EDICS No. SP Abstract. A procedure is developed to design IIR synthesis lters in a multirate lter bank. The lters

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1 Design of Multirate Filter Banks by H 1 Optimization Tongwen Chen Electrical and Computer Engineering University of Calgary Calgary, Alberta TN 1N4 Canada chent@enel.ucalgary.ca Bruce Francis Electrical and Computer Engineering University of Toronto Toronto, Ontario M5S 1A4 Canada francis@control.utoronto.ca EDICS No. SP.4.3 Abstract A procedure is developed to design IIR synthesis lters in a multirate lter bank. The lters minimize the `-induced norm of the error system between the multirate lter bank and a desired pure time-delay system. This criterion is reduced to one of H1 optimization, for which there is ready-made software. 1 Introduction The object of study in this paper is the multirate lter bank in Figure 1. In this discrete-time x(k) H 0 (z) # " F 0 (z) ^x(k) -H 1 (z) - # - " - F 1 (z) - j -? Figure 1: Multirate lter bank. system, H 0 (z), H 1 (z), F 0 (z), and F 1 (z) are transfer functions of linear time-invariant (LTI) lters, # denotes the downsampler (subsampler, decimator) by a factor of two, and " denotes the upsampler (expander, interpolator) by the same factor. This multirate signal processing system is important, for example, in subband coding, and has been studied a great deal; see, for example, the recent books [], [8], [9], and [19] and their references. The system operates as follows: The input signal x(k) is rst processed by the analysis lters H 0 (z) and H 1 (z). Typically, these are frequency-selective, H 0 (z) being lowpass and H 1 (z) highpass. The outputs of these lters can therefore be subsampled without loss of information. They are This work was supported by the Natural Sciences and Engineering Research Council (Canada). 1

2 then quantized (not shown) and transmitted over a communication channel (also not shown). The upsamplers and synthesis lters F 0 (z) and F 1 (z) are then required to reconstruct x(k), ideally with, according to standard practice, at most a time-delay error. The reconstructed signal is denoted ^x(k), so perfect reconstruction is said to be achieved [19] if for some integer m 0 ^x(k) = x(k? m): That is, the desired system from x(k) to ^x(k) is LTI with transfer function T d (z) = z?m : Note that in general the system from x(k) to ^x(k) is time-varying because of the presence of the down- and upsamplers. In this paper we take the viewpoint that the analysis lters have already been designed for good coding of the input, and the synthesis lters are now to be designed to achieve as close to perfect reconstruction as possible. To dene the degree of closeness, consider the error system of Figure. Thus e(k) is the reconstruction error between the desired output, x(k? m), and the actual output, - T d (z) x(k? m) x(k) H 0 (z) # " F 0 (z) -H 1 (z) - # - " - F 1 (z) - j-? j ^x(k)?? - e(k) Figure : Error system. ^x(k). Let ` denote the Hilbert space of square-summable discrete-time signals dened over the time set Zof all integers. The inner product and corresponding norm on ` are hx; yi = X k x(k)y(k); kxk = X k jx(k)j! 1= : Then J := sup kek ; kxk =1 the `-induced norm from x(k) to e(k), is the worst-case norm of e(k) when x(k) is normalized. This quantity, J, is taken to be the performance measure of the multirate lter bank. A small value of J means that the error e(k) is small uniformly over all inputs x(k); J = 0 means that perfect reconstruction has been achieved. Our precise design problem statement is as follows:

