gusucode.com > 基于MATLAB实现的两个语音的盲信号分离（BSS）程序源码 > 利用MATLAB实现的两个语音的盲信号分离（BSS）/code/jadeR.m
function B = jadeR(X,m)
% B = jadeR(X, m) is an m*n matrix such that Y=B*X are separated sources
% extracted from the n*T data matrix X.
% If m is omitted, B=jadeR(X) is a square n*n matrix (as many sources as sensors)
%
% Blind separation of real signals with JADE. Version 1.8. May 2005.
%
% Usage:
% * If X is an nxT data matrix (n sensors, T samples) then
% B=jadeR(X) is a nxn separating matrix such that S=B*X is an nxT
% matrix of estimated source signals.
% * If B=jadeR(X,m), then B has size mxn so that only m sources are
% extracted. This is done by restricting the operation of jadeR
% to the m first principal components.
% * Also, the rows of B are ordered such that the columns of pinv(B)
% are in order of decreasing norm; this has the effect that the
% `most energetically significant' components appear first in the
% rows of S=B*X.
%
% Quick notes (more at the end of this file)
%
% o this code is for REAL-valued signals. An implementation of JADE
% for both real and complex signals is also available from
% http://sig.enst.fr/~cardoso/stuff.html
%
% o This algorithm differs from the first released implementations of
% JADE in that it has been optimized to deal more efficiently
% 1) with real signals (as opposed to complex)
% 2) with the case when the ICA model does not necessarily hold.
%
% o There is a practical limit to the number of independent
% components that can be extracted with this implementation. Note
% that the first step of JADE amounts to a PCA with dimensionality
% reduction from n to m (which defaults to n). In practice m
% cannot be `very large' (more than 40, 50, 60... depending on
% available memory)
%
% o See more notes, references and revision history at the end of
% this file and more stuff on the WEB
% http://sig.enst.fr/~cardoso/stuff.html
%
% o This code is supposed to do a good job! Please report any
% problem to cardoso@sig.enst.fr
% Copyright : Jean-Francois Cardoso. cardoso@sig.enst.fr
verbose = 1 ; % Set to 0 for quiet operation
% Finding the number of sources
[n,T] = size(X);
if nargin==1, m=n ; end; % Number of sources defaults to # of sensors
if m>n , fprintf('jade -> Do not ask more sources than sensors here!!!\n'), return,end
if verbose, fprintf('jade -> Looking for %d sources\n',m); end ;
% to do: add a warning about complex signals
% Mean removal
%=============
if verbose, fprintf('jade -> Removing the mean value\n'); end
X = X - mean(X')' * ones(1,T);
%%% whitening & projection onto signal subspace
% ===========================================
if verbose, fprintf('jade -> Whitening the data\n'); end
[U,D] = eig((X*X')/T) ; %% An eigen basis for the sample covariance matrix
[Ds,k] = sort(diag(D)) ; %% Sort by increasing variances
PCs = n:-1:n-m+1 ; %% The m most significant princip. comp. by decreasing variance
%% --- PCA ----------------------------------------------------------
B = U(:,k(PCs))' ; % At this stage, B does the PCA on m components
%% --- Scaling ------------------------------------------------------
scales = sqrt(Ds(PCs)) ; % The scales of the principal components .
B = diag(1./scales)*B ; % Now, B does PCA followed by a rescaling = sphering
%% --- Sphering ------------------------------------------------------
X = B*X; %% We have done the easy part: B is a whitening matrix and X is white.
clear U D Ds k PCs scales ;
%%% NOTE: At this stage, X is a PCA analysis in m components of the real data, except that
%%% all its entries now have unit variance. Any further rotation of X will preserve the
%%% property that X is a vector of uncorrelated components. It remains to find the
%%% rotation matrix such that the entries of X are not only uncorrelated but also `as
%%% independent as possible'. This independence is measured by correlations of order
%%% higher than 2. We have defined such a measure of independence which
%%% 1) is a reasonable approximation of the mutual information
%%% 2) can be optimized by a `fast algorithm'
%%% This measure of independence also corresponds to the `diagonality' of a set of
%%% cumulant matrices. The code below finds the `missing rotation ' as the matrix which
%%% best diagonalizes a particular set of cumulant matrices.
%%% Estimation of the cumulant matrices.
% ====================================
if verbose, fprintf('jade -> Estimating cumulant matrices\n'); end
%% Reshaping of the data, hoping to speed up things a little bit...
