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Chebyshev Filter Properties
C. Sidney Burrus
This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License †
1 Chebyshev Filter Properties
The Butterworth lter does not give a su ciently good approximation across the complete passband in many cases. The Taylor's series approximation is often not suited to the way speci cations are given for lters. An alternate error measure is the maximum of the absolute value of the dierence between the actual lter response and the ideal. This is considered over the total passband. This is the Chebyshev error measure and was de ned and applied to the FIR lter design problem. For the IIR lter, 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 lter, and simple design formulas result, just as for the Butterworth lter. The design of Chebyshev lters 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 [4], [7]. The form for the magnitude squared of the frequency-response function for the Chebyshev lter is
|F (jω) |^2 =
1 + ε^2 CN (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 ((1)).
1.1 CHEBYSHEV POLYNOMIALS
The Chebyshev polynomial is a powerful function in approximation theory. Although the function is a polynomial, it is best de ned and developed in terms of trigonometric functions by[4], [5], [1], [7].
CN (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.
CN (ω) = cos (N φ) (3) ∗Version 1.1: Jun 24, 2008 12:10 am GMT- †http://creativecommons.org/licenses/by/2.0/
where
ω = cos (φ) (4) Although this de nition of CN (ω) may not at rst appear to result in a polynomial, the following recursive relation derived from ((4)) shows that it is a polynomial.
CN +1 (ω) = 2ωCN (ω) − CN − 1 (ω) (5) From ((2)), it is clear that C 0 = 1 and C 1 = ω, and from ((5)), it follows that
C 2 = 2ω^2 − 1 (6)
C 3 = 4ω^3 − 3 ω (7)
C 4 = 8ω^4 − 8 ω^2 + 1 (8) etc. Other relations useful for developing these polynomials are
C^2 N (ω) = (C 2 N (ω) + 1) / 2 (9)
CM N (ω) = CM (CN (ω)) (10) where M and N are coprime. These are remarkable functions [7]. They oscillate between +1 and -1 for − 1 < ω < 1 and go monotoni- cally to +/- in nity 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 ((3)) and ((4)). For − 1 < ω < 1 , a plot of CN (ω) can be made using the concept of Lissajous gures. Example plots for C 0 , C 1 , C 2 , C 3 , and C 4 are shown in Figure 1.
Analog Chebyshev Filter
Magnitude Response
Frequency, ω
passband ripple
Figure 2: Fifth Order Chebyshev Filter Frequency Response
The approximation parameters must be clearly understood. The passband ripple is de ned to be the dierence 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 de nitions appear in the literature. Most digital [4], [5], [3], [2] and analog [7] lter design books use the de nition just stated. Approximation literature, especially concerning FIR lters, 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 Figure 2, 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 ((1)) determines the ripple in the passband and the order N determines the rate that the response goes to zero as ω goes to in nity. Pole Locations A method for nding the pole locations for the Chebyshev lter transfer function is next developed. The details of this section can be skipped and the results in ((22),(24)) used if desired. From ((1)), it is seen that the poles of F F (s) occur when
1 + ε^2 C N^2 (s/j) = 0 (11)
or
CN = ±
j ε
From ((4)), de ne φ = cos−^1 (ω) with real and imaginary parts given by
φ = cos−^1 (ω) = u + jv (13) This gives,
CN = cos (N φ) = cos (N u) cosh (N ν) − jsin (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 (15) which implies
cos (N u) = 0 (16) which in turn implies that u takes on values of
u = uk = (2k + 1) π/ 2 N, k = 0, 1 , ...N − 1 (17) For these values of u, sin (nu) = ± 1 , we have
sinh (N ν) = 1/ε (18) which requires ν to take on a value of
ν = ν 0 =
sinh−^1 (1/ε)
/N (19)
Using s = jω gives
s = jω = jcos (φ) = jcos (u + jν) = jcos ((2k + 1) π/ 2 N + jν 0 ) (20) which gives the location of the N poles in the s plane as
sk = σk + jωk (21) where
σk = −sinh (ν 0 ) cos (kπ/ 2 N ) (22)
ωk = cosh (ν 0 ) sin (kπ/ 2 N ) (23) for N values of k where
k = ± 1 , ± 3 , ± 5 , · · · , ± (N − 1) for N even (24)
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 lter. For N even, the form is
F (s) =
k
s^2 − 2 σks + (σ k^2 + ω^2 k)
δ = 1 − 10 −a/^20 = 1 −
1 + ε^2
In some cases, stopband performance is not given in terms of degree of atness 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 εG^2
cosh−^1 (ωs)
The design of a Chebyshev lter 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 ((32)).
