Time-local Fourier analysis with a scalable, phase-modulated analyzing function: the S-transform with a complex window

Authors:
C. R. Pinnegar;L. Mansinha
Affiliations:
Department of Earth Sciences, University of Western Ontario London, Ont., Canada N6A 5B7;Department of Earth Sciences, University of Western Ontario London, Ont., Canada N6A 5B7
Venue:
Signal Processing
Year:
2004

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Abstract

The S-transform was originally defined as a method of determining the local spectrum of a time series, through the use of a translating, real Gaussian window that dilates to accomodate the different cycle durations of different frequencies. The S-transform "wavelet" is obtained by multiplying this real window with the complex Fourier sinusoid. Since the Fourier sinusoid has time-invariant frequency, the S-transform is consequently unsuitable for resolving waveforms whose frequency changes with time. This problem can be addressed by introducing a complex Gaussian window, with a user designed, complex phase function. The phase function modulates the frequency of the Fourier sinusoid to give a specific waveform, leading to better time frequency localization of similar waveforms on the time series. The complex-window S-transform is similar to a wavelet transform, but has the fixed phase reference of the Fourier transform.