Polarization Modulation Design for Reduced RF Chain Wireless · cross-polarized components...

Polarization Modulation Design for Reduced RF Chain Wireless

Hemadeh, I. A., Xiao, P., Kabiri, Y., Xiao, L., Fusco, V., & Tafazolli, R. (2020). Polarization Modulation Design forReduced RF Chain Wireless. IEEE Transactions on Communications.https://doi.org/10.1109/TCOMM.2020.2979455

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IEEE TRANSACTIONS ON COMMUNICATIONS 1

Polarization Modulation Design forReduced RF Chain Wireless

Ibrahim A. Hemadeh , Member, IEEE, Pei Xiao , Senior Member, IEEE, Yasin Kabiri , Member, IEEE,Lixia Xiao, Member, IEEE, Vincent Fusco , Fellow, IEEE, and Rahim Tafazolli , Senior Member, IEEE

Abstract— In this treatise, we introduce a novel polarization1

modulation (PM) scheme, where we capitalize on the recon-2

figurable polarization antenna design for exploring the polar-3

ization domain degrees of freedom, thus boosting the system4

throughput. More specifically, we invoke the inherent properties5

of a dual polarized (DP) antenna for transmitting additional6

information carried by the axial ratio (AR) and tilt angle7

of elliptic polarization, in addition to the information streams8

transmitted over its vertical (V) and horizontal (H) components.9

Furthermore, we propose a special algorithm for generating an10

improved PM constellation tailored especially for wireless PM11

modulation. We also provide an analytical framework to compute12

the average bit error rate (ABER) of the PM system. Further-13

more, we characterize both the discrete-input continuous-output14

memoryless channel (DCMC) capacity and the continuous-input15

continuous-output memoryless channel (CCMC) capacity as16

well as the upper and lower bounds of the CCMC capacity.17

The results show the superiority of our proposed PM system18

over conventional modulation schemes in terms of both higher19

throughput and lower BER. In particular, our simulation results20

indicate that the gain achieved by the proposed Q-dimensional21

PM scheme spans between 10dB and 20dB compared to the22

conventional modulation. It is also demonstrated that the PM23

system attains between 54% and 87.5% improvements in terms24

of ergodic capacity. Furthermore, we show that this technique25

can be applied to MIMO systems in a synergistic manner in26

order to achieve the target data rate target for 5G wireless27

systems with much less system resources (in terms of bandwidth28

and the number of antennas) compared to existing MIMO29

techniques.30

Index Terms— 5G, wireless networks, MIMO, dual-polarized,31

polarization modulation, index modulation, spatial modulation,32

polarization, MPSK, MQAM, practical implementations, channel33

modulation, hard-decision detection.34

Manuscript received May 8, 2019; revised September 30, 2019 andDecember 6, 2019; accepted January 23, 2020. This work was supported bythe U.K. Engineering and Physical Sciences Research Council (EPSRC) underGrant EP/N020391/1. The authors also would like to acknowledge the supportof the University of Surrey 5GIC (http://www.surrey.ac.uk/5gic) membersfor this work. A U.K. patent “Wireless Data Transmission using PolarisedElectromagnetic Radiation” (reference number GB1812108.7) related to thiswork was filed on July 25, 2018. The associate editor coordinating thereview of this article and approving it for publication was M. Di Renzo.(Corresponding author: Ibrahim A. Hemadeh.)

Ibrahim A. Hemadeh, Pei Xiao, Yasin Kabiri, Lixia Xiao, andRahim Tafazolli are with the Institute for Communication Systems (ICS),University of Surrey, Guildford GU2 7XH, U.K. (e-mail: [email protected]).

Vincent Fusco is with the School of Electronics, Electrical Engineering andComputer Science, Queen’s University Belfast, Belfast BT7 1NN, U.K.

Color versions of one or more of the figures in this article are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCOMM.2020.2979455

I. INTRODUCTION 35

MULTIPLE-INPUT multiple-output (MIMO) techniques 36

are capable of providing unprecedented improve- 37

ments for wireless communication systems in terms of 38

capacity [1], [2]. Explicitly, MIMO systems are capable of 39

attaining an enhanced bit error rate (BER) performance as well 40

as an improved throughput in comparison to single-antenna 41

implementations, provided that each of the transmitted signals 42

has a unique signature at each of the receive antenna elements 43

(AEs). In the context of spatial transmission schemes, multiple 44

AEs are spaced sufficiently apart in order to experience 45

independent fading. Typically, array elements are placed 10λ 46

apart from each other at the base station, where λ represents 47

the carrier wavelength. However, it is often impractical to 48

accommodate multiple AEs, especially in small hand-held 49

devices [3]. One solution is to communicate at high frequency 50

bands, such as the millimeter-wave (mmWave) band [4], which 51

allows fitting a high number of AEs within a relatively small 52

area, while still providing an independent fading. However, 53

it would still be a challenging task to obtain a unique spatial 54

signature of distinct AEs in a highly dense and closely 55

spaced antenna arrays due to the dominant line-of-sight (LOS) 56

component. An alternative way of overcoming this problem 57

is to separate the transmitted signals over the polarization 58

domain, which can be achieved by using dual-polarized AEs 59

(DP-AEs) [5], [6]. In particular, by employing DP-AEs the 60

number of transmit and receive AEs can be doubled in 61

comparison to uni-polarized AEs (UP-AEs). 62

In a nutshell, a single DP-AE constitutes a pair of 63

co-located and orthogonally-polarized vertical (V) and hori- 64

zontal (H) components. These are typically referred to as the 65

VH components and come in different shapes and forms [7]. 66

The orthogonality of the V and H components offers a new 67

means of spatial separation, namely over the polarization 68

dimension, providing a near nil spatial correlation at both the 69

transmitter and the receiver [8], [9]. By invoking the addi- 70

tional degrees-of-freedom (DoF) offered by cross-polarized 71

components, the spectral efficiency of a MIMO system can be 72

further enhanced [10]. Note that the communication between 73

cross-polarized components instigates channel depolarization, 74

which impacts the cross-channel gains. This can be measured 75

by the cross-polar discrimination (XPD) [11]. 76

Polarization [12] is a key element of defining the electro- 77

magnetic (EM) wave propagation in addition to the frequency, 78

time, amplitude and phase elements [12]. It is characterized 79

by the variations of the direction and the amplitude of an EM 80

wave with respect to time. 81

0090-6778 © 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.

https://orcid.org/0000-0002-1422-0333

https://orcid.org/0000-0002-7886-5878

https://orcid.org/0000-0002-7888-6775

https://orcid.org/0000-0002-4041-4027

https://orcid.org/0000-0002-6062-8639

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2 IEEE TRANSACTIONS ON COMMUNICATIONS

TABLE I

NOMENCLATURE

Several technologies have been long utilizing the con-82

cept of polarization, namely in optical fiber communica-83

tions [13], satellite communications [14] as well as in radar84

applications [15], however it has recently started to gain85

some interest in wireless communications as presented by86

Shafi et al. in [16] and the references therein. For instance,87

the polarization effect was considered in the development88

of various technologies, such as for the 2D and 3D spatial89

channel model (SCM) for the third-generation partnership90

project (3GPP) and 3GPP2 model [17], [18], the indoor91

communications operating at the 60 GHz band [19] as well92

as for the mmWave channel models presented in [4], [20].93

Moreover, several studies focused mainly on the polarization94

effect in DP-based MIMO systems [6], [21].95

The effect of polarization on spatial multiplexing was96

investigated by Bolcskei et al. in [22], where a two-input97

two-output (TITO) (2 × 2)-element DP system was presented98

and a closed-form average BER (ABER) expression was99

formulated. The results showed that even with high spatial100

fading correlation, a DP implementation is capable of attain-101

ing enhanced multiplexing gain. This was later extended by102

Nabar et al. in [23] to include both transmit diversity as well103

as spatial multiplexing. In [24], Anreddy and Ingram suggested104

that the BER performance of antenna selection with DP-AE105

outperforms that with UP-AE.106

Polarization shift keying (POLSK) was first theorized by107

Benedetto and Poggiolini in [13] for optical communications108

and was later applied to wireless communications systems109

by Dhanasekaran in [25]. Here, information is transmitted by110

switching on and off the V and H components of a DP-AE.111

This approach was later combined with spatial modulation112

(SM) [26]–[28] by Zafari et al. in the DP-SM scheme [29],113

which has the advantage of using a single transmit RF chain114

and multiple DP-AEs. More specifically, DP-SM switches on a115

single DP-AE and activates one of its orthogonal components116

(V or H) for transmitting a single complex symbol. This117

allows DP-SM to implicitly convey the implicit information 118

of the activated component index. It was shown in [30] that 119

the DP-SM system outperforms the conventional UP-based 120

SM scheme, while doubling the number of transmit antennas. 121

DP-SM was later investigated again by Zafari et al. in [30] 122

over correlated Rayleigh and Rician fading channels. In [31], 123

Zhang et al. extended the philosophy of using a single RF 124

chain with DP-AEs in the polarization shift keying (PolarSK) 125

scheme. PolarSK employs a single transmit RF chain with an 126

improved design for transmitting a single PolarSK symbol, 127

which is a combination of complex symbols as well as a 128

specific polarization angle. Furthermore, Park and Clerckx 129

proposed utilizing DP-AEs for multi-user transmission in a 130

massive MIMO structure [32], where by employing DP-AEs 131

the number of transmitting ports is doubled. 132

In this treatise, we propose a novel polarization modulation 133

(PM) scheme, which invokes the polarization characteristics 134

of DP-AEs for transmitting an extra information over the 135

polarization dimension in addition to a pair of complex 136

symbols, while maintaining a reduced number of RF chains. 137

In particular, at each DP-AE, the PM system selects one out 138

of multiple polarization configurations that is jointly applied 139

to the V and H components for shaping the transmitted 140

signal’s polarization pattern. The polarization configurations 141

applied are predefined at the transmitter and are known to 142

the receiver. Accordingly, the transmitted signal conveys both 143

the complex symbols and the polarization pattern applied. 144

In fact, each polarization pattern can shape the signal car- 145

rying the complex symbols differently and hence, we refer to 146

the polarization patterns as the space-polarization dispersion 147

matrices. 148

In PM, a space-polarization dispersion matrix disperses a 149

pair of complex symbols over the space and polarization 150

dimensions, in a similar manner to space-time dispersion 151

matrices [33], [34]. Space-polarization dispersion matrices 152

are represented by (2 × 2)-element diagonal matrices, since 153

they configure two orthogonal components (V and H) over 154

a single time slot. Having used a matrix representation of 155

the polarization configurations, space-polarization dispersion 156

matrices can be generated based on a fixed criterion [35]–[37] 157

for optimizing the performance of the PM system [38]–[40]. 158

Against this background, the novel contributions of this treatise 159

are as follows: 160

1) We propose the novel concept of polarization modula- 161

tion, which invokes the polarization characteristics of 162

DP-AEs (i.e. magnitude and angle) for achieving an 163

improved transmission rate as well as an enhanced BER 164

performance. 165

2) We formulate a closed-form generalized ABER expres- 166

sion of the PM system with Rayleigh fading as well as 167

with Rician fading channels. 168

3) We characterize both the discrete-input continuous- 169

output memoryless channel (DCMC) capacity and the 170

continuous-input continuous-output memoryless chan- 171

nel (CCMC) capacity of our PM system. Furthermore, 172

we provide the upper and lower bounds of CCMC 173

capacity. 174

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HEMADEH et al.: POLARIZATION MODULATION DESIGN FOR REDUCED RF CHAIN WIRELESS 3

TABLE II

LIST OF SYMBOLS

4) We conceive an efficient space-polarization matrix opti-175

mization technique for optimizing the PM constellation.176

To be specific, the optimized matrix set is generated177

based on the random search method, which aims for178

minimizing the maximum achievable ABER as well as179

maximizing the DCMC capacity.180

The remainder of the treatise is organized as follows.181

In Section II, we introduce our PM system, which182

includes both the transmission and detection mechanisms.183

Fig. 1. Dual-polarized antenna element with an elliptic polarization state.

Next, a DCMC and CCMC achievable capacities are pre- 184

sented and the lower and upper bounds of the CCMC capac- 185

ity are developed in Section III. In Section IV, we derive 186

the closed-form ABER expression. Then, the improved 187

PM-constellation generation technique is introduced in 188

Section V. Section VI provides the numerical results, while 189

the conclusions are drawn in Section VII. 190

II. PROPOSED POLARIZATION MODULATION 191

In this contribution we consider an (Nt ×Nr)-element 192

MIMO system with Nt/2 being the number of DP-AEs 193

employed at the transmitter and Nr/2 the number of DP-AEs 194

employed at the receiver. The transmitter is equipped with N tc 195

RF-chains, each of which is connected to a single DP-AE. 196

A single DP-AE constitutes both a vertical and a horizontal 197

component and hence, the number of transmit antennas Nt is 198

twice that of N tc . In what follows, we present our PM transmis- 199

sion scheme, which is capable of conveying information bits 200

by invoking the polarization characteristics of multi-polarized 201

AEs. This approach opens a new dimension for implicit 202

information transfer, while maintaining traditional amplitude- 203

phase modulation (APM) complex symbol communication. 204

A. The Concept of PM 205

Let us now consider the DP-AE depicted in Figure 1, which 206

constitutes a pair of co-located horizontally-and vertically- 207

polarized ports. The trace of the EM field polarization ellipse 208

emitted by the DP-AE is shaped by the conjoint characteristics 209

of its vertical and horizontal components, which could form 210

a linear, circular and more generally an elliptic polarization, 211

as shown Figure 1. The resultant radio wave ellipse can be 212

represented both by the axial ratio (AR) and by the tilt angle τ . 213

The AR represents the major axis (OA) to minor axis (OB) 214

ratio defined as 215

AR =OA

OB, (1) 216

as seen in Figure 1. Furthermore, the major and minor axes 217

of Equation (1) of the polarization ellipse can be expressed 218

as [12], [41] 219

OA=

�12

�E2

x+E2y +�E4

x+E4y +2E2

xE2y cos (2δL)

�, (2) 220

and 221

OB=

�12

�E2

x+E2y−�E4

x+E4y +2E2

xE2y cos (2δL)

�, (3) 222

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Fig. 2. PM transmitter block diagram.

respectively, where (Ex, Ey) define the EM field vector223

components with a time-phase difference angle δL =δx − δy.224

Likewise, the angle τ , which describes the tilt angle with225

respect to the principal axis, as depicted in Figure 1 is given by226

τ =12

arctan�

2ExEy

E2x − E2

y

cos (δL)�. (4)227

In this regard, we adjust both the AR and τ components of228

DP-AEs in order to produce Q distinct polarization traces (or229

shapes), which can be used for implicitly transferring log2 (Q)230

bits over each DP-AE, while still transmitting a pair of APM231

complex symbols at the V and H components.232

It is worth mentioning here that Q is always an integer233

power of 2, which is comparable to the size of a conventional234

APM constellation L. Hence, when a single polarization235

shape is applied (e.g. Q = 1 with all vertical, horizontal or236

slant), no information will be transmitted over the polarization237

domain. Furthermore, the maximum value of Q is not fixed238

and can be adjusted according to the system requirements.239

However, choosing the number of polarization shapes depends240

mainly on the antenna specifications, which is represented by241

its AR and tilt angle ranges.242

To further illustrate the mechanism of our proposed243

PM scheme, let us consider the PM constellation depicted244

in Figure 2, which is formed of a 4PSK constellation as well245

as a Q = 4 polarization states. Given that a pair of QPSK246

symbols can be transmitted at the V and H components of247

the DP-AE, which conveys a total of 4 bits per channel use248

(bpcu), an additional log2 (Q) =2 bits can be transmitted249

by switching between the four distinct polarization traces of250

Figure 2. This allows the system to apply a dual transmission251

mechanism, using the conventional APM symbols as well as252

the polarization information. In what follows, we detail our253

PM encoding scheme at the transmitter.254

B. PM System Model255

The PM transmitter block diagram is depicted in Figure 3.256

The B-sized input bit stream of Figure 3 is divided into N tc257

parallel BPM -sized sub-streams, where the ntc-th sub-stream258

at the ntc-th RF chain of BPM bits is fed into the nt

c-th PM259

encoder for generating the ntc-th PM symbol transmitted at the260

ntc-th DP-AE, given that nt

c =1, . . . , N tc . The PM encoder of261

Figure 3 will be detailed further in Section II-E. In a nutshell,262

the BPM -sized sub-stream constitutes the pair of information263


denoting the polarization information as well as the APM 264

symbols information. More explicitly, the first log2 (Q) bits 265

of BPM are used to select one out of Q polarization config- 266

urations, which configures the V and H components of the 267

ntc-th DP-AE, while the remaining 2 log2 (L) bits are invoked 268

to modulate a pair of L-PSK symbols. The total number of bits 269

transmitted by a PM system equipped with N tc PM encoders 270

is given by 271

B = N tc · log2

�L2Q. (bits) (5) 272

Now, the symbol S(ntc)∈ C2×1 transmitted at the nt

c-th 273

DP-AE can be expressed as 274

S(ntc) = A

(ntc)

q X(nt

c)lv ,lh

, (6) 275

where A(nt

c)q =

A

ntc

q,v 00 A

ntc

q,h

�∈ C2×2 denotes the polarization 276

shaping matrix, which configures the ntc-th DP-AE polariza- 277

tion using the q-th polarization information selected from 278

{Aq}Q1 . Moreover, Aq,v =aq,ve

jθq,v and Aq,h =aq,hejθq,h 279

represent the V and the H polarization information, which 280

are associated with moduli |aq,v| and |aq,h| as well as argu- 281

ments θq,v and θq,h, respectively.1 The polarization matri- 282

ces {Aq}Q1 are constructed under the power constraint of 283

trace�AqA

Hq

=1. Furthermore, X

(ntc)

lv ,lh=�x

ntc

l,v xnt

c

l,h

�T∈ 284

C2×1 is the APM symbol vector, where xntc

l,v and xntc

l,h represent 285

the pair of L-PSK symbols transmitted at the (2ntc − 1)-th V 286

component and at the (2ntc)-th H component of the nt

c-th DP- 287

AE, respectively, given that l =1, . . . , L. Hence, the ntc-th PM 288

symbol vector can be expressed as 289

S(ntc) =

A

ntc

q,v 00 A

ntc

q,h

� x

ntc

l,v

xnt

c

l,h

�=

A

ntc

q,v · xntc

l,v

Ant

c

q,h · xntc

l,h

�, (7) 290

while the (Nt × 1)-element PM symbol vector S has the 291

following form: 292

S =�S(1) · · · S(Nt

c)�T. (8) 293

1��aq,h

�� and |aq,v | are equivalent to Ex and Ey in Equations (1-4),respectively, while θq,h and θq,v characterize δx and δy of the differenceangle δL presented in Section II-A.