- The order N is determined by the desired atness at ω = ∞ or a maximum-allowed response for frequencies above ωs using ((34)).
- ν 0 is calculated from ε and n using ((19)), and the scale factors sinh (ν 0 ) and cosh (ν 0 ) are then determined.
- The pole locations are calculated from ((22)) or ((29)). This can be done by scaling the poles of a Butterworth prototype lter.
- These pole locations are combined in ((27)) and ((28)) to give the nal lter transfer function.
This process is easily programmed for computer aided design as illustrated in Program 8 in the appendix. If the design procedure uses ((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 speci cations. Direct use of the order with ε from ((32)) 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 ((34)) for a new value of ε, then cause ((34)) to be an equation with the speci ed ωs and G. This gives a lter that exactly meets the stopband speci cations and gives a smaller passband ripple than originally requested. A similar set of alternatives exists for the elliptic-function lter. Example 7-2. The Design of a Chebyshev Lowpass Filter. The design speci cations 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 ((32)) implies
ε = 0. 484322 (35) Given G = 0. 2 and ωs = 1. 6 , equation ((34)) implies an order of N = 3. From ε and N , ν 0 is 0.49074 from ((19)) and
sinh (ν 0 ) = 0. 510675 (36)
cosh (ν 0 ) = 1. 122849 (37) These multipliers are used to scale the root locations of the example third-order Butterworth lter to give
F (s) =
(s + 0.51067) (s + 0.25534 + j 0 .97242) (s + 0. 25534 − j 0 .97242)
F (s) =
(s + 0.51067) (s^2 + 0. 510675 s + 1.010789)
F (s) =
s^3 + 102135s^2 + 1. 271579 s + 0. 516185
The frequency response is shown in Figure 3
Design of a Chebyshev Filter
Magnitude Response
Frequency ω
Ripple d = 0.
Figure 3: Example Design of a Third Order Chebyshev Filter Frequency Response
2 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 lter or the Chebyshev II lter. This error measure uses a Taylor's series for the passband just as for the Butterworth lter and minimizes the maximum error over the total stopband. It reverses the types of approximation used in the preceding section. A fth-order example is illustrated in c and Figure 4c. Rather than developing the approximation directly, it is easier to modify the results from the regular Chebyshev lter. First, the frequency variable ω in the regular Chebyshev lter, described in ((1)), is replaced by 1 /ω, which interchanges the characteristics at ω equals zero and in nity and does not change the performance at ω equals unity. This converts a Chebyshev lowpass lter into a Chebyshev highpass lter as illustrated in Figure 4 moving from the rst to second frequency response.
the locations are
σ' k =
σk σ^2 k + ω^2 k
ω' k =
ωk σ^2 k + ω k^2
Although this gives a straightforward formula for calculating the location of the poles and zeros of the inverse- Chebyshev lter, they do not lie on a simple geometric curve as did those for the Butterworth and Chebyshev lters. Note that the conditions for a Taylor's series approximation with preset zero locations are satis ed. A partially factored form for the Butterworth lter and for the Chebyshev lter can be written for the inverse-Chebyshev lter using the zero locations from ((45)) and the pole locations from the regular Chebyshev lter. For N even, this becomes
F (s) =
k
s^2 + ω^2 zk
k (s (^2) − 2 (σk/ (σ^2 k +^ ω
2 k))^ s^ + 1/^ (σ
2 k +^ ω
2 k)^ (49) 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^2 zk
(s + 1/sinh (ν 0 ))
k (s (^2) − 2 (σk/ (σ^2 k +^ ω
2 k))^ s^ + 1/^ (σ
2 k +^ ω
2 k)^ (50) for k = 2, 4 , 6 , · · · , N − 1 Because of the relationships between the locations of the poles of the Butterworth, Chebyshev, and inverse-Chebyshev lters, it is easy to write a design program with many common calculations. That is illustrated in the program in the appendix.
