Convex-Programming Design of Linear and Nonlinear Phase Wideband Blocking Filters

Blocking filters are commonly used in array processing to excise targets from data when suitable target-free training data is not available. A drawback is the reduction of the effective size of the array by the order of the filter, motivating a search for the smallest filter that meets the design specifications. Common linear-phase approaches lead to convex optimization and efficient global solutions, but can be inefficient when the phase of the blocking filter is unimportant. Although direct magnitude-response optimization is nonconvex in general, for narrowband linear arrays optimal-magnitude blocking filters can be found using constrained optimization followed by spectral factorization. This approach cannot be directly applied to wideband arrays, as finite-support spectral factors do not generally exist in higher dimensions. Instead, a procedure is presented in which an approximate multidimensional spectral factor of an optimized spectral density is used as a target response in a second optimization stage. Linear and nonlinear-phase solutions are found and compared for both narrowband and wideband notch filters, with the nonlinear phase outperforming the linear phase in both notch width and passband error. The various optimizations are performed using second-order cone programming, an efficient class of convex optimization.