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a0_ULA_SNR.m
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a0_ULA_SNR.m
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% Version 1.0: (05/09/2022)
% written by Y. Park
% System framework as defined in Noiselab DOA estimation
% Version 2.0: (06/08/2022)
% written by Y. Park
% AP Covariance (JASA 2022)
% Mark Wagner, Yongsung Park, & Peter Gerstoft
% MPL/SIO/UCSD
% noiselab.ucsd.edu
% Citation
% M. Wagner, Y. Park, and P. Gerstoft, “Gridless DOA estimation and root-MUSIC for non-uniform linear arrays,” IEEE Trans. Signal Process. 69, 2144–2157 (2021).
% M. Wagner, P. Gerstoft, and Y. Park, “Gridless DOA estimation via alternating projections,” in Proc. IEEE ICASSP (2019), pp. 4215–4219.
% Y. Park and P. Gerstoft, “Alternating pro jections gridless covariance-based estimation for DOA,” in Proc. IEEE ICASSP (2021), pp. 4385–4389.
% Y. Park and P. Gerstoft, “Gridless sparse covariance-based beamforming via alternating projections including co-prime arrays,” J. Acoust. Soc. Am. 151(6), 3828-3837 (2022).
%%
clear; clc; %close all;
addpath([cd,'/_common'])
errCut = 10; % Maximum RMSE cut-off.
dbstop if error;
% Case: ULA Multi- snapshot
Nsnapshot = 20;
% anglesTrue = [-65; -2; 3]; % +- .5 Line 68
xAxes = [-10 0 10 20 30 40]; % SNR list
for xaxis = 1:length(xAxes)
% Number of Monte-Carlo simulations
Nsim = 100;
for nsim=1:Nsim
rngN = (xaxis-1)*Nsim + nsim; rng(rngN);
disp(' ')
disp(['SNR',num2str(xAxes(xaxis)),'#Sim : ',num2str(nsim)])
% Environment parameters
c = 1500; % speed of sound
f = 200; % frequency
lambda = c/f; % wavelength
% ULA-horizontal array configuration
Nsensor = 20; % number of sensors
d = 1/2*lambda; % intersensor spacing
q = (0:1:(Nsensor-1))'; % sensor numbering
xq = q*d; % sensor locations
% signal generation parameters
SNR = xAxes(xaxis);
% total number of snapshots
% Nsnapshot = 10;
% range of angle space
thetalim = [-90 90];
% Angular search grid
theta_separation = .25;
theta = (thetalim(1):theta_separation:thetalim(2))';
Ntheta = length(theta);
% Design/steering matrix (Sensing matrix)
sin_theta = sind(theta);
sensingMatrix = exp(-1i*2*pi/lambda*xq*sin_theta.')/sqrt(Nsensor);
% Generate received signal
anglesTrue = [-65; -2; 3];
anglesTrue = anglesTrue + rand(size(anglesTrue)) - 0.5;
disp(['True DOAs : ',num2str(anglesTrue.')])
% source_amp = [ 7; 7; 7; 4; 4; 13];
anglesTracks = repmat(anglesTrue,[1,Nsnapshot]);
sinAnglesTracks = sind(anglesTracks);
Nsource = numel(anglesTrue);
receivedSignal = zeros(Nsensor,Nsnapshot);
for snapshot = 1:Nsnapshot
% Source amplitude
% source_amp(:,snapshot) = 6*rand(size(anglesTrue)) + 4;
source_amp(:,snapshot) = 10*ones(size(anglesTrue));
% source_amp(:,snapshot) = [10; 7; 4; 7];
Xsource = source_amp(:,snapshot).*exp(1i*2*pi*rand(Nsource,1)); % random phase
% Represenation matrix (steering matrix)
transmitMatrix = exp( -1i*2*pi/lambda*xq*sinAnglesTracks(:,snapshot).' );
% Received signal without noise
receivedSignal(:,snapshot) = sum(transmitMatrix*diag(Xsource),2);
% add noise to the signals
rnl = 10^(-SNR/20)*norm(Xsource);
nwhite = complex(randn(Nsensor,1),randn(Nsensor,1))/sqrt(2*Nsensor);
e = nwhite * rnl; % error vector
receivedSignal(:,snapshot) = receivedSignal(:,snapshot) + e;
% for CRB
crnl(snapshot) = rnl;
cX(:,snapshot) = Xsource;
end
%% CRB
% CRB-YP Van Trees Book Eq.(8.106) & (8.110)
vanTreeV = exp( -1i*2*pi/lambda*xq*sinAnglesTracks(:,snapshot).' );
vanTreeD = (-1i*2*pi/lambda*xq*cosd(anglesTracks(:,snapshot)).') ...
