ASYMPTOTICS OF OPTIMAL FILTERS The Asymptotics of Optimal (Equiripple) Filters Jianhong Shen and Gilbert Strang Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 39-437 Dedicated to the memory of Wolfgang Fuchs Abstract For equiripple lters, the relation among the lter length N +, the transition bandwidth!, and the optimal passband and stopband errors p and s has been a secret for more than twenty years. This paper is aimed to solve this mystery. We derive the exact asymptotic results in the weight-free case p = s =, which enables us to interpret and improve the existing empirical formulas. Our main results are nally combined into formula (7). In the transition band, the lter response is discovered to be asymptotically close to a scaled error function. The main tools are potential theory in the complex plane and asymptotic analysis. Keywords Asymptotics, equiripple lters, Kaiser's formula, Green's function, optimality. optimal (in the minimax sense). The algorithm is directly available in MATLAB as remez.m and is very widely used. The designer begins with a passband (ending at frequency! p ) and a stopband (starting at! s ) and an acceptable error. This paper considers rst the weight-free case with equal errors in the passband and stopband: p = s =. The transition bandwidth! =! s?! p is critical to the relation of the lter length N + = n + to the distance from an ideal one-zero response. A useful formula derived experimentally by Kaiser [7] suggests an appropriate lter length. There are similar formulas in Rabiner and Gold [] and Vaidyanathan [3]. Kaiser's is the I. Introduction T is a familiar (and happy) fact that the equirip- property of an optimal lowpass lter Iple suggests a good algorithm for designing that lter. This is the Remez-Parks-McClellan algorithm (see Cheney [] and Parks and McClellan []), which iteratively pushes down the error at its maximum point. Eventually the error has equal magnitudes and alternating signs at N + points. Since no polynomial of degree simplest and most characteristic: N ' log?? 3 :34! () For this value of N, the Remez algorithm yields the frequency response H(!) closest to the ideal \one-zero function" F (!) on the union of passband j!j! p and stopband j?!j?! s. The code outputs the coecients h[]; ; h[n] of this optimal lowpass lter, for which the error is approximately (See Fig. ). Our paper analyzes this relation of to N (or n). N can have N + sign changes, this equiripple lter cannot be improved at all N + points. It is The error decays exponentially, e?n = p n, and
ASYMPTOTICS OF OPTIMAL FILTERS actually g(). Since our intervals are real, the critical point is also real (and it lies in the transition.8.6.4. ω s ω p transition band. 3 3 Fig.. The frequency response of the optimal FIR lowpass lter with N + = coecients and! p = :44;! s = :56. the key problem is to compute the exponent = (! p ;! s ). The leading term of is controlled by! =! s?! p and our asymptotic result is close to Kaiser's experiments for small, see Eq.(5): N ' log?? log log? :7! This asymptotic result is later modied to the semiempirical formula (7), which applies to a wide range of practical parameters and is hence recommended to replace Kaiser's empirical formula. Kaiser also discovered a nearly optimal family of lters based on the I {sinh function. An empirical formula similar to () was also established by Kaiser [7] for this family. The constant in the denominator becomes slightly smaller, which increases N. ω p δ ω s This family was analyzed theoretically by Wolfgang Fuchs in [5] and we return to it in Section 5. The fundamental tool in the analysis is the Green's function g(z), which solves Laplace's equation on the complement of two intervals (with a pole at innity). This function has a unique critical point, and is band). But our problem is emphatically one of complex and not real analysis. The oscillations of a real polynomial prove that an equiripple lter is optimal, but for more information we must go deeper into the complex plane! We give references to fundamental work of Walsh [4] and Widom [5] and Fuchs [4]. The virtue of complex analysis (Appendix D) is to permit contours of integration to be deformed. Then the leading term in an integral with a large parameter can be computed by the method of steepest descent. The Green's function has an elementary form only in the symmetric case, when! s +! p =. Then the critical frequency is! c = =, at the center of the transition band. Our analysis is most complete in this symmetric case. Our task in all other cases (when g(z) becomes an elliptic function) is to recapture the same form, in which N! plays such a key role. We also present early results on nearly optimal lters, for which n is of the same order as the optimal error sequence. Unlike equiripple lters, nearly optimal lters may have closed forms and allow fast algorithms. For the symmetric and weight-free case, we propose an explicit set of interpolation points. This leads to the discovery of the asymptotic behavior of optimal lters in the transition band. The frequency response is close to an error function. The limit as n! is the ideal brick wall lter with cuto at the critical frequency! c. Our paper has been organized as follows. Section introduces an important result due to Fuchs in approximation theory. The complete asymptotic relation among design parameters in the symmetric case! p +! s = is derived in section 3. For the non-
ASYMPTOTICS OF OPTIMAL FILTERS 3 symmetric case, theoretical results as well as a MAT- LAB algorithm for the crucial geometric constant are described in section 4. Asymptotic analysis is also carried out for the case of narrow transition band. In section 5, our results are compared with that of Fuchs on Kaiser's window family of lters. In section 6, we show the numerical comparison between our asymptotic formula and Kaiser's empirical one. Section 7 describes the asymptotic behavior of optimal lters in the transition band. Some proofs are included in the appendix. II. Leading Order For n A. Leading Order for General Problem We now present Fuchs' result on polynomial approximation on several domains in the complex plane. Let K be a compact domain with disjoint simply connected components K ; ; K m. Our problem is to approximate by polynomials the function f(z) that equals h i (z) on the component K i. (The h i (z) are entire functions and not all identical.) The minimum error in the maximum norm is n when the polynomials have degree at most n: n = min max pp n zk jf(z)? p(z)j: [4] gives details). Our case will automatically have q =, since h (z) = and h (z) =. The exponent is a geometric constant, entirely determined by K. For m =, is Green's logarithmic radius of the unique critical point of K c. Its meaning will be explained immediately. B. Potential Theory in the Complex Plane Let G(z; s) be the Green's function for the Laplacian on the complement K c, which is completely characterized by the following properties: G(z; s) is harmonic over K c except at z = s, where G(; s) behaves like? ln jz?sj (or ln jzj when s = ). For any xed s, G(z; s) goes to zero as z approaches @K c, the boundary of K c. We are particularly interested in g(z) = G(z; ). For any z K c, g(z) is called its Green's logarithmic radius, and is denoted by jzj K. The function g has exactly m? critical points ordered by j j K j j K j m? j K inside the domain K c (Nevanlinna []). A critical point of g (or of K c ) means that the gradient at is zero. Geometrically, the level line of g through is self-intersected at (see Fig. ). Then in Fuchs' theorem is given by Here P n denotes the space of all polynomials of degree not greater than n. =j I j K : f can be continued analytically on jzj K < j I j K but not on jzj K < j I+ j K : (3) Theorem (Fuchs) There exist a non-negative integer q, a positive number, and two positive constants A? and A +, such that A? n q? exp(?n) n A + n q? exp(?n): () Remark The nonnegative integer q is determined by the objective function f(z) and domain K together. It is the multiplicity of a particular critical point as a zero of a dierence h i (z)? h j (z) (Fuchs Since the h i are not identical, I does exist. When m =, must be jj K = g() at the unique critical point. Remark For optimal polynomial approximation, real analysis yields the famous \Alternation Theorem" (the equiripple property and exchange algorithm, see Cheney [] and Rabiner and Gold []). The deeper asymptotic problems require complex analysis and potential theory. We recommend the
