奇异值分解中考虑频率因素的矩阵维数

doi:10.3901/JME.2019.16.007

机械工程学报 ›› 2019, Vol. 55 ›› Issue (16): 7-16.doi: 10.3901/JME.2019.16.007

奇异值分解中考虑频率因素的矩阵维数

赵学智, 邵啟鹏, 叶邦彦, 陈统坚

华南理工大学机械与汽车工程学院广州 510640

收稿日期:2019-01-26 修回日期:2019-07-02 出版日期:2019-08-20 发布日期:2019-08-20
作者简介:赵学智,男,1970年出生,博士,教授,博士研究生导师。主要研究方向为信号处理与机械故障诊断。E-mail:mezhaoxz@scut.edu.cn
基金资助:
国家自然科学基金（51375178、51875216）和广东省自然科学基金（No.2018A030310017）资助项目。

Matrix Dimension Considering Frequency Factor in Singular Value Decomposition

ZHAO Xuezhi, SHAO Qipeng, YE Bangyan, CHEN Tongjian

School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510640

Received:2019-01-26 Revised:2019-07-02 Online:2019-08-20 Published:2019-08-20

摘要/Abstract

摘要： Hankel矩阵的维数对于奇异值分解的信号处理效果有非常重要的影响，传统的维数没有考虑信号中的频率成分，这是不合理的。通过对频率因素的分析，提出一种确定矩阵维数的最小公倍数法，将原始信号中各频率成分的周期的最小公倍数作为基数，Hankel矩阵的行数和列数必须同时为这个基数的整数倍，并在这一必要条件下使Hankel矩阵的维数最大，由此通过优化得到了最佳的矩阵行、列数。对模拟信号和转子振动信号的处理实例结果表明，与传统的最大维数法相比，在最小公倍数法确定的矩阵维数下，奇异值分解的计算量要小得多，但是却可以获得波形误差更小的信号分解结果。

关键词: 矩阵维数, 频率周期, 奇异值分解, 信号分离, 最小公倍数

Abstract: The dimension of Hankel matrix has a very important influence on the signal processing effect of singular value decomposition (SVD). The traditional matrix dimension does not consider the frequency components in the signal and this is unreasonable. A least common multiple method is put forward to determine the matrix dimension based on the analysis for the frequency factor, and in this method, the least common multiple of the periods of the all frequency components in the original signal is used as a base number, and the row and column number of Hankel matrix must be the integer multiple of this base number, under this necessary condition, the dimension of Hankel matrix should be maximized, and then the optimal row number and column number are obtained by the optimization computation. The processing examples of simulation signal and rotor vibration signal are provided, which show that, under the matrix dimension determined by the least common multiple method, the calculation amount of SVD is much smaller, but waveform error of the decomposition results is much smaller than the ones of the traditional maximum dimension method.

Key words: frequency period, least common multiple, matrix dimension, signal separation, singular value decomposition

中图分类号:

TN911

赵学智, 邵啟鹏, 叶邦彦, 陈统坚. 奇异值分解中考虑频率因素的矩阵维数[J]. 机械工程学报, 2019, 55(16): 7-16.

ZHAO Xuezhi, SHAO Qipeng, YE Bangyan, CHEN Tongjian. Matrix Dimension Considering Frequency Factor in Singular Value Decomposition[J]. Journal of Mechanical Engineering, 2019, 55(16): 7-16.

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奇异值分解中考虑频率因素的矩阵维数

Matrix Dimension Considering Frequency Factor in Singular Value Decomposition

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