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A Bessel filter (not shown) has an even poorer transition in the frequency domain, but maintains the best phase fidelity of a waveform.
Different applications emphasize different design requirements, leading to different choices among these (and other) optimizations, or requiring a filter of a higher order.
========,2,Example implementations.
A popular circuit implementing a second order active R-C filter is the Sallen-Key design, whose schematic diagram is shown here.
This topology can be adapted to produce low-pass, band-pass, and high pass filters.
An N order FIR filter can be implemented in a discrete time system using a computer program or specialized hardware in which the input signal is subject to N delay stages.
The output of the filter is formed as the weighted sum of those delayed signals, as is depicted in the accompanying signal flow diagram.
The response of the filter depends on the weighting coefficients denoted "b", "b", ... "b".
For instance, if all of the coefficients were equal to unity, a so-called boxcar function, then it would implement a low-pass filter with a low frequency gain of N+1 and a frequency response given by the sinc function.
Superior shapes for the frequency response can be obtained using coefficients derived from a more sophisticated design procedure.
========,2,Mathematics of filter design.
LTI system theory describes linear "time-invariant" (LTI) filters of all types.
LTI filters can be completely described by their frequency response and phase response, the specification of which uniquely defines their impulse response, and "vice versa".
From a mathematical viewpoint, continuous-time IIR LTI filters may be described in terms of linear differential equations, and their impulse responses considered as Green's functions of the equation.
Continuous-time LTI filters may also be described in terms of the Laplace transform of their impulse response, which allows all of the characteristics of the filter to be analyzed by considering the pattern of poles and zeros of their Laplace transform in the complex plane.
Similarly, discrete-time LTI filters may be analyzed via the Z-transform of their impulse response.
Before the advent of computer filter synthesis tools, graphical tools such as Bode plots and Nyquist plots were extensively used as design tools.
Even today, they are invaluable tools to understanding filter behavior.
Reference books had extensive plots of frequency response, phase response, group delay, and impulse response for various types of filters, of various orders.
They also contained tables of values showing how to implement such filters as RLC ladders - very useful when amplifying elements were expensive compared to passive components.
Such a ladder can also be designed to have minimal sensitivity to component variation a property hard to evaluate without computer tools.
Many different analog filter designs have been developed, each trying to optimise some feature of the system response.
For practical filters, a custom design is sometimes desirable, that can offer the best tradeoff between different design criteria, which may include component count and cost, as well as filter response characteristics.
These descriptions refer to the "mathematical" properties of the filter (that is, the frequency and phase response).
These can be "implemented" as analog circuits (for instance, using a Sallen Key filter topology, a type of active filter), or as algorithms in digital signal processing systems.
Digital filters are much more flexible to synthesize and use than analog filters, where the constraints of the design permits their use.
Notably, there is no need to consider component tolerances, and very high Q levels may be obtained.
FIR digital filters may be implemented by the direct convolution of the desired impulse response with the input signal.
They can easily be designed to give a matched filter for any arbitrary pulse shape.
IIR digital filters are often more difficult to design, due to problems including dynamic range issues, quantization noise and instability.
Typically digital IIR filters are designed as a series of digital biquad filters.
All low-pass second-order continuous-time filters have a transfer function given by All band-pass second-order continuous-time have a transfer function given by where
***LIST***.
========,1,preface.
The Essenes (in Modern Hebrew: , "Isiyim"; Greek: Ἐσσηνοί, Ἐσσαῖοι, or Ὀσσαῖοι, "Essenoi, Essaioi, Ossaioi") were a sect of Second Temple Judaism that flourished from the 2nd century BCE to the 1st century CE which some scholars claim seceded from the Zadokite priests.
Being much fewer in number than the Pharisees and the Sadducees (the other two major sects at the time), the Essenes lived in various cities but congregated in communal life dedicated to asceticism (some groups practised celibacy), voluntary poverty, and daily immersion.
Many separate but related religious groups of that era shared similar mystic, eschatological, messianic, and ascetic beliefs.
These groups are collectively referred to by various scholars as the "Essenes."
