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.
The development of network analysis needed to take place before network synthesis was possible. The theorems of Gustav Kirchhoff and others and the ideas of Charles Steinmetz phasors and Arthur Kennelly complex impedance  laid the groundwork. Foster ,  A Reactance Theorem , in which Foster introduces the idea of a driving point impedance , 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. Foster's reactance theorem stipulates necessary and sufficient conditions for realisability: that the reactance must be algebraically increasing with frequency and the poles and zeroes must alternate. Wilhelm Cauer expanded on the work of Foster  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.
Cauer generalised this to any 2-element kind one-port network, finding there was an isomorphism between them. He also found ladder realisations [note 14] of the network using Thomas Stieltjes ' 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 double-tuned amplifiers are a common way of widening bandwidth without sacrificing selectivity. 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 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. 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.
Image filters continued to be designed up to that point and many remained in service into the 21st century. The computational difficulty of the network synthesis method was addressed by tabulating the component values of a prototype filter 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 Sidney Darlington. 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 largely superfluous. 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 Lagrangian mechanics , Cauer formed the matrix equation,. Here [ R ],[ L ] and [ D ] 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 driving point impedance by the method of Lagrange multipliers ;. From stability theory Cauer found that [ R ], [ L ] and [ D ] must all be positive-definite matrices for Z p s to be realisable if ideal transformers are not excluded.
Realisability is only otherwise restricted by practical limitations on topology. This follows from the Poisson integral representation of these functions. Brune coined the term positive-real 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 Sidney Darlington states that any positive-real function Z s can be realised as a lossless two-port terminated in a positive resistor R. No resistors within the network are necessary to realise the specified response.
As for equivalence, Cauer found that the group of real affine transformations ,. 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 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 rational function , becoming ever closer to the ideal the higher the order of the polynomial. The first to address this problem was Stephen Butterworth using his Butterworth polynomials.
Independently, Cauer used Chebyshev polynomials , initially applied to image filters, and not to the now well-known ladder realisation of this filter. Butterworth filters are an important class [note 15] of filters due to Stephen Butterworth  which are now recognised as being a special case of Cauer's elliptic filters. Butterworth discovered this filter independently of Cauer's work and implemented it in his version with each section isolated from the next with a valve amplifier 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 Butterworth polynomials.
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  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. 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 other via an ideal transformer perfectly matching them.
Versions of this theory are due to Sidney Darlington , 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. 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 network terminated at both ends. The procedure consists of the following steps:. Darlington additionally used a transformation found by Hendrik Bode that predicted the response of a filter using non-ideal components but all with the same Q. 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. Elliptic filters are filters produced by the insertion-loss method which use elliptic rational functions 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 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. Sergei Schelkunoff 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 ripple in passband and stopband , Chebyshev filter ripple only in passband , reverse Chebyshev filter ripple only in stopband and Butterworth filter no ripple in either band.
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;. Zeroes of F correspond to zero loss and the poles of F correspond to transmission zeroes. J sets the passband ripple height and the stopband loss and these two design requirements can be interchanged. The zeroes and poles of F and J can be set arbitrarily. The nature of F determines the class of the filter;. A Chebyshev response simultaneously in the passband and stopband is possible, such as Cauer's equal ripple elliptic filter.
Darlington relates that he found in the New York City library Carl Jacobi 's original paper on elliptic functions, published in Latin in 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.
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, 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. Edward Norton , around , designed a mechanical filter for use on phonograph recorders and players. Norton designed the filter in the electrical domain and then used the correspondence of mechanical quantities to electrical quantities to realise the filter using mechanical components. Mass corresponds to inductance , stiffness to elastance and damping to resistance. The filter was designed to have a maximally flat frequency response.
In modern designs it is common to use quartz crystal filters , especially for narrowband filtering applications. The signal exists as a mechanical acoustic wave while it is in the crystal and is converted by transducers between the electrical and mechanical domains at the terminals of the crystal. 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. William Herschel — , for instance, constructed an apparatus with two tubes of different lengths which attenuated some frequencies but not others. Joseph-Louis Lagrange — 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 analogy 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 did not show the multiple passbands that are a characteristic of all distributed-element filters. Mason starting in Transversal filters are not usually associated with passive implementations but the concept can be found in a Wiener and Lee patent from which describes a filter consisting of a cascade of all-pass sections.
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 discrete-time filter applications such as television, radar and high-speed data transmission. The filters were introduced during WWII described  by Dwight North and are often eponymously referred to as " North filters ".
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 Gaussian signals from a noisy background.
Practical Filter Design Challenges and Considerations for Precision ADCs
An early paper on this was published during WWII by Norbert Wiener with the specific application to anti-aircraft fire control analogue computers. Rudy Kalman Kalman filter later reformulated this in terms of state-space smoothing and prediction where it is known as the linear-quadratic-Gaussian control 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. LC filters at low frequencies become awkward; the components, especially the inductors, become expensive, bulky, heavy, and non-ideal. 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 active filters that can have complex poles and zeros.
In the s, Sallen—Key active RC filters were made with vacuum tube amplifiers; these filters replaced the bulky inductors with bulky and hot vacuum tubes. Transistors offered more power-efficient active filter designs. Later, inexpensive operational amplifiers 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.
Gradually, the low frequency active RC filter was supplanted by the switched-capacitor filter 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 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 digital filters 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 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.
From Wikipedia, the free encyclopedia. This article is about the history and development of passive linear analogue filters used in electronics. For linear filters in general see Linear filter.
