1 内容介绍
由于大大提高了空间分辨率和阵列增益，大规模天线阵列的使用可以为无线系统带来能量和/或频谱效率的显着提高。大规模多输入多输出 (MIMO) 领域的最新工作表明，当基站 (BS) 上的天线数量增加时，用户信道会去相关，因此可以在用户间干扰很小的情况下实现强信号增益。由于这些结果依赖于渐近，因此研究常规系统模型在这种渐近状态下是否合理是很重要的。本文考虑了一种新的系统模型，该模型在 BS（配备大型天线阵列）和单天线用户设备 (UE) 处结合了通用收发器硬件损伤。与理想硬件的传统情况相反，我们表明硬件损伤会在信道估计精度和每个 UE 的下行链路/上行链路容量上产生有限的上限。令人惊讶的是，容量主要受 UE 的硬件限制，而大规模阵列中损伤的影响逐渐消失，用户间干扰（特别是导频污染）变得可以忽略不计。此外，我们证明了大规模 MIMO 提供的巨大自由度可用于降低发射功率和/或容忍更大的硬件损伤，从而允许使用廉价且节能的天线元件。
2 仿真代码
%This Matlab script can be used to generate Figure 4, in the article:
%
%Emil Bj鰎nson, Jakob Hoydis, Marios Kountouris, M閞ouane Debbah, 揗assive
%MIMO Systems with Non-Ideal Hardware: Energy Efficiency, Estimation, and
%Capacity Limits,?To appear in IEEE Transactions on Information Theory.
%
%Download article: http://arxiv.org/pdf/1307.2584
%
%This is version 1.0 (Last edited: 2014-08-26)
%
%License: This code is licensed under the GPLv2 license. If you in any way
%use this code for research that results in publications, please cite our
%original article listed above.
%
%Please note that the channels are generated randomly, thus the results
%will not be exactly the same as in the paper.
%Initialization
close all;
clear all;
%%Simulation parameters
rng('shuffle'); %Initiate the random number generators with a random seed
%%If rng('shuffle'); is not supported by your Matlab version, you can use
%%the following commands instead:
%randn('state',sum(100*clock));
N = 50; %Number of BS antennas
%Compute normalized channel covariance matrix R according to the exponential
%correlaton model in Eq. (17).
correlationFactor = 0.7;
R = toeplitz(correlationFactor.^(0:N-1));
%Maximal pilot length
Bmax = 10;
%Define the level of hardware impairments at the UE and BS
kappatUE = 0.05^2;
kapparBS = 0.05^2;
%Range of SNRs in simulation (we have normalized sigma2 to 1)
SNRdB = [5 30]; %In decibel scale
SNR = 10.^(SNRdB/10); %In linear scale
%%Initialize Monte Carlo simulations
%Number realizations in Monte Carlo simulations
nbrOfMonteCarloRealizations = 100000;
%Generate random realizations
h = sqrtm(R)*(randn(N,nbrOfMonteCarloRealizations)+1i*randn(N,nbrOfMonteCarloRealizations))/sqrt(2); %Generate channel realizations
etatBS = (randn(1,nbrOfMonteCarloRealizations,Bmax)+1i*randn(1,nbrOfMonteCarloRealizations,Bmax))/sqrt(2); %Generate distortion noise at transmitter (to be scaled by kappatUE)
etarUE = ( repmat(abs(h),[1 1 Bmax]) .* (randn(N,nbrOfMonteCarloRealizations,Bmax)+1i*randn(N,nbrOfMonteCarloRealizations,Bmax)))/sqrt(2); %Generate distortion noise at receiver (to be scaled by kapparBS)
nu = (randn(N,nbrOfMonteCarloRealizations,Bmax)+1i*randn(N,nbrOfMonteCarloRealizations,Bmax))/sqrt(2); %Generate receiver realizations
%Placeholders for storing simulation results
normalizedMSE_corr_distortion = zeros(length(SNR),Bmax); %Normalized MSE for estimator in Eq. (15) for fully correlated distortion noise
