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Chebyshev or Equal Ripple Error
Approximation Filters
C. Sidney Burrus
This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License †
If one poses the FIR lter design problem by requiring the maximum error over certain bands of fre- quencies be minimized, we call the resulting lter a Chebyshev lter or an equal ripple lter. The fact that the minimization of the Chebyshev or L∞ error results in an equal ripple error comes from the alternation theorem. This very powerful theorem allows one to minimize the Chebyshev error by directly constructing an equal ripple approximation with the proper number of ripples. That is the basis of several very eective algorithms, including the Remez exchange algorithm. There are several ways one could pose the Chebyshev FIR lter design problem. For a simple length-N linear phase, lowpass lter with a transition band, if one considers the length N, the passband ripple δp, the stopband ripple δs, and the transition bandwidth ∆ = ωs − ωp, then one can x or constrain any three of them and minimize the fourth. Or, as Parks and McClellan do, x the band edges, ωp and ωs, and the ratio of δp and δs and minimize one of them. The Chebyshev error measure is often used for approximation in digital lter design. This is particularly true when the signals and/or noise are narrow band or single frequency or when one wants to minimize worst case possibilities. Theoretical justi cation for its use has been given by Weisburn, Parks, and Shenoy [93]. For FIR lter design, the Parks-McClellan formulation of the lter design problem and application of the Remez exchange algorithm is most commonly used [48], [49]. It is a particularly interesting and powerful method that should be studied and understood to be fully utilized. Linear programming was used earlier [88], [26], [64] but dropped out of favor when the Parks-McClellan algorithm was introduced. It is now becoming more popular again because of more powerful computers, better algorithms [82], [6], and linear programming's ability to allow a variety of constraints [80]. Still another approach to achieving a Chebyshev approximation is to minimize the pth^ power of the error using a large value of p or to use an iterative scheme that solves a weighted least squared error with the weights at each stage determined by the error of the previous stage [15]. Still another design method that produces an equal ripple error approximation uses a constrained least squared error criterion [77], [76] which results in a Chebyshev solution if tight constraints are imposed. The early work by Herrmann and Sch¨ussler [27], [29] and the algorithm by Hofstetter, Oppenheim, and Siegel [31], [32] posed and solved a similar problem but they had only approximate control of ωo (or ωp or ωs) and always achieved the extra ripple" design. Given the proper speci cations, the Parks-McClellan algorithm could design any lter that the Hofstetter-Oppenheim-Siegel algorithm could, but the opposite is not true. This seems to be one of the reasons the Hofstetter-Oppenheim-Siegel algorithm is not commonly used. ∗Version 1.2: Nov 27, 2008 7:39 am US/Central †http://creativecommons.org/licenses/by/2.0/
1 The Linear Phase FIR Filter Chebyshev Approximation Problem
The Chebyshev error is de ned as the maximum dierence between the actual and desired response over a band or several bands of frequencies. This is
= max ω∈Ω
|A (ω) − Ad (ω) | (1)
where Ω is the union of the bands of frequencies that the approximation is over [17], [20]. The approximation problem in lter design is to choose the lter coe cients to minimize . One way to minimize is to set up the frequency response in four equations for the four types of linear phase FIR lters as done in Equation? from FIR Digital Filters^1 , Equation? from FIR Digital Filters^2 , and the corresponding sine expressions. An alternative approach [48] uses the fact that all four can be obtained from the odd-length, even-symmetry type 1 and uses only Equation? from FIR Digital Filters^3. From one of these frequency response representations together with powerful Alternation Theorem several optimization schemes can be developed. If the amplitude response for odd L is expressed as a sum of R cosine terms
A (ω) =
R∑− 1
n=
a (n) cos (ωn) (2)
or for even L
A (ω) =
∑^ R
n=
a (n) cos (ω (n − 1 /2)) (3)
with R = M + 1 = L+1 2 for odd length-L and R = L/ 2 for even length-L, as derived in Equation? from FIR Digital Filters^4 and Equation? FIR Digital Filters^5 , then Theorem 1 If A (ω) is the linear combination of R cosine functions given in (2) or (3), the necessary and su cient conditions for A (ω) to be the least Chebyshev error approximation to Ad (ω) over ω ∈ Ω are: The error function, ε (ω) = A (ω) − Ad (ω) have at leastR + 1 extremal frequencies in Ω. The extremal frequencies are ordered points ω 1 < ω 2 < · · · < ωR+1 such that ε (ωk) = −ε (ωk+1) and maxω∈Ω|ε (ω) | = |ε (ωk) | for k = 1, 2 , · · · , R + 1. The alternation theorem [48], [59] states that the minimum Chebyshev error has at least R + 1 extremal frequencies. This is stated mathematically by
A (ωk) = Ad (ωk) + (−1)kδ (4) for k = 0, 1 , 2 , · · · , R, where the ωk are the ordered extremal frequencies where the equal ripple error has maximum value. In other words, the optimal solution to the linear phase FIR lter design problem will have an equal ripple error function with a required number of ripples. How all of these characteristics relate can be rather complicated and good designs require experience [28]. When applied to other approximation problems, care must be taken to ensure the approximating functions satisfy the Haar conditions" or other restrictions [17], [49], [20], [59].
