function [A,S]=jade(X,m)
% Source separation of complex signals with JADE.
% Jade performs `Source Separation' in the following sense:
% X is an n x T data matrix assumed modelled as X = A S + N where
%
% o A is an unknown n x m matrix with full rank.
% o S is a m x T data matrix (source signals) with the properties
% a) for each t, the components of S(:,t) are statistically
% independent
% b) for each p, the S(p,:) is the realization of a zero-mean
% `source signal'.
% c) At most one of these processes has a vanishing 4th-order
% cumulant.
% o N is a n x T matrix. It is a realization of a spatially white
% Gaussian noise, i.e. Cov(X) = sigma*eye(n) with unknown variance
% sigma. This is probably better than no modeling at all...
%
% Jade performs source separation via a
% Joint Approximate Diagonalization of Eigen-matrices.
%
% THIS VERSION ASSUMES ZERO-MEAN SIGNALS
%
% Input :
% * X: Each column of X is a sample from the n sensors
% * m: m is an optional argument for the number of sources.
% If ommited, JADE assumes as many sources as sensors.
%
% Output :
% * A is an n x m estimate of the mixing matrix
% * S is an m x T naive (ie pinv(A)*X) estimate of the source signals
%
%
% Version 1.5. Copyright: JF Cardoso.
%
% Last updated: Joachim Behar - 12/05/2013
%
% See notes, references and revision history at the bottom of this file
if size(X,1)>size(X,2)
X=X';
end
[n,T] = size(X);
%% source detection not implemented yet !
if nargin==1, m=n ; end;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% A few parameters that could be adjusted
nem = m; % number of eigen-matrices to be diagonalized
seuil = 1/sqrt(T)/100;% a statistical threshold for stopping joint diag
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% whitening
%
if m<n, %assumes white noise
[U,D] = eig((X*X')/T);
[puiss,k]=sort(diag(D));
ibl = sqrt(puiss(n-m+1:n)-mean(puiss(1:n-m)));
bl = ones(m,1) ./ ibl ;
W = diag(bl)*U(1:n,k(n-m+1:n))';
IW = U(1:n,k(n-m+1:n))*diag(ibl);
else %assumes no noise
IW = sqrtm((X*X')/T);
W = inv(IW);
end;
Y = W*X;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Cumulant estimation
R = (Y*Y' )/T ;
C = (Y*Y.')/T ;
Yl = zeros(1,T);
Ykl = zeros(1,T);
Yjkl = zeros(1,T);
Q = zeros(m*m*m*m,1) ;
index = 1;
for lx = 1:m ; Yl = Y(lx,:);
for kx = 1:m ; Ykl = Yl.*conj(Y(kx,:));
for jx = 1:m ; Yjkl = Ykl.*conj(Y(jx,:));
for ix = 1:m ;
Q(index) = ...
(Yjkl * Y(ix,:).')/T - R(ix,jx)*R(lx,kx) - R(ix,kx)*R(lx,jx) - C(ix,lx)*conj(C(jx,kx)) ;
index = index + 1 ;
end ;
end ;
end ;
end
%% If you prefer to use more memory and less CPU, you may prefer this
%% code (due to J. Galy of ENSICA) for the estimation the cumulants
%ones_m = ones(m,1) ;
%T1 = kron(ones_m,Y);
%T2 = kron(Y,ones_m);
%TT = (T1.* conj(T2)) ;
%TS = (T1 * T2.')/T ;
%R = (Y*Y')/T ;
%Q = (TT*TT')/T - kron(R,ones(m)).*kron(ones(m),conj(R)) - R(:)*R(:)' - TS.*TS' ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%computation and reshaping of the significant eigen matrices
[U,D] = eig(reshape(Q,m*m,m*m));
[la,K] = sort(abs(diag(D)));
%% reshaping the most (there are `nem' of them) significant eigenmatrice
M = zeros(m,nem*m); % array to hold the significant eigen-matrices
Z = zeros(m) ; % buffer
h = m*m;
for u=1:m:nem*m,
Z(:) = U(:,K(h));
M(:,u:u+m-1) = la(h)*Z;
h = h-1;
end;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% joint approximate diagonalization of the eigen-matrices
%% Better declare the variables used in the loop :
B = [ 1 0 0 ; 0 1 1 ; 0 -i i ] ;
Bt = B' ;
Ip = zeros(1,nem) ;
Iq = zeros(1,nem) ;
g = zeros(3,nem) ;
G = zeros(2,2) ;
vcp = zeros(3,3);
D = zeros(3,3);
la = zeros(3,1);
K = zeros(3,3);
angles = zeros(3,1);
pair = zeros(1,2);
c = 0 ;
s = 0 ;
%init;
encore = 1;
V = eye(m);
% Main loop
while encore, encore=0;
for p=1:m-1,
for q=p+1:m,
Ip = p:m:nem*m ;
Iq = q:m:nem*m ;
% Computing the Givens angles
g = [ M(p,Ip)-M(q,Iq) ; M(p,Iq) ; M(q,Ip) ] ;
[vcp,D] = eig(real(B*(g*g')*Bt));
[la, K] = sort(diag(D));
angles = vcp(:,K(3));
if angles(1)<0 , angles= -angles ; end ;
c = sqrt(0.5+angles(1)/2);
s = 0.5*(angles(2)-j*angles(3))/c;
if abs(s)>seuil, %%% updates matrices M and V by a Givens rotation
encore = 1 ;
pair = [p;q] ;
G = [ c -conj(s) ; s c ] ;
V(:,pair) = V(:,pair)*G ;
M(pair,:) = G' * M(pair,:) ;
M(:,[Ip Iq]) = [ c*M(:,Ip)+s*M(:,Iq) -conj(s)*M(:,Ip)+c*M(:,Iq) ] ;
end%% if
end%% q loop
end%% p loop
end%% while
%%%estimation of the mixing matrix and signal separation
A = IW*V;
S = V'*Y;
return ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Note 1: This version does *not* assume circularly distributed
% signals as 1.1 did. The difference only entails more computations
% in estimating the cumulants
%
%
% Note 2: This code tries to minimize the work load by jointly
% diagonalizing only the m most significant eigenmatrices of the
% cumulant tensor. When the model holds, this avoids the
% diagonalization of m^2 matrices. However, when the model does not
% hold, there is in general more than m significant eigen-matrices.
