<p>Optimization problems over the cone of nonnegative trigonometric matrix polynomials are common in signal processing. Traditionally, those problems are formulated and solved as semidefinite programs (SDPs). However, the SDP formulation increases the dimension of the problem, resulting in large problems that are challenging to solve. In this paper we propose first-order methods that circumvent the SDP formulation and instead optimize directly within the space of trigonometric matrix polynomials. Our methods are based on a particular Bregman proximal operator. We apply our approach to two fundamental signal processing applications: to rectify a power spectrum that fails to be nonnegative and for graphical modeling of Gaussian time series. Numerical experiments demonstrate that our methods are orders of magnitude faster than an interior-point solver applied to the corresponding SDPs.</p>