Design of High Frequency Integrated Analogue Filters, Yichuang Sun, ,
For electronic filters in general see Electronic filter. Linear analog electronic filters Network synthesis filters. Image impedance filters. Constant k filter m-derived filter General image filters Zobel network constant R filter Lattice filter all-pass Bridged T delay equaliser all-pass Composite image filter mm'-type filter. Simple filters. See also: L-carrier.
However, electronic tunability ranges of the OTA There has been a great interest to resistorless designs in the based filters are restricted by the limited bandwidth of the literature due to the difficulties in the implementation of transconductance gain of the OTA. Furthermore, stability resistors in integrated circuits IC , such as high tolerance, conditions and the linearity of the OTA which depends on parasitic components, and large chip area.
In this paper, some Electronically tunable circuits attracted considerable trade-offs in the electronically tunable filters are investi- attention in the design of analog integrated circuits because gated. In addition, the tunability ranges of some first and tolerances of the electronic components in IC realization in second order OTA-C and OTA-RC filters are comparatively practice are unacceptably high and thus fine-tuning is examined. Moreover, an OTA-C all-pass filter circuit is necessary.
Tunability is more important in advanced presented. Operation of the presented all-pass filter is verified weak. The OTA-based circuits are most suitable from experimentally. Tunability The OTA is a commercially available active component which has been used widely in many applications.
More- over, the OTA is a simpler element compared to similar B. Then, the OTA circuits S. Minaei have been shown to be potentially advantageous for the Department of Electronics and Communications Engineering, Dogus University, Acibadem, Kadikoy , Istanbul, Turkey design of many first and second order high-frequency analog e-mail: sminaei dogus. For example, first order all-pass filter circuit in  includes two OTAs, one resistor and one capacitor. The O. However, the all-pass filters in [2, 3] are not in e-mail: cicekogl boun.
However, many circuits require additional and one triple output OTA. Also, many second order OTA- active element for cascading which increases the power based filters have been presented in the literature [5—8]. The consumption and chip area. In addition, the filter circuit reported in  employs off: Analog design without resistors obviously removes three OTAs. In  both resistor and OTA are used in the resistor thermal noise but introduces transistor flicker same circuit. Many other examples can be found in  and noise. However the amount of the noise is also dependent .
In addition, OTA has been an inspiration for developing on the circuit topology and needs further investigation new active elements. Recently, an element called current which is kept out of the scope of this study. An ideal OTA is an infinite bandwidth voltage- Grounded or floating capacitor trade-off: A trade-off controlled current source, with an infinite input and output exists between convenience for IC implementation and impedance.
The output current of an OTA is given as high-frequency operation. The transconductance parameter gm of the OTA vents roll-off at high frequencies. On the other hand a can be controlled by an external current. Assuming ideal grounded capacitor has less parasitic elements compared to OTA, routine analysis of the proposed circuit gives the the floating one in the IC implementation. Therefore, keeping the control current sufficiently higher than the amplitude of the output signal current may improve the linearity of the active element.
This may limit the tunability range of the filter. On the other hand, due to the parasitic capacitances of the active element parasitic poles are created and this alters the ideal transfer function of the circuit.
This may lead to two important problems. First, the order of the filter may be increased and some undesired high order terms appear in the transfer function. The effects of these undesired terms may be very strong on the tunability range for the filter. Secondly, the altered transfer function can be unstable for some passive element values.
There- fore, the stability conditions may also decrease the tunability range of the circuit. Therefore, these effects and xp is the parasitic pole frequency. Furthermore, are examined under two different subsections. In this sec- since the proposed circuit uses a dual-output OTA, both tion, two OTA-based second-order filters reported in  of the output currents of the OTA should be equal for and  are chosen for investigation. In other words the gains of the current mirrors employed at the output stage of 3.
However, in practice due to in second order filters the mismatches between current-mirror transistors, these gains can be considered as a1 and a2 instead of unity for The OTA-RC circuit in  is shown in Fig. Routine the first and second outputs of the OTA, respectively. Comparison of 5 and 3 shows that the circuit in 1 b has a more complicated dependency to the output current mismatches of the OTA. In order to operate the circuits in Fig. On the other hand, the limited bandwidth of the gm does not impose any restriction with respect to the stability as it is seen in 3 and 5.
In addition, employing a in their transfer functions. In order to operate the circuit as large number of OTAs in a circuit increases the number of a second-order filter the following conditions should be feedback loops and may deteriorate the circuit stability. Therefore, we apply Fig. The transfer function of this Routh-Hurwitz stability criteria on Eq. Vo gm1 gm2 In this way, we can find minimum allowable value of xp. This means that the frequency-dependent gm of the In this way, we can compare tunability range of the circuit OTA does not cause stability problem in the circuit for the in Fig.
As it can be in the simulations . Here the transconductance gain seen from Fig. The circuit of Fig. The result is shown in Fig. It can be transistors . In order to show the tunability assumptions. Dual and triple output OTAs of results of the gain and phase responses are depicted in Fig.
The phase and the gain responses are depicted in Fig. To illustrate the time-domain performance and linearity of the circuits in Fig. The filters are constructed with capacitor value of 60 pF and control current of 52 lA. The THD values for linearity comparison for various input Fig. The two OTAs in. The control current should be kept sufficiently higher It seems that the presented circuit can be useful when the than the output signal amplitude for this circuit, restricting trade-offs above are taken into consideration.
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In contrast to the tunability range. The limited bandwidth of the gm grounded capacitor advantage of the circuit in , the decreases tunability range of these first order filters as given floating capacitor of the presented filter bypasses a section of in Eq. On the other hand, the limited bandwidth cies.