带高斯系数的一元三角多项式的精确SOHS分解
Exact SOHS decompositions of trigonometric univariate polynomials with Gaussian coefficients
AI总结：
针对高斯整数系数一元三角多项式的正定性验证需求，提出三种混合数值-符号SOHS精确分解算法，证明其多项式复杂度，并通过基准测试和FIR滤波器设计验证了性能。
摘要：
证明三角多项式的正定性对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，单位圆上为正的任意一元三角多项式都可分解为复系数的埃尔米特平方。本文聚焦高斯整数系数（即实部和虚部均为整数）的多项式情形。我们从理论和实践层面设计、分析并比较了三种混合数值-符号算法，用于计算单位圆上为正的高斯系数一元三角多项式的加权埃尔米特平方和（SOHS）分解。前两种算法的数值步骤分别依赖复根隔离和半定规划，借助补偿技术可得到精确的埃尔米特平方和分解。第三种算法同样基于复半定规划，是对Peyrl和Parrilo提出的舍入投影算法的改进。针对这三种算法，我们证明了其比特复杂度和输出规模估计：复杂度与输入多项式的次数呈多项式关系，与系数的最大比特长度呈线性关系。我们在随机选取的基准测试上比较了它们的性能，并进一步设计了一个经过验证的有限脉冲响应（FIR）滤波器。
英文摘要：
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-Fejéz 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.
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