function [tfr,t,f] = tfrcw(x,t,N,g,h,sigma,trace);%TFRCW	Choi-Williams time-frequency distribution.%	[TFR,T,F]=TFRCW(X,T,N,G,H,SIGMA,TRACE) computes the Choi-Williams  %	distribution of a discrete-time signal X, or the%	cross Choi-Williams representation between two signals. % %	X     : signal if auto-CW, or [X1,X2] if cross-CW.%	T     : time instant(s)          (default : 1:length(X)).%	N     : number of frequency bins (default : length(X)).%	G     : time smoothing window, G(0) being forced to 1. %	                                 (default : Hamming(N/10)).%	H     : frequency smoothing window, H(0) being forced to 1.%	                                 (default : Hamming(N/4)). %	SIGMA : kernel width             (default : 1)%	TRACE : if nonzero, the progression of the algorithm is shown%                                        (default : 0).%	TFR   : time-frequency representation. When called without %         output arguments, TFRCW runs TFRQVIEW.%	F     : vector of normalized frequencies.%%	Example:%	 sig=fmlin(128,0.05,0.3)+fmlin(128,0.15,0.4);  %	 g=window(9,'Kaiser'); h=window(27,'Kaiser'); %	 t=1:128; tfrcw(sig,t,128,g,h,3.6,1);% %	See also all the time-frequency representations listed in%	 the file CONTENTS (TFR*)%	F. Auger, May-August 1994, July 1995.%	Copyright (c) 1996 by CNRS (France).%%	------------------- CONFIDENTIAL PROGRAM -------------------- %	This program can not be used without the authorization of its%	author(s). For any comment or bug report, please send e-mail to %	f.auger@ieee.orgif (nargin == 0), error('At least 1 parameter required');end;[xrow,xcol] = size(x);if (xcol==0)|(xcol>2), error('X must have one or two columns');endif (nargin <= 2), N=xrow;elseif (N<0), error('N must be greater than zero');elseif (2^nextpow2(N)~=N), fprintf('For a faster computation, N should be a power of two\n');end;hlength=floor(N/4); hlength=hlength+1-rem(hlength,2); glength=floor(N/10);glength=glength+1-rem(glength,2);if (nargin == 1), t=1:xrow; g = window(glength); h = window(hlength); sigma = 1.0; trace = 0;elseif (nargin == 2)|(nargin == 3), g = window(glength); h = window(hlength); sigma = 1.0; trace = 0;elseif (nargin == 4), h = window(hlength); sigma = 1.0; trace = 0;elseif (nargin == 5), sigma = 1.0; trace = 0;elseif (nargin == 6), trace = 0;end;[trow,tcol] = size(t);if (trow~=1), error('T must only have one row'); end; [grow,gcol]=size(g); Lg=(grow-1)/2;if (gcol~=1)|(rem(grow,2)==0), error('G must be a smoothing window with odd length'); end;[hrow,hcol]=size(h); Lh=(hrow-1)/2; h=h/h(Lh+1);if (hcol~=1)|(rem(hrow,2)==0),  error('H must be a smoothing window with odd length');end;if (sigma<=0.0), error('SIGMA must be strictly positive'); end;normfac = 16.0*pi/sigma; spreadfac = 16.0/sigma;taumax = min([round(N/2),Lh]); tau = 1:taumax; points = -Lg:Lg;CWKer = exp(-kron( points.' .^2, 1.0 ./ (spreadfac*tau.^2)));CWKer = diag(g) * CWKer;tfr= zeros (N,tcol) ;  if trace, disp('Choi-Williams distribution'); end;for icol=1:tcol, ti= t(icol); taumax=min([ti+Lg-1,xrow-ti+Lg,round(N/2)-1,Lh]); if trace, disprog(icol,tcol,10); end; tfr(1,icol)= x(ti,1) .* conj(x(ti,xcol)); for tau=1:taumax,  points= -min([Lg,xrow-ti-tau]):min([Lg,ti-tau-1]);  g2 = CWKer(Lg+1+points,tau); g2=g2/sum(g2);  R=sum(g2 .* x(ti+tau-points,1) .* conj(x(ti-tau-points,xcol)));  tfr(  1+tau,icol)=h(Lh+tau+1)*R;  R=sum(g2 .* x(ti-tau-points,1) .* conj(x(ti+tau-points,xcol)));  tfr(N+1-tau,icol)=h(Lh-tau+1)*R; end; tau=round(N/2);  if (ti<=xrow-tau)&(ti>=tau+1)&(tau<=Lh),  points= -min([Lg,xrow-ti-tau]):min([Lg,ti-tau-1]);  g2 = CWKer(Lg+1+points,tau); g2=g2/sum(g2);  tfr(tau+1,icol) = 0.5 * ...   (h(Lh+tau+1)*sum(g2 .* x(ti+tau-points,1) .* conj(x(ti-tau-points,xcol)))+...    h(Lh-tau+1)*sum(g2 .* x(ti-tau-points,1) .* conj(x(ti+tau-points,xcol)))); end;end; clear CWKer;if trace, fprintf('\n'); end;tfr= fft(tfr); if (xcol==1), tfr=real(tfr); end ;if (nargout==0), tfrqview(tfr,x,t,'tfrcw',g,h,sigma);elseif (nargout==3), f=(0.5*(0:N-1)/N)';end;