# Improving Spectral Efficiency via Pilot Assignment and Subarray Selection
Under Realistic XL-MIMO Channels

Gabriel Avanzi Ubiali, Taufik Abrão, José Carlos Marinello
This work was partly supported by the National Council for Scientific and
Technological Development (CNPq) of Brazil under Grants 310681/2019-7 and
163568/2021-9 (scholarship), and in part by State University of Londrina
(UEL), PR, Brazil.G. A. Ubiali and T. Abrão are with the Electrical
Engineering Department, State University of Londrina, PR, Brazil. Rod. Celso
Garcia Cid - PR445, s/n, Campus Universitário, Po.Box 10.011. CEP: 86057-970.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>C. Marinello is with
Electrical Engineering Department, Federal University of Technology PR,
Cornélio Procópio, PR, Brazil. CEP: 86300-000<EMAIL_ADDRESS>
###### Abstract

The main requirements for 5G and beyond connectivity include a uniform high
quality of service, which can be attained in crowded scenarios by extra-large
MIMO (XL-MIMO) systems. Another requirement is to support an increasing number
of connected users in (over)crowded machine-type communication (mMTC). In such
scenarios, pilot assignment (PA) becomes paramount to reduce pilot
contamination and consequently improve spectral efficiency (SE). We propose a
novel quasi-optimal low-complexity iterative pilot assignment strategy for XL-
MIMO systems, based on a genetic algorithm (GA). The proposed GA-based PA
procedure turns the quality of service more uniform, taking into account the
normalized mean-square error (NMSE) of channel estimation from each candidate
of the population. The simulations reveal that the proposed iterative
procedure minimizes the channel estimation NMSE averaged over the UEs. The
second procedure is the subarray (SA) selection. In XL-MIMO systems, commonly
a UE is close to a SA antenna subset such that a sufficient data rate can be
achieved if only a specific SA serves that UE. Thus, a SA selection procedure
is investigated to make the system scalable by defining the maximum number of
UEs each SA can help. Hence, the SA clusters are formatted based on the PA
decision. Furthermore, we introduce an appropriate channel model for XL-MIMO,
which considers a deterministic LoS component with a distance-dependent
probability of existence combined with a stochastic spatially correlated
Rayleigh NLoS fading component. The developed simulations and analyses rely on
this suitable channel model under realistic assumptions of pilot contamination
and correlated channels.

###### Index Terms:

Extra-large-scale antenna array, non-stationary channels, channel model,
spectral efficiency, pilot assignment, subarray selection

##  I Introduction

One of the main enabling technologies for achieving high data rates in
wireless networks is Massive multiple-input multiple-output (mMIMO). The
massive number of channel observations, at the AP antennas, increases the
array gain and the ability of using the spatial domain to discriminate the
desired signal from the interfering signals, in the uplink (UL), and to
transmit the signals very directively, in the downlink (DL). Indeed, the UEs
that are close to one of the APs, i.e., located at a cell-center area, will
experience high data rates; however, the UEs that are relatively distant from
any AP, i.e., located at a cell-edge area, may have bad quality of service
(QoS) due to low signal-to-noise-ratio (SNR) and to the inter-cell
interference, which comes from the UEs in the neighboring cells, due to the
reuse of pilot sequences and to the inability of an AP to cancel the
interference from the other cells’ UEs, since it only estimates the channel of
the UEs it is serving.

Thus, despite the current cellular networks can achieve high peak data rates,
the QoS could result in unreliable due to significant data rate variations
within the cell. However, wireless access is supposed to be ubiquitous since
payments, navigation, and entertainment rely on wireless connectivity. It is
more important for future mobile networks to provide more uniform high rates
over the coverage area than to increase the peak rates. More uniform high
rates can be achieved under large antenna arrays dimensionality, the so-called
extra-large MIMO (XL-MIMO) networks, which can serve relatively large areas
such as stadiums and shopping malls densely populated. XL-MIMO networks can
turn the QoS more uniform across the entire cell area by achieving moderate to
high macro-diversity and reducing the average distance from a UE to the
closest AP antenna.

An XL-MIMO antenna array is split into subarray (SA) arrangements. As each UE
is physically close to a subset of SAs only, the configuration where all the
SAs serve all the UEs is impractical and unnecessary. Limiting the set of SAs
that serve each UE can negatively affect the QoS, but if the SA selection is
made judiciously, the system can still provide suitable SE, while also
reducing the computational complexity and the amount of signaling and control
data that needs to be transferred from the SAs to the CPU and vice-versa.
Furthermore, SA selection procedure is essential in XL-MIMO systems to
guarantee scalability, i.e., to assure that the complexity will not go to
infinity as the number of UEs grows unbounded.

Knowledge of the UE–SA channel responses is paramount to coherent processing.
The most efficient way to estimate the channels is to consider a TDD scheme
where we only need to transmit pilots in the uplink. Depending on the users’
mobility, the size of the normalized coherence block may be smaller than the
number of UEs. The mutually orthogonal pilot sequences must be reused among
the UEs in this case. Most of XL-MIMO papers consider perfect channel
estimation [1, 2, 3, 4, 5, 6], which is unrealistic. During the UL pilot
transmission, the estimation error is generated by two sources: the background
noise and the signals coming from the pilot-sharing UEs. The last one results
in so-called coherent interference. In this paper, we develop our analysis
assuming imperfect channel estimations to obtain realistic numerical
simulation results.

The XL-MIMO channel can be described as the composition of a deterministic
line-of-sight (LoS) path and a stochastic non-LoS (NLoS) component. Due to the
low probability of LoS occurrence, generally, the LoS component modeling is
not distinguished between suburban and urban macro scenarios; indeed, it is
selectable for the urban micro scenario only [7]. Usually, research papers in
mMIMO only consider the NLoS channel component. On the other hand, papers in
CF-mMIMO usually consider either LoS or NLoS component. However, due to the
uncertainties in the propagation environment in urban areas, it is impossible
to accurately determine the presence of a LoS link[8], and a probabilistic LoS
model is necessary. Furthermore, the LoS component contributes considerably to
the total channel power[9], since it is associated to a shorter propagation
distance and to a slower signal attenuation with the distance. Particularly,
in XL-MIMO systems, the distance between a UE and the BS antenna array may be
small relative to the XL-array length, so the LoS probability should not be
neglected. In this paper, we introduce a realistic channel model that
incorporates a deterministic LoS component, with a distance-dependent
probability of existence, and a spatially correlated NLoS component. Shadow
fading correlation between two different links is also considered.

Depending on the mobility, the size of the coherence block may be smaller than
the number of UEs. In this case, the mutually orthogonal pilots must be reused
among the UEs and can be appropriately assigned to them to limit the pilot
contamination[10]. The most simple and naive approach is to give pilots
randomly. However, it cannot prevent closely located UEs from being assigned
the same pilot, leading to high interference and system performance
degradation. On the other extreme, the exhaustive search, which tries all
possible assignments and chooses the best one, leads to a computational
complexity that presents a combinatorial growth with the number of active UEs.
Thus, only suboptimal methods are feasible in practice. One approach is to
optimize a utility function concerning one UE at a time, considering each UE
once or iterating until convergence [10]. Several suboptimal algorithms
consider a golden rule that closely located UEs should not be assigned the
same pilot.

Works on cellular mMIMO usually consider that two UEs within the same cell are
assigned mutually orthogonal pilot sequences. If the pilot reuse factor is
one, the performance of cell-edge UEs may be degraded by pilot-sharing UEs of
adjacent cells. In this scenario, the pilot contamination problem can be
reasonably relieved greedily by simply assigning the best pilots, i.e., the
ones that result in the most negligible pilot contamination to the cell-edge
UEs[11]. In [12], the problem of pilot assignment in multi-cell mMIMO systems
is tackled by using a deep reinforcement learning (DRL) scheme. The distance
and the angle-of-arrival (AoA) information of the users are deployed to define
the cost function, representing the pilot contamination effects. The DRL
algorithm is applied to find an effective pilot assignment strategy by
tracking the changes in the environment, learning the near-optimal pilot
assignment, and achieving comparable performance to that of the optimum
(genie) PA performed by exhaustive search.

However, due to the increased density of UEs in typical applications of XL-
MIMO systems and the scarcity of pilots, crucial tasks, such as pilot
assignment, network access, and user scheduling, become challenging [13]. In
our work, we consider a crowded XL-MIMO scenario where the number of UEs in
the coverage area is more significant than the number of orthogonal pilot
sequences, representing a scenario with pilot scarcity. A greedy algorithm
would assign the orthogonal pilots randomly to some of the UEs, and then
assign pilots to the remaining UEs to minimize the pilot contamination, each
UE at a time. Works in [2, 13, 14, 15] address the problem of random access
and pilot assignment in XL-MIMO, considering uncorrelated Rayleigh fading and
no subarray selection. Specifically, a joint random access and scheduling
protocol was proposed in [13], taking advantage of the different visibility
regions (VRs) that arise in the UE and XL-MIMO link, besides seeking UEs with
non-overlapping VRs to be scheduled in the same payload data pilot resource.
The difference to our work is that we focus on the pilot assignment problem
and develop an efficient PA algorithm for a probabilistic LoS/NLoS XL-MIMO
channel model based on a heuristic evolutionary genetic algorithm (GA)
approach and build up the analysis considering the employment of SA selection
to make the receive combining scalable.

The contribution of this work is threefold.

