see in Eq. 2, the
+kinetic inductance is quadratically dependent on (I/Ic)2,
+and so the nonlinearity must be dominated by the part
+of the conductor with the lowest Ic. This is confirmed by
+measurements of our circuits, where we find that nanowire-
+embedded resonators show a much higher degree of non-
+linearity in their S21 response than bare resonators (see
+Supplemental Information). The quadratic nature of the
+nonlinearity suggests that a strongly nonlinear response is
+a consequence of the magnitude of the rf current Ires in the
+resonator reaching a significant fraction of the nanowire
+critical current Inw
+c . Our NbN resonator readout method
+therefore gives us an indirect readout of whether the
+current in the nanowire is close to Inw
+c .
+Figure 3 shows the S21 response of the nanowire-
+embedded resonator at δΦ = 0 and δΦ = 1/2, both
+measured in the low-power limit with an estimated res-
+onator photon population of ⟨n⟩ ≈ 50. In both cases, the
+response is linear and well fitted by Eq. 1. We calculate,
+using the relation Is = dE(Φ)/dΦ that the maximum
+induced screening current in the nanowires Is(δΦ = 1/2)
+is 120 nA, an order of magnitude less than Inw
+c . Crucially,
+the lack of nonlinearity of the resonance at δΦ = 1/2,
+along with the fact that Qi remains a significant fraction
+of its zero-field value, means that the nanowires are not
+being driven close to their critical current by the applied
+flux. By a similar argument, we know that the nanowires
+are not being driven through Tc by a local heating process,
+as this would also result in nonlinearity of the resonance
+at δΦ = 1/2 due to dissipation in the highly resistive
+normal metal.
+-Is the nanowire a constriction Josephson-junction?
+‘Dayem-bridge’ Josephson-junction SQUIDs are com-
+monly embedded within CPW resonators [45, 46] to
+provide a flux-tunable resonant frequency.
+However,
+when the SQUID loop has a large inductance, one ob-
+serves hysteretic tuning, characterised by the parameter
+βL = 2LIc/Φ0. When βL ≳ 1, the SQUID behaviour be-
+comes hysteretic with applied flux and the resonator will
+exhibit discontinuous jumps in the resonant frequency,
+as observed in [47, 48]. Our device does not undergo
+any discontinuous jumps, and the tuning over a single
+flux-quantum is symmetric, so βL < 1. Given this and the
+known loop inductance, we can set an upper bound on the
+critical current of any Josephson junction of IJJ
+c
+< 100 nA.
+This bound is 10× smaller than the expected transport
+critical current of our nanowires and also less than Is(δΦ0).
+FIG. 4. Normalised tunnelling flux ∆φ+
+N (defined in the main
+text), as a function of the winding number N.
+The solid
+line corresponds to half a flux quantum, ∆φ+
+N = 0.5.
+In-
+set: Magnetic-field dependence of resonant frequency of the
+nanowire-embedded resonator up to an applied field of 2.4
+mT. The black dotted lines mark transitions between winding
+number states (φ+
+N), and the red dotted lines mark the energy
+minimum of each winding number state (φmin,N).
+Therefore, it is unrealistic to conclude that the flux-tuning
+we observe is a consequence of a dc SQUID formed of
+Dayem-bridge Josephson junctions. We also note that
+the closeness of the fit shown in Fig. 2 suggests there is
+no contribution to the flux-dependent inductance from a
+Josephson junction, which would add a 1/ cos Φ term to
+Eq. 2.
+-Is this a thermal or quantum process? Figure 4 shows
+that tunnelling always occurs at degeneracy. Following
+Petkovi´c et al. [30], ∆φ+
+N is defined as ∆φ+
+N = φ+
+N−φmin,N,
+where φ+
+N is the normalised flux φ = Φ/Φ0 at which tun-
+nelling from state N to state N + 1 occurs, and φmin,N is
+the flux that minimises the loop free-energy for a particu-
+lar winding number N. Our data shows the periodicity,
+defined in this way, to be half a flux quantum for all values
+of the winding number except N = 1 (we attribute the
+latter exception to enhanced flux-focusing at low magnetic
+field). This is in contrast to [30], where a characteristic
+dependence of ∆φ+
+N on N is shown to be a defining fea-
+ture of thermally-activated phase-slips. By following the
+method of [30], we calculate that ∆φ+
+N ≈ 300 would be
+required in order for the energy barrier to phase slips
+in our nanowire to be tuned to ≲ kBT.
+Correspond-
+ingly, we can estimate (see Supplemental Information)
+that ΓQPS = 35 MHz for our nanowire, and we calculate
+a temperature-dependent ΓTAPS that is below 1 Hz for
+T < 1.5 K. Therefore, at our measurement temperature,
+quantum tunnelling of flux is overwhelmingly more likely
+than a thermal transition. For comparison, the inverse
+experimental timescale is 1/τE ≈ 0.2 Hz since, for each
+
+0.5
+0.4
+3.380
+Resonant Frequency (GHz)
+3.378
+0.3
++z
+3.374
+0.2
+3.372
+3.370
+0.1
+3.368
+3.366
+0.0
+0.0
+0.5
+1.0
+1.5
+2.0
+Applied Field (mT)
+0
+2
+4
+6
+8
+10
+12
+14
+16
+Winding NumberN6
+setpoint of the magnetic field, it takes the VNA approxi-
+mately 5 s to collect S21 data across the resonance. Be-
+cause kBT/h > ΓQPS ≫ 1/τE — and to our knowledge
+this is the first reported study in this regime — when
+our device bias is swept through the degeneracy point,
+a single quantum phase-slip always occurs before we are
+able to observe the system in a higher-energy metastable
+state.
+IV.
+CONCLUSIONS
+We have used a Ne FIB to fabricate NbN nanowires
+with widths down to 25 nm embedded within CPW res-
+onators.
+We observe periodic modulation of resonant
+frequency and intrinsic quality factor, which is consis-
+tent with quantum tunnelling of individual flux quanta
+mediated by quantum phase-slip, occurring when states
+of different winding number become degenerate. This
+behaviour has been observed in resonators with intrinsic
+quality factor, Qi, up to 2.7 × 104 at 300 mK, which to
+our knowledge is the highest quality factor measured in
+quantum phase-slip experiments — note that the losses
+here are significantly lower than suggested by comparable
+reports [10]. We estimate that the QPS rate is of the
+order 10–100 MHz, which means that the tunnelling of
+a single flux-quantum is effectively deterministic on the
+timescale of microseconds. This result shows the suitabil-
+ity of the Ne FIB process for fabricating QPS devices. We
+also suggest that an incoherent QPS device with a high
+QPS rate such as ours could be promising for classical
+digital logic processing applications, where the quantum
+nature of the flux tunnelling implies a reduction in heat
+dissipation compared with current state-of-the-art devices,
+opening up a route to resolving an important roadblock
+to the upscaling of qubit control electronics.
+ACKNOWLEDGMENTS
+The authors thank O. W. Kennedy for useful dis-
+cussions. The authors gratefully acknowledge funding
+from the United Kingdom Engineering and Physical
+Sciences Research Council, Grant Nos. EP/L015242/1,
+EP/J017329/1, and EP/T001062/1.
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+Joint analysis constraints on the physics of the first
+galaxies with low frequency radio astronomy data
+Harry T. J. Bevins1,2,∗, Stefan Heimersheim3, Irene Abril-Cabezas3, Anastasia Fialkov2,3,
+Eloy de Lera Acedo1,2, William Handley1,2, Saurabh Singh4, and Rennan Barkana5,6
+1 Astrophysics Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge, CB3 0HE, UK
+2Kavli Institute for Cosmology, Madingley Road, Cambridge CB3 0HA, UK
+3 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
+4Raman Research Institute, C V Raman Avenue, Sadashivanagar, Bangalore 560080, India
+5School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv, 69978, Israel
+6Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540, USA and
+∗ htjb2@cam.ac.uk
+(Dated: January 10, 2023)
+Observations of the first billion years of cosmic history are currently limited. We demonstrate
+the synergy between observations of the sky-averaged 21-cm signal from neutral hydrogen and
+interferometric measurements of the corresponding spatial fluctuations. By jointly analysing data
+from SARAS3 (redshift z ≈ 15−25) and limits from HERA (z ≈ 8 and 10), we produce the tightest
+constraints to date on the astrophysics of galaxies 200 million years after the Big Bang. We disfavour
+at 95% confidence scenarios in which power spectra are ≥ 126 mK2 at z = 25 and the sky-averaged
+signals are ≤ −277 mK.
+PROBING THE INFANT UNIVERSE
+The infant Universe, corresponding to cosmic time between 100 and 700 million years after the Big Bang, remains
+largely unexplored. This epoch in cosmic history covers the birth of primordial stars, formation of the very first black
+holes and assembly of the earliest galaxies. The nature of the first bright objects to form and the exact timing of these
+events are yet to be constrained by observations. Theoretical studies and numerical simulations suggest that first
+stars form between z ∼ 20 − 60, i.e. around 35 − 200 million years after the Big Bang[1–3], and, thus, are currently
+out of reach of the modern telescopes.
+The newly launched James Webb Space Telescope (JWST) [4–6], with an increased sensitivity over previous in-
+struments such as the Hubble Space Telescope (HST), is advancing the observational frontier [7] with the prospect of
+infrared observations of the brightest early galaxies out to z ≈ 20 [4]. A complementary view of the infant Universe
+will be offered by radio telescopes such as the upcoming Square Kilometre Array (SKA) [8–10] which will probe
+the astrophysical processes at early times by measuring the redshifted 21-cm line of neutral hydrogen gas located
+in-between the first star forming regions. This signal, sensitive to the processes of star formation, cosmic heating and
+reionziation, can inform us about the state of the Universe between 100 and 1000 million years after the Big Bang
+[11–13]) and provide an insight into the properties of the first sources of light [8–10]. This joined effort across the broad
+wavelength range will lift the veil on the period between the formation of the Cosmic Microwave Background (CMB),
+when the Universe was approximately 300, 000 years old, and the end of the Epoch of Reionization (EoR).
+The field of 21-cm cosmology is rapidly evolving with the very first, yet unconfirmed, detection of the sky-averaged
+(or global) 21-cm signal made with the EDGES Low Band antenna and reported in 2018 [14]. This tentative detection
+is much deeper than what is predicted by conventional theoretical modelling [15–19] and, thus, is hard to interpret,
+and the cosmological nature itself of the EDGES absorption feature has been disputed by a number of works [20–24]
+which suggest the existence of unaccounted for systematics in the data. However, if this signal is of cosmological
+origin, the redshift range of the detected absorption feature implies an onset of efficient star formation at z ∼ 22
+followed (with some delay) by a strong heating of the neutral gas. To explain the anomalously strong absorption,
+exotic mechanisms need to be invoked such as cooling of neutral gas via interactions with charged cold dark matter
+[25–32] or production of strong radio background in addition to the CMB [33–36], e.g. by radio-luminous galaxies
+such as considered here.
+Although the EDGES signal is the only detection of the high-redshift 21-cm signal reported to date, advances have
+been made by other existing radio telescopes with upper limits reported by both the interferometers such as PAPER
+[37], MWA [38, 39], LOFAR [40, 41], AARTFAAC [42] and HERA [43–45] on the 21-cm power spectrum, which
+quantifies the variation in the 21-cm brightness field as a function of angular scale and time, and on the global 21-cm
+signal by SARAS [46–49], EDGES High Band [50–52] and LEDA [53]. These upper limits are becoming increasingly
+constraining and have already being used to put limits on the properties of high-redshift luminous objects [44–49, 54].
+The ongoing and upcoming experiments including SARAS [55], MIST [56], REACH [57], LOFAR [58], NenuFAR [59],
+arXiv:2301.03298v1  [astro-ph.CO]  9 Jan 2023
+
+2
+HERA [60], the SKA and future proposed space based missions such as DAPPER and FARSIDE [61] aim to further
+improve our understanding of the infant Universe.
+Existing upper limits on the 21-cm signal provide the first very weak constraints on the astrophysical sources at a
+broad range of redshifts. In this paper, we take the current tightest publicly available upper limits to date from the
+HERA interferometer on the magnitude of the 21-cm power spectrum at redshifts z ≈ 8 and ≈ 10 which provides a
+window to the EoR when ultraviolet photons emitted by first massive galaxies efficiently ionize the neutral hydrogen
+gas [44], and the SARAS3 radiometer [55] that probes the sky-averaged 21-cm emission at higher redshifts between
+z ≈ 15 − 25 when the first stars and X-ray emitting objects are expected to have formed in small galaxies during
+the Cosmic Dawn. We combine these two data sets for the first time to improve constraints on the properties of the
+first galaxies and the state of the neutral gas. We develop the methods to perform the joint analysis using a machine
+learning enhanced Bayesian workflow and pave the way for future discoveries. We find that when considered together,
+these two experiments provide a better leverage on theoretical scenarios that bridge across the wide redshift range
+compared to the constraints from each individual experiment. In synergy, the two experiments leave only 64.9+0.3
+−0.1%
+of the explored broad theoretical parameter space to be consistent with the joint data set in comparison to 92.3+0.3
+−0.1%
+for SARAS3 and 79.0+0.5
+−0.2% for HERA alone. We use the joint analysis to limit star formation efficiency, minimum
+halo mass for star formation, X-ray luminosity of early emitters and the radio luminosity of early galaxies. The joint
+analysis disfavours at 68% confidence a combination of galaxies with X-ray emission that is ≲ 33 and radio emission
+that is ≳ 32 times as efficient as present day galaxies.
+In Methodology we review the data and the specifics of the experiments incorporated in our analysis. This is followed
+by a discussion about the synergies between the power spectrum and sky-averaged 21-cm experiments in the same
+section. We present the implications of our work for the astrophysical constraints and the validity of the EDGES
+absorption feature as a sky-averaged 21-cm signal in Results. This is followed by a summary in Conclusion. Additional
+information about the methodology and additional results can be found in Supplementary materials.
+METHODOLOGY
+The SARAS3 radiometer experiment took 15 hours of data, integrated in the frequency range 55 − 85 MHz, from a
+lake in Southern India. After corrections were made for the receiver noise temperature, radio frequency interference
+and emission from the water beneath the antenna, the data was appropriately scaled by the total efficiency of the
+system to produce a measurement of the average sky temperature which is expected to include the Galactic and
+extra-galactic foregrounds as well as the 21-cm signal. SARAS3 reported a null detection of the EDGES absorption
+feature with a confidence of 95.3% [55] and more recently the data has been used to place constraints on the properties
+of the first galaxies which we discuss in more detail below [49].
+SARAS2, the previous iteration of the instrumentation recorded data at higher frequencies (lower redshifts) of
+110 − 200 MHz and was found to contain a non-smooth systematic structure possibly caused by emission from the
+ground that the antenna was placed on. The data has been used to derive constraints on galaxies in the infant
+Universe initially by fitting the foreground and systematic structure together with high order polynomials [46, 47] and
+subsequently by modelling the two components separately [48]. In Supplementary materials we combine constraints
+from the latter with constraints from SARAS3 and HERA.
+To date, interferometric experiments have only observed upper limits of the cosmological 21-cm power spectrum,
+which still allow for a large range of realistic astrophysical scenarios. The tightest constraints are derived from the
+data from the HERA interferometer [43], followed by MWA in the redshift range z = 6.5−8.7 [38] and LOFAR, which
+provide the tightest upper limits at redshifts z ∼ 9.1 [40] and z ∼ 9.3 − 10.6 [41]. The HERA telescope is a radio
+interferometer located in the Karoo Desert of South Africa [60]. Already, this first public data release delivered the
+strongest constraints on the 21-cm power spectrum to date. The publicly available HERA data[62] from the analysis
+of Internal Data Release 2 that we use in this paper is based on 18 nights of observations, with 39 antennas operating
+at science quality level. For our analysis, we employ the publicly available spherically averaged power spectra derived
+from this data [43], in the wavenumber range k = 0.128 hMpc−1 to 0.960 hMpc−1 and from the two bands focusing
+on redshifts z = 7.9 and 10.3. In Supplementary materials we consider the implications of including the upper limits
+on the power spectrum from MWA and LOFAR in our joint analysis.
+In 21-cm cosmology, data analysis efforts are increasingly employing Bayes theorem
+P(θ|D, M) = L(θ)π(θ)
+Z
+,
+(1)
+to derive constraints on the astrophysical scenarios of the early Universe. Here the likelihood, L(θ), represents the
+
+3
+probability that we observe the data, D, from SARAS3 or HERA, given a particular model, M. The prior, π(θ),
+represents our assumed knowledge before we consider any data and the evidence, Z, is a normalisation constant. The
+posterior, P(θ|D, M), tells us which parts of the parameter space, θ, given the data and chosen model, are more
+probable than others. The evaluation of Bayes theorem is usually performed with Nested Sampling or Markov Chain
+Monte Carlo (MCMC) algorithms (see Supplementary materials). In many 21-cm experiments, θ is composed of
+parameters that describe instrumental effects, θI, foregrounds, θfg, and the astrophysical processes that influence the
+21-cm signal, θ21. We typically refer to the set of θI and θfg as the nuisance parameters. Since we are only interested
+in the 21-cm signal, we work with the marginal or nuisance-free posteriors and nuisance-free likelihoods, L(θ21), which
+can be estimated using normalising flows and the marginal Bayesian statistics code margarine [63, 64]. We give an
+overview of this in Supplementary materials however we note that it allows for efficient combination of the HERA and
+SARAS3 constraints. We detail the astrophysical processes included in the modelling, and the definition of θ21, next.
+In order to realistically model the range of time covered by the Cosmic Dawn and Epoch of Reionization, we need
+a consistent modelling of the cosmological and astrophysical processes from redshift 60, when star formation might
+have began, all the way to redshift 5 at the end of the EoR. To that end we employ semi-numerical simulations
+[18, 65–68] that have previously been used in the HERA [44], SARAS2 [48] and SARAS3 [49] analyses as well
+as similar analysis of the LOFAR [54] and EDGES High-Band [52] data. We model the three-dimensional 21-cm
+field as a function of cosmic time during the infant Universe taking into account important astrophysical process
+including the Wouthuysen-Field (WF) effect [68–70], heating of the intergalactic medium by X-ray [67], Ly-α [18]
+and CMB photons [35], multiple scattering of Ly-α, relative velocity between dark matter and gas [65], feedback of
+Lyman-Werner radiation on star formation [66], and radio emission from galaxies [36]. The key parameters in the
+astrophysical model are: the star formation efficiency, f∗, which quantifies the percentage of the baryonic mass in
+the star forming halos that is converted into stars; the minimum virial circular velocity, Vc, which is proportional
+to the cube root of the halo mass, M; the X-ray production efficiency, fX, which is directly proportional to the
+X-ray luminosity per star formation rate, LX/SFR measured in erg s−1 M−1
+⊙ yr, between 0.2 and 95 keV; the CMB
+optical depth, τ; finally, we model the contribution of high redshift radio-luminous galaxies to the radio background
+by specifying a radio production efficiency, fradio, which is proportional to the radio luminosity per star formation
+rate, Lr, measured in W Hz−1 M−1
+⊙
+yr at 150 MHz, and normalized such that it has a value of one for the present
+day population of radio galaxies [36]. The X-ray spectral energy density is modelled based on a population of X-ray
+binaries as in [71]. In our Bayesian analysis, θ21, therefore, comprises the set of parameters {f∗, Vc, fX, τ, fradio} or
+equivalently {f∗, M, LX/SFR, τ, Lr/SFR}.
+We explore wide prior ranges on all the parameters in an attempt to let the data inform us about the high-redshift
+astrophysical processes. Specifically, the model for radio-luminous galaxies that we employ here is not conventionally
+considered in 21-cm cosmology where typically the CMB is assumed to be the only source of radio background photons.
+Here we expect that early galaxies will contribute to the radio background, thus increasing the amplitude of both the
+sky-averaged 21-cm signal [72] and the power spectrum [35, 36].
+Both the sky-averaged 21-cm signal and the power spectrum rely on the same underlying physics, and constraints
+from experiments targeting the different probes can be effectively combined to improve our knowledge of the infant
+Universe. The semi-numerical simulations take of order a few hours to produce a signal per parameter set, which
+is impractical for nested sampling or MCMC runs. We therefore train neural networks on outputs of the detailed
+simulations. Typically, these networks take of order a few tens of milliseconds to evaluate meaning they are much
+more well suited for computationally intensive fitting algorithms. The specific emulators used in this analysis and
+more details regarding the signal modelling can be found in Supplementary materials.
+RESULTS
+Although the constraints presented here are weak, through the novelty of combining the previously reported HERA
+and SARAS3 constraints we produce the tightest constraints to date on the properties of the infant Universe as
+detailed below. This is the first time a joint analysis between a global signal data and interferometric limits has been
+attempted.
+To visualize the importance of combining the constraining power of HERA and SARAS3 we show, in the top panel
+of Fig. 1, constraints on the ratio of the spin temperature of neutral hydrogen and background radiation temperature.
+The background radiation temperature is the sum of the CMB temperature and radio background from galaxies.
+The ratio determines the maximum absorption of the sky-averaged 21-cm signal, the smaller the ratio the larger the
+signal can be. The SARAS3 limits on the 21-cm signal (grey markers) correspond to lower limits on Ts/Tr, with the
+corresponding 1 and 2 σ contours shown as lines and extrapolated out of the SARAS3 band. Our simulations provide
+
+4
+FIG. 1. Key results from the joint analysis. Top Panel: The ratio of the spin temperature of neutral hydrogen, Ts, and
+the radio background temperature, Tr, as a function of redshift for the joint HERA and SARAS3 analysis in green. We show the
+HERA and SARAS3 68% and 95% confidence constraints in blue and grey respectively as triangles at the relevant redshifts and
+solid and dashed lines. As a guideline, we show the ratio for Tr = TCMB and assuming adiabatically cooled gas in an expanding
+Universe in the absence of any heating but with saturated coupling between Ts and the gas kinetic temperature (dashed black
+line). Bottom Left: The 2D PDF from the joint analysis on the minimum virial circular velocity, Vc, in combination with the
+star formation efficiency, f∗, marginalising over fradio, fX and τ. The solid black line shows the 68% contour, approximated by
+the pink dashed line, and the black dashed line shows the 95% contour. The joint analysis disfavours low values of Vc and high
+f∗ corresponding to efficient star formation. Bottom right: The constraint on the X-ray and radio luminosities from the joint
+analysis marginalising over Vc, f∗ and τ. The joint analysis disfavours at 68% confidence low X-ray efficiencies in combination
+with high radio production efficiencies.
+a natural link between the power spectra and the global quantities, e.g. Ts/Tr, meaning that we can use the limits on
+the power spectrum from HERA to derive an equivalent constraint on Ts/Tr. These constraints are shown in Fig. 1
+(blue markers and lines). The joint constraint, as shown by the green contours, provides the strongest constraints on
+the ratio, and in particular at redshifts z = 15 − 20, gives better constraints than either experiment alone. Further,
+the combination of the two experimental data sets improves the constraints at intermediate redshifts over a pure
+extrapolation of each one of the sets of constraints. As a guideline, the dashed black line in the figure shows the
+ratio for Tr = TCMB, i.e. in the absence of radio emission from galaxies, and assuming adiabatically cooled gas in
+an expanding Universe in the absence of any astrophysical heating sources but with saturated coupling between the
+21-cm spin temperature and the gas kinetic temperature. This limit is often used in the literature to give context to
+the constraints (e.g. [44, 45]). However, we note that for the models tested here the gas does not cool adiabatically
+because of the CMB and Ly-α heating at early times, and we have an excess radio background above the CMB.
+Next, we explore the functional constraints in the T21−z and ∆21−z planes as shown in Fig. 2. These are calculated
+by taking the samples of θ21 output from our fits and using the neural network emulators to produce corresponding
+global signals and power spectra. Using our theoretical models we can easily map between the constraints on the
+
+0.0
+-0.5
+log(T/Tr)
+SARAS3 + HERA
+.1.0
+HERA
+SARAS3
+HERA Band
+-1.5
+SARAS3 Band
+-2.0
+10
+15
+20
+25
+30
+Z
+26
+log(Lr /SFR)
+1.5
+1.0
+22
+-2
+38
+40
+42
+log(f*)
+log(Lx/SFR)5
+FIG. 2. Functional constraints on T21 and ∆2. The functional prior (purple), SARAS3 (grey), HERA (blue) and joint
+(green) posteriors for the sky-averaged 21-cm signal (top row) and power spectrum (bottom row). The yellow shaded region
+shows the SARAS3 band and the dashed yellow lines show the HERA redshifts.
+The functional prior and posteriors are
+calculated by taking representative samples from the corresponding probability distributions for the astrophysical parameters
+and generating the corresponding signals using neural networks. We see that by combining the constraining power of HERA and
+SARAS3, we significantly reduce the 3σ (lightest shaded regions) constraints on the magnitude of both signals from −2630 mK
+to −1770 mK at z = 15 for the global signal and 103.7 mK2 to 103.2 mK2 for the power spectrum at z = 25. The figure is
+produced with fgivenx [74].
+power spectra and global signals. We see that for both the power spectra and the sky-averaged signals, although more
+clearly for the latter, the range of plausible models is reduced by our joint analysis. The signals that are inconsistent
+with the data typically have strong power spectra and corresponding deep absorption trough in the global signal, and
+belong to scenarios with weak X-ray heating and strong radio luminosity. Further, we see that the 2σ region for the
+functional constraints correspond to signals with the power spectra ≲ 102.1 mK2 at z = 25, which via our modelling
+maps to global 21-cm signals shallower than −277 mK. We note that the 3σ limit on the power spectrum at z = 25 of
+≲ 103.2 mK2 is approximately equivalent to the expected sensitivity of NenuFAR from 1000 hours of observations [73].
+For the global 21-cm signal 3σ limit on the magnitude reduces from ≈ −2630 mK from HERA to ≈ −1770 mK from
+the joint analysis. Remarkably, we find that for the sky-averaged signal, the 2σ limit is very close to the minimum
+depth of the ‘standard astrophysical’ models, ≈ −165 mK [18], where the radio background is equated to the CMB,
+the contributions from radio galaxies and X-ray heating sources are assumed to be negligible, while the CMB and
+Ly-α heating are present. The relatively tight constraints on the global signal and power spectrum suggest that future
+improved measurements will allow us to dig deeper into the models with weak excess radio background radiation. It
+is clear from our analysis, that the joint constraint improves limits on the range of plausible global signals and power
+spectra.
+We now interpret the constraints on the temperatures in terms of constraints on the properties of the first galaxies.
+The key constraints from the joint analysis are shown in the bottom panels of Fig. 1 including the limits on the high-
+redshift star formation which drives the high-redshift portion of the 21-cm signal via the process of Ly-α coupling
+(constraints in the planes Vc − f∗) and the constraints on the luminosity of X-ray and radio sources (Lr − LX) which
+primarily regulates the depth of the absorption trough. The full marginalised 1D and 2D posteriors corresponding
+to the joint analysis are shown in Fig. S1 and the key numerical results are summarized in Tab. I alongside the
+individual constraints from SARAS3 and HERA. We find that the combination of the two experiments leads to
+
+Prior
+SARAS3
+HERA
+Joint
+0
+[mK]
+-2500
+T21
+0
+0
+SARAS3
+L250
+-250
+-250
+-5000
+HERA
+10
+20
+30
+10
+20
+30
+10
+20
+30
+6
+[mK²]
+4
+2
+10
+20
+30 10
+20
+30 10
+20
+30 10
+20
+30
+Z
+Z
+Z
+Z
+1g
+2g
+3g6
+SARAS3
+HERA
+SARAS3 + HERA
+Signal
+Sky-averaged
+Power Spectrum
+Both
+z
+≈ 15 − 25
+≈ 8 & ≈ 10
+≈ 8, ≈ 10 &
+≈ 15 − 25
+Lr/SFR
+≳ 1.55 × 1025
+≳ 4.00 × 1024
+≳ 3.31 × 1024
+LX/SFR
+–
+≲ 7.60 × 1039
+≲ 3.71 × 1039
+Lr/SFR
+&
+LX/SFR
+≳ 1.00 × 1025 &
+≲ 1.09 × 1042
+≳ 4.00 × 1024 &
+≲ 7.60 × 1039
+≳ 3.16 × 1023 &
+≲ 1.00 × 1042
+M
+4.40 × 105 ≲ M
+≲ 1.10 × 107
+–
+2.55 × 105 ≲ M
+≲ 7.04 × 106
+f∗
+≳ 0.05
+–
+≳ 0.06
+f∗ & M
+≳ 0.03 &
+≲ 8.53 × 108
+–
+≳ 0.02 &
+≲ 4.50 × 107
+TABLE I. Key parameter constraints from SARAS3, HERA and the joint analysis. Here the SARAS3 and HERA
+limits are taken from the respective papers. In the top two rows we show the type of signal targeted by each set of analysis and the
+corresponding redshifts. The joint analysis produces improved constraints on the radio and X-ray backgrounds while retaining
+the constraining power of SARAS3 on the star formation properties of early galaxies. Lr is measured in W Hz−1 M−1
+⊙
+yr at a
+reference frequency of 150 MHz, LX in erg s−1 M−1
+⊙
+yr, and is calculated by integrating X-ray spectral distribution of sources
+between 0.2 and 95 keV assuming an X-ray SED consistent with that for X-ray binaries [71]. The halo mass, M, is measured
+in solar masses. These constraints are derived from Kernel Density estimates (KDE) of the 1D and 2D posterior distributions.
+stronger constraints in the two-dimensional probability distribution of Lr − LX than either of the two experiments
+individually. Where HERA constrains the population of high redshift radio-luminous galaxies to be ≲ 400 times
+brighter in the radio band than the current population, the combination of the data sets constrains the galaxies to be
+≲ 300 times brighter when marginalising over the other parameters (Vc, f∗, fX and τ). Similarly, HERA disfavours at
+68% confidence galaxies with an X-ray luminosity ≲ 0.25 times the present day value in combination with the radio
+luminosity of galaxies in the early universe that is ≳ 400 times the present day value. The joint analysis provides a
+stronger constraint, ruling out scenarios where the X-ray luminosity is ≲ 33 times the present day value and the radio
+luminosity of the first galaxies is ≳ 32 times the present day value at 68% confidence.
+We find comparable constraints on f∗ and minimum mass of star forming halos, M, as was found with SARAS3
+alone [49] when combining the data sets. Marginalising over the radio and X-ray luminosities, we disfavour at 68%
+confidence galaxies in which ≳ 2% of the gas is converted into stars and the minimum mass for star forming halos is
+≲ 45 million solar masses.
+Further, we explore the structure of the global 21-cm signals which are consistent with the data from SARAS3,
+HERA and the two sets together. In Supplementary materials and Constraints on Phemenological Parameters, we
+see that when considered individually, the SARAS3 and HERA experiments allow for astrophysically motivated
+sky-averaged 21-cm signals that have a minimum temperature, location and width in agreement with the EDGES
+detection, while the joint analysis rules out models with a depth that is consistent with EDGES at greater than a 2σ
+significance. The joint analysis has a preference for shallower (lower values of |Tmin|) and narrower signals with higher
+central frequencies, as can be seen in the corresponding 1D PDFs, which is driven largely by the HERA constraints.
+We also consider the impact of SARAS2 on our joint constraints as well as MWA and LOFAR in Supplementary
+materials.
+The latter two data sets have no effect on our results and the inclusion of SARAS2 leads to weaker
+constraints on the star formation properties. We focus in the main text on SARAS3 and HERA, due to the uncertainty
+in the modelling of and presence of systematic structures in the SARAS2 data which is discussed in more detail in
+Supplementary materials.
+CONCLUSION
+Through a combination of constraints on fluctuations and the sky-averaged 21-cm signal of neutral hydrogen, we
+have improved our understanding of the first galaxies that formed in the infant Universe between 200 and 700 million
+years after the Big Bang. This is the first time the data from the two different 21-cm probes have been combined
+
+7
+to derive constraints on the astrophysical properties of the early galaxies. Even though the existing constraints are
+weak, we develop novel methodology and outline the approach which will become increasingly more useful as the
+next-generation experiments deliver stronger observational constraints.
+Considering a wide space of plausible astrophysical models including high-redshift sources of ultraviolet, X-ray and
+radio photons which affect the 21-cm signal, we calculate corresponding sky-averaged spectra as well as the power
+spectra of fluctuations. Using an upper limit on the fluctuations from the HERA interferometer and non-detection of
+the global 21-cm signal by the SARAS3 radiometer, we find that only 64.9+0.3
+−0.1% of the explored theoretical parameter
+space is consistent with the joint SARAS3 and HERA constraint, which is a significant improvement over the individual
+values of 92.3+0.3
+−0.1% and 78.7 ± 0.2% respectively.
