a thesis drexel university mikhail aleksandrovich

Compact Multifunctional Dipole Antenna Array for MIMO Systems

A Thesis

Submitted to the Faculty

of

Drexel University

by

Mikhail Aleksandrovich Chernyavskiy

in partial fulfillment of the

requirements for the degree

of

Master of Science in Electrical Engineering / Telecommunications

June 2012

ii

Dedications

To my father

and my grandmother,

for their love and support.

iii

Acknowledgments

I would like to thank my advisor, Dr. Kapil Dandekar for his guidance, advice, and

support these past three years. I am very grateful to Prathap for being a great mentor

to me throughout his time in the lab, to Guillermo and David for their help with my

measurements, to Magda and Kevin for never being too busy for me, and to John

for his excellent advice. A big thank you to all past and present members of DWSL.

It was truly a pleasure to work with each and every one of you. Last but not least,

thanks to all of my friends for everything.

iv

Table of Contents

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Thesis Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 MIMO Wireless Communications . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Channel Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Antennas for MIMO Systems . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Diversity Combining Techniques . . . . . . . . . . . . . . . . . . . . 15

2.2.3 Reconfigurable & Multifunctional Antennas . . . . . . . . . . . . . . 16

3. Antenna Array Design and Simulation . . . . . . . . . . . . . . . . 20

3.1 Design Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Antenna Array Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

v

3.3 Antenna Array Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Antenna Array Construction . . . . . . . . . . . . . . . . . . . . . . . . 27

4. Antenna Array Measurement Results . . . . . . . . . . . . . . . . 30

4.1 Measured Antenna Parameters . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Channel Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.1 Channel Measurement Results . . . . . . . . . . . . . . . . . . . . . 36

5. Discussion and Future Research . . . . . . . . . . . . . . . . . . . . 41

5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Appendix A: Table of Symbols . . . . . . . . . . . . . . . . . . . . . . . 44

Appendix B: Table of Acronyms . . . . . . . . . . . . . . . . . . . . . . 45

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

vi

List of Tables

3.1 Antenna array structural parameters. . . . . . . . . . . . . . . . . . . . . 25

4.1 Area of the proposed antenna array and reference antenna array. . . . . . 35

4.2 Average measured percentage capacity improvement achievable with thestacked antenna array at 2.484 GHz with respect to the λ and λ/2 sepa-rations of the reference dipoles. . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Average measured percentage capacity improvement achievable with thestacked antenna array at 5.805 GHz with respect to the 2λ and λ separa-tions of the reference dipoles. . . . . . . . . . . . . . . . . . . . . . . . . 39

vii

List of Figures

2.1 Block diagrams of (a) SISO and (b) MIMO communication systems . . . 6

2.2 Three antenna diversity techniques: (a) spatial diversity, (b) pattern di-versity, and (c) polarization diversity. . . . . . . . . . . . . . . . . . . . 13

3.1 (a) Schematic and (b) prototype of the triband antenna from [32]. . . . 22

3.2 Schematics of the proposed antenna array design: (a) front view, (b) backview, and (c) perspective view. . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Simulated return loss and isolation plots of the two antennas in the pro-posed antenna array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4 Simulated radiation patterns in dBi of the two antennas in the proposedantenna array at 2.484 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.5 Simulated 3D radiation patterns of the two antennas in the proposed an-tenna array at 2.484 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.6 (a) Front and (b) back of the prototype of the proposed antenna array. . 29

4.1 Measured return loss and isolation plots of the two antennas in the pro-posed antenna array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Measured radiation patterns in dBi of the two antennas in the proposedantenna array at (a) 2.484 GHz, and at (b) 5.805 GHz. . . . . . . . . . 32

4.3 Floor plan of test environment. Test locations of TX and RX are indicated. 34

4.4 Comparison of the proposed antenna array (left) and the reference dipoles(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.5 CDF of capacity of the stacked antenna array and the reference dipolesassuming 10 dB SNR, measured at (a) 2.484 GHz, and at (b) 5.805 GHz. 38

4.6 CDF of capacity of the stacked antenna array as a function of SNR at2.484 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

viii

AbstractCompact Multifunctional Dipole Antenna Array for MIMO Systems

Mikhail Aleksandrovich ChernyavskiyKapil R. Dandekar, Ph.D.

A compact, stacked, multi-frequency dipole antenna array is designed and presented

for use in Multiple Input Multiple Output (MIMO) wireless communication systems.

The array consists of two dual band frame-printed dipoles occupying the same physical

space. Each antenna can operate in the 2.4 and 5 GHz bands for wireless local area

network (WLAN) applications. The array can be used in a 2 × 2 MIMO link at either

the transmitter (TX) or receiver (RX) or both.

The lack of spatial diversity that arises from having co-located antennas is coun-

terbalanced by the pattern diversity resulting from the mutual coupling between the

two antenna elements. This system takes advantage of the otherwise undesirable mu-

tual coupling within the antenna array elements by producing pattern diversity from

the shift in the radiation patterns.

The proposed antenna array was simulated and manufactured and its radiation

characteristics were tested. Channel measurements were also taken using the antenna

array and the WARP software defined radio platform. The proposed antenna array

has radiation characteristics and measured channel capacity comparable to that of

specialized antennas operating in each of the frequency bands, while providing the

added benefit of size reduction.

1

Chapter 1: Introduction

1.1 Motivation

Wireless communication has become an integral part of people’s daily lives and a

critical business tool. Wireless systems offer convenient and reliable connectivity

that allows for user mobility. In addition, wireless communication allows for net-

work access to be introduced to areas where it would be traditionally be difficult to

connect to a wired network, since wireless networks are easier to deploy. Because of

these attractive characteristics, wireless communication is currently at the forefront

of telecommunications research.

Modern communication systems require multiple antennas that support several

frequency bands in a compact space [1]. Multiple input multiple output (MIMO)

wireless communication [2] is a promising technology that plays an important role

in new and upcoming mobile communication systems. MIMO techniques combine

signals from multiple antennas to exploit the multipath in wireless channels and

enable higher capacity, better coverage, and increased reliability without using extra

spectrum and power resources [1]. The main advantage of MIMO systems is that

they provide the ability to form parallel orthogonal transmission channels, even in

rich scattering environments [3].

