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A wide variety of optimization problems involving nonnegative polynomials or trigonometric polynomials can be formulated as convex optimization problems by expressing (or relaxing) the constraints using sum-of-squares representations. The semidefinite programming problems that result from this formulation are often difficult to solve due to the presence of large auxiliary matrix variables. In this paper we extend a recent technique for exploiting structure in semidefinite programs derived from sum-of-squares expressions to multivariate trigonometric polynomials. The technique is based on an equivalent formulation using discrete Fourier transforms and leads to a very substantial reduction in the computational complexity. Numerical results are presented and a comparison is made with general-purpose semidefinite programming algorithms. As an application, we consider a two-dimensional FIR filter design problem.