# PSA Based Power Control for Cell-Free Massive MIMO under LoS/NLoS Channels

Ashish Pratap Singh Ribhu Chopra organization=Department of Electronics and
Electrical Engineering, Indian Institute of Technology, Guwahati (emails:
<EMAIL_ADDRESS>ribhu@outlook.com), postcode=781039,
state=Assam, country=India

###### Abstract

A primary design goal of the cell-free (CF) massive MIMO architecture is to
provide uniformly good coverage to all the user equipments (UEs) connected to
the network. However, it has been found that this requirement may not be
satisfied in case the channels between the access points (APs) and the UEs are
mixed LoS/NLoS. In this paper, we try to address this issue via the use of
appropriate power control in both the uplink and downlink of a CF massive MIMO
system under mixed LoS/NLoS channels. We find that simplistic power control
techniques, such as channel inversion-based power control perform sub-
optimally as compared to max-min power control. As a consequence, we propose a
particle swarm algorithm (PSA) based power control algorithm to optimize the
performance of the system under study. We then use numerical simulations to
evaluate the performance of the proposed PSA-based solution and show that it
results in a significant improvement in the fairness of the underlying system
while incurring a lower computational complexity.

###### keywords:

Cell-Free Massive MIMO Line-of-Sight (LoS) Particle Swarm Algorithm (PSA)

##  1 Introduction

Using a large number of antennas at the base station to serve a much smaller
number of User Equipments (UEs) over the same time-frequency resources, known
as massive multiple input multiple output (mMIMO) has gained much research
interest over the past decade and a half [1]. It has been shown that subject
to the availability of channel state information (CSI) at the base station
(BS), mMIMO systems could offer high spectral and energy efficiencies with
simple signal processing [2]. Due to these advantages, mMIMO has been widely
accepted as a key enabling technology for the fifth generation of wireless
systems (5G) [3, 4]. However, it has been shown that mMIMO fails to provide
uniformly good coverage to all the UEs in the system [5]. Consequently, a key
goal in beyond 5G (B5G) wireless systems is to provide uniformly good coverage
to all the UEs, and it has been found that a distributed mMIMO system can
achieve this goal [6, 7]. One such architecture, dubbed cell free (CF) mMIMO
[8] proposes a distributed mMIMO system, wherein the BS is replaced with
multiple access points (APs) spread all across the area under service. These
APs are connected to a central processing unit (CPU) via a backbone network to
serve all UEs over the same time-frequency resource [9]. Since the UEs are now
proximal to the APs, CF-mMIMO can provide a high likelihood of coverage [10].

It is important to note that the high likelihood of coverage in a CF-mMIMO
system implies a high likelihood of the existence of a line of sight (LoS)
path between the APs and the UEs. However, unlike the presently studied models
in the literature that either consider an LoS link to be deterministically
present [11-13] or deterministically absent [5, 9, 14-18], the presence of LoS
link may depend on various physical parameters such as the physical locations
of the APs and the UEs, the presence and density of the blockages, etc.. [19].
Therefore, the exact characterization and optimization of the performance of a
CF-mMIMO system with probabilistic line-of-sight components become necessary
[20]. In this paper, our aim is to build on the previous work in this
direction and develop power control techniques to optimize the performance of
a CF-mMIMO system in the presence of probabilistic LoS components.

The performance of CF-mMIMO systems under probabilistic LoS/NLoS channels has
been discussed in detail for various receive architectures in [20]. It has
been shown that in the presence of probabilistic LoS channels, MMSE-based data
detection achieves near-optimal performance, whereas conjugate beamforming-
based data detection performs much worse even with the availability of
accurate CSI at the CPU. It has also been shown that, at moderate AP
densities, even under perfect/accurate CSI a fraction of UEs (approximately
equal to the fraction of UEs without an LoS path) experience a SINR well below
$0$ dB, indicating a “capture" of the system by the UEs with LoS paths. This
in turn implies the need for appropriate power control for all UEs to achieve
equitable performance.

In the context of power control in CF-mMIMO systems, the authors in [9] have
done an extensive comparative max-min fairness analysis of CF-mMIMO with
centralized mMIMO. In [5], a power control algorithm was developed to maximize
energy efficiency considering hardware and backhaul power consumption. In
[21], authors exploited the max-min fairness optimization problem in rician
fading by adapting the power and the AP–weighting receiver filter coefficients
and solves two sub-problems, receiver filter coefficients through a
generalized eigenvalue problem and the power control problem by the bisection
search method with linear programming. The authors in [14] have considered
max-min power control in rich scattering-based CF-mMIMO systems and developed
a heuristic algorithm-based power control strategy that achieves near-optimal
performance. In [17], authors propose a bisection algorithm-based max-min
power control strategy for downlink CF-mMIMO systems with rich scattering.

In most of the above-mentioned articles, the power allocation optimization
problem is either solved through the sequential convex approximation (SCA)
approach or by using convex approximation combined with the geometric
programming (GP) approach. Since both approaches rely on second-order cone
programming (SOCP), which requires high computational complexity and creates
scalability issues for CF mMIMO systems, this way, to reduce computational
complexity and run time, several works have investigated alternative
optimization problems. The authors in [22] solved the max-min optimization
problem using a low-complexity method called as accelerated project gradient
(APG). Furthermore, in [23], a power allocation problem is solved for
maximizing energy efficiency using the first-order method for non-convex
programming. Additionally, the alternative optimization problem can also
depend on deep learning. In [24], authors performed the power optimization to
maximize the network’s sum rate using a heuristic sub-optimal scheme named as
deep convolution neural networks (DCNN). Again, a deep learning-powered max-
min-based uplink power allocation strategy was developed in [25]. Further, in
[26], authors proposed a power control algorithm in the downlink, which solves
the max-min UE fairness problem by modeling a deep neural network (DNN) with
supervised training. In [27], it has been shown that using centralized and
decentralized deep neural networks (DNNs) substantially reduces the complexity
and processing time of the max-min power control optimization problem. Authors
in [28] solved the uplink max-min fairness power allocation optimization
problem using meta-heuristics.