X = X';
dimsymm = (m*(m+1))/2; % Dim. of the space of real symm matrices
nbcm = dimsymm ; % number of cumulant matrices
CM = zeros(m,m*nbcm); % Storage for cumulant matrices
R = eye(m); %%
Qij = zeros(m); % Temp for a cum. matrix
Xim = zeros(m,1); % Temp
Xijm = zeros(m,1); % Temp
Uns = ones(1,m); % for convenience
%% I am using a symmetry trick to save storage. I should write a short note one of these
%% days explaining what is going on here.
%%
Range = 1:m ; % will index the columns of CM where to store the cum. mats.
for im = 1:m
Xim = X(:,im) ;
Xijm= Xim.*Xim ;
%% Note to myself: the -R on next line can be removed: it does not affect
%% the joint diagonalization criterion
Qij = ((Xijm(:,Uns).*X)' * X)/T - R - 2 * R(:,im)*R(:,im)' ;
CM(:,Range) = Qij ;
Range = Range + m ;
for jm = 1:im-1
Xijm = Xim.*X(:,jm) ;
Qij = sqrt(2) *(((Xijm(:,Uns).*X)' * X)/T - R(:,im)*R(:,jm)' - R(:,jm)*R(:,im)') ;
CM(:,Range) = Qij ;
Range = Range + m ;
end ;
end;
%%%% Now we have nbcm = m(m+1)/2 cumulants matrices stored in a big m x m*nbcm array.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% The inefficient code below does the same as above: computing the big CM cumulant matrix.
%% It is commented out but you can check that it produces the same result.
%% This is supposed to help people understand the (rather obscure) code above.
%% See section 4.2 of the Neural Comp paper referenced below. It can be found at
%% "http://www.tsi.enst.fr/~cardoso/Papers.PS/neuralcomp_2ppf.ps",
%%
%%
%%
%% if 1,
%%
%% %% Step one: we compute the sample cumulants
%% Matcum = zeros(m,m,m,m) ;
%% for i1=1:m,
%% for i2=1:m,
%% for i3=1:m,
%% for i4=1:m,
%% Matcum(i1,i2,i3,i4) = mean( X(:,i1) .* X(:,i2) .* X(:,i3) .* X(:,i4) ) ...
%% - R(i1,i2)*R(i3,i4) ...
%% - R(i1,i3)*R(i2,i4) ...
%% - R(i1,i4)*R(i2,i3) ;
%% end
%% end
%% end
%% end
%%
%% %% Step 2; We compute a basis of the space of symmetric m*m matrices
%% CMM = zeros(m, m, nbcm) ; %% Holds the basis.
%% icm = 0 ; %% index to the elements of the basis
%% vi = zeros(m,1); %% the ith basis vetor of R^m
%% vj = zeros(m,1); %% the jth basis vetor of R^m
%% Id = eye (m) ; %% convenience
%% for im=1:m,
%% vi = Id(:,im) ;
%% icm = icm + 1 ;
%% CMM(:, :, icm) = vi*vi' ;
%% for jm=1:im-1,
%% vj = Id(:,jm) ;
%% icm = icm + 1 ;
%% CMM(:, :, icm) = sqrt(0.5) * (vi*vj'+vj*vi') ;
%% end
%% end
%% %% Now CMM(:,:,i) is the ith element of an orthonormal basis for_ the space of m*m symmetric matrices
%%
%% %% Step 3. We compute the image of each basis element by the cumulant tensor and store it back into CMM.
%% mat = zeros(m) ; %% tmp
%% for icm=1:nbcm
%% mat = squeeze(CMM(:,:,icm)) ;
%% for i1=1:m
%% for i2=1:m
%% CMM(i1, i2, icm) = sum(sum(squeeze(Matcum(i1,i2,:,:)) .* mat )) ;
%% end
%% end
%% end;
%% %% This is doing something like \sum_kl [ Cum(xi,xj,xk,xl) * mat_kl ]
%%
%% %% Step 4. Now, we can check that CMM and CM are equivalent
%% Range = 1:m ;
%% for icm=1:nbcm,
%% M1 = squeeze( CMM(:,:,icm)) ;
%% M2 = CM(:,Range) ;
%% Range = Range + m ;
%% norm (M1-M2, 'fro' ) , %% This should be a numerical zero.