(^1) "FIR Digital Filters", (34) http://cnx.org/content/m16889/latest/#uid1009 (^2) "FIR Digital Filters", (40) http://cnx.org/content/m16889/latest/#uid11009 (^3) "FIR Digital Filters", (34) http://cnx.org/content/m16889/latest/#uid1009 (^4) "FIR Digital Filters", (34) http://cnx.org/content/m16889/latest/#uid1009 (^5) "FIR Digital Filters", (40) http://cnx.org/content/m16889/latest/#uid11009
M = (L-1)/2;
C = cos(2pi(f[0:M])); %Freq response matrix CC = [C, -ones(LF,1); -C, -ones(LF,1)]; %LP matrix AD = [Ad; -Ad]; c = [zeros(M+1,1);1]; %Cost function x0 = [zeros(M+1,1);max(AD)+1]; %Starting values x = lp(c,CC,AD,[],[],x0); %Call the LP d = x(M+2); %delta or deviation a = x(1:M+1); %Half impulse resp. h = [a(M+1:-1:2);2a(1);a(2:M+1)]./2; %Impulse response
This program has numerical problems for lters longer than 10 or 20 and is fairly slow. The lp() function uses an algorithm that seems not well suited to the equations required by lter design. It would be nice to have Meteor written in Matlab, both to show how the Simplex algorithm works, and to have an e cient LP lter design system in Matlab. The above program has been tested using Karmarkar's algorithm [6], [66], [82] as implemented in Matlab by Lang [35]. It proved to be robust and reliable for lengths up to 100 or more. It was faster than the Matlab function but slower than Meteor or CPlex. Its use should be further investigated. Direct use of quadratic programming and other optimization algorithms seem promising [22], [39], [52], [55], [53], [54], [57], [91], [90], [94]
3 Chebyshev Approximations using the Exchange Algorithms
A very e cient algorithm which uses the results of the alternation theorem is called the Remez exchange algorithm. Remez [65], [17], [59] showed that, under rather general conditions, an algorithm that takes a starting estimate of the location of the extremal frequencies and exchanges them with a new set calculated at each iteration will converge to the optimal Chebyshev approximation. The e ciency of this algorithm comes from nding the optimal solution by directly constructing a function that satis es the alternation theorem rather than minimizing the Chebyshev error as done by the linear programming technique. The Remez exchange algorithm has proven to be well suited to the design of linear phase FIR lters [44], [47], [30]. A particularly useful FIR lter design implementation of the Remez exchange is called the Parks- McClellan algorithm and is described in [49], [63], [62], [48]. It has been implemented in Fortran in [50], [62], [18], [48] and in Matlab in a program at the end of this material. The Matlab program is particularly helpful in understanding how the algorithm works, however, because it does not use any special tricks, it is limited to lengths of 60 or so. Extensions and details can be found in [45], [10], [21], [78], [33], [24], [25], [71], [73], [72], [5]. This is a robust, e cient algorithm that signi cantly changed DSP when Parks and McClellan rst described it in 1972 and has undergone important improvements. Examples are illustrated in [62], [46].
