A Novel Scheme for the Design of Approximate Hilbert Transform Pairs of Orthonormal Wavelet Bases

In designing the Hilbert transform pairs of orthonormal wavelet bases, several authors have shown that the requirements of the equal magnitude responses and the half-sample phase offset are the necessary and sufficient conditions on the scaling filters of conjugate quadrature filter (CQF) banks. In this paper, we first show that the scaling filters of CQF banks with equal magnitude responses cannot approach the half-sample phase offset in the high-frequency range properly. Then, a new design scheme, making a tradeoff between the requirements of the equal magnitude responses and the half-sample phase offset, is presented. The design scheme can make the scaling filter pairs approximately satisfy the requirement of the half-sample phase offset over the full frequency range while the magnitude responses of each pair remain nearly equal. As a result, the corresponding orthonormal wavelet bases can approximate to Hilbert transform pairs better. Some orthogonal wavelet bases we designed demonstrate the superiority of our scheme.