基于Fourier变换离散化的连续小波变换频域算法

对于固定的尺度,小波变换是待分析信号与小波基函数的线性卷积。当小波基函数的Fourier变换有显式表达式时,利用其Fourier变换进行线性卷积称为小波变换的频域计算方法。由于线性卷积的长度大于信号的长度,因此,选取线性卷积中的哪一部分作为小波变换的系数也是一个亟需回答的问题。本文利用Fourier变换的离散化和离散Fourier变换的关系由小波变换时域算法推导了小波变换频域算法,证明了时域算法与频域算法的等价性;解释了这两种方法分别应该选取线性卷积中的哪一部分作为小波变换的系数;分析了频域算法产生边界效应的原因;给出了频域算法中参数的选取方法,以便克服边界效应。时间复杂度分析以及数值实验均表明了频域算法至少比时域算法减少了1/3的运行时间。

THE FREQUENCY DOMAIN ALGORITHM OF CONTINUOUS WAVELET TRANSFORM BASED ON THE DISCRETIZATION OF FOURIER TRANSFORM

For a fixed scale, continuous wavelet transform(CWT) is a linear convolution of signal and wavelet function. Because the length of a linear convolution is larger than that of a signal, which part of linear convolution should be chosen as the wavelet coefficients is a question that need an answer. If the Fourier transform of wavelet function has explicit expression, the expression can be exploited to calculate the linear convolution, which method is known as the frequency-domain algorithm of CWT. In this paper, the frequency-domain algorithm of CWT is deduced from the time-domain algorithm of CWT by exploiting the relationship of the discretization of Fourier transform and the discrete Fourier transform. Four results are given. Firstly, the equivalent relation of these two algorithms are proved; Secondly, for time-domain method, the middle coefficients of linear convolution, while for frequency-domain method, the first coefficients of linear convolution are equal, and are the right wavelet coefficients. Thirdly, how to choose the parameter of the frequency domain algorithm is given by the theoretical derivation, which can conquer the boundary effect of frequency-domain method. In the end, the time complexity of the frequency domain algorithm of CWT is lower than that of the time domain algorithm, which is demonstrated by theoretical analysis and numerical experiments.