The E ects of Finite Antenna Separation on Signal ...users.cecs.anu.edu.au/~Haley.Jones/supervision/anindita_thesis.pdf · The E ects of Finite Antenna Separation on Signal Correlation

The Effects of Finite Antenna

Separation on Signal Correlation

in Spatial Diversity Receivers

Anindita Saha

B.E. Mangalore University

August 2003

A thesis submitted for the degree of Master of Philosophy

of The Australian National University

Department of Telecommunications EngineeringResearch School of Information Sciences and Engineering

The Australian National University

i

Declaration

The contents of this thesis are the results of original research and have not been

submitted for a higher degree to any other university or institution.

A part of the work in this thesis has been published

Haley M.Jones, Anindita Saha and Thushara D. Abhayapala. The Effect Of

Finite Antenna Separation On The Performance Of Spatial Diversity Receivers. In

Proceedings of International Symposium on Signal Processing and its Applications,

Paris, France 2003.

This thesis is the result of my work performed jointly with Dr T.D. Abhaya-

pala and Dr H. M. Jones. All sources used in the thesis have been furthermore

acknowledged.

Anindita Saha

Department of Telecommunications

Research School of Information Sciences and Engineering

The Australian National University

Canberra ACT 0200

Australia.

iii

Acknowledgments

The work in this thesis is a culmination of my efforts strongly backed by the

support of a lot of people to whom I am deeply grateful.

I owe a great deal to my supervisors, Dr Thushara Abhayapala and Dr Haley

Jones for guiding and training me through my entire tenure as a student in RSISE.

I deeply admire their patience in putting up with my inexperience with research

and helping me understand the technical concepts involved in my work.

I would like to thank the students and staff at RSISE for providing me with all

the technical, administrative and emotional support that I asked for. At this point

I would like to mention Tony’s help in answering my quick and often silly questions

even when he was really busy with his thesis.

I am deeply grateful to my parents, brother and uncle for providing me with

the resources and moral support to come to Australia and fulfilling my dream of

completing postgraduate studies here together with experiencing a world different

from home. The encouragement and love from them kept me going during those

days when things just did not look heartening.

I owe a special thanks to my friends at Fenner Hall who made my stay here a

truly multicultural experience. I will cherish all the wonderful times I spent with

them and am grateful to them for pulling me out of difficult times and sharing my

happy moments.

v

Abstract

The importance of the spatial aspects of the wireless communication channel

has received increasing recognition in the recent years. In particular, antenna ar-

rays have been established to be effective combatants of the fading effect of the

wireless channel. They work by exploiting either the spatial, temporal, frequency

or polarization diversity characteristics offered by the channel. Usually the channel

is modelled as imposing either a Rayleigh, Rician or Nakagami fading envelopes

on the transmitted signal such that, at points separated in space, the signal fading

characteristics are independent, though identically distributed. That is, the actual

points at which we may receive the signal (and place antennas) are considered ir-

relevant, as long as they are spatially separated ‘enough’.

In this thesis we consider the effects of the actual spatial separation between

antennas on the receiver output SNR and BER when popular spatial diversity tech-

niques such as maximal ratio combining and equal gain combining are employed.

The analysis is carried out using a spatial channel model where the channel vector

is separated into the product of a deterministic matrix and a random vector. The

deterministic matrix captures the physical configuration of the antenna elements

and the random vector characterizes the wireless environments. The performance

of the system is compared with the standard independent Rayleigh fading model.

The results obtained by the analysis show the degradation of the system per-

formance due to correlation effects that manifest due to closely spaced antennas.

As implied above, such correlation effects are most often neglected in performance

analysis.

Closed form expressions for the average BER for different array configurations

are derived for MRC using BPSK signal modulation. The theoretical results show

close similarity with those obtained by simulation, highlighting the effect of finite

antenna separation on the performance of diversity combining schemes. It is found

that a uniform circular array with a certain radius provides better performance

than a uniform linear array of the same aperture.

Contents

List of Figures xiii

1 Introduction 1

1.1 The mobile wireless channel . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Antenna arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Questions to be answered in this thesis . . . . . . . . . . . . . . . . 6

1.4 Thesis motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Overview of the mobile wireless channel and spatial diversity tech-

niques 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Wireless signal propagation . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Propagation mechanisms . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Pathloss due to large scale fading . . . . . . . . . . . . . . . 11

2.3 Stochastic channel modelling . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Impulse response characterization of a multipath channel . . 13

2.3.2 Stochastic channel characterization via the autocorrelation

function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Fading models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 Statistical modelling of independently faded signal envelopes 15

2.4.2 Small scale fading manifestations . . . . . . . . . . . . . . . 18

2.5 Mitigating losses in SNR using diversity . . . . . . . . . . . . . . . 20

2.5.1 Diversity methods . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5.2 Spatial diversity combining techniques . . . . . . . . . . . . 24

2.5.3 Analysis of diversity techniques . . . . . . . . . . . . . . . . 25

2.5.4 Hybrid combining . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

vii

viii Contents

3 Correlation effects in diversity schemes 29

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Impact of independent fading on diversity combining . . . . . . . . 29

3.3 Correlated fading effects on diversity systems . . . . . . . . . . . . . 31

3.3.1 Dual diversity systems . . . . . . . . . . . . . . . . . . . . . 32

3.3.2 Antenna arrays with greater than two elements . . . . . . . 35

3.4 Factors influencing spatial correlation . . . . . . . . . . . . . . . . . 36

3.4.1 Mutual coupling . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4.2 Angular spread . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4.3 Antenna spacing . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5 Spatial channel modelling . . . . . . . . . . . . . . . . . . . . . . . 40

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Performance of EGC and MRC under finite antenna separation 43

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Spatial channel model . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2.1 The SIMO model . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.2 Analyzing the channel matrix . . . . . . . . . . . . . . . . . 46

4.2.3 Comments on the model . . . . . . . . . . . . . . . . . . . . 48

4.3 Effect of introducing ‘space’ into diversity systems . . . . . . . . . . 49

4.3.1 SNR performance of MRC and EGC with finite antenna sep-

aration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.2 Performance of BER using the spatial model . . . . . . . . . 58

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 Effects of spatial correlation on diversity receivers 63

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 Spatial correlation effects between adjacent antenna elements . . . 64

5.3 Average SNR using diversity incorporating spatial correlation . . . 68

5.4 Error probability of MRC using BPSK modulation . . . . . . . . . 72

5.5 Covariance matrices for different array configurations . . . . . . . . 75

5.5.1 Uniform linear array . . . . . . . . . . . . . . . . . . . . . . 77

5.5.2 Uniform circular array . . . . . . . . . . . . . . . . . . . . . 81

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6 Conclusion 87

6.1 Summary of main results . . . . . . . . . . . . . . . . . . . . . . . . 87

Contents ix

6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

List of Figures

1.1 Multipath in a wireless environment is the result of a number of

delayed versions of the transmitted wave arriving at the receiver due

to reflection from various structures. A line-of-sight (LOS) path is

included in this illustration. . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Illustration of the beamforming lobes at the switched beam array

showing sectors from where the users signals are detected. . . . . . 5

2.1 The effect of increasing transmitter and receiver separation d, for

various pathloss exponents, n, in (2.2). . . . . . . . . . . . . . . . . 12

2.2 Illustration of the fluctuations of a signal when subjected to Rayleigh

fading with 10 multipaths. The signal envelope for a mobile device

appears below the threshold more frequently than a stationary de-

vice indicating greater chance of signal loss. . . . . . . . . . . . . . 16

2.3 Small scale fading manifestations. . . . . . . . . . . . . . . . . . . . 19

2.4 Illustration of a) frequency diversity, b) time diversity and c) space

diversity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Illustration of switched/selection schemes in spatial diversity. . . . . 23

2.6 Illustration of general gain combining diversity schemes where w1, w2, ..wP

are the weights added to the respective branches. . . . . . . . . . . 25

3.1 Comparison of improvement in diversity gain using MRC, EGC and

SC for increasing number of antennas in an independent Rayleigh

fading environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Illustration of signals arriving at the receiver from a uniform ring of

scatterers with radius R, where θ is the angle of arrival and φ is the

angular spread of the signal. . . . . . . . . . . . . . . . . . . . . . 37

xi

xii List of Figures

4.1 Spatial scattering model under consideration for a flat fading SIMO

system. rR is the radius of the sphere enclosing the receiver array

and the scatterers are distributed outside the ball of radius rRS which

is in the farfield of the receiver array. A(ϕ) represents the gain of the

complex scattering environment for signals arriving at the receiver

scatter-free region from direction ϕ from a single transmitter. . . . 44

4.2 Illustration of the highpass character of the Bessel function when

m = 10 and m = 100. Also shown is the low pass character of J0(·). 46

4.3 Comparison of the average output SNR with increasing number of

receiver antennas. The performance of MRC and EGC schemes when

the antennas are separated by a distance of λ/2 is compared with

independent Rayleigh fading. Also shown is the SNR gain for the

case where correlation is maximised between antennas for both MRC

and EGC, i.e., when d = 0. . . . . . . . . . . . . . . . . . . . . . . 50

4.4 Performance of MRC when the separation ‘d’ between the antennas

in a ULA is reduced such that d = λ, λ/2, λ/10 and 0. . . . . . . . 51

4.5 Comparison of the performance of MRC and EGC with the Rayleigh

fading model when the number of antennas are increased in a ULA

with a constant aperture D = λ. The case where D = 0 is also

shown to represent maximized correlation. . . . . . . . . . . . . . . 52

4.6 Comparison of the performance of MRC and EGC when the aperture

of the ULA is decreased i.e. D = λ/2, λ/10 and 0. . . . . . . . . . . 53

4.7 Comparison of the performance of MRC and EGC when the radius

‘R’ of the UCA is varied such that R = λ/2, λ/10 and 0. . . . . . . 54

4.8 Illustration of the effect of increasing the separation distance be-

tween 2 antennas in steps of λ/4. . . . . . . . . . . . . . . . . . . . 55

4.9 Comparison of BER performance for the independent Rayleigh fad-

ing case when the number of antennas is increased for MRC. . . . . 57

4.10 BER performance of MRC when the number of antennas is increased

in a ULA with a constant aperture of λ. . . . . . . . . . . . . . . . 58

4.11 BER performance of a dual diversity MRC system with varying array

aperture compared with the independent fading case. . . . . . . . . 59

4.12 BER performance of MRC with varying aperture in a ULA with 2

antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.13 Comparison of BER performance with varying radius for a UCA. . 61

List of Figures xiii

5.1 Comparison of spatial correlation versus separation for a diffuse scat-

tering field, limited in the azimuth with the source centered around

π/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2 Illustration of the effective SNR when calculated using the lower

bound in equation (5.29). . . . . . . . . . . . . . . . . . . . . . . . 69

5.3 Illustration of the effect of spatial correlation on the BER perfor-

mance of a dual diversity system in a 2 dimensional isotropic diffuse

field with varying ULA aperture D. . . . . . . . . . . . . . . . . . . 76


mance when 3 antennas are employed in a diffuse, isotropic field. . . 77

5.5 Illustration of the effect of increasing antenna array aperture on

average BER with a 2 antenna array for different average SNR values. 78

5.6 Illustration of the effect of increasing array aperture at a constant

average SNR of 10dB. . . . . . . . . . . . . . . . . . . . . . . . . . 79


mance for a ULA with 2 antennas angular spreads of 20 and 5 and

array apertures of D = λ/2 and λ/5. . . . . . . . . . . . . . . . . . 80

5.8 BER performance when 3 antennas are used in a ULA with the

energy arriving within a beamwidth of 30 and 10 when the aperture

is λ/2. Also shown the performance degradation when the aperture

is reduced to λ/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81


mance for a uniform circular array in a diffused isotropic field when

the radius R is decreased. . . . . . . . . . . . . . . . . . . . . . . . 82


mance for a uniform circular array in a diffuse isotropic field with 3

receiver antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83


mance of a dual diversity UCA with angular spreads of 20 and 5

and radii of R = λ/5 and λ/10. . . . . . . . . . . . . . . . . . . . . 84

Chapter 1

Introduction

Recent exponential trends in the growth of the number of mobile device users has

fuelled the development of sophisticated and economical wireless devices. The idea

of ubiquitous access to information anywhere, anyplace and anytime, characterizes

the information systems of the 21st century. Increasing demands for untethered

and lightweight communication devices has triggered the need for higher power and

bandwidth efficiencies together with improved quality of service. Research into the

optimal use of the available radio spectrum has lead to the development of various

modulation and coding schemes together with the application of adaptive antenna

arrays to exploit the time, space and frequency varying environment.

There are several major characteristics of the mobile wireless channel which

affect signal propagation and detection. The characteristics we consider in this

chapter include multipath, Doppler shift, co-channel interference and noise. In

general these characteristics are considered to have an adverse effect on detection

performance. However, some characteristics, such as multipath, can be seen to

offer a natural form of signal diversity which can be exploited. This natural signal

diversity has been taken advantage of by using antenna arrays. However, it is most

often assumed that the signals at the different antennas are uncorrelated. In real-

ity this is not always the case, motivating us to explore the effects of correlation

between signals at closely spaced antennas.

In this introductory chapter we review the most significant characteristics of

the mobile wireless channel leading into a discussion of the role of antenna arrays

in wireless systems and, subsequently, our motivation for considering the effects of

the actual geometry of antenna arrays on signal detection. We consider the most

1

2 Introduction

LOS

BaseStation

Figure 1.1: Multipath in a wireless environment is the result of a number of delayedversions of the transmitted wave arriving at the receiver due to reflection fromvarious structures. A line-of-sight (LOS) path is included in this illustration.

important relevant aspects of mobile wireless channels and correlation effects in

greater detail in subsequent chapters.

1.1 The mobile wireless channel

The most challenging aspect of studying wireless communication systems is charac-

terizing the hostile channel. Unlike wired or fixed radio link transmission the mo-

bile wireless environment provides the additional challenge of a constantly changing

channel, due primarily to the relative movement of the communication devices and,

to a lesser extent, other articles in the environment. In this section we consider

several aspects of the mobile wireless channel. It should be understood that mobil-

ity may be considered an inherent part of the wireless channel even if the receiver

and transmitter are fixed. That is, we have no control over the movement of other

items in the channel (e.g., people, wind, moving leaves etc).

Multipath propagation channels

The transmitted signal may arrive at the receiver via several propagation mech-

anisms including reflection, refraction, diffraction and/or scattering due to atmo-

spheric conditions, buildings and local terrain features. The resultant signal at

the receiver is a complex summation of the time, space and frequency dispersed

1.1 The mobile wireless channel 3

copies of the transmitted signal with or without the presence of a direct signal.

This is referred to as multipath propagation. An illustration of multiple copies of

the transmitted signal arriving at the mobile device is shown in Figure 1.1. The

multipath waves combine vectorially to produce a received signal which is a result

of constructive and/or destructive interference of the original signal which causes a

phenomenon commonly referred to as multipath fading. The presence of multipath

fading requires the transmitter to use more power to achieve the required signal to

noise ratio (SNR) levels compared with that used in non fading channels.

Doppler shift

The relative motion between the transmitter and the receiver in the case of mobile

devices causes variations in the channel parameters. One consequence of the rela-

tive motion is the shift in the frequency of the signal experienced at the receiver.

The frequency shift is directly proportional to the relative speed of the moving

device and is called the Doppler shift. The shift is positive when the relative mo-

tion is towards the receiver and negative when the relative motion is away from

the receiver. As a consequence of the Doppler shift, the bandwidth of the received

signal is increased. This phenomenon causes adjacent symbols of the transmitted

signal to overlap, resulting in intersymbol interference (ISI), and a high bit error

rate at the output.

Channel distortions due to co-channel interference and noise

Cellular systems are designed such that a geographical location is divided into

smaller areas referred to as ‘cells’. In order to improve spectral efficiency, users in

different cells separated by a given minimum distance use the same set of channel

frequencies. A major concern of the frequency reuse concept is interference among

the different users using the same frequencies. This form of interference is called

co-channel interference (CCI). A received signal distorted by CCI can be a highly

degraded version of the transmitted signal. In such cases the receiver may not

be robust enough to distinguish the desired signal from the interfering signals and

background noise. Further, obstacles in the environment as well as atmospheric

conditions, as discussed earlier in this section, can significantly degrade the trans-

mitted signal.

Efforts to recover the transmitted signal from the degraded version with greatest

4 Introduction

fidelity have lead to techniques such as channel coding, equalization and diversity.

