ON FRAMES WITH STABLE OVERSAMPLED FILTER BANKS

Transcription

1 ON FRAMES WITH STABLE OVERSAMPLED FILTER BANKS Li Chai,, Jingxin Zhang, Cishen Zhang and Edoardo Mosca Department of Electrical and Computer Systems Engineering Monash University, Clayton, VIC3800, Australia {lichai, Department of Automation, Hangzhou Dianzi University, Hang Zhou, , PR China School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore Dipartimento di Sistemi e Informatica, Universityá di Firenze, Italy ABSTRACT This paper studies the frames corresponding to stable oversampled filter banks (FBs) For this class of frames, we present explicit and numerically efficient formulae to compute the tightest frame bounds, to obtain the dual frame and to construct a paraunitary FB for a given non-paraunitary FB The derivation uses the well developed techniques from modern control theory, which results in the formulae that involve only algebraic matrix manipulation and can be performed efficiently and reliably without the approximation required in the existing methods 1 INTRODUCTION Recently oversampled FBs with redundant signal expansions, as shown in Fig 1, have attrted much attention Their advantages include increased design freedom, enhanced noise reduction, and improved capity for signal and information representation 1, 2, 3 A great deal of research has been devoted to the analysis and design of oversampled FBs, see 2, 5, 6, 7, 8, 9, 10, 11 and the references therein In 1, 2, an elegant frame-theoretic approh is presented to the analysis and design of oversampled FBs It is revealed that perfect reconstruction (PR) oversampled FBs are equivalent to a particular class of frames in l 2 (R), and that the frame bounds are important numerical properties of FBs which can be used to analyze and design oversampled FBs These results have provided a powerful theoretical guide to the analysis and design of oversampled FBs To apply these important This work is supported in part by the National Natural Science Foundation of China under grant and , and in part by a research grant from Australia Research Council n 0(Mk) x(k) ˆx(k) e(k) H 0(z) M M F 0(z) H 1(z) H N 1(z) M M n 1(Mk) n N 1(Mk) z d M M F 1(z) F N 1(z) Fig 1 Oversampled filter bank with N>M results in prtice, the efficient methods must be developed to compute the frame bounds for given FBs and to construct the FBs for implementing frames It is suggested in 1 that in prtice, the tightest frame bounds have to be estimated (approximately) by sampling a complex matrix function S(e jθ ) on the unit circle (θ 0, 2π)) and by performing eigen-analysis on the samples It is also suggested in 1 that the minimum norm PR synthesis FB is prtically difficult to obtain since it involves the inversion of a transfer matrix, therefore it needs to be approximated by the truncated Neumann series These approximation based methods are suitable for the generic frame and wavelets where the ext solutions in general cannot be to obtained 3, 4 However, oversampled FBs is a special class of frames, as will be shown in this paper, we can get explicit formaulae for their analysis and construction without approximation We provide in this paper explicit and numerically

2 efficient formulae to compute frame bounds and the PR synthesis FB corresponding to dual frame The derivations are based on the well-developed techniques in modern control theory 12 The main advantage of our formulae is that they provide directly computable formulation with the effort not exceeding algebraic matrix manipulations and not requiring any approximation Both FIR and IIR FBs are studied in a unified framework The charterization of all PR synthesis FBs in state spe form is also presented The paper is organized as follows Section 2 introduces notations and reviews some basic concepts on transfer matrices and state spe realization, oversampled FBs and their relation to frames The explicit formulae to compute the tightest frame bounds, to obtain the dual frame and to construct a paraunitary FB for a given non-paraunitary FB