# Joint AP-UE Association and Power Factor Optimization for Distributed
Massive MIMO

Mohd Saif Ali Khan*, Samar Agnihotri* and Karthik R.M.+

Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>*School of Computing & EE, Indian Institute of
Technology Mandi, HP, India  +Ericsson India Global Services Pvt. Ltd.,
Chennai, TN, India

###### Abstract

The uplink sum-throughput of distributed massive multiple-input-multiple-
output (mMIMO) networks depends majorly on Access point (AP)-User Equipment
(UE) association and power control. The AP-UE association and power control
both are important problems in their own right in distributed mMIMO networks
to improve scalability and reduce front-haul load of the network, and to
enhance the system performance by mitigating the interference and boosting the
desired signals, respectively. Unlike previous studies, which focused
primarily on addressing these two problems separately, this work addresses the
uplink sum-throughput maximization problem in distributed mMIMO networks by
solving the joint AP-UE association and power control problem, while
maintaining Quality-of-Service (QoS) requirements for each UE. To improve
scalability, we present an l1-penalty function that delicately balances the
trade-off between spectral efficiency (SE) and front-haul signaling load. Our
proposed methodology leverages fractional programming, Lagrangian dual
formation, and penalty functions to provide an elegant and effective iterative
solution with guaranteed convergence. Extensive numerical simulations validate
the efficacy of the proposed technique for maximizing sum-throughput while
considering the joint AP-UE association and power control problem,
demonstrating its superiority over approaches that address these problems
individually. Furthermore, the results show that the introduced penalty
function can help us effectively control the maximum front-haul load.

###### Index Terms:

Distributed Massive MIMO, AP-UE Association, Power Control, Uplink Massive
MIMO, Sum Spectral Efficiency

##  I Introduction

The increasing number of users and their demands for low latency and data-
intensive services in cellular networks require adoption of technologies that
can provide higher capacity, efficiency, and reliability in the networks.
Massive MIMO (mMIMO) has emerged as an appealing solution in this scenario,
leveraging large number of antennas to offer spatial multiplexing and channel
diversity, leading to increased spectral efficiency, improved coverage, and
enhanced interference management due to favorable propagation [1]. Moreover,
mMIMO systems support higher user densities, making them well-suited for high-
demand scenarios. However, in mMIMO based cellular systems, two key challenges
need to be addressed: inter-cell interference and large variations in
achievable QoS for end users [2]. Inter-cell interference stems from
overlapping coverage zones of neighboring cells where simultaneous
transmissions can cause interference to users in adjacent cells. QoS
variations occur due to wide variations in physical channels between UEs and
the Base Station (BS).

Distributed mMIMO offers a new paradigm that builds on the advantages of mMIMO
in cellular networks, yet addresses the accompanying challenges [3, 4, 5].
Unlike traditional cellular networks, distributed mMIMO deploys a large number
of access points (APs) throughout the coverage region. This departure from
typical cell definition eliminates inter-cell interference, a persistent
problem in cellular networks. Furthermore, UEs’ closeness to these distributed
APs may substantially enhance the coverage probability, improving overall QoS
for users.

In the original notion of distributed mMIMO [3], in the absence of traditional
cell boundaries, it is assumed that all APs simultaneously serve all UEs over
the same time and frequency resources. This assumption makes the system
impractical, especially when the number of UEs increases dramatically, leading
to substantial increase in the computational overhead and front-haul signaling
load associated with Channel State Information (CSI) and data transfer [6].
This leads to the introduction of scalable distributed mMIMO, as in [7, 6],
where each UE is served by a subset of APs. This is justified as more than 95%
received signal strength is concentrated in a few nearby APs. Thereby, the AP-
UE association is one of the important aspects of distributed mMIMO networks
to reduce the computation costs and front-haul load. Power control is another
important aspect to achieve efficient utilization of resources in any
communication network. Beyond its role in enhancing system energy efficiency,
as emphasized in [5], efficient power control plays a crucial role in pro-
actively minimizing interference, thereby improving the spectral efficiency
(SE). Thus, the implementation of appropriate power control measures is highly
desirable. So far these AP-UE association and power control problems have been
handled separately. However, we argue that jointly addressing power control
and AP-UE association problems is more effective in mitigating interference,
boosting signal strength, and enhancing achievable throughput.