Channel codes are utilized in digital data transmission to introduce redundancies

into the transmitted bit stream. These redundant bits help to provide effective

detection, by helping reduce bit errors at the receiver. Equalization is another

technique employed to reduce the bit error rate (BER). An equalizer acts like a

filter which attempts to undo the transfer function of the channel. Since the mo-

bile channel is constantly changing with time, receivers generally employ adaptive

equalization to estimate the changing channel parameters.

Diversity techniques

Diversity techniques are used to exploit multipath in order to improve system per-

formance. By employing or exploiting diversity in space, time, frequency or polar-

ization, the receiver is supplied with independently faded replicas of the transmitted

signal. Diversity is exploited by providing two or more, preferably uncorrelated,

channels or diversity branches at the transmitter (multiple input, single output),

receiver (single input, multiple output) or at both (multiple input, multiple out-

put). The probability that the combined signal from the multiple branches is above

a given threshold for signal recovery is increased when compared with employing a

single branch. Though antenna diversity is widely used at base stations for mobile

communications, its use in hand-held mobile device has not attracted commercial

attention because of constraints in available space and the potential for coupling

between such closely spaced antennas.

An appropriate combination of the aforementioned signal recovery techniques

can be used to further improve the performance of a given system. For example,

the RAKE receiver [44, pg 391] exploits multipath diversity by using an array

of correlators to detect the multipath signals. The signals are selectively delayed

and coherently added to give a relatively higher received signal SNR. In the next

section we discuss one of the most commonly used structures for exploiting and/or

employing signal diversity, the antenna array.

1.2 Antenna arrays

An antenna array is a group of sensors arranged in a particular geometry appropri-

ate to the application of interest. The most commonly used geometries are linear

1.2 Antenna arrays 5

desired signal

1 2

3

4

5

desired signal

Figure 1.2: Illustration of the beamforming lobes at the switched beam array show-ing sectors from where the users signals are detected.

arrays where the sensors are placed in a straight line and circular arrays where the

sensors are placed on the circumference of a ring. Antenna arrays form the basis

of many diversity systems and have long been employed in radar and navigation

applications [7]. With the development of low cost and reliable digital signal pro-

cessors, antenna arrays are now employed at the base stations of cellular mobile

systems [30].

The spacing between the sensors in an array is of particular interest because of the

correlation and coupling that exists between the signals arriving at the different

sensors. Most array configurations assume that the sensors are separated by a dis-

tance of at least λ/2 [63], where λ is the carrier wavelength. This assumption is

derived from the fact that the first null of the sinc(·) function appears at a spac-

ing of λ/2 which essentially decorrelates the two signals arriving at the sensors.

However, this assumption is true for isotropic scattering 1 environments only [54].

Telatar [56] showed that using antenna arrays with decorrelated elements, both

at the transmitter and the receiver, increases the probability of receiving indepen-

dently faded signals which improves system performance while providing a substan-

tial growth in the capacity of the system. Various works [17] [7] [65] [64] have since

exploited the spatial dimension by using a finite but large number of antennas to

find the limits of capacity and other performance measures of the mobile wireless

1Isotropic scattering implies that the signal power arrives at the receiver with equal powerfrom all directions i.e., it is uniformly distributed from all directions due to the scatterers beingrandomly and uniformly distributed in the space around the receiver.

6 Introduction

channel.

Employing antenna arrays in order to enhance received power levels while keeping

the Signal to Interference and Noise Ratio (SINR) low, has led to the development

of smart antennas for mobile applications. The term smart antenna [9] is often

used to describe two categories of antenna array systems.

1. Switched arrays : Switched arrays have several fixed predetermined beam

patterns. The predetermined beam patterns tend to be characterized by a

main lobe2with a given beamwidth. The main lobes of different patterns

tend to not overlap, effectively dividing the surrounding space into sectors as

shown in Figure 1.2. The antenna system chooses the sector according to the

direction from which the strongest signal from the desired user is detected.

The disadvantage of such a system is that if an interfering signal is within

the main beamwidth of the desired signal, it cannot be effectively removed

using nulling, and other techniques must be employed.

2. Adaptive arrays: Adaptive arrays continuously update their beam patterns

based on changes in the mobile user’s position and that of interfering signal.

The antenna system attempts to place the main lobe in the direction of the

user while placing a null in the direction of the interferer by using sophisti-

cated signal-processing algorithms such as Sample Matrix Inversion (SMI),

Least Mean Square algorithm (LMS) and the Constant Modulus Algorithm

(CMA). Antenna systems employing techniques of adaptive signal processing

have shown to provide substantial improvements when used with the time di-

vision multiple access (TDMA) and code division multiple access (CDMA) [9]

schemes.

1.3 Questions to be answered in this thesis

In this section we formulate certain questions identified during the course of survey-

ing the wide amount of literature on the performance of spatial diversity combining

schemes which we attempt to answer in this thesis

1. Does the performance of a given spatial diversity combining technique im-

prove linearly with the number of antennas when the antenna aperture is

restricted?

2The segment of the antenna radiation pattern which contains maximum energy.

1.4 Thesis motivation 7

2. To what extent does correlation due to spatial constraints affect the system

performance at the output?

3. For a given region in space does a particular array configuration perform

better than all others?

1.4 Thesis motivation

As pointed out earlier, most channel models used in the analysis of the received

signal in MIMO systems utilize the assumption of independent signals whose en-

velopes are modelled using the Rayleigh, Rician or Nakagami distributions. These

models do not characterize the physical aspects of the antenna arrays employed

and hence do not provide any practical insight into the effect of the relative posi-

tions of the sensors in the array. The assumption of receiving independently faded

signals at the receiver by these models is valid when the antennas are placed several

wavelengths apart. However with the shrinking size of mobile devices, the effect

of relative sensor spacing on performance at the output of the receiver is signifi-

cant. The motivation of this thesis is to investigate the importance of the spatial

dependence of signals at the receiver when multiple antennas are employed at the

receiver.

Insights into understanding the effects of spatial dependence of the signals at mul-

tiple receive antennas utilizing diversity techniques are given in this thesis. A

recently developed spatial channel model [42] for MIMO systems which incorpo-

rates the spatial separation of sensors is used for analysis. The effects of varying

separation distances and angular spreads in different array configurations such as

the uniform linear array (ULA) and the uniform circular array (UCA) on the per-

formance of the system are investigated in this thesis.

1.5 Thesis overview

We consider the spatial dependence of signals at different sensors in an antenna

array on the performance of two widely used diversity combining schemes namely

Maximal Ratio Combining (MRC) and Equal Gain Combining (EGC). Theoreti-

cal and simulation results which consider performance with respect to separation

between sensors are presented.

8 Introduction

The thesis is structured as follows.

Chapter 2 describes the theory and parameters involved in mobile wireless

channel modelling. Some statistical channel models commonly employed to ana-

lyze system performance criteria are described. An overview of the various diversity

schemes employed in MIMO systems and their effect on system performance is in-

cluded in this chapter with an emphasis on spatial diversity techniques.

In Chapter 3 a literature review of the effect of correlated signals at the out-

put of the receiver is provided, laying the foundation for the work presented in

the following chapters. Also discussed are a few spatial channel models used for

performance analysis based on the angles of arrival of the signals at the receiver.

A novel SIMO spatial channel model is presented in Chapter 4. The simula-

tion results obtained by studying the effect of finite antenna separations on the

performance of spatial diversity systems are presented in this chapter. The results

obtained are compared with those for independently faded signals.

In Chapter 5, equations for the correlation between closely spaced antennas, the

combined average SNR at the receiver output for MRC and EGC and average BER

using BPSK for MRC when different array configurations are employed are derived.

In Chapter 6 we present our conclusions, a summary of the work presented in

this thesis and suggestions for future research.

Chapter 2

Overview of the mobile wireless

channel and spatial diversity

techniques

2.1 Introduction

Analysis of the wireless communication channel poses a greater challenge than fixed

channels because of its random nature. Natural and artificial structures prevent

the signal from reaching the destination directly (line of sight) and the speed of

the mobile transmitter or receiver causes a change in the characteristics of the

intervening channel. Modelling this constantly changing channel has evoked great

interest in recent years. The fundamental theory behind the different propagation

mechanisms together with a brief description of the manifestations of fading effects

is summarized in this chapter.

We also provide an overview of the various diversity techniques which exploit the

multipath nature of the wireless channel. We emphasise spatial diversity techniques

which have received a lot of attention in recent years in combating SNR losses due

to multipath fading.

2.2 Wireless signal propagation

The topographical variations in the signal path contribute to distortions in the

received signal. The different propagation models used to describe the statistics

of the received signal are usually based on the principal propagation mechanisms

affecting the transmitted signal.

9

10 Overview of the mobile wireless channel and spatial diversity techniques

2.2.1 Propagation mechanisms

The basic propagation mechanisms that contribute to effective signal strength at

the receiver are described using the physics of reflection, diffraction and scattering

of the propagating wave [44].

• Reflection: When the wavelength of the propagating wave is very small

compared with the dimensions of the obstructing object, reflection occurs.

The “2-ray” model is often used to predict the signal strength over large

transmission distances. This is a ground reflection model which assumes the

presence of a direct path between the transmitter and the receiver together

with one earth reflected propagation path. Signal distortion due to multipath

can be characterized by the superposition of many reflected rays with the

direct path, extending the concept of the 2-ray model [6]. In the multipath

case reflections can be from objects other than the ground.

• Diffraction: When the propagating wave encounters an object with sharp

edges, it tends to ‘bend’ around the object giving rise to secondary waves.

This phenomenon is termed diffraction. The presence of tall buildings or

mountains can prevent the propagating signal from reaching the receiver. In

such cases the receiver is said to be ‘shadowed’ from the transmitted signal.

The obstacle is often treated as a diffracting knife edge and signal propagation

around these obstacles is explained using knife edge diffraction models [24]

[44].

• Scattering: When the dimension of an object encountered by the propa-

gating wave is smaller than its wavelength, the phenomenon of scattering

occurs. Scattering causes the transmitted signal power to be reradiated in

different directions, causing signal distortion. The most commonly encoun-

tered scatterers during signal propagation are lamp posts, street signs and

foliage.

The resultant signal strength is usually a combination of the above phenom-

ena [5] causing random variations in the amplitude, phase and angle of arrival of the

desired signal. For example, the double knife edge diffraction model is used in [67]

to determine the signal attenuation due to two diffracting structures together with

losses due to ground reflections to predict the field strength at a given receiving

point.

2.2 Wireless signal propagation 11

Deterministic channel modelling using the ray tracing technique or the knife

edge model can be used to accurately determine channel characteristics for a given

physical setting. These techniques require detailed site specific information such

as building databases or employing architecture drawings. Hence, these models

cannot be generalized for a general physical environment and often fail to quantify

the signal modulation caused by the small scale manifestations of the channel.

2.2.2 Pathloss due to large scale fading

The line of sight, free space propagation model describes the received signal strength

as a function of the distance from the transmitter. Traditionally this model is used

to describe satellite communications or microwave radio links. The pathloss factor

indicates the signal attenuation due to propagation through free space. From the

Friis [49] free space equation, the path loss factor Lfs(d), assuming an isotropic1

antenna at the receiver, is expressed as

Lfs(d) =[4πd

λ

]2

(2.1)

where d is the distance between the transmitter and the receiver and λ is the wave-

length of the propagating signal. None of the characteristics of the mobile wireless

channel discussed so far in this thesis are captured by (2.1) rendering it highly in-

effective as a tool for analyzing such channels. A more detailed equation describing

pathloss in mobile wireless channels is required.

Fading, a term used to denote the random fluctuations of the signal ampli-

tude and phase due to multipath, is normally classified into large scale fading and

small scale fading [44]. Large scale fading characterizes the behaviour of the signal

over large transmitter/receiver separations, taking into account the various terrain

features of the propagating environment while small scale fading deals with the

dramatic changes in the received signal due to small changes in the position of the

receiver. A general path loss model [44, pg 139] [30] which gives the mean path loss

factor L(d) due to large scale fading in a realistic wireless environment is usually

expressed in decibels as

L(d) = Lfs(d0)(dB) + 10n log[ d

d0

]

+Xσ(dB) (2.2)

1a theoretical antenna which transmits/receives equally in all directions.


100 10170

75

80

85

90

95

100

105

110

115

Separation between the transmitter and receiver in Kilometers

Pat

hlos

s in

dB

n = 2n = 3n = 4

Figure 2.1: The effect of increasing transmitter and receiver separation d, for vari-ous pathloss exponents, n, in (2.2).

where d02 is the received power reference point such that d > d0, n is the path-

loss exponent and Xσ is a zero mean Gaussian random variable that models the

variations in the average received power about L(d). The values of n and Xσ are

determined experimentally [30] since they are site specific. For example, in urban

areas where the propagating wave encounters many obstructions, the value of n

is higher than in areas where a strong guided wave may be present. In [5] some

typical measured values of the aforementioned parameters for outdoor and indoor

propagations are given.

2.3 Stochastic channel modelling

Unlike large scale fading where the signal parameters show variations over a large

period of time, small scale fading incorporates the rapid and random behaviour of

the signal over small travel distances. In order to characterize a signal which is a

sum of many variations in the channel, stochastic processes are used. Commonly

used stochastic models such as the Rayleigh fading model, Rician fading model and

Nakagami-m fading model which are used to describe the signal envelope due to

small scale fading are discussed in Section 2.4.

2d0 is a point located in the far-field of the transmit antenna

2.3 Stochastic channel modelling 13

2.3.1 Impulse response characterization of a multipath chan-

nel

If τ denotes the time taken for a multipath signal to reach the receiver after a

certain travel distance, the signal r(t) at the receiver can be written as

r(t) = a(t)s(t− τ)e−iφc(t−τ) (2.3)

where s(t) is the complex baseband signal modulated onto a carrier wave with

phase φc and a(t) is the gain of the propagation path.

The channel impulse response for a linear time invariant system which is a

convolution of the channel transfer function h(t) with the transmitted signal is

defined as

r(t) =

∫ ∞

−∞h(t− τ)s(τ)dτ. (2.4)

Due to the effect of multipath, the wireless channel is time varying and hence

the received signal may be represented as a convolution of the time varying impulse

response [44] of the channel with the transmitted signal

r(t) =

∫ ∞

−∞h(t− τ, t)s(τ)dτ = h(τ, t) ⊗ s(τ). (2.5)

In the presence of multipath, the receiver ‘sees’ the sum of multiple paths.

The resultant signal at the receiver is therefore a combination of constructive and

destructive interference of the incident waves. The overall channel impulse response

is, thus, the effective sum of all of the individual multipath responses given by

h(t− τ, t) =N∑

n=1

an(t)eiφn(t)δ(τ − τn) (2.6)

where N is the total number of multipaths, an is the amplitude of the nth compo-

nent which is usually modelled as a Rayleigh or Rician or Nakagami-m distributed

random variable, φn is the phase shift modelled as a uniformly distributed ran-

dom variable and δ(·) is the unit impulse function which determines the specific

multipath component at time τ and an excess delay of τn.


2.3.2 Stochastic channel characterization via the autocor-

relation function

Autocorrelation is the most commonly used function to characterize a stochastic

signal. The general definition of the autocorrelation function of a given time varying

channel with impulse response h(t) is [43]

Rh(t1, t2) = E[h(t1)h∗(t2)] (2.7)

where E[·] denotes ensemble average and (·)∗ denotes complex conjugation. How-

ever, in the study of wireless communication, the channel is generally assumed to

be wide sense stationary (WSS) [48]. Under the assumption of WSS, the autocor-

relation function is defined as a function of the time difference 4t = t1− t2. Hence,

for a WSS channel the autocorrelation function is defined as

Rh(4t) = E[h(t)h∗(t+ 4t)]. (2.8)

Most processes in communication systems are modelled as zero mean processes. In

such cases the autocorrelation function is termed the autocovariance function. If

the autocovariance function is normalized against the mean power of the process,

the function is defined as the unit autocovariance function. When the autocorrela-

tion function is defined such that it includes all of the random dependencies of the

channels, i.e., space, time and frequency separations, the term joint autocorrelation

function is used to describe the overall process, provided the channel is WSS with

respect to space, frequency and time [14].

2.4 Fading models

Statistical models are commonly employed to characterize stochastic channels when

the number of multipaths are large [15]. We briefly discuss fading models used to

describe wireless channels.