.* exp( -1i*2*pi/lambda*xq*sinAnglesTracks(:,snapshot).' ); % D Eq.(8.100)
Xs = cX;
Pn = mean(diag(((e.*sqrt(crnl/crnl(1)))*(e.*sqrt(crnl/crnl(1)))')/Nsnapshot));
vanTreeSf = diag(diag(Xs*Xs'/Nsnapshot)); % S_f
% H Eq.(8.101) where P_V Eq.(8.96)
H = vanTreeD'...
*(eye(Nsensor) - vanTreeV/(vanTreeV'*vanTreeV)*vanTreeV')...
*vanTreeD;
% det. CRB Eq.(8.110)
CRB = real(H .* (vanTreeSf.'));
CRB = eye(size(Xs,1)) / CRB * (Pn / Nsnapshot / 2);
outputsCRBd(xaxis,nsim) = mean(diag(CRB));
%% Conventional beamforming (CBF)
Ryy = receivedSignal*receivedSignal' / Nsnapshot;
Pcbf = zeros(numel(theta),1);
for ii=1:length(theta)
Pcbf(ii) = sensingMatrix(:,ii)'*Ryy*sensingMatrix(:,ii)/(sensingMatrix(:,ii)'*sensingMatrix(:,ii));
end
% Pcbf = sensingMatrix' * receivedSignal;
% plot(theta,mean(Pcbf.*conj(Pcbf),2)/max(mean(Pcbf.*conj(Pcbf),2)),'k:','linewidth',1,'displayname','CBF')
% plot(theta,abs(Pcbf)/max(abs(Pcbf)),'k:','linewidth',1.5,'displayname','CBF')
[~, Ilocs] = findpeaks(abs(Pcbf),'SORTSTR','descend','Npeaks', Nsource);
% DoA_error = errorDOA(theta(Ilocs),anglesTrue);
DoA_error = errorDOAcutoff(theta(Ilocs),anglesTrue,errCut);
disp(['RMSE CBF : ',num2str(sqrt(mean(power(DoA_error,2))))])
if nsim==1 && xaxis==1, outputsCBF = []; end
outputCBF = struct('theta',theta(Ilocs),'error',DoA_error);
outputsCBF = [outputsCBF; outputCBF];
%% root-MUSIC
spRmusic = rmusic_1d(Ryy, Nsource, 2*pi*d/lambda);
% DoA_error = errorDOA(-rad2deg(spRmusic.x_est),anglesTrue);
DoA_error = errorDOAcutoff(-rad2deg(spRmusic.x_est),anglesTrue,errCut);
disp(['RMSE root-MUSIC : ',num2str(sqrt(mean(power(DoA_error,2))))])
if nsim==1 && xaxis==1, outputsrMUSIC = []; end
outputrMUSIC = struct('theta',-rad2deg(spRmusic.x_est),'error',DoA_error);
outputsrMUSIC = [outputsrMUSIC; outputrMUSIC];
%% AP-ULA
MAX_IT = 1000;
NeigAP = Nsource;
tol = 1e-3;
[T,iAPM] = AP_Gridless( receivedSignal,q,NeigAP,MAX_IT,tol,zeros(Nsensor),0 );
[t_est,~] = wagner_decomp( q, NeigAP, T ); %decompose
t_est = -t_est*lambda/d;
DoA_est_deg = asin(t_est)/pi*180;
DoA_error = errorDOAcutoff(DoA_est_deg,anglesTrue,errCut);
disp(['RMSE AP-Snapshot ULA: ',num2str(sqrt(mean(power(DoA_error,2))))])
if nsim==1 && xaxis==1, outputsAPula = []; end
outputAPula = struct('theta',DoA_est_deg,'error',DoA_error);
outputsAPula = [outputsAPula; outputAPula];
%% AP-Gridless
