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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: ibrahimhemadeh@gmail.com).
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
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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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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
Fig. 3. PM transmitter block diagram.
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
IEEE P
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8 IEEE TRANSACTIONS ON COMMUNICATIONS
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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HEMADEH et al.: POLARIZATION MODULATION DESIGN FOR REDUCED RF CHAIN WIRELESS 9
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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10 IEEE TRANSACTIONS ON COMMUNICATIONS
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.
IEEE P
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HEMADEH et al.: POLARIZATION MODULATION DESIGN FOR REDUCED RF CHAIN WIRELESS 11
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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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
IEEE P
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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: ibrahimhemadeh@gmail.com).
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
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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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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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4 IEEE TRANSACTIONS ON COMMUNICATIONS
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
Fig. 3. PM transmitter block diagram.
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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6 IEEE TRANSACTIONS ON COMMUNICATIONS
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
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8 IEEE TRANSACTIONS ON COMMUNICATIONS
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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HEMADEH et al.: POLARIZATION MODULATION DESIGN FOR REDUCED RF CHAIN WIRELESS 9
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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10 IEEE TRANSACTIONS ON COMMUNICATIONS
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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12 IEEE TRANSACTIONS ON COMMUNICATIONS
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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HEMADEH et al.: POLARIZATION MODULATION DESIGN FOR REDUCED RF CHAIN WIRELESS 13
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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14 IEEE TRANSACTIONS ON COMMUNICATIONS
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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16 IEEE TRANSACTIONS ON COMMUNICATIONS
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