The baseband received signal r(t), in the absence of any LOS path, ignoring

the presence of additive noise, is often statistically denoted as the sum of all of the

multipath waves as

r(t) =N∑

i=1

ai cos(ωct+ ωdi+ φi) (2.9)

2.4 Fading models 15

where ai and φi are the amplitude and phase of the ith multipath signal and φi is

uniformly distributed between [0, 2π], ωc is the angular carrier frequency, N is the

total number of multipaths and ωdiis the Doppler shifted angular frequency of the

ith path, given by

ωdi=ωcv

ccos(αi) (2.10)

where c is the velocity of the propagating wave and v is the velocity of the mobile

device.

We see from (2.9) that the received signal has random variations in both ampli-

tude and phase. The motion of the mobile device causes further degradation of the

signal envelope due to the introduction of a Doppler shift. The random behaviour

of the signal together with the variations in the channel parameters gives rise to

small scale fading manifestations which are discussed in the next section.

2.4.1 Statistical modelling of independently faded signal

envelopes

The independent Rayleigh fading model

The signal in (2.9) can be written as

r(t) = I(t) cos(ωct) +Q(t) cos(ωct) (2.11)

where I(t) and Q(t) are the in-phase and quadrature components of the signal

given by

I(t) =N∑

i=1

ai cos(ωdi+ φi) (2.12)

and

Q(t) =N∑

i=1

ai sin(ωdi+ φi). (2.13)

The envelope of the received signal r(t) can therefore be written as

r =√

[I(t)]2 + [Q(t)]2 (2.14)

where r = |r(t)| and | · | is the absolute value operator.


0 20 40 60−1.5

−1

−0.5

0

0.5

1

1.5

2Rayleigh faded signal for a stationary device

rf si

gnal

(vol

ts)

time(ms)0 20 40 60

0

0.5

1

1.5

2Rayleigh faded signal envelope for a stationary device

enve

lope

(vol

ts)

time(ms)

0 20 40 600

0.5

1

1.5

2

Rayleigh faded signal envelope for a mobile device moving at 25 m/s

enve

lope

(vol

ts)

time(ms) 0 20 40 600

0.5

1

1.5

2

2.5

Rayleigh faded signal envelope for a mobile device moving at 50 m/s

enve

lope

(vol

ts)

time(ms)

Figure 2.2: Illustration of the fluctuations of a signal when subjected to Rayleighfading with 10 multipaths. The signal envelope for a mobile device appears belowthe threshold more frequently than a stationary device indicating greater chanceof signal loss.

When N is very large, the central limit theorem tells us that the in-phase and

quadrature components are Gaussian distributed [43] and hence the probability

distribution function (pdf) of the signal envelope follows a Rayleigh distribution.

The Rayleigh faded envelope is given by

p(r) =

rσ2 exp[−r2

2σ2 ] for r ≥ 0

0 otherwise.

(2.15)

In (2.15 ) the term 2σ2 denotes the predetection mean power of the multipath signal.

The phase of the fading signal is defined as

φ = tan−1[Q(t)

I(t)

]

(2.16)


which has a uniform distribution with a pdf given by

p(φ) =1

2π, 0 ≤ φ ≤ 2π. (2.17)

The Rician fading model

The Rayleigh fading model is a special case of Rician fading which expresses the

fading environment as a combination of a stationary wave component together with

incoherently fluctuating signal components. The stationary wave is called the direct

path or LOS wave, while the incoherent component is comprised of the multipath

waves. The pdf of the received envelope for the Rician fading distribution is given

by

p(r) =

rσ2 exp[−r2+A2

2σ2 ]Io

[

rAσ2

]

for r ≥ 0 and A ≥ 0

0 otherwise

(2.18)

where A is the peak magnitude of the significant LOS component and is called the

specular component. Io(·) is the zeroth order modified Bessel function of the first

kind [43]. The Rician distribution is usually described as a ratio of the power in the

LOS component to the power in the multipath signal. This parameter is termed

the ‘K factor’ and is given by

K =A2

2σ2. (2.19)

The Nakagami-m distribution model

The Nakagami-m Distribution which has received significant attention in recent

years in modelling signal envelopes, is a generalized form of both the Rayleigh and

Rician distributions. The application of this distribution is based on the parameter

m which is called the fading figure. The fading figure is usually a positive number

greater than or equal to 0.5. The practical values of m range from m = 1 to

m = 15 [53]. In general, a small value of m indicates a severe fading environment.

It can be observed that form = 1 the Nakagami distribution reduces to the Rayleigh

distribution function. The pdf of the Nakagami distribution is

p(r) =

2Γ(m)

(

mΩ

)m

r2m−1exp[

−mr2

Ω

]

for r ≥ 0 and m ≥ 1/2

0 otherwise

(2.20)


where Ω = E[r2] is the second moment of the random variable, m = Ω2

E[(r2−Ω)2]and

Γ(·) is the gamma function3.

The Rayleigh, Rician and Nakagami distribution functions are generally used

to estimate system performance measures such as the average SNR and BER at the

receiver when combining schemes are used. The underlying assumption when using

these models is that the signals arriving at each diversity branch are independent

and identically distributed. Such analyses also assume that there is no consequence

of varying the separation between any two neighbouring antennas in an array. An

investigation of the actual consequences of the separation is carried out in Chapters

4 and 5.

2.4.2 Small scale fading manifestations

It was seen in the previous section that the two main factors causing small scale

fading are the time delay between the multiple copies of the received signal and the

Doppler shift introduced in the signal frequency due to movement of the receiver.

This prompts the study of small scale fading manifestations in both the frequency

and time domains. We now briefly discuss the various fading manifestations that

occur in the mobile wireless channel. Figure 2.3 gives a detailed illustration of the

fading manifestations due to small scale fading.

Frequency Selective and Frequency non-selective Fading

When the time span for receiving multipath components of one symbol exceeds the

symbol period or when the coherence bandwidth4 f0 is less than the bandwidth of

the signal, the different frequency components of the signal will be affected differ-

ently in gain and phase. A channel where the frequency components of the signal

have different gains and phases is called a frequency selective channel [48] [44].

Frequency selective fading gives rise to channel induced ISI.

If all of the multipath components of the symbol lie well within the symbol

period or if the channel is narrowband i.e., the transmitted signal is narrow in

bandwidth compared to the channel’s fading bandwidth, frequency non-selective

fading occurs. A frequency non-selective channel is often termed a flat fading

channel. In a flat fading channel all the frequency components of the transmitted

3The gamma function is defined as Γ(x) =∫

∞

0tx−1e−tdt, x > 0.

4The frequency separation for which the signals are still strongly correlated is called thecoherence bandwidth.


indicates Fourier Transforms

Small scale fading

Time spreadingof the signal of the channel

Time−delaydomain

Frequencydomain

Time domain Doppler−shiftdomain

selective fading

fadingFlatFrequency

Frequencyselective fading

Flat fading

Fastfading

Slow fading

Fast fading

Slow fading

Time variance

indicates Duals

Figure 2.3: Small scale fading manifestations.

signal undergo the same random attenuation and phase shift through the channel.

A flat fading channel is, hence, a channel where all the multipath components

arrive at the receiver at almost the same time.

Fast and Slow Fading

When the coherence time5 of the channel is very small compared to the symbol

period, the channel changes rapidly when a symbol is propagating. This leads to

distortion of the baseband signal. A similar condition occurs when the Doppler

spread is greater than the channel bandwidth. This form of distortion leads to the

condition called fast fading.

When the effects of Doppler spread are negligible or the channel remains static

over a symbol propagation time i.e., the coherence time is large compared to the

symbol propagation time, slow fading occurs.

In our analysis in the following chapters of this thesis we assume the channel to be

frequency non-dispersive such that the signals are not affected over the propagation

period, i.e., the channel is assumed to exhibit flat and slow fading.

5The time duration over which two received signals have strong amplitude correlation is definedas the coherence time.


Techniques to overcome fading effects

Some of the techniques that can be employed to reduce the effects of fast fading

and frequency selectiveness in a channel are listed below.

1. Adaptive equalization is used to effectively attenuate frequencies with large

amplitudes while amplifying those with small amplitudes. The equalizer, in

effect, acts as an inverse channel filter and provides a flat frequency response

with linear phase [44].

2. Error correcting codes [43] provide improvement in performance by reducing

the required SNR for a desired error performance.

3. Spread spectrum techniques such as Frequency Hopped /Spread Spectrum

(FH/SS) where the hopping frequency is chosen to be greater than the symbol

period is employed to prevent frequency selective fading [43]. By adding signal

redundancies the symbol rate can also be increased to mitigate effects of fast

fading. The performance of FH/SS is improved by error-correction coding

and diversity.

4. Orthogonal Frequency Division Multiplexing (OFDM) [43] which divides a

high symbol rate sequence into a number of sequence groups with lower sym-

bol rates is a scheme used to mitigate frequency selective fading effects.

5. Pilot signals [33] are employed to provide information about the channel to

the receiver, thus helping with channel estimation which improves system

performance under fading effects.

2.5 Mitigating losses in SNR using diversity

In recent years exploitation of multipath has given rise to the use of antenna diver-

sity techniques at the receiver to reduce SNR losses. Though the system com-

plexity increases, diversity systems provide performance improvements without

additional requirements of power or bandwidth. The concept of diversity arises

from the probability that all components (e.g. multipaths) of a transmitted signal

will not undergo fading simultaneously. Combining independently faded signals

thus overcomes fading effects of multipath, considerably improving system perfor-

mance [17] [25].

2.5 Mitigating losses in SNR using diversity 21

Let r1, r2, ...rn, ...., rN be the signal replicas due to N multipath components. If

each of these replicas are similar (e.g. only small variations in phase among the N

components), the composite signal r = r1 + r2 + ...rn + ...rN will outperform the

individual components, assuming the presence of uncorrelated noise. If, further,

the signals are weighted by a proportional channel gain based on the fluctuations

of the individual signal components, the signal quality will be superior to the case

where a single component is received.

r = w1r1 + w2r2 + . . .+ wnrn + . . .+ wNrN =N∑

n=1

wnrn. (2.21)

Equation (2.21) depicts the resultant signal r due to a generalized linear diversity

combiner where wn is the weighting factor proportional to the gain of the signal

component rn.

Two or more copies of the same signal can be obtained at the receiver by several

methods such as time, frequency, angle, polarization or spatial diversity.

2.5.1 Diversity methods

Time diversity

Time diversity is achieved when the same information signal is transmitted across

the channel over two or more well separated time slots. This diversity method is

commonly used in commercial continuous wave (CW) stations. In [3] Alamouti

proposed a 2-branch transmit diversity scheme for base stations with performance

similar to a 2-branch receive diversity scheme. The separation distance between

time slots which ensures independently faded signals should be at least equal to

the channel coherence time t0. Time diversity provides no benefit in applications

where there is no mobility, i.e., transmitter and receiver are both stationary [24]

since the coherence time is inversely proportional to the Doppler spread which is a

function of the speed of the moving device.

Frequency diversity

In frequency diversity the information signal is transmitted simultaneously on more

than one carrier frequency. Each of these narrowband channels are separated by a

bandwidth of more than the coherence bandwidth f0 of the channel which ensures

that the individual diversity bands are unaffected by frequency selective fading.


t

coherence time

bandwidthcoherence

decorrelation distance

combining logic

narrow band channels

time slots

sensors

a)

b)

c)

f

Figure 2.4: Illustration of a) frequency diversity, b) time diversity and c) spacediversity.

A basic hindrance in implementing this form of diversity is the high bandwidth

requirement. Orthogonal frequency division multiplexing (OFDM) is a technique

that uses the basic idea of frequency diversity and has received considerable atten-

tion in recent years because of its efficiency in conserving bandwidth [51].

Space diversity

Space diversity is the most widely used diversity method [66] because of the relative

simplicity in implementation and no additional requirements in bandwidth. Spatial

diversity is achieved by using multiple antennas either at the transmitter, the re-

ceiver or both. In these systems the multiple replicas of the transmitted signal are

combined using various techniques from antennas separated by at least their decor-

relation distance6. The different spatial diversity schemes and their performance

benefits are discussed later in this chapter.

6The minimum separation distance for which the signals are independent or not correlated isthe decorrelation distance.


P

monitor and control logic to select the bestsignal branch

selected signal to thedemodulator

1 2 p

Figure 2.5: Illustration of switched/selection schemes in spatial diversity.

Angle diversity

Angle diversity is obtained by directing beams in certain different directions such

that the signals associated with each of the directions are uncorrelated. Angular

diversity is also termed beamforming, where the individual antenna responses are

weighted and linearly combined [19]. In conventional beamforming the main lobe,

which is the area of high gain is placed in the direction of the desired signal while

nulls in the beampattern are directed at the interfering signals.

Polarization diversity

When the desired signal is transmitted using two orthogonal polarizations the

method is called polarization diversity. The vertical and horizontal components

of the signal are independent, providing diversity benefits [24]. This form of di-

versity is similar to spatial diversity because the polarized waves are transmitted

on two separate antennas. A comparative study of the relationships between cross

polarization discrimination, signal correlation and polarization diversity gains for

different cellular environments and antenna inclinations was carried out in [32].

It was shown that diversity gains of polarization schemes depend on the mean

difference between the signals at two receiver branches and that horizontal space

diversity performs better than horizontal/vertical polarization diversity.


2.5.2 Spatial diversity combining techniques

Based on the combining scheme employed at the receiver, spatial diversity tech-

niques can be classified into switching schemes and gain combining schemes. While

switching schemes are practically easier to implement, gain combining techniques

provide better performance at the output. Gain combining schemes require spe-

cial “phase control” circuitry if combined during pre-detection. Phase control is

essential to equalize the phases of all of the multipath signals before summation.

However, the need for co-phasing does not arise if the signals are combined after

signal detection i.e., if post-detection combining is employed.

Scanning diversity

In scanning diversity a selection device scans thorough all channels in sequence and

chooses the signal that is above a given pre-selected threshold. The signals from

all other channels are ignored. This process continues until the chosen signal drops

below the threshold level. The switching or the selection device then sequentially

scans all the channels again until it finds another signal above the threshold. Also

referred to as switched diversity, this diversity system does not require a separate

receiver for each channel. However, the performance of this scheme is poor when

compared to other combining schemes [7].

Selection combining

Selection combining (SC) is the classical form of diversity combining. Selection

combining is also a switched technique. In SC the signal which has the highest

SNR amongst all the branches is selected for the system output while the other

signals are disregarded. The simplicity of implementation of this scheme makes it

a popular practical technique for modern cellular systems [30].

Maximal ratio combining

Maximal ratio combining (MRC) is often referred to as optimum combining [13]

because it yields the highest SNR at the output when compared to all other com-

bining techniques. In MRC the output signal is the weighted sum of the individual

branches. The weights are chosen to be proportional to the gain of the individual

signal and inversely proportional to the mean square noise in the channel. Im-

plementation of MRC is cumbersome due to the additional circuitry required in


z

Combining Logic

Combiner output

1 2 p P

Receiver Elements

w 1 w ww 2 p P

Figure 2.6: Illustration of general gain combining diversity schemes wherew1, w2, ..wP are the weights added to the respective branches.

order to measure the channel gains. If combining is performed using MRC at pre-

detection, additional phase-control circuitry would be needed to estimate the phase

of each branch signal.

Equal gain combining

Equal gain combining (EGC) is the simplest diversity technique in which all indi-

vidual signals are added together after co-phasing. In EGC systems the weights

are all set to one with the requirement that the channel gains are approximately

constant. This is usually achieved by using an automatic gain controller (AGC)

in the system [7]. EGC systems are of great practical importance because of the

simplicity of implementation.

2.5.3 Analysis of diversity techniques

The statistical distribution of narrowband fading channel envelopes are used in

calculating the performance of diversity combining techniques. Depending on the

type of environment, an uncorrelated Rayleigh, Rician or Nakagami-m distributed

signal envelope [7] is used.

If the signal envelope rn, of the nth branch of the diversity system has a variance

of σ2n, and is Rayleigh distributed, then its pdf is given by


p(rn) =rn

σ2n

e− r2

n2σ2

n . (2.22)

The instantaneous SNR γn, of the nth received envelope under the assumption of

independent fading and additive white gaussian noise has an exponential distribu-

tion [50] and its pdf p(γn) is given by

p(γn) =1

Γn

eγnΓn (2.23)

where Γn is the average SNR of the nth diversity branch. The combined pdf of the

SNR p(γ) is calculated depending on the type of scheme employed.