3 Given causal, stable, FIR or IIR analysis lters H 0 (z) and H 1 (z) and given a tolerable time delay m; Design causal, stable, IIR synthesis lters F 0 (z) and F 1 (z) to minimize J. The optimum performance measure is therefore J opt = inf J F 0 (z); F 1 (z) = inf sup F 0 (z); F 1 (z) kxk =1 kek ; where the inmum is over all causal, stable, IIR synthesis lters. Thus the idea is to regard T d (z) as a model ideally to be matched by the multirate lter bank; the design problem as posed is therefore one of `-induced norm model-matching. Observe that if F 0 (z) F 1 (z) 0, then J equals the induced norm from x(k) to x(k? m), which equals 1. Thus J opt 1. Also, since T d (z) depends on the time delay m, so does J opt and we may write J opt (m). The function J opt (m) can be shown to be a non-increasing function of m. One result of this paper is that, under mild conditions on the analysis lters, J opt (m) converges to 0 as m tends to 1. In words: If we are willing to tolerate a large enough time delay, synthesis lters can be designed to give reconstruction that is arbitrarily close to perfect. We remark that this paper treats the -channel multirate lter bank for simplicity of notation and block diagrams. The extension to the M-channel case is routine in theory, that is, the theorems extend in the obvious way, while numerical problems in accurately computing lters will undoubtedly play a dominant role at some point. Now some words on the literature. Vaidyanathan's book [19] gives a comprehensive treatment of multirate lter banks, with a detailed historical survey of the literature up to 199. The subject is a very active one at present: In references [1], [], [7], [11], [1], and [13], the constraint of perfect reconstruction is placed on the system; in [5] and [15] this is relaxed to a weaker constraint (nearly allpass system with small aliasing). The idea of optimal model matching is central in modern control theory and, in particular, L -induced norm model-matching in continuous time has been thoroughly studied (e.g., [3]). The optimal model-matching approach to multirate lter design was advocated by Shenoy in [17], as was the possibility of using H 1 optimization. Shenoy, Burnside, and Parks [18] formulate the problem of `-induced norm model matching for discrete-time periodic systems. They showed that the problem can be converted to a frequency-domain one of H 1 norm optimization of the alias component matrix of the error system. They also observe that one could equivalently convert to this problem by blocking of signals to get an LTI error system. Finally, they show how several multirate lter design problems can be formulated and solved this way. They do not treat the multirate lter bank problem, however. This introduction concludes with three examples to illustrate some aspects of the design procedure developed in the remainder of the paper. It turns out that the orders of the synthesis lters, F 0 (z) and F 1 (z), increase linearly with m. So if m must be large for J opt (m) to reach a pre-specied small value, then high-order synthesis lters result, and in this sense the design problem is considered to be hard. The rst example is easy, the second moderate, and the third hard. The examples were 3

4 performed using MATLAB and the -Analysis and Synthesis Toolbox the programs are available from the authors by request. It is emphasized that the examples are chosen to illustrate the new procedure; the analysis lters are not necessarily meant to be realistic. Example 1: H 0 (z) from a Butterworth lter Digital IIR lters are traditionally obtained by transforming analog ones. Let G(s) be the third-order analog Butterworth lter with cuto frequency 1 rad/s, G(s) = 1 s 3 + s + s + 1 ; and take H 0 (z) to be the bilinear transformation of G(s) (without prewarping). Thus H 0 (z) is a third-order lowpass lter with cuto frequency =. Taking H 1 (z) = H 0 (?z), we have H 1? e j! = H 0 e j(!?), so H 1 (z) is third-order highpass. The magnitude Bode plots of these two lters are shown in Figure 3. The function J opt (m) is graphed in Figure 4 (solid curve). It has the curious Figure 3: Example 1: jh 0 j (solid) and jh 1 j (dash) in db versus!=. property that J opt (m) = J opt (m + 1) when m is even. The graph decreases relatively rapidly, with, for example, J opt (9) = 0:0141. This means that for the time delay m = 9, the `-norm of e(k) is at most 1.41% of the `-norm of x(k), for every x(k). For this value of m, optimal synthesis lters were computed and their Bode plots are shown in Figure 5. In general the MATLAB program gives order F 0 = order F 1 (m + order H 0 + order H 1 )? 1: (1) In this case the right-hand side equals ( )? 1 = 9, but the program actually gives both these lters of order 0 (the program cancels uncontrollable or unobservable states); the orders can be reduced with negligible error to 11, that is, the `-induced norm is still to four signicant gures. (Order reduction is discussed in the Conclusion.) 4

5 Figure 4: J opt (m) versus m: Example 1 (solid), Example (dash), Example 3 (dot) Figure 5: Example 1: jf 0 j (solid) and jf 1 j (dash) in db versus!=. 5