Table 1: Notations used in paper Notation | Definition
---|---
$l_{m}$ | Height of the $m$-th AP
$l_{k}^{{}^{\prime}}$ | Height of the $k$-th UE
$\delta_{mk}$ | Index variable for LoS channel between the
|  $m$-th AP and $k$-th UE
$P_{mk}$ | Probability of existence of an LoS channel
| between the $m$-th AP and $k$-th UE
$\bar{h}_{mk}$ | LoS channel gain between the $m$-th AP and
|  $k$-th UE
$\dot{h}_{mk}$ | Fast fading coefficient for the NLoS channel
| between the $m$-th AP and $k$-th UE
$x_{mk}$ |  $3$-D link distance between the $m$-th AP and
|  $k$-th UE
$d_{mk}$ | Horizontal ($2$-D) distance between the $m$-th
| AP and $k$-th UE
$N$ | Number of antennas at the APs
$d$ | Spacing between the antennas at all the APs
$M$ | Number of APs in the system
$K$ | Number of UEs in the system
$d_{0}$ | Reference distance between any transceiver pair
$\eta$ | Pathloss exponent
$\alpha$ | Fraction of built-up area in the network
$\gamma$ | Average altitude of Building/blockages
$\mu$ | Average number of blockages per unit area
$\lambda_{c}$ | Carrier Wavelength
$\mathbb{E}[.]$ | Expectation operation
$\mathbf{0}_{N}$ | Null vector of length $N$
$\textbf{I}_{N}$ | Identity matrix of order $N$

In this paper, we focus on improving the fairness of a CF-mMIMO system with
mixed LoS/NLoS channels having moderate AP densities via power control. In
this context, we first formulate the max-min power control problem for a CF-
mMIMO system and demonstrate the optimality of its brute force solution in a
two UE case. We then use one meta-heuristic, the particle swarm algorithm
(PSA), to obtain a lower complexity and with (near-)optimal solution for this
problem. The primary motivation behind using the PSA algorithm is that it
incurs lower computational complexity, low run time, and a (near) real-time
adaptive approach that doesn’t need offline training. These advantages of
using PSA make it a viable, scalable, and efficient optimization technique to
be employed in CF mMIMO networks. Our precise contributions are listed as
follows.

  1. 1.

We derive the rates achievable by the CF-mMIMO system with probabilistic LoS
channels and use Jain’s Fairness Index to quantify the fairness for both
uplink and downlink. We then use these to pose the power control problem for
the underlying system.

  2. 2.

We consider a two UE test case and evaluate the performance of brute force-
enabled max-min power control against channel inversion-based power control.

  3. 3.

Motivated by the significant gap in the performance of the two algorithms, and
the high computational complexity of the brute force solution of the max-min
power control, we develop a PSA-based power control strategy for the
underlying system.

  4. 4.

Via extensive numerical simulations, we show that our proposed PSA-based power
control strategy results in near-optimal performance in the two UE case, and
in significant improvement in the fairness in the 64 UE case as compared to
channel inversion-based power control.

Therefore, the key takeaway of this work is that fairness can be achieved in
CF-mMIMO systems with mixed LoS/NLoS using appropriate power control.
Moreover, our proposed PSA approach can achieve near-optimal performance in
this regard.

The rest of the paper is organized as follows. Section 2 states the system and
channel model. Section 3 describes our proposed system’s performance analysis
in both uplink and downlink. The description of Two UE Testing with
performance comparison between channel inversion-based power control and
without power control has been given in Section 4. Furthermore, in Section 5,
the performance analysis of our proposed system using PSA-based power control
has been done. In section 6, the computational complexity analysis of all
considered power control strategies has been done. Next, the section 7
demonstrates all the simulation results with comprehensive discussions. In
Section 8 brief conclusion to this paper has been made, and finally, in the
end, references to the paper have been mentioned. The notation used in this
paper is summarized in Table 1, and the simulation parameters used in this
paper are listed in Table 2.

##  2 System and Channel Models

Figure 1: The System Model

We consider a CF-mMIMO system comprising $M$ APs, each equipped with $N$
antennas, and the network serving a total of $K<<MN$ single antenna UEs. The
heights of the $m$th AP and the $k$th UE are assumed to be $l_{m}$ and
$l^{\prime}_{k}$, respectively. This system operates in the time division
duplexed (TDD) mode, with an overall frame duration of
$T=\tau_{p}+\tau_{u}+\tau_{d}$ channel uses or slots. Out of these, the first
$\tau_{p}$ slots are utilized for CSI acquisition at the APs, the next
$\tau_{u}$ slots are for uplink data transmission, and final $\tau_{d}$ slots
are used for downlink data transmission. Similar to most other works on CF-
mMIMO [4, 8, 13], we assume the uplink and downlink channels to be reciprocal,
negating the need for any downlink training. Also, for simplicity, we assume
the pilot duration ($\tau_{p}$) to be long enough to avoid any pilot
contamination and assume the availability of accurate CSI at all the APs.
Here, we would like to note that the problem of CSI acquisition with limited
pilots in a CF-mMIMO system with mixed LoS/NLoS channels is a non-trivial
extension of the current literature on CSI acquisition (see [29] and the
references therein) in CF-mMIMO systems and is beyond the scope of this work.

Table 2:  Simulation Parameters Parameters |  Descriptions |  Values
---|---|---
$d_{0}$ |  Reference distance |  $1$ m
$f_{c}$ |  Carrier Frequency |  $3.5$ GHz
$l_{m}$ |  AP antenna height |  $10$ m
$l^{\prime}_{k}$ |  UE antenna height |  $1.5$ m
$MN$ |  AP antenna density |  $1024/\text{km}^{2}$
$\mu$ |  Average number of blockages per unit area |  $300/\text{km}^{2}$
$\alpha$ |  Fraction of built-up area in the network |  $0.5$
$\gamma$ |  Average altitude of buildings/blockages |  $20$
$\eta$ |  Propagation Exponent |  $3$
$P$ |  Number of PSA Particles |  $50$
$w$ |  Weight Coefficient |  $1$
$c_{1}$ |  Cognitive Coefficient |  $2$
$c_{2}$ |  Social Coefficient |  $2$
$Q$ |  Number of iterations |  $10000$

We note that an LoS path may or may not exist between an AP and a UE. This is
due to the random nature of the underlying blockages. Consequently, the
channel between the $m$th AP and the $k$th UE,
$\textbf{h}_{mk}\in\mathbb{C}^{N\times 1}$, is represented as,

$\textbf{h}_{mk}=\delta_{mk}\bar{\textbf{h}}_{mk}+\sqrt{\beta_{mk}}\dot{\textbf{h}}_{mk},$
(1)

where $\delta_{mk}$ is a binary-valued random variable that indicates the
presence or the absence of an LoS component, $\bar{\textbf{h}}_{mk}$ denotes
the LoS channel gain, $\beta_{mk}$ denotes the slow fading path loss component
of the NLoS channel, and $\dot{\textbf{h}}_{mk}$ represents the fast fading
component of the NLoS channel. More specifically, the fast fading NLoS
channel, $\dot{\textbf{h}}_{mk}$, is assumed to consist of independent and
identically distributed (i.i.d) zero-mean circularly symmetric complex
Gaussian (ZMCSCG) entries each having a unit variance, i.e.,
$\dot{\textbf{h}}_{mk}\sim\mathcal{CN}(0,\textbf{I}_{N})$. The slow fading
coefficient of the NLoS component, $\beta_{mk}$, is modeled as
$\beta_{mk}=\left(\frac{d_{mk}}{d_{0}}\right)^{-\eta}$, with $d_{mk}$ being
the horizontal (two dimensional) distance between the $m$th AP and the $k$th
UE and $d_{0}$ being the reference distance. On the other hand,
$\delta_{mk}=1$ represents the presence of an LoS component and
$\delta_{mk}=0$ indicates its absence, with
Pr$\left\\{\delta_{mk}=1\right\\}=P_{mk}$. We model $P_{mk}$ according to the
International Telecommunication Union (ITU) blockage model [29, 30]

