Chebyshev Filter Properties The Butterworth filter does not give a sufficiently good approximation across the complete passband in many cases. The Taylor's series approximation is often not suited to the way specifications are given for filters. An alternate error measure is the maximum of the absolute value of the difference between the actual filter response and the ideal. This is considered over the total passband. This is the Chebyshev error measure and was defined and applied to the FIR filter design problem. For the IIR filter, the Chebyshev error is minimized over the passband and a Taylor's series approximation at ω=∞ is used to determine the stopband performance. This mixture of methods in the IIR case is called the Chebyshev filter, and simple design formulas result, just as for the Butterworth filter. The design of Chebyshev filters is particularly interesting, because the results of a very elegant theory insure that constructing a frequency-response function with the proper form of equal ripple in the error will result in a minimum Chebyshev error without explicitly minimizing anything. This allows a straightforward set of design formulas to be derived which can be viewed as a generalization of the Butterworth formulas , . The form for the magnitude squared of the frequency-response function for the Chebyshev filter is | F ( j ω ) | 2 = 1 1 + ϵ 2 C N ( ω ) 2 where CN(ω) is an Nth-order Chebyshev polynomial and ϵ is a parameter that controls the ripple size. This polynomial in ω has very special characteristics that result in the optimality of the response function ().

CHEBYSHEV POLYNOMIALS The Chebyshev polynomial is a powerful function in approximation theory. Although the function is a polynomial, it is best defined and developed in terms of trigonometric functions by, , , . C N ( ω ) = cos ( N cos - 1 ( ω ) ) where CN(ω) is an Nth-order, real-valued function of the real variable ω. The development is made clearer by introducing an intermediate complex variable φ. C N ( ω ) = cos ( N φ ) where ω = cos ( φ ) Although this definition of CN(ω) may not at first appear to result in a polynomial, the following recursive relation derived from () shows that it is a polynomial. C N + 1 ( ω ) = 2 ω C N ( ω ) - C N - 1 ( ω ) From (), it is clear that C0=1 and C1=ω, and from (), it follows that C 2 = 2 ω 2 - 1 C 3 = 4 ω 3 - 3 ω C 4 = 8 ω 4 - 8 ω 2 + 1 etc. Other relations useful for developing these polynomials are C N 2 ( ω ) = ( C 2 N ( ω ) + 1 ) / 2 C M N ( ω ) = C M ( C N ( ω ) ) where M and N are coprime. These are remarkable functions . They oscillate between +1 and -1 for -1<ω<1 and go monotonically to +/- infinity outside that domain. All N of their zeros are real and fall in the domain of -1<ω<1, i.e., CN is an equal ripple approximation to zero over the range of ω from -1 to +1. In addition, the values for ω where CN reaches its local maxima and minima and is zero are easily calculated from () and (). For -1<ω<1, a plot of CN(ω) can be made using the concept of Lissajous figures. Example plots for C0, C1, C2, C3, and C4 are shown in .

Chebyshev Polynomials for N = 0, 1, 2, 3, and 4 The filter frequency-response function for N=5 is given in showing the passband ripple in terms of the parameter ϵ.