+Using the newly developed methodology we place the tightest constraints to date on the properties of cosmic
+gas, such as the spin temperature of the 21-cm hydrogen line (closely related, but not equal, to the gas kinetic
+temperature) and the radio background temperature, Tr, as well as on the radio and X-ray luminosities of the first
+galaxies disfavouring at 68% confidence galaxies that are approximately 32 times more efficient radio emitters than
+present galaxies and simultaneously are less than 33 times bright in the X-ray band. This work reports an increased
+degree of confidence over a wider range of redshifts than previous works which typically extrapolate outside the
+redshifts targeted by individual experiments, while here we interpolated between the observations of SARAS3 at
+z = 15 − 25 and the HERA limits at lower redshifts z ∼ 8 and 10.
+In this work we also considered the addition of interferometric data sets from MWA and LOFAR to our analysis,
+which led to a negligible improvement in the results. Similarly to HERA, these experiments probe the physics of the
+EoR covering a similar redshift range to HERA (between z ≈ 6 − 10), with the current HERA data providing the
+tightest constraints on our models. Of the current global or sky-averaged 21-cm experiments only SARAS2, SARAS3
+and EDGES were able to place limits on the astrophysics of the infant Universe. Our main focus is on the SARAS3
+limits (although we also consider SARAS2), as there are concerns surrounding the cosmological nature of the signal
+reported in EDGES and a degree of uncertainty in the modelling of systematics in the SARAS2 data. We note that
+the SARAS2 data covers a similar redshift range as the HERA data and based on our analysis does not lead to a
+significant improvement in our constraints.
+The new methodology developed in this paper will allow for synergies between the upcoming observations, e.g. of
+the power spectrum from cosmic dawn measured by the NenuFAR [59], HERA, LOFAR or the SKA [8], as well as
+the measurements of the global signal by the wide-band REACH experiment covering the redshift range z ≈ 7 − 28
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+
+12
+ACKNOWLEDGMENTS
+HTJB acknowledges the support of the Science and Technology Facilities Council (STFC) through grant number
+ST/T505997/1. WJH and AF were supported by Royal Society University Research Fellowships. EdLA was supported
+by the STFC through the Ernest Rutherford Fellowship. RB acknowledges the support of the Israel Science Foundation
+(grant No. 2359/20), The Ambrose Monell Foundation and the Institute for Advanced Study.
+AUTHOR CONTRIBUTION
+HTJB lead the data analysis and the writing of the manuscript. SH provided the power spectrum emulator, assisted
+with the writing and data analysis. IAC performed the combined analysis of HERA, LOFAR and MWA data and
+assisted with the writing of the manuscript. AF generated the idea, supervised the project and contributed to the
+writing of the article. EdLA supervised the work and helped with the revision of the article. WJH supervised the
+analysis, helped with revision of the article and provided advice on the Bayesian methodology. The astrophysical
+signal models were provided by AF and RB and the SARAS3 data was provided by SS. RB and SS provided comments
+on the manuscript and assisted with revisions.
+
+13
+SUPPLEMENTARY MATERIALS
+Modelling the Thermal History of the Infant Universe: Power Spectrum and Global Signal Synergies
+At high redshifts around z ≈ 20−30, the first stars begin to form and produce Ly-α photons that interacts with the
+baryonic matter, predominantly composed of neutral hydrogen, in the Universe. Neutral hydrogen atoms absorb and
+remit ambient Ly-α photons in a process known as the Wouthuysen-Field (WF) effect [69, 70] that drives the relative
+number of atoms with aligned and anti-aligned proton and electron spins. This process couples the spin temperature
+of the neutral hydrogen, Ts, which describes the distribution of hydrogen atom spins, to the gas temperature, Tk,
+which is cooling at a faster rate than the radio background as the Universe expands. Further, interactions between
+the neutral hydrogen and Ly-α emission result in the transfer of kinetic energy that raises the gas temperature, and
+coupled spin temperature, in a process known as Ly-α heating [76, 77] preventing the gas from cooling adiabatically.
+However, despite the heating, the dominant WF effect produces an absorption feature in the sky-averaged 21-cm
+signal, which is measured relative to the radio background. Further, it leads to a peak in the power spectrum at high
+z ≈ 25 on angular scales corresponding to the effective horizon of the Ly-α emission and the distribution of galaxies,
+which disappears when the coupling becomes saturated [78].
+The intensity and spatial fluctuations of the Ly-α
+emission evolve with the population of early galaxies and consequently it is dependent on their star formation rate
+and the minimum halo mass for star formation. We parameterize these quantities with the star formation efficiency,
+f∗, which quantifies the percentage of the baryonic mass in the star forming halos that is converted into stars, and
+the minimum virial circular velocity, Vc, which is proportional to the cube root of the halo mass, M.
+At intermediate redshifts of z ≈ 10 − 20 the gas is further heated by X-ray binaries [67, 71], continuing Ly-α
+heating, CMB heating [79] and heating through structure formation. X-ray heating is dependent on processes such as
+X-ray production in high redshift galaxies and black hole binary formation. This means that it has to be separately
+parameterized and in our semi-numerical simulations, we model the X-ray production efficiency, fX, which is directly
+proportional to the X-ray luminosity per star formation rate, LX/SFR measured in erg s−1 M−1
+⊙ yr, between 0.2 and
+95 keV. The heating affects the redshift of the minimum and depth of the sky-averaged 21-cm signal. If sufficiently
+efficient it can raise the temperature of the gas, and coupled 21-cm brightness temperature, above the radio background
+resulting in emission in the sky-averaged signal at low redshifts. The heating rate has a direct impact on how fast
+the brightness temperature transforms from absorption to 0 K or emission. In the power spectrum, due to the non-
+uniformity of the heating, the various mechanisms can produce a peak in the signal at around z ≈ 15. Although this
+redshift is model dependent (see [78]) and in some cases when heating is done by hard X-rays (energies > 1 keV with
+long mean free paths) X-ray heating is smooth and no peak is imprinted in the power spectrum [67, 68].
+Finally, at more recent times, z ≈ 5 − 15, ultraviolet emission from the first massive galaxies begins to ionize the
+neutral hydrogen, stripping the abundant gas of its electrons. This reduces the sky-averaged 21-cm signal and when
+reionization is complete, the signal disappears. The process produces a peak in the power spectrum at the scales
+corresponding to the typical size of ionized bubbles, but again the signal is destroyed once reionization is complete.
+The process of reionization is highly dependent on the ionizing efficiency of sources, ζ, which in the models explored
+here is normalized by the CMB optical depth, τ. While τ has been weakly constrained by cosmological experiments
+such as Planck [80], we treat it as a free parameter and note that 21-cm cosmology offers a means by which to break
+degeneracies between τ and other cosmological parameters.
+Throughout our analysis, we explore a broad range of radio luminosities that produce radio excesses above the
+CMB of between ≈ 0.5 − 270 times the CMB temperature at z = 20 and ≈ 1 − 32000 times the CMB at z = 10.
+While moderate radio emission might be expected [81], the extreme values of radio production efficiencies are usually
+invoked to explain the anomalously deep EDGES signal [14, 33–36, 72, 81] but struggle to explain the rapid star
+formation and rapid heating of the gas [82] that is implied by the shape of the EDGES signal.
+Combining Constraints with margarine
+We use the marginal Bayesian statistics code margarine [63] to combine constraints from different data sets.
+margarine uses neural networks known as Masked Autoregressive Flows (MAFs) to model the probability distribu-
+tion, P(θ|D, M), of a set of samples, here θ21, marginalising over nuisance parameters describing the instrumental
+systematics, θI, and foregrounds, θfg in the process. It does this by shifting and scaling a base standard normal
+distribution, making the probability of the distribution easily tractable, to replicate the target samples, where the
+shifting and scaling are determined by the outputs of the MAFs.
+
+14
+This can subsequently be used to calculate a nuisance-free likelihood given a prior distribution and the Bayesian
+evidence using
+L(θ21) ≡
+�
+L(θ21, α)π(θ21, α)dα
+�
+π(θ21, α)dα
+= P(θ21|D, M)Z
+π(θ21)
+,
+(2)
+where α = {θI, θfg} [64]. In instances when the prior is flat then the following is true up to a normalisation constant
+L(θ21) ≈ P(θ21|D, M).
+(3)
+With margarine we can subsequently evaluate log(L(θ21)) for any set of θ21 for any existing posterior distribution,
+P(θ|D, M), from previous analysis of a data set like HERA or SARAS3
+θ = {θI, θfg, θ21} → {θ21} → margarine → log(P(θ21|D, M) → log(L(θ21)),
+(4)
+where the log is base 10. The likelihood evaluations can then be combined,
+log(Ljoint(θ21)) = log(LHERA(θ21)) + log(LSARAS3(θ21)),
+(5)
+as discussed in the main text, to be sampled over using MCMC methods or in our case nested sampling implemented
+with polychord [83, 84].
+MCMC sampling methods approximate the unnormalised posterior distribution by directly sampling the product
+P(θ) ≈ L(θ)π(θ) and do not provide estimates of the evidence, Z. The family of algorithms typically use random
+walkers to traverse the parameter space, with points accepted and rejected based on some probabilistic criteria in an
+effort to explore the space fully. The HERA analysis [44] of the excess radio background models explored here used
+the emcee [85] implementation of MCMC sampling [44].
+Nested sampling [86] numerically approximates the integral
+Z =
+�
+L(θ)π(θ)δθ,
+(6)
+which can be derived from equation 1 and the requirement that the posterior must integrate to 1, by evolving a series
+of live points to higher and higher likelihood values. By approximating the evidence, the algorithm produces samples
+on the normalised posterior distribution, which we subsequently use to determine preferred and disfavoured regions
+of the parameter space. A comprehensive review of nested sampling can be found in [87].
+Emulating Signals from the EoR and CD
+As described in the main text, semi-numerical simulations typically have a runtime of multiple hours for a given set
+of astrophysical parameters, thus performing parameter inference with the simulation directly is very costly. However,
+as both the sky-averaged signal and the power spectrum change smoothly when we vary parameters, we can interpolate
+values at intermediate parameters from existing simulations. An increasingly common practice to achieve this are the
+fast and precise neural network based emulators we discuss in the following sections.
+The sky-averaged 21-cm signal
+To emulate the sky-averaged 21-cm signal in models with an excess radio background from high redshift radio-
+luminous galaxies we use the publicly available emulator globalemu [88] trained on sets of astrophysical simulations.
+The simulations include the effects of Ly-α and CMB heating. The mean free path of ionizing photons, Rmfp, is fixed
+at 40 Mpc and the and X-ray heating is powered by a population of X-ray binaries with a realistic spectral energy
+distribution [67]. The models correspond to the parameterisation detailed above and in [36].
+The emulator has previously been used in the individual analysis of data from SARAS2 [48] and SARAS3 [49]
+and detailed in the corresponding papers. We therefore only briefly summarise the accuracy of the neural network
+here. The original set of simulations comprise approximately 10,000 models however the explored range of τ, the
+CMB optical depth, is large and when training our neural network we filter out models that have a value of τ in the
+range given by Planck ±3σ. This results in a training set of approximately 4,300 models and the network is tested
+on approximately 500 models. A root mean squared error (RMSE) of 5.11 mK is found and a 95 percentile RMSE of
+20.53 mK indicating a high level of accuracy (see Table M.2 in [49]).
+
+15
+The 21-cm power spectrum
+For comparing models with the HERA measurements we use the same 21-cm power spectrum emulator used in the
+HERA analysis [44], based on the same suit of simulations of the sky-averaged signal emulator used in the SARAS2
+and SARAS3 analysis. Based on an input of the 5 model parameters, this emulator returns the 21-cm power spectrum
+with a relative accuracy of 20% at the wave numbers and redshifts observed by HERA. For HERA the impact of τ is
+more significant because the data corresponds to lower redshifts and so the full prior range on the parameter is used
+for training. This results in a training set of ∼8,000 simulations and another 2,000 independent samples for testing.
+Full details and tests of this emulator can be found in the HERA analysis paper [44], Appendix B.
+We note that when performing our joint analysis we use the narrow prior on τ defined by Planck, since the SARAS3
+posterior is not defined in the original analysis for values outside this range.
+Temperatures
+To emulate physical properties such as the spin temperature and temperature of the radio background (as seen in
+Figure 1) we use a globalemu-style emulator. In this framework, the emulator takes in the five astrophysical pa-
+rameters and a single redshift and returns a single corresponding temperature. In practice, this means that vectorised
+calls have to be made to emulate the spin temperature, Ts(z), and the radio background temperature, Tr(z), as a
+function of redshift, but the method is found to be more accurate, quicker and allows for interpolation at a range of
+different redshifts. We use the same training and test sets as used for the 21-cm power spectrum emulator, and a
+similar architecture of 4 layers, with a reduced size of 100, 30, 10, and 5 nodes per layer. This allows us to emulate
+the spin temperature Ts within ±6% accuracy (95% confidence interval), and the radio background temperature Tr
+within ±4% accuracy (95% confidence interval).
+Constraints on the Astrophysical Parameters
+Fig. S1 shows the 1D and 2D marginal posteriors found for the joint analysis of SARAS3 and HERA. The graph
+shows that the combined constraining power of the two experiments leads to a strong constraint on the combination
+of the radio and X-ray luminosities per star formation rate for an early population of galaxies when marginalising
+over the other astrophysical parameters. We also see constraints in the plane of Vc − f∗ when marginalizing over the
+radio background and X-ray heating params. The constraints are summarized in detail in the main text.
+Constraints on Phemenological Parameters
+In order to explore the structure of the global 21-cm signal we look at the minimum temperature, Tmin, the
+corresponding central redshift, z0, and an approximate full width at half max, ∆z, of each absorption trough as
+defined in the top right of Fig. S2. More specifically, for each parameter set θ21 (either from the prior parameter
+range, or from the posterior ranges consistent with SARAS3, HERA, and the joint analysis) we generate a global
+signal using the neural network emulator globalemu [88]. We then measure the values of Tmin, z0, and ∆z for
+each signal producing probability distributions for each parameter corresponding to the constraints from each data
+and showing the extent of the prior (see results in Fig. S2). We compare these distributions with the values used to
+parameterise the phemenological EDGES flattened Gaussian signal with the EDGES 99% confidence ranges shown
+by the black crosses in the corner plot in Fig. S2.
+Constraints on the Radio and X-ray Backgrounds
+The product f∗fradio is proportional to the total radio background created by radio-luminous galaxies, and, equiv-
+alently, f∗fX is a proxy for the total X-ray background created by the early population of X-ray sources. This is
+demonstrably true for our model parameterisation as the star formation rate is proportional to f∗ and the radio lumi-
+nosity and X-ray luminosities per star formation rate are proportional to fradio and fX respectively. Both X-ray and
+radio backgrounds are responsible for regulating the depth of the absorption feature, and they can also be observed
+independently by other telescopes (e.g. observations of the unresolved X-ray background by Chandra [89] and of the
+
+16
+FIG. S1. Parameter constraints from the joint anlaysis. The astrophysical parameter constraints on models with excess
+radio background in addition to the CMB derived when combining an upper limit on the 21-cm power spectrum at z ≈ 8
+and ≈ 10 from HERA with data from the 21-cm sky-averaged experiment SARAS3 in the band z ≈ 15 − 25. Through the
+combination of these two data sets probing different statistical properties of the 21-cm signal at different redshifts we are able
+to improve constraints on the radio and X-ray luminosities of early radio-luminous galaxies and maintain constraints provided
+by SARAS3 on the star formation properties of these early galaxies. Lr is measured in units of W Hz−1 M−1
+⊙
+yr at 150 MHz
+and LX is in units of erg s−1 M−1
+⊙
+yr calculated between 0.2 and 95 keV assuming a realistic SED of an early X-ray binary
+population. The pink dashed lines approximate regions that are disfavoured with 68% confidence.
+low-frequency radio background by ARCADE2/LWA [90, 91]. In Fig. S3 we show constraints on the values of f∗ fX
+and f∗ fradio achieved by SARAS3, HERA and the joint analysis. Since these combinations of the parameters regu-
+late the absorption depth of the global 21-cm signal, we can also condition our prior on the astrophysical parameters
+to produce signals with the same central frequency and depth as the absorption feature found in the EDGES data
+[18, 35, 49]. In each panel of Fig. S3, we show black contours corresponding to these EDGES-like physical signals. We
+see that while HERA and SARAS3 allow for combinations of fX f∗ and fradio f∗ that could partially explain EDGES,
+the combination of the two experiments, which produces a tighter constraint on the X-ray and radio luminosities of
+early galaxies, disfavours a large portion of the EDGES-like parameter space (i.e. most of the EDGES-like parameter
+space is beyond the 95% contours of the joint constraints, while it is well within the 95% contours for SARAS3 and
+HERA individually). This demonstrates further the power of combining different data sets. However, we note that the
+explored theoretical signals do not fit EDGES data well as none of them closely reproduces the flattened Gaussian-like
+feature found in the data [14].
+
+95% Confidence
+68% Confidence
+1.5
+1.0
+log(PR)
+42.5
+40.0
+37.5
+0.075
+0.050
+25.0
+22.5
+2
+5
+5.
+0
+5.
+0.050
+0.075
+25.0
+0.
+4
+T
+log( FR
+log(sFR)17
+FIG. S2. Phenomenological constraints. The triangular plots shows the prior (purple) and posteriors (grey for SARAS3,
+blue for HERA, green for joint) of the features of a typical global absorption signal: the central redshift, z0, the corresponding
+minimum temperature, Tmin, and the width of the signal, ∆z, as is depicted in the top right corner. Darker shaded regions
+show 1σ constraints, lighter shaded regions show 2σ constraints. Overlaid on the posterior distributions are the 99% confidence
+intervals, black crosses, reported for the corresponding phemenological parameterisation of the EDGES absorption feature in
+[14]. Note that this is not the same as the physical EDGES-like distribution explored in Fig. S3. We see that individually
+the experiments allow for signals with depths that are consistent with EDGES. However, the combination of the two data
+sets disfavours these signals with greater than 2σ significance. We do not disfavour signals with the same width or central
+frequency as EDGES, but note that the joint analysis indicates a preference for shallower and narrower signals with higher
+central redshifts as can be seen in the 1D PDFs.
+The impact of SARAS2
+Previous analysis of the SARAS2 data revealed some weak constraints, most notably in the plane of LX − Lr
+in agreement with HERA and SARAS3, on the properties of galaxies in the infant Universe [48]. SARAS2 is at
+much lower redshifts than SARAS3 but overlaps with the redshifts probed by HERA having recorded observations
+in the band z ≈ 7 − 12. The data is contaminated by a sinusoidal systematic and a number of different models
+were fitted to this feature. The model corresponds to a signal introduced prior to the antenna possibly from ground
+emission or some unknown component of the foreground and separately a signal introduced in the system electronics
+potentially from cable reflections.
+The sinusoidal systematic was fitted alongside a signal model generated with
+globalemu and a foreground model that is conditioned to be smooth preventing it fitting out non-smooth systematics
+or signals in the data [24]. Here we take the best fitting model, with a systematic from ground emission or a non-
+smooth component from the foreground, with the highest evidence from the original analysis [48] and combine the
+corresponding constraints on the astrophysical parameters Vc, f∗, fX, fradio and τ with the joint constraints from
+HERA and SARAS3 to assess the impact.
+As can be seen in Fig. S4, the addition of SARAS2 to our analysis washes out the constraint in the plane f∗ − Vc.
+
+Physical
+EDGES
+0
+Az
+K
+m.
+-250
+(Z0, Tmin)
+-500
+(Z0, Tmin)
+10
+20
+30 10
+20
+30
+之
+之
+Prior
+[mK]
+SARAS3
+500
+HERA
+SARAS3+HERA
+-1000
+16
+8
+8
+16
+24
+-1000
+-500
+8
+16
+Tmin [mK]
+Az
+2018
+FIG. S3. Background constraints. The figure shows constraints on the radio and X-ray backgrounds, parameterized by
+f∗fradio and f∗fX respectively, from SARAS3 (grey), HERA (blue) and the joint analysis (green). The SARAS3 and HERA
+posteriors are based on the results presented in [49] and [44] respectively. We show each distribution individually on the top
+row, overlaid pairs of distributions for comparison in the middle, and all three on the same figure in the bottom row. In
+all panels, we show 68% and 95% contours (black solid and dashed lines respectively) for physical signal models that have
+similar depths and central frequencies as the EDGES absorption feature as defined by the inequality in [35]. These physical
+EDGES-like models have previously been explored in the literature in [35, 36]. We note that while individually both HERA
+and SARAS3 allow for astrophysically motivated signal models that could explain the depth of the EDGES feature, together
+they rule the corresponding parameter space out with approximately greater than 2σ confidence, although some EDGES-like
+signals are still viable. We stress again, that the explored physical models cannot fully explain the shape of the EDGES signal.
+One possible explanation for this is that the addition of SARAS2, while constraining the properties that affect the
+signal at low redshifts, increases the envelope of possible models at higher redshifts, where star formation is more
+important, that are plausible even given the constraints from the SARAS3 data.
+Despite this, we note that we
+maintain the constraint in the plane LX − Lr when we add SARAS2 into our analysis.
+We can quantify the impact of SARAS2 on our analysis by looking at the percentage of the astrophysical prior volume
+which is consistent with the different combinations of the three different data sets. To calculate this percentage, we use
+margarine to calculate the marginal Kullback-Liebler divergence, D, between the flat prior on the five astrophysical
+parameters in the set θ21 and the corresponding posteriors. The KL divergence is related to the percentage via
+% = 100 × exp(−D) ≈ 100 × VP
+Vπ
+,
+(7)
+
+Individual Constraints
+SARAS3
+HERA
+Joint
+104
+radio.
+103
+f 102
+101
+10-2
+100
+10-2
+100
+10-2
+100
+fxf*
+fxf*
+fxf*
+Comparison of Pairs
+SARAS3 vs HERA
+SARAS3 vs Joint
+HERA vs Joint
+104
+*
+103
+102
+101
+100
+10-2
+100
+10-2
+100
+10-2
+fxf*
+fxf*
+fxf*
+Comparison with Joint
+SARAS3 vs HERA vs Joint
+SARAS3
+104
+HERA
+Joint
+103
+EDGES-like
+Signals
+102
+101
+10-2
+100
+fxf*19
+HERA + SARAS3
+1.0
+1.5
+log(Vc)
+37.5
+40.0
+42.5
+log( LX
+SFR)
+0.050
+0.075
+τ
+−2
+−1
+log(f∗)
+22.5
+25.0
+log( Lr
+SFR)
+1.0
+1.5
+log(Vc)
+37.5
+40.0
+42.5
+log( LX
+SFR)
+0.050
+0.075
+τ
+22.5
+25.0
+log( Lr
+SFR)
+HERA + SARAS3 + SARAS2
+95% Confidence
+68% Confidence
+1.0
+1.5
+37.5
+40.0
+42.5
+0.050
+0.075
+−2
+−1
+log(f∗)
+22.5
+25.0
+1.0
+1.5
+log(Vc)
+37.5
+40.0
+42.5
+log( LX
+SFR)
+0.050
+0.075
+τ
+22.5
+25.0
+log( Lr
+SFR)
+0
+max
+FIG. S4. The impact of SARAS2 data on the astrophysical constraints. We show the joint posterior distributions for
+HERA and SARAS3 on the left panel (identical to Figure S1, but shown here for comparison) and for HERA, SARAS3 and
+SARAS2 on the right panel. SARAS2 covers the band z ≈ 7 − 12 and therefore has some overlap with HERA but not with
+SARAS3. The addition of SARAS2 to the joint analysis washes out the constraint on star formation properties, Vc and f∗,
+because it leads to increased uncertainty in the structure of the signals at high redshifts. However, we still see a consistent
+disfavouring of a population of radio galaxies with high radio and low X-ray luminosities. The one dimensional posteriors for
+τ appear to be in disagreement, however, we note that these are basically flat. We exclude SARAS2 from our main results in
+the text because of uncertainty in the modelling of systematics in the data.
+where Vπ is the prior volume and VP is the posterior volume. This quantity is useful as it quantifies the constraining
+power of the different data sets in all five dimensions, including correlations that may not be visible in the one and
+two dimensional projections used to produce the corner plots in this paper and in the literature.
+We show in Fig. S5 the percentage of the astrophysical parameter prior volume that is consistent with different
+combinations of the data sets discussed in this work (including additional interferometric measurements of the power
+spectrum discussed in Other Power Spectrum Experiments). We see that the combination of either or both of the
+SARAS data sets with HERA lead to a percentage consistency with the data of ≈ 63 − 65% and this is likely
+dominated by HERA. Individually, HERA allows for ≈ 80% of the astrophysical parameter space, SARAS2 for
+≈ 90% and SARAS3 for ≈ 92%. Due to the uncertainty in the modelling of the systematics in the SARAS2 analysis,
+we leave SARAS2 out of the main results.
+Other Power Spectrum Experiments
+In Fig. S6, we show the projected posteriors derived using HERA data alone (left panel) and HERA, MWA and
+LOFAR together (right panel). We note that the constraints from the different interferometers are all at low redshifts
+between z ≈ 6 − 10 and varying wavenumbers or angular scales.
+These are detailed in Tab. S1 along with the
+constraints from the individual experiments on key parameters.
+We derived parameter constraints from the MWA and LOFAR data using the approach taken in the orginal HERA
+analysis. Specifically, we take the measured upper limits, the mean power spectrum and uncertainty, and treat it as a
+measurement of cosmological signal plus systematics. As in HERA [44] we take this uncertainty to be Gaussian and
+marginalize over a uniform prior on the systematics, yielding the likelihood
+
+20
+60
+65
+70
+75
+80
+85
+90
+95
+100
+% Prior Consistent with Data
+SARAS2 +
+HERA
+SARAS2 + SARAS3 +
+HERA
+SARAS3 +
+HERA
+SARAS3 +
+SARAS2
+HERA + MWA +
+LOFAR
+HERA
+SARAS2
+SARAS3
+FIG. S5. Constraining power of different data sets. The percentage of the wide astrophysical parameter prior that is
+found to be consistent with the different data sets and different combinations of data sets explored in this work. A lower
+value indicates a better set of constraints, although a difference of a few percent does not necessarily translate into significant
+differences in the parameter constraints as can be seen when comparing the results from HERA and HERA + LOFAR + MWA.
+HERA
+LOFAR
+MWA
+LOFAR + MWA +
+HERA
+z
+≈ 8 & ≈ 10
+9.1 & 9.3 − 10.6
+6.5 − 8.7
+Discrete and contin-
+uous ranges of z be-
+tween 6.5 − 10.6
+k [h Mpc−1]
+0.128 − 0.960
+0.075−0.432 & 0.053 0.070 − 3.000
+Discrete and continu-
+ous ranges of k
+Lr/SFR
+≳ 4.00 × 1024
+≳ 1.20 × 1025
+≳ 1.58 × 1025
+≳ 4.00 × 1024
+LX/SFR
+≲ 7.60 × 1039
+≲ 8.70 × 1038
+≲ 1.16 × 1039
+≲ 1.58 × 1040
+Lr/SFR
+&
+LX/SFR
+≳ 4.00 × 1024 & ≲
+7.60 × 1039
+≳ 3.16 × 1025 & ≲
+1.00 × 1040
+≳ 1.00 × 1025 & ≲
+1.00 × 1040
+≳ 4.00 × 1024 & ≲
+1.58 × 1040
+TABLE S1. Constraints from interferometers. The table shows the various constraints on the radio and X-ray luminosities
+for HERA, MWA, LOFAR and the combination of all three along with their respective wavenumbers and redshift ranges. The
+joint analysis only marginally improves our understanding of the infant universe.
+L(θ21) =
+Nd
+�
+i
+1
+2
+�
+1 − erf
+�di − mi(θ21)
+√2σi
+��
+,
+(8)
+where Nd represents the number of data points, di and σi correspond to the mean and standard deviation in each
+data point, and mi(θ21) is the model prediction for that redshift and wave number. Thus a model prediction m ≫ d
+gives L ≈ 0 while m ≪ d gives a constant. This likelihood is effectively a step function that disfavours models above
+a given amplitude. A full discussion of its derivation can be found in [44].
+We performed this joint analysis using a full analytic likelihood approach (independent of margarine) since there
+are no nuisance parameters describing the systematics. We find that each of the experiments disfavours individually
+similar regions of the Lr − LX plane. However, the joint analysis does not improve the results derived from HERA
+data alone (as we summarise in Tab. S1 and is further illustrated in Fig. S5) which motivates our decision to only use
+HERA in the main text.
+
+21
+FIG. S6. The impact of MWA and LOFAR on the parameter constraints. Projected posterior distribution functions
+(PDFs) for the 5 simulation parameters, obtained by assuming flat priors and combining different observations: HERA alone
+(left, as in [44]) and LOFAR, HERA and MWA (right). Solid (dashed) lines correspond to regions containing the highest 68%
+(95%) probability. We see that HERA constraints are not significantly improved by adding the published limits on the power
+spectrum from other interferometers.
+
+HERA
+HERA + MWA + LOFAR
+95% Confidence
+68% Confidence
+1.5
+1.5
+0
+1.0
+1.0
+max
+6'68 >
+< 40.2
+42.5
+42.5
+40.0
+40.0
+37.5
+37.5
+0.075
+0.075
+0.050
+> 24.6
+0.050
+> 24.6
+25.0
+25.0
+22.5
+22.5
+37.5
+40.0
+42.5
+0.050
+0.075
+25.0
+5.
+5
+5
+40.0
+42.5
+0.050
+0.075
+22.5
+3
+2
+5
+2
+0.
+2
+<
+log(f*)
+log(f*)
+log(sFR)

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+Draft version January 13, 2023
+Typeset using LATEX default style in AASTeX62
+Intensity Interferometry observations of the Hα envelope of γ Cas with M´eO and a portable telescope
+Nolan Matthews,1 Jean-Pierre Rivet,2 David Vernet,3 Mathilde Hugbart,1 Guillaume Labeyrie,1
+Robin Kaiser,1 Julien Chab´e,4 Cl´ement Courde,4 Olivier Lai,2 Farrokh Vakili,2 Olivier Garde,5, 6 and
+William Guerin1
+1Universit´e Cˆote d’Azur, CNRS, Institut de Physique de France, France
+2Universit´e Cˆote d’Azur, Observatoire de la Cˆote d’Azur, CNRS, Laboratoire Lagrange, France
+3Universit´e Cˆote d’Azur, Observatoire de la Cˆote d’Azur, CNRS, UMS Galil´ee, France
+4Universit´e Cˆote d’Azur, Observatoire de la Cˆote d’Azur, CNRS, Laboratoire G´eoazur, France
+52SPOT (Southern Spectrocopic Project Observatory Team)
+6Observatoire de la Tourbi`ere - 38690 Chˆabons - France
+(Received January 13, 2023)
+ABSTRACT
+We report on observations of the extended environment of the bright Be star γ-Cas performed using
+intensity interferometry measurements within its Hα emission line. These observations were performed
+using a modified version of the I2C intensity interferometry instrument installed onto the 1.54 meter
+M´eO optical metrology telescope and a portable 1-meter telescope (T1M). In order to better constrain
+the extent of the H α envelope, observations were performed for two different positions of the T1M
+telescope, corresponding to an intermediate and long baselines in which the extended region was
+partially and fully resolved. We find that the observed data are consistent with past interferometric
+observations of γ-Cas. These observations demonstrate the capability to equip optical telescopes of
+different optical designs with intensity interferometry capabilities and illustrate the potential to scale
+a similar system onto many additional telescopes.
+Keywords: stars: emission-line, Be — instrumentation: high angular resolution — techniques: inter-
+ferometric
+1. INTRODUCTION
+Recognized as the first stellar object displaying emission line spectra (Secchi A. 1867), γ-Cas is the prototype of the
+Be stellar class. The emission line features originate from radiative processes with up to X-ray energies (Smith et al.
+2012) occurring in an extended disc surrounding the star. The disc formation is primarily attributed to mass ejection
+from the central star enabled from a combination of strong radiative pressure, and low effective surface gravity near
+the equatorial latitudes. The latter is a consequence of the extremely high rotation rate that is nearly critical, in which
+the outward centrifugal force is equal to the inward gravitational force.