A significant amount of work has been undertaken to address power control and
AP-UE association problems in distributed mMIMO systems. However, it is worth
noting that much of it tends to focus entirely on either power control [3, 5,
8, 6], or AP-UE association [6, 9, 10]. In [5] and [11], the authors focus on
a downlink system, where the inherent coupling between AP-UE association and
power control makes the joint analysis less complex in contrast to the uplink
scenario, which lacks such natural coupling between the two. This highlights
the importance of customized methodologies for extending joint optimization
techniques from downlink to uplink distributed mMIMO systems. While [12]
introduces a deep learning-based joint approach for power control and AP-UE
association in uplink scenarios. It has high computational cost of extensive
training and lacks the explainability associated with traditional models. In
[13] and [14], a novel approach is adopted by formulating an optimization
problem from the perspective of maximizing the minimum Signal-to-Interference-
Noise ratio for joint power control and AP-UE association weights for the
uplink distributed mMIMO. However, there is an inherent difficulty in
allocating APs to UEs within the defined optimization challenges and it
prompted the authors to suggest different methods for AP-UE scheduling.

In this work, we address the challenging nature of uplink distributed mMIMO
systems by formulating a mixed-integer non-convex sum-throughput maximization
problem that incorporates joint AP-UE association and power control, subject
to the QoS requirements of each UE in terms of the SE. To navigate the trade-
off between the SE and the front-haul load, we introduce an l1-penalty
approach [15]. This strategy allows us to control the trade-off between the SE
and the front-haul load in the network as per the operational requirements.
Our formulation incorporates an important consideration for preserving QoS, a
minimum threshold of the SE for each UE, thus offering equitable connectivity
experience for all users. The optimization problem is solved by using an
iterative approach with guaranteed convergence to a solution, by leveraging
the fractional programming and the Lagrangian dual formulations.

Organization: The paper is organised as follows: Section II presents the
system model outlining the key components and assumptions used in this study
and subsection II-C details the problem formulation. Section III introduces
our proposed solution and establishes its convergence. In Section IV, we
numerically evaluate the performance of the proposed scheme and establish its
effectiveness with respect to existing schemes. Finally, Section V concludes
the paper by summarizing our work and discussing its future directions.

##  II System Model

We consider a system consisting of $T$ single antenna UEs and $M$ multi-
antenna APs, where each AP has $A$ antennas such that $T<<MA$. The APs and UEs
are distributed uniformly over the given coverage area and APs are jointly
serving all users over the same time and frequency resources. In order to help
in the signal processing and coordination among APs, they are connected with a
CPU via a front-haul connection. We consider the TDD mode of operation where
the AP-UE channel vector is estimated using the uplink transmission. To model
our channel, we have considered the block fading model with a coherence block
of length $L_{c}$. Let $L_{p}$ be the length of orthogonal pilots such that
$L_{p}$ length of coherence block is used for uplink pilot training. The
Rayleigh fading coefficient representing the small-scale fading is denoted by
$\textbf{h}_{mt}\in\mathbb{C}^{\textit{A}\times 1}$, whereas the large scale-
fading coefficient (LSFC) that incorporates both path-loss and shadowing
effects is denoted by $\beta_{mt}$. The channel vector
$\textbf{g}_{mt}\in\mathbb{C}^{\textit{A}\times 1}$, that incorporates both
the small-scale fading as well as large scale fading, is defined as
$\textbf{g}_{mt}=\beta_{mt}^{1/2}\textbf{h}_{mt}$. We assume that
$\textbf{h}_{t,m}{\sim}\mathcal{N}_{\mathbb{C}}(0,1)$ is independent and
identically distributed (i.i.d.) random vector.

