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Certifying the positivity of trigonometric polynomials is of first importancefor design problems in discrete-time signal processing. It is well known fromthe Riesz-Fej\'ez spectral factorization theorem that any trigonometricunivariate polynomial positive on the unit circle can be decomposed as aHermitian square with complex coefficients. Here we focus on the case ofpolynomials with Gaussian integer coefficients, i.e., with real and imaginaryparts being integers. We design, analyze and compare, theoretically andpractically,three hybrid numeric-symbolic algorithms computing weighted sums ofHermitian squares decompositions for trigonometric univariate polynomialspositive on the unit circle with Gaussian coefficients. The numerical steps thefirst and second algorithm rely on are complex root isolation and semidefiniteprogramming, respectively. An exact sum of Hermitian squares decomposition isobtained thanks to compensation techniques. The third algorithm, also based oncomplex semidefinite programming, is an adaptation of the rounding andprojection algorithm by Peyrl and Parrilo. For all three algorithms, we provebit complexity and output size estimates that are polynomial in the degree ofthe input and linear in the maximum bitsize of its coefficients. We comparetheir performance on randomly chosen benchmarks, and further design a certifiedfinite impulse filter.

Exact SOHS decompositions of trigonometric univariate polynomials with Gaussian coefficients