Many wireless communication standards require operation in multiple frequency

bands. For example, an IEEE 802.11n (WiFi) device requires at least two antennas

Chapter 1: Introduction 2

operating in the 2.4 GHz and 5 GHz bands. At the same time, consumer devices

like laptops, tablets, and smartphones continue to get smaller and thinner, leaving

less and less room for antennas. These constraints provide a need for a low profile

antenna array that is compatible with MIMO systems and able to operate on multiple

frequencies [1].

This thesis offers one possible solution to this problem by presenting a compact

multifunctional antenna array for MIMO communication systems comprised of two

dual band frame-printed dipoles. The antenna array was designed, simulated, con-

structed, and tested.

1.2 Related Work

Current research in the area of antennas for MIMO systems has been focused on

electrically reconfigurable [4–13] and multimode antennas [14–16]. These antennas

include spiral antennas [7, 15], dipole hybrids [4–6, 14, 15], patch antennas [8, 9, 15],

and pixel antennas [13] that can reconfigure their radiation patterns, polarizations,

and frequency operating bands.

Patch antennas are often used as reconfigurable antennas. A reconfigurable multi-

port circular patch antenna was designed such that it can excite different electromag-

netic modes by varying its radius with PIN diode switches [8]. Another circular patch

antenna array consisting of two stacked circular disks presented in [9] is capable of

changing the shape of its radiation pattern by selecting a pair of feed points con-

nected to the two antennas, which again excite different EM modes. The authors of

[10] demonstrate a compact pattern reconfigurable U-slot patch antenna.


Many somewhat more exotic antennas have also been investigated in the reconfig-

urable antenna literature. The authors of [7] propose a single arm Archimedean spiral

antenna whose arm length can be reconfigured in length using PIN diode switches,

exciting different radiation patterns in the process. A reconfigurable leaky wave an-

tenna was designed making possible dynamic changes to the array radiation pattern

[11]. A reconfigurable 2D fractal tree antenna is shown in [12]. A pixel antenna utiliz-

ing microelectromechanical switches (MEMS) capable of reconfiguring its radiation

modes and operating frequency has been designed for narrowband MIMO systems

[13]. Even a cubic antenna is investigated in [16].

Dipoles have been incorporated into many different reconfigurable antenna appli-

cations. A wideband reconfigurable MIMO antenna, a combination of a reconfigurable

balanced dipole and a two-port chassis antenna, has been demonstrated in [4]. The

authors of [5] present a compact reconfigurable antenna array consisting of two hybrid

monopole/dipole elements. Each element can operate in either monopole or dipole

mode. A circular polarization spiral-dipole antenna has been proposed in [15]. A

dipole antenna is loaded with spirals at both of its ends to generate omnidirectional

left-hand or right-hand circular polarization.

More traditional dipoles are presented in [6, 14]. A linear printed dipole array is

oriented in a fashion that introduces pattern diversity in [14]. An antenna array of

two printed dipoles, in which each of the dipoles can be reconfigured in length using

PIN diode switches can be seen in [6]. The switch configuration can be modified in

accordance to changes in the environment.


1.3 Thesis Contribution

This thesis contributes to the development of an optimal antenna system for MIMO

communications. The proposed antenna array is composed of two dipole antennas,

each operating in multiple frequency bands, creating frequency diversity. The array

is constructed in a way that introduces pattern diversity to the system. Most im-

portantly, while typical MIMO antennas are separated to achieve decorrelation, the

proposed array is compact for potential use in mobile devices.

1.4 Thesis Organization

Chapter 2 provides background information regarding topics such as MIMO wire-

less communications, channel capacity, diversity, and antennas for MIMO systems.

Chapter 3 presents the design of the proposed compact dipole antenna array and the

simulation results. Chapter 4 discusses the measurement results and evaluates the

antenna as part of a system. Chapter 5 concludes this thesis by providing a summary

of the work, a discussion of the research in context with existing designs, as well as

proposals for future work.

5

Chapter 2: Background

2.1 MIMO Wireless Communications

In a conventional radio communication system, one transmit (TX) and one receive

(RX) antenna are used to transmit information over a communication channel. This

is referred to as a single input single output (SISO) system. A block diagram of a

SISO system can be seen in Figure 2.1a. If a simplified channel is assumed to be

time and frequency invariant, the channel is denoted by a scalar h. The scalar signal

model is given as:

y = hx+ n (2.1)

where y is the received signal, x is the transmitted signal, and n is complex additive

white Gaussian noise (AWGN) with zero mean.

In a noise-limited scenario, the spectral efficiency of a channel is fundamentally

limited by the Shannon-Nyquist criterion [17], expressed as:

C = log2

(1 + SNR · |h|2

)(2.2)

= log2

(1 +

Pt |h|2

σ2n

)(2.3)

where h is the transfer function from the TX to the RX, SNR denotes the signal to

noise ratio at the RX, Pt is the transmitted power, and σ2n is the noise variance. The

channel capacity can increase logarithmically with an increase in transmit power.

Chapter 2: Background 6

(a)

(b)

Figure 2.1: Block diagrams of (a) SISO and (b) MIMO communication systems

If a link in a wireless communication system is equipped with multiple antenna

elements at both the transmitting and the receiving end, the system becomes a mul-

tiple input multiple output (MIMO) system. A block diagram of a MIMO system

can be seen in Figure 2.1b. The channel response for a narrowband non-frequency

selective MIMO system with Nr receive antennas and Nt transmit antennas is now


denoted by a channel matrix H ∈ CNr×Nt ,

H =

h11 h12 · · · h1Nt

h21 h22 · · · h2Nt

......

. . ....

hNr1 hNr2 · · · hNrNt

(2.4)

where hij is the transfer function, or scalar SISO channel, between the i-th RX an-

tenna and the j-th TX antenna [17]. The vector signal model is given as:

y = Hx + n (2.5)

where y ∈ CNr×1 is the received signal vector at the RX antennas, x ∈ CNt×1 is the

transmitted signal vector for the TX antennas, and n ∈ CNr×1 is the AWGN vector

at the RX antennas [6].

