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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 Engineering
Research 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. Abhayapala 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 arrays 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 irrelevant, 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 techniques 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 performance 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 techniques
9
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Wireless signal propagation . . . . . . . . . . . . . . . . . . . . . .
9
2.3
2.4
2.5
2.6
2.2.1
Propagation mechanisms . . . . . . . . . . . . . . . . . . . .
10
2.2.2
Pathloss due to large scale fading . . . . . . . . . . . . . . .
11
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
Fading models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.4.1
Statistical modelling of independently faded signal envelopes
15
2.4.2
Small scale fading manifestations . . . . . . . . . . . . . . .
18
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
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
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
3.4
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4.2
Spatial channel model . . . . . . . . . . . . . . . . . . . . . . . . .
44
4.3
4.2.1
The SIMO model . . . . . . . . . . . . . . . . . . . . . . . .
45
4.2.2
Analyzing the channel matrix . . . . . . . . . . . . . . . . .
46
4.2.3
Comments on the model . . . . . . . . . . . . . . . . . . . .
48
Effect of introducing ‘space’ into diversity systems . . . . . . . . . .
49
4.3.1
SNR performance of MRC and EGC with finite antenna separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
Performance of BER using the spatial model . . . . . . . . .
58
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.3.2
4.4
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
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
5.6
6 Conclusion
6.1
Summary of main results . . . . . . . . . . . . . . . . . . . . . . . .
87
87
Contents
ix
6.2
88
Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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
Illustration of the beamforming lobes at the switched beam array
showing sectors from where the users signals are detected. . . . . .
5
The effect of increasing transmitter and receiver separation d, for
various pathloss exponents, n, in (2.2). . . . . . . . . . . . . . . . .
12
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 device 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
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
1.2
2.1
2.2
3.2
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(ϕ̂) represents the gain of the
complex scattering environment for signals arriving at the receiver
scatter-free region from direction ϕ̂ from a single transmitter. . . .
44
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
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
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
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
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
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
Illustration of the effect of increasing the separation distance between 2 antennas in steps of λ/4. . . . . . . . . . . . . . . . . . . .
55
Comparison of BER performance for the independent Rayleigh fading 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
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
List of Figures
Comparison of spatial correlation versus separation for a diffuse scattering field, limited in the azimuth with the source centered around
π/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Illustration of the effective SNR when calculated using the lower
bound in equation (5.29). . . . . . . . . . . . . . . . . . . . . . . .
5.3 Illustration of the effect of spatial correlation on the BER performance of a dual diversity system in a 2 dimensional isotropic diffuse
field with varying ULA aperture D. . . . . . . . . . . . . . . . . . .
5.4 Illustration of the effect of spatial correlation on the BER performance when 3 antennas are employed in a diffuse, isotropic field. . .
5.5 Illustration of the effect of increasing antenna array aperture on
average BER with a 2 antenna array for different average SNR values.
5.6 Illustration of the effect of increasing array aperture at a constant
average SNR of 10dB. . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Illustration of the effect of spatial correlation on the BER performance for a ULA with 2 antennas angular spreads of 20◦ and 5◦ and
array apertures of D = λ/2 and λ/5. . . . . . . . . . . . . . . . . .
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. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Illustration of the effect of spatial correlation on the BER performance for a uniform circular array in a diffused isotropic field when
the radius R is decreased. . . . . . . . . . . . . . . . . . . . . . . .
5.10 Illustration of the effect of spatial correlation on the BER performance for a uniform circular array in a diffuse isotropic field with 3
receiver antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11 Illustration of the effect of spatial correlation on the BER performance of a dual diversity UCA with angular spreads of 20◦ and 5◦
and radii of R = λ/5 and λ/10. . . . . . . . . . . . . . . . . . . . .
xiii
5.1
66
69
76
77
78
79
80
81
82
83
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 reality 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
Base
Station
Figure 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.
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 characterizing the hostile channel. Unlike wired or fixed radio link transmission the mobile 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 mobility 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 mechanisms including reflection, refraction, diffraction and/or scattering due to atmospheric 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 relative 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 motion 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 transmitted 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 mobile 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 performance. By employing or exploiting diversity in space, time, frequency or polarization, 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 output). 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 appropriate to the application of interest. The most commonly used geometries are linear
1.2 Antenna arrays
5
desired signal
1
2
3
desired signal
4
5
Figure 1.2: Illustration of the beamforming lobes at the switched beam array showing 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 processors, 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 distance 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 spacing 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 independently faded signals which improves system performance while providing a substantial 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
1
Isotropic scattering implies that the signal power arrives at the receiver with equal power
from all directions i.e., it is uniformly distributed from all directions due to the scatterers being
randomly 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 lobe2 with 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 sophisticated 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 division 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 surveying 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 improve linearly with the number of antennas when the antenna aperture is
restricted?
2
The 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 envelopes 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 positions 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 significant. 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 multiple receive antennas utilizing diversity techniques are given in this thesis. A
recently developed spatial channel model [42] for MIMO systems which incorporates 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 performance 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). Theoretical 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 analyze system performance criteria are described. An overview of the various diversity
schemes employed in MIMO systems and their effect on system performance is included in this chapter with an emphasis on spatial diversity techniques.
In Chapter 3 a literature review of the effect of correlated signals at the output 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 simulation 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 propagating 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 encountered scatterers during signal propagation are lamp posts, street signs and
foliage.
The resultant signal strength is usually a combination of the above phenomena [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 Lf s (d), assuming an isotropic1
antenna at the receiver, is expressed as
Lf s (d) =
h 4πd i2
(2.1)
λ
where d is the distance between the transmitter and the receiver and λ is the wavelength 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 ineffective 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 amplitude 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) = Lf s (d0 )(dB) + 10n log
1
hdi
d0
+ Xσ (dB)
a theoretical antenna which transmits/receives equally in all directions.
(2.2)
12
115
110
105
Pathloss in dB
100
95
90
85
80
75
70
0
10
n=2
n=3
n=4
1
Separation between the transmitter and receiver in Kilometers
10
Figure 2.1: The effect of increasing transmitter and receiver separation d, for various pathloss exponents, n, in (2.2).
where d0 2 is the received power reference point such that d > d0 , n is the pathloss 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.
2
d0 is a point located in the far-field of the transmit antenna
2.3 Stochastic channel modelling
2.3.1
13
Impulse response characterization of a multipath channel
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) =
Z
∞
−∞
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) =
Z
∞
−∞
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
X
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 component 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 random variable and δ(·) is the unit impulse function which determines the specific
multipath component at time τ and an excess delay of τn .
14
2.3.2
Stochastic channel characterization via the autocorrelation 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. However, in the study of wireless communication, the channel is generally assumed to
be wide sense stationary (WSS) [48]. Under the assumption of WSS, the autocorrelation 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 autocorrelation 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
X
i=1
ai cos(ωc t + 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 ωdi is the Doppler shifted angular frequency of the
ith path, given by
ωc v
cos(αi )
(2.10)
c
where c is the velocity of the propagating wave and v is the velocity of the mobile
device.
ω di =
We see from (2.9) that the received signal has random variations in both amplitude 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(ωc t) + Q(t) cos(ωc t)
(2.11)
where I(t) and Q(t) are the in-phase and quadrature components of the signal
given by
I(t) =
N
X
ai cos(ωdi + φi )
(2.12)
N
X
ai sin(ωdi + φi ).
(2.13)
i=1
and
Q(t) =
i=1
The envelope of the received signal r(t) can therefore be written as
r=
p
[I(t)]2 + [Q(t)]2
where r = |r(t)| and | · | is the absolute value operator.
(2.14)
16
Rayleigh faded signal for a stationary device
2
Rayleigh faded signal envelope for a stationary device
2
1.5
1
envelope(volts)
rf signal(volts)
1.5
0.5
0
−0.5
1
0.5
−1
−1.5
0
20
time(ms)
40
Rayleigh faded signal envelope
for a mobile device moving at 25 m/s
2
2.5
0
20
time(ms)
40
60
Rayleigh faded signal envelope
for a mobile device moving at 50 m/s
2
envelope(volts)
1.5
envelope(volts)
0
60
1.5
1
0.5
0
1
0.5
0
20
time(ms)
40
60
0
0
20
time(ms)
40
60
Figure 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 device indicating greater chance
of 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) =





