Dual Polarized Omnidirectional Array Element for MIMO Systems

Dual Polarized Omnidirectional Array Element for

MIMO Systems

ALEJANDRO NIETO GONZALEZ

Master of Science ThesisStockholm, Sweden 2005-02-04

IR-SB-EX-0502

Abstract

The capacity of a Multiple-Input Multiple-Output (MIMO) system dependson the channel matrix as well as the Signal-to-Noise ratio. A novel dual-polarized planar channel model with one or two scattering rings is used tocompare the capacity of different antenna arrays. The correlation betweenchannel matrix elements of the presented channel model and the statisticalKronecker model is also compared. We find that the Kronecker model isequivalent to the “two-ring” model, but not to the “one-ring” model.

A dual-polarized omni-directional antenna is presented which provideslow correlation between channel elements, in order to achieve high capacity.The capacity simulations show that although linear polarized TX and RXantennas provide higher capacity when they are polarization matched, dual-polarized antennas provide better performance when an arbitrary orientationis considered.

i

Acknowledgements

I would like to thank my parents for their support during my studies, evenwhen I told them that I wanted to come to Sweden to do my last year ofuniversity. And that year turned on a year and a half and this MSc Thesis.I would also like to thank Ines Cabrera Molero, without whom I would neverhave come to Sweden.

Finally, I would like to thank Ph.D. Bjorn Lindmark at the AntennaSystem Technology Group at the Department of Signals, Sensors and Systemsat the Royal Institute of Technology (Stockholm, Sweden), for providing methe opportunity to perform this MSc Thesis and for his guidance.

iii

Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Channel models . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Mutual coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Dual-polarized antennas in MIMO communications . . . . . . 41.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Channel Analysis 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 One-ring model . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Two-ring model . . . . . . . . . . . . . . . . . . . . . . 82.2.3 Mutual coupling . . . . . . . . . . . . . . . . . . . . . 9

2.3 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 Equal received power . . . . . . . . . . . . . . . . . . . 112.3.2 Equal transmitted power . . . . . . . . . . . . . . . . . 12

2.4 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.1 Kronecker model . . . . . . . . . . . . . . . . . . . . . 122.4.2 Comparison with a one-ring model . . . . . . . . . . . 162.4.3 Comparison with a one-disc model . . . . . . . . . . . 192.4.4 Comparison with a two-ring model . . . . . . . . . . . 19

2.5 Dual Polarized Antennas . . . . . . . . . . . . . . . . . . . . . 232.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 242.5.2 Cross-polarization ratio and scatterer coefficients . . . 25

2.A Analytical Study of the Correlation . . . . . . . . . . . . . . . 28

3 Antenna Design 333.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Loop antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 Constant current square loop . . . . . . . . . . . . . . 343.3 Planar antenna . . . . . . . . . . . . . . . . . . . . . . . . . . 36

v

vi CONTENTS

3.3.1 Square loop . . . . . . . . . . . . . . . . . . . . . . . . 383.3.2 Monopole . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Capacity Simulations 434.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Ideal electric and magnetic dipoles . . . . . . . . . . . . . . . 444.3 2 × 2 Two ring . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 4 × 4 Two ring . . . . . . . . . . . . . . . . . . . . . . . . . . 484.5 2 × 2 One ring . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Conclusions and Future Work 535.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Chapter 1

Introduction

1.1 Background

High data rate wireless communications is currently of interest in wirelesslocal area networks (WLAN). Nowadays, the WLAN offer peak rates of 50Mb/s, while the wired LANs provide until 10 Gb/s. Designing wireless net-works that provide such rates constitutes a significant and research challenge.One way of providing higher capacities than the classical wireless networksis using multiple antennas at both ends of the communication link. SuchMultiple-Input Multiple-Output (MIMO) systems offer the possibility to ex-ploit a multi-path environment to achieve a capacity enhancement [1]. It isknown that a system consisting of nt transmit antennas and nr receive anten-nas provides a capacity proportional to min(nt, nr) with no increase of thebandwidth or transmit power, providing a sufficient rich multipath scatter-ing. This rich scattering must ensure a independent spatial fading betweeneach transmit and receive element of the antennas.

In most MIMO capacity analysis, the antenna elements are ideal isotropicor quarter-wavelength dipoles [2]. For achieving a substantial increase incapacity the correlation between the elements should be low [1] [3]. Thisconstrains the practical design of the Multi-element arrays (MEA), becausethe elements must be enough separated. Another way of obtaining this lowcorrelation is to use dual-polarized arrays [4], which avoid undesired antennaspacings. It is well known that the two polarization states of electromag-netic waves can be used to transmit separate channels of information. Thisproperty has been used in microwave transmission systems and as a diversitytechnique for wireless fading channels [5]. Recently it has been investigatedthe possibility of using six polarization channels instead of two in a multi-path environment [6], although this point will not be analyzed in this thesis.

1

2 CHAPTER 1. INTRODUCTION

h11

h22

h12

h21

11

22

TXRX

Figure 1.1: 2×2 Channel

However, any capacity increase that results from using dual-polarization maybe diminished by the consequences of a reduced radiation pattern [7]. A fo-cused radiation pattern would result in fewer multipath components beingreceived and a thus a reduction of the rich scattering, which is so importantfor obtaining high capacities. For that reason it would be very useful to havea dual-polarized omni-directional antenna. Such an antenna would have aloop with a dipole in the center, thereby providing orthogonal polarizationstates and zero mutual coupling.

This project will study the capacity of a system consisting in this kindof antennas at both ends of the wireless communication in comparison withclassical MEAs. A practical antenna design is also to be developed.

1.2 Channel models

It is known that the capacity1 a MIMO communication systems is given bythe following formula [1]:

C = log2 det

(Inr +

ρ

nt

·HHH

)b/s

Hz(1.1)

where I is the identity matrix, ρ is the average SNR at each receiver branch,nt is the number of transmitters, H is the channel matrix and H denotes theHermitian transpose (complex conjugate transpose). To study the capacityof such a system an accurate characterization of the channel matrix must bedone. Fig. 1.1 depicts a 2×2 channel.

1The correct term is mutual information between transmitter and receiver, but capacityis widely used.

1.3. MUTUAL COUPLING 3

Many MIMO channel models have been developed, and they can generallybe divided in three classes [8]: ray-tracing, scattering and correlation models.In the ray tracing the propagation, diffraction, reflections and scattering aremodelled [9]. Each propagation path through the channel is followed. Thisgives a very good prediction, but the complexity becomes very high. Thecorrelation model generates the channel by multiplying a complex Gaussianindependent and identically distributed (i.i.d.) by the square roots of thecorrelation matrices at the receiver and the transmitter [10] [11] [12]. Thisis very simple to implement, although it is not very clear which is the phys-ical background of this model. The scattering model assumes a particulardistribution of the scatters and generates channel realizations based on theinteraction of the scatters and the planar wavefronts [13] [14] [15]. The prob-lems of this model are to generate the scatterers distribution (usually onlyone bounce is considered) and the resulting simulation time. A summary ofthe different channel models is presented in [16].

In this thesis we are going to deal with three models: the widely used cor-relation model and two planar physical scattering models (“one ring model”and “two ring model”).

1.3 Mutual coupling

Mutual coupling has been a subject of interest in MIMO communications,since it influence the channel matrix, and thus, the capacity of the system.Although is a well known subject in antenna theory [17] it has not beenstudied so much from a signal processing perspective.

Some papers [18] [19] [20] state that mutual coupling provides an en-hancement of MIMO communications, since it decorrelates the antenna el-ements. This happens because the antenna patterns of the elements areinfluenced by mutual coupling (“pattern diversity”). This result in two op-posite effects [18]: increased element gain and channel decorrelation. Thesecond effect influence more and the performance of the system is higher.

There are other studies [21] [11] which state that the mutual coupling doesnot increase the capacity. These analysis suggest that the power losses causedby the mutual coupling reduce the capacity. There is also the affirmationof [22] that the pattern diversity is just an extra correlation between theantennas, and thus the capacity under effect of mutual coupling is lower.

This controversy can be observed also in [23], which shows that the mutualcoupling decorrelates the antenna elements (and thus, increases the capacity)when the elements are placed is short distances, but the correlation becomeshigher if the elements are far away.


More recent studies propose a complete characterization of the system [24]using network theory analysis, and a conclusion using this technique [25] isthat mutual coupling can increase the capacity if a proper matching is used,although usually is impractical, and self-impedance matching is not enough.

