Symmetric tight wavelet frames with rational coefficients

In this paper we present a necessary and sufficient condition to construct a symmetric framelet system a(z), b1(z) and b2(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 b1(z) and b2(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.