Analogue filter - Knowledge (XXG)

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iterative process. Image filters, in their simplest form, consist of a chain of repeated, identical sections. The design can be improved simply by adding more sections and the computation required to produce the initial section is on the level of "back of an envelope" designing. In the case of network synthesis filters, on the other hand, the filter is designed as a whole, single entity and to add more sections (i.e., increase the order) the designer would have no option but to go back to the beginning and start over. The advantages of synthesised designs are real, but they are not overwhelming compared to what a skilled image designer could achieve, and in many cases it was more cost effective to dispense with time-consuming calculations. This is simply not an issue with the modern availability of computing power, but in the 1950s it was non-existent, in the 1960s and 1970s available only at cost, and not finally becoming widely available to all designers until the 1980s with the advent of the desktop personal computer. Image filters continued to be designed up to that point and many remained in service into the 21st century.

1849:, Wilhelm Cauer and others all working more or less independently and is often taken as synonymous with network synthesis. Butterworth's filter implementation is, in those terms, an insertion-loss filter, but it is a relatively trivial one mathematically since the active amplifiers used by Butterworth ensured that each stage individually worked into a resistive load. Butterworth's filter becomes a non-trivial example when it is implemented entirely with passive components. An even earlier filter which influenced the insertion-loss method was Norton's dual-band filter where the input of two filters are connected in parallel and designed so that the combined input presents a constant resistance. Norton's design method, together with Cauer's canonical LC networks and Darlington's theorem that only LC components were required in the body of the filter resulted in the insertion-loss method. However, ladder topology proved to be more practical than Cauer's canonical forms. 1836:. This gives Butterworth the credit for being both the first to deviate from image parameter theory and the first to design active filters. It was later shown that Butterworth filters could be implemented in ladder topology without the need for amplifiers. Possibly the first to do so was William Bennett (1932) in a patent which presents formulae for component values identical to the modern ones. Bennett, at this stage though, is still discussing the design as an artificial transmission line and so is adopting an image parameter approach despite having produced what would now be considered a network synthesis design. He also does not appear to be aware of the work of Butterworth or the connection between them. 1088:. The advantage of network synthesis over previous methods is that it provides a solution which precisely meets the design specification. This is not the case with image filters, a degree of experience is required in their design since the image filter only meets the design specification in the unrealistic case of being terminated in its own image impedance, to produce which would require the exact circuit being sought. Network synthesis on the other hand, takes care of the termination impedances simply by incorporating them into the network being designed. 2184:. The outputs of the various sections are summed in the proportions needed to result in the required frequency function. This works by the principle that certain frequencies will be in, or close to antiphase, at different sections and will tend to cancel when added. These are the frequencies rejected by the filter and can produce filters with very sharp cut-offs. This approach did not find any immediate applications, and is not common in passive filters. However, the principle finds many applications as an active delay line implementation for wide band 608:(FDM). FDM requires the sending end to be transmitting at different frequencies for each individual communication channel. This demands individual tuned resonators, as well as filters to separate out the signals at the receiving end. The harmonic telegraph achieved this with electromagnetically driven tuned reeds at the transmitting end which would vibrate similar reeds at the receiving end. Only the reed with the same resonant frequency as the transmitter would vibrate to any appreciable extent at the receiving end. 1142:' continued fraction expansion. This work was the basis on which network synthesis was built, although Cauer's work was not at first used much by engineers, partly because of the intervention of World War II, partly for reasons explained in the next section and partly because Cauer presented his results using topologies that required mutually coupled inductors and ideal transformers. Designers tend to avoid the complication of mutual inductances and transformers where possible, although transformer-coupled 2590:. The order of a filter is the order of the highest order polynomial of the two and will equal the number of elements (or resonators) required to build it. Usually, the higher the order of a filter, the steeper the roll-off of the filter will be. In general, the values of the elements in each section of the filter will not be the same if the order is increased and will need to be recalculated. This is in contrast to the image method of design which simply adds on more identical sections 883:

understood that although superficially similar, they are really quite different. The ladder construction is essential to the Campbell filter and all the sections have identical element values. Modern designs can be realised in any number of topologies, choosing the ladder topology is merely a matter of convenience. Their response is quite different (better) than Campbell's and the element values, in general, will all be different.

2072: 744: 732: 724: 430:. Network synthesis put filter design on a firm mathematical foundation, freeing it from the mathematically sloppy techniques of image design and severing the connection with physical lines. The essence of network synthesis is that it produces a design that will (at least if implemented with ideal components) accurately reproduce the response originally specified in 2278:

more power than an equivalent LC filter. Inexpensive digital technology has largely supplanted analogue implementations of filters. However, there is still an occasional place for them in the simpler applications such as coupling where sophisticated functions of frequency are not needed. Passive filters are still the technology of choice at microwave frequencies.

482:) found its way into filter design ahead of electrical resonance. Resonance can be used to achieve a filtering effect because the resonant device will respond to frequencies at, or near, to the resonant frequency but will not respond to frequencies far from resonance. Hence frequencies far from resonance are filtered out from the output of the device. 624: 491: 879: 541:(Lord Kelvin) who, in 1853, postulated that there was inductance present in the circuit as well as the capacitance of the jar and the resistance of the load. This established the physical basis for the phenomenon – the energy supplied by the jar was partly dissipated in the load but also partly stored in the magnetic field of the inductor. 971:(1934–1937). Rather than enumerate the behaviour (transfer function, attenuation function, delay function and so on) of a specific circuit, instead a requirement for the image impedance itself was developed. The image impedance can be expressed in terms of the open-circuit and short-circuit impedances of the filter as 389:. The frequency dependence of electrical response was known for capacitors and inductors from very early on. The resonance phenomenon was also familiar from an early date and it was possible to produce simple, single-branch filters with these components. Although attempts were made in the 1880s to apply them to 1127:, that is, the impedance that is connected to the generator. The expression for this impedance determines the response of the filter and vice versa, and a realisation of the filter can be obtained by expansion of this expression. It is not possible to realise any arbitrary impedance expression as a network. 856:, a glance at the circuit diagram of this filter is enough to see its relationship to a loaded transmission line. The cut-off phenomenon is an undesirable side-effect as far as loaded lines are concerned but for telephone FDM filters it is precisely what is required. For this application, Campbell produced 1904:

elliptic functions. Cauer had some contact with Darlington and Bell Labs before WWII (for a time he worked in the US) but during the war they worked independently, in some cases making the same discoveries. Cauer had disclosed the Chebyshev approximation to Bell Labs but had not left them with the proof.

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that operated in the discrete time domain rather than the continuous time domain. All of these filter technologies require precision components for high performance filtering, and that often requires that the filters be tuned. Adjustable components are expensive, and the labor to do the tuning can be

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Darlington considers the topology of coupled tuned circuits to involve a separate approximation technique to the insertion-loss method, but also producing nominally flat passbands and high attenuation stopbands. The most common topology for these is shunt anti-resonators coupled by series capacitors,

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The approximation problem in network synthesis is to find functions which will produce realisable networks approximating to a prescribed function of frequency within limits arbitrarily set. The approximation problem is an important issue since the ideal function of frequency required will commonly be

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Today, it is often preferred to carry out filtering in the digital domain where complex algorithms are much easier to implement, but analogue filters do still find applications, especially for low-order simple filtering tasks and are often still the norm at higher frequencies where digital technology

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uality of a resonating circuit. It is roughly proportional to the number of oscillations, which a resonator would support after a single external excitation (for example, how many times a guitar string would wobble if pulled). One definition of Q factor, the most relevant one in this context, is the

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like a high-Q resonant circuit. Obviously, these are contradictory requirements to be met with a single resonant circuit. The solution to these needs was founded in the theory of transmission lines and consequently the necessary filters did not become available until this theory was fully developed.

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and other parameters. These developments took place in the 1920s and filters produced to these designs were still in widespread use in the 1980s, only declining as the use of analogue telecommunications has declined. Their immediate application was the economically important development of frequency

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The open-circuit impedance of a two-port network is the impedance looking into one port when the other port is open circuit. Similarly, the short-circuit impedance is the impedance looking into one port when the other is terminated in a short circuit. The open-circuit impedance of the first port in

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The insertion-loss method of designing filters is, in essence, to prescribe a desired function of frequency for the filter as an attenuation of the signal when the filter is inserted between the terminations relative to the level that would have been received were the terminations connected to each

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to calculate the number of loading coils needed on his loaded lines, the device that led to his electrical filter development. Lagrange, Godfrey, and Campbell all made simplifying assumptions in their calculations that ignored the distributed nature of their apparatus. Consequently, their models

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This "poles and zeroes" view of filter design was particularly useful where a bank of filters, each operating at different frequencies, are all connected across the same transmission line. The earlier approach was unable to deal properly with this situation, but the poles and zeroes approach could

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and only coupled to the first electromagnetically (i.e., no circuit connection). Hertz showed that the response of the second circuit was at a maximum when it was in tune with the first. The diagrams produced by Hertz in this paper were the first published plots of an electrical resonant response.