3.1 The Basic Parks-McClellan Formulation and Algorithm
Parks and McClellan formulated the basic Chebyshev FIR lter design problem by specifying the desired amplitude response A (ω) and the transition band edges, then minimizing the weighted Chebyshev error over the pass and stop bands. For the basic lowpass lter illustrated in Figure 1, the pass band edge ωp and the stop band edge ωs are speci ed, the maximum passband error is related to the maximum stop band error by δp = K δs and they are minimized.
Length−15 Optimal Chebyshev FIR Filter
Normalized Frequency
Amplitude Response, A
passband
transitionband
stopband
Figure 1: Amplitude Response of a Length-15 Optimal Chebyshev Filter
Notice that if there is no transition band, i.e. ωp = ωs, that δp + δs = 1 and no minimization is possible. While not the case for a least squares approximation, a transition band is necessary for the Chebyshev approximation problem to be well-posed. The eects of a small transition band are large pass and stopband ripple as illustrated in Figure 3b. The alternation theorem states that the optimal approximation for this problem will have an error function with R + 1 extremal points with alternating signs. The theorem also states that there exists R + 1 frequencies such that, if the Chebyshev error at those frequencies are equal and alternate in sign, it will be minimized over the pass band and stop band. Note that there are nine extremal points in the length- example shown in Figure 1, counting those at the band edges in addition to those that are interior to the pass and stopbands. For this case, R = (L + 1) / 2 which agree with the example. Parks and McClellan applied the Remez exchange algorithm [49] to this lter design problem by writing R + 1 equations using Equation? from FIR Digital Filters^7 and Equation? from Design of IIR Filters by Frequency Transformations^8 evaluated at the R + 1 extremal frequencies with R unknown cosine parameters
(^7) "FIR Digital Filters", (34) http://cnx.org/content/m16889/latest/#uid1009 (^8) "Properties of IIR Filters", (1) http://cnx.org/content/m16898/latest/#uid1
It is interesting to note that at each iteration, the approximation is optimal over that set of extremal frequencies and δ increased over the previous iteration. At convergence, δ has increased to the maximum error over Ω and that is the minimum Chebyshev error. At each iteration, the exchange of a proper set of extremal frequencies with alternating signs of the errors is always possible. One can show there will never be too few and if there are too many, one uses those corresponding to the largest errors. In step 4 it is suggested that the amplitude response A (ω) be calculated over a dense grid in the pass and stopbands and in step 5 the local extremes are found by searching over this dense grid. There are more accurate methods that use bisection methods and/or Newton's method to nd the extremal points. In step 2 it is suggested that the simultaneous equation of (12) be solved. Parks and McClellan [50] use a more e cient and numerically robust method of evaluating δ using a form of Cramer's rule. With that δ, an interpolation method can be used to nd a (n). This is faster and allows longer lters to be designed than with the linear algebra based approach described here. For the low pass lter, this formulation always has an extremal frequency at both pass and stop band edges, ωp and ωs, and at ω = 0 and/or at ω = π. The extra ripple lter has R + 2 extremal frequencies including both zero and pi. If this algorithm is started with an incorrect number of extremal frequencies in the stop or pass band, the iterations will correct this. It is interesting and informative to plot the frequency response of the lters designed at each iteration of this algorithm and observe how the correction takes place. The Parks-McClellan algorithm starts with xed pass and stop band edges then minimizes a weighted form of the pass and stop band error ripple. In some cases it may be more appropriate to x one of the ripples and minimize the other or to x both ripples and minimize the transition band width. Indeed Sch¨ussler, Hofstetter, Tufts, and others [29], [27], [31], [32] formulated some of these ideas before Parks and McClellan developed their algorithm. The DSP group at Rice has developed some modi cations to these methods and they are presented below.