2
r
exp[ −r
]
σ2
2σ 2
for r ≥ 0
0
(2.15)
otherwise.
In (2.15 ) the term 2σ 2 denotes the predetection mean power of the multipath signal.
The phase of the fading signal is defined as
φ = tan−1
h Q(t) i
I(t)
(2.16)
2.4 Fading models
17
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

h i
rA
r
−r 2 +A2

exp[
]I
for r ≥ 0 and A ≥ 0

o σ2
2σ 2
 σ2
(2.18)
p(r) =


 0
otherwise
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
.
2σ 2
(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 for m = 1 the Nakagami distribution reduces to the Rayleigh
distribution function. The pdf of the Nakagami distribution is
p(r) =




2
Γ(m)


 0
m
m
Ω
r2m−1 exp
h
−mr 2
Ω
i
for r ≥ 0 and m ≥ 1/2
otherwise
(2.20)
18
where Ω = E[r 2 ] is the second moment of the random variable, m =
Γ(·) is the gamma function3 .
Ω2
E[(r 2 −Ω)2 ]
and
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 differently 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
R∞
The gamma function is defined as Γ(x) = 0 tx−1 e−t dt, x > 0.
4
The frequency separation for which the signals are still strongly correlated is called the
coherence bandwidth.
3
2.4 Fading models
19
Small scale fading
Time variance
of the channel
Time spreading
of the signal
Time−delay
domain
Frequency
selective
fading
Time domain
Frequency
domain
Fast
fading
Flat
fading
Doppler−shift
domain
Slow
fading
Fast
fading
Frequency
selective
fading
Slow
fading
Flat
fading
indicates Duals
indicates Fourier Transforms
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.
5
The time duration over which two received signals have strong amplitude correlation is defined
as the coherence time.
20
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 symbol 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 diversity techniques at the receiver to reduce SNR losses. Though the system complexity 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 performance [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 = w 1 r1 + w 2 r2 + . . . + w n rn + . . . + w N rN =
N
X
wn rn .
(2.21)
n=1
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.
22
a)
narrow band channels
coherence
bandwidth
f
b)
time slots
coherence time
t
c)
sensors
decorrelation distance
combining logic
Figure 2.4: Illustration of a) frequency diversity, b) time diversity and c) space
diversity.
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 attention 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 receiver or both. In these systems the multiple replicas of the transmitted signal are
combined using various techniques from antennas separated by at least their decorrelation distance 6 . The different spatial diversity schemes and their performance
benefits are discussed later in this chapter.
6
The minimum separation distance for which the signals are independent or not correlated is
the decorrelation distance.
1
23
2
p
P
monitor and control
logic to select the best
signal branch
selected signal to the
demodulator
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 diversity 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.
24
2.5.2
Spatial diversity combining techniques
Based on the combining scheme employed at the receiver, spatial diversity techniques 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 special “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 combining 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. Implementation of MRC is cumbersome due to the additional circuitry required in
25
Receiver Elements
1
2
p
P
wp
wP
z
w1
w2
Combining Logic
Combiner output
Figure 2.6: Illustration of general gain combining diversity schemes where
w1 , w2 , ..wP are the weights added to the respective branches.
order to measure the channel gains. If combining is performed using MRC at predetection, 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 individual 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 σn2 , and is Rayleigh distributed, then its pdf is given by
26
2
p(rn ) =
rn − 2σrn2
e n.
σn2
(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 distribution [50] and its pdf p(γn ) is given by
p(γn ) =
1 Γγn
e n
Γn
(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
γ
F (γ) = P (γ1 , ...., γN ≤ γ) = [1 − e− Γ ]N .
(2.24)
Differentiating the above equation with respect to γ gives the pdf for the output
SNR for SC [19] as
i
γ N −1
N −γ h
e Γ 1 − e− Γ
.
(2.25)
Γ
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
p(γ) =
Pe =
Z
∞
P(γ)p(γ)dγ
(2.26)
0
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 ) =
γ N −1 γ
1
eΓ .
(N − 1)! ΓN
(2.27)
27
The BER for ideal coherent binary phase shift keying (BPSK) is given by [43]
1
√
PBPSK = erfc( γ).
2
(2.28)
Using (2.27) and (2.28) in (2.26), the probability of bit error can be calculated
as [50]
−1 1 − ν N N
X
N − 1 + n 1 + ν n
PBPSK =
(2.29)
2
n
2
n=0
where
ν=
r
Γ
.
1+Γ
(2.30)
Similar calculations are used to calculate error rates using different modulation 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 Combining (H-S/EGC) or Hybrid Selection/Maximal Ratio Combining (H-S/MRC),
28
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 HS/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 nonoptimal combining.
2.6
Summary
In this chapter the fundamental fading mechanisms encountered in wireless channel 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. Diversity 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 complete 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
10
9
8
10 log<γ>/Γ dB
7
6
5
4
3
2
MRC
EGC
SC
1
0
1
2
3
4
5
6
Number of antennas
7
8
9
10
Figure 3.1: Comparison of improvement in diversity gain using MRC, EGC and SC
for increasing number of antennas in an independent Rayleigh fading environment.
PSC =
N
X
1
k=1
PMRC = N,
k
,
π
PEGC = 1 + (N − 1) ,
4
N
X
1
P(H-S/MRC) = 1 +
.
i
i=l+1
(3.1)
(3.2)
(3.3)
(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 performance 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 distributed. This can be caused due to factors such as different fading statistics at
different beams of a multiple beamformer or varied shadowing effects at different 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 probability [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 fading 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 combining 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 techniques [24] [46] [31] [34] using the joint statistics of the Rayleigh/Rician/Nakagamim distributions.
The correlation coefficient
The power correlation coefficient ρ of two Rayleigh distributed random variables
r1 and r2 with variances of σr21 and σr22 , respectively, [12] is given by
h
i
E r12 − E[r12 ] r22 − E[r22 ]