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had also noted the retardation effect a few years earlier in 1849 and came to a similar conclusion as Faraday. However, there was not so much interest in Germany in underwater and underground cables as there was in Britain, the German overhead cables did not noticeably suffer from retardation and

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have no problem implementing ultra-precise (and stable) values, so no tuning or adjustment is required. Digital filters also don't have to worry about stray coupling paths and shielding the individual filter sections from one another. One downside is the digital signal processing may consume much

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signal right up until the publication of Carson's 1922 paper. Another advance concerned the nature of noise, Carson and Zobel (1923) treated noise as a random process with a continuous bandwidth, an idea that was well ahead of its time, and thus limited the amount of noise that it was possible to

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expanded on the work of Foster (1926) and was the first to talk of realisation of a one-port impedance with a prescribed frequency function. Foster's work considered only reactances (i.e., only LC-kind circuits). Cauer generalised this to any 2-element kind one-port network, finding there was an

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along a metal bar. This model incorporates only resistance and capacitance, but that is all that was needed in undersea cables dominated by capacitance effects. Kelvin's model predicts a limit on the telegraph signalling speed of a cable but Kelvin still did not use the concept of bandwidth, the

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in 1847 published his important work on conservation of energy in part of which he used those principles to explain why the oscillation dies away, that it is the resistance of the circuit which dissipates the energy of the oscillation on each successive cycle. Helmholtz also noted that there was

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to convert to and from the electrical domain. Indeed, some of the earliest ideas for filters were acoustic resonators because the electronics technology was poorly understood at the time. In principle, the design of such filters can be achieved entirely in terms of the electronic counterparts of

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coefficient). For applications such as a mains filters, the awkwardness must be tolerated. For low-level, low-frequency, applications, RC filters are possible, but they cannot implement filters with complex poles or zeros. If the application can use power, then amplifiers can be used to make RC

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LC filters at low frequencies become awkward; the components, especially the inductors, become expensive, bulky, heavy, and non-ideal. Practical 1 H inductors require many turns on a high-permeability core; that material will have high losses and stability issues (e.g., a large temperature

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Darlington's insertion-loss method is a generalisation of the procedure used by Norton. In Norton's filter it can be shown that each filter is equivalent to a separate filter unterminated at the common end. Darlington's method applies to the more straightforward and general case of a 2-port LC

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Once computational power was readily available, it became possible to easily design filters to minimise any arbitrary parameter, for example time delay or tolerance to component variation. The difficulties of the image method were firmly put in the past, and even the need for prototypes became

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in their transfer function as an approximation to the ideal filter response and the result is called a Chebyshev approximation. This is the same Chebyshev approximation technique used by Cauer on image filters but follows the Darlington insertion-loss design method and uses slightly different

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unachievable with rational networks. For instance, the ideal prescribed function is often taken to be the unachievable lossless transmission in the passband, infinite attenuation in the stopband and a vertical transition between the two. However, the ideal function can be approximated with a

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Image filters continued to be used by designers long after the superior network synthesis techniques were available. Part of the reason for this may have been simply inertia, but it was largely due to the greater computation required for network synthesis filters, often needing a mathematical

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Campbell's sketch of the low-pass version of his filter from his 1915 patent showing the now ubiquitous ladder topology with capacitors for the ladder rungs and inductors for the stiles. Filters of more modern design also often adopt the same ladder topology as used by Campbell. It should be

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of these pairs of poles and zeroes. Any circuit which has the requisite poles and zeroes will also have the requisite response. Cauer pursued two related questions arising from this technique: what specification of poles and zeroes are realisable as passive filters; and what realisations are

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The filters designed by Campbell were named wave filters because of their property of passing some waves and strongly rejecting others. The method by which they were designed was called the image parameter method and filters designed to this method are called image filters. The image method

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The exact date Campbell produced each variety of filter is not clear. The work started in 1910, initially patented in 1917 (US1227113) and the full theory published in 1922, but it is known that Campbell's filters were in use by AT&T long before the 1922 date (Bray, p.62, Darlington,

2043:'s original paper on elliptic functions, published in Latin in 1829. In this paper Darlington was surprised to find foldout tables of the exact elliptic function transformations needed for Chebyshev approximations of both Cauer's image parameter, and Darlington's insertion-loss filters. 2218:

Control systems have a need for smoothing filters in their feedback loops with criteria to maximise the speed of movement of a mechanical system to the prescribed mark and at the same time minimise overshoot and noise induced motions. A key problem here is the extraction of

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Norton's mechanical filter together with its electrical equivalent circuit. Two equivalents are shown, "Fig.3" directly corresponds to the physical relationship of the mechanical components; "Fig.4" is an equivalent transformed circuit arrived at by repeated application of

318:, filters have been of crucial importance in a number of technological breakthroughs and have been the source of enormous profits for telecommunications companies. It should come as no surprise, therefore, that the early development of filters was intimately connected with 2206:(S/N) at the expense of pulse shape. Pulse shape, unlike many other applications, is unimportant in radar while S/N is the primary limitation on performance. The filters were introduced during WWII (described 1943) by Dwight North and are often eponymously referred to as " 2264:

enabled other active RC filter design topologies. Although active filter designs were commonplace at low frequencies, they were impractical at high frequencies where the amplifiers were not ideal; LC (and transmission line) filters were still used at radio frequencies.

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also had FDM systems. Separate pairs were used for the send and receive signals. The Siemens and GEC systems had six channels of telegraph in each direction, the AT&T system had twelve. All of these systems used electronic oscillators to generate a different

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By the 1890s electrical resonance was much more widely understood and had become a normal part of the engineer's toolkit. In 1891 Hutin and Leblanc patented an FDM scheme for telephone circuits using resonant circuit filters. Rival patents were filed in 1892 by

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So far, the investigation had been on the natural frequency of transient oscillation of a resonant circuit resulting from a sudden stimulus. More important from the point of view of filter theory is the behaviour of a resonant circuit when driven by an external

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to invent many improved forms. Carson and Zobel steadily demolished many of the old ideas. For instance the old telegraph engineers thought of the signal as being a single frequency and this idea persisted into the age of radio with some still believing that

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which is the best known of the filter topologies. It is for this reason that ladder topology is often referred to as Cauer topology (the forms used earlier by Foster are quite different) even though ladder topology had long since been in use in image filter

960:(or m-type filter). The particular problems Zobel was trying to address with these new forms were impedance matching into the end terminations and improved steepness of roll-off. These were achieved at the cost of an increase in filter circuit complexity. 691:

was required to prevent telegraph clicks being heard on the telephone line. From the 1920s onwards, telephone lines, or balanced lines dedicated to the purpose, were used for FDM telegraph at audio frequencies. The first of these systems in the UK was a

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At this early stage the idea of signal bandwidth, and hence the need for filters to match to it, was not fully understood; indeed, it was as late as 1920 before the concept of bandwidth was fully established. For early radio, the concepts of Q-factor,

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and then scaling the frequency and impedance and transforming the bandform to those actually required. This kind of approach, or similar, was already in use with image filters, for instance by Zobel, but the concept of a "reference filter" is due to

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Generally, for insertion-loss filters where the transmission zeroes and infinite losses are all on the real axis of the complex frequency plane (which they usually are for minimum component count), the insertion-loss function can be written as;

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significant. Tuning the poles and zeros of a 7th-order elliptic filter is not a simple exercise. Integrated circuits have made digital computation inexpensive, so now low frequency filtering is done with digital signal processors. Such

1375:. Here , and have associated energies corresponding to the kinetic, potential and dissipative heat energies, respectively, in a mechanical system and the already known results from mechanics could be applied here. Cauer determined the 2160:(1736–1813) studied waves on a string periodically loaded with weights. The device was never studied or used as a filter by either Lagrange or later investigators such as Charles Godfrey. However, Campbell used Godfrey's results by 2433:

to both conductors of the telephone line will not be heard at the telephone receiver which can only detect voltage differences between the conductors. The telegraph signal is typically recovered at the far end by connection to the

1885:. Darlington used this transformation in reverse to produce filters with a prescribed insertion-loss with non-ideal components. Such filters have the ideal insertion-loss response plus a flat attenuation across all frequencies. 1187:

Realisability (that is, which functions are realisable as real impedance networks) and equivalence (which networks equivalently have the same function) are two important questions in network synthesis. Following an analogy with

1607: 644:(i.e. combine) the wider bandwidth of telephone channels (as opposed to telegraph) without either an unacceptable restriction of speech bandwidth or a channel spacing so wide as to make the benefits of multiplexing uneconomic. 1293: 2052:

less commonly, by inductors, or in the case of a two-section filter, by mutual inductance. These are most useful where the design requirement is not too stringent, that is, moderate bandwidth, roll-off and passband ripple.

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at intervals along the line. Campbell found that as well as the desired improvements to the line's characteristics in the passband there was also a definite frequency beyond which signals could not be passed without great

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of the filters described in this article. Analogue filters are most often used in wave filtering applications, that is, where it is required to pass particular frequency components and to reject others from analogue

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in the US noted that a steel needle placed close to the discharge does not always magnetise in the same direction. They both independently drew the conclusion that there was a transient oscillation dying with time.

409:. Image filter theory grew out of transmission line theory and the design proceeded in a similar manner to transmission line analysis. For the first time filters could be produced that had precisely controllable 2538:

The terms wave filter and image filter are not synonymous, it is possible for a wave filter to not be designed by the image method, but in the 1920s the distinction was moot as the image method was the only one

322:. Transmission line theory gave rise to filter theory, which initially took a very similar form, and the main application of filters was for use on telecommunication transmission lines. However, the arrival of 1503:(1931), who worked with Cauer in the US prior to Cauer returning to Germany. A well known condition for realisability of a one-port rational impedance due to Cauer (1929) is that it must be a function of 534:. Wollaston was attempting to decompose water by electric shock but found that both hydrogen and oxygen were present at both electrodes. In normal electrolysis they would separate, one to each electrode. 510:, and is, in fact, an early form of capacitor. When a Leyden jar is discharged by allowing a spark to jump between the electrodes, the discharge is oscillatory. This was not suspected until 1826, when 1908:

provided this and a generalisation to all equal ripple problems. Elliptic filters are a general class of filter which incorporate several other important classes as special cases: Cauer filter (equal

896:. This exactly corresponds to the way the properties of a finite length of transmission line are derived from the theoretical properties of an infinite line, the image impedance corresponding to the 1025: 2165:

did not show the multiple passbands that are a characteristic of all distributed-element filters. The first electrical filters that were truly designed by distributed-element principles are due to

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Distributed-element filters are composed of lengths of transmission line that are at least a significant fraction of a wavelength long. The earliest non-electrical filters were all of this type.

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which starts with a given network and by applying the various electric circuit theorems predicts the response of the network. The term was first used with this meaning in the doctoral thesis of

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From the work of Heaviside (1887) it had become clear that the performance of telegraph lines, and most especially telephone lines, could be improved by the addition of inductance to the line.

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and some other English scientists tried to keep acoustic and electric terminology separate and promoted the term "syntony". However it was "resonance" that was to win the day. Blanchard, p.422

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for this class of function and proved that it was a necessary and sufficient condition (Cauer had only proved it to be necessary) and they extended the work to LC multiports. A theorem due to

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also played a part in the development of the theory, and proved to be a more useful idea than network terminals. The first milestone on the way to network synthesis was an important paper by

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converting sound to and from an electrical signal. It is no great leap from this view of the harmonic telegraph to the idea that speech can be converted to and from an electrical signal.

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and anti-resonators respectively. Both the loaded line and FDM were of great benefit economically to AT&T and this led to fast development of filtering from this point onwards.

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Heaviside's model of the transmission line. L, R, C and G in all three diagrams are the primary line constants. The infinitesimals δL, δR, δC and δG are to be understood as Lδ

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Lord Kelvin's model of the transmission line accounted for capacitance and the dispersion it caused. The diagram represents Kelvin's model translated into modern terms using

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are not usually associated with passive implementations but the concept can be found in a Wiener and Lee patent from 1935 which describes a filter consisting of a cascade of

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stipulates necessary and sufficient conditions for realisability: that the reactance must be algebraically increasing with frequency and the poles and zeroes must alternate.

