by William Chou...Figure 1.4: Blueprint for metamaterial antenna [8] 1.2 Metamaterial Antenna This...

Beamforming Based MIMO Processing With

Closely Spaced Antennas

by

William Chou

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Graduate Department of Computer and Electrical EngineeringUniversity of Toronto

Copyright c© 2011 by William Chou

Abstract

Beamforming Based MIMO Processing With

Closely Spaced Antennas

William Chou

Master of Applied Science

Graduate Department of Computer and Electrical Engineering

University of Toronto

2011

When antennas are placed closely spaced together, the mutual coupling and spatial

correlation effects undermine the advantages provided by multiple input and multiple

output (MIMO) antennas. In this thesis, we compare and analyze the performance of

digital beamforming, fixed radio frequency (RF) beamforming and element based pat-

terning with closely spaced antenna systems.

In the case where only one RF-chain is available, we have demonstrated performance

improvements using RF beamforming-based MIMO processing instead of element-based

MIMO processing with closely spaced metamaterial antennas. The result indicates that

even without mutual coupling, antenna based MIMO processing is greatly impacted when

moving from rich to correlated scattering environments.

In the second half of the thesis, we investigate the switch and examine receiver com-

bining (SEC) technique. We derive the switching rate of SEC and show that even though

it has the same outage probability as traditional selection combining, it has a significantly

lower switching rate.

ii

Acknowledgements

I would like to offer my sincerest gratitude to the following people:

• Professor Adve for his support and advice in my academic and social life. Without

him, my Master study would not have been as rewarding and exciting.

• Professors Eleftheriades, Sarris and Hum for their contributions on the ideas and

research results for this thesis.

• Mohammad Memarian, Neeraj Sood, K.V. Srinivas and Derek Zhou for their sim-

ulation data and results used in this research.

• friends in Communication Group and undergraduate study (Devin Lui, Adam

Tenenbaum, Hassan Masoom, Amir Aghaei, Ehsan Karamad, Mohammad Ma-

hanta, Sanam Sadr, Kianoush Hosseini, Gokul Sridharan, Helia Mohammadi, Jeff

Lee, Patrick Li, Lei Zhang, Gerry Chen, Daniel Huynh, Jacky Mak and Wallace

Wee) who have made my Graduate School life fun and interesting.

• my parents and sister for their spiritual and emotional support.

Without the above people, my life would not have been as meaningful and complete

Finally, I would also like to thank Natural Sciences and Engineering Research Council

of Canada (NSERC), Research in Motion (RIM) and Rogers for their funding support

throughout my Master’s program.

iii

Contents

1 Introduction 1

1.1 Challenges with Closely Spaced Antennas . . . . . . . . . . . . . . . . . 3

1.2 Metamaterial Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Beamforming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Receiver Combining Techniques . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Background 10

2.1 Spatial Correlation Derivation . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Mutual Coupling Derivation . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Beamforming Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 No Beamforming . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.2 RF Beamforming . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.3 Digital Beamforming . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Combining Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.1 Switching Rate for Switch or Stay Combining . . . . . . . . . . . 17

2.5 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5.1 Correlated MIMO Channel Model . . . . . . . . . . . . . . . . . . 18

2.5.2 Ring Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

iv

2.5.3 Doppler Frequency Shifts . . . . . . . . . . . . . . . . . . . . . . . 20

3 Beamforming with Closely Spaced Antennas 22

3.1 Capacity vs. Inter-element Distance . . . . . . . . . . . . . . . . . . . . . 22

3.2 Comparing Beamforming Systems with Multiple RF Chains Available . . 25

3.2.1 Digital Beamforming vs. No Beamforming . . . . . . . . . . . . . 25

3.2.2 RF Beamforming . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Single RF Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.1 Diversity Combining Beamformers . . . . . . . . . . . . . . . . . 44

3.3.2 RF Beamforming Selection vs Antenna Element Selection . . . . . 52

3.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Switch and Examine Combining Techniques 61

4.1 Switch and Examine Algorithm . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 SEC Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.1 Correlated Rayleigh Fading Channel . . . . . . . . . . . . . . . . 66

4.3 Switching Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3.1 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4 SEC Switching Rate Monte Carlo vs. Theoretical Closed Form Solution . 76

4.4.1 Comparison at Different Normalized Sample Rates . . . . . . . . 76

4.5 Performance against SC . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.5.1 Two Receiver Antennas without Beamforming . . . . . . . . . . . 80

4.5.2 Six Receiver Antennas without Beamforming . . . . . . . . . . . . 82

4.5.3 Six Receiver Antennas with Beamforming . . . . . . . . . . . . . 84

4.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Conclusions and Future Work 87

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

v

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2.1 Finding Optimal Beam Pattern . . . . . . . . . . . . . . . . . . . 88

5.2.2 Incorporating Mutual Coupling . . . . . . . . . . . . . . . . . . . 89

5.2.3 Further Analysis on SEC . . . . . . . . . . . . . . . . . . . . . . . 91

A Transmitter and Receiver Mutual Impedance Expression 94

B Derivation of Mutual Information for Digital Beamforming 96

C Transitional Probability Derivations for SSC System with Two Inde-

pendent Branches 98

C.1 Group 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

C.2 Group 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

C.3 Group 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

C.4 Group 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

C.5 Group 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

C.5.1 Solution for the Second Term in Equation (C.20) . . . . . . . . . 103

C.5.2 Solution for the First Term in Equation (C.20) . . . . . . . . . . . 103

C.6 Group 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

D Histograms for Metamaterial Antenna Based on Ray-tracing Simula-

tion 106

D.0.1 Transmitter Backward and Receiver Forward . . . . . . . . . . . . 106

D.0.2 Transmitter Forward and Receiver Forward . . . . . . . . . . . . 108

D.0.3 Transmitter and Receiver in Vertical Direction . . . . . . . . . . . 108

Bibliography 112

vi

Chapter 1

Introduction

As society’s demand for mobile information access continues to grow, the next generation

of wireless technology needs to reliably provide higher data rate and longer link range

without additional bandwidth or transmit power. Therefore, MIMO (Multiple-input

and Multiple-output) systems, which utilize multiple antennas at transmitters and/or

receivers (see Figure 1.1), have attracted a lot of attention due to this ability to improve

reliability (via diversity) or data rates (via multiplexing) [1]. These systems have been

widely used in the wireless standards such as WiFi, WiMax, HSPA+ and LTE [2].

One of the main benefits which MIMO systems provide is diversity gain. The per-

formance of a SISO (Single-Input, Single-Output) system is severely undermined by the

fading of the signal caused by the multipath propagation and the Doppler shifts. By uti-

lizing multiple antennas spaced far enough apart at the receiver, each receiving antenna

receives a different copy of the signal. Thus, the probability that the received signals are

all fading is reduced. This gain provided by multiple antennas is called diversity gain [1].

Another benefit which MIMO systems provide is multiplexing. With multiple an-

tennas, MIMO systems can decompose the MIMO channels into parallel channels and

multiplex different data steams onto these different channels. This gain provided by mul-

tiplexing different data streams is called multiplexing gain. However, there is a trade-off

1

Chapter 1. Introduction 2

Tx Rx

Figure 1.1: An example MIMO system with two transmitting (Tx) and two receiving

(Rx) antennas

between diversity and multiplexing gains in a MIMO system [1]. Having full diversity

gain means all the parallel channels in the system are used to send redundant data over

multiple channels and thereby trades off data rate for lower bit error rate performance.

On the other hand, having full multiplexing gain means each parallel channel in the

MIMO system is used for transmitting different data streams. Unless powerful channel

coding is employed, this system will have poor performance. The benefits and detailed

analysis of MIMO systems and the diversity-multiplexing trade-off are provided in [1,3].

Crucially, the many benefits of MIMO systems are best realized when the fading

channels between transmit-receive pairs are statistically independent [4, 5]. This is be-

cause the key to the MIMO gains is the statistical independence in the fading between

different transmitters and receivers. However, in the context of cellular networks, one of

the greatest challenges in designing MIMO systems is to realize the multiple antennas

in small handheld devices. In this case, the advantages offered by MIMO are reduced

due to the mutual coupling and spatial correlation effects between the closely spaced

antennas [6].


Figure 1.2: Illustration of mutual coupling

1.1 Challenges with Closely Spaced Antennas

The first factor that affects the performance in a MIMO system is mutual coupling. It

is caused by the re-radiation from the currents induced by the incoming signal. This,

in turn, induces currents on other antennas. Figure 1.2 illustrates the cause of mutual

coupling when the antennas are in receiving mode. The figure was obtained from [7] and

the conceptual sequence is as follows:

I Incident wave (0) reaches “Antenna m”

II The incident wave causes current flows (1) in “Antenna m”

III Part of the incident wave is re-scattered (2) into free space

IV This re-scattered wave (3) is transmitted into “Antenna n” and thereby introduces

mutual coupling


The effect of mutual coupling mentioned above is, in general, inversely proportional

to the distance between the antennas. In a handheld device which has a strict size

constraint, the inter-element distance between the antennas needs to be small. Therefore,

this increases the mutual coupling between antennas and introduces higher correlation

between the MIMO channels.

Mutual coupling does not only occur during receiving mode. The effects discussed

above also happen in the transmitting mode. The work in [7] has a more detailed analysis

of mutual coupling in the transmitting mode.

The second factor which affects the performance of a MIMO system is spatial corre-

lation. It is the correlation between the received signals at two antennas. Normally, in

independent identically distributed channels, fading is assumed to be so severe that the

path from one transmitting antenna to a receiving antenna is completely independent

from the path of the transmitting antenna to another receiving antenna. In practice,

this is not the case since there are always some residual correlation between the received

signals. The mutual coupling effect described in the previous section is also a part of spa-

tial correlation. However, even without mutual coupling, the small spacing between the

antennas will still cause the signals to be correlated. The degree of correlation depends

on the channel and physical location of the scatterers.

Spatial correlation, as does mutual coupling, generally falls off with inter-antenna

spacing. Given a spacing, the correlation is determined by the scattering environment.

In a rich scattering environment, with many sources of multipath, the correlation falls

off rapidly. At the other extreme, in perfect line-of-sight conditions between transmitters

and receivers, any two antennas are perfectly correlated independent of distance.


Figure 1.3: Metamaterial antenna [8]

Figure 1.4: Blueprint for metamaterial antenna [8]

1.2 Metamaterial Antenna

This thesis is motivated by the potential use of closely spaced metamaterial antennas [9]

in a handheld device. These antennas can be very closely spaced with very little mutual

coupling. Metamaterials are artificially engineered structures which have unusual elec-

tromagnetic properties such as negative index of refraction [10]. These properties allow

the antennas to have smaller size and higher directivity. Details regarding the physics

and analysis of metamaterial antenna can be found in [11]. Figure 1.3 is a photograph of

the prototype of metamaterial antenna and Figure 1.4 shows the blueprint of the proto-

type. As shown in Figure 1.4, the distance between the antennas is merely λ0/13 (where


λ0 is the wavelength of the electromagnetic wave). These antennas were designed for an

operating frequency of 2.5 GHz [9]. As we will show later on, this distance is a lot smaller

than the λ/2 inter-element distance that is currently used in many designs. According

to [9], metamaterial antennas can be placed in extremely close proximity such as λ/13

without suffering too much mutual coupling between antennas. Even though there is no

mutual coupling between antennas, the close physical locations of the antennas can still

contribute to spatial correlation.

The motivation behind our research is to find a compensation scheme that reduces

the spatial correlation between the metamaterial antennas. However, due to the diffi-

culty in mathematically analyzing the performance of the metatmaterial antennas, we

have simplified the system to use closely spaced dipole antennas instead of metamate-

rial antennas. This thesis provides the research results comparing the performance of

different compensation schemes to reduce the correlation between closely spaced dipoles.

Applying our correlation compensation schemes on the metamaterial antennas is left as

future work.

1.3 Beamforming

As mentioned before, this thesis is dedicated to find the schemes which compensate for

the spatial correlation in closely spaced dipoles. We apply the concept of beamforming

to examine whether pattern diversity provides greater benefits than antenna diversity in

closely spaced dipoles.

There are two types of beamforming techniques to achieve pattern diversity: RF

beamforming and digital beamforming. RF beamforming applies different phase shifts

to each antenna to generate different antenna array patterns before the downconversion

of the RF signals [12]. RF beamforming therefore combines the signals of an antenna

array before the RF chain. Thus, by finding the “adequate” beamforming patterns, we


may reduce the correlation between the beamforming patterns and thereby improving

the performance of the MIMO system.

Digital beamforming, on the other hand, combines the downconverted and digitized

signals; an appropriate choice of digital weights can provide the same beamforming pat-

terns as RF beamforming. By applying beamforming pattern digitally, this allows for

greater flexibility in adjusting the beamforming pattern in real time [13]. However, as

we will see, the impact on the receiver noise terms is quite different. In this thesis, we

will derive and compare the improvement in the mutual information provided by these

two beamforming techniques.

1.4 Receiver Combining Techniques

Besides investigating the effect of beamforming in this thesis, we also consider the per-

formance of different diversity combining schemes. As mentioned previously, multiple

antennas at a receiver allow for diversity gains. The key to realize these gains is to com-

bine the received signals in a useful manner. There have been many works on comparing

the performance of different receiver combining schemes [14–16]. In considering the di-

versity gain for MIMO systems, it is well accepted that maximal ratio combining (MRC)

is optimal in terms of signal-to-noise ratio (SNR). However, the motivating application

for this work is an array of closely spaced elements within a handheld device. In this

regard, it appears unlikely that the handheld would be able to sustain more than a single

RF chain; in practice, using multiple RF-chains is expensive and power inefficient [17].

MRC requires as many RF chains as elements or beamforming patterns. Therefore, a

more practical approach is to perform selection combining (SC) which scans all branches

and selects the one that has the highest SNR. However, the action of scanning every

branch requires training and becomes too time consuming and leads to many switching

operations. In this regard, switch or stay combining (SSC) provides an alternative to


minimize the system overhead. With SSC, a branch switch only occurs when the current

branch SNR is below a chosen threshold; it does not need to keep track the instantaneous

SNR of previous switched branch. However, in the case in which the threshold SNR is

almost always larger than the signal SNR, SSC will perform many pointless switching

operations which therefore defeats its original purpose. To resolve this, another simi-

lar technique called switch and examine combining (SEC) is introduced in [14]. This

technique functions similarly as SSC with the difference being that it checks the signal

strength at every branch before switching. If every branch is below the threshold, no

switching is performed and an outage will then be declared. In this thesis, we first ana-

lyze the switching rate of SEC in detail and then compare its performance against SC in

a closely spaced antenna beamforming system.

1.5 Objectives

The main objective of this thesis is to analyze and evaluate the performance of applying

RF beamforming, digital beamforming, and element based combining in closely spaced

dipole antennas systems with SC and SEC combining techniques.

We focus on answering the following three questions:

1. Without the constraint on single RF-chain, do RF beamforming or digital beam-

forming techniques provide better performance than element based combining sys-

tem in a closely spaced dipole antennas system?

2. With the constraint on having single RF-chain, is RF beamforming SC better than

element based SC in a closely spaced dipole antenna system.

3. How much performance improvement does SEC provide over SC in both beam-

forming and element based systems?

Bit error rate (BER) and mutual information (capacity) are used as the figures of


merit to answer the first two questions. Probability of outage and switching rate are

used as the performance indicators for the third question.

1.6 Outline

In Chapter 2, we first review the related published works analyzing mutual coupling and

spatial correlation. Next, the models for both RF and digital beamforming are presented.

Furthermore, the prior work on SC, SSC and SEC is introduced. This chapter ends with

the introduction of the system models that are used throughout the thesis.

Chapter 3 answers the first two questions in the previous section. This chapter first

shows the undermining effect that mutual coupling and spatial correlation has on capac-

ity. Next, this chapter compares the performance between digital and RF-beamforming

in a multiple RF-chain system. Then, a performance comparison between digital beam-

forming, RF beamforming and element-based processing under the constraint of a single

RF-chain is provided.

Chapter 4 answers the last question in the previous section. It first explains in detail

the algorithm for SEC. Next, it derives the switching rate of SEC in closed form. The

derived closed form solution is then verified with results of using Monte Carlo simulations.

The chapter concludes with a comparison between beamforming and element-based SEC

systems in terms of the switching rate and probability of outage.

Finally, this thesis wraps up in Chapter 5 with conclusions and suggestions for future

work.

