Certifying the positivity of trigonometric polynomials is of first importance for design problems in discrete-time signal processing. It is well known from the Riesz-Fejér spectral factorization theorem that any trigonometric univariate polynomial non-negative on the unit circle can be decomposed as a Hermitian square with complex coefficients. Here we focus on the case of polynomials with Gaussian integer coefficients, i.e., with real and imaginary parts being integers.

A semialgebraic set is a set described by a boolean combination of real polynomial inequalities in several variables. A linear matrix inequality (LMI) is a condition expressing that a symmetric matrix whose entries are affine-linear combinations of variables is positive semidefinite. We call solution sets of LMIs spectrahedra and their linear images semidefinite representable.