LMI characterization for the convex hull of trigonometric curves and applications

In this paper, we develop a new linear matrix inequality (LMI) technique, which is practical for solutions of the general trigonometric semi-infinite linear constraint (TSIC) of competitive orders. Based on the new full LMI characterization for the convex hull of a trigonometric curve, it is shown that the semi-infinite optimization problem involving TSIC can be solved by an LMI optimization problem with additional variables of dimension just n, the order of the trigonometric curve. Our solution method is very robust which allows us to address almost all practical filter design problems. Unlike most previous works involving several complex mathematical tools, our derivation arguments are based on simple results of the convex analysis and some formal elementary transforms. Furthermore, many filter/filterbank design problems can be reformulated as the optimization of linear/convex quadratic objectives over the TSIC. Based on this reformulation, these problems can be equivalently reduced to LMI optimization problems with the minimal size. Our examples of designing up to 1200-tap filters verifies the viability of our formulation.