正弦信号非零奇异值的变化规律研究

在信号的奇异值分解中,非零奇异值是信号的重要特征参数,其变化对特征提取结果有重要影响。研究了非零奇异值的变化特性,发现随着矩阵维数的增大,正弦信号的非零奇异值并不是单调上升的,而是在上升过程中存在周期性波动。进一步研究发现,这种波动和原信号的频率密切相关,原信号的频率越大,非零奇异值的波动速度就越快。从理论上对非零奇异值的这种变化规律进行了分析,证明了非零奇异值波动的快慢是由正弦信号的频率决定的,而其波动基频是原始信号频率的两倍,并通过数值模拟实例进行了验证。

Study on Variation Law of Non-zero Singular Values of Sinusoidal Signal

In the singular value decomposition (SVD) of signal, non-zero singular values are the important feature parameters of signal, and the variation of non-zero singular values has the important influence on the feature extraction results. The variation characteristic of non-zero singular values is studied, and it is found that with the increase of the matrix dimension, the non-zero singular values of the sinusoidal signal do not rise monotonically, and there is periodic fluctuation in the rising process of the singular values. Furthermore, the fluctuation of non-zero singular values is closely related to the frequency of the original signal, and the greater the frequency of the original signal, the faster the fluctuation speed of the non-zero singular values will be. The fluctuation of non-zero singular values is analyzed theoretically, and it is proved that the fluctuation speed of non-zero singular values is determined by the frequency of sinusoidal signal. In addition, the fundamental frequency of fluctuation is twice as much as that of the original signal frequency, and this characteristic is verified by the numerical simulation examples.