If selection combining is considered, then the branch with the highest SNR

is chosen. Because the branches are assumed to fade independently, the ordered

statistics of the branch SNRs are also independent. Therefore the cumulative

distribution function (cdf) of the output SNR can be written as

F (γ) = P (γ1, ...., γN ≤ γ) = [1 − e−γΓ ]N . (2.24)

Differentiating the above equation with respect to γ gives the pdf for the output

SNR for SC [19] as

p(γ) =N

Γe−

γΓ

[

1 − e−γΓ

]N−1

. (2.25)

The pdf of the output SNR can similarly be found for other combining schemes.

Once p(γ) is known for a particular combining scheme, the probability of bit error

Pe can be calculated using

Pe =

∫ ∞

0

P(γ)p(γ)dγ (2.26)

where P(γ) is the BER for the particular digital modulation scheme being used.

For example, in the case of MRC where the output SNR is the sum of all of the

individual branch SNRs, the pdf of the SNR can be written using the chi-square

distribution with 2N degrees of freedom as

p(γMRC) =1

(N − 1)!

γN−1

ΓNe

γΓ . (2.27)


The BER for ideal coherent binary phase shift keying (BPSK) is given by [43]

PBPSK =1

2erfc(

√γ). (2.28)

Using (2.27) and (2.28) in (2.26), the probability of bit error can be calculated

as [50]

PBPSK =(1 − ν

2

)NN−1∑

n=0

(

N − 1 + n

n

)

(1 + ν

2

)n

(2.29)

where

ν =

√

Γ

1 + Γ. (2.30)

Similar calculations are used to calculate error rates using different modula-

tion schemes such as differential phase shift keying (DPSK), M-ary quadrature

amplitude modulation (MQAM), quadrature phase shift keying (QPSK) and so

on. BER expressions for diversity systems using M-ary phase shift keying (MPSK)

were calculated in [57] and [1]. Analysis of the BER for MRC using quadrature

amplitude modulation (QAM) was carried out in [27]. These calculations prove

to be useful when choosing a particular modulation scheme best suited to a given

set of performance parameters provided each branch is independently faded. In

chapter 5 we consider the performance of diversity techniques when fading may

not be independent in all diversity branches, due to the placement of antennas in

finite space.

2.5.4 Hybrid combining

Although it has been proven that performance benefits are proportional to the

order of diversity, the utilization of all of the diversity branches at the receiver is

not practical. The main limitation on employing diversity in a mobile handset,

for example, is not because of the constraint of the handset size, but the cost and

power consumption of the receiver electronics for each antenna [64].

Efficiency of hybrid schemes

The motivation to reduce the complexity of combining schemes while retaining the

advantages has lead to employment of certain hybrid schemes [61] [37] which select

the “best” L branches out of the M available branches and combine them using

either EGC or MRC. These schemes are called Hybrid Selection/Equal Gain Com-

bining (H-S/EGC) or Hybrid Selection/Maximal Ratio Combining (H-S/MRC),


respectively.

The performance of the H-S/MRC scheme for an independent Rayleigh faded

environment is upper bounded by MRC and lower bounded by SC [61]. The H-

S/MRC scheme performs better than H-S/EGC, highlighting the optimality of the

MRC scheme [11]. Increasing the number of branches in EGC does not necessarily

provide higher gains. This is because the noise present in the branches may not

be counter balanced by the combiner under severe fading conditions due to non-

optimal combining.

2.6 Summary

In this chapter the fundamental fading mechanisms encountered in wireless chan-

nel together with some of the basic channel models employed to characterize the

wireless channel were briefly described. The statistical description of the channel

parameters using the commonly used distribution functions were outlined. Di-

versity combining techniques which have received considerable attention in recent

years to combat multipath fading were introduced to provide a foundation for the

detailed analysis of the gain combining schemes in the following chapters.

Chapter 3

Correlation effects in diversity

schemes

3.1 Introduction

Using multiple antenna systems to achieve diversity benefits both at the transmitter

and the receiver has resulted in improvements in system performance proportional

to the number of antennas used in the array. Most of the theoretical work focussing

on systems employing diversity either at the transmitter or the receiver [65] [3] [28]

uses the assumption that the fades between the antenna elements are independent

and identically distributed. However, in realistic propagation environments com-

plete statistical independent fading can rarely be achieved [45] due to factors such

as, physical configuration of the antenna array or the nature of the scatterers in

the system. In this chapter the effects of correlation on system performance when

more than one antenna is used in an antenna array system is discussed.

3.2 Impact of independent fading on diversity

combining

Under the assumption of independent Rayleigh fading at each antenna and equal

instantaneous branch SNRs, the ratio P of the instantaneous SNR to the average

SNR at a particular branch for the different diversity combining schemes with N

antennas is given by [19] [11]

29

30 Correlation effects in diversity schemes

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Number of antennas

10 lo

g<γ>

/Γ d

B

MRCEGCSC

Figure 3.1: Comparison of improvement in diversity gain using MRC, EGC and SCfor increasing number of antennas in an independent Rayleigh fading environment.

PSC =N∑

k=1

1

k, (3.1)

PMRC = N, (3.2)

PEGC = 1 + (N − 1)π

4, (3.3)

P(H-S/MRC) = 1 +N∑

i=l+1

1

i. (3.4)

We see from (3.1) - (3.4) and Figure 3.1 that when the antennas in the array receive

independently faded signals, a significant improvement in the output SNR perfor-

mance is achieved with each additional antenna. We also see that MRC provides

the best performance results with uncorrelated signals at each diversity branch,

because of the inherent optimality of the scheme.

At times the instantaneous average SNR at each branch is not identically dis-

tributed. This can be caused due to factors such as different fading statistics at

different beams of a multiple beamformer or varied shadowing effects at differ-

ent branches [60]. As of consequence of these phenomenons, the statistics of the

3.3 Correlated fading effects on diversity systems 31

branch SNRs are no longer independent. However, the branch SNR variables can

be transformed into a new set of conditionally independent SNR variables [62] by

using the virtual branch technique. When the virtual branch technique is applied,

closed form equations for the combiner output SNR [61] and the symbol error prob-

ability [62] for the H-S/MRC can be derived concisely showing the improvement of

system performance as the number of branches combined using MRC are increased.

We now consider cases where the independence of the branch signals is lost due

to the presence of correlation between them due to spatial or temporal properties

of the channel.

3.3 Correlated fading effects on diversity systems

The study of the relative effects of correlation between signals arriving at different

array elements dates back to the early 1950’s [7]. The problem of correlated fad-

ing for a two branch selection combining scheme was among the first correlation

problems in diversity systems to be considered. Correlated fading was shown to

affect the diversity gain1 of the system. Amongst the initial works of gain combin-

ing schemes, Pierce and Stein [39] studied the BER for MRC and EGC for BPSK

modulation in a correlated Rayleigh fading channel and showed that an increase

in correlation had adverse effects on the BER of the system. Numerous studies

have since reported the results of non-independent fading for various diversity tech-

niques [24] [46] [31] [34] using the joint statistics of the Rayleigh/Rician/Nakagami-

m distributions.

The correlation coefficient

The power correlation coefficient ρ of two Rayleigh distributed random variables

r1 and r2 with variances of σ2r1

and σ2r2

, respectively, [12] is given by

ρ =E[

(

r21 − E[r2

1])(

r22 − E[r2

2])

]

√

E[

(

r21 − E[r2

1])2]

E[

(

r22 − E[r2

2])2]

. (3.5)

Correlation between two random signals can also be measured in terms of the

envelope correlation ρe. The importance of finding the envelope correlation arises

1The diversity gain is defined as the reduction in the required average SNR for a given BER.


from the fact that measurement data is expressed in terms of the envelope correla-

tion while most analytical work is performed using the power correlation coefficient.

The envelope correlation between two random signals r1 and r2 is defined as

ρe =E[

(

r1 − E[r1])(

r2 − E[r2])

]

√

E[

(

r1 − E[r1])2]

E[

(

r2 − E[r2])2]

. (3.6)

The envelope correlation coefficient for a pair of correlated Rayleigh distributed

signals can also be calculated from the power correlation using the relationship [12]

ρe =(1 +

√ρ)ε(

2ρ1/4

1+√

ρ

)

− π2

2 − π2

(3.7)

where ε(·) is the complete elliptical integral of the second kind [20, pg 852] , which

is defined using the Jacobi elliptical function [20, pg 857] .

The correlation coefficients defined above are used for theoretical analysis of

the performance of diversity schemes. Various analytical tools used to find closed

form expressions for the output average SNR and BER in terms of the correlation

coefficient when diversity schemes are employed at the receiver are studied next.

3.3.1 Dual diversity systems

The potential of using two antennas, known as dual diversity reception, on hand

held mobile devices has prompted a large amount of research [21] [36] [16] [31] [29].

The most common approach used for analysis of the performance of correlated dual

branch diversity systems is calculation of the output SNR using the joint pdf of the

two received signals [36] [22]. We now consider calculation of the SNR and BER for

dual diversity MRC and SC combining schemes for the correlated Rayleigh chan-

nel.

For correlated random signals the bivariate Rayleigh distribution, which is a spe-

cial case of the Nakagami-m distribution [38], gives the joint pdf in terms of the

correlation coefficient ρ as

p(r1, r2|ρ) =4r1r2

Ω1Ω2(1 − ρ)exp(

− Ω2r21 + Ω1r

22

Ω1Ω2(1 − ρ)

)

× I0

( 2√ρr1r2√

Ω1Ω2(1 − ρ)

)

(3.8)


where r1 ≥ 0, r2 ≥ 0, Ω1 = E[r21] , Ω2 = E[r2

2] and I0(·) is the modified Bessel

function of the first kind.

If SC is used at the receiver, the cumulative distribution function (cdf) of the re-

ceived signal, can be written using the infinite series representation of the bivariate

pdf [53] as

F (r1, r2|ρ) = (1 − ρ)∞∑

k=0

ρkΥ(

k + 1,r21

Ω1(1 − ρ)

)

Υ(

k + 1,r22

Ω2(1 − ρ)

)

(3.9)

where Υ(α, x) =∫ x

0tα−1e−tdt, Re α > 0 is the incomplete Gamma function.

The average SNR at the output of the dual selection combiner can be found

using (3.8) and (3.9) [21].

A closed form expression for the pdf of the output SNR of a dual MRC combiner

[12] can also be determined using the bivariate Rayleigh distribution as

p(m) =me

(

−m2(σ21+σ2

2)

4σ21σ2

2(1−ρ)

)

√

(σ22 − σ2

1)2 + 4ρσ2

1σ22

×[

eκ − e−κ]

(3.10)

where

m =√

r21 + r2

2 (3.11)

and

κ =m2√

(σ22 − σ2

1)2 + 4ρσ2

1σ22

4(1 − ρ)σ21σ

22

. (3.12)

Equation (3.10) indicates the pdf for the general case of the MRC combiner with

unequal branch SNRs and can be simplified to obtain closed form expressions for

cases with equal average branch powers with or without correlation. By using

(3.10) in (2.26) the BER for coherent modulation schemes can be derived [12].

Analytical results for the performance of a dual diversity system in the presence


of correlation can also be derived by first transforming the correlated random signals

and representing them as independent random signals [16]. The transformation

results can then be used in expressions derived for independent fading. This method

provides a simplified tool when compared with the aforementioned methods to

obtained closed form expressions for the pdf of the combiner output SNR and BER

for MRC and SC i.e., the signals at the output of the transformed system are

defined by

r = Tr (3.13)

where,

r =

[

r1

r2

]

(3.14)

and

T =1√2

[

1 1

−1 1

]

. (3.15)

By performing this transformation the average SNRs of the transformed branches

become

Γ1 = (1 + ρ)Γ (3.16)

and

Γ2 = (1 − ρ)Γ. (3.17)

Now, using the above transformation, the output SNR for a correlated dual branch

MRC scheme can easily be written, using the pdf of the same scheme for indepen-

dent branches as

p(x) =1

2Γρ

(

e−x

Γ(1+ρ) − e−x

Γ(1−ρ)

)

(3.18)

where x = r21 + r2

2.

The BER for the same scheme using BPSK is represented as

Pb =1

2

[

1 −√

(1 + ρ)Γ

(1 + ρ)Γ + 1−√

(1 − ρ)Γ

(1 − ρ)Γ + 1+

√

(1 − ρ2)Γ2

(1 − ρ2)Γ2 + 2Γ

]

. (3.19)

Using the above expressions it was shown in [16] that the output BER severely de-

grades with the increase of the correlation coefficient, ρ i.e., when compared with

uncorrelated fading and fading with ρ = 0.9, an extra 10 dB power is required to


maintain a BER of 10−3. The theoretical methods presented above provide ex-

pressions to analyze the performance of the dual diversity system using the joint

pdf of the combiner output incorporating the correlation coefficient. The results

of these methods show degradation of the BER when ρ is increased but provide

no insight into what causes it to increase. Also these results cannot be extended

for cases of correlated fading where more than two diversity branches are employed.

We observed from Figure 3.1, when fading is independent system performance

improves as the number of branches is increased due to diversity gain. The effect

of correlated fading on the diversity gain of antenna systems where more than two

antennas are used is now discussed.

3.3.2 Antenna arrays with greater than two elements

The Nakagami-m distribution, which provides greater flexibility of comparing dif-

ferent fading environments, was used in [37] to study the effect of correlated fading

on the average BER of H-S/MRC scheme. However, the analytical expression de-

rived for this scheme is only valid for cases where the correlation coefficient between

any pair of signals arriving at the antennas is equal, i.e., the antenna array is lim-

ited to configurations such as the vertices of an equilateral triangle or a regular

tetrahedron. In the presence of correlation and a fading parameter of m = 2 it was

shown in [37] that choosing more branches to perform MRC for a given diversity

order improved the symbol error probability of QPSK, indicating the advantage of

employing more diversity branches at the receiver.

In [60] the concept of the virtual branch technique was employed to transform

the random variables representing correlated branch signals with unequal SNRs

and arbitrary Nakagami fading parameters, into conditionally independent and

identically distributed signals and then the performance of the system was evalu-

ated. The transformation was carried out under the assumption that the individual

branch SNRs were indexed in increasing order of the Nakagami m parameter. Un-

der these assumptions it was shown that as the fading parameter increased i.e., a

situation where fading is severe, correlation further degraded the symbol error rate.

For standard diversity techniques, the usual approach to evaluate performance

is to consider fading correlation between adjacent antennas [2] with an arbitrary

number of antennas in the array. This approach was used in [45] to evaluate the


performance of in adaptive arrays in mobile communications. The degradation

due to correlation was found not to be significant when the signals were optimally

combined using MRC.

If a small region of space contains a large number of antennas, consideration of

correlation only between adjacent antennas cannot fully describe the correlation

between all antennas in the system. In order to account for the correlation due to

each pair of antennas in an arbitrary antenna array, a fading correlation matrix can

be used [8] [34]. The correlation matrix is of size N ×N , where N is the number

of antennas in the array. Each element in this matrix is given by

ρn,m = E[rnr∗m] n,m = 1, 2, ..., N. (3.20)

For realistic situations the entries of the correlation matrix must include parame-

ters which actually cause correlation between antennas to provide an insight into

choosing the optimal array configuration for a given scattering environment. The

physical parameters which cause correlation are discussed below.

3.4 Factors influencing spatial correlation

The main factors which contribute to signal correlation in an antenna array system

are

• antenna configuration

• physical properties of the antenna elements

• scattering environment surrounding the antenna system.

The manifestation of the physical parameters on the correlation due to mutual

coupling, angular spread and spatial separation are now briefly discussed.

3.4.1 Mutual coupling

A voltage in an array element produces an induced current. The existence of

this induced current creates an electromagnetic field around the element which

affects the surrounding elements. This results in a co-dependency between nearby

elements of an array which is called mutual coupling. The correlation that is present

between the signals of nearby array elements can actually decrease in the presence

of mutual coupling [52]. In fact, the separation between elements can be reduced

3.4 Factors influencing spatial correlation 37

d

R

θ

ψ

Figure 3.2: Illustration of signals arriving at the receiver from a uniform ring ofscatterers with radius R, where θ is the angle of arrival and φ is the angular spreadof the signal.

to half the usual decorrelation distance, in the presence of mutual coupling, with

no noticeable increase in signal correlation. This phenomenon can be explained

by the fact that a slow wave structure is created in the presence of the induced

electromagnetic field [54] which decreases the effective wavelength of the signal.

The effect of mutual coupling on closely spaced antennas is not taken into account

in the work presented in this thesis. Under the consideration that mutual coupling

reduces spatial correlation [34] the results that are obtained in the later chapters

can be considered as upper bounds on the correlation between compactly placed

antennas.

