HISTORY OF NETWORK SYNTHESISGeneral network synthesis as we know it now was rst developed for RLC circuits. Vacuum tube circuits were relatively simple circuitry (interstage circuits in vacuum tubeampliers, simple feedback in oscillators and regenerativeradioreceivers, etc.). There were no operational ampliers, transistors, gyrators, impedance converters, etc. Thus network synthesis theory for RLC circuits is an ancestor from which more general synthesis theories are descended. Since 1962, there have been important advances in RLC theory but they have been relatively few except for computer applications. Most of the important advances in circuit theory have concerned circuits of other kinds, such as active, switched, or digital circuits. Circuit Analysis determines characteristics of given circuits. Network Synthesis is the inverse. It determines circuitswith given (desired) characteristics. Development of circuit analysis was a prerequisite for the development of network synthesis. First the performance of the individual components had to be formulated (modeled, in modern terms). The result was Ohms law for resistors, current derivatives for inductors, voltage derivatives for capacitors, etc. Then the operation of interconnected components had to be formulated. For linear time-invariant RLC circuits, Kirchhoffs laws led to simultaneous linear differential equations in voltages and/or currents, with constant coefcients. A mathematical theoryof such differential equation was of course well known, and also applications to classical dynamics. An understanding of the differential equations as they apply to circuits included such concepts as steady state versus transient responses, the superposition theorem, Thevenins and Nortons theorems, and the reciprocity theorem. The use of complex numbers madecomputations of steady-state responses much easier, and the frequency variable was important in the analysis of transientsand the later development of synthesis techniques. The development of submarine cables had an important inuence on the development of circuit theory. Initially, transmission lines such as submarine cables were poorly understood.Misconceptions led to serious practical difculties. An important early contributor to an understanding of transmission lines, and other aspects of circuit theory, was Oliver Heaviside. A good biography of Heaviside was published recently by Nahin. Initially, Heavisides work was not generally accepted and he was pretty much forgotten until the latter 1920s. Inhis course on transients in tuned coupled circuits, in 1927 or 1928, G. W. Pierce did not include the Heaviside expansiontheorem. In 1929, people at MIT were reading a new book on Heavisides operational calculus. But George Campbell was always interested in Heaviside, and was a very important, very early (born 1870) contributor to the reduction to practice of Heavisides mathematics. An understanding of transmission lines brought in such concepts as propagation constants (attenuation and phase), matched impedances, andreections at nonmatching impedances. Lumped loading (initially proposed by Heaviside) was an ancestor of image parameter lters.

Grounded versus balanced lines and phantom circuits brought in the concepts Filter theory evolved rstfrom loaded lines. A line combines the effects of distributed inductance and capacitance. (Inductive loading increases the effective inductance, which is usually much lower than the capacitance.) Distributed loading increases the distributed inductance itself (e.g., by magnetic material wrapped around the conductors). Lumped loading approximatesthe effect by lumped inductors inserted in the line at uniformly spaced points. Heavyside proposed both distributed andlumped loading. His proposals were not generally accepted, because it was thought that the increased inductance wouldretard the transmission of signals. Actually, of course, it does retard the transmission, but this is only an increase in delay, with a decrease in delay distortion. The result is a lumped LC low-pass lter (with all its transmission zeros at innitefrequencies). It has a ladder conguration: alternate series branches in the line and shunt branches across the line.Campbell also disclosed all-pass lattice sections (1920 1922). Articial lines and attenuators were developed quite early and at least some used ladder congurations. The lter theories included the concept of lter sections, image impedances, and image attenuation and phase. In these terms, the lters derived from loaded lines may be described as tandemconnections of identical elementary sections. Since the image impedances were matched between sections, the image attenuation and phase of the overall lter were the sums of those of the individual sections. Otto Zobel invented simple lter sections with the same image impedances and pass- and stopbands as the original iterative lters, but with differentimage attenuation and phase characteristics. In particular, his m-derived sections gave sharper cutoffs, and attenuation peaks at arbitrary nite frequencies. By dividing his derived sections in the middle, he obtained image impedances at theends of lters which varied less over passband frequencies, thereby reducing terminal reection losses. At the same time as the development of lter theory, the general concept of two-ports was introduced in Germany and France. Correspondingimpedance, admittance, and chain matrices (matrices of frequency functions) were introduced and used to computecorresponding matrices of series, parallel, and cascade connected two-ports. There was now a scientic basis for developing a technique. for lter design. Additional lter sections were devised, not only for low-pass lters but also for high-pass, bandpass, and bandstop lters. For a while, a circuit theorist might seek to raise his visibility by inventing stillanother lter section. Over the years, a large body of art was added to the science, the result was a large assortment of practical lters meeting many practical requirements. In the 1930s a more systematic approach to image parameter lter theory was introduced by Bode (1934) and Piloty (1937 1938) and by Cauer. Here one looks at the relation between theopen- and short-circuit impedances of the whole lter and the image impedances and attenuation and phase functions.

MULTIPLE RESONANCE NETWORKS

The concept of multiple resonance networks generalizesthe concepts of the double resonance andtripleresonance networks found in the literature. In thesereferences, the networks are composed of lossless inductors, transformers, and capacitors, and designed in such a way that all the energy initially stored in a capacitor is, atthe closing of a switch, transferred to another capacitor in the circuit through a linear transient. At a certain instant, all the energy is concentrated in this other capacitor, andcan be used for some purpose. Networks with thisfunction can be found in pulsed power systems for physics research, where the energy is transferred from a largecapacitancecharged at low voltage to a small capacitance, that becomescharged at high voltage, with the sameenergy. Thedouble resonance case is long known, having been usedin early radio transmitters, electrotherapeutic devices, andin the generation of high voltages for demonstrations about electricity, as the Teslacoil. The usual system is shown in fig. 1.

The secondary capacitor is usually formed by the distributed capacitance of the secondary windings of thetransformer and the distributed capacitance of a terminal electrode. The relations for optimumdesign can be obtained as:

The constantsk and l are two positive integers, with l-kodd, that define the operation mode. They determine thetwonatural oscillation frequencies of the network with the switch closed, that appear as:

The triple resonance system was developed more recently, for pulsed power applications, as a solution for the insulation problem caused by the transformer in the double resonance system. As shown in fig. 2, a third inductor and another capacitor were added, with the result being that at the end of the energy transfer transient all the output voltage is over the third inductor, with no voltages or currents in the transformer.

Where k, l, and m are three successive integers with odd differences that define the operating mode, and also multiply a basic frequency to produce the three natural oscillation frequencies of the complete circuit with the switch closed:

With the formulas shown, complete energy transfer occurs / seconds after the closure of the switch.

