Sharper Symmetric Self-Hilbertian wavelets

Symmetric Self-Hilbertian (SSH) wavelets are building blocks to many dual-tree complex wavelet transform systems. The SSH wavelets, and the corresponding scaling filters, within a pair are time-reverse versions of each other, and the complex wavelet function has conjugate symmetry. Previously reported methods for designing these wavelets focus mainly on achieving the best approximation to the Hilbert transform, where all design parameters are optimized with respect to the analytic quality. The exception is a technique proposed by the author (Signal Processing 2012) that allows the control of the frequency response sharpness of the corresponding scaling filter. However the technique is practical only when the number of free-parameters is small due to the high computational load otherwise. This paper proposes novel techniques that are based on the orthogonal lattice and are practical with any number of free-parameters. Higher analytic quality Hilbert pairs with sharper response can be obtained when there are more free-parameters. Three strategies for optimizing the lattice parameters to give high quality filters are presented here.