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which is a meaning also used in this article. A 2-terminal network amounts to a single impedance (although it may consist of many elements connected in a complicated set of

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had been investigated by researchers from a very early stage, it was at first not widely understood by electrical engineers. Consequently, the much more familiar concept of

1499:) to be realisable if ideal transformers are not excluded. Realisability is only otherwise restricted by practical limitations on topology. This work is also partly due to 1167:. Darlington (1939), was also the first to tabulate values for network synthesis prototype filters, nevertheless it had to wait until the 1950s before the Cauer-Darlington 2260:

amplifiers; these filters replaced the bulky inductors with bulky and hot vacuum tubes. Transistors offered more power-efficient active filter designs. Later, inexpensive

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is to start with a required filter response and produce a network that delivers that response, or approximates to it within a specified boundary. This is the inverse of

397:. Network analysis was not yet powerful enough to provide the theory for more complex filters and progress was further hampered by a general failure to understand the 3278:

Cauer, W, "Die Verwirklichung der Wechselstromwiderstände vorgeschriebener Frequenzabhängigkeit" ("The realisation of impedances of specified frequency dependence"),

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embrace it by specifying a constant impedance for the combined filter. This problem was originally related to FDM telephony but frequently now arises in loudspeaker

450:, respectively. In particular they are used in combinations, such as LC, to mean, for instance, a network consisting only of inductors and capacitors. Z is used for 1507:

that is analytic in the right halfplane (σ>0), have a positive real part in the right halfplane and take on real values on the real axis. This follows from the

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problem. Kalman started an interest in state-space solutions, but according to Darlington this approach can also be found in the work of Heaviside and earlier.

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remove by filtering to that part of the noise spectrum which fell outside the passband. This too, was not generally accepted at first, notably being opposed by

206: 2355:) and can also be described as a one-port network. For networks of more than two terminals it is not necessarily possible to identify terminal pairs as ports. 1760:{\displaystyle \mathbf {} ={\begin{bmatrix}1&0\cdots 0\\T_{21}&T_{22}\cdots T_{2n}\\\cdot &\cdots \\T_{n1}&T_{n2}\cdots T_{nn}\end{bmatrix}}} 1832:

which made calculation of component values easy since the filter sections could not interact with each other and each section represented one term in the

3946: 2611:. The implication of finite polynomials is that the impedance, when realised, will consist of a finite number of meshes with a finite number of elements 1198: 330:

is still impractical, or at least, less cost effective. Wherever possible, and especially at low frequencies, analogue filters are now implemented in a

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The term "image parameter method" was coined by Darlington (1939) in order to distinguish this earlier technique from his later "insertion-loss method"

575:(1887) experimentally demonstrated the resonance phenomena by building two resonant circuits, one of which was driven by a generator and the other was 1916:), Chebyshev filter (ripple only in passband), reverse Chebyshev filter (ripple only in stopband) and Butterworth filter (no ripple in either band). 1828:. Butterworth discovered this filter independently of Cauer's work and implemented it in his version with each section isolated from the next with a 655:

from adjacent channels in any given channel. What is required is a much more sophisticated filter that has a flat frequency response in the required

3222: 2282: 1027:. Since the image impedance must be real in the passbands and imaginary in the stopbands according to image theory, there is a requirement that the 199: 3873: 2134:, especially for narrowband filtering applications. The signal exists as a mechanical acoustic wave while it is in the crystal and is converted by 426:. The mathematical bases of network synthesis were laid in the 1930s and 1940s. After World War II, network synthesis became the primary tool of 588:

As mentioned earlier, it was acoustic resonance that inspired filtering applications, the first of these being a telegraph system known as the "

1389: 1077: 795:(1854) found the correct mathematical description needed in his work on early transatlantic cables; he arrived at an equation identical to the 651:

far from the point of resonance. This means that if telephone channels are squeezed in side by side into the frequency spectrum, there will be

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with similar ideas, priority eventually being awarded to Pupin. However, no scheme using just simple resonant circuit filters can successfully

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At the turn of the century as telephone lines became available, it became popular to add telegraph onto telephone lines with an earth return

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This article is about the history and development of passive linear analogue filters used in electronics. For linear filters in general, see

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general (except for symmetrical networks) is not equal to the open-circuit impedance of the second and likewise for short-circuit impedances

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cancel in the passband and correspond in the stopband. The behaviour of the filter can be entirely defined in terms of the positions in the

2156:(1738–1822), for instance, constructed an apparatus with two tubes of different lengths which attenuated some frequencies but not others. 2339:

A terminal of a network is a connection point where current can enter or leave the network from the world outside. This is often called a

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and others. Its purpose was to simultaneously transmit a number of telegraph messages over the same line and represents an early form of

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signal: there is a sudden peak in the circuit's response when the driving signal frequency is at the resonant frequency of the circuit.

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sets the passband ripple height and the stopband loss and these two design requirements can be interchanged. The zeroes and poles of

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waves. While there are few applications for such devices in mechanics per se, they can be used in electronics with the addition of

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Darlington, S, "A history of network synthesis and filter theory for circuits composed of resistors, inductors, and capacitors",

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for each telegraph signal and required a bank of band-pass filters to separate out the multiplexed signal at the receiving end.

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eased the computation difficulty because sections could be isolated and iterative processes were not then generally necessary.

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The resonant frequency is very close to, but usually not exactly equal to, the natural frequency of oscillation of the circuit

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Helmholtz explained why the oscillation decayed but he had not explained why it occurred in the first place. This was left to

3838: 3762: 974: 775:(1827) who established that resistance in a wire is proportional to its length. The Ohm model thus included only resistance. 415: 2036:

A Chebyshev response simultaneously in the passband and stopband is possible, such as Cauer's equal ripple elliptic filter.

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resonant circuit, but that rapidly falls in response (much faster than 6 dB/octave) at the transition from passband to

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of an infinite chain of identical filter sections and then terminating the desired finite number of filter sections in the

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in 1866. Maxwell explained resonance mathematically, with a set of differential equations, in much the same terms that an

255:; the combining and later separation of multiple telephone conversations onto a single channel; the selection of a chosen 2459: 1905: 697: 647:

The basic technical reason for this difficulty is that the frequency response of a simple filter approaches a fall of 6

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equivalent to each other. The results of this work led Cauer to develop a new approach, now called network synthesis.

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terminated in a positive resistor R. No resistors within the network are necessary to realise the specified response.

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Incidentally, the harmonic telegraph directly suggested to Bell the idea of the telephone. The reeds can be viewed as

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Maurice Hutin, Maurice Leblanc, "Êtude sur les Courants Alternatifs et leur Application au Transport de la Force",

2827: 2147: 1794:, becoming ever closer to the ideal the higher the order of the polynomial. The first to address this problem was 1158:

The computational difficulty of the network synthesis method was addressed by tabulating the component values of a

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The development of network analysis needed to take place before network synthesis was possible. The theorems of

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Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS2000)

2385:

This image is from a later, corrected, US patent but patenting the same invention as the original French patent

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Analogue filters have played an important part in the development of electronics. Especially in the field of

3931: 2344: 1833: 1799: 1376: 1124: 873: 135: 3804: 1926: 3936: 3884: 2298: 2232: 1864: 1143: 828: 821: 809: 779:

noted that signals were delayed and elongated along a cable, an undesirable form of distortion now called

562: 531: 405: 147: 107: 3845: 454:, any 2-terminal combination of RLC elements and in some sections D is used for the rarely seen quantity 3909: 2261: 2220: 2203: 2177: 2157: 2120: 1803: 1535: 1311: 1112: 889: 627:

Hutin and Leblanc's multiple telegraph filter of 1891 showing the use of resonant circuits in filtering.

597: 522: 439: 2080:, the purpose being to remove the series resonant circuit from the body of the filter leaving a simple 1806:, initially applied to image filters, and not to the now well-known ladder realisation of this filter. 3831:

A History of Engineering and Science in the Bell System: Volume 5: Communications Sciences (1925–1980)

3846:"On discontinuities connected with the propagation of wave-motion along a periodically loaded string" 2579: 2422: 1380: 1307: 1189: 934: 921: 908: 849: 813: 648: 479: 471: 451: 2965:

James E. Brittain, "The Introduction of the Loading Coil: George A. Campbell and Michael I. Pupin",

2587: 2575: 2405: 2088: 1909: 1821: 1795: 1100: 904: 780: 669: 550: 546: 538: 3689:"An analysis of the factors which determine signal/noise discrimination in pulsed carrier systems" 3049: 2477: 2430: 2124: 1815: 915:. This gave the AT&T engineers a new insight into the way their filters were working and led 589: 475: 64: 2766: 2425:

connection which is common to all the telegraph lines on a route. Telephone lines are typically

438:

Throughout this article the letters R, L, and C are used with their usual meanings to represent

152: 2504:

independently made a similar discovery which he was not allowed to publish immediately because

3862: 3834: 3758: 3709: 3672: 3607: 3563: 3442: 3422: 3376: 3307: 3198: 2918: 2876: 2659: 2604: 2583: 2513: 2501: 2348: 2308: 2066: 1846: 1791: 1516: 1335: 1164: 1108: 1096: 1073: 967:(1930), and further developed by several other investigators including Piloty (1937–1939) and 953: 768: 673: 554: 346: 339: 319: 315: 308: 303: 299: 283: 245: 240:. Amongst their many applications are the separation of an audio signal before application to 233: 114: 46: 32: 2227:

with the specific application to anti-aircraft fire control analogue computers. Rudy Kalman (

2095:

recorders and players. Norton designed the filter in the electrical domain and then used the

3819: 3786: 3778: 3287: 3087: 2974: 2454:

At least, Ohm described the first model that was in any way correct. Earlier ideas such as

2153: 1528: 1508: 1481: 1159: 1139: 1116: 1092: 957: 912: 857: 805: 637: 398: 363: 335: 119: 69: 1845:

other via an ideal transformer perfectly matching them. Versions of this theory are due to

2674:

Joseph Henry, "On induction from ordinary electricity; and on the oscillatory discharge",

2443: 2418: 2181: 2166: 2071: 1894: 1829: 1825: 1168: 1104: 1055: 1028: 930: 893: 853: 845: 784: 684: 287: 84: 74: 3103: 2455: 1881:

that predicted the response of a filter using non-ideal components but all with the same

