The Bi-Gaussian S-Transform

Abstract

The S-transform kernel is derived from the kernel of the Fourier transform through the introduction of a scalable, translating window. The width of the window is a function of inverse frequency. In effect the S-transform is a method of spectral localization, with some similarities to wavelet transforms but using the concept of frequency rather than the concept of scale. An important property of the S-transform is that it collapses into the Fourier transform when integrated over the time axis; this property requires the window to satisfy a normalizing condition. The window which has been used in the majority of previous S-transform research is the symmetric Gaussian window introduced by Stockwell, Mansinha, and Lowe [IEEE Trans. Signal Process., 44 (1996), pp. 998--1001]. One problem with the use of a Gaussian, however, is degradation of time resolution in the time-frequency spectrum due to the long front taper. In this paper, a bi-Gaussian window is introduced, constructed through the welding of two half Gaussian windows. The asymmetry of the bi-Gaussian introduces asymmetry in the resultant time-frequency spectrum, with time resolution better in the "front" direction, as compared with the "back" direction. The bi-Gaussian S-transform is better at resolving the sharp onset of events in a time series.