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Chebyshev Filter Properties

Module by: C. Sidney Burrus

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=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 Entry 4, Entry 7.

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 | F ( j ω ) | 2 = 1 1 + ϵ 2 C N ( ω ) 2 (1)

where CN(ω)CN(ω) is an Nth-order Chebyshev polynomial and ϵϵ is a parameter that controls the ripple size. This polynomial in ωω has very special characteristics that result in the optimality of the response function (Equation 1).

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 byEntry 4, Entry 5, Entry 1, Entry 7.

C N ( ω ) = cos ( N cos - 1 ( ω ) ) C N ( ω ) = cos ( N cos - 1 ( ω ) ) (2)

where CN(ω)CN(ω) is an Nth-order, real-valued function of the real variable ωω. The development is made clearer by introducing an intermediate complex variable φφ.

C N ( ω ) = cos ( N φ ) C N ( ω ) = cos ( N φ ) (3)

where

ω = cos ( φ ) ω = cos ( φ ) (4)

Although this definition of CN(ω)CN(ω) may not at first appear to result in a polynomial, the following recursive relation derived from (Equation 4) shows that it is a polynomial.

C N + 1 ( ω ) = 2 ω C N ( ω ) - C N - 1 ( ω ) C N + 1 ( ω ) = 2 ω C N ( ω ) - C N - 1 ( ω ) (5)

From (Equation 2), it is clear that C0=1C0=1 and C1=ωC1=ω, and from (Equation 5), it follows that

C 2 = 2 ω 2 - 1 C 2 = 2 ω 2 - 1 (6)
C 3 = 4 ω 3 - 3 ω C 3 = 4 ω 3 - 3 ω (7)
C 4 = 8 ω 4 - 8 ω 2 + 1 C 4 = 8 ω 4 - 8 ω 2 + 1 (8)

etc.

Other relations useful for developing these polynomials are

C N 2 ( ω ) = ( C 2 N ( ω ) + 1 ) / 2 C N 2 ( ω ) = ( C 2 N ( ω ) + 1 ) / 2 (9)
C M N ( ω ) = C M ( C N ( ω ) ) C M N ( ω ) = C M ( C N ( ω ) ) (10)

where M and N are coprime.

These are remarkable functions Entry 7. They oscillate between +1 and -1 for -1<ω<1-1<ω<1 and go monotonically to +/- infinity outside that domain. All NN of their zeros are real and fall in the domain of -1<ω<1-1<ω<1, i.e., CNCN is an equal ripple approximation to zero over the range of ωω from -1 to +1. In addition, the values for ωω where CNCN reaches its local maxima and minima and is zero are easily calculated from (Equation 3) and (Equation 4). For -1<ω<1-1<ω<1, a plot of CN(ω)CN(ω) can be made using the concept of Lissajous figures. Example plots for C0C0, C1C1, C2C2, C3C3, and C4C4 are shown in Figure 1.

Figure 1: Chebyshev Polynomials for N = 0, 1, 2, 3, and 4
figIIR6.png

The filter frequency-response function for N=5N=5 is given in Figure 2 showing the passband ripple in terms of the parameter ϵϵ.

Figure 2: Fifth Order Chebyshev Filter Frequency Response
figIIR7.png

The approximation parameters must be clearly understood. The passband ripple is defined to be the difference between the maximum and the minumum of |F||F| over the passband frequencies of 0<ω<10<ω<1. There can be confusion over this point as two definitions appear in the literature. Most digital Entry 4, Entry 5, Entry 3, Entry 2 and analog Entry 7 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||||F|-|Fd||, where |Fd||Fd| is the center line in the passband of Figure 2, which |F||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 (Equation 1) determines the ripple in the passband and the order NN determines the rate that the response goes to zero as ωω 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 (Equation 22,Equation 24) used if desired.

