# Antenna Selection in Polarization Reconfigurable MIMO (PR-MIMO)
Communication Systems ††thanks: Paul S. Oh and Sean S. Kwon are co-first
authors with the equal contribution. ††thanks: Part of this work was presented
at IEEE Globecom 2015 [1].

Paul S. Oh†, Student Member, IEEE, Sean S. Kwon†, Member, IEEE and Andreas F.
Molisch∗, Fellow, IEEE
†Department of Electrical Engineering, California State University, Long Beach
∗ Department of Electrical and Computer Engineering, University of Southern
California
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Adaptation of a wireless system to the polarization state of the propagation
channel can improve reliability and throughput. This paper in particular
considers polarization-reconfigurable multiple-input - multiple-output (PR-
MIMO) systems, where both transmitter and receiver can change the (linear)
polarization orientation at each element of their antenna arrays. We first
introduce joint polarization pre-post coding to maximize bounds on the
capacity and the maximum eigenvalue of the channel matrix. For this we first
derive approximate closed form equations of optimal polarization vectors at
one link end, and then use iterative joint polarization pre-post coding to
pursue joint optimal polarization vectors at both link ends. Next we
investigate the combination of PR-MIMO with hybrid antenna selection / maximum
ratio transmission (PR-HS/MRT), which can achieve a remarkable improvement of
channel capacity and symbol error rate (SER). Further, two novel schemes of
element-wise and global polarization reconfiguration are presented for PR-
HS/MRT. Comprehensive simulation results indicate that the proposed schemes
provide 3 – 5 dB SNR gain in PR-MIMO spatial multiplexing and approximately 3
dB SNR gain in PR-HS/MRT, with concomitant improvements of channel capacity
and SER.

###### Index Terms:

Hybrid selection, reconfigurable polarization, multi-input multi-output
(MIMO), Maximum ratio transmission (MRT), channel capacity, symbol error rate,

##  I Introduction

An electromagnetic waves that propagates from the transmitter (Tx) to the
receiver (Rx) of a wireless communications system is characterized, inter
alia, by its polarization, i.e., orientation of the field vector. Interaction
of a wave (multipath component, MPC) with environmental objects may change the
orientation, and different MPCs may experience different changes in the
orientation. This effect can reduce performance, e.g., if the (fixed)
polarization of the Rx antenna is mis-matched to the polarization of the
arriving field. However, it can also be an advantage, since the different
polarizations offer degrees of freedom that can be exploited for diversity
and/or spatial multiplexing. Therefore, numerous investigations have been made
into use of the polarization domain [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
1, 14, 15]. It has been demonstrated that utilizing the polarization domain
may increase channel capacity and spectral efficiency and improve symbol error
rate (SER) [16, 17, 18, 1, 19, 20]. For this reason, impact of polarization on
the wireless communication systems has been regarded as a promising research
topic [21, 22, 23, 24, 25, 1, 2, 5].

Most of the above papers assume that the Tx and Rx have co-located dual-
polarized antennas, such that two antenna ports are available for each antenna
element at a distinct spatial location. Compared to the case where the same
number of antennas with only a single polarization per antenna is available,
this leads to an increase in diversity and capacity, although the gain depends
significantly on the cross-polarization discrimination (XPD) [16]. However,
the co-located dual-polarized antennas imply doubling of number of antenna
ports, with the consequent increase in the number of RF (up/down-conversion)
chains and increased baseband processing. To mitigate the complexity issues,
this paper considers polarization reconfigurable multi-input multi-output (PR-
MIMO), where each antenna element has only a single port, but whose
polarization orientation can be adapted through use of switches and parasitic
elements [13, 12, 11, 26, 27]. The performance analysis and transceiver
schemes in this paper are based on these types of antennas. By optimally
employing the polarization reconfigurable antennas in conjunction with MIMO
spatial multiplexing, maximum-ratio transmission (MRT), and hybrid antenna
selection (HS), this paper will show that the system performance is
substantially enhanced compared to that of a conventional scheme with the same
number of antenna ports. Throughout the paper, we will assume knowledge of
channel state information (CSI), including polarization information, at both
Tx and Rx. The methods for acquiring such CSI have been amply discussed in the
literature.

The two main methods for exploiting MIMO systems are spatial multiplexing, and
diversity. In the former case, independent data streams are transmitted and
received through spatial parallelized MIMO channels. The parallelization is
created via precoding and postcoding based on singular value decomposition
(SVD) of the channel matrix. In this paper, we propose and analyze a MIMO
system that exploits spatial multiplexing with polarization reconfigurable
antenna elements and the functionality of polarization precoding/postcoding,
which we denominate _polarization reconfigurable MIMO_ (_PR-MIMO_). We will
show that the adjustment of the polarization vectors at Tx and Rx antenna
elements can improve the capacity of a spatial multiplexing system.

On the other hand, the PR-MIMO system can focus on Tx and Rx diversity for the
improvement of the link-quality in terms of signal-to-noise ratio (SNR). For
example, $N_{t}$ Tx antennas employ maximal-ratio transmission (MRT) to
simultaneously transmit weighted replica of the single bit stream. The ideal
weights are obtained by SVD of the channel impulse matrix. In the similar
manner, $N_{r}$ Rx antennas utilize maximal-ratio combining (MRC), where the
weighted received signals are linearly combined to increase the effective SNR.
In overall, a diversity degree of $N_{t}\cdot N_{r}$ can be obtained.

However, a system employing both MRT and MRC requires $N_{t}$ and $N_{r}$
complete RF chains, respectively; this high degree of hardware complexity is
undesirable in particular when numerous antennas are involved. For instance,
5G base stations, BSs, (called next-generation Node B (gNB)), may contain at
least 64 elements in a single panel as described in [28]. To lower hardware
complexity, antenna selection can be used, and is foreseen in current cellular
and WiFi schemes [29]. This paper will analyze a situation where the gNB uses
hybrid selection and MRT (HS/MRT); whereas, the user equipment (UE) or the Rx
applies MRC. HS/MRT scheme selects $L_{t}$ out of $N_{t}$ Tx antennas, which
reduce the hardware complexity by lowering the number of RF chain from $N_{t}$
to $L_{t}$.

The benefits of hybrid selection and spatial multiplexing are demonstrated in
[29, 30, 31, 32, 33]. On the other hand, polarization diversity is not taken
into account in the majority of previous research works. Although there are
previous reports that consider polarization diversity with antenna selection,
they consider fixed antenna polarization such as dual-polarized antennas
in[34] or tri-polarized antennas in [35]. In contrast, we exploit polarization
reconfigurable hybrid antenna selection and MRT (PR-HS/MRT) which
significantly outperforms that of the conventional scheme of the HS/MRT
system.

The primary contributions of this paper can be summarized as follows:

  * •

we introduce and characterize the antenna selection with polarization
reconfigurable antennas on both link ends;

  * •

we provide a closed-form approximation of the optimal Tx/Rx-polarization
vectors in the PR-MIMO system at one link end to achieve channel capacity
beyond that of a conventional MIMO communication system (with the same number
of ports). We also propose iterative joint polarization pre/post-coding for
joint optimal polarization vectors;

  * •

we propose two novel schemes of PR-HS/MRT with polarization reconfigurable
antenna elements, and provide analysis of the system performance in terms of
channel capacity, SER and the distribution of channel gain;

  * •

we perform a statistical analysis of the effect of polarization reconfigurable
antennas on the distribution of channel gain;

  * •

we validate the significant improvement of the system performance in terms of
channel capacity and SER caused by the proposed PR-MIMO and PR-HS/MRT schemes.

The remainder of this paper is organized as follows. Section II presents the
PR-MIMO system model focusing on polarization reconfigurable antennas. In
Section III, the iterative joint polarization precoding/postcoding scheme
utilizing optimal Tx/Rx-polarization vectors is proposed and described in
detail. PR-HS/MRC is presented in Section IV; the statistical analysis of the
polarization reconfigurable channel is described in Section V. Section VI
derives the upper and lower bounds of achievable SNR. Extensive simulation
results demonstrate significant improvement of the system performance in terms
of channel capacity, SNR and statistics of channel gain in Section VII.
Finally, Section VIII concludes the paper.

##  II System Model

The fundamental block diagram of the PR-MIMO system is shown in Fig. 1, where
the Tx and the Rx have $N_{\rm t}$ and $N_{\rm r}$ antenna elements,
respectively. Each antenna element is polarization reconfigurable with the
polarization vector $\vec{p}_{{\rm Tx},j}$ and $\vec{p}_{{\rm Rx},i}$ in which
$j\in\\{1,...,N_{\rm t}\\}$; $i\in\\{1,...,N_{\rm r}\\}$. The polarization
vector is adjusted according to the channel state information (CSI); perfect
CSI at all antenna elements is assumed to be available at the Rx (CSIR) as
well as the Tx (CSIT). It is noteworthy that the CSI is available for both
orthogonal polarization directions; it can be obtained through training
schemes similar to those employed in antenna selection systems (see, e.g.,
[36]). The impact of imperfect CSI is outside the scope of this paper and will
be analyzed in future work.

In addition to the polarization precoding/postcoding, the PR-MIMO system
contains precoding/postcoding that allows optimum exploitation of the spatial
degree of freedom; it is well-known from conventional (non-polarization
reconfigurable) spatial multiplexing systems with perfect CSI [37] that linear
precoding/postcoding based on the (SVD) of the effective channel impulse
matrix maximizes sum-rate. It is intuitive that exploiting polarization
reconfigurable antenna elements with polarization precoding/postcoding - on
top of the standard SVD-based spatial precoding/postcoding - can achieve
higher capacity than single-polarization or fixed-polarization antenna
elements (with the same number of data streams or RF up/down-conversion
chains). Mathematical derivations for this intuition are presented below.

The effective channel impulse response matrix in Fig. 1 can be expressed as

$H^{\rm eff}=\begin{bmatrix}\vec{p}_{{\rm Rx},1}^{~{}T}H_{11}\vec{p}_{{\rm
Tx},1}&\ldots&\vec{p}_{{\rm Rx},1}^{~{}T}H_{1N_{\rm t}}\vec{p}_{{\rm
Tx},N_{\rm t}}\\\ \vdots&\ddots&\vdots\\\ \vec{p}_{{\rm Rx},N_{\rm
r}}^{~{}T}H_{N_{\rm r}1}\vec{p}_{{\rm Tx},1}&\ldots&\vec{p}_{{\rm Rx},N_{\rm
r}}^{~{}T}H_{N_{\rm r}N_{\rm t}}\vec{p}_{{\rm Tx},N_{\rm t}}\end{bmatrix},$
(1)

where the operation $(\cdot)^{T}$ is the transpose of a given vector or
matrix, and the dimension of $H^{\rm eff}$ is $N_{\rm r}\times N_{\rm t}$.
Further, we denominate $H_{ij}$ as “polarization-basis matrix” which is

$H_{ij}=\begin{bmatrix}h_{ij}^{\rm vv}&h_{ij}^{\rm vh}\\\ h_{ij}^{\rm
hv}&h_{ij}^{\rm hh}\\\ \end{bmatrix},$ (2)

where $h_{ij}^{\rm xy}$ with ${\rm x\in\\{v,h\\}}$; ${\rm y\in\\{v,h\\}}$ is
the XY-channel impulse response from the Y-polarization Tx antenna to the
X-polarization Rx antenna. For instance, $h_{ij}^{\rm hv}$ is the HV-channel
impulse response from the vertically polarized (V-Pol) Tx antenna to the
horizontally polarized (H-Pol) Rx antenna. We assume here a flat-fading
channel, such that the $h_{ij}^{\rm xy}$ are complex scalars. Lastly,
$\vec{p}_{{\rm Tx},j}$ and $\vec{p}_{{\rm Rx},i}$ are, respectively, the Tx-
polarization vector at the $j$th Tx antenna and the Rx-polarization vector at
the $i$th Rx antenna, and they are expressed as

$\displaystyle\vec{p}_{{\rm Tx},j}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}p_{{\rm Tx},j}^{\rm v}\\\ p_{{\rm Tx},j}^{\rm
h}\\\ \end{bmatrix}=\begin{bmatrix}\cos\theta_{j}\\\ \sin\theta_{j}\\\
\end{bmatrix},$ (3) $\displaystyle\vec{p}_{{\rm Rx},i}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}p_{{\rm Rx},i}^{\rm v}\\\ p_{{\rm Rx},i}^{\rm
h}\\\ \end{bmatrix}=\begin{bmatrix}\cos\theta_{i}\\\ \sin\theta_{i}\\\
\end{bmatrix}.$ (4)

Here, we call the angles $\theta_{j}$ and $\theta_{i}$ Tx- and Rx-polarization
angles, respectively. It is worth mentioning that Tx- and Rx-polarization
vectors are unit vectors so that the overall signal power is preserved after
polarization precoding and postcoding.

Figure 1: PR-MIMO system with polarization reconfigurable antennas.