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Observe in (7) that an additional means of information294

transmission is introduced by adjusting the joint configurations295

of the moduli and arguments of the diagonal vector of A(nt

c)q .296

Given that the coefficients of A(nt

c)q constitute the polarization297

information, {Aq}q1 can be constructed using one of the three298

following modes:299

• The AR mode, where the polarization information is300

explicitly transmitted over the AR component, which is301

represented by the moduli of Aqdenoted by |aq,v| and302

|aq,h|. In the AR mode, no information is conveyed303

over the tilt component (e.g. θq,v =θv and θq,h =θh304

∀ {Aq}Qq=1), where θv and θh are constant angle values.305

• The Tilt mode, where the polarization information is306

explicitly transmitted over the tilt component designated307

by the arguments θq,v and θq,h of Aq , while having308

static moduli (e.g. |aq,v| =av and |aq,h| =ah ∀ {Aq}Qq=1),309

where av and ah are constant real numbers.310

• The tilted-AR mode, where information is conveyed over311

an amalgam of both the tilt and the AR components,312

which is characterized by the general representation of313

A(nt

c)q in (7). In this mode, every polarization shaping314

matrix in {Aq}Qq=1 has a unique signature constituted by315

a specific combination of AR (i.e. |aq,v| and |aq,h|) and316

tilt angles (i.e. θq,v and θq,h).317

The PM system may also reduce to the conventional spatial318

multiplexing (MUX) system [22], [42], when no information319

is transmitted over the polarization dimension (e.g. Q = 1).320

In this treatise, we refer to a PM system as321

PM(AR/Tilt/TAR/MUX, N tc ,

Nr

2 , Q, L − QAM/PSK)322

and to the PM encoder as PM(AR/Tilt/TAR/MUX, Q,323

L − QAM/PSK), where AR, Tilt, TAR and MUX represent324

the AR modulation, tilt modulation, tilted-AR modulation as325

well as the basic QAM/PSK multiplexing modulation without326

any polarization,2 respectively.327

It should be also noted that by using the Tilt mode, where328

the polarization information is explicitly transmitted over the329

tilt component the system converges to the PolarSK system330

proposed in [31], namely when associated with N tc =1 and331

the PSK modulation. Hence, PolarSK is a special case of our332

PM scheme.333

Now, having generated the space-polarization block, the PM334

symbol vector S of (8) is transmitted over a frequency-flat and335

slow fading channel and received by the Nr

2 DP-AEs at the336

receiver. In general, the vector-based system model can be337

expressed as338

Y = HS + V , (9)339

where H∈ CNr×Nt denotes the channel matrix and340

V ∈ CNr×1 is the zero-mean additive white Gaussian341

noise (AWGN) vector, each element of which obeys342

CN (0, N0), given that N0 is the noise power.343

2In the case of using MUX, no information is transmitted over the

polarization domain, that is Q = 1, A(nt

c)q =I2 and log2 (Q) = 0 bits.

C. Channel Model 344

In this regards, H describes the DP channel matrix that 345

combines both the spatial separations and the XPD depolar- 346

ization effects and it is defined as [5], [6], [43] 347

H =

⎡⎢⎣

H1,1 · · · H1,Ntc

... Hnr/2,ntc

...HNr/2,1 · · · HNr/2,Nt

c

⎤⎥⎦ , (10) 348

where Hnr/2,ntc∈ C2×2 designates the TITO channel matrix 349

between the ntc-th and nr/2-th transmit and receive DP-AEs, 350

respectively. In particular, each TITO channel model can be 351

expressed as 352

Hnr/2,ntc

=

hvv

nr/2,ntc

√Xhvhnr/2,nt

c√Xhhvnr/2,nt

chhh

nr/2,ntc

�, (11) 353

where X denotes the XPD, which is a combination of the 354

cross-polar ratio (XPR) and the cross-polar isolation (XPI) as 355

defined in [6]. More specifically, the X parameter indicates the 356

cross-attenuation between the co-polarized channels (vv, hh) 357

and the cross-polarized channels (hv, vh). XPD is defined as 358

the ratio of the power of co-polarized channels to the power 359

of cross-polarized channels over V and H, expressed as [44] 360

ϕ−1v = E

��hvvi,j

��2� /E ��hvhi,j

��2� , (12) 361

ϕ−1h = E

��hhhi,j

��2� /E ��hhvi,j

��2� , (13) 362

respectively, where hvh/hvi,j denotes the channel fading 363

coefficient including the cross-attenuation effect, 364

E��hvv

i,j

��2� =E��hhh

i,j

��2� =1, E

��hvhi,j

��2� =ϕv and 365

E

��hhvi,j

��2� =ϕh. By assuming equal cross-attenuation [22] 366

(e.g. ϕv = ϕh=ϕ and 0 ≤ ϕ ≤ 1), the XPD parameter can be 367

expressed as X =ϕ. In what follows, we express the inverse 368

of the XPD in dBs as X−1dB =−10 logX dB. 369

To expound a little further on the channel model, the SISO 370

channel presented in (11) can be defined as 371

Hnr/2,ntc

= H � χ, (14) 372

where χ =�

1√X√X 1

�, � denotes the Hadamard element- 373

by-element product and H represents the UP-based channel, 374

which can be defined as 375

H =

�K

K + 1HLOS +

�1

K + 1HNLOS, (15) 376

and hence 377

Hnr/2,ntc=

�K

K + 1χ � HLOS +

�1

K + 1χ � HNLOS, 378

(16) 379

given that K is the K-Rician factor, HLOS is the LOS 380

channel component and HNLOS is the NLOS Rayleigh fading 381

channel. 382

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Fig. 4. PM encoder block diagram.

D. Detection383

Having generated the PM symbol vector S, we now intro-384

duce the ML detector of our PM scheme. In an uncoded385

scenario, the PM detector aims to detect both the APM386

symbols as well as the polarization information of the transmit387

DP-AEs, where both {Aq}Q1 and {xl}L1 denoting the PM388

constellation S are available at the receiver.389

The ML detector’s main function is to maximize the a390

posteriori probability by invoking the conditional probability391

of receiving Y given that Si is transmitted defined by [45]392

p (Y |Si ) =1

(πN0)Nr

exp

�−�Y − HSi�2

N0

�, (17)393

where Si∈ S represents the transmitted symbol vector under394

the assumption that all symbols in set S are equi-probable395

with p (Si) =1/2B ∀Si ∈ S. Hence, the ML detector may be396

formulated as397 �q, l�

= arg min∀q,l

�Y − HSi�2 (18)398

= arg min∀q,l

�Y − HAiXi�2, (19)399

= arg min∀q,l

��Y −Nt

c�nt

c=1

HntcA(k)

q X(k)l

��2

, (20)400

with Hntc∈ CNr×2 being the nt

c-th sub-channel between the401

ntc-th DP-AE and the Nr/2 receive AE, which denotes the402

ntc and nt

c + 1 column vectors of H . Furthermore, q and l403

denotes the estimated values of q and l, which designate the404

selected sets of q and l information, respectively.405

E. Practical Considerations406

In this section, we present a discussion on the feasibility407

of the PM system in practical implementations, namely in the408

context of the PM encoder design as well as of its hardware409

considerations. In order to invoke the polarization character-410

istics of a DP-AE, a phase-shifter and a power amplifier are411

required at its front-end. However, more complications may412

arise in the construction of the transmitter if maintaining a dual413

stream transmission per DP-AE were required. For instance,414

a straightforward approach is to implement two distinct RF415

chains; one for the V port and the other for the H port of each 416

DP-AE, and hence a total of�2Nt

2

RF chains are required. 417

1) PM Encoder Design: In order to retain a dual data 418

stream transmission with a reduced RF-chain implementa- 419

tion, we propose the PM encoder architecture of Figure 4. 420

In this figure, the BPM input bits are divided into three 421

parts for constructing the PM symbol vector. More specif- 422

ically, the first part is used to select the q-th phase-shifter 423

combination ∠Aq =�θq,v, θq,h, while the second part is 424

used to generate the phases of the APM symbols pair ∠L − 425

APM =�φl,v, φl,h, as shown in Figure 4. A multiplier is 426

employed to combine both phases and generate the ntc-th PM 427

symbol’s phase ∠S(ntc) =�φl,v + θq,v, θq,h + φl,h. Further- 428

more, the third part is used to produce the (ql)-th power 429

arrangement �|xl,vaq,v| , |xq,hal,h|, which configures the vari- 430

able power amplifiers to match the (ql)-th PM symbol’s 431

moduli, as portrayed in Figure 4. Observe in Figure 4 that by 432

entirely relying on phase-based modulation schemes, the two 433

variable gain power amplifiers can be replaced with a single 434

power amplifier connected at the front-end of the encoder, 435

which improves the encoder’s power efficiency. This can be 436

achieved with the aid of reconfigurable antennas, which are 437

capable of continuously tuning both the AR and the tilt angle 438

of the transmitted signal [46]. In what follows, we consider 439

the PM encoder of Figure 4, which produces a pair of APM 440

symbols amalgamated with the polarization information of 441

the DP-AE. 442

2) Hardware Considerations: The PM encoder design 443

requires the switching and DP-AE controlling units presented 444

in Figure 4 for the sake of maintaining a dual-stream trans- 445

mission, which increases the hardware complexity of the 446

transmitter. This is one of the noticeable drawbacks of the PM 447

encoder design, when compared to conventional RF implemen- 448

tations. However, by comparing the architecture of a single 449

switching-aided RF-chain of Figure 4 to a pair of end-to-end 450

RF chains, which are required to operate a couple of AEs 451

(e.g. two DP-AE ports), the hardware requirements become 452

less demanding. For instance, it has been shown in [47] 453

that the most expensive component (in terms of cost and 454

power consumption) in switch-aided transmitters, comparable 455

to our PM design, is the RF chain (see [48] for details). This 456

excludes the additional switching modules, serial-to-parallel 457

(S/P) converters and the RF switches of our PM encoder. 458

Nonetheless, the practical implementations of the PM system 459

require further investigation, albeit the evident cost-power 460

consumption and complexity design trade-off. 461

We note here that the design of Figure 4 may be relaxed 462

by transmitting a single APM symbol rather than two symbols 463

over the DP-AE ports. However, this would reduce the achiev- 464

able throughput B of Equation (5) to (N tc · log2 (LQ)) bits. 465

The implementation of DP-AEs using the above-mentioned 466

architecture is worthwhile investigating, hence in what fol- 467

lows we characterize both the capacity as well as the BER 468

performance of the PM system. 469

III. PM SYSTEM CAPACITY 470

In this section, we present both the DCMC capacity and 471

the ergodic CCMC capacity of our PM system. Furthermore, 472

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we formulate the upper and lower bounds of the ergodic473

CCMC capacity.474

A. DCMC Capacity475

The DCMC capacity of our PM system, which designates476

the mutual information expressing the number of error-free477

bits that can be decoded at the PM receiver, can be formulated478

as [49]479

CDCMC480

= maxp(S)

I (S; Y )481

= max{p(S)}∀q,l

� +∞�−∞

· · ·+∞�

−∞482

p (Y |Si) p (Si) log2

�p (Y |Si)�

∀S∈S p (Y |S i) p (S i)

�dY ,483

(21)484

which can be maximized by using equi-probable p (Si).485

Next, by relying on the system’s conditional probability of486

Equation (17), the DCMC capacity can be now formulated487

as [49]488

CDCMC =B− b�q,l

E

⎡⎣log2

⎧⎨⎩�

q,l

exp�ψ |Si

⎫⎬⎭⎤⎦ , (22)489

where b = 1(2B) and ψ is given as490

ψ = −�H (Si − S i) + V �2 + �V �2 , (23)491

with S i being the transmitted symbol vector having%q, l&

492

indices. Unfortunately, there is no closed-form formulation493

available for Equation (22) and hence, we rely on numerical494

averaging procedures for evaluating the DCMC capacity.495

B. Ergodic CCMC Capacity496

On the other hand, the ergodic CCMC capacity of a MIMO497

system including our PM system is provided for maximizing498

the mutual information in a MIMO channel, which can be499

denoted as the maximum number of bits in an error-free500

continuous transmission and it is defined as [50]501

CCCMC = maxp(S)

H (Y ) −H (Y |S ) , (24)502

where H (Y ) and H (Y |S ) denote the destination entropy503

and the entropy of Y given S, respectively, which can be504

written as505

CCCMC = E

'log2

��INr +ρ

Nt

�HHH

��(. (25)506

C. Ergodic Capacity Bounds507

In order to clearly show the effect of XPD on the achievable508

capacity of the PM system, in what follows we examine the509

bounds of CCCMC of (25) at the ultimate minimum XPD510

(i.e. X−1dB → 0) and the ultimate maximum XPD (X−1

dB → ∞),511

given K = 0.512

At X−1dB → 0: The XPD provided in Equation (11) attains its 513

maximum (X=1) and the system transforms to a conventional 514

UP-based MIMO system. Hence, closed-form of Equation (25) 515

at X=1 can be expressed as [51] 516

CX−1dB →0≥μ log2

⎡⎣1+

ρ

Ntexp

⎛⎝ 1μ

μ�j=1

K−j�p=1

1p−γ⎞⎠⎤⎦, (26) 517

given that μ=min (Nt,Nr), K =max (Nt,Nr) and 518

γ≈0.577215 is Euler’s constant. This can be obtained 519

by relying on 520

E

'ln�� 1Nt

�HHH

��(

=Nr�j=1

E {ln Ωj} −Nr lnNt, (27) 521

given that 522

E {ln Ωj} = ψ (Nt − j − 1) =K−j−1�

p=1

1p− γ, (28) 523

where Ωj∼χ22(Nt−j+1). 524

Here, CX−1dB →0 represents the upper bound of the capac- 525

ity CCCMC , since no cross polarization attenuation exists 526

between the V and H components, and hence no degradation 527

in the achievable capacity is incurred. 528

At X−1dB → ∞: The cross V/H channels attenuation of (11) 529

becomes infinitesimally low (i.e.√X =0) and the row vectors 530

hvnr/2 and hh

nr/2 of H in (10) denoting the V and H receive 531

AE channels at the nr/2-th received DP-AE, respectively, are 532

then expressed as 533�hv

nr/2

hhnr/2

�534

=

· · · hvv

nr/2,ntc

0 hvvnr/2,nt

c+1 0 · · ·· · · 0 hhh

nr/2,ntc

0 hhhnr/2,nt

c· · ·

�. 535

(29) 536

Observe in (29) that the resultant power of��hv

nr/2

��2 reduces 537

by half, which transforms the Chi-squared variable Ωj of (27) 538

into Ω�j∼ χ2

2(Nt−j2 +1), where E

-ln Ω�

j

.=ψ�

Nt−j2 − 1

. 539

Hence, the ergodic capacity reduces to 540

CX−1dB →∞ ≥ μ log2

⎡⎣1 +

ρ

Ntexp

⎛⎝ 1μ

μ�j=1

K−j2�

p=1

1p− γ

⎞⎠⎤⎦ . 541

(30) 542

The capacity CX−1dB →∞ of (30) denotes the lower bound 543

of the achievable capacity given a total V/H communication 544

blockage. Therefore, the CCMC capacity at any XPD level is 545

bounded by CX−1dB

→0 and CX−1dB

→∞ as 546

CX−1dB →∞ ≤ CX−1

dB≤ CX−1

dB →0. (31) 547

It is clearly seen in (31) that as the XPD attenuation 548

increases towards infinity the achievable capacity CX−1dB

549

decreases towards the lower bound (30). However, as the 550

XPD attenuation approaches zero the achievable capacity 551

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CX−1dB

approaches its maximum level, which is equivalent to552

a (Nt ×Nr)-element3 UP-based system.553

It is worth noting that the DCMC capacity as seen554

in Equation (22) is affected by the design of the set of555

space-polarization dispersion matrices {Aq}Qq . However, the556

ergodic capacity provided in Equation (24) is only restricted557

by the transmit power, bandwidth as well as the XPD level.558

IV. ABER ANALYSIS559

The average BER for the PM system is generally formulated560

using the general MIMO upper-bounding technique given561

by [52]562

BER =�q=1

�q=1

�l=1

�l=1

Dh

�q, l, q, l

log2 (B)

P�S → S

, (32)563

where Dh

�q, l, q, l

denotes the hamming distance between564

the bit-mapping of S and S and P�S→S

is the average565

pairwise error probability (APEP). The APEP in fact is the566

average probability E

/P�S→ S |H

0, which determines567

the probability that a PM symbol S is erroneously detected as568

S given H and can be expressed as [52], [53]569

P�S → S |H

= P

��H �S − S

+ V

�� < ��V �� 570

= Q

⎛⎝1

�H��2

2N0

⎞⎠ , (33)571

where � = S − S and Q (·) denotes the Q-function defined572

in [54] as573

Q (x) =1π

π/2�0

exp�− x2

2 sin2 θ

�dθ, (34)574

and subsequently the PEP representation of (33) can now be575

expressed as576

P�S → S |H

=

1π

π/2�0

exp�− γ

2 sin2 θ

�dθ, (35)577

Now, by averaging Equation (35) over [0,∞] the legitimate578

range of the random variable γ, the unconditional PEP can be579

formulated as [55]580

P�S → S

=

1π

π/2�0

Φγ

�− 1

2 sin2 θ

�dθ, (36)581

where Φ (·) denotes the moment-generating function (MGF)582

of γ.583

In case of implementing UP-AEs, where no cross attenua-584

tion exists between V and H (X−1dB =0 dB), our PM system585

reduces to an ordinary spatial multiplexing system, which can586

be evaluated based on Appendix B of [56]. However, when587

introducing DP-AEs, a new parameter X denoting the DP-AE588

3It should be equipped with double the number of DP-AEs (i.e.�2Nt

2× 2Nr

2

�-element).

polarization effects arises and hence should be considered for 589

the ABER formulation. 590

Let us consider �ntc

=S(ntc)−S

(ntc) the symbol difference 591

at the ntc-th transmit DP-AE, which can be expressed as 592

�ntc

=��nt

c,v

�ntc,h

�=

⎡⎣�A

ntc

q,v · xntc

l,v −Ant

c

q,v · xntc

l,v

�A

ntc

q,h · xntc

l,h −Ant

c

q,h· xnt

c

l,h

⎤⎦ , (37) 593

where �ntc,v and �nt

c,h denote the symbol difference at 594

the vertical and horizontal components of the ntc-th trans- 595

mit DP-AE, respectively. Given α=�H��2and using Equa- 596

tion (37), α can be rewritten as 597

α =

��Nt

c�nt

c=1

Nr�nr=1

Hnr,ntc�nt

c

��2

, (38) 598

α =

��Nt

c�nt

c=1

Nr2�

nr2 =1

H nr2 ,nt

c�nt

c

��2

, (39) 599

where H nr2 ,nt

cis the TITO sub-channel between the 600

ntc-th transmit DP-AE and the nr/2-th receive DP-AE defined 601

in (11). Hence, α appears in the following form 602

α =

��Nt

c�nt

c=1

Nr2�

nr2 =1

hvv

nr2 ,nt

c

√Xhvhnr2 ,nt

c√Xhhvnr2 ,nt

chhh

nr2 ,nt

c

��ntc,v

�ntc,h

��2

. 603

(40) 604

Now, by using the norm representation of �AI×J�2 = 605�Ii=1

�Jj=1 |ai,j |2, Equation (40) can be rewritten as [57] 606

α =Nt

c�nt

c=1

⎛⎝ Nr

2�nr2 =1

��ntc,vh

vvnr2 ,nt

c+√X�nt

c,hhvhnr2 ,nt

c

��2 607

+

Nr2�

nr2 =1

��hhhnr2 ,nt

c�nt

c,h +√X�nt

c,vhhvnr2 ,nt

c

��2⎞⎠ . (41) 608

Each element of the MIMO channel matrix H of (10) is 609

assumed to be an i.i.d random variable, and hence (41) can be 610

reformulated as 611

α =Nt

c�nt

c=1

12

��ntc,v

��2 + X ��ntc,h

��2 2 34 5

Υv

ς21,Nr612

+Nt

c�nt

c=1

12

��ntc,h

��2 + X ��ntc,v

��2 2 34 5

Υh

ς22,Nr, (42) 613

and 614

γ =1

2N0

�Υvς

21,Nr

+ Υhς22,Nr

, (43) 615

with ς2i,Nr∼χ2

Nrbeing a noncentral chi-squared random vari- 616

able (RV)4 with Nr degrees of freedom and noncentrality 617

parameter of K . 618

4In NLOS (i.e. K = 0) ς2i reduces to a Chi-squared distributed randomvariable.