3.4.2 Angular spread

When a signal arrives at an antenna in the array at an angle θ from the broadside2

of the array, θ is called the angle of arrival (AOA) of the signal. In a wireless

environment with local scatterers the received wave is dispersed giving rise to an

angular spread ψ. The angular spread is therefore defined as the angle over which

the signal arrives at the receiver antenna element. Measurements have shown that

for larger angular spreads, the required distance for spatial decorrelation is less

than in case of a small angular spread [64]. In a rich scattering environment, where

the angular spread may be considered to be 360, a quarter wavelength separation

between antennas is sufficient to provide decorrelation between signals. However,

for high base stations where the angular spread is generally just a few degrees, a

2for a given arrangement of antennas, the broadside is perpendicular to the line or axis of thearray.


separation of about 10λ− 20λ is required [64].

The effective correlation between signals received at two different points in space

can be described in terms of the angular separation between the two signals. The

correlation coefficient between two received signals separated by a distance d and

with and AOA of θ can be defined as [54]

ρ12 = E[eikd cos θ] (3.21)

where k is the wavenumber and d is the distance between the antennas.

If the energy arrives at the array from a restricted azimuth field [22], i.e., (θ −ψ/2, θ + ψ/2) the spatial correlation can be defined, based on the angular spread

as

ρ12 =∞∑

m=−∞eim( π

2−θ)sinc(m

ψ

2)Jm(kd) (3.22)

The diversity gain of a system reduces when the beamwidth reduces. For example

in Figure 3.2, when there is no angular spread such that ψ = 0, the value of the

correlation coefficient is maximized. In [58] it was shown that, for a uniform cir-

cular array, the fading correlation between the elements increased as the angular

spread decreased indicating degradation in the performance of the system.

3.4.3 Antenna spacing

The maximum advantage from using diversity combining techniques is achieved by

separating the antenna elements by the decorrelation distance [66]. A separation

of half a wavelength between elements is most commonly assumed [55], though the

distance may vary depending on the factors mentioned above.

Definition of correlation based on spatial separation

The autocorrelation function described in Section 2.3 can be used to characterize

the channel at two points separated in space. When defined in terms of the positions

of the two points, the autocorrelation function is called the ‘spatial correlation’

3.4 Factors influencing spatial correlation 39

function and is written as

Rh(4x) = E[h(x)h∗(x+ 4x)] (3.23)

where 4x is the spatial separation between the two points.

The correlation coefficient is the normalized form of (3.23) and is defined as

ρ(4x) =Rh(4x)

√

E[h(x1)h∗(x1)]E[h(x2)h∗(x2)]. (3.24)

When considering omni-directional diffuse scattering in the 3 dimensional space,

the spatial correlation coefficient is given by the sinc(·) function

ρ(4x) = sinc(k4x). (3.25)

In (3.25), when 4x = 0, we get ρ(0) = 1, indicating maximum correlation between

elements which are not physically separated in space. Also the first null of the

sinc(·) function appears at λ/2 which is most often assumed to be the decorrelation

distance between antennas [55]. However in the case of the 2 dimensional diffuse

field, i.e., a height invariant field where the signals are assumed to arrive from a

uniform ring of scatterers in the horizontal plane, the spatial correlation coefficient

is defined in terms of the zeroth order Bessel function [50]

ρ(4x) = J0(k4x). (3.26)

The assumption of decorrelation at λ/2 does not hold when the correlation coeffi-

cient is defined as in (3.26).

Several closed form expressions for the spatial correlation function for both 2

dimensional and 3 dimensional space for the different scattering environments are

derived in [55]. It was shown that to derive benefits from diversity systems with

sensors placed closer than λ/2 the scattering must be omni-directional. The effect

of placing sensors in close proximity to each other is discussed in the next two

chapters.


3.5 Spatial channel modelling

It was shown in the previous section that the spatial properties of the channel

have an enormous impact on the performance of antenna array systems. Recent

spatial channel models have been built on the concepts of classical models which

accounted for the amplitude and time varying characteristics of the channel. They

also incorporate the distribution of scatterers which affects the AOA properties

of the channel. Some channel models which include the AOA properties of the

channel are listed below. For a detailed description of the these channel models,

the reader is referred to [15].

• Lee’s model and Macrocell Model where the scatterers are effectively evenly

distributed on a circular ring around the receiver;

• Discrete Uniform Distribution Model and Uniform Sectored Distribution model

where the scatterers are assumed to lie within a narrow beamwidth centered

about the LOS of the receiver;

• The Gaussian Angle of Arrival model which is a special case of the Gaussian

wide sense stationary uncorrelated scattering model, where a single cluster

of scatterers is used to model the covariance matrix;

• Geometrically Based Elliptical Model where the scatterers are distributed

uniformly in an ellipse with the transmitter and the receiver located at the

foci of the ellipse.

Though most of these channel models incorporate the spatial aspects of the

channel, few actually highlight the physical significance of employing different array

configurations at the receiver.

Applications of spatial channel models

The spatial channel models discussed above, as mentioned, provide means to model

the received signal vector covariance matrix based on the AOA properties of the

channel. By using these models the system performance for different angular

spreads can be predicted. For example, in the case of a uniform circular receiver

array with randomly distributed scatterers, the angular power distribution is Lapla-

cian and it was shown in [58] that the fading correlation decreases with increase in

angular spread .

3.6 Summary 41

The one-ring scatterer model was used in [18] under the assumption that the

angles of departure follow a Laplacian distribution and the AOA are uniformly

distributed to obtain the channel capacity autocorrelation function. The Gaussian

Angle of Arrival model was used to find a closed form expression for the power

correlation coefficient when a linear array of vertical omnidirectional antennas was

used at the receiver [34]. Employing the scenario where the receiver is surrounded

by local scatterers while the transmitter is unobstructed, an analytical expression

for the average symbol error probability for MPSK for an arbitrary number of di-

versity antennas was obtained in [8]. Using this expression it was seen that the

performance improves as scattering angle widens when the signal comes from the

broadside, when fading is correlated.

3.6 Summary

In this chapter different analytical methods used to evaluate the performance of

correlation effects on dual and multiple antenna diversity systems was studied. It

was seen that most of the analytical methods provide little insight into the physical

factors causing correlation. Most results are also limited to a fixed set of channel

realizations.

Several ways of defining correlation between elements of an antenna array were

presented. Brief descriptions of channel models which use the AOA to describe the

spatial properties of the channel were also given. Such models are used to analyze

effective correlation and its effect on multiple antenna systems.

It was seen that in order to derive maximum advantage of spatial diversity at the

receiver the physical properties of the antenna array system and the surrounding

scatterers need special consideration and must be modelled effectively. We consider

these factors in detail in the next two chapters with simulation results in Chapter

4 and theoretical results in Chapter 5.

Chapter 4

Performance of EGC and MRC

under finite antenna separation

4.1 Introduction

Relatively recent research efforts have highlighted the potential contribution of the

spatial domain in improvement of the performance of wireless communication sys-

tems. Such promising results have lead to the widespread employment of spatial

diversity (i.e., multiple antenna) systems in place of the traditional single antenna

system. It was shown in [56] and [17] that simultaneous transmission of data via

multiple channels provides a significant boost to system capacity, improving system

performance. However, these results are based on the assumption that the signals

fade independently such that the parallel channels between the transmitter and the

receiver are practically uncorrelated. These results are rather unrealistic because

they do not take into account the effects of the actual antenna separation distances

within the array. It was seen in the previous chapter that reducing antenna spacing

causes the correlation between sensors to increase. We study the effects of this in-

creased correlation due to closely spaced antennas on the performance of diversity

systems in this chapter.

In [40] a novel MIMO spatial channel model was introduced. This model incor-

porates the actual antenna array geometry as well as tractable parameterizations

of the complex spatial signal scattering. We utilize the SIMO version of this model

to gain insights into the effects of finite antenna separations on the performance of

gain combining diversity schemes at the output of the receiver. Correlation effects

are inherently included in this model. The results obtained using this model are

43

44 Performance of EGC and MRC under finite antenna separation

x

ϕ

ϕ

RSrrRA( )

Receivers

Transmitter

Scatterers

Figure 4.1: Spatial scattering model under consideration for a flat fading SIMOsystem. rR is the radius of the sphere enclosing the receiver array and the scat-terers are distributed outside the ball of radius rRS which is in the farfield of thereceiver array. A(ϕ) represents the gain of the complex scattering environment forsignals arriving at the receiver scatter-free region from direction ϕ from a singletransmitter.

compared with the performance of the independent Rayleigh fading model which

assumes zero correlation between elements irrespective of the antenna positions.

4.2 Spatial channel model

The performance of a given diversity combining scheme at the receiver depends

on how the signals are manipulated by the channel. Most channel models used

for performance analysis account for amplitude and time varying properties of

the channel, neglecting the manipulation of the spatial aspects of the transmitted

signal. The SIMO model that we use in this chapter separates the channel vector

into the product of a deterministic matrix and a random vector. The deterministic

matrix incorporates the spatial geometric configuration of the receiver antennas

and the random vector characterizes the complex scattering environment. Channel

models such as those in [63] [66] characterize the entire channel as random. We

show the deficiency of these models by demonstrating that they effectively ignore

valuable signal information provided by knowledge of the antenna array geometry.

We now introduce the SIMO model which exploits this information.

4.2 Spatial channel model 45

4.2.1 The SIMO model

Consider a SIMO communication system with P receiver antennas located in a

scatter free ball of radius rR as shown in Figure 4.1. The antennas are located at

positions zp, p ∈ 1, 2, . . . , P . We assume that the scatterers are distributed outside

a ball of radius rRS(> rR) where rRS is in the farfield of the receiver antennas.1

The channel is assumed to exhibit flat and frequency non-selective fading where

the propagation delay between different multipaths is less than the symbol period

and all frequency components are affected in the same way throughout the channel.

Let there be one transmitter antenna where u is the transmitted baseband sig-

nal during a signaling interval from the transmitter, and let r = [r1, r2, . . . , rp]′ be

the vector of the received signals at the P receivers where [·]′ denotes transpose.

The signal at the pth receiver can be written as

rp =

∫

Ω

A(ϕ)ue−ikzp·ϕdΩ(ϕ) + np (4.1)

where A(ϕ) is the complex gain of the signal entering the receiver scatterer-free

ball from direction ϕ, k is the wave number given by 2πf/c, f being the carrier

frequency and c the speed of the propagating wave, i =√−1 and np is the additive

white Gaussian noise (AWGN) with variance Np. The factor e−ikzp·ϕ represents free

space wave propagation inside the scatterer-free receiver region. The integration in

(4.1) is over the unit sphere for the 3 dimensional multipath case where dΩ(ϕ) is

a surface element of the unit sphere Ω. For the 2 dimensional case, the integration

is over the unit circle.

From (4.1) the received signals under the assumption of independent noise at

each branch, can be represented in vector form as

r = hu+ n (4.2)

where n = [n1, n2, . . . , nP ]′ and h is the length P channel vector with pth element

hp =

∫

Ω

A(ϕ)e−ikzp.ϕdΩ(ϕ). (4.3)

In order to gain a deeper understanding of the structure of the channel vector, in

1The Rayleigh distance [35] gives the approximation for the farfield distance from the arrayorigin as d = 2l2/λ, where l is the array dimension and λ is the transmitted signal wavelength.


100 101 102 103−0.5

0

0.5

1

Argument z

J 0(z)

100 101 102 103−0.4

−0.2

0

0.2

0.4

Argument z

J 10(z

)

100 101 102 103−0.2

−0.1

0

0.1

0.2

Argument z

J 100(z

)

Figure 4.2: Illustration of the highpass character of the Bessel function when m =10 and m = 100. Also shown is the low pass character of J0(·).

the next section we use modal analysis to reduce the integral expression of h. The

analysis is restricted to 2 dimensional space2, however the results can be extended

to 3 dimensional space.

4.2.2 Analyzing the channel matrix

We begin our analysis of the channel matrix by using the Jacobi-Anger expansion

[10, pg 67] for a 2 dimensional propagation environment. We can therefore express

the Fourier series expansion of e−ikzp·ϕ as

e−ikz·ϕ =∞∑

m=−∞

[

Jm(k|z|)e−im(ϕz−π/2)]

eimϕ (4.4)

where Jm(·) is the mth order Bessel function3 of the first kind, z = (z, ϕz) and

ϕ = (1, ϕ) in the polar coordinate system.

2This models the situation in 3 dimensional space where the multipath propagation is restrictedto the horizontal plane, having no component arriving at significant elevations. It is assumed herethat the signal field is height invariant.

3Note that Jm(·) = (−1)mJm(·)

4.2 Spatial channel model 47

Bessel functions Jm(·) have a spatial high pass character for |m| ≥ 1, i.e.,

Jm(·) increases monotonically as indicated in Figure (4.2), until its maximum at

arguments around O(m) before slowly decaying. It was shown in [26] that Jm(z) ≈0 for m > dze/2e, where d.e is the ceiling operator and e ≈ 2.7183. Using this

result (4.4) can be truncated to

e−ikz.ϕ =

MR∑

m=−MR

[Jm(k|z|)e−im(ϕz−π/2)]eimϕ (4.5)

where

MR = dkerR/2e + 4 (4.6)

with a relative truncation error

εM(z) ,

∣

∣

∣e−ikz.ϕ −

M∑

m=−M

[Jm(k|z|)e−im(ϕz−π/2)]eimϕ∣

∣

∣

|e−ikz.ϕ| (4.7)

≤ 0.16127e−4 (4.8)

where 4 is an integer [26]. Therefore, the relative truncation error is not more

than 16.1% once M = dkerR/2e. It thereafter decreases exponentially to zero as

M increases. For example, when 4 = 1 we get εM(z) ≤ 0.0593 and when 4 = 2

the truncation error is εM(z) ≤ 0.0218.

Therefore using (4.1) and (4.5) the signal received at the pth receiver can be

written as

rp =

MR∑

m=−MR

Jm(k|zp|)e−im(ϕp−π/2)hmu+ np (4.9)

where

hm =

∫

Ω

A(ϕ)e−imϕd(ϕ). (4.10)

Now the channel vector h in (4.9) can be expressed as

h = JRhs (4.11)

where JR is referred to as the receiver configuration matrix, given by


JR =

J−MR(kz1)e

iMR(ϕ1−π2) . . . JMR

(kz1)e−iMR(ϕ1−π

2)

.... . .

J−MR(kzp)e

iMR(ϕp−π2) . . . JMR

(kzp)e−iMR(ϕp−π

2)

(4.12)

and

hs = [h−M , ...., h0, ...., hM ]′. (4.13)

Since the scattering function A(ϕ) is periodic in ϕ we can expand it using the 2

dimensional Fourier series. Therefore A(ϕ) can be written as

A(ϕ) =∞∑

m=−∞hme

−imϕ. (4.14)

We also normalize the scattering gain function by assuming

∫ 2π

0

E[|A(ϕ)|2]dϕ = 1 (4.15)

By substituting (4.14) into (4.15) we obtain

∞∑

m=−∞E[|hm|2] =

1

2π. (4.16)

4.2.3 Comments on the model

The spatial channel has been decomposed into two regions, namely, the scatterer-

free region which consists of a fixed number of antennas at known positions for a

given configuration and a complex scattering media which can be modelled as a

rich scattering environment.

The channel model discussed above has the following features

1. The conventional channel vector h is now expressed in terms of the config-

uration of the receiver antenna array. It is seen that JR provides a better

insight into the specific positions and orientations of the receiver antennas

which can be used in the analysis of various performance parameters of the

system for different array geometries. The essential random behaviour of the

wireless channel is retained and characterized by hs.

2. Since we assume a random scattering environment the components of the

4.3 Effect of introducing ‘space’ into diversity systems 49

vector hs can be modelled as complex Gaussian random variables.

The novelty of the model is, hence, the explicit incorporation of the array

geometry in addition to the essential randomness of the channel. The channel model

described in this section will be used to analyze the effects of spatial dependence

of the signals on the diversity combining schemes in the next section.

4.3 Effect of introducing ‘space’ into diversity

systems

In Figure 3.1 it was seen that increasing the number of antennas, in an independent

Rayleigh fading channel, provides a significant boost to the system performance

when diversity combining schemes are used at the receiver. It was also seen that

MRC performed the best when compared to EGC and SC. However, the plot in Fig-

ure 3.1 does not convey any indication about the performance of diversity systems

when physical antenna configurations with finite separation between the elements

is considered. We show that this criteria could be crucial when designing a multiple

antenna receiver system with spatial constraints.