Fifth Order Chebyshev Filter Frequency Response The approximation parameters must be clearly understood. The passband ripple is defined to be the difference between the maximum and the minumum of |F| over the passband frequencies of 0<ω<1. There can be confusion over this point as two definitions appear in the literature. Most digital , , , and analog filter design books use the definition just stated. Approximation literature, especially concerning FIR filters, use one half this value which is a measure of the maximum error, ||F|-|Fd||, where |Fd| is the center line in the passband of , which |F| oscillates around. The Chebyshev theory states that the maximum error over that band is minimum and that this optimal approximation function has equal ripple over the pass band. It is easy to see that e in () determines the ripple in the passband and the order N determines the rate that the response goes to zero as ω goes to infinity. Pole Locations A method for finding the pole locations for the Chebyshev filter transfer function is next developed. The details of this section can be skipped and the results in (,) used if desired. From (), it is seen that the poles of FF(s) occur when 1 + ϵ 2 C N 2 ( s / j ) = 0 or C N = ± j ϵ From (), define φ=cos-1(ω) with real and imaginary parts given by φ = cos - 1 ( ω ) = u + j v This gives, C N = cos ( N φ ) = cos ( N u ) cosh ( N ν ) - j sin ( N u ) sinh ( N ν ) = ± j ϵ which implies the real part of CN is zero. This requires cos ( N u ) cosh ( N ν ) = 0 which implies cos ( N u ) = 0 which in turn implies that u takes on values of u = u k = ( 2 k + 1 ) π / 2 N , k = 0 , 1 , . . . N - 1 For these values of u, sin(nu)=±1, we have sinh ( N ν ) = 1 / ϵ which requires ν to take on a value of ν = ν 0 = ( sinh - 1 ( 1 / ϵ ) ) / N Using s=jω gives s = j ω = j cos ( φ ) = j cos ( u + j ν ) = j cos ( ( 2 k + 1 ) π / 2 N + j ν 0 ) which gives the location of the N poles in the s plane as s k = σ k + j ω k where σ k = - sinh ( ν 0 ) cos ( k π / 2 N ) ω k = cosh ( ν 0 ) sin ( k π / 2 N ) for N values of k where k = ± 1 , ± 3 , ± 5 , ⋯ , ± ( N - 1 ) for N even k = 0 , ± 2 , ± 4 , ⋯ , ± ( N - 1 ) for N odd A partially factored form for F(s) can be derived using the same approach as for the Butterworth filter. For N even, the form is F ( s ) = ∏ k 1 s 2 - 2 σ k s + ( σ k 2 + ω k 2 ) for k=1,3,5,⋯,N-1. For N odd, F(s) has a single real pole and, therefore, the form F ( s ) = 1 sinh ( ν 0 ) F ( s ) = ∏ k 1 s 2 - 2 σ k s + ( σ k 2 + ω k 2 ) for k=2,4,6,...,N-1 This is a convenient form for the cascade and parallel realizations. A single formula for both even and odd N is σ = - sinh ( ν 0 ) sin ( ( 2 k + 1 ) π / 2 N ) ω k = cosh ( ν 0 ) cos ( ( 2 k + 1 ) π / 2 N ) for N values of k where k=0,1,2,⋯,N-1 Note the similarity to the pole locations for the Butterworth filter. Cross multiplying, squaring, and adding the terms in (,) gives ( σ k sinh ( ν 0 ) ) 2 + ( ω k cosh ( ν 0 ) ) 2 = 1 This is the equation for an ellipse and shows that the poles of a Chebyshev filter lie on an ellipse similar to the way the poles of a Butterworth filter lie on a circle, , , , , . Summary This section has developed the classical Chebyshev filter approximation which minimizes the maximum error over the passband and uses a Taylor's series approximation at infinity. This results in the error being equal ripple in the passband. The transfer function was developed in terms of the Chebyshev polynomial and explicit formulas were derived for the location of the transfer function poles. These can be expressed as a modification of the pole locations for the Butterworth filter and are implemented in the appendix. It is possible to develop a theory for Chebyshev passband approximation and arbitrary zero location similar to the Taylor's series result in ().