% In this case, this code still `works' but is no longer equivalent to
% the minimization of a well defined contrast function: this would
% require the diagonalization of *all* the eigen-matrices. We note
% (see the companion paper) that diagonalizing **all** the
% eigen-matrices is strictly equivalent to diagonalize all the
% `parallel cumulants slices'. In other words, when the model does
% not hold, it could be a good idea to diagonalize all the parallel
% cumulant slices. The joint diagonalization will require about m
% times more operations, but on the other hand, computation of the
% eigen-matrices is avoided. Such an approach makes sense when
% dealing with a relatively small number of sources (say smaller than
% 10).
%
%
% Revision history
%-----------------
%
% Version 1.5 (Nov. 2, 97) :
% o Added the option kindly provided by Jerome Galy
% (galy@dirac.ensica.fr) to compute the sample cumulant tensor.
% This option uses more memory but is faster (a similar piece of
% code was also passed to me by Sandip Bose).
% o Suppressed the useles variable `oui'.
% o Changed (angles=sign(angles(1))*angles) to (if angles(1)<0 ,
% angles= -angles ; end ;) as suggested by Iain Collings
% <i.collings@ee.mu.OZ.AU>. This is safer (with probability 0 in
% the case of sample statistics)
% o Cosmetic rewriting of the doc. Fixed some typos and added new
% ones.
%
% Version 1.4 (Oct. 9, 97) : Changed the code for estimating
% cumulants. The new version loops thru the sensor indices rather than
% looping thru the time index. This is much faster for large sample
% sizes. Also did some clean up. One can now change the number of
% eigen-matrices to be jointly diagonalized by just changing the
% variable `nem'. It is still hard coded below to be equal to the
% number of sources. This is more economical and OK when the model
% holds but not appropriate when the model does not hold (in which
% case, the algorithm is no longer asymptotically equivalent to
% minimizing a contrast function, unless nem is the square of the
% number of sources.)
%
% Version 1.3 (Oct. 6, 97) : Added various Matalb tricks to speed up
% things a bit. This is not very rewarding though, because the main
% computational burden still is in estimating the 4th-order moments.
%
% Version 1.2 (Mar., Apr., Sept. 97) : Corrected some mistakes **in
% the comments !!**, Added note 2 `When the model does not hold' and
% the EUSIPCO reference.
%
% Version 1.1 (Feb. 94): Creation
%
%-------------------------------------------------------------------
%
% Contact JF Cardoso for any comment bug report,and UPDATED VERSIONS.
% email : cardoso@sig.enst.fr
% or check the WEB page http://sig.enst.fr/~cardoso/stuff.html
%
% Reference:
% @article{CS_iee_94,
% author = "Jean-Fran\c{c}ois Cardoso and Antoine Souloumiac",
% journal = "IEE Proceedings-F",
% title = "Blind beamforming for non {G}aussian signals",
% number = "6",
% volume = "140",
% month = dec,
% pages = {362-370},
% year = "1993"}
%
%
% Some analytical insights into the asymptotic performance of JADE are in
% @inproceedings{PerfEusipco94,
% HTML = "ftp://sig.enst.fr/pub/jfc/Papers/eusipco94_perf.ps.gz",
% author = "Jean-Fran\c{c}ois Cardoso",
% address = {Edinburgh},
% booktitle = "{Proc. EUSIPCO}",
% month = sep,
% pages = "776--779",
% title = "On the performance of orthogonal source separation algorithms",
% year = 1994}
%_________________________________________________________________________
% jade.m ends here