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-Fej\'ez spectral factorization theorem that any trigonometric univariate polynomial positive 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. We design, analyze and compare, theoretically and practically,three hybrid numeric-symbolic algorithms computing weighted sums of Hermitian squares decompositions for trigonometric univariate polynomials positive on the unit circle with Gaussian coefficients. The numerical steps the first and second algorithm rely on are complex root isolation and semidefinite programming, respectively. An exact sum of Hermitian squares decomposition is obtained thanks to compensation techniques. The third algorithm, also based on complex semidefinite programming, is an adaptation of the rounding and projection algorithm by Peyrl and Parrilo. For all three algorithms, we prove bit complexity and output size estimates that are polynomial in the degree of the input and linear in the maximum bitsize of its coefficients. We compare their performance on randomly chosen benchmarks, and further design a certified finite impulse filter.

翻译：证明三角多项式非负性对于离散时间信号处理中的设计问题至关重要。根据Riesz-Fejéz谱分解定理，任何在单位圆上正定的一元三角多项式均可表示为具有复系数的埃尔米特平方。本文重点关注高斯整数系数（即实部与虚部均为整数）多项式的情形。我们设计、分析并比较了三种混合数值符号算法，用于计算单位圆上具有高斯系数的正定三角一元多项式的加权埃尔米特平方和分解。前两种算法分别依赖于复根隔离和半定规划，通过补偿技术获得精确的埃尔米特平方和分解。第三种算法基于复半定规划，是对Peyrl和Parrilo的舍入投影算法的改进。对于所有三种算法，我们证明了其位复杂度和输出规模估计均与输入多项式的次数呈多项式关系，与其系数的最大位长呈线性关系。我们通过随机选取的基准测试对比了算法性能，并进一步设计了一个认证有限冲激响应滤波器。