Discrete Time-frequency Distributions Research Articles

Improved Discrete Definition of Quadratic Time-Frequency Distributions

Computation of a time-frequency distribution (TFD) requires a discrete version of the continuous distribution. This discrete TFD (DTFD) should be free from aliasing and conserve all the important mathematical properties of the continuous distribution. Existing DTFD definitions, however, poorly approximate this ideal. One popular definition, the generalized DTFD (GDTFD), is alias free but does not retain all the desirable properties from the continuous distribution. Another definition, the so-called alias-free GDTFD (AF-GDTFD), retains most properties yet is not always alias free. We propose a new DTFD definition, based on the GDTFD, that retains all desirable properties and is always alias free.

The Zak transform and decimated time-frequency distributions

In this paper, the interrelation between the Zak transform and the generalized discrete time-frequency distribution (GDTFD) is examined. Starting with the discrete Zak transform, its definition is broadened to include arbitrary windows creating the windowed Zak transform (WZT). The WZT is then combined with the spectrogram. It is demonstrated that the spectrogram based upon the WZT, called the Zak spectrogram (ZS), is a generalization of the standard spectrogram. Next, building upon the idea of the weighted spectrogram, the weighted ZS is used to produce a new class of GDTFD, called the decimated GDTFD (D-GDTFD). The decimated GDTFD is similar to the GDTFD, except it trades bandwidth for computational speed and requires significantly less storage in order to be implemented. The reduction in discrete bandwidth is from 2/spl pi/ for the GDTFD to 2/spl pi//m for the D-GDTFD. An important attribute of the DGDTFD is that it requires significantly less storage than the GDTFD. The D-GDTFD requires only 1/m/sup 2/ of the storage of the GDTFD. An example using the binomial distribution is given to illustrate the connection between the D-GDTFD and the GDTFD. Throughout the paper, examples of multirate systems are given, which could be used to implement the building blocks of the DGDTFD. This makes the D-GDTFD practical to implement using slower and less expensive elements.

The discrete-time phase derivative as a definition of discrete instantaneous frequency and its relation to discrete time-frequency distributions

We compare two definitions of the instantaneous frequency of a discrete signal: the two-point symmetric phase difference and the discrete-time phase derivative. The phase derivative definition avoids the pitfalls associated with the two-point definition and is equal to the first moment w.r.t. frequency of a properly defined, alias-free time-frequency distribution of the signal, which is consistent with the continuous-time case. >

Fast implementations of generalized discrete time-frequency distributions

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.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Computing time-frequency distributions (signal analysis)

Recently, numerous strategies have been proposed for computing discrete time-frequency distributions such as the Wigner distribution. The author describes an efficient and straightforward strategy for computing time-frequency distributions that are members of Cohen's class. The strategy is based on the work by A.H. Nuttall (Naval Underwater Syst. Center, Tech. Rep., 8533, April 1989) which has resolved the questions concerning aliasing and required sampling rate for the Wigner distribution.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>