$P_{mk}=(1-\omega)^{\sqrt{\alpha\mu d_{mk}}},$ (2)

where
$\omega\triangleq\sqrt{\frac{\pi}{2}}\frac{\gamma}{l_{m}-l_{k}^{{}^{\prime}}}\left[\text{erf}\left(\frac{l_{m}}{\gamma\sqrt{2}}\right)-\text{erf}\left(\frac{l_{k}^{{}^{\prime}}}{\gamma\sqrt{2}}\right)\right]$,
and erf(z)$\triangleq\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}dt$, $\gamma$
is the average altitude of blockages, $\alpha$ is the fraction of the built up
area, and $\mu$ is the average number of blockages per unit area. Finally, the
LoS channel gain between the $m$th AP and the $k$th UE takes the form

$\bar{\textbf{h}}_{m,k}=\textbf{a}(\theta_{mk})\sqrt{G_{m}G_{k}}\left(\frac{l^{\prime}_{k}l_{m}}{4\pi
x_{mk}}\right)e^{\iota 2\pi\frac{x_{mk}}{\lambda_{c}}},$ (3)

with $x_{mk}\triangleq\sqrt{d_{mk}^{2}+(l_{m}-l^{\prime}_{k})^{2}}$
representing the three-dimensional distance between the respective APs and
UEs, $\theta_{mk}$ being the azimuth angle between the AP and the UE,
$\lambda_{c}$ denoting the carrier wavelength, $G_{m}$ and $G_{k}$ being the
gains associated with the antennas at the $m$th AP and $k$th UE respectively,
and $\textbf{a}(\theta$) representing the array response vector (for an angle
$\theta$) at the AP such that, $\textbf{a}(\theta)=[1,e^{\iota
2\pi\frac{d}{\lambda_{c}}sin(\theta)},.........,e^{\iota
2(N-1)\pi\frac{d}{\lambda_{c}}sin(\theta)}]^{T},$ with $\iota=\sqrt{-1}$.

##  3 Performance Analysis

###  3.1 Uplink

We now evaluate the uplink performance of the system under consideration in
the presence of accurate CSI at the CPU. We assume that all the UEs
simultaneously transmit data to the APs over the same time-frequency resource
and the $k$th UE transmits the symbol $q_{k}$, satisfying
$\mathbb{E}\left[|q_{k}|^{2}\right]=1$. We can write the uplink signal
received at the $m$th AP as,

$\textbf{y}_{u,m}=\sum_{k=1}^{K}\sqrt{\zeta_{u,k}}\textbf{h}_{mk}q_{k}+\sqrt{N_{0}}\textbf{w}_{u,m},$
(4)

where $\zeta_{u,k}$ is the uplink power control coefficient for the $k$th UE,
and $\textbf{w}_{u,m}\sim\mathcal{CN}(\textbf{0}_{N},\textbf{I}_{N})$. Letting
$\boldsymbol{\zeta_{u}}=[\zeta_{u,1},....,\zeta_{u,K}]^{T}$,
$\textbf{q}=[q_{1},.....,q_{K}]^{T}$,
$D_{\boldsymbol{\zeta}_{u}}=\text{diag}(\boldsymbol{\zeta}_{u})$,
$\textbf{H}_{m}=[{\mathbf{h}_{m1},\ldots,\mathbf{h}_{mK}}]$, and
$\mathbf{H}=\left[\mathbf{H}^{T}_{1},\ldots,\mathbf{H}^{T}_{M}\right]^{T}$, we
can define the MMSE combining matrix at the CPU as,
$\textbf{V}=(\textbf{HH}^{H}+N_{0}\textbf{I}_{MN})^{-1}\textbf{H}$. Then,
letting
$\textbf{y}_{u}=[\textbf{y}_{u,1},\textbf{y}_{u,2},.....,\textbf{y}_{u,M}]^{T}$,
we can write the combined signal vector
$\textbf{r}=\textbf{V}^{H}\textbf{y}_{u}$, at the CPU as,

$\textbf{r}=\textbf{H}^{H}(\textbf{HH}^{H}+N_{0}\textbf{I}_{MN})^{-1}\textbf{H}\sqrt{D_{\boldsymbol{\zeta_{u}}}}\textbf{q}\\\
+\sqrt{N_{0}}\textbf{H}^{H}(\textbf{HH}^{H}+N_{0}\textbf{I}_{MN})^{-1}\textbf{w},$
(5)

whose $k$th component, representing the $k$th UEs uplink data stream, can be
written as,

$\displaystyle r_{k}$
$\displaystyle=\textbf{h}_{k}^{H}(\textbf{HH}^{H}+N_{0}\textbf{I}_{MN})^{-1}\textbf{h}_{k}\sqrt{\zeta_{u,k}}q_{k}$
$\displaystyle\qquad+\sum_{l=1,l\neq
k}^{K}\textbf{h}_{k}^{H}(\textbf{HH}^{H}+N_{0}\textbf{I}_{MN})^{-1}\textbf{h}_{l}\sqrt{\zeta_{u,l}}q_{l}$
$\displaystyle\qquad+\sqrt{N_{0}}\textbf{h}_{k}^{H}(\textbf{HH}^{H}+N_{0}\textbf{I}_{MN})^{-1}\textbf{w}$
(6) $\displaystyle=f_{kk}\sqrt{\zeta_{u,k}}q_{k}+\sum_{l=1,l\neq
k}^{K}f_{kl}\sqrt{\zeta_{u,l}}q_{l}+z_{u,k}.$ (7)

Here,
$f_{kk}=\textbf{h}_{k}^{H}(\textbf{HH}^{H}+N_{0}\textbf{I}_{MN})^{-1}\textbf{h}_{k}$
represents the effective channel coefficient for the $k$th UE’s data stream,
$f_{kl}=\textbf{h}_{k}^{H}(\textbf{HH}^{H}+N_{0}\textbf{I}_{MN})^{-1}\textbf{h}_{l}$
represents the interference channel coefficient from the $l$th UE’s stream and
$z_{u,k}=\sqrt{N_{0}}\textbf{h}_{k}^{H}(\textbf{HH}^{H}+N_{0}\textbf{I}_{MN})^{-1}\textbf{w}$
represents the effective noise term. Consequently, the uplink rate achievable
by the $k$th UE can be written as,

$R_{u,k}=\mathbb{E}\left[\log_{2}\left(1+\frac{|f_{kk}|^{2}\zeta_{u,k}}{\sum_{l=1,l\neq
k}^{K}\mathbb{E}[|f_{kl}|^{2}]\zeta_{u,l}+\text{var}(z_{u,k})}\right)\right].$
(8)

Here, we need to choose the power control coefficients, $\zeta_{u,k}$, to
optimize the system performance. Like most other works on cell-free massive
MIMO [16, 34, 35], which do not solve the expectations and use simulation
techniques to evaluate the rates achievable by the system. We also do the same
to evaluate the rates achievable by our considered system. To start with, we
consider two approaches for power control, viz. channel inversion-based power
control and max-min fairness-based power control, both are listed as follows,

####  3.1.1 Channel Inversion based Power Control

Figure 2: Comparison of Jain’s Fairness Index in uplink

In this case, the uplink power control coefficient for the $k$th UE,
$\zeta_{u,k}$, is defined as the inverse of the overall mean squared channel
power, that is,

$\zeta_{u,k}=\frac{1}{\sum_{m=1}^{M}|\bar{\textbf{h}}_{mk}|^{2}+\sum_{m=1}^{M}\beta_{mk}}.$
(9)

This means that the UEs with better channel conditions would receive less
power, while those with poor channel conditions would receive more power. This
is a simple power control technique that has been used in cellular massive
MIMO as an off the shelf solution, and acts as a good starting point for the
problem of power control in cell free massive MIMO.