%% end;
%%
%% end; %% End of the demo code for the computation of cumulant matrices
%%
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% joint diagonalization of the cumulant matrices
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Init
if 0, %% Init by diagonalizing a *single* cumulant matrix. It seems to save
%% some computation time `sometimes'. Not clear if initialization is really worth
%% it since Jacobi rotations are very efficient. On the other hand, it does not
%% cost much...
if verbose, fprintf('jade -> Initialization of the diagonalization\n'); end
[V,D] = eig(CM(:,1:m)); % Selectng a particular cumulant matrix.
for u=1:m:m*nbcm, % Accordingly updating the cumulant set given the init
CM(:,u:u+m-1) = CM(:,u:u+m-1)*V ;
end;
CM = V'*CM;
else, %% The dont-try-to-be-smart init
V = eye(m) ; % la rotation initiale
end;
%% Computing the initial value of the contrast
Diag = zeros(m,1) ;
On = 0 ;
Range = 1:m ;
for im = 1:nbcm,
Diag = diag(CM(:,Range)) ;
On = On + sum(Diag.*Diag) ;
Range = Range + m ;
end
Off = sum(sum(CM.*CM)) - On ;
seuil = 1.0e-6 / sqrt(T) ; % A statistically scaled threshold on `small' angles
encore = 1;
sweep = 0; % sweep number
updates = 0; % Total number of rotations
upds = 0; % Number of rotations in a given seep
g = zeros(2,nbcm);
gg = zeros(2,2);
G = zeros(2,2);
c = 0 ;
s = 0 ;
ton = 0 ;
toff = 0 ;
theta = 0 ;
Gain = 0 ;
%% Joint diagonalization proper
if verbose, fprintf('jade -> Contrast optimization by joint diagonalization\n'); end
while encore, encore=0;
if verbose, fprintf('jade -> Sweep #%3d',sweep); end
sweep = sweep+1;
upds = 0 ;
Vkeep = V ;
for p=1:m-1,
for q=p+1:m,
Ip = p:m:m*nbcm ;
Iq = q:m:m*nbcm ;
%%% computation of Givens angle
g = [ CM(p,Ip)-CM(q,Iq) ; CM(p,Iq)+CM(q,Ip) ];
gg = g*g';
ton = gg(1,1)-gg(2,2);
toff = gg(1,2)+gg(2,1);
theta = 0.5*atan2( toff , ton+sqrt(ton*ton+toff*toff) );
Gain = (sqrt(ton*ton+toff*toff) - ton) / 4 ;
%%% Givens update
if abs(theta) > seuil,
%% if Gain > 1.0e-3*On/m/m ,
encore = 1 ;
upds = upds + 1;
c = cos(theta);
s = sin(theta);
G = [ c -s ; s c ] ;
pair = [p;q] ;
V(:,pair) = V(:,pair)*G ;
CM(pair,:) = G' * CM(pair,:) ;
CM(:,[Ip Iq]) = [ c*CM(:,Ip)+s*CM(:,Iq) -s*CM(:,Ip)+c*CM(:,Iq) ] ;
On = On + Gain;
Off = Off - Gain;
%% fprintf('jade -> %3d %3d %12.8f\n',p,q,Off/On);
end%%of the if
end%%of the loop on q
end%%of the loop on p
if verbose, fprintf(' completed in %d rotations\n',upds); end
updates = updates + upds ;
end%%of the while loop
if verbose, fprintf('jade -> Total of %d Givens rotations\n',updates); end
%%% A separating matrix
% ===================
B = V'*B ;
%%% Permut the rows of the separating matrix B to get the most energetic components first.
%%% Here the **signals** are normalized to unit variance. Therefore, the sort is
%%% according to the norm of the columns of A = pinv(B)
if verbose, fprintf('jade -> Sorting the components\n',updates); end
A = pinv(B) ;
[Ds,keys] = sort(sum(A.*A)) ;
B = B(keys,:) ;
B = B(m:-1:1,:) ; % Is this smart ?
% Signs are fixed by forcing the first column of B to have non-negative entries.
if verbose, fprintf('jade -> Fixing the signs\n',updates); end
b = B(:,1) ;
signs = sign(sign(b)+0.1) ; % just a trick to deal with sign=0
B = diag(signs)*B ;
return ;
% To do.
% - Implement a cheaper/simpler whitening (is it worth it?)
%
% Revision history:
%
%- V1.8, May 2005
% - Added some commented code to explain the cumulant computation tricks.
% - Added reference to the Neural Comp. paper.
%
%- V1.7, Nov. 16, 2002
% - Reverted the mean removal code to an earlier version (not using
% repmat) to keep the code octave-compatible. Now less efficient,
% but does not make any significant difference wrt the total
% computing cost.
% - Remove some cruft (some debugging figures were created. What
% was this stuff doing there???)
%
%
%- V1.6, Feb. 24, 1997
% - Mean removal is better implemented.
% - Transposing X before computing the cumulants: small speed-up
% - Still more comments to emphasize the relationship to PCA
%
%- V1.5, Dec. 24 1997
% - The sign of each row of B is determined by letting the first element be positive.
%
%- V1.4, Dec. 23 1997
% - Minor clean up.
