%TFDEMO5 Affine class time-frequency distributions.%	Time-Frequency Toolbox demonstration.%%	See also TFDEMO.%	O. Lemoine - July 1996. %	Copyright (c) CNRS.clc; zoom on; clf; echo on;% The Affine class : presentation%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~% This class gathers all the quadratic time-frequency representations % which are covariant by translation in time and dilation. The WVD is% an element of the affine class, provided that we introduce an % arbitrary non-zero frequency nu0, and identify the scale with the % inverse of the frequency : a=nu0/nu.% The choice of an element in the affine class can be reduced to the % choice of an affine correlation kernel PI(t,nu). When PI is a % two-dimensional low-pass function, it plays the role of an affine% smoothing function which tries to reduce the interferences generated % by the WVD.%% The scalogram %"""""""""""""""%  A first example of affine distribution is given by the scalogram,% which is the squared modulus of the wavelet transform. It is the affine% counterpart of the spectrogram. As illustrated in the following example,% the tradeoff between time and frequency resolutions encountered with the% spectrogram is also present with the scalogram.%  We analyze a signal composed of two gaussian atoms, one with a low % central frequency, and the other with a high one, with the scalogram % (Morlet wavelet) :sig=atoms(128,[38,0.1,32,1;96,0.35,32,1]);clf; tfrscalo(sig);% The result obtained brings to the fore dependency, with regard to the % frequency, of the smoothing applied to the WVD, and consequently of the% resolutions in time and frequency.%% Press any key to continue... pause; clc; clf; set(gca,'visible','off'); % The affine smoothed pseudo Wigner distribution (ASPWVD)%"""""""""""""""""""""""""""""""""""""""""""""""""""""""""%  One way to overcome the tradeoff between time and frequency resolutions% of the scalogram is, as for the smoothed-pseudo-WVD, to use a smoothing% function which is separable in time and frequency. The resulting% distribution is called the affine smoothed pseudo WVD. It allows a % flexible choice of time and scale resolutions in an independent manner % through the choice of two windows g and h. %%  As for the SPWVD, the ASPWVD allows a continuous passage from the % scalogram to the WVD, under the condition that the smoothing functions % g and h are gaussian. The time-bandwidth product then goes from 1 % (scalogram) to 0 (WVD), with an independent control of the time and % frequency resolutions. This is illustrated in the following example :	load movsc2wvpausemovie(M,5);% Here again, the WVD gives the best resolutions (in time and in frequency),% but presents the most important interferences, whereas the scalogram gives% the worst resolutions, but with nearly no interferences ; and the affine% smoothed-pseudo WVD allows to choose the best compromise between these two% extremes.%% Press any key to continue... pause; clc; close% The localized bi-frequency kernel (or affine Wigner) distributions%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%  A useful subclass of the affine class consists in characterization% functions which are perfectly localized on power laws or logarithmic laws% in their bi-frequency representation. The corresponding time-scale % distributions are known as the localized bi-frequency kernel distributions.% % The Bertrand distribution%"""""""""""""""""""""""""""%  If we further impose to these distributions the a priori requirements of% time localization and unitarity, we obtain the Bertrand distribution. This% distribution satisfies many properties, and is the only localized% bi-frequency kernel distribution which localizes perfectly the hyperbolic% group delay signals. To illustrate this property, consider the signal % obtained using the file gdpower.m (taken for k=0), and analyze it with % the file tfrbert.m :sig=gdpower(128);tfrbert(sig,1:128,0.01,0.22,128,1);% Note that the distribution obtained is well localized on the hyperbolic% group delay, but not perfectly : this comes from the fact that the file% tfrbert.m works only on a subpart of the spectrum, between two bounds fmin% and fmax.%% Press any key to continue... pause; clc;% The D-Flandrin distribution %"""""""""""""""""""""""""""""%  If we now look for a localized bi-frequency kernel distribution which is% real, localized in time and which validates the time-marginal property, % we obtain the D-Flandrin distribution. It is the only localized % bi-frequency kernel distribution which localizes perfectly signals having % a group delay in 1/sqrt(nu). This can be illustrated as following :sig=gdpower(128,1/2);tfrdfla(sig,1:128,0.01,0.22,128,1);% Here again, the distribution is almost perfectly localized.