.
ρ= r h
i h 2
i
2
2 2
2 2
E r1 − E[r1 ] E r2 − E[r2 ]
(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
1
The diversity gain is defined as the reduction in the required average SNR for a given BER.
32
from the fact that measurement data is expressed in terms of the envelope correlation while most analytical work is performed using the power correlation coefficient.
The envelope correlation between two random signals r1 and r2 is defined as
h
i
E r1 − E[r1 ] r2 − E[r2 ]
ρe = r h
.
2 i h
2 i
E r1 − E[r1 ] E r2 − E[r2 ]
(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]
(1 +
ρe =
√
2
2ρ1/4
√
1+ ρ
− π2
ρ)ε
−
π
2
(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 channel.
For correlated random signals the bivariate Rayleigh distribution, which is a special case of the Nakagami-m distribution [38], gives the joint pdf in terms of the
correlation coefficient ρ as
2√ρr r
Ω r2 + Ω r2 4r1 r2
1 2
1 2
2 1
p(r1 , r2 |ρ) =
× I0 √
exp −
Ω1 Ω2 (1 − ρ)
Ω1 Ω2 (1 − ρ)
Ω1 Ω2 (1 − ρ)
(3.8)
33
where r1 ≥ 0, r2 ≥ 0, Ω1 = E[r12 ] , Ω2 = E[r22 ] 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 received signal, can be written using the infinite series representation of the bivariate
pdf [53] as
F (r1 , r2 |ρ) = (1 − ρ)
where Υ(α, x) =
Rx
0
∞
X
k=0
ρk Υ k + 1,
r22
r12
Υ k + 1,
Ω1 (1 − ρ)
Ω2 (1 − ρ)
(3.9)
tα−1 e−t dt, 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
where
p(m) = p
me
2 +σ 2 )
−m2 (σ1
2
2 σ 2 (1−ρ)
4σ1
2
(σ22 − σ12 )2 + 4ρσ12 σ22
m=
and
κ=
m2
p
h
κ
× e −e
q
r12 + r22
(σ22 − σ12 )2 + 4ρσ12 σ22
.
4(1 − ρ)σ12 σ22
−κ
i
(3.10)
(3.11)
(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
34
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
1
T= √
2
"
1 1
−1 1
and
#
(3.14)
#
.
(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 independent branches as
p(x) =
where x = r12 + r22 .
x
x
1 − Γ(1+ρ)
e
− e− Γ(1−ρ)
2Γρ
(3.18)
The BER for the same scheme using BPSK is represented as
"
1
Pb =
1−
2
s
(1 + ρ)Γ
−
(1 + ρ)Γ + 1
s
(1 − ρ)Γ
+
(1 − ρ)Γ + 1
s
#
(1 − ρ2 )Γ2
.
(1 − ρ2 )Γ2 + 2Γ
(3.19)
Using the above expressions it was shown in [16] that the output BER severely degrades 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
35
maintain a BER of 10−3 . The theoretical methods presented above provide expressions 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 different 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 derived 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 limited 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 evaluated. The transformation was carried out under the assumption that the individual
branch SNRs were indexed in increasing order of the Nakagami m parameter. Under 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
36
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[rn rm
]
n, m = 1, 2, ..., N.
(3.20)
For realistic situations the entries of the correlation matrix must include parameters 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
R
ψ
θ
d
Figure 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.
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
2
for a given arrangement of antennas, the broadside is perpendicular to the line or axis of the
array.
38
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 =
∞
X
π
ψ
eim( 2 −θ) sinc(m )Jm (kd)
2
m=−∞
(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 circular 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
Rh (4x)
ρ(4x) = p
.
∗
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 coefficient 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.
40
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 Laplacian 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 diversity 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 systems. 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 increased 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 incorporates 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
Scatterers
^
ϕ
A( ^
ϕ)
rR
rRS
x
Transmitter
Receivers
Figure 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(ϕ̂) represents the gain of the complex scattering environment for
signals arriving at the receiver scatter-free region from direction ϕ̂ from a single
transmitter.
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
4.2.1
45
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 signal during a signaling interval from the transmitter, and let r = [r1 , r2 , . . . , rp ]0 be
the vector of the received signals at the P receivers where [·]0 denotes transpose.
The signal at the pth receiver can be written as
rp =
Z
A(ϕ̂)ue−ikzp ·ϕ̂ dΩ(ϕ̂) + np
(4.1)
Ω
where A(ϕ̂) is the complex gain of the signal entering the receiver scatterer-free
ball from direction ϕ̂, 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 ·ϕ̂ 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(ϕ̂) 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 ]0 and h is the length P channel vector with pth element
hp =
Z
A(ϕ̂)e−ikzp .ϕ̂ dΩ(ϕ̂).
(4.3)
Ω
In order to gain a deeper understanding of the structure of the channel vector, in
1
The Rayleigh distance [35] gives the approximation for the farfield distance from the array
origin as d = 2l2 /λ, where l is the array dimension and λ is the transmitted signal wavelength.
46
J0(z)
1
0.5
0
−0.5
0
10
1
10
0.4
Argument z
2
10
2
10
2
10
10
3
J10(z)
0.2
0
−0.2
−0.4
0
10
1
10
0.2
Argument z
10
3
J100(z)
0.1
0
−0.1
−0.2
0
10
1
10
Argument z
10
3
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 ·ϕ̂ as
e
−ikz·ϕ̂
=
∞
X
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
ϕ̂ = (1, ϕ) in the polar coordinate system.
2
This models the situation in 3 dimensional space where the multipath propagation is restricted
to the horizontal plane, having no component arriving at significant elevations. It is assumed here
that the signal field is height invariant.
3
Note that Jm (·) = (−1)m Jm (·)
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.ϕ̂
MR
X
=
[Jm (k|z|)e−im(ϕz −π/2) ]eimϕ
(4.5)
m=−MR
where
MR = dkerR /2e + 4
(4.6)
with a relative truncation error
M (z) ,
M
X
−ikz.ϕ̂
−
[Jm (k|z|)e−im(ϕz −π/2) ]eimϕ e
m=−M
≤ 0.16127e−4
|e−ikz.ϕ̂ |
(4.7)
(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 M (z) ≤ 0.0593 and when 4 = 2
the truncation error is M (z) ≤ 0.0218.
Therefore using (4.1) and (4.5) the signal received at the pth receiver can be
written as
MR
X
Jm (k|zp |)e−im(ϕp −π/2) hm u + np
(4.9)
rp =
m=−MR
where
hm =
Z
A(ϕ̂)e−imϕ d(ϕ).
(4.10)
Ω
Now the channel vector h in (4.9) can be expressed as
h = J R hs
where JR is referred to as the receiver configuration matrix, given by
(4.11)
48

and

π
π
J−MR (kz1 )eiMR (ϕ1 − 2 ) . . . JMR (kz1 )e−iMR (ϕ1 − 2 )


..
...