CAUER NETWORKWilhelm Cauer (June 24, 1900 April 22, 1945) was a German mathematician and scientist. He is most noted for his work on the analysis and synthesis of electrical filters and his work marked the beginning of the field of network synthesis. Prior to his work, electronic filter design used techniques which were essentially approximation methods, as no exact answers were produced for the behaviour of the filter under real conditions. This required a certain amount of experience on the part of the designer to choose suitable sections to include in the design. Cauer placed the field on a firm mathematical footing, providing tools that could produce exact solutions to a given specification for the design of an electronic filter.

Network synthesis The major part of Cauer's legacy is his contribution to the network synthesis of passive networks. Indeed, he is considered the founder of the field. He treated network synthesis as being the inverse problem of network analysis. Whereas network analysis asks what is the response of a

given network, network synthesis on the other hand asks what are the networks that can produce a given desired response. Cauer solved this problem by comparing electrical quantities and functions to their mechanical equivalents. Then, realizing that they were completely analogous, applying the known Lagrangian mechanics to the problem. According to Cauer, there are three major tasks that network synthesis has to address. The first is the ability to determine whether a given transfer function is realizable as an impedance network. The second is to find the canonical (minimal) forms of these functions and the relationships (transforms) between different forms representing the same transfer function. Finally, it is not, in general, possible to find an exact finite-element solution to an ideal transfer function - such as zero attenuation at all frequencies below a given cutoff frequency and infinite attenuation above. The third task is therefore to find approximation techniques for achieving the desired responses. Initially, the work revolved around one-port impedances. The transfer function between a voltage and a current amounting to the expression for the impedance itself. A useful network can be produced by breaking open a branch of the network and calling that the output. Realizations Following on from Foster, Cauer generalized the relationship between the expression for the impedance of a one-port network and its transfer function. He discovered the necessary and sufficient condition for realizations of a one-port impedance. That is, those impedance expressions that could actually be built as a real circuit. In later papers he made generalizations to multiport networks. Transformation Cauer discovered that all solutions for the realization of a given impedance expression could be obtained from one given solution by a group of affine transformations. He generalized Foster's ladder realization to filters which included resistors (Foster's were reactance only) and discovered an isomorphism between all two-element kind networks. He identified the canonical forms of filter realization. That is, the minimal forms, which includes the ladder networks obtained by Stieltjes's continued fraction expansion.

CAUER NETWORK Cauer filters have equal maximum ripple in the passband and the stopband. The Cauer filter has a faster transition from the passband to the stopband than any other class of network synthesis filter. The term Cauer filter can be used interchangeably with elliptical filter, but the general case of elliptical filters can have unequal ripples in the passband and stopband. An elliptical filter in the limit of zero ripple in the passband is identical to a Chebyshev Type 1 filter. An elliptical filter in the limit of zero ripple in the stopband is identical to a Chebyshev Type 2 filter. An elliptical filter in the limit of zero ripple in both passbands is identical to a Butterworth filter. The filter is named after Wilhelm Cauer and the transfer function is based on elliptic rational functions. Driving point impedance:

Low-pass filter implemented as a ladder (Cauer) topology

The driving point impedance is a mathematical representation of the input impedance of a filter in the frequency domain using one of a number of notations such as Laplace transform(sdomain) or Fourier transform( j-domain). Treating it as a oneport network, the expression is expanded using continued fraction or partial fraction expansions. The resulting expansion is transformed into a network (usually a ladder network) of electrical elements. Taking an output from the end of this network, so realised, will transform it into a two-port network filter with the desired transfer function. Not every possible mathematical function for driving point impedance can be realized using real electrical components. Wilhelm Cauer (following on from R. M. Foster ) did much of the early work on what mathematical functions could be realised and in which filter topologies. The ubiquitous ladder topology of filter design is named after Cauer. There are a number of canonical forms of driving point impedance that can be used to express all (except the simplest) realizable impedances. The most well known ones are:

Cauer's first form of driving point impedance consists of a ladder of shunt capacitors and series inductors and is most useful for low-pass filters. Cauer's second form of driving point impedance consists of a ladder of series capacitors and shunt inductors and is most useful for high-pass filters.

Foster's first form of driving point impedance consists of parallel connected LC resonators and is most useful for band-pass filters. Foster's second form of driving point impedance consists of series connected LC antiresonators and is most useful for band-stop filters.

ELLIPTIC FILTER An elliptic filter (also known as a Cauer filter, named after Wilhelm Cauer) is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband. The amount of ripple in each band is independently adjustable, and no other filter of equal order can have a faster transition in gain between the passband and the stopband, for the given values of ripple (whether the ripple is equalized or not). Alternatively, one may give up the ability to independently adjust the passband and stopband ripple, and instead design a filter which is maximally insensitive to component variations. As the ripple in the stopband approaches zero, the filter becomes a type I Chebyshev filter. As the ripple in the passband approaches zero, the filter becomes a type II Chebyshev filter and finally, as both ripple values approach zero, the filter becomes a Butterworth filter. The gain of a lowpass elliptic filter as a function of angular frequency is given by:

where Rn is the nth-order elliptic rational function (sometimes known as a Chebyshev rational function) and 0 is the cutoff frequency is the ripple factor is the selectivity factor The value of the ripple factor specifies the passband ripple, while the combination of the ripple factor and the selectivity factor specify the stopband ripple. Properties of Elliptic Filters

In the passband, the elliptic rational function varies between zero and unity. The passband of the gain therefore will vary between 1 and .

In the stopband, the elliptic rational function varies between infinity and the discrimination factor Ln which is defined as:

The gain of the stopband therefore will vary between 0 and

.

In the limit of the elliptic rational function becomes a Chebyshev polynomial, and therefore the filter becomes a Chebyshev type I filter, with ripple factor Since the Butterworth filter is a limiting form of the Chebyshev filter, it follows that in the limit of , becomes a Butterworth filter and such that the filter

In the limit of , and such that 0 = 1 and becomes a Chebyshev type II filter with gain

, the filter

Poles and zeroes

Log of the absolute value of the gain of an 8th order elliptic filter in complex frequency space (s=+j) with =0.5, =1.05 and 0 = 1. The white spots are poles and the black spots are zeroes. There are a total of 16 poles and 8 double zeroes. What appears to be a single pole and zero near the transition region is actually four poles and two double zeroes as shown in the expanded view below. In this image, black corresponds to a gain of 0.0001 or less and white corresponds to a gain of 10 or more.

An expanded view in the transition region of the above image, resolving the four poles and two double zeroes.

