目录
一.介绍
图中“RF Chain” 全称为Radio Frequency Chain，代表射频链路。
此MIMO预编码包含了基带预编码W（改变幅度和相位）和射频预编码F（改变相位）。用户k端收到的信号为：
上式中代表基站到用户的信道矩阵，F代表射频预编码，W代表基带预编码，s代表发射信号向量，
代表加性高斯白噪声。
发射流的最大数量为K，则：
用户k端受到的信干噪比（SINR）为：
上式子中P代表基站的发射功率。其他参数的意义上面已解释。
系统的总频谱效率为：
上式中E[]代表求数学期望。 其他参数的意义上面已解释。
依据基础的迫零算法进行改进，将K个射频链路输出与个发射天线进行耦合，仅仅进行相位控制，然后在基带上进行低维多流处理，最终降低用户间的干扰，本方案创新性提出相位-迫零算法（Phased-ZF)，优点：
•极大降低计算复杂度；
•性能逼近原始ZF（迫零）算法；
对该预编码的两步，总结来看为如下：
二. 对比本方案优化后的迫零算法与原始的迫零算法
三. 源代码
一共八个文件，两个主运行文件，六个函数文件。注意最后的两个文件为实际产生图像的代码部分。
（1）
% BD precoder design subfunction r = CalBDPrecoder(G, K) --%
% Input: H, K*Nr x Nt, aggregate channel,
% NsUE, # streams per user, K x 1
% Output: r, Nt x K*Lr BD precoder
function r = CalBDPrecoder(H, NsUE) % no waterfilling
if nargin <= 1
NsUE = ones(size(H,1)); % essentially ZF precoding with column normalization
end
[NR, Nt] = size(H);
K = length(NsUE);
Nr = NR/K;
F = [];% BD precoder initialization
%***** Standard BD design based on paper "ZF methods for DL spatial multiplexing..." ***%
for iK = 1 : K
Gj = H(((iK-1)*Nr+1):(iK*Nr),:);
G_tilde = H;
G_tilde(((iK-1)*Nr+1):(iK*Nr),:) = []; % Delete iK-user's channel, G_tilde
[U, S, V] = svd(G_tilde);
rv = rank(G_tilde);
Htmp = Gj*V(:, (rv + 1):Nt );
Lr = rank(Htmp);% # of data streams accommodated by iK user
[Ut, St, Vt] = svd(Htmp);
Ftmp = V( : , (rv + 1):Nt)*Vt(: , 1:NsUE(iK));
F = [F Ftmp];
end
r = F;（2）
% Function to calculate achievable rates
% r = CalRate(H, W, K);
% Input: H, channel instantialization, (KNr) x Nt
% W, precoding matrix, column normalized, Nt x Ns
% NsUE, a vector showing # streams of each UE, K x 1
% Incov, input covariance, i.e., power allocation, Ns x Ns
% Output: r, achievable rate
function r = CalRate(Incov, H, W, NsUE)
Incov = diag(Incov); % vectorize
if nargin <= 3
NsUE = ones(size(W,2), 1);% set default, single stream per user
end
K = length(NsUE);% # of users
Nr = size(H, 1)/K;% # antenna per user, assuming equal
Ns = sum(NsUE);% total # data streams
rate = 0;
for iK = 1 : K
st = sum(NsUE(1:(iK-1)))+1;
ed = sum(NsUE(1:iK));
Hj = H( (Nr*(iK-1)+1) : (Nr*iK), :);% user channel of i
Wj = W(:, st:ed);% precoder for user i
covj = diag(Incov(st:ed));
Pj = Hj*Wj*covj*(Hj*Wj)'; % signal power term
Wintf = W;% precoder for interferers
covtp = Incov;
Wintf(:, st:ed) = [];
covtp(st:ed) = [];
cov_intf = diag(covtp);
Pintf = (Hj*Wintf)*cov_intf*(Hj*Wintf)';% interference power term
rate = rate + log2(det( eye(Nr) + inv( eye(Nr) + Pintf ) * Pj ));
end
r = rate;（3）
% expintn exponential intergral of the order n
% r = expintn(x, n) gives integral from 1 to inf of (exp(-xt)/t^n)dt, for x > 0
function r = expintn_noMaple(x, n)
if n >= 2
sum = 0;
for m = 0 : (n-2)
sum = sum + (-1)^m*factorial(m)/x^(m+1);
end
minuend = x^(n-1) * (-1)^(n+1)/factorial(n-1) * expint(x);
subtractor = x^(n-1) * (-1)^(n+1)/factorial(n-1) * exp(-x) * sum;
r = minuend - subtractor;
else
r = expint(x);
end（4）
% expintn(x, n) is an an exponential integral of order n with input x,
% computed using maple
function r = expintn(x, n)
cmd_str = sprintf('evalf[16](Ei(%d, %g));', n, x);