##  III Polarization Pre-post Coding at the Polarization reconfigurable
Antenna

###  III-A Polarization Precoding and Postcoding with Optimal Tx- and Rx-
polarization

The Tx and the Rx can utilize SVD-based precoding and postcoding under the
assumption of full CSIT and CSIR. The combination of SVD-based precoding and
postcoding achieves the MIMO channel capacity for a given channel matrix via
constructing parallel channels [37]. On the other hand, the MIMO communication
system with polarization reconfigurable antennas in Fig. 1 can tune of the
effective channel impulse response matrix $H^{\rm eff}$ itself in (1) by
either polarization precoding at the Tx or polarization postcoding at the Rx;
or joint polarization pre-post coding at both ends. We first focus on
polarization precoding in this section.

The effective channel impulse response matrix can be decomposed by SVD as

$H^{\rm eff}=W\Sigma U^{{\dagger}},$ (5)

where $\Sigma$ is a diagonal matrix containing singular values, and $W$ and
$U^{{\dagger}}$ are unitary matrices composed of the left and right singular
vectors, respectively [37]. In this paper, $(\cdot)^{\dagger}$ is the
Hermitian transpose operation. The channel capacity with SVD-based precoding
and postcoding is

$C=\sum_{k=1}^{R_{H^{\rm
eff}}}\log_{2}\left(1+\frac{P_{k}}{{\sigma_{n}}^{2}}{\sigma_{k}}^{2}\right),$
(6)

where $R_{H^{\rm eff}}$ is the rank of the matrix $H^{\rm eff}$, and $P_{k}$
is the power allocated to the $k$th eigenmode. Further, $\sigma_{k}$ is the
$k$th singular value of the effective channel impulse response matrix $H^{\rm
eff}$, and ${\sigma_{n}}^{2}$ is the noise power. Capacity maximization is
achieved by a power allocation $P_{k}$ that satisfies waterfilling conditions

$P_{k}=\max\left(0,\epsilon-\frac{{\sigma_{n}}^{2}}{{\sigma_{k}}^{2}}\right){\rm~{}s.t.~{}}P=\sum_{k=1}^{R_{H^{\rm
eff}}}P_{k}{~{}},$ (7)

i.e., the threshold $\epsilon$ is determined by the constraint of the total
transmitted power P. We assume the total power is independent of the number of
antennas.

Following [38], we use Jensen’s inequality to obtain

$\displaystyle C$ $\displaystyle\leq$ $\displaystyle R_{H^{\rm
eff}}{~{}}\log_{2}\left(1+\frac{1}{R_{H^{\rm eff}}}\sum_{k=1}^{R_{H^{\rm
eff}}}\frac{P_{k}}{{\sigma_{n}}^{2}}{\sigma_{k}}^{2}\right)$ (8)
$\displaystyle=$ $\displaystyle R_{H^{\rm
eff}}{~{}}\log_{2}\left(\frac{1}{R_{H^{\rm
eff}}}\frac{\epsilon}{{\sigma_{n}}^{2}}\sum_{k=1}^{R_{H^{\rm
eff}}}{\sigma_{k}}^{2}\right),$ (9)

where high SNR is assumed; all $P_{k}$ is greater than zero, i.e.,
$P_{k}=\epsilon-{\sigma_{n}}^{2}/{\sigma_{k}}^{2}>0$ in (7). Further,
${\sigma_{k}}^{2}$ is the eigenvalue of $\left(H^{\rm eff}(H^{\rm
eff})^{\dagger}\right)$; therefore [39],

$\displaystyle\sum_{k=1}^{R_{H^{\rm eff}}}{\sigma_{k}}^{2}$ $\displaystyle=$
$\displaystyle{\rm Tr}\left(H^{\rm eff}(H^{\rm
eff})^{\dagger}\right)=\sum_{n,m}\left|h_{nm}^{\rm eff}\right|^{2}.$ (10)

This is the sum of squared envelopes of all channel impulse response elements
in $H^{\rm eff}$. This quantity is not only important for the channel
capacity, but will also play an important role in the performance assessment
of PR-HS/MRT systems, since the bounds employed there make use of it.

It is worth emphasizing that each element of $H^{\rm eff}$ is affected by the
Tx- and Rx-polarization as implied in (1). Hence, the polarization vectors at
polarization reconfigurable antenna elements impact constructively or
destructively the MIMO channel capacity itself, even though SVD-based
precoding/postcoding reaches the MIMO channel capacity for the given full
CSIR/CSIT. The $j$th Tx polarization-agile antenna affects the $j$th column in
(1); thus, the sum of squared singular values in (10) can be written as

$\displaystyle\sum_{k=1}^{R_{H^{\rm eff}}}{\sigma_{k}}^{2}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N_{\rm t}}\sum_{i=1}^{N_{\rm r}}\vec{p}_{{\rm
Tx},j}^{~{}T}\left(H_{ij}^{{\dagger}}\vec{p}_{{\rm Rx},i}\vec{p}_{{\rm
Rx},i}^{~{}T}H_{ij}\right)\vec{p}_{{\rm Tx},j}$ (11) $\displaystyle=$
$\displaystyle\sum_{j=1}^{N_{\rm t}}\vec{p}_{{\rm Tx},j}^{~{}T}H_{{\rm
Tx},j}^{\rm PD}\vec{p}_{{\rm Tx},j}{~{}},$

where the “ _Tx-polarization-determinant matrix_ ” for the $j$th Tx
polarization-agile antenna, $H_{{\rm Tx},j}^{\rm PD}$, is defined as

$\displaystyle H_{{\rm Tx},j}^{\rm PD}$ $\displaystyle\triangleq$
$\displaystyle\sum_{i=1}^{N_{\rm r}}H_{ij}^{\dagger}\vec{p}_{{\rm
Rx},i}\vec{p}_{{\rm Rx},i}^{~{}T}H_{ij}{~{}}.$ (12)

The Tx-polarization vector at each Tx polarization-agile antenna is
independent of those at other Tx-polarization antennas; therefore, the optimal
Tx-polarization vector at the $j$th Tx polarization-agile antenna,
$\vec{p}_{{\rm Tx},j}$, is the one which maximizes $\vec{p}_{{\rm
Tx},j}^{~{}T}H_{{\rm Tx},j}^{\rm PD}\vec{p}_{{\rm Tx},j}$ in (11).

From the viewpoint of the linear-algebraic approach, $\vec{p}_{{\rm
Tx},j}^{~{}T}H_{{\rm Tx},j}^{\rm PD}\vec{p}_{{\rm Tx},j}$ is positive semi-
definite; thus, the equation, $\vec{p}_{{\rm Tx},j}^{~{}T}{~{}}H_{{\rm
Tx},j}^{\rm PD}{~{}}\vec{p}_{{\rm Tx},j}=c_{j}$ corresponds to the ellipse as
portrayed in Fig. 2, where $x$ and $y$ coordinates correspond to the elements
of $\vec{p}_{{\rm Tx},j}$, i.e., $p_{{\rm Tx},j}^{\rm v}$ and $p_{{\rm
Tx},j}^{\rm h}$, respectively [39]. Geometrically, the principal axes of the
ellipse are along eigenvectors of the matrix $H_{{\rm Tx},j}^{\rm PD}$,
$\overrightarrow{e}_{1}$ and $\overrightarrow{e}_{2}$, and the distances from
the origin to the ellipse along the principal axes are
$c_{j}/\sqrt{\lambda_{1}^{\rm PD}}$ and $c_{j}/\sqrt{\lambda_{2}^{\rm PD}}$,
where $\lambda_{1}^{\rm PD}$ and $\lambda_{2}^{\rm PD}$ are the eigenvalues of
$H_{{\rm Tx},j}^{\rm PD}$.

Figure 2: Polarization-determinant ellipse and polarization-vector unit
circle.

Our objective at this stage is to estimate the optimal Tx-polarization vector
at each Tx polarization-agile antenna element, $\vec{p}_{{\rm Tx},j}\in
j=\\{1,\cdots,N_{\rm t}\\}$, which maximizes the column-sum of element-wise
squared envelopes in $H^{\rm eff}$, i.e., $\vec{p}_{{\rm Tx},j}^{~{}T}H_{{\rm
Tx},j}^{\rm PD}\vec{p}_{{\rm Tx},j}$. On the other hand, the Tx-polarization
vector $\vec{p}_{{\rm Tx},j}$ is on the unit circle as presented in (3);
therefore, the ellipse must have, at least, one intersection or contact point
with the unit circle; whereas, at the same time it must make $\vec{p}_{{\rm
Tx},j}^{~{}T}H_{{\rm Tx},j}^{\rm PD}\vec{p}_{{\rm Tx},j}=c_{j}$ as large as it
can. Hence, the optimal Tx-polarization vector $\vec{p}_{{\rm Tx},j}^{\rm
opt}$ and corresponding optimal Tx-polarization angle $\theta_{j}^{\rm opt}$
described in (3) are as

$\displaystyle\vec{p}_{{\rm Tx},j}^{\rm~{}opt}$ $\displaystyle=$
$\displaystyle\arg\max\limits_{\vec{p}_{{\rm Tx},j}}{~{}}\vec{p}_{{\rm
Tx},j}^{~{}T}H_{{\rm Tx},j}^{\rm PD}\vec{p}_{{\rm Tx},j}={~{}}\vec{e}_{2},$
(13) $\displaystyle\theta_{j}^{\rm opt}$ $\displaystyle=$
$\displaystyle\arctan(\vec{e_{2}}).$ (14)

Notice that $\vec{e}_{2}$ is the eigenvector corresponding to $\lambda_{2}$,
which is the maximum eigenvalue of the Tx-polarization-determinant matrix,
$H_{{\rm Tx},j}^{\rm PD}$. In this manner, each Tx polarization-agile antenna
element can perform polarization precoding with the optimal Tx-polarization
vector.

In a completely analogous manner, the optimal RX-polarization vector can be
derived; we just now have to employ the row-sum (instead of the column-sum) of
element-wise squared envelopes in $H^{\rm eff}$. The optimum polarization
vector can be shown to be

$\displaystyle\vec{p}_{{\rm Rx},i}^{\rm~{}opt}$ $\displaystyle=$
$\displaystyle\arg\max\limits_{\vec{p}_{{\rm Rx},i}}{~{}}\vec{p}_{{\rm
Rx},i}^{~{}T}H_{{\rm Rx},i}^{\rm PD}\vec{p}_{{\rm Rx},i}={~{}}\vec{e}_{2},$
(15) $\displaystyle\theta_{i}^{\rm opt}$ $\displaystyle=$
$\displaystyle\arctan(\vec{e_{2}}),$ (16)

where the “ _Rx-polarization-determinant matrix_ ” for the $i$th Rx
polarization-agile antenna, $H_{{\rm Rx},i}^{\rm PD}$, is defined as

$\displaystyle H_{{\rm Rx},i}^{\rm PD}$ $\displaystyle\triangleq$
$\displaystyle\sum_{j=1}^{N_{\rm t}}H_{ij}\vec{p}_{{\rm Tx},j}\vec{p}_{{\rm
Tx},j}^{~{}T}H_{ij}^{\dagger}{~{}}.$ (17)

###  III-B Joint Polarization Pre-post Coding for Polarization Matching

The Tx-polarization-determinant matrix at the $j$th Tx polarization-agile
antenna, $H_{{\rm Tx},j}^{\rm PD}$, depends on the Rx-polarization vectors
$\vec{p}_{{\rm Rx},i}$ as shown in (12), and vice versa in (17). Further, Tx-
and Rx-polarization mismatching will deteriorate the system performance in
terms of the channel capacity in this paper. For those reasons, joint
polarization pre-post coding is required to maximize PR-MIMO channel capacity.