1.4 Dual-polarized antennas in MIMO com-

munications

The idea of using electric and magnetic dipoles (dipole and loop) simulta-neously for a wireless communication is not new. In [26] an investigation ofco-located elements is done by placing the dipole and the loop in a planar,stack configuration. The mutual coupling between the elements is studiedand results demonstrate that the system is useful for increasing the capacity.However, the polarization used in this MEA is the same in both elements(no dual-polarization is used).

A deeper investigation about MIMO capacity is done in [27], with a prac-tical design and measurements. Still, in this study the electric and magneticmonopoles are also co-located, and thus the polarizations are not orthogonal.

There are also investigations on dual-polarized antennas. Two stud-ies [4] [28] which are very similar point out the benefits of using dual-polarizedantennas in MIMO communications rather than co-polarized: less spacing be-tween elements and less correlation, although also indicate that the antennadesign is more complicated. However, they lack of a better channel modeland they consider isotropic elements rather than a practical antenna design.

Some other studies propose better channel models for dual-polarized ele-ments [29] [30], but always avoiding to consider concrete antennas.

There is also a paper [31] that examine the antenna structure that thisthesis is going to study, with the electric and magnetic dipole momentsaligned. Though, the study just analyze the mutual coupling between theantenna elements, which result to be very low, without considering the struc-ture in a MIMO communication system.

Finally, a recent antenna design has been presented in [32] which is dual-polarized and omni-directional. Our design is very similar to the one pre-sented there.

1.5 Outline

In Chapter 2 the channel models are presented. A comparison between thedifferent models, physical and statistical, is presented. At the end of the

1.5. OUTLINE 5

chapter a dual-polarized channel model is presented, which includes the crosspolarization ratio.

In Chapter 3 the antenna design is done using CST Microwave Studio2.Before designing the dual-polarized omni-directional antenna the fundamen-tal aspects are studied.

In Chapter 4 the designed antenna and the channel model are used to-gether to simulate the capacity under different channel conditions and com-paring the results to typical MEAs.

Finally, Chapter 5 present the conclusions of the thesis, along with sug-gestions for future work.

2www.cst.com

Chapter 2

Channel Analysis

2.1 Introduction

In MIMO communications there are nt transmit antennas and nr receiveantennas, where each antenna has a different signal to transmit. Undernarrow band conditions, the relation between the transmit data s ∈ Cnt andthe receive data x ∈ Cnr is commonly modelled by the matrix equation

x = Hs + n (2.1)

where n ∈ Cnr is white zero-mean complex Gaussian noise with Enn∗ = I.H is a complex nr × nt channel matrix containing elements that model theattenuation and phase of the channel between each transmit and receiveantenna respectively.

The mutual information for s having a covariance matrix Rss is givenby [2]

I = log2 det

(Inr +

ρ

nt

·HRssHH

)b/s

Hz(2.2)

where I is the identity matrix, ρ is the average SNR at each receiver branch,nt is the number of transmitters, H is the channel matrix and H denotes theHermitian transpose (complex conjugate transpose).

Acquiring channel knowledge at the transmitter is in general very difficultin practical systems. In this thesis we study the capacity of the systemwhen there is no transmitter knowledge. In the absence of channel stateinformation at the transmitter it is reasonable to choose s spatially white,i.e., Rss = Int . This implies that the signals transmitted from each antennaare independent and equal powered. The mutual information achieved withthis covariance matrix, which we will call from now capacity, is given by [1]:

7

8 CHAPTER 2. CHANNEL ANALYSIS

C = log2 det

(Inr +

ρ

nt

·HHH

)b/s

Hz(2.3)

To study the capacity of such a system an accurate characterization of thechannel matrix must be performed.

2.2 Physical Model

2.2.1 One-ring model

One commonly used model to create the channel matrix is a 2D physical“one-ring model”. A model similar to the one described in [13] and in [15] isconsidered.

In the model, the base station (BS) is unobstructed by local scatterers(as it is usually elevated) and the subscriber unit (SU) is surrounded byscatterers. This situation fits well a macro-cellular environment. Only onebounce is considered here. The transmit array and the receive array areseparated by the distance D. The scatterers are uniformly distributed in aring of radius R whose center is situated in the receive array. The scatterersare modeled by a complex coefficient α. As in [14], the scattering coefficientsare normal complex random variables, with zero mean and unit variance.The channel parameters hi,j connecting the transmitting element j and thereceive element i is thus

hi,j =L∑

l=1

αle−j 2π

λ(DBjSl

+DSlMi)

|(DBjSlDSlMi

)| ej(θj,l) ei(θi,l) (2.4)

where DBjSlis the distance from the base station j to the scatterer l, DSlMi

is the distance from the scatterer l to the mobile antenna i. The angle θj,l isfrom the base station j to the scatterer l and θi,l is the angle from the mobilestation i to the scatterer l. e is the antenna pattern. L is the total numberof scatterers. An illustration of the model is shown in Fig. 2.1.

2.2.2 Two-ring model

This model is similar to the previous one, but both transmitters and receiversare surrounded by scatterers [33]. This model is more accurate in indoor orpicocell environments. We assume that each path between transmit andreceive antenna consist of a single bounce at each scattering ring. Thus, thechannel is generated by

2.2. PHYSICAL MODEL 9

B1

B2

M1

M2

h11

h21

h12

h22

DB2S1

Scatterer 1

DS1M2

D R

Figure 2.1: Illustration of the “one-ring” model.

hi,j =L′∑

l′=1

L∑

l=1

αlα′l′e−j 2π

λ(DBjSl

+DSlSl′+DSl′Mi)

|DBjSlDSlSl′DSl′Mi

| ej(θj,l) ei(θi,l′) (2.5)

where the total number of scatterers in each ring is L and L′ and the param-eters DBjSl

, DSlSl′ and DSl′Miare defined in Fig. 2.2. The angle θj,l is from

the base station j to the scatterer l in the first ring and θi,l′ is the angle fromthe mobile station i to the scatterer l′ in the second ring.

2.2.3 Mutual coupling

In the previous models we include the antenna pattern e. Lets now defineformally the pattern and include the effects of the mutual coupling. For thecalculation of the antenna pattern we need the directivity of the antennas inthe presence of the others, with the same load as the generator impedance,and the S-parameters of the array.

We define the antenna pattern of the antenna n as the electric far field atone meter from the antenna when all the generators are switched off, exceptgenerator n. If we have an array of m antennas, the pattern of the antennan is:


B1

B2

M1

M2

h11

h21

h12

h22

DS1S1’

Scatterer 1’

DS1’M2

D R

Scatterer 1

DB2S1

Figure 2.2: Illustration of the “two-ring” model.

En = Eff,n(r = 1, θ, φ)|an,ai6=n=0(2.6)

We have assumed that the antennas are well matched, and thus the re-flected power in the loads is zero (ai6=n=0). Under this assumption we havethe following input power, reflected power and radiated power:

Pin = |an|2 (2.7)

Prefl =m∑

k=1

|bk|2 (2.8)

Prad = |an|2 −m∑

k=1

|bk|2 (2.9)

By the definition of the S-parameters and by the previous assumption wecan express

bk = Sknan (2.10)

and therefore the radiated power is

Prad = |an|2(1−

∑|Skn|2

)(2.11)

2.3. NORMALIZATION 11

The radiation pattern can also be expressed as an integral of the antennapattern

Prad =η

2

∫

Ω

|En|2dΩ (2.12)

By combining (2.12) and (2.11) we conclude that the pattern of the an-tenna n is

En = an

√1−

(1−

∑|Skn|2

)en(θ, φ)

√Dn(θ, φ) (2.13)

where Dn is the directivity of the antenna n. Therefore the pattern to includein the channel model is

en(θ, φ) = |En| = an

√1−

(1−

∑|Skn|2

)√Dn(θ, φ) (2.14)

2.3 Normalization

Ir order to make fair comparisons between different scenarios a proper nor-malization of the channel matrix must be performed.

2.3.1 Equal received power

A normalization, which removes the effects of the path loss, is (e.g. [15])

E||H||2F = ntnr (2.15)

where || · ||F means the Frobenius matrix norm.To achieve this normalization there are N realizations of the channel

matrix, Hn. The normalization given by ( 2.15) is performed according to

N∑n=1

||Hn||2F = Nntnr (2.16)

and therefore, if we let Hi be the un-normalized matrix, the new normalizedmatrix is given by

Hi =Hi

(∑N

n=1 ||Hn||2F 1nrntN

)1/2(2.17)

This normalization creates a Rayleigh channel and it is used for comparingthe correlation between the channel elements of the different models.