601: 266:

Passive linear electronic analogue filters are those filters which can be described with

907:, also working for AT&T, began to develop a new way of looking at signals using the 338:

in order to avoid the wound components (i.e. inductors, transformers, etc.) required by

3750: 2439: 2303: 2274: 2224: 2207: 2197: 2131: 852:

such as the loading coils. This naturally led Campbell (1910) to produce a filter with

804:. The mathematical model of the transmission line reached its fullest development with 572: 359: 295: 260: 3050:

Synthesis of reactance 4-poles which produce prescribed insertion loss characteristics

2343:

in the literature, especially the more mathematical, but is not to be confused with a

1979:

filter) or an odd (resulting in an symmetric filter) function of frequency. Zeroes of

956:(or k-type filter) to distinguish Campbell's filter from later types, notably Zobel's 3925: 3364: 2426: 2352: 2313: 2249: 2228: 2185: 1512: 1327: 1176: 1134: 1085: 1046: 968: 938: 796: 776: 736: 701: 633: 576: 427: 291: 256: 129: 79: 28: 1870:

find the driving point impedance from the short-circuit and open-circuit impedances,

743: 2373: 2293: 1878: 1081: 964: 836: 706: 677: 641: 527: 515: 511: 241: 2574:

A class of filters is a collection of filters which are all described by the same

1288:{\displaystyle \mathbf {} =s^{2}\mathbf {} +s\mathbf {} +\mathbf {} =s\mathbf {} } 3856: 2578:, for instance, the class of Chebyshev filters are all described by the class of 2032:

is a maximally flat approximation the result is a stopband maximally flat filter,

2018:

is a maximally flat approximation the result is a passband maximally flat filter,

1138:

isomorphism between them. He also found ladder realisations of the network using

948:

Several improvements were made to image filters and their theory of operation by

370:

corresponding to the energy in inductors, capacitors and resistors respectively.

3589: 3033: 2505: 2500:

Campbell has publishing priority for this invention but it is worth noting that

2404:

ratio of resonant frequency to bandwidth of a circuit. It arose as a measure of

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filter applications such as television, radar and high-speed data transmission.

1853:

network terminated at both ends. The procedure consists of the following steps:

841: 801: 792: 788: 731: 593: 566: 507: 447: 367: 252: 237: 185: 3790: 3091: 2138:

between the electrical and mechanical domains at the terminals of the crystal.

3522: 3375:, page 85, National Academy of Sciences (U.S.), National Academies Press 2004 3017: 2599:

A rational impedance is one expressed as a ratio of two finite polynomials in

2462:

were either incorrect, or inadequately described. See, for example. p.603 of;

2435: 2421:

with only a single conductor provided, the return path is achieved through an

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from a noisy background. An early paper on this was published during WWII by

1899:

Elliptic filters are filters produced by the insertion-loss method which use

302:

which may not have a passive topology – both of which may have the same

17: 2112: 2108: 1319: 878: 861: 844:. This was a result of the loading coils and the line capacitance forming a 772: 723: 713: 652: 467: 455: 431: 350: 271: 2831:, United States patent US0838545, filed 9 May 1894, issued 18 December 1906 860:

to the same ladder topology by replacing the inductors and capacitors with

3195:

A century of electrical engineering and computer science at MIT, 1882–1982

1781:), that is, all the transformed networks are equivalents of the original. 672:

and tuning sufficed. This was all to change with the developing theory of

2442:. The return path is via an earth connection as usual. This is a form of 2394: 1913: 1456:{\displaystyle Z_{\mathrm {p} }(s)={\frac {\det \mathbf {} }{s\,a_{11}}}} 925: 800:

limit was entirely explained in terms of the dispersion of the telegraph

664: 660: 656: 410: 279: 275: 1146:

are a common way of widening bandwidth without sacrificing selectivity.

3896:

Microwave Filters, Impedance-Matching Networks, and Coupling Structures

3291: 2116: 1873:

expand the driving point impedance into an LC (usually ladder) network.

693: 623: 490: 249: 2656:

Speech science primer: physiology, acoustics, and perception of speech

2025:

is a Chebyshev approximation the result is a reverse Chebyshev filter,

963:

A more systematic method of producing image filters was introduced by

924:(FM) transmission could be achieved with a smaller bandwidth than the 3910:"The use of continued fractions in the design of electrical networks" 558: 2978: 2973:, No. 1 (Jan., 1970), pp. 36–57, The Johns Hopkins University Press 2268:

Gradually, the low frequency active RC filter was supplanted by the

478:(which in turn, can be explained in terms of the even more familiar 2795:

Heinrich Hertz, "Electric waves", p.42, The Macmillan Company, 1893

1824:(1930) which are now recognised as being a special case of Cauer's 326:

techniques greatly enhanced the degree of control of the designer.

622: 349:

using mechanical components which filter mechanical vibrations or

506:, invented in 1746. The Leyden jar stores electricity due to its 3391:

J. Zdunek, "The network synthesis on the insertion-loss basis",

3133:

Theory and Design of Uniform and Composite Electric Wave Filters

2100: 2097:

correspondence of mechanical quantities to electrical quantities

494:

A 1915 example of an early type of resonant circuit known as an

2952:

Heaviside, O, "Electromagnetic Induction and its propagation",

2748:

William Grove, "An experiment in magneto–electric induction",

2429:

with two conductors per circuit. A telegraph signal connected

2039:

Darlington relates that he found in the New York City library

2011:

is a Chebyshev approximation the result is a Chebyshev filter,

1857:

determine the poles of the prescribed insertion-loss function,

739:

elements, but this was not the actual approach used by Kelvin.

2464:*John C. Shedd, Mayo D. Hershey, "The history of Ohm's law", 1820:

Butterworth filters are an important class of filters due to

2702:

Uber die Erhaltung der Kraft (On the Conservation of Force)

933:(who ironically, actually succeeded in reducing noise with 727:

Ohm's model of the transmission line was simply resistance.

38:

Filter used in signal processing on continuous-time signals

808:. Heaviside (1881) introduced series inductance and shunt 3852:, ser. 5, vol. 45, no. 275, pp. 356–363, April 1898. 3135:, Bell System Technical Journal, Vol. 2 (1923), pp. 1–46. 1511:

representation of these functions. Brune coined the term

848:, an effect that is only apparent on lines incorporating 3916:, volume 35, pages 463–498, 1929 (full text available). 3669:

Introduction to System Design Using Integrated Circuits

2875:, pp. 84–85, Institution of Electrical Engineers, 2001 2091:, around 1930, designed a mechanical filter for use on 1877:

Darlington additionally used a transformation found by

1534:

As for equivalence, Cauer found that the group of real

1084:(1930) and apparently arose out of a conversation with 1020:{\displaystyle \scriptstyle Z_{i}={\sqrt {Z_{o}Z_{s}}}} 502:

Resonance was noticed early on in experiments with the

3499:

Butterworth, S, "On the Theory of Filter Amplifiers",

3393:

Proceedings of the Institution of Electrical Engineers

2767:

On Mr Grove's experiment in magneto–electric induction

1630: 1345: 978: 820:

and the distributed-element parameters are called the

286:) and are designed to operate on continuously varying 3796:

Blanchard, J, "The History of Electrical Resonance",

2252:

