Multi-Antenna Systems

By

Ahmad Nasrallah 730030

Baha`a Almashaqbeh

Mohannad Abu Al-Nadi 731688

Yazan Al-Najar 731505

Hamzeh Al-Sayyed 734575

Supervised by

Dr. Abdelkarim Bayati

Hashemite University

Department of Electrical Engineering

2010/2011

Abstract

The use of multi antenna at the transmitter and receiver, also called multiple input

multiple output (MIMO), is a standard method for achieving the performance of

digital communication system, (MIMO) technique can improve the data rate and the

system performance with no additional power nor bandwidth.

In this thesis the multiple-input Multiple-output (MIMO) system will be investigated

using the QAM modulation technique with number of transmitters NT and number of

receivers NR showing the system performance and Bit error rate (BER), and

comparing it with the Single input Single output (SISO) we assuming that we have

Rayleigh channel

with large diversity order (no correlation between the two

transmitters), and also we add an additive with Gaussian noise (AWGN) to the

transmitted symbols.

i

List of abbreviations

NT : Number of transmitters.

NR : Number of receivers.

Ts : Symbol period.

Tb: Bit period.

a : Symbol vector.

α : Envelope of Rayleigh distribution.

ɸ : Phase of Rayleigh distribution.

: Detected symbol vector.

: Zero forcing symbol detection.

H: channel matrix gain.

H+ : Moore-Penrose Pseudo-Inverse of H.

: Hermitian transpose of H.

Q( ): Quantizer.

r : received vector.

SNR: Signal or symbol to noise ratio.

No : additive white noise power.

Pa: symbol power.

W : Weighting Matrix

ii

List of Figures

Figure 1, Illustration of MIMO system.

Figure 2, diagram representation for MIMO algorithm.

Figure 3, signal constellation.

Figure 4, System performance.

Figure 5, Envelope of Rayleigh distribution.

iii

Table of Contents

Abstract ……..…………………………………………………i

List of Abbreviation ….…………………………………..……ii

List of Figures ………….………………………………...……iii

Introduction ……………..………………………………..……v

1. The MIMO System …..…………………………….………..1

2. Channel Model for Multi Antenna System ………..…..……3

2.1 Slow frequency-nonselective model

2.2 Channel Estimation

3. Detection algorithm ………………………………………...6

3.1 Zero Forcing Technique

3.2 Linear Least Square (LLS) Technique

4. System Evaluation Performance …………………………....7

5. Channel Capacity for MIMO System ………………………9

Appendix A ( QAM Modulation Scheme ) …………………..10

Appendix B ( Rayleigh Channel ) ……………………………11

Appendix C ( MATLAB Code ) …………………………...…12

iv

Introduction

Recent research on wireless communication systems has shown that using multiple

antennas at both transmitter and receiver offers the possibility of wireless

communication at higher data rates compared to single antenna systems. The

information-theoretic capacity of these multiple-input multiple-output (MIMO)

channels was shown to grow linearly with the smaller of the numbers of transmit and

receive antennas in rich scattering environments.

There are several channel model in wireless communication system, we will assume

the simplest channel model, slow frequency nonselective (slow flat) channel in our

discussion.

The earliest ideas in this field go back to work by A.R. Kaye and D.A. George (1970)

and W. van Etten (1975, 1976). Jack Winters and Jack Salz at Bell Laboratories

published several papers on beamforming related applications in 1984 and 1986.

We assume that there is no correlation between the transmitters; in other word, there

is no correlation between the elements of the channel matrix.

In the last we introduce brief information about the capacity and diversity of multi

antenna communication system.

We will demonstrate bit error rate (BER) for various number of transmitter and

receiver using simulation MATLAB, for code [See Appendix C].

v

1. The MIMO Systems

Techniques that use arrays of multiple transmit and receive antennas that offers high

capacity to present and future wireless communications systems, by increase the data

rate using it’s multiple antenna to separate the data streams into multi-streams each

transmitted by single antenna in sequence . Multiple-input multiple-output (MIMO)

systems provide for a linear increase of capacity with the number of antenna elements,

which is a significant increase over single-input single-output (SISO) systems. To

evaluate the performance of MIMO systems, the MIMO channel must be

appropriately modeled. As we said it is common to model the MIMO channel

assuming an independent quasi-static flat Rayleigh fading at all antenna components.

