Hilbert Transform Pairs of Tight Frame Wavelets with Integer Dilation Factor

Abstract

This paper aims to design a pair of tight frame wavelets with dilation $$M$$ M , where all the generators form Hilbert transform pairs. To ensure the primal and dual filterbanks satisfy the perfect reconstruction condition simultaneously, the periods of phase functions should be $$\frac{2\pi }{M}$$ 2 ? M based on our results. According to the sufficient and necessary condition of phase functions, we not only give a linear phase solution for them but also establish their formulae. Moreover, by analyzing the symmetric properties of the dual tight frame, it is shown that as long as their primal tight frames are symmetric those duals are symmetric as well. Finally, we construct an optimization model with minimizing the error to search for the optimal filter coefficients. A design example is given to illustrate the results.