Gourab's M.tech Thesis

102
PERFORMANCE ANALYSIS OF ALAMOUTI CODED MIMO SYSTEMS IN RAYLEIGH FADING CHANNEL By GOURAB MAITI 09/ ECE/ 402 Under the Supervision of ANIRUDDHA CHANDRA Thesis submitted in the partial fulfillment of the requirement for the degree of Master of Technology in Telecommunication Engineering DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING NATIONAL INSTITUTE OF TECHNOLOGY, DURGAPUR WEST BENGAL 713209, INDIA May, 2011 A-PDF Merger DEMO : Purchase from www.A-PDF.com to remove the watermark

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This is a M.Tech thesis of Gourab Maiti. The thesis is on "Performance of Alamouti Coded MIMO Systems in Rayleigh Fading Channel".

Transcript of Gourab's M.tech Thesis

Page 1: Gourab's M.tech Thesis

PERFORMANCE ANALYSIS OF ALAMOUTI CODED

MIMO SYSTEMS IN RAYLEIGH FADING CHANNEL

By

GOURAB MAITI

09/ ECE/ 402

Under the Supervision of

ANIRUDDHA CHANDRA

Thesis submitted in the partial fulfillment of the

requirement for the degree of

Master of Technology in

Telecommunication Engineering

DEPARTMENT OF ELECTRONICS AND COMMUNICATION

ENGINEERING

NATIONAL INSTITUTE OF TECHNOLOGY, DURGAPUR

WEST BENGAL – 713209, INDIA

May, 2011

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Dedicated to

My Parents and Elder Brother

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NATIONAL INSTITUTE OF TECHNOLOGY, DURAPUR

WEST BENGAL – 713209

Certificate of Recommendation

This is to recommend that the work in the thesis entitled “Performance Analysis of Alamouti Coded MIMO systems in Rayleigh fading Channel” has been carried out by Mr. Gourab Maiti under my supervision and may be accepted in partial fulfillment of the requirement for the degree of Master of Technology in Telecommunication Engineering, at department of Electronics & Communication Engineering, NIT Durgapur.

Aniruddha Chandra

Assistant Professor

Department of Electronics and

Telecommunication Engineering,

NIT Durgapur

Gautam Kumar Mahanti

Professor and Head

Department of Electronics and

Telecommunication Engineering,

NIT Durgapur

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NATIONAL INSTITUTE OF TECHNOLOGY, DURAPUR

WEST BENGAL – 713209

Certificate of Approval

The foregoing thesis is hereby approved as a creditable study of engineering subject to warrant its acceptance as a prerequisite to obtain the degree for which it has been submitted. It is understood that by this approval the unsigned don’t necessarily endorse or approve any statement made, opinion expressed or conclusion drawn therein but approved the thesis only for the purpose for which it is submitted.

*Only in case the thesis is approved

Project Guide

External Examiner

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ACKNOWLEDGEMENT

I would like to acknowledge many people who helped me during the course of

this work.

First, I would like to thank my thesis supervisor, Assistant Professor Aniruddha

Chandra, for providing me with the right balance of guidance and independence in my

research. I am greatly indebted to him for his full support, constant encouragement and

advice both in technical and non-technical matters. His broad of expertise and superb

intuition have been a source of inspiration to me over the past two years. Her detailed

comments have greatly influenced my technical writing, and are reflected throughout the

presentation of this dissertation.

I would like to thank my friends: Pradipta Sarkar, Subhranil Koley and too many

to be listed here for their friendship, help and cheerfulness in this 2 years course. In

addition, I gratefully acknowledge the financial support of UGC.

Last, but certainly not the least, I would like to acknowledge the commitment,

sacrifice and support of my parents and elder brother, who have always motivated me. In

reality this thesis is partly theirs too.

May, 2011

Gourab Maiti

Roll No. 09/ ECE/ 402

Department of Electronics and

Telecommunication Engineering,

NIT Durgapur, West Bengal

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Abstract

Current and future wireless systems or standards like cellular mobile phones,

wireless local area network (WLAN), bluetooth, 4G all has to support multiple mode of

operations like voice, image, text, and video data, that require high data rate with low

error rate and wider coverage. Unfortunately, radio bandwidth and transmitted power are

among the most severely limited parameters during design. First of all, the radio

spectrum is a scarce resource that must be allocated to many different applications and

systems. For this reason spectrum allocation is controlled by regulatory bodies both

regionally and globally. Also mobile phones and other portable devices must be small,

low-power, and lightweight, so transmitted power is also restricted due to small battery

size. Again, wireless systems operate over a complex and harsh time-varying radio

channel which introduces severe multipath fading and shadowing, rendering the link

budget expensive for a typical capacity, outage probability and error rate requirements.

On the other hand, one resource that is growing at a very rapid rate is that of

processing power. Moore’s Law, which asserts a doubling of processor capabilities every

18 months, has been found to be quite accurate over the past 30 years, and its accuracy

promises to continue for at least a decade. Given these circumstances, there has been

considerable research effort in recent years aimed at development of novel signal

transmission techniques and advanced receiver signal processing methods that allow

significant increase in wireless capacity without an increase in the transmitted bandwidth

and power. Diversity combining is such a sophisticated spectral and power efficient fade

mitigation technique, which is required to improve radio link performance.

Apart from diversity, for higher data rate in limited bandwidth we considered M-

ary modulation schemes. Specially M-ary phase shift keying (MPSK) and M-ary

quadrature amplitude modulation (MQAM) are considered for their certain benefits like

spectral efficiency.

The objective of this thesis is to asses the performance, of systems over wireless

fading channels, when diversity techniques (transmit/ receive/ both) are employed. The final

goal is to provide the researchers or system designers an insight to make comparison and

tradeoff studies among the various systems employing diversity so as to determine the

optimum choice in the face of his or her available constraints. Extensive Monte Carlo

simulations were performed to validate the theoretical expressions.

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Contents Acknowledgements v

Abstract vi

List of Figures x

List of Acronyms xiii

Chapter 1 Introduction 1-4

1.1 Motivation 1

1.2 Thesis Objectives 2

1.3 Thesis Outline 3

Chapter 2 Background Materials 5-29

2.1 Introduction 5

2.2 Wireless Channel 6

2.2.1 Mobile Radio Propagation 7

2.3 Digital Modulation Schemes 9

2.3.1 M-ary Phase Shift Keying (PSK) 9

2.3.2 M-ary Quadrature Amplitude Modulation (MQAM) 11

2.3.3 Comparison among different M-ary Schemes 12

2.4 Performance Metrics 12

2.4.1 Capacity 12

2.4.2 Outage Probability 13

2.4.3 Symbol Error Rate (SER) 13

2.5 Receiver Diversity Schemes 15

2.5.1 Diversity Combining 16

2.5.2 Combining Methods 17

2.6 Multiple Input Multiple Output (MIMO) Systems 20

2.6.1 Narrowband MIMO Model 20

2.7 Space Time Coding (STC) 21

2.7.1 Space Time Block Code (STBC) 22

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2.7.2 Performance Comparison of Diversity-on-Receive and 26

Diversity-on-Transmit Schemes

2.8 Literature Survey 27

2.9 Chapter Summary 29

Chapter 3 Multi branch Switch-and- 30-45

Examine Combining in

Alamouti Coded MIMO Systems

3.1 Introduction 30

3.2 System Model and Description 31

3.3 Analysis of Performance Metrics 33

3.3.1 Capacity 33

3.3.2 Outage Probability 37

3.3.3 Symbol Error Rate (SER) 39

3.4 Chapter Summary 45

Chapter 4 Transmit Antenna Selection in 46-58

Alamouti Coded MISO Systems

4.1 Introduction 46

4.2 System Model and Description 46

4.3 Analysis of Performance Metrics 49

4.3.1 Capacity 49

4.3.2 Outage Probability 51

4.3.3 Symbol Error Rate (SER) 53

4.4 Chapter Summary 58

Chapter 5 Joint Transmit and Receive 59-80

Antenna Selection in Alamouti

Coded MIMO Systems

5.1 Introduction 59

5.2 System Model and Description 60

5.3 Analysis of Performance metrics 62

5.3.1 Capacity 62

Contents

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5.3.2 Outage Probability 66

5.3.3 Symbol Error Rate (SER) 69

5.4 Chapter Summary 80

Chapter 6 Comparative Studies and 81-85

Discussions

6.1 Summary of Contribution 81

6.1.1 Comparative study among different Schemes 81

6.2 Limitations 84

6.3 Future Scopes 85

Bibliography 86-88

Publications Based on Thesis Work 89

Contents

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List of Figures

Figure 2.1 Loss, Shadowing and Multipath versus Distance 9

Figure 2.2 Signal space diagram for coherent BPSK 10

Figure 2.3 Signal space diagram for octa-phase shift keying 11

Figure 2.4 Signal space diagram for M-ary QAM for M=16 12

Figure 2.5 Pre-Detection Receiver 16

Figure 2.6 Several types of combining (a) MRC, (b) SC, (c) SSC, (d) SEC 19

Figure 2.7 MIMO Systems 20

Figure 2.8 Block diagram of orthogonal space-time block encoder 23

Figure 2.9 Transmission Matrix 24

Figure 2.10 System model of Alamouti Scheme 25

Figure 2.11 Comparison of average signal-to-noise ratio vs. bit error rate 27

performance of coherent BPSK over flat Rayleigh fading

channel for three configurations

Figure 3.1 Transmission model of a 2×L MIMO system employing 32

Alamouti code at transmitter and pre-detection switch

and examine combining at receiver

Figure 3.2 Capacity curves for Alamouti based SEC system with fixed 35

threshold (th = 3 dB) for different numbers of Rx antennas

Figure 3.3 Capacity curves for Alamouti based SEC system with optimum 36

threshold (as found from Table I) for different numbers of Rx

antennas

Figure 3.4 Outage probability curves for Alamouti based SEC system with 38

fixed threshold (th = 2 dB) for different numbers of Rx antennas

Figure 3.5 Outage probability curves for Alamouti based SEC system with 38

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optimum threshold for different numbers of Rx antennas

Figure 3.6 SER curves for Alamouti based SEC system with fixed 42

threshold (th = 3 dB) for different M and for different

numbers of Rx antennas

Figure 3.7 Optimum BER curves for Alamouti based SEC system with 42

fixed threshold (th = 3 dB) for BPSK (M=2) and for

different numbers of Rx antennas

Figure 3.8 SER curves for Alamouti based SEC system with fixed 45

threshold (th = 3 dB) for different M and for different

numbers of Rx antennas

Figure 4.1 Transmission model of Ltx1 MISO system employing Alamouti 47

code at transmitter

Figure 4.2 Capacity curves for Alamouti based MISO system for different 51

number of transmit antennas

Figure 4.3 Outage probability curves of Alamouti based MISO system for 53

different number of transmit antennas

Figure 4.4 SER curves for Alamouti based MISO system using MPSK for 55

different number of transmit antennas

Figure 4.5 SER curves of Alamouti based MISO system using MQAM for 58

different number of transmit antenna

Figure 5.1 Transmission model of a Lt×L MIMO system employing 61

Alamouti code at transmitter and pre-detection switch and

examine combining at the receiver

Figure 5.2 Capacity curves for Alamouti coded TAS employing SEC 66

system with fixed threshold (th = 3 dB) for different numbers

of Rx antennas

Figure 5.3 Outage probability curves for Alamouti coded TAS employing 68

List of Figures

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SEC system with fixed threshold (th = 2 dB) for different

numbers of Rx antennas

Figure 5.4 SER curves for Alamouti coded TAS employing SEC system 76

with fixed threshold (th = 3 dB) for M= 4, 8 and for different

numbers of Rx antennas

Figure 5.5 SER curves for Alamouti coded TAS employing SEC system 80

with fixed threshold (th = 3 dB) for M=4 and for different

numbers of Rx antennas

Figure 6.1(a) Capacity curves for Alamouti based different schemes with a 82

fixed threshold (th = 3 dB) for different numbers of total

antennas

Figure 6.1(b) Outage probability curves for Alamouti based different schemes 83

with same switching threshold and target threshold dBoth 3=γ=γ

for different numbers of total antennas

Figure 6.1(c) SER curves for Alamouti based different schemes with a fixed 83

threshold (th = 3 dB) using 4-PSK for different numbers of

total antennas

Figure 6.1(d) SER curves for Alamouti based different schemes with a fixed 84

threshold (th = 3 dB) using 4-QAM for different numbers

of total antennas.

List of Figures

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List of Acronyms

Sl

No.

Notation Name of the function Expression Reference

1. )(zerf Error function due

z

u∞

π

22

(8.250.1) [23]

2. )(zerfc Complementary error function ( )zerf−1 (8.250.4) [23]

3.

)(zQ

Q-function duez

u

∞ −

π

2

2

2

1

(4.1) [9]

4. )(1 zE Exponential integration dtet

z

t∞

−−1 (5.1.1) [24]

5. )(zPq Poisson function z

q

v

v

ev

z −−

=

1

0 !

(26.4.21) [24]

6. ( )zΓ Gamma function dxex

xz∞

−−

0

1 (6.1.1) [24]

7. ( )za,γ Incomplete Gamma function dxex

zxa

−−

0

1 (8.350.1) [23]

8. ( )za,Γ Complementary Incomplete Gamma

function dxex

z

xa∞

−−1 (8.350.2) [23]

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Chapter 1

Introduction

By definition, the term wireless communication designates any radio

communication link between two terminals of which one or, both are either stationary or,

non-stationary. As an example, in common cellular systems the base station is fixed

while users carrying mobile stations are on the move. Apart from cellular telephony

which is quite familiar nowadays, other applications of wireless communications include

cordless technology, wireless LANs (e.g. HIPERLAN), personal area networks (e.g.

Bluetooth), wireless local loops (WLL) etc. Generally, wireless technologies provide the

last-mile solution, i.e., they are used in the last hop (to/ from the subscriber) in a network.

In recent years, we are experiencing huge growth rates in wireless and mobile

communication system due to the various important factors: advances in

microelectronics, high speed intelligent networks, positive user response and an

encouraging regulatory climate worldwide. For wireless communication, to achieve a

high data rate and a strong reliable signal at receiver, the number of cells should be

increased and the frequency reuse should be maximized. But the allocated area and the

spectrum is limited and/ or restricted which results in increased interference, cross talk

and performance degradation. Thus the most challenging task in current wireless

communication scenario is to achieve higher data rate, higher link reliability and wider

coverage with these limited spectrum bandwidth and improve the link performance which

may be realized through adopting diversity and different modulation schemes.

1.1 Motivation

Current wireless systems like cellular mobile phones, wireless local area network

(WLAN), bluetooth, mobile low earth orbit (LEO) satellite etc. all require very high data

rate (>100 mbps), lower delay, greater transmission reliability and wider coverage. But

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the limitations are fading, limited available spectrum and battery life of wireless portable

devices.

Diversity combining is such sophisticated spectral and power efficient fade

mitigation technique which is used to improve radio link performance (diversity gain),

higher transmission rate (multiplexing gain) and for wider coverage (low outage

probability). Diversity, where signal replicas are obtained through the use of either

temporal, frequency, spatial or polarization spacing, is an effective technique to mitigate

the multipath fading.

Also for higher data rate over wireless channel M-ary modulation schemes are

frequently used. The coherent M-ary schemes provide better error performance or require

lesser signal to noise ratio (SNR) to achieve a target symbol error rate when compared to

their non-coherent or differentially coherent counterparts. Out of coherent schemes, M-

ary phase shift keying (MPSK) and M-ary quadrature amplitude modulation (MQAM) is

often preferred over M-ary frequency shift keying (MFSK) as it is bandwidth inefficient.

Thus among M-ary modulation schemes we have selected MPSK and MQAM as

the desirable modulation schemes that are incorporated in our system models for their

certain benefits, discussed above.

1.2 Thesis Objectives

The main objective of the thesis is to study the performance analysis of Alamouti

coded multiple input multiple output (MIMO) systems in Rayleigh fading channel. To

tackle the problem, we have subdivided our main objective into the following three

different goals:

(1) Performance analysis of multibranch switch-and-examine combining in Alamouti

coded MIMO systems in Rayleigh fading channel.

(2) Performance analysis of transmit antenna selection in Alamouti coded MISO systems

in Rayleigh fading channel.

Chapter 1: Introduction

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(3) Performance analysis of joint transmit and receive antenna selection in Alamouti

coded MIMO systems in Rayleigh fading channel.

Thus the objective is to analyze such systems one by one, develop analytical

expressions for different performance metrics and verify the derived relations through

comprehensive simulation studies.

1.3 Thesis Outline

The rest of the thesis is organized as follows. The primary goal of chapter 2 is to

introduce basic concepts, models and notations that will be used throughout the thesis.