%% Press any key to continue... pause; clc;% The active Unterberger distribution%"""""""""""""""""""""""""""""""""""""%  Finally, the only localized bi-frequency kernel distribution which% localizes perfectly signals having a group delay in 1/nu^2 is the active% Unterberger distribution :sig=gdpower(128,-1);tfrunter(sig,1:128,'A',0.01,0.22,172,1);% Press any key to continue... pause; clc;% Relation with the ambiguity domain%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%  When the signal under analysis can not be considered as narrow-band% (i.e. when its bandwidth B is not negligible compared to its central% frequency nu0), the narrow-band ambiguity function is no longer appropriate% since the Doppler effect can not be approximated as a frequency-shift. We% then consider a wide-band ambiguity function (WAF). It corresponds to % the wavelet transform of the signal x, whose mother wavelet is the signal% x itself. It is then an affine correlation function, which measure the % similarity between the signal and its translated (in time) and dilated % versions. To see how it behaves on a practical example, let us consider an% Altes signal :	sig=altes(128,0.1,0.45);clf; ambifuwb(sig);% The WAF is maximum at the origin of the ambiguity plane.  %% Press any key to continue... pause; clc  % Interference structure%~~~~~~~~~~~~~~~~~~~~~~~~%  The interference structure of the localized bi-frequency kernel % distributions can be determined thanks to the following geometric % argument : two points (t1,nu1) and (t2,nu2) belonging to the trajectory % on which a distribution is localized interfere on a third point % (ti,nui) which is necessarily located on the same trajectory.%  To illustrate this interference geometry, let us consider the case of a% signal with a sinusoidal frequency modulation :[sig,ifl]=fmsin(128);% The file plotsid.m allows one to construct the interferences of an affine% Wigner distribution perfectly localized on a power-law group-delay% (specifying k), for a given instantaneous frequency law (or the% superposition of different instantaneous frequency laws). For example, if% we consider the case of the Bertrand distribution (k=0),plotsid(1:128,ifl,0);% we obtain an interference structure completely different from the one% obtained for the Wigner-Ville distribution (k=2) :%% press any key to continue... pause;plotsid(1:128,ifl,2);% For the active Unterberger distribution (k=-1), the result is the% following : %% press any key to continue... pause;plotsid(1:128,ifl,-1); % Press any key to continue... pause; clc% The pseudo affine Wigner distributions%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%   The affine Wigner distributions show great potential as flexible% tools for time-varying spectral analysis. However, as some distributions of% the Cohen's class, they present two major practical limitations : first the% entire signal enters into the calculation of these distributions at every% point (t,nu), and second, due to their nonlinearity, interference% components arise between each pair of signal components. To overcome these% limitations, a set of (smoothed) pseudo affine Wigner distributions has% been introduced.%  Here are two examples of such distributions, analyzed on a real % echolocation signal from a bat :load bat; N=128;sig=hilbert(bat(801:7:800+N*7)');% The affine smoothed pseudo Wigner distribution %------------------------------------------------figure(1); tfrwv(sig); figure(2); tfrspaw(sig,1:N,2,24,0,0.1,0.4,N,1); % On the left, the WVD presents interference terms because of the% non-linearity of the frequency modulation. On the right, the affine% frequency smoothing operated by the affine smoothed pseudo Wigner% distribution almost perfectly suppressed the interference terms.%% Press any key to continue... pause; clc% The pseudo Bertrand distribution%----------------------------------figure(1); tfrbert(sig,1:N,0.1,0.4,N,1);figure(2); tfrspaw(sig,1:N,0,32,0,0.1,0.4,N,1); % The first plot represents the Bertrand distribution. The approximate% hyperbolic group delay law of the bat signal explains the good result% obtained with this distribution (compared to the WVD). However, it% remains some interference terms, which are almost perfectly canceled% on the second plot (pseudo Bertrand distribution).%% Press any key to end this demonstrationpause; close;echo off