are provided in Section 3 The charterization of all PR synthesis FBs in state spe form is also presented in Section 3 Concluding remarks are given in Section 4 2 PRELIMINARIES This sections reviews some preliminaries, in particular, the concepts of transfer matrices, state spe realization, frames and oversampled filter banks Please refer 1, 2, 12, 17 for more details The notation used in this paper is standard The real and complex numbers are denoted by R and C respectively The set of n-dimensional real (complex) vectors is denoted by R n (C n ) R m n (C m n ) denotes the m n real (complex) matrix set l 2 (Rn ) denotes the Hilbert spe of all square norm sumable sequences (x i ) i=0 writtenascolumnvectors x 0 x 1 x 2 T, where x i R n Similarly l 2 (R n ) denotes the Hilbert spe of all square norm summable doubly infinite sequences (x i ) i= writtenascolumnvectors x 1 x 0 x 1 T where x i R n The underline x 0 means that x 0 appears at the zero position Definition 1 E(z) = i= E iz i C N M is BIBO stable if i= E i is bounded E(z) is called causal if E i =0fori<0, and called anticausal if E i =0for all i 0 Definition 2 For any rational causal transfer matrix E(z) = i=0 E iz i C N M, if the matrices A R n n,b R n M,C R N n and D R N M are such that E(z) =D C(zI A) 1 B, then (A, B, C, D) is called a state spe realization of E(z) and denoted as The realization is minimal if the dimension of A is minimal Definition 3 For any rational anti-causal transfer matrix E(z) = 0 i= E iz i C N M, if the matrices A R n n,b R n M,C R N n and D R N M are such that E(z) = C(z 1 I A) 1 B D, then (A, B, C, D) is called a state spe realization of E(z) and denoted as Definition 4 A rational transfer matrix N(z) is called inner if N(z) is stable and N (z)n(z) =I for all z = e jθ,θ 0, 2π) The following lemma from 16 is very useful for computing the cascaded discrete-time systems with state spe description Lemma 1 Assume the compatibility of the operations Then A A1 B 1 A2 B 1 B 1 C 2 B 1 D 2 2 = 0 A 2 B 2 C 1 D 1 C 2 D 2 C 1 C 2 D 1 D 2 A1 B 1 A2 B 2 = C 1 D 1 C 2 D 2 A1 A 1 YB 2 B 1 D 2 A 2 B 2 C 1 D 1 D 2 C 1 YB 2 D 1 C 2 C 1 YA 2 0 where Y is given by the Sylvester equation A 1 YA 2 Y B 1 C 2 =0 (1) Lemma 2 For causal stable E(z) C N M with a minimal realization, assume that I B has full column rank for all θ 0, 2π) andd has full column rank Then E(z) =N(z)M(z) 1 with N(z) inner, where M N A BF BW 1 2 : = F C DF W 1 2 DW 1 2 (2) W = D D B XB (3) F = W 1 (B XA D C) (4) and X = X 0 is the unique stabilizing solution to A XA X C C (A XB C D)W 1 (B XA D C)=0 (5)

3 For an N-channel oversampled filter bank with decimation ftor M shown in Fig 1, assume that the analysis and synthesis filters, denoted by H k (z) and F k (z), k=0, N 1, are all linear and stable, and have the following form H k (z) = n= h k nz n and F k (z) = n= f k nz n where h k n andf k n are impulse response coefficients of H k (z) andf k (z), respectively Note that H k (z) is called FIR if there exists a finite number K such that h k n = 0 for all n K, and called IIR otherwise Definition 5 A sequence {ϕ k } k= in a Hilbert spe l 2 (R) is said to be a frame if there are constant α, β > 0 such that α x 2 k= <x,ϕ k > 2 β x 2 (6) for all x l 2 (R), where <x,ϕ k > denotes the inner product of x and ϕ k Let U denote a forward shift operator on l 2, that is, U x 1 x 0 x 1 T = x 1 x 0 x 1 T Definition 6 The operator T is said to be (k 1,k 2 )-shift invariant if TU k1 = U k2 T Lemma 3 The analysis operator E : l 2 (R) l 2 (R N ) and the synthesis operator R : l 2 (R N ) l 2 (R) are (M,1)-shift-invariant and (1,M)-shift-invariant respectively The frame operator S : l 2 (R) l 2 (R) ofe is defined as Sx = E Ex An FB satisfies PR with zero delay if and only if RE = I For such FB, ˆxn = N 1 k=0 m= <x,h k,m >f k,m n In the language of frame theory, this corresponds to an expansion of the input signal into f k,m n 1,2 Itis well-known that {h k,m } is a frame if and only if PR is hieved for any x l 2 (R) So the PR condition indicates the invertibility of the analysis