IEEE P

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By substituting γ of (43) into (35), the PEP can be619

formulated as620

P�S→ S |H

=

1π

π/2�0

exp

�−��

Υvς21,Nr

4N0 sin2 θ

+

�Υhς

22,Nr

4N0 sin2 θ

��dθ,621

(44)622

and hence after averaging it over [0,∞], Equation (44) can be623

expressed as624

P�S → S

=

1π

π/2�0

ΦΥvς21,Nr

�1

4N0 sin2 θ

�625

·ΦΥhς22,Nr

�1

4N0 sin2 θ

�dθ. (45)626

A. Rayleigh Fading, K = 0627

In the case of considering a Rayleigh fading channel628

(e.g. K = 0), Equation (53) can be rewritten as [56]629

P�S → S

=

1π

π/2�0

L=26l=1

�sin2 (θ)

sin2 (θ) + cl

�Nr2

dθ, (46)630

where c1 = Υv

2N0, c2 = Υh

2N0and the MGF of the chi-squared631

RV ς2l is defined by632

Φaς2l(−s) = (1 + 2as)−

Nr2 . (47)633

The closed-form solution of (46) can be formulated using634

two approaches. Following the solution provided in Appen-635

dix 5A.9 in [58], the first closed-form solution of (46) can be636

expressed as637

P�S → S

=

12

L=2�l=1

Nr/2�k=1

Jkl

�1 −�

clcl + 1

638

·k�

j=0

�2jj

�1

[4 (1 + cl)]j

�, (48)639

given that640

Jkl =

'd

Nr2

−k

dxNr2 −k

7L=2n=1n �=l

�1

1+cnx

Nr2(��

x=− 1cl�

Nr

2 − k!c

Nr2 −k

l

. (49)641

For the special case of using a single DP-AE receiver642

(e.g. Nr

2 =1), Equation (48) reduces to643

P�S → S

644

=12

L=2�l=1

��1 −�

clcl + 1

�645

·L=26n=1n �=l

�2jj

�cl

[(cl − cn)]

�. (50)646

In the second approach, the closed-form of the PEP given 647

in (46) can be formulated as 648

P�S → S

649

=12π

(c1c2)−Nr

2 · β�

12, Nr +

12

�650

·F1

�Nr +

12,Nr

2,Nr

2, Nr + 1;−c−1

1 ,−c−12

�, (51) 651

which is detailed in Appendix A, where β (·, ·) denotes the 652

Beta function and F1 (α, β, β�, γ;x, y) the confluent hyperge- 653

ometric function of two variables (Equation (61)). 654

In the high SNR-regime (i.e. N0 � 1), Equation (51) can 655

be written as 656

P�S → S

≤ 1

2π

�ΥvΥh

16N20

�−Nr2

β

�12, Nr +

12

�, (52) 657

where F1

�Nr + 1

2 ,Nr

2 ,Nr

2 , Nr + 1; 0, 0=1 at c1→∞ and 658

c2→∞. Hence, the achievable diversity gain defined by the 659

slope of P�S → S

is equivalent to Nr. 660

Note here that Equation (46) simplifies to Equation (36) 661

when X−1dB =0 dB (i.e. Υv =Υh) and hence, Equation (46) 662

can be solved using ( [56], Equation (64)). Additionally, it can 663

be seen in (52) that the XPD level does not have any effect 664

on the achievable diversity order of the PM system. 665

B. Rician Fading, K > 0 666

When considering a Rician fading channel (e.g. K >0), 667

Equation (45) can be written as [59] 668

P�S → S

669

=1π

π/2�0

L=26l=1

�sin2 (θ)

sin2 (θ) + cl670

· exp�− Kcl

sin2 (θ) + cl

��Nr2

dθ, (53) 671

where the MGF of the noncentral chi-squared RV ς2l is defined 672

as [56] 673

Φaς2l(−s) = (1 + 2as)−

Nr2 exp

�−KNr

2· s

1 + 2as

�. (54) 674

There is no closed-form of Equation (53) and hence, it can 675

be evaluated numerically. Note here that at K = 0 the problem 676

reduces to Equation (46). 677

However, by using the Q-function approximation proposed 678

in [60], the APEP of Equation (45) can be approximated as 679

P�S → S

680

≈112

�ΦΥvς2

1,Nr

�1

4N0

�· ΦΥhς2

2,Nr

�1

4N0

��Nr2

681

+14

�ΦΥvς2

1,Nr

�1

3N0

�· ΦΥhς2

2,Nr

�1

3N0

��Nr2

, (55) 682

which is detailed in Appendix B. 683

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The PM system is comparable to a spatial multiplexing684

system, which suffers from a degraded performance in the685

presence of a LOS component, as a result of the correlation686

fading effect. To overcome this issue in a DP-based MIMO,687

we employ our PM system by relying on a single transmit688

DP-AE (N tc =1) at high XPDs, yielding E

��hvhi,j

��2��1 and689

E

��hhvi,j

��2��1.690

V. SPACE-POLARIZATION IMPROVED CONSTELLATION691

In this section, we introduce our PM improved-constellation692

generation procedure. Observe in Equation (6) that the polar-693

ization configuration matrix Aq disperses the PSK/QAM694

complex symbols of X l over the spatial and polarization695

dimensions at a single time slot, in a conceptually similar696

manner to space-time dispersion matrices [33], [34], [37]. This697

opens a new prospect for designing the polarization shape of698

PM constellations.699

In a nutshell, the polarization shaping matrices {Aq}Q1 may700

be randomly generated so that the performance of the system701

is improved. In this regard, the shaping matrices may be702

constructed so that the unconditional PEP of Equation (46) is703

minimized, while retaining the maximum achievable diversity704

order. Hence, the optimal set of Q unit polarization vectors705

Aopt can be constructed by conducting a Random Search (RS)706

that aims at minimizing the maximum PEP as707

Aopt = argAimin

/maxP

�S → S

0, (56)708

which translates to709

Aopt =argAimax {min (c1c2)}=argAi

max {min (ΥvΥh)} ,710

(57)711

which can be rewritten as712

Aopt = max {min ��} . (58)713

It is worth emphasizing here that the construction of714

{Aq,h, Aq,v} designating the H and V configurations of715

{Aq}Q1 , respectively, should fall within the polarization shap-716

ing capabilities of the DP-AE, namely its AR range (1) and its717

Tilt angle range (4). Additionally, multiple transmit AEs are718

spaced sufficiently far apart in order to experience independent719

fading hence, random search is performed using a single720

transmit DP-AE, where the Aopt set produced is used at each721

DP-AE.722

In what follows we present the generation process of723

Aopt satisfying (58) using a TITO (2 × 2)-element system.724

We first generate a random set of (1 × 2)-element unit vectors725

denoting the diagonal vectors of the (2 × 2)-element matrix set726

Ai={Aq}Q1 . The vector set generated should obey the Rank727

Criterion (i.e. rank(��H) = 1 ∀ q, q ∈ Q) in order to guar-728

antee a normalized power space-polarization set. Next, we cal-729

culate the minimum Euclidean distance dmin={min ��}.730

The random search continues by repeating both steps, while731

retaining the Ai set having the maximum dmin. The algorithm732

presented above is summarized in Algorithm 1. Furthermore,733

an example is provided in Appendix C to ease understanding.734

Note that by obtaining the minimum distance 735

dmin=max{min ��} in (58) the PEP P (�H (S−S) + V � 736

<�V �) of (33) is minimized, and hence the DCMC exponent 737

ψ = − �H(Si − S i) +V �2+�V �2 of (23) is subsequently 738

minimized, which improves the achievable DCMC capacity. 739

Algorithm 1 Polarization Shaping Algorithmminimum distance: κ = 0initialize Aopt;

Start: (i = 1 :106 loops)Loop: Generate Q random (2 × 1)-element unit vectors

{aq}Q1

Ai = {Aq = diag2 (aq)}Q1

compute S, S and � ∀q, l1, l2if�rank

��H

= 1

Compute OA, OB and τ using {Aq}Qq=1

if (OA, OB and τ doesn’t match the DP-AE range)GOTO Loop

else GOTO LoopCompute di

min =min {��}if(di

min >κ)Apply Aopt=Ai

GOTO LoopReturn Aopt

End

VI. SIMULATION RESULTS 740

In this section, we present our Monte Carlo simulation 741

results with a minimum of 106 bits per SNR value as well as 742

the theoretical analysis of our PM system. In our simulations 743

we assume perfect CSI at the receiver side for invoking the 744

ML optimum detector of Equation (18). Furthermore, multiple 745

DP-AEs are spaced sufficiently far apart in order to experience 746

independent fading. We choose the polarization shaping matrix 747

set {Aq}Qq=1 by selecting several AR and τ values based on 748

the discussion presented in Section II-A. Particularly, Table III 749

shows the main PM systems used in our simulations with 750

Q=4 as follows5: three AR systems (i.e. AR-1,…, AR-3), 751

two Tilt systems (i.e. Tilt-1, Tilt-2) and four TAR systems 752

(i.e. TAR-1,…, TAR-4). Additionally, all plots showing the 753

performance of PM-systems associated with the RS-aided 754

constellation presented in Section V are labeled as TAR-RS. 755

The TAR-RS system used below is presented in Appendix C. 756

Note here that the tuning capabilities of DP-AEs over the 757

AR and the tilt angle vary from one antenna to another. For 758

instance, the reconfigurable DP-AE presented in [46] utilizes 759

a maximum AR of 35 dB and a tilt angle spanning between 760

30◦ and 100◦. 761

A. Comparison Fairness 762

In this contribution we define fair comparison as follows: a 763

fair performance comparison between a DP-based system and 764

a UP-based system is attained by employing an equivalent 765

number of AEs in both systems. To expound a little further, 766

5Other systems with various Q configuration are used.

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TABLE III

AR AND TILT ANGLES OF VARIOUS PM SYSTEMS DESIGNED FOR PROVIDING Q = 4 SPACE-POLARIZATION

CONFIGURATIONS DENOTED BY�aq,h , aq,v) AND

�θq,h , θq,v), RESPECTIVELY

Fig. 5. DCMC capacity comparison between various PM systems attaining4 bpcu by relying on the AR, Tilt and TAR configurations with differentpolarization shapes at an XPD of X−1

dB =10 dB.

consider a PM system that is equipped with a single transmit767

DP-AE. This system would require a single RF chain for768

transmitting a single PM symbol, and hence it is comparable to769

a UP-based system having a single UP-AE. By increasing the770

number of UP-AEs to match the number of ports in a single771

DP-AE (e.g. use UP-AEs) an additional RF chain is required,772

which negates fairness.773

Furthermore, in MIMO implementations, AE spacing has774

to be on the order of ten wavelengths, in order to experience775

independent channel fading. In DP-based MIMOs, the V and H776

components of each DP-AE are separated over the polarization777

dimension, where Nt/2 AEs only require to be spaced far778

apart. However, adding Nt UP-AEs would require double the779

area of a DP-based system. In what follows, we refer to any780

simulated system as (M ×N), where M and N denote the781

number of transmit and receive AEs (DP or UP), respectively.782

B. DCMC Capacity783

Based on the unified capacity metric provided in Equa-784

tion (22), Figure 5 depicts the DCMC capacity curves of our785

PM system designed for achieving a normalized throughput 786

of 4 bpcu. Here, we employed (1 × 1) DP-AEs with various 787

PM configurations. More specifically, Figure 5 shows the 788

DCMC curves of the AR-1-3, Tilt-1-2 and TAR-1-3 systems 789

detailed in Table III as well as of TAR-RS and TAR-RS-1PSK, 790

where TAR-RS-1PSK is a symbol-free RS-based PM�TAR,1, 791

1, Q = 16, 1PSK

system (i.e. polarization information only). 792

We also characterize the conventional (1 × 1) UP-AE-based 793

16QAM and 16PSK systems. It can be observed in Figure 5 794

that TAR-based PM systems outperform all the other PM 795

configurations, while the RS-based systems achieve the high- 796

est throughput. For instance, TAR-RS outperforms PolarSK 797

(i.e. Tilt-PM) by 2.8 dB and conventional 16QAM and 16PSK 798

by 3.7 dB and 6 dB, respectively. This verifies the discussion 799

presented in Section V, where constructing the optimal Aopt 800

under the constraint of maximizing dmin=max {min ��} 801

could further improve the achievable capacity of the PM 802

system. 803

In order to characterize the effect of the XPD on the PM sys- 804

tem, Figure 6 portrays the 3D surface of the achievable capac- 805

ity of a PM�TAR,1, 1, Q = 4, BPSK

system with respect 806

to XPD and SNR. Furthermore, the achievable throughput at 807

X−1dB =0 dB is projected onto the (SNR, Capacity)-plane for 808

the sake of comparison. As seen in Figure 6, the achievable 809

throughput degrades as the XPD increases, which can be 810

clearly seen at high XPDs. To expound a little further, Figure 7 811

showcases the projected 3D surface of Figure 6 onto the (SNR, 812

Capacity)-plane between X−1dB =0 dB and X−1

dB =30 dB. It can 813

be seen from the figure that a maximum degradation of 3.5 dB 814

is observed in the DCMC capacity between X−1dB =0 dB 815

and X−1dB =30 dB. However, the degradation in the achiev- 816

able capacity becomes marginal at high XPDs, especially at 817

X−1dB >15 dB. 818

C. CCMC Capacity 819

To investigate the ergodic CCMC capacity of our PM 820

system, the capacities of three PM systems are illustrated by 821

the 3D surfaces drawn in Figure 8, namely for the (1 × 1), 822

(2 × 2) and (4 × 4) DP-AEs MIMO arrangements. One can 823

observe in Figure 8 that the CCMC capacity is affected both 824

by the transmission power as well as the XPD level. 825

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Fig. 6. A 3D representation of the DCMC capacity of aPM(TAR, 1, 1, Q = 4, BPSK) system with respect to SNR and XPD.

Fig. 7. The 2D projection of Figure 6 onto the (SNR, DCMC)-plane.

Figures 9(a)-(c) depict the 2D projection of the 3D surfaces826

of Figure 8 onto the (SNR, CCMC)-plane at an XPD of827

X−1dB =10 dB. The theoretical upper and lower bounds of828

equations (26) and (30), respectively, are also shown in each829

figure. Furthermore, the capacity of an equivalent number830

of UP-AEs is shown for the sake of comparison, where the831

capacity improvement of DP-based systems is shown addi-832

tionally by the red curve. It can be observed in Figure 9 that833

DP-AE implementations substantially boost the capacity of a834

MIMO system, achieving between 87.5% and 54% capacity835

improvement over an SNR range spanning between −10 dB836

and 40 dB, respectively, for all three systems considered.837

Moreover, we note that the DP-based capacity curves por-838

trayed in Figure 9 are confined within the upper and lower839

bounds described in Section III, which are separated 3 dB840

Fig. 8. A 3D representation of the ergodic CCMC capacity of three PMsystems in terms of SNR (dB) and XPD (dB), namely for the (1 × 1), (2 × 2)and (4 × 4) DP-AEs arrangements.

apart. In fact, the simulated curves (through Monte Carlo) of 841

the DP-based systems in Figures 9(a)-(c) and the lower bound 842

analysis (CX−1dB

→∞) precisely match at X−1dB =10 dB. 843

To examine the effect of the XPD on the achievable CCMC 844

capacity, let us assume that we project Figure 8 onto the (XPD, 845

CCMC)-plane at SNR of 12 dB, 16 dB and 20 dB, as portrayed 846

in Figure 10. The figure shows that as the XPD level increases 847

the ergodic capacity decreases, however it remains relatively 848

constant after a specific value of XPD, as for example at an 849

XPD of X−1dB =16 dB at SNR of 12 dB. Furthermore, one can 850

observe that at an even high XPD level, the maximum loss in 851

CCMC capacity is less than 1.4 bps/Hz. 852

The novel polarization modulation technique presented in 853

this paper constitutes a viable solution to significantly boost 854

the data transmission rate for future wireless systems. In what 855

follows, we present the BER performance of our PM system. 856

D. BER Simulation 857

In Figure 11 we compare the achievable BER performance 858

of (1 × 1) and (2 × 2) PM systems,6 which achieve a through- 859

put of 4 and 8 bpcu, respectively, at an XPD of X−1dB =10 dB 860

and K = 0. Moreover, the BER performance of their UP-AE 861

(1 × 1) and (2 × 2) counterparts 16PSK are included for 862

comparison,7 while the dashed curves represent the theoretical 863

6A (1 × 1)-DP-AE implementation is equivalent to a (2 × 2) UP system,since the V and H components transmit over separate polarization dimensions.