3.2 Examples of the Parks-McClellan Algorithm
Here we look at several examples of lters designed by the Parks-McClellan algorithm. The examples here are length-15 with that shown in Figure 2a having a passband 0 < f < 0. 3 , a transition band 0. 3 < f < 0. 5 , and a stopband 0. 5 < f < 1. The number of cosine terms in the frequency response formula is R = 8, therefore, the alternation theorem says we must have at least R + 1 extremal points. There are four in the passband, counting the one at zero frequency, the minimum, the maximum, and the minimum at the bandedge. There are ve in the stopband, counting the ones at the bandedge and at f = 1. So, the number is nine which is at least R + 1. However, in Figure 2c, there are ten extremal points but that is also at least 9, so it also is optimal. For a low pass lter, the maximum number of extremal points is R + 2 and that is what this lter has. This special case is called the maximum ripple" case.
Optimal Chebyshev FIR Filter
Normalized Frequency
Amplitude Response, A −2 −1 0 1 2
Zero Location
Real part of z
Imaginary part of z
Maximum Ripple Chebyshev Filter
Normalized Frequency
Amplitude Response, A −2 −1 0 1 2
Zero Location
Real part of z
Imaginary part of z
Figure 2: Amplitude Response of Length-15 Optimal Chebyshev Filters
It is possible to have ripples that do not touch the maximum value and, therefore, are not considered extremal points. That is illustrated in Figure 3a. The eects of a narrow transitionband are illustrated in Figure 3c. Note the zero locations for these lters and how they relate to the amplitude response.
0 0.2 0.4 0.6 0.8 1
−1.
−
−0.
0
1
a. Optimal Chebyshev Bandpass Filter
Normalized Frequency
Amplitude Response, A
−2 −1 0 1 2 −1.
−
−0.
0
1
b. Zero Location
Real part of z
Imaginary part of z
Figure 4: Amplitude Response of Length-31 Optimal Chebyshev Bandpass Filter
3.3 The Modi ed Parks-McClellan Algorithm
If one wants to x the pass band ripple and minimize the stop band ripple [73], equation (12) is changed so that the pass band ripple is added to the appropriate top part of the vector Ad of the desired response and the unknown stop band is kept in the lower part of the last column of the cosine matrix C.
Ad (ω 0 ) Ad (ω 1 ) .. . Ad (ωp) Ad (ωs) .. . Ad (ωR)
δp −δp .. . ±δp 0 .. . 0
cos (ω 0 0) cos (ω 0 1) · · · cos (ω 0 (R − 1)) 0 cos (ω 1 0) cos (ω 1 1) · · · cos (ω 1 (R − 1)) 0 .. .
cos (ωp0) cos (ωp1) · · · cos (ωp (R − 1)) 0 cos (ωs0) cos (ωs1) · · · cos (ωs (R − 1)) 1 .. .
cos (ωR0) cos (ωR1) · · · cos (ωR (R − 1)) ± 1
a (0) a (1) a (2) .. . a (R − 1) δs
Iteration of this equation will keep the pass band ripple δp xed and minimize the stop band ripple δs. A problem with convergence occurs if one of the δ's becomes negative during the iterations. A modi cation to the basic exchange has been developed to give reliable convergence [73].
3.4 The Hofstetter, Oppenheim, and Siegel Algorithm
This algorithm [31], [32], [73] came into existence in order to design the lters posed by Herrmann and Schu¨ssler [29], [27] where both the pass and stop band ripple sizes, δp and δs, are xed and the location of
the transition band is not directly controlled. This problem results in a maximum ripple design which, for the lowpass lter, requires extremal frequencies at both ω = 0 and ω = π but does not use either pass or stop band frequencies ωp or ωs. This results in R extremal frequencies giving R equations to nd the R values of a (n).