Josephus records that Essenes existed in large numbers, and thousands lived throughout Roman Judaea.
The Essenes have gained fame in modern times as a result of the discovery of an extensive group of religious documents known as the Dead Sea Scrolls, which are commonly believed to be the Essenes' library—although not conclusive.
These documents preserve multiple copies of parts of the Hebrew Bible untouched from possibly as early as 300 BCE until their discovery in 1946.
Some scholars dispute the notion that the Essenes wrote the Dead Sea Scrolls.
Rachel Elior questions even the existence of the Essenes.
The first reference is by the Roman writer Pliny the Elder (died ) in his "Natural History".
Pliny relates in a few lines that the Essenes do not marry, possess no money, and had existed for thousands of generations.
Unlike Philo, who did not mention any particular geographical location of the Essenes other than the whole land of Israel, Pliny places them in Ein Gedi, next to the Dead Sea.
A little later, Josephus gave a detailed account of the Essenes in "The Jewish War" (), with a shorter description in "Antiquities of the Jews" () and "The Life of Flavius Josephus" ().
Claiming first hand knowledge, he lists the "Essenoi" as one of the three sects of Jewish philosophy alongside the Pharisees and the Sadducees.
He relates the same information concerning piety, celibacy, the absence of personal property and of money, the belief in communality, and commitment to a strict observance of Sabbath.
He further adds that the Essenes ritually immersed in water every morning, ate together after prayer, devoted themselves to charity and benevolence, forbade the expression of anger, studied the books of the elders, preserved secrets, and were very mindful of the names of the angels kept in their sacred writings.
Pliny, also a geographer, located them in the desert near the northwestern shore of the Dead Sea, where the Dead Sea Scrolls were discovered.
========,2,Name.
Josephus uses the name "Essenes" in his two main accounts but some manuscripts read here "Essaion"; "holding the Essenes in honour"; "a certain Essene named Manaemus"; "to hold all Essenes in honor"; "the Essenes").
In several places, however, Josephus has "Essaios", which is usually assumed to mean "Essene" ("Judas of the "Essaios" race"; "Simon of the "Essaios" race"; "John the "Essaios""; "those who are called by us "Essaioi""; "Simon a man of the "Essaios" race").
Josephus identified the Essenes as one of the three major Jewish sects of that period.
Philo's usage is "Essaioi", although he admits this Greek form of the original name that according to his etymology signifies "holiness" to be inexact.
Pliny's Latin text has "Esseni".
Gabriele Boccaccini implies that a convincing etymology for the name Essene has not been found, but that the term applies to a larger group within Palestine that also included the Qumran community.
It was proposed before the Dead Sea Scrolls were discovered that the name came into several Greek spellings from a Hebrew self-designation later found in some Dead Sea Scrolls, "'osey hatorah", "observers of torah".
Although dozens of etymology suggestions have been published, this is the only etymology published before 1947 that was confirmed by Qumran text self-designation references, and it is gaining acceptance among scholars.
It is recognized as the etymology of the form "Ossaioi" (and note that Philo also offered an O spelling) and "Essaioi" and "Esseni" spelling variations have been discussed by VanderKam, Goranson, and others.
In medieval Hebrew (e.g.
Sefer Yosippon) "Hassidim" ("the pious ones") replaces "Essenes".
While this Hebrew name is not the etymology of "Essaioi"/"Esseni", the Aramaic equivalent "Hesi'im" known from Eastern Aramaic texts has been suggested.
Others suggest that Essene is a transliteration of the Hebrew word "chitzonim" (chitzon=outside), which the Mishna (e.g.
Megila 4:8) uses to describe various sectarian groups.
Another theory is that the name was borrowed from a cult of devotees to Artemis in Asia Minor, whose demeanor and dress somewhat resembled those of the group in Judaea.
Flavius Josephus in Chapter 8 of "The Jewish War" states:
========,2,Location.
According to Josephus, the Essenes had settled "not in one city" but "in large numbers in every town".
Philo speaks of "more than four thousand" "Essaioi" living in "Palestine and Syria", more precisely, "in many cities of Judaea and in many villages and grouped in great societies of many members".