We begin in chapter 2 with a brief overview on the current and future requirements of

wireless services and some methods to fulfill that criteria in section 2.1. The next section

2.2 briefly discusses on wireless channel, specifically large scale fading and small scale

fading. Section 2.3 tells us about the digital modulation schemes mainly MPSK and

MQAM, their constellation diagrams and a brief comparison, whereas section 2.4 is

devoted to performance metrics, i.e. capacity, outage probability and symbol error rate

(SER). Section 2.5 talks about different types of receiver diversity schemes. Under

section 2.6 we discuss about MIMO systems. Section 2.7 tells us about the space-time

code (STC) used in MIMO systems. The next section, section 2.8 provides a brief

literature survey i.e. works on diversity, MIMO and STC on last ten years. Lastly the

chapter concludes with a chapter summary in section 2.9.

The primary goal of chapter 3 is to analyze the system employing Alamouti

coding, a type of diversity, at the transmitter side and multibranch switch-and-examine

combining (SEC) at the receiver side.

In chapter 4 we derive the performance metrics of a system employing transmit

antenna selection and Alamuti code.

In chapter 5, we show how a system performs if we employ both transmit antenna

selection and Alamouti code at the transmitter side and SEC as receiver diversity.

Chapter 1: Introduction

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The thesis ends with chapter 6, which consists of a comparative study among the

schemes that are presented in chapters 3, 4 and 5. Also some limitationss we have

discussed that should be kept in mind when we are adopting such schemes. We end the

chapter with future scopes.

Chapter 1: Introduction

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Chapter 2

Background Materials

2.1 Introduction

Current wireless systems require higher transmission rate with lower delay,

higher link reliability and wider coverage. The traditional resources that have been used

to add capacity to wireless systems are radio bandwidth and transmitter power.

Unfortunately, these two resources are among the most severely limited parameters

during design: radio bandwidth because of the very tight situation with regard to useful

radio spectrum, and transmitter power because mobile radio and other portable devices

must be small, low-power, and lightweight, which restrict their capabilities. Also,

wireless systems operate over a complex and harsh time-varying radio channel which

introduces severe multipath fading and shadowing, rendering the link budget expensive

for a typical symbol error rate (SER)/ bit error rate (BER) requirement.

Given these circumstances, there has been considerable research effort in recent

years aimed at development of novel signal transmission techniques and advanced

receiver signal processing methods that allow significant increase in wireless capacity

without an increase in the transmitted bandwidth and power. Diversity combining is such

a sophisticated spectral and power efficient fade mitigation technique, which are used to

improve radio link performance.

Diversity, where signal replicas are obtained through the use of either temporal,

frequency, spatial, or polarization spacing, is an effective technique to mitigate the

multipath fading. For example, an information bit can be transmitted simultaneously from

two antennas (linked by some form of coding), and then the signals can be combined

coherently at the receiver. If one of the spatial subchannels experiences a deep fade, it

may be possible to recover the information from the signal on the other spatial

subchannel. For each additional diversity branch, the chance of the combined signals

being severely attenuated decreases.

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The rest of the chapter is organized as follows. Fading in wireless channel is

described in section 2.2. Section 2.3 presents the different types of digital modulation

schemes addressed in this thesis followed by analytical expression for the theoretical

performance metrics (capacity, outage probability and error probability of corresponding

modulation schemes) in additive white Gaussian channel and wireless channel, in section

2.4. Section 2.5 is devoted to different diversity schemes. The next section, section 2.6

describes multiple input multiple output (MIMO) systems which is followed by space-

time coding (STC), a type of transmit diversity used in MIMO systems, in section 2.7.

Lastly, we present a brief literature survey on diversity and STC in section 2.8 and

conclude the chapter with a short summary in section 2.9.

2.2 Wireless Channel

Impairments in the propagation channel have the effect of disturbing the

information carried by the transmitted signal. Additive noise and multiplicative fading are

the two of several reasons for channel disturbances. The focus of this section is to

characterize the wireless channel by identifying the parameters of the corruptive elements

that distort the information carrying signal as it penetrates the propagation medium.

Basically, in idealized free-space model, the attenuation of radio frequency (RF)

energy between transmitter and receiver behaves according to an inverse-square law. The

received power expressed in terms of transmitted power is attenuated by a factor, ( )dLs ,

known as path loss or free space loss. When the receiving antenna is isotropic, this factor

is expressed as [1]

( )2

4

λ

π=

ddLs (2.1)

where d is the distance between the transmitter and the receiver, and λ is the wavelength

of the propagating signal.

In a wireless mobile communication system, a signal can travel from transmitter

to receiver over multiple reflective paths; this phenomenon is referred to as multipath

Chapter 2: Background Materials

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propagation. The effect can cause fluctuations in the received signal’s amplitude, phase,

and angle of arrival, giving rise to the terminology multipath fading.

2.2.1 Mobile Radio Propagation

Fading effects that characterize the mobile communication can be of two types:

large-scale and small-scale fading. Large-scale fading represents the average signal

power attenuation or path loss due to motion over large areas. This phenomenon is

affected by prominent terrain contours (hills, forests, billboards, clumps of buildings,

etc.) between the transmitter and receiver. The receiver is often represented as being

“shadowed” by such prominences. This is described in terms of a log-normally

distributed variation about the mean. Small-scale fading refers to the dramatic changes in

signal amplitude and phase that can be experienced as a result of small changes (as small

as a half-wavelength) in the spatial separation between a receiver and transmitter. Small-

scale fading often described by Rayleigh fading, because if the multiple reflective paths

are large in number and there is no line-of-sight signal component, the envelope of the

received signal is statistically described by a Rayleigh PDF. When there is a dominant

nonfading signal component present, such as a line-of sight propagation path, the

smallscale fading envelope is described by a Rician PDF [2].

Large Scale Fading

For the mobile radio application, the mean path loss, ( )dLp , as a function of

distance, d, between the transmitter and receiver is proportional to an nth power of d

relative to a reference distance 0d [2]

( )n

pd

ddL

α

0

(2.2)

( )dLp is often stated in decibels, as shown below

( )( ) ( )( ) ( )00 log10 ddndBdLdBdL sp += (2.3)

The reference distance 0d corresponds to a point located in the far field of the antenna.

Typically, the value of 0d is taken to be 1 km for large cells, 100 m for microcells, and 1

Chapter 2: Background Materials

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m for indoor channels. ( )dLp is the average path loss (over a multitude of different sites)

for a given value of d. The value of the exponent n ( )42 ≤≤ n depends on the frequency,

antenna heights, and propagation environment. In free space, n = 2. Measurements have

shown that for any value of d, the path loss ( )dLp is a random variable having a log-

normal distribution about the mean distant-dependent value ( )dLp [3]. Thus, path

loss ( )dLp can be expressed in terms of ( )dLp plus a random variable σX , as follows [2]:

( )( ) ( )( ) ( ) ( )dBXddndBdLdBdL sp σ++= 00 log10 (2.4)

where σX denotes a zero-mean Gaussian random variable (in decibels) with standard

deviation σ (also in decibels).

Small Scale Fading

When the received signal is made up of multiple reflective rays without any

significant line-of-sight component, the envelop amplitude due to small scale fading has a

Rayleigh probability density function (PDF), expressed as

( )

σ−

σ=

otherwise 0

0rfor 2

exp22

rr

rp (2.5)

where r is the envelope amplitude of the received signal, and 22σ is the predetection mean

power of the multipath signal. The Rayleigh faded component is sometimes called the

random or scatter or diffuse component.

Figure 2.1 illustrates the ratio of received-to-transmit power in dB versus log-

distance for the combined effect of path loss, shadowing and multipath. Where Pr, Pt are

the received power and transmitted power respectively.

Chapter 2: Background Materials

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Figure 2.1 Path loss, shadowing and multipath versus distance.

2.3 Digital Modulation Schemes

In digital passband transmission, the incoming data stream is modulated onto a

carrier (generally sinusoidal) with fixed frequency limits imposed by a bandpass channel

of interest. There are three basic signaling schemes and they are amplitude-shif keying

(ASK), frequency-shift keying (FSK) and phase-shift keying (PSK). In this section we

will discuss only about two digital modulation schemes that we used later as our

modulation schemes (a) phase shift keying (PSK) and (b) quadrature amplitude

modulation (QAM).

2.3.1 M-ary Phase Shift Keying

In this section we will focus on coherent PSK schemes like binary phase shift

keying (BPSK) and M-ary phase shift keying (MPSK).

Binary Phase Shift keying (BPSK)

In a coherent binary PSK [4] system, the pair of signals ( )ts1 and ( )ts2 used to

represent binary symbols 1 and 0, respectively, are defined as

( ) ( )

( ) ( ) ( )tfT

Etf

T

Ets

tfT

Ets

c

b

bc

b

b

c

b

b

π−=π+π=

π=

2cos2

2cos2

2cos2

2

1

(2.6)

Chapter 2: Background Materials

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where, bTt ≤≤0 , bT is a single bit period cf is carrier frequency and bE is the

transmitted signal energy per bit. Figure 2.2 illustrates the signal space diagram of

coherent BPSK. Where ( )tϕ is the basis function.

M-ary Phase Shift Keying (MPSK)

In case of M-ary PSK [4], the carrier takes on one of the M possible values,

namely, ( ) Mii π−=θ 12 , where i=1, 2,…, M. Accordingly, during each signaling

interval of duration T, one of the M possible signals

( ) ( ) ,12

2cos2

π+π= i

Mtf

T

Ets ci i =1, 2,….,M (2.7)

is sent where E is the signal energy per symbol. The signal constellation of M-ary PSK is

two dimensional. The M message points are equally spaced on a circle of radius E and

center at origin, as illustrated in Figure 2.3 for the case of octaphase shift keying (M=8).

The baseband message signals are denoted by si where i = 1, 2,.., 8.

0

Region 2Z Region 1Z

Decision

Boundary

(1)

Message

Point 1

(0)

Message

Point 2

Threshold

bE+ bE− ( ) ( )tf

Tt c

b

π=ϕ 2cos2

Figure 2.2 Signal space diagram for coherent BPSK.

Figure 2.3 Signal space diagram for octa-phase shift keying.

Message

Point s1

s2

s3

s4

s5

s6

s7

E−

E−

E

E 0

s8

Mπ Decision

region ( ) ( )tcf

Tt π=ϕ 2cos

21

( ) ( )tcfT

t π=ϕ 2sin2

2

Chapter 2: Background Materials

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2.3.2 M-ary Quadrature Amplitude Modulation (MQAM)

For MQAM [4], the information bits are encoded in both the amplitude and phase

of the transmitted signal. Thus, whereas both MPAM and MPSK have one degree of

freedom (amplitude or phase) in which the information bits are encoded, MQAM has two

degrees of freedom. As a result, MQAM is more spectrally-efficient than MPAM and

MPSK, in that it can encode the most number of bits per symbol for a given average

energy. The transmitted M-ary QAM signal for symbol k, is defined as

( ) ( ) ( ) ,...., , T; kttfbT

Etfa

T

Ets ckckk 2100 ,2sin

22cos

2 00 ±±=≤≤π−π= (2.8)

Where ka and kb are inphase and quadrature amplitude of the signal. 0E is the transmitted

symbol energy The signal ( )tsk consists of two phase-quadrature carriers with each one

being modulated by a set of discrete amplitudes, hence the name quadrature amplitude

modulation.

Depending on the number of possible symbols M, we may distinguish two distinct

QAM constellation: square constellation where the number of bits per symbol is even and

cross constellation where the number of bits per symbol is odd.

With an even number of bits per symbol, we may write

ML =

where, L is a positive integer.

Figure 2.4 shows the constellation diagram of 16-QAM. Zi are the decision

regions and si denotes the baseband message signals where i = 1, 2,.., 16.

Chapter 2: Background Materials

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12

2.3.3 Comparison among M-ary Schemes

MPSK is bandwidth efficient compared to MFSK. MPSK has circular and

MQAM has square/ rectangular constellation diagram. So the constellation diagram

reveals that the distance between message points in case of MPSK is smaller than the

distance between the message points of MQAM. Accordingly, in an AWGN channel, M-

ary QAM outperforms the corresponding M-ary PSK in error performance for M >4.

2.4 Performance Metrics

2.4.1 Capacity

The growing demand for wireless communication makes it important to determine

the capacity limits of these channels. These capacity limits dictate the maximum data

rates that can be transmitted over wireless channels with asymptotically small error

probability. In this section we first look at the well-known formula for capacity of a time-

invariant AWGN channel. We next consider capacity of time-varying flat-fading

channels where only the fading distribution is known at the transmitter and receiver.

Capacity in AWGN Channel

Consider a discrete-time AWGN channel with channel input/ output relationship

( ) ( ) ( )iii tntxty += , where ( )ix is the channel input , ( )iy is the corresponding channel

S14 S13 S12 Z13 Z14 Z16

Z5 Z6 Z7

( ) ( )tcfT

t π=ϕ 2cos2

1

( ) ( )tcfT

t π=ϕ 2cos2

2

0010 0011

0001 0000

0101 0111

0110 0100

1101

1111 1110

1100

1011

23d− 2d− 2d

23d−

2d−

2d

23d

23d

1001

1010 1000

Figure 2.4 Signal space diagram for M-ary QAM for M=16.

Z1 Z2 Z3 Z4

Z8

Z9 Z10 Z11 Z12

Z17

S1 S2 S3 S4

S5 S6 S7 S8

S9 S9 S10 S11

S15

( ) ( )tfT

t cπ=ϕ 2sin2

2

Chapter 2: Background Materials

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13

output, and ( )in is a white Gaussian noise random variable (RV) at si iTt = where i = 0, 1,

….. . Assume a channel bandwidth B and transmit power P. The channel SNR is constant

and given by BNP 0=γ , where 0N is the power spectral density of the noise. The

capacity ( )γC of this channel is given by Shannon’s well-known formula:

( ) ( )γ+=γ 1log2BC (2.9)

where, the capacity units are bits/second (bps).

Capacity of Wireless Channel

Shannon capacity of a fading channel with receiver CSI for an average power

constraint P (i.e. P denotes the average transmit signal power) can be obtained from

integrating Shannon capacity for an AWGN channel given by ( )γ+1log2B , with SNR γ ,

averaged over the distribution of γ , i.e.,

( ) ( ) γγγ+= ∞

dpBC0

2 1log (2.10)

where, ( )γp is the PDF of the received instantaneous SNR at the receiver, corresponding

to the wireless channel.

2.4.2 Outage Probability

The outage probability [5], outP , of the combiner is defined as the probability that

its output SNR γ falls below a given target threshold oγ , [ ]oγ<γPr , and therefore can be

obtained from cumulative distribution function (CDF) ( )oF γγ at oγ=γ . So the outage

probability expression can be obtained from the following:

[ ] ( )γ

γγ=γ<γ=o

oout dpP0

Pr (2.11)

where, ( )γp is the PDF of the instantaneous received SNR at the combiner.

2.4.3 Symbol Error Rate (SER)

M-ary Phase Shift Keying (MPSK)

For MPSK the SER in AWGN channel can be given by

Chapter 2: Background Materials

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14

( )

( )

θ

θ

πγ−

π=γ

−π

dM

PM

M

s

1

0

22 sinsinexp1

(2.12)

Now, in a mobile radio environment, we have an additional effect to consider, namely,

the fluctuation of amplitude and phase of the received signal due to multipath

propagation effects. To be specific, consider the transmission of data over a Rayleigh

fading channel, for which the low-pass complex envelop of the received signal modified

as follows:

( ) ( ) ( ) ( )twtsjtx ~~exp~ +ϕ−α= (2.13)

where, ( )ts~ is the complex envelop of the transmitted signal, α is the Rayleigh

distributed random variable describing the attenuation in transmission, ϕ is the

uniformly distributed random variable describing the phase-shift in transmission and ( )tw~

is a complex-valued white Gaussian noise process. It is assumed that the channel is flat.

So the average probability of error is used as a performance metric when cs TT ≈ . Where

sT is one symbol period and cT is coherent time. Thus, we can assume that received

SNR γ (which has Chi-square distribution) is roughly constant over a symbol time. For

fading channel, the SER, ( )γsP , becomes conditional on the fading SNR γ , which may

be obtained from (3.21) by replacing η with γ . Then the average probability of error is

computed by integrating the error probability in AWGN over the fading distribution:

( ) ( )∞

γ γγγ=0

dpPP ss (2.14)

substituting equation (2.12) in equation (2.14) we get,

( )

( )

θγ

θ

πγ−γ

π=

−π

=γγ dd

MpP

M

M

s22

1

0 0

sinsinexp1

(2.15)

when M= 2, i.e. in case of BPSK it simplifies to

( ) ( ) γγγ= γ

dpQPs0

2 (2.16)

Chapter 2: Background Materials

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15

where, in AWGN channel the BER of BPSK is given by, ( ) ( )γ=γ 2QPs and the Q

function, also known as Gaussian probability integral, is defined as

( ) ( ) ( )∞

−π=z

duuzQ 2exp21 2 .