Ad (ω 0 )
Ad (ω 1 )
Ad (ωp− 1 )
Ad (ωs+1)
Ad (ωR− 1 )
δp
−δp
±δp
δs
±δs
cos (ω 0 0) cos (ω 0 1) · · · cos (ω 0 (R − 1))
cos (ω 1 0) cos (ω 1 1) · · · cos (ω 1 (R − 1))
cos (ωp− 1 0) cos (ωp− 1 1) · · · cos (ωp− 1 (R − 1))
cos (ωs+10) cos (ωs+11) · · · cos (ωs+1 (R − 1))
cos (ωR− 1 0) cos (ωR− 1 1) · · · cos (ωR− 1 (R − 1))
a (0)
a (1)
a (2)
a (R − 1)
This algorithm is iterated as a multiple exchange, keeping the number of ripples in the pass and stop band constant, to give an optimal extra ripple lter. The location and width of the transition band is controlled only by the choice of how the number of initial ripples are divided between the pass and stop band. The nal lter may not have the transition located where you want it. Indeed, no solution may exist with the desired location of the transition band. The designs produced by the HOS algorithm are always maximum ripple but this comes with a loss of accurate control over the location of the transition band. The algorithm is not, strictly speaking, an optimization algorithm. It is an interpolation algorithm. The Chebyshev error is not minimized, the designed amplitude interpolates the speci ed error ripples. However, although not directly minimized, the transition band width of these designs seems to be minimized [63], [51], [62]. Extra or maximum ripple designs seem to be e cient in using all the zeros to produce small ripple size and narrow transition bands, however, the loss of accurate control over the location of the transition bands becomes even more problematic with multiple transition band designs. Perhaps some compromise methods can be devised that use some of the e ciency of the maximum ripple approximations with some of the control of other methods. The next two design methods are of that type.
3.5 The Shpak and Antoniou Algorithm
Shpak and Antoniou [78] propose decoupling the size of the pass and stopband ripple sizes in order to have control over the pass and stop band edges and have an extra ripple design. The Parks-McClellan design has the ripple sizes related with a xed weight δp = K δs, the modi ed Parks-McClellan design xes one ripple size and minimizes the other, the Hostetter, Oppenheim, and Siegel design xes both ripple sizes but cannot set the transition band edges. The Shpak-Antoniou design xes the transition band edges and gives a maximum ripple design with minimum ripple but the relationship of the pass and stopband ripple is uncontrolled. This method has two ripple sizes, δp and δs, appended to the a (n) vector similar to the single δ used in (12) or (13). This allows controlling an additional extremal frequency and results in an extra ripple approximation. This can become somewhat complicated for multiple transition bands but seems very exible [5].
3.7 Estimations of , the Length of Optimal Chebyshev FIR Filters
All of the design methods discussed so far have assumed that N ,the length of the lter, is given as part of the seci cations. In many cases, perhaps even most, N is a parameter that we would like to minimize. Often speci cations are to meet certain pass and stopband ripple speci cations with given pass and stopband edges and with the shortest possible lter. None of our methods will do that. Indeed, it is not clear how to do that kind of optimization other than by some sort of search. In other words, design a set of lters of dierent lengths and choose the one that meet the speci cations with minimum length. Fortunately, emperical formulas have been derived that give a good estimate of the relationship of the length of an optimal Chebyshev FIR lter for given pass and stopband ripple and transition band edges [62], [63]. Kaiser's formula is
N =
− 20 log 10
δpδs
14 .6 (fs − fp)
and it is fairly accurate for average lter speci cations (neither wide nor narrow bands).
3.8 Examples of Optimal Chebyshev Filters
In order to better understand the nature of an optimal Chebyshev and to see the power of the Parks- McClellan algorithm, we present the design of a length-21 linear phase FIR bandpass lter. To see the eects of the design speci cations, we will x the two pass band edges and the upper stop band edge, then look at the eects of varying the lower stop band edge. The Matlab program that generated the designs is:
% ChebyPlot9.m generates Chebyshev figures. % Change in opt frequency response as band edge is changed, csb 1/26/ N = 20; M = [0 0 1 1 0 0]; W = [7.5 10 7.5]; ff = [0:512]/512; k=0; %for fk = .10:.02:. % k = k+1; clf; for k = 1: fk = .1 + .02(k-1); F = [0 fk .35 .8 .85 1]; b = firpm(N,F,M,W); %clf; axis([0 1 0 1.2]); AA = abs(fft(b,1024)); AA = AA(1:513); dd = max(AA(1:50)); ddd = dd(W(1)/W(2)); subplot(3,2,k); plot(ff,AA,'r'); hold; plot([0 F(2) F(2) F(5) F(5) 1],[dd dd 0 0 dd dd],'b'); plot([0 F(3) F(3) F(4) F(4) 1],[0 0 1-ddd 1-ddd 0 0],'b'); plot([0 F(3) F(3) F(4) F(4) 1],[0 0 1+ddd 1+ddd 0 0],'b'); title('L-21 Chebyshev Filter, f_s = 0.1'); ylabel('Magnitude |H(\omega)|'); pause; end; hold off;
The results are shown in Figures Figure 5 and Figure 6.