JR = 
.


iMR (ϕp − π2 )
−iMR (ϕp − π2 )
J−MR (kzp )e
. . . JMR (kzp )e
hs = [h−M , ...., h0 , ...., hM ]0 .
(4.12)
(4.13)
Since the scattering function A(ϕ) is periodic in ϕ̂ we can expand it using the 2
dimensional Fourier series. Therefore A(ϕ̂) can be written as
A(ϕ) =
∞
X
hm e−imϕ .
(4.14)
m=−∞
We also normalize the scattering gain function by assuming
Z
2π
E[|A(ϕ)|2 ]dϕ = 1
(4.15)
0
By substituting (4.14) into (4.15) we obtain
∞
X
m=−∞
4.2.3
E[|hm |2 ] =
1
.
2π
(4.16)
Comments on the model
The spatial channel has been decomposed into two regions, namely, the scattererfree 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 configuration 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 Figure 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 antennas 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 independent 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
50
38
36
Average SNR in dB
34
32
30
28
26
MRC Rayleigh
EGC Rayleigh
MRC d= λ/2
EGC d= λ/2
d=0
24
22
1
2
3
4
5
6
Number of antennas
7
8
9
10
Figure 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.
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 following equations.
The SNR for the MRC combining scheme was calculated using
γMRC =
1
2
P
X
2
wp h p p=1
P
X
p=1
Np
(4.17)
51
38
36
Average SNR in dB
34
32
30
28
d=λ
d = λ/2
d = λ/10
d=0
26
24
22
1
2
3
4
5
6
Number of antennas
7
8
9
10
Figure 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.
where Np is the noise variance and wp is the weight of the pth branch
wp =
h∗p
.
Np
(4.18)
The SNR for the EGC combining scheme was calculated using
γEGC =
1
2
P i
hX
2
h p p=1
P
X
.
(4.19)
Np
p=1
The average SNR of the two schemes was calculated as the ensemble average of the
instantaneous SNRs
ΓMRC = E[γMRC ]
(4.20)
52
38
36
Average SNR in dB
34
32
30
28
26
MRC Rayleigh
EGC Rayleigh
MRC D = λ
EGC D = λ
D=0
24
22
1
2
3
4
5
6
Number of antennas
7
8
9
10
Figure 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.
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.
53
38
36
Average SNR in dB
34
32
30
28
26
MRC D = λ/2
EGC D = λ/2
MRC D = λ/10
EGC D = λ/10
D=0
24
22
1
2
3
4
5
6
Number of antennas
7
8
9
10
Figure 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.
4.3.1
SNR performance of MRC and EGC with finite antenna 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 linear 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 constant 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 independently. A separation of λ/2 is commonly referred to as the decorrelation
4
The region in space over which the energy is collected is called the aperture.
54
38
36
Average SNR in dB
34
32
30
28
26
MRC R = λ/2
EGC R = λ/2
MRC R = λ/10
EGC R = λ/10
R=0
24
22
1
2
3
4
5
6
Number of antennas
7
8
9
10
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 correlation 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 significant fall in the SNR performance. A loss of about 1 dB is seen in the average
24
55
MRC
EGC
22
SNR in db
20
18
16
14
12
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 5
Separation distance in λ
Figure 4.8: Illustration of the effect of increasing the separation distance between
2 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 antennas 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 independent, 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.
56
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 indication 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 employed 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
57
−2
10
1 antenna
2 antennas
3 antennas
5 antennas
−3
Average BER
10
−4
10
−5
10
−6
10
0
2
4
6
8
10
12
Average SNR in dB
14
16
18
20
Figure 4.9: Comparison of BER performance for the independent Rayleigh fading
case when the number of antennas is increased for MRC.
schemes for a UCA with constant radius are similar to those of a ULA with constant 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 observed 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 independent 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
58
−2
10
1 antenna
2 antennas
3 antennas
5 antennas
−3
Average BER
10
−4
10
−5
10
−6
10
−7
10
0
2
4
6
8
10
12
Average SNR in dB
14
16
18
20
Figure 4.10: BER performance of MRC when the number of antennas is increased
in 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 probability 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
59
−2
10
Independent fading
D = λ/2
D = λ/5
D=0
−3
Average BER
10
−4
10
−5
10
−6
10
0
2
4
6
8
10
12
Average SNR in dB
14
16
18
20
Figure 4.11: BER performance of a dual diversity MRC system with varying array
aperture 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 subjected 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 misleading 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-
60
−2
10
D=λ
D = λ/2
D = λ/5
D=0
−3
Average BER
10
−4
10
−5
10
−6
10
0
2
4
6
8
10
12
Average SNR in dB
14
16
18
20
Figure 4.12: BER performance of MRC with varying aperture in a ULA with 2
antennas.
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 independent 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 antennas 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 antennas 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 increasing 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
−2
10
R = λ/5 (2 antennas)
R = 0 (2 antennas)
−3
Average BER
10
−4
10
−5
10
−6
10
−7
10
0
2
4
6
8
10
12
Average SNR in dB
14
16
18
20
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-
62
