Gram pair parameterization of multivariate sum-of-squares trigonometric polynomials

In this paper we propose a parameterization of sum-of-squares trigonometric polynomials with real coefficients, that uses two positive semidefinite matrices twice smaller than the unique matrix in the known Gram (trace) parameterization. Also, we formulate a Bounded Real Lemma for polynomials with support in the positive orthant. We show that the new parameterization is clearly faster and thus can replace the old one in several design problems.