M-ary Quadrature Amplitude Modulation (MQAM)

Similarly, for AWGN channel the SER of MQAM can be given by

( )( )

( )θ

θ−

γ−

π

−θ

θ−

γ−

π=γ

π

π

dMM

dMM

Ps

4

02

2

2

02

sin12

3exp

11

4

sin12

3exp

11

4

(2.17)

So, in case of fading channel, to estimate the average probability of error we have to

average the conditional probability of error ( )γsP over all possible values of γ i.e.,

( ) ( )∞

γ γγγ=0

dpPP ss (2.18)

Substituting equation(2.17) in equation (2.18) we get,

( ) ( )( )

( )( )

θγ

θ−

γ−γ

π

−θγ

θ−

γ−γ

π=γ

π

=γγ

π

=γγ

ddM

pM

ddM

pM

Ps

4

0 02

2

2

0 02

sin12

3exp

11

4

sin12

3exp

11

4

(2.19)

2.5 Receiver Diversity Schemes

Rayleigh fading and log normal shadowing both induce a very large power

penalty on the performance of modulation over wireless channels. One of the most

powerful techniques to mitigate the effects of fading is to use diversity-combining of

independently fading signal paths. Diversity-combining uses the fact that independent

signal paths have a low probability of experiencing deep fades simultaneously. Thus, the

idea behind diversity is to send the same data over independent fading paths. These

Chapter 2: Background Materials

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16

independent paths are combined in some way such that the fading of the resultant signal

is reduced. This section focuses on common techniques at the receiver to achieve

diversity.

Diversity to mitigate the effects of shadowing due ot buildings and objects is

called macrodiversity. On the other hand, diversity techniques that mitigate the effect of

multipath fading are called microdiversity, and that is the focus of this section.

2.5.1 Diversity Combining

There are many methods for combing the signals that are received on the

disparate diversity branches, and several ways of categorizing them. Diversity combining

that takes place at RF is called pre-detection combining, while diversity combining that

takes place at baseband is called post-detection combining. Here, implementation of pre-

detection combining is studied.

Figure 2.5 shows a receiver system employing pre-detection combining. The RF

signals that are received by the different antenna branches are first processed by

combiner, and then applied to a diversity combiner.

If the signal ( )tSm is transmitted, the signals on the different diversity branches

are

( ) ( ) ( )tntShtr kmkk += ; k=1,2,……, L (2.20)

where, ( )kkk jh θ−α= exp is the fading gain associated with the kth

branch. For ideal case,

all ( )kk jθ−α exp are independent and identically distributed (i.i.d.) random variables.

The AWGN process ( )tnk independent from branch to branch. Usually L is referred to as

the diversity order.

Figure 2.5 Pre-detection receiver.

)(1 tr

)(2 tr

)(trL

r~

1~r

Lr~

2~r

Diversity

Combiner

Combiner

Combiner

Combiner

Chapter 2: Background Materials

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17

The fading gains of the various diversity branches typically have some degree of

correlation, and the degree of correlation depends on the type of diversity being used and

the propagation environment. Branch correlation reduces the achievable diversity gain.

Nevertheless, to simplify analysis, the diversity branches are usually assumed to be

uncorrelated [6].

2.5.2 Combining Methods

Whatever may be the diversity technique being used, (example- space, time,

frequency etc.) ideally we must get L (>1) uncorrelated faded replicas of the original

signal. An important part of a diversity system is the way in which these L branches are

used by the receiver. There are several possible combining methods employed in

receivers, among which the most common techniques are:

(1) Maximal Ratio Combining (MRC)

(2) Selection Combining (SC)

(3) Dual Branch Switch-and-Stay Combining (SSC)

(4) Multi branch Switch-and-Examine Combining (SEC)

Maximal Ratio Combining (MRC)

In MRC (shown in Figure 2.6(a)) the output of the combiner is just a weighted

sum of the different fading paths or branches. Combining of more than one branch signal,

requires co-phasing, where the phase iθ of the ith branch is removed through the

multiplication by ( )ii jθ−α exp which is obtained from a channel estimator. This phase

removal requires coherent detection of each branch to determine its phase iθ . Without co-

phasing, the branch signals would not add up coherently in the combiner, so the resulting

output could still exhibit significant fading due to constructive and destructive addition of

the signals in all the branches.

It has the advantage of producing an output with an acceptable SNR even when

none of the individual received branch signal is acceptable. Equal gain combining (EGC)

can be thought as a special case of maximal ratio combining where all branch gains are

set equal. That accounts for the name equal gain. The possibility of producing an

Chapter 2: Background Materials

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18

acceptable signal from a number of unacceptable inputs is still retained, and performance

is marginally inferior to MRC [7, 8].

Selection Combining (SC)

Selection Diversity is the simplest diversity technique. A block diagram of this

method is similar to that shown in Figure 2.6(b). In selection combining (SC), the

combiner outputs the signal on the branch with the highest SNR. Since only one branch is

used at a time, SC often requires just one receiver that is switched into the active antenna

branch. However, a dedicated receiver on each antenna branch may be needed for

systems that transmit continuously in order to simultaneously and continuously monitor

SNR on each branch. With SC the path output from the combiner has an SNR equal to

the maximum SNR of all the branches [5].

In case of MRC or EGC they need channel state information (CSI) from all the

received signals, so if the demodulator uses a noncoherent or differential detection

algorithm, i.e. the receiver does not come with an inbuilt synchronization circuitry, SC is

an ideal match. Implementation of MRC or EGC would require extra co-phasor circuit

blocks which may be avoided only when the demodulation is coherent type. When the

noises and interferences are correlated, selection/ switched combining becomes more

competitive. Also SC simplifies the receiver design.

Dual Branch Switch-and-Stay Combining (SSC)

In case of SC, that transmit continuously may require a dedicated receiver on each

branch to continuously monitor branch SNR. A simpler type of combining, called

threshold combining/ switch-and-stay combining (SSC), avoids the need for a dedicated

receiver on each branch by scanning each of the branches in sequential order and

outputting the first signal with SNR above a given threshold Tγ . The block diagram is

shown in Figure 2.6(c). As in SC, since only one branch output is used at a time, co-

phasing is not required. Thus, this technique can be used with either coherent or

differential modulation.

Once a branch is chosen, as long as the SNR on that branch remains above the

desired switching threshold Tγ , the combiner outputs that signal. If the SNR on the

Chapter 2: Background Materials

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19

selected branch falls below the threshold, the combiner switches to another branch. Since

the SSC does not select the branch with the highest SNR, its performance is between that

of no diversity and ideal SC [9].

Multi Branch Switch-and-Examine Combining (SEC)

Because only two paths are involved at most in the diversity combining decision

of SSC schemes, this scheme cannot benefit in diversity from additional paths when these

paths are i.i.d. or equicorrelated and identically distributed. In this case, one should rather

implement an SEC type of combining (shown in Figure 2.6(d)) for which it is assumed

that if the current path is not of acceptable quality, then the combiner switches and

examines the quality of the next available path. This switching–examining process is

repeated until either an acceptable path is found or all available diversity paths have been

1 L 1 L

1 2 1 2

Control

Unit

Selection

Out

(b)

Weights

Out

Weights

and Phase

Estimation

Phase

(a)

(c)

Out

Switching

Logic

Selection

Threshold

SNR

Figure 2.6 Several types of combining- (a) MRC, (b) SC, (c) SSC, (d) SEC.

(d)

Switching

Logic

Threshold

SNR

Selection

Out

L

Chapter 2: Background Materials

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20

examined. In the latter case, the combiner either settles on the last examined path or

connects to the receiver the path with the best quality among all examined paths [9].

2.6 Multiple Input Multiple Output (MIMO) Systems

In this section we consider systems with multiple antennas at the transmitter and

receiver, which are commonly referred to as multiple input multiple output (MIMO)

systems. The multiple antennas can be used to increase data rates through multiplexing or

to improve performance through diversity. In MIMO systems the transmit and receive

antennas can both be used for diversity gain. Multiplexing is obtained by exploiting the

structure of the channel gain matrix to obtain independent signaling paths that can be

used to send independent data.

2.6.1 Narrowband MIMO Model

Here we consider a narrowband MIMO channel. A narrowband point-to-point

communication system of tM transmit and rM receive antennas is shown in Figure 2.7.

This system can be represented by the following discrete time model:

+

=

rttrr

t

r MMMMM

M

M n

n

x

x

hhh

hh

y

y

.

.

.

.

.

.

.

.

......... . . . . . . . .

........

.

.

.

. 11

1

1111

(2.21)

1x

tMx

2x

rMy

1y

2y

11h

trMMh

Figure 2.7 MIMO systems.

Chapter 2: Background Materials

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21

or simply as y =Hx + n. Here x represents tM -dimensional transmitted symbol, n is rM

-dimensional noise vector, and H is tr MM × -dimensional matrix of channel gains ijh

representing the gain from transmit antenna j to receive antenna i.

When both the transmitter and receiver have multiple antennas, there is

performance gain called multiplexing gain [5] and diversity gain. The multiplexing gain

of a MIMO system results from the fact that a MIMO channel can be decomposed into a

number R of parallel independent channels. By multiplexing independent data onto these

independent channels, we get an R-fold increase in data rate in comparison to a system

with just one antenna at the transmitter and receiver. This increased data rate is called the

multiplexing gain. The diversity gain can be defined as the increase ain signal-to-noise

ratio due to some diversity scheme, or how much the transmission power can be reduced

when a diversity scheme is introduced, without a performance loss.

2.7 Space Time Coding (STC)

Since a MIMO channel has input-output relationship y = Hx + n, the symbol

transmitted over the channel each symbol time is a vector rather than a scalar, as in

traditional modulation for the SISO channel. Moreover, when the signal design extends

over both space (via the multiple antennas) and time (via multiple symbol times), it is

typically referred to as a space-time code.

Space-time codes are designed for quasi-static channels where the channel is

constant over a block of U symbol times, and the channel is assumed unknown at the

transmitter. Under this model the channel inputs and outputs become matrices, with

dimensions corresponding to space (antennas) and time. Let X denote the UM t ×

channel input matrix with ith column xi equal to the vector channel input over the ith

transmission time. Let Y denote the UM r × channel output matrix with ith column yi

equal to the vector channel output over the ith transmission time, and let N denote the

UM r × noise matrix with ith column ni equal to the receiver noise vector on the ith

transmission time. With this matrix representation the input-output relationship over all U

blocks becomes

Chapter 2: Background Materials

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22

Y=HX + N (2.22)

As with ordinary channel codes, STC employ redundancy for the purpose of

providing protection against channel fading, noise and interference. They may also be

used to minimize the outage probability or equivalently, maximize the outage capacity.

STC may themselves be classified into two types space-time trellis code (STTC)

and space-time block code (STBC) depending on how the transmission over wireless

channel takes place.

2.7.1 Space Time Block Code (STBC)

In space-time block code (STBC), by contrast, transmission of signal takes place

in blocks. The code is defined by a transmission matrix, the formulation of which

involves three parameters:

• The number of transmitted symbols denoted by l

• The number of transmission antennas, denoted by tN , which defines

the size of the transmission matrix

• The number of time slots in a data block, denoted by m

With m time slots involved in transmission of l symbols, the ratio l/m defines the rate of

the code, which is denoted by k.

For efficient transmission, the transmitted symbols are expressed in complex

form. Moreover, in order to facilitate the use of linear processing to estimate the

transmitted symbols at the receiver and thereby simplify the receiver design,

orthogonality is introduced into the design of transmission matrix. Here we may identify

two different design procedures:

(1) Complex Orthogonal Design: In this case the transmission matrix is square, satisfying

the condition for complex orthogonality in both spatial and temporal sense.

(2) Generalized Complex Orthogonal Design: In this case the transmission matrix is non-

square, satisfying the condition for complex orthogonality only in the temporal sense; the

code rate is less than unity.

Chapter 2: Background Materials

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23

Figure 2.8 shows the baseband diagram of space-time block encoder, which

consists of two functional units: a mapper (may be M-ary PSK or M-ary QAM) and a

block encoder itself. The mapper takes the incoming binary data stream kb , 1±=kb , and

generates a new sequence of blocks, with each block made up of multiple symbols that

are complex. All the symbols of a particular column of a transmission matrix are pulse

shaped and then modulated into a suitable form for simultaneous transmission over the

channel by the transmit antennas. The block encoder converts each of complex symbol

produced by the mapper into an l -by- tN transmission matrix S here l and tN are

temporal dimension and spatial dimension, respectively, of transmission matrix. The

individual element of transmission matrix S are made up of complex symbols, say, ks ,

generated by mapper, their complex conjugates *ks , and linear combination of ks and *

ks ,

where asterisk denotes the complex conjugate.

Alamouti Code

The Alamouti Code is a orthogonal space-time block code. That is, it uses two

transmit antennas ( )2=tN and a single receive antenna, as shown in Figure 2.10, and may

be defined by following three functions [10, 11] as:

• The encoding and transmission sequence of information symbols at the

transmitter

• The combining scheme at the receiver

• The decision rule for maximum likelihood detection (MLD)

I. The Encoding and Transmission Sequence: Let 0S and 1S denote the complex

symbols (signals) produced by the mapper which are to be transmitted over the wireless

channel. Signal over the channel proceeds as follows:

Constellation

Mapper

Block

Encoder

kb ks Transmit

Antennas

Figure 2.8 Block diagram of orthogonal space-time block encoder.

Chapter 2: Background Materials

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24

• At some arbitrary time t, antenna 0 (Tx 0) and antenna 1(Tx 1) transmits

0S and 1S simultaneously

• At time t+T, where T is symbol duration, signal transmission is switched with

*1S− and *

0S are transmitted from Tx 0 and Tx 1 respectively

The two-by-two space-time block code, is formally written in matrix form [11] as

The transmission matrix S is a complex orthogonal matrix, in that it satisfies the

condition for orthogonality in both spatial and temporal sense. Orthogonal in spatial

sense means [11]

( )

+=′

1

0

0

12

1

2

0 SSSS (2.23)

where, S´ is the Hermitian transpose of S. The same result also holds for the S´S which is

proof of orthogonality in the temporal sense.

The channel at time t can be modeled by a complex multiplicative distortion

( )th0 for Tx 0 and ( )th1 for Tx 1. Assuming that fading is constant over two consecutive

symbol periods, we can write

( ) ( )

111

000

)()( hTthth

hTthth

=+=

=+= (2.24)

The received symbol can then be expressed as

S =

− *0

*1

10

SS

SS

Time

Space

Figure 2.9 Transmission

Chapter 2: Background Materials

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25

( )

( ) 1*01

*101

011000

nShShTtrr

nShShtrr

++−=+=

++== (2.25)

where. 0r and 1r are the received signal at time t and t+T and 0

n and 1

n are complex

random variables representing receiver noise and interference.

II. The Combining Scheme: The combiner builds the following two combined

signals that are transmitted to the MLD

*

100*1

*1

*110

*00

~

rhrhS

rhrhS

−=

+= (2.26)

substituting equation (2.25) in equation (2.26) we get,

( )( ) 0

*1

*101

21

201

*110

*00

21

200

~

~

nhnhSS

nhnhSS

+−α+α=

++α+α= (2.27)

*1

0

S

S

− *

0

1

S

S

0h 1h

Receive

Antenna

Transmit

antenna 0

Transmit

antenna 1

( )000 exp θα= jh

Channel

Estimator

Combiner

Maximum Likelihood Detector

( )111 exp θα= jh

Noise 1

0

n

n

0h

1h

0

~S 1

~S

0S

1S

Figure 2.10 System model of Alamouti scheme [10].

Chapter 2: Background Materials

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26

III. The Maximum Likelihood Decision Rule: The combined signals are then sent

to the MLD which, for each of the signal 0S and 1S , uses decision rule and produces the

estimates 0S

and 1S

.

2.7.2 Performance Comparison of Diversity-on-Receive and Diversity-

on-Transmit Schemes

Figure 2.11 presents both theoretical and simulation comparing the bit error rate

(BER) performance of coherent BPSK over an uncorrelated Rayleigh fading channel for

three different schemes [13]:

(a) No diversity (one transmit antenna and one receive antenna)

(b) The MRC (one transmit antenna and two receive antennas)

(c) The Alamouti code (two transmit antennas and one receive antenna)

It is assumed that the total transmit power is same for all three schemes, and in the

case of two diversity schemes (b) and (c), there is perfect knowledge of channels at the

receiver(s).

From the Figure 2.11, we see that the performance of Alamouti code is 3dB

worse, compared with the maximal-ratio combining for the same number of total

antenna(s). This 3dB penalty is incurred because the simulation assumes that each

transmit antenna in case of Alamouti scheme (c) radiates half the energy in order to

ensure the same total radiated power as with one transmit antenna as in MRC case (b). If

each transmit antenna in Alamouti coding scheme is allowed to radiate the same energy

as the single transmit antenna for MRC, the performance would be identical.

Chapter 2: Background Materials

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27

0 5 10 1510

-4

10-3

10-2

10-1

100

Average Signal-to-Noise Ratio (dB)

Bit E

rror R

ate

(B

ER

)

Theoretical

Simulation(No Diversity)

(2 Tx, 1 Rx (Alamouti))

(1 Tx, 2 Rx (MRC))

Figure 2.11 Comparison of average signal-to-noise ratio vs. bit error rate performance of coherent BPSK

over flat Rayleigh fading channel for three configurations

2.8 Literature Survey

The use of multiple antennas for wireless communication systems has gained

overwhelming interest during the last decade - both in academia and industry. Multiple

antennas can be utilized in order to accomplish a multiplexing gain, a diversity gain, or

an antenna gain, thus enhancing the data rate, the error performance, or the signal-to-

noise-ratio of wireless systems, respectively. With an enormous amount of yearly

publications, the field of multiple-antenna systems, often called MIMO systems, has

evolved rapidly. To date, there are numerous papers on the performance limits of MIMO

systems, and an abundance of transmitter and receiver concepts has been proposed. The

objective of this literature survey is to provide a comprehensive overview of this exciting

research field. To this end, the last thirteen years of research efforts are recapitulated,

with focus on spatial multiplexing and spatial diversity techniques.