7We use the same number of AEs for both systems (i.e. DP-AE and UP-AE)in order to maintain fairness. The (1 × 1) DP-AE system for instance hastwo input ports, while the (1 × 1) UP-AE has a single input port, while bothrequire a single RF chain implementation. In case two UP-AEs are used tocompare with the (1 × 1) DP-AE system, two RF chain are required, whichleads to unfairness in the number of RF components as well as in the requiredtransmitted power.

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Fig. 9. The 2D projection of Figure 8 onto the (SNR, CCMC)-planeat XPD of X−1

dB =10 dB of the following DP/UP systems: a) (1 × 1);b) (2 × 2); c) (4 × 4), which further include the upper and lower capacitybounds. Furthermore, the capacity improvement is shown by the red curve.

upper bounds developed in Section IV. Figure 11(a) shows864

the performance of PM�AR,Q = 4,BPSK

, PM

�Tilt,Q =865

4,BPSK, PM

�TAR,Q = 4,BPSK

and PM

�TAR,Q =866

4,BPSK-RS8 systems associated with (1 × 1)-DP-AEs.867

8The RS here features the improved RS-aided constellation provided inSection V.

Fig. 10. A 2D projection of Figure 8(a) onto the (XPD, CCMC)-planeshowing the effect of XPD on the attainable CCMC capacity at SNR of 12 dB,16 dB and 20 dB.

Here, log2 (4)= 2 bits are used to activate one out of 868

Q = 4 space-polarization matrices, while the remaining 869

2 log2 (2) =2 bits are modulated to a pair of BPSK sym- 870

bols. The performance of the PM�TAR,Q = 16,1PSK

- 871

RS system is also shown in Figure 11(a), where the whole 872

BPM =log2 (16)= 4 bits are used to switch between Q = 16 873

polarization shapes. It is shown in 11(a) that the PM system 874

outperforms the conventional (1 × 1)-DP-AE by 10 dB, 15 dB 875

and 19 dB at a BER of 10−5, when employing the AR, Tilt and 876

TAR configurations, respectively. Furthermore, the improved 877

constellation PM systems provide further BER enhancements 878

of 2 dB and 4 dB by using the PM�TAR,Q = 16,1PSK

-RS 879

and PM�TAR,Q = 4,BPSK

-RS systems, respectively. Note 880

here that the theoretical model presented in Section IV matches 881

perfectly with the Monte Carlo simulations. 882

In Figure 11(b), the BER performance of the 883

above-mentioned PM systems is shown with a (2 × 2)- 884

DP-AE implementation. As seen in the Figure, the PM 885

system outperforms the conventional (2 × 2)-element 886

multiplexing system by 0.5 dB, 5.4 dB 9.5 dB, when 887

employing the AR, Tilt and TAR configurations, and by 888

12.7 dB and 15 dB by using the PM�TAR,Q = 16,1PSK

-RS 889

and PM�TAR,Q = 4,BPSK

-RS systems, respectively. 890

In Figures 12(a)-(b), we show the performance of a 891

PM(TAR, Q = 4, BPSK)-RS system (i.e. TAR-RS) transmit- 892

ting over a Rician fading channel at 4 bpcu, while employing 893

a single transmit DP-AE at a high XPD of X−1dB =30 dB. 894

Furthermore, Figures 12(a)-(b) include both the exact and 895

approximated theoretical bounds presented in equations (53) 896

and (55), respectively. In particular, in Figure 12(a) we inves- 897

tigate the effect of the Rician factor on the performance of 898

the PM system at K = 0, 5, 10 and 15, when associated with 899

(1 × 1)-element implementation. We notice in Figure 12(a) 900

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Fig. 11. The BER performance of (1 × 1) and (2 × 2) DP-AE PM systemsassociated with Q = 4 and BPSK achieving a throughput of 4 and 8 bpcu at anXPD of X−1

dB =10 dB and K = 0 compared with their UP-AE counterparts.

that upon increasing K the BER performance of the PM901

system improves. As indicated by Equation (53), the ABER902

improves exponentially with the value of K . Furthermore,903

the exact theoretical model of (53) matches perfectly with904

the Monte Carlo simulations, while the approximate model905

of (55) is marginally shifted at K = 0 and K = 5, which906

perfectly overlaps at low BER values. On the other hand,907

Figure 12(b) shows the performance of the simulated PM908

system with a different number of receive DP-AEs, namely909

Nr/2=1, 2, 4 and 8 at K = 5. We notice in the figure that the910

approximate theoretical bound developed in Section IV tends911

to accurately match both the exact bound and the Monte Carlo912

simulations as the number of receive AEs increase.913

A comparison between multiple configurations of 4-level914

(Q = 4) PM systems with (1 × 2)-elements is illustrated in915

Figure 13, which all achieve a spectral efficiency of 4 bpcu at916

X−1dB =10 dB and K=0. To expound further, the PM systems917

under study are: AR-1-3, Tilt-1-2 and TAR-1-4 as well as918

TAR-RS, as detailed in Table III. As observed in Figure 13,919

the achievable performance of all systems spans over 35 dBs920

of SNR at a BER of 10−5. In all cases, it can be observed921

that TAR-based systems exhibit the best BER performance922

compared to AR-based and Tilt-based systems. This can be923

attributed to the multi-dimensional structure of TAR-based924

PM, where polarization information is dispersed over both the925

AR and the tilt contrary to other configurations (e.g. AR and926

Tilt) that exploit either of them.927

In Figure 14, we compare the BER performance of our PM928

system with its DP-AE-based counterparts. More specifically,929

we compare the BER performance of PM(TAR, 1, 1, Q =930

2, BPSK) with that of a DP-SM(1, 1, QPSK) system [30]931

as well as that of a PolarSK(Nr/2=1, Q = 2, BPSK)932

Fig. 12. The BER performance of a PM(TAR, Q = 4, BPSK)-RSsystem (TAR-RS) transmitting 4 bpcu over a Rician fading channel: a) with(1 × 1)-elements at K = 0, 5, 10 and 15; b) with Nr/2 =1, 2, 4 and 8 atK = 5 and X−1

dB =30 dB.

system [31], where each exhibits a transmission rate of 3 bpcu 933

over Rayleigh fading channel (i.e. K = 0) at an XPD of 934

X−1dB =10 dB. Figure 14 further shows the performance of 935

the improved-constellation PM(TAR, 1, 1, Q = 2, BPSK)-RS 936

and PM(TAR, 1, 1, Q = 8, 1PSK)-RS systems as well as the 937

performance of UP-AE-based SM and Quadrature SM (QSM) 938

systems associated with Nt =2 AEs. It can be observed from 939

Figure 14 that our PM system outperforms PolarSK, DP-SM 940

and the conventional SM by 2 dB, 1.2 dB, 22 dB, respectively. 941

To elaborate further on the effect of the level of XPD on 942

the BER performance, the BER performance of a (2 × 2)-DP- 943

AE PM system associated with an PM(TAR, Q = 2, BPSK) 944

encoder at different XPD levels is presented in Figure 15. 945

More specifically, we show the BER performance of the 946

system at an XPD spanning between X−1dB =0 dB and 947

X−1dB =30 dB with a step of 5 dB, where the theoretical bound- 948

aries are shown exclusively at X−1dB =0 dB and X−1

dB =30 dB. 949

Figure 15 demonstrates that the performance of the PM system 950

is directly affected by the XPD level, where it improves as the 951

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Fig. 13. The BER performance of a (1 × 2)-DP-AE PM systems associatedwith Q = 4 AR, Tilt and TAE configurations, which achieve a throughputof 4 bpcu at an XPD of X−1

dB =10 dB and K = 0.

Fig. 14. BER comparison of a static TAR-based PM system, two RS-aidedTAR-based PM systems, equivalent throughput DP-SM [30] and PolarSK [31]systems as well as UP-AE-based SM and QSM systems.

XPD decreases due to the increased polarization diversity gain.952

However, it can be seen in the figure that the performance953

is marginally affected when X−1dB ≥25 dB. Furthermore,954

the theoretical boundaries presented in Figure 15 confirms the955

precision of the XPD parameter provided in equations (46)956

and (51).957

It can be observed in Figures 11-15 that the theoretical958

boundaries provided in Section IV match the Monte Carlo959

Fig. 15. Performance comparison between TAR-based PM systems at anXPD spanning between X−1

dB =0 dB and X−1dB =30 dB.

simulations for all PM configurations, namely for the AR, 960

Tilt, TAR and MUX configurations over various antenna 961

arrangements. In what follows, we present our conclusion. 962

VII. CONCLUSION 963

In this treatise, we have introduced a novel modulation 964

technique referred to as the polarization modulation, which 965

invokes the polarization characteristics of a DP-AE for data 966

transmission. More specifically, a block of information in a PM 967

system is formed by dispersing a pair of PSK/QAM symbols 968

into the space- and polarization- dimension with the aid of 969

Q polarization shaping matrices {Aq}Qq=1. The polarization 970

shaping matrix may adjust the AR, Tilt or Tilted-AR of the 971

EM matrix with the aid of a single RF-chain per DP-AE. 972

The polarization shaping matrices can be selected empirically, 973

however we have proposed a special algorithm for generating 974

an improved-constellation tailored for the PM modulation. 975

Furthermore, we provided a theoretical analysis for the DCMC 976

and CCMC capacity as well as for the BER performance of the 977

PM system. It has been shown that by invoking the polarization 978

dimension, the ergodic capacity of a DP-based MIMO system 979

can be improved by 54% to 87.5% compared to UP-based 980

MIMO. Similarly, the DCMC capacity of our PM system was 981

improved by up to 6 dB in comparison to systems relying 982

on UP-AE. Furthermore, the simulation results indicated that 983

the gain achieved by our proposed PM system relying on 984

Q-state polarization levels spans between 10dB and 20dB 985

over UP-AE-based conventional systems. Our simulation also 986

showed that by utilizing the proposed improved-constellation 987

algorithm the DCMC capacity and BER performance of our 988

PM system have significantly improved. 989

APPENDIX A 990

The derivation of Equation (51) can be formulated by 991

substituting u =sin2 (θ) and dθ= du

2√

u(1−u)into Equation (46), 992

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yielding993

P�S → S

=

1π

1�0

�u

u+ c1

�−Nr2�

u

u+ c1

�−Nr2

994

du

28u (1 − u)

du, (59)995

and996

P�S → S

=

(c1c2)−Nr

2

2π

1�0

uNr− 12 (1 − u)−

12997

�1 +

1c1u

�−Nr2�

1 +1c2u

�−Nr2

. (60)998

Now, by relying on the confluent hypergeometric function999

of two variables given as (Section 9.18 [61])1000

F1 (α, β, β�, γ;x, y) =Γ (c)

Γ (a) Γ (c− a)1001

1�0

zα−1 (1 − z)γ−α−1 (1 − xz)−β (1 − yz)−β�dz, (61)1002

the closed-form expression of (60) can be expressed as shown1003

in Equation (51), where Γ (·) denotes the Gamma function.1004

APPENDIX B1005

Instead of using the Q-function defined in Equation (34),1006

we can simply use the approximation defined by [60] as1007

Q (x) ≈112e−

x22 +

14e−

2x23 . (62)1008

By plugging (62) into (33), we arrive at1009

P�S → S |H

≈

112e−

γ2 +

14e−

2γ3 . (63)1010

Given Nr receive AEs, the SNR of the nr/2-th channel1011

denoting the channel received at the nr/2-th AE is given1012

by γnr2

= 12N0

{Υv + Υh}Nr=2, where {Υv + Υh}Nr=2 is1013

equivalent to Equation (42) with N tc =1 and Nr =2.1014

Now, the average of PEP can be expressed as1015

P�S → S

1016

≈

� ∞

0

. . .

� ∞

02 34 5Nr/2

Nr/26nr2 =1

�exp�−γ

nr2

2

�fγ�γnr

2

1017

+ exp�−2γnr

2

3

�fγ�γnr

2

�dγ1 · · ·dγNr

2. (64)1018

By using the definition of the MGF function in ( [56],1019

Equation (21)), the close-form expression of P�S → S

can1020

be formulated as shown in Equation (55).1021

APPENDIX C 1022

Here, we provide an example of an RS-based PM�TAR, 1023

Q = 4, BPSK

system using the technique presented in 1024

Section V, which is referred to as TAR-RS in Section VI. 1025

Consider a PM system that relies on a set of BPSK symbols 1026

X l ={−1,+1} and on a randomly generated set {Aq}Qq for 1027

data transmission, which can be formulated as follows: 1028

A1 1029

=�−0.331952 + 0.686751i 0

0 −0.631246 + 0.140389i

�, 1030

(65) 1031

A2 1032

=�−0.853098 + 0.0743741i 0

0 0.0196869 + 0.516047i

�, 1033

(66) 1034

A3 1035

=�−0.493946− 0.228332i 0

0 −0.797398 + 0.260841i

�, 1036

(67) 1037

A4 1038

=�−0.160197− 0.557432i 0

0 −0.43818− 0.686735i

�, 1039

(68) 1040

where q = 1,. . . ,Q = 4. By using Equations (1-4) These 1041

configurations can be translated to the following parameters: 1042

Eh = {0.762771, 0.856334, 0.544168, 0.579994} , (69) 1043

Ev = {0.646669, 0.516423, 0.838976, 0.814621} , (70) 1044

θh = {115.798, 175.017, −155.191, −106.034} , (71) 1045

θv = {167.461, 87.8153, 161.886, −122.54} , (72) 1046

and Finally, 1047

τ = {130.29, 121.088, 57.0821, 54.55} , (73) 1048

and 1049

ARdB = {42.1995, 38.6012, 27.1086, 52.4841} . (74) 1050

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The Netherlands: Elsevier, 2007.1241

Ibrahim A. Hemadeh (Member, IEEE) received1242

the B.Eng. degree(Hons.) in computer and commu-1243

nications engineering from the Islamic University of1244

Lebanon, Lebanon, in 2010, and the M.Sc. degree1245

(Hons.) in wireless communications and the Ph.D.1246

degree in electronics and electrical engineering from1247

The University of Southampton, U.K., in 2012 and1248

2017, respectively. In 2017, he joined the Southamp-1249

ton Next Generation Wireless Group, The Univer-1250

sity of Southampton, as a Post-Doctoral Researcher.1251

In 2018, he joined the 5G Innovation Centre (5GIC),1252

University of Surrey. He is currently working as a Staff Engineer in the1253

industry. His research interests include millimeter-wave communications,1254

multi-functional multiple input multiple output (MIMO), multi-dimensional1255

(time-space and frequency) transceiver designs, channel coding, and multi-user1256

MIMO.1257

Pei Xiao (Senior Member, IEEE) worked at1258

Newcastle University and Queen’s University1259

Belfast. He also held positions at Nokia Networks,1260

Finland. He is currently a Professor of wireless1261

communications with the Institute for Communi-1262

cation Systems, Home of 5G Innovation Centre1263

(5GIC), University of Surrey. He is the Technical1264

Manager of 5GIC, leading the research team in the1265

new physical layer work area, and coordinating/1266

supervising research activities across all the1267

work areas within 5GIC (www.surrey.ac.uk/5gic/1268

research). He has published extensively in the fields of communication theory,1269

RF and antenna design, signal processing for wireless communications. He is1270

an Inventor on more than ten recent 5GIC patents addressing bottleneck1271

problems in 5G systems.1272

Yasin Kabiri (Member, IEEE) received the M.Eng. 1273

degree (Hons.) in electronics and communication 1274

engineering from the University of Birmingham, 1275

Birmingham, U.K., in 2012, and the Ph.D. degree 1276

from the School of Electronic and Electrical Engi- 1277

neering, University of Birmingham, in 2015. During 1278

his Ph.D., he has developed an approach called 1279

injection matching theory which can be used for 1280

making small, wide band, and reconfigurable anten- 1281

nas with high efficiency. He was also a Research 1282

Fellow with the 5G Innovation Center, Guildford, 1283

U.K., with a focus on 5G antennas. He holds multiple patents in the field 1284

and has contributed in major grant applications. He is currently working 1285

as a principal RF and microwave engineer in industry section. His research 1286

interests include RF and microwave, phased array and beam steerable antenna, 1287

mmwave system, satellite communication, electrically small antenna, active 1288

antennas, and microwave filters. 1289

Lixia Xiao (Member, IEEE) received the B.E., M.E., 1290

and Ph.D. degrees from the University of Electronic 1291

Science and Technology of China (UESTC) in 2010, 1292

2013, and 2017, respectively. She is currently a 1293

Research Fellow with the Department of Electrical 1294

Electronic Engineering, University of Surrey. Her 1295

research is in the field of wireless communications 1296

and communication theory. In particular, she is very 1297

interested in signal detection and performance analy- 1298

sis of wireless communication systems. 1299

Vincent Fusco (Fellow, IEEE) is currently a Per- 1300

sonal Chair of high frequency electronic engineering 1301

with QUB. He has authored more than 500 scientific 1302

articles in major journals and referred international 1303

conferences, and 2 textbooks. He holds patents 1304

related to self-tracking antennas and has contributed 1305

invited articles and book chapters. His research focus 1306

on advanced microwave through millimetre wave 1307

wireless. His current research interests include phys- 1308

ical layer secure active antenna techniques. In 2012, 1309

he was awarded the IET Senior Achievement Award, 1310

the Mountbatten Medal. 1311

Rahim Tafazolli (Senior Member, IEEE) is 1312

currently a Professor of mobile and personal 1313

communications and the Director of the Institute 1314

of Communication Systems, 5G Innovation Centre, 1315

University of Surrey. He has been active in research 1316

for more than 20 years and published more than 1317

500 research articles. In 2018, he was appointed as 1318

a Regius Professor in electronic engineering for the 1319

recognition of his exceptional contributions to digital 1320

communications technologies more than the past 1321

30 years. He is a fellow of IET and Wireless World 1322

Research Forum. He served as the Chairman for EU Expert Group on Mobile 1323

Platform (e-mobility SRA) and Post-IP Working Group in e-mobility, and the 1324

past Chairman for WG3of WWRF. He has been a technical advisor to many 1325

mobile companies. He has lectured, chaired, and been invited as a keynote 1326

speaker to a number of IEE and IEEE workshops and conferences. He is 1327

nationally and internationally known in the field of mobile communications. 1328

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IEEE TRANSACTIONS ON COMMUNICATIONS 1

Polarization Modulation Design forReduced RF Chain Wireless

Ibrahim A. Hemadeh , Member, IEEE, Pei Xiao , Senior Member, IEEE, Yasin Kabiri , Member, IEEE,Lixia Xiao, Member, IEEE, Vincent Fusco , Fellow, IEEE, and Rahim Tafazolli , Senior Member, IEEE

Abstract— In this treatise, we introduce a novel polarization1

modulation (PM) scheme, where we capitalize on the recon-2

figurable polarization antenna design for exploring the polar-3

ization domain degrees of freedom, thus boosting the system4

throughput. More specifically, we invoke the inherent properties5

of a dual polarized (DP) antenna for transmitting additional6

information carried by the axial ratio (AR) and tilt angle7

of elliptic polarization, in addition to the information streams8

transmitted over its vertical (V) and horizontal (H) components.9

Furthermore, we propose a special algorithm for generating an10

improved PM constellation tailored especially for wireless PM11

modulation. We also provide an analytical framework to compute12

the average bit error rate (ABER) of the PM system. Further-13

more, we characterize both the discrete-input continuous-output14

memoryless channel (DCMC) capacity and the continuous-input15

continuous-output memoryless channel (CCMC) capacity as16

well as the upper and lower bounds of the CCMC capacity.17

The results show the superiority of our proposed PM system18

over conventional modulation schemes in terms of both higher19

throughput and lower BER. In particular, our simulation results20

indicate that the gain achieved by the proposed Q-dimensional21

PM scheme spans between 10dB and 20dB compared to the22

conventional modulation. It is also demonstrated that the PM23

system attains between 54% and 87.5% improvements in terms24

of ergodic capacity. Furthermore, we show that this technique25

can be applied to MIMO systems in a synergistic manner in26

order to achieve the target data rate target for 5G wireless27

systems with much less system resources (in terms of bandwidth28

and the number of antennas) compared to existing MIMO29

techniques.30

Index Terms— 5G, wireless networks, MIMO, dual-polarized,31

polarization modulation, index modulation, spatial modulation,32

polarization, MPSK, MQAM, practical implementations, channel33

modulation, hard-decision detection.34

Manuscript received May 8, 2019; revised September 30, 2019 andDecember 6, 2019; accepted January 23, 2020. This work was supported bythe U.K. Engineering and Physical Sciences Research Council (EPSRC) underGrant EP/N020391/1. The authors also would like to acknowledge the supportof the University of Surrey 5GIC (http://www.surrey.ac.uk/5gic) membersfor this work. A U.K. patent “Wireless Data Transmission using PolarisedElectromagnetic Radiation” (reference number GB1812108.7) related to thiswork was filed on July 25, 2018. The associate editor coordinating thereview of this article and approving it for publication was M. Di Renzo.(Corresponding author: Ibrahim A. Hemadeh.)

