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1 SOE FUNDAENTAL POPETIES OF SE FILTE BANKS. Kvanc hcak, Pierre oulin, Kannan amchandran University of Illinois at Urbana-Chamaign Beckman Institute and ECE Deartment 405 N. athews Ave., Urbana, IL ihai Anitescu Argonne National Laboratories Argonne, IL ABSTACT We design lter banks that are best matched to inut signal statistics in -channel subband coders, using a broad class of rate{distortion criteria. We resent fundamental roerties and analytical exressions for minimum mean-squared error (SE) lter banks, without constraints on lter length, under otimal bit allocation requirements. We also investigate a constrained{length version of this roblem, which is alicable to ractical coding scenarios. While the otimal lter banks are nearly erfect-reconstruction at high rates, we show that SE FI lter banks enjoy a signicant advantage (in the SE sense) over otimal erfect{reconstruction FI lter banks at all rates.. INTODUCTION We consider {channel subband coders with analysis lters fh i(f); 0 i < g and synthesis lters f Hi(f); 0 i < g. Fig. shows an equivalent reresentation of the codec in terms of the analysis and synthesis olyhase matrices H(f) and H(f). The roblem of interest here is to design the lters so as to otimize the rate{distortion erformance of subband coders that use uniform scalar quantizers in each channel. The distortion measure is mean{squared reconstruction error. We exlore fundamental roerties of SE lter banks (for which the erfect{reconstruction (P) is not imosed a riori []). We also resent analytical exressions for the resulting otimal lter banks in terms of bit rate and second-order inut signal statistics... Basic odel for Signal and uantization Noise The inut to the subband coder is assumed to be real{ valued and wide{sense stationary with zero mean and sectral density S(f). Throughout, we assume that S(f) is bounded away from zero. The total bit budget is bits er samle, to be allocated to the quantizers in each channel. uantizer i in channel i oerates on a signal y i(n) with variance P i, is scalar and uniform, and is allocated i bits, where i =. We make the assumtion that the quantization noise is additive, white and indeendent of the signal; and that the quantization noise sources in dierent channels are mutually indeendent. This is a standard model which is valid at high bit rates, but not at low Work suorted by NSF grant IP and DOE contract W-3-09-Eng-38. bit rates. Each quantizer is assumed to have a distortion{ rate function i D( i), where in this context distortion is quantization noise variance. The following roerties are required : D(:) is strictly ositive, strictly monotonic decreasing, and strictly convex. For Theorems. and. to hold, we also require lnd(:) to be concave. The standard exonential model D( i) = i i for the rate{distortion function satises all of the assumtions above. We rst state the design criterion for P lter banks in Sec.. and extend it to SE lter banks in Sec Design Criterion for P Systems Under the assumtions above, the reconstruction error ^ is a cyclostationary rocess with eriod. For P systems, the goal is to minimize the exected mean{squared error (SE) [] Ej^ j = n=0 D( i) i jj h ijj () where jj h ijj reresents the amlication factor for white noise assed through synthesis lter h i. In order to nd the otimal bit allocation given the lter banks, the otimization roblem () with bit rate constraints is transformed into the Lagrange otimization roblem inimize P D(i) i jj h ijj P i where is the Lagrange multilier. The solution satises the condition, i jj h ijj dd() d i = ; 0 i < : () Hence, for otimal bit allocation, the sloe of the distortion{ rate function D(i) i jj hijj at the encoder's oerating oint must be the same for all i. It has been assumed here that is large enough so that the ositivity constraints i 0 are all inactive. Due to the strict convexity of D(:), the otimal bit allocation roblem has a unique solution. Let S(f) be the sectral density matrix for the olyhase vector, inut to H(f) in Fig.. We have i = 0:5 0:5 (HSH y ) ii df; jj h ijj = 0:5 0:5 ( H y H) ii df:

2 - H (f) -.3. Design Criterion for SE Filter Banks In the SE lter bank aroach, the P constraints are relaxed so as to trade o systematic reconstruction errors (due to lack of P) against quantization noise. The solution is nearly identical to the P solution at high bit rates (low quantization noise), but notable imrovements over P designs have been obtained using numerical simulations at lower bit rates []. Here, we seek analytical exressions for the lters and bit allocation f ig that jointly minimize the SE. Using the model in Sec.., signal and quantization noise are assumed to be indeendent, so the SE is the sum of the noise term () and a signal term. The SE is E = P D( i) (HSH y ) ii ( H y H) ii T r ( HH I ) S ( HH I ) y ; (3) to be minimized over H; H, and f ig. allocation condition is still given by (). The otimal bit - ^ H (f) - - y(n) Figure : Polyhase reresentation of {channel subband coder and decoder using analysis lters H i(f) and synthesis lters Hi(f). The olyhase comonents of H i(f) (res. Hi(f)) are contained in row (res. column) i of the olyhase matrix H(f) (res. H(f).). FUNDAENTAL POPETIES OF SE FILTE BANKS We have recently roven that otimal (in the sense ()) P lter banks enjoy two fundamental roerties: total decorrelation of subband channels, and sectral majorization [, 3]. These roerties were reviously known to aly only to araunitary lter banks, in which case the solution is a rincial{comonent lter bank (PCFB) [4, 5, 6]. We now show that these fundamental roerties hold even if the P conditions are relaxed, and the cost function (3) is used. The roof of the two theorems below uses variational techniques and arallels the roof of Theorem.3 and Lemma.6 in []. Theorem. (Total Decorrelation is Necessary for Otimality.) The system H; H, f ig minimizes (3) only if the matrices HSH y and H y H are diagonal, and f ig satises (). Theorem. (Sectral ajorization is Necessary for Otimality.) Let H; H, f ig be a minimizer of (3), and = H y H. Without loss of generality, assume that 0. The normalized sectral densities w i i S y;ii(f) for the subband signals satisfy the sectral majorization roerty: w 0 0 S y;00(f) S y;;(f); 8f; w where w i 4 = D( i). Likewise, the normalized quantities w i jj h i jj ii(f) satisfy the sectral majorization roerty w 0jj h 00(f) 0jj w jj h ;(f); 8f: jj The articular lter bank structure shown in Fig. was shown to be otimal in the P class [, 3], in which case G i(f) = =G i(f). The rst block in this structure, U(f), is a PCFB. While at this oint we are not able to ascertain

3 - - U(f) z(n) 0 - y(n) G 0(f) G (f) - U(f) ^ Figure : Cascade of a rincial{comonent lter bank U(f) and a set of relters G i(f) and ostlters Gi(f) around each quantizer. This system satises the total decorrelation and sectral majorization roerties. whether the structure in Fig. is also otimal for SE lter banks, we are able to comute the best lters G i(f) and Gi(f). We then rove some imortant roerties of the resulting lter bank. We can write the cost function (3) in terms of the re{ and ostlters G i(f) and Gi(f) as E = P D(i) jg i Sij j Gij P j GiG i j S i; (4) where S i(f) is the sectral density for the signal z i(n) in Fig.. Analytical exressions for the otimal lters G i(f), G i(f) deend on the bit rates f ig. Theorem.3 For any f ig, the lters that minimize the SE (3) for the system in Fig. are jg i(f)j = c i Pi(f)S =4 i (f); j Gi(f)j = c i Pi(f)S =4 i (f) (5) where c i are arbitrary ositive scaling factors, and the roduct lters P i(f) = G i(f) Gi(f), 0 i <, are given by P i(f) = max 0; D(i) D( i) The