A convex optimization model for finding non-negative polynomials

This paper presents a convex optimization model for the problem of finding some polynomials for which certain linear combinations are non-negative polynomials. This model is then applied to solve several filter design problems. We first reformulate some low-pass filter design problems, with finite or infinite impulse response, as optimization problems over non-negative (real or complex) polynomials whose feasibility problems can be solved by applying our model. The whole optimization problems are then solved by using a combination of a bisection search procedure on an appropriate parameter and our convex optimization model to solve the feasibility problems. Some numerical examples illustrate the method.