Wireless systems operate over a complex and harsh time-varying radio channel

which introduces severe shadowing and multipath fading, causing a larger error rate and

Chapter 2: Background Materials

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28

smaller coverage compared to the wired channel. To avoid these circumstances, there has

been a considerable research effort was aimed at the development of receive diversity

techniques that allows significant increase in wireless capacity and link reliability without

an increase in the transmitted power and bandwidth. Receive diversity uses the fact that

independent signal paths have a low probability of experiencing deep fades

simultaneously. Till now there has been a lot of works on different receive diversity

schemes such as SC, MRC, EGC, SSC, SEC etc [5, 9].

But the problem is, at the same time, the remote units i.e. the wireless devices

supposed to be small, light weight pocket communicators keeping the link reliability

level efficient. So in this case implementing receive diversity is physically impracticable.

In 1998 Siavash M. Alamouti gave the proposal about a simple transmit diversity scheme

called Alamouti coding [10] which gives the same performance as MRC but the cost for

this scheme is added complexity at the receiver side i.e. receiver should know the pure

channel state information.

However for further performance improvement in 2005 W. Li and N. C. Beaulieu

gave a proposal, combined Alamouti coding at the transmitter side with various receive

diversity schemes (SC, SSC and MRC) and evaluated the error performances in Rayleigh

fading channel [12]. Recently in 2010 Y. N. Trivedi and A. K. Chaturvedi proposed a

scheme Alamouti scheme with transmit antenna selection, which is a very much effective

scheme [14]. They evaluated the error performance and outage probability in Rayleigh

fading channel.

In recent years work is going on, employing, both transmit antenna selection

(TAS) and receive antenna selection (RAS). A work incorporating both TAS and MRC in

Rayleigh fading channel [15] is provide by D. Haccouna, M. Torabi, W. Ajib in 2010.

Again in 2011 A. F. Coskun and O. Kucur gave an analytical performance on joint TAS

and RAS in Nakagami-m fading channel [16].

So we can conclude the research work till date may be grouped into the following

three categories, performance analysis with

(i) receive diversity

Chapter 2: Background Materials

Page 42: Gourab's M.tech Thesis

29

(ii) transmit diversity

(iii) both transmit diversity and receive diversity

According to the current interest of research works and requirements in this

domain, we tried to evaluate the analytical expression of performance metrics (capacity,

outage probability and SER) employing both receive diversity and transmit diversity in

Rayleigh fading channel.

2.9 Chapter Summary

The main objective of this chapter was to elaborate on different diversity schemes

and Alamouti coding which are used to avoid the current wireless systems drawbacks i.e.

higher error probability, lower coverage. Also we observed the comparison between

receive diversity and transmit diversity.

Also we can combine the Alamouti coding with receive diversity or transmit

diversity to see how the systems perform in presence of wireless fading environment and

these are illustrated in next consecutive chapters.

Chapter 2: Background Materials

Page 43: Gourab's M.tech Thesis

Chapter 3

Multi branch Switch-and-Examine Combining in

Alamouti Coded MIMO Systems

3.1 Introduction

Major 4G wireless standards, like WiMax and LTE, have already adopted the

MIMO capability as an integral part of their air interface specifications [17]. Use of

multiple antennas at transmitter (Tx) and receiver (Rx) results in additional diversity/

multiplexing/ array gain, enhanced channel capacity, and fewer errors during

transmission. A simple MIMO configuration, with 2 Tx antennas, may be realized through

Alamouti coding [10]. On the other hand, at the receiver side, traditional combining

schemes may be used to realize diversity.

Recently, there has been an upsurge of literature concerning performance analysis

of Alamouti coded MIMO systems with some sort of receiver diversity [12, 18-22].

Although many variants of receiver diversity combining algorithms exist, it has been

focused largely on MRC or SC. In this chapter we have focused on multibranch SEC as

our receive diversity and took an approach to combine Alamouti coding at the transmitter

side with SEC to evaluate the numerical performance metrics like capacity, outage

probability and SER using MPSK and MQAM.

In previous chapter we have discussed the Alamouti code which is used at the

transmitter side. In this chapter we will see the how the performance metrics may vary if

the receive diversity SEC is combined with Alamouti code.

The remainder of this chapter is organized as follows. The system model under

study, is presented in Section 3.2. Next, section, Section 3.3 presents analysis of

performance metrics (capacity, outage probability and SER for MPSK and MQAM). The

chapter finally ends with some concluding remarks in Section 3.4.

Page 44: Gourab's M.tech Thesis

31

3.2 System Model and Description

The system model with 2 Tx and L Rx antennas is shown in Figure 3.1. Let s1 and

s2 denote the equivalent baseband signals corresponding to two successive information

bits which are sent using a 2×1 Alamouti code [10]. For a slow fading channel it may be

assumed that the channel transfer function remains constant over two consecutive symbol

intervals, and accordingly the received signals on nth branch in these two intervals can be

expressed as

nnnn nshshr 122111 ++= (3.1a)

nnnn nshshr 212212 ++−= ∗∗ (3.1b)

where ∗∗21 ,ss are the complex conjugates of 21, ss , ( )mnmnmn jh θα= exp ,2,1; ∈m

Ln ,,2,1 ∈ is the complex channel gain between the mth Tx antenna and the nth Rx

antenna with α and θ being the random amplitude and phase variations respectively, and

the additive noise nmn is a zero-mean circularly symmetric complex Gaussian random

variable (RV) having a variance N0.

At the receiver, the space time (ST) combiners attached to each branch process

the signal to produce an output pair nn yy 21 , given by

*221

*11

ˆˆnnnnn rhrhy += (3.2a)

nnnnn rhrhy 1*2

*212

ˆˆ +−= (3.2b)

where mnh is an estimate of mnh . If the channel estimator produces CSI, it can be shown

that

( ) mnmnnmn wsy +α+α= 22

21 ; 2,1∈m (3.3)

by substituting equation (3.1) in equation (3.2) and using the definition of hmn. As the RV

wmn has a variance of 2N0, the instantaneous SNR available at the ST combiner output

would be

( )22

21

2nnn α+α

η=γ , Ln ,,2,1 ∈ (3.4)

Chapter 3: Multibranch SEC in Alamouti Coded MIMO Systems

Page 45: Gourab's M.tech Thesis

32

where, ( )0NE=η is the SNR in additive white Gaussian noise (AWGN) channel and E

is the symbol energy. For a Rayleigh fading channel, the distribution of L

nm

,2

1,1 ==α is [9]

Figure 3.1 Transmission model of a 2×L MIMO system employing Alamouti code at transmitter and pre-

detection switch and examine combining at receiver.

( )

Ω

α−

Ω

α=αα

2

exp2

f ; Ω=α2E 0≥α (3.5)

Accordingly, the PDF of nγ will follow a central chi-square distribution with four

degrees of freedom

( )

γ

γ−

γ

γ=γγ

n

n

n

nnf

2exp

42

; nnE γ=γ ; 0≥γn (3.6)

and the corresponding CDF would be

( )

γ

γ−

γ

γ+−=γγ

n

n

n

nnF

2exp

211 ; 0≥γ n (3.7)

which can be derived by expressing the CDF, ( ) ( ) γγ= γγx

nn dfxF 0 , with an incomplete

gamma function [23, (8.350.1)], and further reducing the same with [23, (8.352.1)].

For SEC, the diversity combiner operates in discrete time fashion, i.e. the branch

switching occurs at time uTt = , where u is any integer. As the ST combiners give out the

pair nn yy 21 , after every T2 amount of time, a parallel to serial conversion (not shown in

Switched combiner

S

p

a

c

e

Time

s1 -s2*

t t + T

s2 s1*

s1, s2

r11, r21

n11, n21

ST Combiner y11, y21

r1L, r2L

n1L, n2L

ST Combiner y1L, y2L

h11

h21

h1L

h2L

Switching

logic

Channel

Estimator

Threshold

SNR

L

y1p, y2q 1, 2

Transmitter

Decision

device

Chapter 3: Multibranch SEC in Alamouti Coded MIMO Systems

Page 46: Gourab's M.tech Thesis

33

Figure 3.1) is necessary before the output can be fed to the combiner. The channel

estimator estimates the current SNR in different branches at every uTt = . Using the

information, the switching logic block triggers the selector to switch from the current

branch to the next branch if SNR in current branch falls below some threshold value

(generally found from a table that stores the optimum thresholds for different SNR).

Let us assume that the pth branch is selected during the two signaling intervals of

interest. The output of the combiner pp yy 21 , is then hard-decoded

( )pp yyEss 2121 ,sgnˆ,ˆ ℜ= (3.8)

to produce an estimate of the original signal pair 21, ss .

3.3 Analysis of Performance Metrics

3.3.1 Capacity

In order to find the average capacity for Alamouti coded MIMO systems with

SEC, we need to average ( )γC over the PDF ( )γγ SECf , of the combiner output SNR, i.e.

( )∞

γ γγγ=0

,)( dfCC SEC (3.9)

where, ( )γγ SECf , is the PDF of γ at SEC output. Assuming independent and identically

distributed (IID) fading, ( )γγ SECf , can be expressed as [9, (9.341)]

( )( ) ( )[ ]

( ) ( )[ ]

γ≥γγγ

γ<γγγ

=γ−

=γγ

−γγ

γ

th

L

j

j

th

th

L

th

SECFf

Ff

f;

;

1

0

1

, (3.10)

where, )(γγf and )(γγF are given by equations (3.6) and (3.7) respectively. Substituting the

value of )(γC and ( )γγ SECf , in equation (3.9) we get

( )[ ] ( )[ ] ( ) ( )[ ] ( ) γγωγ+γγω

γ−γ= ∞−

γ−

−γ dFeBdFFeBC

L

j

j

th

L

j

j

th

L

th

th

0

1

02

0

1

0

12 loglog (3.11)

Chapter 3: Multibranch SEC in Alamouti Coded MIMO Systems

Page 47: Gourab's M.tech Thesis

34

where, ( ) ( ) ( )γγ+=γω f1ln . The first integration may be readily solved through

integration by parts, taking, ( )γ+γ= 1lnu and ( )γγ−= 2expv

( )

γ

γ−γγ+

γ=

γ 2exp1ln

4

021

thI

γ

γ+−

γ×

γ

γ−+

γ

γ−−γ+

γ

γ−

+

γ

γ−=

)1(22

2exp

21

2exp)1ln(

2exp1

21

11th

thth

thth

EE

(3.12)

where, ( ) ( )∞ − −= x dtttxE exp1

1 ; 0>x is the exponential integral of first order [24,

(5.1.1)]. To solve the second integral we make use of the following result [15]

( )

( ) ( )∞

−λµ−µ+

−λ

µ

0

1exp)1ln(

!1dxxxx

( ) ( ) ( ) ( )−λ

=−λλ µ−µ+µµ−=

1

11

1

qqq PP

qEP (3.13)

where ( ) ( ) )exp(!1

0

xvxxPq

v

vq −=

= is the Poisson CDF. Thus, the second integral

γ−

γ+

γ

γ−=

γ

γ

γ−γγ+

γ=

2222

2exp)1ln(

4

1112

022

PPEP

dI

+

γ

γ

γ−= 1

22exp

21 1E (3.14)

Substituting equations (3.12) and (3.14) in equation (3.11) we get

( )[ ] ( )[ ] ( )[ ] 2

1

021

1

0

12 loglog IFeBIFFeBC

L

j

j

th

L

j

j

th

L

th −

−γ γ+

γ−γ= (3.15)

Figure 3.2 shows a plot of equation (3.15) for 3=γ th dB, i.e. the capacity of an

Alamouti based SEC system in Rayleigh fading channel for a fixed threshold. For, L = 2,

Chapter 3: Multibranch SEC in Alamouti Coded MIMO Systems

Page 48: Gourab's M.tech Thesis

35

SEC operation becomes identical with dual branch SSC system. Further, for larger values

of L, the capacity increases only when average SNR ( )γ is close to the thγ value.

In order to exploit the capacity advantage throughout the SNR axis, the combiner

needs to operate with optimum switching threshold ( )∗γ th which may be found by

differentiating equation (3.15) with respect to thγ , and setting the result to zero, i.e.

0=γ∂∂ ∗γ=γ thththC . A closed-form expression for ∗γ th is, however, unattainable and

numerical minimization technique was used to tabulate ∗γ th (shown in table 3.1) for each

value of average SNR γ .

0 5 10 15

100.1

100.2

100.3

100.4

100.5

100.6

Average Signal-to-Noise Ratio (dB)

Capacity (B

its/s

/Hz)

Theoretical

Simulation

(L=6)

(L=3)

(L=2)

Figure 3.2 Capacity curves for Alamouti based SEC system with fixed threshold (th = 3 dB) for different

numbers of Rx antennas.

The corresponding plot of optimum capacity, for both theoretical and simulated

values is given in Figure 3.3. The results show that capacity values increase with

additional Rx antennas throughout the whole SNR range.

Chapter 3: Multibranch SEC in Alamouti Coded MIMO Systems

Page 49: Gourab's M.tech Thesis

36

0 5 10 15

100.1

100.2

100.3

100.4

100.5

100.6

100.7

Average Signal-to-Noise Ratio (dB)

Capacity (B

its/s

/Hz)

Theoretical

Simulation

(L=6)

(L=3)

(L=2)

Figure 3.3 Capacity curves for Alamouti based SEC system with optimum threshold (as found from Table

I) for different numbers of Rx antennas.

Table 3.1 Optimum switching threshold for capacity as a function of increasing average SNR per branch

for different Rx antennas.

γ (dB) Optimum common switching threshold

L=2 (SSC) L=3 L=6

0 -0.51 0.22 1.27

1 0.45 1.16 2.23

2 1.43 2.12 3.22

3 2.38 3.07 4.17

4 3.29 4.02 5.13

5 4.23 4.98 6.12

6 5.18 5.96 7.08

7 6.14 6.93 8.06

8 7.10 7.88 9.03

9 8.06 8.86 10.01

10 9.02 8.83 10.98

11 9.99 10.8 11.98

12 10.97 11.78 12.97

13 11.94 12.76 13.96

14 12.92 13.75 14.97

15 13.90 14.75 15.98

Chapter 3: Multibranch SEC in Alamouti Coded MIMO Systems

Page 50: Gourab's M.tech Thesis

37

3.3.2 Outage Probability

In this case outage probability outP is a function of both target threshold 0γ and

switching threshold thγ . The outage probability can be calculated from the PDF ( )γγ SECf , ,

as discussed in chapter 2

( ) ( ) γγ=γ<γ= γ

γ dfpP SECout

0

0,0 (3.16)

Inserting equation (3.10) in equation (3.16) we obtain

( )[ ] ( )[ ] ( ) ( )[ ] ( )γ

γ

γ−

−γ γγγ+γγ

γ−γ=01

00

1

0

1

th

L

j

j

th

thL

j

j

th

L

thout dfFdfFFP (3.17)

Solving the integrals through integration by parts

( )

γ

γ−

γ+γ

γ−=γγ=

γth

th

thdfI

2exp

2

21

03 (3.18)

And

( )

γ

γ−

+

γ

γ−=γγ=

γ

γ

000

4

2exp1

21

th

dfI (3.19)

Substituting equations (3.18) and (3.19) in equation (3.17), the final outage probability

expression becomes

( )[ ] ( )[ ] ( )[ ] 4

1

0

1

0

1IFFFP

L

j

j

th

L

j

j

th

L

thout −

=

+γγ γ+γ−γ= (3.20)

Figure 3.4 shows the outage probability performance of Alamouti coded SEC in

Rayleigh fading channel for a fixed switching threshold of 2=γ th dB and a target

threshold of 0γ =3 dB. The horizontal axis (x-axis) is normalized with respect to target

threshold.

Like the capacity case, the outage probability also attains its minimum value when

the combiner operates with optimum switching threshold ∗γ th , which may be obtained by

setting 0=γ∂∂ ∗γ=γ thththoutP . It can be easily shown that, for 0γ=γ th ,

( ) *

00,min outththoutout PPP =γγ= γ=γ, and we get the optimum performance. The

corresponding plot for 30 =γ=γ th dB is shown in Figure 3.5.

Chapter 3: Multibranch SEC in Alamouti Coded MIMO Systems

Page 51: Gourab's M.tech Thesis

38

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

10-4

10-3

10-2

10-1

100

Normalized Average Signal-to-Noise Ratio (dB)

Outa

ge P

robability

Theoretical

Simulation

(L=6)

(L=3)

(L=2)

Figure3.4 Outage probability curves for Alamouti based SEC system with fixed threshold (th = 2 dB) for

different numbers of Rx antennas.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

10-4

10-3

10-2

10-1

100

Normalized Average Signal-to-Noise Ratio (dB)

Outa

ge P

robability

Theoretical

Simulation

(L=6)

(L=3)

(L=2)

Figure 3.5 Outage probability curves for Alamouti based SEC system with optimum threshold for different

numbers of Rx antennas.