L−21 Chebyshev Filter, fs = 0.
Magnitude |H(
ω
L−21 Chebyshev Filter, fs = 0.
L−21 Chebyshev Filter, fs = 0.
Magnitude |H(
ω
L−21 Chebyshev Filter, fs = 0.
L−21 Chebyshev Filter, fs = 0.
Magnitude |H(
ω
Normalized Frequency: f 0 0.5^1
L−21 Chebyshev Filter, fs = 0.
Normalized Frequency: f
Figure 5: Amplitude Response of Length-21 Optimal Chebyshev Bandpass Filter with various Stop Band Edges
4 Chebyshev Approximation using Approximation
It is possible to approximate the eects of Chebyshev approximation by minimizing the pth^ power of the error. For large p this is close to the results of a true Chebyshev approximation. This is a variation on a method called Lawson's method. This approach is described in [13], [14], [15] using the iterative reweighted least squared (IRLS) error method and looks attractive in that it can use dierent p in dierent frequency bands. This would allow, for example, a least squared error approximation in the passband and a Chebyshev approximation in the stopband. The IRLS method can also be used for complex Chebyshev approximations [86].
5 Characteristics of Optimal Chebyshev Filters
Examples of expected and unexpected results of optimality. Rabiner's work will be used here. The non- unique designs for certain multiband designs will be illustrated.
6 Complex Chebyshev Approximation
Algorithms that directly use the alternation theorem, such as the standard Remez multiple exchange al- gorithm, are di cult to apply to the complex approximation or 2-D approximation problem because the concept of alternation" is di cult to de ne and the number of ripples in an optimal solution is more dif- cult to determine [92], [84], [83], [9], [40], [41], [85]. Work has been done on the complex approximation problem at Rice by Parks and Chen [16] and by Burrus, Barreto, and Selesnick [15], [7], at Erlangen by Schuessler, Preuss, Schulist, and Lang [60], [61], [67], [68], [70], [69], at MIT by Alkhairy et al [3], [4], at USC by Tseng and Gri ths [86], [87], at Georgia Tech by Karam and McClellan [34], at Cornell by Burnside and Parks [11], and by Potchinkov and Reemtsen at Cottbus [53], [54], [57], [58], [56]. The work done by Adams which uses an implementation of a constrained quadratic programming algorithm might be useful here [1], [2]. Lang has extended and further developed this constrained approach [36], [37], [38] and Selesnick is applying it to IIR lter design [75]. Tseng gives a good summary of complex approximation in [87].
7 Conclusions and Discussions of Chebyshev Design
By adding the Chebyshev lter design methods described above to the Parks-McClellan algorithm, one has a rather complete set of approaches to equal ripple lter designs that allows a wide variety of speci cations. The new exchange algorithm which minimizes the transition band width while allowing the speci cation of the center or either edge of the transition band edge may t many design environments better than the traditional Parks-McClellan. An alternative approach which speci es the pass and stop band peak error yet has no zero weighted transition band will be presented in [74], [77]. Matlab programs are available for the Parks-McClellan algorithm, the modi ed Parks-McClellan algorithm, the Hofstetter-Oppenheim-Siegel algorithm, the new minimum transition band design algorithm, and the constrained least squares algorithm. They are written with a common format and notation to easily see how they are programmed and how they are related. This book generally presents the lowpass case. The bandpass and multi-band cases use the same ideas but are a bit more complicated and are discussed in more detail in the references.
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