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mmWave_channelEst_angularDomain_1bit_CRBvsSNR.m
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mmWave_channelEst_angularDomain_1bit_CRBvsSNR.m
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% Copyright (C) 2018,2023 Mitsubishi Electric Research Laboratories (MERL)
%
% SPDX-License-Identifier: AGPL-3.0-or-later
% angular-domain mmWave channel estimation with 1-bit ADC: Cramer-Rao bound
%
% SNR is defined on a per receive antenna per pilot basis
%
% Refer to
%
% P. Wang, J. Li, M. Pajovic, P.T. Boufounos, P.V. Orlik,
% "On Angular-Domain Chanel Estimation for One-Bit Massive MIMO Systems with Fixed and Time-Varying Thresholds",
% 2017 Asilomar Conference on Signals, Systems and Computers (ACSSC), Pacific Grove, CA, November, 2017.
%
clc;clear;close all;
%% initialize parameters
nMC = 100; % # of Monte-Carlo runs; in the paper we used 100 runs;
N_tx = 8; % # of Tx antennas
N_rx = 16; % # of Rx antennas
SNRset = [-10:2.5:30]; % [dB]
snrSet = 10.^(SNRset/10);
M = 4; % # of channel paths
K = 100; % # of pilots; in the paper we used 100 pilots;
nTQ = 50; % # of time-varying thresholds averaged for each Monte-Carlo run; in the paper we used 50;
%% compute angular-domain one-bit CRB with fixed zero-threshold and time-varying thresholds
% channel matrix: load pre-saved channel path gain matrix
load('hTemp.mat');
hMatrix = diag(hTemp);
hVec = hMatrix(:);
for lmc = 1:nMC
% AoA and AoD are uniformly selected from four clusters (see)
% clusters: [-50,-45], [-10,-5],[30,36], [65,75]
thetaVec = [-50+5*rand(1,1), -10+5*rand(1,1),30+6*rand(1,1), 65+10*rand(1,1)]; % Rx (AoA)
% clusters: [15,18], [45,50], [-20,-16], [-72,-70],
phiVec = [15+3*rand(1,1), 45+5*rand(1,1), -20+4*rand(1,1), -72+2*rand(1,1) ]; % Tx (AoD)
% steering matrices
A_ms = zeros(N_tx, M);
A_bs = zeros(N_rx, M);
indTx = (0:N_tx-1);
indRx = (0:N_rx-1);
A_ms = 1/sqrt(N_tx) * exp(1i*pi*indTx(:)*sind(phiVec));
A_bs = 1/sqrt(N_rx) * exp(1i*pi*indRx(:)*sind(thetaVec));
for lsnr = 1:length(SNRset) % iterate over Monte-Carlo runs
snr = snrSet(lsnr);
S = exp(1i*pi*(randi(4,N_tx,K)-1)/2)/sqrt(N_tx); % normalized pilots signals; [Nt x K]
Gamma = kron(S.'*conj(A_ms),A_bs);
%% CRB
GammaR = real(Gamma);
GammaI = imag(Gamma);
hR = real(hVec);
hI = imag(hVec);
% I/Q channels
yR = GammaR*hR-GammaI*hI;
yI = GammaR*hI+GammaI*hR;
% compute noise variance according to SNR
nvar = sum(abs(yR).^2+abs(yI).^2)/snr/N_rx/K;
[lmc,10*log10(sum(abs(yR).^2+abs(yI).^2)/(nvar*N_rx*K))]
% optimal threshold: known noise variance
thresR0 = GammaR*hR-GammaI*hI;
thresI0 = GammaR*hI+GammaI*hR;
tempFIM1 = zeros(4*M,4*M);
tempCRB1 = zeros(4*M,4*M);