Chapter 3: Multibranch SEC in Alamouti Coded MIMO Systems

Page 52: Gourab's M.tech Thesis

39

3.3.3 Symbol Error Rate (SER)

M-ary Phase Shift Keying (MPSK)

After discussing capacity and outage probability, in this section, we would derive

the expression of SER for MPSK in conjunction with our 2 x L MIMO system. For

MPSK modulation, the SER in AWGN channel is given by equation (2.12) and for fading

channel it is given by equation (2.14) as discussed in previous chapter.

Multi-branch Switch and Examine Combining (SEC)

With the assumption of statistical independence between fading and noise, the

average SER ( )sP of alamouti coded SEC can be calculated by averaging the conditional

error probability ( )γγ SECf , over the underlying fading random variable ( ) γ as

( ) ( )∞

γ γγγ=0

, dfPP SECss (3.21)

where ( )γγ SECf , is as mentioned in (3.10). Interchanging the integration limit, we get

( )

−π

=γγ θγ

γθ

π

−γπ

=M

M

SECs ddM

fP

)1(

0 02

2

,sin

sin

exp1

( )

( ) ( ) θ

γ

γ

θ

π

−γ

γ−γ

γ

θ

π

−γγγπ

=

γ

−γ

−π

ddM

PP

dM

P

thL

j

j

th

L

th

M

M

L

j

j

th

02

2

1

0

1

)1(

0 02

2

1

02

sin

sin2

exp

sin

sin2

exp41

(3.22)

Now,

2

2

2

02

2

sin

sin2

1sin

sin2

exp

θ

π

γ

θ

π

−γ∞ M

dM

(3.23)

Chapter 3: Multibranch SEC in Alamouti Coded MIMO Systems

Page 53: Gourab's M.tech Thesis

40

again,

2

2

2

2

2

02

2

sin

sin2

sin

sin2

,2sin

sin2

exp

θ

π

γ

θ

π

γ=γ

γ

θ

π

−γγ

MMd

Mth

th

(3.24)

Substituting equations (3.23) and (3.24) in equation (3.24) we get

( )[ ] [ ] [ ]

θ

θΞ

θΛγ−θΛ−θΞ

×

γ−γ

+θθΞγγπ

=

−π

−γ

=

−π

γ

M

M

th

L

j

j

th

L

th

L

j

M

M

j

ths

d

FFdFP

)1(

0

1

0

11

0

)1(

02

)(

)()()(

)()( )(41

(3.25)

where ( )221)( γ+=θΞ D , ( )[ ] )(2exp)( θΞγγ+−=θΛ thD , and ( ) θπ= 22 sinsin MD .

Figure 3.6 shows the SER performance of MPSK for Alamouti coded multi-

branch SEC system over Rayleigh fading channel for a fixed threshold of 3=γ th dB and

different M values.

For BPSK (M = 2) the expression given in equation (3.25) reduces to

( )[ ] ( )[ ] ( )[ ] ( ) ( ) ( )[

( )( ) ( )

γξ−

π

γ

+γ−γξ×

γΘ−γγ−γ+γΘ−γ= γ

−γ

thth

th

thth

L

j

j

th

L

the

Q

QFFFP

exp2

12

2112

1

2

0

1

(3.26)

where ( ) ( ) )2()3(2 +γ+γ+γγ=γΘ and ( ) γ+γγ=γξ )2(thth .

Like capacity and outage probability the BER may be substantially improved if

the combiner operates with optimum threshold ( )∗γ th which may be found by

differentiating (3.26) with respect to thγ , and setting the result to zero, i.e.

0=γ∂∂ ∗γ=γ thththeP . A closed-form expression for ∗γ th is, however, unattainable and

numerical minimization technique was used to tabulate ∗γ th for each value of average

channel SNR γ .

Chapter 3: Multibranch SEC in Alamouti Coded MIMO Systems

Page 54: Gourab's M.tech Thesis

41

The corresponding BER plot, for both theoretical and simulated optimum values

(given in table 3.2) is given in Figure 3.7. The results show that BER values decrease

with additional Rx antennas throughout the SNR range.

Dual-branch Switch and Stay Combining (SSC)

Substituting L = 2 in equation (3.26), one can obtain the error probability for

Alamouti coded SSC scheme

( )[ ] ( )[ ] ( ) ( )[ ( ) ( )( )

( )

γξ−

π

γ

+γ−

γξγΘ−γγ−+γΘ−γ= γγ

thth

ththththe QQFFP

exp2

1

2 2112

1

(3.27)

Table 3.2 Optimum Switching Threshold as A Function of Increasing Average SNR per Branch for

Different Rx Antennas.

γ (dB) Optimum common switching threshold

L=2 (SSC) L=3 L=6

0 -1.43 -0.62 0.57

1 -0.62 0.20 1.43

2 0.16 1.00 2.24

3 0.90 1.77 3.05

4 1.60 2.50 3.80

5 2.30 3.20 4.54

6 2.92 3.86 5.25

7 3.52 4.50 5.93

8 4.08 5.09 6.60

9 4.60 5.65 7.20

10 5.09 6.19 7.79

11 5.55 6.69 8.35

12 5.98 7.16 8.89

13 6.38 7.60 9.40

14 6.75 8.04 9.90

15 7.10 8.44 10.4

Chapter 3: Multibranch SEC in Alamouti Coded MIMO Systems

Page 55: Gourab's M.tech Thesis

42

0 5 10 1510

-4

10-3

10-2

10-1

100

average Signal-to-Noise Ratio (dB)

Sym

bol E

rror R

ate

(S

ER

)

Theoretical

Simulation

L=6

L=3L=2

M=2

M=4

M=8

M=16

Figure 3.6 SER curves for Alamouti based SEC system with fixed threshold (th = 3 dB) for different M and

for different numbers of Rx antennas.

0 5 10 1510

-4

10-3

10-2

10-1

100

Average Signal-to-Noise Ratio (dB)

Bit E

rror R

ate

(B

ER

)

Theoretical

Simulation

(L=6)

(L=3)

(L=2)

Figure 3.7 Optimum BER curves for Alamouti based SEC system with fixed threshold (th = 3 dB) for

BPSK (M=2) and for different numbers of Rx antennas.

Chapter 3: Multibranch SEC in Alamouti Coded MIMO Systems

Page 56: Gourab's M.tech Thesis

43

M-ary Quadrature Amplitude Modulation (MQAM)

Here we have considered QAM square constellation i.e. ML = , where L is a

positive integer. We know the SER of MQAM in AWGN channel is given by equation

(2.17). For fading channel, the SER, ( )γsP , becomes conditional on the fading SNR γ ,

which may be obtained from equation (2.18).

With the assumption of statistical independence between fading and noise, the

average SER ( )sP can be calculated by averaging the conditional error probability ( )γsP

over the underlying fading random variable ( ) γ as

( ) ( )∞

γ γγγ=0

, dfPP SECss (3.28)

where ( )γγ SECf , is as mentioned in equation (3.10). Interchanging the integration limit,

we get

( )( )

( )( )

θγ

θ−

γ−γ

π−

θγ

θ−

γ−γ

π=

π

=γγ

π

=γγ

ddM

fM

ddM

fM

P

SEC

SECs

4

02

0,

2

2

02

0,

sin12

3exp

11

4

sin12

3exp

11

4

(3.29)

Now substituting ( )γγ SECf , from equation (3.10) in equation (3.29) we get,

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) θ

γβγ−γ

γ−γ

+γβγ−γ

γ

π−θ

γβγ−γ×

γ−γ+γβγ−γγ

π=

γ

γ

−γ

γ

π

γ

γ

π

−γ

γ

ddfFF

dfFM

ddf

FFdfFM

P

thL

j

j

th

L

th

L

j

j

th

th

L

j

j

th

L

th

L

j

j

ths

0

1

0

1

0

4

0

1

0

2

0

2

0

1

0

1

0

1

0

exp

exp1

14

exp

exp1

14

(3.30)

where, ( ) [ ]θ−=β 2sin123 M .

Now ,

( ) ( )∞

γ γβγ−γ0

exp df

Chapter 3: Multibranch SEC in Alamouti Coded MIMO Systems

Page 57: Gourab's M.tech Thesis

44

2

2

2

sin)1(2

32

14

θ−+

γ

γ=

M

( )θΝγ

=2

4 (3.31)

and

( ) ( )γ

γ γβγ−γth

df0

exp

2

222 sin)1(2

32

sin)1(2

32,2

4

θ−+

γ

γ

θ−+

γγ

γ=

MMth

( )

( )θΝ

θΝ

γγ

γ= th,2

42

(3.32)

where, ( )2

2sin)1(2

321

θ−+

γ=θΝ

M.

Substituting equations (3.31) and (3.32) in equation (3.30) we get the ultimate expression

of MQAM as follows

( ) ( ) ( ) ( )

( )( ) ( ) ( )

( ) ( ) ( )

( )

θθΝ

θΝ

γγ

γ−γ

+θθΝγγ

π−

θθΝ

θΝ

γγ

×

γ−γ

+θθΝγγ

π=

π

−γ

π

π

−γ

π

4

0

1

0

1

4

0

1

02

22

0

1

0

12

0

1

02

,2

411

4,2

411

4

dFF

dFM

d

FFdFM

P

thL

j

j

th

L

th

L

j

j

thth

L

j

j

th

L

th

L

j

j

ths

(3.33)

Figure 3.8 shows the SER performance of MQAM for Alamouti coded multi-

branch SEC system over Rayleigh fading channel for a fixed threshold of 3=γ th dB and

different M values.

Chapter 3: Multibranch SEC in Alamouti Coded MIMO Systems

Page 58: Gourab's M.tech Thesis

45

0 5 10 1510

-4

10-3

10-2

10-1

100

Average Signal-to-Noise Ratio (dB)

Sym

bol E

rror R

ate

(S

ER

)

Theoretical

Simulation

L=6L=3

L=2

M=4

M=16

M=64

Figure 3.8 SER curves for Alamouti based SEC system with fixed threshold (th = 3 dB) for different M and

for different numbers of Rx antennas.

3.4 Chapter Summary

Closed-form analytical expressions for capacity, outage probability, and SER

have been obtained for a 2 x L MIMO system employing Alamouti code and SEC

diversity. For verification of the derived expressions, extensive Monte Carlo simulations

were carried out. It was found that the theoretical values (represented by solid lines) show

excellent match with the simulation results (represented by black dots). Graphical plots

indicate that if the SEC system is run with optimum threshold, it can outperform the SSC

system by taking advantage of the extra diversity branches.

Also a approach may be taken incorporating transmit antenna selection (TAS)

with Alamouti coding to analyze the system performance and it is discussed in next

chapter.

Chapter 3: Multibranch SEC in Alamouti Coded MIMO Systems

Page 59: Gourab's M.tech Thesis

Chapter 4

Transmit Antenna Selection in Alamouti Coded MISO

Systems

4.1 Introduction

In transmit diversity there are multiple transmit antennas with the transmit power

divided among these antennas. Transmit diversity is desirable in systems such as cellular

systems where more space, power, and processing capability is available on the transmit

side compared to the receive side. Transmit diversity design depends on whether or not

the complex channel gain is known at the transmitter or not. When this gain is known, the

system is very similar to receiver diversity. However, without this channel knowledge,

transmit diversity gain requires a combination of space and time diversity via a novel

technique called the Alamouti scheme. Antenna selection (AS) schemes in space-time

coded multiple input multiple output (STC-MIMO) systems are well documented in the

literature [22, 25-26]. In this chapter we have considered a multiple input single output

(MISO) system equipped with Lt transmit antennas in spatially uncorrelated Rayleigh

fading channels.

In previous chapter it is shown how the performance metrics may vary with

receive diversity. In this chapter we tried to analyze the numerical performances

employing transmit diversity and with that, incorporating Alamouti coding [10] at the

transmitter side.

The rest of the chapter is organized as follows. Section 4.2 describes the system

model and in section 4.3 we present analysis of performance metrics and it ends with a

conclusion in section 4.4.

4.2 System Model and Description

Page 60: Gourab's M.tech Thesis

47

We considered a MISO system equipped with Lt transmit antennas, where (Lt

2). The block diagram of system model is shown in Figure 4.1. We consider such a

Antenna Selection (AS) scheme, wherein all the Lt antennas are divided into two groups

with L1 and L2 antennas such that L1 + L2 = Lt. Assuming perfect channel state

information (CSI), the best single antenna within each group is selected to employ

Alamouti coding at the transmitter side.

The channel fading coefficients, between ith

transmit antenna and the receive

antenna, denoted by ih for tLi ≤≤1 , are identically distributed circularly symmetric

complex Gaussian random variables with zero mean and unit variance. We assume that

the channels remain constant for a block of at least two data symbols. Let us denote the

low pass equivalent received signals for two consecutive instants as 1y and 2y . Then

using the well-known Alamouti transmit diversity [10], the received signal can be

represented as

+

−=

*2

1

*2

1

***2

1

n

n

s

s

hh

hh

y

y

uv

vu (4.1)

where, ( )0,0~ NCNni for 21 ≤≤ i is additive noise, 1s and 2s are data symbols taken

from M-ary modulation schemes with average power 2sE . We assume that perfect CSI

is available at the receiver and based on which the receiver selects two transmit antennas

with indices VU , such that

Channel Estimator

+ Maximum Likelihood Decoder

21 , yy

21ˆ ,ˆ SS

Noise (Gaussian)

Receiver

S P A C E

Group-u

Group-v

Transmitter

*1S 2S

*2S−

1S 21 S ,S

t+T t

Time

2Lh

1Lh

2h

1h

11+Lh

21+Lh

Figure 4.1 Transmission model of Ltx1 MISO system employing Alamouti code at transmitter.

Chapter 4: Transmit Antenna Selection in Alamouti Coded MISO Systems

Page 61: Gourab's M.tech Thesis

48

2

11

2

11

maxarg

maxarg

vLv

uLu

hV

hU

≤≤

≤≤

=

=

(4.2)

The receiver feeds back the indices of the selected antennas to the transmitter via

a noiseless link without any delay. At the receiver, the resulting decision variables for

both the symbols have been given by

[ ]

[ ]

−=

=

*2

1*2

*2

1*1

ˆ

ˆ

y

yhhs

y

yhhs

uv

vu

(4.3)

where, 1s and 2s are the decision variables for data symbols 1s and 2s respectively.

As both the symbols 1s and 2s are independent and equally likely, we consider any

one symbol and derive the PDF of received SNR. The instantaneous (with respect to

fading) SNR nγ can be represented [10] as

nn h γ=γ2

(4.4)

where, [ ]TVU hhh = and γ denotes 02 NEs , where sE is the symbol energy and 02N is

the variance of noise vector. Now the PDF of γ can be obtained as follows.

For convenience, let us denote the channel power gain2

ih as iX , where. tLi ≤≤1 .

Then each iX is a chi-squared distributed variable with two degrees of freedom. As

all iX are equally distributed, we can represent the PDF ( )xf X and the cumulative

distribution function (CDF) ( )xFX as [27]

( ) 0 , ≥= −xexf

xX (4.5)

( ) xX exF

−−=1 (4.6)

Further since all iX are independent, the PDF of UX can be expressed using order

statistics [28] as

( ) ( )[ ] ( ) 0 ,1

11 ≥=−

UUX

L

UXUUX xxpxFLxf

Chapter 4: Transmit Antenna Selection in Alamouti Coded MISO Systems

Page 62: Gourab's M.tech Thesis

49

( ) ( )−

=

+−−

−=

11

0

111 1

1L

n

Uxnne

n

LL (4.7)

Similarly for the other group of 2L antennas, the PDF of VX is same as equation (4.7) by

replacing 1L with 2L . Let us represent the resulting channel power gain VU XX + or 2

h

asY . Then the PDF ( )yfY can be determined [29] as

( ) ( ) ( )dttyftfyfVU X

y

XY −= 0

( ) ( )

( ) ( )

0 ,

11111

11

11

0

12

0

21121

0

121

−×

−+

−=

+−+−

=

≠=

++−−

=

ymn

ee

m

L

n

Lye

p

L

p

LLL

ynym

L

n

L

nmm

nmypP

p

(4.8)

where 21,min LLP = . Finally the PDF ( )nTASf γγ can be represented using equations (4.4)

and (4.8) as

( )( )

( ) 0 ,

expexp

1

11

2

exp11

2

1

0

1

0

211

0

2121 1 2

≥γ

Λ

γ−−

Λ

γ−

×

−+

γ

Λ

γ−γ

γ=γ

+

=

=

nn

n

m

n

nm

L

n

L

nm

m

P

p n

p

nn

n

n

mn

m

L

n

L

p

L

p

LLLf

TAS

(4.9)

where, ( ))1(2 +γ=Λ ini .