normalizedMSE_uncorr_distortion = zeros(length(SNR),Bmax); %Normalized MSE for estimator in Eq. (15) for uncorrelated distortion noise
normalizedMSE_ideal = zeros(length(SNR),Bmax); %Normalized MSE for LMMSE estimator with ideal hardware
%Go through SNR values
for m = 1:length(SNR)
%Output the progress of the simulation
disp(['SNR: ' num2str(m) '/' num2str(length(SNR))]);
%Compute the pilot signal for the given SNR value
d = sqrt(SNR(m));
%Go through different pilot lengths
for B = 1:Bmax
%Compute matrix A in the LMMSE estimator (see Eq. (9))
A_LMMSE = conj(d) * R(1:N,1:N) / (abs(d)^2*(1+kappatUE)*R(1:N,1:N) + abs(d)^2*kapparBS*diag(diag(R(1:N,1:N)))+eye(N));
%Compute matrix A in the LMMSE estimator (see Eq. (9)) for ideal hardware
A_ideal = conj(d) * R(1:N,1:N) / (abs(d)^2*R(1:N,1:N) +eye(N));
%Placeholders for storing squared estimation errors at current SNR
errors_corr_distortion = zeros(nbrOfMonteCarloRealizations,1);
errors_uncorr_distortion = zeros(nbrOfMonteCarloRealizations,1);
errors_ideal = zeros(nbrOfMonteCarloRealizations,1);
%Go through all Monte Carlo realizations
for k = 1:nbrOfMonteCarloRealizations
%Compute received signals
z_corr = h(1:N,k) * ( d + abs(d)*sqrt(kappatUE)*etatBS(1,k,1) ) + abs(d)*sqrt(kapparBS)*etarUE(1:N,k,1) + sum(nu(1:N,k,1:B),3)/B;
z_uncorr = h(1:N,k) * ( d + abs(d)*sqrt(kappatUE)*sum(etatBS(1,k,1:B),3)/B ) + abs(d)*sqrt(kapparBS)*sum(etarUE(1:N,k,1:B),3)/B + sum(nu(1:N,k,1:B),3)/B;
z_ideal = h(1:N,k) * d + sum(nu(1:N,k,1:B),3)/B;
%Compute channel estimates
hhat_corr = A_LMMSE*z_corr;
hhat_uncorr = A_LMMSE*z_uncorr;
hhat_ideal = A_ideal*z_ideal;
%Compute the squared norms of the channel estimation errors
errors_corr_distortion(k) = norm(hhat_corr - h(1:N,k)).^2/N;
errors_uncorr_distortion(k) = norm(hhat_uncorr - h(1:N,k)).^2/N;
errors_ideal(k) = norm(hhat_ideal - h(1:N,k)).^2/N;
end
%Compute normalized MSEs as the average of the squared norms of the
%estimation errors over the Monte Carlo realizations
normalizedMSE_corr_distortion(m,B) = mean(errors_corr_distortion);
normalizedMSE_uncorr_distortion(m,B) = mean(errors_uncorr_distortion);
normalizedMSE_ideal(m,B) = mean(errors_ideal);
end
end
%Plot Figure 4 from the paper
figure; hold on; box on;
for m = 1:length(SNR)
plot(1:Bmax,normalizedMSE_corr_distortion(m,:),'ro-','LineWidth',1);
plot(1:Bmax,normalizedMSE_uncorr_distortion(m,:),'bd-.','LineWidth',1);
plot(1:Bmax,normalizedMSE_ideal(m,:),'k*--','LineWidth',1);
end
set(gca,'YScale','log');
xlabel('Pilot Length (B)');
ylabel('Relative Estimation Error per Antenna');
legend('Fully-Correlated Distortion Noise','Uncorrelated Distortion Noise','Ideal Hardware','Location','NorthEast');
3 运行结果
4 参考文献
[1] Bjornson E , Hoydis J , Kountouris M , et al. Massive MIMO Systems With Non-Ideal Hardware: Energy Efficiency, Estimation, and Capacity Limits[J]. IEEE Transactions on Information Theory, 2013, 60(11):7112-7139.