From (Equation 1), it is seen that the poles of FF(s)FF(s) occur when

1 + ϵ 2 C N 2 ( s / j ) = 0 1 + ϵ 2 C N 2 ( s / j ) = 0 (11)

or

C N = ± j ϵ C N = ± j ϵ (12)

From (Equation 4), define φ=cos-1(ω)φ=cos-1(ω) with real and imaginary parts given by

φ = cos - 1 ( ω ) = u + j v φ = cos - 1 ( ω ) = u + j v (13)

This gives,

C N = cos ( N φ ) = cos ( N u ) cosh ( N ν ) - j sin ( N u ) sinh ( N ν ) = ± j ϵ C N = cos ( N φ ) = cos ( N u ) cosh ( N ν ) - j sin ( N u ) sinh ( N ν ) = ± j ϵ (14)

which implies the real part of CNCN is zero. This requires

cos ( N u ) cosh ( N ν ) = 0 cos ( N u ) cosh ( N ν ) = 0 (15)

which implies

cos ( N u ) = 0 cos ( N u ) = 0 (16)

which in turn implies that uu takes on values of

u = u k = ( 2 k + 1 ) π / 2 N , k = 0 , 1 , . . . N - 1 u = u k = ( 2 k + 1 ) π / 2 N , k = 0 , 1 , . . . N - 1 (17)

For these values of uu, sin(nu)=±1sin(nu)=±1, we have

sinh ( N ν ) = 1 / ϵ sinh ( N ν ) = 1 / ϵ (18)

which requires νν to take on a value of

ν = ν 0 = ( sinh - 1 ( 1 / ϵ ) ) / N ν = ν 0 = ( sinh - 1 ( 1 / ϵ ) ) / N (19)

Using s=jωs=jω gives

s = j ω = j cos ( φ ) = j cos ( u + j ν ) = j cos ( ( 2 k + 1 ) π / 2 N + j ν 0 ) s = j ω = j cos ( φ ) = j cos ( u + j ν ) = j cos ( ( 2 k + 1 ) π / 2 N + j ν 0 ) (20)

which gives the location of the NN poles in the ss plane as

s k = σ k + j ω k s k = σ k + j ω k (21)

where

σ k = - sinh ( ν 0 ) cos ( k π / 2 N ) σ k = - sinh ( ν 0 ) cos ( k π / 2 N ) (22)
ω k = cosh ( ν 0 ) sin ( k π / 2 N ) ω k = cosh ( ν 0 ) sin ( k π / 2 N ) (23)

for NN values of kk where

k = ± 1 , ± 3 , ± 5 , , ± ( N - 1 ) for N even k = ± 1 , ± 3 , ± 5 , , ± ( N - 1 ) for N even (24)
k = 0 , ± 2 , ± 4 , , ± ( N - 1 ) for N odd k = 0 , ± 2 , ± 4 , , ± ( N - 1 ) for N odd (25)

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 ) F ( s ) = k 1 s 2 - 2 σ k s + ( σ k 2 + ω k 2 ) (26)

for k=1,3,5,,N-1k=1,3,5,,N-1. For NN odd, F(s)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 ) F ( s ) = 1 sinh ( ν 0 ) F ( s ) = k 1 s 2 - 2 σ k s + ( σ k 2 + ω k 2 ) (27)

for k=2,4,6,...,N-1k=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 ) σ = - sinh ( ν 0 ) sin ( ( 2 k + 1 ) π / 2 N ) (28)
ω k = cosh ( ν 0 ) cos ( ( 2 k + 1 ) π / 2 N ) ω k = cosh ( ν 0 ) cos ( ( 2 k + 1 ) π / 2 N ) (29)

for NN values of kk where k=0,1,2,,N-1k=0,1,2,,N-1

Note the similarity to the pole locations for the Butterworth filter. Cross multiplying, squaring, and adding the terms in (Equation 28,Equation 29) gives

( σ k sinh ( ν 0 ) ) 2 + ( ω k cosh ( ν 0 ) ) 2 = 1 ( σ k sinh ( ν 0 ) ) 2 + ( ω k cosh ( ν 0 ) ) 2 = 1 (30)

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 circleEntry 4, Entry 5, Entry 3, Entry 6, Entry 1, Entry 7.