MAX_IT = 1000;
NeigAP = Nsource;
tol = 1e-3;
% [T,iAPM] = AP_Gridless( receivedSignal,q,NeigAP,MAX_IT,tol,zeros(Nsensor),0 ); % for AP-ULA
[T,iAPM] = AP_Gridless( receivedSignal,q+.1,NeigAP,MAX_IT,tol,zeros(Nsensor),0 ); % for AP-Gridless
[t_est,~] = wagner_decomp( q, NeigAP, T ); %decompose
t_est = -t_est*lambda/d;
DoA_est_deg = asin(t_est)/pi*180;
DoA_error = errorDOAcutoff(DoA_est_deg,anglesTrue,errCut);
disp(['RMSE AP-Snapshot : ',num2str(sqrt(mean(power(DoA_error,2))))])
if nsim==1 && xaxis==1, outputsAPsnapshot = []; end
outputAPsnapshot = struct('theta',DoA_est_deg,'error',DoA_error);
outputsAPsnapshot = [outputsAPsnapshot; outputAPsnapshot];
%% AP-Covariance
max_iter = 1000;
Nalg = Nsource;
tol = 1e-4;
if exist('Tu_init','var') == 0
Tu_init = rand(Nsensor) + 1i*rand(Nsensor);
Z_init = rand(Nsensor) + 1i*rand(Nsensor);
end
[ Tu,iAP,~,~,~ ] = APCOVv1p00(receivedSignal,q,Nalg,max_iter,tol,Tu_init,Z_init);
[t_est,~] = wagner_decomp( q, Nalg, Tu ); %decompose
t_est = -t_est*lambda/d;
while(1)
t_est(t_est>1) = t_est(t_est>1) - 2;
t_est(t_est<-1)= t_est(t_est<-1) + 2;
if sum(t_est>1 | t_est<-1) == 0, break; end
end
DoA_est_deg = asin(t_est)/pi*180;
clear Tu_init Z_init
DoA_error = errorDOAcutoff(DoA_est_deg,anglesTrue,errCut);
disp(['RMSE AP-Covariance : ',num2str(sqrt(mean(power(DoA_error,2))))])
if nsim==1 && xaxis==1, outputsAPcov = []; end
outputAPcov = struct('theta',DoA_est_deg,'error',DoA_error);
outputsAPcov = [outputsAPcov; outputAPcov];
%% SBL
options = SBLSet();
options.convergence.error = 10^(-3);
options.Nsource = ceil(Nsensor/2);
options.gamma_range=10^-20;
[gamma, reportSBL] = SBL_v4( sensingMatrix, receivedSignal, options );
[~,peak_SBL] = findpeaks(gamma,'SORTSTR','descend','Npeaks', Nsource);
DoA_error = errorDOAcutoff(theta(peak_SBL),anglesTrue,errCut);
disp(['RMSE SBL : ',num2str(sqrt(mean(power(DoA_error,2))))])
if nsim==1 && xaxis==1, outputsSBL = []; end
outputSBL = struct('theta',theta(peak_SBL),'error',DoA_error);
outputsSBL = [outputsSBL; outputSBL];
end
end
%% Figure
for ind=1:length(xAxes)
totETcbf = [];
totETrmu=[];
totETAPula=[];
totETAPsnapshot=[];
totETAPcov=[];
totETsbl = [];
for index=1:Nsim
totETcbf = [totETcbf;outputsCBF((ind-1)*Nsim+index).error];
totETrmu = [totETrmu;outputsrMUSIC((ind-1)*Nsim+index).error];
totETAPula = [totETAPula;outputsAPula((ind-1)*Nsim+index).error];