With a simple MIMO channel system consisting of NT transmit antennas and NR

receive antennas, so we can describe the channel matrix as

(1.1)

Where

The h11 symbol means that the channels distribution from the first transmit antenna to

the first receiver antenna which -in our case - independent for any other hij in the

channel matrix fig1.1 shows the MIMO system using transmitters and receivers

devices

1

Figure.1: Illustration of MIMO system

2

2. Channel Model for Multi Antenna System

2.1 slow frequency-nonselective model

Assume a as symbol vector transmitted by NT transmitters, into a wireless channel

with dimension NR× NT, NR is the number of receivers, for MIMO system, NT > 1 and

NR > 1.

In multi antenna systems the general channel model is as following

(2.1)

Where Hij(t;τ) is the path gain of the jth

transmitter with the ith

receiver, Hij the channel

gains are identically distributed and statistically independent from channel to channel

If the channel is flat fading the general expression can be reduced as follow

(2.2)

If the channel is slow fading the general expression can be reduced as follow

(2.3)

if the channel assumed to be slow fading and flat fading the last expression will be

reduced to a constant matrix usually matrix with complex elements (Rayleigh

Channel).

In our estimation we assume the channel to be flat and slow fading .

3

For a digital signal a is transmitted over a fading multipath channel H, and the

channel is assumed to be a Rayleigh distribution channel [see Appendix B], thermal

noise is generated at the receiver and it is modeled by additive white Gaussian noise

(AWGN) n, the received symbol can be expressed as

r = Ha + n (2.4)

, 0 < t < Ts

The block diagram representation for MIMO algorithm is shown in figure.2.

Figure.2: diagram representation for MIMO algorithm

4

2.2 Channel Estimation

The detectors require knowledge on the channel impulse response (CIR), which can be

provided by a separate channel estimator. Usually the channel estimation is based on

the known sequence of bits, which is unique for a certain transmitter and which is

repeated in every transmission burst. Thus, the channel estimator is able to estimate

CIR for each burst separately by exploiting the known transmitted bits and the

corresponding received samples.

In MIMO systems the transmitted signal will take many paths to the receiver so the

channel estimation will yield a matrix that containing each path’s envelop and phase

(complex matrix)

Least-squares (LS) channel estimation

The LS channel estimates are found by minimizing the following squared error

quantity

2 (2.6)

(2.7)

Where ( ) +

and ( )-1

denote the Hermitian and inverse matrices, respectively.

EX: for 4×2 MIMO channel using MATLAB simulation.

The generated H was :

H = -0.4326 + 0.3273i -1.1465 - 0.5883i

-1.6656 + 0.1746i 1.1909 + 2.1832i

0.1253 - 0.1867i 1.1892 - 0.1364i

0.2877 + 0.7258i -0.0376 + 0.1139i

The Estimated H is :

H_estim = -0.4276 + 0.3290i -1.1373 - 0.5869i

-1.6618 + 0.1764i 1.1953 + 2.1770i

0.1311 - 0.1864i 1.1869 - 0.1436i

0.2920 + 0.7244i -0.0292 + 0.1193i 5

3. Detection algorithm

There are many detection algorithms can be used to recover the transmitted signal

again from received vector.

3.1 Zero forcing technique

Zero-Forcing (ZF) receiver is a low-complexity linear detection algorithm that outputs

Where H+ denotes the Moore-Penrose Pseudo-Inverse of H, which is a generalized

inverse that exists even when H is not a square matrix.

3.2 Linear Least Square (LLS) Technique

The LLS detection technique outputs the estimate data by

Where is a linear estimator given by

= Wr (3.4)

Where W is chosen to minimize

(3.5)

For the model here, where H and n (noise vector) are Gaussian, the LLSE estimator

matrixes given by

Where W is the weighting matrix and:

SNR denote the signal-to-noise ratio

N0 denote the noise value

INR an NR× NR identity matrix 6

( 3.3)

( 3.1)

( 3.2)

4. System Performance

We evaluate the performance of MIMO systems using a QAM modulation scheme by

simulation MATLAB, QAM Modulation introduced in [Appendix A].