###  II-A Uplink Pilot Training

We assume that during uplink channel estimation, each UE
$t\in\mathcal{T}\overset{\Delta}{=}\\{1,...,T\\}$ transmits the pilot sequence
$\mathbf{\Psi}_{t}\in\mathbb{C}^{L_{p}\times 1}$, such that
$||\mathbf{\Psi}_{t}||^{2}=p_{p}L_{p}$, where $p_{p}$ represents the
normalized signal-to-noise power. The received signal
$\textbf{y}^{pilot}_{m}\in\mathbb{C}^{\textit{A}\times L_{p}}$ at the $m^{th}$
AP is given by :

$\displaystyle\textbf{y}^{pilot}_{m}=\sum_{t\in\mathcal{T}}\textbf{g}_{mt}\mathbf{\Psi}_{t}+\textbf{n}_{m},$

where $\textbf{n}_{m}\in\mathbb{C}^{\textit{A}\times L_{p}}$ is the complex
additive white Gaussian noise matrix of the AP $m$ consists of complex i.i.d.
elements. The MMSE estimate
$\hat{\textbf{g}}_{mt}\in\mathbb{C}^{\textit{A}\times 1}$ of $\textbf{g}_{mt}$
is given by [3]:

$\displaystyle\hat{\textbf{g}}_{mt}=\frac{\beta_{mk}}{\sum_{t^{\prime}=1}^{T}p_{p}L_{p}\beta_{mt^{\prime}}\left|\Psi^{H}_{t}\Psi_{t^{\prime}}\right|^{2}+1}\textbf{y}^{pilot}_{m}\mathbf{\Psi}_{t}.$

The mean-square of the estimate $\hat{\textbf{g}}_{mt}$ for any antenna
element given by, as in [3]:

$\displaystyle\gamma_{mt}=\left(\frac{p_{p}\tau\beta_{mt}^{2}}{\sum_{t^{\prime}=1}^{T}p_{p}\tau\beta_{mt^{\prime}}\left|\Psi^{H}_{t}\Psi_{t^{\prime}}\right|^{2}+1}\right).$

###  II-B Uplink Data Transmission

Let $x_{t}$ be the uplink payload signal transmitted from the UE $t$, such
that $\mathbb{E}\\{|x_{t}|^{2}\\}=1$. The uplink normalized signal-to-noise
power is denoted by $p_{u}$. The uplink signal received at the $m^{th}$ AP is
given by:

$\displaystyle\textbf{y}^{u}_{m}=\sqrt{p_{u}}\sum_{t\in\mathcal{T}}\textbf{g}_{mt}\sqrt{\eta^{u}_{t}}x_{t}+\textbf{n}_{m}.$

To detect the signal of the UE $t$ in a centralised manner, the AP $m$ from
$\mathcal{M}_{t}$ send $\hat{\textbf{g}}_{mt}^{*}\textbf{y}^{u}_{m}$ to the
CPU, where
$\mathcal{M}_{t}\subseteq\mathcal{M}\overset{\Delta}{=}\\{1,...,M\\}$ is the
subset of APs serving the UE $t$. The received signal at CPU for the UE $t$
is, as in [3]:

$\displaystyle\hat{\textbf{y}}^{u}_{t}=\sum_{m\in\mathcal{M}_{t}}\sum_{a=1}^{A}[\hat{\textbf{g}}_{mt}^{*}]_{a}[\textbf{y}^{u}_{m}]_{a},$

where $[.]_{a}$ represents the $a^{th}$ element of the vector $[.]$.
Decomposing the $\hat{\textbf{y}}^{u}_{t}$ as in [3], results in:

$\displaystyle\hat{\textbf{y}}^{u}_{t}=\text{DS}_{t}\ x_{t}+\text{BU}_{t}\
x_{t}+\sum_{t^{\prime}=1,t^{\prime}\neq t}^{T}\text{UI}_{tt^{\prime}}\
x_{t^{\prime}}+n_{t},$