The zeroes of the gain of an elliptic filter will coincide with the poles of the elliptic rational function, which are derived in the article on elliptic rational functions. The poles of the gain of an elliptic filter may be derived in a manner very similar to the derivation of the poles of the gain of a type I Chebyshev filter. For simplicity, assume that the cutoff frequency is equal to unity. The poles (pm) of the gain of the elliptical filter will be the zeroes of the denominator of the gain. Using the complex frequency s = + j this means that:

Defining js = cd(w,1 / ) where cd() is the Jacobi elliptic cosine function and using the definition of the elliptic rational functions yields:

where K = K(1 / ) and Kn = K(1 / Ln). Solving for w

where the multiple values of the inverse cd() function are made explicit using the integer index m. The poles of the elliptic gain function are then:

As is the case for the Chebyshev polynomials, this may be expressed in explicitly complex form

where n is a function of and and xm are the zeroes of the elliptic rational function. n is expressible for all n in terms of Jacobi elliptic functions, or algebraically for some orders, especially orders 1,2, and 3. For orders 1 and 2 we have

where

The algebraic expression for 3 is rather involved. The nesting property of the elliptic rational functions can be used to build up higher order expressions for n:

where Lm = Rm(,).

Minimum Q-factor elliptic filters

The normalized Q-factors of the poles of an 8-th order elliptic filter with =1.1 as a function of ripple factor . Each curve represents four poles, since complex conjugate pole pairs and positive-negative pole pairs have the same Qfactor. (The blue and cyan curves nearly coincide). The Q-factor of all poles are simultaneously minimized at Qmin=1/Ln=0.02323...

Elliptic filters are generally specified by requiring a particular value for the passband ripple, stopband ripple and the sharpness of the cutoff. This will generally specify a minimum value of the filter order which must be used. Another design consideration is the sensitivity of the gain function to the values of the electronic components used to build the filter. This sensitivity is inversely proportional to the quality factor (Q-factor) of the poles of the transfer function of the filter. The Q-factor of a pole is defined as:

and is a measure of the influence of the pole on the gain function. For an elliptic filter, it happens that, for a given order, there exists a relationship between the ripple factor and selectivity factor which simultaneously minimizes the Q-factor of all poles in the transfer function:

This results in a filter which is maximally insensitive to component variations, but the ability to independently specify the passband and stopband ripples will be lost. For such filters, as the order increases, the ripple in both bands will decrease and the rate of cutoff will increase. If one decides to use a minimum-Q elliptic filter in order to achieve a particular minimum ripple in the filter bands along with a particular rate of cutoff, the order needed will generally be greater than the order one would otherwise need without the minimum-Q restriction. An image of the

absolute value of the gain will look very much like the image in the previous section, except that the poles are arranged in a circle rather than an ellipse. They will not be evenly spaced and there will be zeroes on the axis, unlike the Butterworth filter, whose poles are also arranged in a circle. Comparison with other linear filters Here is an image showing the elliptic filter next to other common kind of filters obtained with the same number of coefficients:

As is clear from the image, elliptic filters are sharper than all the others, but they show ripples on the whole bandwidth. Cauer I & II Realizations for LC Networks Assuming we have the following general equation for Z(s), Z(s) = a5s5 + a3s3 + a1s (1.0) B4s4+b2s+b0 The polynomials numerator and denominator are arranged in a descending power of s. If Z(s) is not in a factor form, factor out and use continuous partial fraction to obtain the values of the elements.

Network used to represent both Cauer I and Cauer II realizations

In a LC network, if we use Cauer I to obtain the input impedance Z(s), the inductor will be found in theseries branches of the network. The capacitors will be in the shunt branches of the network. Besides, the first element of the network will be a series inductor if Z() =; otherwise, if Z() = 0 the first element will be a shunt capacitor. If the first element is a series inductor, we use Z(s) to obtain our Cauer realization. If the first element is a shunt capacitor, we use Y(s) to obtain our realization. In a Cauer II network, capacitors are placed in the series branches and inductors shunting to ground the network. If the first element is a series capacitor, Z(0) =, the circuit will be built in Z(s). On the other hand, if the first element is a shunt inductor, Z(0) = 0, the expansion is done on Y(s).Also, the behavior of the input impedance at high frequencies of s is fundamental for the last element of the circuit. For example, when Z()= the last element is an inductor. At the same time, when Z()=0 the last element is a capacitor. General equation of Cauer II is given in (2.0), Y(s) = b4s4 + b2s2 + b0 A5s5 + b3s3 + a1s Synthesis of RC Networks In a first Cauer canonical form for RC impedance, we arrange ZRC(s) so that the numerator and denominator polynomials appear in ascending order of s. To obtain the values of the elements in the network we expand the general equation in continued-fraction forms. The second Cauer canonical form for RC impedance is obtained after arranging the numerator and denominator polynomials in ascending order of s. We expand the resulting function in a continued fraction to obtain the values of the components in the network. (2.0)

FOSTER NETWORKRonald Martin Foster (3 October 1896 2 February 1998), was a Bell Labs mathematician whose work was of significance regarding electronic filters for use on telephone lines. He published an important paper, A Reactance Theorem, which quickly inspired Wilhelm Cauer to

begin his program of network synthesis filters which put the design of filters on a firm mathematical footing. He is also known for the Foster census of cubic symmetric graphs and the 90-vertex cubic symmetric Foster graph. Foster's reactance theorem Foster's reactance theorem is an important theorem in the fields of electrical network analysis and synthesis. The theorem states that the reactance of a passive, lossless two-terminal (one-port) network always monotonically increases with frequency. The proof of the theorem was first presented by Ronald Martin Foster in 1924. History The theorem was developed at American Telephone & Telegraph as part of ongoing investigations into improved filters for telephone multiplexing applications. This work was commercially important, large sums of money could be saved by increasing the number of telephone conversations that could be carried on one line. The theorem was first published by Campbell in 1922 but without a proof.[11] Great use was immediately made of the theorem in filter design, it appears prominently, along with a proof, in Zobel's landmark paper of 1923 which summarized the state of the art of filter design at that time. Foster published his paper the following year which includes his canonical realization forms. Cauer in Germany grasped the importance of Foster's work and used it as the foundation of network synthesis. Amongst Cauer's many innovations was to extend Foster's work to all 2element-kind networks after discovering an isomorphism between them. Cauer was interested in finding the conditions for realisability of a rational one-port network from its polynomial function (the condition of being a Foster network is not a necessary and sufficient condition, for that, see positive-real function) and the reverse problem of which networks were equivalent, that is, had the same polynomial function. Both of these were important problems in network theory and filter design Explanation Reactance is the imaginary part of the complex electrical impedance. The specification that the network must be passive and lossless implies that there are no resistors (lossless), or amplifiers or energy sources (passive) in the network. The network consequently must consist entirely of inductors and capacitors and the impedance will be purely an imaginary number with zero real part. Other than that, the theorem is quite general, in particular, it applies to distributed element circuits although Foster formulated it in terms of discrete inductors and capacitors. Foster's theorem applies equally to the admittance of a network, that is the susceptance (imaginary part of admittance) of a passive, lossless one-port monotonically increases with frequency. This result may seem counterintuitive since admittance is the reciprocal of impedance, but is easily proved. If an impedance,

where, is impedance is reactance is the imaginary unit then the admittance is given by

where, is admittance is susceptance If X is monotonically increasing with frequency then 1/X must be monotonically decreasing. 1/X must consequently be monotonically increasing and hence it is proved that B is increasing also. It is often the case in network theory that a principle or procedure apply equally to impedance or admittance as they do here. It is convenient in these circumstances to use the concept of immittance which can mean either impedance or admittance. The mathematics are carried out without stating which it is or specifying units until it is desired to calculate a specific example. Examples