Ibrahim A. Hemadeh, Pei Xiao, Yasin Kabiri, Lixia Xiao, andRahim Tafazolli are with the Institute for Communication Systems (ICS),University of Surrey, Guildford GU2 7XH, U.K. (e-mail: [email protected]).

Vincent Fusco is with the School of Electronics, Electrical Engineering andComputer Science, Queen’s University Belfast, Belfast BT7 1NN, U.K.

Color versions of one or more of the figures in this article are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCOMM.2020.2979455

I. INTRODUCTION 35

MULTIPLE-INPUT multiple-output (MIMO) techniques 36

are capable of providing unprecedented improve- 37

ments for wireless communication systems in terms of 38

capacity [1], [2]. Explicitly, MIMO systems are capable of 39

attaining an enhanced bit error rate (BER) performance as well 40

as an improved throughput in comparison to single-antenna 41

implementations, provided that each of the transmitted signals 42

has a unique signature at each of the receive antenna elements 43

(AEs). In the context of spatial transmission schemes, multiple 44

AEs are spaced sufficiently apart in order to experience 45

independent fading. Typically, array elements are placed 10λ 46

apart from each other at the base station, where λ represents 47

the carrier wavelength. However, it is often impractical to 48

accommodate multiple AEs, especially in small hand-held 49

devices [3]. One solution is to communicate at high frequency 50

bands, such as the millimeter-wave (mmWave) band [4], which 51

allows fitting a high number of AEs within a relatively small 52

area, while still providing an independent fading. However, 53

it would still be a challenging task to obtain a unique spatial 54

signature of distinct AEs in a highly dense and closely 55

spaced antenna arrays due to the dominant line-of-sight (LOS) 56

component. An alternative way of overcoming this problem 57

is to separate the transmitted signals over the polarization 58

domain, which can be achieved by using dual-polarized AEs 59

(DP-AEs) [5], [6]. In particular, by employing DP-AEs the 60

number of transmit and receive AEs can be doubled in 61

comparison to uni-polarized AEs (UP-AEs). 62

In a nutshell, a single DP-AE constitutes a pair of 63

co-located and orthogonally-polarized vertical (V) and hori- 64

zontal (H) components. These are typically referred to as the 65

VH components and come in different shapes and forms [7]. 66

The orthogonality of the V and H components offers a new 67

means of spatial separation, namely over the polarization 68

dimension, providing a near nil spatial correlation at both the 69

transmitter and the receiver [8], [9]. By invoking the addi- 70

tional degrees-of-freedom (DoF) offered by cross-polarized 71

components, the spectral efficiency of a MIMO system can be 72

further enhanced [10]. Note that the communication between 73

cross-polarized components instigates channel depolarization, 74

which impacts the cross-channel gains. This can be measured 75

by the cross-polar discrimination (XPD) [11]. 76

Polarization [12] is a key element of defining the electro- 77

magnetic (EM) wave propagation in addition to the frequency, 78

time, amplitude and phase elements [12]. It is characterized 79

by the variations of the direction and the amplitude of an EM 80

wave with respect to time. 81

0090-6778 © 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.

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TABLE I

NOMENCLATURE

Several technologies have been long utilizing the con-82

cept of polarization, namely in optical fiber communica-83

tions [13], satellite communications [14] as well as in radar84

applications [15], however it has recently started to gain85

some interest in wireless communications as presented by86

Shafi et al. in [16] and the references therein. For instance,87

the polarization effect was considered in the development88

of various technologies, such as for the 2D and 3D spatial89

channel model (SCM) for the third-generation partnership90

project (3GPP) and 3GPP2 model [17], [18], the indoor91

communications operating at the 60 GHz band [19] as well92

as for the mmWave channel models presented in [4], [20].93

Moreover, several studies focused mainly on the polarization94

effect in DP-based MIMO systems [6], [21].95

The effect of polarization on spatial multiplexing was96

investigated by Bolcskei et al. in [22], where a two-input97

two-output (TITO) (2 × 2)-element DP system was presented98

and a closed-form average BER (ABER) expression was99

formulated. The results showed that even with high spatial100

fading correlation, a DP implementation is capable of attain-101

ing enhanced multiplexing gain. This was later extended by102

Nabar et al. in [23] to include both transmit diversity as well103

as spatial multiplexing. In [24], Anreddy and Ingram suggested104

that the BER performance of antenna selection with DP-AE105

outperforms that with UP-AE.106

Polarization shift keying (POLSK) was first theorized by107

Benedetto and Poggiolini in [13] for optical communications108

and was later applied to wireless communications systems109

by Dhanasekaran in [25]. Here, information is transmitted by110

switching on and off the V and H components of a DP-AE.111

This approach was later combined with spatial modulation112

(SM) [26]–[28] by Zafari et al. in the DP-SM scheme [29],113

which has the advantage of using a single transmit RF chain114

and multiple DP-AEs. More specifically, DP-SM switches on a115

single DP-AE and activates one of its orthogonal components116

(V or H) for transmitting a single complex symbol. This117

allows DP-SM to implicitly convey the implicit information 118

of the activated component index. It was shown in [30] that 119

the DP-SM system outperforms the conventional UP-based 120

SM scheme, while doubling the number of transmit antennas. 121

DP-SM was later investigated again by Zafari et al. in [30] 122

over correlated Rayleigh and Rician fading channels. In [31], 123

Zhang et al. extended the philosophy of using a single RF 124

chain with DP-AEs in the polarization shift keying (PolarSK) 125

scheme. PolarSK employs a single transmit RF chain with an 126

improved design for transmitting a single PolarSK symbol, 127

which is a combination of complex symbols as well as a 128

specific polarization angle. Furthermore, Park and Clerckx 129

proposed utilizing DP-AEs for multi-user transmission in a 130

massive MIMO structure [32], where by employing DP-AEs 131

the number of transmitting ports is doubled. 132

In this treatise, we propose a novel polarization modulation 133

(PM) scheme, which invokes the polarization characteristics 134

of DP-AEs for transmitting an extra information over the 135

polarization dimension in addition to a pair of complex 136

symbols, while maintaining a reduced number of RF chains. 137

In particular, at each DP-AE, the PM system selects one out 138

of multiple polarization configurations that is jointly applied 139

to the V and H components for shaping the transmitted 140

signal’s polarization pattern. The polarization configurations 141

applied are predefined at the transmitter and are known to 142

the receiver. Accordingly, the transmitted signal conveys both 143

the complex symbols and the polarization pattern applied. 144

In fact, each polarization pattern can shape the signal car- 145

rying the complex symbols differently and hence, we refer to 146

the polarization patterns as the space-polarization dispersion 147

matrices. 148

In PM, a space-polarization dispersion matrix disperses a 149

pair of complex symbols over the space and polarization 150

dimensions, in a similar manner to space-time dispersion 151

matrices [33], [34]. Space-polarization dispersion matrices 152

are represented by (2 × 2)-element diagonal matrices, since 153

they configure two orthogonal components (V and H) over 154

a single time slot. Having used a matrix representation of 155

the polarization configurations, space-polarization dispersion 156

matrices can be generated based on a fixed criterion [35]–[37] 157

for optimizing the performance of the PM system [38]–[40]. 158

Against this background, the novel contributions of this treatise 159

are as follows: 160

1) We propose the novel concept of polarization modula- 161

tion, which invokes the polarization characteristics of 162

DP-AEs (i.e. magnitude and angle) for achieving an 163

improved transmission rate as well as an enhanced BER 164

performance. 165

2) We formulate a closed-form generalized ABER expres- 166

sion of the PM system with Rayleigh fading as well as 167

with Rician fading channels. 168

3) We characterize both the discrete-input continuous- 169

output memoryless channel (DCMC) capacity and the 170

continuous-input continuous-output memoryless chan- 171

nel (CCMC) capacity of our PM system. Furthermore, 172

we provide the upper and lower bounds of CCMC 173

capacity. 174

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TABLE II

LIST OF SYMBOLS

4) We conceive an efficient space-polarization matrix opti-175

mization technique for optimizing the PM constellation.176

To be specific, the optimized matrix set is generated177

based on the random search method, which aims for178

minimizing the maximum achievable ABER as well as179

maximizing the DCMC capacity.180

The remainder of the treatise is organized as follows.181

In Section II, we introduce our PM system, which182

includes both the transmission and detection mechanisms.183

Fig. 1. Dual-polarized antenna element with an elliptic polarization state.

Next, a DCMC and CCMC achievable capacities are pre- 184

sented and the lower and upper bounds of the CCMC capac- 185

ity are developed in Section III. In Section IV, we derive 186

the closed-form ABER expression. Then, the improved 187

PM-constellation generation technique is introduced in 188

Section V. Section VI provides the numerical results, while 189

the conclusions are drawn in Section VII. 190

II. PROPOSED POLARIZATION MODULATION 191

In this contribution we consider an (Nt ×Nr)-element 192

MIMO system with Nt/2 being the number of DP-AEs 193

employed at the transmitter and Nr/2 the number of DP-AEs 194

employed at the receiver. The transmitter is equipped with N tc 195

RF-chains, each of which is connected to a single DP-AE. 196

A single DP-AE constitutes both a vertical and a horizontal 197

component and hence, the number of transmit antennas Nt is 198

twice that of N tc . In what follows, we present our PM transmis- 199

sion scheme, which is capable of conveying information bits 200

by invoking the polarization characteristics of multi-polarized 201

AEs. This approach opens a new dimension for implicit 202

information transfer, while maintaining traditional amplitude- 203

phase modulation (APM) complex symbol communication. 204

A. The Concept of PM 205

Let us now consider the DP-AE depicted in Figure 1, which 206

constitutes a pair of co-located horizontally-and vertically- 207

polarized ports. The trace of the EM field polarization ellipse 208

emitted by the DP-AE is shaped by the conjoint characteristics 209

of its vertical and horizontal components, which could form 210

a linear, circular and more generally an elliptic polarization, 211

as shown Figure 1. The resultant radio wave ellipse can be 212

represented both by the axial ratio (AR) and by the tilt angle τ . 213

The AR represents the major axis (OA) to minor axis (OB) 214

ratio defined as 215

AR =OA

OB, (1) 216

as seen in Figure 1. Furthermore, the major and minor axes 217

of Equation (1) of the polarization ellipse can be expressed 218

as [12], [41] 219

OA=

√12

[E2

x+E2y +√E4

x+E4y +2E2

xE2y cos (2δL)

], (2) 220

and 221

OB=

√12

[E2

x+E2y−√E4

x+E4y +2E2

xE2y cos (2δL)

], (3) 222

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respectively, where (Ex, Ey) define the EM field vector223

components with a time-phase difference angle δL =δx − δy.224

Likewise, the angle τ , which describes the tilt angle with225

respect to the principal axis, as depicted in Figure 1 is given by226

τ =12

arctan(

2ExEy

E2x − E2

y

cos (δL)). (4)227

In this regard, we adjust both the AR and τ components of228

DP-AEs in order to produce Q distinct polarization traces (or229

shapes), which can be used for implicitly transferring log2 (Q)230

bits over each DP-AE, while still transmitting a pair of APM231

complex symbols at the V and H components.232

It is worth mentioning here that Q is always an integer233

power of 2, which is comparable to the size of a conventional234

APM constellation L. Hence, when a single polarization235

shape is applied (e.g. Q = 1 with all vertical, horizontal or236

slant), no information will be transmitted over the polarization237

domain. Furthermore, the maximum value of Q is not fixed238

and can be adjusted according to the system requirements.239

However, choosing the number of polarization shapes depends240

mainly on the antenna specifications, which is represented by241

its AR and tilt angle ranges.242

To further illustrate the mechanism of our proposed243

PM scheme, let us consider the PM constellation depicted244

in Figure 2, which is formed of a 4PSK constellation as well245

as a Q = 4 polarization states. Given that a pair of QPSK246

symbols can be transmitted at the V and H components of247

the DP-AE, which conveys a total of 4 bits per channel use248

(bpcu), an additional log2 (Q) =2 bits can be transmitted249

by switching between the four distinct polarization traces of250

Figure 2. This allows the system to apply a dual transmission251

mechanism, using the conventional APM symbols as well as252

the polarization information. In what follows, we detail our253

PM encoding scheme at the transmitter.254

B. PM System Model255

The PM transmitter block diagram is depicted in Figure 3.256

The B-sized input bit stream of Figure 3 is divided into N tc257

parallel BPM -sized sub-streams, where the ntc-th sub-stream258

at the ntc-th RF chain of BPM bits is fed into the nt

c-th PM259

encoder for generating the ntc-th PM symbol transmitted at the260

ntc-th DP-AE, given that nt

c =1, . . . , N tc . The PM encoder of261

Figure 3 will be detailed further in Section II-E. In a nutshell,262

the BPM -sized sub-stream constitutes the pair of information263


denoting the polarization information as well as the APM 264

symbols information. More explicitly, the first log2 (Q) bits 265

of BPM are used to select one out of Q polarization config- 266

urations, which configures the V and H components of the 267

ntc-th DP-AE, while the remaining 2 log2 (L) bits are invoked 268

to modulate a pair of L-PSK symbols. The total number of bits 269

transmitted by a PM system equipped with N tc PM encoders 270

is given by 271

B = N tc · log2

(L2Q). (bits) (5) 272

Now, the symbol S(ntc)∈ C2×1 transmitted at the nt

c-th 273

DP-AE can be expressed as 274

S(ntc) = A

(ntc)

q X(nt

c)lv ,lh

, (6) 275

where A(nt

c)q =

[A

ntc

q,v 00 A

ntc

q,h

]∈ C2×2 denotes the polarization 276

shaping matrix, which configures the ntc-th DP-AE polariza- 277

tion using the q-th polarization information selected from 278

{Aq}Q1 . Moreover, Aq,v =aq,ve

jθq,v and Aq,h =aq,hejθq,h 279

represent the V and the H polarization information, which 280

are associated with moduli |aq,v| and |aq,h| as well as argu- 281

ments θq,v and θq,h, respectively.1 The polarization matri- 282

ces {Aq}Q1 are constructed under the power constraint of 283

trace(AqA

Hq

)=1. Furthermore, X

(ntc)

lv ,lh=[x

ntc

l,v xnt

c

l,h

]T∈ 284

C2×1 is the APM symbol vector, where xntc

l,v and xntc

l,h represent 285

the pair of L-PSK symbols transmitted at the (2ntc − 1)-th V 286

component and at the (2ntc)-th H component of the nt

c-th DP- 287

AE, respectively, given that l =1, . . . , L. Hence, the ntc-th PM 288

symbol vector can be expressed as 289

S(ntc) =

[A

ntc

q,v 00 A

ntc

q,h

] [x

ntc

l,v

xnt

c

l,h

]=

[A

ntc

q,v · xntc

l,v

Ant

c

q,h · xntc

l,h

], (7) 290

while the (Nt × 1)-element PM symbol vector S has the 291

following form: 292

S =[S(1) · · · S(Nt

c)]T. (8) 293

1��aq,h

�� and |aq,v | are equivalent to Ex and Ey in Equations (1-4),respectively, while θq,h and θq,v characterize δx and δy of the differenceangle δL presented in Section II-A.

