On Properties and Structure of the Analytic Singular Value Decomposition
On Properties and Structure of the Analytic Singular Value Decomposition
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解析奇异值分解的性质和结构
DOI：
10.1109/tsp.2024.3387726
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发表时间：
2024
影响因子：
5.4
通讯作者：
J. McWhirter
中科院分区：
文献类型：
--
作者：
Stephan Weiss;I. Proudler;Giovanni Barbarino;Jennifer Pestana;J. McWhirter
本文研究了一个矩形矩阵$\boldsymbol{\mathit{A}}（z）$的奇异值分解（SVD），该矩阵是一个至少包含单位圆的环上的解析函数。这样的矩阵出现，例如，作为表示宽带多输入多输出系统的传递函数的矩阵。我们的分析是基于适用于连续时间系统的解析SVD的研究结果，并在解析特征值分解。使用这些，我们建立了两个潜在的重叠的情况下，SVD因素的分析被拒绝。首先，从结构的角度来看，复用系统需要过采样的复用因子，以承认一个解析解。其次，从代数的角度来看，我们的状态下，任何奇异值的谱零点需要额外的过采样的两个因素，如果一个解析solution is to be found. In所有其他情况下，一个解析矩阵承认一个解析SVD，从而奇异值是唯一的置换，和左，右奇异向量通过联合模糊度w.r.t.耦合。任意全通函数。我们演示了一些国家的最先进的多项式矩阵分解算法近似这个解决方案，激励需要专用的算法。
We investigate the singular value decomposition (SVD) of a rectangular matrix $\boldsymbol{\mathit{A}}(z)$ of functions that are analytic on an annulus that includes at least the unit circle. Such matrices occur, e.g., as matrices of transfer functions representing broadband multiple-input multiple-output systems. Our analysis is based on findings for the analytic SVD applicable to continuous time systems, and on the analytic eigenvalue decomposition. Using these, we establish two potentially overlapping cases where analyticity of the SVD factors is denied. Firstly, from a structural point of view, multiplexed systems require oversampling by the multiplexing factor in order to admit an analytic solution. Secondly, from an algebraic perspective, we state under which condition spectral zeros of any singular value require additional oversampling by a factor of two if an analytic solution is to be found. In all other cases, an analytic matrix admits an analytic SVD, whereby the singular values are unique up to a permutation, and the left- and right-singular vectors are coupled through a joint ambiguity w.r.t. an arbitrary allpass function. We demonstrate how some state-of-the-art polynomial matrix decomposition algorithms approximate this solution, motivating the need for dedicated algorithms.
影响因子：
5.4
作者：
Joanne A. Foster;J. McWhirter;M. Davies;J. Chambers
通讯作者：
Joanne A. Foster;J. McWhirter;M. Davies;J. Chambers