Plot of the reactance of an inductor against frequency

Plot of the reactance of a capacitor against frequency

Plot of the reactance of a series LC circuit against frequency

Plot of the reactance of a parallel LC circuit against frequency

The following examples illustrate this theorem in a number of simple circuits. Inductor The impedance of an inductor is given by,

is inductance is angular frequency so the reactance is,

which by inspection can be seen to be monotonically (and linearly) increasing with frequency. Capacitor The impedance of a capacitor is given by,

is capacitance so the reactance is,

which again is monotonically increasing with frequency. The impedance function of the capacitor is identical to the admittance function of the inductor and vice versa. It is a general

result that the dual of any immittance function that obeys Foster's theorem will also follow Foster's theorem. Series resonant circuit A series LC circuit has an impedance that is the sum of the impedances of an inductor and capacitor,

At low frequencies the reactance is dominated by the capacitor and so is large and negative. This monotonically increases towards zero (the magnitude of the capacitor reactance is becoming smaller). The reactance passes through zero at the point where the magnitudes of the capacitor and inductor reactances are equal (the resonant frequency) and then continues to monotonically increase as the inductor reactance becomes progressively dominant. Parallel resonant circuit A parallel LC circuit is the dual of the series circuit and hence its admittance function is the same form as the impedance function of the series circuit,

The impedance function is,

At low frequencies the reactance is dominated by the inductor and is small and positive. This monotonically increases towards a pole at the anti-resonant frequency where the susceptance of the inductor and capacitor are equal and opposite and cancel. Past the pole the reactance is large and negative and increasing towards zero where it is dominated by the capacitance.

Poles and zeroes

Plot of the reactance of Foster's first form of canonical driving point impedance showing the pattern of alternating poles and zeroes. Three anti-resonators are required to realise this impedance function.

A consequence of Foster's theorem is that the poles and zeroes of any passive immittance function must alternate with increasing frequency. After passing through a pole the function will be negative and is obliged to pass through zero before reaching the next pole if it is to be monotonically increasing. Not all networks obey Foster's theorem, those that do may be referred to as Foster networks and those that do not are non-Foster networks. A circuit containing an amplifier may well not be a Foster network. In particular, it is possible to simulate negative capacitors and inductors with negative impedance converter circuits. These circuits will have an immittance function of frequency with a negative slope. With the addition of a scaling factor, the poles and zeroes of an immittance function completely determine the frequency characteristics of a Foster network. Two Foster networks that have identical poles and zeroes will be equivalent circuits in the sense that their immittance functions will be identical. Another consequence of Foster's theorem is that the plot of a Foster immittance function on a Smith chart must always travel around the chart in a clockwise direction with increasing frequency.

Realisation

Foster's first form of canonical driving point impedance realisation. If the polynomial function has a pole at =0 one of the LC sections will reduce to a single capacitor. If the polynomial function has a pole at = one of the LC sections will reduce to a single inductor. If both poles are present then two sections reduce to a series LC circuit.

Foster's second form of canonical driving point impedance realisation. If the polynomial function has a zero at =0 one of the LC sections will reduce to a single capacitor. If the polynomial function has a zero at = one of the LC sections will reduce to a single inductor. If both zeroes are present then two sections reduce to a parallel LC circuit.

A one-port passive immittance consisting of discrete elements (that is, not a distributed element circuit) is described as rational in that in can be represented as a rational function of s,

where, is immittance are polynomials with real, positive coefficients is the Laplace operator, which can be replaced with AC signals. when dealing with steady-state

This is sometimes referred to as the driving point impedance because it is the impedance at the place in the network at which the external circuit is connected and "drives" it with a signal.

Foster in his paper describes how such a lossless rational function may be realised in two ways. Foster's first form consists of a number of series connected parallel LC circuits. Foster's second form of driving point impedance consists of a number of parallel connected series LC circuits. The realisation of the driving point impedance is by no means unique. Foster's realisation has the advantage that the poles and/or zeroes are directly associated with a particular resonant circuit, but there are many other realisations. Perhaps the most well known is Cauer's ladder realisation from filter design.

PASSIVE FILTERS BESSEL FILTERIn electronics and signal processing, a Bessel filter is a type of linear filter with a maximally flat group delay (maximally linear phase response). Bessel filters are often used in audio crossover systems. Analog Bessel filters are characterized by almost constant group delay across the entire passband, thus preserving the wave shape of filtered signals in the passband. The filter's name is a reference to Friedrich Bessel, a German mathematician (17841846), who developed the mathematical theory on which the filter is based. The filters are also called BesselThomson filters in recognition of W. E. Thomson, who worked out how to apply Bessel functions to filter design. The transfer function

A plot of the gain and group delay for a fourth-order low pass Bessel filter. Note that the transition from the pass band to the stop band is much slower than for other filters, but the group delay is practically constant in the passband. The Bessel filter maximizes the flatness of the group delay curve at zero frequency.

A Bessel low-pass filter is characterized by its transfer function:

where n(s) is a reverse Bessel polynomial from which the filter gets its name and 0 is a frequency chosen to give the desired cut-off frequency. The filter has a low-frequency group delay of 1 / 0. Bessel polynomials

The roots of the third-order Bessel polynomial are the poles of filter transfer function in the s plane, here plotted as crosses.

