Lecture 5

MIMO Channel Equalisation MATLAB Programming Conclusions 

DSP Applications for Wireless Communications: 

Linear Equalisation of MIMO Channels 

Dr. Waleed Al-Hanafy 

waleed alhanafy@yahoo.com 

Faculty of Electronic Engineering, Menoufia Univ., Egypt 

Digital Signal Processing (ECE407) — Lecture no. 5 

August 10, 2011 

Dr. Waleed Al-Hanafy Digital Signal Processing (ECE407) — Lecture no. 5 

Linear Equalisation of MIMO Channels


Overview 

1 MIMO Channel Equalisation 

2 MATLAB Programming 

3 Conclusions 




System Model 

v 

s 

y ˜s 

H W H 

y = Hs+v (1) 

y — noisy N r-dimensional received vector 

s — N t-dimensional transmit vector ∈ S N is assumed to be a 

spatially-uncorrelated and uniformly distributed complex random vector process 

with zero-mean and variance σ 2 s (i.e. R ss = E [ ss H] = σ 2 sI Nt 

v — noise vector with dimension N r ×1 drawn from CN ( 0,σ 2 v) 

, or 

equivalently R vv = E [ vv H] = σ 2 vI Nr 

H — N r ×N t flat-fading channel with entries h ij are assumed i.i.d. complex 

Gaussian random variables with zero-mean and unit-variance E [ |h ij| 2] = 1, i.e., 

h ij ∈ CN (0,1) 




Zero-Forcing (ZF) Equaliser 

The ZF approach is a very simple linear filter scheme that is accomplished by 

computing the Moore-Penrose pseudo-inverse of the channel matrix, H + . Accordingly, 

nulling in the ZF criterion is equivalent to completely cancelling the interference 

contributed by streams of other users (transmitting antennas). However, ZF receivers 

generally suffer from noise enhancement. The ZF filter is therefore given by 

( −1H 

WZF H = H+ = H H) H H , (2) 

and its output as 

˜s ZF = W H ZF y = s+ (H H H) −1H H v (3) 

The error covariance matrix is therefore given by 

Φ ZF = E 

[ 

(˜s ZF −s)(˜s ZF −s) H] ( −1 

= σv 

2 H H) H (4) 

Noticeably, by referring to (4), it is evident that small eigenvalues of H H H will lead to 

significant errors due to noise amplification. This, in fact, represents the main 

drawback of the ZF filter design as it disregards the noise term from the overall design 

and focuses only on perfectly removing the interference term from signal s 




Minimum Mean Square Error (MMSE) Equaliser 

An improved performance over ZF can be achieved by considering the noise term in 

the design of the linear filter W H . This is achieved by the MMSE equaliser, whereby 

the filter design accounts for a trade-off between noise amplification and interference 

suppression. The MMSE filter is obtained by solving for error minimisation of the error 

criterion defined by [ ] ( [ 

ϕ = E e H e = tr E ee H]) (5) 

where the error vector e d = s−˜s = s−W H y. Minimisation of ϕ leads to the 

Wiener-Hopf equation 

WMMSE H Ryy = Rsy (6) 

This equation can also be obtained directly by invoking the orthogonality principle 

which states that the estimate ˜s achieves minimum mean square error if the error 

sequence e is orthogonal to the observation y, i.e., their cross-correlation matrix has to 

be the zero matrix E [ ey H] = 0. After some algebraic manipulation, the linear 

MMSE-sense filter is given by 

( ) −1 

WMMSE H = H H H+ σ2 v 

σs 

2 I Nt H H (7) 




MMSE Equaliser (cont’d) 

Similar to (4) the error covariance matrix using the MMSE filter in (7) is given as 

( ) −1 

Φ MMSE = σv 

2 H H H+ σ2 v 

σs 

2 I Nt (8) 

Obviously by comparing (8) and (4), the error rate of the MMSE solution Φ MMSE is 

less than its ZF counterpart Φ ZF specifically at low SNR defined as 

[ ] 

