SVD和小波变换的信号处理效果相似性及其机理分析  

The Similarity of Signal Processing Effect between SVD and Wavelet Transform and Its Mechanism Analysis

作　　者：赵学智[1] 叶邦彦[1] 

机构地区：[1]华南理工大学机械工程学院,广东广州510640

出　　处：《电子学报》2008年第8期1582-1589,共8页Acta Electronica Sinica

基　　金：国家自然科学基金(No.50305005);广州市科技计划项目(No.2008B12080633)

摘　　要：指出当采用Hankel矩阵时,利用SVD可以获得与小波变换十分相似的信号处理效果.证明了在Hankel矩阵方式下SVD可将原始信号分解为一系列分量信号的简单线性叠加,而分量信号实质上反映了与左、右正交矩阵形成的m维和n维空间的规范正交基的相似程度.从空间基的角度和Hankel矩阵本身的特点分析了SVD和小波变换在信号处理机理上的相似性,并研究了这两种方法获得的结果之间的正交性差异.指出采用Hankel矩阵时,SVD也具有类似于小波变换那样的奇异性检测能力,但是与小波相比,SVD的奇异性检测具有两个特点:一是各分量的消失矩阶数逐次增加,第n个SVD分量具有n-1阶消失矩,因而各分量可以检测出具有不同奇异性指数的奇异点;二是所有SVD分量中指示奇异点位置的脉冲宽度始终保持不变,而且这个宽度由所构造的Hankel矩阵的列数决定.It＇s pointed out that singular value decomposition （SVD） has the very similar signal processing effect as wavelet transform when Hankel matrix is used. It＇ s proved that original signal can be decomposed into the linear superpositon of a series of component signals by SVD using Hankel matrix, and what the component signals reflect essentially are the similarity extent of original signal to standard orthogonal bases of m and n dimension space constituted by left and right orthogonal matrix in SVD. The similarity mechanism of signal prneessing of SVD and wavelet transform is analyzed from the angle of basis of vector space and characteristic of Hankel matrix. The orthogonality difference of component signals got by SVD and wavelet transform is studied theoretically. It＇s pointed out that SVD also has the singularity detection ability when Hankel matrix is used, but compared with wavelet trans-form,SVD has two characteristics in singularity detection, one is that the vanishing moment of SVD component signals is increased progressively and the n＇ th SVD component signal has the n-1＇th order vanishing moment so that singular point with different Lip index can also be detected, the other is that the width of the impulse that indicates the position of singular point will keep the same in all SVD component signals and this width is determined by the column number of Hankel matrix.

关 键 词：SVD 小波变换 HANKEL矩阵 信号处理 相似性 

分 类 号：TN911.7[电子电信—通信与信息系统] TH165.3[电子电信—信息与通信工程]