  * i)

We introduce a realistic XL-MIMO spatially correlated channel model, since
uncorrelated channels appear under extremely strict physical requirements.
Since the XL-MIMO antennas are not co-located, sometimes the UEs will be very
close to some SAs, such that the channel modeling must be a composition of a
sum of NLoS links and an LoS link, with a distance-dependent probability of
occurrence. To the best of our knowledge, all the works on XL-MIMO have
considered only one of these two components.

  * ii)

We propose a GA-based PA strategy for crowded XL-MIMO systems, mitigating
pilot contamination by approaching the optimal PA solution w.r.t. minimizing
the average normalized mean-square error (NMSE) of the channel estimates among
all the active UEs.

  * iii)

We provide extensive numerical results corroborating the effectiveness and
efficiency of the proposed SA selection and PA methodology for XL-MIMO
systems, exploring crowded scenarios by considering both the LoS and NLoS
channel components. All the analysis and obtained results of this paper are
based on the realistic assumption of imperfect channel estimation, the
employment of a scalable receive combiner, and a proper SA selection, which is
also essential to make the XL subarrays scalable.

The remainder of this paper is organized as follows. Section II describes the
proposed channel model for XL-MIMO. Section III deals with the XL-MIMO system
model, describing the signal model for the UL pilot and data transmissions and
the adopted channel estimator and receive combiner. Section IV presents the
considered strategies for PA in XL-MIMO and the adopted SA selection
algorithm. Finally, numerical results corroborating our findings are provided
in Section V, and final remarks and conclusions are presented in Section VI.

##  II XL-MIMO Channel Model

We consider an XL-MIMO antenna array with a $M$-antenna ULA, divided into $L$
SAs containing $N$ antennas each, such that $M=LN$, serving $K$ UEs. The
antennas in each SA are separated by a distance $d$. If $N$ is small, then
each SA will experience spatial stationarity, i.e., all its $N$ antennas will
experience the same condition of existence or absence of the LoS path
component. Each SA is connected to a central processing unit (CPU), which is
responsible for the SA cooperation. The height of the $l$th SA and the $k$th
UE are denoted by $z_{l}^{\text{SA}}$ and $z_{k}^{\text{UE}}$, respectively,
and $d_{kl}$ denotes the distance from the $k$th UE to the first antenna of
the $l$th SA. Fig. 1 illustrates the XL-MIMO antenna array (yellow bar)
highlighting the azimuth and elevation angles associated to the LoS path
between a given UE $k$ and a given SA $l$, represented by the red bar.

(a)

(b)

Figure 1: 3-D XL-MIMO representation, showing a) the ULA and the azimuth and
elevation angles corresponding to the LoS path between UE $k$ and SA $l$ and
b) the antenna spacing $d$ (inside SA $l$) and the distance between UE $k$ and
SA $l$.

The associated channel model with LoS and the NLoS components can be expressed
as:

$\textbf{h}_{kl}=\alpha_{kl}\textbf{h}_{kl}^{\text{LoS}}+\textbf{h}_{kl}^{\text{NLoS}},$
(1)

where
$\textbf{h}_{kl}^{\text{LoS}},\textbf{h}_{kl}^{\text{NLoS}},\textbf{h}_{kl}\in\mathbb{C}^{N}$
denote, respectively, the LoS, the NLoS and the probabilistic LoS/NLoS channel
vectors between UE $k$ and SA $l$, while $\alpha_{kl}$ is a Bernoulli random
variable indicating the presence ($\alpha_{kl}=1$) or absence
($\alpha_{kl}=0$) of a LoS component between the $k$th UE and the $l$th SA.

The LoS channel vector between the $k$th UE and the $l$th SA is given by[8]

$\textbf{h}_{kl}^{\text{LoS}}=\sqrt{G_{k}G_{l}}\frac{z_{k}^{\text{UE}}z_{l}^{\text{SA}}}{4\pi
d_{kl}}\sqrt{\mathcal{X}_{kl}^{\text{LoS}}}e^{-j2\pi\frac{d_{kl}}{\lambda}}\textbf{a}(\overline{\varphi}_{kl},\overline{\theta}_{kl}),$
(2)

where $G_{k}$ and $G_{l}$ denote the antenna gain at the $k$th UE and the
$l$th SA, respectively; $\lambda=c/f$ is the wavelength, being $c$ the speed
of light and $f$ the carrier frequency; $\mathcal{X}_{kl}^{\text{LoS}}$ is the
shadow fading of the LoS component; and
$\textbf{a}(\overline{\varphi}_{kl},\overline{\theta}_{kl})$ is the array
response vector of the ULA, regarding to the LoS ray. The azimuth and
elevation angles of the LoS path between the $k$th UE and the $l$th SA are
denoted by $\overline{\varphi}_{kl}$ and $\overline{\theta}_{kl}$,
respectively. The array response vector, as a function of the LoS azimuth and
elevation angles between UE $k$ and SA $l$, respectively
$\overline{\varphi}_{kl}$ and $\overline{\theta}_{kl}$, is given by

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\textbf{a}(\overline{\varphi}_{kl},\overline{\theta}_{kl})=\left[1,e^{-j2\pi\frac{d}{\lambda}\frac{\text{sin}(\overline{\varphi}_{kl})}{{\text{cos}(\overline{\theta}_{kl})}}},\dots,e^{-j2(N-1)\pi\frac{d}{\lambda}\frac{\text{sin}(\overline{\varphi}_{kl})}{{\text{cos}(\overline{\theta}_{kl})}}}\right]^{\text{T}},}$
(3)

where $d$ is the antenna spacing inside the $l$-th SA, Fig. 1(b). Indeed, the
array response vector $\textbf{a}(\cdot,\cdot)$ in eq. (3) accounts for the
phase differences among the signal that arrives in different antennas of the
same uniform linear SA (ULSA). These variations appear when the azimuth angle
is not zero, i.e., when the wavefront is not parallel to the ULA.

Near-field vs. Far-field. An important observation on XL-MIMO arrays is that
the wavefront propagation could result in spherical, since some users may be
in the near-field distance w.r.t. the entire XL-MIMO array, i.e., $d_{kl}\ll
d_{\rm Rayl}$, being $d_{\rm Rayl}$ the Rayleigh distance [16]. Also, such
conditions will result in different propagation distances for different SAs.
On the other hand, since the SA aperture is small compared to the UE-SA
propagation distances, one can assume that all users are in the far-field
propagation from each SA perspective, such that we can assume that the
wavefront propagation is planar to the SA point of view. The implication is
that the difference between the propagation distance for two antennas
belonging to the same SA is negligible, such that the corresponding large-
scale fading (LSF) coefficients are nearly the same for all antenna elements
of the same SA. However, even this negligible difference is sufficient to
produce phase differences among these antennas. Such phase differences are
accounted by the array response vector in (3).

The LSF channel gain for the LoS component in each XL-MIMO ULSA is determined
by the normalized trace as

$\beta_{kl}^{\text{LoS}}=\frac{1}{N}||\textbf{h}_{kl}^{\text{LoS}}||^{2}=\underbrace{\frac{G_{k}G_{l}(z_{k}^{\text{UE}}z_{l}^{\text{SA}})^{2}}{(4\pi)^{2}}}_{\beta_{0}}\frac{\mathcal{X}_{kl}^{\text{LoS}}}{d_{kl}^{2}}=\frac{\beta_{0}}{d_{kl}^{2}}\mathcal{X}_{kl}^{\text{LoS}},$
(4)

where $\beta_{0}$ is the average channel gain at the reference distance of 1
m, and $\beta_{0}/d_{kl}^{2}$ is the path-loss of the LoS component. Replacing
(4) in (2) gives

$\textbf{h}_{kl}^{\text{LoS}}=\sqrt{\beta_{kl}^{\text{LoS}}}e^{-j2\pi\frac{d_{kl}}{\lambda}}\textbf{a}(\overline{\varphi}_{kl},\overline{\theta}_{kl}).$
(5)

On the other hand, the NLoS component describes a practical spatially
correlated multi path environment. The NLoS channel vector can be obtained as
the superposition of $B$ physical signal paths, each one reaching the ULSA as
a plane wave from particular azimuth and elevation angles $\varphi_{kl}^{(b)}$
and $\theta_{kl}^{(b)}$ [17]:

$\textbf{h}_{kl}^{\text{NLoS}}=\sum_{b=1}^{B}g_{kl}^{(b)}\textbf{a}(\varphi_{kl}^{(b)},\theta_{kl}^{(b)}),$
(6)

where $g_{kl}^{(b)}$ is an i.i.d. complex random variable with zero mean and
variance $\mathbb{E}\\{|g_{kl}^{(b)}|^{2}\\}$, accounting for the gain and
phase rotation of the $b$th physical NLoS path, while $\varphi_{kl}^{(b)}$ and
$\theta_{kl}^{(b)}$ are, respectively, the SA azimuth and elevation angles
formed by the $b$th NLoS path between UE $k$ and SA $l$.