4.3 Analysis of Performance Metrics

4.3.1 Capacity

As discussed in section 2.4.1 the capacity in Rayleigh fading channel can be

obtained by averaging the capacity in AWGN channel given by ( ) ( )γ+=γ 1log2BC over

Chapter 4: Transmit Antenna Selection in Alamouti Coded MISO Systems

Page 63: Gourab's M.tech Thesis

50

the PDF of received instantaneous SNR given by equation (2.10). Therefore the capacity

expression of the transmit antenna selection in Alamouti coded MISO system is given by,

( ) ( )∞

γ γγ+=0

2 1logTAS

fBC

( ) ( ) γγγ+= γ

dfeBTAS

02 1lnlog (4.10)

where, ( )γγTASf is given by equation (4.9).

( ) ( ) γγγ+ γ

dfTAS

0

1ln

( )

( ) ( )

γ

Λ

γ−−

Λ

γ−

γ+−

−+

γ

Λ

γ−γγ+

γ

γ=

∞+

=

=

=

dmnm

L

n

L

dp

L

p

LLL

nmnmL

n

L

m

P

p p

nm

expexp

1ln 111

exp1ln211

2

0

1

0

1

0

21

1

0 0

2121

1 2

(4.11)

now,

( ) γγγ+Λ

γ−∞

de p

0

1ln

using the equation (3.13) from chapter 3

( ) γγγ+Λ

γ−∞

de p

0

1ln

+

Λ

Λ−Λ=

Λ1

111 1

1

2

pp

p Eep

(4.12)

again,

( ) γγ+Λ

γ−∞

de m

0

1ln

ΛΛ=

Λ

m

m Ee m1

1

1

(4.13)

Substituting the equations (4.12), (4.13) and (4.11) we get the ultimate expression of

capacity of Alamouti coded MISO system in Rayleigh fading channel using AS scheme

Chapter 4: Transmit Antenna Selection in Alamouti Coded MISO Systems

Page 64: Gourab's M.tech Thesis

51

( )

ΛΛ−

Λ

Λ

×

−−

−+

+

Λ

×

Λ−Λ

γ

γ=

Λ

Λ

+−

=

=

Λ−

=

n

n

m

m

nmL

n

L

mp

p

p

P

p

EeEe

mnm

L

n

LE

ep

L

p

LLLeBC

nm

nm

p

11

11

111

1

11

211

2log

1

1

1

1

1

0

1

0

211

1

21

0

21212

1 2

(4.14)

where, B is the bandwidth of the channel and 0 ; 11 >=

∞−−

xdtetEx

t is the exponential

integral of first order [24].

Figure 4.2 shows the plot of both theoretical (shown by black continuous line) and

simulation (shown by black dots) values of capacity of the Alamouti based MISO system

in Rayleigh fading channel.

0 5 10 150.9214

1.9214

2.9214

3.9214

4.9214

5.4043

Average Signal-to-Noise Ratio (dB)

Capacity (B

its/s

/Hz)

Theoretical

Simulation

(L1=1, L

2=1)

(L1=2, L

2=2)

Figure 4.2 Capacity curves for Alamouti based MISO system for different number of transmit antennas for

different numbers of transmit antennas..

4.3.2 Outage Probability

Chapter 4: Transmit Antenna Selection in Alamouti Coded MISO Systems

Page 65: Gourab's M.tech Thesis

52

As discussed in section 2.4.2 the outage probability expression can be obtained by

integrating the PDF (equation 4.9) of received instantaneous SNR over the range over the

range [ ]0,0 γ , where 0γ is the target threshold.

( ) ( ) γγ=γ<γ= γ

γ dfpPTASout

0

00

( )

∞+

=

=

=

γ

Λ

γ−−

Λ

γ−

−+

γ

Λ

γ−γ

γ

γ=

≠0

1

0

1

0

21

1

0 0

2121

expexp

111

exp211

2

1 2

dmnm

L

n

L

dp

L

p

LLL

nmnmL

n

L

m

P

p p

nm

(4.15)

now,

γ

Λ

γ−γ

dp0

exp

γ

γµ−

+

µ−=

0

1

1 pp

p (4.16)

where, ieiΛ

γ−

0

.

again,

γ

Λ

γ−−

Λ

γ−

0

expexp dnm

+

µ−−

+

µ−=

1

1

1

1

nm

nm (4.17)

Substituting equations (4.16) and (4.17) in equation (4.15) we get,

( )

+

µ−−

+

µ−

+

γ

γµ−

+

µ−

+

=

=

=

+

=

≠1

1

1

11

11

1

1

1

11

1

0

1

0

21

1

0

0

21

21

1 2

nmmn

m

L

n

L

pp

p

L

p

L

LLP

nmL

n

L

m

nm

P

p

pp

out

nm

(4.18)

Figure 4.3 shows the plot of both theoretical (shown by black continuous line) and

simulation (shown by black dots) values of outage probability of the Alamouti based

MISO system in Rayleigh fading channel.

Chapter 4: Transmit Antenna Selection in Alamouti Coded MISO Systems

Page 66: Gourab's M.tech Thesis

53

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

-4

10-3

10-2

10-1

100

Normalized Average Signal-to-Noise Ratio

Outa

ge P

robability

Theoretical

Simulation

(L1=2, L

2=2)

(L1=1, L

2=1)

Figure 4.3 Outage probability curves of Alamouti based MISO system for different number of transmit

antennas.

4.3.3 Symbol Error Rate (SER)

M-ary Phase shift Keying (MPSK)

As discussed in section 2.4.3 the SER sP expression of Almaouti coded MISO

system using M-ary PSK in Rayleigh fading channel can be obtained by averaging the

conditional error probability ( )γsP over the underlying fading random variable γ as

( ) ( )∞

γ γγγ=0

dfPPTASss (4.19)

where, ( )γsP and ( )γγTASf is given by equations (2.12) and (4.9). Interchanging the

integration limit we get,

( )( )( )

θγγπ

=

−π

γθ

π−∞

=γγ ddefP

M

MM

s TAS

1

0

sin

sin

0

2

2

1

Chapter 4: Transmit Antenna Selection in Alamouti Coded MISO Systems

Page 67: Gourab's M.tech Thesis

54

( )( )

( )

( )( )

−π

+−

=

=

=

−π

γ

θ

π−

Λ

γ−−

Λ

γ−

×−

−+

γ

γ

θ

π+

Λ−γ

γ

γ=

M

M

nm

nmL

n

L

m

P

p

M

M

p

s

dM

mn

m

L

n

L

dM

p

L

p

LLLP

nm

1

0 02

2

1

0

1

0

21

1

0

1

0 02

22121

sin

sinexp

expexp

111

sin

sin1exp

211

2

1 2

(4.20)

now,

( )

( )2

2

20

sin

sin1

sin

sin1

12

2

θ

π+

Λ

=γγ∞

γ

θ

π+

Λ−

M

de

p

M

p (4.21)

and,

( )

( )

θ

π+

Λ

=γ∞ γ

θ

π+

Λ−

2

20

sin

sin1

sin

sin1

12

2

Mde

m

M

m (4.22)

Now applying the result from [9, (5A.17), (5A.15)] on (4.21) and (4.22) and substituting

the result in equation (4.20) we get,

( ) ( )( ) ( )

( )( ) ( )

( ) ( )

α+

π

++−

α+

π×

++−

+

α

++

α+

π

++

π−

−=

−−

=

=

+−

=

n

n

nm

m

mL

n

L

m

nm

p

P

pppp

p

s

K

K

n

K

K

mmn

m

L

n

L

KKpK

p

L

p

L

LL

M

MP

nm

11

1

0

1

0

211

1

0

1

223

21

21

tan211

1tan

2

11

11

11

2

tan2sin

32tan2112

11

1

1 2

(4.23)

where, ( ) ( ) [ ]12sin 2 +πγ= iMK i and ( ) ( )MKK iii π+=α cot1 .

A special case when we will take M = 2, i.e. for BPSK equation (4.23) simplifies to

Chapter 4: Transmit Antenna Selection in Alamouti Coded MISO Systems

Page 68: Gourab's M.tech Thesis

55

( )

+

σ−−

+

σ−

+

γ

σ−

+

σ−

+

=

=

≠=

+

=

1

1

1

1

111

1

1

1

11

2

11

0

12

0

21

31

0

21

21

nmmn

m

L

n

L

pp

p

L

p

L

LLP

nmL

n

L

nmm

nm

ppP

pb

(4.24)

where, ( )1++γγ=σ ii .

Figure 4.4 shows the plot of both theoretical (shown by black continuous line) and

simulation (shown by black dots) values of SER of the Alamouti based MISO system

using MPSK in Rayleigh fading channel.

0 5 10 1510

-4

10-3

10-2

10-1

100

Average Signal-to-Noise Ratio (dB)

Sym

bol E

rro R

ate

(S

ER

)

Theoretical

Simulation

(M=2)

(M=4)

(M=8)

(M=16)

(L1=2, L

2=2)

(L1=1, L

2=1)

Figure 4.4 SER curves for Alamouti based MISO system using MPSK for different number of transmit

antennas and different M.

M-ary Quadrature Amplitude Modulation (MQAM)

As discussed in section 2.4.3 the SER sP expression of Almaouti coded MISO

system using M-ary QAM in Rayleigh fading channel can be given by

Chapter 4: Transmit Antenna Selection in Alamouti Coded MISO Systems

Page 69: Gourab's M.tech Thesis

56

( )( )

( )( )

θγ

θ−

γ−γ

π

−θγ

θ−

γ−γ

π=

π

=γγ

π

=γγ

ddM

fM

ddM

fM

P

TAS

TASs

4

02

0

2

2

02

0

sin12

3exp

11

4

sin12

3exp

11

4

(4.25)

where, ( )γγTASf is given by equations (4.9).

or,

( )( )

( )

( )( )

( )

π

+−

=

=

π

=

π

+−

=

=

π

=

γ

θ−−

Λ

γ−−

Λ

γ−

×−

−+γ

γ

θ−+

Λ−γ

γ

γ

π

γ

θ−−

Λ

γ−−

Λ

γ−

×−

−+γ

γ

θ−+

Λ−γ

γ

γ

π=

4

0 02

1

0

1

0

214

0 02

1

0

2121

2

2

0 02

1

0

1

0

212

0 02

1

0

2121

sin12

3exp

expexp

111

sin12

31exp

211

2

11

4

sin12

3exp

expexp

111

sin12

31exp

211

2

11

4

1 2

1 2

dMmn

m

L

n

Ld

M

p

L

p

LLL

M

dMmn

m

L

n

Ld

M

p

L

p

LLL

MP

nm

nmL

n

L

mp

P

p

nm

nmL

n

L

mp

P

ps

nm

nm

now,

2

2

0

sin)1(2

31

sin)1(2

31

12

θ−+

Λ

=γγ∞

γ

θ−+

Λ−

M

de

p

Mp (4.27)

again,

(4.26)

Chapter 4: Transmit Antenna Selection in Alamouti Coded MISO Systems

Page 70: Gourab's M.tech Thesis

57

θ−+

Λ

=γ∞ γ

θ−+

Λ−

2

0

sin)1(2

31

sin)1(2

31

12

M

de

m

Mm (4.28)

Now applying the result from [9, (5A.9), (5A.12)] on (4.27) and (4.28) and substituting it

in (4.26) we get the expression of SER of MQAM in Rayleigh fading channel

( )( )( )

( )

( )( )( ) ( )

( )( ) ( )

( ) ( )

+

αα−

+

αα

+

α

−+

α−

++

α

−−

+

α−

+

α

++++

α

−=

−−

=

=

+

−−

=

=

=

+

=

1

1tan

1

1tan

111

2

tan2sin

32tan2112

11

41

11

11

111

32112

11

11

12

11

1

0

1

0

211

11

02

21

21

21

0

1

0

21

1

02

21

21

1 2

1 2

nm

mn

m

L

n

L

K

pK

p

L

p

L

LL

Mnmmn

m

L

n

L

KpK

p

L

p

L

LLM

P

nnmm

L

n

L

m

nm

p-

pp

P

pp

p

L

n

L

m

nm

nm

P

pp

p

p

s

nm

nm

where, ( )( ) [ ]1143 −+γ= MiK i and ( )1+=α iii KK .

Figure 4.5 shows the plot of both theoretical (shown by black continuous line) and

simulation (shown by black dots) values of SER of the Alamouti based MISO system

using MQAM in Rayleigh fading channel.

(4.29)

Chapter 4: Transmit Antenna Selection in Alamouti Coded MISO Systems

Page 71: Gourab's M.tech Thesis

58

0 5 10 1510

-4

10-3

10-2

10-1

100

Average Signal-to-Noise Ratio (dB)

Sym

bol E

rror ra

te (S

ER

)

Theoretical

Simulation

(M=4)

(M=16)

(M=64)

(L1=2,L

2=1)

(L1=1,L

2=1)

Figure 4.5 SER curves of Alamouti based MISO system using MQAM for different number of transmit

antenna for different M.

4.4 Chapter Summary

We have considered a MISO system equipped with tL transmit antennas assuming

spatially independent Rayleigh fading channels. We consider such a AS scheme, wherein

two out of tL transmit antennas are selected. For Alamouti transmit diversity, we have

derived the exact closed-form expressions for the capacity, probability of outage and SER

for MPSK and MQAM from direct PDF approach. We have verified our analytical results

with the simulations.

Also we can combine both TAS and Alamouti coding at the transmitter side and

SEC as receive diversity to see how the system performs, which is discussed in the next

chapter.

Chapter 4: Transmit Antenna Selection in Alamouti Coded MISO Systems

Page 72: Gourab's M.tech Thesis

Chapter 5

Joint Transmit and Receive Antenna Selection in

Alamouti Coded MIMO Systems

5.1 Introduction

Multi-antenna systems have attracted great attention for the system capacity and

error performance enhancements that they provide. Nevertheless, they suffer from

hardware and signal processing complexity. Transmit and/or receive antenna selection

(TAS and/or RAS) have been suggested to maintain the advantages of multi-antenna

systems with lower complexity [21, 22]. By performing the signal transmission and/or

reception through a selected antenna subset that maximizes the instantaneous received

SNR, full-diversity transmission can be achieved with reduced signal processing

complexity.

Recently there has been an upsurge in literature concerning performance analysis

of TAS in Alamouti coded MIMO systems with some sort of receiver diversity [18-20].

Although many variants of receiver diversity combining algorithms exist, most of the

literatures focus on MRC or SC. By contrast, the use of SEC as the receive diversity, as

already discussed in chapter 3, has not been attempted before.

In chapter 3 we evaluated the numerical performances of a Alamouti coded L×2

MIMO system equipped with multibranch SEC, instead of MRC or SC at the receiver

side. Again in chapter 4 we showed the numerical performance of transmit antenna

selection in Alamouti coded MISO systems. In the current chapter we have combined both

TAS at the transmitter side and receive diversity SEC at the receiver side to see the

improvement in performance metrics. We have also employed Alamouti coding at the

transmitter side.

The remainder of this chapter is organized as follows. The system model under

study is presented in section 5.2 followed by expressions for PDF and CDF of the

Page 73: Gourab's M.tech Thesis

60

instantaneous SNR at each branch of the receiver. Next, using the model in section 5.2,

analysis of capacity, outage probability, and SER using MPSK and MQAM modulation

schemes, are described in section 5.3. The chapter finally ends with some concluding

remarks in section 5.4.

5.2 System Model and Description

The system model, with )2(>tL transmit (Tx) antennas and L receive (Rx)

antennas, is shown in Figure 5.1. Let s1 and s2 denote the equivalent baseband signals

corresponding to two successive information bits which are sent using a 2×1 Alamouti

code [10]. For a slow fading channel it may be assumed that the channel transfer function

remains constant over two consecutive symbol intervals, and accordingly the received

signals on nth branch in these two intervals can be expressed as

njninn nshshr 1211 ++= (5.1a)

njninn nshshr 2122 ++−= ∗∗ (5.1b)

where ∗∗21 ,ss are the complex conjugates of 21, ss , ( )mnmnmn jh θα= exp ,,; jim ∈

1,..,2,1, Li ∈ 2,..,2,1, Lj ∈ ; Ln ,,2,1 ∈ is the complex channel gain between the mth

Tx antenna and the nth Rx antenna with α and θ being the random amplitude and phase

variations respectively, and the additive noise nmn is a zero-mean circularly symmetric

complex Gaussian random variable (RV) having a variance N0. We assume that the perfect

CSI is known to the receiver and based on which, it selects the two transmit antennas with

indices VU , , one from each group, such that [14]

2

1 1maxarg i

LihU

≤≤= (5.2a)

2

2 1maxarg j

LjhV

≤≤= (5.2b)

At the receiver, the ST combiners attached to each branch process the signal to

produce an output pair nn yy 21 , given by

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 74: Gourab's M.tech Thesis

61

*21

*1 njnninn rhrhy

+= (5.3a)

njnninn rhrhy 1**

22

+−= (5.3b)

where mnh

is an estimate of mnh .

Figure 5.1 Transmission model of a Lt×L MIMO system employing Alamouti code at transmitter and pre-

detection switch and examine combining at the receiver.