+Due to its bright stellar magnitude and characteristic stellar size, optical interferometry has been extensively used
+to study the disk emission of γ-Cas. The extended atmosphere of γ-Cas was first resolved with the I2T interferometer
+(Thom et al. 1986) and subsequently by the GI2T interferometer showing that the Hα region could be fit by a
+disk model and was in Keplerian motion (Mourard et al. 1989). Observations by Quirrenbach A. et al. (1997) with
+the MkIII interferometer demonstrated that the emission-line region were not compatible with circularly symmetric
+models and required the assumption of an elongated profile. Density and velocity relationships in the equatorial plane
+were constrained and accounted for by a radiative wind driven model in Stee et al. (1995). Subsequently, spectro-
+interferometric measurements of the envelope size were performed across both the Hα and Hβ lines, as well as in the
+near-by continuum emission (Stee et al. 1998) leading to a measurement of the disk mass and opening angle (Stee et
+Corresponding author: Nolan Matthews
+nolankmatthews@gmail.com
+arXiv:2301.04878v1  [astro-ph.IM]  12 Jan 2023
+
+2
+al. 2003). In addition, the Navy Precision Optical Interferometer (NPOI) was used to characterize the disc geometry
+and further confirmed the oblateness of the disc (Tycner et al. 2006). The CHARA interferometric array measured
+the disk extent in the K’ photometric band for the first time, found to be slightly smaller than previous observations
+in Hα (Gies. D. R. et al.
+2007). Finally, high sensitivity spectro-interferometric measurements with CHARA were
+performed in the near-infrared, as well across the Hα line and near-by continuum suggesting a larger disk size than
+prior measurements and linking the origin of X-ray emission to a compact binary companion due to the absence of
+one-armed spiral structures or secondary star (Stee et al. 2012).
+In this work we present the first known intensity interferometry (II) measurements of the extended atmosphere of
+γ-Cas using a modified version of our intensity interferometry instrument (I2C) installed onto the 1.54-meter telescope
+of the M´eO laser ranging facility and a mobile 1-meter telescope (hereafter T1M), both located on the Calern Plateau
+site of the Observatoire de la Cˆote d’Azur. While the I2C instrument shares similarities with past II observations
+using the telescopes of the Centre P´edagogique Plan`ete Univers (C2PU) (Guerin et al. 2017, 2018; Rivet et al. 2020;
+de Almeida et al. 2022), there were several modifications required to outfit these telescopes with II capabilities, and
+to be compatible with each other in an interferometric mode. The experimental setup is thus described in Section 2
+with additional details also presented in Matthews et al. (2022). The observations and results are shown in Section 3
+with an analysis presented in Section 4. Finally, we discuss the results and present an outlook for future intensity
+interferometry measurements in Section 5.
+2. EXPERIMENTAL SETUP
+2.1. Principles of Intensity Interferometry
+An intensity interferometer correlates the intensity fluctuations of starlight between separated telescopes in order
+to measure the squared visibility. For two telescopes with a projected baseline r between them, the second order
+coherence function is
+g(2)(r, τ) = ⟨I1(t)I2(r, t + τ)⟩
+⟨I1⟩⟨I2⟩
+(1)
+where I1 and I2 are the intensities recorded at each of the two telescopes, τ is the relative time-lag between the signals,
+and the brackets indicate an average over time t. The Siegert relation (Siegert 1943; Ferreira et al. 2020) relates the
+second-order coherence function g(2) to the first order coherence function g(1) by
+g(2)(r, τ) = 1 + |g(1)(r, τ)|2
+(2)
+where the first-order coherence function can be separated into spatial and temporal components
+g(1)(r, τ) = V (r)g(1)(τ)
+(3)
+where V (r) is the interferometric visibility of the source, given by the Fourier transform of the source sky brightness
+distribution. For an unresolved point-like source V (r) = 1, and the resulting second order coherence function will
+depend only on the temporal component g(1)(τ) given by the Fourier transform of the measured light spectral den-
+sity (Wiener 1930; Khintchine 1934). For linearly polarized thermal light at zero optical path delay g(1)(τ = 0) = 1
+where for time-lags much greater than the coherence or correlation time the first order coherence function should be
+equal to zero such that there is a “bunching peak” centered about zero optical path delay with an effective temporal
+width given by the coherence time. The peak amplitude at zero time-lag of g(2) thus measures the squared visibility at
+some projected baseline assuming that the instrumental resolving time is shorter than the light coherence time. The
+coherence time can be defined by the integral of the squared first-order coherence function (Mandel, L., & Wolf, E.
+1995),
+Tc =
+�
+|g(1)(τ)|2dτ =
+�
+|s(ν)|2dν,
+(4)
+which is equal to the integral of the squared normalized spectral density s(ν) by Parseval’s theorem. For visible light
+with a bandpass of ∆λ ∼ 1 nm the corresponding coherence time is of order 1 ps, much shorter than what can be
+achieved with conventional detectors. In this case, a measurement averages over many coherence times and reduces
+the value of the g(2) peak amplitude at τ = 0 by a factor of ∼ Tc/Td where Td is the effective time-resolution of the
+detector. The amplitude of the g(2) peak therefore measures this loss of contrast times the squared visibility. The
+squared visibility can be extracted by dividing the value of g(2)(r) − 1 peak amplitude measured between telescopes
+
+3
+Figure 1.
+Photographs of the coupling assemblies mounted on the Nasmyth arm of the M´eO telescope (left) and at the
+Newtonian focus of the T1M portable telescope (right).
+to the g(2)(r = 0) − 1 peak amplitude measured at zero-baseline under the assumption that the profile of the g(2)(τ)
+peak is constant. In practice, we measure the ratio of the area of the g(2)(r) − 1 peak to the area of the g(2)(r = 0) − 1
+peak for the squared visibility,
+|V (r)|2 =
+� �
+g(2)(r, τ) − 1
+�
+dτ
+� �
+g(2)(r = 0, τ) − 1
+�
+dτ .
+(5)
+The denominator, or equivalently the area of the g(2) peak at zero baseline, corresponds to the coherence time that can
+be calculated from the measured spectrum as given by Equation 4. The equivalence between the coherence time from
+intensity interferometry and spectral measurements assumes that the spectral resolution is narrower than any spectral
+lines within the instrumental bandpass. Since intensity interferometry measurements probe the intrinsic spectrum it
+is a useful method for characterizing narrow spectral lines present in, for example, studies of light scattering off of
+atomic clouds (Dussaux et al. 2016), with potential applications in astrophysics (Tan & Kurtsiefer 2017).
+2.2. Telescopes
+The II observations presented in this paper were performed by outfitting two telescopes located on the Calern Plateau
+site of the Observatoire de la Cˆote d’Azur with individual coupling assemblies (CAs). A CA is mounted near the focus
+of each telescope, both shown in Figure 1. The first facility was the 1.54 m diameter M´eO (M´etrologie Optique)
+telescope primarily used for satellite laser ranging (Bertrand, B. et al. 2021), lunar ranging measurements (Bourgoin,
+A. et al. 2021) and low Earth orbit satellite laser communication (Giggenbach, D. et al. 2022). The optical design
+is based upon a Ritchey-Chr´etien configuration on an altitude-azimuth mount with a primary focal length of 31.0 m
+giving an approximate focal ratio of f/20.1. In typical operation the light is brought to a Coud´e focus, but for II
+observations the light is redirected to the CA along the Nasmyth arm using a removable 45 degree mirror. In addition,
+a f=150 mm lens is inserted before the CA in order to decrease the effective focal length.
+The second facility is the portable 1 m diameter T1M, a Newtonian telescope on a Dobson-type fully motorized
+azimuthal mount. The portability of the telescope enables configurable baselines to expand the accessible coverage of
+the uv-plane where the telescope can be disassembled and moved in just a few hours. The telescope has a primary
+focal length of 3 m, and a Barlow lens is included in order to expand the effective length at the input of the CA.
+While both telescopes are azimuthal, there will be a relative field rotation due to the Newtonian versus Nasmyth
+optical designs. However, each CA utilizes polarizing filters that must be aligned with respect to one another, although
+not with respect to the target as astrophysical polarization effects are not investigated. To compensate, the M´eO
+telescope CA is mounted into a rotation stage that orients the CA such that the polarization axes are aligned. The
+
+地4
+Figure 2. Schematic of the experimental setup. The light collected by both telescopes are brought to individualized coupling
+assemblies (CA), shown in detail in the right inset.
+A tip-tilt corrects the beam with respect to transverse displacements.
+The converging beam is collimated using a diverging lens (L1). A dichroic (D) reflects short wavelengths to a guiding camera
+(GC) used in a closed loop with the tip-tilt. The transmitted light passes through a narrowband filter (NF) centered on Hα.
+A polarizing beam splitter (PBS) separates the light into orthogonal polarizations where each polarization is injected into a
+graded-index multimode fiber (GRIN-MMF). A linear polarizer (P) is included on the reflected arm to improve polarization
+purity. Not shown is the rotation stage used for the M´eO telescope, and additional focal reducers/extenders, which are described
+in the text. The light for each polarization mode for each telescope is split by a 50/50 fiber beamsplitter, and passed to single
+photon resolving detectors. The photon arrival times are recorded by the TDC that also produces intensity correlations.
+stage is actively controlled throughout the observation where the amount of rotation is determined from the target
+sky position.
+The relative position of both telescopes must be known with a precision less than a few centimeters for optical path
+delay corrections. For M´eO, the position was previously determined to millimeter accuracy in terrestrial coordinates
+due to its use in geodetic surveys. To determine an absolute position of the mobile T1M telescope, geodetic markers
+were installed by the National Institute of Geographic and Forest Information at the ground level for several positions
+and their positions were measured from differential GPS methods. The T1M was installed above these markers and
+the offset between the marker and the T1M reference point was estimated. The estimated cumulative error on the
+reference position is ±1.5 cm in all directions.
+2.3. Instrumental Setup
+The primary function of the CAs (shown in Figure 2) is to perform spectral/polarization filtering, and fiber injection.
+The current version includes an automated tip-tilt device that provides stable fiber injection over several hours without
+manual intervention (Matthews et al. 2022). The detector signal output is fed to a time-to-digital converter (TDC)
+that measures photon arrival times, and produces the correlation between all relevant pairs of detectors. Across both
+telescopes, there are 4 independent measurements of the zero baseline correlation g(2)(r = 0, τ) enabled by using fiber
+splitters in each polarization mode. These allow normalization of the spatial correlations across telescopes to measure
+visibilities. Spatial intensity correlations are obtained by calculating the correlation across telescopes for all pairs of
+detectors in the same polarization mode corresponding to a total of 8 measurements of g(2)(r, τ). During one night, a
+time delay monitoring system was used on the electronic cables connecting the detectors from the M´eO telescope to
+
+Coupling Assembly (CA)
+MéO
+T1M
+IP
+D
+NF
+△α ~ 1nm PBS
+L2
+GRIN-MMF
+TDC
+GC5
+the TDC. The drift throughout the whole night was significantly less than the characteristic jitter of the detectors of
+∼ 500 ps.
+3. OBSERVATIONS
+The observations of γ-Cas were performed between the nights of January 17th, 2022 to January 21st, 2022 in the short
+baseline configuration, and then from January 24, 2022 to January 27th, 2022 in the longer baseline configuration. At
+the beginning of each night images were recorded in both telescopes of the visual binary system γ-Ari. The measured
+position angle of the binary for both telescopes, and thus of the polarization axes, were found to be always within 5
+degrees, corresponding to a loss of visibility of less than 1%.
+3.1. Temporal Intensity Correlations
+The coherence time obtained from zero baseline intensity correlations were compared to expected values from the
+spectral throughput. These temporal intensity correlation functions were computed for each polarization state and
+for each telescope by summing all individually acquired correlations acquired over the entire observation. Each of
+the resulting correlation functions were then shifted by the instrumental delay and then co-added together.
+The
+resulting correlation function is displayed on the left in Figure 3. The peak is fit by a Gaussian with free parameters
+for the amplitude and width. The coherence time, given by the integral of the peak, is extracted via the fit values.
+Before fitting, there is a choice of the time-lag range to fit over and the number of time-lags to bin. Here, we fit
+and show the data over a range of ± 20 ns binned into 50 ps bins. Under these parameters, we find an amplitude
+of (1.43±0.05)×10−3 and a full-width at half-maximum of 885±40 ps corresponding to a measured coherence time
+of 1.35 ± 0.05 ps.
+Systematics of the fitting process were studied by fitting the data varying the fit range from
+±10 ns to ±40 ns and additionally the binning size from 10 ps to 80 ps. Within these parameters we find a maximum
+difference of 0.015 ps in the extracted coherence time, notably less than the measurement error, In the previously quoted
+coherence time, the correlations from different polarization states and telescopes were co-added and subsequently fit.
+This procedure requires that shape of the bunching peak in each correlation, given by the temporal response of the
+detectors, are similar. To test systematics, each correlation function for both polarization states and telescopes were
+fit by a Gaussian to extract the coherence time. Each individual fit was within 1 σ of the quoted coherence time, and
+furthermore the weighted mean of fits (1.35±0.05 ps) is in perfect agreement with a single fit of co-added correlations
+indicating that within our measurement precision there are no significant systematics that preclude us from combining
+individual zero-baseline correlations from separate detector pairs.
+The coherence time measured from the zero-baseline correlations can then be compared to expectation from the
+recorded spectrum. The spectral transmission of the Hα filter was measured in the laboratory with a high resolution
+spectrograph (Matthews et al. 2022). Spectra of γ-Cas with resolution R=25000 were recorded contemporaneously with
+Figure 3. The left plot shows the measured zero-baseline correlation in the blue points with a Gaussian fit shown by the dashed
+black line. The right image displays the measured spectrum of γ-Cas, along with the theoretical (black) and measured (blue)
+spectral transmission of the filter.
+
+2.0
+1.5
+)-1(×103)
+1.0
+(2)
+0.5
+6
+0.0
+-0.5
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+Time-lag (ns)100
+NominalTransmission
+MeasuredTransmission
+ Transmission (a.u.)
+GammaCasSpectra
+80
+60
+Normalized
+40
+20
+0
+655.5
+656.0
+656.5
+657.0
+657.5
+658.0
+Wavelength (nm)6
+Table 1. Spatial intensity cor-
+relation results.
+Baseline Range
+Peak Area
+m
+ps
+0.0
+1.35±0.05
+3.8 < r ≤ 13.0
+1.40±0.17
+13.0 < r ≤ 18.0
+0.83±0.14
+18.0 < r ≤ 21.3
+0.31±0.08
+32.0 < r ≤ 37.8
+0.07±0.04
+our observations using a Whoppshel echelle spectrograph provided through collaboration with the 2SPOT1 association
+of amateur astronomers. This is especially important as the width of the temporally variable emission line is narrower
+than the filter bandpass thus affecting the coherence time. The right side of Figure 3 shows the filter transmission,
+and the emission line spectra. Through Equation 4 we extract the expected coherence time to be 1.41 ps which is
+1.2 σ larger than the value from zero-baseline correlations when taking only the measurement uncertainty and thus in
+fair agreement. In contrast, the coherence time that would be expected for a flat stellar spectrum would be 1.16 ps.
+This is considerably less than the measured value by 3.8 σ illustrating the importance of including the emission line
+profile in calculations of the coherence time. The general agreement of the coherence time measured between intensity
+interferometry and spectral measurements indicate that there are no systematic effects arising from the presence of
+unidentified narrow spectral lines due to a lack of spectral resolution.
+3.2. Spatial Intensity Correlations
+The spatial intensity correlations correspond to the correlations between all detector pairs on separate telescopes
+that are observing in the same polarization mode corresponding to a total of 8 cross-correlations across telescopes. All
+computed correlations are shifted in time by instrumental and geometrical delays, and then co-added together. The
+full data set corresponds to a wide projected baseline and position angle range and so the data was divided into smaller
+baseline ranges. For each sub-division, we compute the averaged correlation function and then fit a Gaussian function
+to the resulting bunching peak. The measured areas of the g(2) peak for each subdivision are presented in Table 1.
+Squared visibilities are extracted by computing the ratio of the integral of the Gaussian peak of the cross-correlation
+to the computed value from the measured spectrum.
+4. ANALYSIS OF RESULTS
+The reduced II data resulted in 4 measurements of the squared visibility, each averaged over a range of baselines
+required to significantly resolve a bunching peak. In turn, the limited sampling does not allow any reasonable in-
+dependent visibility modeling. Nevertheless, it is interesting to compare the measured values to past results. Past
+interferometric observations generally characterize the angular brightness distribution of γ-Cas with a parameterized
+geometrical model. A common assumption is a two-component system consisting of a photosphere and disk, with
+some flux ratio between them. The photosphere is typically approximated as a uniform disk. This is an oversimplified
+model of Be stars as it does not take into strong temperature gradients and equatorial flattening from the near critical
+rotation (Domiciano de Souza et al. 2002). However, to resolve these effects requires an angular resolution at the
+characteristic diameter of the photosphere (∼ 0.5 mas for γ-Cas (Stee et al. 1998)), whereas the effective resolution
+for our observations (1.22λ/D) at the largest baseline is ∼ 8.3 mas. Furthermore, these observations were conducted
+within the Hα line in which the disk emission is much stronger, such that photospheric contributions are significantly
+minimized. For the disk emission, several geometric models were tested, including elongated uniform disks, Gaussian
+disks, and uniform rings. Tycner et al. (2006); Stee et al. (2012) showed that a Gaussian disk profile best described
+the extended emission relative to the other assumptions.
+1 www.2spot.org
+
+7
+Table 2. Reported Gaussian disk fit values in prior γ-Cas observations. θGD is the
+full-width at half-maximum, φ is the position angle, and r is the axial ratio.
+Observatory
+Ref.
+θGD
+φ
+r
+(mas)
+(◦)
+MkIII
+Quirrenbach A. et al. (1997)
+3.47±0.02
+19±2
+0.70±0.02
+NPOI
+Tycner et al. (2006)
+3.59±0.04
+31.2±1.2
+0.58±0.03
+CHARA
+Stee et al. (2012)
+4.4±0.4
+19±5
+0.74
+Figure 4. The left image shows the uv-plane coverage, over-plotted on expected squared visibilities formed from a Gaussian
+disk model of γ-Cas using reported parameters from Stee et al. (2012). Each of the colors represent the range of sampled points
+averaged together in order to measure squared visibilities, as correspondingly plotted on the right. Additionally, we plot the
+expected squared visibilities for each of the models in Table 2 at our sampled uv-plane points.
+This two component model of a uniform disk + elongated Gaussian disk was applied to γ-Cas data in three prior
+reported observations using the MkIII interferometer (Quirrenbach A. et al. 1997), NPOI (Tycner et al. 2006), and
+CHARA (Stee et al. 2012). The reported parameters for these observations are summarized and presented in Table 2.
+Figure 4 shows our squared visibilities, along with expected values obtained from a Gaussian disk model from the
+previous observations. The comparison of our results with the models produced using reported values from literature
+tends to align with the values given by Tycner et al. (2006) and Quirrenbach A. et al. (1997) over Stee et al. (2012)
+who suggest a smaller extent of the Hα region. Within our measurement precision this is not strongly conclusive
+and can also be a result of instrumental differences. Already, Stee et al. (2012) noted the larger angular extent could
+be explained in that they used high resolution spectro-interferometry, in contrast to the other observations including
+our own, that utilize narrowband filters. The reasoning is that the filters detect more of the less resolved continuum
+resulting in an effectively smaller angular extent.
+5. DISCUSSION AND OUTLOOK
+We reported here on II measurements of the extended Hα emitting region of γ-Cas. The observed angular extent
+of the emission was found to be consistent with past direct interferometry measurements. Following our previous
+observations of Rivet et al. (2020) and de Almeida et al. (2022) this extends the work of II measurements in emission
+lines to another system and complements recent on-sky results of other intensity interferometry facilities (Acciari
+et al. 2020; Abeysekara et al. 2020; Horch et al. 2022).
+Future improvements to the system will aim to improve
+the sensitivity.
+The most significant gain comes from simultaneously performing II correlations in many spectral
+channels that can be co-added to improve the signal to noise ratio by a factor of the square root of the number of
+channels (Trippe et al. 2014). One could also imagine recording many independent spectral channels across the Hα
+line in order to perform intensity spectro-interferometry to test the discrepancy seen in past observations between
+
+Model Visibilities and (u,v)Sampling
+1.0
+40
+0.8
+20
+(m)
+0.6
+0
+>
+0.4
+20
+-40
+0.2
+-40
+-20
+0
+20
+40
+u (m)MkillI
+NPOI
+1.0
+ CHARA
+0.8
+0.6 -
+0.4 -
+0.2
+0.0
+0
+5
+10
+15
+20
+25
+30
+35
+Baseline (m)8
+those using filtered bandpasses, and those in dispersed light. Furthermore, higher sensitivity observations, paired with
+polarimetric capabilities would allow for better constraints, if not a direct measure of, radiative processes displaying
+polarized emission in the disk, as was attempted by Rousselet-Perraut et al. (1997).
+These observations were performed using two facilities: the T1M and M´eO that had not been used for interfero-
+metric observations prior to this report. The portability of the T1M provides the capability to optimize the baseline
+configuration for a given target, or similarly perform multiple configurations for a given target as done here. This
+technical accomplishment also illustrates the potential for performing up to 4-telescope II measurements on the Calern
+Plateau enabling 6 simultaneous baselines by including the 2 additional C2PU telescopes.
+We acknowledge the financial support of the R´egion PACA (project I2C), the French National Research Agency
+(ANR, project I2C, ANR-20-CE31-0003), OCA, Doeblin federation, UCA science councils grants, and the LABEX
+Cluster of Excellence FIRST-TF (ANR-10-LABX-48-01), within the program Investissements d’Avenir operated by the
+ANR. The authors would like to thank Jacques Belin and Damien Pesce for their installment of reference markers, and
+to the members of the M`eO team including Hervey Mariey, Mourad Aimar, Herv´e Viot, Gr´egoire Martinot-Lagarde,
+Julien Scariot, Nicolas Maurice, Duy-H`a Phung, and Nils Raymond for their assistance during the observations.
+REFERENCES
+Abeysekara, A. U., Benbow, W., Brill, A., et al. 2020,
+Nature Astronomy, 4, 1164.
+doi:10.1038/s41550-020-1143-y
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+Improving Inference Performance of Machine
+Learning with the Divide-and-Conquer Principle
+Alex Kogan
+Oracle Labs
+alex.kogan@oracle.com
+Abstract
+Many popular machine learning models scale poorly when
+deployed on CPUs. In this paper we explore the reasons
+why and propose a simple, yet effective approach based on
+the well-known Divide-and-Conquer Principle to tackle this
+problem of great practical importance. Given an inference
+job, instead of using all available computing resources (i.e.,
+CPU cores) for running it, the idea is to break the job into
+independent parts that can be executed in parallel, each with
+the number of cores according to its expected computational
+cost. We implement this idea in the popular OnnxRuntime
+framework and evaluate its effectiveness with several use
+cases, including the well-known models for optical character
+recognition (PaddleOCR) and natural language processing
+(BERT).
+1
+Introduction
+We live in the era of unprecedented attention to machine
+learning (ML) from researchers and practitioners alike. New
+ML models across a variety of domains (or modalities, such
+as video, images and text) are proposed nearly daily, the
+models grow bigger and more sophisticated, and their com-
+ponents are continuously revised to achieve better accuracy
+scores on various tasks. While lots of attention is given to
+training efficiency and prediction accuracy, seemingly less
+effort is focused on making sure those models perform well
+when deployed in practice, i.e., during inference [11]. As we
+demonstrate in this paper, some models scale poorly (and at
+times, even worse!) when the number of available cores in a
+CPU-based deployment is increased.
+Why does not the inference on CPUs scale? There are a
+variety of reasons, and we devote the entire section of this
+paper to look into some of them. Briefly, they range from
+the micro-level, such as the use of non-scalable operators
+inside ML architectures, to macro-level, such as employing
+ML architectures that process input iteratively.
+To mitigate those scalability challenges, one might con-
+sider redesigning their ML architecture or reimplementing its
+non-scalable operations with a more efficient version. Such
+approaches, however, require either substantial ML domain
+specific expertise, exceptional engineering skills and famil-
+iarity with ML frameworks used for inference, significant
+investments (e.g., to retrain a new model, with a potential
+risk to the accuracy metrics), or all of the above.
+In this paper, we take a different approach and propose to
+leverage the poor scalability of ML models by applying the
+Divide-and-Conquer Principle, a well-known algorithm de-
+sign technique in Computer Science [8]. Specifically, instead
+of allocating all available computing resources (CPU cores)
+to the entire problem, we propose to divide the problem into
+smaller chunks1, let the framework decide how the comput-
+ing resources should be allocated among those chunks and
+then run their respective computations in parallel. We argue
+that in many use cases, such a division is natural and requires
+only trivial changes in the user code. We also describe a sim-
+ple mechanism that allocates computing resources based on
+the expected computational intensity (or weight) of each
+chunk.
+Consider, for instance, a model for solving a natural lan-
+guage processing (NLP) task such as tweet classification. Our
+approach allows efficient batching of inference requests of
+various sizes, eliminating the need for padding (a common,
+but wasteful solution to deal with batches of requests of
+variable size) and letting the framework allocate comput-
+ing resources proportionally to the length of each sequence.
+We implement the aforementioned allocation mechanism in
+OnnxRuntime [24], a popular framework for training and
+inferencing ML models, and extend its inference API to al-
+low user code to invoke parallel inference on multiple inputs.
+We demonstrate the effectiveness of our approach with sev-
+eral use cases, including highly popular models for image
+processing (PaddleOCR [14]) and NLP tasks (BERT [10]).
+The remainder of this paper is organized as following.
+In Section 2 we elaborate on various reasons for why the
+inference (on CPUs) commonly does not scale well. Next,
+we describe in Section 3 the concept and implementation
+details of the Divide-and-Conquer Principle as it applies to
+inference. Following that, we present in Section 4 several
+use cases of ML models where this principle can be applied,
+along with the performance evaluation of its benefits. We
+discuss related work in Section 5 and conclude the paper in
+Section 6.
+1We note that unlike the classical Divide-and-Conquer Principle [8], we
+divide the problem only once, although it might be possible in some cases to
+divide it recursively into increasingly smaller chunks that can be executed
+by one thread each.
+1
+arXiv:2301.05099v1  [cs.LG]  12 Jan 2023
+
+2
+Why is Inference Slow?
+There are numerous reasons for this lack of scalability. In
+this section we survey some of them.
+2.1
+Not “enough” work
+One reason is simply because the amount of computation
+required by a model during inference is not “enough” for
+efficient parallelization. As noted by Aminabadi et al. [3],
+kernel implementations of various ML operations are often
+geared towards training, which tends to consist of sizable
+batches of large inputs (e.g., sentences of 512 tokens). During
+inference, however, the batches tend to be much smaller, and
+often include just one input (e.g., for real-time / interactive
+inference). Besides, the inputs themselves can be small, e.g.,
+a tweet or chatbot interaction consisting of just a few words.
+Consider, for instance, highly popular Transformer-based [28]
+models for NLP tasks, such as BERT [10] or GPT-3 [5], which
+rely mostly (but not solely) on matrix multiplication prim-
+itives. Those primitives are known to scale well for large
+matrices [11, 22, 30]. However, when the actual input to
+the model during inference is short, matrix multiplications
+involve smaller and therefore, less amendable to efficient
+parallelization, matrices [11, 23, 30].
+2.2
+Non-Scalable Operators
+Another reason for poor scalability of some ML models is
+the use of non-scalable (and often, sequential) operators in
+their architecture. Typically, the overhead of those operators
+would be negligible compared to other, more scalable parts
+of the model. Yet, as the number of cores increases and fol-
+lowing directly from the Amdahl’s Law [2], their negative
+impact of non-scalable operators on the overall inference per-
+formance would grow. Considering again the Transformer-
+based [28] models mentioned above, Dice and Kogan have
+observed that while matrix multiplication scales well, at least
+for long inputs, other operations such as layer normaliza-
+tion and softmax do not, contributing to the overall poor
+scalability of those models [11]. In this paper, we consider
+a vision-based model, which employs sequentially imple-
+mented functions for internal format conversions, which
+similarly cause the entire model not to scale.
+We note that some of those cases could be considered a
+performance bug in the underlying ML framework, which
+could be fixed by reimplementing the respective operators
+with more efficient (and parallel) alternatives. This, however,
+requires lots of engineering effort, which includes perfor-
+mance analysis and deep understanding of corresponding
+framework implementation details. Besides, some of the ML
+operators, such as layer normalization [4], require careful
+coordination among computing threads (e.g., to compute
+variance and standard deviation of all the hidden units in a
+layer and then use those statistics to normalize the values
+of the units) and therefore do not lend themselves naturally
+for efficient parallelization.
+2.3
+Framework Overhead
+Somewhat related to the prior point, an ML framework might
+add small but measurable overhead in invoking model op-
+erations. Most popular ML frameworks, such as PyTorch,
+Tensorflow or OnnxRuntime, support multiple backends for
+executing ML operations, targeting different hardware archi-
+tectures (CPU, GPU, TPU), utilizing different BLAS libraries
+(MKL, OpenBLAS, oneDNN, etc.), different threading infras-
+tructure (Intel TBB, pthreads, custom implementation, etc.),
+etc. Dispatching appropriate kernel (implementation) for
+every operator is efficient, but is sequential and requires
+non-trivial amount of work, especially when the model is
+executed interactively [11] (the default execution mode in
+PyTorch). This overhead becomes substantial as the actual
+execution time of the kernels reduces with the increased
+number of cores.
+In addition to the above, various kernels might require
+specific memory layout for its input parameters (tensors),
+and the framework would add appropriate dummy operators
+for input/output conversion or data preparation [30]. As we
+demonstrate later in this paper, these operators might add
+substantial overhead as well.
+2.4
+Model Architecture
+Quite often the high-level architecture of an ML model itself
+plays a substantial role in causing inference not to scale. For
+instance, some ML models, especially ones built for video
+and image processing (e.g., [14, 21, 29]), are composed as a
+multi-phase pipeline. The first phase of the pipeline would
+typically identify the points of interest in the input (e.g., text
+boxes in an image or a moving object in a video), while sub-
+sequent phases would process those points (iteratively or as
+a batch) to solve the predefined problem (e.g., identify text
+in the boxes or classify the moving object in the video). The
+inference latency of such models might grow linearly with
+the number of objects identified in the first phase. Further-
+more, if even one phase of the pipeline does not scale well,
+the scalability of the entire pipeline is impaired.
+2.5
+Padding
+Batching multiple inputs and processing them at once is a
+well-known way of improving inference throughput [1, 3, 9,
+15, 32]. In fact, multiple serving system for machine learning
+models (such as TensorFlow Serving [16] or TorchServe [7])
+include tunable parameters that configure how long an in-
+ference server can wait in order to batch as many input
+requests as possible. However, when inputs in a batch do not
+have exactly the same shape, they need to be padded to be
+processed efficiently, since underlying kernels typically an-
+ticipate batches of homogeneous inputs. The padding leads
+2
+
+to reduced computational efficiency, since it is treated by ker-
+nels as the rest of the input, even though the corresponding
+output produced by the model is dismissed.
+3
+Divide-and-Conquer Principle Applied
+to Inference
+In this section, we describe the application of the Divide-
+and-Conquer Principle [8] to the inference of ML models at
+the conceptual level and as a concrete realization by imple-
+menting it in the OnnxRuntime framework. We note that
+applying this principle does not directly address the reasons
+for poor scalability detailed in the previous section. In fact,
+the advantage of our approach is that one does not have to
+identify and/or fix any scalability bottlenecks in their models
+to rip the benefits of its underlying idea.
+3.1
+Concept
+The basic idea is pretty straightforward — consider a compu-
+tation job 𝐽, which can be broken into𝑘 independent parts, 𝑗1,
+𝑗2, ..., 𝑗𝑘, which can be executed in parallel. Assume we have
+an oracle assigning relative weight 𝑤𝑖 ∈ (0, 1] correspond-
+ing to, e.g., the number of required floating point operations
+(FLOPs) or single-thread latency of the computation job part
+𝑗𝑖. Finally, assume we have 𝐶 computing cores available. We
+strive to allocate to each part the number of cores relative
+to its weight, namely, we assign 𝑐𝑖 = 𝑚𝑎𝑥{1, ⌊𝑤𝑖 ∗𝐶⌋} cores
+for the part 𝑗𝑖. This effectively means allocating 𝑐𝑖 worker
+threads for 𝑗𝑖 since we later create one worker thread per core
+(as common in ML frameworks, including in OnnxRuntime).
+Note that �𝑘
+𝑖=1 𝑐𝑖 might be larger than C. This is obvious
+when the number of job parts, 𝑘, is larger than C, but it is
+possible even when 𝑘 ≤ 𝐶. This does not create a problem
+other than implying that some job parts will be run after
+other job parts have finished (rather than running them all
+in parallel). At the same time, due to the rounding-down
+(floor) function intended to reduce the above possibility of
+oversubscription, some unallocated cores might remain. To
+avoid this waste of available resources, we sort all the job
+parts by their remaining unallocated weight, i.e., by 𝑤𝑖 ∗𝐶 −
+⌊𝑤𝑖 ∗ 𝐶⌋, and assign one core to each part in the descending
+order, up until all cores are allocated. The C++-like pseudo-
+code for the entire algorithm is given in Listing 1.
