Time-varying and finite field filter banks

Citation

Phoong, See-May (1996) Time-varying and finite field filter banks. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/1bzz-ba31. https://resolver.caltech.edu/CaltechETD:etd-12222007-114654

Abstract

Filter banks find many applications in signal processing. This thesis deals with four different problems in filter banks. First we find a new application of filter banks: Filter bank convolver. We prove two filter bank convolution theorems which tell us how to do the convolution in the subbands. Applying the multirate technique to the problem of convolution, we obtain a significant improvement in the accuracy of the convolutional result when the computation is done with finite precision. The derivation also leads to a low sensitivity robust structure for FIR filters. In the second part, a new class of two-channel biorthogonal filter banks is proposed. We successfully design IIR filter banks which achieve the following desired properties simultaneously: (i) Perfect reconstruction (PR); (ii) causality and stability; (iii) near linear-phase; (iv) frequency selectivity. Two classes of causal stable maximally flat IIR wavelets are derived and closed form formulas are given. We also provide a novel mapping of the proposed 1D framework into 2D. The mapping preserves: (i) PR; (ii) stability in the IIR case and linear phase in the FIR case; (iii) frequency selectivity; (iv) low complexity. In the third part, the theory of paraunitary (PU) filter banks is extended to the case of GF(q) with prime q. We show that finite field PU filter banks are very different from real or complex PU filter banks. Despite all the differences, we are able to prove a number of factorization theorems. All unitary matrices in GF(q) are factorizable in terms of Householder-like matrices. The class of first-order PU matrices, the lapped orthogonal transform in finite fields, can always be expressed as a product of degree-one or degree-two building blocks. Finally the theory of conventional LTI filter banks is extended to the time-varying case. We develop a polyphase representation method for time-varying filter bank (TVFB). Using the proposed polyphase approach, we are able to show some unusual properties which are not exhibited by the conventional LTI filter banks. For example, we can show that for a PR TVFB, the losslessness of analysis bank does not always imply that of the synthesis bank, and a PR TVFB in general will only generate a discrete-time frame, rather than a basis, for the class of finite energy signals. The class of lossless TVFB is studied in detail. We show that all lossless linear time-varying systems are invertible and provide explicit construction of the inverse. The interplay between invertibility, uniqueness and losslessness of the inverse is investigated. The factorizability of lossless TVFB is addressed and we show that there are factorizable and unfactorizable examples.

Thesis Files