ASYMPTOTICS OF OPTIMAL FILTERS 4.5 tion symbol (x p ; x s ) or (! p ;! s ). To determine the.4 z plane leading order of n, we have to compute explicitly,.3 which is the task of the next two sections. Lemma. leads to the following theorem in terms of logarithms.. K s σ K p Its proof has been placed in Appendix A.. Theorem For long equiripple lters ( n ), the. asymptotic error satises.3.4 n ' ln? n? (x p ; x s ) ln ln? n (5).5.8.6.4...4.6.8 Fig.. The level lines and critical point of the Green's function associated to a domain K = [?;?b] [ [a; ]. Here we show the case a = b = :. By symmetry =. classical monographs by Walsh [4] and Henrici [6]. Appendix D contains a short survey on the connection between optimal polynomial approximation and potential theory. C. Leading Order for n It is natural to work in the x = cos! domain. Let x p = cos! p and x s = cos! s. The passband and stopband become K p = [x p ; ]; K s = [?; x s ]. Then K = K p [ K s and is the unique critical point of K c. Lemma There exist two positive constants A? and A + such that for all n A? n? exp(?njjk ) n A + n? exp(?njjk ) (4) Proof: Use Theorem for this special case of m =. By equation (3), = jj K. On the other hand, h and h. Therefore z = is a zero of order q = of h (z)? h (z). Remark 3 Our K has only two free parameters x p and x s (or! p and! s ). So we will also use the func- Remark 4 The empirical formulas (Kaiser [7], Rabiner and Gold [], and Vaidyanathan [3]) only catch the leading term ln n?. They do not capture the correct or the double logarithm term due to the factor n?= (which is overshadowed by the exponential term in all experiments). III. Symmetric Case In the next section, we shall see that (x p ; x s ) generally has no description by elementary functions. In the symmetric case x p + x s =, or! p +! s =, the Green's function simplies and can be computed explicitly. Several elementary properties will be useful (referred to as Property,,3 later):. (Unit disk) The Green's function for the domain jwj with source s = is? ln jwj.. (Conformal equivalence) Suppose w = f(z) is a conformal mapping from a domain K z onto a domain K w. Assume that f is continuous up to the boundary and f(@k z ) @K w. Let z be an interior point of K z and w = f(z ). Suppose g (w) is the Green's function for K w corresponding to source w. Then g (f(z)) is the Green's function of K z corresponding to source z. 3. (Pullback by covering mapping ) In Property, suppose that f is an analytic mapping, but z is the only preimage of w and all the other condi-
ASYMPTOTICS OF OPTIMAL FILTERS 5 tions still hold. Then g (f(z))=d is still the corresponding Green's function provided that z is the (d? )-multiple zero or pole of f (z). Lemma (Green's function: symmetric case) Suppose! p +! s =. Then x p =?x s = a >. The Green's function g(z) for K c corresponding to source s = is? ln? a [ z? a? p (z? a ) (z? ) ]? Here the square root has K as its branch line and takes a positive value at z =. Proof: Dene (Z) =? a [Z? a? p (Z? a ) (Z? ) ]? Here Z = z folds K into a single interval I = [a ; ] in the Z{plane. The inverse Joukowski transform w = (Z) maps the complement of I onto the unit disk D w in the w{plane, and maps Z = to w =. Let f(z) = (z ). Then this lemma is a direct conclusion from Properties and 3 with d =. Lemma 3 Suppose x p =?x s = a. Then the exponent in the error formula is Proof: = ln + x p? x p = ln + cos! p = ln cot! p? cos! p (6) By symmetry, the unique critical point for K c must be =. Therefore = g() = ln + a? a : The combination of (5) and (6) can be used for design problems when! p +! s =. Notice that even in the symmetric case, is not strictly linear in!. However, Kaiser's idea of linear approximation to as shown in the denominator of his formula () is good for most applications. The estimated coecient :34 can be improved by our asymptotic analysis. We now look for a theoretical formula in the symmetric case that is similar to Kaiser's. Suppose!. Noticing! p = =?!=, we have by (6) ' x p = cos(=?!=) '!=: In (5), we replace by!= and rewrite it in terms of decibels by changing the logarithm to base. By ignoring the O() term (compared with logarithms of n? ), we obtain the following. Theorem 3 (Asymptotic Relation of N to n ) Assume that! p +! s = and! p is close to =. Then the order is related to the ripple height n by N = n ' log n?? log log n? : (7) (5 log e)! Remark 5 (a) Numerically ln cot(! p =) is close to!= except when! is close to (see Fig. 3). most applications,! is small. is a satisfactory approximation to. For Hence!= In fact, when! = =4, the relative error is only (?!=)= ' :6%. (b) Kaiser's linear coecient :34 is larger than our corrected value 5 log e ' :7. The relative error is (:34? :7)=:7 ' 7%. This slope deviation can be detected in Figures 5 and 6. (c) Since the second leading term for n is a double logarithm, the number \3" in Kaiser's formula is not correct theoretically. However, it does reveal the fact that the second leading term changes very slowly. Practically we only deal with n ranging from? to?6. Then the double logarithm in (7) goes from to 6 (and 3 is inside this range).
