FAST IMPLEMENTATIONS OF GENERALIZED DISCRETE TIME-FREQUENCY DISTRIBUTIONS

Cohen's class of time-frequency distributions (TFD's) have significant potential for the analysis of complex signals. In order to evaluate the TFD of a signal using its samples, discrete-time TFD's (DTFD's) have been defined as the Fourier transform of a smoothed discrete autocorrelation. Existing algorithms evaluate real-valued DTFD's using FFT's 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. We introduce two approaches for reducing the computation time of DTFD's. First, we define approximations to real-valued DTFD's, using spectrograms, that admit fast, space-saving evaluations. Frequency downsampling reduces the computation time of these approximations. Next, we define DTFD's that admit fast evaluations over sparse sets of time-frequency samples. A single STFT is calculated in order for DTFD time-frequency samples to be evaluated at an additional, fixed cost per sample.