Filter design using sparse signal restoration techniques

The paper proposes the design of linear phase filters using the optimization techniques based on methods for sparse signal restoration. The filter response is treated as weighted sum of the phase shifted versions of the signal. The filter design problem is converted into form Ax ≈ b, where A has rows given by [1 cos(-ω) cos(-2ω) ...], corresponding to a given frequency ω. x corresponds to the filter coefficients and b = R(ω) cos φ (ω), where R stands for the magnitude and φ the phase of the desired response. The problem is formulated into an L1 term and and L2 term and solved iteratively using ISTA (Iterative Soft Thresholding Algorithm [1]) to get the filter coefficients. The visible outcome of the approach is the considerable reduction in the non-zero coefficients.