ASYMPTOTICS OF OPTIMAL FILTERS 6 5 4.5 4 3.5 3.5.5.5 β ω /...3.4.5.6.7.8.9 ω/π Fig. 3. = ln cot!p '!?! '!. The dashed line corresponds to and the solid line is the true (! p) with! p =. The horizontal axis shows!=. [3] as functions of x p and x s. k = s r = exp s = exp (x p? x s ) ( + x p )(? x s )? K c(k) K (k)? K i(k) K (k) (8) (9) : () The elliptic functions sn(u; k), K c (k), K i (k), and K (k) are dened in Appendix B. B. Green's Function and Let g A (w) denote the Green's function for the annulus A r corresponding to the source s. Then by IV. General Case In the non-symmetric case, (x p ; x s ) is no longer an elementary function. In this section, we rst describe the theoretical approach to determine, and then create a numerical algorithm using MATLAB to compute it. A. Conformal Equivalence to Annulus Lack of symmetry (! p +! s 6= ) makes K c a nontrivial doubly connected domain (DCD). Hence one has to turn to the general theory. A famous theorem says that any DCD is conformally equivalent to an annulus A r : r < jwj < (see Nehari [9]). The \modulus" r is uniquely determined by the DCD. In our case, this conformal mapping can be obtained in closed form using elliptic functions. The inverse mapping z = f(w) is (Kober [8]) + x p?? x p Here < s <, and f(s) =. sn ( K ln w; k) + sn ( K ln s; k) sn ( K ln w; k)? sn ( K ln s; k): The three parameters r; s; k are given by Freund Property, g(z) = g A (f? (z)) is the Green's function for K c corresponding to s =. Let and A denote the unique critical points of g(z) and g A (w). Then = f( A ) since f preserves level lines. Hence (x p ; x s ) = g() = g A ( A ). Dene = ln s= ln r (; ). For any c inside the unit circle, the symbol [c] = [c](w) denotes the Mobius transform of the unit disk associated with c: [c](w) = w? c c w? : Then g A (w) is given by Akhiezer [] as X ln jwj? ln j[s]j? ln j[r j s]j + ln j[ rj s ]j : j= The partial sum from to J of this innite series converges on A r with rate O(r J ). For small r this is quite satisfactory. However when the transition band is narrow, r dened by (8) is close to. So the following form of g A (w) is much better numerically: ln jwj? ln s? ln js + (w)j + ln js? (w)j () Now the partial sums of S + and S? from?j to J give greater accuracy O(r J ): S + (w) = S? (w) = X j=? X j=? r j?w rs r j?ws r j () j : (3)
ASYMPTOTICS OF OPTIMAL FILTERS 7 Our MATLAB code uses this form for g A (w). Theoretically, the unique critical point A can be located as the zero of the gradient vector rg A. This generally requires substantial computation. The following theorem changes it to a one-dimensional optimization problem. Theorem 4 ( by Optimization) Consider g A (x) = ln(?x)? ln s? ln S + (x) + ln S? (x) (4) for? x?r. Then (x p ; x s ) = max g A (x). Proof: By denition, g A (x) and g A (?) = g A (?r) =. Hence g A (x) reaches its maximum value inside (?;?r). On the other hand, since g A (w) is symmetric with respect to y (w = x + iy), @g A =@y must be zero along (?;?r). Therefore @g A (w)=@x = immediately implies a critical point of g A. Since there is only one critical point A, it must yield the maximum of g A (x). Hence (x p ; x s ) = g A ( A ) = max g A (x): function g A on the annulus A r. Our last program betak.m applies the minimization fmin to?g A (x) dened in (4) and nally nds. We distinguish betak from MATLAB's beta. These MAT- LAB functions are available upon request. D. Asymptotics for narrow transition When we compute numerically, we don't know its exact behavior as a function of! p and! s. compare with earlier empirical formulas, we apply asymptotic analysis to when the transition bandwidth is narrow (! ) and xed. In practice, this narrow transition is preferred. We measure! p and! s from the mid-frequency! m = (! p +! s ):! p =! m?! and! s =! m +! : Since! is xed, (! p ;! s ) becomes a function only of! m and is denoted by (! m ). Theorem 5 The leading term of (! m ) is (=) in the range! min(! m ;?! m ). Practically, the range can be taken as (see Fig. 4): To! <! m <?!: C. Algorithm and MATLAB Code The complete elliptic function called ellipk in MATLAB can be used to compute K c and K. For the incomplete elliptic function K i, we apply MATLAB integration quad8 to the function for_call, which is simply =? m sin x with m = k. Set = sin? ( is dened in Appendix B (iii).) Then \K i = quad8('for call'; ; ; e? 4; [ ]; m); " The proof is in Appendix C. It is Kaiser's empirical formula () that led to our discovery of Theorem 5. In turn, our asymptotic result provides a theoretical support to the form of his empirical formula. The transition bandwidth! is crucial and the position! m of the transition band has small eect. Our analysis gives the correct constant in the leading term, and also the next term. With the help of Theorem 5, Theorem 4 generalizes to the non-symmetric case. computes K i to the precision?4. This yields r and s from (9) and () (by RS.m ). Then Green.m uses (){(3) to compute the Green's Theorem 6 If! min(! m ;?! m ), then N = n ' log n?? log log n? : (5) (5 log e)!
