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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
Dedicated to
My Parents and Elder Brother
iii
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
iv
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
v
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
vi
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.
vii
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
viii
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
ix
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
x
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
xi
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
xii
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
xiii
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]
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
2
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
3
(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
4
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
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.
6
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
7
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
8
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
9
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
10
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
11
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
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
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
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
=θ
∞
=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
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
=θ
π
=θ
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
=θ
∞
=γγ
π
=θ
∞
=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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
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
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
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−
=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
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+
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
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
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
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−
=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 −
=γ
−
=
+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
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
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
=θ
∞
=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
+γ
γ
θ
π
+γ
−γγγπ
=
γ
=γ
−
=γ
−γ
−π
=θ
∞
=γ
−
=γ
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
=γ
γ
θ
π
+γ
−γ∞ M
dM
(3.23)
Chapter 3: Multibranch SEC in Alamouti Coded MIMO Systems
40
again,
2
2
2
2
2
02
2
sin
sin2
sin
sin2
,2sin
sin2
exp
θ
π
+γ
γ
θ
π
+γ
γ=γ
γ
θ
π
+γ
−γγ
=γ
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
−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
−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
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
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
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
=θ
∞
=γγ
π
=θ
∞
=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
∞
γ
π
−
=γ
γ
γ
π
−
=γ
−γ
∞
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
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
−γ
π
−
=γ
π
−
=γ
−γ
π
−
=γ
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
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
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
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
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
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
+
−
=
−
≠
=
−
=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
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
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
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−
=µ
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
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
=θ
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
54
( )( )
( )
( )( )
−π
=θ
∞
+−
=
−
=
−
=
−π
=θ
∞
=γ
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
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
56
( )( )
( )( )
θγ
θ−
γ−γ
−
π
−θγ
θ−
γ−γ
−
π=
π
=θ
∞
=γγ
π
=θ
∞
=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
=θ
∞
+−
=
−
=
π
=θ
∞
=γ
−
=
π
=θ
∞
+−
=
−
=
π
=θ
∞
=γ
−
=
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
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
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
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
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
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
+
−
=
−
≠
=
−
=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
62
( ) [ ]
( )
Λ
γ−−
Λ−
Λ
γ−−Λ
−−
−
−+
γ+Λ
Λ
γ−Λ−Λ
γ
−
−
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−
=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
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+−
−
−
+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
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
γ−γ+Λ+
Λ
γ+−
Λ
ΛΛ+
Λ
γ−γ+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
1
0
1
0
21
112
221
0
1211
1 2
(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+−
−
−
+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
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
−
=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
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 total 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
−
=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
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
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+
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 total antennas
Chapter 5: Joint Transmit and Receive Antenna Selection in Alamouti Coded MIMO Systems
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
=θ
∞
=γγ θγ
γθ
π
−γπ
=M
M
s ddM
fPSECTAS
)1(
0 02
2
sin
sin
exp1
,
( ) ( )
( )
( ) ( ) ( )
( )
−π
=θ
γ
=γγ
−
=γ
−γ
−π
=θ
∞
=γγ
−
=γ
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
70
( )
γ
γθ
π
−
−
Λ
γ−−
Λ
γ−
×−
−
−
+γ
γ
θ
π
−Λ
−γ
γ
−
−
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
+Λ
=γ
γ
θ
π
−Λ
−γ∞
=γ
Md
M
pp
(5.30)
again,
γ
γθ
π
−
Λ
γ−−
Λ
γ−
∞
=γ
dM
nm2
2
0 sin
sin
expexpexp
θ
π
+Λ
−
θ
π
+Λ
=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
71
( )
θ
π
+Λ
−
θ
π
+Λ
×−
−
−
+
θ
π
+Λ
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
γ
θ
π
−Λ
−γ
γ
−
−
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
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
72
θ
π
+Λ
γ
θ
π
+Λ
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
=γ
dMth
nm2
2
0 sin
sin
expexpexp
θ
π
+Λ
γ
θ
π
−Λ
−−−
θ
π
+Λ
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
73
( )
θ
π
+Λ
γ
θ
π
−Λ
−−−
θ
π
+Λ
γ
θ
π
−Λ
−−
×−
−
−
+
θ
π
+Λ
γ
θ
π
+Λ
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
74
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( )( )
( )( )
( )
θ
θΦ
γθΦ−−−
θΦ
γθΦ−−
−
−
−
−+θ
θΦ
θΦγγ
γ
−
−
γ
γ−γ
π
+
θ
θΦ−
θΦ−
−
−
−
+θ
θΦ
γ
−
−
γγ
π=
−π
=θ
−
=
−
=
+−π
=θ
−
=
−
=γ
−γ
−π
=θ
−
=
−
≠
=
+
−π
−
=
−
=γ
≠
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
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
−γ
−
−−
=
−
≠
=
+
−
=
−
=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
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 total 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
=θ
∞
=γγ
π
=θ
∞
=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
77
( ) ( )( )
( ) ( )
( )( )
π
=θ
γ
=γγ
−
=γ
−γ
π
=θ
∞
=γγ
−
=γ
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
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
79
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( )( )
( )( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( )( )
( )( )
θ
θΓ
γθΓ−−−
θΓ
γθΓ−−
−
−
−
−+θ
θΓ
θΓγγ
γ
−
−
γ
γ−γ+
θ
θΓ−
θΓ−
−
−
−
+θ
θΓ
γ
−
−
γγ
−
π−
θ
θΓ
γθΓ−−−
θΓ
γθΓ−−
−
−
−
−+θ
θΓ
θΓγγ
×
γ
−
−
γ
γ−γ+
θ
θΓ−
θΓ−
−
−
−
+θ
θΓ
γ
−
−
γγ
−
π=
π
=θ
−
=
−
=
+π
=θ
−
=
−
=γ
−γ
π
−
=
−
≠
=
+
π
−
=
−
=γ
π
=θ
−
=
−
=
+π
=θ
−
=
−
=γ
−γ
π
−
=
−
≠
=
+
π
−
=
−
=γ
≠
≠
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
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 = 4,16 and for different numbers of total 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
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)
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
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
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
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
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.