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On orthonormal wavelets and paraunitary filter banks

Soman, Anand K. and Vaidyanathan, P. P. (1993) On orthonormal wavelets and paraunitary filter banks. IEEE Transactions on Signal Processing, 41 (3). pp. 1170-1183. ISSN 1053-587X. http://resolver.caltech.edu/CaltechAUTHORS:SOMieeetsp93a

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Abstract

Binary tree-structured filter banks have been employed in the past to generate wavelet bases. While the relation between paraunitary filter banks and orthonormal bases is known to some extent, there are some extensions which are either not known, or not published so far. In particular it is known that a binary tree-structured filter bank with the same paraunitary polyphase matrix on all levels generates an orthonormal basis. First, we generalize the result to binary trees having different paraunitary matrices on each level. Next, we prove a converse result: that every discrete-time orthonormal wavelet basis can be generated by a tree-structured filter bank having paraunitary polyphase matrices. We then extend the concept of orthonormal bases to generalized (i.e., nonbinary) tree structures, and see that a close relationship exists between orthonormality and paraunitariness in this case too. We prove that a generalized tree structure with paraunitary polyphase matrices produces an orthonormal basis. Since not all bases can be generated by tree-structured filter banks, we prove that if an orthonormal basis can be generated using a tree structure, it can be generated specifically by a paraunitary tree.


Item Type:Article
Additional Information:© Copyright 1993 IEEE. Reprinted with permission. Manuscript received June 9, 1991; revised May 5, 1992. This work was supported in part by NSF Grant MIP 8919196, Tektronix Inc., and Rockwell International. The authors wish to thank T. Chen, graduate student at the California Institute of Technology, for several useful comments.
Subject Keywords:digital filters, filtering and prediction theory, matrix algebra, trees (mathematics), wavelet transforms
Record Number:CaltechAUTHORS:SOMieeetsp93a
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:SOMieeetsp93a
Alternative URL:http://dx.doi.org/10.1109/78.205722
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:8715
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:10 Sep 2007
Last Modified:26 Dec 2012 09:41

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