Chapter 2

Background

This chapter introduces some of the background information required such as previous

work on analyzing spatial correlation, mutual coupling and switching algorithms. The

system model representations for RF and digital beamforming are provided as well. Fi-

nally, this chapter concludes with the different channel models that are used throughout

the thesis.

2.1 Spatial Correlation Derivation

Consider a two dipole antennas receiving system. Let s1 be the signal received at dipole

antenna #1 and let s2 be the signal received at another dipole antenna distance d away

from antenna #1. The work in [18] has derived the spatial correlation, ρ, between these

two dipoles to be:

ρ =E[s1s

∗2]

√

E[|s1|2]√

E[|s2|2]

=

∫ π

0

∫ 2π

0

ejkd cosφ sin θfθ(θ)fΦ(φ)dφdθ (2.1)

where E[·] denotes the expected value and ∗ denotes the complex conjugate. φ is the

azimuthal angle and θ is the elevation angle. fΦ(φ) and fθ(θ) are the probability distri-

butions of the received signal energy in azimuthal and elevation plane respectively.

10

Chapter 2. Background 11

In the case that most energy arrives from an elevation angle of π/2 (fθ(θ) = δ(θ−π/2))

and is uniformly distributed in azimuth (fΦ(φ) = 1/2π), the correlation can be written

in closed form as ρ = J0(kd) where J0 is the zeroth order Bessel function of first kind.

These assumptions have been proven to be valid for an urban environment in which the

base station is at a considerable height over the surrounding scatterers and the receiver.

In the case which the above assumptions are not valid, the correlation can only be found

through Equation (2.1).

To represent spatial correlation under the above assumptions in matrix form, let H

be a channel with N transmitters and M receivers. Thus, H is of size M × N . When

the transmitters are uncorrelated, the receiver spatial correlation is as follows:

E[HHH ] =

1 J0(kd) · · · J0((M − 1)kd)

J0(kd) 1 · · · J0((M − 2)kd)

......

......

J0((M − 1)kd) J0((M − 2)kd) · · · 1

In the case in which the assumptions made on fΦ(φ) and fθ(θ) are not valid, each entry

of the above matrix has to be found through Equation (2.1).

DefineΨR as receiver spatial correlation matrix defined as E[HHH ]. Similarly, letΨT

be the transmitter spatial correlation. In the case when the receivers are uncorrelated,

ΨT = E[HHH] (2.2)

Thus, based on the channel model defined in [19], the channel can be represented as

follows:

H =√ΨRHu

√ΨT (2.3)

where the square root means the square root of the matrix (not the square root of the

individual entry). Hu represents an uncorrelated wireless channel. Here, “uncorrelated”

means the path from one transmitting antenna to a receiving antenna is completely


independent from the path of the transmitting antenna to another receiving antenna.

Thus, E[Hu(Hu)H ] = IM and E[(Hu)HHu] = IN . IM and IN are identity matrices with

the dimension of M and N respectively.

2.2 Mutual Coupling Derivation

Figure 2.1: Mutual coupling system model

Figure 2.1 is the system model representation of mutual coupling introduced in [6].

The H term represents the entire communication channel block which contains both

mutual coupling and spatial correlation. Hij is the wireless propagation path from jth

transmitter antenna to ith receiver antenna. In Figure 2.1, ZSn is the source impedance

in the nth transmitter and ZLm is the load impedance in the mth receiver. ZTx and ZRx

are the mutual impedance matrices at the transmitter and the receiver respectively. In

an uncoupled system, ZTx and ZRx only have self-impedance and therefore are diagonal

matrices. The details on computing ZTx and ZRx are included in Appendix A.


Let ZT and ZR represent the mutual coupling matrix of the transmitter and the

receiver respectively. These matrices are related to the source/load impedance matrices

(Zs/ZL) and the impedance matrices ZTx and ZRx as:

ZT =ZTx(ZTx + Zs)

−1

CT

(2.4)

ZR =ZL(Z

Rx + ZL)−1

CR

(2.5)

where CT and CR are the normalization constants. [6] has defined CT = ZTx11 /(Z

Tx11 +Zs1)

and CR = (ZRx11 )

∗/(ZL1+(ZRx11 )

∗) under the assumption that each transmitting/receiving

antenna has the same source/load impedance.

2.3 Beamforming Techniques

2.3.1 No Beamforming

Figure 2.2: No beamforming system

We begin with the standard approach of no beamforming. With N transmitters and

M receivers, the received signal at any symbol instant can be written as

y = Hx+ n (2.6)

where H represents the M × N channel including correlation and mutual coupling. x

is a transmit data vector of length N and n is additive white Gaussian noise (AWGN).

Figure 2.2 represents the no beamforming system.


2.3.2 RF Beamforming

Figure 2.3: RF beamforming system

Figure 2.3 presents the system model for RF beamforming. WH represents the phase

shifts applied onto each receiving antenna to create the desired beamforming patterns.

Note that these phase shifts are applied before the down conversion to baseband. WH

has the dimension of Mb ×M where Mb is the number of beamforming patterns. Thus,

based on Fig 2.3, the received signal vector, y, can be written as follows:

y = WHHx+ n (2.7)

As mentioned in the introduction, RF beamforming involves applying different phase

shifts to each antenna to generate different antenna array patterns. Currently, a common

argument on the RF-beamforming system model described in Figure 2.3 is that such sys-

tem is impractical since noise source impacts the signal after the phase shifts are applied.

It is therefore not possible for the system to remain noiseless until after beamforming

patterns are generated. [20] and [21] have investigated this issue in detail. The work

in [20] considers the noise source caused by background radiation that are picked up by

the receiver antennas. This background radiation includes cosmic radiation, noise from a

cover of thick clouds, and man-made noise scattered from different sources; these sources

are referred to as sky noise. In the case when sky noise dominates, [20] proves that the

channel capacity cannot be improved by any internal coupling network.

The work in [21] considers the case which the receiver amplifier noise in the circuit

dominates over the sky noise. In such a case, the author demonstrates that optimum


capacity can be achieved by using a lossless imbedding matrix introduced in [22] to decou-

ple the amplifier noise. Moreover, [21] mentions that amplifier noise typically dominates

except when the interference from other users is the main source of noise. Thus, the

research results from [20] and [21] demonstrate the validity in using Figure 2.3 as system

model for RF beamforming.

2.3.3 Digital Beamforming

Figure 2.4: digital beamforming system

Figure 2.4 presents the system model representation for digital beamforming. As

mentioned in the introduction, digital beamforming mimics RF-beamforming by injecting

digitized weights onto the received signals after down conversion. Since amplifier noise

is the main source of noise as mentioned in [21], these digitized weights are also applied

to the amplifier noise in the system. The model in Equation (2.7) changes to:

y = WHHx+WHn (2.8)

2.4 Combining Schemes

There are three typical types of combining techniques: Selection Combining (SC), Maxi-

mal Ratio Combining (MRC) and Equal Gain Combining (EGC). The choice of technique

depends essentially on the complexity restriction put on the communication system and

the amount of channel state information (CSI) available at the receiver.


As shown in [15], MRC is the optimal combining scheme in the absence of interference.

Maximizing SNR, MRC achieves full diversity order. However, since MRC requires the

knowledge of channel phase, this scheme is not practical for noncoherent detection and

requires as many RF-chains as antenna elements.

A simpler technique is EGC, which co-phases the signals on each branch and then

combines them with equal amplitude. The work in [1] has demonstrated that the perfor-

mance of EGC is quite close to MRC. The performance difference is the trade-off for the

reduced complexity of applying equal gain. EGC requires as many RF-chains as MRC.

Since the two former combining techniques (MRC and EGC) require full or partial

CSI (amplitude and phase of the channel), an even simpler combining scheme is SC.

It chooses the branch that has the highest SNR. Because only one branch is used at a

time, SC requires just one RF-chain that is switched into the active antenna branch.

This reduces the complexity and amount of CSI required at the receiver with the cost of

system performance. A detailed comparison of the performance between these combining

schemes is presented in [1] and [15].

In the standard SC scheme introduced in [14], the instantaneous received signal en-

velope is monitored. The element with the highest receive SNR is always selected. A

simpler version of SC change the element only when the instantaneous SNR fall below a

predetermined threshold level. If this second branch is also in a fade, [23] has summarized

two different ways to deal with such event. One of the more popular schemes is to switch

to this second branch regardless whether this branch is in a fade [24]. This method is

Switch or Stay Combining (SSC). Another method is to stay at the first branch since the

second branch is fading as well. This method is Switch and Examine Combining (SEC).

In the first method, the switching decision is based only on the current channel estimate.

With the SSC algorithm, a branch switch only occurs when the current branch SNR

is below a chosen threshold; it does not need to keep track the instantaneous SNR of

previous switched branch. However, for SSC to achieve the same performance as SC, an


optimal threshold SNR needs to be calculated first. The work in [25] has developed a

scheme which finds the optimal threshold SNR based on measuring the average SNR of

the signal. Such method requires a training signal to be sent first to calculate the average

SNR and the timing delay of the system might be even greater than SC. Thus, in this

thesis, we focus on the case in which the threshold of SSC is set to fixed level.

The details of the algorithm for SEC and the prior research on the switching rate for

SSC will be discussed later in detail in Chapter 4.

2.4.1 Switching Rate for Switch or Stay Combining

A key parameter of selection-based combining scheme is the switching rate. Although

many works, e.g., [15,26–28] have analyzed the performance on SSC based on parameters

such as BER, probability of outage, level crossing rate and average fading duration, there

is not much work which details the switching rate advantage that SSC has over SC. The

work in [29] presented a Markov chain based analytical framework for the performance

comparison for SC and SSC. The result from this paper has demonstrated that SSC has

lower switching rate than SC for TDMA systems in a slow fading environment. However,

the definition of “switching rate” defined in [29] is the probability of switching branches

upon start up of each time slot in TDMA system under the assumption that the each

time slot length is greater than the coherence time. This definition of “switching rate”

is different from the definition in [30]. The work in [30] compares the switching rates of

SSC and SC in time varying Rician and Nakagami fading environments. However, the

SSC that is used in [24] is similar to the SSC developed by [26] in which a switch only

occurs at downward slope of receiving SNR. This mechanism is not as efficient as SEC

and so far, according to our knowledge, there has not been any research performed on

the switching rate for SEC. Therefore, in a later chapter, we derive the theoretical closed

form solution for SEC switching rate and compare it against the switching rate for SC.


2.5 System Model

In this section, we present the system model to be used in later chapters.

2.5.1 Correlated MIMO Channel Model

Based on equations (2.2), (2.4) and (2.5), the overall communication channel, H, in

Figure 2.1 can be defined as

H = ZR

√

ΨRHu√

ΨTZT (2.9)

where Hu is uncorrelated channel matrix defined in Section 2.1.

By substituting the above channel into Shannon’s capacity equation defined in [31], we

have the following mutual information equation which includes both spatial correlation

and mutual coupling:

I = log2(det(IM +Es

σ2NHHH))

= log2(det(IM +Es

σ2NZR

√

ΨRHu√

ΨTZTZTH(√

ΨT)H(Hu)H(

√

ΨR)HZR

H))

(2.10)

where IM is the M × M identity matrix. Es is the total transmitting power and σ2 is

the noise power.

2.5.2 Ring Model

In this thesis, to model multipath propagation with fading correlation, the “one-ring”

model illustrated in Figure 2.5 introduced by [14] is used. The receivers, Rx, are sur-

rounded by a ring of 1200 scatterers uniformly distributed around the ring. The horizontal

distance from transmitter to the centre of the receivers, D, is set to be 10000λ and the

radius of the ring is set to be 1000λ to ensure that the far field approximation applies.

β is the azimuth angle with the reference labeled in Figure 2.5.


r2

r1

Tx

Rx

β

D

Figure 2.5: One-ring model

The channel between a chosen transmitter and the mth receiver is given by [20]

(HNB)m =jη

2π

Ns∑

n=1

κne−j 2π

λ(r1n+r2nm)

r1nr2nm(2.11)

where r1n and r2nm are respectively the distance from the transmitter and themth receiver

antenna to the scatterer. Ns is the total number of scatterers, which is 1200 in this thesis.

η is the intrinsic impedance and κn is the scattering coefficient of nth scatterer and is

modeled as a complex Gaussian random variable with zero mean and unit variance. For

rich scattering case, the scatterers are uniformly distributed over the ring.

To model RF beamforming using the one-ring model, each entry in the channel matrix

between the mth beamforming pattern and the transmitter can be represented as follows:

(HRF )m =jη

2π

Ns∑

n=1

κne−j 2π

λ(r1n+r2n)

r1nr2ngm(βn) (2.12)

where gm(βn) is the mth beamforming pattern applied to the nth scatterer and r2n

is the distance from nth scatterer to the centre of the subarray. Note that the use of

closely spaced antennas and scatterers provides a natural approach to model inter-channel

correlation.

To model digital beamforming, the digital weight matrix, WH , can be applied onto

the Equation (2.11). The digital beamforming channel for mth digital beamforming


pattern can be modeled as follow:

(HDB)m = wHmHNB (2.13)

where wHm is the 1×M weight vector for the mth digital beamforming pattern.

Correlated Environment

Figure 2.6: Location of the scatterer in correlated scattering environment

The ring of scatterers model can also be used to model environments with greater

correlation. In correlated scattering environment, the one-ring model mentioned in the

previous section still applies. The only difference is the location of the scatterers in this

environment are concentrated around β = 90◦ (almost line of sight) instead of being

uniformly scattered, to reduce the angular spread of the transmitted signal. Figure 2.6

indicates one possible set of locations of the scatterers.

2.5.3 Doppler Frequency Shifts

The above model equations only considers the slow fading case in which the amplitude and

phase change imposed by the channel are constant over the period of use. These equations


will not be useful when comparing the switching rates between SC and SEC in later

chapter since the amplitude of the signal is always constant. Thus, the channel equations

above need to incorporate the model described in [14] which includes the Doppler spread

of the channel. By setting the receiver in Fig 2.5 to be moving, Equation (2.11) can be

modified as follows:

(HDoppler)m(t) =jη

2π

N∑

n=1

κn

exp (−j 2πλ(r1n + r2nm))

r1nr2nmexp (−j2π(fdt cos(βn + π/2)))

(2.14)

where fd is the maximum Doppler frequency shift and t is the overall sample time defined

as the kth sample time (t = kTs, Ts = sampling period). fd is derived as vmax

λwhere vmax

is the speed of the receiver.

To normalize the maximum Doppler frequency term in the above equation, the fd

and the Ts terms can be combined into one variable. Let the combined term be T ,

Equation (2.14) can be expressed as:

(HDoppler)m(kT/fd) =jη

2π

N∑

n=1

κn

exp (−j 2πλ(r1n + r2nm))

r1nr2nmexp (−j2π(kT cos(βn + π/2)))

(2.15)

Similar approach can be applied to RF-beamforming. The following is the channel

equation with Doppler frequency shifts for RF-beamforming:

(HRF Doppler)m(kT/fd) =jη

2π

N∑

n=1

κn

exp (−j 2πλ(r1n + r2n))

r1nr2nexp (−j2π(kT cos(βn + π/2)))gm(βn)

(2.16)

Chapter 3

Beamforming with Closely Spaced

Antennas

This chapter compares the performance between the cases of RF beamforming, digital

beamforming and element based processing (no beamforming) in a closely spaced antenna

system. Section 3.1 demonstrates the decrease in channel capacity due to mutual coupling

and spatial correlation. Section 3.2 compares the performance of the above three cases

when multiple RF-chains are available. Section 3.3 performs the same comparisons as

Section 3.2 for the scenario which only one RF-chain is available.

3.1 Capacity vs. Inter-element Distance

Figure 3.1 demonstrates the effect mutual coupling and spatial correlation has on mu-

tual information in a 2× 2 MIMO system. Note that mutual information is the figure of

merit here instead of capacity since we are assuming the transmitters have no information

regarding to the channel and thus the total power is divided equally among the transmit-

ters. The result in Figure 3.1 is generated by applying Equation (2.10). Table 3.1 details

the parameters that are used to generate Figure 3.1. In the case of no mutual coupling,

we let ZT = ZR = I. In the “uncorrelated” case, ΨT = ΨR = I as well. This graph

22

Chapter 3. Beamforming with Closely Spaced Antennas 23

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 24.6

4.7

4.8

4.9

5

5.1

5.2

5.3

5.4

5.5

5.6

Inter−element distance (λ)

Mu

tua

l In

form

atio

n (

b/s

/Hz)

Mutual CouplingSpatial Correlation OnlyUncorrelated

Figure 3.1: Mutual coupling and spatial correlation effects on mutual information


focuses on mutual coupling and spatial correlation on the receiver side. In the standard

cellular system since the base station is typically very large and has wide gap between

antennas, the transmitters are assumed to be 10λ apart such that no mutual coupling

and spatial correlation occurs.