that can have complex poles and zeros. In the 1950s,

1929: 1610: 1591:{\displaystyle \mathbf {} ^{T}\mathbf {} \mathbf {} } 1547: 1392: 1344: 1201: 977: 3078:Carson, J. R., "Notes on the Theory of Modulation" 2722:William Thomson, "On transient electric currents", 2099:to realise the filter using mechanical components. 2235:smoothing and prediction where it is known as the 2202:The purpose of matched filters is to maximise the 1960: 1759: 1590: 1455: 1367: 1287: 1019: 3891:, vol. 20, no. 4, pp. 405–414, October 1941. 3604:RF Design Guide: Systems, Circuits, and Equations 2676:Proceedings of the American Philosophical Society 1860:from that find the complex transmission function, 1618: 1612: 1584: 1578: 1573: 1567: 1556: 1550: 1430: 1424: 1281: 1275: 1264: 1258: 1250: 1244: 1233: 1227: 1209: 1203: 393:, these designs proved inadequate for successful 2658:, p.113, Lippincott Williams & Wilkins 2006 1420: 1175:largely superfluous. Furthermore, the advent of 937:) and was only finally settled with the work of 911:of Heaviside which in essence is working in the 3002:, p 62, Institute of Electrical Engineers, 2002 2582:. For realisable linear passive networks, the 2481:Siemen's ideas were not accepted. (Hunt, p.65.) 1480:to which the one-port is to be connected. From 1368:{\displaystyle \scriptstyle s=\sigma +i\omega } 3785:, vol. 50, iss. 5, pp. 848–855, May 1962 3781:, "Summary of the history of circuit theory", 3724:Jack L. Bowers, "R-C bandpass filter design", 3104:Transient Oscillation in Electric Wave Filters 2893:Die galvanische Kette, mathematisch bearbeitet 378:There are three main stages in the history of 3914:Bulletin of the American Mathematical Society 2130:In modern designs it is common to use quartz 680:are based, as explained in the next section. 207: 8: 3805:"Life and work of Wilhelm Cauer (1900–1945)" 3593:, filed 31 May 1929, issued 17 February 1931 3044: 3042: 3000:Innovation and the Communications Revolution 2812: 2810: 2397:is a dimensionless quantity enumerating the 787:(1853) established that this was due to the 696:installation between London and Manchester. 561:, and was also aware of the earlier work of 3325: 3323: 3321: 3319: 3157: 3155: 2654:L. J. Raphael, G. J. Borden, K. S. Harris, 2468:, pp.599–614, December 1913 ISSN 0161-7370. 2056:Other notable developments and applications 1123:, in which Foster introduces the idea of a 498:which uses Leyden jars for the capacitance. 3526:, filed 29 June 1929, issued 15 March 1932 3180: 3178: 3176: 2917:, p.333, Cambridge University Press, 1989 835:implemented this idea (1899) by inserting 557:in 1868 in connection with experiments on 294:which are not analogue in implementation ( 214: 200: 42: 3816:IEEE Transactions on Circuits and Systems 3671:, pp.252–254, New Age International 1992 3654: 3652: 3650: 3456: 3454: 3021:, filed 15 July 1915, issued 22 May 1917. 2512:, p.725, Cambridge University Press 2004 1983:correspond to zero loss and the poles of 1949: 1930: 1928: 1740: 1724: 1709: 1680: 1667: 1655: 1625: 1611: 1609: 1577: 1566: 1560: 1549: 1546: 1444: 1439: 1423: 1417: 1398: 1397: 1391: 1343: 1274: 1257: 1243: 1226: 1220: 1202: 1200: 1008: 998: 992: 983: 976: 466:Early filters utilised the phenomenon of 345:It is possible to design linear analogue 31:. For electronic filters in general, see 3545: 3543: 3541: 3477: 3475: 3419:Electronic filter analysis and synthesis 3350: 3348: 3346: 3344: 3145: 3143: 3141: 3127: 3125: 2641: 2639: 2637: 2635: 2633: 2631: 2629: 2627: 2570: 2568: 2283:Multiple feedback topology (electronics) 2070: 1111:) laid the groundwork. The concept of a 877: 742: 730: 722: 489: 3880:, vol. 98, no. 1, pp. 20–29, 2002. 3306:, p.5-20, Technical Publications, 2007 3029: 3027: 2623: 2329: 1519:states that any positive-real function 888:essentially consists of developing the 816:in all. This model is now known as the 458:, which is the inverse of capacitance. 45: 3644:, United States patent US2024900, 1935 3225:IEEE biography, retrieved 13 June 2009 2547: 2545: 2335: 2333: 2231:) later reformulated this in terms of 1999:can be set arbitrarily. The nature of 3885:"Electrical and mechanical analogies" 3562:, p.355, New Age International 1986, 1961:{\displaystyle {\frac {1}{1+JF^{2}}}} 1484:Cauer found that , and must all be 7: 3243:Foster, R M, "A Reactance Theorem", 3036:, Quadrivium, retrieved 26 June 2009 2123:. The filter was designed to have a 2003:determines the class of the filter; 1192:, Cauer formed the matrix equation, 3833:, AT&T Bell Laboratories, 1984 2508:was still ongoing. (Thomas H. Lee, 1987:correspond to transmission zeroes. 1975:is either an even (resulting in an 1802:. Independently, Cauer (1931) used 380:passive analogue filter development 3874:"On Shannon and Shannon's formula" 3800:, vol. 23, pp. 415–433, 1944. 3587:E. L. Norton, "Sound reproducer", 3439:CRC handbook of electrical filters 3193:Karl L. Wildes, Nilo A. Lindgren, 3054:Journal of Mathematics and Physics 1399: 791:present in the transmission line. 25: 3947:History of electronic engineering 3861:, Cornell University Press, 2005 2828:Multiple Telegraphy and Telephony 2237:linear-quadratic-Gaussian control 1473:is the complement of the element 783:but then called retardation, and 526:evidence of oscillation from the 416:intercity and international lines 414:division multiplexing for use on 3560:Network theory and filter design 1615: 1581: 1570: 1553: 1527:) can be realised as a lossless 1427: 1278: 1261: 1247: 1230: 1206: 3818:, vol. 31, pp. 3–13, 1984 3803:Cauer, E; Mathis, W; Pauli, R, 3606:, pp.81–84, Artech House, 1995 3437:John T. Taylor, Qiuting Huang, 3223:Arthur E. Kennelly, 1861 – 1939 606:frequency division multiplexing 395:frequency-division multiplexing 270:(linear); they are composed of 2576:class of mathematical function 2417:Telegraph lines are typically 1411: 1405: 232:are a basic building block of 1: 3889:Bell System Technical Journal 3798:Bell System Technical Journal 3302:Atul P. Godse, U. A. Bakshi, 3245:Bell System Technical Journal 3108:Bell System Technical Journal 3102:Carson, J R and Zobel, O J, " 1867:at the terminating resistors, 1183:Realisability and equivalence 1150:Image method versus synthesis 553:heard of the phenomenon from 268:linear differential equations 90:Optimum "L" (Legendre) filter 3757:, pp. 10–11, Springer, 2009 3704:Nadav Levanon, Eli Mozeson, 3060:, pp.257–353, September 1939 2510:Planar microwave engineering 2254:Sallen–Key active RC filters 1095:and others and the ideas of 514:in France, and later (1842) 470:to filter signals. Although 358:mechanical quantities, with 3755:Analog Filters using MATLAB 3732:, pages 131–133, April 1947 3640:N Wiener and Yuk-wing Lee, 3631:Fagen & Millman, p. 108 3304:Electronic Circuit Analysis 3110:, vol 2, July 1923, pp.1–29 2466:The Popular Science Monthly 2214:Filters for control systems 2142:Distributed-element filters 1901:elliptic rational functions 1863:from that find the complex 1306:matrices of, respectively, 943:thermal noise power formula 812:into the model making four 3968: 3791:10.1109/JRPROC.1962.288301 3535:Matthaei et al., pp.85–108 3092:10.1109/JRPROC.1922.219793 3034:"History of Filter Theory" 2943:, pp.139–140, Boston, 1925 2682:, pp.193–196, 17 June 1842 2280: 2195: 2148:Distributed-element filter 2145: 2064: 1892: 1813: 1486:positive-definite matrices 1129:Foster's reactance theorem 1065: 871: 797:conduction of a heat pulse 771:was probably described by 767:The earliest model of the 711: 26: 3421:, p.2, Artech House 1994 3280:Archiv für Elektrotechnic 3197:, p.157, MIT Press, 1985 2319:Network synthesis filters 2270:switched-capacitor filter 1068:Network synthesis filters 1062:Network synthesis filters 423:Network synthesis filters 263:and rejection of others. 142:Bridged T delay equaliser 58:Network synthesis filters 3894:Matthaei, Young, Jones, 3829:Fagen, M D; Millman, S, 3824:10.1109/TCS.1984.1085415 3811:, Perpignan, June, 2000. 