利用奇异值分解（Singular value decomposition，SVD）进行信号处理的关键在于矩阵的构造，为利用SVD分离信号中的不同频率成分，提出一种变矩阵结构递推SVD算法，其思想是在SVD递推分解过程中逐次改变矩阵的结构，每进行一次SVD分解，矩阵的结构就规律性地变化一次，由此形成对信号中不同频率成分的适应性，从而达到将其分离出来的目的。推导出这种变结构SVD的信号分解算法，证明了这种算法可以将原始信号分解为一系列分量信号的线性组合。进一步从理论上分析了这种算法的信号分离机理，证明了对于一些特定的频率结构，这种变结构SVD算法可以实现对原信号中单个频率分量的逐次分离。最后通过对模拟信号和工程实际信号的分离实例证实了变结构SVD算法良好的信号分离效果，并与小波分析和多分辨SVD方法进行了比较，结果表明变结构SVD的信号分离结果优于这两种方法。
The key question of applying singular value decomposition (SVD) to signal processing is the construction of the matrix. In order to separate the different frequency components from the original signal through SVD, a recursive SVD algorithm with variable matrix structure is proposed, whose idea is to change the structure of the matrix in the process of SVD recursion decomposition, each time when the SVD is carried out, the structure of the matrix to be decomposed will change regularly to adapt to the different frequency components in the signal, so that the different frequency components can be separated. The signal decomposition algorithm of the variable structure SVD is deduced, and it is proved that the original signal can be decomposed into a linear combination of a series of component signals by this algorithm. Furthermore, the signal separation mechanism of this algorithm is analyzed theoretically, and it is proved that for some frequency structure, variable structure SVD can separate the each single frequency component successively from the original signal. Finally, the separation examples of the simulation signal and actual engineering signal are provided, which demonstrate the good signal separation effect of the variable structure SVD algorithm. The comparison with wavelet analysis and multi-resolution SVD method shows that variable structure SVD can achieve the better signal separation results than these two methods.
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