%TFDEMO4 Cohen's class time-frequency distributions.%	Time-Frequency Toolbox demonstration.%%	See also TFDEMO.%	O. Lemoine - May 1996. %	Copyright (c) CNRS.clc; zoom on; clf; echo on;% The Wigner-Ville distribution%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~% A time-frequency energy distribution which is particularly interesting % is the Wigner-Ville distribution (WVD),which satisfies a large number % of desirable mathematical properties. Let us see what we obtain on two % particular synthetic signals :% - the first signal is the academic linear chirp signal :sig=fmlin(128);% If we choose a 3-dimension plot to represent it, we can see that the WVD% can take negative values, and that the localization obtained in the% time-frequency plane for this signal is almost perfect.tfrwv(sig);% Press any key to continue... pause; clc;% - the second one illustrates the Doppler effect, which expresses the % dependence of the frequency received by an observer from a transmitter% on the relative speed between the observer and the transmitter : [fm,am,iflaw]=doppler(128,50,13,10,200);sig=am.*fm;tfrwv(sig);% Looking at this time-frequency distribution, we notice that the energy is% not distributed as we could expect for this signal. Although the signal% term is well localized in the time-frequency plane, numerous other terms% (the interference terms, due to the bilinearity of the WVD) are present at% positions in time and frequency where the energy should be null. %% Press any key to continue... pause; clc; close% Interference geometry of the WVD%""""""""""""""""""""""""""""""""""% The rule of interference construction of the WVD can be summarized as% follows : two points of the time-frequency plane interfere to create a% contribution on a third point which is located at their geometrical% midpoint. Besides, these interference terms oscillate perpendicularly to% the line joining the two points interfering, with a frequency% proportional to the distance between these two points.%  This can be seen on the following example : we consider two atoms in% the time-frequency plane, analyzed by the WVD, whose relative distance % is increasing from one realization to the other, and then decreasing. % The WVDs were calculated and saved on the file movieat.mat. We load them % and run the sequence using the function movie :load movwv2atpause;movie(M,5);% We can notice, from this movie, the evolution of the interferences% when the distance between the two interfering terms changes, and in% particular the change in the period and the direction of the oscillations.%% Press any key to continue... pause; clc; % The pseudo-WVD%""""""""""""""""%   As the analyzed signal is not known from -infinity to +infinity in% pratice, one often consider a windowed version of the WVD, called the% pseudo-WVD. The time-windowing operated has the effect of smoothing the% WVD in frequency. Thus, because of their oscillating nature, the % interferences will be attenuated in the pseudo-WVD compared to the WVD.% However, the consequence of this improved readability is that many % properties of the WVD are lost.% If we consider a signal composed of four gaussian atoms, each localized% at a corner of a rectangle,sig=atoms(128,[32,.15,20,1;96,.15,20,1;32,.35,20,1;96,.35,20,1]);% and compute its WVDtfrwv(sig);% we can see the four signal terms, along with six interference terms (two of% them are superimposed). If we now compute the pseudo-WVD,figuretfrpwv(sig);% we can note the important attenuation of the interferences oscillating% perpendicularly to the frequency axis, and in return the spreading in% frequency of the signal terms.%% Press any key to continue... pause; clc; close; % Importance of the analytic signal%"""""""""""""""""""""""""""""""""""%   Due to the quadratic nature of the WVD, its discrete version may be % affected by a spectral aliasing, in particular if the signal x is % real-valued and sampled at the Nyquist rate. A solution to this problem % consists in using the analytic signal. Indeed, as its bandwidth is half the% one of the real signal, the aliasing will not take place in the useful% spectral domain [0,1/2] of this signal. This solution presents a second% advantage : since the spectral domain is divided by two, the number of% components in the time-frequency plane is also divided by two. % Consequently, the number of interference terms decreases significantly. % Here is an illustration : we first consider the WVD of the real part of % a signal composed of two atoms :sig=atoms(128,[32,0.15,20,1;96,0.32,20,1]);tfrwv(real(sig));% We can see that four signal terms are present instead of two, due to the% spectral aliasing. Besides, because of the components located at negative% frequencies (between -1/2 and 0), additional interference terms are% present. %% Press any key to continue...pause; % If we now consider the WVD of the same signal, but in its complex% analytic form,tfrwv(sig);% the aliasing effect has disappeared, as well as the terms corresponding to% interferences between negative- and positive- frequency components.