r = double(maple(cmd_str));（5）
% Generate a simplified MU-MIMO channel based on uniform linear array (ULA)
% "The capacity optimality of beam steering in large mmWave MIMO channels"
% [H, Gain, At] = GenChannelSimp (Nt, K, Ncls)
% Input: Nt, number of TX antennas
% K, number of single-antenna UEs
% Ncls, number of clusters, single path per cluster
% d, normalized antenna spacing
% Output: H, generated MU-MIMO channel
% Gain, complex gain of each path
% At, concatenation of array response vectors
function [H, Gain, At] = GenChannelSimp(Nt, K, Ncls, d)
if nargin <= 3
d = 1/2;
end
% spread = pi/12;% [-15, 15]
spread = pi;
ang = (2*rand(Ncls, K)-1)*spread; % generated path angles, uniform distribution
Gain = (randn(Ncls, K) + j*randn(Ncls, K))/sqrt(2);
At = zeros(Nt, Ncls, K);
H = zeros(K, Nt);
for ik = 1 : K
tmp = 0 : (Nt-1);
tmp = tmp(:);
kron1 = j*2*pi*d*tmp;
kron2 = sin(ang(:, ik));
kron2 = kron2.';
kronr = kron(kron1, kron2);
At(:, :, ik) = 1/sqrt(Nt) * exp(kronr);
H(ik, :) = sqrt(Nt/Ncls) * Gain(:, ik).' * At(:, :, ik)';
end（6）
% Quant(B, W) quantizes phases of each element in W up to B bits of precision
function r = Quant(B, W)
delta = 2*pi/2^B; % quantization interval
r = zeros(size(W, 1), size(W, 2));% ininitialize quantized matrix
for i1 = 1 : size(W, 1)
for i2 = 1 : size(W, 2)
ph = phase(W(i1, i2)); % ph in [-pi, pi]
phq = floor(ph/delta)*delta +(mod(ph, delta) > delta/2)*delta ;% quantized phase
r(i1, i2) = exp(j*phq);
end
end
r = 1/sqrt(size(W, 1)) * r;
end（7）
% Massive MU-MIMO under chain limitations, single-antenna UE
% Hybrid precoding with ZF at baseband and phase reversal at analog
% Compare full-complexity ZF (FC-ZF) vs Hybrid
tic; clear; clc
Nt = 8;
K = 2; % UE number
B1 = 1; % quantized analog beamforming, up to B bits of precision
B2 = 2;
SNR = 0 : 5 : 30;
nSNR = length(SNR);
channNum = 1e3;
rateZF = zeros(nSNR, 1); % FC-ZF
rateZFA = zeros(nSNR, 1);
rateHyb = zeros(nSNR, 1);% W = ZF at baseband, F = PR at analog
rateHybA = zeros(nSNR, 1);
rateQHyb1 = zeros(nSNR, 1);% Quantized PR, ZF at baseband
rateQhyb2 = zeros(nSNR, 1);
for isnr = 1 : nSNR
P = 10^(SNR(isnr)/10);
% ====================================================================
% ===================== Analytical result ============================
% ====================================================================
% ========== ZF analytical results =============
s = 0;
for k = 1 : (Nt)
rho = P/K;
s = s + exp(1/rho)*log2(exp(1)) * expintn_noMaple(1/rho, k); % use maple for expintn()
% s = s + exp(1/rho)*log2(exp(1)) * expintn_noMaple(1/rho, k); % if no maple installed
end
rateZFA(isnr) = K*s;
% ========= Hybrid precoding analytical results ==========
rateHybA(isnr) = K*log2(1 + (pi/4) * (P*Nt)/K );
% ====================================================================
% ===================== Simulation result ============================
% ====================================================================
for ichannel = 1 : channNum
H = (randn(K, Nt) + j*randn(K, Nt))/sqrt(2);
% ============= ZF preocidng, numerical ================
