Fast implementations of generalized discrete time-frequencydistributions

Cohen's class of time-frequency distributions (TFDs) have significant potential for the analysis of complex signals. In order to evaluate the TFD of a signal using its samples, discrete-time TFDs (DTFDs) have been defined as the Fourier transform of a smoothed discrete autocorrelation. Existing algorithms evaluate real-valued DTFDs using FFTs of the conjugate-symmetric autocorrelation. Although the computation required to smooth the autocorrelation is often greater than that for the FFT, there are no widely applicable fast algorithms for this part of the processing. Since the FFT is relatively inexpensive, downsampling is ineffective for reducing computation. If the DTFD needs only to be evaluated at a few frequencies for each time instant, the cost per time-frequency sample can be extremely high. The authors introduce two approaches for reducing the computation time of DTFDs. First, they define approximations to real-valued DTFDs, using spectrograms, that admit fast, space-saving evaluations. Frequency downsampling reduces the computation time of these approximations. Next, they define DTFDs that admit fast evaluations over sparse sets of time-frequency samples. A single short time Fourier transform is calculated in order for DTPD time-frequency samples to be evaluated at an additional, fixed cost per sample