%% Press any key to continue...pause; clc; % The Cohen's class%~~~~~~~~~~~~~~~~~~~%  The Cohen's class gather all the time-frequency energy distributions which% are covariant by translations in time and in frequency. It can be expressed% as a 2-D correlation between a function PI(t,nu) and the WVD of the % analyzed signal. The WVD is the element of the Cohen's class for which PI % is a double Dirac, and the spectrogram is th element for which PI is the % WVD of the short time window h. We consider in the following other elements% of this class% % The smoothed pseudo-WVD%"""""""""""""""""""""""""% If we consider a separable smoothing function PI(t,nu)=g(t)H(-nu) (where % H(nu) is the Fourier transform of a smoothing window h(t)), we allow a % progressive and independent control, in both time and frequency, of the % smoothing applied to the WVD. The obtained distribution is known as the % smoothed-pseudo WVD. % For example,let's consider a signal composed of two components : the first % one is a complex sinusoid (normalized frequency 0.15) and the second one % is a Gaussian signal shifted in time and frequency :  sig=sigmerge(fmconst(128,.15),amgauss(128).*fmconst(128,0.4),5); % If we display the WVD, the pseudo-WV and the smoothed-pseudo-WVD of it,tfrwv(sig);  figure; tfrpwv(sig); figure; tfrspwv(sig);% we can make the following remarks : from the WVD, we can see the two% signal terms located at the right positions in the time-frequency plane,% as well as the interference terms between them. As these interference % terms oscillate globally perpendicularly to the time-axis, the frequency% smoothing done by the pseudo-WVD degrades the frequency resolution without% really attenuating the interferences. On the other hand, the time-smoothing% carried out by the smoothed-pseudo-WVD considerably reduces these% interferences.%% Press any key to continue...pause; clc; close; close; close%   An interresting property of the smoothed-pseudo WVD is that it allows a% continuous passage from the spectrogram to the WVD, under the condition% that the smoothing functions g and h are gaussian. This is clearly % illustrated by the function movsp2wv.m, which considers different % transitions, on a signal composed of four atoms :	load movsp2wvpausemovie(M,5);% This movie shows the effect of a (time/frequency) smoothing on the% interferences and on the resolutions : the WVD gives the best resolutions% (in time and in frequency), but presents the most important interferences,% whereas the spectrogram gives the worst resolutions, but with nearly no% interferences ; and the smoothed-pseudo WVD allows to choose the best% compromise between these two extremes.%% Press any key to continue...pause; clc; % The narrow-band ambiguity function%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%  The narrow-band ambiguity function, often used in radar signal processing,% is the two-dimensional Fourier transform of the WVD. This property can be % used to attenuate some of the interference terms. Indeed, in the case of % a multi-component signal, the elements of the AF corresponding to the % signal components (denoted as the AF-signal terms) are mainly located % around the origin, whereas the elements corresponding to interferences % between the signal components (AF-interference terms) appear at a distance% from the origin which is proportional to the time-frequency distance% between the involved components. This can be noticed on a simple example :% We apply consider a signal composed of two linear FM signals with % gaussian amplitudes :N=64; sig1=fmlin(N,0.2,0.5).*amgauss(N);sig2=fmlin(N,0.3,0).*amgauss(N);sig=[sig1;sig2]; % Let us first have a look at the WVD :tfrwv(sig);% We have two distinct signal terms, and some interferences oscillating in% the middle. %% Press any key to continue...pause;% If we look at the ambiguity function of this signal,clf; ambifunb(sig);% we have around the origin (in the middle of the image) the AF-signal % terms, whereas the AF-interference terms are located away from the origin.