Joint optimization of the pre-post coding is difficult to obtain in closed
form (and also difficult to implement); therefore, we propose an iterative
approach where one iteration is a sequential loop of polarization precoding;
then polarization postcoding. In the $q$th iteration stage,
$\vec{p}^{{\rm~{}opt},(q)}_{{\rm Tx},j}$ is updated to
$\vec{p}^{{\rm~{}opt},(q+1)}_{{\rm Tx},j}$ based on
$\vec{p}^{{\rm~{}opt},(q)}_{{\rm Rx},i}$ according to (12) – (14). Then, in
turn, $\vec{p}^{{\rm~{}opt},(q)}_{{\rm Rx},i}$ is updated to
$\vec{p}^{{\rm~{}opt},(q+1)}_{{\rm Rx},i}$ based on the updated
$\vec{p}^{{\rm~{}opt},(q)}_{{\rm Tx},j}$ following (15) – (17). If initial
values of Rx and Tx polarization vectors are

$\displaystyle\vec{p}_{{\rm Tx},j}^{\ (0)}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}p_{{\rm Tx},j}^{\rm v}\\\ p_{{\rm Tx},j}^{\rm
h}\\\ \end{bmatrix}=\begin{bmatrix}1\\\ 0\\\ \end{bmatrix},$ (18)
$\displaystyle\vec{p}_{{\rm Rx},i}^{\ (0)}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}p_{{\rm Rx},i}^{\rm v}\\\ p_{{\rm Rx},i}^{\rm
h}\\\ \end{bmatrix}=\begin{bmatrix}1\\\ 0\\\ \end{bmatrix},$ (19)

the joint optimization iteration is expressed as

$\displaystyle\vec{p}_{{\rm Tx},j}^{\rm~{}opt,(q+1)}$ $\displaystyle=$
$\displaystyle\arg\max\limits_{\vec{p}_{{\rm Tx},j}}{~{}}\vec{p}_{{\rm
Tx},j}^{~{}T}H_{{\rm Tx},j}^{\rm PD,(q+1)}\vec{p}_{{\rm Tx},j}$ (20)
$\displaystyle H_{{\rm Tx},j}^{\rm PD,(q+1)}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N_{\rm r}}H_{ij}^{\dagger}\vec{p}_{{\rm
Rx},i}^{\rm~{}(q)}\vec{p}_{{\rm Rx},i}^{~{}T(q)}H_{ij}{~{}}.$ (21)
$\displaystyle\vec{p}_{{\rm Rx},i}^{\rm~{}opt,(q+1)}$ $\displaystyle=$
$\displaystyle\arg\max\limits_{\vec{p}_{{\rm Rx},i}}{~{}}\vec{p}_{{\rm
Rx},i}^{~{}T}H_{{\rm Rx},i}^{\rm PD,(q+1)}\vec{p}_{{\rm Rx},i}$ (22)
$\displaystyle H_{{\rm Rx},i}^{\rm PD,(q+1)}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N_{\rm t}}H_{ij}^{\dagger}\vec{p}_{{\rm
Tx},j}^{\rm~{}(q)}\vec{p}_{{\rm Tx},j}^{~{}T(q)}H_{ij}{~{}},$ (23)
$\displaystyle h^{\rm eff}_{\rm opt,ij}=(\vec{p}_{{\rm
Rx},i}^{\rm~{}opt,(q+1)})^{T}H_{ij}\vec{p}_{{\rm Tx},j}^{\rm~{}opt,(q+1)}~{},$
(24)

where q is the iteration index. Note that while each step increases the
capacity, the procedure is not guaranteed to reach the global optimum.
However, we will see in Section VII that the capacity resulted from the
iterative joint pre-post coding is in a close agreement with the one achieved
by brute-force numerical search over all pre-post coding vectors.

##  IV Antenna Selection with Reconfigurable Polarization

Figure 3: System Model of Antenna Selection with Reconfigurable Antennas

This section delineates the communication system that performs polarization
reconfigurable hybrid selection (PR-HS) and MRT with the selected polarization
reconfigurable Tx antenna elements as depicted in Fig. 3. The Tx selects
$L_{t}$ out of $N_{t}$ polarization reconfigurable Tx antenna elements, and
performs MRT with the associated weights $\\{u_{1};u_{2},...,u_{L_{t}}\\}$. At
the same time, conventional MRC (without selection) is applied for multiple
polarization reconfigurable antennas with corresponding weights
$\\{w^{*}_{1},w^{*}_{2},...,w^{*}_{N_{r}}\\}$ at the Rx in Fig. 3. Each
antenna element is polarization reconfigurable and adjusted to an optimal
polarization angle based on the joint polarization pre-post coding proposed
and described in Section III. In the same fashion as the PR-MIMO system, this
section assumes perfect CSI. That is, both the Tx and Rx have perfect
knowledge of the polarization-basis matrix in (1) – (2).

The partial channel impulse response matrix that represents the channels
between selected polarization reconfigurable Tx antenna elements and Rx
antenna elements is defined as $\widetilde{H}^{\rm eff}_{k}$, where the index
$k$ stands for the $k$-th subset of $L_{t}$ number of antenna selection.
Further, the column index matrix $C_{\rm Index}$ indicates the list of all
possible combination of selected Tx antenna elements. That is,

$C_{\rm Index}=\begin{bmatrix}\alpha_{11}&\ldots&\alpha_{1L_{t}}\\\ &\vdots\\\
\alpha_{\binom{N_{t}}{L_{t}}1}&\ldots&\alpha_{\binom{N_{t}}{L_{t}}L_{t}}\end{bmatrix},$
(25)

where each row represents a subset of polarization reconfigurable Tx antenna
indices corresponding to the selected antenna elements.

Each $\widetilde{H}^{\rm eff}_{k}$ consists of $L_{t}$ selected columns of
$H^{\rm eff}$, associated with the selected polarization reconfigurable Tx
antenna elements, and is described as

$\displaystyle\widetilde{H}_{k}^{\rm
eff}={{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}$
(26) $\displaystyle{}\begin{bmatrix}\vec{p}_{{\rm
Rx},1}^{~{}T}H_{1\alpha_{k1}}\vec{p}_{{\rm
Tx},\alpha_{k1}}&\ldots&\vec{p}_{{\rm
Rx},1}^{~{}T}H_{1\alpha_{kL_{t}}}\vec{p}_{{\rm Tx},\alpha_{kL_{t}}}\\\
\vdots&\ddots&\vdots\\\ \vec{p}_{{\rm Rx},N_{\rm r}}^{~{}T}H_{N_{\rm
r}\alpha_{k1}}\vec{p}_{{\rm Tx},\alpha_{k1}}&\ldots&\vec{p}_{{\rm Rx},N_{\rm
r}}^{~{}T}H_{N_{\rm r}\alpha_{kL_{t}}}\vec{p}_{{\rm
Tx},\alpha_{kL_{t}}}\end{bmatrix},{{}{}}$

where $k$ corresponds to the $k$-th row of the column index matrix $C_{\rm
Index}$. The achievable effective SNR of PR-HS/MRT is obtained based on SVD of
$\widetilde{H}^{\rm eff}_{k}$ in (26) as

$\displaystyle\gamma_{\rm PR-HS/MRT}^{\rm
eff}=\underset{k}{\mathrm{max}}\big{(}{\widetilde{\sigma}_{k,\max}}^{~{}2}\big{)},$
(27)

where $\widetilde{\sigma}_{k,\mathrm{max}}^{~{}2}$ is a maximum eigenvalue of
$\widetilde{H}_{k}^{\rm eff}{\widetilde{H}_{k}^{{\rm eff}{\dagger}}}$ [29].
The objective is selecting the column index $k$ that maximize
$\widetilde{\sigma}_{k,\mathrm{max}}^{~{}2}$. For further theoretical
derivation,

$\displaystyle\widetilde{H}_{k}^{\rm eff}\widetilde{H}_{k}^{\rm
eff\dagger}=_{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
(28) $\displaystyle\begin{bmatrix}&\vec{p}_{\rm Rx,1}^{~{}T}H^{\rm PD}_{\rm
Rx,1}\vec{p}_{\rm Rx,1}&\ldots&\vec{p}_{\rm Rx,1}^{~{}T}H^{\rm Off}_{\rm
Rx,1N_{r}}\vec{p}_{\rm Rx,N_{r}}\\\ &\vdots&\ddots&\vdots\\\ &\vec{p}_{\rm
Rx,N_{r}}^{~{}T}H^{\rm Off}_{\rm Rx,N_{r}1},\vec{p}_{\rm
Rx,1}&\ldots&\vec{p}_{\rm Rx,N_{r}}^{~{}T}H^{\rm PD}_{\rm
Rx,N_{r}},\vec{p}_{\rm Rx,N_{r}}\end{bmatrix},$

where

$H^{\rm Off}_{{\rm Rx},ij}=\sum_{j=1}^{L_{\rm t}}H_{i\alpha_{ij}}\vec{p}_{{\rm
Tx},\alpha_{ij}}\vec{p}_{{\rm
Tx},\alpha_{ij}}^{{~{}}T}H_{j\alpha_{ij}}^{\dagger}$ (29)

is the polarization determinant matrix of off-diagonal components. The
algorithm to maximize the diagonal components of the matrix is described by
(16). As will be described in Section VI, the algorithm described in Section
III-B maximizes the lower bound of
$\widetilde{\sigma}_{k,\mathrm{max}}^{~{}2}$, which significantly improve the
achievable effective SNR in (27). We propose two schemes of PR-HS/MRT to
efficiently achieve polarization reconfigurable antenna selection. One of the
primary contributions of this paper is enhancing the effective SNR performance
based on two proposed schemes in this section.

###  IV-A Scheme-1: Element-Wise Polarization Reconfiguration

Polarization pre-post coding in Section III is accomplished with
$\widetilde{H}_{k}^{\rm eff}$ for each $k$ in the scheme of element-wise
polarization reconfiguration. Based on the adjusted polarization for each
$\widetilde{H}_{k}^{\rm eff}$, the best $k$ in terms of the greatest
$\widetilde{\sigma}_{k,\mathrm{max}}^{~{}2}$, is selected as $L_{t}$ number of
Tx antenna elements. We denominate Scheme-1 as _element-wise (EW)_
polarization reconfiguration, since optimal polarization vectors of every
$\widetilde{H}_{k}^{\rm eff}$ for
$k=\big{\\{}1,2,...,\binom{N_{t}}{L_{t}}\big{\\}}$ is estimated. The EW Rx
optimal polarization vectors are defined as

$\displaystyle\vec{p}_{{\rm Rx},i}^{{~{}}\rm EW}$ $\displaystyle=$
$\displaystyle\underset{\vec{p}_{{\rm Rx},i}}{\arg\max}{~{}}\vec{p}_{{\rm
Rx},i}^{{~{}}T}H^{\rm EW}_{{\rm Rx},i}\vec{p}_{{\rm Rx},i}{~{}},$ (30)
$\displaystyle H^{\rm EW}_{{\rm Rx},i}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{L_{\rm t}}H_{i\alpha_{ij}}\vec{p}_{{\rm
Tx},\alpha_{ij}}\vec{p}_{{\rm
Tx},\alpha_{ij}}^{{~{}}T}H_{i\alpha_{ij}}^{\dagger}{~{}},$ (31)

where (31) is EW Rx-polarization-determinant matrix. In a similar manner, the
optimal polarization vectors at the Tx are described as

$\displaystyle\vec{p}_{{\rm Tx},\alpha_{ij}}^{{~{}}\rm EW}$ $\displaystyle=$
$\displaystyle\underset{\vec{p}_{{\rm
Tx},\alpha_{ij}}}{\arg\max}{~{}}\vec{p}_{{\rm Tx},\alpha_{ij}}^{{~{}}T}H^{\rm
EW}_{{\rm Tx},\alpha_{ij}}\vec{p}_{{\rm Tx},\alpha_{ij}},$ (32) $\displaystyle
H^{\rm EW}_{{\rm Tx},\alpha_{ij}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N_{\rm r}}H_{i\alpha_{ij}}^{\dagger}\vec{p}_{{\rm
Rx},i}\vec{p}_{{\rm Rx},i}^{{~{}}T}H_{i\alpha_{ij}}$ (33)

where (33) is EW Tx-polarization-determinant matrix. With (30) and (32), each
$\widetilde{H}^{\rm eff}_{k}$ is tuned with corresponding EW Tx/Rx
polarization vectors. That is,

$\displaystyle\widetilde{H}^{\rm eff}_{{\rm EW},k}{}={}$ (34)
$\displaystyle\begin{bmatrix}(\vec{p}_{{\rm Rx},1}^{~{}\rm
EW})^{T}H_{1\alpha_{k1}}\vec{p}_{{\rm Tx},\alpha_{k1}}^{~{}\rm
EW}\ldots(\vec{p}_{{\rm Rx},1}^{~{}\rm
EW})^{T}H_{1\alpha_{kL_{t}}}\vec{p}_{{\rm Tx},\alpha_{kL_{t}}}^{~{}\rm EW}\\\
\vdots~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\ddots~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
(\vec{p}_{{\rm Rx},N_{\rm r}}^{~{}\rm EW})^{T}H_{N_{\rm
r}\alpha_{k1}}\vec{p}_{{\rm Tx},\alpha_{k1}}^{~{}\rm EW}\ldots(\vec{p}_{{\rm
Rx},N_{\rm r}}^{~{}\rm EW})^{T}H_{N_{\rm r}\alpha_{kL_{t}}}\vec{p}_{{\rm
Tx},\alpha_{kL_{t}}}^{~{}\rm EW}\end{bmatrix}.$

The effective SNR with EW polarization reconfiguration is

$\displaystyle\gamma_{\rm PR-HS/MRT}^{\rm
EW}=\underset{k}{\mathrm{max}}\big{(}\widetilde{\sigma}_{k,{\rm max}}^{\rm
EW}\big{)}^{2},$ (35)

where $\big{(}\widetilde{\sigma}_{k,{\rm max}}^{\rm EW}\big{)}^{2}$ is a
maximum eigenvalue of $\widetilde{H}_{{\rm EW},k}^{\rm eff}\widetilde{H}_{{\rm
EW},k}^{{\rm eff}{~{}}\dagger}$.

