Symmetric tight wavelet frames with rational coefficients

Abstract

In this paper we present a necessary and sufficient condition to construct a symmetric framelet system a ( z ) , b 1 ( z ) and b 2 ( z ) having rational coefficients. Once a low-pass filter a ( z ) is given and such a necessary and sufficient condition is satisfied, a step-by-step algorithm will be used to construct a symmetric framelet filter bank with two high-pass filters b 1 ( z ) and b 2 ( z ) which is of interest in applications such as signal de-noising and image de-blurring, etc. Two new examples will be given to illustrate our results and the algorithm.