A general approach to derivative calculation using wavelet transform

Application of wavelet transform (WT) for derivative calculation has been reported based on the property of specific wavelet function: Haar, Daubechies and Gaussian wavelets. In this work, the underlying principle of the wavelet transform for derivative calculation is investigated, and a general approach is proposed. By theoretical analysis, it can be found that wavelet transform with commonly used wavelet functions can be regarded as a smoothing and a differentiation process, and the order of differentiation is determined by the property of the wavelet function. Derivatives of different simulated signals by using all the commonly used wavelet functions are investigated and compared with the conventional numerical differentiation method. It is shown that differentiation is a common property of all these wavelet functions, and nth-order derivative can be simply obtained through just one transform procedure, instead of repeated transform, by using an appropriate wavelet function. Furthermore, both discrete wavelet transform and continuous wavelet transform show similar characteristic with a given wavelet function.