6 As a test, the system in Figure was simulated with the input x(k) = cos(0:5k) + cos(:5k); the sum of a low-frequency sinusoid (! = 0:5) and a high-frequency sinusoid (! = :5). The result of the simulation is shown in Figure. As can be seen, the reconstruction error is very small relative Figure : Example 1: x(k) (solid) and e(k) (dash) versus k. to the size of the input. This input does not have nite `-norm, since it doesn't converge to zero. It makes sense in this case to look instead at the rms value of the signal: rms(x) := lim N!1 1 N + 1 NX?N 11= x(k) A : For this particular input, rms(x) = 1. The rms value of the error is computed to be It can be proved [10] that the `-induced norm and the rms-induced norm are equal: sup kek = sup rms(e); kxk =1 rms(x)=1 and thus rms(e) J opt rms(x). This inequality evidently holds for the present simulation (0: < 0:0141 1). Example : H 0 (z) an FIR lter In this example H 0 (z) is the FIR linear-phase lter of order 19 obtained using the MATLAB function r1, which uses a Hamming window. Again, H 1 (z) was taken to be H 0 (?z). The magnitude Bode plots of these two lters are shown in Figure 7. (These lters are not power complementary, so the perfect reconstruction solution of [19], Section 5.3., does not apply.) The function J opt (m) is graphed in Figure 4 (dashed curve). It has the properties that J opt (m) = J opt (m + 1) when m is even

7 Figure 7: Example : jh 0 j (solid) and jh 1 j (dash) in db versus!=. and 10, and that J opt (m) = 1 (i.e., the reconstruction error is 100%) for m 9. The graph lies to the right of that in Example 1. This means that for the same level of accuracy, a higher time delay is required in this second example. For the time delay m = 1, J opt = 0:099. For this value of m, optimal synthesis lters were computed having order 77, which were reduced with negligible error to 3. The Bode plots are shown in Figure 8 and the result of the simulation in Figure 9. Again, the reconstruction error is very small relative to the size of the input; the rms value of the error equals Example 3: H 0 (z) from an elliptic lter In this example G(s) is a fth-order analog elliptic lter with cuto frequency 1 rad/s. Then H 0 (z) and H 1 (z) are obtained as in Example 1; their magnitude Bode plots are shown in Figure 10. Note the very narrow transition bands. The graph of J opt (m) is shown in Figure 4 (dotted curve). In this case, J opt (m) = J opt (m + 1) for m odd. Evidently J opt (m) converges much more slowly than in the preceding examples. To get less than % error requires m = 41: J opt (41) = 0: For this value of m, the lter orders are initially 101, which can be reduced to 53 with negligible error. The corresponding plots are in Figures 11 and 1. Again, the reconstruction error is very small; the rms value of the error equals The examples illustrate that synthesis lters can be designed to get arbitrarily close to perfect reconstruction. The tradeo in accuracy is the order of the synthesis lters. Also, the quality of the analysis lters (atness on the passband and stopband, sharpness of cuto) aects the order of the synthesis lters. As a nal remark, the optimal synthesis lters do not necessarily remove all aliasing, that is, the system from x(k) to ^x(k) is not necessarily LTI. Indeed, the reconstruction error would in general 7

8 Figure 8: Example : jf 0 j (solid) and jf 1 j (dash) in db versus!= Figure 9: Example : x(k) (solid) and e(k) (dash) versus k. 8

9 Figure 10: Example 3: jh 0 j (solid) and jh 1 j (dash) in db versus!= Figure 11: Example 3: jf 0 j (solid) and jf 1 j (dash) in db versus!=. 9