+Naturally, the idea described above raises the question of
+how to assign relative weight to a job part 𝑗𝑖. In all our cases
+considered in Section 4, the weight is simply set proportion-
+ally to the size of input tensors. Specifically, let 𝑠𝑖 be the size
+of the input tensor for job part 𝑗𝑖. We set 𝑤𝑖 to
+𝑠𝑖
+�𝑘
+𝑖=1 𝑠𝑖 , essen-
+tially assuming that the amount of computation (expressed
+as the number of required FLOPs) grows roughly linearly
+with the input tensors’ size. In general, however, assigning
+weight can be done with the help of a profiling phase and a
+lightweight classification mechanism, which associates job
+parts of the same (or similar) shape (as the one encountered
+1 vector<int> allocate(vector<Tensor> inputs, int numCores) {
+2
+vector<int> threadAllocation;
+3
+vector<tuple<int, float>> threadUnallocatedWeight;
+4
+int numInputs = inputs.size();
+5
+int allocatedCores = 0;
+6
+int index = 0;
+7
+int totalSize = 0;
+8
+for (auto j_i : inputs) totalSize += j_i.size()
+9
+for (auto j_i : inputs) {
+10
+int numThreadsToUse = 1;
+11
+if (numInputs <= numCores) {
+12
+int size = j_i.size();
+13
+float w_i = ((float)size) / totalSize;
+14
+int numThreadsToUse = floor(w_i * numCores);
+15
+// this may happen due to flooring
+16
+if (numThreadsToUse < 1) numThreadsToUse = 1;
+17
+unallocatedWeight.add(
+18
+make_tuple(index, w_i * numCores - numThreadsToUse));
+19
+}
+20
+threadAllocation.add(numThreadsToUse);
+21
+allocatedCores += numThreadsToUse;
+22
+index++;
+23
+}
+24
+if (allocatedCores < numCores) {
+25
+// sort the vector in decreasing order by
+26
+// comparing the second field in each tuple
+27
+sort(unallocatedWeight, bySecondField);
+28
+int nextToAdjust = 0;
+29
+while (allocatedCores < numCores) {
+30
+// fetch the first field in the `nextToAdjust` tuple
+31
+index =
+32
+unallocatedWeight[nextToAdjust % numInputs].get(0);
+33
+threadAllocation[index]++;
+34
+allocatedCores++;
+35
+nextToAdjust++;
+36
+}
+37
+}
+38
+return threadAllocation;
+39 }
+Listing 1. Thread allocation algorithm
+during the profiling phase) to the relative weight obtained
+during profiling.
+3.2
+Implementation Details
+We extend the API of the InferenceSession class of On-
+nxRuntime with a new prun method. This method is modeled
+3
+
+1 class TextRecognizer(object):
+2
+def __init__(self, args):
+3
+...
+4
+self.predictor = ort.InferenceSession(args.file_path)
+5
+self.postprocess_op = build_post_process(args)
+6
+...
+7
+def __call__(self, img_list):
+8
+img_num = len(img_list)
+9
+for beg_img_no in range(0, img_num, batch_num):
+10
+end_img_no = min(img_num, beg_img_no + batch_num)
+11
+inputs = prepare(img_list, beg_img_no, end_img_no)
+12
+outputs = self.predictor.run(inputs)
+13
+preds = outputs[0]
+14
+rec_result = self.postprocess_op(preds)
+15
+all_results.add(rec_result)
+16
+return all_results
+Listing 2. Original (shortened and edited for clarity)
+TextRecognizer class implementation from PaddleOCR
+1 class TextRecognizer(object):
+2
+def __init__(self, args):
+3
+...
+4
+self.predictor = ort.InferenceSession(args.file_path)
+5
+self.postprocess_op = build_post_process(args)
+6
+...
+7
+def __call__(self, img_list):
+8
+img_num = len(img_list)
+9
+for beg_img_no in range(0, img_num, batch_num):
+10
+end_img_no = min(img_num, beg_img_no + batch_num)
+11
+inputs = prepare(img_list, beg_img_no, end_img_no)
+12
+all_inputs.append(inputs)
+13
+all_outputs = self.predictor.prun(all_inputs)
+14
+for outputs in all_outputs:
+15
+preds = outputs[0]
+16
+rec_result = self.postprocess_op(preds)
+17
+all_results.add(rec_result)
+18
+return all_results
+Listing 3. Modified TextRecognizer class implementation
+(uses prun). Added or modified lines are in red
+after the existing run method used as the main entry point
+when running inference. The main difference is that prun
+accepts a list of inputs (instead of just one) and returns a list
+of outputs.
+Internally, the implementation of prun iterates over the
+list of inputs, calculates their size (after validating those are
+tensors) and corresponding relative weight, and applies the
+allocation algorithm described in Listing 1 to associate the
+number of worker threads with each input (job part). Follow-
+ing that, the implementation creates one worker thread for
+each input, and runs them in parallel. Each worker thread, in
+turn, creates a thread pool of the size calculated by the alloca-
+tion algorithm (the thread pool includes the worker thread it-
+self), and invokes the run method of the InferenceSession
+object with that thread pool. The entire patch of the On-
+nxRuntime codebase to implement the prun functionality
+and other minor internal changes (such as having the run
+method to accept a thread pool as an optional argument in-
+stead of always using the default pool) consisted of around
+200 lines of code.
+On the user side, the code also has to change to make
+use of the new prun API. Those changes, however, are quite
+straightforward. Instead of invoking run for every job, a user
+needs to create a list of job parts and call prun. In addition, the
+user needs to rearrange the post-processing code to iterate
+over the results of prun, and apply any post-processing to
+each returned output (object). As an example of what the
+user code changes entail, we show the original Python code
+(edited for brevity and clarity) of the TextRecognizer class in
+PaddleOCR (Listing 2) alongside the modified version that
+makes use of the new prun API (Listing 3).
+4
+Use Cases
+Before we detail the use cases where the Divide-and-Conquer
+Principle is beneficial and report on our performance find-
+ings, we give a brief summary of our evaluation setup and
+methodology. We run all our experiments on a 16-core AMD-
+based VM in Oracle Cloud (aka OCI VM.Standard.E3.Flex).
+(We also ran some experiments on a newer E4 shape, but
+have not noticed substantial differences). To reduce perfor-
+mance variability, especially as we create separate thread
+pools for the variants that use prun, we use thread binding
+(pinning), for all the evaluated variants. Every experiment
+was repeated 5 times, and we report the mean. We note that
+the standard deviation of all reported results, except for one
+specific case discussed below, was extremely low (typically,
+less than 1% of the mean). For our experiments, we use the
+latest release versions (as of the date of writing this paper)
+of the corresponding software, specifically OnnxRuntime
+v1.11.1 and PaddleOCR v2.5.
+4.1
+Sequential Pipeline
+Our first example of where applying the Divide-and-Conquer
+Principle is extremely useful is PaddleOCR [14]. PaddleOCR
+is a lightweight OCR system, which consists of three parts:
+Text Detection, Text Classification (called Detection Boxes
+Rectify in [14]) and Text Recognition. Each of those parts
+corresponds to a separate ML model.
+4
+
+Figure 1. PaddleOCR 3-phase pipeline (edited version of Figure 2 from [14]).
+The OCR pipeline accepts an image file and passes it first
+through the text detection phase whose objective is to locate
+text areas in the image. The output of this phase is a list of
+potential text boxes’ coordinates. Next, the list is iterated
+over, and each item in that list (i.e., a text box) is sent to
+the text classification model, which decides whether the
+box needs to be transformed into a horizontal rectangle box
+before the actual text recognition takes place. Based on the
+classifier’s decision, each box is altered respectively. Finally,
+the list is iterated over again, and each item is sent to the text
+recognition model for inference, which recognizes the text
+in the given box and produces the actual character sequence
+based on the supplied character dictionary. This process is
+depicted in Figure 1, which is a redacted version of Figure 2
+from [14].
+In our experiments with PaddleOCR, we observe that the
+system does not scale well with the increase in the number
+of available cores. We demonstrate that in Figure 2 depicting
+inference latency as a function of available cores (which
+directly translates into the number of worker threads used
+by the runtime). For all experiments in this section, including
+the one in Figure 2, we use a subset of images from the
+OpenImages dataset [17], selected according to a criterium
+described below.
+In Figure 2, we break the total latency into time spans cor-
+responding to the three phases of the OCR pipeline discussed
+above. As one can notice, the average inference latency goes
+down from 554 ms for 1 thread to 364 ms for 4 cores and then
+back up to 435 ms for 16 cores. Interestingly, the Text Clas-
+sification phase shows negative scalability, where it takes
+27 ms to process an image, on average, with 1 thread, but it
+takes 38 ms to do the same with 16 threads — a slowdown
+of 1.4x. This shows an example of a system where, beyond a
+certain point, adding more threads not only does not help,
+but actually harms performance. Discussing concrete rea-
+sons for the lack of scalability of these specific models is
+not in the scope of this paper. For a curious reader, however,
+we note that a built-in OnnxRuntime profiling tool shows
+inflated execution times for the output reordering operators
+(which are inserted by the framework, along with the input
+reordering operator, to convert the memory layouts of input
+arguments for various kernels).
+We apply the Divide-and-Conquer Principle to the last
+two phases of the OCR pipeline, namely the Text Classifica-
+tion and Recognition. To that end, instead of invoking the
+corresponding models for each text box produced by Text
+Detection, we send all the boxes to the runtime (by invoking
+the prun API) and effectively let the runtime decide how
+many cores / worker threads to allocate each box based on
+its relative size. The required changes to implement this
+functionality in the Text Recognition phase are depicted in
+Listing 3; the changes to the Text Classification phase are
+similar.
+For our performance evaluation, we compare the prun
+implementation as discussed in Section 3 (and depicted in
+Listing 1), which we denote as prun-def on the charts, to
+a few simple variants. The first variant, denoted as prun-1,
+simply allocates one worker thread to each input in the
+list given to prun. The second variant, denoted as prun-eq,
+allocates an equal number of cores for each input (but at
+least one), i.e., sets 𝑐𝑖 = 𝑚𝑎𝑥{1, ⌊𝑘/𝐶⌋}. Our motivation is
+to show that trivial solutions might also be useful in certain
+scenarios (as discussed below), yet they tend to underperform
+compared to prun-def.
+We note that the benefit of prun in this use case is possible
+only when there are at least two text boxes identified in the
+Text Detection phase. Otherwise, the other two phases would
+not be used (if no text boxes detected) or the prun-def vari-
+ant will use the same (maximum) number of cores as the base
+(unmodified) version (if only one text box is detected). As a
+result, the subset of images used for performance evaluation
+in this section includes images with at least two identified
+text boxes. The pie chart in Figure 3 shows the distribution
+of the actual number of boxes detected in the first phase of
+the OCR pipeline for the entire dataset. The total number of
+images in the dataset was 500 – this number was chosen to
+keep the evaluation times reasonably short. (We note that we
+also ran evaluations on a larger dataset that includes images
+with less than two text boxes and confirmed that the use of
+prun does not create any overhead in those cases.)
+5
+
+营养护发器
+ODM OEM
+ODMOEM
+ODMOEM
+ODMOEM
+ODMOEM
+Image
+Text Detection
+Detection Boxes Rectify
+Text Recognition
+(db_mv3_slim,1.4M)
+(dir_cls_mv3_slim, 0.5M)
+(crnn_mv3_slim, 1.6M)
+OutputFigure 2. Inference latency of PaddleOCR with a varying
+number of threads, broken down by the three phases of the
+pipeline.
+In light of the discussion above, we break down the com-
+parison of the latency results by the number of detected
+boxes, as depicted in Figure 4. The latency numbers in this
+figure were collected with 16 cores; we discuss the over-
+all scalability trends later on. We also break down the per-
+formance in two of the phases where we have used prun,
+namely Text Classification (Figure 4 (a)) and Recognition
+(Figure 4 (b)).
+Considering the results in Figure 4, one can notice that,
+as expected, the benefit of prun increases with the number
+of detected text boxes. For instance, when considering the
+total end-to-end latency (Figure 4 (c)), with only two boxes
+prun-def outperforms base by 1.28x. However, with 9 and
+10+ boxes, prun-def outperforms base by 2.33x and 1.81x,
+respectively.
+It is interesting to compare the performance of prun-def
+with other pun-based variants. As one can notice in Fig-
+ure 4 (a), the prun-1 variant produces the lowest latency
+when the number of detected boxes is small. In fact, the base
+variant also performs better than prun-def in this case. We
+attribute this to two factors. First, this specific phase of the
+pipeline shows negative scalability, which can be also seen in
+Figure 2. Therefore, best performance is achieved when fewer
+threads per box is used in this phase, which is what prun-1
+effectively achieves. Second, prun-def (and prun-eq) cre-
+ate and destroy more threads than prun-1 in those cases as
+they create thread pools containing more threads for each
+prun invocation. This adds small, but non-negligible over-
+head given that the the execution time of this phase is short.
+In the future work, we intend to experiment with reusing
+thread pools between prun invocations. As the number of
+detected boxes increases, however, all prun variants allocate
+less threads (or even just 1) per each box, and they allocate
+a similar number of threads for their pools, thus closing the
+gap with the prun-1 variant.
+When the Text Recognition phase is concerned (cf. Fig-
+ure 4 (b)), however, it is apparent from Figure 2 that one
+Figure 3. Distribution of the number of detected text boxes
+in the input dataset.
+can improve its latency by using more than one thread. We
+note that, quantitively, this phase is also far more dominant
+than the Text Classification one. Here, prun-def manages
+to achieve best or close to best result across all counts of
+detected boxes, which translates to overall highly competi-
+tive end-to-end inference performance (cf. Figure 4 (c)). In
+general, the results in Figure 4 call for a dynamic mechanism,
+which would choose the best thread allocation strategy based
+on the given workload and available resources. Devising and
+experimenting with such a strategy is left for future work.
+Finally, we shed more light on how the scalability im-
+proves with the use of prun in Figure 5, where we vary the
+number of cores (and therefore, the total number of worker
+threads) available for OnnxRuntime. Once again, we include
+the latency of each of the two last phases of PaddleOCR
+(denoted as Rec for Text Recognition and Cls for Text Clas-
+sification) along with the end-to-end (Total) latency. We
+include only the results of the base and prun-def variants
+(denoted simply as prun in Figure 5), for clarity.
+Overall, one can notice similar trends to the ones discussed
+above. In the base version, the Text Recognition phase does
+scale up to 4 threads, but then its performance suffers as the
+number of threads increases. The prun variant avoids this
+performance degradation, and in fact, continues to scale up
+to 16 threads. Indeed, when considering the Text Recognition
+phase only, the prun variant outperforms base by more than
+2.4x at 16 threads. However, since both variants have an
+identical Text Detection phase, which according to Figure 2
+subsumes a substantial part of the total latency, the end-to-
+end speedup of prun is only 1.5x at 16 threads.
+4.2
+Batching of Heterogeneous Inputs
+Our next example concerns with the Transformer architec-
+ture [28], which revolutionized the domain of NLP when it
+was introduced in 2017 and has been applied to other do-
+mains since then (e.g., [12, 18]). This architecture consists
+of a stack of layers, each composed of a self-attention block
+followed by a fully connected network [28]. Past work has
+6
+
+700
+Detection
+Recognition
+600
+Classification
+Other
+500
+Latency (ms)
+400
+300
+200
+100
+0
+1
+2
+8
+12
+4
+16
+# threads2 (46.4%)
+3 (18.6%)
+4 (10.2%)
+5
+(6.6%)
+6 (4.2%)
+7 (2.2%)
+8 (2.0%)
+9 (1.2%)
+10+ (8.6%)(a) Text Classification
+(b) Text Recognition
+(c) End-to-End Inference
+Figure 4. The impact of using prun in PaddleOCR.
+Figure 5. Total (end-to-end) inference latency of PaddleOCR
+with a varying number of threads. Also shown the latency of
+Text Classification (Cls) and Text Recognition (Rec) phases
+shown that the majority of computation cycles in Transform-
+ers is spent on (scalable) matrix multiplication operations,
+yet up to one third of the cycles is spent elsewhere (i.e., less
+scalable operations) [11].
+It is well-known that one way to improve the inference
+performance (specifically, throughput) of Transformers is
+through input batching [3, 15, 30]. This strategy works well,
+however, when the inputs have the same length. Otherwise,
+one has either give up on batching, or pad inputs to the same
+length. The latter results in wasted computation cycles, since
+special padding tokens are treated exactly as input tokens by
+the architecture and dismissed at the end of the computation.
+This situation presents an ideal case for applying the
+Divide-and-Conquer Principle. Instead of padding the in-
+puts of various lengths up to the longest input in the batch,
+we can run inference on those inputs (as they are, without
+padding) using the prun API, and let the runtime decide
+how many cores should be used to process each of the in-
+puts. We modify the Transformer benchmark built into the
+OnnxRuntime [25] to implement this strategy.
+To evaluate the effectiveness of the approach described
+above, we set up an experiment where we generate 𝑋 in-
+puts of a length chosen uniformly and randomly out of the
+range [16, 512]. We then compare the pad-batch version in
+which all 𝑋 inputs are padded to the longest length in the
+given batch with the prun version in which the inference
+is invoked with prun on all inputs in the batch. We show
+results with the highly popular BERT model [10] (specifi-
+cally, “bert-based-uncased”). We have also experimented
+with other Transformer-based models (such as “bert-large-
+uncased” or “roberta-base”) measuring similar qualitative
+results.
+We note that this experiment includes inherent amount
+of randomness — a batch of small sentences is as likely to
+be chosen as a batch of long sentences. In an attempt to
+reduce the anticpated high varaince of the results, we opted
+to repeat the experiment 1000 times, and so for each 𝑋, each
+data point is an average of 1000 results. Figure 6 presents the
+throughput results with batches of various sizes (i.e., 𝑋 varies
+from 2 to 8), with error bars depicting the standard deviation
+of the reported mean. Even though prun outperforms the
+pad-batch variant across all batch sizes, the variance in the
+measured results remains exceptionally high.
+As a result, we setup two additional experiments in a
+more controlled way likely to produce more stable results. In
+the first, we simply preset the lengths of various sequences
+in each batch. For instance, a batch denoted as “16-64-256”
+includes three sentences, one is 16, another is 64 and yet
+another is 256 tokens long. We show the results of this exper-
+iment in Figure 7. Here, the prun version easily outperforms
+the pad-batch variant, which has to pad all sequences to the
+longest sequence in a batch. As one might expect, the benefit
+from using prun increases with the number of sentences in
+a batch, as this variant eliminates all the redundant work
+associated with padding.
+In the second experiment, we use a batch of 1 long sen-
+tence (256 tokens long) and 𝑋 short sequences of 16 tokens
+each, where we vary 𝑋 between 0 and 15. We show the
+throughput results of this experiment in Figure 7, along with
+a curve depicting the number of threads allocated by prun
+for the long sequence in the batch.
+There are several interesting observations that can be
+made here. First, when 𝑋=0, i.e., the batch contains only one
+long sentence, both variants employ all available cores to pro-
+cess that batch, producing similar result. This shows that the
+7
+
+140
+base
+120
+prun-1
+prun-def
+100
+Latency (ms)
+prun-eg
+80
+60
+40
+20
+0
+2
+3
+4
+5
+6
+7
+8
+9
+10+
+# detected boxes900
+base
+800
+prun-1
+700
+prun-def
+(ms)
+600
+prun-eg
+500
+atency
+400
+Lhhhhhhh
+300
+200
+100
+0
+2
+3
+4
+5
+6
+8
+9
+10+
+# detected boxes1200
+base
+1000
+prun-1
+prun-def
+(ms)
+800
+prun-eg
+Latency (
+600
+400
+bhhhhhh
+200
+0
+2
+3
+4
+5
+6
+7
+8
+9
+10+
+# detected boxes600
+base (Total)
+base (Rec)
+base (Cls)
+500
+prun (Total)
+prun (Rec)
+prun (Cls)
+Latency (ms)
+400
+300
+200
+100
+0
+1
+2
+4
+8
+12
+16
+# threadsFigure 6. Throughput of inferencing BERT on batches of
+sequences of sizes chosen randomly from the range [16, 512]
+Figure 7. Throughput of inferencing BERT on batches of
+sequences of various preset sizes
+overhead of using prun when the input has only one chunk is
+negligible. Second, the throughput of the pad-batch version
+grows, but modestly with the increase in the number of short
+sequences. This is because, as stated above, a larger batch of
+(padded) sequences helps to achieve better throughput with
+Transformers. At the same time, the throughput growth with
+prun is much more dramatic up to 3 short sequences in a
+batch and then it declines, but stays well above that achieved
+with pad-batch. Both phenomena can be explained with the
+fact that inferencing a sequence of 256 tokens takes about the
+same time with 16 threads as it takes with 13. Thus, adding
+a few short sequences into the batch, each allocated with
+just 1 thread (as they have small relative weight), has negli-
+gible impact on the latency, but improves throughput. With
+more short sequences in a batch, less threads are allocated
+for the long sequence (as can be seen in Figure 7) and its
+inference latency grows. This causes the overall throughput
+to decrease.
+4.3
+Batching of Homogeneous Inputs
+Our last example follows directly from the discussion in
+Section 2 on the lack of scalability in ML models. As al-
+ready mentioned, while Transformers models heavily use
+Figure 8. Throughput of inferencing BERT on a batch con-
+taining one long sentence of 256 tokens and 𝑋 short se-
+quence with 16 tokens each, where 𝑋 varies 0 to 15. In
+addition, we show how many threads are dedicated to the
+inference of the one long sentence in the batch in the prun
+variant
+scalable matrix multiplication operations, they also employ
+less scalable operations. The impact of the latter grows with
+the increase in the number of cores. Therefore, one may
+benefit form the Divide-and-Conquer Principle applied to
+Transformers even when the batch includes inputs of the same
+length.
+As a concrete example, consider a batch of two inputs.
+Instead of using all available cores to process the batch, we
+will use half the cores for each input. Intuitively, the less
+scalable operators create less relative overhead when less
+cores are used and the input sequence is shorter (i.e., contains
+half the tokens compared to the entire batch).
+Figure 9 demonstrates this effect with batches of inputs of
+equal lengths. In addition to the pad-batch variant (which
+we simply call batch here, as no padding is required) and
+prun, we include a no-batch variant, which runs inference
+on each sequence in a given batch one at a time. Note that
+we include the latter to simply demonstrate the benefits of
+batching in general, confirming previous findings [3, 15, 30].
+Each set of bars in Figure 9 corresponds to a batch of 4
+sentences with the given length (from 64 tokens to 512).
+Overall, the prun version yields a more modest (yet non-
+trivial) speedup over batch compared to the case of non-
+homogeneous inputs in Section 4.2. This is expected, since
+in this case the room for improvement (over batch) does not
+include wasted computation related to padding.
+5
+Related Work
+As mentioned in the Introduction, the major focus of the
+ML community has been on improving the accuracy and
+training performance of proposed models, while efficient
+inferencing and serving of those models receives relatively
+less attention. Yet, there have been some notable exceptions
+of work focused specifically on inference performance, and
+8
+
+60
+16
+pad-batch
+#
+14
+ threads f
+Throughput (queries/s)
+50
+prun
+prun
+12
+40
+10
+for
+30
+8
+long
+6
+20
+ sequence
+4
+10
+2
+e
+0
+0
+3 
+5
+0
+2
+9
+12
+15
+# of short sequences in a batch35
+pad-batch A
+30
+Throughput (queries/s)
+prun 
+25
+20
+15.
+10
+5
+0
+2
+4
+6
+8
+# sequences in a batch50
+pad-batch
+45
+prun
+40
+35
+30
+25
+20
+15
+10
+5
+0
+16-256
+16-32-64-128-256
+16-64-256
+sequence lengths in a batchFigure 9. Throughput of inferencing BERT with batches of
+4 sequences of equal size
+we survey the most relevant results hereafter. As an aside, we
+note that many of the results below come from less formal
+blog posts published by various companies, highlighting the
+great practical importance of efficient inference.
+Wang et al. [30] explore various factors that influence in-
+ference performance in TensorFlow, including the choice of
+a specific math library, a thread pool library, availability of
+SIMD (single instruction multiple data) support, etc. They
+identify data preparation as one of the causes for poor scala-
+bility of small matrix multiplication operations, something
+we more generally attribute to framework overhead in Sec-
+tion 2. They come up with a set of guidelines one can use
+to tune TensorFlow settings to achieve better performance
+compared to the one achieved with settings recommended
+by TensorFlow authors or Intel.
+With the tremendous rise in popularity of Transformers,
+several papers and blog posts focus on its inference perfor-
+mance. Dice and Kogan investigate inference performance of
+Transformers on CPUs [11]. Their analysis shows that most
+inference computation cycles are spent in matrix multipli-
+cation operations. Hence, they propose an adaptive matrix
+multiplication optimization aimed at reducing the latency
+of those operations and subsequently improving the overall
+inference performance. Intel engineers describe an effort
+to optimize inference of BERT in Apache MXNet using the
+GluonNLP toolkit, where one of the ideas is to quantize
+the model for better performance with lower precision [31].
+Similar quantization ideas (along with distillation, another
+common method of reducing the size of a model [27]) were
+employed by Roblox to speedup their deployment of BERT
+on CPUs [19]. The same blog post also mentions that elimi-
+nating padding of input sentences has led to better perfor-
+mance (though the authors did that for batches of 1 input
+only). A Microsoft team [26] describes their effort on accel-
+erating BERT with OnnxRuntime through operation fusion
+that helps to reduce the amount of overhead (e.g., memory
+copying) in invoking each kernel individually.
+A few recent papers and projects have looked into the de-
+ficiency of padding of heterogenous inputs. Fang et al. [15]
+propose a sequence-length-aware batch scheduler, which
+aims to batch requests of a similar size, thus reducing the
+cost of zero padding of all requests into one batch. It re-
+quires a profiling phase during which the inference cost of
+various batches is collected. Du et al. [13] propose to care-
+fully redesign the GPU kernels employed by Transformers to
+eliminate most redundant computation associated with zero
+padding. The Effective Transformer project by ByteDance [6]
+aims to dynamically remove and restore padding during dif-
+ferent calculation stages. All those efforts target specifically
+the inferencing Transformers on GPUs, and it is not clear
+how efficient they would be on CPUs and/or with other
+architectures.
+Beyond Transformers, Liu at et. [20] describe NeoCPU, an
+approach for optimizing CNN inference on CPUs. NeoCPU
+proposes a configurable design of an efficient convolution
+operation that can be tuned efficiently to popular CPUs.
+This design is coupled with a scheme for obtaining the best
+memory layout for data in different operations of a CNN
+model, in order to minimize the overhead of transforming
+the data between various individual operations.
+6
+Discussion
+In this paper, we have discussed various reasons for the lack
+of scalability of inferencing ML models. While the reasons
+vary from micro to macro-levels, the common motive is that
+existing ML frameworks are geared towards high perfor-
+mance training. This is expressed by the fact that kernels for
+common operations are typically optimized for large batches
+with long inputs, ignoring relatively small overheads in var-
+ious parts of those frameworks that are immaterial to the
+overall training performance. However, during inference the
+batches tend to be much smaller and contain shorter inputs,
+thus making those overheads more prominent. A somewhat
+similar observation has been made by Aminabadi et al. [3].
+We leverage this poor scalability and describe a simple,
+yet powerful approach, in which the given input is broken
+into chunks and each chunk is processed in parallel, instead
+of using all available resources for the entire input. As we
+demonstrate with a few well-known models, this approach
+improves inference scalability and ultimately can lead to
+over 2x latency and throughput improvements.
+This work offers several directions for future research.
+First, we want to explore more dynamic thread allocation
+strategies, e.g., ones that can better adjust to the cases where
+the weight of a work chunk does not correlate linearly with
+its size and/or where the underlying model performs best
+while running with a single thread. Second, we want to find
+ways to automate splitting the input into chunks that can
+be processed in parallel, lowering the cost (in terms of user
+code changes) of using prun even further. Finally, we want
+9
+
+90
+三
+no-batch
+80
+batch
+(queries/s)
+70
+prun
+60
+50
+Throughput (
+40
+30
+20
+10
+0
+64
+128
+256
+512
+sequence length in a batchto explore other use cases where the use of prun would be
+beneficial, including other ML models that feature a pipeline-
+based architecture (e.g., [21, 29]).
+Acknowledgments
+The author would like to thank Dave Dice for valuable com-
+ments on an early draft of this paper.
+References
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+

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+Lesion-aware Dynamic Kernel for Polyp
+Segmentation
+Ruifei Zhang1, Peiwen Lai1, Xiang Wan2,4, De-Jun Fan3, Feng Gao3,
+Xiao-Jian Wu3, and Guanbin Li1,2⋆
+1 School of Computer Science and Engineering, Sun Yat-sen University,
+Guangzhou, China
+liguanbin@mail.sysu.edu.cn
+2 Shenzhen Research Institute of Big Data, Shenzhen, China
+3 The Sixth Affiliated Hospital, Sun Yat-sen University, Guangzhou, China
+4 Pazhou Lab, Guangzhou, China
+Abstract. Automatic and accurate polyp segmentation plays an essen-
+tial role in early colorectal cancer diagnosis. However, it has always been
+a challenging task due to 1) the diverse shape, size, brightness and other
+appearance characteristics of polyps, 2) the tiny contrast between con-
+cealed polyps and their surrounding regions. To address these problems,
+we propose a lesion-aware dynamic network (LDNet) for polyp segmen-
+tation, which is a traditional u-shape encoder-decoder structure incor-
+porated with a dynamic kernel generation and updating scheme. Specif-
+ically, the designed segmentation head is conditioned on the global con-
+text features of the input image and iteratively updated by the extracted
+lesion features according to polyp segmentation predictions. This simple
+but effective scheme endows our model with powerful segmentation per-
+formance and generalization capability. Besides, we utilize the extracted
+lesion representation to enhance the feature contrast between the polyp
+and background regions by a tailored lesion-aware cross-attention mod-
+ule (LCA), and design an efficient self-attention module (ESA) to cap-
+ture long-range context relations, further improving the segmentation
+accuracy. Extensive experiments on four public polyp benchmarks and
+our collected large-scale polyp dataset demonstrate the superior perfor-
+mance of our method compared with other state-of-the-art approaches.
+The source code is available at https://github.com/ReaFly/LDNet.
+1
+Introduction
+Colorectal Cancer (CRC) is one of the most common cancer diseases around
+the world [17]. However, actually, most CRC starts from a benign polyp and
+gets progressively worse over several years. Thus, early polyp detection and re-
+moval make essential roles to reduce the incidence of CRC. In clinical practice,
+colonoscopy is a common examination tool for early polyp screening. An accurate
+and automatic polyp segmentation algorithm based on colonoscopy images can
+⋆ Corresponding author is Guanbin Li.
+arXiv:2301.04904v1  [eess.IV]  12 Jan 2023
+
+2
+R. Zhang et al.
+greatly support clinicians and alleviate the reliance on expensive labor, which is
+of great clinical significance.
+However, accurate polyp segmentation still remains challenge due to the di-
+verse but concealed characteristics of polyps. Early traditional approaches [12,19]
+utilize hand-craft features to detect polyps, failing to cope with complex scenario
+and suffering from high misdiagnosis rate. With the advance of deep learning
+technology, plenty of CNN-based methods are developed and applied for polyp
+segmentation. Fully convolution network [11] is first proposed for semantic seg-
+mentation, and then its variants [3,1] also make a great breakthrough in the
+polyp segmentation task. UNet [16] adopts an encoder-decoder structure and
+introduces skip-connections to bridge each stage between encoder and decoder,
+supplying multi-level information to obtain a high-resolution segmentation map
+through successive up-sampling operations. UNet++ [27] introduces more dense
+and nested connections, aiming to alleviate the semantic difference of features
+maps between encoder and decoder. Recently, to better overcome the above men-
+tioned challenges, some networks specially designed for polyp segmentation task
+have been proposed. For example, PraNet [6] adopts a reverse attention mech-
+anism to mine finer boundary cues based on the initial segmentation map. AC-
+SNet [23] adaptively selects and integrates both global contexts and local infor-
+mation, achieving more robust polyp segmentation performance. CCBANet [13]
+proposes the cascading context and the attention balance modules to aggregate
+better feature representation. SANet [21] designs the color exchange operation to
+alleviate the color diversity of polyps, and proposes a shallow attention module to
+select more useful shallow features, obtaining comparable segmentation results.
+However, existing methods mainly focus on enhancing the network’s lesion repre-
+sentation from the view of feature selection [13,23,6] or data augmentation [21],
+and no attempts have been made to consider the structural design of the net-
+work from the perspective of improving the flexibility and adaptability of model
+feature learning, which limits their generalization.
