The design of optimal uniform filter banks with specified composite response

In this paper we prove that for suitable design specifications and a wide class of optimization criteria, the optimal complex filter bank with specifications on its composite response is composed of frequency translated versions of a prototype filter. In particular, this holds for the min-max and WMMSE (weighted minimum mean square error) criteria. As a result, a simplified design problem whose solution is an optimal prototype filter is formulated. This prototype is essentially an optimal FIR low-pass filter subject to linear constraints on its impulse response. For the WMMSE criterion, this characterization of the optimal filter bank results in a simplified version of the design method presented in [1]. For the min-max criterion, this characterization implies that there exists an optimal window, by which the window design method results in the optimal low-pass prototype. The optimal window design problem is formulated as a linear programming problem, and an approximate solution is derived using the Remez exchange algorithm. For real filter banks in which each filter is composed of a pair of complex filters, the optimal filter bank is no longer composed of frequency translated versions of prototype filter. However, for efficient implementation, the prototype translation property may be part of the design specifications. For this reason, the optimal WMMSE prototype for a class of real filter banks is derived as well.