Figure.3 show the signal constellation for 4-QAM and with Eb/No equal to 0 dB.

Figure.3: signal constellation

For 4-QAM and with Eb/No from 0dB to 10 dB, figure.3 shows the performance of

three communication systems, one is SISO, and the other are:

1) NT = 2 and NR = 4

2) NT = 2 and NR = 2

7

Figure.3: System performance

From figure.3, we can see that the BER is improved in the case of (NT = 2) and

(NR = 4), but it increase in the case of (NT = 2) and (NR = 2), and this leads to a

basic concept in MIMO communication system, " the number of receiver must be

larger than number of transmitter to improve the system performance".

8

5. Channel Capacity for MIMO System

For a single input-single output communication system, also called SISO, and based

on shannon's capacity theorem, capacity is a measure of the maximum transmission

rate for reliable communication in a given channel, this means that we can increase

the transmission rate to some number called channel capacity with no loss of

reliability, shannon's channel coding theorem says that " the basic limitation that noise

causes in a communication channel is not on the reliability of communication but on

the speed of communication.

In fact, the capacity of a wireless communication system depends on a many factor,

noise, number of transmitter and receiver, multipath components, etc…, some of these

factor are fixed, where as other can be determine to maximize the performance of a

communication system like number of transmitter and receiver, this is the main idea

of MIMO systems.

For a single input-single output communication system, and an additive white

Gaussian noise ( AWGN ) channel, the channel capacity C can be expressed by

(4.1)

Where is the average signal to noise ratio at the receiver, and expressed as

, and is the average symbol power.

According to Foschini and Gans, the capacity of a MIMO wireless communication

system for NT transmitter and NR receiver, If the noise is AWGN, such that the entries

of n random variables with equal variance No, and In the particular case, where the

transmitter does not know the channel H, the transmitted power is equally distributed

among the transmit antennas, In this case, the capacity expression reduces to

(4.2)

Where is the identity matrix with diminution NR × NR .

9

Appendix A

QAM Modulation Scheme

In the digital QAM, a finite number of at least two phases and at least two amplitudes

are used. PSK modulators are often designed using the QAM principle, but are not

considered as QAM since the amplitude of the modulated carrier signal is constant.

QAM is used extensively as a modulation scheme for digital telecommunication

systems.

The transmitted signal in QAM is

. m=1,2,3,……,M

Where & are the information-bearing signal amplitude of the quadrature

carriers and g(t) is the signal pulse.

Now, from last equation of , we can find the ortho-normal functions are:

Where is the Baseband Signal Energy.

Now, the transmitted signal can be written in term of the orthonormal function as:

Which results in the in vector representations of the form

10

Appendix B

Rayleigh Channel

Multipath signal at the receiver can be expressed as a summation of two Gaussian

variables (X+jY) (i.e.: envelop of the received signal and the phase) after make the

following transformation we will get Rayleigh PDF as envelop and the phase as

uniform distribution

Rayleigh PDF

,

Rayleigh distribution for a given mean and variance is shown in Figure.4.

Figure.5: Envelope of Rayleigh distribution 11

Appendix C

MATLAB Code

function multi_test(not,nor,bitt,bitd)

%---------------------------------------------------------------------------------------------

% Varible Defintion

clc

bitn = 500; % number of integer using in number of training data

bitn_d = 10000; % number of integer using in number of sending data

EbNo_min = 0; % Minimum Signal to noise ratio

EbNo_step = 1; % Signal to noise ratio step

EbNo_max = 10 % Maximum Signal to noise ratio

M = 4; % number of symbols

k = log2(M); % number of bits per symbol

iter = 25; % number of iteration in computing BER

if not==1 && nor==1

nob=bitt;

nob_d=bitd;

else

nob = not^2*log2(M)*bitn; % number of training data

nob_d = not^2*log2(M)*bitn_d; % number of sending data

end

12

%---------------------------------------------------------------------------------------------

u = 1; % number of SNR point axis

ber_sum = 0; % initialization for summation of bir error rate

%---------------------------------------------------------------------------------------------