####  3.1.2 Max-Min Power Control

In this case, the uplink power control coefficients for all the UEs are chosen
to maximize the minimum rate achieved by each UE, that is, $\zeta_{u,k}$ are
chosen to satisfy

$\begin{split}&\underset{\zeta_{u,k}}{\text{max}}\hskip
5.0pt{\text{min}}\hskip 5.0ptR_{u,k}\\\ &\text{s.t}\hskip
10.00002pt\zeta_{u,k}\geq 0,\hskip 14.22636pt\forall\hskip
5.0ptk=1,......,K.\end{split}$ (10)

Finally, Jain’s fairness index for the uplink rates can be expressed as

$J(R_{u})=\frac{\left(\sum_{k=1}^{K}R_{u,k}\right)^{2}}{K\sum_{k=1}^{K}(R_{u,k})^{2}}.$
(11)

Here, the simulation setup considers $64$ UEs with both the UEs, and the APs
uniformly distributed over a 1 $\text{km}^{2}$ area. We plot Jain’s Fairness
index for the system in the uplink as a function of the number of APs with and
without channel inversion-based power control in Fig. 2. Although, we can
observe a dip in the system performance both with and without channel
inversion based power control, for a smaller number of APs. This can be
attributed to the availability of LoS paths to only a small number of users
that ends up increasing the rate disparity among the users instead of
alleviating it.

###  3.2 Downlink

We now evaluate the system performance in the downlink. Here, we assume that
all $M$ APs serve all $K$ UEs simultaneously in the presence of accurate CSI
at the CPU. Let $s_{k}$ be the intended downlink symbol for the $k$th UE, such
that $\mathbb{E}\left[|s_{k}|^{2}\right]=1$, and
$\textbf{s}=\left[s_{1},s_{2},....,s_{K}\right]^{T}$. Let
$\textbf{A}_{m}=\left(\textbf{H}^{*}\textbf{H}^{T}+N_{0}\textbf{I}_{MN}\right)^{-1}\textbf{H}^{*}_{m}$
be the RZF precoding matrix at the $m$th AP, then the signal received by the
$k$th UE in the downlink can be expressed as,

$y_{d,k}=\sum_{m=1}^{M}\sqrt{\zeta_{d,k}}\textbf{h}_{mk}^{T}\textbf{A}_{m}\textbf{s}+\sqrt{N_{0}}w_{d,k},$
(12)

with $\zeta_{d,k}$ being the downlink power control coefficient for the $k$th
UE, and $w_{d,k}\sim\mathcal{CN}(0,1)$ being the AWGN. We can equivalently
write,

$\displaystyle y_{d,k}$
$\displaystyle=\sqrt{\zeta_{d,k}}\textbf{h}_{k}^{T}\left(\textbf{H}^{*}\textbf{H}^{T}+N_{0}\textbf{I}_{MN}\right)^{-1}\textbf{h}_{k}^{*}s_{k}$
$\displaystyle\qquad+\sum_{l=1,l\neq
k}^{K}\sqrt{\zeta_{d,l}}\textbf{h}_{k}^{T}\left(\textbf{H}^{*}\textbf{H}^{T}+N_{0}\textbf{I}_{MN}\right)^{-1}\textbf{h}_{l}^{*}s_{l}$
$\displaystyle\qquad+\sqrt{N_{0}}w_{d,k}.$ (13)
$\displaystyle=\sqrt{\zeta_{d,k}}g_{kk}s_{k}+\sum_{l=1,l\neq
k}^{K}\sqrt{\zeta_{d,l}}g_{kl}s_{l}+\sqrt{N_{0}}w_{d,k}.$ (14)

Here,
$g_{kk}=\textbf{h}_{k}^{T}\left(\textbf{H}^{*}\textbf{H}^{T}+N_{0}\textbf{I}_{MN}\right)^{-1}\textbf{h}_{k}^{*}$
represents the effective channel coefficient of the $k$th UE’s desired signal,
and
$g_{kl}=\textbf{h}_{k}^{T}\left(\textbf{H}^{*}\textbf{H}^{T}+N_{0}\textbf{I}_{MN}\right)^{-1}\textbf{h}_{l}^{*}$
is the coefficient for the interference channel between the $k$th UE and the
$l$th UE’s data stream. Hence, the downlink rate achievable by the $k$th UE
takes the form,

$R_{d,k}=\mathbb{E}\left[\log_{2}\left(1+\frac{|g_{kk}|^{2}\zeta_{d,k}}{\sum_{l=1,l\neq
k}^{K}\mathbb{E}[|g_{kl}|^{2}]\zeta_{d,l}+N_{0}}\right)\right].$ (15)

Similar to the uplink case, we can either use channel inversion-based power
control or max-min fairness-based power control, with the values of
$\zeta_{d,k}$ in the two cases being given as follows.

####  3.2.1 Channel Inversion based Power Control

Figure 3: Comparison of Jain’s Fairness Index in downlink

$\zeta_{d,k}=\frac{1}{\sum_{m=1}^{M}|\bar{\textbf{h}}_{mk}|^{2}+\sum_{m=1}^{M}\beta_{mk}}.$
(16)

####  3.2.2 Max-Min Power Control

$\zeta_{d,k}$ are chosen to satisfy

$\begin{split}&\underset{\zeta_{d,k}}{\text{max}}\hskip
5.0pt{\text{min}}\hskip 5.0ptR_{d,k}\\\ &\text{s.t}\hskip
10.00002pt\zeta_{d,k}\geq 0,\hskip 14.22636pt\forall\hskip
5.0ptk=1,......,K.\end{split}$ (17)

The consequent Jain’s Fairness index in the downlink can be expressed as,

$J(R_{d})=\frac{\left(\sum_{k=1}^{K}R_{d,k}\right)^{2}}{K\sum_{k=1}^{K}(R_{d,k})^{2}}.$
(18)

We now plot Jain’s fairness index both with and without power control in the
downlink as a function of the number of APs in Fig. 3. The general trend of
achieving a better fairness using channel inversion based power control
motivates us to explore other power control algorithms.