ASYMPTOTICS OF OPTIMAL FILTERS 8 for his windows when < :: β.5.4.3. N ' log?? 8 : :85! (We have converted from f to! = f.) Fuchs, Kaiser, and Landau [5] proved that for large window parameter, Fig. 4...5.5.5 3 ω m Theorem 5: (! m) ' (=). Each solid horizontal line represents (! m) when! is xed. From the bottom to the top,! = : : : : :. The segments bounded by the diagonal dashed lines show the practical range inside which (! m) ' (=). In practice, this yields good results for! =4 and! m [!;?!] by Remark 5 and Theorem 5. V. Kaiser's filters are near optimal Besides the equiripple lters, another popular way of designing FIR lters is the window method. The ideal one-zero lowpass lter is IIR. In the frequency domain, we convolve this ideal response with the window response. The frequency response of a window is often a damped wave. The narrowness of the main lobe and side lobes determines the quality of the resulting FIR lter. Kaiser used the non-linearly scaled zeroth{order modied Bessel function to create a family of windows with good properties. They are of limited duration in the time domain and have most of their energy concentrated at low frequency. (This is the core idea of modern wavelet analysis.) Most important, these lters are nearly optimal: the error sequence has the same order as that of the optimal approximation. First, Kaiser [7] established an empirical formula ' 8 N?! exp (?! 4 N): Comparing with our Eq.() Kaiser's windows are indeed nearly optimal (but not exactly, since '! is only an approximation). Similar to the way we have proved Theorem, Fuchs showed that N ' log?? log log? : (5 log e)! This is exactly (5). The Chebyshev optimal lter is completely characterized by the equiripple property. The underlying mechanism of Fuchs' result is that for large, the side-lobes have approximately the same L norms (same areas). This makes Kaiser's lters near equiripple and hence near optimal. VI. Numerical Experiments We use the MATLAB function remez.m to compute the minimal error N corresponding to each N. The result is then used to test Kaiser's empirical formula and our asymptotic formula. A. Narrow Transition For narrow transition (this practically extends to! =4), Theorem 6 gives the rst two leading terms of N. However, to make Eq. (5) accurate even for small N, we have to know the constant A n appearing in the proof of Theorem. This means that we have to add a constant term (independent of n ) in the numerator of Eq. (5). Finding A n is a mathematically open problem, but our numerical
ASYMPTOTICS OF OPTIMAL FILTERS 9 experiments indicate that we can take this constant term as log. Then the following formula applies to all N: N = n ' log ( n )?? log log n? : (5 log e)! (6) even for wide transition: N = n ' log ( n )?? log log n? ( log e) ln cot?! : 4 (7) The experiments for wide transition are plotted in Fig. 6, with on the horizontal axis and N on the This is very accurate for small!. Our experiments have! = :; :4; ; : and! m = =. For each!, rst we use remez.m to vertical. 8 6 TBW=.6pi,.5pi,.4pi,.3pi,.pi compute the N{ relation exactly. With this result we test the predictions by Kaiser's empirical formula () and our asymptotic formula (6). The test results are plotted in Fig. 5. It shows that Eq. (6) is more accurate. 4 8 Real Kaiser s 6 Asymptotic 4 6 5 4 3 8 7 6 5 TBW=.pi,.8pi,.6pi,.4pi,.pi Real Kaiser s Asymptotic Fig. 6. Comparison of Kaiser's formula and formula (7). The ve sets of curves now correspond to wider transitions! = (: : : : :6). 4 3 So nally, we would recommend Eq. (7) for all design problems with either wide or narrow transition!, and symmetric or non-symmetric bands. Fig. 5. Comparison of Kaiser's formula and formula (6). There are ve sets of curves in the plot, one for each!. From right to left,! = (: : : : :). Each set contains three lines solid, dotted, and dashed, corresponding to the real N{ relation, Kaiser's empirical prediction, and the asymptotic prediction by Eq.(6). C. One Example We compare the accuracy of the formulas through a real design problem. Suppose that! p = :5, and! s = :54: We want an equiripple lter whose passband and stopband errors are p = s = = :. By Kaiser's formula (), the lter length should be N K = 7. The exchange algorithm B. Modied to Include Wide Transition For wide transition, say! ' :5, both Kaiser's formula and formula (6) assume that is a linear function of!. Generally we need the original (! m ) ' (=) = ln cot(?!)=4 in the denominator. Then the following formula is very accurate H K = remez(n K ; [ :5 :54 ]; [ ]) gives the impulse response of the equiripple lter H K. The actual ripple height is K = :55. Hence the relative design error is r K = j? Kj = 7:5%:
ASYMPTOTICS OF OPTIMAL FILTERS The corresponding data using our asymptotic formulas (7) or (6) are: N A = 8; A = :9; r A = 4%: VII. Response in the transition band This section describes the asymptotic behavior of N the equiripple lter response H opt (!) inside the transition band! p j!j! s as the lter length N + = n + increases. This behavior reveals the convergence of impulse responses to the ideal - lter with passband j!j! c. Alan Oppenheim brought this problem to our attention. The results can be stated simply, but the mathematics behind them is more complicated. For proofs we refer to our forthcoming paper \Nearly Optimal Approximation on Two Intervals." A. Nearly Optimal Filters A family of FIR lters H N (!) of length N +, is said to be nearly optimal if its error sequence e N = kh N (!)? I(!)k is of the same order as the optimal error sequence. This means that e N C N for a xed C. Nearly optimal lters serve two purposes. Unlike equiripple lters, they may have closed forms and allow direct mathematical analysis. Their properties should give an approximation to their counterparts (the optimal equiripple lters). Second, by relaxing the optimality, we may have a better design algorithm, such as direct interpolation. For the symmetric case! p +! s =, we do nd such an interpolation scheme. Theorem 7 Suppose! p +! s = and x p = cos! p = a >. Dene k points x j ; j = ; ; ; k by x j = + a +? a cos j? k : Let p n (x) denote the unique polynomial of degree n = k? interpolating at each x + j and at each x? j. Let N = n and dene H N (!) = p n (cos!): Then H N (!) is a sequence of nearly optimal lters. B. Asymptotics in the Transition Band With the help of H N (!) just constructed, we nd the following asymptotic form of the equiripple lter H opt N (!) in the transition band. Theorem 8 Let! m =! p +! s be the mid frequency in the transition band. For! =! s?! p, the leading term of H opt N (!) on! p!! s is given by H opt N (!) erf r N 4!! m?! : (8)! s?! m Here!= is the geometric constant appearing in previous sections and the error function erf(x) is dened by erf(x) = Z x? e?t dt: Practically, this approximation is very satisfactory for a wide range of transition bandwidths. Fig. 7 shows the case of! = :, for both symmetric and non-symmetric bands. Computational experiment guided by our error function formula leads to the following semiempirical formula for the weighted case. In minimizing the maximum deviation from the ideal lter, the stopband error is weighted by W. In practice, W can be. The optimal lter with heights p = W s is still denoted by H opt (!). Then the leading term N approximation in the transition band is: r! H opt N N (!) ' erf! m?!? S N (W ) ; 4! s?! m with S N (W ) = ln W + ln ln W : N
ASYMPTOTICS OF OPTIMAL FILTERS N=3, ω s =.55π, ω p =.45π N=64, ω s =.55π, ω p =.45π Acknowledgments.9.8.9.8 The authors would like to thank Jim Kaiser and.7.6.7.6 Alan Oppenheim for their interest in this problem..5.4.5.4 APPENDIX..3.3.. real optimal erf fitting.. real optimal erf fitting A. Proof of Theorem (Section II).5.5 θ=ω ω m / ω p ω m.9.8 N=3, ω s =.75π, ω p =.65π.5.5 θ=ω ω m / ω p ω m.9.8 N=64, ω s =.75π, ω p =.65π Suppose n = A n n? exp(?n), with A? A n A + by Lemma. Taking the natural logarithm yields ln n? =? ln A n + ln n + n: (9).7.6.5.4.3.. real optimal erf fitting.5.5 θ=ω ω / ω ω m p m.7.6.5.4.3.. real optimal erf fitting.5.5 θ=ω ω / ω ω m p m Fig. 7. Closeness to the error function. The four windows show the scaled transition band: =!?!m. For example,! p now corresponds to =. The solid lines! p?!m represent the optimal equiripple lters, and the dashed lines show the error function (8). For the top two,! s = :55 and! p = :45 with N = 3; 64. For the bottom two,! s = :75 and! p = :65 with N = 3; 64. The tting