2.3.2 Equal transmitted power

This normalization is used to compare in a fair way the different antennasin the model. When the power is fixed in transmission the effect of thedifferent antenna patterns influences the capacity. Using equal power receivedthis effect is dismissed, and only the correlation between the antennas isimportant. The variance of the scatterer coefficient α is normalized in orderto have a concrete Signal-to-Noise ratio when a 1x1 system with verticaldipoles is used.

2.4 Correlation

A comparison between the correlation of the channel coefficients of the sta-tistical model and the physical model is performed.

2.4.1 Kronecker model

As pointed out in [16], a statistical model for MIMO communications iscreated by a channel matrix whose elements are zero mean complex Gaussian,and the covariance matrix can be approximated by the Kronecker productof the covariance matrices seen from both ends, i.e., RH = RTX

⊗ RRX,

where RH is the channel covariance matrix, RTXand RRX

are the covariancematrices at the transmitter and receiver respectively, and ⊗ denotes theKronecker product. Thus, the correlated channel matrix can be written as

H = R1/2RX

GR1/2TX

(2.18)

where G is a zero mean, unit variance complex gaussian matrix. RRXhas

the following appearance

RRX=

VarV1 CovV1, V2 . . . CovV1, VnrCovV2, V1 VarV2 CovV2, Vnr

.... . .

CovVnr , V1 CovVnr , V2 . . . VarVnr

(2.19)

where Vi is the received signal at antenna i. The corresponding matrix atthe transmitter side RTX

has, by reciprocity, a similar appearance but thematrix has the size nt × nt. From the matrix above it is clear to see thatthe matrix elements correspond to the covariance between different antennaelements.

2.4. CORRELATION 13

Now it is important to derive the expression for the covariance betweenthe different antenna elements. It is well known that the electric field gener-ated by an antenna is given by

Ei =e−jkR

Rfi(θ, φ) ei(θ, φ) (2.20)

where fi is the angular dependence and ei expresses the polarization. How-ever, it is more interesting the far-field of the antenna

Ei = limR→∞

REi = fi(θ, φ) ei(θ, φ) (2.21)

Now, we assume that the far-field function of antenna i is denoted as Ei,and it has components in θ and φ, i.e.,

Ei(θ, φ) = [Ei,θ(θ, φ), Ei,φ(θ, φ)]T (2.22)

where T denotes transpose of a vector. Likewise, the distribution of theincident field is given by

a(θ, φ, t) = [aθ(θ, φ, t), aφ(θ, φ, t)]T (2.23)

where the time dependence t is introduced. The unit of the incident field isV/m/sr. The received open circuit voltage on an antenna element is the sumof the contributions from all directions seen by the antenna. This is

V i(t) =

∫

Ω

Ei(θ, φ) · a(θ, φ, t)dΩ (2.24)

where Ω = [θ, φ] and dΩ = sin θdθdφ. Now we define the signal received atthe antenna terminal as a scaled voltage,

Vi(t) =1√2η

∫

Ω

Ei(θ, φ) · a(θ, φ, t)dΩ (2.25)

where η is the characteristic impedance of free space. Since we have theexpressions for the received signals now it is possible to derive the covariancebetween the antenna elements (see [34] for more details). We assume thesignals to be zero mean.


CovVi, Vj = E(Vi − Vi)(Vj − Vj)∗ = EViV

∗j

=1

2ηE

∫

Ωi

Ei(Ωi)Ta(Ωi, t)dΩi(

∫

Ωj

Ej(Ωj)Ta(Ωj, t)dΩj)

∗

=1

2η

∫

Ωi

∫

Ωj

TrEi(Ωi)Ej(Ωj)HEa(Ωi, t)a(Ωj, t)

HdΩidΩj

(2.26)

We assume zero correlation between the radiation coming from two di-rections Ωi and Ωj except when Ωi = Ωj. With this assumption, we canwrite

Ea(Ωi)a(Ωj)H =

[Eaθ(Ωi)aθ(Ωj)

∗ Eaθ(Ωi)aφ(Ωj)∗

Eaφ(Ωi)aθ(Ωj)∗ Eaφ(Ωi)aφ(Ωj)

∗]

=

[Eaθ(Ωi)aθ(Ωi)

∗ Eaθ(Ωi)aφ(Ωi)∗

Eaφ(Ωi)aθ(Ωi)∗ Eaφ(Ωi)aφ(Ωi)

∗]

δ(Ωi − Ωj)

= δ(Ωi − Ωj)J(Ωi)′ (2.27)

and the integration of (2.26) can be reduced to

CovVi, Vj =1

2η

∫

Ωi

∫

Ωj

TrEi(Ωi)Ej(Ωj)Hδ(Ωi − Ωj)J

′(Ωi)dΩidΩj

=1

2η

∫

Ω

TrEi(Ω)Ej(Ω)HJ′(Ω)dΩ (2.28)

The matrix J′ is the covariance matrix of the incident fields. If we assumethat the polarizations are independent with equal angular distribution S(Ω)(∫

ΩS(Ω)dΩ = 1), the covariance of the incident field is

J′(Ω) = S(Ω)1

χ + 1

[χ 00 1

](2.29)

where χ is the cross-polar discrimination or XPD of the environment and isdefined as the ratio between the vertical, Pθ, and the horizontal, Pφ, poweron the antenna in the two polarizations at any given angle θ,φ. If the XPD isassumed to be one, i.e., there is equal power in both polarizations, then 2.28is simplified to

CovVi, Vj =1

2η

∫

Ω

Ei(Ω) · E∗j(Ω)S(Ω)dΩ (2.30)

2.4. CORRELATION 15

With this approach, the variance of a signal is then the power radiatedby the antenna element when the angular distribution of the incident field isone for all directions, as an antenna radiates in all directions. Hence,

VarVi = CovVi, Vi =1

2η

∫

Ω

Ei(Ω) · E∗i (Ω)S(Ω)dΩ = S ≡ 1

=1

2η

∫

Ω

(|Ei,θ(Ω)|2 + |Ei,φ(Ω)2|)dΩ = Prad,i (2.31)

It is now interesting to define the complex correlation coefficient for theantennas i and j, that is equal to the complex correlation between the re-ceived signals Si and Sj,

ρij =E(Vi − Vi)(Vj − Vj)

∗√E|Vi − Vi|2E|Vj − Vj|2

=EViV

∗j √

E|Vi|2E|Vj|2

=CovVi, Vj√

VarViVarVj(2.32)

If we now look to the correlation matrix of an array of two elements inthe transmitter side we have

RTX=

[1 ρt

ρ∗t 1

](2.33)

where ρt is the correlation between elements 1 and 2 of the transmitter array.Likewise, we have

RRX=

[1 ρr

ρ∗r 1

](2.34)

for the receiver side. If we now perform the Kronecker product of (2.33)and (2.34), the correlation matrix of a 2× 2 MIMO system can be expressedas (see [35])

RH = RTX⊗RRX

=

1 ρr ρt ρrρt

ρ∗r 1 ρ∗rρt ρt

ρ∗t ρrρ∗t 1 ρr

ρ∗rρ∗t ρ∗t ρ∗r 1

,

1 ρr ρt s1

ρ∗r 1 s2 ρt

ρ∗t s∗2 1 ρr

s∗1 ρ∗t ρ∗r 1

(2.35)


Parameter ValueR 200λD 3000λnumber of scatterers 100number of transmitters 2number of receivers 2

Table 2.1: Parameters used in the simulation for the “one-ring” model.

The Kronecker product assumes that the correlation between diagonalelements is expressed as the product of the transmit and receive correlations,i.e., s1 = ρrρt and s2 = ρ∗rρt. This assumption does not always hold. It istrue for a symmetric scenario as shown in the next sections.

2.4.2 Comparison with a one-ring model

A comparison between the statistical model and the physical model is per-formed when the antennas are isotropic and the distance between the el-ements is varied. The parameters of the physical model are presented inTable 2.1.

The statistical model is generated for incident waves confined in a 2Dplane (θ = π/2, and thus, S(Ω) = δ(θ − π/2)Sφ(φ)). The incident waves atthe receiver are uniformly distributed in the plane (Sφ(φ) = 1 ∀φ) since thereceivers are surrounded by scatterers. As shown in [34] the correlation ofthe receiver antennas is

ρr = J0(kd) (2.36)

where k is the wave number, d is the distance between receivers and J0 is thezeroth order first kind Bessel function.