Transmitter & receiver dipole antenna length 0.5λ

Transmitter & receiver dipole antenna radius 0.01λ

Transmitter power with respect to receiver noise power 10dB

Noise variance 1

Table 3.1: System specifications for Figure 3.1

As shown in Figure 3.1, the mutual information in the “uncorrelated” case is inde-

pendent of inter-element distance. “spatial correlation only” is the case in which only

spatial correlation exists and all mutual coupling between the dipole antennas is com-

pletely eliminated. Both “uncorrelated” and “spatial correlation only” are the ideal cases

which treats the dipoles as point sources. Even though these cases are impractical, they

provide us the upper bounds on performance. The “mutual coupling” case is the case

which both mutual coupling and spatial correlation exist. As can be seen in Figure 3.1,

both the “mutual coupling” and “spatial correlation only” cases converge to the “uncor-

related” case as the inter-element distance increases. As is clear from the figure, at low

inter-element spacings, the mutual coupling and spatial correlation significantly reduces

mutual information.


3.2 Comparing Beamforming Systems with Multiple

RF Chains Available

In this section, we compare the performance between digital beamforming, RF beamform-

ing and element based (no beamforming) systems in the case that multiple RF-chains are

available. We first analyze the performance of digital beamforming with different beam

patterns against the antenna based (no beamforming) system. We will then investigate

the performance of RF-beamforming under the same scenario.

3.2.1 Digital Beamforming vs. No Beamforming

This section first derives the mutual information equations for digital beamforming with

a fixed beamformer. The system model described in Section 2.5.1 is used here.

Based on the mutual information equations in [31], the mutual information for digital

beamforming is defined as:

I(x; y) = log2det (Σy)

det (Σn)(3.1)

where n = WHn is the effective noise after digital beamforming; Σy and Σn are the

Gaussian covariance matrix of the output and the weighted noise terms respectively.

Thus, based on Equation (2.8), the mutual information equation for digital beamform-

ing under the assumption that the channel information is unavailable to the transmitters

can be represented as follow:

I = log2 det

(

WHW

det(WHW)+

Es

Nσ2 det(WHW)WHHHHW

)

(3.2)

The detailed derivation from Equation (3.1) to Equation (3.2) is shown in Appendix B.

Thus, combining the above equation with Equation (2.9), Equation (2.10) can be


modified as:

I = log2

(

det

(

WHW

det (WHW)+

Es

σ2N det (WHW)WHZR

√

ΨRHu√

ΨTZTZTH(√

ΨT)H(Hu)H(

√

ΨR)HZR

HW

))

(3.3)

In the case of no mutual coupling, ZR = ZT = I. Equation (3.3) can be simplified as

I = log2

(

det

(

WHW

det (WHW)+

Es

σ2N det (WHW)WH

√

ΨRHuΨT(H

u)H(√

ΨR)HW

))

(3.4)

Beamforming Patterns

The following beamforming patterns are used to compare the mutual information per-

formance in digital beamforming in systems (N = 3, M = 10) with λ/2 and λ/10

inter-element spacing:

• Two beamforming patterns with main lobes at 45◦ and 135◦ generated by two

five antenna sub-arrays:

Figure 3.2a shows the beamforming patterns when inter-element spacing is λ/2.

Figure 3.3a shows the beamforming patterns when inter-element spacing is λ/10.

Figure 3.3a shows that due to the small spacing between the antennas, the beamwidths

are wider and there is more overlap between the beamforming patterns. Thus, the

correlation is expected to be higher as the inter-element distance decreases. The

weight matrix is given by:

W =1√5

1 ejγ e2jγ e3jγ e4jγ 0 0 0 0 0

0 0 0 0 0 e−5jγ e−6jγ e−7jγ e−8jγ e−9jγ

T

(3.5)

where γ represents the phased shift applied onto the digital beamforming weights

and γ = 2πλd cos(π

4). “T” denotes the transpose.


Using such beamforming weights, the digital beamforming noise (the WHn term

in Equation (2.8)) still remains uncorrelated (E[

WHnnHW]

= σ2I2). I2 is a 2×2

identity matrix.

• Two overlapping beam patterns with main lobes at 135◦ generated by five anten-

nas:

Figure 3.2b shows the beam pattern at λ/2 inter-element spacing. Figure 3.3b shows

the beamforming pattern at λ/10 inter-element spacing. Since the beamforming

patterns fully overlap together in this case, the magnitude of the correlation be-

tween the beamforming patterns is expected to be unity.

Wcorr =1√5

1 ejγ e2jγ e3jγ e4jγ 0 0 0 0 0

0 0 0 0 0 e5jγ e6jγ e7jγ e8jγ e9jγ

T

(3.6)

Using Wcorr as the beamforming weight matrix, the digital beamforming noise is

also uncorrelated (E[

WHnnHW]

= σ2I2).

• Butler Matrix (10 orthogonal weight vectors)

In [32], J. Butler has developed the “Butler Matrix” to mimic the discrete Fourier

transform (DFT) electronically. Using the Butler matrix as the weight matrix

allows each beamforming pattern to be orthogonal to other beamforming pat-

terns [33]. A detailed explanation of the realization of a Butler Matrix can be

found in [34]. Figure 3.2c and Figure 3.3c show all possible beamforming patterns

at λ/2 and λ/10 inter-element spacings respectively. In Figure 3.3c, beside Beam-

formers #1, #2, and #10, all other beamformers have very small magnitude due to

the narrow visible spectrum caused by the small inter-element spacing. The weight

matrix is given by:


WBM =1√10

1 1 · · · 1 1

1 ω · · · ω8 ω9

1 ω2 · · · ω16 ω18

......

......

...

1 ω8 · · · ω64 ω72

1 ω9 · · · ω72 ω81

T

(3.7)

where ω is ej2π10 .

WBM also gives uncorrelated digital beamforming noise since E[

WHBMnnHWBM

]

=

σ2I10 where I10 is the 10× 10 identity matrix.

• Narrower beamforming patterns at 45◦ and 135◦ by utilizing all 10 antennas:

By utilizing more antennas, the beamforming patterns here have a narrower main

lobe and lower side lobes than the previous case that used the same angles. Fig-

ure 3.2d demonstrates the beamforming pattern when inter-element spacing is λ/2.

Figure 3.3d shows the beamforming patterns when inter-element spacing is λ/10.

W10Ant =1√10

1 ejγ e2jγ e3jγ e4jγ e5jγ e6jγ e7jγ e8jγ e9jγ

1 e−jγ e−2jγ e−3jγ e−4jγ e−5jγ e−6jγ e−7jγ e−8jγ e−9jγ

T

(3.8)

However, even though utilizing all ten antennas gives narrower main lobes and lower

side lobes, it makes the digital beamforming noise of the system to be correlated

(E[

WH10Antnn

HW10Ant

]

6= σ2I10).


0.5

1

1.5

2

2.5

30

210

60

240

90

270

120

300

150

330

180 0

Mainlobe at 135°

Mainlobe at 45°

(a) Main lobes at 45◦ and 135◦

0.5

1

1.5

2

2.5

30

210

60

240

90

270

120

300

150

330

180 0

Mainlobe at 135°

Mainlobe at 135°

(b) Both main lobes at 135◦

Figure 3.2: Different antenna patterns with inter-element spacing of λ/2


1

2

3

4

30

210

60

240

90

270

120

300

150

330

180 0

Beamformer #2Beamformer #1Beamformer #3Beamformer #4Beamformer #5Beamformer #6Beamformer #7Beamformer #8Beamformer #9Beamformer #10

(c) Butler matrix

1

2

3

4

30

210

60

240

90

270

120

300

150

330

180 0

Mainlobe at 135°

Mainlobe at 45°

(d) Utilizing all 10 antennas



1

2

3

30

210

60

240

90

270

120

300

150

330

180 0

Mainlobe at 135°

Mainlobe at 45°

(a) Main lobes at 45◦ and 135◦

1

2

3

30

210

60

240

90

270

120

300

150

330

180 0

Mainlobe at 135°

Another Mainlobe at 135°

(b) Both main lobes at 135◦



1

2

3

4

30

210

60

240

90

270

120

300

150

330

180 0

Beamformer #1Beamformer #2Beamformer #3Beamformer #4Beamformer #5Beamformer #6Beamformer #7Beamformer #8Beamformer #9Beamformer #10

(c) Butler matrix

1

2

3

4

30

210

60

240

90

270

120

300

150

330

180 0

Mainlobe at 135°

Mainlobe at 45°

(d) Utilizing all 10 antennas



Mutual Information Comparisons for λ/2 Spacing

Figure 3.4 and Figure 3.6 demonstrate the change in mutual information with respect

to transmitting SNR with λ/2 and λ/10 inter-element spacings. Table 3.2 lists the

specifications for the simulation environment.

Number of transmitting antennas 3

Number of receiving antennas 10

Number of beamforming patterns 2 or 10

Transmitter & receiver antenna length 0.5λ

Transmitter & receiver antenna radius 0.01λ

Noise variance 1

Channel realizations 100000

Table 3.2: System specifications for Figure 3.4 and Figure 3.6

Figure 3.4 shows the mutual information comparisons for different digital beamform-

ing patterns when inter-element spacing is λ/2. “Mutual Coupling” is the case which

has both mutual coupling and spatial correlation presented in Equation (3.3). “No Mu-

tual Coupling” is the case in which only spatial correlation is present. “Uncorrelated”

is the case in which neither spatial correlation nor mutual coupling is present. Since in

this thesis, our focus is based on assuming the dipole antennas to behave like perfect

metamaterial antennas which remove mutual coupling completely, digital beamforming

technique is performed on the “No Mutual Coupling” case. Even though in practice,

beamforming patterns cannot be treated independently from mutual coupling, we tem-

porarily neglect mutual coupling to keep the problem tractable. In this thesis, we focus

on the investigation of the effects that different beamforming patterns have on spatial

correlation. The analysis which combines mutual coupling into beamforming patterns

will be left as future work.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

2

3

4

5

6

7

8

9

10

11

Transmitter Power with respect to Recv Noise (dB)

Mu

tua

l In

form

atio

n (

bits

/s/H

z)

Mutual CouplingNo Mutual CouplingUncorrelatedbeam Steering (45 & 135)beam Steering (135 & 135 )beam Steering (DFT)beam Steering (10 Antennas)

Figure 3.4: Mutual information for digital beamforming systems with λ/2 inter-element

spacing

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25


Mu

tua

l In

form

atio

n (

bits

/s/H

z)

Mutual CouplingNo Mutual CouplingUncorrelatedbeam Steering (45 & 135)beam Steering (135 & 135)beam Steering (DFT)beam Steering (10 Antennas)

Figure 3.5: Mutual information for digital beamforming systems with λ/10 inter-element Spac-

ing


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

2

3

4

5

6

7

8

9

10

11


Mu

tua

l In

form

atio

n (

bits/s

/Hz)

Mutual CouplingNo Mutual CouplingUncorrelatedbeam Steering (45 & 135)beam Steering (135 & 135)beam Steering (DFT)beam Steering (10 Antennas)

Figure 3.6: Mutual information for digital beamforming systems with λ/10 inter-element spac-

ing (zoomed-in of Figure 3.5)


“Beam Steering DFT” is the case which the Butler matrix is used. “Beam Steering

(45◦ and 135◦)” is the case which W is used. “Beam Steering (135◦ and 135◦)” is the

beamforming case in which overlapping beamforming patterns are used. Finally, “Beam

Steering 10 Antennas” is the beamforming case where W10Ant is used.

In Figure 3.4, the “Uncorrelated” case is independent of inter-element distance and

has the highest mutual information. “No Mutual Coupling” has the second largest mutual

information. The “DFT” case has same mutual information as “No Mutual Coupling”

case since WBm only rotates the channel matrix without changing the eigenvalues of the

channel. Thus, this result demonstrates the data processing inequality theorem [31].

The other three beamforming methods have lower mutual information than the “Beam

Steering DFT” case since these beamforming methods only have two beamforming pat-

terns (RF-chains) whereas the “Beam Steering DFT” case has ten beamformers (RF-

chains). Thus, it is not fair to compare “Beam Steering DFT” against the other three

digital beamforming systems. The main motivation of showing “Beam Steering DFT”

here is to demonstrate the data processing inequality theorem.

As shown in Figure 3.4, “Beam Steering (135◦ and 135◦)” and “Beam Steering (45◦

and 135◦)” have relatively same capacity due to their similar receive SNR and correlation.

Even though, based on intuition, “Beam Steering (135◦ and 135◦)” should have the worst

capacity due to its overlapping patterns, the correlation caused by λ/2 inter-element

spacing is not large enough to cause any significant change. Finally, since ”Beam Steering

10 Antennas” utilizes all ten antennas, its received signal has higher correlation and

thereby decreases capacity. Also, its correlated noise caused by utilizing all ten antennas

contributes to the correlation of the system.

Mutual Information Comparison for λ/10

Figure 3.6 shows the mutual information of the same beamforming patterns mentioned

above at λ/10 inter-element spacing. In this plot, the “Mutual Coupling” case has the


smallest mutual information and the graph is not linear due to the inverse matrix in

Equation (2.5). However, since it has more RF-chains than the two RF-chains cases, it

has higher capacity slope gain than “Beam Steering (45◦ and 135◦)”, “Beam Steering

(135◦ and 135◦)” and “Beam Steering 10 Antennas”.

In the same figure, “Beam Steering 10 Antennas” has a higher mutual information

than the other two beamforming cases due to its higher antenna SNR gain by utilizing all

ten antennas even though it has the highest correlation. “Beam Steering 45◦ and 135◦”

has higher capacity than “Beam Steering 135◦ and 135◦” due to its lower correlation.

Figure 3.5 is the zoomed-in plot of Figure 3.6. It has the same axis as Figure 3.4

to compare the mutual information changes when going from λ/2 to λ/10 inter-element

spacing. Comparing Figure 3.4 with Figure 3.6, we see the reduction in mutual informa-

tion for both “No Mutual Coupling” and “Mutual Coupling” cases as the inter-element

distance decreases. This is due to the increase in the correlation between the individual

antennas. Surprisingly, we see an increase in mutual information for the ‘Beam Steering

(135◦ and 135◦)”, “Beam Steering (45◦ and 135◦)” and “Beam Steering 10 Antennas”

as the inter-element distance between antennas decreases. This increase of mutual in-

formation is due to the increase of SNR. As the inter-element distance decreases, the

beamwidth of the beamforming patterns become wider and covers larger angle of arrival.

Even though wider beamwidth increases the correlation, it increases the receiving SNR

as well. In here, we have demonstrated the impact that different beamforming patterns

have on the performance of the digital beamforming system. To optimize the perfor-

mance of digital beamforming, we need to find beamforming patterns which maximize

receiving SNR while minimize the correlation between the beamforming patterns.

Summary

The findings in Figure 3.4, Figure 3.5 and Figure 3.6 illustrate three key factors that

affect the mutual information of the system:


1. Receive SNR: Receive SNR is the diagonal terms in channel correlation matrix

(HHH). With larger diagonal values, the determinant term in the mutual infor-

mation equation grows larger. In addition, given a fixed total receiver SNR, the

determinant (mutual information) can be further improved by evenly distributing

receive SNR among all the receivers [35].

2. Correlation: The correlation of the channel is represented in the non-diagonal

terms in the channel correlation matrix (HHH). The determinant can be maxi-

mized by having zeros on the non-diagonal terms. A detailed discussion of correla-

tion and capacity on be found in [19,36,37].

3. Noise Correlation: Besides noise power, the correlation of the noise between

different channels may also affect the capacity. As demonstrated in the example of

“Beam Steering (10 Antennas)”, utilizing all ten antennas causes the noise to be

coloured and undermines the capacity.

Moreover, digital beamforming cannot improve mutual information due to the data

processing inequality.

3.2.2 RF Beamforming

In the previous section, the data processing inequality indicated that digital beamforming

cannot improve capacity. This demonstrates the futility in using digital beamforming to

improve mutual information. However, mutual information might be improved using

RF-beamforming. This is because the RF beamformer operates before the main source

of noise in the low noise amplifier (LNA).

The derivation of mutual information for RF beamforming is very straight-forward.