WtZF = H'*inv(H*H');
WZF = WtZF*inv(sqrt(diag(diag(WtZF'*WtZF)))); % normalized columns
rateZF(isnr) = rateZF(isnr) + CalRate(P/K*eye(K), H, WZF);
% ============ Hybrid, numerical ===============
F = 1/sqrt(Nt)*exp(j.*angle(H))';
Fb = CalBDPrecoder(H*F);% baseband, same as inverse with column normalization
wt = F*Fb;% aggregate precoder
WPR = wt*inv(sqrt(diag(diag(wt'*wt))));
rateHyb(isnr) = rateHyb(isnr) + CalRate((P/K)*eye(K), H, WPR);% ZF-PRP
% ============= Quantized ZF-PR precoding ============
FQPR1 = Quant(B1, F);% analog RF
wt = FQPR1*CalBDPrecoder(H*FQPR1);
WQPR1 = wt*inv(sqrt(diag(diag(wt'*wt))));
rateQHyb1(isnr) = rateQHyb1(isnr) + CalRate((P/K)*eye(K), H, WQPR1);
WtQPR2 = Quant(B2, F);
wt = WtQPR2*CalBDPrecoder(H*WtQPR2);
WQPR2 = wt*inv(sqrt(diag(diag(wt'*wt))));
rateQhyb2(isnr) = rateQhyb2(isnr) + CalRate((P/K)*eye(K), H, WQPR2);
end
isnr
end
rateZF = rateZF/channNum;
rateHyb = rateHyb/channNum;
rateQHyb1 = rateQHyb1/channNum;
rateQhyb2 = rateQhyb2/channNum;
LineWidth = 1.5;
MarkerSize = 6;
figure
plot(SNR, abs(rateZF), 'k-x', 'LineWidth', LineWidth, 'MarkerSize', MarkerSize)
hold on
plot(SNR, abs(rateZFA), 'bo', 'LineWidth', LineWidth, 'MarkerSize', MarkerSize);
hold on
plot(SNR, abs(rateHyb),'r-*', 'LineWidth', LineWidth, 'MarkerSize', MarkerSize)
hold on
plot(SNR, abs(rateHybA), 'gd', 'LineWidth', LineWidth, 'MarkerSize', MarkerSize)
hold on
plot(SNR, abs(rateQHyb1), 'b-^', 'LineWidth', LineWidth, 'MarkerSize', MarkerSize)
hold on
plot(SNR, abs(rateQhyb2), 'b-v', 'LineWidth', LineWidth, 'MarkerSize', MarkerSize)
hold off
legend('FC-ZF, simulation', 'FC-ZF, analytical', ...
'PZF, simulation', 'PZF, analytical', ...
'Quantized PZF, B = 1','Quantized PZF, B = 2');
xlabel('SNR (dB)')
ylabel('Spectral Efficiency (bps/Hz)')
title(sprintf('Nt = %d, K = %d', Nt, K))
grid
% saveas(gcf, sprintf('MassiveCompareScheme-Nt%d-K%d', Nt, K));
% saveas(gcf, sprintf('MassiveCompareScheme-Nt%d-K%d-PhaseOfZf', Nt, K));
toc（8）
% DESCRIPTION
% mmWave massive MU-MIMO under chain limitations, single-antenna UE
% Hybrid precoding with ZF at baseband and Phase Reversal (PR) Preocidng at analog
% Compare full-complexity ZF (FC-ZF), Hybrid, QHybrid, and B-MIMO numerically
tic; clear all; clc
Nt = 128;
K = 4; % UE number
Np = 10; % number of paths per user
B1 = 1; % quantized analog beamforming, up to B bits of precision
B2 = 2;
SNR = -30 : 5 : 0;
nSNR = length(SNR);
channNum = 1e3;
rateZF = zeros(nSNR, 1); % FC-ZF
rateHyb = zeros(nSNR, 1);% W = ZF at baseband, F = PR at analog
rateHybQ1 = zeros(nSNR, 1);% Quantized hybrid precoding
rateHybQ2 = zeros(nSNR, 1);
rateBMIMO = zeros(nSNR, 1);% Multiuser beamspace MIMO precoder (BMIMO)
for isnr = 1 : nSNR
P = 10^(SNR(isnr)/10);
for ichannel = 1 : channNum
[H, Gain, At] = GenChannelSimp(Nt, K, Np, 0.5); % mmWave channel
% H = K x Nt, Gain = Np x K, At = Nt x Np x K
% ============= ZF preocidng, numerical ================
WtZF = H'*inv(H*H');
WZF = WtZF*inv(sqrt(diag(diag(WtZF'*WtZF)))); % normalized columns
rateZF(isnr) = rateZF(isnr) + CalRate(P/K*eye(K), H, WZF);