The NLoS LSF coefficient can be calculated as

$\beta_{kl}^{\text{NLoS}}=\frac{1}{N}\mathbb{E}\\{||\textbf{h}_{kl}^{\text{NLoS}}||^{2}\\}=\sum_{b=1}^{B}\mathbb{E}\\{|g_{kl}^{(b)}|^{2}\\},$
(7)

and is given by

$\beta_{kl}^{\text{NLoS}}=\frac{\beta_{0}}{d_{kl}^{\gamma}}\mathcal{X}_{kl}^{\text{NLoS}},$
(8)

where $\gamma$ is the path-loss exponent of the NLoS component, and
$\mathcal{X}_{kl}^{\text{NLoS}}$ is the shadow fading of the NLoS component.
Considering the law of large numbers, the sum of the $B$ random NLoS paths
approaches a normal distribution[17]:

$\textbf{h}_{kl}^{\text{NLoS}}\sim\mathcal{N}_{\mathbb{C}}(\mathbf{0},\textbf{R}_{kl}),$
(9)

with the positive semi-definite spatial correlation matrix $\textbf{R}_{kl}$
composed by the elements:

$[\textbf{R}_{kl}]_{m,n}=\beta_{kl}^{\text{NLoS}}\int\int
e^{j2(m-n)\pi\frac{d}{\lambda}\frac{\text{sin}(\varphi)}{\text{cos}(\theta)}}f_{kl}(\varphi,\theta)d\varphi
d\theta,$ (10)

where $f_{kl}(\varphi,\theta)$ is the joint probability density function
(jPDF) of $\varphi$ and $\theta$, which depends on the azimuth and elevation
angles of the LoS link ($\varphi_{kl}$ and $\theta_{kl}$) and on the angular
standard deviations ($\sigma_{\varphi}\geq 0$ and $\sigma_{\theta}\geq 0$).
The integration interval for $\varphi$ and $\theta$ is
$[-180^{\circ},180^{\circ}]$ and $(-90^{\circ},90^{\circ})$, respectively. As
in [10], we also consider the joint Gaussian distribution:

$f_{kl}(\varphi,\theta)=\frac{1}{2\pi\sigma_{\varphi}\sigma_{\theta}}\exp\left\\{-\frac{(\varphi-\overline{\varphi}_{kl})^{2}}{2\sigma_{\varphi}^{2}}-\frac{(\theta-\overline{\theta}_{kl})^{2}}{2\sigma_{\theta}^{2}}\right\\}.$
(11)

The proof of (9) and (10) can be found in Appendix A. From (10), notice that
$\text{tr}(\textbf{R}_{kl})=N\beta_{kl}^{\text{NLoS}}$, which matches with the
definition in (7).

Separating the large-scale and the small-scale fading, the NLoS channel vector
can be described as

$\textbf{h}_{kl}^{\text{NLoS}}=\sqrt{\beta_{kl}^{\text{NLoS}}}\cdot\ddot{\textbf{h}}_{kl},$
(12)

where $\ddot{\textbf{h}}_{kl}$ represents the spatially correlated small-scale
Rayleigh fading following a complex normal distribution[18, 10, 17]. Note that
the double-integral in (10) computes
$\mathbb{E}\left\\{e^{j2(m-n)\pi\frac{d}{\lambda}\frac{\text{sin}(\varphi)}{\text{cos}(\theta)}}\right\\}$
w.r.t. randomly located multipath components distributed according to the
joint PDF in (11).

In practical XL-MIMO panel scenarios, as the antennas of different SAs are
sufficiently distant, we assume the channel vectors of other SAs are
independently distributed, which means that
$\mathbb{E}\\{\ddot{\textbf{h}}_{kl}\ddot{\textbf{h}}_{kj}^{\text{H}}\\}=\mathbf{0}_{N\times
N}$ for $l\neq j$, where $(.)^{\text{H}}$ denotes the Hermitian operator.

Correlation between $F_{kl}$ and $F_{ij}$ (Shadowing). The shadow fading can
be modeled as a log-normal r.v., in dB scale as $F_{kl}=10\leavevmode\nobreak\
\text{log}_{10}(\mathcal{X}_{kl})$, following the distribution [19]

$F_{kl}\sim\mathcal{N}\left[0,\sigma_{\text{SF}}^{2}(1-e^{-d_{kl}/\delta})^{2}\right].$
(13)

Note that the variance of the r.v. $F_{kl}$ in (13) increases with the
distance and converges to $\sigma_{\text{SF}}^{2}$ when $d_{kl}>>\delta$,
where $\delta$ denotes the decorrelation distance. The shadow fading is
typically modeled as independent links, although in reality proximate links
present spatially correlated shadowing, since the signal is impacted by almost
the same set of objects in the propagation environment. Ignoring such
correlations results in over-estimating the diversity in XL-MIMO systems and
may produce significant differences in network performance[19]. The cross-
correlation between $F_{kl}$ and $F_{ij}$ is given by:

$\displaystyle\mathbb{E}\\{F_{kl}F_{ij}\\}=\frac{\sigma_{\text{SF}}^{2}}{2}\frac{(1-e^{-d_{kl}/\delta})(1-e^{-d_{ij}/\delta})}{\sqrt{(1+e^{-d_{kl}/\delta})(1+e^{-d_{ij}/\delta})}}(e^{-d_{kj}/\delta}+e^{-d_{il}/\delta}+e^{-D_{ki}^{\text{UE}}/\delta}+e^{-D_{lj}^{\text{SA}}/\delta}).$
(14)

where $d_{kl}$ is the distance between the $k$th UE and the $l$th SA,
$D_{ki}^{\text{UE}}$ is the distance between UE $k$ and UE $i$, and
$D_{lj}^{\text{SA}}$ is the distance between SA $l$ and SA $j$ [20, 19]. Note
that the closer the $k$th UE is from the $i$th UE, the more correlated their
shadow fading, $F_{kl}$ and $F_{ij}$, will be. Also, the closer the $l$th SA
is from the $j$th SA, the more correlated $F_{kl}$ and $F_{ij}$ will be.

Finally, from (1) and (9), the channel vector follows a complex normal
distribution with mean
$\overline{\textbf{h}}_{kl}=\alpha_{kl}\textbf{h}_{kl}^{\text{LoS}}$ and
covariance matrix

$\displaystyle\mathbb{C}\text{ov}(\textbf{h}_{kl})=$
$\displaystyle\leavevmode\nobreak\
\mathbb{E}\\{\textbf{h}_{kl}\textbf{h}_{kl}^{\text{H}}\\}-\overline{\textbf{h}}_{kl}\overline{\textbf{h}}_{kl}^{\text{H}}$
$\displaystyle=$ $\displaystyle\leavevmode\nobreak\
\mathbb{E}\\{(\overline{\textbf{h}}_{kl}+\textbf{h}_{kl}^{\text{NLoS}})(\overline{\textbf{h}}_{kl}+\textbf{h}_{kl}^{\text{NLoS}})^{\text{H}}\\}-\overline{\textbf{h}}_{kl}\overline{\textbf{h}}_{kl}^{\text{H}}$
$\displaystyle=$ $\displaystyle\leavevmode\nobreak\
\mathbb{E}\\{\textbf{h}_{kl}^{\text{NLoS}}(\textbf{h}_{kl}^{\text{NLoS}})^{\text{H}}\\}$
$\displaystyle=$ $\displaystyle\leavevmode\nobreak\ \textbf{R}_{kl},$ (15)

such that

$\textbf{h}_{kl}\sim\mathcal{N}_{\mathbb{C}}(\overline{\textbf{h}}_{kl},\textbf{R}_{kl}).$
(16)

Notice that the correlation matrix of $\textbf{h}_{kl}^{\text{NLoS}}$ is also
$\textbf{R}_{kl}$, since $\textbf{h}_{kl}^{\text{NLoS}}$ is a zero mean
vector. Indeed, as demonstrated in (15), the covariance matrix of
$\textbf{h}_{kl}$ is also $\textbf{R}_{kl}$; however, its correlation matrix
is
$\textbf{R}_{kl}+\overline{\textbf{h}}_{kl}\overline{\textbf{h}}_{kl}^{\text{H}}$,
since it is a non-zero mean vector.

Both $\overline{\textbf{h}}_{kl}$ and $\textbf{R}_{kl}$ describe the
macroscopic propagation effects, i.e., path-loss and shadowing, whereas the
distribution in (16) describes the small-scale fading. If there is a LoS path
between UE $k$ and SA $l$, then we say that SA $l$ is inside the visibility
region (VR) of UE $k$. Otherwise, the received power is much weaker as the
signal is coming only from NLoS paths, and then we say that SA $l$ is not in
the VR of UE $k$.

Finally, the probabilistic LoS/NLoS LSF channel gain is defined as

$\beta_{kl}=\frac{1}{N}\mathbb{E}\\{||\textbf{h}_{kl}||^{2}\\},$ (17)

and, by replacing (1) into (17), and recalling that
$||\textbf{h}_{kl}^{\text{LoS}}||^{2}=N\beta_{kl}^{\text{LoS}}$ and
$\mathbb{E}\\{||\textbf{h}_{kl}^{\text{NLoS}}||^{2}\\}=\text{tr}(\textbf{R}_{kl})=N\beta_{kl}^{\text{NLoS}}$,
we come to

$\beta_{kl}=\alpha_{kl}\beta_{kl}^{\text{LoS}}+\beta_{kl}^{\text{NLoS}}.$ (18)

In the following sections, we describe the XL-MIMO system model and the
procedures for both PA and SA selection. An overview of the operations
involved in such processes is shown in Fig. 2: first, the pilots are assigned
to the $K$ UEs based on the channel statistics (channel average and covariance
matrices). Second, the SA selection is performed, also based on the channel
statistics. Third, the instantaneous channels are estimated in each coherence
block, based on the pilot decisions and the received signals during the UL
pilot transmission. Fourth, the combining vectors are computed, based on the
channel statistics and the pilot decisions (which are necessary to obtain the
channel estimation covariance matrices) and on the channel estimates. Fifth,
the signals transmitted by the UEs are estimated, based on the combining
vectors and on the received signal at the ULA during the UL data transmission.
Finally, when evaluating the system performance, we can compute the SINR based
on eq. (35), based on the channel estimates, channel statistics, pilot
decisions, SA clusters and combining vectors.

