Similarity of signal processing effect between Hankel matrix-based SVD and wavelet transform and its mechanism analysis

Abstract

It is pointed out that signal processing effect of singular value decomposition (SVD) is very similar to that of wavelet transform when Hankel matrix is used. It is proved that a signal can be decomposed into the linear sum of a series of component signals by Hankel matrix-based SVD, and essentially what the component signals reflect are projections of original signal on the orthonormal bases of m-dimensional and n-dimensional vector spaces. The similarity mechanism of signal processing between SVD and wavelet transform is analyzed from the angle of basis of vector space and characteristic of Hankel matrix. The orthogonality of the component signals got by SVD and wavelet transform is also studied. It is discovered that singularity of signal can also be detected by Hankel matrix-based SVD, and compared with wavelet transform, there are two characteristics in SVD for singularity detection, one is that the order of vanishing moment of SVD component signals is increased progressively and the one of the nth SVD component signal is ‘ n−1’, so singular points with different Lip index can all be detected, the other is that the width of impulse indicating the position of singularity will always keep the same throughout all SVD components and this width is determined by the column number of Hankel matrix.