常用小波及其时频特性

小波分析以其良好的时频局部化效果，能够用来分析时间序列不同频率的非稳定能量特征。目前，有许多可供选择的小波函数，这些函数的时频特征各不相同，根据研究目的选择合适的分析小波函数就显得非常重要。针对常用的几个实数和复数小波函数，以Nino3海面季度平均温度时间序列(1871-1996，5oS~5oN, 90o~150oW)为例，分别研究它们的时频特征，希望为实际应用过程中选择合适的分析小波提供依据。研究发现，消失矩越大，时域小波越窄，时间分辨率增强，波动幅度越大；而频域小波与时域相反。复数的Morlet小波具有良好的频率分析能力，而Mexican hat小波(Gauss函数的二阶导数)则具有优秀的时间分辨能力。相对小波的总能量谱，复数小波的实部或虚部的能量谱能够明显提高分析结果的时间分辨率，但并不影响小波的频域分辨力。

Frequently-used wavelets and their time-frequency properties

The wavelet analysis has a good ability of time-frequency localization and can be used to analyze time series that contain nonstationary powers at different frequencies.There are many wavelets,which have different time-frequency properties respectively.So it is very important to choose a proper wavelet matching the aim of a study.In the article,several frequently-used real and complex wavelets are investigated using time series of the Nino3 sea surface temperature(Nino3 SST,18711996;5oS5oN,90o150oW),to supply the idea of wavelet choosing.It is found that with an increasing vanishing moment there is a narrower time domain and higher time resolution with bigger amplitude of wavelet,while the frequency resolution is in contrary to the time domain wavelet.The complex Morlet wavelet has an excellent frequency resolution.And Mexican hat wavelet,which is the second derivative of Gauss function,has a good time resolution.In addition,compared with total wavelet power,the power of the real or imaginary part of wavelet can significantly improve time resolution while it does not affect frequency resolution.