3708:, p.24, Wiley-IEEE 2004 2756:, pp.184–185, March 1868 2704:, G Reimer, Berlin, 1847 952:. Zobel coined the term 898:characteristic impedance 719:Transmission line theory 3642:Electric network system 3441:, p.20, CRC Press 1997 3367:, "Sidney Darlington", 3086:, No 1, pp.57–64, 1922 2913:Thomas William Körner, 2777:, pp. 360–363, May 1868 2730:, pp.393–405, June 1853 2700:Hermann von Helmholtz, 1865:reflection coefficients 1834:Butterworth polynomials 1800:Butterworth polynomials 1377:driving point impedance 1144:double-tuned amplifiers 1125:driving point impedance 874:composite image filters 592:". Versions are due to 108:Image impedance filters 75:Elliptic (Cauer) filter 3850:Philosophical Magazine 3783:Proceedings of the IRE 2967:Technology and Culture 2895:, Riemann Berlin, 1827 2862:Lundheim (2002), p. 23 2771:Philosophical Magazine 2765:James Clerk Maxwell, " 2750:Philosophical Magazine 2724:Philosophical Magazine 2299:Composite image filter 2262:operational amplifiers 2085: 2078:a well known transform 1962: 1761: 1592: 1536:affine transformations 1457: 1369: 1298:where ,, and are the 1289: 1021: 890:transmission constants 884: 822:primary line constants 818:telegrapher's equation 764: 740: 728: 628: 532:William Hyde Wollaston 499: 298:), and there are many 148:Composite image filter 3693:RCA Labs. Rep. PTR-6C 3590:U.S. patent US1792655 3578:Matthaei et al., p.95 3523:U.S. patent 1,849,656 3417:Michael Glynn Ellis, 3408:Matthaei et al., p.83 3080:Procedures of the IRE 3018:U.S. patent 1,227,113 2873:History of telegraphy 2853:Blanchard, pp.426–427 2842:La Lumière Electrique 2804:Blanchard, pp.421–423 2786:Blanchard, pp.416–421 2713:Blanchard, pp.416–417 2691:Blanchard, pp.415–416 2580:Chebyshev polynomials 2204:signal-to-noise ratio 2158:Joseph-Louis Lagrange 2074: 1963: 1840:Insertion-loss method 1804:Chebyshev polynomials 1762: 1593: 1458: 1370: 1290: 1171:first came into use. 1022: 945:is well known today. 881: 746: 734: 726: 626: 598:Alexander Graham Bell 523:Hermann von Helmholtz 493: 125:General image filters 95:Linkwitz–Riley filter 3741:Darlington, pp.12–13 3518:Transmission network 3516:William R. Bennett, 3469:Cauer et al., pp.6–7 3369:Biographical Memoirs 3338:Cauer et al., pp.1,6 3013:Electric wave-filter 3011:George A, Campbell, 2825:M Hutin, M Leblanc, 2588:polynomial functions 2127:frequency response. 1927: 1608: 1545: 1390: 1381:Lagrange multipliers 1342: 1199: 1190:Lagrangian mechanics 975: 922:frequency modulation 909:operational calculus 814:distributed elements 569:is described today. 486:Electrical resonance 480:mechanical resonance 472:electrical resonance 452:electrical impedance 3898:, McGraw-Hill 1964. 3507:, 1930, pp. 536–541 3363:Irwin W. Sandberg, 3286:, pp.355–388, 1926 2586:must be a ratio of 2178:Transversal filters 2173:Transversal filters 1822:Stephen Butterworth 1796:Stephen Butterworth 1121:A Reactance Theorem 551:James Clerk Maxwell 539:Sir William Thomson 374:Historical overview 132:(constant R) filter 3844:Godfrey, Charles, 3622:Mason, pp. 409–410 3490:Darlington, pp.7–8 3292:10.1007/BF01662000 3269:Darlington, pp.4–6 3251:, pp.259–267, 1924 3213:Matthaei, pp.83–84 3119:Lundheim, pp.24–25 3069:Matthaei, pp.49–51 2989:Darlington, pp.4–5 2478:Werner von Siemens 2408:in radio receivers 2169:starting in 1927. 2086: 2061:Mechanical filters 1958: 1906:Sergei Schelkunoff 1816:Butterworth filter 1810:Butterworth filter 1757: 1751: 1588: 1453: 1365: 1364: 1285: 1017: 1016: 885: 765: 741: 729: 694:Siemens and Halske 674:transmission lines 629: 619:Early multiplexing 590:harmonic telegraph 584:Acoustic resonance 500: 476:acoustic resonance 401:nature of signals. 347:mechanical filters 320:transmission lines 316:telecommunications 300:electronic filters 65:Butterworth filter 49:electronic filters 3952:Electronic design 3883:Mason, Warren P, 3558:Vasudev K Aatre, 3501:Wireless Engineer 3395:, p.283, part 3, 3260:Cauer et al., p.1 3184:Cauer et al., p.4 3170:Cauer et al., p.6 2937:Electrical Papers 2871:K. G. Beauchamp, 2605:rational function 2584:transfer function 2502:Karl Willy Wagner 2349:transfer function 2309:Electronic filter 2182:all-pass sections 2067:Mechanical filter 1956: 1847:Sidney Darlington 1798:(1930) using his 1792:rational function 1517:Sidney Darlington 1451: 1379:by the method of 1336:complex frequency 1165:Sidney Darlington 1109:complex impedance 1097:Charles Steinmetz 1074:network synthesis 1056:crossover filters 1014: 954:constant k filter 858:band-pass filters 850:lumped components 769:transmission line 555:Sir William Grove 324:network synthesis 304:transfer function 290:. There are many 234:signal processing 224: 223: 115:Constant k filter 33:Electronic filter 16:(Redirected from 3959: 3766: 3748: 3742: 3739: 3733: 3722: 3716: 3702: 3696: 3685: 3679: 3665: 3659: 3658:Darlington, p.11 3656: 3645: 3638: 3632: 3629: 3623: 3620: 3614: 3600: 3594: 3592: 3585: 3579: 3576: 3570: 3556: 3550: 3547: 3536: 3533: 3527: 3525: 3514: 3508: 3497: 3491: 3488: 3482: 3479: 3470: 3467: 3461: 3460:Darlington, p.12 3458: 3449: 3435: 3429: 3415: 3409: 3406: 3400: 3389: 3383: 3361: 3355: 3352: 3339: 3336: 3330: 3329:Belevitch, p.850 3327: 3314: 3300: 3294: 3276: 3270: 3267: 3261: 3258: 3252: 3241: 3235: 3232: 3226: 3220: 3214: 3211: 3205: 3191: 3185: 3182: 3171: 3168: 3162: 3161:Belevitch, p.851 3159: 3150: 3147: 3136: 3129: 3120: 3117: 3111: 3100: 3094: 3076: 3070: 3067: 3061: 3048:S. Darlington, " 3046: 3037: 3031: 3022: 3020: 3009: 3003: 2996: 2990: 2987: 2981: 2963: 2957: 2950: 2944: 2931: 2925: 2915:Fourier analysis 2911: 2905: 2902: 2896: 2889: 2883: 2869: 2863: 2860: 2854: 2851: 2845: 2838: 2832: 2823: 2817: 2816:Blanchard, p.425 2814: 2805: 2802: 2796: 2793: 2787: 2784: 2778: 2763: 2757: 2746: 2740: 2739:Blanchard, p.417 2737: 2731: 2720: 2714: 2711: 2705: 2698: 2692: 2689: 2683: 2672: 2666: 2652: 2646: 2643: 2612: 2597: 2591: 2572: 2563: 2559: 2553: 2549: 2540: 2536: 2530: 2527: 2521: 2498: 2492: 2488: 2482: 2475: 2469: 2452: 2446: 2440:line transformer 2415: 2409: 2392: 2386: 2383: 2377: 2371: 2365: 2362: 2356: 2337: 2221:Gaussian signals 2154:William Herschel 1967: 1965: 1964: 1959: 1957: 1955: 1954: 1953: 1931: 1912:in passband and 1889:Elliptic filters 1826:elliptic filters 1770:is invariant in 1766: 1764: 1763: 1758: 1756: 1755: 1748: 1747: 1732: 1731: 1717: 1716: 1688: 1687: 1672: 1671: 1660: 1659: 1621: 1597: 1595: 1594: 1589: 1587: 1576: 1565: 1564: 1559: 1509:Poisson integral 1482:stability theory 1462: 1460: 1459: 1454: 1452: 1450: 1449: 1448: 1434: 1433: 1418: 1404: 1403: 1402: 1374: 1372: 1371: 1366: 1294: 1292: 1291: 1286: 1284: 1267: 1253: 1236: 1225: 1224: 1212: 1160:prototype filter 1140:Thomas Stieltjes 1117:Ronald M. Foster 1093:Gustav Kirchhoff 1078:network analysis 1029:poles and zeroes 1026: 1024: 1023: 1018: 1015: 1013: 1012: 1003: 1002: 993: 988: 987: 958:m-derived filter 913:frequency domain 806:Oliver Heaviside 638:John Stone Stone 602:Ernest Mercadier 399:frequency domain 364:potential energy 288:analogue signals 278:and, sometimes, 216: 209: 202: 120:m-derived filter 70:Chebyshev filter 43: 21: 3967: 3966: 3962: 3961: 3960: 3958: 3957: 3956: 3942:Analog circuits 3922: 3921: 3905: 3903:Further reading 3858:The Maxwellians 3855:Hunt, Bruce J, 3775: 3770: 3769: 3749: 3745: 3740: 3736: 3723: 3719: 3703: 3699: 3686: 3682: 3666: 3662: 3657: 3648: 3639: 3635: 3630: 3626: 3621: 3617: 3601: 3597: 3588: 3586: 3582: 3577: 3573: 3557: 3553: 3549:Darlington, p.8 3548: 3539: 3534: 3530: 3521: 3515: 3511: 3498: 3494: 3489: 3485: 3481:Darlington, p.7 3480: 3473: 3468: 3464: 3459: 3452: 3436: 3432: 3416: 3412: 3407: 3403: 3390: 3386: 3362: 3358: 3354:Darlington, p.9 3353: 3342: 3337: 3333: 3328: 3317: 3301: 3297: 3277: 3273: 3268: 3264: 3259: 3255: 3242: 3238: 3234:Darlington, p.4 3233: 3229: 3221: 3217: 3212: 3208: 3192: 3188: 3183: 3174: 3169: 3165: 3160: 3153: 3149:Darlington, p.5 3148: 3139: 3130: 3123: 3118: 3114: 3101: 3097: 3077: 3073: 3068: 3064: 3047: 3040: 3032: 3025: 3016: 3010: 3006: 2997: 2993: 2988: 2984: 2979:10.2307/3102809 2964: 2960: 2954:The Electrician 2951: 2947: 2934: 2932: 2928: 2912: 2908: 2904:Hunt, pp. 62–63 2903: 2899: 2890: 2886: 2870: 2866: 2861: 2857: 2852: 2848: 2839: 2835: 2824: 2820: 2815: 2808: 2803: 2799: 2794: 2790: 2785: 2781: 2764: 2760: 2747: 2743: 2738: 2734: 2721: 2717: 2712: 2708: 2699: 2695: 2690: 2686: 2673: 2669: 2653: 2649: 2644: 2625: 2620: 2615: 2598: 2594: 2573: 2566: 2560: 2556: 2550: 2543: 2537: 2533: 2528: 2524: 2499: 2495: 2489: 2485: 2476: 2472: 2463: 2453: 2449: 2444:phantom circuit 2416: 2412: 2393: 2389: 2384: 2380: 2372: 2368: 2363: 2359: 2338: 2331: 2327: 2290: 2285: 2275:digital filters 2256:were made with 2245: 2243:Modern practice 2216: 2200: 2194: 2175: 2167:Warren P. Mason 2150: 2144: 2132:crystal filters 2103:corresponds to 2084:ladder network. 