###  IV-B Scheme-2: Global Polarization Reconfiguration

This scheme determines the polarization pre-post coding in Section III with
the full $N_{r}\times N_{t}$ channel impulse response matrix regardless of
$L_{t}$, the number of Tx antenna elements to be selected. Based on the global
effective channel impulse response matrix $H^{\rm eff}$ rather than any
partial channel impulse response matrix $\widetilde{H}_{k}^{\rm eff}$, $L_{t}$
number of Tx antenna elements are selected to maximize the effective SNR
caused by MRT. Thus, the optimal polarization vectors for both Rx and Tx are
aligned with (13) – (16). We denominate Scheme-2 as Global polarization
reconfiguration; for the consistency, the optimal Rx and Tx polarization
vectors of Scheme-2 are, respectively, expressed as

$\displaystyle\vec{p}_{{\rm Rx},i}^{\rm~{}G}=\vec{p}_{{\rm
Rx},i}^{\rm~{}opt}{~{}},$ (36) $\displaystyle\vec{p}_{{\rm
Tx},j}^{\rm~{}G}=\vec{p}_{{\rm Tx},j}^{\rm~{}opt}{~{}}.$ (37)

The partial channel impulse response matrix associated with the selected
$L_{t}$ number of polarization reconfigurable Tx antenna elements,
$\widetilde{H}^{\rm eff}_{{\rm G},k}$ can be described as

$\displaystyle{}\widetilde{H}^{\rm eff}_{{\rm G},k}{}={}$ (38)
$\displaystyle{}\begin{bmatrix}{}(\vec{p}_{{\rm Rx},1}^{~{}\rm
G})^{T}H_{1\alpha_{k1}}\vec{p}_{{\rm Tx},\alpha_{k1}}^{~{}\rm
G}\ldots(\vec{p}_{{\rm Rx},1}^{\rm~{}G})^{T}H_{1\alpha_{kL_{t}}}\vec{p}_{{\rm
Tx},\alpha_{kL_{t}}}^{~{}\rm G}\\\
\vdots~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\ddots~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
(\vec{p}_{{\rm Rx},N_{\rm r}}^{~{}\rm G})^{T}H_{N_{\rm
r}\alpha_{k1}}\vec{p}_{{\rm Tx},\alpha_{k1}}^{~{}\rm
G}\cdot\cdot(\vec{p}_{{\rm Rx},N_{\rm r}}^{\rm~{}G})^{T}H_{N_{\rm
r}\alpha_{kL_{t}}}\vec{p}_{{\rm Tx},\alpha_{kL_{t}}}^{~{}\rm G}\end{bmatrix}.$

The effective SNR in Scheme-2 is

$\displaystyle\gamma_{\rm PR-HS/MRT}^{\rm
G}=\underset{k}{\mathrm{max}}\big{(}\widetilde{\sigma}_{k,{\max}}^{\rm
G}\big{)}^{2},$ (39)

where $\big{(}\widetilde{\sigma}_{k,{\rm max}}^{\rm G}\big{)}^{2}$ is a
maximum eigenvalue of $\widetilde{H}_{{\rm G},k}^{\rm eff}\widetilde{H}_{{\rm
G},k}^{{\rm eff}{~{}}\dagger}$.

###  IV-C Comparison of Two Schemes

The effective SNR’s provided by PR-HS/MRT in Scheme-1 and Scheme-2 are
proportional to $\big{(}\widetilde{\sigma}_{k,{\rm max}}^{\rm EW}\big{)}^{2}$
and $\big{(}\widetilde{\sigma}_{k,{\rm max}}^{\rm G}\big{)}^{2}$,
respectively, for the given $k$, i.e., for the same set of $L_{t}$ Tx antenna
elements.

$\big{(}\widetilde{\sigma}_{k,{\rm max}}^{\rm EW}\big{)}^{2}$ is a maximum
eigenvalue of $\widetilde{H}_{{\rm EW},k}^{\rm eff}\widetilde{H}_{{\rm
EW},k}^{{\rm eff}{~{}}\dagger}$, which is described as

$\displaystyle{}\widetilde{H}_{{\rm EW},k}^{\rm eff}\widetilde{H}_{{\rm
EW},k}^{{\rm eff}{~{}}\dagger}{}={}$ (40)
$\displaystyle{}\begin{bmatrix}(\vec{p}_{\rm Rx,1}^{\rm~{}EW})^{T}H^{\rm
EW}_{\rm Rx,1}\vec{p}_{\rm
Rx,1}^{\rm~{}EW}~{}~{}~{}~{}\ldots~{}~{}(\vec{p}_{\rm
Rx,1}^{\rm~{}EW})^{T}H^{\rm Off}_{\rm Rx,1N_{r}}\vec{p}_{\rm
Rx,N_{r}}^{\rm~{}EW}\\\
\vdots~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\ddots~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
(\vec{p}_{\rm Rx,N_{r}}^{\rm~{}EW})^{T}H^{\rm Off}_{\rm
Rx,N_{r}1},\vec{p}_{\rm Rx,1}^{\rm~{}EW}\ldots(\vec{p}_{\rm
Rx,N_{r}}^{\rm~{}EW})^{T}H^{\rm EW}_{\rm Rx,N_{r}},\vec{p}_{\rm
Rx,N_{r}}^{\rm~{}EW}\end{bmatrix}$

On the other hand, $\big{(}\widetilde{\sigma}_{k,{\rm max}}^{\rm
G}\big{)}^{2}$ is a maximum eigenvalue of $\widetilde{H}_{{\rm G},k}^{\rm
eff}\widetilde{H}_{{\rm G},k}^{{\rm eff}{~{}}\dagger}$ that is expressed as

$\displaystyle\widetilde{H}_{{\rm G},k}^{\rm eff}\widetilde{H}_{{\rm
G},k}^{{\rm eff}{~{}}\dagger}{}={}$ (41)
$\displaystyle\begin{bmatrix}(\vec{p}_{\rm Rx,1}^{\rm~{}G})^{T}H^{\rm G}_{\rm
Rx,1}\vec{p}_{\rm Rx,1}^{\rm~{}G}~{}~{}~{}\ldots~{}~{}(\vec{p}_{\rm
Rx,1}^{\rm~{}G})^{T}H^{\rm Off}_{\rm Rx,1N_{r}}\vec{p}_{\rm
Rx,N_{r}}^{\rm~{}G}\\\
\vdots~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\ddots~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\vdots\\\
(\vec{p}_{\rm Rx,N_{r}}^{\rm~{}G})^{T}H^{\rm Off}_{\rm Rx,N_{r}1},\vec{p}_{\rm
Rx,1}^{\rm~{}G}\ldots(\vec{p}_{\rm Rx,N_{r}}^{\rm~{}G})^{T}H^{\rm G}_{\rm
Rx,N_{r}},\vec{p}_{\rm Rx,N_{r}}^{\rm~{}G}\end{bmatrix},$

where the _global Rx-polarization-determinant matrix_ , $H^{\rm G}_{\rm Rx,i}$
is

$\displaystyle H^{\rm G}_{\rm Rx,i}=\sum_{j=1}^{L_{\rm
t}}H_{i\alpha_{ij}}\vec{p}_{{\rm Tx},\alpha_{ij}}\vec{p}_{{\rm
Tx},\alpha_{ij}}^{{~{}}T}H_{i\alpha_{ij}}^{\dagger}{~{}}.$ (42)

The diagonal components of (40) are maximized with EW polarization
reconfiguration, since each $\widetilde{H}^{\rm eff}_{{\rm EW},k}$ is tuned
according to its optimal polarization vectors, as described in (30) – (33).
Hence, effective SNR in (35) is maximized. Meanwhile, polarization vectors in
global polarization reconfiguration are optimized for the full channel impulse
response matrix; therefore, diagonal components of (41) are not maximized for
$L_{t}<N_{t}$. That is, those diagonal elements are aligned with (13) – (16),
which maximizes diagonal elements of $H^{\rm eff}(H^{\rm eff})^{\dagger}$.
Nonetheless, global polarization reconfiguration also satisfactorily increases
the effective SNR of the system even when applied to each $\widetilde{H}^{\rm
eff}_{{\rm G},k}$. Further, as $L_{t}$ approaches $N_{t}$,
$\widetilde{H}_{{\rm EW},k}^{\rm eff}$ approaches $\widetilde{H}_{{\rm
G},k}^{\rm eff}$. Although EW polarization reconfiguration outperforms global
polarization reconfiguration, the latter offers an advantage in the complexity
and the corresponding computation time.

##  V Effect of polarization on the channel

###  V-A Random Polarization Vectors

One of the primary contributions of this paper is the statistical analysis of
the polarization reconfigurable wireless channels for the scenarios of PR-MIMO
and PR-HS/MRT. This section analyzes the impact of deploying polarization
reconfigurable antenna elements on the conventional MIMO system. Random
polarization is first considered; then, the impact of the proposed
polarization pre-post coding on the MIMO channel is scrutinized. The elements
of polarization-basis matrix, described in (2), are modeled as independent and
identically distributed (i.i.d.) complex Gaussian random variables with zero
mean and unit variance. Each element of $H^{\rm eff}$ can be expressed as

$\displaystyle h_{ij}^{\rm eff}$ $\displaystyle=$
$\displaystyle\vec{p}^{~{}T}_{{\rm Rx},i}\begin{bmatrix}h^{\rm vv}_{ij}&h^{\rm
vh}_{ij}\\\ h^{\rm hv}_{ij}&h^{\rm hh}_{ij}\end{bmatrix}\vec{p}_{{\rm
Tx},j}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (43)
$\displaystyle=$ $\displaystyle h^{\rm
vv}_{ij}\cos(\theta_{j})\cos(\theta_{i})+h^{\rm
hv}_{ij}\sin(\theta_{i})\cos(\theta_{j})$ $\displaystyle+h^{\rm
vh}_{ij}\cos(\theta_{i})\sin(\theta_{j})+h^{\rm
hh}_{ij}\sin(\theta_{j})\sin(\theta_{i}).$ $\displaystyle=$
$\displaystyle\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4}~{}.$

For convenience, we abbreviate the four terms in (43) as $\beta_{1}$,
$\beta_{2}$, $\beta_{3}$ and $\beta_{4}$, respectively, in the order of the
expression in (43). Considering the random movement and rotation of the UE
along with depolarization of the wireless channel itself [20, 25], the
probability density function (pdf) of the polarization angles,
$f_{\Theta}(\theta)$ can be expressed as

$\displaystyle
f_{\Theta}(\theta)=\left\\{\begin{array}[]{ll}\frac{1}{2\pi}~{},&\quad\rm if\
\theta\ \in\left[-\pi,\pi\right].\\\ \\\ 0~{},&\quad\rm
otherwise.\end{array}\right.$ (47)

The random variable, $\Theta\sim U(-\pi,\pi)$, can be transformed to
$Y=\cos(\Theta)$ and $X=\sin(\Theta)$, where we are interested in the pdf’s of
$Y$ and $X$. The cumulative distribution functions (cdf’s) of $\Theta$ and
$Y=\cos(\Theta)$, i.e., $F_{\Theta}$ and $F_{Y}$, respectively, are as

$\displaystyle
F_{\Theta}(\theta)=\dfrac{\theta}{\pi}{~{}},{~{}~{}~{}}F_{Y}(y)=\dfrac{\arccos(y)}{\pi}{~{}}.$
(48)

Based on (48) the pdf of $Y$, $f_{Y}$ is derived; in the similar fashion, the
pdf of $X=\sin(\Theta)$, $f_{X}$ is obtained as

$\displaystyle
f_{Y}(y)=\dfrac{1}{\pi}\dfrac{1}{\sqrt{1-y^{2}}},{~{}~{}~{}}f_{X}(x)=\dfrac{1}{\pi}\dfrac{1}{\sqrt{1-x^{2}}}{~{}}.$
(49)

The distributions of $Y=\cos(\Theta)$ and $X=\sin(\Theta)$ are portrayed in
Fig. 4, where $\Theta$ is independently generated $10^{6}$ times. The curve
fitting of the empirical distribution exhibits excellent agreement with the
theoretical analysis in (49).

Figure 4: Distribution of the elements in polarization vector:
$Y=\cos(\Theta)$ and $X=\sin(\Theta)$

.

