%TFDEMO6 Reassigned time-frequency distributions.%	Time-Frequency Toolbox demonstration.%%	See also TFDEMO.%	O. Lemoine - August 1996. %	Copyright (c) CNRS.clc; zoom on; clf; echo on; % The reassignment of the spectrogram%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%   The reassignment method, originally introduced in an attempt to% improve the spectrogram, moves each value of the spectrogram computed % at any point (t,nu) to another point which is the center of gravity % of the signal energy distribution around (t,nu). %   Let us have a look at the readability improvement obtained by the % reassigned spectrogram on an example of multicomponent signal ; the % result is compared to the "ideal" representation based on the knowledge % of the instantaneous frequency law of each component :N=128; [sig1 ifl1]=fmsin(N,0.15,0.45,100,1,0.4,-1);[sig2 ifl2]=fmhyp(N,[1 .5],[32 0.05]);sig=sig1+sig2;tfrideal([ifl1 ifl2]);figure; tfrrsp(sig);% The improvement given by the reassignment method is obvious : compared to% the spectrogram, the two components obtained with the reassigned spectrogram % are much better localized and almost perfectly concentrated, and there are % very few cross-terms.%% Press any key to continue... pause; clc; close;% Reassignment of other quadratic time-frequency distributions%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%   The reassignment method has been extended to many other time-frequency% distributions, and in particular to the distributions of the Cohen's class% and of the affine class. We present here some examples, on a signal composed% of 3 components : a sinusoidal frequency modulation followed by a pure tone% together with a linear chirp : [sig1 ifl1]=fmsin(60,0.15,0.35,50,1,0.35,1);[sig2 ifl2]=fmlin(60,0.3,0.1);[sig3 ifl3]=fmconst(60,0.4);sig=[sig1;zeros(8,1);sig2+sig3];iflaw=zeros(128,2);iflaw(:,1)=[ifl1;NaN*ones(8,1);ifl2];iflaw(:,2)=[NaN*ones(68,1);ifl3];% We first plot the instantaneous frequency laws (obtained by tfrideal) % and the WVD of this signal :tfrideal(iflaw);figure; tfrwv(sig);% With the WVD, the signal components are well localized, but the numerous% cross-terms make the figure hardly readable. %% Press any key to continue... pause; clc; % If we now consider the smoothed pseudo-WVD and its reassigned version tfrrspwv(sig);% we can see that the smoothing done by the SPWVD almost completly suppress% the cross terms, but the signal components localization becomes% coarser. The improvement given by the reassignment method is obvious : all% components are much better localized, leading to a nearly ideal% representation.%% Press any key to continue... pause; clc; % The next distributions we consider are the scalogram and the Morlet % scalogram figure(1); tfrrsp(sig);figure(2); tfrrmsc(sig);% These two distributions present nearly no cross terms, except at% the bottom of the sinusoid and around time t=64. But the time and% frequency resolutions are not good, especially at low frequencies in the% case of the scalogram. The reassignment method improves considerably these% localizations, and the reassigned spectrogram is even perfectly% concentrated for the chirp components. The result obtained with the% modified scalogram is less good, especially at low frequencies where the% time-resolution is really inadequate. %% Press any key to continue... pause; clc; % Finally, we represent the pseudo-Page and the pseudo Margenau-Hill% distributions with their reassigned version :  figure(1); tfrrppag(sig);figure(2); tfrrpmh(sig);% These representations (before reassignment) are hardly readable since some% cross-terms are superimposed on the signal components. Their modified% versions give much better localized signal components, but less% concentrated than in the case of the spectrogram or the SPWVD.%% Press any key to continue...pause; clc; close% Connected approaches%~~~~~~~~~~~~~~~~~~~~~~%  Connections of the reassignment method has been found with other% techniques which extract relevant informations from the time-frequency% plane. %% Friedman's instantaneous frequency density%""""""""""""""""""""""""""""""""""""""""""""%  A first example is the instantaneous frequency density : so as to take% advantage of the phase structure of the short-time Fourier transform% (STFT), Friedman simply computed at each time t the histogram of the% frequency displacements of the spectrogram. The resulting time-frequency% representation is no more an energy distribution, and could be derived % as well from any other reassigned distribution.%  Here is an example of this instantaneous frequency density obtained with% the pseudo-WVD of the previous signalt=1:2:127; [tfr,rtfr,hat]=tfrrpwv(sig,t);friedman(tfr,hat,t,'tfrrpwv',1);% Although some cross terms are still present, the localization of the% components is quite good, especially for the chirp components.%% Press any key to continue...pause; clc; % Extraction of ridges and skeleton%"""""""""""""""""""""""""""""""""""%  This method extracts from any reassigned time-frequency distribution% (given its displacement operators) some particular sets of curves % deduced from the stationary points of their phase.%  For example, let us extract the ridges from the spectrogram of the previous% signal :[tfr,rtfr,hat]=tfrrsp(sig); ridges(tfr,hat);% The result is interesting : apart from some ``gaps'' present in particular% on the sinusoidal frequency modulation, this method concentrates and% localizes nearly ideally the signal in the time-frequency plane, even when% there are two components present at the same time (or at the same% frequency).echo off