Approximate message passing (AMP) for Massive MIMO detection, MATLAB codes provided

December 4, 2015 Uncategorized No comments

Note: We have recently proposed to combine LMMSE with AMP (and not using AMP alone); see Section VII in How to use AMP in Massive MIMO?

In this exposition, we want to emphasize the approximate message passing (AMP) has superior complexity when serving Massive MIMO uplink detection problems, although AMP was initially proposed for solving LASSO [DMM09]. Regarding expository details about why AMP works, please see [BM11].

The Massive MIMO architecture is to serve tens of users by employing hundreds of antennas, $\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{w}$ where the channel $\mathbf{H}\in\mathbb{C}^{m\times n}$ has its elements sampled from $\mathrm{\mathcal{N}}_{\mathbb{C}}(0,1/m)$ , $m\gg n$ , $\mathbf{y}\in\mathbb{C}^{m}$ is the received signal, AWGN noise components $w_{i}$ are i.i.d with $\mathrm{\mathcal{N}}_{\mathbb{C}}(0,\sigma^{2})$ ; regarding the transmitted $\mathbf{x}$ , we only assume that it’s zero mean and finite variance $\sigma_{s}^{2}$ .

Before incorporating the AMP algorithm, we should be well aware of two facts: 1. directly using maximum a priori (MAP) $\arg\max p(\mathbf{x}|\mathbf{y})$ or MMSE estimation $\mathbb{E}_{p(\mathbf{x}|\mathbf{y})}(X)$ to work with the exact prior degrade the necessity of employing AMP, because achieving a full diversity requires an extremely large set of constellation points, in which AMP works slowly while doing the moment matching process, not to mention problems about its inability to converge to the lowest fixed point. 2. In the CDMA multiuser detection theory [Verdu98, etc.], their “MMSE” detector does not mean the one working with exact prior , but rather the one assuming a Gaussian prior.

So we use a proxy prior for detecting $\mathbf{x}$ , i.e., assuming that $x_{i}\sim\mathrm{\mathcal{N}}_{\mathbb{C}}(0,\sigma_{s}^{2})$ , even though it may be inexact. In this occurrence, we have the signal power $\sigma_{s}^{2}=2$ in QPSK, $\sigma_{s}^{2}=10$ in 16QAM, etc. So the target function becomes:

$\min\left\Vert \mathbf{y}-\mathbf{H}\mathbf{x}\right\Vert ^{2},s.t.x_{i}\sim\mathrm{\mathcal{N}}_{\mathbb{C}}(0,\sigma_{s}^{2})$

The AMP algorithm to solve the above problem only requires three lines (see [BM11], or our next paper which delivers a universal version of this algorithm):

$\begin{equation} \mathbf{r}^{t}=\mathbf{y}-\mathbf{H}\mathbf{x}^{t-1}+\frac{n}{m}\frac{\sigma_{s}^{2}}{\sigma_{s}^{2}+\alpha^{t-1}}\mathbf{r}^{t-1}\label{eq:26-1-1-1-1-1-1} \end{equation}$

$\begin{equation} \alpha^{t}=\sigma^{2}+\frac{n}{m}\frac{\alpha^{t-1}\sigma_{s}^{2}}{\sigma_{s}^{2}+\alpha^{t-1}}\label{eq:beta-1-1-1-1-1-1-1} \end{equation}$

$\begin{equation} \mathbf{x}^{t}=\frac{\sigma_{s}^{2}}{\sigma_{s}^{2}+\alpha^{t}}(\mathbf{H}^{*}\mathbf{r}^{t}+\mathbf{x}^{t-1})\label{eq:-1-1-1-2-1-1} \end{equation}$ where the initialization is to let $\mathbf{r}^{0}=\mathbf{0}$ , $\mathbf{x}^{0}=\mathbf{0}$ , $\alpha^{0}=\sigma_{s}^{2}$ . In terms of complexity, it only costs $2mn\times\mathrm{(\#Iteration)}$ . Also, according to the second equation of the algorithm, it is converging extremely fast. On the contrary, MMSE has complexity $O(mn^{2})$ . It is noteworthy that known approximation methods to MMSE, such as Richardson’s method or Newman series approximation, both fall behind the complexity-performance trade-off of AMP according to our simulations.

Now we share simulation results as well as MATLAB source codes.

%   AMP detector in Massive MIMO
%   written by Shanxiang Lyu (s.lyu14@imperial.ac.uk)
%   Last updated on 04/12/2015
function main()
clc;clear; close all;
m=128;% # of received antennas
n=16;% # of users
SNRrange=[1:20];
count=0;
for s=SNRrange
SNRdb=s;
    for monte=1:1000
    x=(2*randi([0,1],n,1)-ones(n,1))+sqrt(-1)*(2*randi([0,1],n,1)-ones(n,1));
    sigmas2=2;%signal variance in QPSK
    H=1/sqrt(2*m)*randn(m,n)+sqrt(-1)/sqrt(2*m)*randn(m,n);
    sigma2=2*n/m*10^(-SNRdb/10); %noise variance in control by SNR in DB
    w=sqrt(2*sigma2)*randn(m,1)+sqrt(-1)*sqrt(2*sigma2)*randn(m,1);
    y=H*x+w; %channel model

    %iterAMP is # of iterations in AMP
    iterAMP1=2;
    xhat1=AMP(y,H,sigma2,sigmas2,iterAMP1,m,n);
     iterAMP2=4;
    xhat2=AMP(y,H,sigma2,sigmas2,iterAMP2,m,n);

     x_mmse=(sigma2/sigmas2*eye(n)+H'*H)^(-1)*H'*y;
     x_mmse=sign(real(x_mmse))+sqrt(-1)*sign(imag(x_mmse));
    errorAMP1(monte)=sum(x~=xhat1);
    errorAMP2(monte)=sum(x~=xhat2);
    errorMMSE(monte)=sum(x~=x_mmse);
    end
    count=count+1;
serAMP1(count)=mean(errorAMP1);
serAMP2(count)=mean(errorAMP2);
serMMSE(count)=mean(errorMMSE);
end
figure(1)% plot the SER
semilogy(SNRrange,serAMP1,'-+r', SNRrange,serAMP2,'-pk',SNRrange, serMMSE,'-ob');
grid on;
legend(['AMP iteration=' int2str(iterAMP1)], ['AMP iteration=' int2str(iterAMP2)], 'MMSE');
xlabel('SNR (dB)'); ylabel('SER');
title(['BER performance comparison in system m= ' int2str(m)  '  n=' int2str(n)]);
end
function xhat=AMP(y,H,sigma2,sigmas2,iterAMP,m,n)
%   AMP detector in Massive MIMO
%   written by Shanxiang Lyu (s.lyu14@imperial.ac.uk)
%   Last updated on 04/12/2015
    r=zeros(m,1);
    xhat=zeros(n,1);
    alpha=sigmas2;%initial estimation variance
        for t=1:iterAMP
        r=y-H*xhat+(n/m)*sigmas2/(sigmas2+alpha)*r;
        alpha=sigma2+(n/m)*sigmas2*alpha/(sigmas2+alpha);
        xhat=(sigmas2/(sigmas2+alpha))*(H'*r+xhat);
        end
    xhat=sign(real(xhat))+sqrt(-1)*sign(imag(xhat));
end

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Approximate message passing (AMP) for Massive MIMO detection, MATLAB codes provided

Shanxiang Lyu

Ph.D. of post-quantum cryptography

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