The mean and variance of (49) and (49) are 0 and 1/2, respectively;
furthermore, the variance of $h^{\rm eff}$ is

$\displaystyle{\rm Var}(h^{\rm eff}_{ij})={\rm Var}(\beta_{1})+{\rm
Var}(\beta_{2})+{\rm Var}(\beta_{3})+{\rm Var}(\beta_{4}).$ (50)

Owing to the page limit, the extensive description of deriving expectation and
variance of (49) as well as (50) is provided in Appendix A. Three terms in
each $\beta_{i}$ for $i\in\\{1,2,3,4\\}$ are independent of each other. Thus,
the variance of the real and imaginary parts in $\beta_{i}$ is
$\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}=\frac{1}{8}$; consequently,
$h^{\rm eff}_{ij}$ has unit variance as described in detail in Appendix A. It
is noteworthy that the mean and variance of reconfigurable antennas with
random channel polarization are the same as those of conventional channels
without polarization reconfigurable antenna elements. Furthermore, the squared
envelope of $h^{\rm eff}_{ij}$, with random polarization, follows chi-square
distribution with 2 degrees of freedom.

###  V-B Optimal Polarization Vectors

It is worth emphasizing that combination of antenna selection and polarization
reconfiguration, i.e., PR-HS can provide significant improvement in channel
gain, i.e., the squared envelop of $h^{\rm eff}_{ij}$. The proposed
polarization pre-post coding scheme is based on the closed-form derivation for
the optimal polarization vectors at one end; whereas, the globally optimal
polarization vectors at both ends of the Tx and Rx are achieved by the
iterative methodology. Hence, the complete analysis to reach the closed-form
description is unfeasible. However, the comprehensive simulation results in
Fig. 5 show that the proposed PR-HS scheme improves the system performance in
terms of SNR and the associated distribution of channel gain.

Empirical distributions of channel gain, i.e., $h^{\rm eff}_{ij}$ are depicted
in Fig. 5, where the PR-MIMO system applied PR-HS antenna selection with
$N_{t}=8$ and $L_{t}\in\\{1,2,...,8\\}$. In each channel realization, the
average of $h^{\rm eff}_{ij}$ exhibits significantly improved distribution in
terms of mean, variance and the overall tendency of the distributions.

As $L_{t}$ increases to reach $N_{t}=8$, the empirical distribution converge
to chi-square distribution with 4 degrees of freedom, which is the normal case
with conventional system without proposed schemes and algorithms in this
paper. The scenario that $L_{t}=3$ still has substantial benefit of PR-HS
scheme. The distribution is not chi-square distributed, since the selection
process leads to ordered statistics. However, we can observe empirically that
the resulting distribution has a mean similar to that of a chi-square
distribution with 8 degrees of freedom; however, lower/better variance. In the
scenario that $L_{t}=1$, the mean of the empirical distribution is similar to
that of chi-square distribution with 12 degrees of freedom. In contrast, the
variance of the empirical distribution is lower than chi-square distribution
with 12 degrees of freedom, which will be helpful for the gNB to accomplish
stable link adaptation including adaptive modulation and coding.

Without loss of generality, an element in $H^{\rm eff}$ has the following
description of its squared envelop.

$\displaystyle|h^{\rm eff}_{11}|^{2}=\vec{p}_{{\rm
Rx},1}^{~{}T}\big{(}H_{11}\vec{p}_{{\rm Tx},1}\vec{p}_{{\rm
Tx},1}^{~{}T}H_{11}^{\dagger}\big{)}\vec{p}_{{\rm Rx},1}~{}.$ (51)

From the Analysis of (51), we can have an intuition for the mathematical
interpretation of the aforementioned simulation results. However, owing to the
page limit, Appendix B provides further analysis on (51).

Figure 5: Empirical distribution of $|h^{\rm eff}_{ij}|^{2}$ in PR-HS; the
comparison of chi-square distribution with 4, 5, 8, 12 degrees of freedom.

##  VI Statistics of Achievable SNR in PR-HS/MRT

###  VI-A Upper and Lower Bounds

The analytical solution of (27) cannot be easily obtained; however, for the
$k$-th set of $L_{t}$ Tx antenna elements, the closed form analysis of upper
and lower bounds of the effective SNR can be described as

$\displaystyle\dfrac{1}{{\rm
min}(L_{t},N_{r})}\sum_{m}\big{(}\widetilde{\sigma}^{\rm
EW}_{k,m}\big{)}^{2}\leq\big{(}\widetilde{\sigma}^{\rm EW}_{k,{\rm
max}}\big{)}^{2}\leq\sum_{m}\big{(}\widetilde{\sigma}^{\rm
EW}_{k,m}\big{)}^{2},$ (52)

i.e., the achievable SNR for a polarization reconfigurable channel matrix is
lower bounded by the average of the nonzero eigenvalues of
$\widetilde{H}_{{\rm EW},k}^{\rm eff}\widetilde{H}_{{\rm EW},k}^{{\rm
eff}{~{}}\dagger}$ and upper bounded by the sum of the nonzero eigenvalues of
$\widetilde{H}_{{\rm EW},k}^{\rm eff}\widetilde{H}_{{\rm EW},k}^{{\rm
eff}{~{}}\dagger}$. We take into account EW polarization reconfiguration
(Scheme-1), since it outperforms global polarization reconfiguration
(Scheme-2). The upper bound of achievable SNR in PR-HS scheme is

$\displaystyle\gamma_{\rm UpperBound}$ $\displaystyle=$
$\displaystyle\underset{k}{\mathrm{max}}\bigg{(}\sum_{m}\big{(}\widetilde{\sigma}^{\rm
EW}_{k,m}\big{)}^{2}\bigg{)}$ $\displaystyle=$
$\displaystyle\underset{k}{\mathrm{max}}\bigg{(}\sum_{i}\sum_{j}|\widetilde{h}^{\rm
eff}_{{\rm EW},k,ij}|^{2}\bigg{)}.$

In a similar manner, the lower bound is described as,

$\displaystyle\gamma_{\rm LowerBound}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\min(L_{t},N_{r})}\underset{k}{\mathrm{max}}\bigg{(}\sum_{m}\big{(}\widetilde{\sigma}^{\rm
EW}_{k,m}\big{)}^{2}\bigg{)}$ $\displaystyle=$
$\displaystyle\dfrac{\underset{k}{\mathrm{max}}\bigg{(}\sum_{i}\sum_{j}|\widetilde{h}^{\rm
eff}_{{\rm EW},k,ij}|^{2}\bigg{)}}{\min(L_{t},N_{r})}~{},$

The joint statistics of the ordered SNRs can be described as

$\displaystyle\rho_{\gamma(j)}(\gamma_{(1)},\gamma_{(2)},...,\gamma_{(N_{t})}){}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$
(55)
$\displaystyle=\left\\{\begin{array}[]{ll}N_{t}!\prod\limits_{i=1}^{N_{t}}\dfrac{1}{\Gamma(N_{r})}\gamma_{(j)}^{\rm
N_{r}-1}\\\ \times{\rm exp}\big{(}-\gamma_{(j)}\big{)},&{\rm
for}~{}\gamma_{(1)}<\gamma_{(2)}<...\gamma_{(N_{t})}\\\ \\\ 0,&\rm
otherwise\end{array}\right.$

where $\Gamma(.)$ is the Euler Gamma Function [29]. We utilize $L_{t}$ out of
$N_{t}$ variables $\gamma_{(i)}$; choose the $k$-th row of the column index
matrix $C_{\rm Index}$ in (25) that provides maximum achievable SNR in PR-
HS/MRT. The desired $\gamma_{\rm UpperBound}$ can be written in terms of the
ordered SNRs as

$\gamma_{\rm UpperBound}=\sum_{i=1}^{L_{t}}\gamma_{(j)}.$ (56)

$\gamma_{\rm UpperBound}$ in (56) is the sum of random variables; the
corresponding characteristic function is useful for further analysis. It is
described as [29]

$\displaystyle\Phi(j\nu){}={}$ (57)
$\displaystyle{~{}~{}~{}~{}~{}~{}}\dfrac{N_{t}!}{[\Gamma(N_{r})]^{N_{t}}}\int_{0}^{\infty}d\gamma_{(1)}\gamma_{(1)}^{N_{r}-1}e^{-\gamma_{(1)}}e^{-j\nu\Xi(L_{t}-1)\gamma_{(1)}}$
$\displaystyle{~{}~{}~{}~{}~{}~{}}\times\int_{0}^{\gamma_{(1)}}d\gamma_{(2)}\gamma_{(2)}^{N_{r}-1}e^{-\gamma_{(2)}}e^{-j\nu\Xi(L_{t}-1)\gamma_{(2)}}...~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle{~{}~{}~{}~{}~{}~{}}\times\int_{0}^{\gamma_{(N_{t}-1)}}d\gamma_{(N_{t})}\gamma_{(N_{t})}^{N_{r}-1}e^{-\gamma_{(N_{t})}}e^{-j\nu\Xi(L_{t}-1)\gamma_{(N_{t})}},~{}$

where $\Xi(x)$ is the Heaviside step function

$\displaystyle\Xi(x)=\left\\{\begin{array}[]{ll}1,&\quad\rm if\ x\leq 0,\\\
\\\ 0,&\quad\rm otherwise.\end{array}\right.$ (61)

In the following, we abbreviate the expression $1+j\nu\Xi(L_{t}-i)$ as
$a_{i}$, dropping the dependence on $\nu$ for notational convenience. The rest
of derivation is refered from [29].

The characteristic function of $\gamma_{\rm UpperBound}$ is finally provided
as

$\displaystyle\Phi(j\nu)$
$\displaystyle=\dfrac{N_{t}!}{[\Gamma(N_{r})]^{N_{t}}}\bigg{[}d^{N_{t}-1}(N_{r}-1)!a_{1}^{-N_{r}}$
(62)
$\displaystyle+\sum_{\rho=1}^{N_{t}-1}\sum_{t=0}^{(N_{t}-\rho+1)(N_{r}-1)}\hat{c}_{\rho,t}^{(N_{t}-1)}t!(b_{p}^{N_{t}})^{-(t+1)}\bigg{]}.$

###  VI-B Channel Capacity and Symbol Error Rate

The instantaneous channel capacity for each channel realization is given by

$\displaystyle C(\gamma_{\rm PR-HS/MRT}^{\rm x})=\rm
log_{2}\big{[}1+\bar{\Gamma}(\gamma_{\rm PR-HS/MRT}^{\rm x})\big{]},$ (63)

where ${\rm x}\in\\{\rm EW,G\\}$ and $\bar{\Gamma}$ is the average SNR of a
single-input single-output (SISO) channel. Lower and upper bounds of the
system can be applied to this equation by substituting $\gamma_{\rm
LowerBound}$ and $\gamma_{\rm UpperBound}$ for $\gamma_{\rm PR-HS/MRT}$ in
(63). In the case of quadrature phase shift keying (QPSK) modulation, the
instantaneous SER is

$\displaystyle SER_{n}=1-\bigg{[}1-Q\bigg{(}\sqrt{\bar{\Gamma}(\gamma_{\rm PR-
HS/MRT}^{\rm x})_{n}}\bigg{)}\bigg{]}^{2},$ (64)

where $n=1,2,....N$ and $N$ is total number of channel realizations. Finally,
average SER is represented as

$\displaystyle\overline{SER}=\dfrac{1}{N}\sum_{n=1}^{N}SER_{n}~{}.$ (65)

##  VII Numerical Experiments and Results

###  VII-A Simulations in PR-MIMO

In this section we provide evaluations of our closed-form equations for
optimum polarization vectors and the associated PR-MIMO channel capacity. We
also compare the results to brute-force numerical optimization, where we step
through all possible (discretized) values of the Tx/Rx-polarization angle and
choose the optimal values that correspond to the maximum capacity for each PR-
MIMO channel realization. The step width of the brute-force numerical search
are, respectively, $1^{\circ}$ in Figs. 6 and 7; $5^{\circ}$ in Figs. 8 and 9;
$10^{\circ}$ in Figs. 10 and 11. Unless stated otherwise, we consider
independent and identically distributed (i.i.d.) Rayleigh fading channels with
a cross-polarization discrimination, ${\rm XPD}=0{\rm~{}dB}$.

We first investigate the impact of polarization reconfiguration in several
deterministic channels. The PR-MIMO channel capacity in a $2\times 2$ system
with polarization reconfigurable antenna elements is shown for varying Tx-
polarization angles in Figs. 6 and 7. The relatively low SNR regime such as 5
dB and the very high SNR regime such as 30 dB are shown in Figs. 6 and 7,
respectively. In both scenarios, the channel capacity obtained by the
polarization precoding at the Tx exhibits negligible difference from that
yielded by the numerical result as indicated in both figures. For the 5 dB SNR
in Fig. 6, the theoretically derived optimal Tx-polarization angles themselves
have insignificant differences from numerically derived optimal Tx-
polarization angles. In contrast, in the very high SNR regime in Fig. 7, the
differences between theoretically and numerically obtained optimal Tx-
polarization angles are considerable. This is due to the fact that the
approximation (8) is less accurate at higher SNRs. However, the difference in
capacity is still remarkably small. In case of polarization postcoding at the
Rx, similar results are attained owing to the symmetry described in Sections
III-A; therefore, the results are omitted here. It is worth mentioning that
optimal Tx- or Rx-polarization vectors are not necessarily orthogonal, which
corresponds to $90^{\circ}$ difference in Tx- or Rx-polarization angles, as
described in Figs. 6 and 7.