However, the incident waves at the transmitter are not uniformly dis-tributed since the waves come from the scatterers at the receiver. Hence, thescattering occurs within a angular sector ∆φ centered around φ0 = 0. Theresultant radiation is (see Appendix 2.A)

Sφ(φ) =P0

π

1√R2 − φ2D2

(2.37)

where P0 is the power scattered, R is the ring radius and D is the distance be-tween the transmitter and the receiver. Thus, the correlation of the transmitantennas is (see Appendix 2.A)

2.4. CORRELATION 17

0 0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Distance

Co

rre

latio

n b

etw

ee

n c

ha

nn

el co

eff

icie

nts

h11

h21

1

2

1

2

Physical model

Statistical model

Figure 2.3: Correlation between channel elements (h11 and h21) when theantenna separation is varied (in λ). In the Kronecker model this is equal toρr (correlation of the two antennas surrounded by scatterers). The physicalmodel was generated with 10000 channel realizations.

ρt = J0

(kdR

D

)(2.38)

The correlation between the channel matrix elements h11 and h21, whichin the Kronecker model is equal to ρr, is represented in Fig. 2.3. Bothstatistical and physical model match perfectly.

It is also interesting to look at the correlation between h11 and h12, since itrepresents the transmit correlation (ρt) in the statistical model. As Figure 2.4show, the statistical and the physical give the same results.

However, the previous correlations just take into account three antennasof the system, and the difference between the models is higher in the anti-diagonal of the correlation matrix (i.e., when the correlation involves the fourantennas). If we observe the correlation between the elements h11 and h22 thetwo models produce different results (Fig. 2.5). This is pointed out in [35],where it is shown that having zero correlation at transmitter and receiver ina one-ring model does not produce a i.i.d. channel matrix, although that isthe result if a Kronecker product is used (see (2.35)).


0 0.5 1 1.5 2 2.5 3

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

Distance

Corr

ela

tion b

etw

een c

hannel coeffic

ients

h11

h12

1

2

1

2

Physical model

Statistical model

Figure 2.4: Correlation between channel elements (h11 and h12) when theantenna separation is varied (in λ). In the Kronecker model this is equal toρt (correlation of the two antennas far away from a ring of scatterers). Thephysical model was generated with 10000 channel realizations.

0 0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Distance

Co

rre

latio

n b

etw

ee

n c

ha

nn

el co

eff

icie

nts

Physical model

Statistical model

h11

h22

1

2

1

2

Figure 2.5: Correlation between channel elements (h11 and h22) when the an-tenna separation is varied (in λ). This case is asymmetric. In the Kroneckermodel this is equal to ρtρr. The physical model was generated with 10000channel realizations.

2.4. CORRELATION 19

Parameter ValueRtrasmit 200λRreceive 200λD 3000λnumber of scatterers in transmission 100number of scatterers in reception 100number of transmitters 2number of receivers 2

Table 2.2: Parameters used in the simulation for the “two-ring” model.

2.4.3 Comparison with a one-disc model

The correlation between the channel signals strongly depends on the situationof the scatterers. In order to see the influence of the scatterers position theyare now placed inside the ring. Thus, we have the scatterers placed insidea disc. We call his new distribution “one-disc” model. Still assuming onebounce, the angular distribution of the radiation is (see Appendix 2.A)

Sφ(φ) =2P0

π

√R2 − φ2D2

R2(2.39)

where P0 is the power scattered, R is the radius of the disc and D the distancefrom the disc to the transmitters. The correlation of the transmit antennasis (see Appendix 2.A)

ρt =J1

(kdRD

)kdRD

(2.40)

where J1 is the first order first kind Bessel function. Fig. 2.6 shows a perfectmatch between the statistical model (Equation (2.40)) and the simulatedchannel by the physical model.

2.4.4 Comparison with a two-ring model

Now a double-bounce system, with rings of scatterers both in transmitterand receiver, is considered. The parameters are shown in Table 2.2.

In the statistical model the waves are coming from everywhere in bothtransmitter and receiver (Sφ(φ) = 1 ∀φ), as they are surrounded by scatter-ers. Hence, (2.36) is valid for both ρt and ρr.

Figure 2.7 depicts the correlation between h11 and h21. The statisticaland the physical model match perfectly, as well as the correlation betweenh11 and h12 (Figure 2.8). This was also the case in the one-ring model.


0 0.5 1 1.5 2 2.5 3

0.8

0.85

0.9

0.95

1

1.05

Distance

Co

rre

latio

n b

etw

ee

n e

lem

en

ts

Physical model

Statistical model

h11

h12

1

2

1

2

Figure 2.6: Correlation between channel elements (h11 and h12) when theantenna separation is varied (in λ). One-disc model (the scatterers are in-side the circle). In the Kronecker model this is equal to ρt (correlation oftwo antennas far away from a disc of scatterers). The physical model wasgenerated with 10000 channel realizations.

2.4. CORRELATION 21

0 0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Distance

Co

rre

latio

n b

etw

ee

n c

ha

nn

el co

eff

icie

nts

1

2

1

2

h11

h21

Physical model

Statistical model

Figure 2.7: Correlation between channel elements (h11 and h21) when theantenna separation is varied (in λ). The scatterers are placed in a two-ringfashion. In the Kronecker model this is equal to ρr (two antennas surroundedby a ring of scatterers). The physical model was generated with 10000 chan-nel realizations.


0 0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Distance

Co

rre

latio

n b

etw

ee

n c

ha

nn

el co

eff

icie

nts

1

2

1

2

h11

h12

Physical model

Statistical model

Figure 2.8: Correlation between channel elements (h11 and h12) when theantenna separation is varied (in λ). The scatterers are placed in a two-ringfashion. In the Kronecker model this is equal to ρt (two antennas surroundedby a ring of scatterers). The physical model was generated with 10000 chan-nel realizations.

2.5. DUAL POLARIZED ANTENNAS 23

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

Distance

Co

rre

latio

n b

etw

ee

n c

ha

nn

el co

eff

icie

nts

Physical model

Statistical model

1

2

1

2

h11

h22

Figure 2.9: Correlation between channel elements (h11 and h22) when theantenna separation is varied (in λ). The scatterers are placed in a two-ringfashion. This case is asymmetric. In the Kronecker model this is equal toρtρr. The physical model was generated with 10000 channel realizations.

If we look at the correlation between h11 and h22 (Figure 2.9) the statisti-cal and physical model are the same, unlike the one-ring case. The directions-of-departure (DoD) and directions-of-arrival (DoA) are fully decorrelated andtherefore the diagonal correlations can be written as s = s1 = s2 = ρrρt [35].

2.5 Dual Polarized Antennas

In the previous sections it was assumed that the polarization was equal intransmitter and receiver. Now, if we take dual-polarized antennas the scat-terers have to be modeled in a new way. The scatterers behave differently ifthe incident polarization is vertical or horizontal. There is also cross-couplingfrom vertical polarization to horizontal polarization and viceversa. The Kro-necker model cannot represent now this situation. In previous works [15] [33]a channel model for multi-polarized MIMO systems was presented, both forsingle bounce and double bounce respectively. This channel models are basedon a detailed physical description of the scatterers. Here, we are more inter-ested on a statistical modelling of the interaction between the waves and thescatterers. Therefore our model will be simpler.


2.5.1 Definitions

The approach to simulate the channel is to take physical measurementsand apply the data into the model to create a realistic realization. Themost important parameter to model the polarization diversity is the cross-polarization power discrimination ratio (XPD or χ). The XPD is the powerratio of horizontal and vertical components of the mean incident field whenjust one polarization is used in transmission. Therefore, we have two cases:vertical transmission (χv) and horizontal transmission (χh). Please note thatthe order of the ratios depend on the power transmitted so that the XPD isalways positive if the cross-coupling is reasonable. That is,

χv =Pr,v

Pr,h

∣∣∣∣Pt=Pt,v

=EEr,v|2E|Er,h|2 (2.41)

χh =Pr,h

Pr,v

∣∣∣∣Pt=Pt,h

=E|Er,h|2E|Er,v|2 (2.42)

where Pr,v is the power received in vertical polarization, Pt,v is the powertransmitted in vertical polarization, Er,v is the field received in the verticalpolarization and so on.