Figure 2: System operation phases: i) pilot assignment (four methods were
compared); ii) subarray selection; iii) channel estimation; iv) computation of
the combining vectors; v) signal estimation; vi) SE calculation.

##  III XL-MIMO System Model

###  III-A UL Pilot Transmission

Pilots are $\tau_{\text{p}}$-dimensional vectors, and we can only find at most
$\tau_{\text{p}}$ mutually orthogonal sequences. The set
$\bm{\phi}_{1},\dots,\bm{\phi}_{\tau_{\text{p}}}\in\mathbb{C}^{\tau_{\text{p}}}$,
containing $\tau_{\text{p}}$ orthogonal pilot sequences, is defined such that

$\bm{\phi}_{i}^{\text{H}}\bm{\phi}_{j}=\begin{cases}\tau_{\text{p}},\quad&\text{if}\
i=j,\\\ 0,\quad&\text{if}\ i\neq j.\end{cases}$ (19)

The channels need to be estimated once per coherence block. Therefore,
$\tau_{\text{p}}$ samples must be reserved for uplink pilot signaling in each
coherence block, but the finite length of the coherence blocks imposes the
constraint $\tau_{\text{p}}\leq\tau_{\text{c}}$, where $\tau_{\text{c}}$ is
the number of symbols (samples) that can be transmitted over one coherence
block. Frequently, $K\gg\tau_{\text{c}}$ makes it impossible to assign
mutually orthogonal pilots to all UEs. Therefore, two or more UEs might be
assigned the same pilot. We denote the index of the pilot assigned to UE $k$
as $t_{k}\in\\{1,\dots,\tau_{\text{p}}\\}$ and define the set of UEs using the
same pilot as UE $k$, including itself:

$\mathcal{P}_{k}=\\{i:t_{i}=t_{k},i=1,\dots,K\\}\subset\\{1,\dots,K\\}.$ (20)

The elements of $\sqrt{p_{k}}\bm{\phi}_{t_{k}}$ are transmitted by UE $k$ over
$\tau_{\text{p}}$ consecutive samples at the beginning of the coherence block,
where $p_{k}$ is the uplink transmit power of UE $k$. Then, during the entire
pilot transmission, the received signal
$\textbf{Y}_{l}^{\text{p}}\in\mathbb{C}^{N\times\tau_{\text{p}}}$ at SA $l$ is

$\textbf{Y}_{l}^{\text{p}}=\sum_{i=1}^{K}\sqrt{p_{i}}\textbf{h}_{il}\bm{\phi}_{t_{i}}^{\text{T}}+\textbf{N}_{l}^{\text{p}},$
(21)

where $\textbf{N}_{l}^{\text{p}}\in\mathbb{C}^{N\times\tau_{\text{p}}}$ is the
receiver noise with i.i.d. elements distributed as
$\mathcal{N}_{\mathbb{C}}(0,\sigma_{\text{n}}^{2})$.

Besides, the interference coming from non-pilot-sharing UEs can be eliminated,
by multiplying the received signal $\textbf{Y}_{l}^{\text{p}}$ with the
normalized conjugate of the pilot $\bm{\phi}_{t_{k}}$, resulting in the
processed received pilot signal
$\textbf{y}_{{t_{k}}l}^{\text{p}}\in\mathbb{C}^{N}$, which is a sufficient
statistic for estimating $\textbf{h}_{kl}$:

$\displaystyle\textbf{y}_{{t_{k}}l}^{\text{p}}=\textbf{Y}_{l}^{\text{p}}\bm{\phi}_{t_{k}}^{*}=\sqrt{p_{k}}\tau_{\text{p}}\textbf{h}_{kl}+\sum_{i\in\mathcal{P}_{k}\backslash\\{k\\}}\sqrt{p_{i}}\tau_{\text{p}}\textbf{h}_{il}+\textbf{n}_{{t_{k}}l}^{\text{p}},$
(22)

where
$\textbf{n}_{{t_{k}}l}^{\text{p}}=\textbf{N}_{l}^{\text{p}}\bm{\phi}_{t_{k}}^{*}\sim\mathcal{N}_{\mathbb{C}}(\mathbf{0},\tau_{\text{p}}\sigma_{\text{n}}^{2}\textbf{I}_{N})$.

###  III-B Channel estimation

Channel estimates can be obtained based on the observation
$\textbf{y}_{{t_{k}}l}^{\text{p}}$. The MMSE estimator uses statistical
information, i.e., the channel means, and the covariance matrices to partially
remove the interference from pilot-sharing UEs. Based on the received signal
$\textbf{y}_{{t_{k}}l}^{\text{p}}$, the MMSE estimate of $\textbf{h}_{kl}$ is
defined as [21]:

$\widehat{\textbf{h}}_{kl}=\overline{\textbf{h}}_{kl}+\sqrt{p_{k}}\textbf{R}_{kl}\bm{\Psi}_{{t_{k}}l}^{-1}\left(\textbf{y}_{{t_{k}}l}^{\text{p}}-\overline{\textbf{y}}_{{t_{k}}l}^{\text{p}}\right),$
(23)

where

$\displaystyle\bm{\Psi}_{t_{k}l}=\frac{1}{\tau_{\text{p}}}\text{Cov}(\textbf{y}_{t_{k}l}^{\text{p}})=\sum_{i\in\mathcal{P}_{k}}p_{i}\tau_{\text{p}}\textbf{R}_{il}+\sigma_{\text{n}}^{2}\textbf{I}_{N},$
(24)

and

$\overline{\textbf{y}}_{{t_{k}}l}^{\text{p}}=\sqrt{p_{k}}\tau_{\text{p}}\overline{\textbf{h}}_{kl}+\sum_{i\in\mathcal{P}_{k}\backslash\\{k\\}}\sqrt{p_{i}}\tau_{\text{p}}\overline{\textbf{h}}_{il}.$
(25)

The statistical distributions (the mean vector and covariance matrices) are
assumed perfectly known, since they are constant during hundreds of channel
coherence blocks. In practice, they can be estimated using the sample mean and
sample covariance matrices.

The resulting estimation error ${\bm{\varepsilon}}_{{\bf
h}_{kl}}=\textbf{h}_{kl}-\widehat{\textbf{h}}_{kl}$ can be modeled as a zero-
mean random variable with covariance matrix [21] given by

$\textbf{C}_{kl}=\textbf{R}_{kl}-p_{k}\tau_{\text{p}}\textbf{R}_{kl}\bm{\Psi}_{t_{k}l}^{-1}\textbf{R}_{kl}.$
(26)

The estimate $\widehat{\textbf{h}}_{kl}$ and the estimation error
$\bm{\varepsilon}_{{\bf h}_{kl}}$ are independent random variables distributed
as[21]

$\displaystyle\widehat{\textbf{h}}_{kl}\ $
$\displaystyle\sim\mathcal{N}_{\mathbb{C}}\left(\overline{\textbf{h}}_{kl},\textbf{R}_{kl}-\textbf{C}_{kl}\right),$
(27) $\displaystyle{\bm{\varepsilon}_{{\bf h}_{kl}}}\ $
$\displaystyle\sim\mathcal{N}_{\mathbb{C}}\left(\mathbf{0}_{N},\textbf{C}_{kl}\right).$
(28)

The estimation accuracy is defined by the NMSE of the channel estimate[10].
The NMSE of $\widehat{\textbf{h}}_{k}$ can be computed as

$\displaystyle\text{NMSE}_{k}=\frac{\mathbb{E}\\{||{\bm{\varepsilon}_{{\bf
h}_{k}}}||^{2}\\}}{\mathbb{E}\\{||\textbf{h}_{k}||^{2}\\}}=\frac{\mathbb{E}\\{{\bm{\varepsilon}_{{\bf
h}_{k}}^{\rm H}}\\}\mathbb{E}\\{{\bm{\varepsilon}_{{\bf
h}_{k}}}\\}+\sum\limits_{\begin{subarray}{c}l=1\end{subarray}}^{L}\text{tr}(\textbf{C}_{kl})}{N\sum\limits_{\begin{subarray}{c}l=1\end{subarray}}^{L}\beta_{kl}},$
(29)

where ${\bm{\varepsilon}_{{\bf h}_{k}}}=[{\bm{\varepsilon}_{{\bf
h}_{k1}}}^{\text{T}},\dots,{\bm{\varepsilon}_{{\bf
h}_{kL}}}^{\text{T}}]^{\text{T}}$ is the collective channel estimation error
vector of UE $k$. The term $\mathbb{E}\\{||{\bm{\varepsilon}_{{\bf
h}_{k}}}||^{2}\\}$ is the mean square error (MSE) of the channel estimation
$\widehat{\textbf{h}}_{k}$.

###  III-C UL Data Transmission

During the UL data transmission, the $i$th UE transmits the signal
$s_{i}\in\mathbb{C}$, with power $p_{i}=\mathbb{E}\\{|s_{i}|^{2}\\}$, and the
SAs receive a superposition of the signals sent from the $K$ UEs. The received
signal $\textbf{y}_{l}^{\text{ul}}\in\mathbb{C}^{N}$ at the $l$th SA is then
given by

$\textbf{y}_{l}^{\text{ul}}=\sum_{i=1}^{K}\textbf{h}_{il}s_{i}+\textbf{n}_{l}^{\text{u}},$
(30)

where $\textbf{h}_{il}\in\mathbb{C}^{N}$ denotes the channel vector between
the $k$th UE and the $l$th SA, and
$\textbf{n}_{l}^{\text{u}}\sim\mathcal{N}_{\mathbb{C}}(\textbf{0},\sigma_{\text{n}}^{2}\textbf{I}_{N})$
is the independent additive noise during UL data transmission.

