WLS Design of Sparse FIR Digital Filters

In this paper, we propose a novel algorithm for sparse finite impulse response (FIR) filter designs. The objective of the sparse digital filter design problem considered in this paper is to reduce the number of nonzero-valued filter coefficients, subject to a weighted least-squares (WLS) approximation error constraint imposed on the frequency domain. The proposed design method is inspired by the iterative shrinkage/thresholding (IST) algorithms, which are used in sparse and redundant representation for signals. The basic idea of the proposed design algorithm is to successively transform the original nonconvex problem to a series of constrained subproblems in a simpler form. Despite of their nonconvexity, these subproblems can be efficiently and reliably solved in each iterative step by a numerical approach developed in this paper. Furthermore, it can be demonstrated that the obtained solutions are essentially optimal to their respective subproblems. Since its major part only involves scalar operations, the proposed algorithm is computationally efficient. Three sets of numerical examples are presented in this paper to illustrate the effectiveness of the proposed design algorithm.