Design of FIR filters with discrete coefficients via sphere relaxation

A method for designing FIR digital filters, with each coefficient a sum of signed power-of-2 terms, by sphere relaxation is proposed. To justify the design methodology, we present an analysis showing that the globally optimal design always lies in a vicinity of the optimal FIR filter with continuous coefficients. The design problem at hand is then addressed using a new relaxation method in which the constraints characterizing the binary nature of the design variables are relaxed to a single sphere type constraint on the corresponding continuous variables. This yields a simple nonconvex continuous optimization problem whose solution can be calculated considerably faster than previously reported relaxation methods. Design examples are presented to demonstrate that the proposed algorithm offers near-optimal designs with small fraction of design complexity relative to that required by the existing methods