Now the scatterers cannot be modeled using a single coefficient. Thescatterers interact differently with the incident waves if the field is verticallyor horizontally polarized, and they create cross-coupling between the polar-izations. Therefore, the scatterers are modeled as a 4× 4 matrix coefficient

Es = α Ei =

[Es,v

Es,h

]=

[αvv αhv

αvh αhh

] [Ei,v

Ei,h

](2.43)

where Ei is the incident field to the scatterers and Es is the scattered field.Fig. 2.10 shows how the scatterer works. It is reasonable to state that αvv

and αvh should be correlated, as well as αhh and αhv. For example, in [36]they argue that αvv and αhh are complex gaussian and the other parametersare just a scaling of those. But even then the correlation between the receivedsignals is quite low. Moreover, there is not a physical mechanism that explainproperly why the variables have to be correlated and how much. Therefore,in the following sections the assumption of uncorrelated variables is taken.This assumption gives uncorrelation to the scattered fields Es,v and Es,h.

Now it is important to model properly the scatterer coefficients in differentscenarios, and to relate them with the cross-polarization ratio.


Ei,v

Ei,h

Es,v

Es,h

αhhαvh

αvv

αhv

Figure 2.10: The incident field interacts with the scatterer and produces thescattered field. There is power-leaking from the vertical polarization to thehorizontal polarization and viceversa, represented by the scatterer coefficientsα

2.5.2 Cross-polarization ratio and scatterer coefficients

We will consider two different cases: one ring model and two ring model. Af-ter deriving the relation between the XPD and the scatterer coefficients someassumptions will be taken in order to completely characterize the coefficients.The assumptions will be explained later.

One ring model

In the one ring model it is assumed a single bounce. Therefore the field atthe receive antennas (Er) can be expressed as a function of the transmittedfield (Et) and the scatterer coefficients

Er = α Et (2.44)

If the the transmit antennas are vertically polarized (and so Et,h = 0)then we can express the vertical cross-polarization ratio as

χv =E|Er,v|2E|Er,h|2 =

Eαvvα∗vv|Et,v|2

Eαvhα∗vh|Et,v|2 =E|αvv|2E|αvh|2 (2.45)

Likewise, the horizontal cross-polarization ratio is

χh =E|αhh|2E|αhv|2 (2.46)


DBjSl

Scatterer lScatterer l’

DSlSl’

DSl’Mi

x

y

z

Es,v

Es,hEi,v

Figure 2.11: The two different scatterer coefficients in the two ring model.The other parameters of the model are also shown.

In order to completely characterize the coefficients some assumptions haveto be taken. Here, all the coefficients are assumed to be independent complexgaussian variables with zero mean. Therefore, the variances of αvh and αhv

are

σ2vh =

σ2vv

χv

(2.47)

σ2hv =

σ2hh

χh

(2.48)

Two ring model

In this case two bounces are assumed. The first scatterer coefficient matrixis denoted as α, while the second one as α′. Fig. 2.11 depicts the model.Equation (2.5) is still valid when using the adequate scatterer coefficient.Consequently, the field at the receiver antennas can be expressed as

Er = α′ α Et (2.49)

If the transmitted field is only vertically polarized, the cross-polarizationratio is

χv =E|Er,v|2E|Er,h|2 =

E|α′vvαvv + α′hvαvh|2|Et,v|2E|α′vhαvv + α′hhαvh|2|Et,v|2 (2.50)

Now we assume that all the coefficients are independent and with zeromean. This is rather a general assumption, as we could have constant ampli-tude distributions as long as the phase is uniformly distributed over [0, 2π].We have


Parameter Valueχv 10 dBχh 3 dBσ2

vv 1σ2

hh 1

Table 2.3: Parameters of simulation

χv =E|α′vv|2E|αvv|2+ E|α′hv|2E|αvh|2E|α′vh|2E|αvv|2+ E|α′hh|2E|αvh|2 (2.51)

Likewise, the horizontal cross-polarization ratio is

χv =E|α′hh|2E|αhh|2+ E|α′vh|2E|αhv|2E|α′hv|2E|αhh|2+ E|α′vv|2E|αhv|2 (2.52)

Now we assume that all the coefficients are complex gaussian with zeromean and that the two scatterer coefficient matrices have the same param-eters, i.e., αhh and α′hh have σ2

hh variance, αhv and α′hv have σ2hv, and so on.

We can express

σ2vh =

σ4vv

σ2vv+σ2

hh

2χv +

√(σ2

vv+σ2hh)2

4χ2

v + σ4vv

χv

χh

(2.53)

σ2hv =

σ2vv + σ2

hh

2χv −

√(σ2

vv + σ2hh)

2

4χ2

v + σ4vv

χv

χh

(2.54)

To test that the equations work well let’s do a numerical example. First,we set some conditions of the system, described in Table 2.3. Then wecalculate the values of σ2

vh and σ2hv using (2.53) and (2.54). After that we

simulate the system and check the empirical values of χ for a different numberof realizations. The results can be checked in Fig. 2.12. When the numberof realizations is around 1000, the values obtained are accurate with thesimulation parameters.

However, there is a problem with this model. If you want to set a sys-tem with one polarization in transmission (either vertical or horizontal) anddual-polarized receiver, to completely characterize the parameters in order toperform simulations both χv and χh have to be measured, even though oneof them seem to be useless (why measuring with both vertical and horizontaltransmit antennas if only one of them is going to be used in practice?). Theonly way to solve this problem is to assume that the parameters are quite


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-5

0

5

10

15

20

25

30

Realization number

χ i

n d

B

χv = 10 dB

χh

= 3 dB

Figure 2.12: χ values obtained in the simulation. When the number ofrealizations is increased the values become closer to the values used in thesimulation.

similar (χv ≈ χh) and therefore the variances of the crossed parameters canbe approximated by

σ2vh ≈

σ4vv

σ2vv+σ2

hh

2χv +

√(σ2

vv+σ2hh)2

4χ2

v + σ4vv

=σ2

vv + σ2hh

2χh −

√(σ2

vv + σ2hh)

2

4χ2

h + σ4vv (2.55)

σ2hv ≈

σ2vv + σ2

hh

2χv −

√(σ2

vv + σ2hh)

2

4χ2

v + σ4vv

=σ4

vv

σ2vv+σ2

hh

2χh +

√(σ2

vv+σ2hh)2

4χ2

h + σ4vv

(2.56)

and then use the expression suitable to the measurements.

2.A Analytical Study of the Correlation

In this appendix (2.37) and (2.38) are shown, as well as (2.39) and (2.40).

2.A. ANALYTICAL STUDY OF THE CORRELATION 29

D

R

∆

f(x)

Transmitters

y

x

φ

Figure 2.13: One ring model with two transmitters.

A transmitter is situated at a distance D from a ring of scatterers of radiusR. The ring can be approximated by two circumferences of radius R − ∆and R, as it is shown in Fig. 2.13. The inner circumference is denoted as S0

and the outer one as S1. The equations that describe the circumferences are

S0 : x2 + y2 = (R−∆)2

S1 : x2 + y2 = R2 (2.57)

If D >> R the amount of ring seen from the transmitter along the ydirection is

f(x) = 2|y0 − y1| = 2(√

R2 − x2 −√

(R−∆)2 − x2)

(2.58)


The density of power scattered is defined as the power scattered by thering in the direction of the transmitters per unit of area of the ring; that is

ρ =P0

A=

P0

2πR∆− π∆2(2.59)

where P0 is the power radiated by the ring and A is the area of the ringcreated by the two circumferences.

If we now define the intensity of radiation as the amount of power radiatedin the direction of the transmitters seen along the y axis from a certain pointx, we have:

I = f(x)ρ = 2P0

√1− x2

R−

√(1− ∆

R)2 − x2

R

2π∆− π∆2

R

(2.60)

If we let ∆ → 0 we get

lim∆→0

I =P0

π√

R2 − x2(2.61)

The angle created from the transmitter to the ring is φ = arctan( xD

) ' xD

,and thus x ' φD. If we substitute that expression in the radiation intensity,the angular distribution of the radiation is

Sφ(φ) =P0

π√

R2 − φ2D2(2.62)

For the analysis of the correlation of two transmitters is more convenientto place the transmitters in the y axis. The transmitters are separated by adistance d, and placed in x = 0. Therefore,

E1 = E0ejkd

2 = E0ej 2π

λd2

sin(θ) sin(φ)

E2 = E0e−jkd

2 = E0e−j 2π

λd2

sin(θ) sin(φ)(2.63)

The correlation between the two transmitters is

ρt = C

∫

Ω

Ei(Ω) · E∗j(Ω)S(Ω)dΩ (2.64)

The waves are confined in the plane θ = π/2, and thus S(Ω) = δ(θ −π/2)Sφ(φ). Then

ρt = C

∫ 2π

0

|E0(φ)|2ejkd sin(φ)Sφ(φ)dφ (2.65)

2.A. ANALYTICAL STUDY OF THE CORRELATION 31

The antennas’ pattern are constant over ∆φ, and as the correlation isnormalized to one when the distance between the elements is zero we caneliminate the constat C. Therefore,

ρt ∝∫ φ0+∆φ/2

φ0−∆φ/2

ejkd sin(φ)Sφ(φ)dφ =

∫ ∆φ/2

−∆φ/2

ejkd sin(φ)Sφ(φ)dφ (2.66)

If now (2.61) is introduced in (2.66), a change of variables from φ to y isperformed and φ is approximated by y/D we get

ρt ∝∫ R

−R

ejkdy

D√R2 − y2

dy ∝ J0

(kdR

D

)(2.67)

This result is the same as the one obtained in [35]. However here thesolution is found by studying the correlation of the two transmitters and theprevious reference study the correlation of the channel coefficients.