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 ω=∞. 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 ω=∞. The slope of the response near the transition from pass to stopband at ω=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 0<ω<1. There can be confusion over this point as two definitions appear in the literature. Most digital , , and analog filter design books use the definition just stated. Approximation literature, especially concerning FIR filters, use half this value which is a measure of the maximum error, ||F|-|Fd||, where |Fd| is the center line in the passband around which |F| oscillates. The following formulas relate the passband ripple δ, the passband ripple a in positive dB, and the transfer function parameter ϵ. a = 10 log ( 1 + ϵ 2 ) = - 20 log ( 1 - δ ) , ϵ = 2 δ - δ 2 1 - 2 δ + δ 2 = 10 a / 10 - 1 , δ = 1 - 10 - a / 20 = 1 - 1 1 + ϵ 2 In some cases, stopband performance is not given in terms of degree of flatness at ω=∞, but in terms of a maximum allowed magnitude G in the stopband above a certain frequency ωs, i.e., G>|F|>0 for 1<ωs<ω<∞. For a given ϵ, this will determine the order as the smallest positive integer satisfying N ≥ cosh - 1 ( 1 - G 2 ϵ G 2 ) cosh - 1 ( ω s ) The design of a Chebyshev filter involves the following steps: The maximum-allowed passband variation must be in the form of δ or a. From this, the parameter ϵ is calculated using (). The order N is determined by the desired flatness at ω=∞ or a maximum-allowed response for frequencies above ωs using (). ν0 is calculated from ϵ and n using (), and the scale factors sinh(ν0) and cosh(ν0) are then determined. The pole locations are calculated from () or (). This can be done by scaling the poles of a Butterworth prototype filter. These pole locations are combined in () and () to give the final filter transfer function. This process is easily programmed for computer aided design as illustrated in Program 8 in the appendix. If the design procedure uses () to determine the order and the right-hand side of the equation is not exactly an integer, it is possible to improve on the specifications. Direct use of the order with ϵ from () gives a stopband gain at ωs that is less than G, or the same design can be viewed as giving the maximum-allowed gain G at a lower frequency than ωs. An alternate approach is to solve () for a new value of ϵ, then cause () to be an equation with the specified ωs and G. This gives a filter that exactly meets the stopband specifications and gives a smaller passband ripple than originally requested. A similar set of alternatives exists for the elliptic-function filter. Example 7-2. The Design of a Chebyshev Lowpass Filter. The design specifications require a maximum passband ripple of δ=0.1 or a=0.91515 dB, and can allow no greater response than G=0.2 for frequencies above ωs=1.6 radians per second. Given δ=0.1 or a=0.91515, equation () implies ϵ = 0 . 484322 Given G=0.2 and ωs=1.6, equation () implies an order of N=3. From ϵ and N, ν0 is 0.49074 from () and sinh ( ν 0 ) = 0 . 510675 cosh ( ν 0 ) = 1 . 122849 These multipliers are used to scale the root locations of the example third-order Butterworth filter to give F ( s ) = 1 ( s + 0 . 51067 ) ( s + 0 . 25534 + j 0 . 97242 ) ( s + 0 . 25534 - j 0 . 97242 ) F ( s ) = 1 ( s + 0 . 51067 ) ( s 2 + 0 . 510675 s + 1 . 010789 ) F ( s ) = 1 s 3 + 102135 s 2 + 1 . 271579 s + 0 . 516185 The frequency response is shown in

Example Design of a Third Order Chebyshev Filter Frequency Response

Inverse-Chebyshev Filter Properties A second form of the mixture of a Chebyshev approximation and a Taylor's series approximation is called the Inverse Chebyshev filter or the Chebyshev II filter. This error measure uses a Taylor's series for the passband just as for the Butterworth filter and minimizes the maximum error over the total stopband. It reverses the types of approximation used in the preceding section. A fifth-order example is illustrated in c and c. Rather than developing the approximation directly, it is easier to modify the results from the regular Chebyshev filter. First, the frequency variable ω in the regular Chebyshev filter, described in (), is replaced by 1/ω, which interchanges the characteristics at ω equals zero and infinity and does not change the performance at ω equals unity. This converts a Chebyshev lowpass filter into a Chebyshev highpass filter as illustrated in moving from the first to second frequency response.

Lowpass Chebyshev to Highpass Chebyshev to Lowpass Inverse Chebyshev This highpass characteristic is subtracted from unity to give the desired lowpass inverse-Chebyshev frequency response illustrated in c. The resulting magnitude-squared frequency- response function is given by F F ( j ω ) = ϵ 2 C N 2 ( 1 / ω ) 1 + ϵ 2 C N 2 ( 1 / ω ) Zero Locations The zeros of the Chebyshev polynomial CN(ω) are easily found by C N ( ω ) = 0 ⇒ N cos - 1 ( ω ) = ( 2 k + 1 ) π / 2 which requires ω k = cos ( ( 2 k + 1 ) π / 2 N ) for k=0,1,⋯N-1, or ω k = sin ( k π / 2 N ) for k=0,±2,±4,...,±(N-1) : N odd k=±1,±3,±5,...,±(N-1) : N even The zeros of the inverse-Chebyshev filter transfer function are derived from () and () to give ω z k = 1 / ( cos ( ( 2 k + 1 ) π / 2 N ) ) The zero locations are not a function of ϵ, i.e., they are independent of the stopband ripple. Pole Locations The pole locations are the reciprocal of those for the regular Chebyshev filter. If the poles for the inverse filter are denoted by s k ' = σ k ' + j ω k ' the locations are σ k ' = σ k σ k 2 + ω k 2 ω k ' = ω k σ k 2 + ω k 2 Although this gives a straightforward formula for calculating the location of the poles and zeros of the inverse- Chebyshev filter, they do not lie on a simple geometric curve as did those for the Butterworth and Chebyshev filters. Note that the conditions for a Taylor's series approximation with preset zero locations are satisfied. A partially factored form for the Butterworth filter and for the Chebyshev filter can be written for the inverse-Chebyshev filter using the zero locations from () and the pole locations from the regular Chebyshev filter. For N even, this becomes F ( s ) = ∏ k ( s 2 + ω z k 2 ) ∏ k ( s 2 - 2 ( σ k / ( σ k 2 + ω k 2 ) ) s + 1 / ( σ k 2 + ω k 2 ) for k=1,3,5,⋯,N-1. For N odd, F(s) has a single pole, and therefore, is of the form F ( s ) = ∏ k ( s 2 + ω z k 2 ) ( s + 1 / sinh ( ν 0 ) ) ∏ k ( s 2 - 2 ( σ k / ( σ k 2 + ω k 2 ) ) s + 1 / ( σ k 2 + ω k 2 ) for k=2,4,6,⋯,N-1 Because of the relationships between the locations of the poles of the Butterworth, Chebyshev, and inverse-Chebyshev filters, it is easy to write a design program with many common calculations. That is illustrated in the program in the appendix.