tempFIM1 = func_crb_knownNVar(thresR0(:), thresI0(:), thetaVec, phiVec, Gamma, hVec, S, A_ms, A_bs, N_tx, N_rx, M, nvar);
tempCRB1 = inv(tempFIM1);
tempcrb1 = diag(tempCRB1);
crb_Theta1(:,lsnr,lmc) = tempcrb1(1:M);
crb_Phi1(:,lsnr,lmc) = tempcrb1(M+1:2*M);
crb_AlphaR1(:,lsnr,lmc) = tempcrb1(2*M+1:3*M);
crb_AlphaI1(:,lsnr,lmc) = tempcrb1(3*M+1:4*M);
% fixed-threshold (zeros): known noise variance
thresR1 = zeros(N_rx*K,1);
thresI1 = zeros(N_rx*K,1);
tempFIM3 = zeros(4*M,4*M);
tempCRB3 = zeros(4*M,4*M);
tempFIM3 = func_crb_knownNVar(thresR1(:), thresI1(:), thetaVec, phiVec, Gamma, hVec, S, A_ms, A_bs, N_tx, N_rx, M, nvar);
tempCRB3 = inv(tempFIM3);
tempcrb3 = diag(tempCRB3);
crb_Theta3(:,lsnr,lmc) = tempcrb3(1:M);
crb_Phi3(:,lsnr,lmc) = tempcrb3(M+1:2*M);
crb_AlphaR3(:,lsnr,lmc) = tempcrb3(2*M+1:3*M);
crb_AlphaI3(:,lsnr,lmc) = tempcrb3(3*M+1:4*M);
% time-varying thresholds
nThres = 8; % # of levels
hMaxR = max(abs(yR));
thresSetR = linspace(-hMaxR, hMaxR, nThres);
hMaxI = max(abs(yI));
thresSetI = linspace(-hMaxI, hMaxI, nThres);
for ll = 1:nTQ % iterate over different realization of time-varying thresholds
indR2 = randi(nThres,N_rx*K,1); % uniformly select one level as the threshold
thresR2 = thresSetR(indR2);
indI2 = randi(nThres,N_rx*K,1); % uniformly select one level as the threshold
thresI2 = thresSetI(indI2);
% unknown noise variance
tempFIM5 = zeros(4*M+1,4*M+1);
tempCRB5 = zeros(4*M+1,4*M+1);
tempFIM5 = func_crb_unknownNVar(thresR2(:), thresI2(:), thetaVec, phiVec, Gamma, hVec, S, A_ms, A_bs, N_tx, N_rx, M, nvar);
tempCRB5 = inv(tempFIM5);
tempcrb5 = diag(tempCRB5);
crb_Theta5(:,lsnr,ll,lmc) = tempcrb5(1:M);
crb_Phi5(:,lsnr,ll,lmc) = tempcrb5(M+1:2*M);
crb_AlphaR5(:,lsnr,ll,lmc) = tempcrb5(2*M+1:3*M);
crb_AlphaI5(:,lsnr,ll,lmc) = tempcrb5(3*M+1:4*M);
crb_nVar5(:,lsnr,ll,lmc) = tempcrb5(4*M+1:4*M+1);
% known noise variance
tempFIM6 = zeros(4*M,4*M);
tempCRB6 = zeros(4*M,4*M);
tempFIM6 = func_crb_knownNVar(thresR2(:), thresI2(:), thetaVec, phiVec, Gamma, hVec, S, A_ms, A_bs, N_tx, N_rx, M, nvar);
tempCRB6 = inv(tempFIM6);
tempcrb6 = diag(tempCRB6);
crb_Theta6(:,lsnr,ll,lmc) = tempcrb6(1:M);
crb_Phi6(:,lsnr,ll,lmc) = tempcrb6(M+1:2*M);
crb_AlphaR6(:,lsnr,ll,lmc) = tempcrb6(2*M+1:3*M);
crb_AlphaI6(:,lsnr,ll,lmc) = tempcrb6(3*M+1:4*M);
end
end
end
% --------------------- Remarks --------------------------------
%
% a) you can also use 'func_crb_unknownNVar.m' to compute FIMs for
% 1) optimal threshold with unknown noise variance
% 2) zero threshold with unknown noise variance
% and you will find out corresponding FIMs are singular (as we noted in the
% paper) due to the amplitude/noise variance ambiguity.
%
% b) CRBs for time-varying thresholds are non-singular for both known and
% unknown noise variance.