We define the set $\mathcal{M}_{k}$, containing the SAs that serve the $k$th
UE, and the diagonal matrix $\textbf{D}_{kl}\in\mathbb{C}^{N\times N}$, which
indicates whether the $l$th SA serves the $k$th UE or not:

$\displaystyle\textbf{D}_{kl}=\begin{cases}\textbf{I}_{N},\quad&\text{if}\leavevmode\nobreak\
l\in\mathcal{M}_{k},\\\ \mathbf{0}_{N},\quad&\text{if}\leavevmode\nobreak\
l\notin\mathcal{M}_{k}.\end{cases}$ (31)

If $\alpha_{kl}=1$ and $\alpha_{kj}=0$, i.e., if UE $k$ has LoS to SA $l$ but
not to SA $j$, a proper SA selection scheme will very likely give preference
to SA $l$ over $j$ in serving UE $k$.

Also, let’s define a receive combining vector
$\textbf{v}_{kl}\in\mathbb{C}^{N}$ that SA $l$ could use if it serves UE $k$,
and the collective combining vector
$\textbf{v}_{k}=[\textbf{v}_{k1}^{\text{T}}\dots\textbf{v}_{kL}^{\text{T}}]^{\text{T}}$.
The collective UL data signal $\textbf{y}^{\text{ul}}\in\mathbb{C}^{LN}$ is
given by

$\textbf{y}^{\text{ul}}=\begin{bmatrix}\textbf{y}_{1}^{\text{ul}}\\\ \vdots\\\
\textbf{y}_{L}^{\text{ul}}\end{bmatrix}=\sum_{i=1}^{K}\textbf{h}_{i}s_{i}+\textbf{n},$
(32)

where
$\textbf{h}_{i}=[\textbf{h}_{i1}^{\text{T}}\dots\textbf{h}_{1L}^{\text{T}}]^{\text{T}}\in\mathbb{C}^{LN}$
is the $i$th UE collective channel, and
$\textbf{n}=[\textbf{n}_{1}^{\text{T}}\dots\textbf{n}_{l}^{\text{T}}]^{\text{T}}\in\mathbb{C}^{LN}$
is the collective noise vector along the antennas of all SAs.

In order to obtain an estimate of $s_{k}$, the CPU computes the inner product
between the effective receive combining vector $\textbf{D}_{k}\textbf{v}_{k}$,
where $\textbf{D}_{k}=\text{diag}(\textbf{D}_{k1},\dots,\textbf{D}_{kL})$, and
the received UL data signals at the serving SAs:

$\displaystyle\widehat{s}_{k}$
$\displaystyle=\textbf{v}_{k}^{\text{H}}\textbf{D}_{k}\textbf{y}^{\text{ul}}$
(33)
$\displaystyle=\textbf{v}_{k}^{\text{H}}\textbf{D}_{k}\widehat{\textbf{h}}_{k}s_{k}+\textbf{v}_{k}^{\text{H}}\textbf{D}_{k}{\bm{\varepsilon}_{{\bf
h}_{k}}}s_{k}+\sum\limits_{\begin{subarray}{c}i=1\\\ i\neq
k\end{subarray}}^{K}\textbf{v}_{k}^{\text{H}}\textbf{D}_{k}\textbf{h}_{i}s_{i}+\textbf{v}_{k}^{\text{H}}\textbf{D}_{k}\textbf{n}.$
(34)

Notice that the desired signal
$\textbf{v}_{k}^{\text{H}}\textbf{D}_{k}\textbf{h}_{k}s_{k}$ was divided into
two parts in equation (34): the desired signal over the estimated channel,
$\textbf{v}_{k}^{\text{H}}\textbf{D}_{k}\widehat{\textbf{h}}_{k}s_{k}$, and
the desired signal over the unknown channel,
$\textbf{v}_{k}^{\text{H}}\textbf{D}_{k}{\bm{\varepsilon}_{{\bf
h}_{k}}}s_{k}$. Only the first one can be utilized for signal detection. The
second one is not useful since only the distribution of the estimation error
is known, and thus it is treated as interference or noise when computing the
instantaneous effective uplink SINR of UE $k$, given by

$\text{SINR}_{k}^{\text{ul,c}}=\frac{p_{k}|\textbf{v}_{k}^{\text{H}}\textbf{D}_{k}\widehat{\textbf{h}}_{k}|^{2}}{\sum\limits_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}^{K}p_{i}|\textbf{v}_{k}^{\text{H}}\textbf{D}_{k}\widehat{\textbf{h}}_{i}|^{2}\leavevmode\nobreak\
+\textbf{v}_{k}^{\text{H}}\textbf{Z}_{k}\textbf{v}_{k}+\sigma_{\text{n}}^{2}||\textbf{D}_{k}\textbf{v}_{k}||^{2}},$
(35)

where

$\textbf{Z}_{k}=\sum_{i=1}^{K}p_{i}\textbf{D}_{k}\textbf{C}_{i}\textbf{D}_{k}.$
(36)

Finally, being $\tau_{\text{u}}/\tau_{\text{c}}$ the fraction of each
coherence block that is used for uplink data transmission, the achievable SE
of UE $k$ in the centralized operation can be expressed as

$\text{SE}_{k}^{\text{ul,c}}=\frac{\tau_{\text{u}}}{\tau_{\text{c}}}\mathbb{E}\left\\{\text{log}_{2}\left(1+\text{SINR}_{k}^{\text{ul,c}}\right)\right\\}.$
(37)

###  III-D Receive Combining

A critical implication of deploying an XL antenna array is that the
interference that affects UE $k$ is mainly generated by the UEs that are
located at its neighborhood[10]. Thus, we will utilize the partial MMSE
(P-MMSE) for receive combining, defined as:

$\textbf{v}_{k}=p_{k}\left(\sum_{i\in\mathcal{S}_{k}}p_{i}\textbf{D}_{k}(\widehat{\textbf{h}}_{i}\widehat{\textbf{h}}_{i}^{\text{H}}+\textbf{C}_{i})\textbf{D}_{k}+\sigma_{\text{n}}^{2}\textbf{I}_{LN}\right)^{-1}\textbf{D}_{k}\widehat{\textbf{h}}_{k},$
(38)

where only the UEs that are served by at least one SA that also serves UE $k$
are considered in the inverse matrix, i.e., the UEs that have indices in the
set

$\mathcal{S}_{k}=\\{i:\textbf{D}_{k}\textbf{D}_{i}\neq\mathbf{0}_{LN}\\}.$
(39)

In the special case where $\mathcal{S}_{k}$ contains the $K$ UEs, the vector
$\textbf{v}_{k}$ becomes the MMSE receive combiner, which is the optimal
combiner w.r.t. minimizing the MSE of the estimates $\widehat{s}_{k}$, and
also maximizes the SINR of the $k$th UE, as defined in (35). The P-MMSE
combiner does not attain optimal SE, but it is scalable, since both the
computation of the combining vector $\textbf{v}_{k}$ and the estimation of the
channel vectors $\textbf{h}_{i}$, $i\in\mathcal{S}_{k}$, do not result in a
computational complexity that grows without limit when $K\rightarrow\infty$.

##  IV Pilot Assignment and Subarray Selection

Depending on the mobility, the size of the normalized coherence block may be
smaller than the number of UEs, i.e., $\tau_{\text{c}}<K$. The finite length
of the coherence block imposes that $\tau_{\text{p}}<\tau_{\text{c}}$, such
that it is impossible to assign mutually orthogonal pilots to all UEs in
overcrowded networks where $K>\tau_{\text{c}}$, and consequently,
$K>\tau_{\text{p}}$. Furthermore, it may happen that $\tau_{\text{p}}<K$.
Thus, in crowded scenarios, we may need to reuse the $\tau_{\text{p}}$
mutually orthogonal pilots among the XL-MIMO UEs.

If UEs $k$ and $i$ share the same pilot, i.e., $t_{k}=t_{i}$, UE $i$ will
probably cause much more interference on the estimate $\widehat{s}_{k}$ than
the UEs that use other pilot than $t_{k}$. If the LSF channel gain
$\beta_{kl}$ is stronger than $\beta_{il}$, it may be fortunate to assign to
this SA the task of estimating $s_{k}$ but not $s_{i}$. Thus, each SA shall
estimate the signal of one UE only per pilot sequence, namely the one with the
strongest channel, such that each UE will be served by a different subset of
SAs. It is a necessary and sufficient condition for scalability since the
computational complexity will remain finite as $K\rightarrow\infty$[10].

However, in some cases it may happen that the $k$th UE channel is not the
strongest among the UEs that share the pilot $t_{k}$ in any SA, such that the
CPU would not even process its signal. Thus, the SA selection is insufficient
to ensure a uniform quality of service among the XL-MIMO UEs, and a pilot
assignment strategy will be required. A possible way to avoid the impairment
mentioned above is to assign exclusive orthogonal pilots to the worst channel
UEs. Subsection IV-B discusses the proposed pilot assignment strategy.

Both pilot assignment (PA) and subarray selection (SAS) strategies are based
only on the channel statistics, not on the instantaneous channel realizations;
hence, we do not need to run both algorithms in every coherence block. A basic
algorithm for SA selection is to first assign pilots to all UEs and then let
each SA serve only one UE per pilot, namely the one with the strongest LSF
channel gain111Recall that this is a channel statistical information, which
has been assumed known. Indeed, t is assumed that the algorithms can count on
the perfect knowledge of the channel statistics (channel covariance matrix and
the channel mean) since they change much slower than the channel realizations.
among the subset of UEs that have been assigned that pilot. Note that each SA
can make these decisions locally [10].