Figure 6: Channel capacity for varying Tx-polarization; 5 dB SNR. Figure 7:
Channel capacity for varying Tx-polarization; 30 dB SNR.

The PR-MIMO channel capacity with the optimal Tx-polarization angles is
demonstrated to depend on varying Rx-polarization angles in Fig. 8, where we
also consider the $2\times 2$ RP-MIMO system with polarization-agile antenna
elements at both ends of the Tx and the Rx. The channel capacity exhibits
substantial variation from 21.47 bits/sec/Hz to 13.60 bits/sec/Hz depending on
Rx-polarization angles, although Tx-polarization angles are already set to the
optimum obtained by brute-force numerical search as will be shown in Fig. 9,
i.e., $\theta_{{\rm Tx}-1}=80^{\circ}$; $\theta_{{\rm Tx}-2}=50^{\circ}$. This
result obviously shows that the polarization mismatching between the Tx and
the Rx can cause significant deterioration in the capacity of the whole
system, even if one end of the Tx and the Rx is already set to the optimal
polarization. However, the proposed scheme of joint polarization pre-post
coding also shows negligible difference from numerical results in both optimal
Tx- and Rx-polarization angles and channel capacity.

Local optimal Tx- and Rx-polarization vectors at each iteration of joint
polarization pre-post coding are depicted in Fig. 9 considering the same
scenario of channel impulse matrices and SNR as that in Fig. 8. The $2\times
2$ PR-MIMO system is again considered in this figure. Here, one iteration is a
sequential loop of polarization precoding and then polarization postcoding as
described in Section III-B. Tx/Rx-polarization vectors quickly converge;
moreover, these global optimal polarization vectors show relatively small
difference from the numerical optimum.

Figure 8: Channel capacity for varying Rx-polarization with optimal Tx-
polarization; 30 dB SNR. Figure 9: Optimal Tx/Rx-polarization angles for the
number of iterations; 30 dB.

In contrast to Figs. 6 – 9, from this point on, we investigate how much the
joint polarization pre-post coding improves the PR-MIMO channel capacity in a
statistical sense in Figs. 10 – 12. The simulation results are based on the
$10^{4}$ times realizations of i.i.d. Rayleigh fading channels. Figures 10 and
11 depict cumulative density functions (cdf’s) of the $2\times 2$ PR-MIMO
channel capacity at 5 dB SNR and 30 dB SNR, respectively, for the scenarios of
joint polarization pre-post coding with Tx/Rx polarization-agile antennas; and
random Tx/Rx-polarization. It is noteworthy that the cdf resulting from the
fixed Tx/Rx-polarization exhibits exactly the same cdf of random Tx/Rx-
polarization at each channel realization owing to the random generation of the
i.i.d. Rayleigh fading channels. Furthermore, the cdf’s of the optimal and the
worst Tx/Rx-polarization obtained by brute-force numerical search at each
channel realization, are presented in Figs. 10 and 11. We perform five
iterations for the joint polarization pre-post coding at each PR-MIMO channel
realization, which results in the satisfactory convergence of Tx/Rx-
polarization vectors on the global optimal ones as demonstrated in Fig. 9.

Figure 10: cdf’s of channel capacity for the scenarios of joint polarization
pre-post coding with five iterations, random Tx/Rx-polarization; optimal and
the worst Tx/Rx-polarization through brute-force numerical search at 5 dB SNR.
Figure 11: cdf’s of channel capacity for the scenarios of joint polarization
pre-post coding with five iterations, random Tx/Rx-polarization; optimal and
the worst Tx/Rx-polarization through brute-force numerical search at 30 dB
SNR.

Our joint polarization pre-post coding significantly improves the PR-MIMO
channel capacity at both 5 dB SNR and 30 dB SNR. In Fig. 10, the probability
of the PR-MIMO channel capacity less than 4.2 bits/sec/Hz is 0.75 with 5 dB
SNR and random Tx/Rx-polarization; whereas, with the proposed joint
polarization pre-post coding, the probability of the capacity greater than 4.2
bits/sec/Hz is 0.94. For the improvement of the PR-MIMO channel capacity at 30
dB SNR in Fig. 11, the probability of the capacity greater than 20 bits/sec/Hz
is 0.75 in the scenario of utilizing joint polarization pre-post coding, while
that probability is just 0.12 with random Tx/Rx-polarization. It is noteworthy
that the cdf curves of random Tx/Rx-polarization scenarios in both Figs. 10
and 11 can be regarded as the expectation of the cdf curves in a statistical
sense when the joint polarization pre-post coding is not utilized; however,
the practical channel capacity would exhibit substantial variations between
the cdf curves associated with the optimal and the worst Tx/Rx-polarization
obtained by brute-force numerical search in both figures.

The joint polarization pre-post coding achieves significant improvement of PR-
MIMO channel capacity so that its cdf curve approaches the best cdf curve
obtained by brute-force numerical search, in particular, at 5 dB SNR in Fig.
10. The cdf curve corresponding to the proposed scheme also exhibits slight
difference from the best cdf curve obtained from brute-force numerical search
at 30 dB SNR in Fig. 11, compared to the significant difference between that
best-scenario cdf and the cdf of random Tx/Rx-polarization or, more
pronouncedly, between the best- and the worst-scenario cdf’s obtained by the
numerical search.

Finally, we compare channel capacity for varying SNRs and varying number of
polarization-agile antennas in Fig. 12. For each setting of the number of
polarization-agile antennas, Fig. 12 demonstrates three scenarios: joint
polarization pre-post coding at both link ends; Tx-polarization precoding only
at the transmitter; and random Tx/Rx-polarization as the baseline control. In
the high SNR regime, utilizing our joint polarization pre-post coding improves
PR-MIMO channel capacity with around 5 dB, 4 dB, and 3dB SNR gains in the
cases of $2\times 2$, $3\times 3$, and $4\times 4$ PR-MIMO channels,
respectively.

Moreover, it is noteworthy that in a relatively low SNR regime below 9 dB,
$2\times 2$ and $3\times 3$ PR-MIMO systems adopting the proposed joint
polarization pre-post coding accomplishes almost the same channel capacity as,
respectively, $3\times 3$ and $4\times 4$ MIMO systems with random Tx/Rx-
polarization. We also note that the PR-MIMO system with our joint polarization
pre-post coding shows better channel capacity even than the PR-MIMO system
that has one more antenna element at both link ends of the transmitter and
receiver and uses random polarization, in the low SNR regime (below 3 dB).

Figure 12: PR-MIMO channel capacity for the varying SNR in the scenarios of
random Tx/Rx-polarization, Tx-polarization precoding, and joint polarization
pre-post coding.

###  VII-B Simulations in PR-HS/MRT

In this section, we present simulation results focusing on the system
performance when applying the proposed PR-HS/MRT scheme. The results include
improved effective SNR, channel capacity and SER; the comparison of EW and
global polarization reconfiguration schemes in terms of performance and
selected Tx antenna indices; and verification of the theoretically derived SER
in PR-HS/MRT via Monte-Carlo simulations. Unless otherwise stated, following
system parameters will be used: $\bar{\Gamma}={\rm 20~{}dB},N_{t}=8$ and
$N_{r}=2$ (MRC).

Figure 13: cdf curves of channel capacity for PR-HS/MRT with EW and global
polarization reconfiguration schemes, conventional HS/MRT (Random
Polarization), upper and lower bounds; with a variety of $L_{t}$.

Simulation results in Fig. 13 show the impact of PR-HS/MRT schemes on the
channel capacity. Regardless of the number of selected Tx antenna elements,
$L_{t}$, the proposed PR-HS/MRT schemes outperform the conventional HS/MRT
without polarization reconfiguration,i.e., the case of Random Polarization in
the legend of Fig. 13, by around 1 bit/s/Hz. Further, it is verified that EW
polarization reconfiguration (Scheme-1) has better performance than global
polarization reconfiguration (Scheme-2). However, the difference in
performance decreases as $L_{t}$ increases. Hence, global polarization
reconfiguration can also be a good suboptimal scheme considering the lower
complexity and computation time than those of EW polarization reconfiguration,
in particular, when $L_{t}\geq 4$ in the scenario of Fig. 13.

Figure 14: SER curves for PR-HS/MRT with EW and global polarization
reconfiguration schemes and conventional HS/MRT (random polarization); with a
variety of $L_{t}$..

Effective SER in PR-HS/MRT is also significantly lower than that in
conventional HS/MRT, showing approximately 3 dB SNR gain to for $10^{-3}$ SER
in every case of $L_{t}$, as described in Fig. 14. The SER curves are based on
the analytical result in (65) considering QPSK modulation as a baseline.
Besides, the difference in SER performance between EW and global polarization
reconfiguration schemes is inconsiderable for a variety of $L_{t}$ in the
scenario of Fig. 14. This is not surprising, given the straigthforward mapping
between SNR and SER for the chosen modulation.

Figure 15: Validation of analytical SER curves via Monte-Carlo simulation

Monte-Carlo simulation results in Fig. 15 validate the analytical SER curves
based on (65). SER curves are generated via Monte-Carlo simulation, where
$10^{7}$ randomly generated symbols are transmitted based on PR-HS/MRT at the
Tx, and MRC at the Rx. The scenarios of $L_{t}=1,2,5$ and $8$ are considered,
and SER curves resulted in Monte-Carlo simulation shows good agreement with
the analytical ones with with average SER.

Another observation of the selected Tx antenna indices in PR-HS/MRT and
conventional HS/MRT, is given in Tables I – III. Even for a set of selected Tx
antenna elements based on the conventional HS/MRT, the channel capacity and
SER performance is improved via joint polarization pre-post coding after the
selection. However, that selection of Tx antenna elements is different from
the selection based on the proposed PR-HS/MRT. Moreover, selected antenna
indices in EW polarization reconfigurable scheme is also different from those
in global polarization reconfigurable scheme. Tables I – III illustrate the
selected Tx antenna indices in 10 independent channel realizations for
$L_{t}=1,4$ and $6$.

The conventional HS/MRT scheme does not show full-matching of selected antenna
indices with any of two PR-HS/MRT schemes. On the other hand, the two PR-
HS/MRT schemes, i.e., EW and global polarization reconfiguration, have a
considerable number of cases in which their selected antenna indices fully
match. As described in Tables I – III, the greater $L_{t}$ is, the more
selected antenna indices are matched between proposed two PR-HS/MRT schemes.
It is worth emphasizing that estimation of optimal polarization vectors before
the hybrid antenna selection stage is inevitable to have full benefit of joint
polarization pre-post coding.

Random | G | EW | Matching Index
---|---|---|---
7 | 3 | 4 | 0
1 | 5 | 4 | 0
6 | 1 | 7 | 0
6 | 4 | 4 | 1
2 | 4 | 7 | 0
5 | 6 | 6 | 1
7 | 5 | 8 | 0
7 | 4 | 4 | 1
3 | 2 | 7 | 0
3 | 4 | 4 | 1
TABLE I: Selected Tx Antenna Index when $L_{t}=1$ Random | G | EW | Matching Index
---|---|---|---
1 3 7 8 | 1 3 4 8 | 1 3 4 8 | 4
1 2 5 8 | 2 3 5 8 | 2 3 5 8 | 4
3 5 6 7 | 5 6 7 8 | 1 3 6 7 | 2
1 2 3 8 |  1 3 5 7 |  1 3 5 8 | 3
3 4 7 8 | 2 3 5 8 | 1 3 5 8 | 3
2 5 6 8 | 1 2 4 6 |  2 4 6 8 | 3
1 4 5 8 | 1 4 5 6 | 2 4 5 6 | 3
1 2 5 8 |  1 5 7 8 |  1 3 5 8 | 3
2 3 5 8 | 2 3 4 6 | 2 3 4 6 | 4
1 3 6 7 |  1 3 4 8 |  1 3 4 5 | 3
TABLE II: Selected Tx Antenna Index when $L_{t}=4$ Random | G | EW | Matching Index
---|---|---|---
1 2 3 4 7 8 | 1 2 3 5 6 7 | 1 2 3 5 6 7 | 6
1 3 5 6 7 8 | 1 3 5 6 7 8 | 1 3 5 6 7 8 | 6
1 3 4 5 7 8 |  1 2 4 5 7 8 |  1 2 3 4 7 8 | 5
2 3 4 5 6 8 |  1 2 3 5 6 7 |  1 2 3 4 7 8 | 4
1 2 3 4 7 8 | 2 3 4 6 7 8 | 2 3 4 6 7 8 | 6
1 3 4 5 7 8 | 1 2 4 5 6 7 | 1 2 4 5 6 7 | 6
1 2 4 6 7 8 |  1 3 4 6 7 8 |  1 2 4 6 7 8 | 5
2 3 4 5 6 8 | 1 2 3 5 6 8 | 1 2 3 5 6 8 | 6
1 2 3 4 5 7 | 1 2 3 6 7 8 |  2 3 4 5 7 8 | 4
1 2 5 6 7 8 | 2 3 4 5 6 7 | 2 3 4 5 6 7 | 6
TABLE III: Selected Tx Antenna Index when $L_{t}=6$

##  VIII Conclusion

In this paper, we proposed several novel schemes to support PR-MIMO spatial
multiplexing and PR-HS/MRT whose system is composed of multiple polarization
reconfigurable antenna elements at both the Tx and Rx. In the proposed
iterative joint polarization pre-post coding, the local optimum usually
reached the global optimum of Tx/Rx-polarization vectors within five
iterations. The proposed method offers an energy-efficient as well as cost-
efficient method to significantly increase channel capacity in PR-MIMO spatial
multiplexing. Furthermore, we proposed two PR-HS/MRT schemes, i.e., EW and
global polarization reconfiguration, and both schemes remarkably outperformed
the conventional HS/MRT without polarization reconfigurable antennas. The
theoretical analysis along with extensive simulation results demonstrate the
good performance of the schemes proposed in this paper.

