Sparse FIR Filter Design With k-Max Sparsity and Peak Error Constraints

FIR filters have boosted the development of digital signal processing, beamformers, and so on, due to their stability and low coefficient sensitivity. Generally, the design of FIR filters follows two main principles, i.e., the specification on the response error and the low implementation complexity. In this brief, we describe the sparsity of the filter coefficients using the k-maximum function, which equals to
l
₀
-norm under mild conditions and has no restriction on the magnitude of nonzero coefficients. In order to avoid possible violation of specifications on response errors caused by frequency discretization, we estimate the frequencies at which the magnitude of the response error is maximized when constructing linear problems in the proposed algorithm. To address the nonlinearity and nonconvexity of the resulted optimization problem, we transform it into a piecewise linear concave optimization (PLCO) problem. Considering the fact that a PLCO problems is reduced to a linear programming (LP) problem locally, we outline an iterative algorithm by solving a series of LP problems and provide a brief complexity analysis. Numerical experiments on the propose method and some state-of-the-art methods are performed, the result of which shows the excellent performance of the propose method on balancing the sparsity and computational efficiency.