Now lets describe how the one-disc model is derived. The scatterers, instead of placed on a ring, are inside a circumference. Hence, there is onlyone circumference S0, and the equation that describes it is x2 + y2 = R2. Ifwe create a f(x) function similar to (2.58) we get

f(x) = 2√

R2 − x2 (2.68)

and the density of power is

ρ =P0

A=

P0

πR2(2.69)

Thus, the intensity of radiation is then

I = f(x)ρ =2P0

π

√R2 − x2

R2(2.70)

If we follow the same reasoning as for the one-ring case, the correlation is

ρt ∝∫ R

−R

ejkdx

D

√R2 − x2

R2dx ∝ J1

(kdRD

)kdRD

(2.71)

Chapter 3

Antenna Design

3.1 Introduction

The aim of this chapter is to design a dual polarized omni-directional antennafor a MIMO communication system. This is interesting since for achievinghigh capacities in a MIMO system it is necessary to have low correlationbetween the antenna elements of the array. If only one polarization is usedthe spacing between the elements should be high enough to have a low cor-relation. However, with with a dual-polarized antenna undesired antennaspacing can be avoided while having low correlation between the elements.Any capacity increase that results from using dual-polarization may be di-minished by the consequences of a reduced radiation pattern [7]. A focusedradiation pattern would result in fewer multipath components being receivedand a thus a reduction of the rich scattering, which is so important for ob-taining high capacities in MIMO communications. For that reason it is veryconvenient to have a dual-polarized omni-directional antenna. Such an an-tenna would ideally be a small circular loop with a dipole in the center,thereby providing orthogonal polarization states and zero mutual coupling.

3.2 Loop antennas

Small loop antennas (circular or square) are equivalent to an infinitesimalmagnetic dipole whose axis is perpendicular to the plane of the loop. How-ever, this antennas have small radiation resistances that are usually smallerthan their loss resistances. Thus, they are poor radiators and they are seldomemployed for transmission [17, p. 203]. One solution to this problem is toincrease the size of the loop. As the size of the loop increases the maximum ofthe radiation pattern shifts from the plane of the loop to the axis of the loop

33

34 CHAPTER 3. ANTENNA DESIGN

which is perpendicular to its plane. The reason is that the current distribu-tion over the loop is no longer constant. Therefore the loop antenna is notuseful anymore in our system, since the omni-directional pattern disappears.

One way to solve this problem, as pointed by [37, p. 203], is to design afeed system that provide constant current over the loop. Therefore the loopcan be designed to obtain an omni-directional pattern while the radiationimpedance is high enough.

3.2.1 Constant current square loop

Following the previous idea, a square loop with constant current distributionis studied in order to have an omni-directional antenna with φ polarization.This antenna, illustrated in Fig. 3.1, consist in four dipoles placed in a squareand feeded in phase. The parameters of the antenna are the length of theelements and the separation between them. The desired antenna have anomni-directional pattern in θ = π/2 and an input impedance that providesa low Voltage Standing Wave Ratio (VSWR).

Impedance

The antenna has four dipole elements placed in a square. There is mutualcoupling between them. The way to study the input impedance is to ana-

lyze the system with four inputs, then calculate the Z matrix, and finallycalculating the input impedance when all the antennas are feeded in parallel.

Once we have studied the system with four inputs (e.g., with CST Mi-crowave Studio), we have the following relations

V1

V2

V3

V4

=

Z11 Z12 Z13 Z14

Z21 Z22 Z23 Z24

Z31 Z32 Z33 Z34

Z41 Z42 Z43 Z44

I1

I2

I3

I4

(3.1)

The symmetry of the dipoles lead us to

Z11 Z12 Z13 Z14

Z21 Z22 Z23 Z24

Z31 Z32 Z33 Z34

Z41 Z42 Z43 Z44

=

Z11 Z12 Z12 Z14

Z12 Z11 Z14 Z12

Z12 Z14 Z11 Z12

Z14 Z12 Z12 Z11

(3.2)

Now, when all the dipoles are feeded with the same voltage, i.e., thedipoles are connected in parallel, the input impedance of the whole structureis

3.2. LOOP ANTENNAS 35

v1

v2

v4

v3

+

+

+

+

_

_

_

_

I1

I2

I3

I4

l

d

Figure 3.1: Illustration of the four dipole antenna. The voltages are feeded inphase and consequently V1 = V2 = V3 = V4. The parameters of the antennaare the length of the dipole (l) and the separation between them (d).


Parameter Valuedipole length 0.55λ

dipole separation 0.6λfrequency 2 GHz.

Table 3.1: Parameters used in the design

Zin =Zin,i

4=

Vi

Ii

4(3.3)

where Zin,i is the input impedance of one antenna when the others are inparallel. We know, due to the symmetry, how the currents are related (I1 =−I4 and I2 = −I3). Therefore, if we calculate Zin,1 we get

Zin,1 =V1

I1

= Z11 + Z12I2

I1

− Z12I2

I1

− Z14 = Z11 − Z14 (3.4)

and then, the input impedance of the four-dipole antenna is

Zin =Z11 − Z14

4(3.5)

Design

The proposed design, given in Table 3.1, provides a quite good omni-directionalpattern and a good VSWR with a reference impedance of 50Ω.

In Figs. 3.2 and 3.3 we can see the real and the imaginary part of theinput impedance. If the antenna is used in 2 GHz., the designed frequency,the return loss is 14.26 dB, and the VSWR is 1.48, which is quite good.

In Fig. 3.4 we can see the radiation pattern of the antenna for θ = π/2.The pattern is omni-directional.

3.3 Planar antenna

To physically build a dual-polarized antenna a feeding structure for supplyingan in-phase voltage for the four dipoles must be designed. It would be alsonecessary a structure to place the antenna. One suitable way of building theantenna is printed technology. Therefore the dual-polarized antenna wouldconsist in four in-phase feeded printed dipoles to provide the horizontal po-larization (square loop) and a vertical monopole perpendicular to the planarstructure. Research on printed dipole antennas have been reported in [38]and [39]. Moreover, printed dipoles have been also used for creating patterns

3.3. PLANAR ANTENNA 37

Figure 3.2: Real part of the input impedance versus the frequency.

Figure 3.3: Imaginary part of the input impedance versus the frequency.


Figure 3.4: Pattern of the antenna at 2 GHz. in the plane θ = π/2. Thepattern is omni-directional

similar to a magnetic dipole with horizontal polarization, as in [40]. Andin a similar way, the same authors have recently presented a dual-polarizedantenna for measuring the correlation between different polarizations [32].

Based on this previous work, a dual-polarized antenna is designed. Thedesign is not practical. The coaxial feed is 25Ω instead of 50Ω, there arenot microstrip power dividers and the monopole and the square loop are notpresent at the same time. However, the design is enough for extracting thenecessary data to insert into the channel models and calculate the capacity.It is also shown that this kind of antenna can be practically built.

3.3.1 Square loop

The horizontally polarized square loop consists of a substrate, a ground planeand a microstrip feed. The four dipoles extend out of the ground planeon the lower part of the substrate (Fig. 3.5). The ground of the coaxialfeed is linked to the ground plane of the antenna, and the inner conductoris connected to the microstrip line through the substrate. There are fourmicrostrip lines, one for each dipole. All the lines have the same length andthe same number of corners (Fig. 3.6). The impedance of the coaxial is 25Ω.


62 mm = 0.496 λ

7.23 mm

=

0.058 λ

14.46 mm

=

0.116 λ

18 mm

=

0.144 λ

30 mm

=

0.24 λ

Figure 3.5: Front of the loop. The ground plane also creates the four planardipoles of the square loop.

