16 September 2011 Optimal filters with heuristic 1-norm sparsity constraints
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Abstract
We present a design method for sparse optimal Finite Impulse Response (FIR) filters that improve the visibility of a desired stochastic signal corrupted with white Gaussian noise. We emphasize that the filters we seek are of high-order but sparse, thus significantly reducing computational complexity. An optimal FIR filter for the estimation of a desired signal corrupted with white noise can be designed by maximizing the signal-to-noise ratio (SNR) of the filter output with the constraint that the magnitude (in 2-norm) of the FIR filter coefficients are set to unity.1, 2 This optimization problem is in essence maximizing the Rayleigh quotient and is thus equivalent to finding the eigenvector with the largest eigenvalue.3 While such filters are optimal, they are rarely sparse. To ensure sparsity, one must introduce a cardinality constraint in the optimization procedure. For high order filters such constraints are computationally burdensome due to the combinatorial search space. We relax the cardinality constraint by using the 1-norm approximation of the cardinality function. This is a relaxation heuristic similar to the recent sparse filter design work of Baran, Wei, and Oppenheim.4 The advantage of this relaxation heuristic is that the solutions tend to be sparse and the optimization procedure reduces to a convex program, thus ensuring global optimality. In addition to our proposed optimization procedure for deriving sparse FIR filters, we show examples where sparse high-order filters significantly perform better than low-order filters, whereas complexity is reduced by a factor of 10.
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Mehrdad Yazdani, Mehrdad Yazdani, Robert Hecht-Nielsen, Robert Hecht-Nielsen, } "Optimal filters with heuristic 1-norm sparsity constraints", Proc. SPIE 8137, Signal and Data Processing of Small Targets 2011, 813709 (16 September 2011); doi: 10.1117/12.893170; https://doi.org/10.1117/12.893170
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