Pliny locates them "on the west side of the Dead Sea, away from the coast... [above] the town of Engeda".
Some modern scholars and archaeologists have argued that Essenes inhabited the settlement at Qumran, a plateau in the Judean Desert along the Dead Sea, citing Pliny the Elder in support, and giving credence that the Dead Sea Scrolls are the product of the Essenes.
This theory, though not yet conclusively proven, has come to dominate the scholarly discussion and public perception of the Essenes.
========,2,Rules, customs, theology, and beliefs.
The accounts by Josephus and Philo show that the Essenes led a strictly communal lifeoften compared to later Christian monasticism.
Many of the Essene groups appear to have been celibate, but Josephus speaks also of another ""order" of Essenes" that observed the practice of being engaged for three years and then becoming married.
According to Josephus, they had customs and observances such as collective ownership, electing a leader to attend to the interests of the group, and obedience to the orders from their leader.
Also, they were forbidden from swearing oaths and from sacrificing animals.
They controlled their tempers and served as channels of peace, carrying weapons only for protection against robbers.
The Essenes chose not to possess slaves but served each other and, as a result of communal ownership, did not engage in trading.
Josephus and Philo provide lengthy accounts of their communal meetings, meals and religious celebrations.
After a total of three years' probation, newly joining members would take an oath that included the commitment to practice piety towards God (το θειον) and righteousness towards humanity, to maintain a pure lifestyle, to abstain from criminal and immoral activities, to transmit their rules uncorrupted and to preserve the books of the Essenes and the names of the Angels.
Their theology included belief in the immortality of the soul and that they would receive their souls back after death.
Part of their activities included purification by water rituals, which was supported by rainwater catchment and storage.
Ritual purification was a common practice among the peoples of Judea during this period and was thus not specific to the Essenes.
Ritual baths are found near many Synagogues of the period.
Purity and cleanliness was considered so important to the Essenes that they would refrain from defecation on the Sabbath.
According to Joseph Lightfoot the Church Father Epiphanius (writing in the 4th century CE) seems to make a distinction between two main groups within the Essenes: "Of those that came before his [Elxai, an Ossaean prophet] time and during it, the Ossaeans and the Nazarean."
Epiphanius describes each group as following:
If it is correct to identify the community at Qumran with the Essenes (and claim that the community at Qumran are the authors of the Dead Sea Scrolls), then according to the Dead Sea Scrolls the Essenes' community school was called "Yahad" (meaning "community") in order to differentiate themselves from the rest of the Jews who are repeatedly labeled "The Breakers of the Covenant".
========,2,Scholarly discussion.
Josephus and Philo discuss the Essenes in detail.
Most scholars believe that the community at Qumran that allegedly produced the Dead Sea Scrolls was an offshoot of the Essenes; however, this theory has been disputed by some, for example, Norman Golb argues that the primary research on the Qumran documents and ruins (by Father Roland de Vaux, from the "École Biblique et Archéologique de Jérusalem") lacked scientific method, and drew wrong conclusions that comfortably entered the academic canon.
For Golb, the amount of documents is too extensive and includes many different writing styles and calligraphies; the ruins seem to have been a fortress, used as a military base for a very long period of timeincluding the 1st centuryso they could not have been inhabited by the Essenes; and the large graveyard excavated in 1870, just 50 metres east of the Qumran ruins was made of over 1200 tombs that included many women and childrenPliny clearly wrote that the Essenes who lived near the Dead Sea "had not one woman, had renounced all pleasure ... and no one was born in their race".
Golb's book presents observations about de Vaux's premature conclusions and their uncontroverted acceptance by the general academic community.
He states that the documents probably stemmed from various libraries in Jerusalem, kept safe in the desert from the Roman invasions.
Other scholars refute these arguments—particularly since Josephus describes some Essenes as allowing marriage.
Another issue is the relationship between the "Essaioi" and Philo's "Therapeutae" and "Therapeutrides".
He regarded the "Therapeutae" as a contemplative branch of the "Essaioi" who, he said, pursued an active life.