2.1 Inverse-Chebyshev Filter Design Procedures
The natural form for the speci cations of an inverse-Chebyshev lter is in terms of the atness of the response at ω to determine the passband, and a maximum allowable response in the stopband. The lter order and the stopband ripple are the parameters to be determined by the speci cations. The rate of dropo near the transition from pass to stopband is similar to the regular Chebyshev lter. Because practical speci cations often allow more passband ripple than stopband ripple, the regular Chebyshev lter will usually have a sharper dropo than the inverse-Chebyshev lter. Under those conditions, the inverse-Chebyshev lter will have a smoother phase response and less time-domain echo eects. The stopband ripple d is simply de ned as the maximum value that |F (jω) | assumes in the stopband, which is the set of frequencies 1 < ω < ∞. An alternative speci cation 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 ((41))
ε =
δ √ 1 − δ^2
δ =
ε √ 1 + ε^2
b = − 10 log
ε^2 /
1 + ε^2
= − 20 log (d) (53) In some cases passband performance is not given in terms of degree of atness 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/ωp)
The design of an inverse-Chebyshev lter 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 ((51)).
- The order N is determined from the desired atness at ω = 0, or from a minimum allowed response for frequencies up to ωp using ((54)).
- ν 0 and sinh (ν 0 ) and cosh (ν 0 ) are calculated just as for the regular Chebyshev lter.
- The pole locations for the prototype Chebyshev lter are calculated from ((46)) and ((48)) and then "inverted" according to ((41)) to give the inverse- Chebyshev lter pole locations.
- The pole locations are combined in ((41)) to give the nal lter transfer function denominator.
- The zero locations are calculated from ((45)) and combined with the pole locations to give the total transfer function ((49)) or ((50)).
Example Design of an Inverse-Chebyshev Filter A third-order inverse-Chebyshev lowpass lter 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 ν 0 = 0. 99774. The scale factors are sinh = 1. 171717 and cosh = 1. 540429. The prototype Chebyshev lter transfer function is
F (s) =
(s + 1.1717) (s^2 + 1. 1717 s + 2.0404)
The zeros are calculated from ((45)), and the poles of the prototype are inverted to give, from ((50)), the desired inverse- Chebyshev lter transfer function of
F (s) =
s^2 + 4/ 3 (s + 0.85345) (s^2 + 0. 57425 s + 0.490095)
References
[1] B. Gold and C. M. Rader. Digital Processing of Signals. McGraw-Hill, New York, 1969.
[2] Sanjit K. Mitra. Digital Signal Processing, A Computer-Based Approach. McGraw-Hill, New York, third edition, 2006. First edition in 1998, second in 2001.
[3] A. V. Oppenheim and R. W. Schafer. Discrete-Time Signal Processing. Prentice-Hall, Englewood Clis, NJ, second edition, 1999. Earlier editions in 1975 and 1989.
[4] T. W. Parks and C. S. Burrus. Digital Filter Design. John Wiley & Sons, New York, 1987.
[5] L. R. Rabiner and B. Gold. Theory and Application of Digital Signal Processing. Prentice-Hall, Englewood Clis, NJ, 1975.
[6] F. J. Taylor. Digital Filter Design Handbook. Marcel Dekker, Inc., New York, 1983.
[7] M.E. Van Valkenburg. Analog Filter Design. Holt, Rinehart, and Winston, New York, 1982.