2069: 2063: 2058: 2049: 1945: 1935: 1925: 1924: 1897: 1895:Elliptic filter 1891: 1842: 1830:valve amplifier 1818: 1812: 1787: 1776: 1750: 1749: 1736: 1720: 1718: 1705: 1702: 1701: 1696: 1690: 1689: 1676: 1663: 1661: 1651: 1648: 1647: 1636: 1626: 1606: 1605: 1548: 1543: 1542: 1494: 1478: 1471: 1440: 1435: 1419: 1393: 1388: 1387: 1340: 1339: 1216: 1197: 1196: 1185: 1169:elliptic filter 1152: 1105:Arthur Kennelly 1072:The essence of 1070: 1064: 1043: 1036: 1004: 994: 979: 973: 972: 931:Edwin Armstrong 894:image impedance 876: 870: 854:ladder topology 846:low-pass filter 829:George Campbell 785:Michael Faraday 721: 716: 685:phantom circuit 621: 586: 530:experiments of 488: 464: 376: 332:filter topology 309:continuous-time 220: 191: 190: 166: 158: 157: 153:mm'-type filter 110: 100: 99: 85:Gaussian filter 60: 48: 39: 36: 23: 22: 15: 12: 11: 5: 3965: 3963: 3955: 3954: 3949: 3944: 3939: 3934: 3932:Linear filters 3924: 3923: 3918: 3917: 3904: 3901: 3900: 3899: 3892: 3881: 3870: 3853: 3842: 3827: 3812: 3801: 3794: 3774: 3771: 3768: 3767: 3751:Lars Wanhammar 3743: 3734: 3717: 3697: 3680: 3660: 3646: 3633: 3624: 3615: 3602:Vizmuller, P, 3595: 3580: 3571: 3551: 3537: 3528: 3509: 3492: 3483: 3471: 3462: 3450: 3430: 3410: 3401: 3384: 3356: 3340: 3331: 3315: 3295: 3271: 3262: 3253: 3236: 3227: 3215: 3206: 3186: 3172: 3163: 3151: 3137: 3121: 3112: 3095: 3071: 3062: 3038: 3023: 3004: 2991: 2982: 2958: 2945: 2935:Heaviside, O, 2933:Brittain, p.39 2926: 2906: 2897: 2884: 2864: 2855: 2846: 2833: 2818: 2806: 2797: 2788: 2779: 2758: 2741: 2732: 2715: 2706: 2693: 2684: 2667: 2647: 2645:Lundheim, p.24 2622: 2621: 2619: 2616: 2614: 2613: 2592: 2564: 2554: 2541: 2531: 2522: 2493: 2483: 2470: 2447: 2410: 2387: 2378: 2366: 2357: 2328: 2326: 2323: 2322: 2321: 2316: 2311: 2306: 2304:Digital filter 2301: 2296: 2289: 2286: 2250:active filters 2244: 2241: 2225:Norbert Wiener 2215: 2212: 2198:matched filter 2196:Main article: 2193: 2192:Matched filter 2190: 2174: 2171: 2146:Main article: 2143: 2140: 2125:maximally flat 2065:Main article: 2062: 2059: 2057: 2054: 2048: 2045: 2034: 2033: 2026: 2019: 2012: 1969: 1968: 1952: 1948: 1944: 1941: 1938: 1934: 1893:Main article: 1890: 1887: 1875: 1874: 1871: 1868: 1861: 1858: 1841: 1838: 1814:Main article: 1811: 1808: 1786: 1783: 1774: 1768: 1767: 1754: 1746: 1743: 1739: 1735: 1730: 1727: 1723: 1719: 1715: 1712: 1708: 1704: 1703: 1700: 1697: 1695: 1692: 1691: 1686: 1683: 1679: 1675: 1670: 1666: 1662: 1658: 1654: 1650: 1649: 1646: 1643: 1640: 1637: 1635: 1632: 1631: 1629: 1624: 1620: 1617: 1614: 1603: 1599: 1598: 1586: 1583: 1580: 1575: 1572: 1569: 1563: 1558: 1555: 1552: 1492: 1476: 1469: 1464: 1463: 1447: 1443: 1438: 1432: 1429: 1426: 1422: 1416: 1413: 1410: 1407: 1401: 1396: 1363: 1360: 1357: 1354: 1351: 1348: 1296: 1295: 1283: 1280: 1277: 1273: 1270: 1266: 1263: 1260: 1256: 1252: 1249: 1246: 1242: 1239: 1235: 1232: 1229: 1223: 1219: 1215: 1211: 1208: 1205: 1184: 1181: 1177:active filters 1151: 1148: 1066:Main article: 1063: 1060: 1041: 1034: 1011: 1007: 1001: 997: 991: 986: 982: 872:Main article: 869: 866: 720: 717: 620: 617: 585: 582: 573:Heinrich Hertz 487: 484: 463: 460: 436: 435: 419: 402: 387:Simple filters 375: 372: 360:kinetic energy 296:digital filter 292:linear filters 261:radio receiver 222: 221: 219: 218: 211: 204: 196: 193: 192: 189: 188: 183: 178: 173: 167: 165:Simple filters 164: 163: 160: 159: 156: 155: 150: 145: 139: 136:Lattice filter 133: 127: 122: 117: 111: 106: 105: 102: 101: 98: 97: 92: 87: 82: 77: 72: 67: 61: 56: 55: 52: 51: 37: 24: 14: 13: 10: 9: 6: 4: 3: 2: 3964: 3953: 3950: 3948: 3945: 3943: 3940: 3938: 3937:Filter theory 3935: 3933: 3930: 3929: 3927: 3920: 3915: 3911: 3907: 3906: 3902: 3897: 3893: 3890: 3886: 3882: 3879: 3875: 3872:Lundheim, L, 3871: 3868: 3867:0-8014-8234-8 3864: 3860: 3859: 3854: 3851: 3847: 3843: 3840: 3836: 3832: 3828: 3825: 3821: 3817: 3813: 3810: 3806: 3802: 3799: 3795: 3792: 3788: 3784: 3780: 3777: 3776: 3772: 3764: 3760: 3756: 3752: 3747: 3744: 3738: 3735: 3731: 3727: 3721: 3718: 3715: 3714:0-471-47378-2 3711: 3707: 3706:Radar Signals 3701: 3698: 3694: 3690: 3687:D. O. North, 3684: 3681: 3678: 3677:81-224-0386-7 3674: 3670: 3667:B. S. Sonde, 3664: 3661: 3655: 3653: 3651: 3647: 3643: 3637: 3634: 3628: 3625: 3619: 3616: 3613: 3612:0-89006-754-6 3609: 3605: 3599: 3596: 3591: 3584: 3581: 3575: 3572: 3569: 3568:0-85226-014-8 3565: 3561: 3555: 3552: 3546: 3544: 3542: 3538: 3532: 3529: 3524: 3519: 3513: 3510: 3506: 3502: 3496: 3493: 3487: 3484: 3478: 3476: 3472: 3466: 3463: 3457: 3455: 3451: 3448: 3447:0-8493-8951-8 3444: 3440: 3434: 3431: 3428: 3427:0-89006-616-7 3424: 3420: 3414: 3411: 3405: 3402: 3398: 3394: 3388: 3385: 3382: 3381:0-309-08957-3 3378: 3374: 3370: 3366: 3365:Ernest S. Kuh 3360: 3357: 3351: 3349: 3347: 3345: 3341: 3335: 3332: 3326: 3324: 3322: 3320: 3316: 3313: 3312:81-8431-047-1 3309: 3305: 3299: 3296: 3293: 3289: 3285: 3281: 3275: 3272: 3266: 3263: 3257: 3254: 3250: 3246: 3240: 3237: 3231: 3228: 3224: 3219: 3216: 3210: 3207: 3204: 3203:0-262-23119-0 3200: 3196: 3190: 3187: 3181: 3179: 3177: 3173: 3167: 3164: 3158: 3156: 3152: 3146: 3144: 3142: 3138: 3134: 3131:Zobel, O. J., 3128: 3126: 3122: 3116: 3113: 3109: 3105: 3099: 3096: 3093: 3089: 3085: 3081: 3075: 3072: 3066: 3063: 3059: 3055: 3051: 3045: 3043: 3039: 3035: 3030: 3028: 3024: 3019: 3014: 3008: 3005: 3001: 2995: 2992: 2986: 2983: 2980: 2976: 2972: 2968: 2962: 2959: 2956:, 3 June 1887 2955: 2949: 2946: 2942: 2938: 2930: 2927: 2924: 2923:0-521-38991-7 2920: 2916: 2910: 2907: 2901: 2898: 2894: 2888: 2885: 2882: 2881:0-85296-792-6 2878: 2874: 2868: 2865: 2859: 2856: 2850: 2847: 2843: 2837: 2834: 2830: 2829: 2822: 2819: 2813: 2811: 2807: 2801: 2798: 2792: 2789: 2783: 2780: 2776: 2772: 2768: 2762: 2759: 2755: 2751: 2745: 2742: 2736: 2733: 2729: 2725: 2719: 2716: 2710: 2707: 2703: 2697: 2694: 2688: 2685: 2681: 2677: 2671: 2668: 2665: 2664:0-7817-7117-X 2661: 2657: 2651: 2648: 2642: 2640: 2638: 2636: 2634: 2632: 2630: 2628: 2624: 2617: 2610: 2606: 2603:, that is, a 2602: 2596: 2593: 2589: 2585: 2581: 2577: 2571: 2569: 2565: 2558: 2555: 2548: 2546: 2542: 2535: 2532: 2526: 2523: 2519: 2518:0-521-83526-7 2515: 2511: 2507: 2503: 2497: 2494: 2487: 2484: 2479: 2474: 2471: 2467: 2461: 2457: 2451: 2448: 2445: 2441: 2437: 2432: 2428: 2424: 2420: 2414: 2411: 2407: 2402: 2401: 2396: 2391: 2388: 2382: 2379: 2375: 2370: 2367: 2361: 2358: 2354: 2350: 2346: 2342: 2336: 2334: 2330: 2324: 2320: 2317: 2315: 2314:Linear filter 2312: 2310: 2307: 2305: 2302: 2300: 2297: 2295: 2292: 2291: 2287: 2284: 2279: 2276: 2271: 2266: 2263: 2259: 2255: 2251: 2242: 2240: 2238: 2234: 2230: 2229:Kalman filter 2226: 2222: 2213: 2211: 2209: 2208:North filters 2205: 2199: 2191: 2189: 2187: 2186:discrete-time 2183: 2179: 2172: 2170: 2168: 2163: 2159: 2155: 2149: 2141: 2139: 2137: 2133: 2128: 2126: 2122: 2118: 2114: 2110: 2106: 2102: 2098: 2094: 2090: 2089:Edward Norton 2083: 2079: 2073: 2068: 2060: 2055: 2053: 2047:Other methods 2046: 2044: 2042: 2037: 2031: 2027: 2024: 2020: 2017: 2013: 2010: 2006: 2005: 2004: 2002: 1998: 1994: 1990: 1986: 1982: 1978: 1974: 1950: 1946: 1942: 1939: 1936: 1932: 1923: 1922: 1921: 1917: 1915: 1911: 1907: 1902: 1896: 1888: 1886: 1884: 1880: 1872: 1869: 1866: 1862: 1859: 1856: 1855: 1854: 1850: 1848: 1839: 1837: 1835: 1831: 1827: 1823: 1817: 1809: 1807: 1805: 1801: 1797: 1793: 1785:Approximation 1784: 1782: 1780: 1773: 1752: 1744: 1741: 1737: 1733: 1728: 1725: 1721: 1713: 1710: 1706: 1698: 1693: 1684: 1681: 1677: 1673: 1668: 1664: 1656: 1652: 1644: 1641: 1638: 1633: 1627: 1622: 1604: 1601: 1600: 1561: 1541: 1540: 1539: 1537: 1532: 1530: 1526: 1522: 1518: 1514: 1513:positive-real 1510: 1506: 1502: 1498: 1491: 1487: 1483: 1479: 1472: 1445: 1441: 1436: 1414: 1408: 1394: 1386: 1385: 1384: 1382: 1378: 1361: 1358: 1355: 1352: 1349: 1346: 1337: 1333: 1329: 1325: 1321: 1317: 1313: 1309: 1305: 1301: 1271: 1268: 1254: 1240: 1237: 1221: 1217: 1213: 1195: 1194: 1193: 1191: 1182: 1180: 1178: 1172: 1170: 1166: 1161: 1156: 1149: 1147: 1145: 1141: 1136: 1135:Wilhelm Cauer 1132: 1130: 1126: 1122: 1118: 1114: 1110: 1106: 1102: 1098: 1094: 1089: 1087: 1086:Vannevar Bush 1083: 1079: 1075: 1069: 1061: 1059: 1057: 1051: 1048: 1047:complex plane 1044: 1037: 1030: 1009: 1005: 999: 995: 989: 984: 980: 970: 969:Wilhelm Cauer 966: 961: 959: 955: 951: 946: 944: 940: 939:Harry Nyquist 936: 932: 927: 923: 918: 914: 910: 906: 901: 900:of the line. 899: 895: 891: 880: 875: 868:Image filters 867: 865: 863: 859: 855: 851: 847: 843: 838: 837:loading coils 834: 830: 825: 823: 819: 815: 811: 807: 803: 798: 794: 790: 786: 782: 778: 777:Latimer Clark 774: 770: 763:respectively. 