Joint Sparsity and Order Optimization Based on ADMM With Non-Uniform Group Hard Thresholding
This paper proposes a new optimization framework for the joint optimization of sparsity and filter order (JOSFO) for FIR filter design. Since the cost function for JOSFO involves <inline-formula> <tex-math notation="LaTeX">$\ell _{0}$ </tex-math></inline-formula> and non-uniform overlapped group <inline-formula> <tex-math notation="LaTeX">$\ell _{0}$ </tex-math></inline-formula> norms, which are not convex, a global optimal solution is difficult to obtain. To find an approximate solution of the non-convex problem, existing approaches repeat the following steps: 1) approximate the cost function; 2) find candidates of zero coefficients by minimizing the cost function; and 3) set them to zero. On the other hand, this paper directly solves the optimization problem, without any approximation to the cost function, by using the alternating direction method of multipliers with the pseudo-proximity operators of <inline-formula> <tex-math notation="LaTeX">$\ell _{0}$ </tex-math></inline-formula> and non-uniform non-overlapped group <inline-formula> <tex-math notation="LaTeX">$\ell _{0}$ </tex-math></inline-formula> norms. Experimental results show that resulting filters designed by the proposed method have sparser coefficients and lower orders, while satisfying filter specifications, such as an error from a desired frequency response.