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# Paraunitary filter banks over finite fields

Phoong, See-May and Vaidyanathan, P. P. (1997) Paraunitary filter banks over finite fields. IEEE Transactions on Signal Processing, 45 (6). pp. 1443-1457. ISSN 1053-587X. doi:10.1109/78.599956. https://resolver.caltech.edu/CaltechAUTHORS:PHOieeetsp97a

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## Abstract

In real and complex fields, unitary and paraunitary (PU) matrices have found many applications in signal processing. There has been interest in extending these ideas to the case of finite fields. We study the theory of PU filter banks (FBs) in GF(q) with q prime. Various properties of unitary and PU matrices in finite fields are studied. In particular, a number of factorization theorems are given. We show that (i) all unitary matrices in GF(q) are factorizable in terms of Householder-like matrices and permutation matrices, and (ii) 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. If q>2, we do not need degree-two building blocks. While many properties of PU matrices in finite fields are similar to those of PU matrices in complex field, there are a number of differences. For example, unlike the conventional PU systems, in finite fields, there are PU systems that are unfactorizable in terms of smaller building blocks. In fact, in the special case of 2×2 systems, all PU matrices that are factorizable in terms of degree-one building blocks are diagonal matrices. We derive results for both the cases of GF(2) and GF(Q) with q>2. Even though they share some similarities, there are many differences between these two cases.

Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1109/78.599956DOIUNSPECIFIED
ORCID:
AuthorORCID
Vaidyanathan, P. P.0000-0003-3003-7042
Additional Information:© Copyright 1997 IEEE. Reprinted with permission. Manuscript received September 23, 1995; revised February 3, 1997. This work was supported by Office of Naval Research Grant N00014-93-1-0231, funds from Tektronix, Inc., and Rockwell International. The associate editor coordinating the review of this paper and approving it for publication was Dr. Bruce W Suter.
Subject Keywords:Galois fields, band-pass filters, filtering theory, matrix algebra, signal processing, transforms
Issue or Number:6
DOI:10.1109/78.599956
Record Number:CaltechAUTHORS:PHOieeetsp97a
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:PHOieeetsp97a
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:8701
Collection:CaltechAUTHORS