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Splitting a Matrix of Laurent Polynomials with Symmetry and
its Application to Symmetric Framelet Filter Banks

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Bin Han and Qun Mo

## Abstract

Let $M$ be a $2\times 2$ matrix of Laurent polynomials with real
coefficients and symmetry. In this paper, we obtain a necessary
and sufficient condition for the existence of four Laurent
polynomials (or FIR filters) $u_1, u_2, v_1, v_2$ with real
coefficients and symmetry such that
$$
\left[ \begin{matrix} u_1(z) &v_1(z)\\ u_2(z) &v_2(z)
\end{matrix}\right] \left[ \begin{matrix} u_1(1/z) &u_2(1/z)\\
v_1(1/z) &v_2(1/z)\end{matrix}\right]=M(z) \qquad \forall\; z\in
\CC \bs \{0 \}
$$
and $[Su_1](z)[Sv_2](z)=[Su_2](z)[Sv_1](z)$,
where $[Sp](z)=p(z)/p(1/z)$ for a nonzero Laurent polynomial $p$.
Our criterion can be easily checked and a step-by-step algorithm
will be given to construct the symmetric filters $u_1, u_2, v_1,
v_2$. As an application of this result to symmetric framelet
filter banks, we present a necessary and sufficient condition for
the construction of a symmetric MRA tight wavelet frame with two
compactly supported generators derived from a given symmetric
refinable function. Once such a necessary and sufficient condition
is satisfied, an algorithm will be used to construct a symmetric
framelet filter bank with two high-pass filters which is of
interest in applications such as signal de-noising and image
processing. As an illustration of our results and algorithms in
this paper, we give several examples of symmetric framelet filter
banks with two high-pass filters which have good vanishing moments
and are derived from various symmetric low-pass filters including
some $B$-spline filters.

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