Hilbert Transform Pairs Of Tight Frame Wavelets With Arbitrary Dilation Factor

This paper aims to design a pair of tight frame wavelets with arbitrary integer di-lation factor M, where all the generators form Hilbert transform pairs. To ensure the primal and dual filterbanks satisfy the perfect reconstruction condition simultaneous-ly, the periods of phase functions should be2π/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 sym-metric 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. Additionally, we propose a method based on Taylor series for the construction of the symmetric regular scaling filter. The resulting symmetric tight frames are used as the primal tight frames. Finally, we construct an optimization model with minimizing the error to search for the optimal filter coefficients. When the primal tight frames are symmetric, this optimization prob-lem can be expressed in the real domain. A design example is given to illustrate the results.