FIR filter design via semidefinite programming and spectral factorization

We present a semidefinite programming approach to FIR filter design with arbitrary upper and lower bounds on the frequency response magnitude. It is shown that the constraints can be expressed as linear matrix inequalities (LMIs), and hence they can be easily handled by interior-point methods. Using this LMI formulation, we can cast several interesting filter design problems as convex or quasi-convex optimization problems, e.g. minimizing the length of the FIR filter and computing the Chebychev approximation of a desired power spectrum or a desired frequency response magnitude on a logarithmic scale.