improves as the lter length N increases. This experimental expression for the shift S N (W ) has successfully predicted the impulse response of optimal lters in the transition band. They still converge to the ideal lter with cuto frequency!! m? S N (W ) for narrow transition band. VIII. Conclusions We have proved the asymptotic formula (5), which relates all the key parameters in the design of equiripple lters. To be applicable to all cases of transition bandwidth, formula (5) is modied to formula (7). The numerical experiments conrm its accuracy. Filter designers can use this formula! The dominant term on the right is n, which must equal the dominant term on the left. Hence ln? n n. This determines the leading term. To nd the ln? n next term, assume n = + n. By (9), =? ln A n + ln n + n: As n, the dominant term ln n can only be balanced by n, since ln A n is bounded. Hence n '? ln n '? ln ln? n : B. Denitions of elliptic functions (Section IV, A) (i) v = sn(u; k) is the Jacobian elliptic function with modulus < k <, dened by the incomplete elliptic integral: u = Z v dx p (? x )(? k x ) : (ii) K c (k) = sn? (; k) is a complete elliptic integral: K c (k) = Z dx p (? x )(? k x ) : K (k) in the expression of f is dened by K (k) = K c (? k ), also a complete integral. (iii) K i (k) = sn? (; k) and = q +xp. C. Proof of Theorem 5 (Section IV, C) (i) The unique critical point of g(z) = G(z; ) must lie inside (x s ; x p ). ' If the mid-frequency
ASYMPTOTICS OF OPTIMAL FILTERS! m = (! p +! s ) is below =, the stopband K s is longer than the passband K p. Hence the Green's function g(z) grows more slowly near K s. The maximum of g on [x s ; x p ] occurs closer to x p than to x s, so that > x m. (ii) Dene d = + x px s x p + x s and c = d? p d? : Then c [x s ; x p ] and c = x m + O(x ): The linear fractional transform z = F (z) = z? c? cz () maps K = K s [ K p onto a symmetric domain K in the z {plane: K = K s [ K p = [?; x s] [ [x p; ]: Here x p = F (x p ) =?F (x s ) =?x s. (iii) Set s = F () =?=c and = F (). Then is the unique critical point of G (z ; s ). Since the new source s lies inside (?;?) and the new domain is symmetric, its Green's function G (z ; s ) grows more rapidly near K s than K p. Hence the maximum of G (; s ) (x must occur closer to x s, implying that <. s ;x p ) Now denote F (x m ) by x m. Since F preserves the critical point as well as the order on [x s ; x p ], we have by (i) x m < <. But! min(! m ;?! m ) implies that x m = x m? c? cx m ' O(x m? c) = O(x ): So nally we have = O(x ): () (iv) For the symmetric case in Lemma 3, when x p, G (x ; ) ' q (x p)? (x ) + O((x p) ) () for all x in [?x p; x p]. And since z! x p=z maps K onto itself, Property yields Therefore nally, G (z ; s ) = G (x p=z ; x p=s ): (3) (! m ) = G(; ) = G ( ; s ) = G (; s ) + O(x ) [()] = G (;?x pc) + O(x ) [(3)] = G (?x pc; ) + O(x ) q = (x p)? (x pc) + O(x ) [()] = x pp? c + O(x ) = x p? c p? c + O(x )? cx p = x p? x m p? x? x m + O(x ) [()] m x = p + O(x )? x m =! + O(! ) = ( ) + O(! ): (v) The numerical results displayed in Fig. 4 show that practically, in the whole range of [!;?!], (! m ) ' (=) is a satisfactory approximation. D. The connection between polynomial approximation and complex potential theory (Section II, B) We explain how polynomial approximation on a \reasonable" complex domain K is related to the complex potential on the complement K c. Let g(z) and (z) denote the real and complex potentials on K c generated by a source at z = and with the boundary @K c grounded. Then g(z) = Re ln (z), and j(z)j = along @K c. Near z =, (z) = cz + a + b z + Assume the target function f being approximated is analytic in a neighborhood of K. If K is a disk,