###  IV-A Pilot Contamination Effect

Suppose that UEs $k$ and $i$ use the same pilot. Intuitively, the closer the
UE $i$ is to the antenna array, the higher its NLoS LSF channel gain
($\beta_{il}^{\text{NLoS}}$) will be and, therefore, the more it will
interfere with the estimation of $\textbf{h}_{kl}$. It can be verified in (26)
and (29) that the NMSE increases with $\beta_{il}^{\text{NLoS}}$, which is
implicitly included in $\bm{\Psi}_{{t_{k}}l}^{-1}$. Furthermore, if UE $k$ is
close to the ULA, and especially if there is a LoS path, the associated
channel will be stronger, and the relative estimation error will be smaller
since the LoS component changes slowly; therefore, it is assumed to be
perfectly known. On the other hand, yet according to (29), the NMSE of
estimating $\textbf{h}_{k}$ is inversely proportional to
$\sum_{l=1}^{L}\beta_{kl}$.

Moreover, if $\textbf{R}_{kl}$ is invertible, one can write the relation[10]

$\widehat{\textbf{h}}_{il}=\overline{\textbf{h}}_{il}+\sqrt{\frac{p_{i}}{p_{k}}}\textbf{R}_{il}\textbf{R}_{kl}^{-1}(\widehat{\textbf{h}}_{kl}-\overline{\textbf{h}}_{kl}).$
(40)

Notice that the more similar the matrices $\textbf{R}_{kl}$ and
$\textbf{R}_{il}$ are, the more correlated the estimates
$\widehat{\textbf{h}}_{kl}$ and $\widehat{\textbf{h}}_{il}$ will be. Although
the channels $\textbf{h}_{kl}$ and $\textbf{h}_{il}$ are statistically
independent, the MMSE estimates will be fully correlated if the NLoS channels
are spatially uncorrelated, i.e., if
$\textbf{R}_{kl}=\beta_{kl}^{\text{NLoS}}\textbf{I}_{N}$ and
$\textbf{R}_{il}=\beta_{il}^{\text{NLoS}}\textbf{I}_{N}$, since they will be
the same except for a scaling factor[10].

Since two closely located UEs, $k$ and $i$, have approximately the same
azimuth and elevation angles, they have similar spatial correlation
properties, i.e., the matrices $\textbf{R}_{kl}$ and $\textbf{R}_{il}$ can
result quite similar. As a result, if they use the same pilot, the MMSE
estimates of $\textbf{h}_{kl}$ and $\textbf{h}_{il}$ will be correlated, which
was demonstrated analytically and numerically in [10] to cause strong pilot
contamination.

###  IV-B Pilot Assignment (PA) in XL-MIMO

The pilot assignment (PA) is a combinatorial problem with
$\tau_{\text{p}}^{K}$ possible assignments and it is unfeasible to evaluate
all of them in realistic 5G mMIMO network scenarios[10]. Hence, a practical,
suboptimal PA algorithm typically assigns pilots to the UEs iteratively. The
following PA techniques have been considered in this work.

Random PA: is the most naive strategy, consisting of assigning pilots randomly
to all the $K$ UEs. It does not guarantee any pilot contamination control but
has zero computational complexity.

Genie PA: an exhaustive search can be used to obtain the best PA solution
concerning an optimization criterion, for instance, the average channel
estimation NMSE among the $K$ UEs. The closed-form expression given in (29)
can be used as a cost function for evaluating each candidate solution. The
NMSE can be evaluated through the closed-form expression in (29), such that
the cost function is given by

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sum_{k=1}^{K}\frac{\sum\limits_{\begin{subarray}{c}l=1\end{subarray}}^{L}\text{tr}(\textbf{C}_{kl})}{N\sum\limits_{\begin{subarray}{c}l=1\end{subarray}}^{L}\beta_{kl}}}$
(41)

where the term ${\bm{\varepsilon}_{{\bf h}_{kl}}}$ in (29) was removed since
we consider MMSE channel estimator, which is unbiased. The genie PA procedure
is shown in Algorithm 1. However, this method is feasible only if the number
of possible assignments is small. It guarantees optimality, but it is
unfeasible since there are $(\tau_{\text{p}})^{K}$ possible assignments[10].

Algorithm 1 Genie Pilot Assignment

Input: $\textbf{R}_{kl}$, $\forall kl$.

1: Initialize the population $\bm{\Theta}_{\text{G}}$ containing in its
columns all the $A_{\text{G}}$ possible PA candidates.

2: for $i=1,\dots,\textbf{K}$ do

3: Using equation (29), compute
$\frac{1}{N}\sum\limits_{\begin{subarray}{c}k=1\end{subarray}}^{K}\text{NMSE}_{k}$
as if the pilots were assigned according to the $i$th column of the matrix
$\bm{\Theta}_{\text{G}}$, and assign it to the $i$th column of the cost vector
$\textbf{c}_{\text{G}}$.

4: end for

5: Find the pilots $t_{1},\dots,t_{K}$ corresponding to the column of
$\bm{\Theta}_{\text{G}}$ associated to the smallest element of
$\textbf{c}_{\text{G}}$.

Output: pilot assignment $t_{1},\dots,t_{K}$

Greedy PA: is a simple PA strategy consisting in assigning the
$\tau_{\text{p}}$ orthogonal pilots to $\tau_{\text{p}}$ random UEs, in the
first step, and then assigning pilots to the remaining UEs, one after the
other, as to minimize pilot contamination, in the second step. The pilot
contamination faced by a UE $k$ when assigning to it a pilot $t$ can be
quantified as the sum of the NLoS LSF channel gains of the UEs that were
already assigned pilot $t$. It can be considered only the LSF channel gains to
SA $l_{k}$, which is defined as the SA to which the UE $k$ has the strongest
LSF channel gain, as in Algorithm 2, or the sum LSF channel gains among the
$L$ SAs. The greedy algorithm comes with no performance guarantee, but
effectively avoids closely-located UEs from being assigned the same
pilots[10].

Algorithm 2 Greedy Pilot Assignment

Input: LSF coefficients

1: for $k=1,\dots,\tau_{\text{p}}$ do

2: $t_{k}\leftarrow k$

3: end for

4: for $k=\tau_{\text{p}},\dots,K$ do

5:
$l_{k}\leftarrow\underset{l\in\\{1,\dots,L\\}}{\text{arg\leavevmode\nobreak\
max}}\beta_{kl}$

6:
$\tau\leftarrow\underset{t\in\\{1,\dots,\tau_{\text{p}}\\}}{\text{arg\leavevmode\nobreak\
min}}\sum\limits_{\begin{subarray}{c}i=1\\\
t_{i}=t\end{subarray}}^{k-1}\beta_{il_{k}}^{\text{NLoS}}$

7: $t_{k}\leftarrow\tau$

8: end for

Output: pilot assignment $t_{1},\dots,t_{K}$

Genetic Algorithm (GA) PA. Herein we propose to solve the PA problem in XL-
MIMO networks by applying heuristic genetic algorithm (GA). It uses $A$ random
PA candidates per iteration. The candidates are $K$-length vectors, where the
$k$th entry is the index of the pilot assigned to the $k$th UE. Each candidate
is evaluated in terms of a cost function. Here, for a fair comparison, we will
take the same cost function used by the genie algorithm (exhaustive search),
given by (41), i.e., the average channel estimation NMSE among the $K$ UEs.

The $\phi$ best candidates in the population are used to generate the $A$
members of a new population. From the list of the best candidates, $A$ pairs
of candidates are picked at random to generate the $A$ descendants (elements
of the population in the next iteration of the algorithm). The descendants are
generated by applying the crossover operator between their correspondent
parents, with a random crossover point that is comprised in the interval
$[2,\tau_{\text{p}}]$. Then, the mutation operator is applied to modify the
pilot of each UE. The probability of modification is $p_{\text{mut}}$, which
means that the probability of UE $k$ pilot changing from $t_{k}$ to another
pilot is $p_{\text{mut}}$. Specifically, the probability of changing from
$t_{k}$ to each of the other $\tau_{\text{p}}$ is the same. At the end of each
iteration, the best candidate is updated if it is better than the previous
solution (at the previous iteration). The predefined number of iterations is
denoted by $N_{\text{it}}$. The pseudo-code for the heuristic GA-based PA
method is presented in Algorithm 3.

Algorithm 3 Pilot Assignment via Genetic Algorithm

Input: $A$, $\phi$, $p_{\text{mut}}$, $N_{\text{it}}$, $\textbf{R}_{kl}$,
$\forall kl$.

1: Initialize the population $\bm{\Theta}$ with $A$ random candidates.

2: Evaluate the cost of each candidate in $\bm{\Theta}$, according to (29),
forming the vector c.

3: Sort c in ascending order, reorganizing the columns of $\bm{\Theta}$
accordingly.

4: Initialize $c_{\text{best}}=c_{1}$, and
$\bm{\theta}_{\text{best}}=\bm{\theta}_{1}$.

5: for $i=2,\dots,N_{\text{it}}$ do

6: for $a=1,\dots,A$ do

7: Select randomly two of the $\phi$ best candidates in the population
$\bm{\Theta}$ to be the parents of $\bm{\theta}_{a}$;

8: Apply the crossover operator in a random crossover point $\in\\{2,K\\}$.

9: Apply the mutation operator in $\bm{\theta}_{a}$ with probability
$p_{\text{mut}}$.

10: Evaluate the cost function of $\bm{\theta}_{a}$, according to (29), and
assign it to $c_{a}$.

