Meaningful MRA initialization for discrete time series

Though frequently and commonly applied to discrete-time processes, the discrete wavelet transform is de"ned only for, and should therefore strictly speaking be applied only to, continuous-time processes. We provide a method for applying it to intrinsically discrete data, in such a way that the spectrum of discrete second-order processes can be meaningfully studied. The practical step required is a pre"ltering on the discrete data which can be integrated into the initialization phase of the multiresolution analysis. It is shown that, for spectral analysis of discrete-time processes, the discrete wavelet transform, used without this pre"ltering, results in errors which are avoided by the present method. Details and examples are given so that the method can be conveniently used in practice. The particular relevance of this initialization step to the study of scaling processes, such as fractional Gaussian noise, is explained and illustrated. ( 2000 Elsevier Science B.V. All rights reserved.