Projected least-squares algorithms for constrained FIR filter design

Constrained finite-impulse response (FIR) filter design with time- and frequency-domain linear constraints can be generally transformed into a, or a series of, constrained least-squares problems, which can be generally reformulated as positive definite quadratic programming (QP) problems. This paper presents a novel algorithm referred to as a projected least-squares (PLS) algorithm for the positive definite QP problems. The PLS algorithm essentially projects the unconstrained (least-squares) minimization solution successively onto the boundaries of active constraints that are identified by an active-set strategy. The PLS algorithm has been applied to the constrained least-squares design of FIR filters directly, and to the constrained Chebyshev design of FIR filters in an iterative fashion. The PLS algorithm is compared with the most widely used interior-point methods and an active-set method through design examples of low-pass filters with specified passband and stopband ripples, Nyquist filter constraints and step response constraints. All these examples demonstrate the high efficiency of the PLS algorithm.