11: end for

12: Sort c in descending order, reorganizing the columns of $\bm{\Theta}$
accordingly.

13: if $c_{1}>c_{\text{best}}$ then

14: Update $c_{\text{best}}=c_{1}$ and
$\bm{\theta}_{\text{GA}}=\bm{\theta}_{1}$.

15: end if

16: end for

Output: $\bm{\theta}_{\text{GA}}$

After assigning the pilots, the CPU must inform each SA on the designated
pilot for each UE covered by the specific SA. This is not significant in terms
of fronthaul traffic since the pilot decisions change according to the channel
statistics, not on the instantaneous channels basis.

The proposed GA-based PA for XL-MIMO systems is a suboptimal algorithm since
it is an iterative method searching for solutions in a very reduced search
space, capable to approach the genie solution, which uses the entire search
space. Moreover, the heuristic evolutionary GA-based pilot assignment method
has the potential to be much more successful than greedy PA – and mainly
random-based searching methods – in reducing pilot contamination since the
former aggregates some intelligence in selecting potential candidates/solution
beyond the simple cost function evaluation.

###  IV-C Subarray selection in XL-MIMO

The SA selection is tightly connected to the pilot assignment procedure, and
will be done after all the UEs have been assigned to pilots. For that, in
Algorithm 4, as in [10, Algorithm 4.1], each SA serves one and only one UE per
pilot, namely the one with the largest LSF coefficient among the UEs using
that pilot. It makes good practical sense, since each SA can decide locally
which UEs to serve. If the pilot assignment is performed properly, it is
highly probable that every UE will be served by at least one SA.

Algorithm 4 Strongest-UE SA Selection

Input: LSF coefficients and pilot assignment $t_{1},\dots,t_{K}$

1: Initialization: Set $\mathcal{M}_{1}=\dots=\mathcal{M}_{K}=\emptyset$.

2: for $l=1,\dots,L$ do

3: for $t=1,\dots,\tau_{\text{p}}$ do

4:
$k\leftarrow\underset{i\in\\{1,\dots,K\\};t_{i}=t}{\text{arg\leavevmode\nobreak\
max}}\beta_{il}$

5: $\mathcal{M}_{k}\leftarrow\mathcal{M}_{k}\cup\\{l\\}$

6: end for

7: end for

Output: $\mathcal{M}_{1}=\dots=\mathcal{M}_{K}$

##  V Numerical Results

In this section, we present numerical results based on Monte Carlo simulations
(MCS) to show the influence of the different proposed pilot assignment
strategies and channel estimators on the estimation accuracy and the SE in XL-
MIMO systems. The antenna array containing $L$ uniformly spaced SAs is
centered in the origin of the Cartesian plane and has a length of 100 m,
extending along the y-axis, as depicted in Fig. 1. Each SA contains $N=4$
antennas, spaced by a distance of half a wavelength. In each MCS realization,
the $K$ UEs are randomly dropped on a rectangular area, extending in the range
$[\pm 100]$ m, both on the x-axis and on the y-axis. Table I contains the
adopted values for the simulation parameters.

Table I: Adopted Simulation Parameters values Parameter | Value
---|---
Number of subarrays: $L$ | {25,50}
Number of antennas per subarray: $N$ | 4
Height of the antenna array: $z_{l}^{\text{SA}}$ | 10 m
Height of the UEs: $z_{l}^{\text{UE}}$ | 1.5 m
NLoS path-loss attenuation exponent: $\gamma$ | 4
Shadowing standard deviation: $\sigma_{\text{s}}^{2}$ | 3 dB (LoS)
| 4 dB (NLoS)
Azimuth angle standard deviation: $\sigma_{\varphi}$ | $10^{\circ}$
Elevation angle standard deviation: $\sigma_{\theta}$ | $10^{\circ}$
Median channel gain at a distance of 1 m: $\beta_{0}$ | $8.9125\cdot 10^{-4}$
Array length | 100 m
Wavelength: $\lambda$ | 12.5 cm
Separation distance between consecutive antennas in the same SA: $d$ | $\lambda/2$
Number of antennas in each SA: $N$ | 4
LoS Probability between UEk–SAl: $\text{Pr}(\alpha_{kl}=1)$ | Eq. (42)
UE transmit power: $p_{1},\dots,p_{K}$ | 10 dBm
Noise power: $\sigma_{\text{n}}^{2}$ |  $-96$ dBm
Pilot sequences’ length: $\tau_{\text{p}}$ | 4
Number of Monte-Carlo realizations | 1000

The GA-based PA algorithm was also configured to approach the optimal solution
in minimizing the average NMSE. The stopping criterion is based on the number
of iterations $N_{\rm it}$ and the number of candidates in the population is
set as twice the number of UEs; the probability of mutation is adjusted to
$p_{\rm mut}=0.02$ and the number of best candidates is set to be constrained
by $A/2$. The adopted parameter values in the GA-based PA algorithm are listed
in Table II. The proposed GA-based pilot allocation method is compared with
three different procedures: a) genie-PA, b) greedy-PA, and c) random-PA
methods. SA selection step is applied equally for the four PA methods.

Table II: Adopted parameters values for simulating the GA-based PA algorithm Parameter | Value
---|---
Number of iterations: $N_{\text{it}}$ | 15
Number of candidates in the population: $A$ | $2K$
Probability of mutation: $p_{\text{mut}}$ | 2%
Number of best candidates: $\phi$ | $\lceil A/2\rceil$

For the micro-urban scenario, the probability of existing LoS between UE $k$
and SA $l$ is a function of the distance between them and is modeled according
to [22] as:

$\displaystyle\text{Pr}(\alpha_{kl}=1)=\text{min}\left(\frac{18\leavevmode\nobreak\
\text{m}}{d_{kl}},\,1\right)\cdot\left[1-e^{-\frac{d_{kl}}{36\leavevmode\nobreak\
\text{m}}}\right]+e^{-\frac{d_{kl}}{36\leavevmode\nobreak\ \text{m}}}.$ (42)

According to (42), the propagation is always in LoS conditions for $d_{kl}\leq
18\leavevmode\nobreak\ \text{m}$, i.e., $q=1$, while, for
$d_{kl}>18\leavevmode\nobreak\ \text{m}$, the LoS probability decays
exponentially. Considering the LoS probability given by equation (42) and the
adopted simulation parameters for array length and cell dimensions, we
obtained an average LoS probability of 36%. It represents an expected
condition of the XL-MIMO system operating in micro-urban channel scenarios.
From the trustworthy numerical simulations perspective, we cannot disregard
the LoS component in urban crowded channel scenarios since most UEs are close
to the BS array, which is much longer than co-located mMIMO arrays.

###  V-A Achievable Spectral Efficiency

We compare the channel estimation NMSE and the per-user SE obtained with the
proposed GA-based PA strategy and the other three strategies mentioned in
Section IV: random, greedy, and genie PA methods. According to Fig. 2, the
system operation is composed by the following steps: pilot assignment, SA
selection (Alg. 4), channel estimation and receive combining.

First, recall that the task of both the genie and the GA-based PA algorithms
is to find the pilot assignment solution that minimizes the average channel
estimation NMSE among the $K$ UEs; the first does it via exhaustive search,
while the second is a GA approach. Fig. 3 shows the cumulative density
function (cdf) of the average channel estimation NMSE, i.e.,
$\overline{\text{NMSE}}=\frac{1}{K}\sum_{k=1}^{K}\text{NMSE}_{k}$, where
$\text{NMSE}_{k}$ is computed via eq. (29). The simulation setup assumes
$K=6$, $\tau_{\text{p}}=3$, $L=25$ and $N=4$. According to Fig. 3, simulations
confirmed that genie and GA strategies actually provide the smallest average
NMSE values in every channel realization; one can also verify that the GA-
based PA strategy is efficient since it achieves approximately the same
performance as the genie PA strategy but requires less computation (reduced
search space).

Figure 3: Cdf of the average channel estimation NMSE, among the $K$ UEs.

Figs. 4(a) and 4(b) present, respectively, the cdf of the maximum per user
channel estimation NMSE and the cdf of the minimum per user SE, among the $K$
UEs that are simultaneously served by the XL-MIMO array, representing the
worst case in each channel realization, among the $K$ UEs. According to Fig.
4(b), the genie PA strategy, which is simply an exhaustive search for the PA
solution that minimizes the average channel estimation NMSE among the $K$ UEs,
achieved the best result regarding the maximization of the minimum per user SE
among the $K$ UEs. The GA-based PA algorithm achieved almost the same
performance, since this algorithm repeatedly modifies a population, selecting
the best ones in terms of the same optimization criterion as genie PA, i.e.,
the minimization of the average channel estimation NMSE, but using a reduced
search space.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: Cdf of the channel estimation NMSE and the per-user SE, considering
the worst UE in each Monte-Carlo channel realization (a and b), the best UE (c
and d) and all the $K$ UEs (e and f), with $K=6$, $\tau_{\text{p}}=3$, $L=25$
and $N=4$.

Furthermore, one can see that the proposed GA-based PA beats both random and
greedy PA in all channel realizations, which means that it is significantly
improving the performance of the UEs with bad channel conditions, compared
with these two methods. On the other hand, the proposed method is inferior to
the greedy strategy for the UEs with very good channel conditions, according
to Figures 4(c) and 4(d), which presents respectively the cdf of the minimum
per user channel estimation NMSE and the cdf of the maximum per user SE, among
the $K$ UEs. Finally, Figures 4(e) and 4(f) presents, respectively, the cdf of
the per user channel estimation NMSE and the cdf of the per user SE,
considering all the $K$ UEs in each Monte-Carlo channel realization. Again,
one can observe that the GA-based PA outperforms the random and greedy
strategies, except for the UEs with good channel conditions. The achievements
of the proposed PA strategy for XL-MIMO are in line with the requirements of
beyond 5G networks, where making the quality of service more uniform is more
important than improving the peak data rates. Finally, Figure 5 shows that the
proposed PA strategy also provides higher sum rates than the random and the
greedy strategies. The simulation setup of Figs. 4 and 5 is $K=6$,
$\tau_{\text{p}}=3$, $L=25$ and $N=4$.

