| The S-transform is a spectral localization technique which combines elements of the short-time Fourier Transform (STFT) and continuous wavelet transforms (CWT's). The analyzing window of the S-transform is a scaled Gaussian, whose width varies as the inverse of the frequency. This wavelet-like scaling improves the time resolution of high-frequency events, and the frequency resolution of low-frequency events, in comparison with the STFT, while maintaining the absolute phase of each frequency component in contrast with CWT's. Although the scaled Gaussian window has many desirable properties, it is not ideal for every application. This leads to the generalized S-transform, which allows windows with frequency dependent asymmetry, and widths which do not vary exactly inversely with frequency. Several useful examples of windows of this type are presented, for use with one-dimensional time series and two-dimensional images.; At any particular window position, the S-transform describes the local frequency content of the time series. This leads to the concept of a time equivalent of the S-transform. The result is the TT-transform, a time-time distribution whose horizontal time axis denotes the position of the window, and whose vertical axis denotes the time axis of the time series. Columns of the TT-transform resemble windowed time series, except that the higher frequencies tend to cluster more tightly around the window position than the lower frequencies. We present examples of TT-transforms obtained using synthetic time series and synthetic images, and demonstrate that the TT-transform may be used for time-domain filtering. |