% Generation of Data & Symbols & Channel

H = (randn(nor,not)+randn(nor,not)*j); % generation of Rayleigh channel

a = randint(1,nob); % Training Data generation

symbol_n = bi2de(reshape(a,k,nob/k)'); % symbol number,ie(11 = 3)

S = modulate(modem.qammod(M),symbol_n); % QAM modulation

A = reshape(S,not,(length(S)/not)); % Symbols transmitted from each antenna

a_d = randint(1,nob_d); % Data generation

sn_d = bi2de(reshape(a_d,k,nob_d/k)'); % Symbols number

S_d = modulate(modem.qammod(M),sn_d); % Qam modulation

A_d = reshape(S_d,not,(length(S_d)/not)); % Symbols tranmitted

13

for EbNo = EbNo_min:EbNo_step:EbNo_max

SNR = EbNo + 10*log10(k)-10*log10(not);

% symbol to noise ratio for each transmitting antenna

clc

display(['Wait....' num2str(round((EbNo-EbNo_min)/(EbNo_max-

EbNo_min)*100)) '% done for TX='...

num2str(not) ',RX=' num2str(nor) ]);

for i = 1:iter % number of average bit error rate

%---------------------------------------------------------------------------------------------

% Least Square Channel Estimation

R = awgn(H*A,SNR); % received symbols

H_estim = R*A'*inv(A*A'); % The Estemated channel

%---------------------------------------------------------------------------------------------

% Transmirring Data over estimatied channel

R_d = awgn(H*A_d,SNR);

RDATA = pinv(H_estim)*R_d; % Received symbols

RDATA = reshape(RDATA,1,not*length(RDATA));

14

%---------------------------------------------------------------------------------------------

% Symbols Constellation

% Z = scatterplot(RDATA,1,0,'r.');

% grid on

% hold on

% scatterplot(S,1,0,'black*',Z);

%---------------------------------------------------------------------------------------------

% Demodulation and BER Evaluation

S_estim = demodulate(modem.qamdemod(M),RDATA);

% Decision and Demodulation

r_estim = de2bi(S_estim)';

a_estim = reshape(r_estim,1,nob_d); % Estimated Data

% Bit Error Rate Evaluation between Source and estimation Data

[number_of_errors,ber] = biterr(a_d,a_estim);

ber_sum = ber_sum + ber;

end

bit_e_rate(u) = ber_sum/iter; %#ok<AGROW>

ber_sum = 0;

u = u+1;

end

15

%--------------------------------------------------------------------------------------------

% BER Vs Eb/No Curve ( System Performance)

EbNo = EbNo_min:EbNo_step:EbNo_max;

semilogy(EbNo,bit_e_rate); % BER versus SNR per bit

xlabel('SNR per bit in dB');

ylabel('BER');

title('BER vs SNR for Multiantenna');

hold on

sec=['\leftarrow TX=' num2str(not) ',RX=' num2str(nor) ];

text((EbNo_max+EbNo_min)/2,bit_e_rate(round(u/2)),sec ,'FontSize',8,'color','r')

if not~=1 || nor~=1

multi_test(1,1,nob,nob_d)

end

%-----------------------------------------------------------------

% Theorstical BER for SISO

%divorder = 1;

%ber = berfading(SNR,'qam',M,divorder);

%semilogy(SNR-10*log10(k),ber);

%asa=' \leftarrow Theoretical SISO';

%text((EbNo_max+SNR_min)/2,ber(round(u/2)),asa

,'FontSize',8,'color','r')

16

%---------------------------------------------------------------------------------------------

%Symbol Constellation for the highest SNR

%Z = scatterplot(RDATA,1,0,'r.');

%grid on

%scatterplot(S,1,0,'black*',Z);

%title('Symbol Constellation for the highest SNR')

%---------------------------------------------------------------------------------------------

%display('the Old H was :' )

%display(H)

%display('the New H is :')

%display(H_estim)

End

17