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Vector space framework for unification of one- and multidimensional filter bank theory

Chen, Tsuhan and Vaidyanathan, P. P. (1994) Vector space framework for unification of one- and multidimensional filter bank theory. IEEE Transactions on Signal Processing, 42 (8). pp. 2006-2021. ISSN 1053-587X. doi:10.1109/78.301839.

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A number of results in filter bank theory can be viewed using vector space notations. This simplifies the proofs of many important results. In this paper, we first introduce the framework of vector space, and then use this framework to derive some known and some new filter bank results as well. For example, the relation among the Hermitian image property, orthonormality, and the perfect reconstruction (PR) property is well-known for the case of one-dimensional (1-D) analysis/synthesis filter banks. We can prove the same result in a more general vector space setting. This vector space framework has the advantage that even the most general filter banks, namely, multidimensional nonuniform filter banks with rational decimation matrices, become a special case. Many results in 1-D filter bank theory are hence extended to the multidimensional case, with some algebraic manipulations of integer matrices. Some examples are: the equivalence of biorthonormality and the PR property, the interchangeability of analysis and synthesis filters, the connection between analysis/synthesis filter banks and synthesis/analysis transmultiplexers, etc. Furthermore, we obtain the subband convolution scheme by starting from the generalized Parseval's relation in vector space. Several theoretical results of wavelet transform can also be derived using this framework. In particular, we derive the wavelet convolution theorem.

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Vaidyanathan, P. P.0000-0003-3003-7042
Additional Information:© Copyright 1994 IEEE. Reprinted with permission. Manuscript received October 21, 1992; revised November 19, 1993. This work was supported in part by the National Science Foundation grant MIP 8919196 and by matching funds from Rockwell Inc., and Tektronix, Inc. The associate editor coordinating the review of this paper and approving it for publication was Prof. Sergio D. Cabrera.
Subject Keywords:filtering and prediction theory; matrix algebra; multidimensional digital filters; transmultiplexing; vectors; wavelet transforms
Issue or Number:8
Record Number:CaltechAUTHORS:CHEieeetsp94
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9916
Deposited By: Archive Administrator
Deposited On:26 Mar 2008
Last Modified:13 Apr 2022 17:11

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