The transfer function of the Bessel filter is a rational function whose denominator is a reverse Bessel polynomial, such as the following:

The reverse Bessel polynomials are given by:

where

Example

Gain plot of the third-order Bessel filter, versus normalized frequency

Group delay plot of the third-order Bessel filter, illustrating flat unit delay in the passband

The transfer function for a third-order (three-pole) Bessel low-pass filter, normalized to have unit group delay, is

The roots of the denominator polynomial, the filter's poles, include a real pole at s = 2.3222, and a complex-conjugate pair of poles at s = 1.8389 j1.7544, plotted above. The numerator 15 is chosen to give a gain of 1 at DC (at s = 0). The gain is then

The phase is

The group delay is

The Taylor series expansion of the group delay is

Note that the two terms in 2 and 4 are zero, resulting in a very flat group delay at = 0. This is the greatest number of terms that can be set to zero, since there are a total of four coefficients in the third order Bessel polynomial, requiring four equations in order to be defined. One equation specifies that the gain be unity at = 0 and a second specifies that the gain be zero at = , leaving two equations to specify two terms in the series expansion to be zero. This is a general property of the group delay for a Bessel filter of order n: the first n 1 terms in the series expansion of the group delay will be zero, thus maximizing the flatness of the group delay at = 0. Uses of Bessel Filters An audio crossover is an electronic filter that is used in audio applications. Since most speaker drivers can not cover the entire audio spectrum with the correct distortion control and volume range, they use a combination of speakers and drivers to do so. Audio crossovers can split the audio signal into different frequency bands that can be routed to the speakers that are optimized to use these bands while playing music. Audio crossovers also permit multi-band processing and amplification when the signal is split into adjusted bands before they are mixed together as is done in Dolby A noise reduction. The term Bessel refers to a type of filter response, not a type of filter. It features flat group delay in the passband:

This is the characteristic of Bessel filters that makes them valuable to digital designers. Very few filters are designed with square waves in mind. Most of the time, the signals filtered are sine waves, or close enough that the effect of harmonics can be ignored. If a waveform with high harmonic content is filtered, such as a square wave, the harmonics can be delayed with respect to the fundamental frequency if a Butterworth or Chebyshev response is used. The Fourier Series of a square wave is:

This means that a square wave is an infinite series of odd harmonics, or sinewaves, summed together to create the square shape. Obviously, if a square wave is to be transmitted without distortion, all of the harmonics - out to infinity - must be transmitted. This means that the square wave can be high pass filtered without distortion, if the 3 dB point of the filter is significantly lower than the fundamental. If the square wave is low pass filtered, however, the situation changes dramatically. Harmonics will be eliminated, producing distortion in the square wave. It is the job of the designer to decide just how many harmonics must be passed and what can be eliminated. Suppose that the designer wants to keep five harmonics. The resulting waveform looks something like this:

This may be acceptable to the designer - it depends on the timing of the leading and trailing edge of the waveform. The elimination of harmonics will result in rounding of the edges, and therefore delay in the leading and trailing edges of the digital signal. Of more importance, the harmonics that are passed will not be delayed. The Bessel approximation has a smooth passband and stopband response, like the Butterworth. For the same filter order, the stopband attenuation of the Bessel approximation is much lower that that of the Butterworth approximation. The following figures are representative of a low pass filter. The response characteristics are mirror imaged for high pass filters.

The designer can see that there is no ripple in the passband of a Bessel filter, and that it does not have as much rejection in the stop band as a Butterworth filter. The phase response of the three filter types is shown below. The Bessel response has the slowest rate of change of phase.

What Is a Bessel Crossover? The Bessel filter was not originally designed for use in a crossover, and requires minor modification to make it work properly. The purpose of the Bessel filter is to achieve approximately linear phase, linear phase being equivalent to a time delay. This is the best phase response from an audible standpoint, assuming you don't want to correct an existing phase shift. Bessels are historically low-pass or all-pass. A crossover however requires a separate high-pass, and this needs to be derived from the low-pass. There are different ways to derive a high-pass from a low-pass, but here we discuss a natural and traditional one that maximizes the cutoff

slope in the high-pass. Deriving this high-pass Bessel, we find that it no longer has linear phase. Other derivations of the high-pass can improve the combined phase response, but with tradeoffs. Two other issues that are closely related to each other are the attenuation at the design frequency, and the summed response. The traditional Bessel design is not ideal here. We can easily change this by shifting the low or high-pass up or down in frequency. This way, we can adjust the lowpass vs. high-pass response overlap, and at the same time achieve a phase difference between the low-pass and high-pass that is nearly constant over all frequencies. In the fourth order case this is 360 degrees, or essentially in-phase. In fact, the second and fourth order cases are comparable to a Linkwitz-Riley with slightly more rounded cutoff. Bessel Low-Pass and High-Pass Filters The Bessel filter uses a p(s) which is a Bessel polynomial, but the filter is more properly called a Thomson filter, after one of its developers. Still less known is the fact that it was actually reported several years earlier by Kiyasu. Bessel low-pass filters have maximally flat group delay about 0 Hz, so the phase response is approximately linear in the passband, while at higher frequencies the linearity degrades, and the group delay drops to zero). This nonlinearity has minimal impact because it occurs primarily when the output level is low. In fact, the phase response is so close to a time delay that Bessel low-pass and all-pass filters may be used solely to produce a time delay.

Fourth-Order Bessel Magnitude

Fourth-Order Bessel Group Delay

The high-pass output transfer function may be generated in different ways, one of which is to replace every instance of s in the low-pass with 1/s . This "flips" the magnitude response about the design frequency to yield the high-pass. Characteristics of the low-pass with respect to 0 Hz

are, in the high-pass, with respect to infinite frequency instead. A number of other high-pass derivations are possible, but they result in compromised cutoff slope or polar response. This popular method results in the general transfer function (1); (2) is a fourth-order Bessel example.

Note the reversed coefficient order of the high-pass as compared to the low-pass, once it's converted to a polynomial in s, and an added nth-order zero at the origin. This zero has a counterpart in the low-pass, an implicit nth-order zero at infinity! The nature of the response of the high-pass follows from equation (3) below, where s is evaluated on the imaginary axis to yield the frequency response.

The magnitude responses of the low-pass and the high-pass are mirror images of each other on a log-frequency scale; the negative sign has no effect on this. The phase of the low-pass typically drops near the cutoff frequency from an asymptote of zero as the frequency is increased, and asymptotically approaches a negative value. However, in addition to being mirror images on a log-frequency scale, the phase of the high-pass is the negative of the low-pass, which follows from the negative sign in (3). So the phase rises from zero at high frequency, and approaches a positive value asymptotically as the frequency is decreased. This results in offset curves with similar shape. Any asymmetry of the s-shaped phase curve is mirrored between the low-pass and high-pass. Table 1 gives Bessel crossover denominators normalized for time delay and phase match. Note the near-perfect symmetry for the (last three) phase-match cases.