%% Press any key to continue...pause; clc;% Other important energy distributions%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%% The Rihaczek and Margenau-Hill distributions% """""""""""""""""""""""""""""""""""""""""""""%  The Rihaczek distribution, defined as%	Rx(t,nu)=x(t)X*(nu)e^{-j2pi nu t},% is a complex energy density at point (t,nu). This distribution, which % corresponds to the element of the Cohen's class for which % f(xi,tau)=e^{jpi xi tau}, verifies many good properties. However, it is % complex valued, which can be awkward in practice. The real part of the % Rihaczek distribution is also a time-frequency distribution of the % Cohen's class (f(xi,tau)=cos(pi xi tau)), known as the Margenau-Hill% distribution%   The interference structure of the Rihaczek and Margenau-Hill% distributions is different from the Wigner-Ville one : the interference% terms corresponding to two points located on (t1,nu1) and (t2,nu2) are% positioned at the coordinates (t1,nu2) and (t2,nu1). This can be seen on% the following example :sig=atoms(128,[32,0.15,20,1;96,0.32,20,1]);tfrmh(sig);% Thus, the use of the Rihaczek (or Margenau-Hill) distribution for signals% composed of multi-components located at the same position in time or in% frequency is no advised, since the interference terms will then be% superposed to the signal terms.%% Press any key to continue...pause; clc; close% The Choi-Williams distribution% """""""""""""""""""""""""""""""%  An example of reduced interference distribution is given by the% Choi-Williams distribution, defined as% CWx(t,nu)=sqrt(2/pi)\int\int sigma/|tau| e^{-2sigma^2(s-t)^2/tau^2} %	    x(s+tau/2)x*(s-tau/2) e^{-j2pi nu tau} ds dtau % Note that when sigma->+infty, we obtain the WVD. Inversely, the smaller% sigma, the better the reduction of the interferences. %   The "cross"-shape of the parametrization function of the Choi-Williams% distribution implies that the efficiency of this distribution strongly% depends on the nature of the analyzed signal. For instance, if the signal% is composed of synchronized components in time or in frequency, the% Choi-Williams distribution will present strong interferences. This can be% observed on the following example : we analyze four gaussian atoms% positionned at the corners of a rectangle rotating around the center of % the time-frequency plane : load movcw4atpausemovie(M,5);% When the time/frequency supports of the atoms overlap, some % AF-interference terms are not be completly attenuated (those present % around the axes of the ambiguity plane), and the efficiency of the % distribution is quite poor. %% Press any key to continue...pause; clc;% Comparison of the parametrization functions% """"""""""""""""""""""""""""""""""""""""""""%  To illustrate the differences between some of the presented% distributions, we represent their weighting (parametrization) function in% the ambiguity plane, along with the result obtained by applying them on a% two-component signal embedded in white gaussian noise : the signal is the% sum of two linear FM signals, the first one with a frequency going from% 0.05 to 0.15, and the second one from 0.2 to 0.5. The signal to noise % ratio is 10 dB.% On the left-hand side of the figures, the parametrization functions are% represented in a schematic way by the bold contour lines (the weighting% functions are mainly non-zeros inside these lines), superimposed to the% ambiguity function of the signal. The AF-signal terms are in the middle of% the ambiguity plane, whereas the AF-interference terms are distant from the% center. On the right-hand side, the corresponding time-frequency% distributions are represented.paramfun% From these plots, we can conclude that the ambiguity plane is very% enlightening with regard to interference reduction in the case of% multicomponent signals. On this example, we notice that the% smoothed-pseudo-WVD is a particularly convenient and versatile% candidate. This is due to the fact that we can adapt independently the% time-width and frequency-width of its weighting function. But in the% general case, it is interesting to have several distributions at our% disposal since each one is well adapted to a certain type of% signal. Besides, for a given signal, as a result of the different% interference geometries, these distributions offer complementary% descriptions of this signal.%% Press any key to end this demonstrationpause; echo offclose; clc; clf