Since the weight matrix is applied before the noise component in Equation (2.7), the

WHH term can be substituted as a new correlated channel matrix. Thus, combining


Equation (2.7), Equation (2.9) and Equation (2.10), the mutual information equation for

RF beamforming is as follows:

I = log2

(

det

(

IM+

Es

σ2NWHZR

√

ΨRHu√

ΨTZTZTH(√

ΨT)H(Hu)H(

√

ΨR)HZR

HW

))

(3.9)

By assuming the transmitter antennas are uncorrelated and the receiver antennas are

perfect metamaterial antennas which give no mutual coupling, Equation (3.9) can be

simplified as below:

I = log2

(

det

(

IM +Es

σ2NWH

√

ΨRHu(Hu)H(

√

ΨR)HW

))

(3.10)

To maximize the above mutual information equation, WH can be set as (√ΨR)

−1

given that (√ΨR) is invertible. In this way, WH can always fully decorrelate the channel.

The optimal WH given (√ΨR) is not invertible has yet to be found. Even though we

have found a theoretically optimal WH for a specific case, such a solution cannot be

easily implemented since it is hard to create arbitrary phase and magnitude shifts at RF.

Therefore, one of the important questions raised here is that with limited number

of RF-chains, how can BER be minimized?

Optimal Beamforming

In the standard MIMO system model equation defined in Equation (2.6), one can de-

compose H into H = UΛVH using singular value decomposition. The columns of U are

the M eigenvectors of HHH and the columns of V are the N eigenvectors of HHH. The

M ×N matrix Λ is a diagonal matrix of singular values. Each singular value represents

the power of each parallel channel.

With the constraint of K RF-chains, the optimal beamforming scheme is to pick K

eigenvectors that have the largest K singular values. This idea is similarly applied to the


MUSIC algorithm [38]. In this way, most of the power provided by the MIMO channels

can be obtained.

Performance of Optimal Beamforming Technique

Figure 3.7 is the BER plot for the optimal beamforming technique introduced above. In

here, we only consider the case in which only spatial correlation exists. The Alamouti

code described in [39] which achieves transmit diversity is used here. Table 3.3 lists the

parameters used to generate this plot.


Number of receiver antennas 10

Number of RF-chains (beamforms) 2

Receiver antenna inter-element distance λ/2

Noise variance 1


In Figure 3.7, the “Butler Matrix Beamforming” is the beamforming case that picks

the two orthogonal weight vectors in Butler matrix to generate the orthogonal patterns

in Figure 3.8. “Uncorrelated” is the case which has neither mutual coupling nor spatial

correlation. “Spatial Correlation” is the case in which only spatial correlation is present.

Since no beamforming is performed on these two cases, both of them have total of 10

RF-chains at the receiver. “Optimal Beamforming” and “Butler Matrix Beamforming”

only have two RF-chains. The significance of Figure 3.7 is to demonstrate that with fewer

number of RF-chains, the “optimal beamforming” has almost same performance as the

“Spatial Correlation” case which has ten RF-Chains1. The motivation in presenting the

1Note that in here “Optimal Beamforming” has almost same performance as “Spatial Correlation”since with two transmitters, there are only two eigen-vectors which have non-zero eigenvalues. By acquir-ing those two eigenvectors, we obtain the entire transmitter power. Thus, if the number of transmittersis three, there will be a performance gap between “Optimal Beamforming” and “Spatial Correlation”.


“Butler Matrix Beamforming” case is to demonstrate the performance improvement that

“optimal beamforming” has over “Butler Matrix Beamforming” given the same number

of RF-chains.

Figure 3.9 is the BER plot in which the inter-element spacing has been shortened to

λ/10. The specifications are still the same as listed in Table 3.3 with “Receiver Antenna

Inter-element Distance” change to λ/10. Figure 3.10 is the Butler Matrix beamforming

pattern when the inter-element distance is λ/10. As can be seen in this figure, “Beam-

former #2” has very small magnitude due to the narrowing visible spectrum. Thus,

in this case, only one out of the two beamforming patterns will be useful. This cre-

ates the diversity loss for the “Butler Matrix Beamforming” case in Figure 3.9. Also,

Figure 3.9 shows that with smaller inter-element spacing, “Optimal Beamforming” still

performs as well as the “Spatial Correlation” case. Moreover, even though both “Opti-

mal Beamforming” and “Butler Matrix Beamforming” cases have the same number of

RF-chains, “Optimal Beamforming” has significant performance improvement over the

“Butler Matrix Beamforming” case.

To summarize our findings, this sections has discussed the possibility of using RF-

beamforming to compensate for the effect of mutual coupling and correlation. With no

constraint on the number of RF-chains and equal number of transmitters and receivers,

the RF-beamforming pattern can be designed as the inverse of the correlated MIMO

channel matrix to decorrelate the channel. With a constraint on the number of RF-

chains, the optimal beamforming technique discussed above can be applied.

3.3 Single RF Chain

The previous section has discussed the results of performing digital and RF beamform-

ing for systems that allow multiple RF-chains. However, having multiple RF-chains in a

handheld device may be impractical, since, as discussed in [17], multiple RF-chains re-


0 2 4 6 8 10 1210

−4

10−3

10−2

10−1

100

Transmitter Power with Respect to Recv Noise (dB)

BE

R

UncorrelatedSpatial Correlation onlyButler Matrix BeamformingOptimal Beamforming

Figure 3.7: BER for optimal beamforming with λ/2 inter-element spacing

1

2

3

4

30

210

60

240

90

270

120

300

150

330

180 0

Beamformer #1Beamformer #2

Figure 3.8: Butler matrix beamforming patterns for λ/2 inter-element spacing


0 2 4 6 8 10 12 14 16 1810

−4

10−3

10−2

10−1

100


BE

R

UncorrelatedSpatial Correlation onlyButler Matrix beamformingoptimal beamformed Antenna

Figure 3.9: BER for optimal beamforming patterns with λ/10 inter-element spacing

1

2

3

4

30

210

60

240

90

270

120

300

150

330

180 0

Beamformer #1Beamformer #2

Figure 3.10: Butler matrix beamforming patterns for λ/10 inter-element spacing


quire more hardware space and higher power consumption. Thus, this section is dedicated

to the case in which only a single RF chain is available in the receiver of a MIMO sys-

tem. In this section, our system model is based on the “One-ring Model” described in

Section 2.5.2.

3.3.1 Diversity Combining Beamformers

Since in the previous section, we have demonstrated the effect that different beamform-

ing patterns have on mutual information, this section will be dedicated to investigate

the performance of different beamforming patterns in a single RF-chain system. Start-

ing from this chapter, our system model is based on the “One-ring Model” described

in Section 2.5.2. To combine multiple signals from the received antennas, a diversity

combining scheme is needed. Many investigations have analyzed the performance of dif-

ferent receiver combining schemes. The works in [14–16, 40] have the detailed analysis

and simulation of different receiver combining schemes. The following are the different

diversity combining schemes that will be considered:

• Fixed Beamforming toward Transmitter: Always pick the beamforming pattern

that is pointed toward the transmitter

• Selection: Pick the beamformer that has the highest SNR

• Random Selection: Pick beamformer randomly

• Ideal Sectorized Patterns: This case applies the same method as the “Selection”

case with the difference being that the perfect antenna without sidelobes are used.

This is the ideal case of beamforming selection since there are no overlaps between

the beamforming patterns.

• Maximal Ratio Combining (MRC): Set the weight at each scatterer accordingly to


be the conjugate of the channel. In this way, the output SNR is always maximized. 2

• Equal Gain Combining (EGC): Set the magnitude of the weight to be constant and

the phase to be the conjugate of the channel similar to MRC.

• Power Matching: Set the phase of the weight to be constant and the magnitude of

the weight to match the magnitude of the channel. This is the opposite of EGC.

Note that these forms of MRC, EGC and power matching are impossible to implement in

practice. Also, the beamforming patterns of “Ideal Sectorized Patterns” are impossible

to be generated.

Figure 3.11 and Figure 3.12 plot the beamforming patterns mentioned above for the

inter-element distance of λ/2 and λ/10 respectively. The x-axis in the figures is the

angle of arrival (β in Figure 2.5). All the patterns in here are normalized to have the

unit average power. Figure 3.11a includes the “fixed beamforming”, “MRC”, “EGC”,

and “Power Matching” cases mentioned above. Due to the definition of β in Figure 2.5,

“Fixed Pattern”, “MRC” and “Power Matching” have the strongest magnitude at in the

direction of the transmitter (β = π/2). Since ’EGC’ does not change the magnitude of

the weight, its magnitude is constant in both Figure 3.11a and Figure 3.12a.

Figure 3.11b and Figure 3.12b plot the beamforming patterns that form the selection

set for “Selection” and “Random Selection” for λ/2 and λ/10 inter-element distance re-

spectively. Figure 3.11c is the “Ideal Sectorized Patterns” case in which has six equally

divided non-overlapping sectors. This is the most ideal case of beamforming since there

are no side lobes and the main lobes are extremely directive. However, such a beamform-

ing pattern cannot be realized.

The g(βn) in Equation (2.12) for the above beamforming patterns can be represented

as below:

2The MRC mentioned here is the MRC for each scatterer. It’s not the MRC on beamforming patterns.MRC on beamforming patterns will be investigated in Section 3.3.2


0 50 100 150 200 250 300 3500

1

2

3

4

5

6

7

8

9

angle of arrival (Degree)

Be

am

form

ing

Pa

tte

rn P

ow

er

fixed patternMRCEGCPower Matching

(a) Beamforming for MRC, EGC, Fixed Beam, and Power Matching

0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

main lobe at 0 degreemain lobe at 30 degreemain lobe at 60 degreemain lobe at 90 degreemain lobe at 120 degreemain lobe at 150 degree

(b) Beamforming patterns for Selection and Random Selection

Figure 3.11: Different antenna array patterns at λ/2 inter-element distance


0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

First BeamformSecond BeamformThird BeamformFourth BeamformFifth BeamformSixth Beamform

(c) Ideal sectorized patterns



0 50 100 150 200 250 300 3500.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

angle of arrival (degree)

Be

am

form

ing

Pa

tte

rn P

ow

er

fixed patternMRCEGCPower Matching

(a) Beamforming for MRC, EGC, Fixed Beam, and Power Matching

0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

main lobe at 0 degreemain lobe at 30 degreemain lobe at 60 degreemain lobe at 90 degreemain lobe at 120 degreemain lobe at 150 degree

(b) Beamform patterns for Selection and Random Selection



• : Maximal Combining Ratio: g(βn) = κ∗nexp(+j 2π

λ(r1n+r2n))

r1nr2n

• : Equal Gain Combining: g(βn) = exp(j( 6 κn +2πλ(r1n + r2n)))

• : Power Matching: g(βn) = | κn

r1nr2n|

• : Fixed Beam Pattern: g(βn) = exp (−j(2πλd)(cos(βn)− cos(π

2)))

• : Selection: pick the beamformer in Figure 3.11b or Figure 3.12b that has the

highest received SNR

• : Random: randomly pick beamformer in Figure 3.11b or Figure 3.12b

• : Ideal Sectorized Patterns: Pick the sector in Figure 3.11c that gives the highest

received SNR

Capacity & BER vs SNR for Diversity Combining

Figure 3.13 plots the BER for the different diversity combining schemes mentioned above

for λ/2 inter-element distance. Table 3.4 provides the system specifications.


Number of receiver antennas 6

Receiver antenna inter-element distance λ/2

Noise variance 1

Number of scatterers 1200



The difference between “EGC” and “Power Matching” demonstrates the improve-

ment of knowing the phase information instead of only knowing the magnitude. Also,

the “Selection” case has similar performance as “Ideal Sectorized Patterns” since both


beamforming patterns are still diverse enough when the inter-element distance is λ/2.

Furthermore, Figure 3.13 shows some loss of diversity for the “Power Matching”, “Ran-

dom Picking” and the “Fixed Pattern” case. Due to the normalization of the beamform-

ing pattern, “Random Picking” and “Fixed Pattern” have the same performance in this

case.

Figure 3.14 is the BER plot when the inter-element distance is λ/10. Since the

“MRC”, “EGC”, “Power Match” and “Ideal Sectorized Patterns” cases do not depend

on inter-element distance, their performance do not differ from Figure 3.13. As inter-

element distance decreases, Figure 3.14 shows the diversity order for “Selection” has

decreased.

Based on the results above, we can rank the performance of the different beamforming

patterns as follows:

1. MRC

2. EGC

3. Ideal Sectorized Patterns

4. Selection

5. Power Matching

6. Random and Fixed Beamforming

Even though in this section, we have shown that changing inter-element spacing does

not change the above ranking order, this is not the main focus of this section. The

main purpose of this section is to find the receiver diversity combining scheme that has

the best performance while satisfying the constraints of having single RF-chain and is

realizable. Since from the above choices, only “Selection”, “Random Beamforming” and

“Fixed Beamforming” satisfy these constraints, our main focus in the next section will

be on beamforming selection.


0 10 20 30 40 50 60 70 8010

−3

10−2

10−1

100

Transmitting Power with Repect to Recv Noise(dB)

BE

R

fixed patternMRCEGCPower MatchingSelectionRandom PickIdeal Sectorized Patterns

Figure 3.13: BER for diversity combining schemes in λ/2 inter-element spacing

0 10 20 30 40 50 60 70 8010

−3

10−2

10−1

100

Transmitter Power with Respect to Recv Noise(dB)

BE

R

fixed patternMRCEGCPower MatchingBest of 6 beamsRandom PickIdeal Sectorized Patterns

Figure 3.14: BER for diversity combining schemes in λ/10 inter-element spacing


3.3.2 RF Beamforming Selection vs Antenna Element Selection

Many researchers have compared the performance of beamforming selection against an-

tenna selection. For instance, [39, 41] have demonstrated beamforming selection outper-

forms the antenna selection technique when the Line of Sight (LOS) component has the

highest signal strength in the channel. Moreover, [42] shows that in a MIMO system with

more transmit antennas than receive antennas, if the channel is very correlated, serious

degradation of BER performance can be prevented by applying beamforming. However,

these studies only consider the case when the inter-element distance is λ/2. This dis-

tance is still too great to be fit into a small handheld device and also too far apart to

cause significant correlation to impact the performance of beamforming selection against

antenna selection. Therefore, to ensure the practicality and the simplicity of the system,

we consider closely spaced antenna systems. This work also appeared in [43].

The one-ring model described in Section 2.5.2 is used here. Equation (2.11) represents

each entry in the channel matrix. Similarly, Equation (2.12) represents each entry in the

channel matrix for RF-beamforming. As a point of comparison, we also develop the

theoretically optimal MRC cases with beamforming. To focus on the receiver, we use a

single transmit antenna.

Figure 3.11b and Figure 3.12b illustrate the phased array beamforming patterns used

in this section. In many current wireless communication research publications, the inter-

element distance is set to λ/2. In this case, as can be demonstrated by Figure 3.11b

and Figure 3.12b, the beamforming pattern in λ/2 is more diverse than the beamforming

pattern in λ/10 case. Thus, due to the more diverse beamforming patterns in λ/2 inter-

element distance, we expect the simulation result to demonstrate very little impact on

capacity from the lack of correlation.


Capacity Derivation

This section derives the capacity on using SC and MRC for both RF beamforming and

digital beamforming for a single transmitter. Note that capacity is the figure of merit

here instead of mutual information since there is only one transmitter.

The following is the capacity equation for RF beamforming SC:

C = log2

(

1 +Es

Nσ2hRF maxh

HRF max

)

(3.11)

where hRF max represents the channel in HRF which has the highest SNR.

For the digital beamforming case, based on Equation (2.11), the capacity equation

can be represented as

C = log2

(

1 +Es

Nσ2(WHmaxWmax)

WHmaxHHHWmax

)

(3.12)

where WHmax is the digital beamforming weight vector that gives the highest output SNR

in the channel.

For the MRC case in which the Hermitian of the channel is multiplied to the received

signal to achieve maximum output SNR, the capacity for RF beamforming can be ex-

pressed similarly as Equation (3.2) by setting the WH term as HRFH and the channel

matrix as HRF:

C = log2

(

1 +Es

Nσ2(HHRFHRF )

(HHRFHRF )

2

)

. (3.13)

Note that since there is only one transmitter in this case, the determinant term in Equa-

tion (3.2) can be dropped.

For the digital beamforming case, the mutual information can be derived as:

C = log2

(

1 +Es

Nσ2(HHWHWH)(HHH)2

)

, (3.14)

where H represents WHH.