The channel matrix H is the mathematical representation of the transmission path

of the transmitted data, which includes the multipath channel characteristics of the

environment and the antenna configurations of the transmitting and receiving antenna

arrays. Both the multipath channel characteristics and the antenna configurations

play a large role in determining the performance of the MIMO system.

2.1.1 Channel Capacity

Assuming that the flat fading channel is unknown at the transmitter and known at the

receiver, the signal vector at the transmitter is composed of Nt independent signals


with equal power. In this case, the capacity is shown to be [18]:

C = log2

[det

(INr +

SNR

Nt

HH†)]

(2.6)

where INr is an identity matrix of size Nr × Nr, SNR is the mean signal to noise

ratio per receiver branch, det is the determinant, and the superscript † denotes the

complex conjugate transpose.

The Kronecker model is used in this thesis to generate random channels and

thus show the potential of the proposed antenna array [6]. In a spatially correlated

Rayleigh-fading MIMO channel, the channel matrix H is defined by the Kronecker

model as [19]:

H = R1/2r HwR

1/2t (2.7)

where Rr is the receive spatial correlation matrix, Rt is the transmit spatial correla-

tion matrix, and Hw ∈ CNr×Nt is a matrix of complex Gaussian fading coefficients.

In this thesis, a 2× 2 MIMO system is used, i.e. Nr = Nt = 2.

2.1.2 Fading

The signal radiated from a transmitting antenna is reflected, scattered, diffracted,

and/or refracted by the various structures in its path. Mathematically, these factors

cause the signal to experience path loss, shadowing, and fading [17]. Path loss is the

reduction in field strength when an electromagnetic wave propagates though space


and for free space is modeled as [20]:

L = 20 log10

(4πd

λ

)(2.8)

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

and L is the path loss in decibels.

Shadowing is caused by large obstructions that obscure the main signal path

between the transmitter and the receiver. These obstacles will be different for every

path, causing variations with respect to the value given by the path loss model.

Fading refers to rapid fluctuations in the signal and results from the interference

between multiple waves reaching the receiver from the transmitter [17].

Typically, the fading caused by multipath signal propagation is considered to be a

severe problem in wireless communication channels. However, MIMO systems exploit

the multipath signals in order to increase the system capacity without increasing the

system power or bandwidth [21]. The idea behind MIMO is that signals that travel

through complex multipath environments are combined in such a way that the quality

or data rate of communication for each MIMO user will be improved relative to the

SISO case. Since multiple data streams can be transmitted simultaneously on the

same frequency in MIMO systems, the bandwidth efficiency and capacity can be

increased linearly by the number of data streams, i.e. antennas employed, with no

additional overhead [17].

Multipath fading arises from the presence of multiple transmission paths between

the transmitter and receiver [22]. When a signal leaves a transmitter, it can take


many different paths to the receiver, each with its own reflections, diffractions, etc.

There are two different methods to transmit data through the utilization of these

multiple streams.

Spatial Multiplexing

Spatial multiplexing [23] is a scheme where independent data streams are transmitted

simultaneously in parallel channels from each element in an array of antennas. If

N = min(Nt, Nr), the bit stream in question is demultiplexed into N sub-streams,

then modulated and transmitted from each antenna simultaneously. Assuming the

receiver has knowledge of the channel, it can extract the signals, demodulate them,

and then recombine them to yield the original bit stream [23]. This process increases

spectral efficiency because the data stream can be transmitted N times as quickly as

a non-multiplexed data stream.

Space-Time Coding

Space-time coding [23] is an alternative scheme to spatial multiplexing. It uses the

multiple element antennas for diversity gain by encoding a single data stream across

both time and space. In other words, multiple redundant copies of a data stream are

transmitted on the N channels [23]. This process serves to increase link reliability.

Any errors in one of the transmission paths can be fixed through a comparison with

the other paths, therefore space-time coding can be used to improve the quality of

the transmission.


2.2 Antennas for MIMO Systems

In order for a MIMO communication system to have good performance, the antennas

used in the system must be capable of providing a high degree of diversity [24]. It is

also beneficial for the antenna array to have a compact design that is comparable in

size to a single antenna.

Traditionally the antennas in a MIMO system are spaced farther apart to achieve

higher spatial diversity [25]. However, this is not always possible in small mobile

applications. Thus it is often necessary to use antennas with different radiation

patterns or polarizations, or antennas operating at different frequencies in order to

achieve the required levels of diversity [2]. Section 2.2.1 expounds on the different

types of diversity and the benefits of each one.

It is also possible to achieve an increased diversity level by intelligently selecting

or combining the antennas at the transmitter and receiver that provide the highest

levels of diversity at the two nodes. Switching circuitry can be used to select the

group of antennas that provide the optimal system diversity for a given channel [2].

Section 2.2.2 summarizes the various algorithms used for diversity combining and

selection.

2.2.1 Diversity

The principle of diversity is that the receiver should have more than one copy of the

transmitted signal available, with each copy being received through a statistically

independent channel. If the signals are uncorrelated, the fading dips in the signal

will have a small probability of occurring simultaneously and therefore the multiple


signals can be combined at the receiver to make a signal that has a higher mean SNR

than any single branch of the system has by itself. Five categories of diversity are

discussed in detail in the following sections.

Spatial Diversity

Spatial diversity occurs when multiple physically separated antennas are used in the

system. Spatial diversity takes advantage of the random nature of propagation. Many

independent paths exist at any location, so the signals are uncorrelated when the

locations are separated by a certain minimum distance, usually some multiple of

the wavelength [22]. The independence comes from different multipath components

having different amplitudes and phases when arriving at different points in space.

The further apart the antennas are placed, the greater the produced phase difference,

and the smaller the correlation of the signals obtained at the antennas. Figure 2.2a

shows a diagram of two identical antennas with identical radiation patterns separated

in order to produce spatial diversity. Since the antennas must be separated, spatial

diversity is not a viable option for space constrained devices.

Frequency Diversity

Frequency diversity utilizes multiple frequency bands in the transmission of a signal.

