Algebraic theory of optimal filter banks

We approach the problem of characterizing an optimal FIR filter bank from an algebraic point of view. We introduce the concept of majorization ordering to compare the performance of various filter banks in an admissible set /spl Lscr/. Using the properties of this ordering, we show that a principal component filter bank is associated with the greatest element in /spl Lscr/. A greatest element does not necessarily exist in /spl Lscr/ hence one has to deal with the closely related notion of a maximal element. We show by construction that a maximal element always exist in /spl Lscr/. An interesting result of the presented algebraic theory is that the connection between principal component filter banks and filter banks with maximum coding gain is clearly revealed. In fact, we show that coding gain is a Schur (1973) convex function preserving the order of majorization.

[1]  P. P. Vaidyanathan,et al.  Coding gain in paraunitary analysis/synthesis systems , 1993, IEEE Trans. Signal Process..

[2]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[3]  Toby Berger,et al.  Lossy Source Coding , 1998, IEEE Trans. Inf. Theory.

[4]  Mihai Anitescu,et al.  Theory of rate-distortion-optimal, constrained filterbanks-application to IIR and FIR biorthogonal designs , 2000, IEEE Trans. Signal Process..

[5]  Vivek K. Goyal,et al.  Multiple description coding: compression meets the network , 2001, IEEE Signal Process. Mag..

[6]  Stéphane Mallat,et al.  Analysis of low bit rate image transform coding , 1998, IEEE Trans. Signal Process..

[7]  Georgios B. Giannakis,et al.  Principal component filter banks for optimal multiresolution analysis , 1995, IEEE Trans. Signal Process..

[8]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[9]  Karlheinz Brandenbrg,et al.  First Ideas on Scalable Audio Coding , 1994 .

[10]  P. P. Vaidyanathan,et al.  Theory and design of optimum FIR compaction filters , 1998, IEEE Trans. Signal Process..

[11]  David Bull,et al.  Scalable image and video coding algorithms , 1998 .

[12]  P. Caines.  Linear Stochastic Systems , 1988 .

[13]  R. Gallager.  Information Theory and Reliable Communication , 1968 .

[14]  P. Dewilde,et al.  Time-Varying Systems and Computations , 1998 .

[15]  I. Vajda.  Theory of statistical inference and information , 1989 .

[16]  P. Vaidyanathan.  Multirate Systems And Filter Banks , 1992 .

[17]  Benoit M. Macq,et al.  Signal-adapted multiresolution transform for image coding , 1992, IEEE Trans. Inf. Theory.

[18]  P. P. Vaidyanathan,et al.  Theory of optimal orthonormal subband coders , 1998, IEEE Trans. Signal Process..

[19]  Michael Unser.  Extension of the Karhunen-Loeve transform for wavelets and perfect reconstruction filterbanks , 1993, Optics & Photonics.

[20]  David Bull,et al.  Insights into mobile multimedia communications , 1998 .

[21]  K. Popper,et al.  The Logic of Scientific Discovery , 1960 .

[23]  Azriel Rosenfeld,et al.  An introduction to algebraic structures , 1968 .

[24]  P. P. Vaidyanathan,et al.  On existence of FIR principal component filter banks , 1998, Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181).