totETAPsnapshot = [totETAPsnapshot;outputsAPsnapshot((ind-1)*Nsim+index).error];
totETAPcov = [totETAPcov;outputsAPcov((ind-1)*Nsim+index).error];
totETsbl = [totETsbl;outputsSBL((ind-1)*Nsim+index).error];
end
Nout = 0; % Portion of Outliers, (ignore)
totETcbf = sort(abs(totETcbf));
totETrmu = sort(abs(totETrmu));
totETAPula = sort(abs(totETAPula));
totETAPsnapshot = sort(abs(totETAPsnapshot));
totETAPcov = sort(abs(totETAPcov));
totETsbl = sort(abs(totETsbl));
ecbf(ind) = sqrt(mean(power(totETcbf(1:length(totETcbf)-floor(length(totETcbf)*Nout)),2)));
ermusic(ind) = sqrt(mean(power(totETrmu(1:length(totETrmu)-floor(length(totETrmu)*Nout)),2)));
eapula(ind) = sqrt(mean(power(totETAPula(1:length(totETAPula)-floor(length(totETAPula)*Nout)),2)));
eapsnapshot(ind) = sqrt(mean(power(totETAPsnapshot(1:length(totETAPsnapshot)-floor(length(totETAPsnapshot)*Nout)),2)));
eapcov(ind) = sqrt(mean(power(totETAPcov(1:length(totETAPcov)-floor(length(totETAPcov)*Nout)),2)));
esbl(ind) = sqrt(mean(power(totETsbl(1:length(totETsbl)-floor(length(totETsbl)*Nout)),2)));
end
figure; set(gcf,'position',[750,200,700,600]);
hold on;
h1=plot(xAxes,sqrt(mean(outputsCRBd,2)*180/pi*180/pi),'k','linewidth',1.0,'markersize',10,'displayname','det. CRB');
figH = h1;
h3=plot(xAxes,ecbf,'k:','linewidth',1.5,'markersize',10,'displayname','CBF');
figH = [figH,h3];
pcolor = lines;
h5=plot(xAxes,esbl,'m--','linewidth',1.2,'markersize',10,'displayname',...
'SBL','color',pcolor(2,:)); figH = [figH,h5];
h8=plot(xAxes,ermusic,'b-.','linewidth',2,'markersize',10,'displayname',...
'Root-MUSIC'); figH = [figH,h8];
h9=plot(xAxes,eapula,'c-.','linewidth',1.8,'markersize',12,'displayname',...
'AP-ULA'); figH = [figH,h9];
h91=plot(xAxes,eapsnapshot,'linewidth',1.8,'markersize',12,'color',pcolor(7,:),'displayname',...
'AP-Snapshot'); figH = [figH,h91];
h92=plot(xAxes,eapcov,'linewidth',1.8,'markersize',12,'color','r','displayname',...
'AP-Covariance'); figH = [figH,h92];
hold off;
xlabel('SNR~[dB]','interpreter','latex')
ylabel('RMSE~[$^\circ$]','interpreter','latex')
legend(fliplr(figH),'location','northeast','interpreter','latex')
set(gca,'fontsize',18,'yscale','log')
box on; grid on;
axis([min(xAxes) max(xAxes) min(sqrt(mean(outputsCRBd,2)*180/pi*180/pi)) 1])
% axis([0 20 0 10])
%%
rmpath([cd,'/_common'])