Table 1. Bessel Crossovers of Second, Third, and Fourth-Order, Normalized First for Time Delay Design, then for Phase Match at Crossover

Drag Net Bessel Crossover at -3 dB Rane's Bessel crossover is set for phase match between low-pass and high-pass. This minimizes lobing due to driver separation, and also results in a pretty flat combined response. Another popular option is to have the magnitude response -3 dB at the design frequency. If -3 dB is desired at the setting, the frequency settings need to be changed by particular factors. You will need to enter separate values for low-pass and high-pass: multiply the low-pass and high-pass frequencies by the following factors: Crossover Type Third Order (18 dB/octave) Low-Pass High-Pass 0.786 0.708 0.652 1.413

Second-Order (12 dB/octave) 1.272

Fourth Order (24 dB/octave) 1.533

For example, for a second-order low-pass and high-pass set to 1000 Hz, set the low-pass to 1272 Hz and the high-pass to 786 Hz.

BUTTERWORTH FILTERSThe Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the passband so that it is also termed a maximally flat magnitude filter. It was first described in 1930 by the British engineer Stephen Butterworth in his paper entitled "On the Theory of Filter Amplifiers". Original paper Butterworth had a reputation for solving "impossible" mathematical problems. At the time filter design required an amount of designer experience due to limitations of the theory then in use. The filter was not in common use for over 30 years after its publication. Butterworth stated that: "An ideal electrical filter should not only completely reject the unwanted frequencies but should also have uniform sensitivity for the wanted frequencies". Such an ideal filter cannot be achieved but Butterworth showed that successively closer approximations were obtained with increasing numbers of filter elements of the right values. At the time, filters generated substantial ripple in the passband, and the choice of component values was highly interactive. Butterworth showed that a low pass filter could be designed whose cutoff frequency was normalized to 1 radian per second and whose frequency response (gain) was

where is the angular frequency in radians per second and n is the number of reactive elements (poles) in the filter. If = 1, the amplitude response of this type of filter in the passband is 1/2 0.707, which is half power or 3 dB. Butterworth only dealt with filters with an even number of poles in his paper. He may have been unaware that such filters could be designed with an odd number of poles. He built his higher order filters from 2-pole filters separated by vacuum tube amplifiers. His plot of the frequency response of 2, 4, 6, 8, and 10 pole filters is shown as A, B, C, D, and E in his original graph. Butterworth solved the equations for two- and four-pole filters, showing how the latter could be cascaded when separated by vacuum tube amplifiers and so enabling the construction of higherorder filters despite inductor losses. In 1930 low-loss core materials such as molypermalloy had not been discovered and air-cored audio inductors were rather lossy. Butterworth discovered that it was possible to adjust the component values of the filter to compensate for the winding resistance of the inductors. He used coil forms of 1.25 diameter and 3 length with plug in terminals. Associated capacitors and resistors were contained inside the wound coil form. The coil formed part of the plate load resistor. Two poles were used per vacuum tube and RC coupling was used to the grid of the following tube.

Butterworth also showed that his basic low-pass filter could be modified to give low-pass, highpass, band-pass and band-stop functionality. Overview

The Bode plot of a first-order Butterworth low-pass filter

The frequency response of the Butterworth filter is maximally flat (has no ripples) in the passband and rolls off towards zero in the stopband. When viewed on a logarithmic Bode plot the response slopes off linearly towards negative infinity. A first-order filter's response rolls off at 6 dB per octave (20 dB per decade) (all first-order lowpass filters have the same normalized frequency response). A second-order filter decreases at 12 dB per octave, a third-order at 18 dB and so on. Butterworth filters have a monotonically changing magnitude function with , unlike other filter types that have non-monotonic ripple in the passband and/or the stopband. Compared with a Chebyshev Type I/Type II filter or an elliptic filter, the Butterworth filter has a slower roll-off, and thus will require a higher order to implement a particular stopband specification, but Butterworth filters have a more linear phase response in the pass-band than Chebyshev Type I/Type II and elliptic filters can achieve.

Example

A third-order low-pass filter (Cauer topology). The filter becomes a Butterworth filter with cutoff frequency c=1 when (for example) C2=4/3 farad, R4=1 ohm, L1=3/2 henry and L3=1/2 henry.

A simple example of a Butterworth filter is the third-order low-pass design shown in the figure on the right, with C2 = 4/3 F, R4 = 1 , L1 = 3/2 H, and L3 = 1/2 H. Taking the impedance of the capacitors C to be 1/Cs and the impedance of the inductors L to be Ls, where s = + j is the complex frequency, the circuit equations yield the transfer function for this device:

The magnitude of the frequency response (gain) G() is given by

and the phase is given by

Gain and group delay of the third-order Butterworth filter with c=1

The group delay is defined as the derivative of the phase with respect to angular frequency and is a measure of the distortion in the signal introduced by phase differences for different

frequencies. The gain and the delay for this filter are plotted in the graph on the left. It can be seen that there are no ripples in the gain curve in either the passband or the stop band. The log of the absolute value of the transfer function H(s) is plotted in complex frequency space in the second graph on the right. The function is defined by the three poles in the left half of the complex frequency plane.

Log density plot of the transfer function H(s) in complex frequency space for the third-order Butterworth filter with c=1. The three poles lie on a circle of unit radius in the left half-plane.

These are arranged on a circle of radius unity, symmetrical about the real s axis. The gain function will have three more poles on the right half plane to complete the circle. By replacing each inductor with a capacitor and each capacitor with an inductor, a high-pass Butterworth filter is obtained. A band-pass Butterworth filter is obtained by placing a capacitor in series with each inductor and an inductor in parallel with each capacitor to form resonant circuits. The value of each new component must be selected to resonate with the old component at the frequency of interest. A band-stop Butterworth filter is obtained by placing a capacitor in parallel with each inductor and an inductor in series with each capacitor to form resonant circuits. The value of each new component must be selected to resonate with the old component at the frequency to be rejected.

Transfer function

Plot of the gain of Butterworth low-pass filters of orders 1 through 5, with cutoff frequency 0 = 1. Note that the slope is 20n dB/decade where n is the filter order.