+To this end, we design a Lesion-aware Dynamic Network (LDNet) for the
+polyp segmentation task. Inspired by [9,24], we believe that a dynamic kernel
+can adaptively adjust parameters according to the input image, and thus achiev-
+ing stronger feature exploration capabilities in exchange for better segmentation
+performance. Specifically, our unique kernel (also known as segmentation head)
+is dynamically generated basing on the global features of the input image, and
+generates one polyp segmentation prediction in each decoder stage. Accordingly,
+these segmentation results serve as clues to extract refined polyp features, which
+in turn update our kernel parameters with better lesion perception. For some
+complex polyp regions, the dynamic kernel generation and update mechanism
+we designed can step-wisely learn and mine discriminative regional features and
+gradually improve the segmentation results, enhancing the generalization of the
+model. Besides, we design two attention modules, i.e., Efficient Self-Attention
+(ESA) and Lesion-aware Cross-Attention (LCA). The former is used to capture
+global feature relations, while the latter is designed to enhance feature contrast
+between lesions and other background regions, further improving the segmenta-
+
+Lesion-aware Dynamic Kernel for Polyp Segmentation
+3
+Fig. 1. (a) Overview of our LDNet. (b) Illustration of kernel generation and update.
+tion performance. In summary, the contributions of this paper mainly include
+three folds: (1) We design a lesion-aware dynamic network for polyp segmenta-
+tion. The introduction of a dynamic kernel generation and update mechanism
+endows the model with generalizability to discriminate polyp regions with di-
+verse shapes, sizes, and appearances. (2) Our tailored ESA and LCA modules
+enhance the polyp feature representation, which helps to mine concealed polyps
+with low visual contrast. (3) Extensive experiments on four public polyp bench-
+marks and our collected large-scale polyp dataset demonstrate the effectiveness
+of our proposed method.
+2
+Methodology
+The overview of our LDNet is shown in Fig. 1, which is a general encoder-decoder
+structure, incorporated with our designed dynamic kernel scheme and attention
+modules. The Res2Net [8] is utilized as our encoder, consisting of five blocks.
+The generated feature map of each block is denoted as {Ei}5
+i=1. Accordingly,
+five-layer decoder blocks are adopted and their respective generated features are
+defined as {Di}5
+i=1. 1 × 1 convolution is utilized to unify the dimension of Di to
+64, denoted as ¯Di, which are adaptive to subsequent kernel update operations. In
+contrast to previous methods [23,7,21,6] with a static segmentation head, which
+is agnostic to the input images and remains fixed during the inference stage,
+we design a dynamic kernel as our segmentation head. The dynamic kernel is
+essentially a convolution operator used to produce segmentation result, but its
+parameters are initially generated by the global feature E5 of the input, and
+iteratively updated in the multi-stage decoder process based on the current de-
+coder features ¯Di and its previous segmentation result Pi+1, which is employed
+to make a new prediction Pi. For the convenience of expression, we denote the
+sequential updated kernels as {Ki}5
+i=1. Each segmentation prediction is super-
+vised by the corresponding down-sampled Ground Truth, and the prediction P1
+
+ Flow of feature
+ Flow of map
+Flow of kernel
+(U Upsample O Element-wise multiplication (
++) Element-wise addition G Gate
+Encoder-Decoder
+Kernel Generation (KG)
+E4
+E5
+ pooling
+E2 
+E3
+convlx1
+E1
+★
+E5
+Image
+ESA
+ESA
+ESA
+ ESA
+ESA
+Cs × Hs × Ws
+Kernel (Ks)
+IPs
+Adaptive
+LP2
+LP3
+LP4
+64×K×K
+Pooling
+D1 +
+LCA
+D2
+LCA
+D3 ← LCA
+D4 
+LCA
+Ds
+(× )
+Kernel Update (KU)
+convlx1
+convlxl
+convlxl
+convlxl
+convlxl
+Pi+1
+D3
+D4
+Ds
+D;
+D1
+KU
+KU <
+K3
+ KU <
+KU <
+(K5)
+K2
+KG
+G
+I?
+·P5
+P O
+P2.
+P.
+G
++
+GT
+Ki
+Dynamic Kernel Generation and Update
+(a)
+(b)4
+R. Zhang et al.
+of the last decoder stage is the final result of our model. We detail the dynamic
+kernel scheme and attention modules in the following sections.
+2.1
+Lesion-aware Dynamic Kernel
+Kernel Generation Dynamic kernels can be generated in a variety of ways
+and have been successfully applied in many fields [24,9,22,14]. In this paper, We
+adopt a simple but effective method to generate our initial kernel. As shown in
+Fig.1, given the global context feature E5, we first utilize an adaptive average
+pooling operation to aggregate features into a size of K × K, and then perform
+one 1 × 1 convolution to produce the initial segmentation kernel with a reduced
+dimension of 64. To be consistent with the sequence of decoder, we denote our
+initial kernel as K5 ∈ R1×64×K×K. K5 is acted on the unified decoder features
+¯D5 to generate the initial polyp prediction P5.
+Kernel Update Inspired by [24], we design an iterative update scheme based
+on the encoder-decoder architecture to improve our dynamic kernel. Given the
+i-th unified decoder features ¯Di ∈ R64×Hi×Wi and previous polyp segmentation
+result Pi+1 ∈ R1×Hi+1×Wi+1, we first extract lesion features as:
+Fi =
+Hi
+� Wi
+�
+up2(Pi+1) ◦ ¯Di,
+(1)
+where up2 denotes up-sampling the prediction map by a factor of 2 to keep a
+same size with feature map. ‘◦’ represents the element-wise multiplication with
+broadcasting mechanism.
+The essential operation of the kernel update is to integrate the lesion repre-
+sentations extracted by the current decoder features into previous kernel param-
+eters. In this way, the kernel can not only perceive the lesion characteristics to be
+segmented in advance, but gradually incorporate multi-scale lesion information,
+thus enhancing its discrimination ability for polyps. Since the previous polyp
+prediction may be inaccurate, as in [24], we further utilize a gate mechanism
+to filter the noise in lesion features and achieve an adaptive kernel update. The
+formulation is:
+Ki = GF
+i ◦ φ1(Fi) + GK
+i ◦ φ2(Ki+1),
+(2)
+where φ1 and φ2 denote linear transformations. GF
+i
+and GK
+i
+are two gates,
+which are obtained by the element-wise multiplication between the variants of
+Fi and Ki+1 followed by different linear transformation and Sigmoid function
+(σ), respectively:
+Gi = φ3(Fi) ◦ φ4(Ki+1)
+(3)
+GK
+i = σ(φ5(Gi)), GF
+i = σ(φ6(Gi))
+(4)
+The updated kernel Ki is acted on the specific decoder feature to make a new
+prediction Pi. Both of them are sent to the (i−1)-th decoder stage to iteratively
+perform the above update scheme.
+
+Lesion-aware Dynamic Kernel for Polyp Segmentation
+5
+Fig. 2. (a) Illustration of ESA. (b) Illustration of LCA. FF denotes the feed-forward
+layer. We omit the residual addition between the input and output of FF for simplicity.
+2.2
+Attention Modules
+Efficient Self-Attention Self-attention mechanism is first proposed in Trans-
+former [20], and recently has played a significant role in many tasks [5,4] due
+to its strong long-range modeling capability, however is criticized for prohibitive
+computation and memory cost. To overcome these challenges, we borrow the
+idea from [26,28] and design our ESA module. As shown in Fig. 2, we follow
+the component of Transformer but replace the original self-attention with our
+ESA layer, followed by a feed-forward layer and a reshaping operation. We also
+perform a multi-head parallel scheme to further improve the performance. Specif-
+ically, given one encoder feature map Ei ∈ RCi×Hi×Wi, details of our ESA layer
+are formulated as follows:
+ESA(Ei) = φo(concat(head0, ..., headn)),
+(5)
+headj = Attention(φj
+q(Q), φj
+k(K), φj
+v(V)),
+(6)
+where φo, φj
+q, φj
+k, φj
+v denote the linear projections, and n is the number of
+heads. Q ∈ RNi×Ci(Ni = Hi × Wi) is reshaped from the Ei. K, V ∈ RS×Ci
+are obtained by the pyramid pooling operation [26], which includes 1 × 1, 3 × 3,
+5 × 5 adaptive average pooling to down-sample the feature map, followed by
+reshaping and concatenating operations. Thanks to such a sampling process, we
+utilize fewer representative global features to perform the standard attention [20],
+not only introducing global relations to original features, but significantly saving
+the computation overhead (S = 1 × 1 + 3 × 3 + 5 × 5 ≪ Ni). Attention(·) is
+formulated as:
+Attention(q, k, v) = softmax(qkT
+√dk
+)v,
+(7)
+where dk is the dimension of each head, equal to Ci
+n .
+
+6
+R. Zhang et al.
+Lesion-aware Cross-Attention Besides our lesion-aware dynamic kernel, the
+predicted polyp result is also utilized to enhance the features. Specifically, given
+the decoder feature Di ∈ RCi×Hi×Wi and the prediction Pi ∈ R1×Hi×Wi, the ex-
+tracted lesion representations by Equ. 1 (w/o up2) serve as the K and V ∈ R1×Ci
+to perform the cross-attention, which is similar to the above mentioned self-
+attention. Through such an operation, the more similar the region to the lesion,
+the further enhancement of lesion characteristics, which significantly improves
+the feature contrast and benefits to detect concealed polyps.
+3
+Experiments
+3.1
+Datasets
+Public Polyp Benchmarks We evaluate our proposed LDNet on four public
+polyp datasets, including Kvasir-SEG [10], CVC-ClinicDB [2], CVC-ColonDB [19]
+and ETIS [18]. Following the same setting in [21,6], we randomly select 80% im-
+ages respectively from Kvasir-SEG and CVC-ClinicDB and fuse them together
+as our training set, 10% as validation set. The remaining data of Kvasir-SEG
+and CVC-ClinicDB, and other two unseen datasets are used for testing.
+Our Collected Large-Scale Polyp Dataset We also evaluate LDNet on our
+collected polyp dataset, which has 5175 images in total. This dataset is randomly
+split into 60% for training, 20% for validation, and the remaining for testing.
+3.2
+Implementation Details and Evaluation Metrics
+Our method is implemented based on PyTorch framework [15] and runs on an
+NVIDIA GeForce RTX 2080 Ti GPU. We simply set K = 1 in the kernel gener-
+ation and n = 8 in the multi-head attention mechanism. The SGD optimizer is
+utilized to train the model, with batch size of 8, momentum of 0.9 and weight
+decay of 10−5. The initial learning rate is set to 0.001, and adjusted by a poly
+learning rate policy, which is lr = lrinit × (1 −
+epoch
+nEpoch)power, where power = 0.9,
+nEpoch = 80. All images are uniformly resized to 256 × 256. To avoid overfit-
+ting, data augmentations including random horizontal and vertical flips, rota-
+tion, random cropping are used in the training stage. A combination of Binary
+Cross-Entropy loss and Dice loss is used to supervise the training process.
+As in [23,7], eight common metrics are adopted to evaluate polyp segmen-
+tation performance, including Recall, Specificity, Precision, Dice Score, IoU for
+Polyp (IoUp), IoU for Background (IoUb), Mean IoU (mIoU) and Accuracy.
+3.3
+Experiments on the Public Polyp Benchmarks
+We compare our LDNet with several state-of-the-art methods, including UNet [16],
+ResUNet [25], UNet++[27], ACSNet [23], PraNet [6], SANet [21], CCBANet [13],
+on the public polyp benchmarks. As shown in Table 1, our LDNet achieves su-
+perior performance over other methods across four datasets on most metrics.
+
+Lesion-aware Dynamic Kernel for Polyp Segmentation
+7
+Table 1. Comparison with other state-of-the-art methods on four benchmark datasets.
+The best three results are highlighted in red, green and blue, respectively.
+Methods
+Rec
+Spec
+Prec
+Dice
+IoUp
+IoUb
+mIoU
+Acc
+Kvasir
+UNet [16]
+87.04
+97.25
+84.28
+82.60
+73.39
+93.89
+83.64
+95.05
+ResUNet [25]
+84.70
+97.17
+83.00
+80.50
+70.60
+93.19
+81.89
+94.43
+UNet++ [27]
+89.23
+97.20
+85.57
+84.77
+76.42
+94.23
+85.32
+95.44
+ACSNet [23]
+91.35
+98.39
+91.46
+89.54
+83.72
+96.42
+90.07
+97.16
+PraNet [6]
+93.90
+97.33
+89.87
+90.32
+84.55
+95.98
+90.26
+96.75
+CCBANet [13]
+90.71
+98.04
+91.02
+89.04
+82.82
+96.21
+89.52
+97.02
+SANet [21]
+92.06
+98.20
+91.14
+89.92
+83.97
+96.54
+90.26
+97.18
+Ours
+92.72
+98.05
+92.04
+90.70
+85.30
+96.71
+91.01
+97.35
+CVC-
+ClinicDB
+UNet [16]
+88.61
+98.70
+85.10
+85.12
+77.78
+97.70
+87.74
+97.95
+ResUNet [25]
+90.89
+99.25
+90.22
+89.98
+82.77
+98.18
+90.47
+98.37
+UNet++ [27]
+87.78
+99.21
+90.02
+87.99
+80.69
+97.92
+89.30
+98.12
+ACSNet [23]
+93.46
+99.54
+94.63
+93.80
+88.57
+98.95
+93.76
+99.08
+PraNet [6]
+95.22
+99.34
+92.25
+93.49
+88.08
+98.92
+93.50
+99.05
+CCBANet [13]
+94.89
+99.22
+91.39
+92.83
+86.96
+98.79
+92.87
+98.93
+SANet [21]
+94.74
+99.41
+92.88
+93.61
+88.26
+98.94
+93.60
+99.07
+Ours
+94.49
+99.51
+94.53
+94.31
+89.48
+98.95
+94.21
+99.08
+CVC-
+ColonDB
+UNet [16]
+63.05
+98.00
+68.01
+56.40
+47.32
+94.51
+70.92
+94.84
+ResUNet [25]
+59.91
+98.06
+65.29
+54.87
+44.31
+93.77
+69.04
+94.06
+UNet++ [27]
+63.49
+98.59
+77.79
+60.77
+52.64
+95.19
+73.92
+95.48
+ACSNet [23]
+77.38
+99.26
+81.72
+75.51
+67.38
+96.16
+81.77
+96.32
+PraNet [6]
+81.85
+98.54
+78.43
+76.24
+68.29
+96.06
+82.17
+96.26
+CCBANet [13]
+82.34
+98.39
+77.79
+75.36
+66.57
+95.89
+81.23
+96.14
+SANet [21]
+75.21
+99.09
+81.43
+73.50
+65.47
+96.19
+80.83
+96.40
+Ours
+83.46
+98.49
+81.15
+78.43
+70.58
+96.21
+83.39
+96.48
+ETIS
+UNet [16]
+47.33
+96.36
+48.05
+34.81
+28.38
+94.72
+61.55
+94.90
+ResUNet [25]
+49.12
+97.21
+56.85
+38.65
+30.54
+95.27
+62.90
+95.43
+UNet++ [27]
+55.52
+95.40
+59.14
+40.91
+33.86
+93.87
+63.87
+94.07
+ACSNet [23]
+78.31
+98.44
+68.81
+69.44
+60.96
+97.78
+79.37
+97.89
+PraNet [6]
+81.20
+98.73
+72.23
+72.38
+64.07
+98.29
+81.18
+98.38
+CCBANet [13]
+78.70
+97.19
+61.12
+62.63
+53.81
+96.52
+75.17
+96.66
+SANet [21]
+77.08
+99.04
+72.73
+72.26
+63.33
+98.47
+80.90
+98.54
+Ours
+82.83
+98.44
+72.07
+74.37
+66.50
+98.01
+82.26
+98.10
+Table 2. Comparison with other state-of-the-art methods and ablation study on our
+collected dataset.
+Methods
+Rec
+Spec
+Prec
+Dice
+IoUp
+IoUb
+mIoU
+Acc
+UNet [16]
+87.89
+97.27
+87.23
+85.00
+77.48
+93.95
+85.71
+95.64
+UNet++ [27]
+89.88
+97.43
+88.18
+86.92
+79.88
+94.56
+87.26
+96.21
+ACSNet [23]
+92.43
+97.79
+90.94
+90.54
+84.64
+95.75
+90.19
+97.11
+PraNet [6]
+92.86
+97.87
+90.52
+90.64
+84.60
+95.91
+90.25
+97.28
+CCBANet [13]
+91.91
+97.79
+91.32
+90.39
+84.36
+95.73
+90.04
+97.10
+SANet [21]
+92.18
+98.22
+91.67
+90.75
+84.98
+96.02
+90.50
+97.27
+Ours
+93.22
+98.15
+92.16
+91.66
+86.28
+96.39
+91.34
+97.55
+Baseline
+92.02
+97.03
+87.75
+88.30
+81.26
+94.95
+88.11
+96.54
+Baseline+DK
+92.22
+97.58
+90.41
+89.88
+83.86
+95.53
+89.70
+96.92
+Baseline+DK+ESAs
+91.76
+98.25
+92.14
+90.74
+84.91
+95.85
+90.38
+97.14
+
+8
+R. Zhang et al.
+Fig. 3. Visual comparison of polyp segmentation results.
+In particular, on the two seen datasets, i.e., Kvasir and CVC-ClinicDB, the
+proposed LDNet obtains the best Dice and mIoU scores, outperforming other
+methods. On the other two unseen datasets, the LDNet also shows strong gener-
+alization ability and achieves 78.43% and 74.37% Dice scores, 2.19% and 1.99%
+improvements over the second best approaches, further demonstrating the effec-
+tiveness of our approach. Some visualization examples are shown in Fig. 3.
+3.4
+Experiments on the Collected Large-Scale Polyp Dataset
+On our collected large-scale polyp dataset, we compare the LDNet with UNet [16],
+UNet++[27], ACSNet [23], PraNet [6], SANet [21] and CCBANet [13]. As shown
+in Table 2, our method again achieves the best performance, with a Dice of
+91.66% and a mIoU of 91.34%, respectively.
+3.5
+Ablation Study
+We conduct a series of ablation studies on our collected polyp dataset to verify
+the effectiveness of our designed dynamic kernel scheme and attention modules.
+Specifically, we utilize the traditional u-shape structure with a static segmenta-
+tion head as our baseline, and gradually replace the static head with our designed
+dynamic kernels, then further add ESA and LCA modules, denoting as Baseline,
+Baseline+DK, Baseline+DK+ESAs and Ours respectively. As shown in Table 2,
+the introduction of the dynamic kernel significantly enhances the performance
+of the baseline, with a 1.58% improvement of Dice score. With the addition of
+our ESA and LCA modules, the scores of Dice and mIoU are further boosted
+by 0.86% and 0.68%, 0.92% and 0.96%, respectively.
+4
+Conclusion
+In this paper, we propose the lesion-aware dynamic kernel (LDNet) for polyp
+segmentation, which is generated conditioned on the global information and up-
+
+Lesion-aware Dynamic Kernel for Polyp Segmentation
+9
+dated by the multi-level lesion features. We believe that such a dynamic kernel
+can endow our model with more flexibility to attend diverse polyps regions.
+Besides, we also improve the feature representation and enhance the context
+contrast by two tailored attention modules, i.e., ESA and LCA, which is benefi-
+cial for detecting concealed polyps. Extensive experiments and ablation studies
+demonstrate the effectiveness of our proposed method.
+Acknowledgements This work is supported in part by the Chinese Key-Area
+Research and Development Program of Guangdong Province (2020B0101350001),
+in part by the Guangdong Basic and Applied Basic Research Foundation (2020B
+1515020048), in part by the National Natural Science Foundation of China
+(61976250), in part by the Guangzhou Science and technology project (20210202
+0633), and is also supported by the Guangdong Provincial Key Laboratory of
+Big Data Computing, The Chinese University of Hong Kong, Shenzhen.
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+

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+1
+Investigation of the characteristics of the electromagnetic
+induction transparent-like spectrum with counter-
+propagating waves coupling mechanism for waveguide
+and micro-ring coupled system
+Chaoying Zhaoa,b*
+aHangzhou Dianzi University, College of Science, Zhejiang, China bShanxi University, State Key
+Laboratory of Quantum Optics and Quantum Optics Devices, Taiyuan, China
+E-mail: zchy49@163.com
+Abstract
+In this paper, a new counter-propagation waves coupling mechanism is proposed which is
+expected to realize an electromagnetically induced transparency (EIT)-like effect. Comparing the
+traveling waves coupling
+mechanism (see J. Mod. Opt. 2015,62:313-320 [9]) with the
+counter-propagating waves coupling mechanism, we find out that the transparency window
+breadths of transmission spectra are greatly enhanced and the corresponding phase shift spectra
+possess a flat profile or a square profile. Our numerical simulated results are in good agreement
+with the theoretical analysis. The EIT-like effect can significantly reduce the group velocity near
+the edge of the square profile transparent window. We believe that the counter-propagating waves
+coupling mechanism is particularly beneficial for the realization of active manipulation of slow
+light devices (such as delay lines) required in the conventional EIT scheme. In the vicinity of the
+transparency peak, we can obtain a large group delay, may gain more significant potential
+applications in slow-light transmission and optical storage.
+Keywords: electromagnetically induced transparency-like (EIT-like), counter-propagating waves
+coupling, micro-ring, group delay
+1. Introduction
+The Electromagnetic induced transparency (EIT) effect occurs when the atomic medium
+accompany a strong dispersion, which can significantly reduce the group velocity of light.
+* Author to whom any correspondence should be addressed.
+
+2
+EIT is widely used in slow light[1,2], optical delay lines[3], optical storage[4] and low-loss optical
+devices[5]so on. However, the experimental realization of EIT effect in atomic system usually
+requires some complicated conditions, such as temperature, high intensity laser, etc. Therefore, the
+application of EIT effect in practice has been greatly limited. The all-optical analog to EIT based on
+wave-guide and micro-ring coupled system has attracted much more attention in recent years. The
+EIT-like phenomena have been theoretically analyzed in wave-guide coupled micro-ring[6], and
+wave-guide
+coupled
+two
+micro-ring[7].
+Meng
+et.al
+have
+studied
+the
+transmission
+spectra
+of N
+N
+
+weak linearly array[8]. We have investigated the wave-guide coupled double-micro-ring
+systems by using the traveling wave coupling mechanism[9]. In 2008, Zhang et. al. first proposed the
+concept of PIT (plasmon-induced transparency) in metamaterials[10], Metamaterials are a class of
+artificial electromagnetic media, aiming to provide some controllable electromagnetic properties, such
+as room temperature conditions and large operating bandwidth. They pointed out that the EIT-like
+effect can be simulated by adding a special resonance structure (dark state resonance unit). EIT-like
+effect can be explained by the near-field coupling principle between bright and dark resonant units. In
+2021, Zhao et.al have studied dual-band electromagnetically induced transparency (EIT)-like effect in
+terahertz spectrum. The EIT-like effect can significantly reduce the group speed near the transparent
+window. The EIT-like effect can significantly reduce the group speed near the transparent window, may
+gain more sigificant potential applications in slow-light transmission and optical storage [11]. In this
+paper, we want to investigate the transmission spectra and phase shift by using the counter-propagating
+waves coupling mechanism. We find out that the transparency window breadths of transmission
+spectra are greatly enhanced when the traveling wave coupling is substituted by counter-propagating
+waves coupling, and the phase shift spectra possess a flat or square profile, which result in group
+velocity slowdown very large at the edge of the square profile. This paper is organized as follows. In
+Section 2, the transmission of 2
+2
+
+coupled system for the traveling wave coupling and the
+counter-propagating waves coupling is studied in Section 3. The transmission and phase characteristic
+of 2
+2
+
+and3 3
+
+coupled systems containing single micro-ring, doule micro-ring (which is denoted by
+‘Sm’ in the following) are calculated and analyzed in Section 4, and in Section 5, the conclusions are
+given.
+2.The transmission of traveling wave coupling and counter-propagating waves
+
+3
+coupling for 2
+2
+
+system
+For 2
+2
+
+system, Fig.1(a) depicts the schematic structure of the micro-ring resonator, which consists
+of straight wave-guide a2b2, ring a1b1. The optical field from a excitation wave-guide passing
+through the evanescent tail couples into ring. After propagating a round trip, the wave couples back
+to the excitation wave-guide and interferes with transmitted light. At resonance, the wave appears
+destructive interference. The propagation path of light presents ‘clockwise direction’ in ring (see
+Fig.1(a)). As shown in Figure 1(a), the counter-clockwise mode
+1
+1
+j
+a
+e b
+
+
+
+in the micro-ring
+1 1
+a b ,
+is induced by the traveling wave
+(
+)
+0
+j
+t kz
+e
+
+
+
+
+
+
+in the waveguide
+2
+2
+a b from the left to the right,
+where  is the absorption coefficient,
+L c
+
+
+
+is the round trip phase shift, L represents the
+circumference of the micro-ring
+1 1
+a b , c is the phase velocity of micro-ring mode,  is the optical
+wave oscillation frequency.
+While the light circuits clockwise and counterclockwise direction in ring simultaneously (see
+Fig.1(b)). We can analyze the gap parameter makes an influence on the EIT-like spectrum.For Figure
+1.(b), the clockwise mode
+1
+1 2
+j
+a
+e
+b
+
+
+
+
+in the micro-ring
+1 1
+a b , which is induced by the traveling
+wave
+(
+)
+0
+j
+t kz
+e
+
+
+
+
+
+
+in the waveguide
+2
+2
+a b from the left to the right. The counter-clockwise
+mode
+1
+1 2
+j
+a
+e b
+
+
+
+in
+the
+micro-ring
+1 1
+a b
+,
+which
+is
+induced
+by
+the
+traveling
+wave
+(
+)
+0
+j
+t kz
+e
+
+
+
+
+
+
+in the waveguide
+2
+2
+a b from the right to the left.
+(a)
+(b)
+Figure 1 Schematic diagram of 2
+2
+
+system, (a) traveling wave coupling, (b) counter-propagating wave
+coupling.
+Assuming that the light wave is input from the straight waveguide input, when the resonance
+condition is satisfied, the phase before the light is coupled from the straight waveguide to the
+
+a1
+b1
+a2
+b2b1
+ai
+a2
+b24
+microring is
+1
+
+ 
+. According to the optical waveguide coupling theory, the changes in the
+phase before and after a light wave is coupled from one waveguide to another. Then, when the
+light wave is coupled from the straight waveguide to the micro-ring waveguide, its phase
+becomes
+2
+2
+
+
+ 
+
+. According to the resonance relationship, when the light wave meets the
+resonance condition, the light circles in the microring and interferes with each other to generate
+resonance enhancement. The change 2m of the phase before and after the light wave circulates
+in the microring, where the phase is
+3
+2
+2m
+
+
+
+ 
+
+
+. Similarly, the light wave is again
+coupled from the microring to the straight waveguide, and its phase changes again
+2
+
+,
+becoming
+4
+2m
+
+
+
+ 
+
+
+. Comparing the phase changes at the input and output, we find an
+ odd multiple of the difference between
+1
+
+and
+4
+ , so that the two beams interfere and cancel.
+When the intensity of the light field is the same, the intensity of the light field after interference
+elimination is zero, so the output end of the straight waveguide is zero.
+3. The influence of counter-propagation optical waves with phase difference
+Figure 2 show the Schematic diagram to produce the counter-propagating waves.
+Figure 2 Schematic diagram producing the counter-propagating waves.
+As shown in Figure 2, one traveling wave is split into two equal portions of counter-clockwise
+wave and clockwise wave at the point P. Therefore, the superposed induced counter-clockwise
+wave and clockwise wave of the micro-ring assuming (
+) 2
+cos[ ]
+j
+j
+e
+e
+
+
+
+
+
+
+.Taking account
+the absorption coefficient  around the micro-ring, we have the relation
+1
+1
+cos[ ]
+a
+b
+
+
+
+for
+counter-propagating waves, the mode
+1
+1
+1
+(
+)
+2
+cos[ ]
+j
+j
+a
+e
+e
+b
+b
+
+
+
+
+
+
+
+
+
+in the micro-ring is
+
+P
+a1
+b1
+a2
+3
+b25
+induced by the counter-propagating waves
+(
+)
+(
+)
+0 (
+) 2
+j
+t kz
+j
+t kz
+e
+e
+
+
+
+
+
+
+
+
+
+
+in the waveguide
+2
+2
+a b .
+For the traveling wave propagating in waveguide[4], the relation
+1
+1
+j
+a
+e b
+
+
+
+is also valid. In
+general, the coupling system contains one waveguide and multi-micro-rings [7,9].
+In the derivation of traveling waves coupling and counter-propagation waves coupling formula
+cos[ ]
+j
+e
+
+
+
+
+, we have assumed that the path from p to E (see Figure 2) along the clockwise
+and the counter-clockwise waves have the same length. In general, this lengths may be different,
+the initial phases at point E being
+1 ,
+2
+ for the clockwise and the counter-clockwise waves,
+respectively. The superposed induced waves being
+1
+2
+1
+2
+1
+2
+1
+2
+1
+2
+(
+)
+(
+)
+(
+)
+(
+)
+1
+2
+2
+2
+2
+2
+1
+1
+(
+)
+(
+)
+cos[
+]
+2
+2
+2
+j
+j
+j
+j
+j
+j
+e
+e
+e
+e
+e
+e
+
+
+
+
+
+
+
+
+
+
+ 
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+,(1)
+Therefore, the traveling waves coupling and counter-propagating waves coupling may be
+represented by
+2
+1
+cos[
+]
+j
+e
+
+
+
+
+
+,
+1
+1
+2
+(
+) 2
+
+
+
+
+
+,
+2
+1
+2
+(
+) 2
+
+
+
+
+
+, respectively. In the case
+of
+1
+
+
+,
+2
+2
+1
+cos[
+]
+j
+j
+e
+e
+
+
+
+
+
+
+
+
+possesses the form similar to that of the traveling waves
+coupling.
+4.The numerical calculations and analysis of the transmission and phase shift
+spectra for2
+2
+
+system and3
+3
+ system
+In this section, we calculate the transmission and phase shift spectra reduction near resonance
+concern with the traveling waves coupling and the counter-propagating waves coupling.
+In the following, we begin with the traveling wave coupling of 2
+2
+
+system. The transmission
+and the phase shift spectra [5]
+2
+2 2
+1
+i
+trl
+i
+t
+e
+T
+t e
+
+
+
+
+
+
+
+
+,
+2 2
+[
+]
+1
+i
+trl
+i
+t
+e
+Arg
+t e
+
+
+
+
+
+
+
+
+
+,
+(2)
+Figure 3 shows the transmission and the phase shift spectra for
+2 2
+trl
+T  ,
+2 2
+trl
+
+
+, respectively.
+
+6
+-0.004
+-0.002
+0
+0.002
+0.004
+
+1
+1.5
+2
+2.5
+3
+3.5
+T22
+l
+r
+t
+
+-0.004
+-0.002
+0
+0.002
+0.004
+
+-3
+-2
+-1
+0
+1
+2
+3
+22
+l
+r
+t
+
+Figure 3
+The transmission and phase shift spectrawith the parameters
+0.9998
+t 
+,
+1.00006
+ 
+versus in unit of radian.
+And the corresponding
+2 2
+cou
+T  ,
+2 2
+cou
+
+
+for counter-propagating wave coupling being
+2
+2 2
+cos[ ]
+1
+cos[ ]
+cou
+t
+T
+t
+
+
+
+
+
+
+
+
+,
+2 2
+cos[ ]
+[
+]
+1
+cos[ ]
+cou
+t
+Arg
+t
+
+
+
+
+
+
+
+
+
+,
+(3)
+Figure 4 shows the transmission and the phase shift spectra for
+2 2
+cou
+T  ,
+2 2
+cou
+
+
+, respectively.
+-0.1
+-0.05
+0
+0.05
+0.1
+
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+T22
+u
+o
+c
+
+-0.1
+-0.05
+0
+0.05
+0.1
+
+0
+0.5
+1
+1.5
+2
+2.5
+3
+22
+l
+u
+o
+c
+
+Figure 4
+The transmission and phase shift spectra with the parameters
+0.9998
+t 
+,
+1.00006
+ 
+versus in unit of radian.
+Similarly, the transmission and phase shift spectra for the waveguide and two micro-rings coupled
+system [7]
+2
+2
+2
+2
+1
+1
+2
+1
+2
+2
+2
+1
+1
+( )
+( ,
+)
+1
+( )
+i
+i
+r
+a
+e
+r a
+e
+
+
+ 
+
+ 
+ 
+
+
+
+,
+1
+1
+1
+1
+1
+1
+1
+1
+( )
+1
+i
+i
+r
+a e
+r a e
+
+
+ 
+
+
+
+,
+(4)
+Setting
+1
+2
+
+
+
+
+
+, we have
+2
+2 ( , )
+trl
+sm
+T
+
+ 
+
+,
+2
+[
+( , )]
+trl
+sm
+Arg 
+ 
+
+
+,
+(5)
+Figure 5 gives out the transmission and the phase shift spectra for
+2 2
+trl
+T  ,
+2 2
+trl
+
+
+, respectively.