The same signal could be transmitted on both frequencies or the information could

be split up between the frequencies. If the carrier frequencies are separated by more

than the coherence bandwidth [17] of the channel, then their fading can be considered

to be independent, and the probability that the signal simultaneously experiences a

fade at both frequencies is low [17]. If multiband antennas are used in the system, the


(a) (b) (c)

Figure 2.2: Three antenna diversity techniques: (a) spatial diversity, (b) patterndiversity, and (c) polarization diversity.

same antenna can be used to transmit and/or receive at each frequency of interest.

Limitations of frequency diversity are the availability of bandwidth and that the

channel must be frequency-selective [22].

Pattern Diversity

Pattern diversity (or angle diversity) [17] makes use of antennas having different radi-

ation patterns. Each antenna picks up multipath components coming from different

angular directions. The amplitudes and phases of these multipath components will be

different with different antenna patterns. Therefore, their combination will be uncor-

related [22]. Various types of antennas with differing radiation patterns can be used

for the sake of pattern diversity, but it is also possible to produce different radiation

patterns with identical antennas by mounting them close to each other. This effect is

due to strong electromagnetic interactions between the antennas, otherwise known as


mutual coupling [26]. Each antenna’s pattern will be skewed due to the electromag-

netic interactions from the other antenna [17]. Figure 2.2b shows a diagram of two

co-located antennas producing two distinct radiation patterns at different angles.

Polarization Diversity

Because the reflection and diffraction processes depend on polarization, horizontally

and vertically polarized multipath components propagate differently in a wireless

channel [17]. The propagation effects of the channel depolarize the propagating beam,

which leads to the fading of different polarizations being statistically independent [17].

Thus, the depolarized signal can be split into horizontal and vertical polarizations at

the receiver and processed separately to produce diversity. The only limitation of

polarization diversity is that, as opposed to the other diversity schemes, it is only

possible to generate two diversity branches – horizontally and vertically polarized (or

any other two orthogonal polarizations) [22]. Figure 2.2c shows a diagram of two

co-located antennas with two orthogonal polarizations.

Time Diversity

Since the wireless propagation channel is time-variant, signals sent and received at

different times are uncorrelated [17]. If the same data stream is transmitted multiple

times at intervals that exceed the coherence time [22] of the channel, the streams are

subject to independent fading. An advantage of time diversity, or temporal diversity,

is that multiple antennas are not required in the system. However, this diversity

scheme is highly bandwidth-inefficient and requires storage to save the received data

streams for processing [22].


2.2.2 Diversity Combining Techniques

It is necessary to somehow combine the signals arriving at the receiver due to the above

antenna diversity techniques such that the quality of the overall signal is improved.

Selection diversity selects the best signal while discarding all of the other copies of

the signal. Combining diversity, on the other hand, combines all available copies of

the signal using different algorithms. Generally, diversity combining leads to better

performance because all present information is utilized [17]. There are three main

algorithms that are implemented in diversity combining techniques.

Selection Combining

Selection combining [17] is mathematically the simplest diversity combining scheme.

Since the fades in the individual signals do not happen simultaneously, the instan-

taneous SNR is monitored in all branches and the branch with the highest SNR is

selected as the output signal [22]. This method is the easiest to implement but is

inefficient because the useful signal power received on the non-selected branches is

discarded.

Maximal Ratio Combining

In order to improve the output SNR even further, the signals from all branches can

be combined to form the output signal. Maximal ratio combining takes all of the

individual signals and performs a linear combination on them, using appropriate

weighting [22]. In order to maximize the SNR at the output, a branch with higher

SNR will be given a higher weighting [23]. Since the signals are not in phase, they

have to first be multiplied by a complex phasor in order to bring them to zero phase


so they can be combined coherently. The maximal ratio combiner provides the best

performance when compared to the other algorithms but at the cost of increased

complexity.

Equal Gain Combining

Equal gain combining is similar to maximal ratio combining in that all of the branches

are added together. The difference is that all the branches are weighed equally. The

output SNR is better than the SNR of selection combining, but not as good as the

SNR produced by maximum ratio combining. However, it is easier to implement

than the maximum ratio combiner and as the number of antennas or channels in the

system increases, the difference between equal gain combining and maximum ratio

combining decreases [22].

2.2.3 Reconfigurable & Multifunctional Antennas

Using several antennas in an array in combination with the above-mentioned diversity

combining algorithms can often provide the highest level of diversity. However, when

using these algorithms, not all antennas within the array are simultaneously utilized

at any given time. As a result, this becomes an impractical solution for portable

devices where space is a critical constraint.

Reconfigurable or multifunctional antennas provide the opportunity for a single

antenna that incorporates the diversity techniques outlined in Section 2.2.1 in its

design to replace several antennas in a system. These antennas are usually more

compact than the multiple antenna arrays they replace, making them much more

attractive for mobile applications, as well as other applications where space is a con-


straint. Reconfigurable antennas can adaptively tune their radiation characteristics,

polarization, or frequency of operation in response to the fluctuations in the wireless

channel [2]. Various design techniques have been proposed for modifying antenna

attributes [4–13, 27].

The arrangement of currents on an antenna determines the antenna’s radiation

distribution [2]. RF switches, material changes, and structural changes can be em-

ployed to achieve changes in an antenna’s radiation pattern [2]. One very popular

approach is using PIN diode switches to reconfigure the antenna structure. The an-

tenna array in [5] consists of two elements, each of which has two possible modes

by controlling states of three PIN diodes. Similarly, each of the dipoles in the re-

configurable printed dipole array in [6] can change the length of its arms using PIN

diodes to produce different modes with different radiation patterns. The various an-

tennas in [7–10, 12] all use switches to change the physical shape of the antenna and

consequently the radiation pattern.

Beam-steering antennas are a type of pattern reconfigurable antenna that can

sweep their narrow main lobe across a wide range of angles. The reconfigurable leaky

wave antenna in [11] consists of ten unit cells loaded with varactor diodes and two

independent bias networks used to separately tune the varactors and steer the two

beams.

For polarization reconfigurability, the antenna structure, material properties, or

feed configuration have to change in ways that alter current flow on the antenna

[2]. There are different kinds of polarizations [28]: i) various linear polarizations,

ii) right-handed and left-handed circular polarizations, and iii) elliptical polarizations.


