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A constructive algorithm for 2-D spectral factorization with rational spectral factors

机译:具有合理频谱因子的二维频谱分解的构造算法

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Spectral factorization of para-Hermitian polynomial matrices nonnegative on the imaginary axis is known to play a crucial role in signal and system theory. Such factorization, although always feasible in one-dimensional (1-D) case, can be carried out in two-dimensional (2-D) case in a somewhat modified form. In this modified form, even spectral factors of scalar polynomials must be rational matrices (with normal rank one), analytic in the Cartesian products of open right half planes. Despite this constrained form, such spectral factors are important in the context of 2-D systems. The feasibility of such spectral factorization is apparent in view of classical results from algebraic geometry, but a constructive proof of it has not been available in system theoretic literature. By a selection of elementary techniques borrowed from different sources, we give a constructive algorithm for the aforementioned spectral factorization of 2-D para-Hermitian scalar positive definite polynomials, and thus, indirectly provide a constructive proof of the corresponding result for para Hermitian polynomial matrices as well. An example illustrating the main aspect of the algorithm is included. Analog of the result for discrete time systems, and comparisons with known 2-D spectral factorization results of other types are also included.
机译:虚轴上非负的准Hermitian多项式矩阵的频谱分解在信号和系统理论中起着至关重要的作用。这种分解虽然在一维(1-D)情况下总是可行的,但可以在二维(2-D)情况下以某种修改的形式进行。在这种修改形式中,即使标量多项式的谱因子也必须是有理矩阵(具有正一阶),并在开右半平面的笛卡尔积中进行分析。尽管存在这种受约束的形式,但是这种光谱因子在二维系统中仍然很重要。鉴于代数几何的经典结果,这种频谱分解的可行性是显而易见的,但是在系统理论文献中尚无构造性证明。通过选择从不同来源借来的基本技术,我们为二维准Hermitian标量正定多项式的上述频谱分解提供了一种构造算法,从而间接提供了针对Paramiter多项式矩阵的相应结果的构造证明。也一样包括说明算法主要方面的示例。还包括离散时间系统结果的模拟,以及与其他类型的已知2-D光谱分解结果的比较。

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