## Acknowledgment

Part of this work was supported financially by CSULB Foundation Fund,
RS261-00181-10185.

##  Appendix A Expectation and Variance of $Y=\cos\Theta$, $Y=\sin\Theta$ and
$h^{\rm eff}_{ij}$

To find the expectation and variance of $Y=\cos{\Theta}$ and $X=\sin{\Theta}$,
we start from (49). The expectation is found by solving the integral

$\displaystyle E[Y]=\dfrac{1}{\pi}\int_{-1}^{1}\dfrac{dy}{\sqrt{1-y^{2}}}.$
(66)

To solve, we substitute $u=1-y^{2}$, $du=-2ydy$ and $dy=\dfrac{1}{-2x}du$.
Therefore,

$\displaystyle E[Y]$ $\displaystyle=$
$\displaystyle\dfrac{1}{2\pi}\int\dfrac{1}{\sqrt{u}}du$ $\displaystyle=$
$\displaystyle\dfrac{1}{2\pi}\sqrt{u}=\bigg{[}\dfrac{1}{2\pi}\sqrt{1-y^{2}}\bigg{]}\begin{matrix}1\\\
-1\end{matrix}$ $\displaystyle=$ $\displaystyle 0~{}.$

We find variance by solving the integral

$\displaystyle Var(Y)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}y^{2}f_{Y}(y)dy-\big{(}E[Y]\big{)}^{2}$
$\displaystyle=$
$\displaystyle\dfrac{1}{\pi}\int_{-1}^{1}y^{2}\dfrac{1}{\sqrt{1-y^{2}}}dy~{}.$

The second term of (A) disappears because expectation is 0. Substitute
$y=\sin{u}$, $dy=\cos{u}~{}du$, and $u=\arcsin{y}$,

$\displaystyle Var(Y)$ $\displaystyle=$
$\displaystyle\dfrac{1}{\pi}\int\dfrac{\sin^{2}{u}}{\sqrt{1-\sin^{2}{u}}}\cos{u}~{}du$
$\displaystyle=$
$\displaystyle\dfrac{1}{\pi}\int\dfrac{\sin^{2}{u}}{\cos{u}}\cos{u}~{}du$
$\displaystyle=$ $\displaystyle\dfrac{1}{2\pi}\int 1-\cos{2u}du$
$\displaystyle=$
$\displaystyle\dfrac{1}{2\pi}u-\dfrac{1}{4\pi}\sin{2u}=\dfrac{1}{2\pi}(u-\sin{u}\cos{u})$
$\displaystyle=$
$\displaystyle\dfrac{1}{2\pi}\bigg{(}\arcsin(y)-y\sqrt{1-y^{2}}\bigg{)}\begin{matrix}1\\\
-1\end{matrix}$ $\displaystyle Var(Y)$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}~{}.$

To find $Var(h^{\rm eff}_{ij})$, we start from (43)

$\displaystyle h_{ij}^{\rm eff}$ $\displaystyle=$
$\displaystyle\vec{p}^{~{}T}_{{\rm Rx},i}\begin{bmatrix}h^{\rm vv}_{ij}&h^{\rm
vh}_{ij}\\\ h^{\rm hv}_{ij}&h^{\rm hh}_{ij}\end{bmatrix}\vec{p}_{{\rm
Tx},j}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (70)
$\displaystyle=$ $\displaystyle h^{\rm
vv}_{ij}\cos(\theta_{j})\cos(\theta_{i})+h^{\rm
hv}_{ij}\sin(\theta_{i})\cos(\theta_{j})$ $\displaystyle+h^{\rm
vh}_{ij}\cos(\theta_{i})\sin(\theta_{j})+h^{\rm
hh}_{ij}\sin(\theta_{j})\sin(\theta_{i})$ $\displaystyle=$
$\displaystyle\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4}~{}.$

Therefore, the variance is

$\displaystyle Var(h^{\rm eff}_{ij})$ $\displaystyle=$ $\displaystyle
E\big{[}(\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4})^{2}\big{]}$
$\displaystyle-\bigg{(}E[\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4}]\bigg{)}^{2}~{}.$

All $\beta_{i}$’s are independent from each other; hence second term is
addition of expectations of each $\beta_{i}$’s. Since they are zeros, they are
eliminated from the equation.

When first term is foiled, it consists of squares and cross terms of
$\beta_{i}$’s. The cross terms cancel out because elements of polarization
basis matrix are independent from each other. Therefore, the variance is

$\displaystyle Var(h^{\rm eff}_{ij})$ $\displaystyle=$ $\displaystyle
E\big{[}\beta^{2}_{1}+\beta^{2}_{2}+\beta^{2}_{3}+\beta^{2}_{4}\big{]}$
$\displaystyle=$ $\displaystyle
E[\beta^{2}_{1}]+E[\beta^{2}_{2}]+E[\beta^{2}_{3}]+E[\beta^{2}_{4}]$
$\displaystyle=$ $\displaystyle
Var(\beta_{1})+Var(\beta_{2})+Var(\beta_{3})+Var(\beta_{4})~{}.$

Variance of product of independent random variable is product of variance of
the random variables. Note that each three terms of $\beta_{i}$’s, where
$i\in\\{1,2,3,4\\}$ are independent. Therefore, the variance of each
$\beta_{i}$ is product of variances of three terms. Moreover, because each
polarization-basis matrix element consists of real and imaginary part, we
analyze the two parts separately. Both real and imaginary parts of all
$\beta_{i}$’s, therefore, have variance of 1/8. Then, these variances are
added to yield variance of $h^{\rm eff}_{ij}$, (50); that is 1/2 for both real
and imaginary parts. Hence, real and imaginary part of $h^{\rm eff}_{ij}$ has
means of 0 and variances of 1/2, which brings the element back to $H_{\rm o}$.

##  Appendix B Elaboration of Polarization Determinant Matrix

From (51),

$\displaystyle|h^{\rm eff}_{11}|^{2}=\vec{p}_{{\rm
Rx},1}^{~{}T}\big{(}H_{11}\vec{p}_{{\rm Tx},1}\vec{p}_{{\rm
Tx},j}^{~{}T}H_{11}^{\dagger}\big{)}\vec{p}_{{\rm
Rx},1}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (73)
$\displaystyle=\big{[}\cos(\theta_{{\rm Rx},1})~{}\sin(\theta_{{\rm
Rx},1})\big{]}\begin{bmatrix}h^{\rm PD}_{{\rm Rx},11}&h^{\rm PD}_{{\rm
Rx},12}\\\ h^{\rm PD}_{{\rm Rx},21}&h^{\rm PD}_{{\rm
Rx},22}\end{bmatrix}\begin{bmatrix}\cos(\theta_{{\rm Rx},1})\\\
\sin(\theta_{{\rm Rx},1})\end{bmatrix},$

where

$\displaystyle h^{\rm PD}_{{\rm Rx},11}$ $\displaystyle=$
$\displaystyle|h^{\rm vv}_{11}|^{2}\cos^{2}({\theta_{{\rm Tx},1}})$
$\displaystyle+h^{\rm vh}_{11}(h^{\rm vv}_{11})^{*}\sin(\theta_{{\rm
Tx},1})\cos({\theta_{{\rm Tx},1}})$ $\displaystyle+h^{\rm vv}_{11}(h^{\rm
vh}_{11})^{*}\sin(\theta_{{\rm Tx},1})\cos({\theta_{{\rm Tx},1}})$
$\displaystyle+|h^{\rm vh}_{11}|^{2}\sin^{2}({\theta_{{\rm Tx},1}})$
$\displaystyle h^{\rm PD}_{{\rm Rx},12}$ $\displaystyle=$ $\displaystyle
h^{\rm vv}_{11}(h^{\rm hv}_{11})^{*}\cos^{2}({\theta_{{\rm Tx},1}})$
$\displaystyle+h^{\rm vh}_{11}(h^{\rm hv}_{11})^{*}\sin(\theta_{{\rm
Tx},1})\cos({\theta_{{\rm Tx},1}})$ $\displaystyle+h^{\rm vv}_{11}(h^{\rm
hh}_{11})^{*}\sin(\theta_{{\rm Tx},1})\cos({\theta_{{\rm
Tx},1}})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle+h^{\rm vh}_{11}(h^{\rm hh}_{11})^{*}\sin^{2}({\theta_{{\rm
Tx},1}})$ $\displaystyle h^{\rm PD}_{{\rm Rx},21}$ $\displaystyle=$
$\displaystyle h^{\rm hv}_{11}(h^{\rm vv}_{11})^{*}\cos^{2}({\theta_{{\rm
Tx},1}})$ $\displaystyle+h^{\rm hh}_{11}(h^{\rm vv}_{11})^{*}\sin(\theta_{{\rm
Tx},1})\cos({\theta_{{\rm Tx},1}})$ $\displaystyle+h^{\rm hv}_{11}(h^{\rm
vh}_{11})^{*}\sin(\theta_{{\rm Tx},1})\cos({\theta_{{\rm Tx},1}})$
$\displaystyle+h^{\rm hh}_{11}(h^{\rm vh}_{11})^{*}\sin^{2}({\theta_{{\rm
Tx},1}})$ $\displaystyle h^{\rm PD}_{{\rm Rx},22}$ $\displaystyle=$
$\displaystyle|h^{\rm hv}_{11}|^{2}\cos^{2}({\theta_{{\rm Tx},1}})$
$\displaystyle+h^{\rm hh}_{11}(h^{\rm hv}_{11})^{*}\sin(\theta_{{\rm
Tx},1})\cos({\theta_{{\rm Tx},1}})$ $\displaystyle+h^{\rm hv}_{11}(h^{\rm
hh}_{11})^{*}\sin(\theta_{{\rm Tx},1})\cos({\theta_{{\rm Tx},1}})$
$\displaystyle+|h^{\rm hh}_{11}|^{2}\sin^{2}({\theta_{{\rm Tx},1}})~{}.$

Note that the greatest values that we can acquire are the squared enveloped
terms. The elements of polarization determinant matrices can be transformed
into a different form with trigonometric properties. They result as