BER & Capacity Performance

This section compares the BER and the capacity plots for the rich scattering environment

described in the one-ring model in Section 2.5.2. Figure 3.15 and Figure 3.16 represent the

BER and capacity performance for a rich scattering environment with λ/2 inter-element

distance. Figure 3.17 and Figure 3.18 represent the BER and capacity performance for

rich scattering environment with λ/10 inter-element distance. Note the high transmitter

SNR in these figures is due to the large fading term in Equation (2.12).

From Figure 3.15 and Figure 3.16, “RF Beamforming (MRC)” has best performance

in both BER and Capacity since it always maximizes the output SNR [15]. Moreover,

Figure 3.16 indicates that Digital Beamforming SC has the same capacity as RF Beam-

foring SC. Also, due to the lack of correlation for the λ/2 case, there is no significant

difference between beamforming patterns selection and antenna selection.

From Figure 3.17 and Figure 3.18, the performance gap between beamforming SC and

antenna element selection in both BER and capacity is now observable. This suggests

that the correlation between beamforming patterns is less than the correlation between

closely spaced antenna elements.

BER & Capacity Performance for Correlated Scattering Environment

In this section, we compare the performance between selecting amongst beamforming

patterns and selecting amongst antenna elements in the correlated scattering environment

described in the one-ring model in Section 2.5.2.

Figure 3.19 and Figure 3.20 represent the BER and capacity performance for a corre-

lated scattering environment with λ/2 inter-element distance. Figure 3.21 and Figure 3.22

represent the BER and capacity performance for correlated scattering environment with

λ/10 inter-element distance.

In Figure 3.19 and Figure 3.20, the performance of the MRC case converges to that of

the other beamforming cases. This is because in a correlated scattering environment the


30 40 50 60 70 80 9010

−4

10−3

10−2

10−1

100


BE

R

RF beamforming (SC)RF beamforming (MRC)Antenna Element SCDigital Beamform (SC)

Figure 3.15: BER performance for rich scattering environment with λ/2 inter-element

distance

30 40 50 60 70 80 90 100 110 120 1300

5

10

15

20

25

Transmitter with respect to Recv SNR (dB)

Ca

pa

city

(b

its/s

/Hz)

RF Beamforming (SC)RF Beamforming (MRC)Digital Beamforming (SC)Antenna Element SC

Figure 3.16: Capacity performance in rich scattering environment for λ/2 inter-element

distance


30 40 50 60 70 80 90 10010

−4

10−3

10−2

10−1

100


BE

R

RF beamforming (SC)RF beamforming (MRC)Antenna Element SelectionDigital Beamform (SC)

Figure 3.17: BER performance for rich scattering environment with λ/10 inter-element

distance

30 40 50 60 70 80 90 100 110 120 1300

5

10

15

20

25


Ca

pa

city

(b

its/s

/Hz)

RF Beamforming (SC)RF Beamforming (MRC)Digital Beamforming (SC)Antenna Element Selection

Figure 3.18: Capacity performance in rich scattering environment for λ/10 inter-element

distance


signal only arrives from one direction, which means even in beamforming MRC essentially

only one beamformer is useful. This is literally the same as the beamforming selection

case.

From Figure 3.21 and Figure 3.22 shows a significant improvement that the beam-

forming system has over antenna based system. This can be explained in receive SNR

sense. In an extremely correlated case, the transmitter SNR is basically all concentrated

on one beamformer and therefore by selecting the best beamformer, all of the transmitter

SNR can be obtained. However, in antenna element selection case, even with correlated

scattering, the transmitter SNR is equally divided among all six antennas, so basically

selection only takes one-sixth of the transmitter SNR. This demonstrates that in such

an extremely correlated environment beamforming diversity has surpassed the antenna

diversity.

3.3.3 Summary

To summarize the above results, in closely spaced antenna system with only one RF

chain and transmitter, RF beamforming SC has higher capacity than the traditional

antenna element selection method. Since beamforming based MIMO processing has

better performance in both rich and correlated scattering environment, beamforming

based MIMO processing is more robust to different scattering in closely spaced antennas

system.

3.4 Chapter Summary

In this chapter, we have compared the performance between the cases of RF-beamforming,

digital beamforming and element based processing in a closely spaced antenna system.

In the case of multiple RF-chains, due to the data processing inequality theorem,

digital beamforming cannot perform better than the element based beamforming. RF-


30 40 50 60 70 80 90 10010

−4

10−3

10−2

10−1

100


BE

R

RF beamforming (SC)RF beamforming (MRC)Antenna Element SCDigital Beamforming (SC)

Figure 3.19: BER for correlated scattering environment in λ/2 case

30 40 50 60 70 80 90 100 110 120 1300

5

10

15

20

25


Ca

pa

city

(b

its/s

/Hz)

RF Beamforming (SC)RF Beamforming (MRC)Digital Beamforming(SC)Antenna Element SC

Figure 3.20: Capacity for correlated scattering environment in λ/2 case


30 40 50 60 70 80 90 10010

−4

10−3

10−2

10−1

100

Transmitter SNR (dB)

BE

R

RF beamforming (SC)RF beamforming (MRC)Antenna Element SCButler Matrix (SC)Digital Beamform (SC)

Figure 3.21: BER for correlated scattering environment in λ/10 case

30 40 50 60 70 80 90 100 110 1200

5

10

15

20

25


Ca

pa

city

(b

its/s

/Hz)

RF Beamforming (SC)RF Beamforming (MRC)Digital Beamforming(SC)Antenna Element SC

Figure 3.22: Capacity for correlated scattering environment in λ/10 case


beamforming can decorrelate the channel by setting the RF beamforming pattern to

be the inverse of the correlated MIMO channel. However, this method requires the

knowledge of the channel and might be impractical to be applied in handheld system.

In the case of a single RF-chain, we have demonstrated that the RF beamform selec-

tion technique has higher capacity than the traditional antenna element selection method.

Since it has better performance in both rich and correlated scattering environment, it is

more robust to different scattering environments.

Chapter 4

Switch and Examine Combining

Techniques

The previous chapter showed that selection combining, especially beamforming based

selection combining, provides good performance in systems that allow for only a single

RF-chains. In this chapter, we explore the implications of such a scheme in terms of the

rate at which the selection would have to change, i.e., the switching rate. Specifically,

the focus here is on switch and examine combining.

This chapter first explains the algorithm for SEC in detail. Then, it derives the closed

form solution for the switching rate of SEC and matches the theory against Monte Carlo

simulations. Next, this chapter compares the switching rate of SEC against the switching

rate of SC. Finally, the switching rate of SEC in a beamforming system will be compared

against the SEC switching rate in a system without beamforming.

4.1 Switch and Examine Algorithm

As mentioned previously, Selection and Examine Combining (SEC) is similar to SSC

with the only difference being at when the chosen branch is below the threshold, the

system checks whether the other branch is also below the threshold. If the other branch

61

Chapter 4. Switch and Examine Combining Techniques 62

is below the threshold as well, the system will stay at the current branch and declare

an outage. Figure 4.1 and Figure 4.2 show the algorithm for SC and SEC respectively.

In the following section, we derive the switching rate for SEC based on the analytical

framework provided by Markov chains.

4.2 SEC Theoretical Model

To analyze the switching rate for the above SEC algorithm, the Markov chain analytical

framework similar to [29] is applied to the Doppler system model described in [30]. For

ease of explanation, the analysis focuses on two receive antennas.

Figure 4.3 illustrates the Markov chain state diagram. The four states in the Markov

chains are:

1. Branch #1: Branch #1 is selected

2. Branch #2: Branch #2 is selected

3. Outage From Branch #1: Branch #1 is selected and Outage is declared

4. Outage From Branch #2: Branch #2 is selected and Outage is declared

Figure 4.3 can be expanded to support N receivers by having 2N states since every

branch will have a selection state and an outage state.

Each edge in Figure 4.3 represents transition with transitional probability, which is

the probability of transitioning from the ith state to the jth state. Let si[n] represent the

SNR of the ith branch at time instant n and ST is the threshold SNR. The transitional

probability in Figure 4.3 can be defined as follows:

• P11 = p (s1[n] > ST | s1[n− 1] > ST )

• P12 = p (s1[n] < ST , s2[n] > ST | s1[n− 1] > ST )


Check One

Branch Check next

Branch

Switch to

Current Branch

<Current Branch

SNR

Switch to

Next Branch

>Cu

rrent

Branch

SNR

Figure 4.1: SC algorithm

Check the

Level of

Current Branch

Check next

Branch

Stay at Current

Branch &

Declare Outage

<Threshold

Switch Branch> Thresho

ld

<Threshold

Stay

>Th

reshold

Figure 4.2: SEC algorithm


Figure 4.3: Markov chain representation for SEC


• P13 = p (s1[n] < ST , s2[n] < ST | s1[n− 1] > ST )

• P14 = 0

• P21 = p (s1[n] > ST , s2[n] < ST | s2[n− 1] > ST )

• P22 = p (s1[n] < ST , s2[n] > ST | s2[n− 1] > ST )

• P23 = 0

• P24 = p (s1[n] < ST , s2[n] < ST | s2[n− 1] > ST )

• P31 = p (s1[n] > ST | s1[n− 1] < ST , s2[n− 1] < ST )

• P32 = p (s1[n] < ST , s2[n] > ST | s1[n− 1] < ST , s2[n− 1] < ST )

• P33 = p (s1[n] < ST , s2[n] < ST | s1[n− 1] < ST , s2[n− 1] < ST )

• P34 = 0

• P41 = p (s1[n] > ST , s2[n] < ST | s1[n− 1] < ST , s2[n− 1] < ST )

• P42 = p (s2[n] > ST | s1[n− 1] < ST , s2[n− 1] < ST )

• P43 = 0

• P44 = p (s1[n] < ST , s2[n] < ST | s1[n− 1] < ST , s2[n− 1] < ST )

where p(A|B) is the probability of event A given event B. Note that the transition from

state 4 (3) to state 2 (1) does not represent a switch.

The matrix form of the above transitional probability can be written as:

P =

P11 P21 P31 P41

P12 P22 P32 P42

P13 0 P33 0

0 P24 0 P44

(4.1)


To evaluate the above transitional probabilities, the multivariate Rayleigh probability

density function is required.

4.2.1 Correlated Rayleigh Fading Channel

Based on the derivation in [44], the complex P -variate Gaussian pdf is:

p(x1, · · · , xp) =1

(π)P (det (Σ))exp

(

−[X]H(Σ)−1[X])

(4.2)

where X =

x1

...

xP

and Σ is the {P × P} complex covariance matrix.

Therefore, using an approach similar to [45], the P -variate Rayleigh derived from

Equation (4.2) is:

p(V1, · · · , Vp) =

∏Pi=1 Vi

(π)P (det (Σ))

∫ π

−π

· · ·∫ π

−π

exp(

−[V]H(Σ)−1[V])

dφ1 · · · dφP (4.3)

where V =

V1e(jφ1)

...

VP e(jφP )

Vi and φi are defined as the magnitude and phase of xi respectively.

The only closed form equations that have been found are the bivariate and trivariate

complex Rayleigh distribution which have been derived in [46]. The bivariate Rayleigh

pdf is:

p(V1, V2) = 4V1V2 det (S)I0[2|S12|V1V2] exp (−(S11V21 + S22V

22 )) (4.4)

where S is the inverse of the covariance matrix (Σ) of the complex Gaussian random

variables mentioned in Equation (4.2) and S =

S11 S12

S∗12 S22

= Σ−1. I0 is the zeroth


order modified Bessel function of first kind.

By assuming the different Rayleigh fading channels are symmetric,Σ =

σ2 σ2ρ

σ2ρ∗ σ2

where ρ is the correlation between the two channels, Equation (4.4) can be simplified as

p(V1, V2) = 4V1V2

σ4(1− |ρ|2)I0[

2|ρ|V1V2

σ2(1− |ρ|2)

]

exp

(

−(

V 21

σ2(1− |ρ|2) +V 22

σ2(1− |ρ|2)

))

(4.5)

The above equation is also in agreement with the definitions from [47] and [48]. Thus,

as will be shown in the later sections, to derive a close form solution for SEC switching

rate, we have simplified the system to be a receiver with two independent branches.

Adding any more branches will require the switching rate to be calculated numerically.

4.3 Switching Rates

Based on the symmetric assumption mentioned above, the probability equations listed in

Section 4.2 can be grouped into six different groups and the detailed derivation of each

transitional probability can be found in Appendix C.

• Group 1

Based on Equation (C.7) in Section C.1,

P11 = P22

= p (s1[n] > ST | s1[n− 1] > ST )

=

[

1−Q1

(

|ρ|√2ST

σ√

(1− |ρ|2),

√2ST

σ√

(1− |ρ|2)

)

−Q1

( √2ST

σ√

(1− |ρ|2),

|ρ|√2ST

σ√

(1− |ρ|2)

)]

(4.6)

where Q1(a, b) is the Marcum-Q function introduced in [49]. The Marcum-Q func-

tion is defined as follows:

Q(a, b) =

∫ ∞

b

exp (−a2 + x2

2)I0(ax)xdx (4.7)


• Group 2


P12 = P21

= p (s1[n] < ST , s2[n] > ST | s1[n− 1] > ST )

= e−S2T /σ2

[

Q1

( √2ST

σ√

(1− |ρ|2),

|ρ|√2ST

σ√

(1− |ρ|2)

)

−Q1

(

|ρ|√2ST

σ√

(1− |ρ|2),

√2ST

σ√

(1− |ρ|2)

)]

(4.8)

• Group 3


P13 = P24

= p (s1[n] < ST , s2[n] < ST | s1[n− 1] > ST )

=(

1− e−S2T /σ2

)

[

Q1

( √2ST

σ√

(1− |ρ|2),

|ρ|√2ST

σ√

(1− |ρ|2)

)

−Q1

(

|ρ|√2ST

σ√

(1− |ρ|2),

√2ST

σ√

(1− |ρ|2)

)]

(4.9)

• Group 4


P31 = P42

= p (s1[n] > ST | s1[n− 1] < ST , s2[n− 1] < ST )

=

(

e−S2T /σ2

)

[

Q1

( √2ST

σ√

(1−|ρ|2), |ρ|

√2ST

σ√

(1−|ρ|2)

)

−Q1

(

|ρ|√2ST

σ√

(1−|ρ|2),

√2ST

σ√

(1−|ρ|2)

)]

1− e−S2T/σ2

(4.10)

• Group 5



P32 =P41

=

1− e−S2T /σ2 − e−S2

T /σ2

[

Q1

( √2ST

σ√

(1−|ρ|2), |ρ|

√2ST

σ√

(1−|ρ|2)

)

−Q1

(

|ρ|√2ST

σ√

(1−|ρ|2),

√2ST

σ√

(1−|ρ|2)

)]

1− e−S2T/σ2

(

e−S2T /σ2

)

[

Q1

( √2ST

σ√

(1−|ρ|2), |ρ|

√2ST

σ√

(1−|ρ|2)

)

−Q1

(

|ρ|√2ST

σ√

(1−|ρ|2),

√2ST

σ√

(1−|ρ|2)

)]

1− e−S2T/σ2

(4.11)

• Group 6


P33 =P44

=

1− e−S2T /σ2 − e−S2

T /σ2

[

Q1

( √2ST

σ√

(1−|ρ|2), |ρ|

√2ST

σ√

(1−|ρ|2)

)

−Q1

(

|ρ|√2ST

σ√

(1−|ρ|2),

√2ST

σ√

(1−|ρ|2)

)]

1− e−S2T/σ2

2

(4.12)

4.3.1 Verification

The following section compares the predictions of the theoretical derivations against the

results of Monte Carlo computation of the magnitude of a correlated bivariate complex

Gaussian randoms variable over 106 realizations. The covariance (C) of this complex

bivariate Gaussian Distribution has the following form:

C = σ2

1 ρ

ρ∗ 1

where σ2 is the variance of the complex Gaussian and ρ is the complex correlation coef-

ficient.


The following shows the accuracy of the above closed form derivation in comparison

with the Monte Carlo simulation when the |ρ| of the above correlated Gaussian random

variables is 0, 0.5 and 0.9:

• Group 1

Figure 4.4 represents the comparison between the Monte Carlo simulation and the

theoretical closed form equation for the cases when |ρ| = 0, 0.5 and 0.9. As indicated

in Figure 4.4a and Figure 4.4b, the theoretical closed form solution matches the

Monte Carlo simulation perfectly. The mismatch in Figure 4.4c is due to the lack

of samples in the Monte Carlo method when using high threshold.