In order to gain realistic insights into the performance of systems with closely

spaced antennas and the consequence of varying the separation between the anten-

nas in a given array configuration, the spatial channel model is used for analysis.

In this section we present simulation results of the SNR and BER performances

obtained at the receiver output, when the MRC and EGC gain combining schemes

are employed, using the spatial channel model introduced in Section 4.2. The

results are compared with the performance of the independent Rayleigh fading

channel model. The independent fading case represents the upper bound of the

performance where there is no correlation between the signals at different antennas.

The lower bound on performance is obtained by maximizing correlation, when the

separation between antennas is made equal to zero.

Experimental set up for the simulations

Simulations for SNR performance were carried out using the commonly used inde-

pendent Rayleigh fading model described in section 2.4.1 and the spatial channel

model described in section 4.2 for MRC and EGC. Simulations were carried out

by varying both the array aperture and the antenna separation distances within


1 2 3 4 5 6 7 8 9 1022

24

26

28

30

32

34

36

38

Number of antennas

Ave

rage

SN

R in

dB

MRC RayleighEGC RayleighMRC d= λ/2EGC d= λ/2d = 0

Figure 4.3: Comparison of the average output SNR with increasing number ofreceiver antennas. The performance of MRC and EGC schemes when the antennasare separated by a distance of λ/2 is compared with independent Rayleigh fading.Also shown is the SNR gain for the case where correlation is maximised betweenantennas for both MRC and EGC, i.e., when d = 0.

a uniform linear array and by varying the radius of a uniform circular array. The

instantaneous SNR for the two combining schemes were calculated using the fol-

lowing equations.

The SNR for the MRC combining scheme was calculated using

γMRC =1

2

∣

∣

∣

P∑

p=1

wphp

∣

∣

∣

2

P∑

p=1

Np

(4.17)


1 2 3 4 5 6 7 8 9 1022

24

26

28

30

32

34

36

38

Number of antennas

Ave

rage

SN

R in

dB

d = λd = λ/2d = λ/10d = 0

Figure 4.4: Performance of MRC when the separation ‘d’ between the antennas ina ULA is reduced such that d = λ, λ/2, λ/10 and 0.

where Np is the noise variance and wp is the weight of the pth branch

wp =h∗pNp

. (4.18)

The SNR for the EGC combining scheme was calculated using

γEGC =1

2

[

P∑

p=1

∣

∣

∣hp

∣

∣

∣

]2

P∑

p=1

Np

. (4.19)

The average SNR of the two schemes was calculated as the ensemble average of the

instantaneous SNRs

ΓMRC = E[γMRC] (4.20)


1 2 3 4 5 6 7 8 9 1022

24

26

28

30

32

34

36

38

Number of antennas

Ave

rage

SN

R in

dB

MRC RayleighEGC RayleighMRC D = λEGC D = λD = 0

Figure 4.5: Comparison of the performance of MRC and EGC with the Rayleighfading model when the number of antennas are increased in a ULA with a constantaperture D = λ. The case where D = 0 is also shown to represent maximizedcorrelation.

and

ΓEGC = E[γEGC]. (4.21)

The random channel vector hs was modelled as a normalized independent complex

Gaussian random vector of length 2MR +1, where MR is determined by the radius

circle within which the receiver antennas are located. By assuming independent

elements of hs, we essentially model a 2 dimensional isotropic scattering field.

The results obtained from the simulations using the aforementioned parameters

are now presented.


1 2 3 4 5 6 7 8 9 1022

24

26

28

30

32

34

36

38

Number of antennas

Ave

rage

SN

R in

dB

MRC D = λ/2EGC D = λ/2MRC D = λ/10EGC D = λ/10D = 0

Figure 4.6: Comparison of the performance of MRC and EGC when the apertureof the ULA is decreased i.e. D = λ/2, λ/10 and 0.

4.3.1 SNR performance of MRC and EGC with finite an-

tenna separation

The results of the average output SNR performance of a receiver diversity system

using the spatial channel model and the independent Rayleigh fading model with

distance increasing numbers of antennas and varying separation, in a uniform lin-

ear array (ULA) is shown in Figures 4.3 and 4.4. The effect on the performance

of the EGC and MRC schemes of adding antennas while keeping the aperture4 of

the array constant is shown in Figures 4.5 and 4.6. The results obtained when the

radius of a uniform circular array (UCA) is varied is shown in Figure 4.7

In Figure 4.3 the separation between adjacent elements of the ULA is kept con-

stant at λ/2. The output SNR performance using both MRC and EGC exhibits

a slight degradation when compared to the performance when the signals fade in-

dependently. A separation of λ/2 is commonly referred to as the decorrelation

4The region in space over which the energy is collected is called the aperture.


1 2 3 4 5 6 7 8 9 1022

24

26

28

30

32

34

36

38

Number of antennas

Ave

rage

SN

R in

dB

MRC R = λ/2EGC R = λ/2MRC R = λ/10EGC R = λ/10R = 0

Figure 4.7: Comparison of the performance of MRC and EGC when the radius ‘R’of the UCA is varied such that R = λ/2, λ/10 and 0.

distance in the literature, however, in Figure 4.3 we see a fall in performance when

the antennas are separated by a distance of λ/2. This indicates the presence of

correlation between adjacent elements of the array. This can be explained by the

fact that the nulls of the sinc(·) function which represent spatial correlation in 3

dimensional isotropic scattering occur at spacings of λ/2, but for the 2 dimensional

isotropic case where the field is assumed to be height invariant, the spatial corre-

lation is defined by the zeroth order Bessel function J0(·) where the nulls are not

uniformly spaced.

To gain a deeper insight into the effect of correlation in 2 dimensional space,

the results obtained when the separation between the elements is further decreased

is analyzed from Figure 4.4. When the separation between the elements is reduced

from λ to λ/2 the output SNR of the MRC diversity combining system is affected

by a small amount. However when the spacing is reduced to λ/10, there is a sig-

nificant fall in the SNR performance. A loss of about 1 dB is seen in the average


0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 512

14

16

18

20

22

24

Separation distance in λ

SN

R in

db

MRCEGC

Figure 4.8: Illustration of the effect of increasing the separation distance between2 antennas in steps of λ/4.

SNR at the output of the system when the antennas are very closely spaced. A

separation of λ/10 between the antennas indicates a situation where the signals

are highly correlated and provides us insights into the effect on diversity gain due

to drastic reductions in separation between antennas. The loss in SNR due to this

very small antenna separation indicates that increased correlation does take full

advantage of the benefits that can be obtained by employing a particular diversity

scheme.

It is interesting to note that the average SNR performance shows improvement

as the number of antennas is increased, even when the separation between the an-

tennas is zero, where the antennas can be assumed to be stacked in a column. This

indicates that even when the correlation between the antennas is maximum, the

system benefits from a consistent diversity gain because of the assumption of inde-

pendent, uncorrelated noise at each diversity branch. However, it can be observed

that the SNR performance of the system does suffer when correlation is introduced

between the signals arriving at antennas in close proximity to each other.


The effect on SNR performance when antennas are added in a ULA of constant

aperture is seen in Figures 4.5 and 4.6. For an aperture of λ it can be seen that

the performances of both MRC and EGC do not improve, as expected, when more

antennas are added in the fixed aperture. Though the output SNR increases with

the number of antennas, which is attributable to the diversity gain of the system,

we observe that the system performance gradually falls. This effect is a clear indi-

cation of the increased spatial correlation between signals at antennas located very

close to each other. For example, with an aperture of λ, when the dual diversity

case is considered, the distance between the adjacent antennas is λ and, hence, the

system performance is the same as the independent fading case. However, when an

extra antenna is added, the spacing between each antenna reduces to λ/2 and we

see a slight fall in performance. This is in agreement with the results obtained in

Figure 4.3, where the separation between each antenna was kept constant at λ/2.

It can therefore be seen that as the number of antennas is increased in a confined

space, the antenna spacing effectively reduces, increasing correlation between the

elements which adversely affects the system performance.

The results of reduced aperture when the number of antennas are increased

for MRC and EGC is shown in Figure 4.6. When the aperture of the antenna

array is further reduced, the system performance degrades to a greater extent as

expected. Together with the fact that the system performance is adversely affected

with reduced aperture, it can be noted that the performance of both of the gain

combining schemes tend to converge. When the aperture is λ/10 both MRC and

EGC exhibit almost the same performance. At D = 0 the two schemes produce

the same output. Though increasing the number of antennas without any spacing

between them may seem impractical, it provides us with the insight that adding

too many antennas in a small region of space defeats the purpose of employing

different spatial diversity combining schemes. A diversity gain is still obtained in

this situation because the normalized correlation does not become equal to unity

with the assumption of independent additive noise at each branch. That is the

noise tends to cancel itself to a greater extent as more antennas are added.

The results of the output SNR performance when MRC and EGC were em-

ployed for a UCA with constant radii of λ/2 and λ/10 respectively, with increasing

numbers of antennas is shown in Figure 4.7. The performances of the diversity


0 2 4 6 8 10 12 14 16 18 2010−6

10−5

10−4

10−3

10−2

Average SNR in dB

Ave

rage

BE

R

1 antenna2 antennas3 antennas5 antennas

Figure 4.9: Comparison of BER performance for the independent Rayleigh fadingcase when the number of antennas is increased for MRC.

schemes for a UCA with constant radius are similar to those of a ULA with con-

stant aperture. However, since the effective separation between the antennas is

greater in the case of the UCA, compared to that of the ULA, (with 2 antennas

and a radius or aperture of λ/2 the distance between adjacent antennas in the case

of a UCA is twice than that of a ULA) the performance of the UCA is better due

to smaller effective correlation between antennas.

The effect of gradually increasing the separation distance between antennas in

a dual diversity system is shown in Figure 4.8. A steep rise in performance is ob-

served when the separation between the antennas is increased to λ/2. The increase

in performance thereafter is gradual, because of the gain obtained due to indepen-

dent fading.

It is evident from the simulation results that decreasing antenna separation

distances causes the output SNR performance of both MRC and EGC to degrade

due to correlation effects. From the plots where the aperture was reduced to


0 2 4 6 8 10 12 14 16 18 2010−7

10−6

10−5

10−4

10−3

10−2

Average SNR in dB

Ave

rage

BE

R

1 antenna2 antennas3 antennas5 antennas

Figure 4.10: BER performance of MRC when the number of antennas is increasedin a ULA with a constant aperture of λ.

zero, we observed that most of the diversity gain was due to the assumption of

independent noise at each branch which is commonly known as the array gain.

From these results we infer that reduced spacing between antennas manifests as

correlation between their respective signals. This reduces the average output SNR

of diversity systems when compared to the optimal benefits that can be obtained

when the branches receive independently faded signals.

We now consider the effect of signal correlation due to finite antenna spacing on

the output BER.

4.3.2 Performance of BER using the spatial model

The effect of diversity combining in the presence of spatial correlation on the prob-

ability of the average bit error rate (BER) at the receiver output is now considered.

As in the case of the simulations carried out for SNR, the results of the average

BER at the output obtained with the spatial channel model are compared with

those of the independent Rayleigh faded model. For all simulations we used binary


0 2 4 6 8 10 12 14 16 18 2010−6

10−5

10−4

10−3

10−2

Average SNR in dB

Ave

rage

BE

R

Independent fadingD = λ/2D = λ/5D = 0

Figure 4.11: BER performance of a dual diversity MRC system with varying arrayaperture compared with the independent fading case.

phase shift keying (BPSK) as the modulation scheme and MRC as the diversity

technique to combine the output of the individual branches.

In Figure 4.9, the average BER for MRC at the output of the receiver when sub-

jected to independent Rayleigh fading is shown. It is seen that, as the number of

branches is increased, there is a significant improvement in the average BER. This

can be explained by the fact that when the branches are uncorrelated the diversity

gain causes the bit error rate at the output decreases. However, this result is mis-

leading because the scenario is not the same when diversity branches are increased

within a confined space. For example, when the aperture of an ULA is limited to

λ, adding extra antennas does not provide improved BER. We see from Figure 4.10

that the BER does not improve significantly when more antennas are added within

a confined aperture. This indicates that employing additional branches does not

provide any significant diversity advantage in systems where correlation prevails

due to constraints in space.

A comparison of the BER performance for a dual diversity system using inde-


0 2 4 6 8 10 12 14 16 18 2010−6

10−5

10−4

10−3

10−2

Average SNR in dB

Ave

rage

BE

R

D = λD = λ/2D = λ/5D = 0

Figure 4.12: BER performance of MRC with varying aperture in a ULA with 2antennas.

pendent fading and a ULA with fixed apertures of λ/2, λ/5 and zero is shown in

Figure 4.11. As seen in the case of the output SNR performance, the BER for a

spatial separation of λ/2 between the antennas is nearly the same as in the inde-

pendent fading case. However, as the spacing between the antennas is reduced to

λ/5 there is a substantial degradation in system performance. About 2 dB more

power is required to maintain a BER of 10−3 when the distance between the anten-

nas is reduced from λ/2 to λ/5. In order to observe the effect of the diversity gain

on the BER performance, simulations were carried out for an aperture D = 0. We

observe that twice as much carrier power is required to maintain the same BER

when correlation is maximized compared with the independent fading case.

The performance of a dual diversity ULA with separation between the anten-

nas gradually increased from 0 to λ is shown in Figure 4.12. As observed from

Figure 4.11, we observe that the average BER at the output falls with increas-

ing correlation due to reduced spacing. However, we also observe that separating

the antennas further than λ/2 does not provide any further performance benefits.

4.4 Summary 61

0 2 4 6 8 10 12 14 16 18 2010−7

10−6

10−5

10−4

10−3

10−2

Average SNR in dB

Ave

rage

BE

R

R = λ/5 (2 antennas)R = λ/5 (3 antennas)R = λ/5 (5 antennas)R = 0 (2 antennas)

Figure 4.13: Comparison of BER performance with varying radius for a UCA.

This result suggests that once the antennas are decorrelated, further separation

does not provide any advantage to system performance. When designing a receiver

employing diversity for a given modulation scheme, this information is crucial for

optimizing system design.

We expect the performance of the UCA with a radius of λ/5 to provide better

results than a ULA of the same aperture. This result is shown in Figure 4.13. As

seen in the case of the ULA, adding more antennas in the circular array with a fixed

radius does not provide significant improvements in performance when compared

to optimal benefits when fading is independent.

4.4 Summary

In this chapter a recently developed spatial model channel model was exploited to

study the effect of finite antenna separations. The performance of two important

diversity combining schemes, EGC and MRC was compared to the performance of

independent fading where no emphasis is laid on the relative position of the an-


tennas in a given array configuration. The SIMO spatial channel model provided

a tool to study the effect of varying the separation distances between the antennas

through the deterministic part of the channel matrix, while the essential random-

ness of the scattering environment was also captured.

By comparing the performances of the MRC and EGC antenna diversity schemes

for a uniform linear antenna array and the uniform circular array using the SIMO

spatial model with the uncorrelated Rayleigh fading model, it was seen that increas-

ing the number of antennas makes the array dense which causes spatial correlation

between the antennas to significantly limit the system performance. It was also

seen that increasing the number of antennas in a limited region of space defeats

the purpose of employing different diversity schemes at the receiver. The results

obtained in this chapter establish the fact that separation distances between ele-

ments of an antenna array has a significant impact on the system design in wireless

communication receivers deployed with constraints in space.

The simulation results obtained in this chapter gives us a strong indication to the

answers we aim to find in this thesis. In order to confirm our results we establish

theoretical explanations for the simulation results in the next chapter.

Chapter 5

Effects of spatial correlation on

diversity receivers

5.1 Introduction

In the previous chapter we considered the performances of the EGC and MRC

signal combining schemes in a multipath fading channel taking into account corre-

lations between signals at spatially separated antennas. The results were obtained

via simulation. It is usually assumed that antennas are spatially separated enough

that correlations due to their proximity to each other may be ignored. We showed

that such assumptions are not always valid and that correlation due to closely

spaced antennas can have a significant adverse effect on the expected diversity

gains.

In this chapter we present a theoretical basis for the simulation results obtained

in Chapter 4. We derive an equation for the spatial correlation coefficient using

signal representation of the spatial channel model described in the previous chap-

ter. We also express the average SNR for MRC and EGC using the same signal

representation. In particular in this chapter, we seek a closed form solution for the

average BER for MRC signal combining with BPSK signalling. To achieve this

we use a multivariate distribution which allows us to model the correlated fading

channel in the form of a covariance matrix. Eigen-decomposition of the covariance

matrix is used to find closed form equations for the BER of a ULA and a UCA at

the receiver.