10 Figure 1: Example 3: x(k) (solid) and e(k) (dash) versus k. be larger if all aliasing were required to be canceled. However, it is shown in a followup paper [10] that alias distortion, magnitude distortion, and phase distortion are all small if J is small. Conversion to a Problem of H 1 Optimization In this section we convert the synthesis lter bank design problem to one of H 1 optimization. Design of control systems by H 1 optimization is now a standard technique [4]. To review, the Hardy space H 1 consists of all complex-valued functions of z that are analytic and bounded outside the unit disc, that is, for jzj > 1. Thus H 1 is the space of transfer functions of causal LTI systems that are stable (bounded-input, bounded-output stable on `). The norm of a function G(z) in H 1 is its supremal magnitude on the unit circle: kgk 1 = sup 0! G e j! : Thus, kgk 1 equals the peak magnitude on the Bode plot. If G(z) is the transfer function of a stable, causal LTI system with input x(k) and output y(k), then it is a theorem that the `-induced norm equals the H 1 -norm of G(z), that is, sup kyk = kgk 1 : kxk =1 () This denition extends to matrix-valued functions, in which case the magnitude is replaced by the maximum singular value: kgk 1 = sup max hg! e j!i : 10

11 To extend (), let `n denote the direct sum of n copies of `; thus, elements of `n are square-summable vector-valued signals. The norm of a signal x(k) in `n is dened to be kxk = X k x(k) x(k)! 1= ; where denotes complex-conjugate transpose. If G(z) is the transfer function of a stable, causal LTI system with input x(k) of dimension m and output y(k) of dimension p, so that G(z) is p m, then the induced norm from `m to `p equals the H 1-norm of G(z), that is, () holds. As with related multirate problems [19], Figure is simplied by means of polyphase representations. First, bring in the matrix E(z) whose elements are the type-1 polyphase components of the analysis lters: " # " # H0 (z) 1 = E(z ) : H 1 (z) z?1 Next, the matrix R(z) whose elements are the type- polyphase components of the synthesis lters: h i h i F0(z) F1(z) = z?1 1 R(z ): (3) Use of these in Figure together with the \noble identities" leads to the input-output equivalent system in Figure 13 (see Figure in [19]). Now block the input x and output e in Figure 13 as - T d (z) x - # -? z?1 - - # R(z)E(z) - " "? z?1 - - j-?? ^x? j - e Figure 13: Equivalent system in terms of polyphase components. shown in Figure 14. This latter system has -dimensional input and output: " # " # v0 w0 v = ; w = : v 1 w 1 Blocking preserves ` norm; that is, the norm of x in ` equals the norm of v in ` and the norm of e in ` equals the norm of w in `. Thus the `-induced norm of the system in Figure 13 equals the ` -induced norm of the system in Figure 14. Finally, the reason blocking is useful is that the system 11

12 - T d (z) v " hx - # -? z?1 z R(z)E(z) z?1 v " - h-? # -? h? e " - w 0 z # ^x - "? w- 1 # Figure 14: Input and output blocking. v(k) - W (z) - - -? j? - E(z) R(z) w(k) Figure 15: Final system " " z j T d (z) z # - - # Figure 1: The system dening W (z). 1

13 in Figure 14 from v to w is no longer time-varying it is LTI. Indeed, Figure 14 is equivalent to the nal system, shown in Figure 15, where each of the three matrices E(z), R(z), and W (z) is. The matrix W (z) is the transfer matrix from input to output in Figure 1. It has a simple form as a consequence of the fact that T d (z) = z?m. Actually, W (z) has two possible forms, depending on whether m is even or odd. It is routine to derive that W (z) = 8 >< >: z?d " " 0 z z?d 1 0 # # ; if m = d + 1; ; if m = d: We conclude that the original design problem, minimizing the `-induced norm from x to e in Figure, is equivalent to minimizing the ` -induced norm from v to w in Figure 15. But the latter induced norm equals kw? REk 1, the H 1 -norm of the transfer matrix W (z)? R(z)E(z). The preceding can be summarized as follows: Theorem.1 Assume T d (z) = z?m. Let E(z) and R(z) be the transfer matrices corresponding to polyphase representations of, respectively, the analysis and synthesis lters. Dene Then W (z) = J opt = 8 >< >: z?d " " 0 z z?d 1 0 inf sup F 0 (z); F 1 (z) kxk =1 # # ; if m = d + 1; ; if m = d: kek = inf kw? REk 1 : R(z) The latter optimization is over all matrices R(z) that are analytic and bounded outside the unit disc. 3 State-Space Computations The MATLAB programs for H 1 optimization use state-space representations of transfer functions. In this section we present state-space formulas relevant for the design program, which is a procedure with Input: an integer m > 0 and state models of H 0 (z), H 1 (z); Output: J opt (m) and state models of F 0 (z), F 1 (z). To simplify the presentation, we use the packed notation: Thus " # A B C D stands for the transfer matrix D + C(zI? A)?1 B. 13

