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# Euclidean Algorithm for Extension of Symmetric Laurent Polynomial Matrix and Its Application in Construction of Multiband Symmetric Perfect Reconstruction Filter Bank

Department of Mathematics and Statistics, Sam Houston State University, 1901 Ave. I, Huntsville, TX 77341-2206, USA
Received: 9 March 2017 / Revised: 14 April 2017 / Accepted: 17 April 2017 / Published: 20 April 2017
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
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# Abstract

For a given pair of s-dimensional real Laurent polynomials $( a → ( z ) , b → ( z ) )$ , which has a certain type of symmetry and satisfies the dual condition $b → ( z ) T a → ( z ) = 1$ , an $s × s$ Laurent polynomial matrix $A ( z )$ (together with its inverse $A - 1 ( z )$ ) is called a symmetric Laurent polynomial matrix extension of the dual pair $( a → ( z ) , b → ( z ) )$ if $A ( z )$ has similar symmetry, the inverse $A - 1 ( Z )$ also is a Laurent polynomial matrix, the first column of $A ( z )$ is $a → ( z )$ and the first row of $A - 1 ( z )$ is $( b → ( z ) ) T$ . In this paper, we introduce the Euclidean symmetric division and the symmetric elementary matrices in the Laurent polynomial ring and reveal their relation. Based on the Euclidean symmetric division algorithm in the Laurent polynomial ring, we develop a novel and effective algorithm for symmetric Laurent polynomial matrix extension. We also apply the algorithm in the construction of multi-band symmetric perfect reconstruction filter banks. View Full-Text
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0). # Share & Cite This Article

MDPI and ACS Style

Wang, J. Euclidean Algorithm for Extension of Symmetric Laurent Polynomial Matrix and Its Application in Construction of Multiband Symmetric Perfect Reconstruction Filter Bank. Axioms 2017, 6, 9.

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

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