Time–frequency and time–time filtering with the S-transform and TT-transform

The limitations of frequency-domain filtering methods have motivated the development of alternative techniques, in which a filter is applied to a time–frequency distribution instead of the Fourier spectrum. One such distribution is the S-transform, a modified short-time Fourier transform whose window scales with frequency, as in wavelets. Recently it has been shown that the S-transform's local spectra have time-domain equivalents. Since each of these is associated with a particular window position on the time axis, collectively they give a time–time distribution. This distribution, called the TT-transform, exhibits differential concentration of different frequency components, with higher frequencies being more strongly concentrated around the localization position than lower frequencies. This leads to the idea of filtering on the time–time plane, in addition to the time–frequency plane. Examples of time–frequency filtering and time–time filtering are presented.