Hello Ashish,

the WVD is a great method to perform time frequency (TF) analysis. It essentially computes the Fourier transform of the so called ambiguity function (AF) [AF(tau) = x(t+tau/2)x*(t-tau/2) ]. The AF is a general representation of the signal's autocorrelation function. The WVD does not suffer from leakage effects as the STFT does. Hence, the WVD gives you the best spectral resolution.

However, if you have signals with several frequency components, the WVD suffers from the so called cross terms. These cross terms can be (partly) supressed by extending the WVD - this results in the so called smoothed pseudo WVD (SPWVD). Hence, for most pracitcal applications you would (and should) use the SPWVD. There are good papers from Boualem Boashash and a nice book from Leon Cohen (Time frequency analysis) which cover this topic.

Best,

Johannes Krug

Hello Ashish,

the WVD is a great method to perform time frequency (TF) analysis. It essentially computes the Fourier transform of the so called ambiguity function (AF) [AF(tau) = x(t+tau/2)x*(t-tau/2) ]. The AF is a general representation of the signal's autocorrelation function. The WVD does not suffer from leakage effects as the STFT does. Hence, the WVD gives you the best spectral resolution.

However, if you have signals with several frequency components, the WVD suffers from the so called cross terms. These cross terms can be (partly) supressed by extending the WVD - this results in the so called smoothed pseudo WVD (SPWVD). Hence, for most pracitcal applications you would (and should) use the SPWVD. There are good papers from Boualem Boashash and a nice book from Leon Cohen (Time frequency analysis) which cover this topic.

Best,

Johannes Krug

Hello Ashish,

One possibility to avoid the cross terms is to use a modified WVD presented in the paper "Instantaneous frequency and Wigner-Ville Distribution using the Maximum Entropy Method.." proposed by Zoukaneri Ibrahim M and Milton J. Porsani.

DOI http://dx.doi.org/10.1190/segam2013-0919.1

Regards,

Milton Porsani

Hello Johannes Krug

Thanks for your reply.

What are the cross terms?

What is SPWVD?

What source should I follow to get about these terms.

Will you please help me to explain these terms by some example.

Thanks and regards

Ashish

Hello Milton Porsani

What are the cross terms.

I try to download this paper but not able to download. Please help me to download this paper.

Thanks and regards

Ashish

About the cross terms.

When we have a quadratic form (a+bx)^2 expressed in terms of a matricial representation, the terms that appear outside the main diagonal are the so called cross terms. For a given discrete signal there is a corresponding covariance matrix in which the collection of terms located along every cross diagonal represents the kernels of the Wigner-Ville Distribution. The Fourier Transform of every kernel corresponds to the instantaneous Power Spectrum of the signal, which is naturally affected not only by the events (a and b) but also by the cross terms.

Hello Milton Porsani

I really thankful to you for your valuable explanation.

Ashish

Beautiful answers. Les Atlas, Seho Oh, Yunxin Zhao, Robert Marks, published a number of papers from 1988 - 1991 which are worth looking at.

Time-varying spectra need to be analyzed by time-frequency distributions. Some good informations can be found in the monographic number of "IEEE Signal Processing Magazine" April 1992 on Time-frequency analysis.

Hi,

I found the papers from Boualem Boashash (1980's and 90's) very helpful. The book form Leon Cohen "Time frequency analysis" (1995) also covers the whole topic. You can get this cheap in India. Explaing the cross terms goes way too far ;)

Best, Johannes

When not to use WVD

(1)  Signal is multicomponent at a small instant of time (

Hi,

WVD (Weigner- Ville distribution ) is a great tool that provides excellent localisation in time and frequency , unlike short time Fourier transform, which can provide localisation in either time or frequency , not both simultaneously.

It basically does the correlation between the left and right folded parts of the signal to find the overlap between the past and future values of the signal.

However, it suffers from a major issue of interference which arises when the signal comprises of more than 1 component. This interference is due to the cross terms that appear while taking the auto-correlation function of the signal (say for f=f1+f2, auto-correlation function would be comprise of these 4 terms-f1f1, f1f2 ,f2f1 and f2f2). Hence modified WVD are used so as to suppress these cross terms (here f1f2 and f2f1).

WVD has some nice properties such as marginal properties, energy preserved and support properties, which are not there in case of STFT. But the issue is because of interference and also it doesn't guarantee the density to be positive.

Thanks.