Circuits, Systems, and Signal Processing

, Volume 38, Issue 1, pp 191–217 | Cite as

Separation of Single Frequency Component Using Singular Value Decomposition

  • Xuezhi ZhaoEmail author
  • Bangyan Ye


It is proved that for an arbitrary frequency component, no matter how much the frequency and its amplitude are, this frequency component always generates only two nonzero singular values. Based on this relationship, an approach based on singular value decomposition (SVD) is proposed to separate the single frequency, and the condition for SVD to separate the single frequency is that the amplitude of each frequency is not equal to each other. To separate the frequencies with the same amplitude, it is proposed to add the white noise to the original signal, thus the amplitudes of frequencies become different, and the influence of noise on the singular values is studied. Three principles for selecting singular values are proposed. The quantitative relation among the singular values and frequency parameters is obtained, so the place of the nonzero singular values of each frequency in the singular value sequence can be located. Under these conditions, as long as a frequency can be distinguished in the amplitude spectrum of original signal, it can always be separated from the original signal by SVD. Simulation and practical signal separation examples verify the effectiveness of this approach, and compared with the existing methods, SVD has higher frequency separation accuracy.


Singular value decomposition Single frequency component Signal separation Separation condition Noise 



This work is supported by the National Natural Science Foundation of China ((NSFC, Grant No. 51375178) and Natural Science Foundation of Guangdong province (Grant No. S2012010008789). We are grateful to the editor and the anonymous reviewers for their helpful suggestions to improve the quality of the paper.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mechanical and Automotive EngineeringSouth China University of TechnologyGuangzhouPeople’s Republic of China

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