762: 758: 754: 750: 745: 738: 737:infinitesimal 733: 725: 718: 715: 710: 708: 703: 699: 695: 690: 686: 681: 679: 678:image filters 675: 671: 666: 662: 658: 654: 650: 645: 643: 639: 635: 634:Michael Pupin 625: 618: 616: 614: 609: 607: 603: 599: 595: 591: 583: 581: 578: 574: 570: 568: 564: 560: 556: 552: 548: 542: 540: 535: 533: 529: 524: 520: 517: 513: 509: 505: 497: 492: 485: 483: 481: 477: 473: 469: 461: 459: 457: 453: 449: 445: 441: 433: 429: 428:filter design 425: 424: 420: 417: 412: 408: 407: 406:Image filters 403: 400: 396: 392: 388: 385: 384: 383: 381: 373: 371: 369: 365: 361: 356: 352: 348: 343: 341: 337: 333: 327: 325: 321: 317: 312: 310: 305: 301: 297: 293: 289: 285: 281: 277: 273: 269: 264: 262: 258: 257:radio station 254: 251: 247: 243: 239: 236:much used in 235: 231: 230: 217: 212: 210: 205: 203: 198: 197: 195: 194: 187: 184: 182: 179: 177: 174: 172: 169: 168: 162: 161: 154: 151: 149: 146: 143: 140: 137: 134: 131: 130:Zobel network 128: 126: 123: 121: 118: 116: 113: 112: 109: 104: 103: 96: 93: 91: 88: 86: 83: 81: 80:Bessel filter 78: 76: 73: 71: 68: 66: 63: 62: 59: 54: 53: 50: 47:Linear analog 44: 41: 34: 30: 29:Linear filter 19: 18:Analog filter 3919: 3913: 3895: 3888: 3878:Telektronikk 3877: 3857: 3849: 3830: 3815: 3808: 3797: 3782: 3779:Belevitch, V 3773:Bibliography 3754: 3746: 3737: 3729: 3725: 3720: 3705: 3700: 3692: 3683: 3668: 3663: 3641: 3636: 3627: 3618: 3603: 3598: 3583: 3574: 3559: 3554: 3531: 3517: 3512: 3504: 3500: 3495: 3486: 3465: 3438: 3433: 3418: 3413: 3404: 3396: 3392: 3387: 3372: 3368: 3359: 3334: 3303: 3298: 3283: 3279: 3274: 3265: 3256: 3248: 3244: 3239: 3230: 3218: 3209: 3194: 3189: 3166: 3132: 3115: 3107: 3098: 3083: 3079: 3074: 3065: 3057: 3053: 3012: 3007: 2999: 2994: 2985: 2970: 2966: 2961: 2953: 2948: 2940: 2936: 2929: 2914: 2909: 2900: 2892: 2887: 2872: 2867: 2858: 2849: 2844:, 2 May 1891 2841: 2836: 2826: 2821: 2800: 2791: 2782: 2774: 2770: 2761: 2753: 2749: 2744: 2735: 2727: 2723: 2718: 2709: 2701: 2696: 2687: 2679: 2675: 2670: 2655: 2650: 2608: 2600: 2595: 2557: 2534: 2525: 2509: 2496: 2486: 2473: 2465: 2460:Peter Barlow 2456:Barlow's law 2450: 2413: 2399: 2398: 2390: 2381: 2374:Oliver Lodge 2369: 2360: 2340: 2294:Audio filter 2267: 2246: 2217: 2201: 2176: 2151: 2129: 2087: 2081: 2050: 2038: 2035: 2029: 2022: 2015: 2008: 2000: 1996: 1992: 1988: 1984: 1980: 1972: 1970: 1918: 1898: 1882: 1879:Hendrik Bode 1876: 1851: 1843: 1819: 1788: 1778: 1771: 1769: 1533: 1524: 1520: 1504: 1496: 1489: 1474: 1467: 1465: 1331: 1330:network and 1323: 1303: 1299: 1297: 1186: 1173: 1157: 1153: 1133: 1120: 1090: 1082:Yuk-Wing Lee 1071: 1052: 1039: 1032: 965:Hendrik Bode 962: 947: 935:wide-band FM 902: 886: 826: 766: 760: 756: 752: 748: 682: 646: 630: 610: 587: 571: 543: 536: 528:electrolysis 521: 516:Joseph Henry 512:Felix Savary 501: 465: 437: 421: 404: 386: 379: 377: 344: 328: 313: 265: 253:loudspeakers 226: 225: 40: 3726:Electronics 2891:Georg Ohm, 2506:World War I 2431:common-mode 2406:selectivity 2258:vacuum tube 2233:state-space 2136:transducers 2041:Carl Jacobi 905:John Carson 842:attenuation 810:conductance 793:Lord Kelvin 789:capacitance 670:selectivity 659:like a low- 613:transducers 594:Elisha Gray 567:RLC circuit 563:Henry Wilde 508:capacitance 448:capacitance 368:heat energy 355:transducers 311:) signals. 238:electronics 3926:Categories 3908:Fry, T C, 3839:0932764061 3763:0387927670 2618:References 2436:center tap 2419:unbalanced 2281:See also: 2121:resistance 2105:inductance 2093:phonograph 1977:antimetric 1501:Otto Brune 1316:inductance 1312:resistance 950:Otto Zobel 917:Otto Zobel 903:From 1920 862:resonators 781:dispersion 712:See also: 504:Leyden jar 496:Oudin coil 444:inductance 440:resistance 391:telegraphy 342:topology. 272:capacitors 186:RLC filter 144:(all-pass) 138:(all-pass) 2998:Bray, J, 2539:available 2325:Footnotes 2113:elastance 2109:stiffness 1734:⋯ 1699:⋯ 1694:⋅ 1674:⋯ 1642:⋯ 1362:ω 1353:σ 1338:operator 1320:elastance 1308:impedance 773:Georg Ohm 714:L-carrier 689:LC filter 676:on which 653:crosstalk 649:dB/octave 642:multiplex 600:(1870s), 468:resonance 462:Resonance 456:elastance 432:black box 411:passbands 334:which is 280:resistors 276:inductors 246:mid-range 227:Analogue 181:LC filter 176:RL filter 171:RC filter 2427:balanced 2395:Q factor 2288:See also 1914:stopband 1529:two-port 1119:(1924), 926:baseband 833:AT&T 702:AT&T 665:stopband 657:passband 351:acoustic 3397:vol 105 2971:Vol. 11 2347:of the 2162:analogy 2117:damping 1334:is the 1101:phasors 802:symbols 707:carrier 577:tunable 559:dynamos 340:passive 284:passive 250:tweeter 229:filters 3865: 3837: 3761: 3730:vol 20 3712: 3695:, 1943 3675: 3610: 3566: 3505:vol. 7 3445: 3425: 3399:, 1958 3379: 3373:vol 84 3310: 3284:vol 17 3201: 3084:vol 10 3058:vol 18 2921: 2879: 2775:vol 35 2754:vol 35 2662: 2562:design 2516: 2353:meshes 1971:where 1910:ripple 1602:where, 1466:where 1322:of an 1103:) and 941:whose 759:and Gδ 446:, and 434:terms. 336:active 248:, and 3249:vol 3 2941:vol 1 2728:vol 5 2680:vol 2 2458:from 2438:of a 2423:earth 2028:if 1/ 2021:if 1/ 687:. An 259:in a 3863:ISBN 3835:ISBN 3759:ISBN 3710:ISBN 3673:ISBN 3608:ISBN 3564:ISBN 3443:ISBN 3423:ISBN 3377:ISBN 3308:ISBN 3199:ISBN 2919:ISBN 2877:ISBN 2660:ISBN 2514:ISBN 2491:p.5) 2345:pole 2341:pole 2115:and 2101:Mass 1995:and 1488:for 1328:mesh 1318:and 1113:port 1038:and 755:, Cδ 751:, Rδ 700:and 636:and 366:and 242:bass 3820:doi 3787:doi 3288:doi 3106:", 3088:doi 3052:", 2975:doi 2769:", 2607:in 2210:". 2119:to 2111:to 2014:if 2007:if 1421:det 1031:of 831:at 698:GEC 3928:: 3912:, 3887:, 3876:, 3848:, 3807:, 3753:, 3728:, 3691:, 3649:^ 3540:^ 3520:, 3503:, 3474:^ 3453:^ 3371:, 3343:^ 3318:^ 3282:, 3247:, 3175:^ 3154:^ 3140:^ 3124:^ 3082:, 3056:, 3041:^ 3026:^ 3015:, 2969:, 2939:, 2809:^ 2773:, 2752:, 2726:, 2678:, 2626:^ 2567:^ 2544:^ 2520:.) 2332:^ 2107:, 2082:LC 1669:22 1657:21 1538:, 1477:11 1470:11 1446:11 1383:; 1314:, 1310:, 1058:. 824:. 596:, 547:AC 442:, 382:: 362:, 274:, 244:, 3869:. 3841:. 3826:. 3822:: 3793:. 3789:: 3765:. 3290:: 3090:: 2977:: 2609:s 2601:s 2400:q 2030:F 2023:F 2016:F 2009:F 2001:F 1997:J 1993:F 1989:J 1985:F 1981:F 1973:F 1951:2 1947:F 1943:J 1940:+ 1937:1 1933:1 1883:Q 1779:s 1777:( 1775:p 1772:Z 1753:] 1745:n 1742:n 1738:T 1729:2 1726:n 1722:T 1714:1 1711:n 1707:T 1685:n 1682:2 1678:T 1665:T 1653:T 1645:0 1639:0 1634:1 1628:[ 1623:= 1619:] 1616:T 1613:[ 1585:] 1582:T 1579:[ 1574:] 1571:A 1568:[ 1562:T 1557:] 1554:T 1551:[ 1525:s 1523:( 1521:Z 1505:s 1497:s 1495:( 1493:p 1490:Z 1475:A 1468:a 1442:a 1437:s 1431:] 1428:A 1425:[ 1415:= 1412:) 1409:s 1406:( 1400:p 1395:Z 1359:i 1356:+ 1350:= 1347:s 1332:s 1326:- 1324:n 1304:n 1302:x 1300:n 1282:] 1279:Z 1276:[ 1272:s 1269:= 1265:] 1262:D 1259:[ 1255:+ 1251:] 1248:R 1245:[ 1241:s 1238:+ 1234:] 1231:L 1228:[ 1222:2 1218:s 1214:= 1210:] 1207:A 1204:[ 1107:( 1099:( 1042:s 1040:Z 1035:o 1033:Z 1010:s 1006:Z 1000:o 996:Z 990:= 985:i 981:Z 761:x 757:x 753:x 749:x 661:Q 418:. 307:( 282:( 215:e 208:t 201:v 35:. 20:)

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Index