Fir filter design by convex optimization using rank refinement

Finite impulse response filters have been one of the primary topics of digital signal processing since their inception. Consequently, diverse class of design techniques including Chebyshev approximation, Fast Fourier Transform and optimization based methods had been proposed in the literature. With developments in com- putational tools, new design technique tools and formulations on filters including interior-point solvers and semidefinite programming (SDP), emerged. Since FIR filter design problem can be modelled as a quadratically constrained quadratic program, filter design problem can be solved via interior-point based convex op- timization methods such as semidefinite programming. Unfortunately, SDP for- mulation of problem is nonconvex due to positive lower limit constraint in the passband. To overcome that problem, nonconvex problem can be cast into a convex SDP using semidefinite relaxation, which can be solved in polynomial time. Since relaxed formulation does not guarantee rank-1 solution matrix, re- cently proposed directed iterative rank refinement (DIRR) algorithm is used to impose a convex rank-1 constraint. Due to utilization of semidefinite relaxation and DIRR, addition of various constraints, such as phase and group delay masks, in convex manner is made possible. For feasibility type optimization formulations of filter design problem, a convergence rate improved version of DIRR is devel- oped. Proposed techniques are applied on filter design problems with different set of constraints including phase and group delay constraints. Explicit simulations demostrate that the proposed technique is capable of solving nonlinear phase, phase constrained, and group delay constrained filter design problems.