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 randomness 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 increasing 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 elements 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 correlations 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 chapter. 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
5.2
Effects of spatial correlation on diversity receivers
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 distributed 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 between the complex envelopes of the signals rp and rq received at antennas at positions zp = (zp , ϕp ) and zq = (zq , ϕq ) can be written using (3.24) as
ρpq = q
E[rp rq∗ ]
.
(5.1)
E[rp rp∗ ]E[rq rq∗ ]
Using (4.1) the covariance between the two signals rp and rq is given by
E[rp rq∗ ]
=
σu2
Z Z
0
E[A(ϕ̂)A∗ (ϕ̂0 )]e−ikzp ·ϕ̂ eikzq ·ϕ̂ dϕ̂dϕ̂0
(5.2)
where σu2 = E[|u|2 ].
Assuming independence between signals entering the scatterer free ball from
different directions, we have
E[A(ϕ̂)A∗ (ϕ̂0 )] =
(
E[| A(ϕ̂) |2 ] if ϕ̂ = ϕ̂0
0
otherwise.
(5.3)
We define the angular power distribution as
P (ϕ̂) , R
E[| A(ϕ̂) |2 ]
.
E[| A(ϕ̂) |2 ]dϕ̂
(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 (ϕ̂) = E[| A(ϕ̂) |2 ].
(5.5)
Thus, using (5.3) and (5.5), we get a simplified equation for the covariance in (5.2)
E[rp rq∗ ]
=
σu2
Z
P (ϕ̂)e−ik(zp −zq )·ϕ̂ dϕ̂.
(5.6)
Similarly, we can express the denominator terms of the correlation coefficient in
(5.1) as
Z
∗
∗
2
E[rp rp ] = E[rq rq ] = σu P (ϕ̂)dϕ̂.
(5.7)
Now, substituting (5.6) and (5.7) into (5.1), we obtain
ρpq =
Z
P (ϕ̂)e−ik(zp −zq )·ϕ̂ dϕ̂
(5.8)
Now, in [55], the angular power distribution P (ϕ̂) is expressed, using a Fourier
series expansion, as
∞
X
αn einϕ
(5.9)
P (ϕ) =
n=−∞
where
1
αn =
2π
Z
2π
P (ϕ)einϕ dϕ.
(5.10)
0
We use (5.9) and the Jacobi-Anger expansion [10], given by
eikz·ϕ̂ =
∞
X
im Jm (k|z|)eimϕz e−jmϕ
(5.11)
m=−∞
where z = (z, ϕz ) and ϕ̂ = (1, ϕ) in the polar coordinate system, to rewrite the
correlation coefficient in (5.8) as
ρpq =
∞
X
m=−∞
αn im Jm (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 Section 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
66
corralation co−efficient ρ
1
2−D diffuse field
Azimuth limited field φ = π/2
Azimuth limited filed φ = π/3
0.5
0
−0.5
0
0.2
0.4
0.6
0.8
1
1.2
Spatial Separation in λ
1.4
1.6
1.8
2
Figure 5.1: Comparison of spatial correlation versus separation for a diffuse scattering 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, represented by the αn s. Closed form expressions for αn exist for many scattering
distributions [55]. Some of the most commonly used angular power density distributions 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 containing the antennas, the power density distribution and the corresponding Fourier
coefficients are given by
P (ϕ) =
and
αn =
(
1
2π
∀ϕ
(5.13)
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
(
1
if (φ − φ0 ) ≤ ∆
2∆
P (ϕ) =
(5.15)
0 otherwise.
In this case the Fourier coefficients are given by
αn =
sin(n∆) −inφ0
.
e
n∆
(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
In (ς) −inφ0
.
(5.18)
αn =
e
I0 (ς)
In Figure 5.1 the effect of increasing separation distance on the correlation coefficient 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 appropriately 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.
68
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 =
∞
X
m=−∞
Jm (k|zp |)e−im(ϕzp −π/2) hm u + 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[np n∗p ] = Np
(5.21)
and the covariance between the noise at the pth and q th branches, at any time, is
given by
E[np n∗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
30
28
Average SNR in dB
26
24
22
20
Rayleigh Fading
d = λ/2
18
16
2
4
6
8
10
Number of antennas
12
14
16
Figure 5.2: Illustration of the effective SNR when calculated using the lower bound
in equation (5.29).
and
Γp =
statistical mean signal power of the pth branch
= E[γp ].
mean noise power of the branch
(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
70
γEGC =
P
hX
|rp |
1 p=1
P
2 X
i2
(5.25)
Np
p=1
E[γEGC ] =
1
E
2Np P
or
P
ΓEGC
"
P
X
p=1
|rp |
2 #
(5.26)
P
1 XX
E[|rp ||rq |].
=
2Np P p=1 q=1
(5.27)
Using the signal representation in (5.19) the average SNR can be written as
ΓEGC =
1
2Np P
P
X
p,q=1
"
X
E Jm (k|zp |)e−im(ϕzp −π/2) hm ×
m
#
X
0
Jm0 (k|zq |)eim (ϕzq −π/2) hm0 .
m0
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
P
1 XX
Jm (k|zp − zq |)E[| hm |2 ].
≥
2Np P p,q=1 m
(5.28)
If the scattering can be considered uniform over all directions equation (5.28) reduces to
ΓEGC
P
1 X
≥
J0 (k|zp − zq |).
2Np P p,q=1
(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
or
γMRC =
P
X
P
X
rp2
=
n
p=1 p
(5.30)
X
X
0
Jm (k|zp |)e−im(ϕzp −π/2) hm Jm0 (k|zp |)eim (ϕzp −π/2) hm0
m0
m
np
p=1
.
(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
X
p=1
E
"
X
m
Jm (k|zp |)e−im(ϕzp −π/2) hm
X
m0
np
0
Jm0 (k|zp |)eim (ϕzp −π/2) hm0 #
. (5.32)
Again, using the assumption of independent signals we get
ΓMRC =
P
X
X
[Jm (kzp )]2 E[|hm |2 ]
m
Np
p=1
.
(5.33)
It is evident from (5.33) that for a rich isotropic scattering environment
ΓMRC =
P
X
p=1
γp
X
[Jm (kzp )]2 .
(5.34)
m
The second sum in (5.34) is equal to one [20] and thus, for an isotropic scattering 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
72
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 modulation
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]
h 1
i
1
† −1
pr (r) =
exp − r R r
(2π)p/2 (detR)0.5
2
(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