Thus, the impedance of the microstrip lines is 100Ω, to have matching. Thedipoles are matched due to two stubs: the microstrip lines are finished in aJ-hook open-stub balun; the arms of the dipoles have a parallel stub in themiddle. The parameters of the designed antenna are shown in Table 3.2 andin Figs. 3.6 and 3.5.

3.3.2 Monopole

The vertically polarized monopole is located in the middle of the planarstructure for symmetry. Since the coupling to the square loop is virtually

Table 3.2: Parameters of the square loopParameter Value

Substrate height 1.54 mm = 0.012λMicrostrip width 1.34 mm = 0.011

Microstrip thickness 0.01 mm = 0.00008λSubstrate side 100.5 mm = 0.8λ

Substrate permitivity (ε) 2.33


25.5 mm = 0.204 λ

7.23 mm = 0.058 λ

7 mm = 0.056 λ

3.62 mm = 0.028 λ

15 mm = 0.12 λ

Figure 3.6: Back of the loop. The microstrip feeds the four dipoles of thesquare loop, on the other side of the substrate.

Figure 3.7: S11 of the loop


0.496λ

0.554λ

Figure 3.8: Drawing of the dual polarized loop-monopole antenna. Thesubstrate is not shown for clarity.

zero, it is not necessary to modify the design to have the monopole and thesquare loop at the same time since we only need the radiation patterns forour capacity calculations. We have thus simply removed the microstrip feednetwork when calculating the pattern of the monopole and vice versa. Themonopole start in the substrate and the length is slightly less than λ/4 toachieve a match of 15 dB. Fig. 3.8 depicts how the loop and the monopolewould fit together in the same structure.

As seen in Fig. 3.9, the CST Microwave Studio simulation show an almostomni-directional pattern for the square loop in the xy-plane. The verticalpattern is also as expected with a maximum at θ = 90. For the monopole,however, we have a maximum at θ = 130 in the vertical plane, since theground plane is not infinite [37, p. 720]. The directivity at θ = 90 isalmost 3 dB lower which is a problem if we use a planar channel model. Ifwe align the monopole vertically, it would receive considerably less powerusing a planar channel model compared to a 3-D case. In order to make afair comparison between different antenna types, we have therefore used themonopole directivity at θ = 130 for the cases when it is oriented verticallyin the capacity calculations in the ring models.


0 45 90 135 180 225 270 315 360-20

-15

-10

-5

0

5

θ / φ

Dir

ec

tiv

ity

(d

B)

Loop φ=π/2

Loop θ=π/2

Monopole φ=π/2

Monopole θ=π/2

Figure 3.9: Directivity of the dual-polarized antenna.

Chapter 4

Capacity Simulations

4.1 Introduction

To carry out the simulations, the directivity and the S-parameters were ex-tracted from CST. The data was plugged into the channel models and 10000realizations were done in order to obtain accurate results. The capacity iscalculated with the assumption of no knowledge in transmission. That im-plies that the power has to be equally distributed over all TX antennas, asno waterfilling can be employed. The capacity is obtained from (2.3). How-ever, the procedure of obtaining the capacity is also valid for known channels,where an appropriately modified capacity equation is to be used.

The channel models used in the simulation are two ring model and onering model with 100 scatters, and a symmetric cross polarization ratio (χv =χh). The parameters used in the models are given in Table 4.1. The scatterercoefficients are normalized for each value χ so that the average SNR = 20dB for the case of a single dipole at a random transmit and receive location.Thus, we have a equal transmitted power normalization (section 2.3).

We want to compare the MIMO capacity of our designed dual-polarized

Parameter ValueR 200λD 3000λnumber of scatterers 100number of transmitters 2number of receivers 2Target SNR 20 dB

Table 4.1: Parameters used in the simulation.

43

44 CHAPTER 4. CAPACITY SIMULATIONS

z

x

y

z oriented x oriented y oriented

dipoles

dual polarized

omni-directional

crossed dipoles

Figure 4.1: Orientation of the antennas in the simulations.

antenna to a typical single-polarized MEA. The single polarized MEA iscomposed of dipoles. The length of designed dual-polarized antenna is 0.8λ(substrate). Thus, if we want to make a fair comparison between the dualpolarized antenna and two parallel dipoles they must occupy the same space.Therefore, in the simulations, the two dipoles are separated 0.7λ. We do notseparate the dipoles 0.8λ to let some space for the feeding structure. Whencomparing two of the dual-polarized antennas and four dipoles, we distributethe four dipoles in 1.5λ, equally spaced, while the two dual-polarized anten-nas are placed one beside the other, without space between them. We alsouse for the comparisons two crossed dipoles (two co-located and perpendic-ular dipoles), which provide dual-polarization, although the pattern of thehorizontal one is not omni-directional. The ergodic capacity is comparedunder different orientations of the antennas, as seen in Fig. 4.1. The planarchannel model is situated in the xy-plane.

The first simulations are performed with the ideal patterns of electric andmagnetic dipoles, with no mutual coupling between them. By doing this, wehave an optimal situation to compare the rest of the cases. After that, some2x2 cases, with the designed dual-polarized antenna and with the dipoles,are presented. Then, the 4x4 cases are studied. Finally, 2x2 cases in a onering model are analyzed.

4.2 Ideal electric and magnetic dipoles

We compare two ideal vertically polarized dipoles, separated by 0.7λ, both intransmission and reception, and a co-located magnetic and an electric dipole.As a further simplification, we assume isotropic pattern in this case, in orderto study only the effect of the polarization mismatch. The electric dipoles in

4.2. IDEAL ELECTRIC AND MAGNETIC DIPOLES 45

0 2 4 6 8 10 12

2

4

6

8

10

12

χ dB

C b

ps/H

z

Electric dipoles, z → z

Co-located

magnetic/electric

dipoles

Electric dipolesz → x

Figure 4.2: Mean capacity of ideal dipoles. Electric dipoles aligned (blue,solid and dots) and not aligned (blach, dash-dot and circles). Electric andmagnetic dipoles (red, solid and cross)

transmission and reception can be unaligned, i.e., in transmission are verticaland in reception horizontal.

As we can see in Fig. 4.2 the higher capacity is provided by the alignedelectrical dipoles, except when χ = 0 dB. Although the correlation betweenthe channel coefficients is higher in that case compared to the magnetic andelectric dipoles, since the dual polarization decorrelates the signals, the powerreceived is higher. Therefore, the capacity is higher. However, when one ofthe ends has the electric dipoles is placed horizontally, the power receiveddrops, and so does the capacity. The case of electric and magnetic dipole isinvariant to rotation. So even though the capacity is lower than with twoelectric dipoles, the utilization of this configuration in wireless devices is moreconvenient. When χ = 0dB all the antenna configurations receive the samepower, as half of the power transmitted couples to the cross polarization.However, the magnetic and electric dipoles have the lower correlation betweenchannel paths, and therefore the higher capacity.


0 2 4 6 8 10 12

2

4

6

8

10

12

χ dB

C

bp

s/H

z

Vertical dipoles, z → z

Horizontal dipoles x → x

Flipped dipoles z → x

Flipped dipoles z → y

Figure 4.3: Mean capacity. Parallel dipoles both in vertical orientation (blue,solid and dotted), one of the ends flipped (black, dash-dot and circles for sideby side and blue, dotted and triangles for one on top of the other), both endsin horizontal orientation (blue, dash and x-mark).

4.3 2 × 2 Two ring

Let’s consider now a more realistic case, with the patterns of the antennassimulated in CST Microwave Studio. The antennas can be placed in differentorientations: z, perpendicular to the rings plane; x and y, which are on theplane of the rings (see Fig. 2.11).

In Fig. 4.3 it is shown the cases for the parallel dipoles. The highercapacity is provided when the dipoles are aligned and vertical (z → z).The capacity of the parallel dipoles drops when the RX array is rotated(z → y, z → x). The z → y case is worse since the projection of thedipoles in the horizontal plane is in the same place, and thus the correlationbetween the dipoles is one. We have two factors that affect the capacity:polarization mismatch and mean gain in the xy-plane. The latest explainswhy z → z case provides higher capacity than the y → y case. In Fig. 4.4we compare the dual-polarized antenna and the crossed dipoles, again fordifferent orientations. The dual-polarized antenna is only slightly affectedby rotation, but the crossed dipoles do suffer from it since the pattern of ahorizontal dipole is not omni-directional in xy.