##  4  Two UE Testing

We now evaluate the performances of the two power control schemes under a test
case with only two randomly placed UEs in the system.

###  4.1 Uplink

Figure 4: CDF of the minimum achievable uplink SINRs for different power
control strategies.

The uplink achievable rate for the first UE is, The uplink achievable rate for
the first UE is,

$R_{u,1}=\mathbb{E}\left[\log_{2}\left(1+\frac{|f_{11}|^{2}\zeta_{u,1}}{\mathbb{E}[|f_{12}|^{2}]\zeta_{u,2}+var(z_{u,1})}\right)\right],\\\
$ (19)

and for the second UE is,

$R_{u,2}=\mathbb{E}\left[\log_{2}\left(1+\frac{|f_{22}|^{2}\zeta_{u,2}}{\mathbb{E}[|f_{21}|^{2}]\zeta_{u,1}+var(z_{u,2})}\right)\right],$
(20)

such that under channel inversion-based power control,

$\zeta_{u,1}=\frac{1}{\sum_{m=1}^{M}|\bar{\textbf{h}}_{m1}|^{2}+\sum_{m=1}^{M}\beta_{m1}},$
(21)
$\zeta_{u,2}=\frac{1}{\sum_{m=1}^{M}|\bar{\textbf{h}}_{m2}|^{2}+\sum_{m=1}^{M}\beta_{m2}}.$
(22)

Similarly, under max-min power control $\zeta_{u,1}$ and $\zeta_{u,2}$ are
chosen to satisfy

$\begin{split}&\underset{\left\\{\zeta_{u,1},\zeta_{u,2}\right\\}}{\text{max}}\hskip
1.00006pt{\text{min}}\hskip 5.0pt(R_{u,1},\hskip 1.00006ptR_{u,2})\\\
&\text{s.t}\hskip 10.00002pt\zeta_{u,1}\geq 0,\hskip 10.00002pt\zeta_{u,2}\geq
0.\end{split}$ (23)

Considering a system having $256$ APs and $64$ single antenna UEs uniformly
distributed over a 1 $\text{km}^{2}$ area operating at a carrier frequency of
2 GHz, we can now evaluate the relative performance of these two power control
strategies. For this purpose, we have calculated the achievable SINRs for both
cases over 10000 realizations. In Fig. 4, we plot the CDF of minimum uplink
SINR achievable by a UE. We observe that despite its simplicity, channel
inversion-based power control is sub-optimal and results in inferior system
performance, even with interference from just a single UE. This is mainly due
to the fact that while channel inversion based power control, guarantees equal
received SNR (on an average) to all the UEs it does not guarantee equal SINR,
and hence fails. Max min power control, on the other hand has high complexity
but guarantees a fixed minimum QoS to all the users.

###  4.2 Downlink

In this case, the rates achievable by the first and the second UEs are
respectively given as,

$R_{d,1}=\mathbb{E}\left[\log_{2}\left(1+\frac{|g_{11}|^{2}\zeta_{d,1}}{\mathbb{E}[|g_{12}|^{2}]\zeta_{d,2}+N_{0}}\right)\right],$
(24)
$R_{d,2}=\mathbb{E}\left[\log_{2}\left(1+\frac{|g_{22}|^{2}\zeta_{d,2}}{\mathbb{E}[|g_{21}|^{2}]\zeta_{d,1}+N_{0}}\right)\right].$
(25)

where

$\zeta_{d,1}=\frac{1}{\sum_{m=1}^{M}|\bar{\textbf{h}}_{m1}|^{2}+\sum_{m=1}^{M}\beta_{m1}}$
(26)

and

$\zeta_{d,2}=\frac{1}{\sum_{m=1}^{M}|\bar{\textbf{h}}_{m2}|^{2}+\sum_{m=1}^{M}\beta_{m2}},$
(27)

for channel inversion-based power control. Similarly, under max-min power
control $\zeta_{d,1}$ and $\zeta_{d,2}$ are chosen to satisfy

$\begin{split}&\underset{\left\\{\zeta_{d,1},\zeta_{d,2}\right\\}}{\text{max}}\hskip
1.00006pt{\text{min}}\hskip 5.0pt(R_{d,1},\hskip 1.00006ptR_{d,2})\\\
&\text{s.t}\hskip 10.00002pt\zeta_{d,1}\geq 0,\hskip 10.00002pt\zeta_{d,2}\geq
0.\end{split}$ (28)

Figure 5: CDF of the minimum achievable downlink SINRs for different power
control strategies.

We also evaluate the relative performance of the two power control strategies
in downlink. We plot the comparison of the CDFs of the minimum achievable
downlink SINRs for a UE over 10000 realizations in Fig. 5. We observe that
channel inversion-based power control is sub-optimal in downlink as well.
Again, a gap between the SINRs achievable in both uplink and downlink under
the two power control strategies can be observed, necessitating the
development of low-complexity power control techniques for the underlying
system model. In the next section, we describe the use of the particle swarm
algorithm as a candidate solution to our problem.

##  5 Particle Swarm Algorithm Based Power Control

Algorithm 1 PSA based power control for a CF-mMIMO system under LoS/NLoS
channel

Initialize $\mathbf{X}^{gbest}=0$, val($\mathbf{X}^{gbest}$) = 0

for $p$=$1$ to $P$ do

Initialize $\mathbf{X}^{p}\sim(\mathcal{U}[0,1])^{K}$,
$\mathbf{V}^{p}\sim(\mathcal{U}[0,1])^{K}$

$\text{val}(\mathbf{X}^{p})=g(\textbf{X}^{p})$

$\mathbf{X}^{pbest(p)}=\mathbf{X}^{p},\hskip
5.0pt\text{val}(\mathbf{X}^{pbes(p)})=\text{val}(\mathbf{X}^{p})$

if $\text{val}(\mathbf{X}^{pbest(p)})>\text{val}(\mathbf{X}^{gbest})$ then

$\text{val}(\mathbf{X}^{gbest})=\text{val}(\mathbf{X}^{pbest(p)})$

$\mathbf{X}^{gbest}=\mathbf{X}^{pbest(p)}$

end if

end for

for t = $1$ to $Q$ do

for $p$=$1$ to $P$ do

$\mathbf{V}^{p}(t+1)=w\mathbf{V}^{p}(t)+c_{1}\mathbf{r}_{1}\odot\left(\mathbf{X}^{pbest(p)}(t)-\mathbf{X}^{p}(t)\right)+c_{2}\mathbf{r}_{2}\odot(\left(\mathbf{X}^{gbest}(t)-\mathbf{X}^{p}(t)\right)$

