Linear Programming Algorithms for Sparse Filter Design

• Published in: IEEE transactions on signal processing Vol. 58; no. 3; pp. 1605 - 1617
• Main Authors: Baran, T., Wei, D., Oppenheim, A.V.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.03.2010; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithm design and analysis; Algorithms; Applied sciences; Arithmetic; Arrays; Complexity; Computation; Computational efficiency; Constraint optimization; Costs; Data acquisition; Design engineering; Detection, estimation, filtering, equalization, prediction; Exact sciences and technology; FIR digital filters; Impulse response; Information, signal and communications theory; Iterative algorithms; linear arrays; Linear programming; Miscellaneous; Nonlinear filters; Sensor arrays; Signal and communications theory; Signal processing; Signal, noise; Sparse filters; Studies; Telecommunications and information theory; FIR digital filters; Data acquisition; Discrete time systems; Frequency domain method; Iterative method; Linear programming; linear arrays; Polynomial method; Equalization; Algorithm; Computational complexity; Sparse filters; Optimization; Polynomial time; Digital filter; Pulse response; Signal processing; Finite impulse response filter; Metric; Linear antenna arrays; Data communication; Sensor array; Signal detection
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2009.2036471

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.