% - Added a verbose switch
% - Added the sorting of the rows of B in order to fix in some reasonable way the
% permutation indetermination. See note 2) below.
%
%- V1.3, Nov. 2 1997
% - Some clean up. Released in the public domain.
%
%- V1.2, Oct. 5 1997
% - Changed random picking of the cumulant matrix used for initialization to a
% deterministic choice. This is not because of a better rationale but to make the
% ouput (almost surely) deterministic.
% - Rewrote the joint diag. to take more advantage of Matlab's tricks.
% - Created more dummy variables to combat Matlab's loose memory management.
%
%- V1.1, Oct. 29 1997.
% Made the estimation of the cumulant matrices more regular. This also corrects a
% buglet...
%
%- V1.0, Sept. 9 1997. Created.
%
% Main references:
% @article{CS-iee-94,
% title = "Blind beamforming for non {G}aussian signals",
% author = "Jean-Fran\c{c}ois Cardoso and Antoine Souloumiac",
% HTML = "ftp://sig.enst.fr/pub/jfc/Papers/iee.ps.gz",
% journal = "IEE Proceedings-F",
% month = dec, number = 6, pages = {362-370}, volume = 140, year = 1993}
%
%
%@article{JADE:NC,
% author = "Jean-Fran\c{c}ois Cardoso",
% journal = "Neural Computation",
% title = "High-order contrasts for independent component analysis",
% HTML = "http://www.tsi.enst.fr/~cardoso/Papers.PS/neuralcomp_2ppf.ps",
% year = 1999, month = jan, volume = 11, number = 1, pages = "157-192"}
%
%
%
%
% Notes:
% ======
%
% Note 1) The original Jade algorithm/code deals with complex signals in Gaussian noise
% white and exploits an underlying assumption that the model of independent components
% actually holds. This is a reasonable assumption when dealing with some narrowband
% signals. In this context, one may i) seriously consider dealing precisely with the
% noise in the whitening process and ii) expect to use the small number of significant
% eigenmatrices to efficiently summarize all the 4th-order information. All this is done
% in the JADE algorithm.
%
% In *this* implementation, we deal with real-valued signals and we do NOT expect the ICA
% model to hold exactly. Therefore, it is pointless to try to deal precisely with the
% additive noise and it is very unlikely that the cumulant tensor can be accurately
% summarized by its first n eigen-matrices. Therefore, we consider the joint
% diagonalization of the *whole* set of eigen-matrices. However, in such a case, it is
% not necessary to compute the eigenmatrices at all because one may equivalently use
% `parallel slices' of the cumulant tensor. This part (computing the eigen-matrices) of
% the computation can be saved: it suffices to jointly diagonalize a set of cumulant
% matrices. Also, since we are dealing with reals signals, it becomes easier to exploit
% the symmetries of the cumulants to further reduce the number of matrices to be
% diagonalized. These considerations, together with other cheap tricks lead to this
% version of JADE which is optimized (again) to deal with real mixtures and to work
% `outside the model'. As the original JADE algorithm, it works by minimizing a `good
% set' of cumulants.
%
%
% Note 2) The rows of the separating matrix B are resorted in such a way that the columns
% of the corresponding mixing matrix A=pinv(B) are in decreasing order of (Euclidian)
% norm. This is a simple, `almost canonical' way of fixing the indetermination of
% permutation. It has the effect that the first rows of the recovered signals (ie the
% first rows of B*X) correspond to the most energetic *components*. Recall however that
% the source signals in S=B*X have unit variance. Therefore, when we say that the
% observations are unmixed in order of decreasing energy, this energetic signature is to
% be found as the norm of the columns of A=pinv(B) and not as the variances of the
% separated source signals.
%
%
% Note 3) In experiments where JADE is run as B=jadeR(X,m) with m varying in range of
% values, it is nice to be able to test the stability of the decomposition. In order to
% help in such a test, the rows of B can be sorted as described above. We have also
% decided to fix the sign of each row in some arbitrary but fixed way. The convention is
% that the first element of each row of B is positive.
%
%
% Note 4) Contrary to many other ICA algorithms, JADE (or least this version) does not
% operate on the data themselves but on a statistic (the full set of 4th order cumulant).
% This is represented by the matrix CM below, whose size grows as m^2 x m^2 where m is
% the number of sources to be extracted (m could be much smaller than n). As a
% consequence, (this version of) JADE will probably choke on a `large' number of sources.
% Here `large' depends mainly on the available memory and could be something like 40 or
% so. One of these days, I will prepare a version of JADE taking the `data' option
% rather than the `statistic' option.
% JadeR.m ends here.