Non-decimated Wavelet Transform for a Shift-invariant Analysis - Brassarote, Gabriela de Oliveira Nascimento; Souza, Eniuce Menezes de; Monico, João Francisco Galera (Trends in Computational and Applied Mathematics, 2018)
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.