+
+7
+-1.5
+-1
+-0.5
+0
+0.5
+1
+1.5
+
+0.2
+0.4
+0.6
+0.8
+T m
+s
+l
+r
+t
+
+-1.5
+-1
+-0.5
+0
+0.5
+1
+1.5
+
+-1
+-0.5
+0
+0.5
+1
+ m
+s
+l
+r
+t
+
+Figure
+5
+The transmission and the phase shift spectra for
+trl
+sm
+T
+,
+trl
+sm
+
+with parameters
+1
+0.999
+r 
+, 2
+1
+2
+r
+a
+a
+
+
+,
+1
+0.9999
+a 
+,
+2
+0.88
+a 
+versus
+in unit of radian.
+The corresponding formula for counter-propagating waves are
+2
+2 1
+1
+2
+2
+1
+2
+2
+2 1
+1
+2
+( )cos[
+]
+( ,
+)
+1
+( )cos[
+]
+cou
+cou
+cou
+r
+a
+r a
+
+
+
+
+ 
+
+
+
+
+
+
+,
+1
+1
+1
+1
+1
+1
+1
+1
+cos[ ]
+( )
+1
+cos[ ]
+cou
+r
+a
+r a
+
+
+
+
+
+
+
+,
+(6)
+Setting
+1
+2
+
+
+
+
+
+, we have
+2
+2 ( , )
+cou
+cou
+sm
+T
+
+ 
+
+,
+2
+[
+( , )]
+cou
+cou
+sm
+Arg 
+ 
+
+
+,
+(7)
+Figure 6 gives out the transmission spectra and the phase shift spectra for
+2 2
+cou
+T  ,
+2 2
+cou
+
+
+,
+respectively.
+-3
+-2
+-1
+0
+1
+2
+3
+
+0.2
+0.4
+0.6
+0.8
+1
+T m
+s
+u
+o
+c
+
+-3
+-2
+-1
+0
+1
+2
+3
+
+-1
+-0.5
+0
+0.5
+1
+ m
+s
+u
+o
+c
+
+Figure
+6
+The transmission and phase shift spectra for
+cou
+sm
+T
+,,
+cou
+sm
+
+with the parameters
+1
+0.999
+r 
+, 2
+1
+2
+r
+a
+a
+
+
+,
+1
+0.9999
+a 
+,
+2
+0.88
+a 
+versus in unit of radian.
+Figure 7 shows the schematic diagram of 3 3
+
+system, (a) the traveling wave coupling, (b) the
+counter-propagation waves coupling.
+(a)
+(b)
+
+a1
+b1
+a2
+b2
+a3
+b3b1
+a2
+b2
+a3
+b38
+Figure 7. Schematic diagram of3 3
+
+system, (a)the traveling wave coupling, (b)the counter-propagation waves
+coupling.
+The transmission and the phase shift spectra for3 3
+
+system[12].
+2
+1
+3
+1
+3
+3 3
+1
+3
+1
+3
+1
+trl
+t
+B
+B
+B B
+T
+B
+B
+B B
+t
+
+
+
+
+
+
+
+
+
+
+
+
+,
+1
+i
+B
+e 
+
+
+,
+3
+i
+B
+e
+
+
+
+
+,
+1
+3
+1
+3
+3 3
+1
+3
+1
+3
+[
+]
+1
+trl
+t
+B
+B
+B B
+Arg
+B
+B
+B B
+t
+
+
+
+
+
+
+
+
+
+
+
+
+
+,
+(8)
+Figure 8 gives out the transmission and the phase shift spectra for
+3 3
+trl
+T  ,
+3 3
+trl
+
+
+, respectively.
+-4
+-2
+0
+2
+4
+
+0
+0.2
+0.4
+0.6
+0.8
+T33
+l
+r
+t
+
+-4
+-2
+0
+2
+4
+
+0
+1
+2
+3
+4
+5
+6
+ m
+s
+l
+r
+t
+
+Figure
+8
+The
+transmission
+and
+phase
+shift
+spectra
+for
+3 3
+3 3
+,
+trl
+trl
+T 
+
+
+with
+parameters
+0.9998
+t 
+,
+0.99998
+ 
+,
+0
+ 
+,
+0
+ 
+versus in unit of
+4
+10 radian
+The transmission and the phase shift spectra for3 3
+
+system[9].
+2
+1
+3
+1
+3
+3 3
+1
+3
+1
+3
+1
+cou
+t
+B
+B
+B B
+T
+B
+B
+B B
+t
+
+
+
+
+
+
+
+
+
+
+
+
+,
+1
+cos[ ]
+B
+
+
+
+,
+3
+cos[ ]
+B
+
+
+
+,
+1
+3
+1
+3
+3 3
+1
+3
+1
+3
+[
+]
+1
+cou
+t
+B
+B
+B B
+Arg
+B
+B
+B B
+t
+
+
+
+
+
+
+
+
+
+
+
+
+
+,
+(9)
+Figure 9 gives out the transmission and the phase shift spectra for
+3 3
+cou
+T  ,
+3 3
+cou
+
+
+, respectively.
+-1000
+-500
+0
+500
+1000
+
+0
+0.2
+0.4
+0.6
+0.8
+T33
+u
+o
+c
+
+-1000
+-500
+0
+500
+1000
+
+0
+0.5
+1
+1.5
+2
+2.5
+3
+33
+u
+o
+c
+
+Figure 9
+The transmission and phase shift spectra for
+3 3
+3 3
+,
+cou
+cou
+T 
+
+
+with parameters,
+0.9998
+t 
+,
+0.99998
+ 
+,
+0
+ 
+,
+0
+ 
+versus in unit of
+4
+10 radian.
+The numerical calculations shown in Figures.3-6, 8-9, refer to that of the traveling wave coupling,
+the breadths of the transmission spectra for counter-propagating waves coupling are enhanced by a
+
+9
+factor of 0.02 0.001
+20
+
+, 0.3 0.05
+6
+
+, 200 0.8
+250
+
+for the 2
+2
+
+, Sm, 3 3
+
+system,
+respectively.
+5.The analysis of the phase velocity spectra
+We have obtained the phase spectra
+( )
+
+  
+, in the case of counter-propagating waves
+coupling, the phase shift spectra possess a flat (
+( )
+cou
+sm
+
+
+ 
+) or a square profile
+(
+2 2
+3 3
+( )
+,
+cou
+cou
+
+
+
+
+ 
+
+).
+The
+inverse
+of
+the
+phase
+velocity
+is
+given
+by
+3 3
+3 3
+cou
+cou
+g
+g
+c v
+n
+d
+d
+n
+c d
+Ld
+n
+c
+L
+
+
+
+
+
+
+
+
+
+
+
+
+
+[12], where
+L c
+
+
+
+, the group
+delay
+3 3
+cou
+g
+d
+d
+
+
+
+
+
+represents the time delay of narrow-band optical pulses in optical devices.
+The strong phase dispersion around the transparent window can cause a large group delay.
+-400
+-200
+0
+200
+400
+
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+vc
+Figure 10. The phase velocity spectra for3 3
+
+system versus in unit of
+4
+10 radian.
+As shown in Figure 10,
+gv approaches to zero (namely a large group delays
+g
+ are obtained) at the
+edge of the square profile (namely in the vicinity of the transparency peak, see Figure 9).
+The performance characteristics of micro-ring resonator are compared in Table1.
+Table 1 Comparison of EIT-like characteristic parameters based on different resonator
+Micro-ring
+resonator
+microcavity
+structure
+Number of
+EIT
+coupling
+mechanism
+material
+group
+delays
+[9]
+two
+micro-ring
+single
+traveling
+waves
+silica
+---
+[11]
+multiple
+U-shaped
+resonators
+multiple
+dark-light
+/light-light
+mode
+metamaterials
+high
+[13]
+chaotic
+single
+micro-ring
+single
+chaos-assisted
+dynamical
+tunneling
+silica
+---
+[14]
+two
+microtoroid
+single
+traveling
+waves
+silica
+---
+Our work
+single/two
+single
+counter-propa
+gating waves
+silica
+high
+
+10
+6.Loss analysis
+According to the coupling mode theory, optical loss can directly affect the line type and
+characteristics of EIT spectrum of the system. For the dual-coupling microring resonator proposed
+in this paper, the transmission loss is mainly composed of scattering loss 1
+scat
+Q
+caused by the
+roughness of the waveguide side wall, bending loss 1
+bend
+Q
+of the microring waveguide and
+leakage loss 1
+leak
+Q
+caused by the light field entering the substrate. 1
+mat
+Q
+is the absorption loss
+of microring resonator. As the light waves travel in the waveguide, part of the light field is
+absorbed by the silicon material. In addition, the top layer of the optical waveguide device also has
+an oxide coating, which can absorb the light field in different environments. Finite element
+analysis (FEM) was used to conduct mode analysis on the waveguide cross section, and negative
+imaginary part ( k ) was added to the material refraction to obtain the absorption loss of the whole
+system [1]. 1
+leak
+Q
+is leakage loss, which is an important loss mechanism in SOI structure, and is
+generated by the middle spectral field of the waveguide into the silicon dioxide layer and silicon
+substrate. The leakage loss decreases with the increase of silica layer thickness. 1
+scat
+Q
+is the
+scattering loss, which occurs when the side wall of the waveguide is rough. Increasing the ratio of
+the width to the height of the waveguide, reducing the mode overlap at the side of the waveguide,
+and improving the fabrication technology of the waveguide can reduce the scattering loss of the
+system. In practical optical waveguide devices, the roughness of the side wall is larger than that of
+the surface. 1
+bend
+Q
+is the bending loss, is the radiation loss caused by the bending of the optical
+waveguide, and decreases with the increase of the radius of the microring. When the radius of the
+microring is about 30 m
+
+[15], the bending loss is about 0.015
+180
+dB
+[16]. 1
+back
+Q
+represents
+backscattering loss, which is one of the loss sources in waveguides based on SOI structure. Both
+the roughness of the waveguide side wall and the directional coupling can cause backscattering
+losses, which can be suppressed by various ways, such as improving the etching process to reduce
+the roughness of the waveguide surface, and lower Q values [17]. In this paper, except for the
+feedback arm region, the coupling distance of the microring resonator is 0.15 m
+
+. At this time,
+
+11
+the resonance splitting has little influence on the transmission spectrum, so the backscattering can
+be ignored.
+Assuming that the transmission loss of microring resonator mainly includes absorption loss
+(1
+mat
+Q
+), leakage loss (1
+leak
+Q
+) and bending loss (1
+bend
+Q
+). Therefore, the relationship between
+the value of
+tot
+Q
+the system and the loss coefficient a can be expressed as [18]:
+2
+gn
+Q
+a
+
+
+
+(10)
+exp(
+)
+2
+ring
+L
+a
+
+
+
+(11)
+Where,
+gn
+is the group refractive index, which can be obtained by the finite-difference
+time-domain method of full-vector calculation by MODE Solutions software. a is the round-trip
+loss coefficient of light wave in microring waveguide, represents the loss value per unit length
+of optical waveguide, and
+tot
+Q
+is inversely proportional to  . Taking all loss mechanisms into
+consideration, the transmission loss of the optical waveguide in this paper is about9dB cm [19].
+7.Conclusions
+In conclusion, we have demonstrated the transmission spectra and the phase shift spectra of two
+micro-rings coupled system through numerical simulations and theoretical calculations and
+analyzed the causes of resonance peaks and transparent windows leads to the EIT-like effects. As
+we can see in this paper, we can generate a wide transparent windows, a large flat phase shift and
+a large group delay by increasing the number of micro-ring resonators. Therefore, it can be
+potential applied in multi-band filters and multi-band slow light devices (such as delay lines) in
+optical communication field.
+Acknowledgments
+This work was supported by the State Key Laboratory of Quantum Optics and Quantum Optics
+Devices, Shanxi University, Shanxi, China (KF202004，KF202205).
+ORCID iD
+Chaoying Zhao
+https://orcid.org/0000-0003-1116-0790
+References
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+

K9E2T4oBgHgl3EQfVQcE/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf,len=344
+page_content='1  Investigation of the characteristics of the electromagnetic  induction transparent-like spectrum with counter-  propagating waves coupling mechanism for waveguide  and micro-ring coupled system  Chaoying Zhaoa,b*  aHangzhou Dianzi University, College of Science, Zhejiang, China bShanxi University, State Key  Laboratory of Quantum Optics and Quantum Optics Devices, Taiyuan, China  E-mail: zchy49@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='com  Abstract  In this paper, a new counter-propagation waves coupling mechanism is proposed which is  expected to realize an electromagnetically induced transparency (EIT)-like effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' Comparing the  traveling waves coupling  mechanism (see J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' 2015,62:313-320 [9]) with the  counter-propagating waves coupling mechanism, we find out that the transparency window  breadths of transmission spectra are greatly enhanced and the corresponding phase shift spectra  possess a flat profile or a square profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' Our numerical simulated results are in good agreement  with the theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' The EIT-like effect can significantly reduce the group velocity near  the edge of the square profile transparent window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' We believe that the counter-propagating waves  coupling mechanism is particularly beneficial for the realization of active manipulation of slow  light devices (such as delay lines) required in the conventional EIT scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' In the vicinity of the  transparency peak, we can obtain a large group delay, may gain more significant potential  applications in slow-light transmission and optical storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  Keywords: electromagnetically induced transparency-like (EIT-like), counter-propagating waves  coupling, micro-ring, group delay  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' Introduction  The Electromagnetic induced transparency (EIT) effect occurs when the atomic medium  accompany a strong dispersion, which can significantly reduce the group velocity of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  Author to whom any correspondence should be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  2  EIT is widely used in slow light[1,2], optical delay lines[3], optical storage[4] and low-loss optical  devices[5]so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' However, the experimental realization of EIT effect in atomic system usually  requires some complicated conditions, such as temperature, high intensity laser, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' Therefore, the  application of EIT effect in practice has been greatly limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' The all-optical analog to EIT based on  wave-guide and micro-ring coupled system has attracted much more attention in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' The  EIT-like phenomena have been theoretically analyzed in wave-guide coupled micro-ring[6], and  wave-guide  coupled  two  micro-ring[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  Meng  et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='al  have  studied  the  transmission  spectra  of N  N  \uf0b4  weak linearly array[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' We have investigated the wave-guide coupled double-micro-ring  systems by using the traveling wave coupling mechanism[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' In 2008, Zhang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' first proposed the  concept of PIT (plasmon-induced transparency) in metamaterials[10], Metamaterials are a class of  artificial electromagnetic media, aiming to provide some controllable electromagnetic properties, such  as room temperature conditions and large operating bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' They pointed out that the EIT-like  effect can be simulated by adding a special resonance structure (dark state resonance unit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' EIT-like  effect can be explained by the near-field coupling principle between bright and dark resonant units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' In  2021, Zhao et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='al have studied dual-band electromagnetically induced transparency (EIT)-like effect in  terahertz spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' The EIT-like effect can significantly reduce the group speed near the transparent  window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' The EIT-like effect can significantly reduce the group speed near the transparent window, may  gain more sigificant potential applications in slow-light transmission and optical storage [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' In this  paper, we want to investigate the transmission spectra and phase shift by using the counter-propagating  waves coupling mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' We find out that the transparency window breadths of transmission  spectra are greatly enhanced when the traveling wave coupling is substituted by counter-propagating  waves coupling, and the phase shift spectra possess a flat or square profile, which result in group  velocity slowdown very large at the edge of the square profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' In  Section 2, the transmission of 2  2  \uf0b4  coupled system for the traveling wave coupling and the  counter-propagating waves coupling is studied in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' The transmission and phase characteristic  of 2  2  \uf0b4  and3 3  \uf0b4  coupled systems containing single micro-ring, doule micro-ring (which is denoted by  ‘Sm’ in the following) are calculated and analyzed in Section 4, and in Section 5, the conclusions are  given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='The transmission of traveling wave coupling and counter-propagating waves  3  coupling for 2  2  \uf0b4  system  For 2  2  \uf0b4  system, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='1(a) depicts the schematic structure of the micro-ring resonator, which consists  of straight wave-guide a2b2, ring a1b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' The optical field from a excitation wave-guide passing  through the evanescent tail couples into ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' After propagating a round trip, the wave couples back  to the excitation wave-guide and interferes with transmitted light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' At resonance, the wave appears  destructive interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' The propagation path of light presents ‘clockwise direction’ in ring (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' As shown in Figure 1(a), the counter-clockwise mode  1  1  j  a  e b  \uf071  \uf061  \uf03d  in the micro-ring  1 1  a b ,  is induced by the traveling wave  (  )  0  j  t kz  e  \uf077  \uf065  \uf065  \uf02d  \uf02d  \uf03d  in the waveguide  2  2  a b from the left to the right,  where \uf061 is the absorption coefficient,  L c  \uf071  \uf077  \uf03d  is the round trip phase shift, L represents the  circumference of the micro-ring  1 1  a b , c is the phase velocity of micro-ring mode, \uf077 is the optical  wave oscillation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  While the light circuits clockwise and counterclockwise direction in ring simultaneously (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' We can analyze the gap parameter makes an influence on the EIT-like spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='For Figure  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' (b), the clockwise mode  1  1 2  j  a  e  b  \uf071  \uf061  \uf02d  \uf03d  in the micro-ring  1 1  a b , which is induced by the traveling  wave  (  )  0  j  t kz  e  \uf077  \uf065  \uf065  \uf02d  \uf02b  \uf03d  in the waveguide  2  2  a b from the left to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' The counter-clockwise  mode  1  1 2  j  a  e b  \uf071  \uf061  \uf03d  in  the  micro-ring  1 1  a b  ,  which  is  induced  by  the  traveling  wave  (  )  0  j  t kz  e  \uf077  \uf065  \uf065  \uf02d  \uf02d  \uf03d  in the waveguide  2  2  a b from the right to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  (a)  (b)  Figure 1 Schematic diagram of 2  2  \uf0b4  system, (a) traveling wave coupling, (b) counter-propagating wave  coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  Assuming that the light wave is input from the straight waveguide input, when the resonance  condition is satisfied, the phase before the light is coupled from the straight waveguide to the  a1  b1  a2  b2b1  ai  a2  b24  microring is  1  \uf06a  \uf046 \uf03d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' According to the optical waveguide coupling theory, the changes in the  phase before and after a light wave is coupled from one waveguide to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' Then, when the  light wave is coupled from the straight waveguide to the micro-ring waveguide, its phase  becomes  2  2  \uf06a  \uf070  \uf046 \uf03d  \uf02b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' According to the resonance relationship, when the light wave meets the  resonance condition, the light circles in the microring and interferes with each other to generate  resonance enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' The change 2m\uf070 of the phase before and after the light wave circulates  in the microring, where the phase is  3  2  2m  \uf06a  \uf070  \uf070  \uf046 \uf03d  \uf02b  \uf02b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' Similarly, the light wave is again  coupled from the microring to the straight waveguide, and its phase changes again  2  \uf070  ,  becoming  4  2m  \uf06a  \uf070  \uf070  \uf046 \uf03d  \uf02b  \uf02b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' Comparing the phase changes at the input and output, we find an  \uf070 odd multiple of the difference between  1  \uf046  and  4  \uf046 , so that the two beams interfere and cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  When the intensity of the light field is the same, the intensity of the light field after interference  elimination is zero, so the output end of the straight waveguide is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' The influence of counter-propagation optical waves with phase difference  Figure 2 show the Schematic diagram to produce the counter-propagating waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  Figure 2 Schematic diagram producing the counter-propagating waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  As shown in Figure 2, one traveling wave is split into two equal portions of counter-clockwise  wave and clockwise wave at the point P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' Therefore, the superposed induced counter-clockwise  wave and clockwise wave of the micro-ring assuming (  ) 2  cos[ ]  j  j  e  e  \uf071  \uf071  \uf071  \uf02d  \uf02b  \uf03d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='Taking account  the absorption coefficient \uf061 around the micro-ring, we have the relation  1  1  cos[ ]  a  b  \uf061  \uf071  \uf03d  for  counter-propagating waves, the mode  1  1  1  (  )  2  cos[ ]  j  j  a  e  e  b  b  \uf071  \uf071  \uf061  \uf061  \uf071  \uf02d  \uf03d  \uf02b  \uf03d  in the micro-ring is  P  a1  b1  a2  3  b25  induced by the counter-propagating waves  (  )  (  )  0 (  ) 2  j  t kz  j  t kz  e  e  \uf077  \uf077  \uf065  \uf065  \uf02d  \uf02d  \uf02d  \uf02b  \uf03d  \uf02b  in the waveguide  2  2  a b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  For the traveling wave propagating in waveguide[4], the relation  1  1  j  a  e b  \uf071  \uf061  \uf03d  is also valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' In  general, the coupling system contains one waveguide and multi-micro-rings [7,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  In the derivation of traveling waves coupling and counter-propagation waves coupling formula  cos[ ]  j  e  \uf071  \uf071  \uf02d  \uf0ae  , we have assumed that the path from p to E (see Figure 2) along the clockwise  and the counter-clockwise waves have the same length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' In general, this lengths may be different,  the initial phases at point E being  1\uf071 ,  2  \uf071 for the clockwise and the counter-clockwise waves,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' The superposed induced waves being  1  2  1  2  1  2  1  2  1  2  (  )  (  )  (  )  (  )  1  2  2  2  2  2  1  1  (  )  (  )  cos[  ]  2  2  2  j  j  j  j  j  j  e  e  e  e  e  e  \uf071  \uf071  \uf071  \uf071  \uf071  \uf071  \uf071  \uf071  \uf071  \uf071  \uf071 \uf071  \uf071 \uf071  \uf071  \uf071  \uf071  \uf02d  \uf02d  \uf02b  \uf02b  \uf02d  \uf02d  \uf02d  \uf02d  \uf02d  \uf02b  \uf02d  \uf02b  \uf03d  \uf02b  \uf03d  \uf02d  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='(1)  Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' the traveling waves coupling and counter-propagating waves coupling may be  represented by  2  1  cos[  ]  j  e  \uf064  \uf071  \uf064  \uf02d  \uf02d  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  1  1  2  (  ) 2  \uf064  \uf071  \uf071  \uf03d  \uf02d  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  2  1  2  (  ) 2  \uf064  \uf071  \uf071  \uf03d  \uf02b  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' In the case  of  1\uf064  \uf071  \uf03d  ,  2  2  1  cos[  ]  j  j  e  e  \uf064  \uf064  \uf071  \uf064  \uf02d  \uf02d  \uf02d  \uf03d  possesses the form similar to that of the traveling waves  coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='The numerical calculations and analysis of the transmission and phase shift  spectra for2  2  \uf0b4  system and3  3  \uf0b4 system  In this section, we calculate the transmission and phase shift spectra reduction near resonance  concern with the traveling waves coupling and the counter-propagating waves coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  In the following, we begin with the traveling wave coupling of 2  2  \uf0b4  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' The transmission  and the phase shift spectra [5]  2  2 2  1  i  trl  i  t  e  T  t e  \uf071  \uf071  \uf061  \uf061  \uf0b4  \uf02d  \uf03d  \uf02d  ,  2 2  [  ]  1  i  trl  i  t  e  Arg  t e  \uf071  \uf071  \uf061  \uf061  \uf0b4  \uf02d  \uf046  \uf03d  \uf02d  ,  (2)  Figure 3 shows the transmission and the phase shift spectra for  2 2  trl  T \uf0b4 ,  2 2  trl  \uf0b4  \uf046  , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='002  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='004  \uf071  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  T2\uf02a2  l  r  t  \uf048\uf071\uf04c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='002  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='004  \uf071  3  2  1  0  1  2  3  \uf0462\uf02a2  l  r  t  \uf048\uf071\uf04c  Figure 3  The transmission and phase shift spectrawith the parameters  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='9998  t \uf03d  ,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='00006  \uf061 \uf03d  versus\uf071 in unit of radian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  And the corresponding  2 2  cou  T \uf0b4 ,  2 2  cou  \uf0b4  \uf046  for counter-propagating wave coupling being  2  2 2  cos[ ]  1  cos[ ]  cou  t  T  t  \uf061  \uf071  \uf061  \uf071  \uf0b4  \uf02d  \uf03d  \uf02d  ,  2 2  cos[ ]  [  ]  1  cos[ ]  cou  t  Arg  t  \uf061  \uf071  \uf061  \uf071  \uf0b4  \uf02d  \uf046  \uf03d  \uf02d  ,  (3)  Figure 4 shows the transmission and the phase shift spectra for  2 2  cou  T \uf0b4 ,  2 2  cou  \uf0b4  \uf046  , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='1  \uf071  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  T2\uf02a2  u  o  c  \uf048\uf071\uf04c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='1  \uf071  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  3  \uf0462\uf02a2  l  u  o  c  \uf048\uf071\uf04c  Figure 4  The transmission and phase shift spectra with the parameters  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='9998  t \uf03d  ,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='00006  \uf061 \uf03d  versus\uf071 in unit of radian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' the transmission and phase shift spectra for the waveguide and two micro-rings coupled  system [7]  2  2  2  2  1  1  2  1  2  2  2  1  1  ( )  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  )  1  ( )  i  i  r  a  e  r a  e  \uf066  \uf066  \uf074 \uf066  \uf074  \uf066 \uf066  \uf074 \uf066  \uf02d  \uf03d  \uf02d  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  1  1  1  1  1  1  1  1  ( )  1  i  i  r  a e  r a e  \uf066  \uf066  \uf074 \uf066  \uf02d  \uf03d  \uf02d  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  (4)  Setting  1  2  \uf066  \uf066  \uf066  \uf03d  \uf03d  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' we have  2  2 ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' )  trl  sm  T  \uf074  \uf066 \uf066  \uf03d  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  2  [  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' )]  trl  sm  Arg \uf074  \uf066 \uf066  \uf046  \uf03d  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  (5)  Figure 5 gives out the transmission and the phase shift spectra for  2 2  trl  T \uf0b4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  2 2  trl  \uf0b4  \uf046  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  \uf071  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='8  T m  s  l  r  t  \uf048\uf071\uf04c  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  \uf071  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  1  \uf046 m  s  l  r  t  \uf048\uf071\uf04c  Figure  5  The transmission and the phase shift spectra for  trl  sm  T  ,  trl  sm  \uf046  with parameters  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='999  r \uf03d  , 2  1  2  r  a  a  \uf03d  \uf0b4  ,  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='9999  a \uf03d  ,  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='88  a \uf03d  versus\uf071  in unit of radian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  The corresponding formula for counter-propagating waves are  2  2 1  1  2  2  1  2  2  2 1  1  2  ( )cos[  ]  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  )  1  ( )cos[  ]  cou  cou  cou  r  a  r a  \uf074  \uf066  \uf066  \uf074  \uf066 \uf066  \uf074  \uf066  \uf066  \uf02d  \uf03d  \uf02d  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  1  1  1  1  1  1  1  1  cos[ ]  ( )  1  cos[ ]  cou  r  a  r a  \uf066  \uf074  \uf066  \uf066  \uf02d  \uf03d  \uf02d  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  (6)  Setting  1  2  \uf066  \uf066  \uf066  \uf03d  \uf03d  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' we have  2  2 ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' )  cou  cou  sm  T  \uf074  \uf066 \uf066  \uf03d  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  2  [  ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' )]  cou  cou  sm  Arg \uf074  \uf066 \uf066  \uf046  \uf03d  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  (7)  Figure 6 gives out the transmission spectra and the phase shift spectra for  2 2  cou  T \uf0b4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  2 2  cou  \uf0b4  \uf046  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  3  2  1  0  1  2  3  \uf071  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='8  1  T m  s  u  o  c  \uf048\uf071\uf04c  3  2  1  0  1  2  3  \uf071  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  1  \uf046 m  s  u  o  c  \uf048\uf071\uf04c  Figure  6  The transmission and phase shift spectra for  cou  sm  T  ,,  cou  sm  \uf046  with the parameters  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='999  r \uf03d  , 2  1  2  r  a  a  \uf03d  \uf0b4  ,  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='9999  a \uf03d  ,  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='88  a \uf03d  versus\uf071 in unit of radian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  Figure 7 shows the schematic diagram of 3 3  \uf0b4  system, (a) the traveling wave coupling, (b) the  counter-propagation waves coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  (a)  (b)  a1  b1  a2  b2  a3  b3b1  a2  b2  a3  b38  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' Schematic diagram of3 3  \uf0b4  system, (a)the traveling wave coupling, (b)the counter-propagation waves  coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  The transmission and the phase shift spectra for3 3  \uf0b4  system[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  2  1  3  1  3  3 3  1  3  1  3  1  trl  t  B  B  B B  T  B  B  B B  t  \uf06d  \uf064  \uf064  \uf06d  \uf0b4  \uf02d  \uf02d  \uf02b  \uf03d  \uf02d  \uf02d  \uf02b  ,  1  i  B  e \uf071  \uf061  \uf03d  ,  3  i  B  e  \uf071  \uf061  \uf02d  \uf03d  ,  1  3  1  3  3 3  1  3  1  3  [  ]  1  trl  t  B  B  B B  Arg  B  B  B B  t  \uf06d  \uf064  \uf064  \uf06d  \uf0b4  \uf02d  \uf02d  \uf02b  \uf046  \uf03d  \uf02d  \uf02d  \uf02b  ,  (8)  Figure 8 gives out the transmission and the phase shift spectra for  3 3  trl  T \uf0b4 ,  3 3  trl  \uf0b4  \uf046  , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  4  2  0  2  4  \uf071  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='8  T3\uf02a3  l  r  t  \uf048\uf071\uf04c  4  2  0  2  4  \uf071  0  1  2  3  4  5  6  \uf046 m  s  l  r  t  \uf048\uf071\uf04c  Figure  8  The  transmission  and  phase  shift  spectra  for  3 3  3 3  ,  trl  trl  T \uf0b4  \uf0b4  \uf046  with  parameters  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='9998  t \uf03d  ,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='99998  \uf061 \uf03d  ,  0  \uf06d \uf03d  ,  0  \uf064 \uf03d  versus\uf071 in unit of  4  10\uf02d radian  The transmission and the phase shift spectra for3 3  \uf0b4  system[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  2  1  3  1  3  3 3  1  3  1  3  1  cou  t  B  B  B B  T  B  B  B B  t  \uf06d  \uf064  \uf064  \uf06d  \uf0b4  \uf02d  \uf02d  \uf02b  \uf03d  \uf02d  \uf02d  \uf02b  ,  1  cos[ ]  B  \uf061  \uf071  \uf03d  ,  3  cos[ ]  B  \uf061  \uf071  \uf03d  ,  1  3  1  3  3 3  1  3  1  3  [  ]  1  cou  t  B  B  B B  Arg  B  B  B B  t  \uf06d  \uf064  \uf064  \uf06d  \uf0b4  \uf02d  \uf02d  \uf02b  \uf046  \uf03d  \uf02d  \uf02d  \uf02b  ,  (9)  Figure 9 gives out the transmission and the phase shift spectra for  3 3  cou  T \uf0b4 ,  3 3  cou  \uf0b4  \uf046  , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  1000  500  0  500  1000  \uf071  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='8  T3\uf02a3  u  o  c  \uf048\uf071\uf04c  1000  500  0  500  1000  \uf071  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  3  \uf0463\uf02a3  u  o  c  \uf048\uf071\uf04c\uf048\uf071\uf04c  Figure 9  The transmission and phase shift spectra for  3 3  3 3  ,  cou  cou  T \uf0b4  \uf0b4  \uf046  with parameters,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='9998  t \uf03d  ,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='99998  \uf061 \uf03d  ,  0  \uf06d \uf03d  ,  0  \uf064 \uf03d  versus\uf071 in unit of  4  10\uf02d radian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  The numerical calculations shown in Figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='3-6, 8-9, refer to that of the traveling wave coupling,  the breadths of the transmission spectra for counter-propagating waves coupling are enhanced by a  9  factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='001  20  \uf03d  , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='05  6  \uf03d  , 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='8  250  \uf03d  for the 2  2  \uf0b4  , Sm, 3 3  \uf0b4  system,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='The analysis of the phase velocity spectra  We have obtained the phase spectra  ( )  \uf077  \uf046 \uf03d \uf046  , in the case of counter-propagating waves  coupling, the phase shift spectra possess a flat (  ( )  cou  sm  \uf077  \uf046  \uf03d \uf046  ) or a square profile  (  2 2  3 3  ( )  ,  cou  cou  \uf077  \uf0b4  \uf0b4  \uf046  \uf03d \uf046  \uf046  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  The  inverse  of  the  phase  velocity  is  given  by  3 3  3 3  cou  cou  g  g  c v  n  d  d  n  c d  Ld  n  c  L  \uf071  \uf077  \uf074  \uf0b4  \uf0b4  \uf03d  \uf02b  \uf046  \uf03d  \uf02b  \uf046  \uf03d  \uf02b  [12], where  L c  \uf071  \uf077  \uf03d  , the group  delay  3 3  cou  g  d  d  \uf074  \uf077  \uf0b4  \uf03d  \uf046  represents the time delay of narrow-band optical pulses in optical devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  The strong phase dispersion around the transparent window can cause a large group delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  400  200  0  200  400  \uf071  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='6  v\uf048\uf071\uf04c\uf090c  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' The phase velocity spectra for3 3  \uf0b4  system versus\uf071 in unit of  4  10\uf02d radian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  As shown in Figure 10,  gv approaches to zero (namely a large group delays  g  \uf074 are obtained) at the  edge of the square profile (namely in the vicinity of the transparency peak, see Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  The performance characteristics of micro-ring resonator are compared in Table1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='Table 1 Comparison of EIT-like characteristic parameters based on different resonator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='Micro-ring  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='resonator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='microcavity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='structure  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='Number of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='EIT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='coupling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='mechanism  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='material  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='group  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='delays  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='[9]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='two  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='micro-ring  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='single  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='traveling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='waves  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='silica  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='---  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='[11]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='multiple  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='U-shaped  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='resonators  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='multiple  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='dark-light  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='/light-light  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='mode  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='metamaterials  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='high  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='[13]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='chaotic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='single  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='micro-ring  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='single  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='chaos-assisted  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='dynamical  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='tunneling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='silica  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='---  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='[14]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='two  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='microtoroid  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='single  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='traveling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='waves  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='silica  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='---  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='Our work  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='single/two  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='single  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='counter-propa  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='gating waves  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='silica  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='high  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='Loss analysis  According to the coupling mode theory, optical loss can directly affect the line type and  characteristics of EIT spectrum of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' For the dual-coupling microring resonator proposed  in this paper, the transmission loss is mainly composed of scattering loss 1  scat  Q  caused by the  roughness of the waveguide side wall, bending loss 1  bend  Q  of the microring waveguide and  leakage loss 1  leak  Q  caused by the light field entering the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' 1  mat  Q  is the absorption loss  of microring resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' As the light waves travel in the waveguide, part of the light field is  absorbed by the silicon material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' In addition, the top layer of the optical waveguide device also has  an oxide coating, which can absorb the light field in different environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' Finite element  analysis (FEM) was used to conduct mode analysis on the waveguide cross section, and negative  imaginary part ( k ) was added to the material refraction to obtain the absorption loss of the whole  system [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' 1  leak  Q  is leakage loss, which is an important loss mechanism in SOI structure, and is  generated by the middle spectral field of the waveguide into the silicon dioxide layer and silicon  substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' The leakage loss decreases with the increase of silica layer thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' 1  scat  Q  is the  scattering loss, which occurs when the side wall of the waveguide is rough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' Increasing the ratio of  the width to the height of the waveguide, reducing the mode overlap at the side of the waveguide,  and improving the fabrication technology of the waveguide can reduce the scattering loss of the  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' In practical optical waveguide devices, the roughness of the side wall is larger than that of  the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' 1  bend  Q  is the bending loss, is the radiation loss caused by the bending of the optical  waveguide, and decreases with the increase of the radius of the microring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' When the radius of the  microring is about 30 m  \uf06d  [15], the bending loss is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='015  180  dB  [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' 1  back  Q  represents  backscattering loss, which is one of the loss sources in waveguides based on SOI structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' Both  the roughness of the waveguide side wall and the directional coupling can cause backscattering  losses, which can be suppressed by various ways, such as improving the etching process to reduce  the roughness of the waveguide surface, and lower Q values [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' In this paper, except for the  feedback arm region, the coupling distance of the microring resonator is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='15 m  \uf06d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' At this time,  11  the resonance splitting has little influence on the transmission spectrum, so the backscattering can  be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  Assuming that the transmission loss of microring resonator mainly includes absorption loss  (1  mat  Q  ), leakage loss (1  leak  Q  ) and bending loss (1  bend  Q  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' Therefore, the relationship between  the value of  tot  Q  the system and the loss coefficient a can be expressed as [18]:  2  gn  Q  a  \uf070  \uf06c  \uf03d  (10)  exp(  )  2  ring  L  a  \uf061  \uf03d  \uf02d  (11)  Where,  gn  is the group refractive index, which can be obtained by the finite-difference  time-domain method of full-vector calculation by MODE Solutions software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' a is the round-trip  loss coefficient of light wave in microring waveguide,\uf061 represents the loss value per unit length  of optical waveguide, and  tot  Q  is inversely proportional to \uf061 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' Taking all loss mechanisms into  consideration, the transmission loss of the optical waveguide in this paper is about9dB cm [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='Conclusions  In conclusion, we have demonstrated the transmission spectra and the phase shift spectra of two  micro-rings coupled system through numerical simulations and theoretical calculations and  analyzed the causes of resonance peaks and transparent windows leads to the EIT-like effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' As  we can see in this paper, we can generate a wide transparent windows, a large flat phase shift and  a large group delay by increasing the number of micro-ring resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content=' Therefore, it can be  potential applied in multi-band filters and multi-band slow light devices (such as delay lines) in  optical communication field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  Acknowledgments  This work was supported by the State Key Laboratory of Quantum Optics and Quantum Optics  Devices, Shanxi University, Shanxi, China (KF202004，KF202205).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
+page_content='  ORCID iD  Chaoying Zhao  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfVQcE/content/2301.03820v1.pdf'}
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+arXiv:2301.13475v1  [eess.SP]  31 Jan 2023
+A Knowledge-Driven Meta-Learning Method for
+CSI Feedback
+Han Xiao1, Wenqiang Tian1, Wendong Liu1, Zhi Zhang1, Zhihua Shi1, Li Guo1 and Jia Shen1
+1Department of Standardization, OPPO Research Institute, Beijing, China
+Email: {xiaohan1, tianwenqiang, liuwendong1, zhangzhi, szh, v-guoli, sj}@oppo.com
+Abstract—Accurate and effective channel state information
+(CSI) feedback is a key technology for massive multiple-input and
+multiple-output (MIMO) systems. Recently, deep learning (DL)
+has been introduced to enhance CSI feedback in massive MIMO
+application, where the massive collected training data and lengthy
+training time are costly and impractical for realistic deployment.