14 We begin with state representations for the analysis lters, H 0 (z) = and for W (z), W (z) = " # AH0 B H0 ; H C H0 D 1 (z) = H0 " # AW B W : C W D W " # AH1 B H1 ; C H1 D H1 The transfer matrix associated with their polyphase representation is then E(z) = " # AE B E := C E D E 4 A H 0 0 A H0 B H0 A H 0 B H0 0 A H 1 A H1 B H1 A H 1 B H1 C H0 0 D H0 C H0 B H0 0 C H1 D H1 C H1 B H1 Instead of Figure 15, MATLAB requires a system of the form in Figure 17. To make Figures 15 and : v(k) w(k) G(z) R(z) Figure 17: System for MATLAB. 17 equivalent, take " W (z)?i G(z) = E(z) 0 # : This has the realization G(z) = " # AG B G := C G D G 4 A W 0 B W 0 0 A E B E 0 C W 0 D W?I 0 C E D E : The MATLAB function hinfsyn inputs a realization for G(z) and outputs a realization of R(z) with the same order as G(z) that minimizes the H 1 -norm from v to w in Figure 17. (Actually, hinfsyn works in continuous time, so one must do a bilinear transformation of G(z) to G c (s), then run hinfsyn to get R c (s), and do a bilinear transformation to R(z).) Let the realization of R(z) be R(z) = " # AR B R : C R D R 14

15 Then R(z ) = " # AR B R := 4 C R D R Finally, from (3) we have h F 0 (z) F 1 (z) i = 4 h h 0 I 0 A R 0 B R C R 0 D R A R i C i R C R 1 h h : B R D R i i D R For a dimension count, let n H denote the size of the A-matrix in a realization of a transfer matrix H(z). Adding up the dimensions in the above formulas, one gets n F0 = n F1 = (n W + n H0 + n H1 ) + 1: Since n W = m? 1, this yields n F0 = n F1 = (m + n H0 + n H1 )? 1: It may be possible to get lower order synthesis lters if one reduces to minimal realizations at each appropriate stage, that is, if one discards uncontrollable or unobservable states. This happened in Example : 4 Limiting Performance In this section it is proved that, under a mild condition on the analysis lters, arbitrarily good reconstruction is possible if a suciently large time delay can be tolerated. The mild condition is on the transfer matrix E(z). Theorem 4.1 Assume E(e j! ) is nonsingular for all 0!. Then lim m!1 J opt (m) = 0. The proof of this theorem requires a preliminary result. A rational transfer matrix G(z) is said to be stable if it is analytic in jzj 1. Lemma 4.1 Let G(z) be a rational matrix with no poles on the unit circle. Let Y (z) = z?d G(z) and K(d) = inf X(z) ky? Xk 1, where the inmum is over all X(z)s that are analytic and bounded outside the unit disc, that is, K(d) equals the distance from Y (z) to H 1. Then lim d!1 K(d) = 0. Proof By Nehari's theorem [14], [4], K(d) equals the norm of the Hankel operator,? Y, associated with Y (z). To write down? Y explicitly, bring in the series expansion of G(z): G(z) = + G? z + G?1 z + G 0 + G 1 z?1 + G z? + : Then the Hankel operator associated with G(z) is? G = 4 G?1 G? G?3 G? G?3 G?4 G?3 G?4 G?