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Observe in (7) that an additional means of information294

transmission is introduced by adjusting the joint configurations295

of the moduli and arguments of the diagonal vector of A(nt

c)q .296

Given that the coefficients of A(nt

c)q constitute the polarization297

information, {Aq}q1 can be constructed using one of the three298

following modes:299

• The AR mode, where the polarization information is300

explicitly transmitted over the AR component, which is301

represented by the moduli of Aqdenoted by |aq,v| and302

|aq,h|. In the AR mode, no information is conveyed303

over the tilt component (e.g. θq,v =θv and θq,h =θh304

∀ {Aq}Qq=1), where θv and θh are constant angle values.305

• The Tilt mode, where the polarization information is306

explicitly transmitted over the tilt component designated307

by the arguments θq,v and θq,h of Aq , while having308

static moduli (e.g. |aq,v| =av and |aq,h| =ah ∀ {Aq}Qq=1),309

where av and ah are constant real numbers.310

• The tilted-AR mode, where information is conveyed over311

an amalgam of both the tilt and the AR components,312

which is characterized by the general representation of313

A(nt

c)q in (7). In this mode, every polarization shaping314

matrix in {Aq}Qq=1 has a unique signature constituted by315

a specific combination of AR (i.e. |aq,v| and |aq,h|) and316

tilt angles (i.e. θq,v and θq,h).317

The PM system may also reduce to the conventional spatial318

multiplexing (MUX) system [22], [42], when no information319

is transmitted over the polarization dimension (e.g. Q = 1).320

In this treatise, we refer to a PM system as321

PM(AR/Tilt/TAR/MUX, N tc ,

Nr

2 , Q, L − QAM/PSK)322

and to the PM encoder as PM(AR/Tilt/TAR/MUX, Q,323

L − QAM/PSK), where AR, Tilt, TAR and MUX represent324

the AR modulation, tilt modulation, tilted-AR modulation as325

well as the basic QAM/PSK multiplexing modulation without326

any polarization,2 respectively.327

It should be also noted that by using the Tilt mode, where328

the polarization information is explicitly transmitted over the329

tilt component the system converges to the PolarSK system330

proposed in [31], namely when associated with N tc =1 and331

the PSK modulation. Hence, PolarSK is a special case of our332

PM scheme.333

Now, having generated the space-polarization block, the PM334

symbol vector S of (8) is transmitted over a frequency-flat and335

slow fading channel and received by the Nr

2 DP-AEs at the336

receiver. In general, the vector-based system model can be337

expressed as338

Y = HS + V , (9)339

where H∈ CNr×Nt denotes the channel matrix and340

V ∈ CNr×1 is the zero-mean additive white Gaussian341

noise (AWGN) vector, each element of which obeys342

CN (0, N0), given that N0 is the noise power.343

2In the case of using MUX, no information is transmitted over the

polarization domain, that is Q = 1, A(nt

c)q =I2 and log2 (Q) = 0 bits.

C. Channel Model 344

In this regards, H describes the DP channel matrix that 345

combines both the spatial separations and the XPD depolar- 346

ization effects and it is defined as [5], [6], [43] 347

H =

⎡⎢⎣

H1,1 · · · H1,Ntc

... Hnr/2,ntc

...HNr/2,1 · · · HNr/2,Nt

c

⎤⎥⎦ , (10) 348

where Hnr/2,ntc∈ C2×2 designates the TITO channel matrix 349

between the ntc-th and nr/2-th transmit and receive DP-AEs, 350

respectively. In particular, each TITO channel model can be 351

expressed as 352

Hnr/2,ntc

=

[hvv

nr/2,ntc

√Xhvhnr/2,nt

c√Xhhvnr/2,nt

chhh

nr/2,ntc

], (11) 353

where X denotes the XPD, which is a combination of the 354

cross-polar ratio (XPR) and the cross-polar isolation (XPI) as 355

defined in [6]. More specifically, the X parameter indicates the 356

cross-attenuation between the co-polarized channels (vv, hh) 357

and the cross-polarized channels (hv, vh). XPD is defined as 358

the ratio of the power of co-polarized channels to the power 359

of cross-polarized channels over V and H, expressed as [44] 360

ϕ−1v = E

[∣∣hvvi,j

∣∣2] /E [∣∣∣hvhi,j

∣∣∣2] , (12) 361

ϕ−1h = E

[∣∣hhhi,j

∣∣2] /E [∣∣∣hhvi,j

∣∣∣2] , (13) 362

respectively, where hvh/hvi,j denotes the channel fading 363

coefficient including the cross-attenuation effect, 364

E[∣∣hvv

i,j

∣∣2] =E[∣∣hhh

i,j

∣∣2] =1, E

[∣∣∣hvhi,j

∣∣∣2] =ϕv and 365

E

[∣∣∣hhvi,j

∣∣∣2] =ϕh. By assuming equal cross-attenuation [22] 366

(e.g. ϕv = ϕh=ϕ and 0 ≤ ϕ ≤ 1), the XPD parameter can be 367

expressed as X =ϕ. In what follows, we express the inverse 368

of the XPD in dBs as X−1dB =−10 logX dB. 369

To expound a little further on the channel model, the SISO 370

channel presented in (11) can be defined as 371

Hnr/2,ntc

= H � χ, (14) 372

where χ =[

1√X√X 1

], � denotes the Hadamard element- 373

by-element product and H represents the UP-based channel, 374

which can be defined as 375

H =

√K

K + 1HLOS +

√1

K + 1HNLOS, (15) 376

and hence 377

Hnr/2,ntc=

√K

K + 1χ � HLOS +

√1

K + 1χ � HNLOS, 378

(16) 379

given that K is the K-Rician factor, HLOS is the LOS 380

channel component and HNLOS is the NLOS Rayleigh fading 381

channel. 382

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Fig. 4. PM encoder block diagram.

D. Detection383

Having generated the PM symbol vector S, we now intro-384

duce the ML detector of our PM scheme. In an uncoded385

scenario, the PM detector aims to detect both the APM386

symbols as well as the polarization information of the transmit387

DP-AEs, where both {Aq}Q1 and {xl}L1 denoting the PM388

constellation S are available at the receiver.389

The ML detector’s main function is to maximize the a390

posteriori probability by invoking the conditional probability391

of receiving Y given that Si is transmitted defined by [45]392

p (Y |Si ) =1

(πN0)Nr

exp

(−‖Y − HSi‖2

N0

), (17)393

where Si∈ S represents the transmitted symbol vector under394

the assumption that all symbols in set S are equi-probable395

with p (Si) =1/2B ∀Si ∈ S. Hence, the ML detector may be396

formulated as397 ⟨q, l⟩

= arg min∀q,l

‖Y − HSi‖2 (18)398

= arg min∀q,l

‖Y − HAiXi‖2, (19)399

= arg min∀q,l

∥∥∥∥∥∥Y −Nt

c∑nt

c=1

HntcA(k)

q X(k)l

∥∥∥∥∥∥2

, (20)400

with Hntc∈ CNr×2 being the nt

c-th sub-channel between the401

ntc-th DP-AE and the Nr/2 receive AE, which denotes the402

ntc and nt

c + 1 column vectors of H . Furthermore, q and l403

denotes the estimated values of q and l, which designate the404

selected sets of q and l information, respectively.405

E. Practical Considerations406

In this section, we present a discussion on the feasibility407

of the PM system in practical implementations, namely in the408

context of the PM encoder design as well as of its hardware409

considerations. In order to invoke the polarization character-410

istics of a DP-AE, a phase-shifter and a power amplifier are411

required at its front-end. However, more complications may412

arise in the construction of the transmitter if maintaining a dual413

stream transmission per DP-AE were required. For instance,414

a straightforward approach is to implement two distinct RF415

chains; one for the V port and the other for the H port of each 416

DP-AE, and hence a total of(2Nt

2

)RF chains are required. 417

1) PM Encoder Design: In order to retain a dual data 418

stream transmission with a reduced RF-chain implementa- 419

tion, we propose the PM encoder architecture of Figure 4. 420

In this figure, the BPM input bits are divided into three 421

parts for constructing the PM symbol vector. More specif- 422

ically, the first part is used to select the q-th phase-shifter 423

combination ∠Aq =〈θq,v, θq,h〉, while the second part is 424

used to generate the phases of the APM symbols pair ∠L − 425

APM =〈φl,v, φl,h〉, as shown in Figure 4. A multiplier is 426

employed to combine both phases and generate the ntc-th PM 427

symbol’s phase ∠S(ntc) =〈φl,v + θq,v, θq,h + φl,h〉. Further- 428

more, the third part is used to produce the (ql)-th power 429

arrangement 〈|xl,vaq,v| , |xq,hal,h|〉, which configures the vari- 430

able power amplifiers to match the (ql)-th PM symbol’s 431

moduli, as portrayed in Figure 4. Observe in Figure 4 that by 432

entirely relying on phase-based modulation schemes, the two 433

variable gain power amplifiers can be replaced with a single 434

power amplifier connected at the front-end of the encoder, 435

which improves the encoder’s power efficiency. This can be 436

achieved with the aid of reconfigurable antennas, which are 437

capable of continuously tuning both the AR and the tilt angle 438

of the transmitted signal [46]. In what follows, we consider 439

the PM encoder of Figure 4, which produces a pair of APM 440

symbols amalgamated with the polarization information of 441

the DP-AE. 442

2) Hardware Considerations: The PM encoder design 443

requires the switching and DP-AE controlling units presented 444

in Figure 4 for the sake of maintaining a dual-stream trans- 445

mission, which increases the hardware complexity of the 446

transmitter. This is one of the noticeable drawbacks of the PM 447

encoder design, when compared to conventional RF implemen- 448

tations. However, by comparing the architecture of a single 449

switching-aided RF-chain of Figure 4 to a pair of end-to-end 450

RF chains, which are required to operate a couple of AEs 451

(e.g. two DP-AE ports), the hardware requirements become 452

less demanding. For instance, it has been shown in [47] 453

that the most expensive component (in terms of cost and 454

power consumption) in switch-aided transmitters, comparable 455

to our PM design, is the RF chain (see [48] for details). This 456

excludes the additional switching modules, serial-to-parallel 457

(S/P) converters and the RF switches of our PM encoder. 458

Nonetheless, the practical implementations of the PM system 459

require further investigation, albeit the evident cost-power 460

consumption and complexity design trade-off. 461

We note here that the design of Figure 4 may be relaxed 462

by transmitting a single APM symbol rather than two symbols 463

over the DP-AE ports. However, this would reduce the achiev- 464

able throughput B of Equation (5) to (N tc · log2 (LQ)) bits. 465

The implementation of DP-AEs using the above-mentioned 466

architecture is worthwhile investigating, hence in what fol- 467

lows we characterize both the capacity as well as the BER 468

performance of the PM system. 469

III. PM SYSTEM CAPACITY 470

In this section, we present both the DCMC capacity and 471

the ergodic CCMC capacity of our PM system. Furthermore, 472

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we formulate the upper and lower bounds of the ergodic473

CCMC capacity.474

A. DCMC Capacity475

The DCMC capacity of our PM system, which designates476

the mutual information expressing the number of error-free477

bits that can be decoded at the PM receiver, can be formulated478

as [49]479

CDCMC480

= maxp(S)

I (S; Y )481

= max{p(S)}∀q,l

∑ +∞∫−∞

· · ·+∞∫

−∞482

p (Y |Si) p (Si) log2

[p (Y |Si)∑

∀S∈S p (Y |S i) p (S i)

]dY ,483

(21)484

which can be maximized by using equi-probable p (Si).485

Next, by relying on the system’s conditional probability of486

Equation (17), the DCMC capacity can be now formulated487

as [49]488

CDCMC =B− εb∑q,l

E

⎡⎣log2

⎧⎨⎩∑

q,l

exp(ψ) |Si

⎫⎬⎭⎤⎦ , (22)489

where εb = 1(2B) and ψ is given as490

ψ = −‖H (Si − S i) + V ‖2 + ‖V ‖2 , (23)491

with S i being the transmitted symbol vector having⟨q, l⟩

492

indices. Unfortunately, there is no closed-form formulation493

available for Equation (22) and hence, we rely on numerical494

averaging procedures for evaluating the DCMC capacity.495

B. Ergodic CCMC Capacity496

On the other hand, the ergodic CCMC capacity of a MIMO497

system including our PM system is provided for maximizing498

the mutual information in a MIMO channel, which can be499

denoted as the maximum number of bits in an error-free500

continuous transmission and it is defined as [50]501

CCCMC = maxp(S)

H (Y ) −H (Y |S ) , (24)502

where H (Y ) and H (Y |S ) denote the destination entropy503

and the entropy of Y given S, respectively, which can be504

written as505

CCCMC = E

{log2

∣∣∣∣INr +ρ

Nt

(HHH

)∣∣∣∣}. (25)506

C. Ergodic Capacity Bounds507

In order to clearly show the effect of XPD on the achievable508

capacity of the PM system, in what follows we examine the509

bounds of CCCMC of (25) at the ultimate minimum XPD510

(i.e. X−1dB → 0) and the ultimate maximum XPD (X−1

dB → ∞),511

given K = 0.512

At X−1dB → 0: The XPD provided in Equation (11) attains its 513

maximum (X=1) and the system transforms to a conventional 514

UP-based MIMO system. Hence, closed-form of Equation (25) 515

at X=1 can be expressed as [51] 516

CX−1dB →0≥μ log2

⎡⎣1+

ρ

Ntexp

⎛⎝ 1μ

μ∑j=1

K−j∑p=1

1p−γ⎞⎠⎤⎦, (26) 517

given that μ=min (Nt,Nr), K =max (Nt,Nr) and 518

γ≈0.577215 is Euler’s constant. This can be obtained 519

by relying on 520

E

{ln∣∣∣∣ 1Nt

(HHH

)∣∣∣∣}

=Nr∑j=1

E {ln Ωj} −Nr lnNt, (27) 521

given that 522

E {ln Ωj} = ψ (Nt − j − 1) =K−j−1∑

p=1

1p− γ, (28) 523

where Ωj∼χ22(Nt−j+1). 524

Here, CX−1dB →0 represents the upper bound of the capac- 525

ity CCCMC , since no cross polarization attenuation exists 526

between the V and H components, and hence no degradation 527

in the achievable capacity is incurred. 528

At X−1dB → ∞: The cross V/H channels attenuation of (11) 529

becomes infinitesimally low (i.e.√X =0) and the row vectors 530

hvnr/2 and hh

nr/2 of H in (10) denoting the V and H receive 531

AE channels at the nr/2-th received DP-AE, respectively, are 532

then expressed as 533[hv

nr/2

hhnr/2

]534

=

[· · · hvv

nr/2,ntc

0 hvvnr/2,nt

c+1 0 · · ·· · · 0 hhh

nr/2,ntc

0 hhhnr/2,nt

c· · ·

]. 535

(29) 536

Observe in (29) that the resultant power of∣∣∣hv

nr/2

∣∣∣2 reduces 537

by half, which transforms the Chi-squared variable Ωj of (27) 538

into Ω′j∼ χ2

2(Nt−j2 +1), where E

{ln Ω′

j

}=ψ(

Nt−j2 − 1

). 539

Hence, the ergodic capacity reduces to 540

CX−1dB →∞ ≥ μ log2

⎡⎣1 +

ρ

Ntexp

⎛⎝ 1μ

μ∑j=1

K−j2∑

p=1

1p− γ

⎞⎠⎤⎦ . 541

(30) 542

The capacity CX−1dB →∞ of (30) denotes the lower bound 543

of the achievable capacity given a total V/H communication 544

blockage. Therefore, the CCMC capacity at any XPD level is 545

bounded by CX−1dB

→0 and CX−1dB

→∞ as 546

CX−1dB →∞ ≤ CX−1

dB≤ CX−1

dB →0. (31) 547

It is clearly seen in (31) that as the XPD attenuation 548

increases towards infinity the achievable capacity CX−1dB

549

decreases towards the lower bound (30). However, as the 550

XPD attenuation approaches zero the achievable capacity 551

IEEE P

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CX−1dB

approaches its maximum level, which is equivalent to552

a (Nt ×Nr)-element3 UP-based system.553

It is worth noting that the DCMC capacity as seen554

in Equation (22) is affected by the design of the set of555

space-polarization dispersion matrices {Aq}Qq . However, the556

ergodic capacity provided in Equation (24) is only restricted557

by the transmit power, bandwidth as well as the XPD level.558

IV. ABER ANALYSIS559

The average BER for the PM system is generally formulated560

using the general MIMO upper-bounding technique given561

by [52]562

BER =∑q=1

∑q=1

∑l=1

∑l=1

Dh

(q, l, q, l

)log2 (B)

P(S → S

), (32)563

where Dh

(q, l, q, l

)denotes the hamming distance between564

the bit-mapping of S and S and P(S→S

)is the average565

pairwise error probability (APEP). The APEP in fact is the566

average probability E

{P(S→ S |H

)}, which determines567

the probability that a PM symbol S is erroneously detected as568

S given H and can be expressed as [52], [53]569

P(S → S |H

)= P

(∥∥H (S − S

)+ V

∥∥ < ∥∥V ∥∥)570

= Q

⎛⎝√

‖H�‖2

2N0

⎞⎠ , (33)571

where � = S − S and Q (·) denotes the Q-function defined572

in [54] as573

Q (x) =1π

π/2∫0

exp(− x2

2 sin2 θ

)dθ, (34)574

and subsequently the PEP representation of (33) can now be575

expressed as576

P(S → S |H

)=

1π

π/2∫0

exp(− γ

2 sin2 θ

)dθ, (35)577

Now, by averaging Equation (35) over [0,∞] the legitimate578

range of the random variable γ, the unconditional PEP can be579

formulated as [55]580

P(S → S

)=

1π

π/2∫0

Φγ

(− 1

2 sin2 θ

)dθ, (36)581

where Φ (·) denotes the moment-generating function (MGF)582

of γ.583

In case of implementing UP-AEs, where no cross attenua-584

tion exists between V and H (X−1dB =0 dB), our PM system585

reduces to an ordinary spatial multiplexing system, which can586

be evaluated based on Appendix B of [56]. However, when587

introducing DP-AEs, a new parameter X denoting the DP-AE588

3It should be equipped with double the number of DP-AEs (i.e.�2Nt

2× 2Nr

2

�-element).

polarization effects arises and hence should be considered for 589

the ABER formulation. 590

Let us consider �ntc

=S(ntc)−S

(ntc) the symbol difference 591

at the ntc-th transmit DP-AE, which can be expressed as 592

�ntc

=[�nt

c,v

�ntc,h

]=

⎡⎣(A

ntc

q,v · xntc

l,v −Ant

c

q,v · xntc

l,v

)(A

ntc

q,h · xntc

l,h −Ant

c

q,h· xnt

c

l,h

)⎤⎦ , (37) 593

where �ntc,v and �nt

c,h denote the symbol difference at 594

the vertical and horizontal components of the ntc-th trans- 595

mit DP-AE, respectively. Given α=‖H�‖2and using Equa- 596

tion (37), α can be rewritten as 597

α =

∥∥∥∥∥∥Nt

c∑nt

c=1

Nr∑nr=1

Hnr,ntc�nt

c

∥∥∥∥∥∥2

, (38) 598

α =

∥∥∥∥∥∥Nt

c∑nt

c=1

Nr2∑

nr2 =1

H nr2 ,nt

c�nt

c

∥∥∥∥∥∥2

, (39) 599

where H nr2 ,nt

cis the TITO sub-channel between the 600

ntc-th transmit DP-AE and the nr/2-th receive DP-AE defined 601

in (11). Hence, α appears in the following form 602

α =

∥∥∥∥∥∥Nt

c∑nt

c=1

Nr2∑

nr2 =1

[hvv

nr2 ,nt

c

√Xhvhnr2 ,nt

c√Xhhvnr2 ,nt

chhh

nr2 ,nt

c

][�ntc,v

�ntc,h

]∥∥∥∥∥∥2

. 603

(40) 604

Now, by using the norm representation of ‖AI×J‖2 = 605∑Ii=1

∑Jj=1 |ai,j |2, Equation (40) can be rewritten as [57] 606

α =Nt

c∑nt

c=1

⎛⎝ Nr

2∑nr2 =1

∣∣∣�ntc,vh

vvnr2 ,nt

c+√X�nt

c,hhvhnr2 ,nt

c

∣∣∣2 607

+

Nr2∑

nr2 =1

∣∣∣hhhnr2 ,nt

c�nt

c,h +√X�nt

c,vhhvnr2 ,nt

c

∣∣∣2⎞⎠ . (41) 608

Each element of the MIMO channel matrix H of (10) is 609

assumed to be an i.i.d random variable, and hence (41) can be 610

reformulated as 611

α =Nt

c∑nt

c=1

12

(∣∣�ntc,v

∣∣2 + X ∣∣�ntc,h

∣∣2)︸︷︷︸

Υv

ς21,Nr612

+Nt

c∑nt

c=1

12

(∣∣�ntc,h

∣∣2 + X ∣∣�ntc,v

∣∣2)︸︷︷︸

Υh

ς22,Nr, (42) 613

and 614

γ =1

2N0

(Υvς

21,Nr

+ Υhς22,Nr

), (43) 615

with ς2i,Nr∼χ2

Nrbeing a noncentral chi-squared random vari- 616

able (RV)4 with Nr degrees of freedom and noncentrality 617

parameter of K . 618

4In NLOS (i.e. K = 0) ς2i reduces to a Chi-squared distributed randomvariable.