Polarization is usually modified using RF switches or material changes [29]. In [27],

an antenna consisting of a single octagonal microstrip patch has two ports located

on perpendicular sides of the patch that excite two orthogonal polarizations of the

radiated electric field. The antenna uses MEMS switches to select between the two

polarization bases.

Frequency reconfigurable antennas [30] can switch their frequency of operation,

thus implementing frequency diversity without the need for multiple antennas. Multi-

band antennas can operate on multiple frequencies simultaneously, combining multi-

ple elements in order to create antennas that operate in several independent bands.

The reconfigurable dipole-chassis antennas presented in [4] are already multiband

by design, but they can also sweep their operating frequencies by several hundred

megahertz by varying the supplied voltage to four varactor diodes in the matching

circuit.

The best reconfigurable antennas allow for simultaneous changes to multiple radi-

ation characteristics. These reconfigurations are often achieved through pixel-based

approaches such as the pixel antenna in [13]. However the use of a large number of

switches introduces high losses and decreases the radiation efficiency of the antenna

[2].

The contribution of this thesis is a novel compact multifunctional dipole antenna

array for MIMO communication systems. The stacked antenna system consists of two

individual dual band frame-printed dipoles sharing the same physical space on the

board. The antenna array demonstrates both pattern and frequency diversity without

any losses from switches or switching networks. The frequency diversity comes from


both dipoles in the array being designed to resonate at multiple frequencies instead

of reconfiguring their frequency of operation with switches. The pattern diversity is

generated by the mutual coupling effects between the antenna array elements. The

pattern and frequency diversity replace the spatial diversity typically used in MIMO

systems.

20

Chapter 3: Antenna Array Design and Simulation

3.1 Design Motivation

Modern communication standards often necessitate the use of multiple antennas at

multiple frequencies. At the same time, modern consumers demand compact wireless

devices that require the antenna profile to be as small as possible. These require-

ments and constraints raise two challenges. First, the antennas need to be designed

extremely compactly with a low profile. Secondly, the interaction between the antenna

elements conventionally needs to be kept to a minimum to prevent a mutual coupling

effect between the antennas. This is generally done by separating the antennas by at

least a half-wavelength.

However, these two constraints are at odds with one another. If it is necessary to

minimize the profile of the antennas and design them to be as small and compact as

possible due to the imposed space constraint, it will not be feasible to place them far

apart. A possible solution to this problem would be eliminating the second constraint

of no mutual coupling.

Mutual coupling has been shown to be beneficial in MIMO systems [26]. The

presence of other array elements in the near field of each antenna array element will

distort the radiation pattern of each of the array elements in a unique fashion [26].

This distortion will produce pattern diversity in the system and improve the quality

of the communications link by increasing the channel capacity [31].

Chapter 3: Antenna Array Design and Simulation 21

Instead of viewing mutual coupling as a drawback, as is generally done in MIMO

communications, the design proposed in this thesis embraces the coupling between

the two driven antenna array elements. The mutual coupling results in changes in

the source currents on both of the elements, which results in a modification of the

impedance presented at the element terminals [28]. More importantly, it also modifies

the radiation patterns of the antennas in the array due to interactions between each

of the antennas, thus providing pattern diversity to the system. The two dipoles

therefore have two different radiation patterns and can pick up multipath components

arriving from different directions [17].

3.2 Antenna Array Design

The design proposed in this thesis is based on the frame-printed dipole presented in

[32]. The authors of [32] designed a multiband antenna consisting of a set of printed

frame dipoles of different sizes, printed on a double-sided dielectric substrate. The

arms of the dipoles are printed on opposite sides of the substrate and the antennas

are fed from a 50 Ω coaxial cable through a microstrip-to-twinline tapered transition.

The printed frame dipole is constructed by etching off a section of the arms of the

traditional strip dipole. Thus, the characteristics of the frame dipole are similar to

those of the strip dipole. The dipoles are nested within each other and are employed

as resonators to produce a multiband response. Each of the three dipoles operates

at a unique frequency band and the overall triband antenna operates at 1.8 GHz,

2.4 GHz, and 3.5 GHz. The nested dipoles can be considered parallel-connected, so

the off-resonant dipoles will have a higher shunt impedance and will not debase the


performance of the active dipole [32]. A schematic of the antenna presented in [32],

as well as the milled prototype, can be seen in Figure 3.1. The antenna was not

evaluated in a system.

(a)

(b)

Figure 3.1: (a) Schematic and (b) prototype of the triband antenna from [32].

The design proposed in this thesis first modifies the antennas proposed in [32]

in order for the array to operate in the 2.4 GHz and the 5 GHz bands for WLAN


(wireless local area network) applications. This modification is done by changing the

lengths of the dipole arms. Next, two of these antennas are incorporated in the space

that would typically be used by only a single antenna by mirroring the front and back

arms of the dipole and angling the microstrip feedlines in opposite directions in order

to create two separate ports. The width of each feedline is chosen to match each port

to 50 Ω.

To avoid the crossing of the microstrip lines on the back of the proposed antenna

array, a middle board layer containing a segment of one of the microstrip lines is

added and used as a bridge. The microstrip line of one of the antennas begins on

the 3rd layer, is connected to the bridge on the 2nd layer, and is connected back to

the antenna feedline on the 3rd layer after passing over the microstrip line of the

other antenna. Adding the middle board layer produces a three layer board and

results in two virtually identical independent dual band frame dipoles that share

one physical space despite each dipole having its own input port. Schematics of the

proposed antenna can be seen in Figure 3.2. A summary of the main dimensions of

the proposed antenna array and the material properties of the substrate is presented

in Table 3.1.

3.3 Antenna Array Simulation

The antenna array was designed and simulated in HFSS [33], a finite element method

solver for electromagnetic structures. Figure 3.3 shows a plot of the simulated return

loss of the two antennas (S11 and S22). As can be seen from the figure, the antennas

have similar but not identical return loss curves. However, they both radiate in the


(a) (b)

(c)

Figure 3.2: Schematics of the proposed antenna array design: (a) front view,(b) back view, and (c) perspective view.


Table 3.1: Antenna array structural parameters.