+In this paper, a knowledge-driven meta-learning solution for
+CSI feedback is proposed, where the DL model initialized by
+the meta model obtained from meta training phase is able to
+achieve rapid convergence when facing a new scenario during
+the target retraining phase. Specifically, instead of training with
+massive data collected from various scenarios, the meta task
+environment is constructed based on the intrinsic knowledge
+of spatial-frequency characteristics of CSI for meta training.
+Moreover, the target task dataset is also augmented by exploiting
+the knowledge of statistical characteristics of channel, so that
+the DL model initialized by meta training can rapidly fit into
+a new target scenario with higher performance using only a
+few actually collected data in the target retraining phase. The
+method greatly reduces the demand for the number of actual
+collected data, as well as the cost of training time for realistic
+deployment. Simulation results demonstrate the superiority of the
+proposed approach from the perspective of feedback performance
+and convergence speed.
+Index
+Terms—CSI
+feedback,
+meta-learning,
+MIMO,
+knowledge-driven
+I. INTRODUCTION
+Accurate and effective channel state information (CSI)
+feedback has been intensively studied for supporting massive
+multiple-input and multiple-output (MIMO) systems. Along
+with the standardization in the 3rd Generation Partnership
+Project (3GPP), various solutions based on the TypeI and
+enhanced TypeII (eTypeII) codebook have been proposed
+to improve the CSI feedback performance [1]. However, to
+resolve the issues of larger feedback overhead and insufficient
+recovery accuracy, methods for further enhancing the CSI
+feedback are still being actively studied.
+Recently, deep learning (DL) has been introduced for CSI
+feedback enhancement, where the DL model can achieve
+higher CSI recovery accuracy with reduced feedback overhead.
+An autoencoder method of CsiNet for CSI feedback [2] is first
+proposed, where an encoder at the user equipment (UE) com-
+presses the channel matrix and a decoder at the base station
+(BS) recovers the corresponding channel matrix. Subsequently,
+a series of follow-up works are conducted under various condi-
+tions [3]–[7] . However, there are still some challenges for DL-
+based CSI feedback. First, the generalization issue should be
+considered since the DL methods tend to express the scenario-
+specific property. Moreover, plenty of training data of target
+scenario is quite impractical for deployment due to the expense
+and long-time training and collecting data. Meta-learning is
+utilized for CSI feedback in [8] and [9], where the model
+is initialized by the meta model obtained in meta training
+phase with massive CSI samples corresponding to multiple
+various scenarios, and then achieves quick convergence with
+small amount of CSI data in a new target scenario. However,
+the above meta-learning based solutions still require massive
+collected data for the meta training phase. Moreover, in target
+retraining phase the model is retrained on the original small
+amount of data within short time, thus it might suffer from
+performance loss in the new target scenario in comparison with
+models trained on sufficient data. Further, the above methods
+also fail to consider the knowledge of intrinsic characteristics
+of the wireless communication during both phases.
+In this paper, a novel knowledge-driven meta-learning
+method for CSI feedback is proposed. Specifically, instead
+of training with massive CSI data collected from different
+wireless scenarios in meta training phase, one can construct the
+meta task environment by exploring the intrinsic knowledge
+of spatial-frequency characteristic of CSI eigenvector for meta
+training. After the DL model obtains the initialization in meta
+training phase, it is capable of achieving rapid convergence
+by retraining on target task dataset, which is augmented
+from only small amount of actually collected seeded data
+with the assistance of the knowledge of statistical feature of
+wireless channels. Simulation results illustrate the superiority
+of the proposed method from the perspective of feedback
+performance and convergence speed.
+Notations: uppercase and lowercase letters denote scalars.
+Boldface uppercase and boldface lowercase letters denote ma-
+trices and vectors, respectively. Calligraphic uppercase letters
+denote sets. A(:, B) and A(B, :) denote the sub-matrices of
+A that consist of the columns and rows indexed by set B,
+respectively. E{·} denotes expectation and Tr{·} denotes trace.
+AH denotes the Hermitian matrix of A. rand(A, a) denotes
+the random sampling of a samples from set A without re-
+placement. The sets of real and complex numbers are denoted
+by R and C, respectively. | · | denotes the cardinality of a set
+or the absolute value of a scalar.
+
+II. SYSTEM DESCRIPTION
+A. System Model
+A MIMO system with Nt = NhNv transmitting anten-
+nas at BS and Nr receiving antennas at UE is considered,
+where Nh and Nv are the numbers of horizontal and vertical
+antenna ports, respectively. Note that our proposed methods
+are suitable for antennas with either dual or single polariza-
+tion, and that single polarization is considered to illustrate
+the basic principle in this paper. The downlink channel in
+time domain can be denoted as a three-dimensional matrix
+�H ∈ CNr×Nt×Nd, where Nd is the number of paths with
+various delays. By conducting Discrete Fourier transform
+(DFT) over the delay-dimension of the time-domain downlink
+channel matrix �H, the downlink channel in frequency domain
+�H ∈ CNr×Nt×Nsc can be written as
+�H =
+� �H1, �H2, · · · , �HNsc
+�
+,
+(1)
+where Nsc
+is the number of subcarriers, and Hk
+∈
+CNr×Nt, 1 ≤ k ≤ Nsc denotes the downlink channel on the
+kth subcarrier. Normally, the CSI eigenvector feedback is per-
+formed on each subband which consists of Ngran subcarriers
+with Nsc = NgranNsb. Assuming the rank 1 configuration for
+downlink transmission, the corresponding eigenvector for the
+lth subband wl ∈ CNt×1 with ||wl||2 = 1, can be calculated
+by the eigenvector decomposition on the subband as
+
+
+1
+Ngran
+lNgran
+�
+k=(l−1)Ngran+1
+�HH
+k �Hk
+
+ wl = λlwl,
+(2)
+where 1 ≤ l ≤ Nsb and λl represents the corresponding
+maximum eigenvalue for the l-th subband. Therefore, the CSI
+eigenvector for all Nsb subbands can be written as
+W =
+�
+w1, w2, · · · , wNsb
+�
+∈ CNt×Nsb,
+(3)
+wherein total NsbNt complex coefficients need to be com-
+pressed at the UE and then recovered at the BS side.
+Generally, the optimization objective for CSI feedback can
+be given as
+min
+F −ρ(W, W′) = min
+F − 1
+Nsb
+Nsb
+�
+l=1
+� ∥wHw′∥2
+∥w∥2∥w′∥2
+�2
+,
+(4)
+where ρ(·, ·) ∈ [0, 1] denotes the squared generalized cosine
+similarity (SGCS), ∥·∥2 denotes ℓ2 norm, wl and w′
+l represent
+the original and recovered CSI eigenvector of the l-th sub-
+band, respectively, F represents the alternative CSI feedback
+schemes such as TypeI, eTypeII and DL-based autoencoder.
+B. DL-based CSI Feedback
+The architecture of DL-based CSI feedback using autoen-
+coder is introduced in Fig. 1, where the neural network (NN)
+encoder and decoder, fe(·; ΘE) and fd(·; ΘD) with trainable
+parameters Θ = {ΘE, ΘD} are deployed at UE and BS,
+respectively. Thus the DL-based autoencoder fa(·; Θ) with
+trainable parameters Θ = {ΘE, ΘD} can be represented as
+W′ = fd(fe(W; ΘE); ΘD) = fa(W; Θ),
+(5)
+UE
+BS
+W
+W�
+Encoder �e
+Decoder �d
+Bitstream b
+SGCS
+Fig. 1. Illustration of DL-based CSI feedback.
+where the encoder first compresses and quantizes the original
+CSI eigenvector W to a bitstream b of length B. Then the
+decoder uses b to recover W′. During training phase, the
+encoder and decoder are jointly optimized to solve (4) with
+sufficient numbers of CSI eigenvector samples.
+C. Meta-learning based CSI Feedback
+Generally, the goal of meta-learning based CSI feedback
+is to find a good initialization of Θ = {ΘE, ΘD}, so that the
+autoencoder can converge quickly with a small amount of CSI
+samples and a few training steps for a new scenario. Specifi-
+cally, the procedure of meta-learning based CSI feedback can
+be divided into two phases, i.e., the meta training phase and
+target retraining phase.
+During meta training phase, the model is trained over a big
+dataset consisting of T CSI tasks of diverse scenarios, which
+can be defined as meta task environment Tmeta = {T1, ..., TT },
+wherein each task Tj = {Wj
+1, ..., Wj
+T }, 1 ≤ j ≤ T consists
+of |Tj| CSI samples denoted as Wj
+i , 1 ≤ i ≤ Tj. Based on the
+meta task environment Tmeta, meta-learning algorithms can be
+performed to learn the initial parameters �Θ, i.e.,
+min
+�Θ
+ETj⊂Tmeta
+�
+−ρ′(Tj, fa(Tj; Ug
+Tj(�Θ)))
+�
+,
+(6)
+where Ug
+Tj(�Θ) is the operator that updates �Θ for g training
+steps using data sampled from Tj. The initialization �Θ learnt
+in (6) is expected to has the same ability of quick adaptation
+with small amount of data on an unobserved target task Ttarget.
+Secondly, the target retraining phase can be formulated as
+min
+Φ=Ug
+Ttarget (�Θ)
+−ρ′(Ttarget, fa(Ttarget; Φ)),
+(7)
+where Φ denotes the possible parameter sets trained on Ttarget
+after g retraining steps based on the initialization �Θ, which
+indicates that the final parameters on a new target task of
+scenario can be rapidly obtained with only a few retraining
+steps.
+However, the existing meta-leaning based CSI feedback still
+has to face two major challenges, which our knowledge-driven
+meta-learning method aims to solve.
+• During meta training phase, it requires sufficient samples
+to construct the meta task environment Tmeta to solve (6),
+which is extremely costly since it is impractical to collect
+all existing types of wireless scenarios with adequate
+diversity.
+• During target retraining phase, despite the rapid conver-
+gence for solving (7) using small amount of data Ttarget
+based on the initialization �Θ, it is always difficult to
+achieve comparable performance with using large amount
+of CSI data in target scenario.
+
+Knowledge
+Meta
+Training
+Phase 
+Target 
+Retraining
+Phase 
+Initialization
+Meta Task 
+Environment
+Meta Model
+Seeded Data
+Knowledge
+Augmented Data
+Target Scenario
+Target Model
+UE
+BS
+�
+Encoder �
+Decoder �
+Bitstream 
+SGCS
+Fig. 2. Proposed knowledge-driven meta-learning framework for CSI feedback.
+III. KNOWLEDGE-DRIVEN META-LEARNING FOR CSI
+FEEDBACK
+A. Knowledge-driven Meta Training Phase
+1) Spatial-Frequency Characteristic: Generally, consider-
+ing the intrinsic structure of the CSI eigenvector, W ∈
+CNt×Nsb can be decomposed as
+W = SEFH
+(8)
+where S ∈ CNt×Nt is constructed with Nt orthogonal basis
+vectors in spatial domain and F ∈ CNsb×Nsb is constructed
+with Nsb orthogonal basis vectors in frequency domain.
+Specifically, both S and F are unitary matrices, which indicate
+the full-rank spatial-frequency characteristic. The projection
+coefficient matrix E ∈ CNt×Nsb represents that each CSI
+eigenvector W can be completely expressed by the linear
+combination of the orthogonal basis vectors in S and F.
+Obviously, the distribution of the elements in E with relatively
+larger amplitude determines the dominant spatial-frequency
+feature of W given the same S and F, where the dominant
+spatial-frequency features can be considered as the intrinsic
+knowledge and hence can be learnt by the DL model during
+the meta-training phase.
+2) Knowledge-driven Meta Training: Inspired by the intrin-
+sic knowledge of spatial-frequency feature in section III-A1, a
+knowledge-driven algorithm is proposed to solve (6). During
+the meta training phase, the meta task environment Tmeta =
+{T1, ..., TT } consisting of T tasks is firstly established, where
+the construction approach of CSI eigenvector in each task
+explores the CSI decomposition formula in section III-A1.
+Each task usually consists of different CSI eigenvectors from
+specific number of UEs that can be sampled on various number
+of slots. Specifically, denote Nue,j and Nslot,j as the number
+of UEs and slots for the j-th task Tj, 1 ≤ j ≤ T , respectively,
+which can be set as
+Nue,j = rand({1, ..., �
+Nue}, 1),
+(9)
+Nslot,j = rand({1, ..., �
+Nslot}, 1),
+(10)
+where Nslot,j Nue,j = |Tj|, �
+Nue and �
+Nslot denote the maxi-
+mum number of UEs and maximum number of slots of CSI
+that can be generated in one task, respectively.
+Moreover, according to the intrinsic knowledge of spatial-
+frequency feature, to generate the CSI samples in the j-th task
+Tj, P groups of spatial orthogonal basis vector and one group
+of frequency orthogonal basis vector can be firstly given as
+Sp = [sp,1, ..., sp,Nt] ∈ CNt×Nt, 1 ≤ p ≤ P,
+(11)
+F = [f1, ..., fNsb] ∈ CNsb×Nsb,
+(12)
+respectively, where each column of Sp and F is an orthogonal
+basis vector. Specifically, each basis in Sp indicates a beam
+direction in spatial domain, and multiple groups of orthogonal
+basis vectors are designed in order to improve the diversity
+of spatial features. Here we introduce a Schmidt orthogo-
+nalization method for obtaining the basis vector groups Sp
+and F. For each group of spatial orthogonal basis vector
+Sp, 1 ≤ p ≤ P, and the frequency orthogonal basis vector
+group F, the Schmidt orthogonalization can be performed on
+three full-rank random matrices Xh
+p ∈ CNh×Nh ∼ CN(0, 1),
+Xv
+p ∈ CNv×Nv ∼ CN(0, 1) and Xf ∈ CNsb×Nsb ∼ CN(0, 1),
+obtaining the orthogonal matrices Uh
+p, Uv
+p and Uf, respec-
+tively. F = Uf can be utilized as frequency orthogonal basis
+vector. The p-th spatial orthogonal basis vector group can be
+obtained by performing kronecker product, i.e., Sp = Uh
+p⊗Uv
+p
+Next, the method of generating CSI samples for the j-th
+task Tj is introduced. The group index pj for task Tj are first
+randomized by
+{pj} = rand({1, ..., P}, 1),
+(13)
+and the indices of dominant spatial and frequency feature
+vectors are also randomized by
+�Sj = rand({1, ..., Nt}, Ltask),
+(14)
+�Fj = rand({1, ..., Nsb}, Mtask),
+(15)
+respectively, where the parameters Ltask ≤ Nt and Mtask ≤
+Nsb are defined to constrain the degree of feature diversity of
+the task in spatial and frequency domain, respectively.
+For the m-th UE 1 ≤ m ≤ Nue in task Tj, the indices
+of the dominant spatial and frequency feature vectors are also
+randomized by
+�Sm = rand( �Sj, Lm),
+(16)
+�Fm = rand( �
+Fj, Mm),
+(17)
+respectively, where the degree of feature diversity in spatial
+and frequency domain Lm and Mm are both UE-specific, i.e.,
+{Lm} = rand({1, ..., Ltask}, 1),
+(18)
+{Mm} = rand({1, ..., Mtask}, 1),
+(19)
+respectively.
+Similarly for the n-th slot 1 ≤ n ≤ Nslot of the m-th UE in
+task Tj, the dominant spatial and frequency feature vectors are
+respectively selected from the corresponding dominant vectors
+of the UE, so that the feature is maintained for the m-th UE
+but distinguished between different slots, i.e.,
+Sm,n = rand( �Sm, ⌈αLm⌉),
+(20)
+
+Algorithm 1: Knowledge-driven Meta Training Phase
+Initialization: �
+Nue, �
+Nslot, T , α, β, g, ǫ, �Θ;
+Formulate the feature basis using (11) to (12);
+for j = 1, . . . , T do
+Construct structure of Tj using (9), (10) and (13)
+to (15);
+for m = 1, . . . , Nue,j do
+Consturt structure of UE m using (16) to (19);
+for n = 1, . . . , Nslot,j do
+Generate a CSI of slot n using (20) to (23);
+end
+end
+end
+Meta training by iterating (24).
+Fm,n = rand( �
+Fm, ⌈βMm⌉),
+(21)
+where the parameters α ∈ (0, 1] and β ∈ (0, 1] are set to scale
+the diversity of the feature of each slot. Consequently, a CSI
+sample for the n-th slot of the m-th UE in task Tj can be
+generated as
+Wj
+m,n = Spj(:, Sm,n)�EFH(:, Fm,n),
+(22)
+where the elements in �E ∈ C|Sm,n|×|Fm,n| are indepen-
+dently sampled from complex normal distribution CN(0, 1).
+A subband-level normalization should be also performed for
+1 ≤ l ≤ Nsb using
+Wj
+m,n(:, l) =
+Wj
+m,n(:, l)
+||Wj
+m,n(:, l)||2
+.
+(23)
+Through the procedure of (9) to (23) for generating each CSI
+sample of each UE, the meta task environment Tmeta can be
+finally constructed.
+Utilizing the meta task environment Tmeta, the meta training
+procedure can be conducted to solve (6). The parameters of
+the DL model of CSI feedback is randomly initialized by �Θ.
+For the j-th task Tj in the meta task environment Tmeta, �Θ
+can be updated with
+�Θ = �Θ + ǫ(Ug
+Tj(�Θ) − �Θ),
+(24)
+where Ug
+Tj(�Θ) is the operator that updates �Θ for g training
+steps on task Tj, and ǫ denotes the step size of meta training.
+After that, the obtained �Θ can be utilized as initialization for
+further fast retraining on a new target task of scenario. The
+proposed algorithm for knowledge-driven meta training phase
+is summarized in Algorithm 1.
+B. Knowledge-driven Target Retraining Phase
+1) Statistical Feature of Channel: In this part, the statistical
+features of the channel in both spatial domain and time delay
+domain are explored. Specifically, for a specific UE in the
+target scenario, denote the actually collected �
+Nslot channel
+samples in time domain as H = { �H1, ..., �H �
+Nslot}, where each
+channel sample �Ht ∈ CNr×Nt×Nd, 1 ≤ t ≤ �
+Nslot.
+Firstly, the statistical feature in delay domain can be
+described by the power-delay spectrum. Denote
+�H′
+t,d
+∈
+CNr×Nt, 1 ≤ d ≤ Nd as the d-th delay of the t-th channel
+sample �Ht, the power of the d-th delay can be calculated as
+ˆpd =
+1
+NtNr �
+Nslot
+�
+Nslot
+�
+t=1
+|| �H′
+t,d||2
+F,
+(25)
+Secondly, the statistical feature in spatial domain can be
+demonstrated by the self-correlation matrices of the transmit-
+ting and receiving antenna ports, which can be calculated as
+Rtx
+d =
+Nt
+� �
+Nslot
+t=1
+�H′H
+t,d �H′
+t,d
+Tr(� �
+Nslot
+t=1
+�H′H
+t,d �H′
+t,d)
+,
+(26)
+Rrx
+d =
+Nr
+� �
+Nslot
+t=1
+�H′
+t,d �H′H
+t,d
+Tr(� �
+Nslot
+t=1
+�H′
+t,d �H′H
+t,d)
+,
+(27)
+respectively, where the trace operation Tr(·) is performed for
+normalization. Then the kronecker product is implemented
+on the transmitting and receiving self-correlation matrices to
+obtain the joint spatial feature as
+Rd = Rrx
+d ⊗ Rtx
+d ∈ CNtNr×NtNr,
+(28)
+It should be noted that the dataset of the target scenario
+could be very small, and thus it is not sufficient for training
+autoencoder with superior CSI feedback and to ensure recov-
+ery performance, even though it is able to converge quickly
+based on the initialization �Θ obtained by meta training phase.
+Therefore, it is necessary to consider a data augmentation
+seeded by H exploiting the knowledge of statistical features
+in spatial and delay domain.
+The intrinsic knowledge of statistical features of the channel
+for a specific UE can be completely described by ˆpd and
+Rd, 1 ≤ d ≤ Nd. Therefore, to align with the statistical
+features of the collected channel samples, the augmented
+channel for the d-th delay ˆhaug
+d
+should satisfy
+E[ˆhaug
+d
+(ˆhaug
+d
+)H] = ˆpdRd.
+(29)
+2) Knowledge-driven Target Retraining: The knowledge-
+driven target retraining is introduced with data augmentation
+inspired by (29). Firstly, SVD is performed on Rd, i.e.,
+Ud, Dd, Vd = svd(Rd),
+(30)
+where Vd = UH
+d because of Rd = RH
+d .
+Secondly, the augmented channel sample for the d-th delay
+can be generated by conducting
+ˆhaug
+d
+=
+�
+ˆpdUdD
+1
+2
+d n,
+(31)
+where the random vector n ∈ CNtNr×1 ∼ CN(0, 1).
+Next, ˆhaug
+d
+can be reshaped as the channel matrix �Haug
+d
+∈
+CNr×Nt. By concatenating all Nd augmented channel matri-
+ces, the augmented channel sample can be obtained as
+Haug = [ �Haug
+1
+, ..., �Haug
+Nd ],
+(32)
+
+Algorithm 2: Knowledge-driven Target Retraining
+Phase
+Initialization:H, Naug, g′;
+Calculate statistical delay power spectrum using (25);
+for q = 1, . . . , �
+Nue do
+for d = 1, . . . , Ndelay do
+Data augmentation using (26) to (32) and (1)
+to (3);
+end
+end
+Target retraining using (33).
+where Haug ∈ CNr×Nt×Nd. Then the augmented CSI eigen-
+vector sample Waug can be finally obtained by implementing
+(1) to (3) on Haug.
+For each UE, the total Naug channel samples can be
+provided with Naug randomly generated vectors n. Moreover,
+for Nue UEs, we can generate totally NueNaug augmented CSI
+eigenvector samples that can be used to construct the target
+task dataset T aug
+target.
+Based on the target task dataset T aug
+target and the initialization
+�Θ obtained in knowledge-driven meta training phase, (7) can
+be solved with higher SGCS using a few training steps, i.e.,
+Φ = Ug
+Ttarget(�Θ).
+(33)
+The proposed algorithm for knowledge-driven target retraining
+phase can be summarized in Algorithm 2.
+IV. SIMULATION RESULTS
+The
+simulation
+results
+are
+provided in
+this
+section.
+Knowledge-driven scheme in meta training phase (KMeta-
+*) and target retraing phase (*-KAug) are evaluated, where
+‘None’ denotes no knowledge-driven schemes are used. The
+simulation parameters are listed in Table I. CDL-C [10]
+channel model with delay spread 300 ns and random dis-
+tributed UE with speed 300 km/h are utilized as actually
+collected channels. The training was performed three times
+with different random seeds, one of which is shown since the
+TABLE I
+BASIC SIMULATION PARAMETERS
+Parameter
+Value
+System bandwidth
+10MHz
+Carrier frequency
+3.5GHz
+Subcarrier spacing
+15KHz
+Subcarriers number Nsc
+624
+Subband number Nsb
+13
+Horizontal Tx antenna ports per polarization Nh
+8
+Vertical Tx antenna ports per polarization Nv
+2
+Tx antenna ports Nt
+32
+Rx antennas Nr
+4
+Meta task enviroment size T
+8000
+Meta training step size ǫ
+0.25
+Step number per task g
+32
+Spatial diversity degree Ltask
+6
+Frequency diversity degree Mtask
+6
+Spatial diversity scale α
+0.75
+Frequency diversity scale β
+0.75
+0
+500
+1000
+1500
+2000
+Steps
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+SGCS
+None-None
+None-KAug
+KMeta-None
+KMeta-KAug
+eTypeII
+Fig. 3.
+Convergence process of target retraining phase with the number of
+training steps on CDL-C channel ( �
+Nue = 300, �
+Nslot = 10, Naug = 100).
+results are almost equal. Moreover, the Transformer backbone
+for CSI feedback [4] with number of feedback bits B = 64 is
+implemented in evaluation.
+Fig. 3 show the convergence process of target retraining
+phase with the number of training steps. Note that the vertical
+axis represents the best achieved SGCS on the test set within
+the steps. Here we consider the eTypeII codebook and DL-
+based method without meta training and target augmentation
+(None-None) as the baselines. Note that since the existing
+meta-learning methods for CSI require a large amount of
+multi-scenario real data, and our method lever knowledge for
+meta-learning, it is unfair to compare our method with them
+in terms of data cost. In terms of convergence speed, it can
+be noticed that the proposed KMeta-None require fewer train-
+ing steps to achieve convergence than None-None. Even on
+augmented data, KMeta-KAug can also fit more quickly than
+None-KAug. From the perspective of feedback performance,
+the knowledge-driven meta training brings higher SGCS since
+KMeta-None outperforms None-None. Moreover, the methods
+of *-KAug outperform the methods of *-None, which reveals
+that the knowledge-driven target retraining phase can further
+effectively improve the SGCS performance.
+In Fig. 4 and Fig. 5 we compare the SGCS performance
+training 2000 steps on different number of seeded UEs �
+Nue
+and slots �
+Nslot, respectively. It is observed that the proposed
+knowledge-driven method of KMeta-KAug outperforms tra-
+ditional eTypeII codebook and basic DL-based method None-
+None. Specifically, the performance gaps between the methods
+KMeta-* and None-* can respectively demonstrate the gain
+provided by proposed knowledge-driven meta training. The
+gaps between *-KAug and *-None respectively initimate the
+gain obtained from proposed knowledge-driven target retrain-
+ing. Moervoer, in Fig. 5, the performance of None-KAug
+improves as the number of slots �
+Nslot increased, while the
+performance of KMeta-KAug stays almost unchanged, which
+implies that the proposed knowledge-driven target retraining
+requires fewer slots to achieve the performance ceiling when
+it is enhanced by proposed knowledge-driven meta training.
+Table II illustrates the SGCS performance of the proposed
+method and the existing data augmentation methods [4] for
+DL-based CSI feedback including noise injection, flipping,
+
+50
+100
+150
+200
+250
+300
+0.4
+0.5
+0.6
+0.7
+0.8
+SGCS
+None-None
+None-KAug
+KMeta-None
+KMeta-KAug
+eTypeII
+Fig. 4.
+Comparison of SGCS for varying number of seeded UEs �
+Nue on
+CDL-C channel, fixing number of slots per UE �
+Nslot = 10.
