Weighted least mean square design of 2-D FIR digital filters

This paper is concerned with weighted least mean square design of two-dimensional (2-D) zero-phase FIR filters with quadrantally symmetric and antisymmetric frequency responses. The optimal solutions are first characterized by certain integral equations, and the existence, and uniqueness of the weighted least mean square solution for 2-D FIR filter design are then established using contraction mapping, and fixed point theorem. Explicit solution in closed form for discretized weighted least mean square error is obtained for 2-D zero-phase FIR filters, which can be considered as an extension of the methods of Ahmed and Wang (1989) and Tufts and Francis (1970). Further, two numerical algorithms are proposed to iteratively solve the weighted least mean square solution. Convergence of the iterative algorithms is established, and estimates of the convergence rate are also derived. Examples are used to illustrate the effectiveness of the proposed design algorithms