16 and likewise? Y = 4 G?(d+1) G?(d+) G?(d+3) G?(d+) G?(d+3) G?(d+4) G?(d+3) G?(d+4) G?(d+5) : Being rational, the series +G? z +G?1 z (i.e., the anticausal part of G(z)) has a state realization: + G? z + G?1 z = zc(i? za)?1 B; that is, G?(k+1) = CA k B. All the eigenvalues of A lie strictly inside the unit disc, and so A k! 0 as k! 1. We now have? Y = = 4 4 CA d B CA d+1 B CA d+ B CA d+1 B CA d+ B CA d+3 B CA d+ B CA d+3 B CA d+4 B. C CA CA.. 3 h 7 5 Ad B AB A B i : (4) The (observability and controllability) operators o := 4 C CA CA ; c := h B AB A B i are both bounded. So from (4) k? Y k k o k ka d k k c k: The right-hand side converges to 0 as d! 1. Proof of Theorem 4.1 As is well-known, every stable transfer function can be factored into the product of an allpass function (unit magnitude at all frequencies; also called \lossless" function) and a minimum phase function (no zeros in jzj > 1). This fact is true for matrix-valued transfer functions too. The rst step in the proof of the theorem is to factor E(z) into an allpass part, E ap (z), and a minimum phase part, E mp (z). The construction is as follows: The minimum phase factor arises from spectral factorization: E(z)E(z?1 ) T = E mp (z)e mp (z?1 ) T : By the assumption on E(z) and by construction, E mp (z) and E mp (z)?1 are stable. Dening E ap (z) = E mp (z)?1 E(z), we have E ap (z)e ap (z?1 ) T = I, so E ap (z) is unitary on the unit circle. 1

17 Now we have J opt (m) = inf kw? REk 1 R(z) Since E ap (e j! ) is unitary, and so = inf kw? RE mp E ap k 1 R(z) = inf k(w E ap?1? RE mp )E ap k 1 : R(z) k(w E?1 ap? RE mp )E ap k 1 = kw E?1 ap? RE mp k 1 ; J opt (m) = inf kw E ap?1? RE mp k 1 : R(z) Since E mp (z) and E mp (z)?1 are stable, we can dene X(z) = R(z)E mp (z) and equally minimize over X(z) in H 1. Thus J opt (m) = inf kw E ap?1? Xk 1 : X(z) The result now follows from Lemma 4.1. The assumption in Theorem 4.1, nonsingularity of E(e j! ), is not directly in terms of the analysis lters, H 0 (z) and H 1 (z). The next result gives such a connection. Lemma 4. Fix!. The matrix E(e j! ) is nonsingular i H 0 (e j!= )H 1 (?e j!= ) = H 0 (?e j!= )H 1 (e j!= ): Proof By denition of E(z), " # H0 (z) = E(z ) H 1 (z) Replacing z by?z, we have " # H0 (?z) = E(z ) H 1 (?z) " 1 z?1 # : " # 1 :?z?1 Combine the preceding two equations to get " # " # H0 (z) H 0 (?z) 1 z = E(z?1 ) : H 1 (z) H 1 (?z) 1?z?1 Since the right matrix on the right is nonsingular, E(z ) is nonsingular i the matrix on the left is nonsingular, i.e., det E(z ) = 0, H 0 (z)h 1 (?z) = H 0 (?z)h 1 (z); from which the result follows. 17