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By substituting γ of (43) into (35), the PEP can be619

formulated as620

P(S→ S |H

)=

1π

π/2∫0

exp

(−((

Υvς21,Nr

)4N0 sin2 θ

+

(Υhς

22,Nr

)4N0 sin2 θ

))dθ,621

(44)622

and hence after averaging it over [0,∞], Equation (44) can be623

expressed as624

P(S → S

)=

1π

π/2∫0

ΦΥvς21,Nr

(1

4N0 sin2 θ

)625

·ΦΥhς22,Nr

(1

4N0 sin2 θ

)dθ. (45)626

A. Rayleigh Fading, K = 0627

In the case of considering a Rayleigh fading channel628

(e.g. K = 0), Equation (53) can be rewritten as [56]629

P(S → S

)=

1π

π/2∫0

L=2∏l=1

(sin2 (θ)

sin2 (θ) + cl

)Nr2

dθ, (46)630

where c1 = Υv

2N0, c2 = Υh

2N0and the MGF of the chi-squared631

RV ς2l is defined by632

Φaς2l(−s) = (1 + 2as)−

Nr2 . (47)633

The closed-form solution of (46) can be formulated using634

two approaches. Following the solution provided in Appen-635

dix 5A.9 in [58], the first closed-form solution of (46) can be636

expressed as637

P(S → S

)=

12

L=2∑l=1

Nr/2∑k=1

Jkl

[1 −√

clcl + 1

638

·k∑

j=0

(2jj

)1

[4 (1 + cl)]j

], (48)639

given that640

Jkl =

{d

Nr2

−k

dxNr2 −k

∏L=2n=1n �=l

(1

1+cnx

)Nr2}∣∣∣∣

x=− 1cl(

Nr

2 − k)!c

Nr2 −k

l

. (49)641

For the special case of using a single DP-AE receiver642

(e.g. Nr

2 =1), Equation (48) reduces to643

P(S → S

)644

=12

L=2∑l=1

[(1 −√

clcl + 1

)645

·L=2∏n=1n �=l

(2jj

)cl

[(cl − cn)]

]. (50)646

In the second approach, the closed-form of the PEP given 647

in (46) can be formulated as 648

P(S → S

)649

=12π

(c1c2)−Nr

2 · β(

12, Nr +

12

)650

·F1

(Nr +

12,Nr

2,Nr

2, Nr + 1;−c−1

1 ,−c−12

), (51) 651

which is detailed in Appendix A, where β (·, ·) denotes the 652

Beta function and F1 (α, β, β′, γ;x, y) the confluent hyperge- 653

ometric function of two variables (Equation (61)). 654

In the high SNR-regime (i.e. N0 � 1), Equation (51) can 655

be written as 656

P(S → S

)≤ 1

2π

(ΥvΥh

16N20

)−Nr2

β

(12, Nr +

12

), (52) 657

where F1

(Nr + 1

2 ,Nr

2 ,Nr

2 , Nr + 1; 0, 0)=1 at c1→∞ and 658

c2→∞. Hence, the achievable diversity gain defined by the 659

slope of P(S → S

)is equivalent to Nr. 660

Note here that Equation (46) simplifies to Equation (36) 661

when X−1dB =0 dB (i.e. Υv =Υh) and hence, Equation (46) 662

can be solved using ( [56], Equation (64)). Additionally, it can 663

be seen in (52) that the XPD level does not have any effect 664

on the achievable diversity order of the PM system. 665

B. Rician Fading, K > 0 666

When considering a Rician fading channel (e.g. K >0), 667

Equation (45) can be written as [59] 668

P(S → S

)669

=1π

π/2∫0

L=2∏l=1

(sin2 (θ)

sin2 (θ) + cl670

· exp(− Kcl

sin2 (θ) + cl

))Nr2

dθ, (53) 671

where the MGF of the noncentral chi-squared RV ς2l is defined 672

as [56] 673

Φaς2l(−s) = (1 + 2as)−

Nr2 exp

(−KNr

2· s

1 + 2as

). (54) 674

There is no closed-form of Equation (53) and hence, it can 675

be evaluated numerically. Note here that at K = 0 the problem 676

reduces to Equation (46). 677

However, by using the Q-function approximation proposed 678

in [60], the APEP of Equation (45) can be approximated as 679

P(S → S

)680

≈112

(ΦΥvς2

1,Nr

(1

4N0

)· ΦΥhς2

2,Nr

(1

4N0

))Nr2

681

+14

(ΦΥvς2

1,Nr

(1

3N0

)· ΦΥhς2

2,Nr

(1

3N0

))Nr2

, (55) 682

which is detailed in Appendix B. 683

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The PM system is comparable to a spatial multiplexing684

system, which suffers from a degraded performance in the685

presence of a LOS component, as a result of the correlation686

fading effect. To overcome this issue in a DP-based MIMO,687

we employ our PM system by relying on a single transmit688

DP-AE (N tc =1) at high XPDs, yielding E

[∣∣∣hvhi,j

∣∣∣2]�1 and689

E

[∣∣∣hhvi,j

∣∣∣2]�1.690

V. SPACE-POLARIZATION IMPROVED CONSTELLATION691

In this section, we introduce our PM improved-constellation692

generation procedure. Observe in Equation (6) that the polar-693

ization configuration matrix Aq disperses the PSK/QAM694

complex symbols of X l over the spatial and polarization695

dimensions at a single time slot, in a conceptually similar696

manner to space-time dispersion matrices [33], [34], [37]. This697

opens a new prospect for designing the polarization shape of698

PM constellations.699

In a nutshell, the polarization shaping matrices {Aq}Q1 may700

be randomly generated so that the performance of the system701

is improved. In this regard, the shaping matrices may be702

constructed so that the unconditional PEP of Equation (46) is703

minimized, while retaining the maximum achievable diversity704

order. Hence, the optimal set of Q unit polarization vectors705

Aopt can be constructed by conducting a Random Search (RS)706

that aims at minimizing the maximum PEP as707

Aopt = argAimin

{maxP

(S → S

)}, (56)708

which translates to709

Aopt =argAimax {min (c1c2)}=argAi

max {min (ΥvΥh)} ,710

(57)711

which can be rewritten as712

Aopt = max {min ‖�‖} . (58)713

It is worth emphasizing here that the construction of714

{Aq,h, Aq,v} designating the H and V configurations of715

{Aq}Q1 , respectively, should fall within the polarization shap-716

ing capabilities of the DP-AE, namely its AR range (1) and its717

Tilt angle range (4). Additionally, multiple transmit AEs are718

spaced sufficiently far apart in order to experience independent719

fading hence, random search is performed using a single720

transmit DP-AE, where the Aopt set produced is used at each721

DP-AE.722

In what follows we present the generation process of723

Aopt satisfying (58) using a TITO (2 × 2)-element system.724

We first generate a random set of (1 × 2)-element unit vectors725

denoting the diagonal vectors of the (2 × 2)-element matrix set726

Ai={Aq}Q1 . The vector set generated should obey the Rank727

Criterion (i.e. rank(��H) = 1 ∀ q, q ∈ Q) in order to guar-728

antee a normalized power space-polarization set. Next, we cal-729

culate the minimum Euclidean distance dmin={min ‖�‖}.730

The random search continues by repeating both steps, while731

retaining the Ai set having the maximum dmin. The algorithm732

presented above is summarized in Algorithm 1. Furthermore,733

an example is provided in Appendix C to ease understanding.734

Note that by obtaining the minimum distance 735

dmin=max{min ‖�‖} in (58) the PEP P (‖H (S−S) + V ‖ 736

<‖V ‖) of (33) is minimized, and hence the DCMC exponent 737

ψ = − ‖H(Si − S i) +V ‖2+‖V ‖2 of (23) is subsequently 738

minimized, which improves the achievable DCMC capacity. 739

Algorithm 1 Polarization Shaping Algorithmminimum distance: κ = 0initialize Aopt;

Start: (i = 1 :106 loops)Loop: Generate Q random (2 × 1)-element unit vectors

{aq}Q1

Ai = {Aq = diag2 (aq)}Q1

compute S, S and � ∀q, l1, l2if(rank

(��H)

= 1)

Compute OA, OB and τ using {Aq}Qq=1

if (OA, OB and τ doesn’t match the DP-AE range)GOTO Loop

else GOTO LoopCompute di

min =min {‖�‖}if(di

min >κ)Apply Aopt=Ai

GOTO LoopReturn Aopt

End

VI. SIMULATION RESULTS 740

In this section, we present our Monte Carlo simulation 741

results with a minimum of 106 bits per SNR value as well as 742

the theoretical analysis of our PM system. In our simulations 743

we assume perfect CSI at the receiver side for invoking the 744

ML optimum detector of Equation (18). Furthermore, multiple 745

DP-AEs are spaced sufficiently far apart in order to experience 746

independent fading. We choose the polarization shaping matrix 747

set {Aq}Qq=1 by selecting several AR and τ values based on 748

the discussion presented in Section II-A. Particularly, Table III 749

shows the main PM systems used in our simulations with 750

Q=4 as follows5: three AR systems (i.e. AR-1,…, AR-3), 751

two Tilt systems (i.e. Tilt-1, Tilt-2) and four TAR systems 752

(i.e. TAR-1,…, TAR-4). Additionally, all plots showing the 753

performance of PM-systems associated with the RS-aided 754

constellation presented in Section V are labeled as TAR-RS. 755

The TAR-RS system used below is presented in Appendix C. 756

Note here that the tuning capabilities of DP-AEs over the 757

AR and the tilt angle vary from one antenna to another. For 758

instance, the reconfigurable DP-AE presented in [46] utilizes 759

a maximum AR of 35 dB and a tilt angle spanning between 760

30◦ and 100◦. 761

A. Comparison Fairness 762

In this contribution we define fair comparison as follows: a 763

fair performance comparison between a DP-based system and 764

a UP-based system is attained by employing an equivalent 765

number of AEs in both systems. To expound a little further, 766

5Other systems with various Q configuration are used.

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TABLE III

AR AND TILT ANGLES OF VARIOUS PM SYSTEMS DESIGNED FOR PROVIDING Q = 4 SPACE-POLARIZATION

CONFIGURATIONS DENOTED BY�aq,h , aq,v) AND

�θq,h , θq,v), RESPECTIVELY

Fig. 5. DCMC capacity comparison between various PM systems attaining4 bpcu by relying on the AR, Tilt and TAR configurations with differentpolarization shapes at an XPD of X−1

dB =10 dB.

consider a PM system that is equipped with a single transmit767

DP-AE. This system would require a single RF chain for768

transmitting a single PM symbol, and hence it is comparable to769

a UP-based system having a single UP-AE. By increasing the770

number of UP-AEs to match the number of ports in a single771

DP-AE (e.g. use UP-AEs) an additional RF chain is required,772

which negates fairness.773

Furthermore, in MIMO implementations, AE spacing has774

to be on the order of ten wavelengths, in order to experience775

independent channel fading. In DP-based MIMOs, the V and H776

components of each DP-AE are separated over the polarization777

dimension, where Nt/2 AEs only require to be spaced far778

apart. However, adding Nt UP-AEs would require double the779

area of a DP-based system. In what follows, we refer to any780

simulated system as (M ×N), where M and N denote the781

number of transmit and receive AEs (DP or UP), respectively.782

B. DCMC Capacity783

Based on the unified capacity metric provided in Equa-784

tion (22), Figure 5 depicts the DCMC capacity curves of our785

PM system designed for achieving a normalized throughput 786

of 4 bpcu. Here, we employed (1 × 1) DP-AEs with various 787

PM configurations. More specifically, Figure 5 shows the 788

DCMC curves of the AR-1-3, Tilt-1-2 and TAR-1-3 systems 789

detailed in Table III as well as of TAR-RS and TAR-RS-1PSK, 790

where TAR-RS-1PSK is a symbol-free RS-based PM(TAR,1, 791

1, Q = 16, 1PSK)

system (i.e. polarization information only). 792

We also characterize the conventional (1 × 1) UP-AE-based 793

16QAM and 16PSK systems. It can be observed in Figure 5 794

that TAR-based PM systems outperform all the other PM 795

configurations, while the RS-based systems achieve the high- 796

est throughput. For instance, TAR-RS outperforms PolarSK 797

(i.e. Tilt-PM) by 2.8 dB and conventional 16QAM and 16PSK 798

by 3.7 dB and 6 dB, respectively. This verifies the discussion 799

presented in Section V, where constructing the optimal Aopt 800

under the constraint of maximizing dmin=max {min ‖�‖} 801

could further improve the achievable capacity of the PM 802

system. 803

In order to characterize the effect of the XPD on the PM sys- 804

tem, Figure 6 portrays the 3D surface of the achievable capac- 805

ity of a PM(TAR,1, 1, Q = 4, BPSK

)system with respect 806

to XPD and SNR. Furthermore, the achievable throughput at 807

X−1dB =0 dB is projected onto the (SNR, Capacity)-plane for 808

the sake of comparison. As seen in Figure 6, the achievable 809

throughput degrades as the XPD increases, which can be 810

clearly seen at high XPDs. To expound a little further, Figure 7 811

showcases the projected 3D surface of Figure 6 onto the (SNR, 812

Capacity)-plane between X−1dB =0 dB and X−1

dB =30 dB. It can 813

be seen from the figure that a maximum degradation of 3.5 dB 814

is observed in the DCMC capacity between X−1dB =0 dB 815

and X−1dB =30 dB. However, the degradation in the achiev- 816

able capacity becomes marginal at high XPDs, especially at 817

X−1dB >15 dB. 818

C. CCMC Capacity 819

To investigate the ergodic CCMC capacity of our PM 820

system, the capacities of three PM systems are illustrated by 821

the 3D surfaces drawn in Figure 8, namely for the (1 × 1), 822

(2 × 2) and (4 × 4) DP-AEs MIMO arrangements. One can 823

observe in Figure 8 that the CCMC capacity is affected both 824

by the transmission power as well as the XPD level. 825

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Fig. 6. A 3D representation of the DCMC capacity of aPM(TAR, 1, 1, Q = 4, BPSK) system with respect to SNR and XPD.

Fig. 7. The 2D projection of Figure 6 onto the (SNR, DCMC)-plane.

Figures 9(a)-(c) depict the 2D projection of the 3D surfaces826

of Figure 8 onto the (SNR, CCMC)-plane at an XPD of827

X−1dB =10 dB. The theoretical upper and lower bounds of828

equations (26) and (30), respectively, are also shown in each829

figure. Furthermore, the capacity of an equivalent number830

of UP-AEs is shown for the sake of comparison, where the831

capacity improvement of DP-based systems is shown addi-832

tionally by the red curve. It can be observed in Figure 9 that833

DP-AE implementations substantially boost the capacity of a834

MIMO system, achieving between 87.5% and 54% capacity835

improvement over an SNR range spanning between −10 dB836

and 40 dB, respectively, for all three systems considered.837

Moreover, we note that the DP-based capacity curves por-838

trayed in Figure 9 are confined within the upper and lower839

bounds described in Section III, which are separated 3 dB840

Fig. 8. A 3D representation of the ergodic CCMC capacity of three PMsystems in terms of SNR (dB) and XPD (dB), namely for the (1 × 1), (2 × 2)and (4 × 4) DP-AEs arrangements.

apart. In fact, the simulated curves (through Monte Carlo) of 841

the DP-based systems in Figures 9(a)-(c) and the lower bound 842

analysis (CX−1dB

→∞) precisely match at X−1dB =10 dB. 843

To examine the effect of the XPD on the achievable CCMC 844

capacity, let us assume that we project Figure 8 onto the (XPD, 845

CCMC)-plane at SNR of 12 dB, 16 dB and 20 dB, as portrayed 846

in Figure 10. The figure shows that as the XPD level increases 847

the ergodic capacity decreases, however it remains relatively 848

constant after a specific value of XPD, as for example at an 849

XPD of X−1dB =16 dB at SNR of 12 dB. Furthermore, one can 850

observe that at an even high XPD level, the maximum loss in 851

CCMC capacity is less than 1.4 bps/Hz. 852

The novel polarization modulation technique presented in 853

this paper constitutes a viable solution to significantly boost 854

the data transmission rate for future wireless systems. In what 855

follows, we present the BER performance of our PM system. 856

D. BER Simulation 857

In Figure 11 we compare the achievable BER performance 858

of (1 × 1) and (2 × 2) PM systems,6 which achieve a through- 859

put of 4 and 8 bpcu, respectively, at an XPD of X−1dB =10 dB 860

and K = 0. Moreover, the BER performance of their UP-AE 861

(1 × 1) and (2 × 2) counterparts 16PSK are included for 862

comparison,7 while the dashed curves represent the theoretical 863

6A (1 × 1)-DP-AE implementation is equivalent to a (2 × 2) UP system,since the V and H components transmit over separate polarization dimensions.