ρ11 . . . ρ1P
 . .


..
..
R=


ρP 1 . . . ρ P P
73
(5.36)
where ρpq is the correlation coefficient between the pt and q th 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 =
Z
r
v
!
u P
u X
rp pγ (r)dr
Q t2
(5.37)
p=1
where pγ (r) is the joint pdf of the combined SNR at the output and Q(·) is the Q
R∞ 2
function defined as Q(y) = √12π y e−t /2 dt , 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 =
Z
Q
r
√
2rr† p(r)dr.
(5.38)
We now use the definition of the Q(·) function shown in [47]
1
Q(y) =
π
Z
π/2
0
to express Pb as
Pb
=
1
π
R π/2 R
0
1
(2π)p/2 (detR)
h
exp −
h
y2 i
dθ,
2 sin2 θ
rr†
2 sin2 θ
(5.39)
i
exp −
×
h
i
1 † −1
exp
−
r
R
r
drdθ.
0.5
2
r
Using the simplifications shown in [59] by completing the squares inside the exponential and using the matrix inversion lemma [23] an expression for the average
BER can be written as
1
Pb =
π
Z
π/2
0
1
h
det
R
sin2 θ
+I
i dθ.
(5.40)
74
If the correlation matrix R has distinct, non-repeating eigenvalues λ1 , λ2 , . . . , λP
then we can write,
1
h
det
R
sin2 θ
+I
i =
P
λ
−1
X
p
ζp
+1
sin2 θ
p=1
(5.41)
where ζp is the pth residue of the partial-fraction expansion given by
ζp =
Y
p6=q
λp .
λp − λ q
(5.42)
Therefore Pb becomes
1
Pb =
π
Z
0
P
π/2 X
p=1
ζp
λ
−1
p
.
+
1
sin2 θ
(5.43)
The integral in (5.43) is evaluated using the solution shown in the appendix of [4]
1
In (c) =
π
Z
π/2
0
= [P (c)]m
sin2 θ m
dθ
sin2 θ + c
m−1
X m − 1 + k k=0
where
k
(5.44)
[1 − P (c)]k
"
#
r
1
c
1−
.
P (c) =
2
1+c
(5.45)
(5.46)
When m = 1, a closed form equation for the BER [59] can be written using the
above integral as
s
#
"
P
λp
1X
.
Pb =
ζp 1 −
2 p=1
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
1
h
det
R
sin2 θ
−1
−1
λ
λ
3
2
i
=
ζ
+
1
+
1
+
2ζ
3
2
sin2 θ
sin2 θ
+I
75
(5.48)
where λ1 = λ2 are the repeated eigenvalues and
ζ3 =
"
ζ2 =
λ3
.
λ3 − λ 2
λ2
λ2 − λ 3
#2
Hence, Pb can be written as
1
Pb =
π
Z
π/2
0
(
)
λ
λ
−1
−1
3
2
dθ.
ζ3
+ 2ζ2
+1
+1
2
2
sin θ
sin θ
(5.49)
Again, using the result shown for (5.44), we get a closed form expression for P b
with a pair of repeated eigenvalues
#
#
"
"
r
r
1
λ3
λ2
Pb = ζ3 1 −
+ ζ2 1 −
.
2
1 + λ3
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 configurations
In section 5.2 an equation for the normalized correlation coefficient for a 2 dimensional scattering environment was found to be
ρpq =
∞
X
m=−∞
αm Jm (k|zp − zq |)e−imϕpq
(5.51)
76
−1
10
D=λ
D = λ/2
D = λ/5
−2
Average BER
10
−3
10
−4
10
−5
10
−6
10
0
2
4
6
8
10
12
Average SNR in dB
14
16
18
20
Figure 5.3: Illustration of the effect of spatial correlation on the BER performance
of a dual diversity system in a 2 dimensional isotropic diffuse field with varying
ULA 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 separation 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
∞
X
π
(5.52)
ρpq =
eim( 2 −φ0 ) sinc(m∆)Jm (k|zp − zq |)
m=−∞
where ∆ is the angular spread and φ0 is mean AOA.
77
−1
10
D=λ
D = λ/2
D = λ/5
−2
10
−3
Average BER
10
−4
10
−5
10
−6
10
−7
10
−8
10
0
2
4
6
8
10
12
Average SNR in dB
14
16
18
20
when 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 = z p − 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.

t0 t1
t2

 t1 t2
t3


...
T =  t2 t3
.
...
.
 . ...
tn tn+1 tn+2

. . . tn

. . . tn+1 


. . . tn+2  .



. . . t2n
(5.54)
78
−1
10
SNR = 5 dB
SNR = 10 dB
SNR = 15 dB
−2
Average BER
10
−3
10
−4
10
0
0.2
0.4
0.6
0.8
1
1.2
Aperture in λ
1.4
1.6
1.8
2
Figure 5.5: Illustration of the effect of increasing antenna array aperture on average
BER 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
79
−1
10
2 antennas
3 antennas
4 antennas
−2
Average BER
10
−3
10
−4
10
−5
10
0
0.2
0.4
0.6
0.8
1
1.2
Aperture in λ
1.4
1.6
1.8
2
Figure 5.6: Illustration of the effect of increasing array aperture at a constant
average SNR of 10dB.
closely spaced, the effective correlation increases causing a degradation in the output 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 improvement 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 advantage from a dual diversity system. A similar performance result is obtained when
four branches are employed. These results strongly indicate that careful considera-
80
−1
10
D=λ/2,∆=20
D=λ/2,∆=5
D=λ/5,∆=20
D=λ/5,∆=5
−2
Average BER
10
−3
10
−4
10
−5
10
0
2
4
6
8
10
12
Average SNR in dB
14
16
18
20
for a ULA with 2 antennas angular spreads of 20◦ and 5◦ and array apertures of
D = 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 performance 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 performance 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
81
−1
10
D=λ/2,∆=30
D=λ/2,∆=10
D=λ/5,∆=10
−2
10
−3
Average BER
10
−4
10
−5
10
−6
10
−7
10
0
2
4
6
8
10
12
Average SNR in dB
14
16
18
20
Figure 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.
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.
82
−1
10
R=λ
R = λ/5
R = λ/10
−2
Average BER
10
−3
10
−4
10
−5
10
−6
10
0
2
4
6
8
10
12
Average SNR in dB
14
16
18
20
for a uniform circular array in a diffused isotropic field when the radius R is decreased.
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 symmetry such that ρp = ρP −p . Hence, R becomes a P ×P symmetric circulant matrix.
A circulant matrix is defined as

a1 a2

 an a1
A=
 ..
 . ...
a2 a3

a3 . . . a n

a2 . . . an−1 
.
...


