Non-decimated Wavelet Transform for a Shift-invariant Analysis

Abstract

Due to the ability of time-frequency location, the wavelet transform has
been applied in several areas of research involving signal analysis and processing,
often replacing the conventional Fourier transform. The discrete wavelet transform
has great application potential, being an important tool in signal compression,
signal and image processing, smoothing and denoising data. It also presents
advantages over the continuous version because of its easy implementation, good
computational performance and perfect reconstruction of the signal upon inversion.
Nevertheless, the downsampling required in the discrete wavelet transform
calculous makes it shift variant and not appropriated to some applications, such
as for signals or time series analysis. On the other hand, the Non-Decimated Discrete
Wavelet Transform is shift-invariant because it eliminates the downsampling
and, consequently, is more appropriate for identifying both stationary and nonstationary
behaviors in signals. However, the non-decimated wavelet transform has
been underused in the literature. This paper intends to show the advantages of
using the non-decimated wavelet transform in signal analysis. The main theorical
and pratical aspects of the multiscale analysis of time series from non-decimated
wavelets in terms of its formulation using the same pyramidal algorithm of the
decimated wavelet transform was presented. Finally, applications with a simulated
and real time series compare the performance of the decimated and non-decimated
wavelet transform, demonstrating the superiority of non-decimated one, mainly due
to the shift-invariant analysis, patterns detection and more perfect reconstruction
of a signal.