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? Content introduction
Blind source signal separation technology has been widely concerned and applied in the field of radar anti-jamming, so it is very important to study the radar signal after blind source separation. The amplitude and phase of the radar signal after blind source separation are analyzed. Uncertainty, and through simulation analysis, the influence of SNR on blind source separation is obtained. The simulation results show that, under certain conditions of SNR, blind source separation can separate the signal from strong suppression interference.
? Full code
% This code is just a front-end to source separation algorithms. % Purpose: % 1) generate synthetic data % 2) call some source separation algorithm % 3) display the results % The data are CM (constant modulus signals and QAM4. % The mixing matrix is randomly generated. % Comments, bug reports, info requests are appreciated % and should be directed to [email protected] (Jean-Francois Cardoso) % Author : Jean-Francois Cardoso CNRS URA 820 / GdR TdSI / Telecom Paris %=================================================== ======================== clear; close all; clc; N = 4 ; % N = number of sensors (add the relevant lines in S= ...) M = 3 ; % M = number of sources T = 200 ; % sample size NdB = -15 ; % kind of noise level in dB %------------------------------------------------- --------------------------- disp('Each of the following plots shows the COMPLEX PLANE.') disp('Each point is a sample of a source signal, of a sensor output') disp('or a separated signal as indicated.') while 1 disp('____________________________________________________________'); % the source signals S=[... exp(2*i*pi*rand(1,T)) ; % constant modulus random phase exp(2*i*pi*rand(1,T)) ; % constant modulus random phase (2*fix(2*rand(1,T))-1 + i*(2*fix(2*rand(1,T))-1))/sqrt(2) ; % QAM4 ]; % random mixing matrix A=randn(N,M) + j*randn(N,M); disp('Mixing matrix');disp(A); subplot(1,1,1); for is=1:M, subplot(2,2,is); plot(S(is,:),'.');title('One of the source signals'); axis('square'); axis('equal'); axis([-2 2 -2 2]); end; fprintf('\\ Strike any key to mix\\ ');pause; % mixing and noise noiseamp = 10^(NdB/20)/sqrt(2) ; % (the sqrt(2) accounts for real + imaginary powers) X= A*S + noiseamp*(randn(N,T) + i*randn(N,T)); for is=1:min([ N 4]), subplot(2,2,is); plot(X(is,:),'.');title('One of the mixed signals'); axis('square');axis('equal');%axis([-2 2 -2 2]); end; fprintf('\\ Strike any key to unmix\\ ');pause; % Separation fprintf('\\ Identification running ......\\ '); [Ae,Se]=jade(X,M); for is=1:M, subplot(2,2,is); plot(Se(is,:),'.');title('One of the separated signals'); axis('square');axis('equal');axis([-2 2 -2 2]); end; %Performance disp('Performance:'); disp('The global (sepration*mixing) matrix should be close to a permutation'); disp('The following shows its squared entries (rejection levels)'); disp(abs(pinv(Ae)*A).^2); fprintf('\\ Strike any key for a different mixture\\ ');pause; end; %endless loop
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 modeled 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.
%
% See notes, references and revision history at the bottom of this file
[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 of 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
% ([email protected]) 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
% <[email protected]>. 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 : [email protected]
% 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
? Running result
? References
[1] Wang Yu, Li Xiaobo, Mao Yunxiang, et al. Research on Radar Signal Based on JADE Blind Source Separation Algorithm [J]. Modern Defense Technology, 2017, 45(1):6.
[2] Guo Xiaole, Qiu Wei, Li Xiangyang, et al. Research on Main Lobe Anti-Jamming Algorithm Based on JADE Blind Source Separation [J]. Fire Control Radar Technology, 2018, 47(4):5.
[3] Luo Lu, Wang Qing. Application of JADE Algorithm in Blind Signal Separation[J]. China New Technology and New Products, 2010(4):2.
[4] Yang Shixi, Jiao Weidong, Wu Zhaotong. Separation of Statistical Correlation Sources Using JADE Blind Separation Algorithm [J]. Journal of Vibration Engineering, 2003, 16(4):4.