%% plot Fig. 3 of the paper
% AoA (Fig. 3 (a))
figure(100);
subplot(2,2,1)
for lmc =1:nMC
plotb1 = semilogy(SNRset,crb_Theta1(1,:,lmc),'k-+'); hold on; % OQ
plotb2 = semilogy(SNRset,crb_Theta3(1,:,lmc),'m--x'); hold on; % FQ
temp5 = squeeze(mean(crb_Theta5(1,:,:,lmc),3));
temp6 = squeeze(mean(crb_Theta6(1,:,:,lmc),3));
plotc1 = semilogy(SNRset,temp5,'b-o'); hold on; % TQ
plotc2 = semilogy(SNRset,temp6,'r-s'); hold on; % TQ
end
xlim([SNRset(1),SNRset(end)]); grid on;
xlabel('SNR (dB)');
title(sprintf('K=%g, AoA(1)', K));
legend([plotb1(1), plotb2(1), plotc1(1),plotc2(1)],'CQ (known \sigma^2)','FQ (known \sigma^2)','TQ (unknown \sigma^2)','TQ (known \sigma^2)')
subplot(2,2,2)
for lmc =1:nMC
plotb1 = semilogy(SNRset,crb_Theta1(2,:,lmc),'k-+'); hold on; % OQ
plotb2 = semilogy(SNRset,crb_Theta3(2,:,lmc),'m--x'); hold on; % FQ
temp5 = squeeze(mean(crb_Theta5(2,:,:,lmc),3));
temp6 = squeeze(mean(crb_Theta6(2,:,:,lmc),3));
plotc1 = semilogy(SNRset,temp5,'b-o'); hold on; % TQ
plotc2 = semilogy(SNRset,temp6,'r-s'); hold on; % TQ
end
xlim([SNRset(1),SNRset(end)]); grid on;
xlabel('SNR (dB)');
title(sprintf('AoA(2)'));
subplot(2,2,3)
for lmc =1:nMC
plotb1 = semilogy(SNRset,crb_Theta1(3,:,lmc),'k-+'); hold on; % OQ
plotb2 = semilogy(SNRset,crb_Theta3(3,:,lmc),'m--x'); hold on; % FQ
temp5 = squeeze(mean(crb_Theta5(3,:,:,lmc),3));
temp6 = squeeze(mean(crb_Theta6(3,:,:,lmc),3));
plotc1 = semilogy(SNRset,temp5,'b-o'); hold on; % TQ
plotc2 = semilogy(SNRset,temp6,'r-s'); hold on; % TQ
end
xlim([SNRset(1),SNRset(end)]); grid on;
xlabel('SNR (dB)');
title(sprintf('AoA(3)'));
subplot(2,2,4)
for lmc =1:nMC
plotb1 = semilogy(SNRset,crb_Theta1(4,:,lmc),'k-+'); hold on; % OQ
plotb2 = semilogy(SNRset,crb_Theta3(4,:,lmc),'m--x'); hold on; % FQ
temp5 = squeeze(mean(crb_Theta5(4,:,:,lmc),3));
temp6 = squeeze(mean(crb_Theta6(4,:,:,lmc),3));
plotc1 = semilogy(SNRset,temp5,'b-o'); hold on; % TQ
plotc2 = semilogy(SNRset,temp6,'r-s'); hold on; % TQ
end
xlim([SNRset(1),SNRset(end)]); grid on;
xlabel('SNR (dB)');
title(sprintf('AoA(4)'));
% AoD (Fig. 3 (b))
figure(200);
subplot(2,2,1)
for lmc =1:nMC
plotb1 = semilogy(SNRset,crb_Phi1(1,:,lmc),'k-+'); hold on; % OQ
plotb2 = semilogy(SNRset,crb_Phi3(1,:,lmc),'m--x'); hold on; % FQ
temp5 = squeeze(mean(crb_Phi5(1,:,:,lmc),3));
temp6 = squeeze(mean(crb_Phi6(1,:,:,lmc),3));
plotc1 = semilogy(SNRset,temp5,'b-o'); hold on; % TQ
plotc2 = semilogy(SNRset,temp6,'r-s'); hold on; % TQ
end
xlim([SNRset(1),SNRset(end)]); grid on;
xlabel('SNR (dB)');
title(sprintf('K=%g, AoD(1)', K));
legend([plotb1(1), plotb2(1), plotc1(1),plotc2(1)],'CQ (known \sigma^2)','FQ (known \sigma^2)','TQ (unknown \sigma^2)','TQ (known \sigma^2)')
subplot(2,2,2)
for lmc =1:nMC
