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AOA3DLocMPR_BR.m
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AOA3DLocMPR_BR.m
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function [ mprSol ] = AOA3DLocMPR_BR( theta, phi, senPos, Qa, Qs )
% [ mprSol ] = AOA3DLocMPR_BR( theta, phi, senPos, Qa, Qs )
%
% Estimate the source location in MPR by the Bias Reduction (BR) method
%
% Input:
% theta: (M x 1), noisy azimuth angle measurements
% phi: (M x 1), noisy elevation angle measurements
% senPos: (3 x M), noisy positions of sensors, each column is a sensor
% position(3D) and first column is for the reference sensor;
% Qa: (2M x 2M), AOA covariance matrix
% Qs: (3M x 3M), sensor position covariance matrix
%
% Output:
% mprSol: (3 x 1), BR solution in MPR
%
% Reference:
% Y. Sun, K. C. Ho, and Q. Wan, "Eigenspace solution for AOA localization
% in modified polar representation," IEEE Trans. Signal Process.,
% vol. 68, pp. 2256-2271, 2020.
%
% Yimao Sun, K. C. Ho 03-28-2021
%
% Copyright (C) 2020
% Computational Intelligence Signal Processing Laboratory
% University of Missouri
% Columbia, MO 65211, USA
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
iterNum = 2;
[N,M] = size(senPos);
GA1 = [sin(theta),-cos(theta),zeros(M,1)];
Ga1 = -diag(GA1*(senPos));
GA2 = [cos(theta).*sin(phi),sin(theta).*sin(phi),-cos(phi)];
Ga2 = -diag(GA2*(senPos));
G1 = [GA1,Ga1;GA2,Ga2];
H1 = [cos(theta),sin(theta),zeros(M,1)];
h1 = -diag(H1*senPos);
a1 = -reshape(GA1',[],1);
A1 = [H1,h1;zeros(M,N),zeros(M,1);zeros(M*N,N),a1];
H21 = [-sin(theta).*sin(phi), cos(theta).*sin(phi), zeros(M,1)];
H22 = [cos(theta).*cos(phi), sin(theta).*cos(phi), sin(phi)];
h21 = -diag(H21*senPos);
h22 = -diag(H22*senPos);
a2 = -reshape(GA2',[],1);
A2 = [H21, h21;
H22, h22;
zeros(M*N,N), a2];
A = [A1;A2];
C1 = zeros(M,M*N); C2 = zeros(M,M*N);
W1 = inv(Qa);
Q = blkdiag(Qa,Qs);
for it = 1:iterNum
W11 = W1(1:M,1:M);
W12 = W1(1:M,M+1:2*M);
W21 = W1(M+1:2*M,1:M);
W22 = W1(M+1:2*M,M+1:2*M);
k1 = N-1;
k2 = 2*N-1;
QW = kron(ones(k1,k1),Q).* ...
[kron(ones(k2,k2),W11), kron(ones(k2,k2),W12);
kron(ones(k2,k2),W21), kron(ones(k2,k2),W22)];
R = G1'*W1*G1;
Omega = A'*QW*A;
[V,D2] = eig(R,Omega);
[~,IX] = sort(sum(D2));
u_sol = [V(1:N,IX(1)), -V(1:N,IX(1))]/norm(V(1:N,IX(1)),2);
g_sol = [V(N+1,IX(1)), -V(N+1,IX(1))]/norm(V(1:N,IX(1)),2);
J = zeros(1,2);
for i = 1:2
thetaM = zeros(M,1); phiM = zeros(M,1);
for j = 1:M
thetaM(j,1) = atan2(u_sol(2,i)-g_sol(i)*senPos(2,j), u_sol(1,i)-g_sol(i)*senPos(1,j));
phiM(j,1) = atan2(u_sol(3,i)-g_sol(i)*senPos(3,j), norm(u_sol(1:2,i)-g_sol(i)*senPos(1:2,j),2));
end
J(i) = sum((theta-thetaM).^2+(phi-phiM).^2);
end
[~,ind] = min(J);
u_bar = u_sol(:,ind);
g_est = g_sol(ind);
% update W
b1 = sqrt(sum((u_bar(1:2)-g_est*(senPos(1:2,:))).^2,1))'/norm(u_bar,2);
b2 = sqrt(sum((u_bar-g_est*(senPos)).^2,1))'/norm(u_bar,2);
Ba = diag([b1;b2]);
for m = 1:M
thetaTmp = atan2(u_bar(2)-g_est*senPos(2,m), u_bar(1)-g_est*senPos(1,m));
phiTmp = atan2(u_bar(3)-g_est*senPos(3,m), norm(u_bar(1:2)-g_est*senPos(1:2,m),2));
alpha1 = [sin(thetaTmp);-cos(thetaTmp);0];
C1(m,(1:N)+(m-1)*N) = -alpha1'*g_est;
alpha2 = [cos(thetaTmp)*sin(phiTmp);sin(thetaTmp)*sin(phiTmp);-cos(phiTmp)];
C2(m,(1:N)+(m-1)*N) = -alpha2'*g_est;
end
Ca = [C1;C2];
W1 = inv(Ba*Qa*Ba' + Ca*Qs*Ca');
end
theta = atan2(u_bar(2),u_bar(1));
phi = atan2(u_bar(3),sqrt(u_bar(1)^2+u_bar(2)^2));
mprSol = [theta;phi;g_est];