E ‖s‖ 2 2 

SNR = [ ] = tr(Rss) 

E ‖v‖ 2 tr(R = σ2 sN t 

2 vv) σvN 2 = P budget/N 0 (9) 

r 

where P budget is the total transmit power budget and N 0 is the total noise power at 

the receiver. At high SNR, the second term in (8) will vanish, which leads to 

asymptotic error performance similar to the ZF filter. Compared to the ZF filter in (2), 

the MMSE filter in (7) can be viewed as a“regularised”expression by a diagonal matrix 

of entries σ2 v 

σs 

2 , which is equal to the reciprocal of the SNR in (9) for equal numbers of 

transmit and receive antennas. This regularisation introduces a bias that gives a much 

more reliable result than (2) when the matrix is ill-conditioned, A matrix is said to be 

ill-conditioned if its condition number (the absolute ratio between the maximum and 

minimum eigenvalues) is too large., and/or the estimation of the channel is noisy. 




MATLAB Programing 

clc; clear all; 

nt = 4; nr = 4; d = 1000; MinErrs = 100; MaxData = 1e6; chs = 10; 

snr dB = 0:3:21; snr = 10.ˆ(snr dB/10); Lsnr = length(snr); 

berZF = zeros(1,Lsnr); berMMSE = zeros(1,Lsnr); 

for e = 1:chs 

disp([’ch ’ num2str(e) ’/’ num2str(chs)]); 

H = (randn(nr,nt)+sqrt(-1)*randn(nr,nt))/sqrt(2); 

G ZF = inv(H’*H)*H’; 

nErZF = zeros(1,Lsnr); nErMMSE = zeros(1,Lsnr); data = zeros(1,Lsnr); 

for z = 1:Lsnr 

disp([’ step ’ num2str(z) ’/’ num2str(Lsnr) ’: SNR(dB) = ’ num2str(snr dB(z))]); 

G MMSE = inv(H’*H + (1/snr(z))*eye(nt))*H’; 

while((nErZF(z)


MATLAB Programing (cont’d) 

aTempZF = G ZF*r; aTempMMSE = G MMSE*r; 

aHatZF = FnDecodeQAM(aTempZF,’QPSK’,1); 

aHatMMSE = FnDecodeQAM(aTempMMSE,’QPSK’,1); 

[dumy dumy nBerZF dumy] = FnSerBerQAM(a,aHatZF,4,’gray’); 

[dumy dumy nBerMMSE dumy] = FnSerBerQAM(a,aHatMMSE,4,’gray’); 

nErZF(z) = nErZF(z) + nBerZF; 

nErMMSE(z) = nErMMSE(z) + nBerMMSE; 

end 

berZF(z) = berZF(z) + nErZF(z)/data(z); 

berMMSE(z) = berMMSE(z) + nErMMSE(z)/data(z); 

end 

end 

berZF = berZF/chs; berMMSE = berMMSE/chs; 

semilogy(snr dB,berZF,’k-s’,snr dB,berMMSE,’m-o’); grid on 

legend(’ZF’,’MMSE’); grid on; xlabel(’SNR (dB)’); ylabel(’BER’); 

. 




Results 

A 4×4 MIMO system with QPSK transmission is considered. It is clearly noted that 

the MMSE filter outperforms its ZF counterpart and their BER performance converges 

at higher SNR where the regularisation factor σ 2 v/σ 2 s ≪ 1 

Nt = 4, Nr = 4, NumChan = 300 

10 0 SNR [dB] 

ZF 

MMSE 

10 −1 

10 −2 

BER 

10 −3 

10 −4 

0 5 10 15 20 25 30 




Conclusions 

Concluding remarks 

Linear equalisation systems for linear MIMO systems is studied 

Both ZF and MMSE equalisers are derived and analysed 

Simulation results comparisons are shown 

.

Lecture 5

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