plotb1 = semilogy(SNRset,crb_Phi1(2,:,lmc),'k-+'); hold on; % OQ
plotb2 = semilogy(SNRset,crb_Phi3(2,:,lmc),'m--x'); hold on; % FQ
temp5 = squeeze(mean(crb_Phi5(2,:,:,lmc),3));
temp6 = squeeze(mean(crb_Phi6(2,:,:,lmc),3));
plotc1 = semilogy(SNRset,temp5,'b-o'); hold on; % TQ
plotc2 = semilogy(SNRset,temp6,'r-s'); hold on; % TQ
end
xlim([SNRset(1),SNRset(end)]); grid on;
xlabel('SNR (dB)');
title(sprintf('AoD(2)'));
subplot(2,2,3)
for lmc =1:nMC
plotb1 = semilogy(SNRset,crb_Phi1(3,:,lmc),'k-+'); hold on; % OQ
plotb2 = semilogy(SNRset,crb_Phi3(3,:,lmc),'m--x'); hold on; % FQ
temp5 = squeeze(mean(crb_Phi5(3,:,:,lmc),3));
temp6 = squeeze(mean(crb_Phi6(3,:,:,lmc),3));
plotc1 = semilogy(SNRset,temp5,'b-o'); hold on; % TQ
plotc2 = semilogy(SNRset,temp6,'r-s'); hold on; % TQ
end
xlim([SNRset(1),SNRset(end)]); grid on;
xlabel('SNR (dB)');
title(sprintf('AoD(3)'));
subplot(2,2,4)
for lmc =1:nMC
plotb1 = semilogy(SNRset,crb_Phi1(4,:,lmc),'k-+'); hold on; % OQ
plotb2 = semilogy(SNRset,crb_Phi3(4,:,lmc),'m--x'); hold on; % FQ
temp5 = squeeze(mean(crb_Phi5(4,:,:,lmc),3));
temp6 = squeeze(mean(crb_Phi6(4,:,:,lmc),3));
plotc1 = semilogy(SNRset,temp5,'b-o'); hold on; % TQ
plotc2 = semilogy(SNRset,temp6,'r-s'); hold on; % TQ
end
xlim([SNRset(1),SNRset(end)]); grid on;
xlabel('SNR (dB)');
title(sprintf('AoD(4)'));
% Real-part Channel Path Gain (Fig. 3 (c))
figure(300);
subplot(2,2,1)
for lmc =1:nMC
plotb1 = semilogy(SNRset,crb_AlphaR1(1,:,lmc),'k-+'); hold on; % OQ
plotb2 = semilogy(SNRset,crb_AlphaR3(1,:,lmc),'m--x'); hold on; % FQ
temp5 = squeeze(mean(crb_AlphaR5(1,:,:,lmc),3));
temp6 = squeeze(mean(crb_AlphaR6(1,:,:,lmc),3));
plotc1 = semilogy(SNRset,temp5,'b-o'); hold on; % TQ
plotc2 = semilogy(SNRset,temp6,'r-s'); hold on; % TQ
end
xlim([SNRset(1),SNRset(end)]); grid on;
xlabel('SNR (dB)');
title(sprintf('K=%g, Real Amp(1)', K));
legend([plotb1(1), plotb2(1), plotc1(1),plotc2(1)],'CQ (known \sigma^2)','FQ (known \sigma^2)','TQ (unknown \sigma^2)','TQ (known \sigma^2)')
subplot(2,2,2)
for lmc =1:nMC
plotb1 = semilogy(SNRset,crb_AlphaR1(2,:,lmc),'k-+'); hold on; % OQ
plotb2 = semilogy(SNRset,crb_AlphaR3(2,:,lmc),'m--x'); hold on; % FQ
temp5 = squeeze(mean(crb_AlphaR5(2,:,:,lmc),3));
temp6 = squeeze(mean(crb_AlphaR6(2,:,:,lmc),3));
plotc1 = semilogy(SNRset,temp5,'b-o'); hold on; % TQ
plotc2 = semilogy(SNRset,temp6,'r-s'); hold on; % TQ
end
xlim([SNRset(1),SNRset(end)]); grid on;
xlabel('SNR (dB)');
title(sprintf('Real Amp(2)'));
subplot(2,2,3)
for lmc =1:nMC
plotb1 = semilogy(SNRset,crb_AlphaR1(3,:,lmc),'k-+'); hold on; % OQ
plotb2 = semilogy(SNRset,crb_AlphaR3(3,:,lmc),'m--x'); hold on; % FQ
temp5 = squeeze(mean(crb_AlphaR5(3,:,:,lmc),3));
temp6 = squeeze(mean(crb_AlphaR6(3,:,:,lmc),3));