AntennaOutside dipole arm length 14.75 mmInside dipole arm length 9.416 mmOutside dipole arm height 18 mmInside dipole arm height 6 mmDipole arm width 1.5 mmLeft antenna feedline thickness 3 mmRight antenna feedline thickness 1.5 mmFeedline length 48 mmBoard length 70 mmBoard width 75 mmBoard height 1.5748 mmSubstrateDielectric FR-4Dielectric permittivity 4.4Dielectric loss tangent 0.02Dielectric thickness 0.7874 mm

2.4 GHz and the 5 GHz bands as desired. The two antennas have return loss values

of -10.6 and -12.6 dB at 2.484 GHz, respectively, and -14.8 and -12.8 dB at 5.32 GHz,

respectively. The return loss is below the target -10 dB in the bands of interest for

both ports. The isolation between the two antenna ports (S21) is also shown on

Figure 3.3. The isolation is -14.0 dB at 2.484 GHz and -8.0 dB at 5.32 dB, below the

target -10 dB in the 2.4 GHz band but slightly above at the higher frequency.

Figure 3.4 shows the simulated azimuthal radiation patterns of each of the dipoles

in dBi. The radiation patterns are similar but shifted by approximately 45, demon-

strating the desired pattern diversity. The radiation patterns are likely not identical

because of the bridge used in the middle layer of the antenna, as it is the only

structural difference between the two antennas. Simulated 3D radiation patterns are


Figure 3.3: Simulated return loss and isolation plots of the two antennas in theproposed antenna array.

displayed in Figure 3.5. The gain values in these plots are displayed in absolute units,

not dBi, in order to emphasize the differences between the two radiation patterns.

One of the antennas has a slightly higher maximum gain value and the shifted ra-

diation patterns are even more pronounced in Figure 3.5. The simulated antenna

radiation efficiency was found to be 0.435 at 2.484 GHz and 0.368 at 5.32 GHz. The

relatively low efficiency is expected when using a low εr material like FR-4. When

the dielectric permittivity value increases, the radiation efficiency decreases [2].


0°

15°

30°

45°

60°

75°90°105°

120°

135°

150°

165°

±180°

−165°

−150°

−135°

−120°

−105°−90° −75°

−60°

−45°

−30°

−15°

−18−14−10−6−2

Figure 3.4: Simulated radiation patterns in dBi of the two antennas in theproposed antenna array at 2.484 GHz.

3.4 Antenna Array Construction

The proposed antenna array allows for a low-cost manufacturing solution. The anten-

nas were milled using a T-Tech Quick Circuit 5000S-FA milling machine [34] on FR-4

dielectric substrate of 31 mil thickness. After milling, the three layers were connected

using copper interconnects, and the layers were adhered together using a LPKF Mul-

tiPress S multiboard press [35]. The final prototype can be seen in Figure 3.6. The

bridge in the middle layer can be seen clearly through the back of the antenna array,

in Figure 3.6b.


(a)

(b)

Figure 3.5: Simulated 3D radiation patterns of the two antennas in the proposedantenna array at 2.484 GHz.


(a) (b)

Figure 3.6: (a) Front and (b) back of the prototype of the proposed antennaarray.

30

Chapter 4: Antenna Array Measurement Results

4.1 Measured Antenna Parameters

The scattering parameters of the dipoles were measured using an Agilent N5230A

Vector Network Analyzer (VNA). The decision was made to perform 5 GHz measure-

ments at 5.805 GHz instead of 5.32 GHz because i) the S-parameters at 5.805 GHz on

the manufactured antenna appear superior to those at 5.32 GHz and ii) the 5.805 GHz

frequency, corresponding to Channel 161, is also free from outside interference. Since

radiation at either frequency is sufficient for 5 GHz channel measurements, the an-

tenna did not have to be redesigned, but rather the target frequency in the 5 GHz

band could simply be changed since the intended design did not have a specific target

frequency in mind.

Figure 4.1 shows the measured return loss curves of the two antennas. The two

antennas have return loss values of -16.2 and -15.7 dB at 2.484 GHz, respectively,

and -16.9 and -17.3 dB at 5.805 GHz, respectively, both well below the target -10 dB

value. The return loss values are comparable to the results of the simulations. The

other peaks seen in Figure 4.1 are caused by coupling between the antennas. The

isolation between the two antennas is -12.8 dB at 2.484 GHz and -18.5 dB at 5.805 dB,

again both well below the target -10 dB and exceeding the simulated results.

Radiation patterns were measured in the Drexel anechoic chamber. Figure 4.2

shows the measured azimuthal radiation patterns in dBi taken at 2.484 and 5.805 GHz.

Chapter 4: Antenna Array Measurement Results 31

Figure 4.1: Measured return loss and isolation plots of the two antennas in theproposed antenna array.

The measured radiation patterns are similar to the simulated ones at 2.484 GHz, with

a comparable shift between the two patterns – approximately 45. The 5.805 GHz

radiation patterns do not look as similar to simulated results, but still exhibit some

pattern diversity. Unfortunately, accurate gain values were not able to be measured

in the 5 GHz band due to limitations of the anechoic chamber. The chamber is not

able to be calibrated for the higher frequency band.

4.2 Channel Measurements

The antenna array was tested as part of a system by taking channel measurements in

an indoor environment using the Wireless Open-Access Research Platform (WARP),


0°

15°

30°

45°

60°

75°90°105°

120°

135°

150°

165°

±180°

−165°

−150°

−135°

−120°

−105°−90° −75°

−60°

−45°

−30°

−15°

−16−12−8−40

(a)

0°

15°

30°

45°

60°

75°90°105°

120°

135°

150°

165°

±180°

−165°

−150°

−135°

−120°

−105°−90° −75°

−60°

−45°

−30°

−15°

−20−10

0

(b)

Figure 4.2: Measured radiation patterns in dBi of the two antennas in theproposed antenna array at (a) 2.484 GHz, and at (b) 5.805 GHz.


a software defined radio testbed developed by Rice University [36]. A 2 × 2 MIMO

orthogonal frequency division multiplexing (OFDM) implementation of WARP was

used. Measurements were performed on channel 14 of the 802.11 band (centered

at 2.484 GHz) and at channel 161 (centered at 5.805 GHz), each with a 20 MHz

bandwidth. The measurements were taken in Drexel’s Wireless Systems Laboratory

on the 3rd floor of the Bossone Research building in Drexel University. A floor plan of

the environment, along with the locations of the TX and RX can be seen in Figure 4.3.

