Design of Time–Frequency Localized Filter Banks: Transforming Non-convex Problem into Convex Via Semidefinite Relaxation Technique

We present a method for designing optimal biorthogonal wavelet filter banks (FBs). Joint time–frequency localization of the filters has been chosen as the optimality criterion. The design of filter banks has been cast as a constrained optimization problem. We design the filter either with the objective of minimizing its frequency spread (variance) subject to the constraint of prescribed time spread or with the objective of minimizing the time spread subject to the fixed frequency spread. The optimization problems considered are inherently non-convex quadratic constrained optimization problems. The non-convex optimization problems have been transformed into convex semidefinite programs (SDPs) employing the semidefinite relaxation technique. The regularity constraints have also been incorporated along with perfect reconstruction constraints in the optimization problem. In certain cases, the relaxed SDPs are found to be tight. The zero duality gap leads to the global optimal solutions. The design examples demonstrate that reasonably smooth wavelets can be designed from the proposed filter banks. The optimal filter banks have been compared with popular filter banks such as Cohen–Daubechies–Feauveau biorthogonal wavelet FBs, time–frequency optimized half-band pair FBs and maximally flat half-band pair FBs. The performance of optimal filter banks has been found better in terms of joint time–frequency localization.