Daubechies wavelets and Mathematica

Analysis of time series data using wavelets provides both scale (frequency) and position information. In contrast, the Fourier transform provides frequency information only. We discuss the Daubechies formulation of wavelets, with reference to the WaveletTransform package that calculates the filter coefficients for any Daubechies basis to arbitrary precision. Examples of the wavelet transform applied to selected time series are presented to highlight the advantages of wavelets. We indicate an application of wavelets to sampled Barkhausen noise, a nonlinear phenomenon encountered in magnetic systems. The elements of the WaveletTransform package are discussed, with the emphasis being on the calculation of filter coefficients and their application to the discrete wavelet transform (DWT) and its inverse. With the construction of quadrature mirror filters, an efficient implementation of the DWT is possible and is similar in structure to the fast Fourier transform algorithm. © 1995 American Institute of Physics.