SE for these lters is given by E(H; H) = n f j S i(f) > D( i) D( i ) Si(f) df Si(f) : (6) Si(f) df D( i) D( i) S i(f) df; (7) where means integration over the (unique) set F i that o satises F i = Si, and means integration over the comlementary set. Outline of the Proof : By the Cauchy-Schwartz inequality, P (4) is lower bounded by E LB = P D(i) G i Gi Si Gi Gi S i, with equality i G i Si = Since the class of lters considered in Fig. contains the otimal P lter bank, the solution is clearly guaranteed to be at least as good as the otimal P lter bank. At high bit rates, F i =. As our numerical results have shown, F i = at all bit rates for A() rocesses. c i G i. Additionally, c i > 0 and P i(f) 0 (see []). The functional E LB is strictly convex in P i, and its unique minimizer is given by (6). Substituting into E LB, we obtain (7). The otimal SE (7) is uer{bounded by the otimal value E = P D(i) 0:5 0:5 Si(f) df for the II biorthogonal case [], and tends to this limit as. The erformance of FI SE lter banks converges to (7) as the lter length tends to innity. As discussed in [, 3], P lter banks do not enjoy a similar roerty: FI lter banks of arbitrary length must satisfy the constraint det H(f), and the erformance of these lters does not converge to that of II lter banks (for which the constraint det H(f) is not alicable). Even at high bit rates, FI SE lter banks of sucient length can vastly outerform FI biorthogonal lter banks of arbitrary length. While conditions (5),(6) for otimality of the lter bank aly for arbitrary f ig, the otimal f ig do satisfy (). 3. IPOTANT SPECIAL CASES Consider the classical model D( i) = i for the rate{ distortion function with otimal bit allocation. Then () yields the closed{form solution i / i k h ik. (Just before going to ress, we discovered the aer [7] which derives the otimal set of re{ and ostlters for a single{channel roblem. However, their framework aarently does not lend itself to joint otimization of lter banks and bit allocation [7,. 04].) The resulting distortions i k hik D( i) are identical for all channels. The exected SE (3) is then E(H; H) = (HSH y ) ii ( H y H) ii = T r ( HH I ) S ( HH I ) y : (8) For the exonential D( i) model and otimal bit allocation, the general exressions (6),(7) are secialized in Theorem 3. below. Analytical exressions for the otimal lters are given in terms of constants fi g which are solutions to a nonlinear system. If it is known that F i =, then the only constant to be solved for is E q. Theorem 3. The lters that minimize the SE (8) for the system in Fig. are given by (5) with roduct lters Here i = P i(f) = max 0; Eq= i Si(f) ; 0 i < : (9) s Si(f) df Si(f) df E qjf i j; (0) are the normalized variances of the subband signals (using c i = ), Y = E q = i ; () and F i = n f j S i(f) > Eq i o. The SE for these lters

4 is given by E(H; H) = E q Eq i jf i j S i(f) df: () Note that according to (0), there exist two ossible candidates for the solution i in each channel. Each of these solutions is a local extremum for (8). The otimal solution is the one that minimizes (8). For bit rates that are high enough, each sign in (0) must be ositive. {Channel Case Substantial simlications arise if the number of channels is =. The roduct lters are given by P 0(f) = max 0; S0(f) P (f) = max 0; 0 S(f) ; : (3) The otimal lters are again given by (5). Here the normalized variances of the subband signals are (using c i = ) 0 = = S0(f) df jf 0 j S(f) df ( ) jf 0 jjf j S(f) df jf j S0(f) df ( : (4) ) jf 0 jjf j The SE corresonding to these lters is given by E = 0 ( ) (jf 0 j 4 0jF j 4 ) ; S i(f) df: (5) (4) is a nonlinear system in 0;. But if it is known that F =F = 0, (4) (5) no longer contains unknowns. 