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On the Transition Width of Finite Impulse- Response Digital Filters I 1

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On the Transition Width of Finite Impulse- Response Digital Filters I 1 IEEE TRANSACTIONS ON AUDIO AND ELECTROACOUSTICS, VOL. AU-21, NO. 1, FEBRUARY 1973 1 On the Transition Width of Finite Impulse- Response Digital Filters THOMAS W. PARKS, LAWRENCE R. RABINER, and JAMES H. MC CLELLAN Abstract-Several properties of finite-duration impulse-response (FIR) digital filters designed to have the maximum possible number of ripples are discussed and illustrated with examples. Such filters have been called extraripple filters. Among the properties of such filters are as follows. 1) Ertraripple low-pass filters with fixed passband ripple 61 and stopband ripple 62 achieve the local minimum of transition width in the class of linear phase filters with fixed impulse-response duration of N samples. 2) For the case & =62 the minimum transition width is roughly f independent of F,, the passband cutoff frequency. 3) For the case &<a1, the minimum transition width decreases FP FS with increasing bandwidth. Several figures are included to show the relation between the Fig. 1. Frequency response and error curve for anbptimal transition width and bandwidth forlow-pass filters. low-pass filter defining the basic filter parameters. Introduction The problem of designingfinite impulse-response (FIR) low-pass digital filters with linear phase has re- cently been studied from several pointsof view [1]-[6]. W(f)is a weight function specifying the relative magni- Whilethe problem of obtainingoptimum magnitude tude of the error in the passband and stopband and is characteristics has been solved, several questions remain defined by concerning the width of the transition band (stopband edgeminus passband edge) for filters with optimum magnituderesponse. This paper attempts to answer these questions. Fig. 1 shows plots of the frequency response and the i 1, F, i f 50.5 error curve of a typical equiripple low-pass filter. The and g(f) is the actual filter amplitude response whichis passband cutoff frequencyis F, and the stopband cutoff defined as frequency is F,.The maximum deviation in the pass- band, or passband ripple is 61, and the stopband ripple g(f) = d(K) cos (27rfh) (5) is &. The width of the transition band is AF= Fa- F,. kc0 If theduration of thefilter impulse response is N samples (N odd) then, for given F, and F,,a filter with where thed(K) are related to thefilter’s impulse response optimummagnitude characteristics can have either h(k) } by the relation (n+2) or (nf3) maximain the weighted errorcurve d(n - h) = 2h(k), k = 0, 1, . * , n - 1 [S] where d(o) = h(n). (6) N-1 a=- (1) 2 The h(h) obey the symmetry condition and the weighted error curveis defined by h(k) = h(212 - K), K = 0, 1, . * - , n. (7) For the optimum magnitude filter, the maxima of the Manuscript received June 1, 1972. weighted error curve must alternate in sign and be equal T. W. Parksand J. H. McClellan arewith Rice University, in magnitude, i.e., Houston, Tex. L. R. Rabiner is with Bell Laboratories, Mtrray Hill, N. J. 07974. e(FJ = - e(Fi,J, i = 1, 2, . + I n + 1 (8) 2 IEEE TRANSACTIONS OK AGDIO AND ELECTROACOUSTICS, FEBRUAllY 1973 where are the frequenciesat which the error maxima Fi 8,- 8,; a.1 occur. -.‘-II It has been conjectured that filters withn+3 maxima (the so-called extraripple filters) have minimum transi- tionwidth [4]. Thisquestion of transitionwidth for extraripple filters was more fully explored in a paper by Hofstetter et al. [7],which has motivated this paper. By mapping the interval0 sf<3 to the interval - 1_<x 5 1 bythe transformation x=cos 27rf, theseauthors converted (5) froma trigonometric polynomial in f to a standard polynomial in x. They obtained the fre- quency response n P(.) = G(f) Ir=cos (~rf) akXIz- ( 9) k=O 0 PASSBAND CUTOFF FREQUENCY lFpl Usingthis formulation Hofstetter et nl. were ableto design a filter with (n+2) maxima x-itha smaller transi- Fig. 2. Transition width as a function of passband cutoff tion bandwidth than the same length extraripple filter frequency for K=l, N=9, and N=ll. (n+3 maxima). Both filters had the same values for 61 and 62. This“counterexample” precluded any argu- Fig. 2 shows a plot of AF versus F, for optimum ments for the optimalityof extraripple filters according filters of length N=9 and of length N = 11. For this to the authors. example 81 = 62 = 0.1 (i.e., K = 1). Since the filter design Based on these observations a more detailed evalua- parameters areF,, Fs,and K,an iterative procedure was tion of the behavior of transition handwidth as a func- usedto adjust F, (orequivalently