1 內(nèi)容介紹

由于大大提高了空間分辨率和陣列增益，大規(guī)模天線陣列的使用可以為無線系統(tǒng)帶來能量和/或頻譜效率的顯著提高。大規(guī)模多輸入多輸出 (MIMO) 領(lǐng)域的最新工作表明，當(dāng)基站 (BS) 上的天線數(shù)量增加時，用戶信道會去相關(guān)，因此可以在用戶間干擾很小的情況下實現(xiàn)強(qiáng)信號增益。由于這些結(jié)果依賴于漸近，因此研究常規(guī)系統(tǒng)模型在這種漸近狀態(tài)下是否合理是很重要的。本文考慮了一種新的系統(tǒng)模型，該模型在 BS（配備大型天線陣列）和單天線用戶設(shè)備 (UE) 處結(jié)合了通用收發(fā)器硬件損傷。與理想硬件的傳統(tǒng)情況相反，我們表明硬件損傷會在信道估計精度和每個 UE 的下行鏈路/上行鏈路容量上產(chǎn)生有限的上限。令人驚訝的是，容量主要受 UE 的硬件限制，而大規(guī)模陣列中損傷的影響逐漸消失，用戶間干擾（特別是導(dǎo)頻污染）變得可以忽略不計。此外，我們證明了大規(guī)模 MIMO 提供的巨大自由度可用于降低發(fā)射功率和/或容忍更大的硬件損傷，從而允許使用廉價且節(jié)能的天線元件。

2 仿真代碼

%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 運(yùn)行結(jié)果

4 參考文獻(xiàn)

[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.

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