The receiver remained stationary while a TDK PP-02 field probe positioner [37] was

used to sweep the transmitter along a 1.5 m × 1.5 m grid in the horizontal and vertical

directions in 50 cm steps. The channel matrix was measured for each location of the

transmitter for both frequencies. Due to the vertical displacement of the transmitter,

the measurements consisted of a combination of both line-of-sight (LOS) and non-

line-of-sight (NLOS) links.

The printed dipoles described in [38] were used as reference antennas for com-

parison of the proposed design. Since the antennas in [38] are only designed for

2.484 GHz, the design was scaled and analogous dipoles were manufactured for the

5 GHz band. One of the 2.4 GHz dipoles and one of the 5 GHz dipoles joined by

a splitter were used at each port of the 2 × 2 MIMO system and the measurements

were repeated as described above for the proposed antenna array. Because of uniform

power allocation, each of the reference dipoles received half of the total input power

[39]. This was done because each of the two antennas in the proposed antenna array

would be replacing two of the reference dipoles in a communication system.

WARP was used to measure the channel matrix at each location of the transmitter


Figure 4.3: Floor plan of test environment. Test locations of TX and RX areindicated.

for both frequencies and for different separation distances of the reference antennas.

The separation distances used were λ and λ/2 for the 2.4 GHz band and 2λ and λ for

the 5 GHz band. The λ/2 separation was impossible to achieve with the shortened

wavelength corresponding to the 5 GHz band because of the way the antennas were

mounted.

Figure 4.4 shows the proposed antenna array next to the reference dipoles. Ta-

ble 4.1 displays the area of each antenna array. The area of the proposed system is

reduced by approximately 58% with respect to a reference system composed of two

2.4 GHz dipoles and two 5 GHz dipoles separated by λ/2 in order to generate spatial

diversity in the system.


Figure 4.4: Comparison of the proposed antenna array (left) and the referencedipoles (right).

Table 4.1: Area of the proposed antenna array and reference antenna array.

Antenna Area (cm2)Reference dipole array 124.3Proposed antenna array 52.5

Channel capacity is selected as the performance metric because it allows the study

of the antenna array performance independently from the system modulation and the

adopted coding technique [2]. To determine the capacity of the MIMO OFDM link,

a Frobenius normalization of the channel matrix for each subcarrier was computed

in order to remove the differences in path loss among the different channel matrices

while preserving the relative antenna gain effects [6]. The normalization factor is


defined as [6]:

NF =

√‖Href‖2FNtNr

(4.1)

Since the channel was characterized over a broad frequency band, the capacity

of the wideband channel was defined as an average value of the capacities over all

the m subcarriers of the MIMO OFDM system [40]. The capacity was additionally

averaged over the k samples in order to minimize the impact arising from the minor

differences in spatial orientation between the stacked dipole array and the reference

dipoles.

Therefore, an estimator of the Shannon capacity is the average of the capacities

over the m subcarriers and the k samples, and was determined using [10]:

C =1

km

k∑j

m∑i

log2

[det

(INr +

SNR

Nt

HijH†ij

N2Fij

)](4.2)

where m is the total number of subcarriers, k is the total number of samples, and NFij

is the Frobenius norm for each subcarrier and sample. In this experiment, m = 52

subcarriers and k = 3200 samples (200 samples at each of the 16 locations) were used.

There are 64 total subcarriers, but only 52 are used for data transmission. A separate

Frobenius norm was calculated for each frequency band. The optimal solution for the

reconfigurable antenna was the one that guaranteed the highest average capacity.

4.2.1 Channel Measurement Results

Figure 4.5 shows the CDF plots of the capacity of the stacked antenna array and the

reference dipoles in the indoor environment shown in Figure 4.3 for both measured


frequencies assuming an SNR of 10 dB. The proposed antenna array significantly

increases MIMO link capacity with respect to a conventional dipole system. As can

be seen in Figure 4.5a, the proposed antenna array convincingly outperforms the

reference dipoles at both spacings in the 2.4 GHz band. It does not perform as well

in the 5 GHz band, but it is still comparable to the reference dipoles at the higher

frequency while providing a form factor improvement.

Table 4.2 shows the average percentage capacity improvement in the 2.4 GHz band

by using the stacked antenna array with respect to the reference dipoles for different

values of SNR in a MIMO OFDM system. Table 4.3 shows the same in the 5 GHz

band. At an SNR of 10 dB, the proposed antenna array outperforms the optimal

configuration of the reference dipoles by 22% in the 2.4 GHz band, and the capacity

improvement ranges from 33% at 5 dB SNR to 8% at 30 dB SNR. As can be seen, the

proposed antenna array performs better with respect to the reference antennas at low

SNR values. Figure 4.6 shows the capacity of the stacked antenna array measured at

2.484 GHz as a function of SNR.

The performance of the antenna in the 5 GHz band was roughly comparable to the

performance of the reference dipoles, at both separations and at all SNR values. The

median capacity improvement ranges from 3% at 5 dB SNR to -3% at 30 dB SNR.

The performance of the proposed antenna array is much worse in the 5 GHz frequency

band than in the 2.4 GHz band, but even at the higher frequency the performance

is still similar to the reference dipoles. Again, the proposed antenna array performs

better at low SNR values.

It is likely that the pattern diversity of the proposed antenna array was not as


1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Capacity (bps/Hz)

F(x

)

Stacked Antenna ArrayReference Array − λ/2 SeparationReference Array − λ Separation

(a)

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Capacity (bps/Hz)

F(x

)

Stacked Antenna ArrayReference Array − λ SeparationReference Array − 2λ Separation

(b)

Figure 4.5: CDF of capacity of the stacked antenna array and the referencedipoles assuming 10 dB SNR, measured at (a) 2.484 GHz, and at (b) 5.805 GHz.


significant in the 5 GHz band, which contributed to the poor capacity measurement

results in that band. The radiation patterns in the 2.4 GHz band demonstrate pattern

diversity (Figure 4.2a) and therefore the system using the antenna array at that

frequency has good channel capacity results. However, since the radiation patterns

in the 5 GHz band are not as clean and the gain values are unknown (Figure 4.2b),

the channel capacity suffered.