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 ((Reference)).

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=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=ω=. The slope of the response near the transition from pass to stopband at ω=1ω=1 becomes steeper as both the order increases and the allowed passband error ripple increases. The dropoff is more rapid than for the Butterworth filterEntry 7.

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||F| over the passband frequencies of 0<ω<10<ω<1. There can be confusion over this point as two definitions appear in the literature. Most digital Entry 4, Entry 5, Entry 3 and analog Entry 7 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||||F|-|Fd||, where |Fd||Fd| is the center line in the passband around which |F||F| oscillates. The following formulas relate the passband ripple δδ, the passband ripple aa in positive dB, and the transfer function parameter ϵϵ.

a = 10 log ( 1 + ϵ 2 ) = - 20 log ( 1 - δ ) , a = 10 log ( 1 + ϵ 2 ) = - 20 log ( 1 - δ ) , (31)
ϵ = 2 δ - δ 2 1 - 2 δ + δ 2 = 10 a / 10 - 1 , ϵ = 2 δ - δ 2 1 - 2 δ + δ 2 = 10 a / 10 - 1 , (32)
δ = 1 - 10 - a / 20 = 1 - 1 1 + ϵ 2 δ = 1 - 10 - a / 20 = 1 - 1 1 + ϵ 2 (33)

In some cases, stopband performance is not given in terms of degree of flatness at ω=ω=, but in terms of a maximum allowed magnitude GG in the stopband above a certain frequency ωsωs, i.e., G>|F|>0G>|F|>0 for 1<ωs<ω<1<ωs<ω<. For a given ϵϵ, this will determine the order as the smallest positive integer satisfying

N cosh - 1 ( 1 - G 2 ϵ G 2 ) cosh - 1 ( ω s ) N cosh - 1 ( 1 - G 2 ϵ G 2 ) cosh - 1 ( ω s ) (34)

The design of a Chebyshev filter involves the following steps:

  • The maximum-allowed passband variation must be in the form of δδ or aa. From this, the parameter ϵϵ is calculated using (Equation 32).
  • The order NN is determined by the desired flatness at ω=ω= or a maximum-allowed response for frequencies above ωsωs using (Equation 34).
  • ν0ν0 is calculated from ϵϵ and nn using (Equation 19), and the scale factors sinh(ν0)sinh(ν0) and cosh(ν0)cosh(ν0) are then determined.
  • The pole locations are calculated from (Equation 22) or (Equation 29). This can be done by scaling the poles of a Butterworth prototype filter.
  • These pole locations are combined in (Equation 27) and (Equation 28) 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 (Equation 34) 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 ϵϵ from (Equation 32) gives a stopband gain at ωsωs that is less than GG, or the same design can be viewed as giving the maximum-allowed gain GG at a lower frequency than ωsωs. An alternate approach is to solve (Equation 34) for a new value of ϵϵ, then cause (Equation 34) to be an equation with the specified ωsωs and GG. 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=0.1δ=0.1 or a=0.91515a=0.91515 dB, and can allow no greater response than G=0.2G=0.2 for frequencies above ωs=1.6ωs=1.6 radians per second.

Given δ=0.1δ=0.1 or a=0.91515a=0.91515, equation (Equation 32) implies

ϵ = 0 . 484322 ϵ = 0 . 484322 (35)

Given G=0.2G=0.2 and ωs=1.6ωs=1.6, equation (Equation 34) implies an order of N=3N=3. From ϵϵ and NN, ν0ν0 is 0.49074 from (Equation 19) and

sinh ( ν 0 ) = 0 . 510675 sinh ( ν 0 ) = 0 . 510675 (36)
cosh ( ν 0 ) = 1 . 122849 cosh ( ν 0 ) = 1 . 122849 (37)

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 ) (