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 (or Maximum Overlap Discrete Wavelet Transform, Stationary Wavelet Transform, Shift-invariant Discrete Wavelet Transform, Redundant 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 artificial satellite signals is investigated. With this analysis some information and hidden patterns which can not be detected in the time domain, may therefore be explained in the space-frequency domain.
Highlights
The theory of wavelets has been quite widespread, providing major advances in several different areas of the science
The Non-Decimated Wavelet Transform (NDWT) was applied to obtain the wavelet and scaling coefficients, given by 2.2, considering the wavelet family Symmlets with 10 vanishing moments (SYM10)
We propose in this paper to estimate this behavior, which is evidenced at smoother scales of the wavelet periodogram and subtract it from the time series (TS)
Summary
The theory of wavelets has been quite widespread, providing major advances in several different areas of the science. The DWT depends on the choice of downsampling, which corresponds to the choice of an origin in time or space [9] In this application, the multiscale analysis is applied to non-stationary time series (TS) from GPS satellite signals. Even if some Fourier methods could be applied to non-stationary TS ([8]; [10]), when some effects need to be investigated and separated in different scales at once, the wavelets methods became advantageous This is because the wavelets can deal naturally with the severe non-stationarity in time caused by spikes of different magnitudes, identifying their frequency of occurrence, localization in time, and making a reliable approximation of magnitude of this effects
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