Like all filters, the typical prototype is the low-pass filter, which can be modified into a highpass filter, or placed in series with others to form band-pass and band-stop filters, and higher order versions of these. The gain G() of an n-order Butterworth low pass filter is given in terms of the transfer function H(s) as

where

n = order of filter c = cutoff frequency (approximately the -3dB frequency) G0 is the DC gain (gain at zero frequency)

It can be seen that as n approaches infinity, the gain becomes a rectangle function and frequencies below c will be passed with gain G0, while frequencies above c will be suppressed. For smaller values of n, the cutoff will be less sharp. We wish to determine the transfer function H(s) where s = + j (from Laplace transform). Since H(s)H(-s) evaluated at s = j is simply equal to |H(j)|2, it follows that

The poles of this expression occur on a circle of radius c at equally spaced points. The transfer function itself will be specified by just the poles in the negative real half-plane of s. The k-th pole is specified by

and hence;

The transfer function may be written in terms of these poles as

The denominator is a Butterworth polynomial in s. Normalized Butterworth polynomials The Butterworth polynomials may be written in complex form as above, but are usually written with real coefficients by multiplying pole pairs which are complex conjugates, such as s1 and sn. The polynomials are normalized by setting c = 1. The normalized Butterworth polynomials then have the general form

for n even

for n odd To four decimal places, they are

n Factors of Polynomial Bn(s) 1 (s + 1) 2 s2 + 1.4142s + 1 3 (s + 1)(s2 + s + 1) 4 (s2 + 0.7654s + 1)(s2 + 1.8478s + 1) 5 (s + 1)(s2 + 0.6180s + 1)(s2 + 1.6180s + 1) 6 (s2 + 0.5176s + 1)(s2 + 1.4142s + 1)(s2 + 1.9319s + 1) 7 (s + 1)(s2 + 0.4450s + 1)(s2 + 1.2470s + 1)(s2 + 1.8019s + 1) 8 (s2 + 0.3902s + 1)(s2 + 1.1111s + 1)(s2 + 1.6629s + 1)(s2 + 1.9616s + 1) The normalized Butterworth polynomials can be used to determine the transfer function for any low-pass filter cut-off frequency c, as follows

,

where Transformation to other bandforms are also possible. Maximal flatness Assuming c = 1 and G0 = 1, the derivative of the gain with respect to frequency can be shown to be

which is monotonically decreasing for all since the gain G is always positive. The gain function of the Butterworth filter therefore has no ripple. Furthermore, the series expansion of the gain is given by

In other words all derivatives of the gain up to but not including the 2n-th derivative are zero, resulting in "maximal flatness". If the requirement to be monotonic is limited to the passband only and ripples are allowed in the stopband, then it is possible to design a filter of the same order, such as the inverse Chebyshev filter, that is flatter in the passband than the "maximally flat" Butterworth. High-frequency roll-off Again assuming c = 1, the slope of the log of the gain for large is

In decibels, the high-frequency roll-off is therefore 20n dB/decade, or 6n dB/octave (the factor of 20 is used because the power is proportional to the square of the voltage gain; see 20 log rule.) Digital implementation Digital implementations of Butterworth and other filters are often based on the bilinear transform method or the matched Z-transform method, two different methods to discretize an analog filter design. In the case of all-pole filters such as the Butterworth, the matched Z-transform method is equivalent to the impulse invariance method. For higher orders, digital filters are sensitive to quantization errors, so they are often calculated as cascaded biquad sections, plus one first-order or third-order section for odd orders. Comparison with other linear filters Here is an image showing the gain of a discrete-time Butterworth filter next to other common filter types. All of these filters are fifth-order.

The Butterworth filter rolls off more slowly around the cutoff frequency than the Chebyshev filter or the Elliptic filter, but without ripple. The term Butterworth refers to a type of filter response, not a type of filter. It is sometimes called the Maximally Flat approximation, because for a response of order n, the first (2n-1) derivatives of the gain with respect to frequency are zero at frequency = 0. There is no ripple in the passband, and DC gain is maximally flat. The following figures are representative of a low pass filter. The response characteristics are mirror imaged for high pass filters.

The designer can see that there is no ripple in the passband of a Butterworth filter or a Bessel filter. The Butterworth filter, however, has a flatter response in the passband. There is a

continuum of filter characteristics, of which Bessel is one, and Butterworth another; but anything in between is possible - it just wouldn't have a name. The Chebyshev response continues the continuum beyond Butterworth, which is the last characteristic for which there is a flat passband. The phase response of the three filter types is shown below. The Butterworth response is a compromise between Bessel and Chebyshev.

The group delay of the three filters is shown below. The Butterworth response is a compromise between Bessel and Chebyshev.

CHEBYSHEV FILTERChebyshev filters are analog or digital filters having a steeper roll-off and more passband ripple (type I) or stopband ripple (type II) than Butterworth filters. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband. This type of filter is named in honor of Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials.

Because of the passband ripple inherent in Chebyshev filters, the ones that have a smoother response in the passband but a more irregular response in the stopband are preferred for some applications. Type I Chebyshev filters These are the most common Chebyshev filters. The gain (or amplitude) response as a function of angular frequency of the nth-order low-pass filter is

where is the ripple factor, 0 is the cutoff frequency and Tn() is a Chebyshev polynomial of the nth order. The passband exhibits equiripple behavior, with the ripple determined by the ripple factor . In the passband, the Chebyshev polynomial alternates between 0 and 1 so the filter gain will alternate between maxima at G = 1 and minima at . At the cutoff frequency

0 the gain again has the value but continues to drop into the stop band as the frequency increases. This behavior is shown in the diagram on the right. The common practice of defining the cutoff frequency at 3 dB is usually not applied to Chebyshev filters; instead the cutoff is taken as the point at which the gain falls to the value of the ripple for the final time. The order of a Chebyshev filter is equal to the number of reactive components (for example, inductors) needed to realize the filter using analog electronics. The ripple is often given in dB: Ripple in dB = so that a ripple amplitude of 3 dB results from = 1. An even steeper roll-off can be obtained if we allow for ripple in the stop band, by allowing zeroes on the j-axis in the complex plane. This will however result in less suppression in the stop band. The result is called an elliptic filter, also known as Cauer filter.

Poles and zeroes

Log of the absolute value of the gain of an 8th order Chebyshev type I filter in complex frequency space (s = + j) with = 0.1 and 0 = 1. The white spots are poles and are arranged on an ellipse with a semi-axis of 0.3836... in and 1.071... in . The transfer function poles are those poles in the left half plane. Black corresponds to a gain of 0.05 or less, white corresponds to a gain of 20 or more.

For simplicity, assume that the cutoff frequency is equal to unity. The poles (pm) of the gain of the Chebyshev filter will be the zeroes of the denominator of the gain. Using the complex frequency s:

Defining js = cos() and using the trigonometric definition of the Chebyshev polynomials yields:

Solving for

where the multiple values of the arc cosine function are made explicit using the integer index m. The poles of the Chebyshev gain function are then:

Using the properties of the trigonometric and hyperbolic functions, this may be written in explicitly complex form:

where m = 1, 2,..., n and

This may be viewed as an equation parametric in n and it demonstrates that the poles lie on an ellipse in s-space centered at s = 0 with a real semi-axis of length sinh(arsinh(1 / ) / n) and an imaginary semi-axis of length of cosh(arsinh(1 / ) / n). The transfer function The above expression yields the poles of the gain G. For each complex pole, there is another which is the complex conjugate, and for each conjugate pair there are two more that are the negatives of the pair. The transfer function must be stable, so that its poles will be those of the gain that have negative real parts and therefore lie in the left half plane of complex frequency space. The transfer function is then given by

where are only those poles with a negative sign in front of the real term in the above equation for the poles.