• Group 2

Figure 4.5 has demonstrated the accuracy of the above closed form derivation. The

theoretical closed form solution is tightly matched with the Monte Carlo simulation

result.

• Group 3

Figure 4.6 has also shown the closely matched results between the closed form

theoretical calculation and Monte Carlo simulation in Figure 4.6a and Figure 4.6b.

The result beyond ”threshold level = 3” in Figure 4.6 was left out due to the lack

of sample issue which has also occurred in Group 1.

• Group 4

Figure 4.7 has demonstrated the accuracy of the above closed form derivation.

The theoretical closed form solution has tightly matched with the Monte Carlo

simulation.

• Group 5

Figure 4.8 has also shown perfect match between the theoretical closed form with

the Monte Carlo simulation.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ratio of Threshold Level with Respect to Recv Noise

Prob

abilit

y

Monte CarloTheoretical

(a) Monte Carlo vs. Closed Form for |ρ| = 0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Prob

abilit

y

Monte CarloClosed Form

(b) Monte Carlo vs.Closed Form for |ρ| = 0.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Prob

abilit

y


(c) Monte Carlo vs.Closed Form for |ρ| = 0.9

Figure 4.4: Group 1 Monte Carlo vs. simulation


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25


Prob

abilit

y



0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25


Prob

abilit

y


(b) Monte Carlo vs. Closed Form for |ρ| = 0.5

0 1 2 3 4 50

0.02

0.04

0.06

0.08

0.1

0.12


Prob

abilit

y


(c) Monte Carlo vs. Closed Form for |ρ| = 0.9



0 1 2 3 4 50

0.2

0.4

0.6

0.8

1


Prob

abilit

y



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Prob

abilit

y



0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


Prob

abilit

y



Figure 4.6: Group 3 Monte Carlo vs. Simulation


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Prob

abilit

y



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Prob

abilit

y



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Prob

abilit

y





0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25


Prob

abilit

y



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25


Prob

abilit

y



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25


Prob

abilit

y





• Group 6

Figure 4.9 has also demonstrated the perfect match between the theoretical closed

form results and Monte Carlo simulations.

4.4 SEC Switching Rate Monte Carlo vs. Theoreti-

cal Closed Form Solution

In the previous section, we verified the transition probabilities in the Markov Chain. In

this section, we derive the SEC switching rate based on the theoretical closed form solu-

tion and compare it against the switching rate found based on Monte Carlo simulation.

The “one-ring” Model in Equation (2.16) is used in here.

In a Markov chain, the steady state probabilities in a Markov Chain is the eigenvector

corresponding to the unit vector of eigenvalue of its transition matrix P [50].

Let pi be the steady state probability at ith state. The theoretical switching rate in

SEC can be represented as:

P12p1 + P21p2 + P32p3 + P41p4Ts

(4.13)

where Ts is the sample time period defined in Equation (2.16).

4.4.1 Comparison at Different Normalized Sample Rates

This section compares the SEC switching rates computed by theoretical closed form with

the Monte Carlo Simulation. Figure 4.10 shows the switch rates plot computed from these

two types of methods. In this figure, the normalized sampling period is 0.3827 units and

this gives a correlation of 0.1 between the time samples. The antennas here are separated

0.3827λ apart to give 0 spatial correlation. Table 4.1 lists the system specifications used

here.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Prob

abilit

y



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Prob

abilit

y



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Prob

abilit

y





−95 −90 −85 −80 −75 −70 −65 −600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Threshold SNR w/ respect to Transmit SNR (dB)

No

rma

lize

d S

witc

hin

g R

ate


Figure 4.10: Normalized switching rate at 0.3827 normalized sampling period

−95 −90 −85 −80 −75 −70 −65 −600

0.2

0.4

0.6

0.8

1

1.2

1.4


No

rma

lize

d S

witc

hin

g R

ate


Figure 4.11: Normalized switching rate at 0.01 normalized sampling period



Number of transmitter 1

Number of receivers 2

Inter-element distance 0.3827λ

Normalized sample rate (fdTs) 0.3827

Number of time samples 4000

Transmit power with respect to receiver noise 73.6 dB


Figure 4.10 demonstrates the accuracy of the closed form derivation. The switching

rate derived based on the closed form solution almost has the same performance as the

Monte Carlo simulation. The small SNR values are due to the large transmitting SNR

used to compensate for the large fading term in Equation (2.15). The gap between the

closed form case and the Monte Carlo case is mainly due to not using enough samples in

both time and channel realizations. Another cause for this gap is due to the steady state

assumption made in Equation (4.13). In the derivation of Equation (4.13). The steady

state probability of each state is assumed to have occurred after infinite steps in the

Markov chain diagram represented in Figure 4.3. However, in simulation, it is impossible

to create infinite steps. Thus, this introduces inaccuracy between the theoretical closed

form result and the Monte Carlo simulation result.

Figure 4.10 has demonstrated that the theoretical closed form equations is accurate in

comparison with the Monte Carlo simulation. However, the 0.3827 normalized sampling

period used here is too slow to be of any practical use. Thus, in a more practical

system, the normalized sampling period is usually around 0.01. With the same system

specification illustrated in Table 4.1, Figure 4.11 demonstrates the normalized switching

rate at 0.01 normalized sampling period.

From Figure 4.11, the gap between the theoretical derivation and Monte Carlo simula-


tion has become larger. This is due to the numerical error in implementing the theoretical

calculation. When the normalized switching rate is at 0.01, the correlation coefficient be-

tween the time samples is 0.999. This reduces all the√

1− |ρ|2 terms from Equation (4.6)

to Equation (4.12) to almost 0 and due to the MATLAB’s rounding of numbers, many of

these equations are not as accurate as the case of the 0.3827 normalized sampling period

case.

4.5 Performance against SC

This section examines the SEC switch rate and outage probability performance against

SC for both beamforming and non-beamforming cases.

4.5.1 Two Receiver Antennas without Beamforming

Figure 4.12 and Figure 4.13 are the outage probability and the switching rate performance

for both SEC and SSC in a two antennas non-beamforming system. These results were

generated based on Monte Carlo simulation. The system specification is provided in

Table 4.2 with the exception that the normalized sample period is changed to 0.01 for

practicality.

Figure 4.12 and Figure 4.13 have indicated that SEC has significantly lower switching

rate than SC while still maintaining the same probability of outage as SC. SEC has the

same outage probability as SC because SC only declares outage when the branch that has

the maximum SNR falls below threshold. This is fundamentally the same as SEC since

for SEC to declare outage means no branch can be above threshold. SEC has significantly

lower switching rate since it only switches when the current branch is below threshold

whereas SC always switches to the branch that has the highest SNR. Thus, in the case of

low threshold SNR, SEC has significantly lower switching rate than SC since the current

branch is usually above threshold and any further switches is not required. Similarly, in


−95 −90 −85 −80 −75 −70 −65 −600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

babi

lity

of O

utag

e

SECSC

Figure 4.12: Outage probability of SEC and SC in two antennas receiver with no beam-

forming

−95 −90 −85 −80 −75 −70 −65 −600

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Nor

mal

ized

Sw

itchi

ng R

ate

SECSC

Figure 4.13: Switching rate of SEC and SC in two antennas receiver with no beamforming




Number of receiving antennas 2

Inter-element distance 0.1λ

Normalized sample rate (fdTs) 0.01

Number of time sample 400

Transmit power with respect to receiver noise 73.6 dB

Table 4.2: System specifications for the case of two receiver antennas without beamform-

ing

the case of high threshold SNR, SEC has significantly lower switching rate since all the

branches are usually below the threshold and no further switches is necessary.

In the next two sections, we compare the performance between SEC and SC in beam-

forming and no-beamforming systems.

4.5.2 Six Receiver Antennas without Beamforming

Figure 4.14 and Figure 4.15 represent the outage probability and the switching rate of

SEC and SC in a six receiver antennas system without beamforming. The system speci-

fication is the same as Table 4.2 with the exception of the number of receiver antennas

changing to 6.

The findings from Figure 4.14 and Figure 4.15 correlates the results discussed in the

previous section. Also, by comparing the results from Figure 4.12 and Figure 4.14, we

observe that the outage probability decreases as the number of antennas increases. This is

because by increasing the number of antennas, the probability of all six branches staying

below the threshold is lower. Moreover, the switching rates in Figure 4.15 are clearly

higher than the switching rates in Figure 4.13 due to the fact that with more antenna


−95 −90 −85 −80 −75 −70 −65 −600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

ba

bili

ty o

f O

uta

ge

SECSC

Figure 4.14: Outage probability of SEC and SC in six no-beamforminng receiver system

−95 −90 −85 −80 −75 −70 −65 −600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5


No

rma

lize

d S

witc

hin

g R

ate

SECSC

Figure 4.15: Switching rate of SEC and SC in six no-beamforminng receiver system


branches, more choices are available for the system to switch to.

4.5.3 Six Receiver Antennas with Beamforming

Figure 4.16 and Figure 4.17 combine the results from Figure 4.14 and Figure 4.15 to

compare the outage probability and the switching rate performance of SEC and SC in

beamforming and no-beamforming systems. The system specification is the same as pre-

vious section. The beamforming system has the beam pattern presented in Figure 3.12.

Figure 4.16 demonstrates that beamforming case has lower outage probability than

the case that has no beamforming. This result is in agreement with the findings in

Section 3.3.2 on Beamforming Selection being more robust than Antenna Element Se-

lection. However, this comes at the cost of having higher switching rate as illustrated in

Figure 4.17. Beamforming SEC has similar switching rate as the no-beamforming SEC

case. Also, in Figure 4.17, the switching rate for the beamforming SEC case has similar

maximum switching rate as the no-beamforming SEC case. This maximum switching

rate occurs when the outage probability is around 0.2. The result in Figure 4.17 indi-

cate the significant advantage in switching rate by using beamforming SEC instead of

beamforming SC.

4.6 Chapter Summary

In conclusion, in this chapter, we have derived the theoretical closed form for SEC using

Markov Chain. The closed form results for a two independent antennas receiver system

match closely to the Monte Carlo simulation. In addition, in this chapter, we have com-

pared the switching rate and outage probability performance between SEC and SC in

both beamforming and no-beamforming systems. The result indicates that even though

SEC has the same outage probability as SC, it has significantly lower switching rate.

Further, with similar performance in switching rate, beamforming system has lower out-


−95 −90 −85 −80 −75 −70 −65 −600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

ba

bili

ty o

f O

uta

ge

SEC No BeamformSC No BeamformSEC BeamformSC Beamform

Figure 4.16: Outage probability of SEC and SC in both beamforming and mo-

beamforming systems

−95 −90 −85 −80 −75 −70 −65 −600

1

2

3

4

5

6

7


No

rma

lize

d S

witc

hin

g R

ate

SEC No BeamformSC No BeamformSEC BeamformSC Beamform

Figure 4.17: Switching rate of SEC and SC in both beamforming and no-beamforming

systems


age probability than the no-beamforming system. Thus, this result is in agreement with

the finding in the previous chapter that a beamforming system is more robust than the

system that has no beamforming.

Chapter 5

Conclusions and Future Work

5.1 Conclusions

In this thesis, we studied the performance of digital beamforming, RF beamforming and

element-based processing in closely spaced antenna systems. When the antennas are

placed closely together, the mutual coupling and spatial correlation effects undermine

the advantages provided by multiple antennas.

In the case where more than one RF-chain is available, due to the data processing

inequality, digital beamforming can never outperform the case of having no beamforming

in terms of capacity. However, RF-beamforming can outperform the no-beamforming

case given that an adequate number of beamforming patterns are used. For example, by

setting the beamforming pattern as the inverse of the correlated channel matrix, we can

decorrelate the channel to perform as well as the uncorrelated channel case. However, it

is difficult to realize such a beamforming pattern. Thus, finding a realizable beamforming

pattern which decorrelates the channel matrix still remains as future work.

In the case where only one RF-chain is available, RF-beamforming selection based

MIMO processing has the same performance as digital beamforming selection based

MIMO processing. Most important of all, we have demonstrated the performance im-

87

Chapter 5. Conclusions and Future Work 88

provement for using RF beamforming based MIMO processing instead of antenna based

MIMO processing in closely spaced antennas systems. The result indicates that even with-

out mutual coupling, antenna based MIMO processing is greatly impacted when moving

from rich to correlated scattering environment. This suggests the robustness in beam-

forming based MIMO processing and its potential to be utilized in small multi-antenna

devices.

Another important contribution in this thesis is the theoretical closed form solution

derivation for the switching rate in Switch and Examine Combining technique. In this

thesis, based on Markov Chain theory, we have derived the theoretical closed form solu-

tion for SEC and the result matches the Monte Carlo simulation result. Furthermore, by

comparing the switching rate and outage probability performance between SEC and SC

in both beamforming and no-beamforming systems, we observe that even though SEC

has the same outage probability as SC, it has significantly lower switching rate. This

result suggests the potential and the practicality of utilizing RF-beamforming along with

SEC in closely spaced multi-antenna handheld devices.

5.2 Future Work

This section provides some of the possible future work.

5.2.1 Finding Optimal Beam Pattern

One of the key future work which has been emphasized throughout the thesis is the finding

of an optimal beam pattern which decorrelates the channel without enhancing noise. As

discussed in Section 3.2.2, the most optimal beamform is the inverse of the correlated

channel (H−1). However, such beamforming pattern cannot be realized and this is only

true when the channel matrix is a square matrix. Therefore, more investigations are still

required to find a realizable optimal beamforming pattern.


5.2.2 Incorporating Mutual Coupling

As can be noted from the thesis, most of the results were based on the assumption that

mutual coupling has been eliminated by, e.g., using metamaterial antennas. However,

this is only the ideal case. In reality, mutual coupling still exists. Thus, to increase the

accuracy of the model, mutual coupling effect needs to be incorporated into the system.

One way to incorporate mutual coupling into the system is through the model defined

in [6]. The model has been explained in detail in Section 2.2. However, in the thesis, we

temporarily omitted this analysis to keep the system simple. Therefore, at the next stage

of our research, the mutual coupling model described in Section 2.2 will be included into

our system model again.

Beside the theoretical model introduced in [6], another way to model mutual coupling

in metamaterial antenna is through complex computational electromagnetic wave mod-

eling. The followings are the three different orientations of the metamaterial antenna

which have been simulated based on the Ray Tracing technique 1:

Figure 5.1 is the orientation of transmitting in backward position which has the feed-

line pointing to the receiver antenna. Both antenna patches lie in the x − y plane. Let

the furtherest antenna patch in the transmitter be Tx1 and the other one be Tx2. Vise

versa, let the furtherest antenna patch in the receiver be Rv1 and the other one be Rv2.

The channel matrix, H, can be described as follow:

H =

H11 H12

H21 H22

(5.1)

where Hij is the channel between the jth transmitter to the ith receiver.

As shown in Appendix D.0.1, both real and imaginary components of the simulated

channel have the shape of Gaussian distribution with the variance specified in Table D.1

1We would like to thank Neeraj Sood and Professor Costas Sarris from the Electromagnetic group inUniversity of Toronto for providing data based on the Ray Tracing Technique mentioned in [51] usingthe metamaterial antenna structure in Figure 1.4.


Figure 5.1: Transmitter in backward and receiver in forward orientations

and Table D.2.

Figure 5.2 represents the case in which the antenna patch lies in the x− y plane. Ap-

pendix D.0.2 provides the histograms and the variance tables for both real and imaginary

components of the MIMO channels.

Figure 5.3 represents the case in which the both antenna patches lie in the x − z

plane. Appendix D.0.3 provides the histograms and the variance tables for both real and

imaginary components of the MIMO channels.

Figure 5.4 is an example of the BER vs Transmitter SNR plot for the vertical ori-

entation mentioned above. Figure 5.5 is the beamforming pattern used for the RF-

beamforming. The result here is still very preliminary. We are still currently cooperating

with Neeraj Sood and Professor Costa Sarris to analyze the simulation channel.

Moreover, to agree with the findings in Section 3.3.2 on beamforming selection based

MIMO processing has better performance than antenna element selection based MIMO

processing, part of our future plan is to generate new channel data for correlated scat-

tering model similar to Figure 2.6.