63

64 Effects of spatial correlation on diversity receivers

5.2 Spatial correlation effects between adjacent

antenna elements

Correlation is typically used as a measure of the similarity between different signals

or different versions of the same signal. In the context of this thesis we have been

considering correlation between signals received at antennas at different points in

space. The correlation coefficient is the normalised correlation and has a range

between 0 and 1. A value of 0 in the current context indicates independently dis-

tributed fading at different antenna locations, while a value of 1 indicates identical

signals. In this section we determine an expression for the correlation coefficient

governing the signals at two closely spaced antennas as given by the spatial channel

model introduced in Section 4.2.

Using notation introduced in Chapter 4, the normalized spatial correlation be-

tween the complex envelopes of the signals rp and rq received at antennas at posi-

tions zp = (zp, ϕp) and zq = (zq, ϕq) can be written using (3.24) as

ρpq =E[rpr

∗q ]

√

E[rpr∗p]E[rqr∗q ]. (5.1)

Using (4.1) the covariance between the two signals rp and rq is given by

E[rpr∗q ] = σ2

u

∫ ∫

E[A(ϕ)A∗(ϕ′)]e−ikzp·ϕeikzq ·ϕ′

dϕdϕ′ (5.2)

where σ2u = E[|u|2].

Assuming independence between signals entering the scatterer free ball from

different directions, we have

E[A(ϕ)A∗(ϕ′)] =

E[| A(ϕ) |2] if ϕ = ϕ′

0 otherwise.(5.3)

We define the angular power distribution as

P (ϕ) ,E[| A(ϕ) |2]

∫

E[| A(ϕ) |2]dϕ. (5.4)

The normalized scattering gain, as in (4.15), is a more useful entity. Thus, normal-

5.2 Spatial correlation effects between adjacent antenna elements 65

ising the angular power distribution in (5.4), we obtain

P (ϕ) = E[| A(ϕ) |2]. (5.5)

Thus, using (5.3) and (5.5), we get a simplified equation for the covariance in (5.2)

E[rpr∗q ] = σ2

u

∫

P (ϕ)e−ik(zp−zq)·ϕdϕ. (5.6)

Similarly, we can express the denominator terms of the correlation coefficient in

(5.1) as

E[rpr∗p] = E[rqr

∗q ] = σ2

u

∫

P (ϕ)dϕ. (5.7)

Now, substituting (5.6) and (5.7) into (5.1), we obtain

ρpq =

∫

P (ϕ)e−ik(zp−zq)·ϕdϕ (5.8)

Now, in [55], the angular power distribution P (ϕ) is expressed, using a Fourier

series expansion, as

P (ϕ) =∞∑

n=−∞αne

inϕ (5.9)

where

αn =1

2π

∫ 2π

0

P (ϕ)einϕdϕ. (5.10)

We use (5.9) and the Jacobi-Anger expansion [10], given by

eikz·ϕ =∞∑

m=−∞imJm(k|z|)eimϕze−jmϕ (5.11)

where z = (z, ϕz) and ϕ = (1, ϕ) in the polar coordinate system, to rewrite the

correlation coefficient in (5.8) as

ρpq =∞∑

m=−∞αni

mJm(k|zp − zq|)eimϕpq (5.12)

where ϕpq is the angle between the x axis and the line joining points zp and zq. We

can truncate the expression for the correlation coefficient in (5.12) using a similar

argument to that used for truncating the expression for the received signal in Sec-

tion 4.2.2. That is, the spatial high-pass nature of the Bessel functions enables us

to eliminate all but a small number of terms while retaining a good approximation


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.5

0

0.5

1

Spatial Separation in λ

corr

alat

ion

co−e

ffici

ent ρ

2−D diffuse fieldAzimuth limited field φ = π/2Azimuth limited filed φ = π/3

Figure 5.1: Comparison of spatial correlation versus separation for a diffuse scat-tering field, limited in the azimuth with the source centered around π/3.

to (5.12). This is enables us to obtain a closed form expression for the correlation

coefficient.

The closed form expression is dependent upon the scattering distribution, rep-

resented by the αns. Closed form expressions for αn exist for many scattering

distributions [55]. Some of the most commonly used angular power density distri-

butions and their Fourier coefficients αn are reported in [55]. We repeat some of

them here.

2 dimensional isotropic diffuse field

For this simple case where the scattering is over all angles in the plane contain-

ing the antennas, the power density distribution and the corresponding Fourier

coefficients are given by

P (ϕ) =1

2π∀ϕ (5.13)

and

αn =

1 if n = 0

0 otherwise.(5.14)

5.2 Spatial correlation effects between adjacent antenna elements 67

2 dimensional uniform limited angular field

If the energy arrives uniformly from a limited range of angles ±∆ around the mean

AOA φ0, then

P (ϕ) =

12∆

if (φ− φ0) ≤ ∆

0 otherwise.(5.15)

In this case the Fourier coefficients are given by

αn =sin(n∆)

n∆e−inφ0 . (5.16)

Von-Mises angular power density distribution

This is a non-isotropic scattering model where

P (ϕ) =eς cos(φ−φ0)

2πI0(ς)(5.17)

where ς > 0 is the degree of non-isotropy, φ0 is the mean AOA and I0(·) is the zero

order modified Bessel function of the first kind. For this distribution the Fourier

coefficients are given by

αn =In(ς)

I0(ς)e−inφ0 . (5.18)

In Figure 5.1 the effect of increasing separation distance on the correlation coeffi-

cient when an azimuthal source with an angular spread of π/3 is used for values of

φ0 = π/2 and π/3 is illustrated. The effect of spatial separation on ρpq when a 2

dimensional diffuse field is employed is also shown.

Inserting each of the above expressions for αn into (5.12) and truncating ap-

propriately will result in closed form expressions for the correlation coefficients for

the respective channel scattering models. However, this is nontrivial and as it is

not essential to the work in this thesis, has not been shown here.


5.3 Average SNR using diversity incorporating

spatial correlation

In (4.9) we gave an expression for the complex received signal at the pth branch.

We repeat this equation here for reference.

rp =∞∑

m=−∞Jm(k|zp|)e−im(ϕzp−π/2)hmu+ np (5.19)

where u is the transmitted baseband signal over a given transmitting interval and

np is the AWGN at the pth branch.

Since we assume the channel to be a flat fading channel such that the symbol

period is less than the reciprocal of the fading rate, the fading pattern does not

change over the symbol duration. Therefore, the transmitted signal mean power

can be normalized. That is

E[| u |2]=1. (5.20)

The complex noise np is the additive noise at the pth branch. We consider the noise

at each branch to be independent and identically complex Gaussian distributed

with variance Np such that the variance of the noise at the pth branch for a given

time instant is given by

E[npn∗p] = Np (5.21)

and the covariance between the noise at the pth and qth branches, at any time, is

given by

E[npn∗q] = 0. (5.22)

Equation (5.21) uses the assumption of ergodicity of the noise process.

The instantaneous SNR, γp, and the average SNR, Γp, at the pth diversity branch

are defined as

γp =local mean signal power of the pth branch

mean noise power of the branch(5.23)

5.3 Average SNR using diversity incorporating spatial correlation 69

2 4 6 8 10 12 14 1616

18

20

22

24

26

28

30

Number of antennas

Ave

rage

SN

R in

dB

Rayleigh Fadingd = λ/2

Figure 5.2: Illustration of the effective SNR when calculated using the lower boundin equation (5.29).

and

Γp =statistical mean signal power of the pth branch

mean noise power of the branch= E[γp]. (5.24)

These definitions are now used to find the average SNR at the output of the receiver

when EGC and MRC are employed.

Equal gain combining

Equal gain combining is a commonly used gain combining scheme where all the

branch signals are co-phased and added together. In EGC the signals are equally

weighted such that all of the weighting coefficients are set to one. The instantaneous

SNR and average SNR, respectively, at the output, in the case of EGC are given

by the following equations


γEGC =1

2

[

P∑

p=1

|rp|]2

P∑

p=1

Np

(5.25)

E[γEGC] =1

2NpPE

[

[ P∑

p=1

|rp|]2]

(5.26)

or

ΓEGC =1

2NpP

P∑

p=1

P∑

q=1

E[|rp||rq|]. (5.27)

Using the signal representation in (5.19) the average SNR can be written as

ΓEGC = 12NpP

P∑

p,q=1

E

[

∣

∣

∣

∑

m

Jm(k|zp|)e−im(ϕzp−π/2)hm

∣

∣

∣×

∣

∣

∣

∑

m′

Jm′(k|zq|)eim′(ϕzq−π/2)hm′

∣

∣

∣

]

.

Assuming that each of the signals is cophased, and using Jensen’s inequality, the

above equation can be simplified and written as a lower bound of the average SNR

such that

ΓEGC ≥ 1

2NpP

P∑

p,q=1

∑

m

Jm(k|zp − zq|)E[| hm |2]. (5.28)

If the scattering can be considered uniform over all directions equation (5.28) re-

duces to

ΓEGC ≥ 1

2NpP

P∑

p,q=1

J0(k|zp − zq|). (5.29)

Figure 5.2 illustrates the lower bound in (5.29) of the SNR using EGC with the

spatial channel model. A performance comparison is made with the independent

Rayleigh fading model. The effect of correlation which is a consequence of reduced

spacing between the antennas is evident from the plot.

5.3 Average SNR using diversity incorporating spatial correlation 71

Maximal ratio combining

In maximal ratio combining the signals are combined such that the output SNR is

the sum of the SNR of the individual branches. This scheme maximizes the output

SNR. The optimum weights are chosen by applying the Schwarz-inequality [19].

Each branch signal is weighted by the ratio of its complex conjugate to the branch

noise power. The instantaneous combined SNR at the output is written as

γMRC =P∑

p=1

r2p

np

(5.30)

or

γMRC =P∑

p=1

∑

m


∑

m′

Jm′(k|zp|)eim′(ϕzp−π/2)hm′

np

. (5.31)

The average SNR at the output of the system is the ensemble average of the

instantaneous branch SNRs at the output and is given by

ΓMRC =P∑

p=1

E

[

∑

m


∑

m′

Jm′(k|zp|)eim′(ϕzp−π/2)hm′

np

]

. (5.32)

Again, using the assumption of independent signals we get

ΓMRC =P∑

p=1

∑

m

[Jm(kzp)]2E[|hm|2]

Np

. (5.33)

It is evident from (5.33) that for a rich isotropic scattering environment

ΓMRC =P∑

p=1

γp

∑

m

[Jm(kzp)]2. (5.34)

The second sum in (5.34) is equal to one [20] and thus, for an isotropic scatter-

ing environment, the average SNR at the output of the receiver is the sum of the

individual branch SNRs.

We observe that the effective distance between any two antennas in the array is

not captured in (5.33) as was the case for EGC. This can be attributed to the fact


that the MRC technique requires each signal to be multiplied by its own conjugate

gain in order to amplify the system gain, while the influence of the neighbouring

branches on the signal is ignored. To successfully implement MRC, the receiver

must also have perfect knowledge of the channel to obtain optimal benefits of the

scheme. This implies requirement of additional circuitry for implementation of the

system. These factors indicate that MRC is not an efficient practical diversity

combining technique, though it is theoretically optimal.

5.4 Error probability of MRC using BPSK mod-

ulation

The average bit error rate is a widely used tool to measure the performance of a

given system and choice of modulation scheme. In the presence of correlated fading,

analysis of the bit error rate is carried out for dual diversity schemes [22] [16] [36]. It

has been shown by the theoretical and experimental results of these works that the

presence of correlation between the two branches has adverse effects on the overall

system performance. In this section, the error probability of a multiple antenna

system for different antenna configurations is analyzed using the joint statistics of

of the baseband complex multivariate distribution which characterizes the fading

channel using the cross-correlation function of the fading processes. We extend

the results obtained in [59] by finding closed form equations for a uniform circular

array. From the expressions for BER analyse the effects of varying the antenna

aperture for the uniform linear array and the radius of the uniform circular array.

General expression for BER with distinct eigenvalues

Let r = [r1, r2, ...rP ] be the vector of zero mean received signals at the P branches

of a diversity system. Then the joint multivariate pdf of r can be compactly written

as [43, pg 49]

pr(r) =1

(2π)p/2(detR)0.5exp[

− 1

2r†R−1r

]

(5.35)

where r† denotes the Hermitian of r and R is the P × P covariance matrix given

by

5.4 Error probability of MRC using BPSK modulation 73

R =

ρ11 . . . ρ1P

.... . .

ρP1 . . . ρPP

(5.36)

where ρpq is the correlation coefficient between the pt and qth branches and R−1

denotes the inverse of R.

The average bit error rate Pb when using BPSK modulation with MRC is

generally represented as [43]

Pb =

∫

r

Q

(

√

√

√

√2P∑

p=1

rp

)

pγ(r)dr (5.37)

where pγ(r) is the joint pdf of the combined SNR at the output and Q(·) is the Q

function defined as Q(y) = 1√2π

∫∞ye−t2/2dt , y ≥ 0.

In order to express the BER in terms of the signal vector r, an alternative

expression of Pb whose distribution is characterized by R, was given in [59] as

Pb =

∫

r

Q(√

2rr†)

p(r)dr. (5.38)

We now use the definition of the Q(·) function shown in [47]

Q(y) =1

π

∫ π/2

0

exp[

− y2

2 sin2 θ

]

dθ, (5.39)

to express Pb as

Pb = 1π

∫ π/2

0

∫

rexp[

− rr†

2 sin2 θ

]

×1

(2π)p/2(detR)0.5exp[

− 12r†R−1r

]

drdθ.

Using the simplifications shown in [59] by completing the squares inside the expo-

nential and using the matrix inversion lemma [23] an expression for the average

BER can be written as

Pb =1

π

∫ π/2

0

1[

det(

R

sin2 θ+ I)

]dθ. (5.40)


If the correlation matrix R has distinct, non-repeating eigenvalues λ1, λ2, . . . , λP

then we can write,

1[

det(

R

sin2 θ+ I)

] =P∑

p=1

ζp

( λp

sin2 θ+ 1)−1

(5.41)

where ζp is the pth residue of the partial-fraction expansion given by

ζp =∏

p6=q

( λp

λp − λq

)

. (5.42)

Therefore Pb becomes

Pb =1

π

∫ π/2

0

P∑

p=1

ζp

( λp

sin2 θ+ 1)−1

. (5.43)

The integral in (5.43) is evaluated using the solution shown in the appendix of [4]

In(c) =1

π

∫ π/2

0

( sin2 θ

sin2 θ + c

)m

dθ (5.44)

= [P (c)]mm−1∑

k=0

(

m− 1 + k

k

)

[1 − P (c)]k (5.45)

where

P (c) =1

2

[

1 −√

c

1 + c

]

. (5.46)

When m = 1, a closed form equation for the BER [59] can be written using the

above integral as

Pb =1

2

P∑

p=1

ζp

[

1 −√

λp

1 + λp

]

. (5.47)

Equation (5.47) is true when the eigenvalues are not repeated, i.e., the eigenvalues

are distinct. However, in cases where the covariance matrix may have repeated

eigenvalues, the result of (5.47) can be extended as shown below.

Consider a case where the covariance matrix has three eigenvalues, with one

repeated pair. Then we can write (5.41) as

5.5 Covariance matrices for different array configurations 75

1[

det(

R

sin2 θ+ I)

] = ζ3

( λ3

sin2 θ+ 1)−1

+ 2ζ2

( λ2

sin2 θ+ 1)−1

(5.48)

where λ1 = λ2 are the repeated eigenvalues and

ζ3 =

[

λ2

λ2 − λ3

]2

ζ2 =λ3

λ3 − λ2

.

Hence, Pb can be written as

Pb =1

π

∫ π/2

0

ζ3

( λ3

sin2 θ+ 1)−1

+ 2ζ2

( λ2

sin2 θ+ 1)−1

dθ. (5.49)

Again, using the result shown for (5.44), we get a closed form expression for Pb

with a pair of repeated eigenvalues

Pb =1

2ζ3

[

1 −√

λ3

1 + λ3

]

+ ζ2

[

1 −√

λ2

1 + λ2

]

. (5.50)

Similar closed form equations can be written for cases with additional non-distinct

eigenvalues to find Pb.

The eigen-decomposition method discussed in this section can be extended to other

forms of modulation schemes [59] for MRC. In [8] a similar approach was used to

find the symbol error rate for MPSK using MRC. However, analysis is mostly

carried out by assuming that the covariance matrix has distinct eigenvalues.

