Characterization and Sampled-Data Design of Dual-Tree Filter Banks for Hilbert Transform Pairs of Wavelet Bases

Characterization and design of dual-tree filter banks for forming Hilbert transform pairs of wavelet bases are studied. The characterization extends the existing results for quadrature mirror filter banks to general prefect reconstruction filter banks that satisfy only a mild technical assumption regarding the ratio of determinants of the two filter banks. We establish equivalent relationships of Hilbert transform pairs on scaling filters, wavelet filters, or scaling functions. The design of scaling filters of a dual filter bank is formulated as a sampled-data H^infin optimization problem. The wavelet filters are then determined using the relationship on the determinants of the filter banks. We convert the sampled-data problem into an equivalent discrete-time H^infin control problem, which can be solved by standard H^infin control theory. An analytical solution to the sampled-data design problem is obtained for a special case. The sampled-data design approach usually gives infinite impulse response filter. In the case where the primal filter bank is of finite impulse response (FIR), we may truncate the impulse responses to get FIR approximations. They also lead to approximate Hilbert transform pairs. Design examples are presented