Figure 5.2: Transmitter in forward and receiver in forward orientations

5.2.3 Further Analysis on SEC

As mentioned before, the BER performance of SEC depends heavily on how the threshold

level is set. However, even with the most optimal SEC, it can only perform as well as SC

in BER sense. SEC represents a trade off BER with reduction in switching rate. Thus,

more studies are required into finding the threshold level that will suffer the least BER

enhancement while still maintaining an adequate switching rate.

Since the switch rate model for SEC involves the One-Ring model that assumes the

receiver is moving, it would be interesting to compare the SEC switching rate performance

against SC in a randomized path. The results will be more practical to model the scenario

in which the user is using a handheld device while in motion.


Figure 5.3: Transmitter and receiver in vertical orientations

0 5 10 15 20 2510

−4

10−3

10−2

10−1

100

Transmitter with Respect to Recv Noise

BE

R

UncorrelatedNo BeamformingBeamforming

Figure 5.4: BER vs transmitter SNR for the orientation in Figure 5.3


0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

Sum BeamformDifference Beamform

Figure 5.5: Beamforming patterns

Appendix A

Transmitter and Receiver Mutual

Impedance Expression

The mutual impedance for a side-by-side antenna configuration can be computed using

the induced EMF method defined in [7]. From [7], the mutual impedance between two

antennas aligning side-by-side is represented as follows:

R21 =η

4π sin2(kl/2)[2Ci(u0)− Ci(ui)− Ci(u2)] (A.1)

X21 =− η

4π sin2(kl/2)[2Si(u0)− Si(ui)− Si(u2)] (A.2)

where R21 is the resistance and X21 is the reactance of the impedance.

The Ci(), Si(), u0, u1 and u2 terms are defined as follows:

Ci(x) = −∫ ∞

x

cos(τ)

τdτ (A.3)

Si(x) = −∫ x

0

sin(τ)

τdτ (A.4)

u0 =2π

λd (A.5)

94

Appendix A. Transmitter and Receiver Mutual Impedance Expression 95

u1 =2π

λ(√d2 + l2 + l) (A.6)

u2 =2π

λ(√d2 + l2 − l) (A.7)

where d is the distance between the dipole antennas and l is the length of the dipole. k

is the wave number defined as 2π/λ.

The self-impedance in the diagonal of the ZT and ZR matrix is represented in [7] as

follows:

R11 =η

4π sin2(kl)

{

C + ln(kl)− Ci(kl) +1

2sin(kl)[Si(2kl)− 2Si(kl)]

+1

2cos(kl)[C + ln(kl) + Ci(2kl)− 2Ci(kl)]

}

(A.8)

X11 =η

4π sin2(kl)

{

2Si(kl) + cos(kl) [Si(2kl)− 2Si(kl)]

− sin(kl)[2Ci(kl)− Ci(2kl)− Ci(2ka2

l)]

}

(A.9)

where C is a constant and a is the radius of the wire.

The above mutual impedance is only for the case when the dipole antennas are placed

side-by-side with d separation distance. More complicated configuration requires a more

complex numerical computation method such as method of moment (MoM) applied sim-

ilarly in [52].

Appendix B

Derivation of Mutual Information

for Digital Beamforming

From Equation (2.8), the digital beamforming system equation is:

y = WHHx+WHn

By assuming that the noise, n, is AWGN (Additive White Gaussian Noise), the

expected value and the variance of the beamformed noise (WHn) is as follows: Let

n = WHn

E[n] = 0; (B.1)

E[nnH ] = E[WHnnHW]

= σ2WHW (B.2)

where σ2 is the variance of n.

The variance for y (Σy) is as follows:

96

Appendix B. Derivation of Mutual Information for Digital Beamforming97

Let H be ZR

√ΨRH

u√ΨTZT defined in [6]

Σy = E[yyH ]

= E[(WHHx+WHn)(WHHx+WHn)H ]

= E[WHHxxHHHW] + E[WHHxnHW] + E[WHnxHHHW] + E[WHnnHW]

= WHHΣxHHW + σ2WHW (B.3)

where Σx is the variance of the input.

Based on the mutual information derivation provided in [31], the mutual information

for digital beamforming can be derived as follows:

I(x; y) = log2(2πe)m|Σy| − log(2πe)m|Σn|

= log2

( |Σy||Σn|

)

= log2

(

det(WHHΣxHHW + σ2WHW)

det(σ2WHW)

)

= log2

(

det

(

σ2WHW +WHHΣxHHW

det(σ2WHW)

))

(B.4)

In the case that the channel information is available to the transmitter, mutual in-

formation can be maximized by applying the water-filling algorithm [31] which adjusts

the power on each transmitting antenna (Σx). However, in this thesis, since we as-

sume the transmitter has no information to the channel, the total transmitting power is

equally distributed among all transmitting antennas. Therefore, Σx = Es/N where Es is

the total transmitting power and N is the number of transmitters. Substitute this into

Equation (B.4), Equation (B.4) becomes:

I(x; y) = log2

(

det

(

WHW

det(WHW)+

Es

N det(σ2WHW)WHHHHW

))

(B.5)

By substituting the channel defined in [6], we have:

I(x; y) = log2

(

det

(

WHW

det(WHW)

+Es

σ2N det(WHW)ZR

√

ΨRHu√

ΨTZTZTH(√

ΨT)H(Hu)H(

√

ΨR)HZR

H

))

(B.6)

Appendix C

Transitional Probability Derivations

for SSC System with Two

Independent Branches

Based on the discussion in Section 4.2, to find a closed form solution for SEC switching

rate, the SEC system in this section is assumed to be a two independent branches system.

C.1 Group 1

P11 = P22

= p (s1[n] > ST | s1[n− 1] > ST )

=p(s1[n] > ST , s1[n− 1] > ST )

p(s1[n− 1] > ST )

=

∫∞ST

∫∞ST

p(s1[n] , s1[n− 1])ds1[n]ds1[n− 1]∫∞ST

p(s1[n− 1])ds1[n− 1](C.1)

The denominator is basically the CCDF of a Rayleigh Random Variable. According

98

Appendix C. Transitional Probability Derivations for SSC System with Two Independent

to [15],∫ ∞

ST

p(s1[n− 1])ds1[n− 1] = e(−S2T /σ2) (C.2)

The following utilizes Equation (4.5) to solve the numerator part of Equation (C.1):

p(s1[n] > ST , s1[n− 1] > ST ) =

∫ ∞

ST

∫ ∞

ST

p(s1[n] , s1[n− 1])ds1[n]ds1[n− 1]

=

∫ ∞

ST

∫ ∞

ST

4s1[n]s1[n− 1]

σ4(1− |ρ|2) I0

[

2|ρ|s1[n]s1[n− 1]

σ2(1− |ρ|2)

]

exp

(

−(

s1[n]2

σ2(1− |ρ|2) +s1[n− 1]2

σ2(1− |ρ|2)

))

ds1[n]ds1[n− 1]

(C.3)

LetR1 =

√2S1[n]

σ√

(1+|ρ|2)

R2 =√2S1[n−1]

σ√

(1+|ρ|2)

p(s1[n] > ST , s1[n− 1] > ST ) =

∫ ∞√2ST

σ√

(1−|ρ|2)

{

(1− |ρ|2)R1e−1/2(R2

1−|ρ|2R21)

∫ ∞√

2ST

σ√

(1−|ρ|2)

[

R2I0(|ρ|R1R2)e(R22+|ρ|2R2

12

)

]

dR2

dR1

(C.4)

=

∫ ∞√2ST

σ√

(1−|ρ|2)

{

(1− |ρ|2)R1e−1/2(1−|ρ|2)R2

1

Q1

(

|ρ|R1,

√2ST

σ√

(1− |ρ|2)

)}

dR1 (C.5)

=e−S2T /(σ2)

[

1−Q1

(

|ρ|√2ST

σ√

(1− |ρ|2),

√2ST

σ√

(1− |ρ|2)

)

−Q1

( √2ST

σ√

(1− |ρ|2), |ρ|

√2ST

σ√

(1− |ρ|2)

)]

(C.6)

where Equation (C.5) was derived using the definition of Marcum Q-function [48]. Equa-

tion (C.6) was derived based on Eqn.(B.24) in [53].


Equation (C.6) is in agreement with Eqn.(A-7-3) in [48]. Thus, combining Equa-

tion (C.2) and Equation (C.6):

P11 =

[

1−Q1

(

|ρ|√2ST

σ√

(1− |ρ|2),

√2ST

σ√

(1− |ρ|2)

)

−Q1

( √2ST

σ√

(1− |ρ|2),

|ρ|√2ST

σ√

(1− |ρ|2)

)]

(C.7)

C.2 Group 2

P12 = P21

= p (s1[n] < ST , s2[n] > ST | s1[n− 1] > ST )

=p (s1[n] < ST , s2[n] > ST , s1[n− 1] > ST )

p (s1[n− 1] > ST )

= p (s2[n] > ST )

(

p (s1[n] < ST , s1[n− 1] > ST )

p (s1[n− 1] > ST )

)

(C.8)

=

∫ ∞

ST

{p(s2[n])} ds2[n](∫∞ST

∫ ST

0p (s1[n], s1[n− 1])ds1[n]ds1[n− 1]∫∞ST

p (s1[n− 1])ds1[n− 1]

)

(C.9)

where Equation (C.8) is derived from the assumption that branch 1 is independent of

branch 2. The∫∞ST

p (s2[n] > ST )ds2[n] and∫∞ST

p (s1[n−1])ds1[n−1] terms are the CCDF

of Rayleigh Random Variables, which have the solution expressed in Equation (C.2).

The numerator part in Equation (C.9) is solved as follow:

p(s1[n] < ST , s1[n− 1] > ST ) =

∫ ∞

ST

∫ ST

0

p(s1[n] , s1[n− 1])ds1[n]ds1[n− 1]

=

∫ ∞

ST

∫ ∞

ST

4s1[n]s1[n− 1]

σ4(1− |ρ|2) I0[2|ρ|s1[n]s1[n− 1]

σ2(1− |ρ|2) ]

exp (−(s1[n]

2

σ2(1− |ρ|2) +s1[n− 1]2

σ2(1− |ρ|2)))ds1[n]ds1[n− 1]

(C.10)

LetR1 =

√2S1[n]

σ√

(1+|ρ|2)

R2 =√2S1[n−1]

σ√

(1+|ρ|2)


p(s1[n] < ST , s1[n− 1] > ST ) =

∫

√2ST

σ√

(1−|ρ|2)

0

{

(1− |ρ|2)R1e−1/2(R2

1−|ρ|2R21)

∫ ∞√

2ST

σ√

(1−|ρ|2)

[

R2I0(|ρ|R1R2)e(R22+|ρ|2R2

12

)

]

dR2

dR1

=

∫

√2ST

σ√

(1−|ρ|2)

0

{

(1− |ρ|2)R1e−1/2(1−|ρ|2)R2

1

Q1

(

|ρ|R1,

√2ST

σ√

(1− |ρ|2)

)}

dR1

=e−S2T /σ2

[

Q1

( √2ST

σ√

(1− |ρ|2),

|ρ|√2ST

σ√

(1− |ρ|2)

)

−Q1

(

|ρ|√2ST

σ√

(1− |ρ|2),

√2ST

σ√

(1− |ρ|2)

)]

(C.11)

where Equation (C.11) is derived based on Eqn.(B.25) from [53]. Thus, substitute Equa-

tion (C.2) and Equation (C.11) into Equation (C.9):

P12 = e−S2T /σ2

[

Q1

( √2ST

σ√

(1− |ρ|2),

|ρ|√2ST

σ√

(1− |ρ|2)

)

−Q1

(

|ρ|√2ST

σ√

(1− |ρ|2),

√2ST

σ√

(1− |ρ|2)

)]

(C.12)

C.3 Group 3

P13 = P24

= p (s1[n] < ST , s2[n] < ST | s1[n− 1] > ST )

=p (s1[n] < ST , s2[n] < ST , s1[n− 1] > ST )

p (s1[n− 1] > ST )

= p (s2[n] < ST )

(

p (s1[n] < ST , s1[n− 1] > ST )

p (s1[n− 1] > ST )

)

(C.13)

=

∫ ST

0

{p(s2[n])} ds2[n](∫∞ST

∫ ST

0p (s1[n], s1[n− 1])ds1[n]ds1[n− 1]∫∞ST

p (s1[n− 1])ds1[n− 1]

)

(C.14)


Equation (C.14) is similar to Equation (C.9) with the only difference being the∫ ST

0{p(s2[n])} ds2[n] term, which is the CDF of Rayleigh Distribution.

Therefore,

P13 =(

1− e−S2T /(σ2)

)

[

Q1

( √2ST

σ√

(1− |ρ|2),

|ρ|√2ST

σ√

(1− |ρ|2)

)

−Q1

(

|ρ|√2ST

σ√

(1− |ρ|2),

√2ST

σ√

(1− |ρ|2)

)]

(C.15)

C.4 Group 4

P31 = P42

= p (s1[n] > ST | s1[n− 1] < ST , s2[n− 1] < ST )

=p (s1[n] > ST , s1[n− 1] < ST , s2[n− 1] < ST )

p (s1[n− 1] < ST , s2[n− 1] < ST )

=p (s1[n] > ST , s1[n− 1] < ST )

p (s1[n− 1] < ST )(C.16)

=

∫ ST

0

∫∞ST

p (s1[n], s1[n− 1])ds1[n]ds1[n− 1]∫ ST

0p (s1[n− 1])ds1[n− 1]

(C.17)

The denominator is basically the CDF of Rayleigh Distribution. The numerator has

already been solved in Equation (C.11).

Therefore,

P31 =

(

e−S2T /(σ2)

)

[

Q1

( √2ST

σ√

(1−|ρ|2), |ρ|

√2ST

σ√

(1−|ρ|2)

)

−Q1

(

|ρ|√2ST

σ√

(1−|ρ|2),

√2ST

σ√

(1−|ρ|2)

)]

1− e−S2T/(σ2)

(C.18)


C.5 Group 5

P32 =P41

=p (s1[n] < ST , s2[n] > ST | s1[n− 1] < ST , s2[n− 1] < ST )

=p (s1[n] < ST , s2[n] > ST , s1[n− 1] < ST , s2[n− 1] < ST )

p (s1[n− 1] < ST , s2[n− 1] < ST )

=

(

p (s1[n] < ST , s1[n− 1] < ST )

p (s1[n− 1] < ST )

)(

p (s2[n] > ST , s2[n− 1] < ST )

p (s2[n− 1] < ST )

)

(C.19)

=

(

∫ ST

0

∫ ST

0p (s1[n], s1[n− 1])ds1[n]ds1[n− 1]∫ ST

0p (s1[n− 1])ds1[n− 1]

)

(∫ ST

0

∫∞ST

p (s2[n], s2[n− 1])ds2[n]ds2[n− 1]∫ ST

0p (s2[n− 1])ds2[n− 1]

)

(C.20)

C.5.1 Solution for the Second Term in Equation (C.20)

The solution for the numerator of the second term in Equation (C.20) has already been

solved in Equation (C.11). The denominator is the CDF of Rayleigh Distribution. Thus,

the second term has the following closed form solution:

(

e−S2T /(σ2)

)

[

Q1

( √2ST

σ√

(1−|ρ|2), |ρ|

√2ST

σ√

(1−|ρ|2)

)

−Q1

(

|ρ|√2ST

σ√

(1−|ρ|2),

√2ST

σ√

(1−|ρ|2)

)]

1− e−S2T/(σ2)

(C.21)

C.5.2 Solution for the First Term in Equation (C.20)

Similar to the second term mentioned above, the denominator for the first term is also

the CDF of Rayleigh Distribution. The closed form for numerator is derived as follows:


p(s1[n] < ST , s1[n− 1] < ST ) =

∫

√2ST

σ√

(1−|ρ|2)

0

{

(1− |ρ|2)R1e−1/2(R2

1−|ρ|2R21)

∫

√2ST

σ√

(1−|ρ|2)

0

[

R2I0(|ρ|R1R2)e(R22+|ρ|2R2

12

)

]

dR2

dR1

=

∫

√2ST

σ√

(1−|ρ|2)

0

{

(1− |ρ|2)R1e−1/2(1−|ρ|2)R2

1

[

1−Q1

(

|ρ|R1,

√2ST

σ√

(1− |ρ|2)

)]}

dR1

=1− e−S2T /(σ2) − e−S2

T /(σ2)

[

Q1

( √2ST

σ√

(1− |ρ|2),

|ρ|√2ST

σ√

(1− |ρ|2)

)

−Q1

(

|ρ|√2ST

σ√

(1− |ρ|2),

√2ST

σ√

(1− |ρ|2)

)]

(C.22)

By combining Equation (C.21) and Equation (C.22),

P32 =

1− e−S2T /(σ2) − e−S2

T /(σ2)

[

Q1

( √2ST

σ√

(1−|ρ|2), |ρ|

√2ST

σ√

(1−|ρ|2)

)

−Q1

(

|ρ|√2ST

σ√

(1−|ρ|2),

√2ST

σ√

(1−|ρ|2)

)]

1− e−S2T/σ2

(

e−S2T /(σ2)

)

[

Q1

( √2ST

σ√

(1−|ρ|2), |ρ|

√2ST

σ√

(1−|ρ|2)

)

−Q1

(

|ρ|√2ST

σ√

(1−|ρ|2),

√2ST

σ√

(1−|ρ|2)

)]

1− e−S2T/(σ2)

(C.23)


C.6 Group 6

P33 =P44

=p (s1[n] < ST , s2[n] < ST | s1[n− 1] < ST , s2[n− 1] < ST )

=p (s1[n] < ST , s2[n] < ST , s1[n− 1] < ST , s2[n− 1] < ST )

p (s1[n− 1] < ST , s2[n− 1] < ST )

=

(

p (s1[n] < ST , s1[n− 1] < ST )

p (s1[n− 1] < ST )

)(

p (s2[n] < ST , s2[n− 1] < ST )

p (s2[n− 1] < ST )

)

(C.24)

=

(

∫ ST

0

∫ ST

0p (s1[n], s1[n− 1])ds1[n]ds1[n− 1]∫ ST

0p (s1[n− 1])ds1[n− 1]

)

(

∫ ST

0

∫ ST

0p (s2[n], s2[n− 1])ds2[n]ds2[n− 1]∫ ST

0p (s2[n− 1])ds2[n− 1]

)

(C.25)

Due to the assumption on the branches being symmetrical, both terms have the same

closed form expression and the closed form of the numerator in these terms has already

been solved in Equation (C.22).