7We use the same number of AEs for both systems (i.e. DP-AE and UP-AE)in order to maintain fairness. The (1 × 1) DP-AE system for instance hastwo input ports, while the (1 × 1) UP-AE has a single input port, while bothrequire a single RF chain implementation. In case two UP-AEs are used tocompare with the (1 × 1) DP-AE system, two RF chain are required, whichleads to unfairness in the number of RF components as well as in the requiredtransmitted power.

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Fig. 9. The 2D projection of Figure 8 onto the (SNR, CCMC)-planeat XPD of X−1

dB =10 dB of the following DP/UP systems: a) (1 × 1);b) (2 × 2); c) (4 × 4), which further include the upper and lower capacitybounds. Furthermore, the capacity improvement is shown by the red curve.

upper bounds developed in Section IV. Figure 11(a) shows864

the performance of PM(AR,Q = 4,BPSK

), PM

(Tilt,Q =865

4,BPSK), PM

(TAR,Q = 4,BPSK

)and PM

(TAR,Q =866

4,BPSK)-RS8 systems associated with (1 × 1)-DP-AEs.867

8The RS here features the improved RS-aided constellation provided inSection V.

Fig. 10. A 2D projection of Figure 8(a) onto the (XPD, CCMC)-planeshowing the effect of XPD on the attainable CCMC capacity at SNR of 12 dB,16 dB and 20 dB.

Here, log2 (4)= 2 bits are used to activate one out of 868

Q = 4 space-polarization matrices, while the remaining 869

2 log2 (2) =2 bits are modulated to a pair of BPSK sym- 870

bols. The performance of the PM(TAR,Q = 16,1PSK

)- 871

RS system is also shown in Figure 11(a), where the whole 872

BPM =log2 (16)= 4 bits are used to switch between Q = 16 873

polarization shapes. It is shown in 11(a) that the PM system 874

outperforms the conventional (1 × 1)-DP-AE by 10 dB, 15 dB 875

and 19 dB at a BER of 10−5, when employing the AR, Tilt and 876

TAR configurations, respectively. Furthermore, the improved 877

constellation PM systems provide further BER enhancements 878

of 2 dB and 4 dB by using the PM(TAR,Q = 16,1PSK

)-RS 879

and PM(TAR,Q = 4,BPSK

)-RS systems, respectively. Note 880

here that the theoretical model presented in Section IV matches 881

perfectly with the Monte Carlo simulations. 882

In Figure 11(b), the BER performance of the 883

above-mentioned PM systems is shown with a (2 × 2)- 884

DP-AE implementation. As seen in the Figure, the PM 885

system outperforms the conventional (2 × 2)-element 886

multiplexing system by 0.5 dB, 5.4 dB 9.5 dB, when 887

employing the AR, Tilt and TAR configurations, and by 888

12.7 dB and 15 dB by using the PM(TAR,Q = 16,1PSK

)-RS 889

and PM(TAR,Q = 4,BPSK

)-RS systems, respectively. 890

In Figures 12(a)-(b), we show the performance of a 891

PM(TAR, Q = 4, BPSK)-RS system (i.e. TAR-RS) transmit- 892

ting over a Rician fading channel at 4 bpcu, while employing 893

a single transmit DP-AE at a high XPD of X−1dB =30 dB. 894

Furthermore, Figures 12(a)-(b) include both the exact and 895

approximated theoretical bounds presented in equations (53) 896

and (55), respectively. In particular, in Figure 12(a) we inves- 897

tigate the effect of the Rician factor on the performance of 898

the PM system at K = 0, 5, 10 and 15, when associated with 899

(1 × 1)-element implementation. We notice in Figure 12(a) 900

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Fig. 11. The BER performance of (1 × 1) and (2 × 2) DP-AE PM systemsassociated with Q = 4 and BPSK achieving a throughput of 4 and 8 bpcu at anXPD of X−1

dB =10 dB and K = 0 compared with their UP-AE counterparts.

that upon increasing K the BER performance of the PM901

system improves. As indicated by Equation (53), the ABER902

improves exponentially with the value of K . Furthermore,903

the exact theoretical model of (53) matches perfectly with904

the Monte Carlo simulations, while the approximate model905

of (55) is marginally shifted at K = 0 and K = 5, which906

perfectly overlaps at low BER values. On the other hand,907

Figure 12(b) shows the performance of the simulated PM908

system with a different number of receive DP-AEs, namely909

Nr/2=1, 2, 4 and 8 at K = 5. We notice in the figure that the910

approximate theoretical bound developed in Section IV tends911

to accurately match both the exact bound and the Monte Carlo912

simulations as the number of receive AEs increase.913

A comparison between multiple configurations of 4-level914

(Q = 4) PM systems with (1 × 2)-elements is illustrated in915

Figure 13, which all achieve a spectral efficiency of 4 bpcu at916

X−1dB =10 dB and K=0. To expound further, the PM systems917

under study are: AR-1-3, Tilt-1-2 and TAR-1-4 as well as918

TAR-RS, as detailed in Table III. As observed in Figure 13,919

the achievable performance of all systems spans over 35 dBs920

of SNR at a BER of 10−5. In all cases, it can be observed921

that TAR-based systems exhibit the best BER performance922

compared to AR-based and Tilt-based systems. This can be923

attributed to the multi-dimensional structure of TAR-based924

PM, where polarization information is dispersed over both the925

AR and the tilt contrary to other configurations (e.g. AR and926

Tilt) that exploit either of them.927

In Figure 14, we compare the BER performance of our PM928

system with its DP-AE-based counterparts. More specifically,929

we compare the BER performance of PM(TAR, 1, 1, Q =930

2, BPSK) with that of a DP-SM(1, 1, QPSK) system [30]931

as well as that of a PolarSK(Nr/2=1, Q = 2, BPSK)932

Fig. 12. The BER performance of a PM(TAR, Q = 4, BPSK)-RSsystem (TAR-RS) transmitting 4 bpcu over a Rician fading channel: a) with(1 × 1)-elements at K = 0, 5, 10 and 15; b) with Nr/2 =1, 2, 4 and 8 atK = 5 and X−1

dB =30 dB.

system [31], where each exhibits a transmission rate of 3 bpcu 933

over Rayleigh fading channel (i.e. K = 0) at an XPD of 934

X−1dB =10 dB. Figure 14 further shows the performance of 935

the improved-constellation PM(TAR, 1, 1, Q = 2, BPSK)-RS 936

and PM(TAR, 1, 1, Q = 8, 1PSK)-RS systems as well as the 937

performance of UP-AE-based SM and Quadrature SM (QSM) 938

systems associated with Nt =2 AEs. It can be observed from 939

Figure 14 that our PM system outperforms PolarSK, DP-SM 940

and the conventional SM by 2 dB, 1.2 dB, 22 dB, respectively. 941

To elaborate further on the effect of the level of XPD on 942

the BER performance, the BER performance of a (2 × 2)-DP- 943

AE PM system associated with an PM(TAR, Q = 2, BPSK) 944

encoder at different XPD levels is presented in Figure 15. 945

More specifically, we show the BER performance of the 946

system at an XPD spanning between X−1dB =0 dB and 947

X−1dB =30 dB with a step of 5 dB, where the theoretical bound- 948

aries are shown exclusively at X−1dB =0 dB and X−1

dB =30 dB. 949

Figure 15 demonstrates that the performance of the PM system 950

is directly affected by the XPD level, where it improves as the 951

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Fig. 13. The BER performance of a (1 × 2)-DP-AE PM systems associatedwith Q = 4 AR, Tilt and TAE configurations, which achieve a throughputof 4 bpcu at an XPD of X−1

dB =10 dB and K = 0.

Fig. 14. BER comparison of a static TAR-based PM system, two RS-aidedTAR-based PM systems, equivalent throughput DP-SM [30] and PolarSK [31]systems as well as UP-AE-based SM and QSM systems.

XPD decreases due to the increased polarization diversity gain.952

However, it can be seen in the figure that the performance953

is marginally affected when X−1dB ≥25 dB. Furthermore,954

the theoretical boundaries presented in Figure 15 confirms the955

precision of the XPD parameter provided in equations (46)956

and (51).957

It can be observed in Figures 11-15 that the theoretical958

boundaries provided in Section IV match the Monte Carlo959

Fig. 15. Performance comparison between TAR-based PM systems at anXPD spanning between X−1

dB =0 dB and X−1dB =30 dB.

simulations for all PM configurations, namely for the AR, 960

Tilt, TAR and MUX configurations over various antenna 961

arrangements. In what follows, we present our conclusion. 962

VII. CONCLUSION 963

In this treatise, we have introduced a novel modulation 964

technique referred to as the polarization modulation, which 965

invokes the polarization characteristics of a DP-AE for data 966

transmission. More specifically, a block of information in a PM 967

system is formed by dispersing a pair of PSK/QAM symbols 968

into the space- and polarization- dimension with the aid of 969

Q polarization shaping matrices {Aq}Qq=1. The polarization 970

shaping matrix may adjust the AR, Tilt or Tilted-AR of the 971

EM matrix with the aid of a single RF-chain per DP-AE. 972

The polarization shaping matrices can be selected empirically, 973

however we have proposed a special algorithm for generating 974

an improved-constellation tailored for the PM modulation. 975

Furthermore, we provided a theoretical analysis for the DCMC 976

and CCMC capacity as well as for the BER performance of the 977

PM system. It has been shown that by invoking the polarization 978

dimension, the ergodic capacity of a DP-based MIMO system 979

can be improved by 54% to 87.5% compared to UP-based 980

MIMO. Similarly, the DCMC capacity of our PM system was 981

improved by up to 6 dB in comparison to systems relying 982

on UP-AE. Furthermore, the simulation results indicated that 983

the gain achieved by our proposed PM system relying on 984

Q-state polarization levels spans between 10dB and 20dB 985

over UP-AE-based conventional systems. Our simulation also 986

showed that by utilizing the proposed improved-constellation 987

algorithm the DCMC capacity and BER performance of our 988

PM system have significantly improved. 989

APPENDIX A 990

The derivation of Equation (51) can be formulated by 991

substituting u =sin2 (θ) and dθ= du

2√

u(1−u)into Equation (46), 992

IEEE P

roof


yielding993

P(S → S

)=

1π

1∫0

(u

u+ c1

)−Nr2(

u

u+ c1

)−Nr2

994

du

2√u (1 − u)

du, (59)995

and996

P(S → S

)=

(c1c2)−Nr

2

2π

1∫0

uNr− 12 (1 − u)−

12997

(1 +

1c1u

)−Nr2(

1 +1c2u

)−Nr2

. (60)998

Now, by relying on the confluent hypergeometric function999

of two variables given as (Section 9.18 [61])1000

F1 (α, β, β′, γ;x, y) =Γ (c)

Γ (a) Γ (c− a)1001

1∫0

zα−1 (1 − z)γ−α−1 (1 − xz)−β (1 − yz)−β′dz, (61)1002

the closed-form expression of (60) can be expressed as shown1003

in Equation (51), where Γ (·) denotes the Gamma function.1004

APPENDIX B1005

Instead of using the Q-function defined in Equation (34),1006

we can simply use the approximation defined by [60] as1007

Q (x) ≈112e−

x22 +

14e−

2x23 . (62)1008

By plugging (62) into (33), we arrive at1009

P(S → S |H

)≈

112e−

γ2 +

14e−

2γ3 . (63)1010

Given Nr receive AEs, the SNR of the nr/2-th channel1011

denoting the channel received at the nr/2-th AE is given1012

by γnr2

= 12N0

{Υv + Υh}Nr=2, where {Υv + Υh}Nr=2 is1013

equivalent to Equation (42) with N tc =1 and Nr =2.1014

Now, the average of PEP can be expressed as1015

P(S → S

)1016

≈

∫ ∞

0

. . .

∫ ∞

0︸︷︷︸Nr/2

Nr/2∏nr2 =1

(exp(−γ

nr2

2

)fγ(γnr

2

)1017

+ exp(−2γnr

2

3

)fγ(γnr

2

))dγ1 · · ·dγNr

2. (64)1018

By using the definition of the MGF function in ( [56],1019

Equation (21)), the close-form expression of P(S → S

)can1020

be formulated as shown in Equation (55).1021

APPENDIX C 1022

Here, we provide an example of an RS-based PM(TAR, 1023

Q = 4, BPSK)

system using the technique presented in 1024

Section V, which is referred to as TAR-RS in Section VI. 1025

Consider a PM system that relies on a set of BPSK symbols 1026

X l ={−1,+1} and on a randomly generated set {Aq}Qq for 1027

data transmission, which can be formulated as follows: 1028

A1 1029

=[−0.331952 + 0.686751i 0

0 −0.631246 + 0.140389i

], 1030

(65) 1031

A2 1032

=[−0.853098 + 0.0743741i 0

0 0.0196869 + 0.516047i

], 1033

(66) 1034

A3 1035

=[−0.493946− 0.228332i 0

0 −0.797398 + 0.260841i

], 1036

(67) 1037

A4 1038

=[−0.160197− 0.557432i 0

0 −0.43818− 0.686735i

], 1039

(68) 1040

where q = 1,. . . ,Q = 4. By using Equations (1-4) These 1041

configurations can be translated to the following parameters: 1042

Eh = {0.762771, 0.856334, 0.544168, 0.579994} , (69) 1043

Ev = {0.646669, 0.516423, 0.838976, 0.814621} , (70) 1044

θh = {115.798, 175.017, −155.191, −106.034} , (71) 1045

θv = {167.461, 87.8153, 161.886, −122.54} , (72) 1046

and Finally, 1047

τ = {130.29, 121.088, 57.0821, 54.55} , (73) 1048

and 1049

ARdB = {42.1995, 38.6012, 27.1086, 52.4841} . (74) 1050

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Ibrahim A. Hemadeh (Member, IEEE) received1242

the B.Eng. degree(Hons.) in computer and commu-1243

nications engineering from the Islamic University of1244

Lebanon, Lebanon, in 2010, and the M.Sc. degree1245

(Hons.) in wireless communications and the Ph.D.1246

degree in electronics and electrical engineering from1247

The University of Southampton, U.K., in 2012 and1248

2017, respectively. In 2017, he joined the Southamp-1249

ton Next Generation Wireless Group, The Univer-1250

sity of Southampton, as a Post-Doctoral Researcher.1251

In 2018, he joined the 5G Innovation Centre (5GIC),1252

University of Surrey. He is currently working as a Staff Engineer in the1253

industry. His research interests include millimeter-wave communications,1254

multi-functional multiple input multiple output (MIMO), multi-dimensional1255

(time-space and frequency) transceiver designs, channel coding, and multi-user1256

MIMO.1257

Pei Xiao (Senior Member, IEEE) worked at1258

Newcastle University and Queen’s University1259

Belfast. He also held positions at Nokia Networks,1260

Finland. He is currently a Professor of wireless1261

communications with the Institute for Communi-1262

cation Systems, Home of 5G Innovation Centre1263

(5GIC), University of Surrey. He is the Technical1264

Manager of 5GIC, leading the research team in the1265

new physical layer work area, and coordinating/1266

supervising research activities across all the1267

work areas within 5GIC (www.surrey.ac.uk/5gic/1268

research). He has published extensively in the fields of communication theory,1269

RF and antenna design, signal processing for wireless communications. He is1270

an Inventor on more than ten recent 5GIC patents addressing bottleneck1271

problems in 5G systems.1272

Yasin Kabiri (Member, IEEE) received the M.Eng. 1273

degree (Hons.) in electronics and communication 1274

engineering from the University of Birmingham, 1275

Birmingham, U.K., in 2012, and the Ph.D. degree 1276

from the School of Electronic and Electrical Engi- 1277

neering, University of Birmingham, in 2015. During 1278

his Ph.D., he has developed an approach called 1279

injection matching theory which can be used for 1280

making small, wide band, and reconfigurable anten- 1281

nas with high efficiency. He was also a Research 1282

Fellow with the 5G Innovation Center, Guildford, 1283

U.K., with a focus on 5G antennas. He holds multiple patents in the field 1284

and has contributed in major grant applications. He is currently working 1285

as a principal RF and microwave engineer in industry section. His research 1286

interests include RF and microwave, phased array and beam steerable antenna, 1287

mmwave system, satellite communication, electrically small antenna, active 1288

antennas, and microwave filters. 1289

Lixia Xiao (Member, IEEE) received the B.E., M.E., 1290

and Ph.D. degrees from the University of Electronic 1291

Science and Technology of China (UESTC) in 2010, 1292

2013, and 2017, respectively. She is currently a 1293

Research Fellow with the Department of Electrical 1294

Electronic Engineering, University of Surrey. Her 1295

research is in the field of wireless communications 1296

and communication theory. In particular, she is very 1297

interested in signal detection and performance analy- 1298

sis of wireless communication systems. 1299

Vincent Fusco (Fellow, IEEE) is currently a Per- 1300

sonal Chair of high frequency electronic engineering 1301

with QUB. He has authored more than 500 scientific 1302

articles in major journals and referred international 1303

conferences, and 2 textbooks. He holds patents 1304

related to self-tracking antennas and has contributed 1305

invited articles and book chapters. His research focus 1306

on advanced microwave through millimetre wave 1307

wireless. His current research interests include phys- 1308

ical layer secure active antenna techniques. In 2012, 1309

he was awarded the IET Senior Achievement Award, 1310

the Mountbatten Medal. 1311

Rahim Tafazolli (Senior Member, IEEE) is 1312

currently a Professor of mobile and personal 1313

communications and the Director of the Institute 1314

of Communication Systems, 5G Innovation Centre, 1315

University of Surrey. He has been active in research 1316

for more than 20 years and published more than 1317

500 research articles. In 2018, he was appointed as 1318

a Regius Professor in electronic engineering for the 1319

recognition of his exceptional contributions to digital 1320

communications technologies more than the past 1321

30 years. He is a fellow of IET and Wireless World 1322

Research Forum. He served as the Chairman for EU Expert Group on Mobile 1323

Platform (e-mobility SRA) and Post-IP Working Group in e-mobility, and the 1324

past Chairman for WG3of WWRF. He has been a technical advisor to many 1325

mobile companies. He has lectured, chaired, and been invited as a keynote 1326

speaker to a number of IEE and IEEE workshops and conferences. He is 1327

nationally and internationally known in the field of mobile communications. 1328

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