周期广义谐和小波变换及重构

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振动与冲击 ›› 2013, Vol. 32 ›› Issue (7) : 24-29.

论文

周期广义谐和小波变换及重构

孔凡1，李杰1,2

作者信息 +

Periodic generalized harmonic wavelet: transformation and reconstruction

Fan Kong1, Jie Li1, 2

Author information +

文章历史 +

摘要

谐和小波和广义谐和小波皆为在频域上紧支且时域为无穷的正交小波，其频域分辨率很好但时域分辨率较差。虽然谐和小波在无穷时域上具有正交性，但其正交性在有限时域上却无法体现。针对这个缺点，在广义谐和小波的基础上，将广义谐和小波周期化后，进而提出了一种周期广义谐和小波（Periodic Generalized Harmonic Wavelet, PGHW）。PGHW的母小波在时域中可以表达为经平移后的若干谐和项之和，在频域中表现若干函数之和，为一种以待分析信号持时为基本周期且在其上正交的离散广义谐和小波。基于PGHW在频域内的简单性，利用快速Fourier变换（FFT）技术实现了PGHW的快速小波变换及逆变换。文章最后的算例给出了某人工合成地震波的周期广义谐和小波变换及其重构，说明了所提算法的高效性与PGHW的完全重构性。

Abstract

The Periodic Generalized Harmonic Wavelet (PGHW) and the algorithm for its Fast Wavelet Transform (FWT) and inverse fast Wavelet Transform (iFWT) are presented in this paper. The PGHW can be represented as a sum of several translated harmonic terms in the time domain, and of several function in the frequency domain. Compared to the Generalized Harmonic Wavelet (GHW) and the Harmonic Wavelet (HW), PGHW is orthogonal and periodic, while the later lose the orthogonality on a finite time interval. Considering the simplicity of the PGHW in the frequency domain, the FWT and iFWT is developed via the Fast Fourier Transform (FFT) technique. Numerical examples demonstrate the computational efficiency of the algorithm and perfect reconstruction of the PGHW.