4. SIULATION ESULTS An A() rocess with correlation coecient r = 0:8 in the two{band case has been considered to illustrate the analysis above. In this case, the PCFB is the traditional lter bank with ideal low and high ass lters. The otimization roblem (8) for the re{ and ostlters has been solved for various rates and the results have been comared with otimal II biorthogonal lter banks and otimal unconstrained{length FI biorthogonal lter banks [, 3]. At all bit rates, F i =. For = :76, the bit rate in the high{ass channel becomes zero, in which case the criterion (8) becomes clearly invalid. The erformance of the otimal unconstrained{length SE lter banks is very close to otimal II biorthogonal lter banks at very high rates, but imrovements become quite signicant as decreases (Fig. 3). At very high rates, the otimal lter banks are close to the (P) II biorthogonal solution. At lower bit rates, the lters dier signicantly from those otimal P lters. Frequency resonses are shown in Fig. 4 for an A() rocess with correlation coecient r = 0:8, and rate = :9. The scaling factors c 0 and c for all three lter banks have been chosen so that the frequency resonses are the same at f = 0 and at f = 0:5. The remarks at the end of Sec. motivated us to investigate the constrained{length version of this design. A simle rectangular windowing technique was used to design constrained{length FI{SE lterbanks from the otimum unconstrained{length solution. As shown in Fig. 5, the results are excellent at medium bit rates. At = :9, the length{63 FI{SE lter bank outerforms otimal FI biorthogonal lter banks of arbitrary length, and the length{03 FI{SE lter bank outerforms otimal II biorthogonal lter banks. Similar advantages hold at arbitrarily high bit rates, but longer FI lters are needed to break the erformance bounds for FI and II biorthogonal lters. enements in the FI{SE lter design method are likely to yield further imrovements. 5. EFEENCES [] K. Gosse and P. Duhamel, \Perfect{econstruction versus SE Filter Banks in Source Coding," IEEE Trans. SP, Vol. 45, No. 9,. 88 0, Se [] P. oulin,. Anitescu and K. amchandran, \Theory of ate{distortion{otimal, Constrained Filter Banks{Alication to II and FI Biorthogonal Designs," submitted to IEEE Trans. Sig. Proc., 998. [3] P. oulin, K. amchandran and. Anitescu, \Asymtotic Performance of Subband Coders Using Constrained, Signal{Adated FI Filter Banks," Proc. 3nd CISS Conference, Princeton, NJ, arch 998. [4]. Unser, \An Extension of the Karhunen-Loeve Transform for Wavelets and Perfect{econstruction Filterbanks," SPIE Vol. 034, , 993. [5]. K. Tsatsanis and G. B. Giannakis, \Princial Comonent Filter Banks for Otimal ultiresolution Analysis," IEEE SP, Vol. 43, , Aug [6] P.P. Vaidyanathan, \Theory of Otimal Orthonormal Subband Coders," IEEE Trans. Sig. Proc., Vol. 46, No. 6, , June 998. [7] J. Tuqan and P.P. Vaidyanathan, \Statistically Otimum Pre{ and Postltering in uantization," IEEE CAS II, Vol. 44, No., , Dec ean Squared Error (db) Bit rate Figure 3: A() rocess with correlation coecient r = 0:8: Otimum values of SE as a function of overall bit rate for dierent lter design methods. Solid line: II biorthogonal (half-whitening) lters [],[3]; Dotted line: FI biorthogonal lters [],[3]; Dashed line: SE lters.

5 Analysis lowass, A rocess, r = Analysis highass, A rocess, r = magnitude magnitude frequency frequency Figure 4: Frequency resonses of otimal, unconstrained length analysis lters in two-band case, for A() rocess at bit rate = :9. Solid line: II biorthogonal (halfwhitening) lters [],[3]; Dotted line: FI biorthogonal lters [],[3]; Dashed line: SE lters. 3 ean Squared Error (db) FI SE filter length Figure 5: A() rocess, = :9: Convergence of SE (solid curve) for FI{SE lterbanks to -7. db limit (dashed line) for unconstrained{length SE lter banks. Comare with SEs for II biorthogonal lter banks (dash{dotted line, db), and unconstrained{length FI biorthogonal lter banks (dotted line, -6.6 db).