Design of Hilbert transform pairs of orthonormal wavelet bases using Remez exchange algorithm

This paper proposes a new method for designing Hilbert transform pairs of orthonormal wavelet bases proposed by Selesnick. The conventional method located as many zeros as possible at z = - 1 to obtain the maximum number of vanishing moments. In this paper, we specify the number of zeros at z = - 1, and then use the remaining degree of freedom to get the best possible frequency selectivity. The Remez exchange algorithm is applied in the stopband to approximate the equiripple magnitude response. Therefore, a set of filter coefficients can be obtained easily by solving a system of linear equations. Furthermore, the optimal solution is attained through a few iterations. Since the number of zeros at z = -1 can be specified arbitrarily, a new class of Hilbert transform pairs of orthonormal wavelet bases with the specified number of vanishing moments can be generated.