Convex optimization over positive polynomials and filter design

Positive polynomial matrices play a fundamental role in systems and control theory: they represent e.g. spectral density functions of stochastic processes and show up in spectral factorizations, robust control and filter design problems. Positive polynomials obviously form a convex set and were recently studied in the area of convex optimization [1, 5]. It was shown in [2, 5] that positive polynomial matrices can be parametrized using block Hankel and Toeplitz matrices. In this paper, we use this parametrization to derive efficient computational algorithms for optimization problems over positive polynomials. Moreover, we show that filter design problems can be solved using these results. Keywords: convex optimization, positive polynomials, trigonometric polynomials, filter design.