Although we have considered a Lt×L system, but at a moment, at the receiver only

one branch will be selected, so at each and every moment we can view the model as a

MISO system. Therefore the PDF of instantaneous received SNR at the ST combiner

output will follow a central chi-square distribution with four degrees of freedom (as

discussed in chapter 4) and it is given by [14]

( )( )

( ) 0 ,

expexp

1

11

2

exp11

2

1

0

1

0

211

0

2121 1 2

≥γ

Λ

γ−−

Λ

γ−

×

−+

γ

Λ

γ−γ

γ=γ

+

=

=

nn

n

m

n

nm

L

n

L

nm

m

P

p n

p

nn

n

n

mn

m

L

n

L

p

L

p

LLLf

TAS

(5.4)

where, ( ))1(2 +γ=Λ ini and 21,min LLP = also the corresponding CDF will be

Group-u

Group-v Switched combiner

Space

Time

s1 -s2*

t t + T

s2 s1

*

s1, s2

r11, r21

n11, n21 ST Combiner y11, y21

r1L, r2L

n1L, n2L ST Combiner y1L, y2L

h11 h21

h1L h2L

Switching logic

Channel Estimator

Threshold SNR

L

y1p, y2q 1, 2

Transmitter

Decision device

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 75: Gourab's M.tech Thesis

62

( ) [ ]

( )

Λ

γ−−

Λ−

Λ

γ−−Λ

−−

−+

γ+Λ

Λ

γ−Λ−Λ

γ

γ=γ

=

+−

=

n

thn

m

thm

L

n

nmL

nm

m

thp

p

thpp

P

p nn

n

mnm

L

n

L

p

L

p

LLLF

TAS

exp1 exp1

11

11

exp211

2

1

0

21

0

1

21

0

2121

1 2

(5.5)

5.3 Analysis of Performance Metrics

5.3.1 Capacity

In order to find the average capacity of joint TS and RS in Alamouti coded

MIMO systems, we need to average ( )γC over the PDF ( )γγ SECTASf

,of the combiner

output SNR γ , i.e.

( )∞

γ γγγ=0

,)( dfCC

SECTAS (5.6)

where, ( )γγ SECTASf

, is the PDF of γ at SEC output. Assuming independent and identically

distributed (IID) fading, ( )γγ SECTASf

,can be expressed as [9, (9.341)]

( )( ) ( )[ ]

( ) ( )[ ]

γ≥γγγ

γ<γγγ

=γ−

=γγ

−γγ

γ

th

L

j

j

th

th

L

th

TASTAS

TASTAS

SECTAS

Ff

Ff

f;

;

1

0

1

, (5.7)

where, )(γγTASf and )(γγTAS

F are given by equations (5.4) and (5.5) respectively.

Substituting the value of )(γC (given by (2.9)) and ( )γγ SECTASf

, in equation (5.6) we get

( ) ( ) ( ) ( ) ( ) γγωγ+γω

γ−γ=

∞−

γ−

−γ deBFeBFFC

L

j

j

th

L

j

j

th

L

th TAS

th

TASTAS0

1

02

02

1

0

1loglog (5.8)

where, ( ) ( ) ( )γγ+=γω γTASf1ln .

Now,

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 76: Gourab's M.tech Thesis

63

( ) γγω= γ

dIth

01

( )

( ) ( )

expexp

1ln111

exp1ln211

2

0

11

0

12

0

21

0

1

0

21211

γ

Λ

γ−−

Λ

γ−

γ+−

Λ

γ−γγ+

γ

γ=

γ−

=

=

+

γ−

=

dmnm

L

n

L

dp

L

p

LLLI

nmthL

n

L

nm

m

nm

th

p

P

p

(5.9)

now,

( ) γ

Λ

γ−γγ+=′

γ

dIth

p01 exp1ln

This integration may be readily solved through integration by parts, taking ( )γγ+= 1lnu

and ( )pv Λγ−= exp .

( ) ( )

( )

Λ

γ+−

Λ

Λ−ΛΛ

+

Λ

γ−Λ−γ+

Λ

γ−ΛΛ+γ−Λ=′

p

th

pp

pp

p

thpth

p

thppthp

EE

I

11

1exp1

exp1lnexp

11

221

(5.10)

where, ( ) ( )∞ − −= x dtttxE exp1

1 ; 0>x is the exponential integral of first order [24, (5.1.1)].

again,

( )γ

γ

Λ

γ−−

Λ

γ−γ+=′′

th

nm

dI0

1 expexp1ln

This integration may be solved through integration by parts, taking ( )γ+= 1lnu

and ( ) ( ) nmv Λγ−−Λγ−= expexp .

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 77: Gourab's M.tech Thesis

64

( )( )

( )( )

Λ

γ+−

Λ

ΛΛ−

Λ

γγ+Λ+

Λ

γ+−

Λ

ΛΛ+

Λ

γγ+Λ−=′′

n

th

nnn

ththn

m

th

mm

m

m

ththm

EE

EEI

111expexp1ln

111expexp1ln

11n

111

(5.11)

Now substituting equations (5.10) and (5.11) in equation (5.9) we get,

( ) ( )

( )

( )

( )( )

( )

Λ

γ+−

Λ

ΛΛ

Λ

γγ+Λ+

Λ

γ+−

Λ

ΛΛ+

Λ

γγ+Λ−

+

Λ

γ+−

Λ

Λ−ΛΛ+

Λ

γ−Λ

−γ+

Λ

γ−ΛΛ+γ−Λ

γ

γ=

=

=

+

=

n

th

nn

n

ththn

m

th

m

m

m

m

ththm

L

n

L

nm

m

nm

p

th

pp

pp

p

thp

th

p

thppthp

P

p

EE

EE

mnm

L

n

L

EE

p

L

p

LLLI

111exp

exp1ln11

1expexp1ln

)(

)1(11

11

1exp1exp

1lnexp211

2

11n

11

11

0

12

0

21

112

221

0

1211

(5.12)

Now,

( )∞

γγω=0

2 dI

( )

( ) ( )

expexp

1ln111

exp1ln211

2

0

11

0

12

0

21

0

1

0

21212

γ

Λ

γ−−

Λ

γ−

γ+−

Λ

γ−γγ+

γ

γ=

∞−

=

=

+

∞−

=

dmnm

L

n

L

dp

L

p

LLLI

nmL

n

L

nm

m

nm

p

P

p

(5.13)

Let, ( )∞

γ

Λ

γ−γγ+=′

02 exp1ln dI

p

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 78: Gourab's M.tech Thesis

65

To solve the second integral we make use of the following result [15]

( )

( ) ( )∞

−λµ−µ+

−λ

µ

0

1exp)1ln(

!1dxxxx

( ) ( ) ( ) ( )−λ

=−λλ µ−µ+µµ−=

1

11

1

qqq PP

qEP (5.14)

where ( ) ( ) )exp(!1

0

xvxxPq

v

vq −=

= is the Poisson CDF. Thus, the second integral

Therefore,

212 1

11exp

11 p

ppp

EI Λ

+

Λ

Λ

Λ−=′ (5.15)

Again,

( ) γ

Λ

γ−−

Λ

γ−γ+=′′

dInm

expexp1ln0

2

with the help of equation (5.14) we get,

Λ

ΛΛ−

Λ

ΛΛ=′′

nn

n

mm

m EEI11

exp11

exp 112 (5.16)

Now substituting equations (5.15) and (5.16) in equation (5.13) we get,

Λ

ΛΛ

Λ

ΛΛ

−+

Λ

+

Λ

Λ

Λ−

γ

γ=

=

=

+

=

nn

n

mm

m

L

n

L

nm

m

nm

p

ppp

P

p

E

Emnm

L

n

L

Ep

L

p

LLLI

11exp

11exp

)(

)1(11

111

exp1

1211

2

1

1

1

0

1

0

21

21

1

0

21212

1 2

(5.17)

Substituting equations (5.12) and (5.17) in equation (5.8) we get

( )[ ] ( )[ ] ( )[ ]

γ+

γ−γ= −

−γ 2

1

01

1

0

12log IFIFFeBC

L

j

j

th

L

j

j

th

L

th TASTASTAS (5.18)

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 79: Gourab's M.tech Thesis

66

Figure 5.2 shows a plot of equation (5.18) for 3=γ th dB, i.e. the capacity of an

Alamouti coded TAS employing SEC MIMO system in Rayleigh fading channel for a

fixed threshold. For, L = 2, SEC operation becomes identical with dualbranch SSC system.

Further, for larger values of L, the capacity increases only when average SNR ( )γ is close

to the thγ value.

0 5 10 150.9214

1.9214

2.9214

3.9214

4.9214

5.4044

Average Signal-to-Noise Ratio (dB)

Capacity (B

its/s

/Hz)

Theoretical

Simulation

(L1=2,L

2=2,L=2)

(L1=2,L

2=2,L=1)

(L1=1,L

2=1,L=2)

(L1=1,L

2=1,L=1)

Figure 5.2 Capacity curves for Alamouti coded TAS employing SEC system with fixed threshold (th = 3

dB) for different numbers of Rx antennas.

5.3.2 Outage Probability

The outage probability can be calculated by integrating the PDF ( )γγ SECTASf

,, given

by equation (5.7) in the range [ ]0,0 γ over the random variable γ as

( ) ( ) γγ=γ<γ= γ

γ dfpPSECTASout

0

,0

0 (5.19)

Inserting (5.7) in (5.19) and after some simplification we obtain

( )[ ] ( )[ ] ( ) ( )[ ] ( )γ

γ

γ

γ

−γ γγγ+γγ

γ−γ=0

0

1

00

1

0

1dfFdfFFP

TASTAS

th

TASTASTAS

L

j

j

th

L

j

j

th

L

thout (5.20)

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 80: Gourab's M.tech Thesis

67

Solving the integrals through integration by parts

( )γ

γ γγ=th

TASdfI

03

( )

γ

Λ

γ−−

Λ

γ−

×

−+γ

Λ

γ−γ

γ

γ=

γ+

=

=

=

γ

th nmnm

L

n

L

nm

m

P

p

th

p

dmn

m

L

n

Ld

p

L

p

LLL

0

11

0

12

0

211

0 0

2121

expexp

1

11exp

211

2

(5.21)

now,

Λ

γγΛ=γ

Λ

γ−γ

γ

p

thp

th

p

d ,2exp 2

0

(5.22)

again,

γ

γ

Λ

γ−−

Λ

γ−

th

nm

d0

expexp

Λ

γ−−Λ−

Λ

γ−−Λ=

n

thn

m

thm exp1exp1 (5.23)

Therefore substituting equations (5.22) and (5.23) in equation (5.21) we get,

Λ

γ−−Λ−

Λ

γ−−Λ

−+

Λ

γγΛ

γ

γ=

=

=

+−

=

n

thn

m

thm

L

n

L

nm

m

nm

p

thp

P

p mnm

L

n

L

p

L

p

LLLI

exp1exp1

)(

)1(11,2

211

2

11

0

12

0

21221

0

1213

(5.24)

( )γ

γ γγ=0

04 dfI

TAS

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 81: Gourab's M.tech Thesis

68

can be derived in same way as equation (5.21) changing the limit,

Λ

γ−−Λ−

Λ

γ−−Λ

−+

Λ

γγΛ

γ

γ=

=

=

+−

=

n

n

m

m

L

n

L

nm

m

nm

p

p

P

p mnm

L

n

L

p

L

p

LLLI

00

11

0

12

0

210221

0

1214

exp1exp1

)(

)1(11 ,2

211

(5.25)

Substituting equations (5.24) and (5.25) in equation (5.20), the final outage probability

expression becomes

( )[ ] ( )[ ] ( )[ ] 4

1

03

1

0

1IFIFFP

L

j

j

th

L

j

j

th

L

thout TASTASTAS−

−γ γ+

γ−γ= (5.26)

Figure 5.3 shows the outage probability performance of Alamouti coded TAS

employing SEC in Rayleigh fading channel for a fixed switching threshold of 3=γ th dB

and a target threshold of 0γ =3 dB. The horizontal axis (x-axis) is normalized with respect

to target threshold.

0 1 2 3 4 510

-4

10-3

10-2

10-1

100

Normalized Average Signal-to-Noise Ratio (dB)

Outa

ge P

robability

Theoretical

Simulation

(L1=1,L

2=1,L=1)

(L1=1,L

2=1,L=2)

(L1=2,L

2=2,L=1)

(L1=2,L

2=2,L=2)

Figure 5.3 Outage probability curves for Alamouti coded TAS employing SEC system with fixed threshold

(th = 3 dB) for different numbers of Rx antennas

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 82: Gourab's M.tech Thesis

69

5.3.3 Symbol Error Rate (SER)

After discussing about capacity and outage probability, in this section, we will

derive the expression of SER for MPSK and MQAM.

M-ary Phase Shift Keying (MPSK)

With the assumption of statistical independence between fading and noise, the

average SER ( )sP of alamouti coded SEC can be calculated by averaging the conditional

error probability ( )γγ SECTASf

, over the underlying fading random variable ( ) γ as

( ) ( )∞

γ γγγ=0

,dfPP

SECTASss (5.27)

where ( )γγ SECTASf

, is as mentioned in equation (5.7). Interchanging the integration limit,

we get

( )

−π

=γγ θγ

γθ

π

−γπ

=M

M

s ddM

fPSECTAS

)1(

0 02

2

sin

sin

exp1

,

( ) ( )

( )

( ) ( ) ( )

( )

−π

γ

=γγ

−γ

−π

=γγ

θγ

γθ

π

−γ

γ−γ

π

+θγ

γθ

π

−γγπ

=

M

M

L

j

j

th

L

th

M

M

L

j

j

th

ddM

fFF

ddM

fF

th

TASTASTAS

TASTAS

1

0 02

2

1

0

1

1

0 02

2

1

0

sin

sin

exp1

sin

sin

exp1

(5.28)

now,

( ) γ

γθ

π

−γ∞

=γγ d

Mf

TAS0

2

2

sin

sin

exp

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 83: Gourab's M.tech Thesis

70

( )

γ

γθ

π

Λ

γ−−

Λ

γ−

×−

γ

θ

π

−Λ

−γ

γ

γ=

=

=

+

=

dM

mn

m

L

n

L

dM

p

L

p

LLL

nm

L

n

L

nm

m

nm

p

P

p

2

2

0

1

0

1

0

21

02

2

1

0

2121

sin

sin

exp

expexp

111

sin

sin1

exp211

2

1 2

(5.29)

now,

2

2

2

02

2

sin

sin1

1sin

sin1

exp

θ

π

γ

θ

π

−Λ

−γ∞

Md

M

pp

(5.30)

again,

γ

γθ

π

Λ

γ−−

Λ

γ−

dM

nm2

2

0 sin

sin

expexpexp

θ

π

θ

π

=2

2

2

2

sin

sin1

1sin

sin1

1MM

nm

(5.31)

Therefore substituting equations (5.30) and (5.31) in equation (5.29) we get,

( ) γ

γθ

π

−γ∞

=γγ d

Mf

TAS0

2

2

sin

sin

exp

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 84: Gourab's M.tech Thesis

71

( )

θ

π

θ

π

×−

+

θ

π

γ

γ=

=

=

+

=

2

2

2

2

11

0

12

0

21

2

2

2

1

0

2121

sin

sin1

1sin

sin1

1

111

sin

sin1

1211

2

MM

m

L

n

L

M

p

L

p

LLL

nm

L

n

L

nm

m

nm

p

P

p

(5.32)

Again,

( ) γ

γθ

π

−γγ

=γγ d

Mf

th

TAS0

2

2

sin

sin

exp

( )

γ

γθ

π

Λ

γ−−

Λ

γ−

×−

γ

θ

π

−Λ

−γ

γ

γ=

γ

=

=

+

γ

=

dM

mn

m

L

n

L

dM

p

L

p

LLL

th nm

L

n

L

nm

m

nm

th

p

P

p

2

2

0

11

0

12

0

21

02

2

1

0

2121

sin

sin

exp

expexp

111

sin

sin1

exp211

2

(5.33)

now,

γ

γ

γ

θ

π

−Λ

−γth

p

dM

02

2

sin

sin1

exp

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 85: Gourab's M.tech Thesis

72

θ

π

γ

θ

π

γ=

2

2

2

2

2

sin

sin1

sin

sin1

,2MM

p

th

p

(5.34)

again,

γ

γθ

π

Λ

γ−−

Λ

γ−

γ

dMth

nm2

2

0 sin

sin

expexpexp

θ

π

γ

θ

π

−Λ

−−−

θ

π

γ

θ

π

−Λ

−−=

2

2

2

2

2

2

2

2

sin

sin1

sin

sin1

exp1

sin

sin1

sin

sin1

exp1

MM

MM

n

th

n

m

th

m

(5.35)

Substituting equations (5.34) and (5.35) in equation (5.33) we get,

( ) γ

γθ

π

−γγ

=γγ d

Mf

th

TAS0

2

2

sin

sin

exp

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 86: Gourab's M.tech Thesis

73

( )

θ

π

γ

θ

π

−Λ

−−−

θ

π

γ

θ

π

−Λ

−−

×−

+

θ

π

γ

θ

π

γ

×

γ

γ=

=

=

+

=

2

2

2

2

2

2

2

2

1

0

1

0

21

2

2

2

2

2

1

0

2121

sin

sin1

sin

sin1

exp1

sin

sin1

sin

sin1

exp1

111

sin

sin1

sin

sin1

,2

211

2

1 2

MM

MM

m

L

n

L

MM

p

L

p

LLL

n

th

n

m

th

m

L

n

L

nm

m

nm

p

th

p

P

p

Now, substituting equations (5.32) and (5.36) in equation (5.28) we get the ultimate

expression for SER of MPSK in Rayleigh fading channel for the above mentioned system

model, given by

(5.36)

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 87: Gourab's M.tech Thesis

74

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )( )

( )( )

( )

θ

θΦ

γθΦ−−−

θΦ

γθΦ−−

−+θ

θΦ

θΦγγ

γ

γ

γ−γ

π

+

θ

θΦ−

θΦ−

θΦ

γ

γγ

π=

−π

=

=

+−π

=

−γ

−π

=

=

+

−π

=

M

M

n

thn

m

thm

L

m

nmM

M

p

pth

P

p

L

j

j

th

L

th

M

M

nm

L

n

L

nm

m

nm

M

M

p

P

p

L

j

j

ths

d

mnm

L

n

Ld

p

L

p

LLLFF

dmnm

L

n

L

dp

L

p

LLLFP

nm

TASTAS

TAS

1

0

1L

0n

1

0

21

)1(

02

1

0

21211

0

1

)1(

0

1

0

1

0

21

)1(

02

21

0

1211

0

)exp(1)exp(1

)(

)1(11,2

211

2

1

11

)(

)1(11

1211

2

1

1 2

1 2

(5.37)

where, ( ) ( )

θ

π+Λ=θΦ 22 sinsin1

Mii .