The group delay

Gain and group delay of a fifth-order type I Chebyshev filter with = 0.5.

The group delay is defined as the derivative of the phase with respect to angular frequency and is a measure of the distortion in the signal introduced by phase differences for different frequencies.

The gain and the group delay for a fifth-order type I Chebyshev filter with =0.5 are plotted in the graph on the left. It can be seen that there are ripples in the gain and the group delay in the passband but not in the stopband. Type II Chebyshev filters

The frequency response of a fifth-order type II Chebyshev low-pass filter with

Also known as inverse Chebyshev, this type is less common because it does not roll off as fast as type I, and requires more components. It has no ripple in the passband, but does have equiripple in the stopband. The gain is:

In the stopband, the Chebyshev polynomial will oscillate between 0 and 1 so that the gain will oscillate between zero and

and the smallest frequency at which this maximum is attained will be the cutoff frequency o. The parameter is thus related to the stopband attenuation in decibels by:

For a stopband attenuation of 5dB, = 0.6801; for an attenuation of 10dB, = 0.3333. The frequency fC = C/2 is the cutoff frequency. The 3dB frequency fH is related to fC by:

Poles and zeroes

Log of the absolute value of the gain of an 8th order Chebyshev type II filter in complex frequency space (s=+j) with = 0.1 and 0 = 1. The white spots are poles and the black spots are zeroes. All 16 poles are shown. Each zero has multiplicity of two, and 12 zeroes are shown and four are located outside the picture, two on the positive axis,

and two on the negative. The poles of the transfer function will be poles on the left half plane and the zeroes of the transfer function will be the zeroes, but with multiplicity 1. Black corresponds to a gain of 0.05 or less, white corresponds to a gain of 20 or more.

Again, assuming that the cutoff frequency is equal to unity, the poles (pm) of the gain of the Chebyshev filter will be the zeroes of the denominator of the gain:

The poles of gain of the type II Chebyshev filter will be the inverse of the poles of the type I filter:

where m = 1, 2, ..., n . The zeroes (zm) of the type II Chebyshev filter will be the zeroes of the numerator of the gain:

The zeroes of the type II Chebyshev filter will thus be the inverse of the zeroes of the Chebyshev polynomial.

for m = 1, 2, ..., n. The transfer function The transfer function will be given by the poles in the left half plane of the gain function, and will have the same zeroes but these zeroes will be single rather than double zeroes.

The group delay

Gain and group delay of a fifth-order type II Chebyshev filter with = 0.1.

The gain and the group delay for a fifth-order type II Chebyshev filter with =0.1 are plotted in the graph on the left. It can be seen that there are ripples in the gain in the stop band but not in the pass band. Implementation Digital As with most analog filters, the Chebyshev may be converted to a digital (discrete-time) recursive form via the bilinear transform. However, as digital filters have a finite bandwidth, the response shape of the transformed Chebyshev will be warped. Alternatively, the Matched Ztransform method may be used, which does not warp the response. Comparison with other linear filters Here is an image showing the Chebyshev filters next to other common kind of filters obtained with the same number of coefficients (all filters are fifth order):

As is clear from the image, Chebyshev filters are sharper than the Butterworth filter; they are not as sharp as the elliptic one, but they show fewer ripples over the bandwidth. The Chebyshev and Butterworth Responses The Chebyshev response is a mathematical strategy for achieving a faster roll-off by allowing ripple in the frequency response. Analog and digital filters that use this approach are called Chebyshev filters. For instance, analog Chebyshev filters were used in Chapter 3 for analog-todigital and digital-to-analog conversion. These filters are named from their use of the Chebyshev polynomials, developed by the Russian mathematician Pafnuti Chebyshev (1821-1894). This name has been translated from Russian and appears in the literature with different spellings, such as: Chebychev, Tschebyscheff, Tchebysheff and Tchebichef. Figure 20-1 shows the frequency response of low-pass Chebyshev filters with passband ripples of: 0%, 0.5% and 20%. As the ripple increases (bad), the roll-off becomes sharper (good). The Chebyshev response is an optimal trade-off between these two parameters. When the ripple is set to 0%, the filter is called a maximally flat or Butterworth filter (after S. Butterworth, a British engineer who described this response in 1930). A ripple of 0.5% is a often good choice for digital filters. This matches the typical precision and accuracy of the analog electronics that the signal has passed through. The Chebyshev filters discussed in this chapter are called type 1 filters, meaning that the ripple is only allowed in the passband. In comparison,

type 2 Chebyshev filters have ripple only in the stopband. Type 2 filters are seldom used, and we won't discuss them. There is, however, an important design called the elliptic filter, which has ripple in both the passband and the stopband. Elliptic filters provide the fastest roll-off for a given number of poles, but are much harder to design. We won't discuss the elliptic filter here, but be aware that it is frequently the first choice of professional filter designers, both in analog electronics and DSP. If you need this level of performance, buy a software package for designing digital filters. The term Chebyshev refers to a type of filter response, not a type of filter. It is sometimes refered to as an equal-ripple approximation, and sometimes spelled "Tschebysheff" or some variation thereof. It features superior attentuation in the stop band, at the expense of ripple in the passband. Generally the designer will choose a ripple depth of between 0.1dB and 3dB. Chebyshev filter response, therefore, is not limited to a single value of response. The following figures are representative of a low pass filter. The response characteristics are mirror imaged for high pass filters.

The designer can see that there is ripple in the passband of a Chebyshev filter. The number and position of ripples is determinined by the order of the filter. Filters with even orders generate ripple that appears above the 0dB intercept and filters with odd orders generate ripple below the 0dB intercept. The phase response of the three filter types is shown below. The Chebyshev response has the fastest rate of phase change.

The group delay of the three filters is shown below. The Chebyshev response has the longest group delay.

Chebyshev Filter Design Procedures The Chebyshev filter has a passband optimized to minimize the maximum error over the complete passband frequency range, and a stopband controlled by the frequency response being maximally flat at =. The passband ripple and the filter order are the two parameters to be determined by the specifications.

The form for the specifications that is most consistent with the Chebyshev filter formulation is a maximum allowed error in the passband and a desired degree of flatness" at =. The slope of the response near the transition from pass to stopband at =1 becomes steeper as both the order increases and the allowed passband error ripple increases. The dropoff is more rapid than for the Butterworth filter. As stated earlier, the design parameters must be clearly understood to obtain a desired result. The passband ripple is defined to be the difference between the maximum and the minimum of |F| over the passband frequencies of 00 for 1