$\mathbf{X}^{p}(t+1)=\mathbf{X}^{p}(t)+\mathbf{V}^{p}(t+1)$

$\text{val}(\mathbf{X}^{p}(t+1))=g(\mathbf{X}^{p}(t+1))$

if $\text{val}(\mathbf{X}^{p}(t+1))>\text{val}(\mathbf{X}^{pbest(p)})(t)$ then

$\mathbf{X}^{pbest(p)}(t)=\mathbf{X}^{p}(t+1)$

$\text{val}(\mathbf{X}^{pbest(p)}(t))=\text{val}(\mathbf{X}^{p}(t+1))$

end if

if $\text{val}(\mathbf{X}^{pbest(p)}(t))>\text{val}(\mathbf{X}^{gbest}(t))$
then

$\text{val}(\mathbf{X}^{gbest})=\text{val}(\mathbf{X}^{pbest(p)}(t))$

$\mathbf{X}^{gbest}(t)=\mathbf{X}^{pbest(p)}(t)$

end if

end for

end for

Particle Swarm Algorithm (PSA) is a bio-inspired algorithm that solves an
optimization problem by first generating a population of candidate solutions
(particles) and then iteratively improving each particle’s performance [32,
33]. Here we consider a total of ‘$P$’ particles, such that the $p$th
particle’s location is given by the vector $\mathbf{X}^{p}$ and velocity is
given by $\mathbf{V}^{p}$. The algorithm’s operation can be summarized as
follows, the $p$th particle travels through the solution space of power
control coefficients based on the sum of two velocity vectors. These velocity
vectors are in the direction of the local best solution of that particle and
the global best solution seen by the algorithm in general.

To elaborate, the objective function to be optimized, known as the fitness
function, is evaluated for each particle at each instant of time. Following
this, each particle stores the coefficients achieving the best value of the
fitness function as $\mathbf{X}^{pbest(p)}$ and the corresponding value is
given as $\text{val}(\mathbf{X}^{pbest(p)})$. Similarly, the position
corresponding to the best value by all the particles over all time is stored
as $\mathbf{X}^{gbest}$, and the corresponding value of the fitness function
is stored as $\text{val}(\mathbf{X}^{gbest})$.

Following this, in each iteration, and for each particle we generate two
velocity vectors, one pointing towards the vector $\mathbf{X}^{pbest(p)}(t)$,
and another towards $\mathbf{X}^{gbest}(t)$. The velocity of each particle is
then updated using a linear combination of its current velocity with these two
vectors, that is,

$\mathbf{V}^{p}(t+1)=w\mathbf{V}^{p}(t)+c_{1}\mathbf{r}_{1}\odot\left(\mathbf{X}^{pbest(p)}(t)-\mathbf{X}^{p}(t)\right)\\\
+c_{2}\mathbf{r}_{2}\odot(\left(\mathbf{X}^{gbest}(t)-\mathbf{X}^{p}(t)\right)$
(29)

with $\mathbf{r}_{1}\sim\mathcal{U}[0,1]$ and
$\mathbf{r}_{2}\sim\mathcal{U}[0,1]$.

Additionally, the movement in each direction is weighted by a different random
factor to provide a higher degree of freedom. Following this, the position of
each particle is updated as,

$\mathbf{X}^{p}(t+1)=\mathbf{X}^{p}(t)+\mathbf{V}^{p}(t+1).$ (30)

In Algorithm 1, $\odot$ represents the element-wise product of two vectors. We
use $w$ to denote the inertia weight coefficient, $c_{1}$ and $c_{2}$ are the
cognitive and social coefficients, respectively, which act as weights of the
stochastic term regarding the difference vectors between the particle
position.

###  5.1 Uplink Power Control

We can write the minimum rate achievable by a UE as

$\Omega(\boldsymbol{\zeta}_{u})={\text{min}}\hskip
5.0pt\Psi_{k}(\boldsymbol{\zeta}_{u}),$ (31)

where
$\boldsymbol{\zeta}_{u}=[\zeta_{u1},....,\zeta_{u,K}]^{T}\in\left(\mathcal{R}^{K}\right)^{+}$
and

$\Psi_{k}(\boldsymbol{\zeta}_{u})=\left[\log_{2}\left(1+\frac{|f_{kk}|^{2}\zeta_{u,k}}{\sum_{l=1,l\neq
k}^{K}\mathbb{E}[|f_{kl}|^{2}]\zeta_{u,l}+\text{var}(z_{u,k})}\right)\right].$

Now, the optimization problem for providing equitable uplink rates to all the
UEs in the system can be stated as,

$\displaystyle\begin{split}&\underset{\boldsymbol{\zeta}_{u}}{\text{max}}\hskip
5.0pt\Omega(\boldsymbol{\zeta}_{u})\\\ &\text{s.t}\hskip
10.00002pt\boldsymbol{\zeta}_{u}\geq 0.\end{split}$ (32)

We can solve (32), using PSA, according to Algorithm 1.

###  5.2 Downlink Power Control

The minimum downlink rate achievable can be written as

$\Omega(\boldsymbol{\zeta}_{d})={\text{min}}\hskip
5.0pt\Psi_{k}(\boldsymbol{\zeta}_{d}),$ (33)

where
$\boldsymbol{\zeta_{d}}=[\zeta_{d1},....,\zeta_{d,K}]^{T}\in\left(\mathcal{R}^{K}\right)^{+}$
and $f_{k}(\boldsymbol{\zeta}_{d})$ is given by,

$\Psi_{k}(\boldsymbol{\zeta}_{d})=\left[\log_{2}\left(1+\frac{|g_{kk}|^{2}\zeta_{d,k}}{\sum_{l=1,l\neq
k}^{K}\mathbb{E}[|g_{kl}|^{2}]\zeta_{d,l}+N_{0}}\right)\right].$

The optimization problem for providing equitable downlink rates to all the UEs
in the system can be stated as

$\displaystyle\begin{split}&{\underset{\boldsymbol{\zeta}_{d}}{\text{max}}\hskip
5.0pt\Omega(\boldsymbol{{\zeta}}_{d})}\\\ &\text{s.t}\hskip
10.00002pt\boldsymbol{\zeta}_{d}\geq 0,\end{split}$ (34)

We can solve (34), using PSA, according to algorithm 1, similar to the uplink
case.

##  6 Computational Complexity

In this section, we provide the computational complexity analysis of the three
power control techniques discussed previously, viz. channel inversion based
power control, max-min fairness based power control, and our proposed PSA
based power control.

Table 3: Computational Complexity of different Power Control Strategies Power Control Strategy |  Computational Complexity
---|---
Channel Inversion |  $\mathcal{O}(MK)$
Max-Min |  $\mathcal{O}(L^{K})$
PSA |  $\mathcal{O}(M^{3}KP)$

###  6.1 Channel Inversion based Power Control

In this case, the calculation of power control coefficients using equations
(9) and equation (16) requires approximately $MK$ operations, resulting in an
overall computational complexity given by $\mathcal{O}(MK)$.

###  6.2 Max-Min Power control

We have considered here a brute-force enabled max-min fairness based power
control, which involves an exhaustive search over all possible power
allocations to find an optimal solution. The key step for the power control
algorithm is the enumeration of all the possible power levels. In our
approach, we search over $L$ possible power levels (within the power
constraint) per UE, for the $K$ UEs, resulting in a search over $L^{K}$
possible power levels. This results in the computational complexity for max-
min power control being $\mathcal{O}(L^{K})$.