Table 4.2: Average measured percentage capacity improvement achievable withthe stacked antenna array at 2.484 GHz with respect to the λ and λ/2 separationsof the reference dipoles.

SNR (dB) 5 10 20 30Capacity improvement w.r.t. λ/2 separation 33% 22% 13% 8%Capacity improvement w.r.t. λ separation 39% 31% 19% 12%

Table 4.3: Average measured percentage capacity improvement achievable withthe stacked antenna array at 5.805 GHz with respect to the 2λ and λ separationsof the reference dipoles.

SNR (dB) 5 10 20 30Capacity improvement w.r.t. λ separation 3% -1% -2% -3%Capacity improvement w.r.t. 2λ separation 2% -2% -3% -3%


0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Capacity (bps/Hz)

F(x

)

30 dB SNR20 dB SNR10 dB SNR5 dB SNR

Figure 4.6: CDF of capacity of the stacked antenna array as a function of SNRat 2.484 GHz.

41

Chapter 5: Discussion and Future Research

5.1 Discussion

A compact stacked multiband dipole antenna array for MIMO communication systems

is proposed in this thesis. The two dipoles in the antenna array each demonstrate fre-

quency diversity individually while also demonstrating pattern diversity with respect

to one another. The antenna array is compact and eliminates the antenna spacing

typically required in MIMO antenna arrays for spatial diversity. The proposed an-

tenna array has been shown to provide an improvement in capacity when compared

against traditional dipoles. An additional benefit of the proposed design is that by

combining two antenna elements in this fashion, manufacturing costs and valuable

space in mounting the antenna are saved.

All of the antennas described in Section 1.2 exhibit at least one form of diversity.

However, most of these antennas have some drawbacks as well. The antennas in [6–10,

15] all operate in a single frequency band, making them impractical for applications

such as WiFi that use both the 2.4 and the 5 GHz bands. The pixel antenna presented

in [13] can tune both its frequency and its radiation pattern by exciting different

modes of radiation. However, by utilizing hundreds of MEMS switches, the authors

introduce losses into the system. The losses are much smaller than if they had used

diodes, but they still contribute to decreases in the antenna’s radiation efficiency.

Additionally, as the authors themselves state, MEMS are not currently reliable enough

Chapter 5: Discussion and Future Research 42

on PCB to introduce the antenna into industry [13].

The linear array of two printed dipoles fed through a microstrip balun in [14]

is the most similar to the antenna array proposed in this thesis. The dipoles are

already dual band, resonating at 2.3 and 5.2 GHz, and are oriented in a fashion that

introduces pattern diversity through mutual coupling. The authors orient one of the

dipoles upside down and alongside the other one. Due to this antenna positioning,

however, the antenna array becomes rather bulky.

On the other hand, the stacked dipole array proposed in this thesis efficiently

utilizes its board space while exhibiting pattern and frequency diversity, making it

more suited for space constrained wireless devices.

5.2 Future Research

Future research will focus on the improvement of the 5 GHz band of the dipoles

in the proposed array. Currently, the measured radiation patterns in the 5 GHz

band do not match the simulated radiation patterns nearly as well as those in the

2.4 GHz band. As a result, measurements taken in the 5 GHz band have significantly

lower capacities than those taken in the 2.4 GHz band. If the isolation, and thus

the efficiency, of the antennas at higher frequencies is improved, perhaps with better

matching or a matching network, the capacity measurements should improve as well.

It is also worthwhile to consider taking measurements at a different frequency in the

5 GHz band to see if that can improve the capacity.

The efficiency of the antenna array can also be improved by using a material with

a higher dielectric permittivity constant as the antenna substrate instead of FR-4,

Chapter 5: Discussion and Future Research 43

which has an εr of 4.4. For example, Rogers RO3010 material has an εr of 10.2

and would be a good initial candidate to replace FR-4. However, depending on the

material chosen, it is likely that it would no longer be possible to manufacture the

antenna array in-house as many of the higher εr materials are too brittle to be milled.

The antenna structure is designed in such a way that it would also be very easy

to add more frequency bands by nesting extra printed dipoles inside the existing

ones. Therefore, in the future it would be possible to have the antennas in the array

resonate at additional frequencies if their application required them to do so.

44

Appendix A: Table of Symbols

Symbol DescriptionCn×m operator to denote complex matrix with dimension n×mdet(·) determinant operator· inner product operator| · | absolute value operator‖ · ‖F Frobenius norm operator(·)† complex conjugate transpose operatorC channel capacityd separation of the TX and RXH channel matrixHw matrix of complex Gaussian coefficientsh scalar channelIn identity matrix with dimension n× nk total number of samplesL path lossm total number of subcarriersN minimum number of antenna elements between the TX and RXNF Frobenius normalization factorNr number of receive antenna elementsNt number of transmit antenna elementsn complex additive white Gaussian noisen complex additive white Gaussian noise vectorPt transmitted powerRr receive spatial correlation matrixRt transmit spatial correlation matrixS11 voltage reflection coefficientS21 voltage forward transmission coefficientSNR signal to noise ratiox signal at the transmitterx signal vector at the transmittery signal at the receivery signal vector at the receiverεr dielectric permittivityλ wavelengthσ2n variance of the additive white Gaussian noise

45

Appendix B: Table of Acronyms

Acronym DefinitionAWGN Additive White Gaussian NoiseCDF Cumulative Distribution FunctionEM ElectromagneticHFSS High Frequency Structural SimulatorLOS Line-of-SightMEMS Microelectromechanical SystemsMIMO Multiple Input Multiple OutputNLOS Non-Line-of-SightOFDM Orthogonal Frequency Division MultiplexingPCB Printed Circuit BoardRF Radio FrequencyRX ReceiverSISO Single Input Single OutputSNR Signal to Noise RatioTX TransmitterVNA Vector Network AnalyzerWARP Wireless Open-Access Research PlatformWLAN Wireless Local Area Network

46

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a thesis drexel university mikhail aleksandrovich

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