Time-Frequency Spectra with the S-transform
Lalu Mansinha - Watch Now - Duration: 46:43
Most interesting data series consist of signal and noise series that are usually non-stationary, i.e. the properties fluctuate with time.Fourier analysis (FT) of the whole time series provides the spectrum of the whole time series, but is not capable of showing the time variation of the spectrum.The S-transform, also known as the Stockwell Transform (ST), uses a scalable, translating Gaussian window to determine the local spectrum at every point on the time series.The local spectrum supplements the local temporal information in the time series, and aids in the detection of onsets and cessations of events.For a N point time series, the output of the FT is a N point complex spectrum; the output of ST is a N x N, 2D time-frequency matrix, giving a N point spectrum at every point on the time series.Since the original time series contains only N points, the additional N2 - N points is a measure of the non-independent, redundant information computed for the ST.The redundancy is useful in presenting the similarities in the neighbouring local spectra and contributes to the visual continuity and smoothness to the 2-D time-frequency spectrum. The computation and storage of N2 - N additional points is a major drag on the usage of ST. Several approaches to reducing the computational burden are presented.Local spectra aids in analysis of 1D data. In images, the 2D local spectra aids in definition of texture and image segmentation.For 3 and 4 colour images, Trinion ST and Quaternion ST have been defined.In use since 1997, ST has found applications in numerous disciplines, including medical data series and images; power quality disturbance; atmospheric physics; exploration geophysics etc.The ST has been implemented in ARM and Raspberry Pi processors by several researchers.
This presentation is introductory, for the interested practitioner.Mathematical content will be at the absolute minimum.