+10
+20
+30
+40
+50
+60
+0.55
+0.6
+0.65
+0.7
+0.75
+0.8
+0.85
+SGCS
+None-None
+None-KAug
+KMeta-None
+KMeta-KAug
+eTypeII
+Fig. 5. Comparison of SGCS for varying number of slots per UE �
+Nslot on
+CDL-C channel, fixing number of UEs �
+Nue = 300.
+cyclic shift, random shift and rotation. It is observed that
+the proposed method can obtain 0.1953 SGCS performance
+gain in comparison to none augmantation. Specifically, the
+performance gap between proposed method and other com-
+petitors is at least 0.1526, which demonstrates that exploiting
+communication knowledge effectively bring performance gain.
+The link-level block error rate (BLER) performance is
+presented in Fig. 6, where omnidirectional and directional
+antennas is deployed at UE and BS, respectively. The gap
+between KMeta-None and None-None proves the performance
+gain of knowledge-driven meta training phase. The gap be-
+tween None-KAug and None-None proves the performance
+gain of knowledge-driven target retraining phase. Since the
+method of KMeta-Aug outperforms other competitors in terms
+of BLER, the advantages and application potential of proposed
+TABLE II
+COMPARISON OF DIFFERENT AUGMENT SCHEMES
+Scheme
+SGCS
+None
+0.5977
+Noise Injection
+0.6189
+Flipping
+0.6171
+Cyclic Shift
+0.6404
+Random Shift
+0.6225
+Rotation
+0.6178
+Proposed
+0.7930
+Note 1: �
+Nue = 300 and �
+Nslot = 10
+Note 2: All methods augments to 30k samples, except the flipping which
+can only augment to 6k samples due to method limitation.
+-1
+0
+1
+2
+3
+4
+5
+6
+SNR (dB)
+10-3
+10-2
+10-1
+100
+BLER
+KMeta-KAug
+KMeta-None
+None-KAug
+None-None
+eTypeII
+Fig. 6. Link-level BLER performance comparison trained on CDL-C300 for
+different solutions ( �
+Nue = 300, �
+Nslot = 10).
+knowledge-driven approach are well demonstrated.
+V. CONCLUSION
+In this paper, we propose a knowledge-driven meta-learning
+method for CSI feedback, where the meta task environment for
+meta training is constructed based on the intrinsic knowledge
+of spatial-frequency feature of CSI eigenvector. Initialized
+by the knowledge-driven meta training phase, the DL model
+is capable of achieving rapid convergence by retraining on
+the target task dataset, which is augmented from only a few
+actually collected seeded data with the assistance of the knowl-
+edge of statistical feature of wireless channels. Simulation
+results demonstrate the superiority of the approach from the
+perspective of feedback performance and convergence speed.
+REFERENCES
+[1] 3GPP, “3GPP TS 38.214 v17.2.0 3rd Generation Partnership Project;
+technical specification group radio access network; NR; physical layer
+procedures for data (release 17),” Tech. Rep., 2022.
+[2] C.-K. Wen, W.-T. Shih, and S. Jin, “Deep learning for massive MIMO
+CSI feedback,” IEEE Wireless Communications Letters, vol. 7, no. 5,
+pp. 748–751, 2018.
+[3] X. Li and H. Wu, “Spatio-temporal representation with deep neural
+recurrent network in MIMO CSI feedback,” IEEE Wireless Communi-
+cations Letters, vol. 9, no. 5, pp. 653–657, 2020.
+[4] H. Xiao, Z. Wang, D. Li, W. Tian, X. Liu, W. Liu, S. Jin, J. Shen,
+Z. Zhang, and N. Yang, “AI enlightens wireless communication: A trans-
+former backbone for CSI feedback,” arXiv preprint arXiv:2206.07949,
+2022.
+[5] J. Guo, C.-K. Wen, and S. Jin, “CAnet: Uplink-aided downlink channel
+acquisition in FDD massive MIMO using deep learning,” IEEE Trans-
+actions on Communications, vol. 70, no. 1, pp. 199–214, 2021.
+[6] H. Xiao, Z. Wang, W. Tian, X. Liu, W. Liu, S. Jin, J. Shen, Z. Zhang, and
+N. Yang, “AI enlightens wireless communication: Analyses, solutions
+and opportunities on CSI feedback,” China Communications, vol. 18,
+pp. 104–116, 2021.
+[7] W. Liu, W. Tian, H. Xiao, S. Jin, X. Liu, and J. Shen, “EVCsiNet:
+Eigenvector-based CSI feedback under 3GPP link-level channels,” IEEE
+Wireless Communications Letters, vol. 10, no. 12, pp. 2688–2692, 2021.
+[8] J. Zeng, J. Sun, G. Gui, B. Adebisi, T. Ohtsuki, H. Gacanin, and
+H. Sari, “Downlink CSI feedback algorithm with deep transfer learning
+for FDD massive MIMO systems,” IEEE Transactions on Cognitive
+Communications and Networking, vol. 7, no. 4, pp. 1253–1265, 2021.
+[9] B. Tolba, A. H. Abd El-Malek, M. Abo-Zahhad, and M. Elsabrouty,
+“A meta learner autoencoder for channel state information feedback in
+massive MIMO systems,” in 2021 28th International Conference on
+Telecommunications (ICT).
+IEEE, 2021, pp. 1–5.
+[10] 3GPP, “3GPP TR 38.901 v17.0.0 3rd Generation Partnership Project;
+technical specification group radio access network; study on channel
+model for frequencies from 0.5 to 100 GHz (release 17),” Tech. Rep.,
+2022.
+

K9FRT4oBgHgl3EQfEDfx/content/tmp_files/load_file.txt

@@ -0,0 +1,386 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf,len=385
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='13475v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='SP]  31 Jan 2023  A Knowledge-Driven Meta-Learning Method for  CSI Feedback  Han Xiao1, Wenqiang Tian1, Wendong Liu1, Zhi Zhang1, Zhihua Shi1, Li Guo1 and Jia Shen1  1Department of Standardization, OPPO Research Institute, Beijing, China  Email: {xiaohan1, tianwenqiang, liuwendong1, zhangzhi, szh, v-guoli, sj}@oppo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='com  Abstract—Accurate and effective channel state information  (CSI) feedback is a key technology for massive multiple-input and  multiple-output (MIMO) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Recently, deep learning (DL)  has been introduced to enhance CSI feedback in massive MIMO  application, where the massive collected training data and lengthy  training time are costly and impractical for realistic deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  In this paper, a knowledge-driven meta-learning solution for  CSI feedback is proposed, where the DL model initialized by  the meta model obtained from meta training phase is able to  achieve rapid convergence when facing a new scenario during  the target retraining phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Specifically, instead of training with  massive data collected from various scenarios, the meta task  environment is constructed based on the intrinsic knowledge  of spatial-frequency characteristics of CSI for meta training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Moreover, the target task dataset is also augmented by exploiting  the knowledge of statistical characteristics of channel, so that  the DL model initialized by meta training can rapidly fit into  a new target scenario with higher performance using only a  few actually collected data in the target retraining phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' The  method greatly reduces the demand for the number of actual  collected data, as well as the cost of training time for realistic  deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Simulation results demonstrate the superiority of the  proposed approach from the perspective of feedback performance  and convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Index  Terms—CSI  feedback,  meta-learning,  MIMO,  knowledge-driven  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' INTRODUCTION  Accurate and effective channel state information (CSI)  feedback has been intensively studied for supporting massive  multiple-input and multiple-output (MIMO) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Along  with the standardization in the 3rd Generation Partnership  Project (3GPP), various solutions based on the TypeI and  enhanced TypeII (eTypeII) codebook have been proposed  to improve the CSI feedback performance [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' However, to  resolve the issues of larger feedback overhead and insufficient  recovery accuracy, methods for further enhancing the CSI  feedback are still being actively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Recently, deep learning (DL) has been introduced for CSI  feedback enhancement, where the DL model can achieve  higher CSI recovery accuracy with reduced feedback overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  An autoencoder method of CsiNet for CSI feedback [2] is first  proposed, where an encoder at the user equipment (UE) com-  presses the channel matrix and a decoder at the base station  (BS) recovers the corresponding channel matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Subsequently,  a series of follow-up works are conducted under various condi-  tions [3]–[7] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' However, there are still some challenges for DL-  based CSI feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' First, the generalization issue should be  considered since the DL methods tend to express the scenario-  specific property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Moreover, plenty of training data of target  scenario is quite impractical for deployment due to the expense  and long-time training and collecting data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Meta-learning is  utilized for CSI feedback in [8] and [9], where the model  is initialized by the meta model obtained in meta training  phase with massive CSI samples corresponding to multiple  various scenarios, and then achieves quick convergence with  small amount of CSI data in a new target scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' However,  the above meta-learning based solutions still require massive  collected data for the meta training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Moreover, in target  retraining phase the model is retrained on the original small  amount of data within short time, thus it might suffer from  performance loss in the new target scenario in comparison with  models trained on sufficient data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Further, the above methods  also fail to consider the knowledge of intrinsic characteristics  of the wireless communication during both phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  In this paper, a novel knowledge-driven meta-learning  method for CSI feedback is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Specifically, instead  of training with massive CSI data collected from different  wireless scenarios in meta training phase, one can construct the  meta task environment by exploring the intrinsic knowledge  of spatial-frequency characteristic of CSI eigenvector for meta  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' After the DL model obtains the initialization in meta  training phase, it is capable of achieving rapid convergence  by retraining on target task dataset, which is augmented  from only small amount of actually collected seeded data  with the assistance of the knowledge of statistical feature of  wireless channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Simulation results illustrate the superiority  of the proposed method from the perspective of feedback  performance and convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Notations: uppercase and lowercase letters denote scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Boldface uppercase and boldface lowercase letters denote ma-  trices and vectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Calligraphic uppercase letters  denote sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' A(:, B) and A(B, :) denote the sub-matrices of  A that consist of the columns and rows indexed by set B,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' E{·} denotes expectation and Tr{·} denotes trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  AH denotes the Hermitian matrix of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' rand(A, a) denotes  the random sampling of a samples from set A without re-  placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' The sets of real and complex numbers are denoted  by R and C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' | · | denotes the cardinality of a set  or the absolute value of a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' SYSTEM DESCRIPTION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' System Model  A MIMO system with Nt = NhNv transmitting anten-  nas at BS and Nr receiving antennas at UE is considered,  where Nh and Nv are the numbers of horizontal and vertical  antenna ports, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Note that our proposed methods  are suitable for antennas with either dual or single polariza-  tion, and that single polarization is considered to illustrate  the basic principle in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' The downlink channel in  time domain can be denoted as a three-dimensional matrix  �H ∈ CNr×Nt×Nd, where Nd is the number of paths with  various delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' By conducting Discrete Fourier transform  (DFT) over the delay-dimension of the time-domain downlink  channel matrix �H, the downlink channel in frequency domain  �H ∈ CNr×Nt×Nsc can be written as  �H =  � �H1, �H2, · · · , �HNsc  �  ,  (1)  where Nsc  is the number of subcarriers, and Hk  ∈  CNr×Nt, 1 ≤ k ≤ Nsc denotes the downlink channel on the  kth subcarrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Normally, the CSI eigenvector feedback is per-  formed on each subband which consists of Ngran subcarriers  with Nsc = NgranNsb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Assuming the rank 1 configuration for  downlink transmission, the corresponding eigenvector for the  lth subband wl ∈ CNt×1 with ||wl||2 = 1, can be calculated  by the eigenvector decomposition on the subband as  \uf8eb  \uf8ed  1  Ngran  lNgran  �  k=(l−1)Ngran+1  �HH  k �Hk  \uf8f6  \uf8f8 wl = λlwl,  (2)  where 1 ≤ l ≤ Nsb and λl represents the corresponding  maximum eigenvalue for the l-th subband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Therefore, the CSI  eigenvector for all Nsb subbands can be written as  W =  �  w1, w2, · · · , wNsb  �  ∈ CNt×Nsb,  (3)  wherein total NsbNt complex coefficients need to be com-  pressed at the UE and then recovered at the BS side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Generally, the optimization objective for CSI feedback can  be given as  min  F −ρ(W, W′) = min  F − 1  Nsb  Nsb  �  l=1  � ∥wHw′∥2  ∥w∥2∥w′∥2  �2  ,  (4)  where ρ(·, ·) ∈ [0, 1] denotes the squared generalized cosine  similarity (SGCS), ∥·∥2 denotes ℓ2 norm, wl and w′  l represent  the original and recovered CSI eigenvector of the l-th sub-  band, respectively, F represents the alternative CSI feedback  schemes such as TypeI, eTypeII and DL-based autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' DL-based CSI Feedback  The architecture of DL-based CSI feedback using autoen-  coder is introduced in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' 1, where the neural network (NN)  encoder and decoder, fe(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' ΘE) and fd(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' ΘD) with trainable  parameters Θ = {ΘE, ΘD} are deployed at UE and BS,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Thus the DL-based autoencoder fa(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Θ) with  trainable parameters Θ = {ΘE, ΘD} can be represented as  W′ = fd(fe(W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' ΘE);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' ΘD) = fa(W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Θ),  (5)  UE  BS  W  W�  Encoder �e  Decoder �d  Bitstream b  SGCS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Illustration of DL-based CSI feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  where the encoder first compresses and quantizes the original  CSI eigenvector W to a bitstream b of length B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Then the  decoder uses b to recover W′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' During training phase, the  encoder and decoder are jointly optimized to solve (4) with  sufficient numbers of CSI eigenvector samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Meta-learning based CSI Feedback  Generally, the goal of meta-learning based CSI feedback  is to find a good initialization of Θ = {ΘE, ΘD}, so that the  autoencoder can converge quickly with a small amount of CSI  samples and a few training steps for a new scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Specifi-  cally, the procedure of meta-learning based CSI feedback can  be divided into two phases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=', the meta training phase and  target retraining phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  During meta training phase, the model is trained over a big  dataset consisting of T CSI tasks of diverse scenarios, which  can be defined as meta task environment Tmeta = {T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=', TT },  wherein each task Tj = {Wj  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=', Wj  T }, 1 ≤ j ≤ T consists  of |Tj| CSI samples denoted as Wj  i , 1 ≤ i ≤ Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Based on the  meta task environment Tmeta, meta-learning algorithms can be  performed to learn the initial parameters �Θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=',  min  �Θ  ETj⊂Tmeta  �  −ρ′(Tj, fa(Tj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Ug  Tj(�Θ)))  �  ,  (6)  where Ug  Tj(�Θ) is the operator that updates �Θ for g training  steps using data sampled from Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' The initialization �Θ learnt  in (6) is expected to has the same ability of quick adaptation  with small amount of data on an unobserved target task Ttarget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Secondly, the target retraining phase can be formulated as  min  Φ=Ug  Ttarget (�Θ)  −ρ′(Ttarget, fa(Ttarget;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Φ)),  (7)  where Φ denotes the possible parameter sets trained on Ttarget  after g retraining steps based on the initialization �Θ, which  indicates that the final parameters on a new target task of  scenario can be rapidly obtained with only a few retraining  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  However, the existing meta-leaning based CSI feedback still  has to face two major challenges, which our knowledge-driven  meta-learning method aims to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  During meta training phase, it requires sufficient samples  to construct the meta task environment Tmeta to solve (6),  which is extremely costly since it is impractical to collect  all existing types of wireless scenarios with adequate  diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  During target retraining phase, despite the rapid conver-  gence for solving (7) using small amount of data Ttarget  based on the initialization �Θ, it is always difficult to  achieve comparable performance with using large amount  of CSI data in target scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Knowledge  Meta  Training  Phase  Target  Retraining  Phase  Initialization  Meta Task  Environment  Meta Model  Seeded Data  Knowledge  Augmented Data  Target Scenario  Target Model  UE  BS  �  Encoder �  Decoder �  Bitstream  SGCS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Proposed knowledge-driven meta-learning framework for CSI feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' KNOWLEDGE-DRIVEN META-LEARNING FOR CSI  FEEDBACK  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Knowledge-driven Meta Training Phase  1) Spatial-Frequency Characteristic: Generally, consider-  ing the intrinsic structure of the CSI eigenvector, W ∈  CNt×Nsb can be decomposed as  W = SEFH  (8)  where S ∈ CNt×Nt is constructed with Nt orthogonal basis  vectors in spatial domain and F ∈ CNsb×Nsb is constructed  with Nsb orthogonal basis vectors in frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Specifically, both S and F are unitary matrices, which indicate  the full-rank spatial-frequency characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' The projection  coefficient matrix E ∈ CNt×Nsb represents that each CSI  eigenvector W can be completely expressed by the linear  combination of the orthogonal basis vectors in S and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Obviously, the distribution of the elements in E with relatively  larger amplitude determines the dominant spatial-frequency  feature of W given the same S and F, where the dominant  spatial-frequency features can be considered as the intrinsic  knowledge and hence can be learnt by the DL model during  the meta-training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  2) Knowledge-driven Meta Training: Inspired by the intrin-  sic knowledge of spatial-frequency feature in section III-A1, a  knowledge-driven algorithm is proposed to solve (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' During  the meta training phase, the meta task environment Tmeta =  {T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=', TT } consisting of T tasks is firstly established, where  the construction approach of CSI eigenvector in each task  explores the CSI decomposition formula in section III-A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Each task usually consists of different CSI eigenvectors from  specific number of UEs that can be sampled on various number  of slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Specifically, denote Nue,j and Nslot,j as the number  of UEs and slots for the j-th task Tj, 1 ≤ j ≤ T , respectively,  which can be set as  Nue,j = rand({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=', �  Nue}, 1),  (9)  Nslot,j = rand({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=', �  Nslot}, 1),  (10)  where Nslot,j Nue,j = |Tj|, �  Nue and �  Nslot denote the maxi-  mum number of UEs and maximum number of slots of CSI  that can be generated in one task, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Moreover, according to the intrinsic knowledge of spatial-  frequency feature, to generate the CSI samples in the j-th task  Tj, P groups of spatial orthogonal basis vector and one group  of frequency orthogonal basis vector can be firstly given as  Sp = [sp,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=', sp,Nt] ∈ CNt×Nt, 1 ≤ p ≤ P,  (11)  F = [f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=', fNsb] ∈ CNsb×Nsb,  (12)  respectively, where each column of Sp and F is an orthogonal  basis vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Specifically, each basis in Sp indicates a beam  direction in spatial domain, and multiple groups of orthogonal  basis vectors are designed in order to improve the diversity  of spatial features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Here we introduce a Schmidt orthogo-  nalization method for obtaining the basis vector groups Sp  and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' For each group of spatial orthogonal basis vector  Sp, 1 ≤ p ≤ P, and the frequency orthogonal basis vector  group F, the Schmidt orthogonalization can be performed on  three full-rank random matrices Xh  p ∈ CNh×Nh ∼ CN(0, 1),  Xv  p ∈ CNv×Nv ∼ CN(0, 1) and Xf ∈ CNsb×Nsb ∼ CN(0, 1),  obtaining the orthogonal matrices Uh  p, Uv  p and Uf, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' F = Uf can be utilized as frequency orthogonal basis  vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' The p-th spatial orthogonal basis vector group can be  obtained by performing kronecker product, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=', Sp = Uh  p⊗Uv  p  Next, the method of generating CSI samples for the j-th  task Tj is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' The group index pj for task Tj are first  randomized by  {pj} = rand({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=', P}, 1),  (13)  and the indices of dominant spatial and frequency feature  vectors are also randomized by  �Sj = rand({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=', Nt}, Ltask),  (14)  �Fj = rand({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=', Nsb}, Mtask),  (15)  respectively, where the parameters Ltask ≤ Nt and Mtask ≤  Nsb are defined to constrain the degree of feature diversity of  the task in spatial and frequency domain, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  For the m-th UE 1 ≤ m ≤ Nue in task Tj, the indices  of the dominant spatial and frequency feature vectors are also  randomized by  �Sm = rand( �Sj, Lm),  (16)  �Fm = rand( �  Fj, Mm),  (17)  respectively, where the degree of feature diversity in spatial  and frequency domain Lm and Mm are both UE-specific, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=',  {Lm} = rand({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=', Ltask}, 1),  (18)  {Mm} = rand({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=', Mtask}, 1),  (19)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Similarly for the n-th slot 1 ≤ n ≤ Nslot of the m-th UE in  task Tj, the dominant spatial and frequency feature vectors are  respectively selected from the corresponding dominant vectors  of the UE, so that the feature is maintained for the m-th UE  but distinguished between different slots, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=',  Sm,n = rand( �Sm, ⌈αLm⌉),  (20)  Algorithm 1: Knowledge-driven Meta Training Phase  Initialization: �  Nue, �  Nslot, T , α, β, g, ǫ, �Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Formulate the feature basis using (11) to (12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' , T do  Construct structure of Tj using (9), (10) and (13)  to (15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' , Nue,j do  Consturt structure of UE m using (16) to (19);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' , Nslot,j do  Generate a CSI of slot n using (20) to (23);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  end  end  end  Meta training by iterating (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Fm,n = rand( �  Fm, ⌈βMm⌉),  (21)  where the parameters α ∈ (0, 1] and β ∈ (0, 1] are set to scale  the diversity of the feature of each slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Consequently, a CSI  sample for the n-th slot of the m-th UE in task Tj can be  generated as  Wj  m,n = Spj(:, Sm,n)�EFH(:, Fm,n),  (22)  where the elements in �E ∈ C|Sm,n|×|Fm,n| are indepen-  dently sampled from complex normal distribution CN(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  A subband-level normalization should be also performed for  1 ≤ l ≤ Nsb using  Wj  m,n(:, l) =  Wj  m,n(:, l)  ||Wj  m,n(:, l)||2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  (23)  Through the procedure of (9) to (23) for generating each CSI  sample of each UE, the meta task environment Tmeta can be  finally constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Utilizing the meta task environment Tmeta, the meta training  procedure can be conducted to solve (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' The parameters of  the DL model of CSI feedback is randomly initialized by �Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  For the j-th task Tj in the meta task environment Tmeta, �Θ  can be updated with  �Θ = �Θ + ǫ(Ug  Tj(�Θ) − �Θ),  (24)  where Ug  Tj(�Θ) is the operator that updates �Θ for g training  steps on task Tj, and ǫ denotes the step size of meta training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  After that, the obtained �Θ can be utilized as initialization for  further fast retraining on a new target task of scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' The  proposed algorithm for knowledge-driven meta training phase  is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Knowledge-driven Target Retraining Phase  1) Statistical Feature of Channel: In this part, the statistical  features of the channel in both spatial domain and time delay  domain are explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Specifically, for a specific UE in the  target scenario, denote the actually collected �  Nslot channel  samples in time domain as H = { �H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=', �H �  Nslot}, where each  channel sample �Ht ∈ CNr×Nt×Nd, 1 ≤ t ≤ �  Nslot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Firstly, the statistical feature in delay domain can be  described by the power-delay spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Denote  �H′  t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='d  ∈  CNr×Nt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' 1 ≤ d ≤ Nd as the d-th delay of the t-th channel  sample �Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' the power of the d-th delay can be calculated as  ˆpd =  1  NtNr �  Nslot  �  Nslot  �  t=1  || �H′  t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='d||2  F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  (25)  Secondly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' the statistical feature in spatial domain can be  demonstrated by the self-correlation matrices of the transmit-  ting and receiving antenna ports,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' which can be calculated as  Rtx  d =  Nt  � �  Nslot  t=1  �H′H  t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='d �H′  t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='d  Tr(� �  Nslot  t=1  �H′H  t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='d �H′  t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='d)  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  (26)  Rrx  d =  Nr  � �  Nslot  t=1  �H′  t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='d �H′H  t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='d  Tr(� �  Nslot  t=1  �H′  t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='d �H′H  t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='d)  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  (27)  respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' where the trace operation Tr(·) is performed for  normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Then the kronecker product is implemented  on the transmitting and receiving self-correlation matrices to  obtain the joint spatial feature as  Rd = Rrx  d ⊗ Rtx  d ∈ CNtNr×NtNr,  (28)  It should be noted that the dataset of the target scenario  could be very small, and thus it is not sufficient for training  autoencoder with superior CSI feedback and to ensure recov-  ery performance, even though it is able to converge quickly  based on the initialization �Θ obtained by meta training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Therefore, it is necessary to consider a data augmentation  seeded by H exploiting the knowledge of statistical features  in spatial and delay domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  The intrinsic knowledge of statistical features of the channel  for a specific UE can be completely described by ˆpd and  Rd, 1 ≤ d ≤ Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Therefore, to align with the statistical  features of the collected channel samples, the augmented  channel for the d-th delay ˆhaug  d  should satisfy  E[ˆhaug  d  (ˆhaug  d  )H] = ˆpdRd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  (29)  2) Knowledge-driven Target Retraining: The knowledge-  driven target retraining is introduced with data augmentation  inspired by (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Firstly, SVD is performed on Rd, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=',  Ud, Dd, Vd = svd(Rd),  (30)  where Vd = UH  d because of Rd = RH  d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Secondly, the augmented channel sample for the d-th delay  can be generated by conducting  ˆhaug  d  =  �  ˆpdUdD  1  2  d n,  (31)  where the random vector n ∈ CNtNr×1 ∼ CN(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Next, ˆhaug  d  can be reshaped as the channel matrix �Haug  d  ∈  CNr×Nt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' By concatenating all Nd augmented channel matri-  ces, the augmented channel sample can be obtained as  Haug = [ �Haug  1  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=', �Haug  Nd ],  (32)  Algorithm 2: Knowledge-driven Target Retraining  Phase  Initialization:H, Naug, g′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Calculate statistical delay power spectrum using (25);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  for q = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' , �  Nue do  for d = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' , Ndelay do  Data augmentation using (26) to (32) and (1)  to (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  end  end  Target retraining using (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  where Haug ∈ CNr×Nt×Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Then the augmented CSI eigen-  vector sample Waug can be finally obtained by implementing  (1) to (3) on Haug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  For each UE, the total Naug channel samples can be  provided with Naug randomly generated vectors n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Moreover,  for Nue UEs, we can generate totally NueNaug augmented CSI  eigenvector samples that can be used to construct the target  task dataset T aug  target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Based on the target task dataset T aug  target and the initialization  �Θ obtained in knowledge-driven meta training phase, (7) can  be solved with higher SGCS using a few training steps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=',  Φ = Ug  Ttarget(�Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  (33)  The proposed algorithm for knowledge-driven target retraining  phase can be summarized in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' SIMULATION RESULTS  The  simulation  results  are  provided in  this  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Knowledge-driven scheme in meta training phase (KMeta-  ) and target retraing phase (*-KAug) are evaluated, where  ‘None’ denotes no knowledge-driven schemes are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' The  simulation parameters are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' CDL-C [10]  channel model with delay spread 300 ns and random dis-  tributed UE with speed 300 km/h are utilized as actually  collected channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' The training was performed three times  with different random seeds, one of which is shown since the  TABLE I  BASIC SIMULATION PARAMETERS  Parameter  Value  System bandwidth  10MHz  Carrier frequency  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='5GHz  Subcarrier spacing  15KHz  Subcarriers number Nsc  624  Subband number Nsb  13  Horizontal Tx antenna ports per polarization Nh  8  Vertical Tx antenna ports per polarization Nv  2  Tx antenna ports Nt  32  Rx antennas Nr  4  Meta task enviroment size T  8000  Meta training step size ǫ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='25  Step number per task g  32  Spatial diversity degree Ltask  6  Frequency diversity degree Mtask  6  Spatial diversity scale α  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='75  Frequency diversity scale β  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='75  0  500  1000  1500  2000  Steps  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='9  SGCS  None-None  None-KAug  KMeta-None  KMeta-KAug  eTypeII  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Convergence process of target retraining phase with the number of  training steps on CDL-C channel ( �  Nue = 300, �  Nslot = 10, Naug = 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  results are almost equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Moreover, the Transformer backbone  for CSI feedback [4] with number of feedback bits B = 64 is  implemented in evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' 3 show the convergence process of target retraining  phase with the number of training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Note that the vertical  axis represents the best achieved SGCS on the test set within  the steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Here we consider the eTypeII codebook and DL-  based method without meta training and target augmentation  (None-None) as the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Note that since the existing  meta-learning methods for CSI require a large amount of  multi-scenario real data, and our method lever knowledge for  meta-learning, it is unfair to compare our method with them  in terms of data cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' In terms of convergence speed, it can  be noticed that the proposed KMeta-None require fewer train-  ing steps to achieve convergence than None-None.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Even on  augmented data, KMeta-KAug can also fit more quickly than  None-KAug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' From the perspective of feedback performance,  the knowledge-driven meta training brings higher SGCS since  KMeta-None outperforms None-None.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Moreover, the methods  of *-KAug outperform the methods of *-None, which reveals  that the knowledge-driven target retraining phase can further  effectively improve the SGCS performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' 5 we compare the SGCS performance  training 2000 steps on different number of seeded UEs �  Nue  and slots �  Nslot, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' It is observed that the proposed  knowledge-driven method of KMeta-KAug outperforms tra-  ditional eTypeII codebook and basic DL-based method None-  None.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Specifically, the performance gaps between the methods  KMeta-* and None-* can respectively demonstrate the gain  provided by proposed knowledge-driven meta training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' The  gaps between *-KAug and *-None respectively initimate the  gain obtained from proposed knowledge-driven target retrain-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Moervoer, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' 5, the performance of None-KAug  improves as the number of slots �  Nslot increased, while the  performance of KMeta-KAug stays almost unchanged, which  implies that the proposed knowledge-driven target retraining  requires fewer slots to achieve the performance ceiling when  it is enhanced by proposed knowledge-driven meta training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Table II illustrates the SGCS performance of the proposed  method and the existing data augmentation methods [4] for  DL-based CSI feedback including noise injection, flipping,  50  100  150  200  250  300  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='8  SGCS  None-None  None-KAug  KMeta-None  KMeta-KAug  eTypeII  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  Comparison of SGCS for varying number of seeded UEs �  Nue on  CDL-C channel, fixing number of slots per UE �  Nslot = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  10  20  30  40  50  60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='85  SGCS  None-None  None-KAug  KMeta-None  KMeta-KAug  eTypeII  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Comparison of SGCS for varying number of slots per UE �  Nslot on  CDL-C channel, fixing number of UEs �  Nue = 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  cyclic shift, random shift and rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' It is observed that  the proposed method can obtain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='1953 SGCS performance  gain in comparison to none augmantation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Specifically, the  performance gap between proposed method and other com-  petitors is at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='1526, which demonstrates that exploiting  communication knowledge effectively bring performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  The link-level block error rate (BLER) performance is  presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' 6, where omnidirectional and directional  antennas is deployed at UE and BS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' The gap  between KMeta-None and None-None proves the performance  gain of knowledge-driven meta training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' The gap be-  tween None-KAug and None-None proves the performance  gain of knowledge-driven target retraining phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Since the  method of KMeta-Aug outperforms other competitors in terms  of BLER, the advantages and application potential of proposed  TABLE II  COMPARISON OF DIFFERENT AUGMENT SCHEMES  Scheme  SGCS  None  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='5977  Noise Injection  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='6189  Flipping  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='6171  Cyclic Shift  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='6404  Random Shift  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='6225  Rotation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='6178  Proposed  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='7930  Note 1: �  Nue = 300 and �  Nslot = 10  Note 2: All methods augments to 30k samples, except the flipping which  can only augment to 6k samples due to method limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  1  0  1  2  3  4  5  6  SNR (dB)  10-3  10-2  10-1  100  BLER  KMeta-KAug  KMeta-None  None-KAug  None-None  eTypeII  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Link-level BLER performance comparison trained on CDL-C300 for  different solutions ( �  Nue = 300, �  Nslot = 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  knowledge-driven approach are well demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' CONCLUSION  In this paper, we propose a knowledge-driven meta-learning  method for CSI feedback, where the meta task environment for  meta training is constructed based on the intrinsic knowledge  of spatial-frequency feature of CSI eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Initialized  by the knowledge-driven meta training phase, the DL model  is capable of achieving rapid convergence by retraining on  the target task dataset, which is augmented from only a few  actually collected seeded data with the assistance of the knowl-  edge of statistical feature of wireless channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Simulation  results demonstrate the superiority of the approach from the  perspective of feedback performance and convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  REFERENCES  [1] 3GPP, “3GPP TS 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='214 v17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
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+page_content=' NR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Abd El-Malek, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Abo-Zahhad, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Elsabrouty,  “A meta learner autoencoder for channel state information feedback in  massive MIMO systems,” in 2021 28th International Conference on  Telecommunications (ICT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  IEEE, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  [10] 3GPP, “3GPP TR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='901 v17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='0 3rd Generation Partnership Project;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='  technical specification group radio access network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' study on channel  model for frequencies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content='5 to 100 GHz (release 17),” Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}
+page_content=',  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FRT4oBgHgl3EQfEDfx/content/2301.13475v1.pdf'}