Therefore,

P33 =

1− e−S2T /(σ2) − e−S2

T /(σ2)

[

Q1

( √2ST

σ√

(1−|ρ|2), |ρ|

√2ST

σ√

(1−|ρ|2)

)

−Q1

(

|ρ|√2ST

σ√

(1−|ρ|2),

√2ST

σ√

(1−|ρ|2)

)]

1− e−S2T/σ2

2

(C.26)

Appendix D

Histograms for Metamaterial

Antenna Based on Ray-tracing

Simulation

D.0.1 Transmitter Backward and Receiver Forward

Figure D.1 and Figure D.2 are the histograms for the real and the imaginary compo-

nents of the channel in the orientation described by Figure 5.1. The top left and right

histograms are the H11 and H12 described in Equation (5.1). The bottom left and right

histograms are the H21 and H22 respectively. Each histogram show the distribution of the

data to be the shape of a Gaussian Distribution with zero mean and very small variance.

The variance of the data in each histogram are presented in Table D.1 and Table D.2.

Tr 1 Tr 2

Rv 1 5.607× 10−5 4.018× 10−5

Rv 2 3.97× 10−5 1.208× 10−4

Table D.1: Variance for the real component in Channel in Fig D.1

106

Appendix D. Histograms for Metamaterial Antenna Based on Ray-tracing Simulation

−0.1 −0.05 0 0.05 0.10

1

2

3

4x 10

4

−0.05 0 0.050

5000

10000

15000

−0.06 −0.04 −0.02 0 0.02 0.04 0.060

0.5

1

1.5

2x 10

4

−0.1 −0.05 0 0.05 0.10

2000

4000

6000

Figure D.1: Historgrams for the Real Component in the Channel for the Configuration

Described in Figure 5.1

−0.1 −0.05 0 0.05 0.10

1

2

3

4x 10

4

−0.05 0 0.050

5000

10000

15000

−0.1 −0.05 0 0.05 0.10

0.5

1

1.5

2x 10

4

−0.15 −0.1 −0.05 0 0.05 0.10

2000

4000

6000

Figure D.2: Historgrams for the Imagonary Component in the Channel for the Configu-

ration Described in Figure 5.1


Tr 1 Tr 2

Rv 1 5.634× 10−5 3.996× 10−5

Rv 2 4.103× 10−5 1.204× 10−4

Table D.2: Variance for the Imaginary Component in Channel in Fig D.2

D.0.2 Transmitter Forward and Receiver Forward

Figure D.3 and Figure D.4 plot the histograms for the real and imaginary components of

the simulated channel based on the orientation in Figure 5.2. Table D.3 and Table D.4

are the variance for the real and imaginary components of each entry in the channel

matrix.

Tr 1 Tr 2

Rv 1 9.703× 10−5 5.308× 10−5

Rv 2 5.308× 10−5 1.255× 10−3

Table D.3: Variance for the real component in Channel in Fig D.3

Tr 1 Tr 2

Rv 1 9.256× 10−5 5.13× 10−5

Rv 2 5.074× 10−5 1.2684× 10−3

Table D.4: Variance for the Imaginary Component in Channel in Fig D.4

D.0.3 Transmitter and Receiver in Vertical Direction

Figure D.5 and Figure D.6 plot the histograms for the real and imaginary components of

the simulated channel based on the orientation in Figure 5.3. Table D.5 and Table D.5 are

the variance of the real and imaginary components of each entry in the channel matrix.


−0.2 −0.1 0 0.1 0.20

1

2

3x 10

4

−0.1 −0.05 0 0.05 0.10

0.5

1

1.5

2x 10

4

−0.1 −0.05 0 0.05 0.10

0.5

1

1.5

2x 10

4

−0.4 −0.2 0 0.2 0.40

0.5

1

1.5

2x 10

4

Figure D.3: Historgrams for the Real Component in the Channel for the Configuration

Described in Figure 5.2

−0.2 −0.1 0 0.1 0.20

0.5

1

1.5

2

2.5x 10

4

−0.1 −0.05 0 0.05 0.10

0.5

1

1.5

2x 10

4

−0.1 −0.05 0 0.05 0.10

5000

10000

15000

−0.4 −0.2 0 0.2 0.40

0.5

1

1.5

2x 10

4

Figure D.4: Historgrams for the Imaginary Component in the Channel for the Configu-

ration Described in Figure 5.2


−0.1 0 0.10

5000

10000

−0.1 0 0.10

1000

2000

3000

−0.1 0 0.10

1000

2000

3000

−0.2 0 0.20

1000

2000

Figure D.5: Historgram for the Real Component in the Channel for the Configuration

Described in Fig 5.3

−0.1 0 0.10

5000

10000

−0.1 0 0.10

1000

2000

3000

−0.1 0 0.10

1000

2000

3000

−0.2 0 0.20

1000

2000

Figure D.6: Historgram for the Imaginary Component in the Channel for the Configura-

tion Described in Fig 5.3


Tr 1 Tr 2

Rv 1 5.468× 10−5 7.558× 10−5

Rv 2 1.198× 10−4 1.273× 10−3

Table D.5: Variance for the real component in Channel in Figure 5.3

Tr 1 Tr 2

Rv 1 5.461× 10−5 7.685× 10−5

Rv 2 1.21× 10−4 1.277× 10−3

Table D.6: Variance for the Imaginary Component in Channel in Figure 5.3

Bibliography

[1] A. Goldsmith, Wireless Communications. Cambridge University Press, 2005.

[2] A. Ghosh, J. Zhang, J. Andrews, and R. Muhamed, Fundamentals of LTE. Prentice

Hall, 2010.

[3] D. Tse and P. Viswanath, Fundamentals of Wireless Communications. Cambridge

University Press, 2005.

[4] G. Foschini and M. Gans, “On limits of wireless communications in a fading envi-

ronment when using multiple antennas,” Wireless Personal Communications, vol. 6,

pp. 311–335, 1998.

[5] I. Telatar, “Capacity of multi-antenna gaussian channels,” Bell Lab, Technical Mem-

orandum, 1995, http://mars.bell-labs.com/cm/ms/what/mars/index.html.

[6] R. Janaswamy, “Effect of element mutual coupling on the capacity of fixed length

linear arrays,” IEEE Antenna and Wireless Propagation Letters, pp. 157–160, Apr.

2001.

[7] C. Balanis, Antenna Theory. Wiley-Interscience, 2005.

[8] J. Zhu and G. Eleftheriades, “A simple approach for reducing mutual coupling in

two closely spaced metamaterial-inspired monopole antennas,” IEEE Antennas and

Wireless Propagation Letter, vol. 9, pp. 379–382, 2010.

112

Bibliography 113

[9] ——, “A simple approach to reducing mutual coupling in two closely-spaced elec-

trically small antennas,” in Antenna and Propagation Society International Sympo-

sium, July 2010.

[10] G. Eleftheriades, “Enabling RF microwave devices using negative refractive-index

transmission-line (NRI-TL) metamaterials,” IEEE Antennas and Wireless Propaga-

tion Magazine, vol. 49, no. 2, pp. 34–51, Apr 2007.

[11] G. Eleftheriades and K. Balmain, Negative-Refraction Metamaterials. Wiley Inter-

Science, 2005.

[12] K. Jo, Y. Ko, and H. Yang, “RF beamforming considering RF characteristic in

mimo system,” in International Conference on Information and Communication

Technology Convergence, vol. 1, December 2010.

[13] J. Litva and T.Lo, Digital Beamforming in Wireless Communication. Artech House,

1996.

[14] W. Jakes, Microwave mobile communications. IEEE Press, 1993.

[15] M. Simon and M. Alouini, Digital Communication over Fading Channel. John

Wiley & Son, 2000.

[16] R. Prasad, Universal Wireless Personal Communication. Artech House, 1998.

[17] Y. Jiang and M. Varanasi, “The RF-chain limited MIMO system-part I: Optimum

diversity- multiplexing tradeoff,” IEEE Transactions on Wireless Communication,

vol. 8, no. 10, pp. 5238–5247, October 2009.

[18] R. Janaswamy, Radiowave Propagation and Smart Antennas For Wireless Commu-

nication. Kluwer Academic, 2001.

Bibliography 114

[19] C. Chuah, D. Tse, and J. Kahn, “Capacity scaling in MIMO wireless systems under

correlated fading,” IEEE Transactions on Information Theory, vol. 48, pp. 637–650,

Mar. 2002.

[20] M. Gans, “Channel capacity between antenna arrays- part I: Sky noise dominates,”

IEEE Transactions on Communication, vol. 54, no. 11, pp. 1586–1592, November

2006.

[21] ——, “Channel capacity between antenna arrays- part II: Amplifier noise dom-

inates,” IEEE Transactions on Communication, vol. 54, no. 11, pp. 1983–1992,

November 2006.

[22] M. Morris and M. Jensen, “Network model for MIMO systems with coupled antenna

and noisy amplifiers,” IEEE Transactions on Antenna and Propagation, vol. 53, pp.

545–552, Jan 2005.

[23] H. Yang and M. Alouini, “Analysis and comparison of various switched diversity

strategies,” in IEEE Vehicular Technology Conference, vol. 4, Dec 2002, pp. 1948–

1952.

[24] A. Abu-Dayya and N. Beaulieu, “Analysis of switched diversity systems on

generalized-fading channels,” IEEE Trans. Communication, vol. COM-42, pp. 2959–

2966, Nov. 1994.

[25] H. Nae and M. Alouini, “Optimization of multi-branch switched diversity systems,”

IEEE Transactions on Communication, vol. 57, no. 10, pp. 2960–2970, October

2009.

[26] M. Blanco and K. Zdunek, “Performance and optimization of switched diversity

systems for the detection of signals with Rayleigh fading,” IEEE Transactions on

Communication, vol. COM-27, no. 12, pp. 1887–1895, Dec. 1979.

Bibliography 115

[27] F. Adachi, M. Feeney, and J. Parsons, “An evaluation of specific diversity combiners

using signals received by vertically-spaced base-station antennas,” Journal of the

Institution of Electronics and Radio Engineers, vol. 57, no. 6, pp. 219–224, Nov

1987.

[28] L. Yang and M. Alouini, “Average level crossing rate and average outage duration of

switched diversity systems,” in Global Telecommunications Conference, vol. 2, Nov

2002, pp. 1420–1424.

[29] H. Yang and M. Alouini, “Markov chains and performance comparison of switched

diversity systems,” IEEE Transactions on Communication, vol. 52, no. 7, pp. 1113–

1125, July 2004.

[30] N. Beaulieu, “Switching rates of selection diversity and switch-and-stay diversity,”

IEEE Trans. Communication, vol. 56, no. 9, pp. 1409–1413, Sept. 2008.

[31] T. Cover and J. Thomas, Elements of Information Theory. John Wiley & Son,

2006.

[32] J. Butler and R. Lowe, “Beamforming matrix simplifies design of electronically

scanned antenna,” Electronics Design, vol. 1, no. 1, pp. 157–160, 1964.

[33] J. Liberti and T. E. Rappaport, Smart Antennas for Wireless Communication.

Prentice Hall, 1999.

[34] R. Collin, Antenna and Radiowave Propagation. McGraw Hill, 1985.

[35] C. Edwards and D. Penny, Differential Equations and Linear Algebra. Pearson

Prentice Hall, 2005.

[36] M. Jensen and J. Wallace, “A review of antennas and propagation for MIMO wire-

less communication,” IEEE Transactions on Antenna and Propagation, vol. 11, pp.

2810–2824, Nov. 2004.

Bibliography 116

[37] D. Shiu, G. Foschini, M. Gans, and J. Kahn, “Fading correlation and its effect on the

capacity of multielement antenna system,” IEEE Transactions on Communication,

vol. 48, pp. 502–513, Mar. 2000.

[38] R. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE

Transations on Antennas and Propagation, vol. 34, pp. 276–280, 1986.

[39] Y. Choi and S. Alamouti, “Performance analysis and comparisons of antenna and

beam selection diversity,” in IEEE Vehicular Technology Conference, vol. 1, Septem-

ber 2004, pp. 166–170.

[40] G. Stuber, Principles of Mobile Communication. Kluwer Academic, 2001.

[41] F. Wang, M. Bialkowski, and X. Liu, “Antenna selection and transmit beamforming

switching scheme for a MIMO system operating over a varying rician channel,” in

Congress on Image and Signal Processing, vol. 9, October 2009.

[42] K. Kim, K. Ko, and J. Lee, “Mode selection between antenna grouping and beam-

forming for MIMO communication system,” in Conference of Information Sciences

and Systems, vol. 1, March 2009, pp. 506–511.

[43] W. Chou and R. Adve, “RF beamforming with closely spaced antennas,” in Cana-

dian Workshop on Information Theory, vol. 1, May 2011, pp. 174–177.

[44] N. Giri, Multivariate Statistical Analysis. Marcel Dekker Inc, 2004.

[45] R. Mallik, “On multivariate Rayleigh and exponential distribution,” IEEE Trans-

actions on Information Theory, vol. 49, no. 6, pp. 1499–1515, June 2003.

[46] K. Miller, Complex Stochastic Processes. Addison-Wesley, 1974.

[47] W. Davenport and W. Root, An Introduction to the Theory Random Signals and

Noise. McGraw-Hill Book Company, 1958.

Bibliography 117

[48] M. Schwartz, W. Bennett, and S. Stein, Communication systems and techniques.

IEEE Press, 1996.

[49] J. Marcum, “A statistical theory of target detection by pulsed radar,” IRE Trans.

Information Theory, vol. IT-6, pp. 59–267, Apr. 1960.

[50] A. Papoulis and S. Pillai, Probability, Random Variables and Stochastic Processes.

McGraw Hill, 2002.

[51] N. Sood, L.Liang, S. Hum, and C. Sarris, “Ray-tracing based modelling of ultra-

wideband pulse propagation in railway tunnels,” in IEEE Antenna Propagation Sym-

posium, vol. 1, July 2011.

[52] E. Lau, R. Adve, and T. Sarkar, “Mutual coupling compensation based on the mini-

mum norm with applications in direction of arrival estimation,” IEEE Transactions

on Antennas and Propagation, vol. 52, no. 8, pp. 2034–2041, August 2004.

[53] M. Simon, Probability Distributions Involving Gaussian Random Variables. Kluwer

Academic Publisher, 2002.

by William Chou...Figure 1.4: Blueprint for metamaterial antenna [8] 1.2 Metamaterial Antenna This...

Documents

Transcript of by William Chou...Figure 1.4: Blueprint for metamaterial antenna [8] 1.2 Metamaterial Antenna This...