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FIR filter design by iterative convex relaxations with rank refinement

Finite impulse response (FIR) filters have been a primary topic of digital signal processing since their inception. Although FIR filter design is an old problem, with the developments of fast convex solvers, convex modelling approach for FIR filter design has become an active research topic. In this work, we propose a new method based on convex programming for designing FIR filters with the desired frequency characteristics. FIR filter design problem, which is modelled as a non-convex quadratically constrained quadratic program (QCQP), is transformed to a semidefinite program (SDP). By relaxing the constraints, a convex programming problem, which we call RSDP(Relaxed Semidefinite Program), is obtained. Due to the relaxation, solution to the RSDPs fails to be rank-1. Typically used rank-1 approximations to the obtained RSDP solution does not satisfy the constraints. To overcome this issue, an iterative algorithm is proposed, which provides a sequence of solutions that converge to a rank-1 matrix. Conducted experiments and comparisons demonstrate that proposed method successfully designs FIR filters with highly flexible frequency characteristics.