Optimal Design of Constrained FIR Filters Without Phase Response Specifications

Finite impulse response (FIR) filters without phase response specifications have found many applications in signal processing and communications. It is shown in this paper that the constrained
${L}_{{p}}$
magnitude error design of such an FIR filter is equivalent to a constrained
${L}_{{p}}$
approximation of a given response with a phase equal to that of the optimal filter of the original problem. An iterative method is then proposed to compute that phase by converting the nonconvex constrained
${L}_{{p}}$
magnitude error problem into a series of convex constrained
${L}_{{p}}$
elliptic frequency response error subproblems. It is also shown that the solutions of these convex subproblems converge to a Karush–Kuhn–Tucker point of the original problem. The convergence and its model parameter and initial condition dependence are shown through the designs of FIR filters without time-domain constraints. The iterative method is then applied to the minimax design of evidence filters, Nyquist filters, and step response constrained filters, and to the
${L}_{{p}}$
design of pulse shaping filters for ultra-wideband systems. Design examples demonstrate that the proposed method obtains better filters in terms of magnitude error, group delay, and spectrum utilization efficiency than existing methods.