18 The condition H 0 (e j!= )H 1 (?e j!= ) = H 0 (?e j!= )H 1 (e j!= ) for!, or equivalently H 0 (e j! )H 1 (?e j! ) = H 0 (?e j! )H 1 (e j! ) (5) for! =, is very mild. For example, if H 0 (z) is lowpass, H 1 (z) is highpass, and both have cuto frequency =, then jh 0 (e j! )j > jh 0 (?e j! )j and jh 1 (?e j! )j > jh 1 (e j! )j for! < = and hence (5) holds for! < =. In the special case that H 1 (z) = H 0 (?z), as in the three examples, the condition reduces to H 0 (e j!= ) = H 1 (e j!= ) for!, or equivalently H 0 (e j! ) = H 1 (e j! ) for! =. If H 0 (z) is lowpass with cuto frequency =, then jh 0 (e j! )j > jh 1 (e j! )j for! < =; and for! = =, H 0 (e j! ) = H 1 (e j! ) because of phase lag. 5 Conclusion The advantage of the design method proposed in this paper is that one does not have to design all four lters, analysis and synthesis, simultaneously: One can design the analysis lters with regard to coding of the input signal, then design the synthesis lters for good reconstruction. The limitation of the design method is that quite high order synthesis lters may result. In this case one can either reduce the order of the synthesis lters or redesign the analysis lters so as not to be so demanding. There are many ways to reduce the order of a lter, but only a few give a guaranteed error bound on the frequency response, and hence on the `-induced norm. One such way to do order reduction is to use the MATLAB function sysbal (balmr could also be used), which truncates a balanced realization. To state the error bound, suppose G(z) is a stable, causal, IIR or FIR lter of McMillan degree n (i.e., n is the size of the A-matrix in a minimal realization). The Hankel operator associated with G(z) has n nonzero singular values, denoted in decreasing order by 1 ; : : :; n. Suppose G r (z) is the approximant of McMillan degree m < n obtained using sysbal. Then kg? G r k 1 ( m n ): The left-hand side is the maximum frequency-response error, so one should choose m so that the right-hand sum is suciently small. The lters F 0 (z) and F 1 (z) in the three examples in the Introduction were reduced in this way. For example, Figure 18 shows plots of k versus k for the lter F 0 (z). Note that for Example 1, 1 is much smaller than preceding ones, leading to a reduced lter of order 11. Similarly for the other examples. As a nal comment, let us note that FIR synthesis lters may be more desirable in certain applications. In this case one would like to formulate the design problem as one of `-induced norm optimization under the constraint that the synthesis lters be FIR of a specied order. This approach is reported in [1]. 18

19 Figure 18: Hankel singular values k versus k for F 0 (z): Example 1 (solid), Example (dash), Example 3 (dot). Acknowledgement The authors are grateful to Dr. Ram Shenoy for helpful discussions and for an advance copy of [18], and to the reviewers for constructive comments. References [1] H. Caglar and A.N. Akansu. A generalized parametric PR-QMF design technique based on Bernstein polynomial approximation. IEEE Trans. Signal Processing, vol. 41, pp. 314{31, [] N.J. Fliege. Multirate Digital Signal Processing, Wiley, Chichester, [3] B.A. Francis. A Course in H 1 Control Theory, Springer-Verlag, Lecture Notes in Control and Information Sciences, vol. 88, Berlin, [4] M. Green and D.J.N. Limebeer. Linear Robust Control, Prentice-Hall, Englewood Clis, [5] R.D. Koilpillai and P.P. Vaidyanathan. A spectral factorization approach to pseudo-qmf design. IEEE Trans. Signal Processing, vol. 41, pp. 8{9, [] J. Kovacevic and M. Vetterli. Perfect reconstruction lter banks with rational sampling factors. IEEE Trans. Signal Processing, vol. 41, pp. 047{0, [7] Y.C. Lim, R.H. Yang, and S.-N. Koh. The design of weighted minimax quadrature mirror lters. IEEE Trans. Signal Processing, vol. 41, pp. 1780{1789, [8] H.S. Malvar. Signal Processing with Lapped Transforms, Artech House, Boston,

20 [9] Y. Meyer. Wavelets: Algorithms and Applications, SIAM, Philadelphia, [10] S. Mirabbasi, B.A. Francis, and T. Chen. Input-output gains of linear periodic discrete-time systems with application to multirate signal processing. Technical report, Electrical and Computer Engineering, University of Toronto, [11] K. Nayebi, T.P. Barnwell III, and M.J.T. Smith. Nonuniform lter banks: a reconstruction and design theory. IEEE Trans. Signal Processing, vol. 41, pp. 1114{117, [1] K. Nayebi, T.P. Barnwell III, and M.J.T. Smith. Low delay FIR lter banks: design and evaluation. IEEE Trans. Signal Processing, vol. 4, pp. 4-31, [13] K. Nayebi, T.P. Barnwell III, and M.J.T. Smith. On the design of FIR analysis-synthesis lter banks with high computational eciency. IEEE Trans. Signal Processing, vol. 4, pp. 85{834, [14] Z. Nehari. On bounded bilinear forms. Annals of Mathematics, vol. 15, pp. 153{1, [15] T.Q. Nguyen. Near-perfect-reconstruction pseudo-qmf banks. IEEE Trans. Signal Processing, vol. 4, pp. 5{7, [1] S. Ratzla, T. Chen, and S.A. Norman. H 1 -optimal design of FIR QMF banks using convex optimization. Technical report, Electrical and Computer Engineering, University of Calgary, [17] R.G. Shenoy. Analysis of multirate components and application to multirate lter design. Proc. ICASSP, pp. III-11{III-14, [18] R.G. Shenoy, D. Burnside, and T.W. Parks. Linear periodic systems and multirate lter design. IEEE Trans. Signal Processing, vol. 4, pp. 4{5, [19] P.P. Vaidyanathan. Multirate Systems and Filter Banks, Prentice-Hall, Englewood Clis,