. . . an
(5.57)
a1
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
83
0
10
R=λ
R=λ/2
R=λ/5
−1
10
−2
Average BER
10
−3
10
−4
10
−5
10
−6
10
−7
10
0
2
4
6
8
10
12
14
Average branch SNR in dB
16
18
20
for a uniform circular array in a diffuse isotropic field with 3 receiver antennas.
form expression
λp =
P
−1
X
ρp cos(2πmp/P ).
(5.58)
m=0
We can now use (5.58) in (5.47) to explicitly calculate the BER for a UCA with ρ p
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
"
"
#
#!
r
r
1
λ1
λ2
Pb =
ζ1 1 −
+ ζ2 1 −
2
1 + λ1
1 + λ2
(5.59)
where λ1 and λ2 are calculated using (5.58), while the residues ζ1 and ζ2 are calculated using (5.42). Equation (5.59) provides a closed form equation for calculating
84
−1
10
R=λ/5,∆=20
R=λ/10,∆=20
R=λ/10,∆=5
−2
10
−3
10
−4
10
−5
10
−6
10
0
2
4
6
8
10
12
14
16
18
20
of a dual diversity UCA with angular spreads of 20◦ and 5◦ and radii of R = λ/5
and 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 employed 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 employed 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.
85
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 −1
repeated eigenvalue pairs.
2
The average error probability for a UCA with P antennas can therefore be
generalized from (5.47) and written in closed form as
s
s
"
"
"
#!
#
#
r
P
−1
X
λP/2
1
λP
λp
ζp 1 −
+
Pb =
ζP 1 −
+ ζP/2 1 −
2
1 + λP
1 + λP/2
1 + λp
P
p= 2 +1
(5.60)
when P is even and
s
#
"
#
"
r
P
−1
X
1
λP
λp
Pb = ζP 1 −
ζp 1 −
+
2
1 + λP
1 + λp
P +1
p=
(5.61)
2
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 attributed 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 circular 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
86
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 representation 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 studied 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 subjected 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 independent 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 independence 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 different angular spread and separation distances can provide more realistic insights into correlation effects on the performance of diversity systems which
can be studied in detail. Several non-isotropic scattering models were considered 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 configurations for combinations of given modulation schemes, performance measures
and spatially restricted receivers.
Bibliography
[1] V. Aalo and S. Pattaramalai. Average Error Rate for Coherent MPSK Signals
in Nakagami Fading Channels. Electronic Letters, pages 1538–1539, 1996.
[2] J. R. Abeysinghe and J. A. Roberts. Bit Error Rate Performance of Antenna
Diversity Systems with Channel Correlation. In Proc. IEEE GLOBECOM
’95, 1995.
[3] S. M. Alamouti. A Simple Transmit Diversity Technique for Wireless Communication. IEEE Journal on Selected Areas in Communications, pages 1451–
1458, October 1998.
[4] M. S. Alouni and A. Goldsmith. A Unified Approach for Calculating Error
Rates of Linearly Modulated Signals over Generalized Fading Channels. IEEE
Transactions on Communications, pages 1324–1334, 1999.
[5] J. B. Anderson, T. S. Rappaport, and S. Yoshida. Propagation Measurements
and Models for Wireless Communication Channels. IEEE Communications
Magazine, pages 42–49, 1995.
[6] C. H. Bianchi and K. Sivaprasad. A Channel Model for Multipath Interference
On Terrestrial Line-of-Sight Digital Radio. IEEE Transactions On Antennas
and Propagation, pages 891–901, 1998.
[7] D. G. Brennan. Linear Diversity Combining Techniques. In Proc. IRE, pages
1075–1102, 1959.
[8] Y. Cho and J. H. Lee. Effect of Fading Correlation on the SER Performance of
M-ary PSK with Maximal Ratio Combining. IEEE Communications Letters,
pages 199–201, July 1999.
[9] M. Chryssomallis. Smart Antennas. IEEE Antennas and Propagation Magazine, pages 129–136, 2002.
91
92
Bibliography
[10] D. Colton and R. Kress. Inverse Acoustics and Electromagnetic Scattering
theory. Springer, New York, 1997.
[11] C. R. C. M da Silva and M. D. Yacoub. A Generalized Solution for Diversity
Combining Techniques in Fading Channels. IEEE Transactions on Microwave
Theory and Techniques, pages 46–50, 2002.
[12] K. Dietze, Jr. C. B. Dietrich, and W. L. Stutzman. Analysis of a Two Branch
Maximal Ratio and Selection Diversity System with Unequal SNRs and Correlated Inputs for a Rayleigh Fading Channel. IEEE Transactions on Wireless
Communications, pages 274–281, 2002.
[13] X. Dong and N. C. Beaulieu. Optimal Maximum Ratio Combining with Correlated Diversity Branches. IEEE Communications Letters, pages 22–24, 2002.
[14] G. D. Durgin. Theory of Stochastic Local Area Channel Modelling for Wireless Communication. PhD thesis, Virginia Polytechnic Institute and State
University, 2000.
[15] R. B. Ertel, P. Cardieri, K. W. Sowerby, T. S. Rappaport, and J. H. Reed.
Overview of Spatial Channel Models for Antenna Array Communication Systems. IEEE Personal Communications, pages 10–22, 1998.
[16] L. Fang, G. Bi, and A. C. Kot. New Method of Performance Analysis for Diversity Reception with Correlated Rayleigh Fading Signals. IEEE Transactions
on Vehicular Technology, pages 1807–1812, 2000.
[17] G. J. Foschini and M. J. Gans. On Limits of Wireless Communications in
Fading Environment when using Multiple Antennas. Wireless Personal Communications, pages 311– 335, 1998.
[18] A. Giorgetti, M. Chiani, M. Shafi, and P. J. Smith. Characterizing MIMO
Capacity under the influence of Spatial/Temporal Correlation. In Proc. 4th
Australian Communication Theory Workshop 2003.
[19] L. Chand. Godara, editor. Handbook of Antennas in Wireless Communications. CRC Press, 2002.
[20] I. S. Gradshteyn and I. M. Ryzhik. Tables of Integrals, Series and Products.
Academic, fifth edition, 1994.
Bibliography
93
[21] J. T. Y. Ho, R. A. Kennedy, and T. D. Abhayapala. Analytical Expression
for Average SNR of Correlated Dual Selection Diversity System. In Proc. 3rd
Australian Communication Theory Workshop, Canberra, Australia, 2002.
[22] J. T. Y. Ho, R. A. Kennedy, and T. D. Abhayapala. Dual Selection Diversity
SNR Performance in Spatially Correlated Scattering Environments. In Proc.
IEEE International Conference of Telecommunications , Beijing, China, 2002.
[23] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press,
Cambridge, 1985.
[24] W. C. Jakes, editor. Microwave Mobile Communications. Wiley, New York,
1974.
[25] S. S. Jeng, G. T. Okamoto, G. Xu, H. P. Lin, and W. J. Vogel. Experimental
Evaluation of Smart Antenna System Performance for Wireless Communications. IEEE Transactions On Antennas and Propagation, pages 749–757,
1998.
[26] H. M. Jones, R. A. Kennedy, and T. D. Abhayapala. On Dimensionality of
Multipath Fields: Spatial Extent and Richness . In Proc. IEEE Conference
