%TFDEMO7 Extraction of information from the time-frequency plane%	Time-Frequency Toolbox demonstration.%%	See also TFDEMO.%	O. Lemoine - august 1996. %	Copyright (c) CNRS.clc; clf; zoom on; set(gca,'visible','off');echo on;% Information from the interferences%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%  The interference terms present in any quadratic time-frequency% representation, even if they disturb the readability of the% representation, contain some information about the analyzed% signal. The precise knowledge of their structure and construction rule% is useful to interpret the information that they contain. % %   For instance, the interference terms contain some information about the% phase of a signal. Let us consider the pseudo WVD of the superposition of% two constant frequency modulations, with a phase shift between the two% sinusoids. If we compare the pseudo WVD for different phase shifts, we can% observe a time-sliding of the oscillating interferences :pause;load movpwdphmovie(M,10);% Each snapshot corresponds to the pseudo WVD with a different phase% shift between the two components. % % Press any key to continue... pause; %   A second example of phase's signature is given by the influence of a% jump of phase in a signal analyzed by the (pseudo) Wigner-Ville% distribution : for instance, if we consider a constant frequency% modulation presenting a jump of phase in its middle : pause; load movpwjphmovie(M,10);% the pseudo WVD presents a pattern around the jump position which is% all the more important since this jump of phase is close to pi. This% characteristic can be exploited to detect a jump of phase in a signal.%% Press any key to continue... close; pause; clc;  % Renyi information%~~~~~~~~~~~~~~~~~~~%  Another interresting information that one may need to know about an% observed non stationary signal is the number of elementary signals% composing this observation. Third order Renyi information is a possible% solution for measuring this information. %  This can be observed by considering the WVD of one, two and then% four elementary atoms, and then by applying the Renyi information on% them :sig=atoms(128,[64,0.25,20,1]); [TFR,T,F]=tfrwv(sig);R1=renyi(TFR,T,F)pause;sig=atoms(128,[32,0.25,20,1;96,0.25,20,1]); [TFR,T,F]=tfrwv(sig);R2=renyi(TFR,T,F)pause;sig=atoms(128,[32,.15,20,1;96,.15,20,1;32,.35,20,1;96,.35,20,1]);  [TFR,T,F]=tfrwv(sig);R3=renyi(TFR,T,F)% We can see that if R is set to 0 for one elementary atom by% subtracting R1, we obtain a result close to 1 for 2 atoms% (R2-R1=0.99) and close to 2 for 4 atoms ( R3-R1=2.01). If the components % are less separated in the time-frequency plane, the information measure% will be affected by the overlapping of the components or by the % interference terms between them. In particular, it is possible to show that% the Renyi information measure provides a good indication of the time% separation at which the atoms are essentially resolved, more precise than% does the time-bandwidth product.%% Press any key to continue... pause; clc; clf;  % Time-frequency analysis : help to decision%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%  The Wigner-Hough transform% """"""""""""""""""""""""""""%  Applying the Hough transform (detection of lines in images) to the WVD% of a linear chirp signal leads to a new tranform, the Wigner-Hough transform,% which ideally concentrates this kind of signal in a parameter space. Then,% by comparing this 2-dimensional statistic to a threshold, we obtain the % asymptotically optimal detector (in the case of white gaussian noise). % Moreover, the coordinates of the peak detected give estimators of the chirp% parameters which asymptotically reach the Cramer-Rao lower bounds.   % %   Here is an illustration of this decision test : first, we consider% a linear chirp signal embedded in a white gaussian noise, with a 1 dB % signal-to-noise ratio :N=64; sig=sigmerge(fmlin(N,0,0.3),noisecg(N),1);% Now, if we analyze it with the WVD followed by the Hough transform,tfr=tfrwv(sig); contour(tfr,5); grid; htl(tfr,N,N,1);% we obtain, in the parameters' space (rho,theta), a peak representing% the chirp signal, significantly more energetic than the other peaks% corresponding to the noise. The decision test is then very simple : it% consists in applying a threshold on this representation, positioned% according to a detection criterion ; if the peak is higher than the% threshold, then the chirp is said to be present, and the coordinates of% that peak (hat{rho},hat{theta}) provide estimates of the chirp% parameters (the change from (hat{rho},hat{theta}) to% (hat{nu0},hat{beta}) corresponds to the change from polar to% Cartesian coordinates).%% Press any key to continue... pause; clc%   In the case of a multi-component signal, the problem of interference% terms appear. However, due to the oscillating structure of these% terms, the integration operated by the Hough transform on the WVD will % attenuate them. This can be observed on the following example :% we superpose two chirp signals with different initial frequencies and% sweep rates :sig=sigmerge(fmlin(N,0,0.4),fmlin(N,0.3,0.5),1);tfr=tfrwv(sig); contour(tfr,5); gridhtl(tfr,N,N,1);% We can see that the components are well separated in the parameter% space, in spite of the use of a nonlinearity in the WHT. Again, the% coordinates of the two peaks provide estimates of the different% parameters. %% Press any key to continue... pause; clc% Analysis of local singularities%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%  Since they are time-dependent in nature, the wavelet-based% techniques also allow an estimation of the local regularity of a% signal. % %   For instance, we consider a 64-points Lipschitz singularity of % strength H=0, centered at t0=32, sig=anasing(64);% and analyze it with the scalogram (Morlet wavelet with half-length=4),[tfr,t,f]=tfrscalo(sig,1:64,4,0.01,0.5,256,1);% The time-localization of the singularity can be clearly estimated from% the scalogram distribution at small scales :H=holder(tfr,f,1,256,32,1) % This value is a good estimation of H (=0 here)%% Press any key to continue... pause;% If we now consider a singularity of strength H=-0.5,sig=anasing(64,32,-0.5);[tfr,t,f]=tfrscalo(sig,1:64,4,0.01,0.5,256,1);% we notice the different behavior of the scalogram along scales, whose% decrease is characteristic of the strength H. The estimation of the% Holder exponent at t=32 gives :H=holder(tfr,f,1,256,32,1)% which is close to 0.5.pause; clf; clc; %-----------------------------------------------------------------------% Thank you for your attention. We hope that you enjoyed this % demonstration, and that your understanding in time-frequency analysis % has made some progress. Now have a nice time with the Time-Frequency% Toolbox.  %-----------------------------------------------------------------------echo off