4.3. 2 × 2 TWO RING 47

0 2 4 6 8 10 12

2

4

6

8

10

12

χ dB

C

bp

s/H

z

Dualpol, z → z

Flipped dual pol, z → x

Horizontal dual pol, x → x

Crossed dipoles, z → z

Horizontal

crossed dipoles

x → x Flipped

crossed dipoles

z → x

Figure 4.4: Mean capacity. Dual-polarized antenna (red), both vertical ori-ented (solid, cross), both horizontal (dashed, triangle down), one in verticaland one in horizontal (dash-dot, circles). Crossed dipoles (black), both verti-cal (solid, square), one vertical and the other horizontal (dash-dot, diamond),both horizontal (dashed, triangles up).


Fig. 4.5 shows an example of a realistic system, a WLAN case. The TXarray has two dipoles along z, separated again by 0.7λ. Several RX arraysare studied in different orientations. The higher capacity is provided whenthe RX array is also two parallel dipoles, aligned with the transmitter (zorientation). There is also a small capacity gain from the dual polarized omni-directional antenna even if the XPD is 0 dB. That gain is explained since theloop is very omni-directional, and in the horizontal plane has lower directivitythan a dipole. However, if the parallel dipoles are rotated to horizontalorientation, either side by side or one on top of each other, the capacitydrops. If instead of parallel dipoles we use the dual-polarized antenna as RXarray the capacity is lower, but it is not affected so much by rotation. Whenusing crossed dipoles we also obtain lower performance and the rotation ofthe antenna affects the capacity. Thus, we can conclude that the antennawhich is less affected by the orientation and thereby provides higher capacityif the orientation is arbitrary is our proposed dual-polarized omni-directionalantenna.

In the previous capacity simulations we have just studied the best andworst case, with the antennas aligned or perpendicular. The rotation of thedipoles causes two effects. First, the loss of received power, because thepattern of the dipoles has lower performance when is not placed vertically.Second, the rotation lead the dipole to receive power in a different polariza-tion. The cross polarization has been measured in different scenarios, andis usually close to 6 dB [41] [5], so both effects are important in a real sit-uation. In a real environment, the probability to have the dipoles alignedin both transmission and reception is very low. In wireless communications,the devices are moved, flipped and rotated. The results also show that thedual polarized antenna has better performance than the dipoles when theorientation of the antennas is arbitrary. For this antenna the loss of capacityis just caused by the loss of power and not by the cross polarization ratio.

4.4 4 × 4 Two ring

Fig. 4.6 depicts the c.d.f. of the capacity when we fix χv = χh = 6 dB,which is a realistic XPD [41] [5]. The capacities involved are now higher,as we have four antennas in both ends of the communication. The highercapacity is still provided by the dipoles, although now there is more mutualcoupling between them. If we rotate the dipoles in one or both ends of thecommunication system there is a reduction of the capacity, due to the lowerpower received. The dual-polarized antenna and the crossed dipoles have lesscapacity than the vertical dipoles, because they receive less power, although

4.4. 4 × 4 TWO RING 49

0 2 4 6 8 10 12

2

4

6

8

10

12

χ dB

C

bp

s/H

z

Vertical dipoles, z

Flipped dipoles

on top of each other,

y

Flipped

dipoles, x

Flipped crossed dipoles, x

Flipped

dualpol, x

Dualpol

Crossed dipoles, z

Figure 4.5: WLAN capacity. Parallel dipoles (blue), which are aligned (solidand dots), flipped side by side (dash-dot and circles) and flipped one on topof the other (dashed and cross). Dual-polarized antenna (red) in verticalorientation (solid) or horizontal (dash-dot). Crossed dipoles (black), vertical(solid and squares) and horizontal oriented (dash-dot and diamonds).


0 5 10 15 20 25 30

0

0. 1

0. 2

0. 3

0. 4

0. 5

0. 6

0. 7

0. 8

0. 9

1

Capacity bps/Hz

CD

F P

rob

ab

ilit

y (

Ca

pa

cit

y <

ab

sc

isa

)

Dipoles one end flipped

Dipoles both ends horizontal

Dipoles vertica l

Dual polarized antenna

Crossed dipoles

Figure 4.6: Capacity cdf when the antennas are dipoles (either parallel, inblue, or crossed, in black) and the dual-polarized antenna (red). χv = χh =6 dB.

the correlation between the elements is lower.

4.5 2 × 2 One ring

A one ring channel model represents a base station that is placed in a highposition, with no scatterers around, and the subscriber unit surrounded byscatterers. Fig. 4.7 shows the capacity of the dipoles and the dual-polarizedantenna for χv = χh = 6 dB. The best option is to use the dual polarizedantenna. Although the power received is higher with the two parallel andvertically aligned dipoles, the correlation between the channel paths is veryhigh in this model (see section 2.4.2 and specially Fig. 2.4). Even if weseparate the dipoles in transmission (in the base station the size of the arrayis not so important) up to 10λ the capacity is still higher than with the dualpolarized antenna. The decorrelation effect of using dual polarization is veryimportant in this case. Even if we flip the receiver dual-polarized antenna wehave a better performance than when we flip the two dipoles in the receiver.We can compare this situation with the two ring model (Fig. 4.8) where themaximum capacity is provided by the parallel and aligned dipoles. We havehigher capacities in the two ring model due to the lower correlation betweenchannel paths.

4.5. 2 × 2 ONE RING 51

0 2 4 6 8 10 12 14 16

0

0. 1

0. 2

0. 3

0. 4

0. 5

0. 6

0. 7

0. 8

0. 9

1Capacity with dual polarized antenna

Capacity bps/Hz

CD

F P

rob

ab

ilit

y (

Ca

pa

cit

y <

ab

sc

isa

)

Dipoles one end flipped

Dipoles vertical

Dual polarized one end flipped

Dual polarized antenna

Dipoles 10 λ one end flipped

Dipoles 10 λ vertical

Figure 4.7: Capacity c.d.f. when the antennas are parallel dipoles (blue andgreen) and the dual-polarized antenna (red). χv = χh = 6 dB. The dipolesin the receiver are always separated 0.7λ, while in the transmitter they areseparated either 0.7λ (blue) or 10λ (green).

0 2 4 6 8 10 12 14 16 180

0. 1

0. 2

0. 3

0. 4

0. 5

0. 6

0.7

0. 8

0.9

1

Capacity bps/Hz

CD

F P

rob

ab

ility

(C

ap

aci

ty <

ab

scis

a)

Dipoles − one end flippedDipoles − both ends horizontalDipoles − verticalDual polarized − one end flippedDual polarized − both ends horizontalDual polarized antenna

Figure 4.8: Capacity c.d.f. when the antennas are parallel dipoles (blue) andthe dual-polarized antenna (red) under different orientations. χv = χh =6 dB.

Chapter 5

Conclusions and Future Work

5.1 Conclusions

In this thesis we have presented an antenna array element designed for aMIMO system. The element was designed in order to have dual polariza-tion and omnidirectionality which provides a low correlation between thechannel elements and thus a high capacity. Two physical channel modelswere presented which were compared, in terms of correlation, with the Kro-necker model. The data from the designed antenna was then plugged intothe channel models and the capacity was simulated and compared with otherantennas.

The results show that although two parallel dipoles aligned provide highercapacity than our designed dual polarized antenna, when the orientation isarbitrary the capacity drops, and the performance of the dual polarized omni-directional antenna is better. This is true providing a two ring model, i.e.,a pico cellular environment. If we use instead a one ring model, that fitbetter a macro cellular situation, the dual-polarized antenna is better in allsituations. Even with a 10λ separation the performance of the dual-polarizedis better, and if we consider also the orientation of the receive terminal weconclude that the dual-polarized antenna provides a great advantage.

5.2 Future Work

To be able to build this antenna some modifications have to be accomplished.First, the feeding point of the square loop has to be moved from the center ofthe structure. However, the length of the microstrip lines and the number ofcorners should remain equal for all the lines. Otherwise the printed dipoleswould not be fed with equal phase, and the pattern would not be equiv-

53

54 CHAPTER 5. CONCLUSIONS AND FUTURE WORK

alent to a magnetic dipole. The coaxial feeding both the square loop andthe monopole should be a 50Ω standard line. Thus, to avoid having 200Ωmicrostrip lines, which would be too thin for a practical design, we could usequarter wave transformers.

It would also be interesting to use a 3-D scattering model to simulatethe capacity of the antenna. The scatterers would then be distributed inelevation.

Finally, the next thing would be to build the antenna and perform mea-surements, study the capacity and obtain XPD data for the channel modeland future simulations.

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Dual Polarized Omnidirectional Array Element for MIMO Systems

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