A basis for efficient representation of the S-transform
The S-transform is a time-frequency representation known for its local spectral phase properties. A key feature of the S-transform is that it uniquely combines a frequency dependent resolution of the time-frequency space and absolutely referenced local phase information. This allows one to define the meaning of phase in a local spectrum setting, and results in many desirable characteristics. One drawback to the S-transform is the redundant representation of the time-frequency space and the consumption of computing resources this requires (a characteristic it shares with the continuous wavelet transform, the short time Fourier transform, and Cohen's class of generalized time-frequency distributions). The cost of this redundancy is amplified in multidimensional applications such as image analysis. A more efficient representation is introduced here as a orthogonal set of basis functions that localizes the spectrum and retains the advantageous phase properties of the S-transform. These basis functions are defined to have phase characteristics that are directly related to the phase of the Fourier transform spectrum, and are both compact in frequency and localized in time. Distinct from a wavelet approach, this approach allows one to directly collapse the orthogonal local spectral representation over time to the complex-valued Fourier transform spectrum. Because it maintains the phase properties of the S-transform, one can perform localized cross spectral analysis to measure phase shifts between each of multiple components of two time series as a function of both time and frequency. In addition, one can define a generalized instantaneous frequency (IF) applicable to broadband nonstationary signals. This is the first time a channel IF has been integrated in an orthogonal local spectral representation. A direct comparison between these basis functions and complex wavelets is performed, highlighting the advantages of this approach. The relationship between this basis set and the fully redundant S-transform is demonstrated highlighting the ability to arbitrarily sample the time-frequency space. The introduction of this basis set leads to efficient analysis routines that may find use in a wide range of fields.
s-transform wavelets time-frequency analysis