Separation of Single Frequency Component Using Singular Value Decomposition

Abstract

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.