###  6.3 PSA based Power Control

The computational complexity of Particle Swarm Algorithm (PSA) based power
control primarily depends on the number of particles, the dimensionality of
search space, evaluation of fitness function and the number of iterations
[28]. The evaluation of fitness functions (32) and (34) for $P$ particles
requires approximately $M^{3}KP$ operations. Following this, the update of
particle positions and velocities requires $KP$ operations. Consequently, the
computational complexity per iteration is $\mathcal{O}\left(M^{3}KP\right)$.

##  7 Simulation Results

In this section, we present simulation results to evaluate the performance of
our proposed PSA-based power control approach. For the purpose of these
simulations, we consider $M$ APs and $K$ single-antenna UEs distributed
uniformly over a 1 $\text{km}^{2}$ area. We assume that the NLoS propagation
constant $\beta_{mk}$ to take the form
$\beta_{mk}=\text{min}(1,(x_{mk}/d_{0})^{-\eta})$, with $d_{0}$ being the
reference distance, and $\eta$ being the propagation exponent. Unless stated
otherwise, Table 2 lists all the parameters used for these experiments.

###  7.1 The Two UE Case

We first compare the CDFs of the minimum achievable rates of the system having
$256$ APs and $2$ UEs under the three power control strategies discussed in
this paper for both uplink and downlink in Fig. 6 and Fig. 7 respectively.
After keen observation, we see that by the use of PSA-based power control, the
90% likely minimum achievable rate of the considered system can be improved by
about 60% in both uplink downlink directions as compared to channel inversion-
based power control. This results in a performance closer to the max min-based
power control. This significant performance gap between channel inversion and
PSA-based power control motivates the use of PSA-based power control for a
higher number of UEs connected in the network.

Figure 6: CDF of the minimum achievable uplink SINRs for different power
control strategies in $2$ UEs case. Figure 7: CDF of the minimum achievable
downlink SINRs for different power control strategies in $2$ UEs case.

###  7.2 The 64 UE case

We now compare the performance of the PSA-based power control algorithm
against channel inversion-based power control for a $256$ AP and $64$ UE
system. In Fig. 8 and Fig. 9, we plot the CDFs of the minimum achievable rates
for uplink and downlink, respectively. We observe that PSA-based power control
provides about 60% improvement for the 90% likely minimum achievable rate of
the considered system in uplink and more than doubles the same for in the
downlink, as compared to channel inversion-based power control.

Figure 8: CDF of the minimum achievable uplink SINRs for different power
control strategies in $64$ UEs case. Figure 9: CDF of the minimum achievable
downlink SINRs for different power control strategies in $64$ UEs case.

Finally, to quantify the improvement in fairness, we compare Jain’s Fairness
index between the PSA and channel inversion-based power control as a function
of the number of APs for uplink and downlink in Fig. 10 and Fig. 11
respectively. We observe that the use of PSA-based power control results in as
much as 50% improvement in the fairness of the system for lower AP densities.

Figure 10: Comparison of Jain’s Fairness Index in the uplink for $64$ UEs
case. Figure 11: Comparison of Jain’s Fairness Index in downlink for $64$ UEs
case.

##  8 Conclusion

In this paper, we discussed the performance of the CF mMIMO system under a
probabilistic LoS/NLoS channel using different power control strategies at
moderate AP densities. First, we evaluated the performance of our proposed
system with Channel inversion-based power control and then compared its
performance with max-min fairness-based power control. Motivated by the
performance gap, we formulated the PSA-based power control to provide
approximately uniform coverage to all the UEs. Future work could include the
design and use of other meta-heuristic algorithm-based power control
strategies.

