Abstract
This paper introduces some properties of symmetric Laurent polynomials, and then extends Euclidean algorithm to symmetric Laurent polynomials. The new results are used to investigate factorization of polyphase matrix for biorthogonal finite filters. It is shown that there exists one and only one symmetric factorization of the polyphase matrix, and the symmetric factorization can be determined directly and efficiently by Euclidean algorithm for symmetric Laurent polynomials. Finally, symmetric implementation and matrix representation of biorthogonal wavelet transforms are introduced, and the study demonstrates that the symmetric implementation has the least multiplication number in all lifting implementations, and it is equivalent to a matrix transform on finite dimensional vector space.
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