ASYMPTOTICS OF OPTIMAL FILTERS 3 the n-th order Taylor expansion of f at the center can be a close approximation to the optimal polynomial. For general K, as we already see in the Parks-McClellan algorithm, the n-th degree optimal polynomial must interpolate f at a set of points ^S n = f^z ; ; ^z n+ g K (the crossings with and in the stopband and passband). The diculty is that there is no simple way to describe ^Sn, which apparently depends on both K and f. The key idea of nearly optimal approximation is to nd a simple set S n = fz ; ; z n+ g K (for each n), on which interpolation yields a nearly optimal polynomial. The connection arises from the integral representation of the interpolating polynomial f n (z): f n (z) = Z? f(t) q n+ (t) q n+ (t)? q n+ (z) t? z dt: (4) Here q n+ (z) is the polynomial with roots S n, and? can be any contour (containing K) in the interior of which f is analytic. The approximation error in K is, and Z n (z) = q n+ (z)? f(t) q n+ (t) dt t? z ; j n (z)j C(?; f) k=q n+ k? jq n+ (z)j: C(?; f) is a constant independent of n and q n+, and k=q n+ k? is the maximum norm on?. We can choose q n+ (z) such that kq n+ k K = kq n+ k @K =. Then k n k K C(?; f)k=q n+ k?. To minimize n, it is important for k=q n+ (t)k? to be as small as possible. This leads to the following fast growth problem. Among all polynomials of degree n+ and with k k @K =, nd q n+ (z), such that jq n+ (z)j is \as large as possible" for any z K c (thus k=q n+ k? can be as small as possible). There is a natural polynomial of degree n + with this property. It is the so-called Faber polynomial and is constructed from the complex potential (z). For any polynomial q n+ of degree n and kq n+ k @K =, dene h(z) = q n+ (z)= n+ (z). This is analytic on K c and h() is nite. Therefore khk K c = khk @K = kq n+ k @K = ; since j(z)j on @K. This leads to jq n+ (z)j j n+ (z)j; for all z K c. The growth of jq n+ (z)j on the complement of K c is dominated by j n+ (z)j. Suppose near z =, n+ (z) = az n+ + bz n + + c + d=z + : Then the n + -th Faber polynomial F n+ (z) is de- ned as the principal part of n+ (z): F n+ (z) = az n+ + bz n + + c: It can be shown that lim n! kf n(z)? n (z)k K c = : Therefore, jf n+ (z)j ' j n+ (z)j achieves the optimal growth. The roots set S n of F n+ (z) can be used to interpolate a given function f(z). Such an interpolation is guaranteed to be nearly optimal. For example, if K is the unit disk centered at the origin, then F n+ (z) = z n+ and the interpolation is simply the n-th order Taylor expansion around. Another important example is when K is the interval [?; ]. Then F n+ (z) is the n + -th Chebyshev polynomial T n+ (z) = cos((n + ) cos? z). When the complement K c is not simply connected (as in our two-interval case), the analysis is much more complicated but Faber's idea still works. References [] Akhiezer, N.I. Elements of the theory of elliptic functions, American Mathematical Society, Providence, RI, 99
ASYMPTOTICS OF OPTIMAL FILTERS 4 [] Cheney, E.W. Introduction to approximation theory, McGraw-Hill, New York, 966 [3] Freund, R.W. \On polynomial preconditioning and asymptotic convergence factors for indenite Hermitian matrices", Linear Alg. Appl. 54{56, 59{88, 99 [4] Fuchs, W.H.J. \On the degree of Chebyshev approximation on sets with several components", Izv. Akad. Nauk Armyan. SSR, 3, No. 5-6, 396-44, 978; See also \On Chebyshev approximation on several disjoint intervals", in Bernard Aupetit, ed., Complex Approximation, Birkhauser, pp. 67-74, 98 [5] Fuchs, W.H.J., Kaiser, J.F., and Landau, H.J. \Asymptotic behavior of a family of window functions used in non{ recursive digital lter design", Technical Memorandum, Bell Laboratories, 98 [6] Henrici, P. Applied and computational complex analysis, 3, Wiley, New York, 986 [7] Kaiser, J.F. \Nonrecursive digital lter design using the I {sinh window function", Proc. 974 IEEE Symp. Circuits and Syst., {3, April, 974 [8] Kober, H. Dictionary of conformal representations, Dover, New York, 957 [9] Nehari, Z. Conformal mapping, McGraw{Hill, New York, 95 [] Nevanlinna, R. Analytic functions, Springer-Verlag, New York, 97 [] Parks, T.W. and McClellan, J.H. \Chebyshev approximation for nonrecursive digital lters with linear phase", IEEE Trans. on Circuit Theory, CT-9, March 97 [] Rabiner, L.R. and Gold, B. Theory and application of digital signal processing, Prentice-Hall, Englewood Clis, NJ, 975 [3] Vaidyanathan, P.P. Multirate systems and lter banks, Prentice-Hall, Englewood Clis, NJ, 99 [4] Walsh, J.L. Interpolation and approximation by rational functions in the complex domain, American Mathematical Society, Providence, RI, 965 [5] Widom, H. \Extremal polynomials associated with a system of curves in the complex plane", Advances in Mathematics, 3, 7{3, 969