on Acoustics, Speech and Signal Processing, May 2002.
[27] C. Kim, Y. Kim, G. Jeong, and H. Lee. BER Analysis of QAM with MRC
Space Diversity in Rayleigh Fading Channels. In IEEE International Symposium on Personal, Indoor, and Mobile Communications, 1995.
[28] Y. G. Kim and S. W. Kim. Optimum Selection Diversity for BPSK Signals
in Rayleigh Fading Channels. IEEE Transactions on Communications, pages
1715–1718, 2001.
[29] J. S. Kwak and J. H. Lee. Performance Analysis of Optimum Combining
for Dual Antenna Diversity with Multiple Interferers in a Rayleigh Fading
Channel. IEEE Communications Letters, pages 541–543, December 2002.
[30] W. C. Y. Lee. Mobile Cellular Telecommunications Analog and Digital systems.
Mc Graw-Hill, Singapore, second edition.
[31] W. C. Y. Lee. Effects on Correlation between two Mobile Radio Base - Station
Antennas. IEEE Transactions on Communications, pages 1214–1224, 1973.
94
Bibliography
[32] J. J. A. Lempi˙ The Performance of Polarisation Diversity Schemes at a Base
Station in Small/Micro cells at 1800MHz.
[33] S. L. Loyka. Influence of Pilot signal on Directivity of Self-phased Arrays under
conditions of Multipath Propagation. IEEE Transactions on Electromagnetic
Compatibility, pages 12–18, 1998.
[34] J. Luo, J. R. Zeidler, and S. McLaughlin. Performance Analysis of Compact
Antenna Arrays with MRC in Correlated Nakagami Fading Channels. IEEE
Transactions on Vehicular Technology, pages 267–277, 2001.
[35] R. J. Mailloux. Phased Array Antenna Handbook. Artech House, Boston, 1994.
[36] R. K. Mallick, M. Z. Win, and J. H. Winters. Performance of Dual-Diversity
Predetection EGC in Correlated Rayleigh Fading with Unequal Branch SNRs.
IEEE Transactions on Communications, pages 1041–1044, 2002.
[37] R. K. Mallik and M. Z. Win. Analysis of Hybrid Selection/Maximal Ratio
Combing in Correlated Nakagami Fading. IEEE Transcations on Communications, pages 1373– 1382, 2002.
[38] M. Nakagami. The M- Distribution a General Formula of Intensity Distribution of Rapid Fading. W.C.Hoffman, New York, 1960.
[39] J. N. Pierce and S. Stien. Multiple Diversity with Non-Independent Fading.
In Proc. IRE, pages 89–104, 1960.
[40] T. S. Pollock, T. D. Abhayapala, and R. A. Kennedy . Introducing Space
into Space-Time MIMO Capacity Calculations: A New Closed Form Upper
Bound. In Proc. International Conference on Telecommunications, ICT’2003.
[41] T. S. Pollock, T. D. Abhayapala, and R. A. Kennedy . Antenna Saturation Effects of Dense Array MIMO Capacity . In Proc. IEEE Conference on
Acoustics, Speech and Signal Processing, Hong Kong, 2003.
[42] T. S. Pollock, T. D. Abhayapala, and R. A. Kennedy. Fundamental Limits
of MIMO Capacity for Spatially Constrained Arrays. In Proc. 4th Australian
Communication Theory Workshop 2003, pages 7–12.
[43] J. G. Proakis. Digital Communications. Mc Graw-Hill, Singapore, third edition, 1995.
Bibliography
95
[44] T. S. Rappaport. Wireless Communications Principles and Practice. Prentice
Hall, New Jersey, 1996.
[45] J. Salz and J. H. Winters. Effect of Fading Correlation on Adaptive Arrays
in Digital Mobile Radio. IEEE Transactions on Vehicular Technology, pages
1049–1057, November 1994.
[46] D. S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn. Fading Correlation
and its Effects on the Capacity of Multielement Antenna Systems. IEEE
Transactions on Communications, pages 502–512, 2000.
[47] M. K. Simon and D. Divsalar. Some New Twists to Problems Involving the
Gaussian Integral. IEEE Transactions on Communications, pages 200–210,
1998.
[48] B. Skylar. Rayleigh Fading Channels in Mobile Digital Communication Systems Part 1: Characterization. IEEE Communications Magazine, pages 90–
100, 1997.
[49] B. Skylar. Digital Communications Fundamentals and Applications. Prentice
Hall, New Jersey, 2001.
[50] G. L. Stüber. Principles of Mobile Communication. Kluwer Academic Publisher, 2001.
[51] Y. Sun. Bandwidth - Efficient Wireless OFDM. IEEE Journal on Selected
Areas in Communications, pages 2267 –2277, 2001.
[52] T. Svantesson and A. Ranhem. Mutual Coupling Effects on the Capacity
of Multielelment Antenna Systems. In Proc. IEEE Conference on Acoustics,
Speech and Signal Processing, 2001.
[53] C. C. Tan and N. C. Beaulieu. Infinite Series Representation of the Bivariate
Rayleigh and Nakagami-m Distribution. IEEE Transactions on Communications.
[54] P. D. Teal. Real Time Characterization of the Mobile Multipath Channel. PhD
thesis, RSISE, The Australian National University, 2001.
[55] P. D. Teal, T. D. Abhayapala, and R. A. Kennedy. Spatial Correlation for
General Distributions of Scatterers. IEEE Signal Processing Letters, pages
305–308, 2002.
96
Bibliography
[56] I. E. Telatar. Capacity Of Multi Antenna Guassian Channels. Technical
report, AT&T Bell labs, 1995.
[57] C. Tellambura, A. J. Mueller, and V. K. Bhargava. Analysis of M-ary Phase
Shift Keying with Diversity Reception for Land-mobile Satellite Channels.
IEEE Transactions on Vehicular Technology, pages 910–922, 1997.
[58] J. A. Tsai, R. M. Buehrer, and B. D. Woerner. Spatial Fading Correlation
Function of Circular Antenna Arrays with Laplacian Energy Distribution.
IEEE Communications Letters, pages 178– 180, May 2002.
[59] V. V. Veeravalli. On Performance Analysis for Signalling on Correlated Fading
Channels. IEEE Transactions on Communications, pages 1879–1883, 2001.
[60] M. Z. Win, G. Chrisikos, and J. Winters. MRC Performance for M-ary Modulation in Arbitrary Correlated Nakagami Fading Channel. IEEE Communications Letters, pages 301–303, 2000.
[61] M. Z. Win and J. Winters. Analysis of Hybrid Selection/Maximal -Ratio
Combining in Rayleigh fading. IEEE Transactions on Communications, pages
1773–1776, 1999.
[62] M. Z. Win and J. H. Winters. Analysis of Hybrid Selection/MaximalRatio
Combining of Diversity Branches with Unequal SNR in Rayleigh Fading . In
Proc. 49th IEEE Vehicular Technology Conference, 1999.
[63] J. Winters. Optimum Combining in Digital Mobile Radio with Co channel
Interference . IEEE Journal on Selected Areas in Communications, pages
528–539, 1984.
[64] J. Winters. Smart Antennas for Wireless Systems. IEEE Personal Communications, pages 23–27, 1998.
[65] J. Winters. The Diversity Gain of Transmit Diversity in Wireless Systems with
Rayleigh fading. IEEE Transactions on Vehicular Technology, pages 119–123,
February 1998.
[66] J. Winters, J. Salz, and R. D. Gitlin. The Impact of Antenna Diversity on
the Capacity of Wireless Communication Systems . IEEE Transactions on
Communications, pages 1740–1751, 1994.
Bibliography
97
[67] X. Zhao and P. Vainikainen. Multipath Propagation Study Combining Terrain
Diffraction and Refraction. IEEE Transactions On Antennas and Propagation,
pages 1204–1209, 2001.

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