plotc1 = semilogy(SNRset,temp5,'b-o'); hold on; % TQ
plotc2 = semilogy(SNRset,temp6,'r-s'); hold on; % TQ
end
xlim([SNRset(1),SNRset(end)]); grid on;
xlabel('SNR (dB)');
title(sprintf('Real Amp(3)'));
subplot(2,2,4)
for lmc =1:nMC
plotb1 = semilogy(SNRset,crb_AlphaR1(4,:,lmc),'k-+'); hold on; % OQ
plotb2 = semilogy(SNRset,crb_AlphaR3(4,:,lmc),'m--x'); hold on; % FQ
temp5 = squeeze(mean(crb_AlphaR5(4,:,:,lmc),3));
temp6 = squeeze(mean(crb_AlphaR6(4,:,:,lmc),3));
plotc1 = semilogy(SNRset,temp5,'b-o'); hold on; % TQ
plotc2 = semilogy(SNRset,temp6,'r-s'); hold on; % TQ
end
xlim([SNRset(1),SNRset(end)]); grid on;
xlabel('SNR (dB)');
title(sprintf('Real Amp(4)'));
% Imaginary-part Channel Path Gain (Fig. 3 (d))
figure(310);
subplot(2,2,1)
for lmc =1:nMC
plotb1 = semilogy(SNRset,crb_AlphaI1(1,:,lmc),'k-+'); hold on; % OQ
plotb2 = semilogy(SNRset,crb_AlphaI3(1,:,lmc),'m--x'); hold on; % FQ
temp5 = squeeze(mean(crb_AlphaI5(1,:,:,lmc),3));
temp6 = squeeze(mean(crb_AlphaI6(1,:,:,lmc),3));
plotc1 = semilogy(SNRset,temp5,'b-o'); hold on; % TQ
plotc2 = semilogy(SNRset,temp6,'r-s'); hold on; % TQ
end
xlim([SNRset(1),SNRset(end)]); grid on;
xlabel('SNR (dB)');
title(sprintf('K=%g, Imag Amp(1)', K));
legend([plotb1(1), plotb2(1), plotc1(1),plotc2(1)],'CQ (known \sigma^2)','FQ (known \sigma^2)','TQ (unknown \sigma^2)','TQ (known \sigma^2)')
subplot(2,2,2)
for lmc =1:nMC
plotb1 = semilogy(SNRset,crb_AlphaI1(2,:,lmc),'k-+'); hold on; % OQ
plotb2 = semilogy(SNRset,crb_AlphaI3(2,:,lmc),'m--x'); hold on; % FQ
temp5 = squeeze(mean(crb_AlphaI5(2,:,:,lmc),3));
temp6 = squeeze(mean(crb_AlphaI6(2,:,:,lmc),3));
plotc1 = semilogy(SNRset,temp5,'b-o'); hold on; % TQ
plotc2 = semilogy(SNRset,temp6,'r-s'); hold on; % TQ
end
xlim([SNRset(1),SNRset(end)]); grid on;
xlabel('SNR (dB)');
title(sprintf('Imag Amp(2)'));
subplot(2,2,3)
for lmc =1:nMC
plotb1 = semilogy(SNRset,crb_AlphaI1(3,:,lmc),'k-+'); hold on; % OQ
plotb2 = semilogy(SNRset,crb_AlphaI3(3,:,lmc),'m--x'); hold on; % FQ
temp5 = squeeze(mean(crb_AlphaI5(3,:,:,lmc),3));
temp6 = squeeze(mean(crb_AlphaI6(3,:,:,lmc),3));
plotc1 = semilogy(SNRset,temp5,'b-o'); hold on; % TQ
plotc2 = semilogy(SNRset,temp6,'r-s'); hold on; % TQ
end
xlim([SNRset(1),SNRset(end)]); grid on;
xlabel('SNR (dB)');
title(sprintf('Imag Amp(3)'));
subplot(2,2,4)
for lmc =1:nMC
plotb1 = semilogy(SNRset,crb_AlphaI1(4,:,lmc),'k-+'); hold on; % OQ
plotb2 = semilogy(SNRset,crb_AlphaI3(4,:,lmc),'m--x'); hold on; % FQ
temp5 = squeeze(mean(crb_AlphaI5(4,:,:,lmc),3));
temp6 = squeeze(mean(crb_AlphaI6(4,:,:,lmc),3));
plotc1 = semilogy(SNRset,temp5,'b-o'); hold on; % TQ
plotc2 = semilogy(SNRset,temp6,'r-s'); hold on; % TQ
end
xlim([SNRset(1),SNRset(end)]); grid on;
xlabel('SNR (dB)');
title(sprintf('Imag Amp(4)'));