$\displaystyle|h^{\rm eff}_{11}|^{2}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4}\bigg{\\{}|h^{\rm vv}_{11}|^{2}+|h^{\rm
vh}_{11}|^{2}+|h^{\rm hv}_{11}|^{2}+|h^{\rm hh}_{11}|^{2}$
$\displaystyle+\cos{(2\theta_{{\rm Tx},1})}\big{[}|h^{\rm
vv}_{11}|^{2}-|h^{\rm vh}_{11}|^{2}$ $\displaystyle+|h^{\rm
hv}_{11}|^{2}-|h^{\rm hh}_{11}|^{2}\big{]}$ $\displaystyle+\sin{(2\theta_{{\rm
Tx},1})}\big{[}h^{\rm vh}_{11}(h^{\rm vv}_{11})^{*}+h^{\rm vv}_{11}(h^{\rm
vh}_{11})^{*}$ $\displaystyle+h^{\rm hh}_{11}(h^{\rm hv}_{11})^{*}+h^{\rm
hv}_{11}(h^{\rm hh}_{11})^{*}\big{]}\bigg{\\}}$
$\displaystyle+\dfrac{1}{4}\cos{(2\theta_{{\rm Rx},1})}\bigg{\\{}|h^{\rm
vv}_{11}|^{2}+|h^{\rm vh}_{11}|^{2}$ $\displaystyle+|h^{\rm
hv}_{11}|^{2}+|h^{\rm hh}_{11}|^{2}$ $\displaystyle+\cos{(2\theta_{{\rm
Tx},1})}\big{[}|h^{\rm vv}_{11}|^{2}-|h^{\rm
vh}_{11}|^{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle+|h^{\rm hv}_{11}|^{2}-|h^{\rm hh}_{11}|^{2}\big{]}$
$\displaystyle+\sin{(2\theta_{{\rm Tx},1})}\big{[}h^{\rm vh}_{11}(h^{\rm
vv}_{11})^{*}+h^{\rm vv}_{11}(h^{\rm vh}_{11})^{*}$ $\displaystyle+h^{\rm
hh}_{11}(h^{\rm hv}_{11})^{*}+h^{\rm hv}_{11}(h^{\rm
hh}_{11})^{*}\big{]}\bigg{\\}}$ $\displaystyle+\dfrac{1}{4}\sin{(2\theta_{{\rm
Rx},1})}\bigg{\\{}h^{\rm vh}_{11}(h^{\rm vv}_{11})^{*}+h^{\rm vv}_{11}(h^{\rm
vh}_{11})^{*}$ $\displaystyle+h^{\rm hh}_{11}(h^{\rm hv}_{11})^{*}+h^{\rm
hv}_{11}(h^{\rm hh}_{11})^{*}$ $\displaystyle+\cos{(2\theta_{{\rm
Tx},1})}\big{[}h^{\rm vh}_{11}(h^{\rm vv}_{11})^{*}-h^{\rm vv}_{11}(h^{\rm
vh}_{11})^{*}$ $\displaystyle+h^{\rm hh}_{11}(h^{\rm hv}_{11})^{*}-h^{\rm
hv}_{11}(h^{\rm hh}_{11})^{*}\big{]}$ $\displaystyle+\sin{(2\theta_{{\rm
Tx},1})}\big{[}h^{\rm hh}_{11}(h^{\rm vv}_{11})^{*}+h^{\rm hv}_{11}(h^{\rm
vh}_{11})^{*}$ $\displaystyle+h^{\rm vh}_{11}(h^{\rm hv}_{11})^{*}+h^{\rm
hv}_{11}(h^{\rm hh}_{11})^{*}\big{]}\bigg{\\}}~{}.$

Depending on the channel condition, squared envelopes of elements of
polarization-basis matrix are maximized with optimal values of $\theta_{{\rm
Tx},j}$ and $\theta_{{\rm Rx},i}$ in each iteration.

$|h^{\rm eff}_{11}|^{2}$ consists of four squared envelope of the elements in
polarization-basis matrix, i.e., $|h^{\rm vv}_{11}|^{2}$, $|h^{\rm
vh}_{11}|^{2}$, $|h^{\rm hv}_{11}|^{2}$ and $|h^{\rm hh}_{11}|^{2}$. Besides,
$|h^{\rm eff}_{11}|^{2}$ includes cross-term products between the elements in
polarization-basis matrix. Squared envelope terms outcome higher values than
that of the product of the cross terms; therefore, we focus on utilizing the
squared envelope terms rather than the cross terms. Each aforementioned
squared envelope terms follow chi-square distribution with 2 degrees of
freedom; hence, summing them will result degrees of freedom up to 8. However,
the four terms are multiplied by one fourth of $\cos(2\theta_{{\rm Tx},1})$
and one fourth of $\cos(2\theta_{{\rm Rx},1})$; therefore, each four squared
envelope term will be multiplied by a weight less than unity. The weighted
terms are then summed up at the end. Tx and Rx polarization angles are
adjusted to yield greater squared envelope terms. Since squared envelope terms
have weights less than unity and are superimposed, distribution will not
always fit to degrees of freedom of an integer. Such case is represented in
Fig. 5.

## References

  * [1] S.-C. Kwon and A. F. Molisch, “Capacity maximization with polarization-agile antennas in the MIMO communication system,” _Proc. IEEE Global Telecommun. Conf._ , pp. 1–6, Dec. 2015.
  * [2] P. Oh and S. Kwon, “Multi-polarization superposition beamforming with XPD-aware transmit power allocation,” _Proc. IEEE Veh. Technol. Conf._ , pp. 1–6, Dec. 2020.
  * [3] J. Zhang, K. J. Kim, A. A. Glazunov, Y. Wang, L. Ding, and J. Zhang, “Generalized polarization-space modulation,” _IEEE Trans. Commun._ , vol. 68, no. 1, pp. 258–273, Jan. 2020.
  * [4] A. Sousa de Sena, D. Benevides da Costa, Z. Ding, and P. H. J. Nardelli, “Massive MIMO-NOMA networks with multi-polarized antennas,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 12, pp. 5630–5642, 2019.
  * [5] L. A. Gutierrez, S. An, S. Kwon, and H. G. Yeh, “Novel approach of spatial modulation: Polarization-aware OFDM subcarrier allocation,” _Proceedings of IEEE Green Energy and Smart Systems Conference (IGESSC), 2020_ , pp. 1–6, 2020.
  * [6] K. Satyanarayana, T. Ivanescu, M. El-Hajjar, P. . Kuo, A. Mourad, and L. Hanzo, “Hybrid beamforming design for dual-polarised millimetre wave MIMO systems,” _IET Electronics Letters_ , vol. 54, no. 22, pp. 1257–1258, 2018.
  * [7] X. Zhang, B. Zhang, and D. Guo, “Performance of poly-polarization multiplexing in narrow-band wireless communication aided by pre-compensation and multi-notch OPPFs,” _IEEE Commun. Let._ , vol. 6, no. 4, pp. 478–481, Aug 2017.
  * [8] G. Zafari, M. Koca, and H. Sari, “Dual-polarized spatial modulation over correlated fading channels,” _IEEE Trans. Commun._ , vol. 65, no. 3, pp. 1336–1352, Mar. 2017.
  * [9] P.-Y. Qin, S.-L. Chen, and Y. J. Guo, “A compound reconfigurable microstrip antenna with agile polarizations and steerable beams,” _Proceedings of IEEE International Symposium on Antennas and Propagation (ISAP), 2017_ , pp. 1–2, 2017.
  * [10] G. Wolosinski, V. Fusco, and O. Malyuskin, “2-bit polarisation agile antenna with high port decoupling,” _Electronics Letters_ , vol. 52, no. 4, pp. 255–256, 2016.
  * [11] B. Babakhani, S. K. Sharma, and N. R. Labadie, “A frequency agile microstrip patch phased array antenna with polarization reconfiguration,” _IEEE Trans. Antennas Propag._ , vol. 64, no. 10, pp. 4316–4327, 2016.
  * [12] H. Sun and S. Sun, “A novel reconfigurable feeding network for quad-polarization-agile antenna design,” _IEEE Trans. Antennas Propag._ , vol. 64, no. 1, pp. 311–316, 2016.
  * [13] L.-P. Cai and K.-K. M. Cheng, “Continuously tunable polarization agile antenna design using a novel varactor-only signal control device,” _IEEE Trans. Antennas Propag._ , vol. 16, pp. 1147–1150, 2017.
  * [14] Y.-J. Liao, H.-L. Lin _et al._ , “Polarization reconfigurable eccentric annular ring slot antenna design,” _IEEE Trans. Antennas Propag._ , vol. 63, no. 9, pp. 4152–4155, 2015.
  * [15] S.-C. Kwon, “Optimal power and polarization for the capacity of polarization division multiple access channels,” in _Proc. IEEE Global Telecommun. Conf._ , Dec. 2014, pp. 1–5.
  * [16] J. Jootar, J.-F. Diouris, and J. Zeidler, “Performance of polarization diversity in correlated Nakagami-m fading channels,” _IEEE Transactions on Vehicular Technology_ , vol. 55, no. 1, pp. 128–136, 2006.
  * [17] Y. Deng, A. Burr, and G. White, “Performance of MIMO systems with combined polarization multiplexing and transmit diversity,” in _Proc. IEEE Veh. Technol. Conf._ , vol. 2, May 2005, pp. 869–873.
  * [18] R. Nabar, H. Bolcskei, V. Erceg, D. Gesbert, and A. Paulraj, “Performance of multiantenna signaling techniques in the presence of polarization diversity,” _IEEE Trans. Sig. Proc._ , vol. 50, no. 10, pp. 2553–2562, Oct. 2002.
  * [19] S.-C. Kwon and G. Stüber, “Polarization division multiple access on NLoS wide-band wireless fading channels,” _IEEE Trans. Wireless Commun._ , vol. 13, no. 7, pp. 3726–3737, Jul. 2014.
  * [20] S.-C. Kwon, “Geometrical theory, modeling and applications of channel polarization,” _Ph.D. Thesis, Georgia Institute of Technology_ , pp. 1–128, 2014.
  * [21] M. R. Andrews, P. P. Mitra, and R. deCarvalho, “Tripling the capacity of wireless communications using electromagnetic polarization,” _Nature_ , vol. 409, pp. 316–318, Jan. 2001.
  * [22] V. Erceg, P. Soma, D. Baum, and S. Catreux, “Multiple-input multiple-output fixed wireless radio channel measurements and modeling using dual-polarized antennas at 2.5 GHz,” _IEEE Trans. Wireless Commun._ , vol. 3, no. 6, pp. 2288 – 2298, Nov. 2004.
  * [23] M. Shafi, M. Zhang, A. Moustakas, P. Smith, A. Molisch, F. Tufvesson, and S. Simon, “Polarized MIMO channels in 3-D: Models, measurements, and mutual information,” _IEEE J. Sel. Areas Commun._ , vol. 24, no. 3, pp. 514–527, Mar. 2006.
  * [24] M. Landmann, K. Sivasondhivat, J.-I. Takada, I. Ida, and R. Thoma, “Polarisation behaviour of discrete multipath and diffuse scattering in urban environments at 4.5 GHz,” _EURASIP J. Wireless Commun. Netw._ , no. 1, p. 60, Jan. 2007 Article ID 57980.
  * [25] S.-C. Kwon and G. L. Stüber, “Geometrical theory of channel depolarization,” _IEEE Trans. Veh. Technol._ , vol. 60, no. 8, pp. 3542–3556, Oct. 2011.
  * [26] D. Landon and C. Furse, “Recovering handset diversity and MIMO capacity with polarization-agile antennas,” _IEEE Trans. Antennas Propag._ , vol. 55, no. 11, pp. 3333–3340, Nov 2007.
  * [27] O. Karabey, S. Bildik, S. Bausch, S. Strunck, A. Gaebler, and R. Jakoby, “Continuously polarization agile antenna by using liquid crystal-based tunable variable delay lines,” _IEEE Trans. Antennas Propag._ , vol. 61, no. 1, pp. 70–76, Jan 2013.
  * [28] P. Oh and S. Kwon, “Multi-polarization superposition beamforming with xpd-aware transmit power allocation,” in _2020 IEEE 92nd Vehicular Technology Conference (VTC2020-Fall)_ , 2020, pp. 1–6.
  * [29] A. Molisch, M. Win, and J. Winters, “Reduced-complexity transmit/receive-diversity systems,” _IEEE Transactions on Signal Processing_ , vol. 51, no. 11, pp. 2729–2738, 2003.
  * [30] M. Win and J. Winters, “Virtual branch analysis of symbol error probability for hybrid selection/maximal-ratio combining in Rayleigh fading,” _IEEE Transactions on Communications_ , vol. 49, no. 11, pp. 1926–1934, 2001\.
  * [31] ——, “Analysis of hybrid selection/maximal-ratio combining in Rayleigh fading,” in _1999 IEEE International Conference on Communications (Cat. No. 99CH36311)_ , vol. 1, 1999, pp. 6–10 vol.1.
  * [32] D. Gore, R. Heath, and A. Paulraj, “Transmit selection in spatial multiplexing systems,” _IEEE Communications Letters_ , vol. 6, no. 11, pp. 491–493, 2002\.
  * [33] A. Molisch, M. Win, Y.-S. Choi, and J. Winters, “Capacity of mimo systems with antenna selection,” _IEEE Transactions on Wireless Communications_ , vol. 4, no. 4, pp. 1759–1772, 2005.
  * [34] V. R. Anreddy and M. A. Ingram, “Wlc11-6: Antenna selection for compact dual-polarized MIMO systems with linear receivers,” in _IEEE Globecom 2006_ , 2006, pp. 1–6.
  * [35] A. Habib, “Multiple polarized MIMO with antenna selection,” in _2011 18th IEEE Symposium on Communications and Vehicular Technology in the Benelux (SCVT)_ , 2011, pp. 1–8.
  * [36] A. Molisch and M. Win, “MIMO systems with antenna selection,” _IEEE Microwave Magazine_ , vol. 5, no. 1, pp. 46–56, Mar 2004.
  * [37] A. F. Molisch, _Wireless Communications_ , 2nd ed. Wiley and IEEE, 2010.
  * [38] D. Tse and P. Viswanath, _Fundamentals of Wireless Communication_ , 1st ed. Cambridge, 2005.
  * [39] G. Strang, _Linear Algebra and Its Applications_ , 4th ed. Thomson Learning, Inc., 2005.