operator E in a given set For more detail, please refer to 1, 2 For an analysis FB, we will use alternatively its transfer function H k (z), its polyphase matrix E(z) anditsframe {h k,m } to refer to the FB 3 MAIN RESULTS We have known that the analysis operator E is (M,1)- shift-invariant and the synthesis operator R is (1,M)- shift-invariant By the technique of blocking14, 15, we can convert E and R to an equivalent LTI operator To this end, let L M be a blocking operator from l 2 (R) to l 2 (R M ) defined as = L M x 0 x 1 x 2 T x M T x M1 (7) x 2M 1 x 0 x 1 x M 1 Lemma 4 The operator EL 1 M : l2 (R M ) l 2 (R N )is linear time-invariant and has transfer function representation E(z), where E(z) is the polyphase matrix representation from E, that is, E(z) =E i,j (z), where E i,j (z) = m= h imm jz m For a causal and stable EL 1 M,E(z) iscausaland stable and has a minimal state-spe realization E(z) =D C(zI A) 1 B where D R N M,C R N n,b R n M and A R n n The frame operator S is M-periodic but not necessarily stable It is shown in 1 that the frame bounds are given by α = ess inf n(s(e jθ )) θ 0,2π),n=0,,M 1 (8) and β = ess sup θ 0,2π),n=0,,M 1 λ n (S(e jθ )) (9) where λ n (S(e jθ )) denotes the eigenvalues and S(e jθ )= E (e jθ )E(e jθ ) Theorem 1 Given a causal stable analysis set {h k,m n}, let E(z) be its polyphase matrix with a minimal realization Assume that jθ I B has full column rank for all θ 0, 2π), and D has full column rank Let M(z) N(z), Z,F and X be defined by (2)-(5) respectively Then 1) The tightest frame bounds of {h k,m n} is α = M(z) 2 and β = E(z) 2, where M(z) := sup θ 0,2π) M(e jθ ) is the H norm of M(z) 12 2) The state spe realization for the synthesis filter R 0 (z) =E (z)e(z) 1 E (z)

4 corresponding to the dual frame is given by A BF (A BF)Y (C DF) BW 1 D F W 1 D FY(C DF) (A BF) (C DF) W 1 B FY(A BF) (10) 0 where Y is given by the Lyapunov equations Y (A BF)Y (A BF) = BW 1 B (11) 3) The frame {h k,m n} is a tight frame with bound α if and only if W = αi and D C B XA =0 4) If {h k,m n} is not tight, then the frame with E(z) = N(z) is paraunitary with frame bound α =1 Proof 1) It follows directly from equations (8)-(9) and the definition of H norm that β = E(z) 2 For the lower bound, note that S 1 =(E E) 1 =(M 1 N NM 1 )=MM Therefore, we have α = ess inf λ n (S(e jθ )) = ess sup λ n (S 1 (e jθ )) 1 = M(z) 2 2) Note that A BF BW 1 2 F W 1 2 R 0 (z) =E E 1 E = MN = (A BF) (C DF) W 1 2 B W 1 2 D By direct computation following Lemma 1, we can get (10), where equation (1) becomes (11) Due to the spe limitation, we won t go into details here 3) It is shown in 1 that {h k,m n} is tight if and only if E (z)e(z) =αi The result then follows directly from Lemma 2118 (page 552 in 12) that E (z)e(z) =αi is equivalent to W = αi and D C B XA =0 4) Since N is inner, it is obvious that N N = I, that is, N is a tight frame with frame bound 1 Corollary 1 Suppose the same assumptions as in Theorem 1 hold Then the synthesis filter R 0 (z) corresponding to the dual frame is causal if and only if W 1 B FY(A BF) =0 In this case, the state spe realization of R 0 (z)isgiven by A BF (A BF)Y (C DF) BW 1 D F W 1 D FY(C DF) Corollary 2 Suppose the same assumptions as in Theorem 1 hold Then the synthesis filter R 0 (z) corresponding to the dual frame is anticausal if and only if (A BF)Y (C DF) BW 1 D =0 In this case, the state spe realization of R 0 (z)isgiven by (A BF) (C DF) W 1 B FY(A BF) W 1 D FY(C DF) jθ B Remark 1 The rank assumption on is equivalent to E(z) has full column rank on the unit circle since is a minimal realization The full column rank assumption on D can be removed if we allow PR with some delay Remark 2 The solution of the algebraic Riccati equation (5) as well as the H norm of a system can be computed efficiently using software, for example, MAT- LAB See dric and dnorminf for detail Remark 3 Theorem 1 tells us that solutions on algebraic Riccati equations and algebraic matrix manipulations are enough for computing a left inverse filter bank, therefore the inverse of a