A review and tutorial of the fundamental ideas and methods of joint time-frequency distributions is presented. The objective of the field is to describe how the spectral content of a signal changes in time and to develop the physical and mathematical ideas needed to understand what a time-varying spectrum is. The basic gal is to devise a distribution that represents the energy or intensity of a signal simultaneously in time and frequency. Although the basic notions have been developing steadily over the last 40 years, there have recently been significant advances. This review is intended to be understandable to the nonspecialist with emphasis on the diversity of concepts and motivations that have gone into the formation of the field. >

/pdf/time-frequency-distributions-a-review-3lxbcw4fg4.pdf

Time-frequency distributions-a review

A tutorial review of both linear and quadratic representations is given. The linear representations discussed are the short-time Fourier transform and the wavelet transform. The discussion of quadratic representations concentrates on the Wigner distribution, the ambiguity function, smoothed versions of the Wigner distribution, and various classes of quadratic time-frequency representations. Examples of the application of these representations to typical problems encountered in time-varying signal processing are provided. >

Linear and quadratic time-frequency signal representations

A method for time-frequency signal analysis is presented. The proposed method belongs to the general class of smoothed pseudo Wigner distributions. It is derived from the analysis of the Wigner distribution defined in the frequency domain. This method provides some substantial advantages over the Wigner distribution. The well-known cross term effects are reduced or completely removed. The oversampling of signal is not necessary. In addition, the computation time can be significantly shorter. The results are demonstrated on two numerical examples with frequency modulated signals. >

A method for time-frequency analysis

It is shown that a correct use of the Wigner distribution (WD) for time-frequency signal analysis requires use of the analytic signal. This version, often referred to as the Wigner-Ville distribution (WVD), is straightforward to compute, does not exhibit any aliasing problem, and introduces no frequency artifacts. The problems introduced by the use of the Wigner distribution with a real signal are clarified. >

/pdf/note-on-the-use-of-the-wigner-distribution-for-time-2czms2k7oc.pdf

Note on the use of the Wigner distribution for time-frequency signal analysis

Turbulence generated by vertical oscillations of a horizontal monoplanar grid in homogeneous water contained in a tank was used to study certain properties of nearly isotropic turbulence. The cases of sustained oscillations and the removal of forcing after a period of oscillations were considered. The former was used to evaluate the Eulerian‐frequency spectra of nearly isotropic turbulence, and the velocity–time power law and spectral behavior of decaying turbulence were studied using the latter. The experimental observations are compared with available theoretical formulations as well as previous experimental observations.

Oscillating grids as a source of nearly isotropic turbulence

The Wigner-Ville distribution (WVD) is a valuable tool for time-frequency signal analysis. In order to implement the WVD in real time, an efficient algorithm and architecture have been developed which may be implemented with commercial components. This algorithm successively computes the analytic signal corresponding to the input signal, forms a weighted kernel function, and analyzes the kernel via a discrete Fourier transform (DFT). To evaluate the analytic signal required by the algorithm, it is shown that the time domain definition implemented as a finite impulse response (FIR) filter is practical and more efficient than the frequency domain definition of the analytic signal. The windowed resolution of the WVD in the frequency domain is shown to be similar to the resolution of a windowed Fourier transform. A real-time signal processor has been designed for evalution of the WVD analysis system. The system is easily paralleled and can be configured to meet a variety of frequency and time resolutions. The arithmetic unit is based on a pair of high-speed VLSI floating-point multiplier and adder chips.

P. Black

Papers

An efficient real-time implementation of the Wigner-Ville distribution