In designing discrete-time filters, the length of the impulse response is often used as an indication of computational cost. In systems where the complexity is dominated by arithmetic operations, the number of nonzero coefficients in the impulse response may be a more appropriate metric to consider instead, and computational savings are realized by omitting arithmetic operations associated with zero-valued coefficients. This metric is particularly relevant to the design of sensor arrays, where a set of array weights with many zero-valued entries allows for the elimination of physical array elements, resulting in a reduction of data acquisition and communication costs. However, designing a filter with the fewest number of nonzero coefficients subject to a set of frequency-domain constraints is a computationally difficult optimization problem. This paper describes several approximate polynomial-time algorithms that use linear programming to design filters having a small number of nonzero coefficients, i.e., filters that are sparse. Specifically, we present two approaches that have different computational complexities in terms of the number of required linear programs. The first technique iteratively thins the impulse response of a non-sparse filter until frequency-domain constraints are violated. The second minimizes the 1-norm of the impulse response of the filter, using the resulting design to determine the coefficients that are constrained to zero in a subsequent re-optimization stage. The algorithms are evaluated within the contexts of array design and acoustic equalization.

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Linear Programming Algorithms for Sparse Filter Design

In this paper, a novel algorithm is proposed to design sparse FIR filters. It is known that this design problem is highly nonconvex due to the existence of -norm of a filter coefficient vector in its objective function. To tackle this difficulty, an iterative procedure is developed to search a potential sparsity pattern, which is then used to compute the final solution by solving a convex optimization problem. In each iterative step, the original sparse filter design problem is successively transformed to a simpler subproblem. It can be proved that under a weak condition, globally optimal solutions of these subproblems can be attained by solving their dual problems. In this case, the overall iterative procedure converges to a locally optimal solution of the original design problem. The design procedure described above can be repeated for several times to further improve the sparsity of design results. The output of the previous stage can be used as the initial point of the subsequent design. The performance of the proposed algorithm is evaluated by two sets of design examples, and compared to other sparse FIR filter design algorithms.

Peak-Error-Constrained Sparse FIR Filter Design Using Iterative SOCP

Iterative reweighted l1 design of sparse FIR filters

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 signal-to-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.

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

This paper presents an exact algorithm for sparse filter design under a quadratic constraint on filter performance. The algorithm is based on branch-and-bound, a combinatorial optimization procedure that can either guarantee an optimal solution or produce a sparse solution with a bound on its deviation from optimality. To reduce the complexity of branch-and-bound, several methods are developed for bounding the optimal filter cost. Bounds based on infeasibility yield incrementally accumulating improvements with minimal computation, while two convex relaxations, referred to as linear and diagonal relaxations, are derived to provide stronger bounds. The approximation properties of the two relaxations are characterized analytically as well as numerically. Design examples involving wireless channel equalization and minimum-variance distortionless-response beamforming show that the complexity of obtaining certifiably optimal solutions can often be significantly reduced by incorporating diagonal relaxations, especially in more difficult instances. In the case of early termination due to computational constraints, diagonal relaxations strengthen the bound on the proximity of the final solution to the optimum.

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

We consider the problem of designing a sparse FIR filter and show that it can be cast into a problem of determining a sparse solution of a linear system of equations. Previously proposed design algorithms for FIR filter utilize an intelligent search over all possible structures for sparse filter. We propose a new filter design method based on a simpler algorithm for finding a sparse solution of the linear system. Simulation experiments show significant improvements over classical nonsparse methods.

Francesco Palmierl

Papers

Efficient sparse FIR filter design