Theory and design of uniform DFT, parallel, quadrature mirror filter banks

Abstract

-In this paper, the theory of uniform DFI, parallel, quadrature mirror filter (QMF) banks is developed. The QMF equations, i.e., equations that need to be satisfied for exact reconstruction of the input signal, are derived. The concept of decimated filters is introduced, and structures for both analysis and synthesis banks are derived using this concept. The QMF equations, as well as closed-form expressions for the synthesis filters needed for exact reconstruction of the input signal x(n), are also derived using this concept. In general, the reconstructed. signal .C( n) suffers from three errors: aliasing, amplitude distortion, and phase distortion. Conditions for exact reconstruction (i.e., all three distortions are zero, and Z(n) is equal to a delayed version of x(n)) of the input signal are derived in terms of the decimated filters. Aliasing distortion can always be completely canceled. Once aliasing is canceled, it is possible to completely eliminate amplitude distortion (if suitable IIR filters are employed) and completely eliminate phase distortion (if suitable FIR filters are employed). However, complete elimination of all three errors is possible only with some simple, pathalogical stable filter transfer functions. In general, once aliasing is canceled, the other distortions can be minimized rather than completely eliminated. Algorithms for this are presented. The properties of FIR filter banks are then investigated. Several aspects of IIR filter banks are also studied using the same framework.

Frame-theoretic analysis of oversampled filter banks

“…The particular PR synthesis FB corresponding to the parapseudoinverse can be given an 6 We note that on the unit circle, the para-pseudoinverse in (8) interesting frame-theoretic interpretation, which has previously been described for the important class of oversampled FIR FB's in [12]. For given analysis filter impulse responses , consider the particular synthesis filter impulse responses provided by frame theory via (4), i.e., with , or in other words, is the UFBF that is dual to .…”

“…Consequently, a FB corresponding to a UFBF is "diagonal in the polyphase domain" if the UFBF matrix is a diagonal matrix diag with It follows from (8) that the polyphase matrix of the minimum norm synthesis FB is given by diag (19) We can see that the calculation of the minimum norm synthesis FB, which in general requires the inversion of the UFBF Integer oversampled or critically sampled DFT FB's [4], [6], [7], [11], [16], [24], [39], [54], [58] are an important example of FB's that are diagonal in the polyphase domain. The corresponding UFBF type is the important class of Weyl-Heisenberg frames [16], [25], [28], [30], [33], [34], [39], [59], [60].…”

“…In [5] and [6], it has been shown that for critical sampling, a DFT FB with PR and FIR filters in both the analysis and the synthesis section is possible only if all the polyphase filters are pure delays. This leads to filters with poor frequency selectivity.…”

Discrete Zak transforms, polyphase transforms, and applications

“…APPENDIX EIGENVALUES AND EIGENFUNCTIONS OF THE GABOR FRAME OPERATOR We recall from Section III-B that the Gabor frame operator is defined as Interchanging the order of summations and using the Poisson sum formula , it is possible to derive the following alternative expression: (42) (a similar expression has been found for the continuoustime Gabor expansion in [12] and [70]). We now restrict our attention to the cases of integer oversampling or critical sampling, i.e., with Inserting [see (14)] in (42), we obtain This shows that is an eigenfunction of with eigenvalue We shall next prove that the are orthogonal and complete in a generalized sense [53]. According to [53, p. 186], one has to show that for all and where…”

Two-Channel Quadrature Mirror Filter Bank: An Overview

“…A gradient based method was presented by Swaminathan and Vaidyanathan [38], to design linear phase FIR filter bank that minimizes the same objective function as in the method by Johnston as well as by Jain and Crochiere.…”