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 de-noising 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 computation of the discrete wavelet transform 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 shiftinvariant because it eliminates the downsampling and, consequently, is more appropriate for identifying both stationary and non-stationary 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 theoretical and practical aspects of the multi-scale 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.

varies with station location and elevation angle of the satellite. Therefore, each case must be treated individually.

ABSTRACT. Due to the numerous application possibilities, the theory of wavelets has been applied in several areas of research. The Discrete Wavelet Transform is the most known version. However, the downsampling required for its calculation makes it sensitive to the origin, what is not ideal for some applications, mainly in time series. On the other hand, the Non-Decimated Discrete Wavelet Transform is shift invariant, because it considers all the elements of the sample, by eliminating the downsampling and, consequently, represents a time series with the same number of coefficients at each scale. In the present paper, the objective is to present the theorical aspects of the a multiscale/multiresolution analysis of non-stationary time series from non-decimated wavelets in terms of its implementation using the same pyramidal algorithm of the decimated wavelet transform. An application with real time series of the effect of the ionospheric scintillation on GPS satellite signals is investigated. Results obtained in this investigation of the time series of the S4 scintillation index indicate the presence of a pattern that is repeated in the series on consecutive days. Through decomposition in multiscale by wavelets, this periodic daily effect could be identified in smoother scales of the wavelet periodogram, estimated and corrected from the original time series. Therefore, this paper presents an innovative methodology for the analysis of the effects of ionospheric scintillation on GPS signals, and take a first step toward separating the effect of scintillation in relations the other effects that may influence its analysis.

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