Non-decimated Wavelet Transform for a Shift-invariant Analysis

Due to the ability of time-frequency location, the wavelet transform hasbeen applied in several areas of research involving signal analysis and processing,often replacing the conventional Fourier transform. The discrete wavelet transformhas great application potential, being an important tool in signal compression,signal and image processing, smoothing and denoising data. It also presentsadvantages over the continuous version because of its easy implementation, goodcomputational performance and perfect reconstruction of the signal upon inversion.Nevertheless, the downsampling required in the discrete wavelet transformcalculous makes it shift variant and not appropriated to some applications, suchas for signals or time series analysis. On the other hand, the Non-Decimated DiscreteWavelet Transform is shift-invariant because it eliminates the downsamplingand, consequently, is more appropriate for identifying both stationary and nonstationarybehaviors in signals. However, the non-decimated wavelet transform hasbeen underused in the literature. This paper intends to show the advantages ofusing the non-decimated wavelet transform in signal analysis. The main theoricaland pratical aspects of the multiscale analysis of time series from non-decimatedwavelets in terms of its formulation using the same pyramidal algorithm of thedecimated wavelet transform was presented. Finally, applications with a simulatedand real time series compare the performance of the decimated and non-decimatedwavelet transform, demonstrating the superiority of non-decimated one, mainly dueto the shift-invariant analysis, patterns detection and more perfect reconstructionof a signal.