An Algorithm for Calculating the QR and Singular Value Decompositions of Polynomial Matrices
An Algorithm for Calculating the QR and Singular Value Decompositions of Polynomial Matrices
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DOI：
10.1109/tsp.2009.2034325
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发表时间：
2010-03
影响因子：
5.4
通讯作者：
Joanne A. Foster;J. McWhirter;M. Davies;J. Chambers
中科院分区：
文献类型：
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作者：
Joanne A. Foster;J. McWhirter;M. Davies;J. Chambers
本文介绍了一种计算多项式矩阵QR分解（QRD）的新算法。该算法相当于通过应用一系列的仿酉矩阵，如初等延迟矩阵和旋转矩阵，将多项式矩阵变换成上三角形。它表明，该算法也可以用来制定一个多项式矩阵的奇异值分解（SVD），这基本上相当于对角化一个多项式矩阵再次通过应用一系列的仿酉矩阵。示例矩阵用于演示这两种类型的分解。还概述了两种分解收敛性的数学证明。最后，讨论了这种分解在多通道信号处理中的可能应用。
In this paper, a new algorithm for calculating the QR decomposition (QRD) of a polynomial matrix is introduced. This algorithm amounts to transforming a polynomial matrix to upper triangular form by application of a series of paraunitary matrices such as elementary delay and rotation matrices. It is shown that this algorithm can also be used to formulate the singular value decomposition (SVD) of a polynomial matrix, which essentially amounts to diagonalizing a polynomial matrix again by application of a series of paraunitary matrices. Example matrices are used to demonstrate both types of decomposition. Mathematical proofs of convergence of both decompositions are also outlined. Finally, a possible application of such decompositions in multichannel signal processing is discussed.