Figure 5.4 shows the SER performance of MPSK for Alamouti coded TAS

employing multi-branch SEC system over Rayleigh fading channel for a fixed threshold

of 3=γ th dB for M = 4 and M = 8.

For BPSK (M = 2) the expression given in (5.37) reduces to

( ) ( ) γγγ= ∞

γ dfQPSECTASb

0,

2

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 88: Gourab's M.tech Thesis

75

( )

( ) ( )

( )

( )

( )

γ

Λ+γ

Λ+

π−

Λ

γ−γ−Λ

γ

Λ+γ

Λ+

π−

Λ

γ−γ−Λ

−+

γ

Λ+γ×

Λ+

Λ

π−

γ

Λ+γ

Λ+

Λ

π

−γ+Λ

Λ

γ−Λγ−Λ

γ

γ

γ−γ+

Λ+−Λ

Λ+−Λ

−+

Λ+

Λ+

Λ

γ

γγ=

=

=

+

=

−γ

−−

=

=

+

=

th

nnn

ththn

th

mmm

ththm

L

n

L

nm

m

nm

th

p

p

p

th

p

p

p

thp

p

thpthp

P

p

L

j

j

th

L

th

n

n

m

m

L

n

L

nm

m

nm

p

p

p

P

p

L

j

j

thb

erfc

erfc

mnm

L

n

L

erfcp

L

p

L

LLFF

mnm

L

n

L

p

L

p

LLLFP

TASTAS

TAS

11,

2

111

1 exp1

11,

2

111

1exp1

2

1

)(

)1(1111,

2

1

11

111,

2

3

11

1

exp)(111

2

111

111

2

1

)(

)1(11

11

1

2

31

1111

2

2

1

2

1

1

0

1

0

21

2

1

2

2

3

221

0

1

211

0

12

1

2

1

1

0

1

0

21

2

3

221

0

1211

0

1 2

1 2

(5.38)

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 89: Gourab's M.tech Thesis

76

0 5 10 1510

-4

10-3

10-2

10-1

100

Average Signal-to-oise Ratio (dB)

Sym

bol E

rror R

ate

(S

ER

)

Theoretical

Simulation

(L1=2,L

2=2,L=2)

(L1=1,L

2=1,L=2)

(L1=2,L

2=2,L=1)

(L1=1,L

2=1,L=1)

M=4

M=8

Figure 5.4 SER curves for Alamouti coded TAS employing SEC system with fixed threshold (th = 3 dB)

for M= 4, 8 and for different numbers of Rx antennas.

M-ary Quadrature Amplitude Modulation (MQAM)

Here we have considered QAM square constellation i.e. ML = , where L is a

positive integer. For fading channel, the SER, ( )γsP , becomes conditional on the fading

SNR γ , which may be obtained from equation (2.18).

( )( )

( )( )

θγ

θ−

γ−γ

π−

θγ

θ−

γ−γ

π=

π

=γγ

π

=γγ

ddM

fM

ddM

fM

P

SECTAS

SECTASs

4

02

0

2

2

02

0

sin12

3exp

11

4

sin12

3exp

11

4

,

,

(5.39)

where, ( )γγ SECTASf

, is as mentioned in equation (5.7).

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 90: Gourab's M.tech Thesis

77

( ) ( )( )

( ) ( )

( )( )

π

γ

=γγ

−γ

π

=γγ

θγ

θ−

γ−γ

γ−γ

π−

θγ

θ−

γ−γγ

π=

4

0 02

1

0

1

2

2

0 02

1

0

sin12

3exp

11

4

sin12

3exp

11

4

ddM

f

FFM

ddM

fFM

th

TAS

TASTAS

TASTAS

L

j

j

th

L

th

L

j

j

th

(5.40)

now, same way as in MPSK according to equation (5.32) we get,

( )( )

γ

θ−

γ−γ

=γγ d

Mf

TAS0

2sin12

3exp

( )

( )

( ) ( )

θ−+

Λ−

θ−+

Λ

×−

+

θ−+

Λ

γ

γ=

=

=

+

=

22

11

0

12

0

21

2

2

1

0

2121

sin12

311

sin12

311

111

sin12

311

211

2

MM

m

L

n

L

Mp

L

p

LLL

nm

L

n

L

nm

m

nm

p

P

p

(5.41)

and

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 91: Gourab's M.tech Thesis

78

( )( )

γ

θ−

γ−γ

γ

=γγ d

Mf

th

TAS0

2sin12

3exp

( ) ( )

( )

( ) ( )

( ) ( )

θ−+

Λ

γ

θ−−

Λ−−−

θ−+

Λ

γ

θ−−

Λ−−

×−

+

θ−+

Λ

γ

θ−+

Λγ

×

γ

γ=

=

=

+

=

22

22

1

0

1

0

21

2

22

1

0

2121

sin12

31

sin12

31exp1

sin12

31

sin12

31exp1

111

sin12

31

sin12

31,2

211

2

1 2

MM

MM

m

L

n

L

MM

p

L

p

LLL

n

th

n

m

th

m

L

n

L

nm

m

nm

p

th

p

P

p

(5.42)

Now, substituting equations (5.41) and (5.42) in equation (5.40) we get,

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 92: Gourab's M.tech Thesis

79

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )( )

( )( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )( )

( )( )

θ

θΓ

γθΓ−−−

θΓ

γθΓ−−

−+θ

θΓ

θΓγγ

γ

γ

γ−γ+

θ

θΓ−

θΓ−

θΓ

γ

γγ

π−

θ

θΓ

γθΓ−−−

θΓ

γθΓ−−

−+θ

θΓ

θΓγγ

×

γ

γ

γ−γ+

θ

θΓ−

θΓ−

θΓ

γ

γγ

π=

π

=

=

=

−γ

π

=

=

+

π

=

π

=

=

=

−γ

π

=

=

+

π

=

4

0

1L

0n

1

0

214

02

1

0

21211

0

1

4

0

1

0

1

0

21

4

02

21

0

1211

0

22

0

1L

0n

1

0

212

02

1

0

21211

0

1

2

0

1

0

1

0

21

2

02

21

0

1211

0

)exp(1)exp(1

)(

)1(11,2

211

2

11

)(

)1(11

1211

2

11

4)exp(1)exp(1

)(

)1(11,2

211

2

11

)(

)1(11

1211

2

11

4

1 2

1 2

1 2

1 2

d

mnm

L

n

Ld

p

L

p

LLLFF

dmnm

L

n

L

dp

L

p

LLLF

Md

mnm

L

n

Ld

p

L

p

LLLFF

dmnm

L

n

L

dp

L

p

LLLF

MP

n

thn

m

thm

L

m

nm

p

pth

P

p

L

j

j

th

L

th

nm

L

n

L

nm

m

nm

p

P

p

L

j

j

th

n

thn

m

thm

L

m

nm

p

pth

P

p

L

j

j

th

L

th

nm

L

n

L

nm

m

nm

p

P

p

L

j

j

ths

nm

TASTAS

TAS

nm

TASTAS

TAS

where, ( )( )

θ−+

Λ=θΓ

2sin12

311

Mi

i .

Figure 5.5 shows the SER performance of MQAM for Alamouti coded TAS

employing multi-branch SEC system over Rayleigh fading channel for a fixed threshold

of 3=γ th dB for M = 4 and M = 16.

(5.43)

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 93: Gourab's M.tech Thesis

80

0 5 10 1510

-4

10-3

10-2

10-1

100

Average Signal-to-Noise Ratio (dB)

Sym

bol E

rror R

ate

(S

ER

)

Theoretical

Simulation

(L1=2,L

2=2,L=2)

(L1=2,L

2=2,L=1)

(L1=1,L

2=1,L=2)

(L1=1,L

2=1,L=1)

M=16M=4

Figure 5.5 SER curves for Alamouti coded TAS employing SEC system with fixed threshold (th = 3 dB)

for M=4 and for different numbers of Rx antennas.

5.4 Chapter Summary

Closed-form analytical expressions for capacity, outage probability, and SER have

been obtained for a Lt x L MIMO system employing transmit antenna selection, Alamouti

code and SEC as receive diversity. For verification of the derived expressions, extensive

Monte Carlo simulations were carried out. It was found that the theoretical values

(represented by solid lines) show excellent match with the simulation results (represented

by black dots).

Also it is interesting to compare all the schemes that we discussed till now in the

previous chapters to analyze a comparative performance and it is discussed in the next

chapter.

Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems

Page 94: Gourab's M.tech Thesis

Chapter 6

Comparative Studies and Discussions

This final chapter summarizes the main contributions of this dissertation and

discusses further scope of work to extend these results. A summary of these results and

some comparative studies among different schemes is presented in the next section,

section 6.1. Section 6.2 demonstrates the limitations of the adopted systems, whereas

interesting and important future research directions are suggested in section 6.3.

6.1 Summary of Contributions

Collectively, the main contribution of this work is like that, first we have

discussed different receiver diversity schemes and their drawbacks. From them we

selected multi-branch SEC as the best trade-off. Second, we employed Alamouti coding,

a transmit diversity scheme, at the transmitter side to further improve the system

performance. Third, we incorporated TAS scheme. Finally we considered a system where

we employed all the schemes together i.e. TAS and Alamouti coding at the transmitter

side and SEC at the receiver side.

In the next subsection we have discussed a comparative study among the schemes

that we already presented in chapter 2 (section 2.7), chapter 3, chapter 4 and chapter 5.

6.1.1 Comparative Study among different Schemes

In this section we have compared all the performance metrics i.e. capacity, outage

probability and symbol error rate (SER) (shown in figure 6.1) among different schemes

which are as follows:

a. No Diversity (Tx = 1, Rx = 1)

b. Alamouti Code (Tx = 2, Rx = 1)

c. Alamouti Code and SEC combined (Tx = 2, Rx > 1)

Page 95: Gourab's M.tech Thesis

82

d. Transmit Antenna Selection and Alamouti Code in MISO (Tx > 2, Rx = 1)

e. Transmit Antenna Selection, Alamouti Coding and SEC in MIMO (Tx > 2, Rx > 2)

From the following four pictures it is seen that applying Alamouti code (b) at the

transmitter side, gives better performance than no diversity (a) case in all aspects

(Chapter 2). Again if we incorporate receive diversity SEC with Alamouti coding

(chapter 3) then the scheme (c) further improves the performance compared to the

scheme (b). However if we design a model without any receive diversity but at the

transmitter side we employ transmit diversity and Alamouti coding (chapter 4) then it (d)

gives more improved result compared to (c) for the same number of total antennas. Lastly

the scheme (e), where we applied transmit diversity, Alamouti coding and receive

diversity (chapter 5), outperforms compared to scheme (c) but gives better results at low

SNR compared to scheme (d) except the outage probability case.

0 5 10 15

100

Average Signal-to-Noise Ratio (dB)

Capacity (B

its/s

/Hz)

(No Diversity)

(Alamouti Code/ 2x1)

(Ala.+SEC/ 2x6)

(Tx Ant. sel.+Ala./ L1=4,L

2=3,L=1)

(Tx. Ant sel.+Ala.+SEC/ L1=3,L

2=3,L=2)

Figure 6.1 (a) Capacity curves for Alamouti based different schemes with a fixed threshold (th = 3 dB) for

different numbers of total antennas.

Chapter 6: Comparative Studies and Discussions

Page 96: Gourab's M.tech Thesis

83

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

-4

10-3

10-2

10-1

100

Normalized Signal-to-Noise Ratio (dB)

Outa

ge P

robability

(No Diversity)

(Alamouti Code/ 2x1)

(Ala.+SEC/ 2x6)

(Tx Ant.sel+Ala/ L1=4,L

2=3,L=1)

(Tx Ant sel.+Ala.+SEC/ L

1=3,

L2=3,L=2)

Figure 6.1 (b) Outage probability curves for Alamouti based different schemes with same switching

threshold and target threshold dBoth 3=γ=γ for different numbers of total antennas.

0 5 10 1510

-4

10-3

10-2

10-1

100

Average Signal-to-Noise Ratio (dB)

Sym

bol E

rror R

ate

(S

ER

)

(No Diversity)

(Alamouti Code/ 2x1)

(Ala.+SEC/ 2x6)

(Tx Ant. sel+Ala.+SEC/L

1=3,L

2=3,L=2)

(Tx. Ant. sel+Ala./ L1=4,L

2=3,L=1)

(M=4)

Figure 6.1 (c) SER curves for Alamouti based different schemes with a fixed threshold (th = 3 dB) using

4-PSK for different numbers of total antennas.

Chapter 6: Comparative Studies and Discussions

Page 97: Gourab's M.tech Thesis

84

0 5 10 1510

-4

10-3

10-2

10-1

100

Average Signal-to-Noise Ratio (dB)

Sym

bol E

rror R

ate

(S

ER

)(No Diversity)

(Alamouti Code/ 2x1)

(Ala.+SEC/ 2x6)

(Tx. Ant sel+Ala./ L

1=4,L

2=3,L=1)

(Tx. Ant sel.+Ala+SEC/ L

1=3,L

2=3,L=2)

Figure 6.1 (d) SER curves for Alamouti based different schemes with a fixed threshold (th = 3 dB) using

4-QAM for different numbers of total antennas.

6.2 Limitations

In the whole thesis we have considered Alamouti coding, multibranch SEC and

transmit diversity in our system models separately or combined and we saw that the

performance becomes better as we go for more complex systems. However, one has to

remember certain limitations and disadvantages too:

1. Alamouti scheme (2x1) is always 3 dB worse than 2-branch MRC scheme (1x2),

when the total transmit power is kept fixed. So in the former case the power is halved.

2. We have considered that perfect CSI is known to the receiver, which means added

complexity. Also it is tough to estimate the perfect CSI.

3. Our assumption that the channel transfer function (TF) is constant over two

consecutive symbol periods dose not hold for the fast fading scenario.

Chapter 6: Comparative Studies and Discussions

Page 98: Gourab's M.tech Thesis

85

4. We have employed receive diversity, but one should remember that it will limit the

portability of the wireless devices.

6.3 Future Scopes

We conclude with some brief remarks on future extensions of the work presented in

this thesis. Future works can be done on different fields associated with the work discussed in

this thesis as:

1. Performance analysis of the systems considering different propagation environments, i.e.

changing the wireless channel from Rayleigh to Nakagami-m and Rician fading channels.

2. Performance analysis of a system incorporating relay in between transmitter and

receiver.

3. As Alamouti’s transmit diversity provides solely a diversity gain, whereas STTC

gives both diversity gain and coding gain so it is interesting to analyze the

performance of systems employing space-time Trellis code (STTC) at the transmitter.

4. Performance analysis may be done employing channel coding.

.

Chapter 6: Comparative Studies and Discussions

Page 99: Gourab's M.tech Thesis

86

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Publications based on Thesis Work

Conferences:

[1] G. Maiti and A.Chandra, “ Error Probability of Alamouti Coded MIMO Systems with

multibranch Switch-and-Examine Combining” , Proc. IEEE CASCOM PGSPC 2010,

vol. 1, no. 2, Nov 2010, pp. 5-8.

[2] G. Maiti and A. Chandra, “Performance Analysis of Alamouti Coded MIMO Systems

with Switch and Examine Combining”, Proc. IEEE ISCI 2011, Mar. 2011, pp. 764-

769.

Journal:

Manuscript is under preparation. It is on joint transmit and receive antenna selection

in Alamouti coded MIMO systems.