We now use two different array geometries, the uniform linear array and the uniform

circular array, to study the effect on BER of small antenna separations.

5.5 Covariance matrices for different array con-

figurations

In section 5.2 an equation for the normalized correlation coefficient for a 2 dimen-

sional scattering environment was found to be

ρpq =∞∑

m=−∞αmJm(k|zp − zq|)e−imϕpq (5.51)


0 2 4 6 8 10 12 14 16 18 2010−6

10−5

10−4

10−3

10−2

10−1

Average SNR in dB

Ave

rage

BE

R

D = λD = λ/2D = λ/5

Figure 5.3: Illustration of the effect of spatial correlation on the BER performanceof a dual diversity system in a 2 dimensional isotropic diffuse field with varyingULA aperture D.

where ϕpq is the angle of the vector connecting the two points zp and zq and αm

characterizes the power distribution of the given environment.

It is clear from (5.51), that the correlation coefficient ρpq depends on the sep-

aration distance between the antennas in the array and the power distribution of

the environment surrounding the antennas.

We now analyze the average BER at the output of the receiver for two different

array geometries using a 2 dimensional isotropic diffuse field, where the spatial

correlation coefficient is defined as in (5.12). Analysis is also carried out for the

uniform limited azimuth field where the energy arrives from a restricted range of

azimuthal angles (angular spread). The correlation coefficient for this case is given

by

ρpq =∞∑

m=−∞eim( π

2−φ0)sinc(m∆)Jm(k|zp − zq|) (5.52)

where ∆ is the angular spread and φ0 is mean AOA.


0 2 4 6 8 10 12 14 16 18 2010−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Average SNR in dB

Ave

rage

BE

R

D = λD = λ/2D = λ/5

Figure 5.4: Illustration of the effect of spatial correlation on the BER performancewhen 3 antennas are employed in a diffuse, isotropic field.

5.5.1 Uniform linear array

In a uniform linear array (ULA), with the first antenna placed at the origin, the

distance between any two antennas at points zp and zq is given by

d = zp − zq. (5.53)

In the case of a ULA, the elements of the covariance matrix R are of the form

ρpq = ρp+q−2 , where p, q = 1, 2, ...P . Therefore R can be written as a Toeplitz

matrix of size P × P .

The Toeplitz matrix (T) has the following form.

T =

t0 t1 t2 . . . tn

t1 t2 t3 . . . tn+1

t2 t3. . . . . . tn+2

... . . .. . .

tn tn+1 tn+2 . . . t2n

. (5.54)


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210−4

10−3

10−2

10−1

Aperture in λ

Ave

rage

BE

R

SNR = 5 dBSNR = 10 dBSNR = 15 dB

Figure 5.5: Illustration of the effect of increasing antenna array aperture on averageBER with a 2 antenna array for different average SNR values.

In a 2 dimensional isotropic diffuse field, since correlation across space can be

expressed as the zeroth order Bessel function, the elements of the covariance matrix

R are real and can be modelled as

Rpq = J0(kzp − zq) (5.55)

where J0(·) is the zeroth order Bessel function of the first kind and k is the wave

number. We consider every branch has equal average SNR such that Γp = Γq.

For the case where the azimuthal range is restricted, the elements of R are

modelled using (5.52).

We now discuss the plotted results derived from varying parameters associated

with (5.47).

Figures 5.3 and 5.4 show the variation of Pb with the average SNR per branch

when 2 and 3 antennas are used in the antenna array, respectively. The aperture

D of the ULA is varied in both cases. The plots obtained are in close agreement

with the simulation results in section 4.3.2, showing that when the antennas are


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210−5

10−4

10−3

10−2

10−1

Aperture in λ

Ave

rage

BE

R

2 antennas3 antennas4 antennas

Figure 5.6: Illustration of the effect of increasing array aperture at a constantaverage SNR of 10dB.

closely spaced, the effective correlation increases causing a degradation in the out-

put BER. We also observe by comparing Figures 5.3 and 5.4 that adding an extra

branch to the dual diversity system when the aperture is λ requires about 3 dB

lesser power. However at an aperture of λ/5 adding an extra branch cause a 2 dB

loss to a dual diversity system.

The effect of varying the separation between two antennas at a constant average

SNR is shown in Figure 5.5. The plots show the improvement of BER when the

SNR is increased from 5 dB to 15 dB. We note that the BER shows significant im-

provement when the antennas are separated further than λ/3. After this distance

the BER obtained is almost constant irrespective of the actual separation. When

the SNR is kept constant at 10 dB and an extra branch is added between the two

antennas, we see from Figure 5.6 that the BER improves. However, the optimal

performance of the three branch diversity system is obtained when the aperture is

increased to 0.8λ which is twice the aperture required to get the maximum advan-

tage from a dual diversity system. A similar performance result is obtained when

four branches are employed. These results strongly indicate that careful considera-


0 2 4 6 8 10 12 14 16 18 2010−5

10−4

10−3

10−2

10−1

Average SNR in dB

Ave

rage

BE

R

D=λ/2,∆=20D=λ/2,∆=5D=λ/5,∆=20D=λ/5,∆=5

Figure 5.7: Illustration of the effect of spatial correlation on the BER performancefor a ULA with 2 antennas angular spreads of 20 and 5 and array apertures ofD = λ/2 and λ/5.

tion needs to be employed when choosing the number of antennas for the required

performance of a system. Diversity does provide performance improvements in the

presence of correlation, however the optimal advantage of the system is derived

once the branches are effectively decorrelated.

It is interesting to note from Figure 5.6 that the average BER is a better per-

formance indicator than the average SNR at the output. When d = 0, such that

the antennas are stacked one above the other, the system has similar BER per-

formance with diversity, however the average SNR shows significant improvement

with each additional branch. This is because, averaging over time often leads to

ignoring instances when no signal component was received at a particular branch

but the noise still contributes to the system performance. The BER which is the

percentage of received bits in error will suffer even when diversity is employed due

to severe fading at a particular branch.

Plots of the results of varying the angular spread and the separation distance


0 2 4 6 8 10 12 14 16 18 2010−7

10−6

10−5

10−4

10−3

10−2

10−1

Average SNR in dB

Ave

rage

BE

R

D=λ/2,∆=30D=λ/2,∆=10D=λ/5,∆=10

Figure 5.8: BER performance when 3 antennas are used in a ULA with the energyarriving within a beamwidth of 30 and 10 when the aperture is λ/2. Also shownthe performance degradation when the aperture is reduced to λ/5.

when 2 and 3 antennas are employed at the receiver are shown in Figures 5.7 and

5.8, respectively. It is interesting to note that the performance of the dual diversity

system in Figure 5.7 decreased slightly when the angular spread was decreased with

the aperture kept at λ/2. However, a 1 dB loss was observed at a BER of 10−3

when the angular spread was reduced to the same extent for an antenna separation

of λ/5. This indicates that for a smaller angular spread the separation between the

antennas has to be increased in order to decrease correlation [34] [55] [22] which

adversely affects the system performance. This effect is highlighted in Figure 5.8,

where we observe that when the angular spread is 10 the performance is severely

degraded.

5.5.2 Uniform circular array

We now carry out analysis of the BER performance when a uniform circular array

(UCA) with radius R is employed at the receiver.


0 2 4 6 8 10 12 14 16 18 2010−6

10−5

10−4

10−3

10−2

10−1

Average SNR in dB

Ave

rage

BE

R

R = λR = λ/5R = λ/10

Figure 5.9: Illustration of the effect of spatial correlation on the BER performancefor a uniform circular array in a diffused isotropic field when the radius R is de-creased.

The distance between any two antennas in a circular array can be written as

dp = 2R sin(πp/P ) (5.56)

where P is the total number of antennas in the array and p ∈ 1, 2, ...., P .

Due to the circular symmetry in a UCA, the elements of R show circular sym-

metry such that ρp = ρP−p. Hence, R becomes a P×P symmetric circulant matrix.

A circulant matrix is defined as

A =

a1 a2 a3 . . . an

an a1 a2 . . . an−1

... . . .. . .

a2 a3 . . . an a1

. (5.57)

The eigenvalues of a circulant symmetric matrix in a 2 dimensional isotropic

diffuse field are real and symmetric [41] and, hence, can be expressed by the closed


0 2 4 6 8 10 12 14 16 18 2010−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Average branch SNR in dB

Ave

rage

BE

R

R=λR=λ/2R=λ/5

Figure 5.10: Illustration of the effect of spatial correlation on the BER performancefor a uniform circular array in a diffuse isotropic field with 3 receiver antennas.

form expression

λp =P−1∑

m=0

ρp cos(2πmp/P ). (5.58)

We can now use (5.58) in (5.47) to explicitly calculate the BER for a UCA with ρp

given by (5.12).

In the case of circulant matrix, we have λp = λP−p. Therefore (5.47) holds only

for the case of a dual diversity system, with the antennas placed on the end points

of the diameter of the circular array. Thus in a circular array with dual diversity

the BER is explicitly given by

Pb =1

2

(

ζ1

[

1 −√

λ1

1 + λ1

]

+ ζ2

[

1 −√

λ2

1 + λ2

])

(5.59)

where λ1 and λ2 are calculated using (5.58), while the residues ζ1 and ζ2 are calcu-

lated using (5.42). Equation (5.59) provides a closed form equation for calculating


0 2 4 6 8 10 12 14 16 18 2010−6

10−5

10−4

10−3

10−2

10−1

R=λ/5,∆=20R=λ/10,∆=20R=λ/10,∆=5

Figure 5.11: Illustration of the effect of spatial correlation on the BER performanceof a dual diversity UCA with angular spreads of 20 and 5 and radii of R = λ/5and λ/10.

Pb if the array is circular where the eigenvalues can be easily calculated using

(5.58). We use this equation to plot the performance of a dual diversity MRC

system with a circular array configuration.

Figure 5.9 illustrates the average error probability when 2 antennas are em-

ployed in a circular array of radius R. It is noted from Figures 5.3 and 5.9 that for

a given aperture the average BER performance of a UCA is better than n ULA,

which can be attributed to the greater separation between antennas in the UCA,

when compared to the ULA.

To evaluate the performance of the UCA when more than 2 antennas are em-

ployed we find certain closed form solutions using the symmetric properties of the

circulant matrix. Depending on whether the array has odd or even number of

antennas closed form equations for Pb for the two cases can be found.


General equation for the average BER for UCA

For a circular symmetric array with P antennas we observe that, when P is even

there are 2 distinct eigenvalues and P2− 1 pairs of repeated eigenvalues and when

P is odd there is 1 distinct eigenvalue and P−12

repeated eigenvalue pairs.

The average error probability for a UCA with P antennas can therefore be

generalized from (5.47) and written in closed form as

Pb =1

2

(

ζP

[

1 −√

λP

1 + λP

]

+ ζP/2

[

1 −√

λP/2

1 + λP/2

])

+P−1∑

p= P2

+1

ζp

[

1 −√

λp

1 + λp

]

(5.60)

when P is even and

Pb =1

2ζP

[

1 −√

λP

1 + λP

]

+P−1∑

p= P+12

ζp

[

1 −√

λp

1 + λp

]

(5.61)

when P is odd.

Thus, we have expressions for Pb in the case of a UCA with an arbitrary number

of antennas and repeating eigenvalues. Using these results the performance of a

multiple antenna system arranged on the circumference of a circle with radius R

can be easily determined for comparison with other array structures.

We plot (5.61) in Figure 5.10 with P = 3. This plot provides us with insights

into the effect of adding an extra branch in a UCA. As expected, a UCA provides

better performance than a ULA with the same number of antennas. This is at-

tributed to the fact that where the radius R of a UCA is the same as the antenna

separation d for a ULA, the antennas in the UCA are actually separated by a

greater distance providing effective decorrelation between the antennas.

Figure 5.11 is a plot of the average BER at the output of a dual diversity cir-

cular array when the field is uniformly limited in the azimuth. As expected the

UCA with a radius of λ/5 to performs better than the ULA with the same aperture

when the angular spread is reduced.

From the plots obtained for the various scenarios we infer that a UCA of a given

radius (aperture) has better performance than a ULA of the same aperture which


agrees with our simulation results from Chapter 4 . Therefore, the UCA can be

thought of as a better configuration in terms of efficient utilization of space and

reducing the effective correlation between antennas.

5.6 Summary

The principle objective of this chapter was to highlight the effect of correlation on

the performance of antenna arrays employing diversity techniques using analytical

techniques. The correlation coefficient was defined in terms of the signal represen-

tation using the spatial channel model. It was seen that the correlation coefficient

can be expressed in a closed form for various scattering distributions.

The average SNR at the output using MRC and EGC was evaluated. A lower

bound for the average SNR using Jensen’s inequality was found. The multivariate

distribution was used such that the spatial covariance matrix could be utilized

to find closed form equations for the average bit error rates for MRC using the

BPSK modulation technique. We analyzed the case of a uniform linear array

where the eigenvalues may not be distinct. By utilizing the expression for the

eigenvalues of a uniform circular array we found expressions to analyze the BER

for this configuration. The theoretical results obtained in this chapter showed close

similarities with the simulation results that we obtained in the previous chapter.

Chapter 6

Conclusion

We present the summary of our main results and scope for further work that can

be undertaken in order to extend the work in this thesis.

6.1 Summary of main results

The main motivation of this thesis was to bring out the importance of the relative

positions of the antennas in antenna arrays, on the performance of spatial diversity

system. The two popular diversity combining schemes, MRC and EGC were stud-

ied to evaluate the systems output SNR and BER performance in the presence of

correlation due to closely spaced sensors.

We used the SIMO case of a recently developed spatial channel model to study

the effect of correlation between closely spaced sensors when diversity schemes

were employed. Through simulations we compared the performance of the MRC

and EGC antenna diversity combining schemes for a ULA and UCA when sub-

jected correlation due to constraints in space, with the independent Rayleigh fading

model. It was seen from our results that correlation increases when antenna spacing

is reduced. The output SNR gain was seen to be upper bounded by the indepen-

dent fading case and lower bounded by the case where correlation was maximized

by having no spacing between antennas. We also inferred that in the presence of

high correlation between elements, the optimality of MRC is lost and it performs

the same way as EGC. Through our simulations we also saw that the output BER

was a better performance measure when compared to the output average SNR.

This is because the diversity gain, which increases linearly by adding additional

antennas does not dominate the system performance when BER is considered.

87

88 Conclusion

The BER for MRC using BPSK as the modulation technique was examined

analytically in Chapter 5. We used a covariance matrix which characterized the

correlation between elements in the array to study the effect of correlation on the

BER. Closed form equations for the UCA and ULA were found for the output

average BER. The results obtained using simulations showed close resemblance to

the plots obtained using the closed form equations for the BER, proving that spatial

correlation based on separation distances does affect the diversity gain of a system

adversely. Our results both from simulations and theoretical analysis showed that

for the same radius or aperture, the UCA performed better than the ULA due to

the fact that the circular topology allows greater separation between neighboring

elements, reducing correlation between the elements. We also saw that the effective

decorrelation distance decreased with wider angular spreads when the arrays were

subjected to a limited azimuthal field.

6.2 Future work

In practical situations it is rarely possible to achieve complete statistical indepen-

dence of all branches when a diversity receiver is employed. Hence, a detailed

study of the factors contributing to the correlation between the signals at different

array elements is needed in order to accurately predict receiver performance at the

receiver. We consider potential areas for future research arising from the work in

this thesis

• The effect of non-isotropic scattering on the system performance due to dif-

ferent angular spread and separation distances can provide more realistic in-

sights into correlation effects on the performance of diversity systems which

can be studied in detail. Several non-isotropic scattering models were con-

sidered in [55] which would serve as a good starting point for such study.

• We have not considered the frequency selective nature of channels in this

thesis. Inclusion of the effects of frequency selectivity will help in completely

characterizing spatial correlation in the mobile wireless channel.

• Gain combining schemes such as MRC require perfect channel knowledge.

This is impractical. Tailoring of such schemes to provide maximum benefits

in light of insights gained in this thesis into the effects of spatial separation

of the receiver performance would be of sue in practical systems.

6.2 Future work 89

• Using the closed form equation for the eigenvalues of the UCA, theoretical

analysis of the BER for different modulation schemes other than BPSK in

the presence of correlated fading can be found.

• It would be useful to find bounds on the number of antennas that should

be employed to get maximum benefits in a confined region of space. This is

related to the inherent dimensionality of a region of space [26].

• Further, this work can be used to find the most efficient antenna configura-

tions for combinations of given modulation schemes, performance measures

and spatially restricted receivers.

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