rational matrix S(z) is completely avoided The main reason is that the frame operator S = E E generated from oversampled FBs is periodic, and can be converted to an equivalent LTI system by the technique of blocking Remark 4 The anticausal inverses of causal maximallysampled FBs are studied in 17 The problem there is different to ours mainly in the following two aspects: (1) D is square and the inverse (if exists) is unique in 17, however, the inverse is generally non-unique for oversampled FBs (2) We restrict all the filters in the stable set, therefore, our results are more preferred in prtice Based on Theorem 1, we can also provide an explicit parameterization of stable synthesis filter banks with PR in terms of the state spe realization of R(z) This is shown in the next Theorem Theorem 2 For E(z) as in Theorem 1, let R 0 (z) be given by (10) Then all the synthesis polyphase matrices R(z) providing PR can be written as where R(z) =R 0 (z)u(z)(q G(z)G (z)) Q =(C DF)Y (C DF) DW 1 D I

5 is a scalar matrix and G(z) = (A BF) (A BF)Y (C DF) BW 1 D (C DF) 0 Proof Using the charterization in 1 and Lemma 1, we can get the results directly by cumbercume matrix manipulations Due to the spe limitation, we won t give details here 4 CONCLUSION We have provided explicit and numerically efficient formulae to the analysis and design of frames corresponding oversampled FBs Different from the existing results in the literature, these formulae do not involve any approximation We have shown that all the problems related to the frame bounds, the dual frame and construction of tight frame from a general frame can be solved explicitly by state-spe method if the frame are generated by oversampled FBs The key point is that the analysis (or synthesis) operator is periodic other than the general time-varying cases 5 REFERENCES 1 H Bölcskei, F Hlawatsch, and H G Feichtinger, Frame-theoretic analysis of oversampled filter banks, IEEE Trans Signal Processing, vol 46, pp , December Z Cvelković and M Vetterli, Oversampled filter banks, IEEE Trans Signal Processing, vol 46, pp , May I Daubechies, Ten Lectures on Wavelets, Philadelphia, PA: SIAM, K Gröchenig, Acceleration of the frame algorithm, IEEE Trans Information Theory, vol 41, pp , December H Bölcskei and F Hlawatsch, Oversampled cosine modulated filter banks with perfect reconstruction, IEEE Trans Circuits Syst II, vol 45, pp , Aug H Bölcskei and F Hlawatsch, Noise reduction in oversampled filter banks using predictive quantization, IEEE Trans Information Theory, vol 47, pp , January L Gan and K-K Ma, Oversampled linear-phase perfect reconstruction filterbanks: theory, lattice structure and parameterization, IEEE Trans Signal processing, vol 51, pp , Mar J Kliewer and A Mertins, Oversampled consinemodulated filterbanks with arbitrary system delay, IEEETransSignalprocessing,vol46,pp , Apr T Strohmer, Finite and infinite-dimensional models for oversampled filter banks, in Modern Sampling Theory: Mathematics and Applications, J J Benedetto and P J S G Ferreira, Eds Birkhäser, K Eneman and M Moonen, DFT modulated filterbank design for oversampled subband systems, Signal processing, vol 81, pp , Sept T Tanaka and Y Yamashita, The generalized lapped pseudo-biorthogonal transform: oversampled linear-phase perfect reconstruction filterbanks with lattice structures, IEEE Trans Signal processing, vol 52, pp , Feb K Zhou, J C Doyle, and K Glover, Robust Optimal Control, Prentice-Hall, M Vetterli and J Kovačević, Wavelets and Subband Coding, Prentice-Hall, R A Meyer and C S Burrus, A unified analysis of multirate and periodically time-varying digital filters, IEEE Trans on Circuits and Systems, vol 22, pp , P Vaidyanathan, Multirate Systems and Filter Banks, Prentice-Hall, H Shu and T Chen, On causality and anticausality of cascaded discrete-time systems, IEEE Trans on Circuits and Systems, I, vol 43, pp , P Vaidyanathan and T Chen, Role of anticausal inverses in multirate filterbanks-part I: system-theoretic fundamentals, IEEE Trans Signal processing, vol 43, pp , May 1995