AF) toachieve tion of passband cutoff frequency was made. This paper 61 = 62 = 0.1 to within somedesired accuracy. Generally, describesthe relation between transition bandwidth on the order of 10 iterations were required to achieve AF and passband cutoff frequency F, for linear phase the desired resolution(81 = 82= 0.1 within 0.01 percent). filters of fixed duration with optimum magnitude and As seen in Fig. 2, the extraripple filters 1)oth for N=9 fixed values for61 and 62. It will be shown that the extra-and N = 11 correspond to local minima of the curves. ripple filters are local minima on the curveof AF versus (Table I givesthe specifications of theseextraripple F, and the influence of K = 81/62 mill be described. filters.) In between these local minima are points with greatertransition width corresponding to filters with (n+2) alternations. It is interesting to note that extra- local Minimum Property of Extraripple Filfers ripple length 9 filters are essentially optimum length 11 Withinthe class of linearphase filters with fixed filters,and occur between the local minima for the duration N= 2n +1 samples, and with optimum magni- N= 11 curve.It is also interesting to note that there are tude (6, and 62 fixed), the filters with n+3 alternations optimum length 9 filters with smaller transition widths inthe error curve locally haveminimum transition than optimum length 11 filters. However, for a given width AF. This property can be shown by first noting passband edge frequency, F,, length 9 optimum filters that by the resultsin [j],filters with optimum magni- always havegreater transition widths than optimum tudemust have either (n+2) or (n+3) alternations. length 11 filters. Furthermore, if one of the error maxima is missing, it To illustrate several of the above points, filters cor- can only be missing from one of the endpoints (f=O, responding to four of the points in the curves of Fig. 2 +, or equivalently x = i l),for otherwise more than one have been examined in greater detail. These points are alternation point would be lost and the resulting filter denoted in Fig. 2 by the letters A, B, C, and D. Fig. 3 would not have optimum magnitude. Finally, if one of shows the magnitude frequency response and z-plane the endpoint maxima aremissing, the same scaling argu-plots of the zero positions for these four filters. FilterA ment as used in [7]shows that the linear compression is an optimum filter (N=11) with 7 equiripple alterna- to give(nS.3) error maximareducesAF. Thus the extra- tions (n+2) and one alternation which is not equal in ripple filters represent local minima on the curve of AF amplitude to the other ripple peaks. T~USfilter A is versus F,. optimum,but not equiripple. Filter B is alength 11 extraripple filter with 8 equiripple alternations (n+3). Filter C is an optimum length 11 filter with 7 equiripple The Relation Between AF and Fp alternations.Filter D is a length 9 extraripplefilter, In order to better illustrate the detailed behavior of which is also an optimum length11 filter. A Comparison of transition width, several figures are included for variousfilters C and D shows they are essentially the same filter. choices of filter length, and weighting factor K = 61/82. The z-plane plots of Fig. 3 show how the zeros move PARKS et al. : FIR DIGITAL FILTER TRANSITION WIDTH 3 TABLE I Extraripple Filters for N =9 andN = 11 (61 =62 = 0.1) nP Number of passbandripples ns Number of stopband ripples Fp Passband cutofffrequency F, Stopband cutoff frequency AF Width of transition band N np n, FP F6 AF 11 0.0344061 5 0.096799 0.062393 0 1 4 0.042644 0.119961 0.077317 li 2 4 0.121333 0.189137 0.067804 9 0.0837900.2339090.1501192 3 3 3 0.215756 0.284244 0.0684880.2842440.215756 3 11 3 9 3 2 0.0837900.3498810.266091 11 4 2 0.310863 0.378667 0.067804 9 4 9 1 0.380039 0.457356 0.077317 11 5 1 0.403201 0.465594 0.062393 FiLTER A 1.0 N.11 F0=0.2100 f 0 0.5 0.5-Fp Fig. 4. Construction for demonstrating the symmetryof the curve of transition width versus passband cutoff frequency for K = 1. FILTER B as we proceed from filter A to filter D.Filter A has 3 complex conjugate sets of zeros on the unit circle and one quadrouplet of zeros off the unitcircle. Filter B has essentially the same zero position plot as filter A since 0.4- their frequency characteristics are very close.In filter C, a = 0.2 - the pair of zeros on the unit circle closest to - 1 has I m- become a mirror-image real pairof zeros on the negative real axis in the z plane. One zero is approximately at 3 Nil1 t 1.2 - Fp20.2660 - 00 m z =O; its mirror-image partner is almost at z= 2 (actually the magnitudes of these zeros are 0.002 and 495.5, respectively).
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