An Eigenfilter-Based Approach to the Design of Time-Frequency Localization Optimized Two-Channel Linear Phase Biorthogonal Filter Banks

We present a novel eigenfilter-based approach to the design of time-frequency optimized, linear-phase, biorthogonal FIR filter banks. We first design a linear-phase, low-pass analysis filter, followed by a complementary linear-phase, low-pass synthesis filter. The optimality criterion used is uncertainty-based time-frequency localization, where the objective function is a convex combination of time variance and frequency variance of the respective filters. The objective function to be minimized is formulated in a convex-quadratic form and the perfect reconstruction (PR) and vanishing moment (VM) conditions are imposed in the eigen design of filters as a set of linear equality constraints. The PR and VM conditions are expressed in the time domain matrix formulation, so that these can directly be incorporated into the eigenfilter design. Using the Rayleigh principle, the optimal filter is obtained as an eigenvector corresponding to the minimum eigenvalue of the real symmetric positive-definite matrix associated with the optimization criterion. Thus, our formulation reduces the design problem of time-frequency optimal filter banks to an eigenfilter-based problem. Furthermore, the filter banks designed in this manner are found to be regular and are valid candidates for wavelet filter banks, allowing for the construction of linear phase wavelets. We present a few examples to show that the smooth wavelets can be constructed using the proposed method.