% ============ Hybrid precoding, numerical ===============
for ik = 1 : K
ph = - phase(H(ik,:));
ph = ph(:);
F(:,ik) =1/sqrt(Nt)* exp(j.*ph); % analog RF preprocessing
end
Fb = CalBDPrecoder(H*F);% digital baseband, same as inverse with column normalization
wt = F*Fb;% aggregated precoder
WPR = wt*inv(sqrt(diag(diag(wt'*wt))));
rateHyb(isnr) = rateHyb(isnr) + CalRate((P/K)*eye(K), H, WPR);% ZF-PRP
% ============= Quantized hybrid precoding ============
FQPR1 = 1/sqrt(Nt) * Quant(B1, F);% analog RF
wt = FQPR1*CalBDPrecoder(H*FQPR1);
WQPR1 = wt*inv(sqrt(diag(diag(wt'*wt))));
rateHybQ1(isnr) = rateHybQ1(isnr) + CalRate((P/K)*eye(K), H, WQPR1);
WtQPR2 = 1/sqrt(Nt) * Quant(B2, F);
wt = WtQPR2*CalBDPrecoder(H*WtQPR2);
WQPR2 = wt*inv(sqrt(diag(diag(wt'*wt))));
rateHybQ2(isnr) = rateHybQ2(isnr) + CalRate((P/K)*eye(K), H, WQPR2);
% =========== Multiuser Beamspace MIMO precoder (B-MIMO) ============== %
D = dftmtx(Nt);
Hf = H*D;
[maxVal, maxInd] = sort(diag(Hf'*Hf), 'descend'); % sort column with decreasing magnitude
FBMIMO = D(:, maxInd(1:K));
FbBMIMO = pinv(H*FBMIMO);
Wt = FBMIMO*FbBMIMO;% analog x baseband precoding
WBMIMO = Wt*inv(sqrt(diag(diag(Wt'*Wt))));
rateBMIMO(isnr) = rateBMIMO(isnr) + CalRate((P/K)*eye(K), H, WBMIMO);
end
isnr
end
rateZF = rateZF/channNum;
rateHyb = rateHyb/channNum;
rateHybQ1 = rateHybQ1/channNum;
rateHybQ2 = rateHybQ2/channNum;
rateBMIMO = rateBMIMO/channNum;
LineWidth = 1.5;
MarkerSize = 6;
figure
plot(SNR, abs(rateZF), 'k-o', 'LineWidth', LineWidth, 'MarkerSize', MarkerSize)
hold on
plot(SNR, abs(rateHyb),'r-*', 'LineWidth', LineWidth, 'MarkerSize', MarkerSize)
hold on
plot(SNR, abs(rateHybQ1), 'b-^', 'LineWidth', LineWidth, 'MarkerSize', MarkerSize)
hold on
plot(SNR, abs(rateHybQ2), 'b-v', 'LineWidth', LineWidth, 'MarkerSize', MarkerSize)
hold on
plot(SNR, abs(rateBMIMO), 'm-s', 'LineWidth', LineWidth, 'MarkerSize', MarkerSize)
hold off
legend('FC-ZF Precoding', 'Hybrid Precoding', 'Quantized Hybrid Precoding, B = 1',...
'Quantized Hybrid Precoding, B = 2', 'B-MIMO Preocoding');
xlabel('SNR (dB)')
ylabel('Spectral Efficiency (bps/Hz)')
% title(sprintf('Nt = %d, K = %d, Np = %d',Nt, K, Np))
grid
saveas(gcf, sprintf('MainCompareScheme-Nt%d-K%d-Np%d', Nt, K, Np)); % save current figure to file
toc四. 运行结果及分析
4.1 天线数为8
ZF: Zero Forcing 迫零算法 SNR： Signal Noise Ratio 信噪 PZF：Phased-Zero Forcing
对比三种算法性能：原始ZF算法，PZF算法，量化后的PZF算法。
横向对比：三种算法均满足，随着SNB从0增加到30dB，频谱效率从2.166增大到23.75 bps/HZ，符合正常逻辑；
纵向对比：在同一SNR下，性能从最优到最劣，理论ZF值、实际ZF仿真值、理论PZF值、实际PZF仿真值、精度到两位比特的PZF值、精度到1位比特的PZF值；
方案对比：本方案PZF比原始ZF性能略差（相差约0.7bps/Hz），但计算复杂性低得多，更具有实用价值；
量化对比：在PZF方案基础上，进一步约算，取近似比特可进一步形成Quantized PZF. 随着近似比特精度取值从1到2，频谱效率也会增大（约2.07bps/Hz）。性能降低但计算复杂度会更一步降低，增加应用价值；
4.2 天线数为128
图中B-MIMO 全称为Beamspace MIMO 代表的意义是 波束空间MIMO
天线数为128:
横向对比：当SNR从-30增加到0dB，频谱效率整体上从0增加到19.63bps/Hz;
纵向对比：从最优到最劣：原始ZF预编码、本论文的混合预编码、比特近似精度取2的混合预编码、比特精度取1的混合预编码、波束空间MIMO方案；
以牺牲不足1bps/Hz的频谱效率，却能极大降低计算复杂度，降低MIMO对射频硬件的要求；
MATLAB代码运行时间：天线数为4时运行3秒左右，天线数为128时运行50.199667秒；
在信噪比为0dB时，本方案PZF频谱效率为18.35bps/Hz。设定4G基站采用4根天线，平均频谱效率为2.9 bps/Hz。设定5G基站天线数为64，平均频谱效率为10 bps/Hz【数据来源《物联网智库》】。