21 List of Figures 1 Multirate lter bank. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 Error system. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 Example 1: jh 0 j (solid) and jh 1 j (dash) in db versus!=. : : : : : : : : : : : : : : 4 4 J opt (m) versus m: Example 1 (solid), Example (dash), Example 3 (dot). : : : : : : 5 5 Example 1: jf 0 j (solid) and jf 1 j (dash) in db versus!=. : : : : : : : : : : : : : : 5 Example 1: x(k) (solid) and e(k) (dash) versus k. : : : : : : : : : : : : : : : : : : : 7 Example : jh 0 j (solid) and jh 1 j (dash) in db versus!=. : : : : : : : : : : : : : : 7 8 Example : jf 0 j (solid) and jf 1 j (dash) in db versus!=. : : : : : : : : : : : : : : 8 9 Example : x(k) (solid) and e(k) (dash) versus k. : : : : : : : : : : : : : : : : : : : 8 10 Example 3: jh 0 j (solid) and jh 1 j (dash) in db versus!=. : : : : : : : : : : : : : : 9 11 Example 3: jf 0 j (solid) and jf 1 j (dash) in db versus!=. : : : : : : : : : : : : : : 9 1 Example 3: x(k) (solid) and e(k) (dash) versus k. : : : : : : : : : : : : : : : : : : : Equivalent system in terms of polyphase components. : : : : : : : : : : : : : : : : : : Input and output blocking. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 15 Final system. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1 The system dening W (z). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 17 System for MATLAB. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Hankel singular values k versus k for F 0 (z): Example 1 (solid), Example (dash), Example 3 (dot). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 19 1

22 Biographies Tongwen Chen (S'8-M'91) received the B.Sc. degree in Electrical Engineering from Tsinghua University, Beijing, China, in 1984, and the M.A.Sc. and Ph.D. degrees also in Electrical Engineering from the University of Toronto, Toronto, Canada, in 1988 and 1991, respectively. Since 1991, he has been an Assistant Professor in Electrical and Computer Engineering at the University of Calgary, Alberta, Canada. His current research interests include systems and control, digital signal processing, digital control and sampled-data systems, robust control, and multirate systems in control and signal processing. He coauthored the book, Optimal Sampled-Data Control Systems, published in the Communications and Control Engineering Series by Springer in Dr. Chen is a registered Professional Engineer in the Province of Alberta, Canada. Bruce A. Francis was born in Toronto, Canada. He received the B.A.Sc. and M.Eng. degrees in mechanical engineering and the Ph.D. degree in electrical engineering from the University of Toronto in 199, 1970, and 1975, respectively. He has held research and teaching positions at the University of California at Berkeley, the University of Cambridge, Concordia University, McGill University, Yale University, the University of Waterloo, Caltech, and the University of Minnesota. He is presently a Professor of Electrical and Computer Engineering at the University of Toronto. Professor Francis is a Fellow of the IEEE. He was a co-recipient of two Outstanding Paper Awards for papers appearing in the IEEE Transactions on Automatic Control, and a co-recipient of the 1991 IEEE W.R.G. Baker Prize Award. He has co-authored three books, including Optimal Sampled-Data Control Systems with T. Chen, published in His current research is in digital control and signal processing.