References:
$\left[1\right]$ Marzetta, Thomas L.: ‘Fundamentals of massive MIMO’,
Cambridge University Press, 2016
$\left[2\right]$ Chopra, Ribhu and Murthy, Chandra R and Papazafeiropoulos,
Anastasios K.: ‘Uplink performance analysis of cell-free mMIMO systems under
channel aging’, IEEE Communications Letters, 2021, 25, pp. 2206–2210
$\left[3\right]$ Björnson, Emil and Sanguinetti, Luca and Wymeersch, Henk and
Hoydis, Jakob and Marzetta, Thomas L.: ‘Massive MIMO is a reality—What is
next?: Five promising research directions for antenna arrays’, Elsevier,
Digital Signal Processing, 2019, 94, pp. 3–20
$\left[4\right]$ Panwar, Nisha and Sharma, Shantanu and Singh, Awadhesh
Kumar.: ‘A survey on 5G: The next generation of mobile communication’,
Elsevier ,Physical Communication, 2016, 18, pp. 64–84
$\left[5\right]$ Ngo, Hien Quoc and Tran, Le-Nam and Duong, Trung Q and
Matthaiou, Michail and Larsson, Erik G.: ‘On the total energy efficiency of
cell-free massive MIMO’, IEEE Transactions on Green Communications and
Networking, 2017, 2, pp. 25–39
$\left[6\right]$ Yang, Ping and Xiao, Yue and Xiao, Ming and Li, Shaoqian.:
‘6G wireless communications: Vision and potential techniques’, IEEE network,
2019, 33, pp. 70–75
$\left[7\right]$ Larsson, Erik G and Edfors, Ove and Tufvesson, Fredrik and
Marzetta, Thomas L.: ‘Massive MIMO for next generation wireless systems’, IEEE
communications magazine, 2014, 52, pp. 186–195
$\left[8\right]$ Nayebi, Elina and Ashikhmin, Alexei and Marzetta, Thomas L
and Yang, Hong.: ‘Cell-free massive MIMO systems’, Asilomar Conference on
Signals, Systems and Computers, 2015, 49, pp. 695–699
$\left[9\right]$ Ngo, Hien Quoc and Ashikhmin, Alexei and Yang, Hong and
Larsson, Erik G and Marzetta, Thomas L.: ‘Cell-free massive MIMO versus small
cells’, IEEE Transactions on Wireless Communications, 2017, 16, pp. 1834–1850
$\left[10\right]$ Ngo, Hien Quoc and Ashikhmin, Alexei and Yang, Hong and
Larsson, Erik G and Marzetta, Thomas L.: ‘Cell-free massive MIMO: Uniformly
great service for everyone’, IEEE international workshop on signal processing
advances in wireless communications (SPAWC), 2015, 16, pp. 201–205
$\left[11\right]$ Özdogan, Özgecan and Björnson, Emil and Zhang, Jiayi.:
‘Performance of cell-free massive MIMO with Rician fading and phase shifts’,
IEEE Transactions on Wireless Communications, 2019, 18, pp. 5299–5315
$\left[12\right]$ Zhang, Yao and Zhou, Meng and Cao, Haotong and Yang,
Longxiang and Zhu, Hongbo.: ‘On the performance of cell-free massive MIMO with
mixed-ADC under Rician fading channels’, IEEE Communications Letters, 2019,
24, pp. 43–47
$\left[13\right]$ Jin, Si-Nian and Yue, Dian-Wu and Nguyen, Ha H.: ‘Spectral
and energy efficiency in cell-free massive MIMO systems over correlated Rician
fading’, IEEE Systems Journal, 2020, 15, pp. 2822–2833
$\left[14\right]$ Nayebi, Elina and Ashikhmin, Alexei and Marzetta, Thomas L
and Yang, Hong and Rao, Bhaskar D.: ‘Precoding and power optimization in cell-
free massive MIMO systems’, IEEE Transactions on Wireless Communications,
2017, 16, pp. 4445–4459
$\left[15\right]$ Bashar, Manijeh and Cumanan, Kanapathippillai and Burr,
Alister G and Ngo, Hien Quoc and Debbah, Mérouane.: ‘Cell-free massive MIMO
with limited backhaul’, IEEE International Conference on Communications (ICC),
2018, 64, pp. 1–7
$\left[16\right]$ Papazafeiropoulos, Anastasios and Kourtessis, Pandelis and
Di Renzo, Marco and Chatzinotas, Symeon and Senior, John M.: ‘Performance
analysis of cell-free massive MIMO systems: A stochastic geometry approach’,
IEEE Transactions on Vehicular Technology, 2020, 69, pp. 3523–3537
$\left[17\right]$ Zhang, Jiayi and Wei, Yinghua and Björnson, Emil and Han, Yu
and Jin, Shi.: ‘Performance analysis and power control of cell-free massive
MIMO systems with hardware impairments’, IEEE Access, 2018, 6, pp. 55302–55314
$\left[18\right]$ Björnson, Emil and Sanguinetti, Luca.: ‘Making cell-free
massive MIMO competitive with MMSE processing and centralized implementation’,
IEEE Transactions on Wireless Communications, 2019, 19, pp. 77–90
$\left[19\right]$ Atzeni, Italo and Arnau, Jesús and Kountouris, Marios.:
‘Downlink Cellular Network Analysis With LOS/NLOS Propagation and Elevated
Base Stations’, IEEE Transactions on Wireless Communications, 2017, 17, pp.
142–156
$\left[20\right]$ Mukherjee, Sudarshan and Chopra, Ribhu.: ‘Performance
analysis of cell-free massive MIMO systems in LoS/NLoS channels’, IEEE
Transactions on Vehicular Technology, 2022, 71, pp. 6410–6423
$\left[21\right]$ Özdogan, Özgecan and Bjöornson, Emil and Zhang, Jiayi.:
‘Cell-free massive MIMO with Rician fading: Estimation schemes and spectral
efficiency’, Asilomar Conference on Signals, Systems, and Computers, 2018, 52,
pp. 975–979
$\left[22\right]$ Farooq, Muhammad and Ngo, Hien Quoc and others.: ‘A low-
complexity approach for max-min fairness in uplink cell-free massive MIMO’,
IEEE Vehicular Technology Conference (VTC2021-Spring), 2021, 93, pp. 1–6
$\left[23\right]$ Tran, Le-Nam and Ngo, Hien Quoc.: ‘First-order methods for
energy-efficient power control in cell-free massive MIMO’, Asilomar Conference
on Signals, Systems, and Computers, 2019, 53, pp. 848–852
$\left[24\right]$ Bashar, Manijeh and Akbari, Ali and Cumanan,
Kanapathippillai and Ngo, Hien Quoc and Burr, Alister G and Xiao, Pei and
Debbah, Merouane and Kittler, Josef.: ‘Exploiting deep learning in limited-
fronthaul cell-free massive MIMO uplink’, IEEE Journal on Selected Areas in
Communications, 2020, 38, pp. 1678–1679
$\left[25\right]$ Saray, Akbar Mazhari and Ebrahimi, Afshin.: ‘MAX-MIN Power
Control of Cell-Free Massive MIMO System employing Deep Learning’, West Asian
Symposium on Optical and Millimeter-wave Wireless Communications (WASOWC),
2022, 4, pp. 1–4
$\left[26\right]$ Rajapaksha, Nuwanthika and Manosha, KB Shashika and
Rajatheva, Nandana and Latva-Aho, Matti.: ‘Deep learning-based power control
for cell-free massive MIMO networks’, ICC IEEE International Conference on
Communications, 2021, 21, pp. 1–7
$\left[27\right]$ Chakraborty, Sucharita and Björnson, Emil and Sanguinetti,
Luca.: ‘Centralized and distributed power allocation for max-min fairness in
cell-free massive MIMO’, asilomar conference on signals, systems, and
computers, 2019, 53, pp. 576–580
$\left[28\right]$ Conceição, Filipe and Antunes, Carlos Henggeler and Gomes,
Marco and Silva, Vitor and Dinis, Rui.: ‘Max-Min Fairness Optimization in
Uplink Cell-Free Massive MIMO Using Meta-Heuristics’, IEEE Transactions on
Communications, 2022, 70, pp. 1792–1807
$\left[29\right]$ Anubhab Chowdhury and Pradip Sasmal and Chandra R. Murthy
and Ribhu Chopra.: ‘On the Performance of Distributed Antenna Array Systems
With Quasi-Orthogonal Pilots’, IEEE Trans. Veh. Technol., 2022, 71, pp.
3326–3331
$\left[30\right]$ Kim, Dongsun and Lee, Jemin and Quek, Tony QS.: ‘Multi-layer
unmanned aerial vehicle networks: Modeling and performance analysis’, IEEE
Transactions on Wireless Communications, 2019, 19, pp. 325–339
$\left[31\right]$ Mondal, Bishwarup and Thomas, Timothy A. and Visotsky,
Eugene and Vook, Frederick W. and Ghosh, Amitava and Nam, Young-han and Li,
Yang and Zhang, Jianzhong and Zhang, Min and Luo, Qinglin and Kakishima,
Yuichi and Kitao, Koshiro.: ‘3D channel model in 3GPP’, IEEE Communications
Magazine, 2015, 53, pp. 16–23
$\left[32\right]$ Kennedy, James and Eberhart, Russell.: ‘Particle swarm
optimization’, Proceedings of ICNN international conference on neural
networks, 2015, 4, pp. 1942–1948
$\left[33\right]$ Blackwell, Tim.: ‘Particle swarm optimization in dynamic
environments’, Evolutionary computation in dynamic and uncertain environments,
2007, 4, pp. 29–49
$\left[34\right]$ Parida, Priyabrata and Dhillon, Harpreet S and Molisch,
Andreas F.: ‘Downlink performance analysis of cell-free massive MIMO with
finite fronthaul capacity’, IEEE 88th Vehicular Technology Conference (VTC-
Fall), 2018, pp. 1–6
$\left[35\right]$ Zhang, Yao and Cao, Haotong and Zhou, Meng and Yang,
Longxiang.: ‘Cell-free massive MIMO: Zero forcing and conjugate beamforming
receivers’, Journal of Communications and Networks, 2007, 21, pp. 529–538