Elaborating further on the achieved SE, one can infer that more accurate
channel estimates leads to higher SEs. When computing the average of the
channel estimation NMSE among a set of UEs, the worst channel UEs have much
more impact on the average NMSE than the UEs with good or moderate channel
condition. Thus, when pilots are suitably assigned aiming to minimize the
average channel NMSE estimation, which is the case of GA-based and genie PA
schemes, we are moving towards increasing mainly the SE of the worst channel
UEs, which is observed in 4.

Figure 5: Cdf of the sum SE.

Fig. 6 presents the average values for sum SE, per user SE and minimum per
user SE, varying the number of UEs, with $\tau_{\text{p}}=10$, $L=50$ and
$N=4$. From Fig. 6(c), we confirm the effectiveness of the proposed GA scheme
for improving the minimum SE among the $K$ UEs, compared to the greedy and
mainly to the random PA strategies, while it did not improve the average per
user SE and the average sum SE compared to the greedy PA, only compared to the
random approach. Finally, we can observe that the GA-based PA is a suitable
way to make the QoS more uniform in XL-MIMO systems and that the channel
estimation NMSE is promising in terms of performance-complexity trade-off to
evaluate each candidate on the population in each iteration of the Algorithm
3. It reinforces the conjecture that one can make intuitively that the SINR of
the active UEs can be improved considerably by reducing the channel estimate
error as a consequence of lowering the pilot contamination due to an
appropriate pilot assignment and subarray joint selection procedures.

(a)

(b)

(c)

Figure 6: a) Average sum SE, b) average per user SE and c) average minimum per
user SE, as a function of the number of UEs. Parameter values:
$\tau_{\text{p}}=10$, $L=50$ and $N=4$.

##  VI Conclusions

We have investigated the performance of different pilot assignment schemes to
improve the QoS uniformity in XL-MIMO systems under both LoS and NLoS channel
components. A proper PA strategy is paramount in crowded XL-MIMO systems,
where the number of available orthogonal pilots is much smaller than the
number of active UEs. Hence, a realistic XL-MIMO spatially correlated channel
model has been introduced, considering that the XL-MIMO subarray antennas are
not co-located, meaning that a given UE may be close to some SAs and distant
from others. Consequently, in most cases, only some of the SAs are relevant to
the signal detection of a given UE, such that a simple SA selection strategy
based on Strongest-UE was implemented to make the system scalable without
compromising the SE. The proposed GA-based PA method for SA-based XL-MIMO
systems was compared with three other methods: a) the genie PA, which is
unfeasible in most practical scenarios of applications, as it is an exhaustive
search for the best solution in terms of minimizing the channel estimation
errors; b) the greedy PA, which is a low complexity algorithm and was
extensively deployed in the literature; c) and the random PA, the simplest
method, treated as the worst SE performance benchmark. The proposed GA-based
PA strategy for XL-MIMO systems considerably improved the minimum per-user
spectral efficiency compared to the greedy and random PA methods, achieving
nearly-optimal results under affordable computational complexity. This is in
line with the requirements of beyond 5G networks, where making the quality of
service more uniform is more important than improving the peak data rates.
Also, the proposed PA strategy also provided higher sum rates than the random
and greedy strategies. Furthermore, the proposed way for evaluating each
candidate in the GA population via the channel estimation NMSE analytical
expression has performed remarkably well with respect to improving the SE,
especially the minimum per-user SE.

##  Appendix A Proof of Eq. (9) and (10)

For notation simplicity, we will use
$\textbf{a}_{kl}^{(b)}=\textbf{a}(\varphi_{kl}^{(b)},\theta_{kl}^{(b)})$ and
$\textbf{a}_{kl}^{(0)}=\textbf{a}(\overline{\varphi}_{kl},\overline{\theta}_{kl})$.
First of all, it is simple to find that $\textbf{h}_{kl}^{\text{NLoS}}$, as
defined in (6), is a zero mean random vector, since $g_{kl}^{(b)}$ and
$\textbf{a}_{kl}^{(b)}$ are independent and therefore uncorrelated:

$\displaystyle{\mathbb{E}\\{\textbf{h}_{kl}^{\text{NLoS}}\\}=\sum_{b=1}^{B}\mathbb{E}\\{g_{kl}^{(b)}\textbf{a}_{kl}^{(b)}\\}=\sum_{b=1}^{B}\underbrace{\mathbb{E}\\{g_{kl}^{(b)}\\}}_{0}\mathbb{E}\\{\textbf{a}_{kl}^{(b)}\\}=0.}$

Second, the covariance matrix of $\textbf{h}_{kl}^{\text{NLoS}}$ is computed
as

$\displaystyle{\textbf{R}_{kl}}{\leavevmode\nobreak\ =\leavevmode\nobreak\ }$
$\displaystyle{\mathbb{E}\\{\textbf{h}_{kl}^{\text{NLoS}}(\textbf{h}_{kl}^{\text{NLoS}})^{\text{H}}\\}-\mathbb{E}\\{\textbf{h}_{kl}^{\text{NLoS}}\\}\mathbb{E}\\{(\textbf{h}_{kl}^{\text{NLoS}})^{\text{H}}\\}}$
$\displaystyle{=\leavevmode\nobreak\ }$
$\displaystyle{\mathbb{E}\\{\left[\sum_{b=1}^{B}g_{kl}^{(b)}\textbf{a}_{kl}^{(b)}\right]\left[\sum_{b=1}^{B}(g_{kl}^{(b)})^{*}(\textbf{a}_{kl}^{(b)})^{\text{H}}\right]\\}-0\cdot
0}$ $\displaystyle{=\leavevmode\nobreak\ }$
$\displaystyle{\sum_{b=1}^{B}\mathbb{E}\\{|g_{kl}^{(b)}|^{2}\\}\mathbb{E}\\{\textbf{a}_{kl}^{(b)}(\textbf{a}_{kl}^{(b)})^{\text{H}}\\}\leavevmode\nobreak\
+}$
$\displaystyle{\sum_{b=1}^{B}\sum\limits_{\begin{subarray}{c}b^{\prime}=1\\\
b^{\prime}\neq
b\end{subarray}}^{B}\underbrace{\mathbb{E}\\{g_{kl}^{(b)}(g_{kl}^{(b^{\prime})})^{*}\\}}_{0}\mathbb{E}\\{\textbf{a}_{kl}^{(b)}(\textbf{a}_{kl}^{(b^{\prime})})^{\text{H}}\\},}$
(43)

where $\mathbb{E}\\{g_{kl}^{(b)}(g_{kl}^{(b^{\prime})})^{*}\\}=0$ because
$g_{kl}^{(b)}$, $b=1,\dots,B$, are i.i.d. random variables. Since the array
response vector is given by
$[\textbf{a}(\varphi,\theta)]_{n}=e^{-j2(n-1)\pi\frac{d}{\lambda}\frac{\text{sin}(\varphi)}{{\text{cos}(\theta)}}}$,
the matrix $\textbf{a}_{kl}^{(b)}(\textbf{a}_{kl}^{(b)})^{\text{H}}$ can be
calculated as

${[\textbf{a}_{kl}^{(b)}(\textbf{a}_{kl}^{(b)})^{\text{H}}]_{m,n}=\text{exp}\left(j2(m-n)\pi\frac{d}{\lambda}\frac{\text{sin}(\varphi_{kl}^{(b)})}{\text{cos}(\theta_{kl}^{(b)})}\right),}$

The expectation
$\mathbb{E}\\{\textbf{a}_{kl}^{(b)}(\textbf{a}_{kl}^{(b^{\prime})})^{\text{H}}\\}\\}$,
over the angles $\varphi_{kl}^{(b)}$ and $\theta_{kl}^{(b)}$, is the same for
$b=1,\dots,B$, and by definition is

$\displaystyle{[\mathbb{E}\\{\textbf{a}_{kl}^{(b)}(\textbf{a}_{kl}^{(b)})^{\text{H}}\\}]_{m,n}}$
$\displaystyle{=\int\int[\textbf{a}_{kl}^{(b)}(\textbf{a}_{kl}^{(b)})^{\text{H}}]_{m,n}f_{kl}(\varphi,\theta)d\varphi
d\theta}$ $\displaystyle{=\int\int
e^{j2(m-n)\pi\frac{d}{\lambda}\frac{\text{sin}(\varphi)}{\text{cos}(\theta)}}f_{kl}(\varphi,\theta)d\varphi
d\theta.}$

Thus, this term can go out of the summary in (43). Finally, recalling that
$\beta_{kl}^{\text{NLoS}}=\sum_{b=1}^{B}\mathbb{E}\\{|g_{kl}^{(b)}|^{2}\\}$:

$\displaystyle{[\textbf{R}_{kl}]_{m,n}=}\leavevmode\nobreak\ $
$\displaystyle{\beta_{kl}^{\text{NLoS}}\int\int
e^{j2(m-n)\pi\frac{d}{\lambda}\frac{\text{sin}(\varphi)}{\text{cos}(\theta)}}f_{kl}(\varphi,\theta)d\varphi
d\theta.}$

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