Sparse Filter Design Under a Quadratic Constraint: Low-Complexity Algorithms

Abstract

Abstract-This paper considers three problems in sparse filter design, the first involving a weighted least-squares constraint on the frequency response, the second a constraint on mean squared error in estimation, and the third a constraint on signalto-noise ratio in detection. The three problems are unified under a single framework based on sparsity maximization under a quadratic performance constraint. Efficient and exact solutions are developed for specific cases in which the matrix in the quadratic constraint is diagonal, block-diagonal, banded, or has low condition number. For the more difficult general case, a low-complexity algorithm based on backward greedy selection is described with emphasis on its efficient implementation. Examples in wireless channel equalization and minimum-variance distortionless-response beamforming show that the backward selection algorithm yields optimally sparse designs in many instances while also highlighting the benefits of sparse design.

A Branch-and-Bound Algorithm for Quadratically-Constrained Sparse Filter Design

“…One example of this is the use of the Wiener filter as the benchmark in adaptive filtering [15]. In the present context, we have used the algorithm in this paper to evaluate the backward selection algorithm in [13], showing that the latter often produces optimal or near-optimal solutions.…”

“…In a companion paper [13], we formulate a problem of designing filters of maximal sparsity subject to a quadratic constraint on filter performance. We show that this general formulation encompasses the problems of least-squares frequency-response approximation, mean square error estimation, and signal detection.…”

“…We show that this general formulation encompasses the problems of least-squares frequency-response approximation, mean square error estimation, and signal detection. The focus in [13] is on lowcomplexity algorithms for solving the resulting combinatorial optimization problem. Such algorithms are desirable when computation is limited, for example in adaptive design.…”

“…For the general case, a backward greedy selection algorithm is shown empirically to yield optimal or near-optimal solutions in many instances. We refer the reader to [13] for additional background on sparse filter design and a more detailed bibliography.…”

“…A major shortcoming of many low-complexity methods, including the backward selection algorithm in [13] and others (e.g. [3], [4], [8]- [10]), is that they do not indicate how close the resulting designs are to the true optimum.…”