Inverse-Chebyshev Filter Design Procedures The natural form for the specifications of an inverse-Chebyshev filter is in terms of the flatness of the response at ω to determine the passband, and a maximum allowable response in the stopband. The filter order and the stopband ripple are the parameters to be determined by the specifications. The rate of dropoff near the transition from pass to stopband is similar to the regular Chebyshev filter. Because practical specifications often allow more passband ripple than stopband ripple, the regular Chebyshev filter will usually have a sharper dropoff than the inverse-Chebyshev filter. Under those conditions, the inverse-Chebyshev filter will have a smoother phase response and less time-domain echo effects. The stopband ripple d is simply defined as the maximum value that |F(jω)| assumes in the stopband, which is the set of frequencies 1<ω<∞. An alternative specification is the minimum-allowed attenuation over stopband expressed in dB as b. The following formulas relate the stopband ripple δ, the stopband attenuation b in positive dB, and the transfer function parameter ϵ in () ϵ = δ 1 - δ 2 δ = ϵ 1 + ϵ 2 b = - 10 log ( ϵ 2 / ( 1 + ϵ 2 ) ) = - 20 log ( d ) In some cases passband performance is not given in terms of degree of flatness at ω=0, but in terms of a minimum-allowed magnitude G in the passband up to a certain frequency ωp, i.e., 1>|F|>G for 0<ω<ωp<1. For a given ϵ, this requirement will determine the order as the smallest positive integer satisfying N > cosh - 1 ( G / ( ϵ 1 - G 2 ) ) cosh - 1 ( 1 / ω p ) The design of an inverse-Chebyshev filter is summarized in the following steps: The maximum-allowed stopband response must be given in the form of δ or b. From this, the parameter ϵ is calculated using (). The order N is determined from the desired flatness at ω=0, or from a minimum allowed response for frequencies up to ωp using (). ν0 and sinh(ν0) and cosh(ν0) are calculated just as for the regular Chebyshev filter. The pole locations for the prototype Chebyshev filter are calculated from () and () and then "inverted" according to () to give the inverse- Chebyshev filter pole locations. The pole locations are combined in () to give the final filter transfer function denominator. The zero locations are calculated from () and combined with the pole locations to give the total transfer function () or (). Example Design of an Inverse-Chebyshev Filter A third-order inverse-Chebyshev lowpass filter is desired with a maximum-allowed stopband ripple of d=0.1 or b=20 dB. This corresponds to an ϵ of 0.100504 and, together with N=3, results in a ν0=0.99774. The scale factors are sinh=1.171717 and cosh=1.540429. The prototype Chebyshev filter transfer function is F ( s ) = 1 ( s + 1 . 1717 ) ( s 2 + 1 . 1717 s + 2 . 0404 ) The zeros are calculated from (), and the poles of the prototype are inverted to give, from (), the desired inverse- Chebyshev filter transfer function of F ( s ) = s 2 + 4 / 3 ( s + 0 . 85345 ) ( s 2 + 0 . 57425 s + 0 . 490095 )