where

$\displaystyle\text{DS}_{t}\overset{\Delta}{=}\sqrt{p_{u}\eta_{t}^{u}}\mathbb{E}\left\\{\sum_{m\in\mathcal{M}_{t}}\sum_{a=1}^{A}[\hat{\textbf{g}}_{mt}^{*}]_{a}[\textbf{g}_{mt}]_{a}\right\\},$
$\displaystyle\text{BU}_{t}\overset{\Delta}{=}\sqrt{p_{u}\eta_{t}^{u}}\sum_{m\in\mathcal{M}_{t}}\sum_{a=1}^{A}[\hat{\textbf{g}}_{mt}^{*}]_{a}[\textbf{g}_{mt}]_{a}-\text{DS}_{t},$
$\displaystyle\text{UI}_{tt^{\prime}}\overset{\Delta}{=}\sqrt{p_{u}\eta_{t^{\prime}}^{u}}\sum_{m\in\mathcal{M}_{t}}\sum_{a=1}^{A}[\hat{\textbf{g}}_{mt}^{*}]_{a}[\textbf{g}_{mt^{\prime}}]_{a},$
$\displaystyle
n_{t}\overset{\Delta}{=}\sum_{m\in\mathcal{M}_{t}}\sum_{a=1}^{A}[\hat{\textbf{g}}_{mt}^{*}]_{a}[\textbf{n}_{m}]_{a}.$

Treating interference as effective-noise, we get SE for the UE $t$ as:

$\displaystyle\text{SE}^{u}_{t}=w\text{log}_{2}\\!\\!\left(\\!\\!1+\frac{|\text{DS}_{t}|^{2}}{\\!\\!\mathbb{E}\\{|\text{BU}_{t}|^{2}\\}+\\!\\!\\!\\!\\!\sum\limits_{t^{\prime}=1,t^{\prime}\neq
t}^{T}\\!\\!\\!\\!\\!\\!\\!\mathbb{E}\\{|\text{UI}_{tt^{\prime}}|^{2}\\}+\mathbb{E}\\{|n_{t}|^{2}\\}\\!\\!}\right),$

where $w\overset{\Delta}{=}\left(\frac{1-\frac{L_{p}}{L_{c}}}{2}\right)$. The
closed form of the SE is, as in [16]:

$\displaystyle\text{SE}^{u}_{t}=w\text{log}_{2}\left(1+\Gamma^{*}_{t}\right),$
(1)

where $\Gamma^{*}_{t}$ is defined as:

$\displaystyle\Gamma^{*}_{t}\overset{\Delta}{=}\frac{A^{2}p_{u}\eta_{t}^{u}(\sum\limits_{m\in\mathcal{M}}d_{mt}\gamma_{mt})^{2}}{{I}_{t}},$
(2)

with

$\displaystyle\begin{split}{I}_{t}\overset{\Delta}{=}\sum\limits_{t^{\prime}=1,t^{\prime}\neq
t}^{T}\\!\\!\\!\\!\\!A^{2}p_{u}\eta_{t^{\prime}}^{u}\left|\Psi^{H}_{t}\Psi_{t^{\prime}}\right|^{2}(\\!\\!\sum\limits_{m\in\mathcal{M}}\\!\\!\\!d_{mt}\gamma_{mt}\frac{\beta_{mt^{\prime}}}{\beta_{mt}})^{2}+\sum\limits_{t^{\prime}=1}^{T}\\!\\!\\!Ap_{u}\eta_{t^{\prime}}^{u}\\!\\!\\!\sum\limits_{m\in\mathcal{M}}\\!\\!\\!d_{mt}\gamma_{mt}\beta_{mt^{\prime}}+\\!\\!\\!\sum\limits_{m\in\mathcal{M}}\\!\\!\\!Ad_{mt}\gamma_{mt},\end{split}$
(3)

and $d_{mt}$ is the indicator variable indicating the association between the
UE $t$ and the AP $m$, such that $d_{mt}=1$ if the AP $m$ serves the UE $t$
otherwise $d_{mt}=0$.

###  II-C Problem Formulation

Our primary objective is to maximize the total SE of all UEs. Furthermore, we
ensure that each UE gets a minimum QoS. The minimum QoS for all the UEs is
defined as the minimum SE that every UE gets. Further, to prevent the low SE
providing APs from serving the UEs, we introduce the l1-penalty in the
objective function. This not only improves the energy efficiency, but also
prevents the front-haul overloading111Computation of front-haul load in our
framework is expressed mathematically as
$\sum_{t\in\mathcal{T}}d_{mt}\text{SE}^{u}_{t}\ \forall m$. Additionally, the
maximum front-haul is defined as the highest front-haul load among all APs.
This metric provides insight into the peak load on the front-haul
infrastructure across the entire network.. This results in the following
problem formulation:

$\displaystyle\underset{\bm{\eta^{u}},\textbf{D}}{\max~{}}\sum\limits_{t=1}^{T}\left(\text{SE}^{u}_{t}-\alpha||{\textbf{D}_{t}}||_{1}\right),$
(4a)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\text{subject
to :}\ d_{mt}\in\\{0,1\\},\forall m\in\mathcal{M},\ t\in\mathcal{T},$ (4b)
$\displaystyle 0\leq\eta^{u}_{t}\leq 1,\forall t\in\mathcal{T},$ (4c)
$\displaystyle\text{SE}^{u}_{t}\geq\text{SE}^{\text{QoS}}_{t},\forall
t\in\mathcal{T},$ (4d) $\displaystyle\sum\limits_{m=1}^{M}d_{mt}\geq 1,\forall
t\in\mathcal{T},$ (4e)

where $\bm{\eta^{u}}=[\eta^{u}_{1},\eta^{u}_{2},...\eta^{u}_{T}]^{\intercal}$
and D is the AP-UE association matrix with each element as $d_{mt}$.
$\textbf{D}_{t}$ represents the column vector of matrix of D with respect to
the UE $t$. The regularization coefficient $\alpha\geq 0$ is the balancing
factor between the front-haul load and the SE. $\text{SE}^{\text{QoS}}_{t}$
represents the minimum QoS requirement for the UE $t$. The constraint (4b)
represents the binary variable indicating the association between the UEs and
APs. Constraint (4c) represents the uplink power control coefficient.
Constraint (4d) is for the minimum QoS requirement and constraint (4e)
specifies that atleast one AP must be connected to the UE. The objective
function in (4a) is a non-smooth and mixed integer type non-convex function
that makes the overall problem NP-hard. Finding the optimal solution of such
mixed integer type problems requires high computation resources. In the
following section we provide a heuristic solution of problem (4) using a
relaxation approach.

##  III Proposed Joint Solution using Relaxation Technique

Problem (4) is a mixed-integer type problem which has a very high complexity,
so we first convert the discrete binary constraint (4b) into a continuous one,
resulting in:

$\displaystyle d_{mt}\in[0,1],\forall m\in\mathcal{M},\ t\in\mathcal{T}.$ (5)

Remark 1 : The introduction of l1-penalty in the objective function serves to
induce the sparsity in the AP-UE association matrix D. By careful selection of
the trade-off factor $\alpha$, controlled-sparsity is achieved by limiting the
number of APs that serve the UEs. Therefore, the matrix D has non-zero values
only for the chosen AP-UE pairs. After completion of AP-UE association, all
entries in D undergo rounding to the nearest integer values i.e., zero and
one, thus maintaining the feasibility of the optimization problem (4a). In the
forthcoming results section, the SE with and without rounding-off of values in
D are compared to numerically show the effectiveness of our proposed approach.

Then, the optimization problem (4a) can be formulated as:

$\displaystyle\underset{\bm{\eta^{u}},\textbf{D}}{\max~{}}$ $\displaystyle\ \
\sum\limits_{t=1}^{T}\left(w\text{log}_{2}\left(\\!1+\Gamma^{*}_{t}\right)-\alpha||{\textbf{D}_{t}}||_{1}\right),$
(6a) subject to : $\displaystyle\ \ d_{mt}\in[0,1],\forall m\in\mathcal{M},\
t\in\mathcal{T},$ (6b) $\displaystyle\ \ 0\leq\eta^{u}_{t}\leq 1,\forall
t\in\mathcal{T},$ (6c) $\displaystyle\ \
w\text{log}_{2}\left(1+\Gamma^{*}_{t}\right)\geq\text{SE}^{\text{QoS}}_{t},\forall
t\in\mathcal{T},$ (6d) $\displaystyle\ \ \sum\limits_{m=1}^{M}d_{mt}\geq
1,\forall t\in\mathcal{T}.$ (6e)

Considering the non-convexity due to the ratio term inside the log in our
objective function (6a), we introduce the lower-bounding auxiliary variable
$\Gamma_{t}$, resulting in the objective function with an additional
constraint as follows:

$\displaystyle\underset{\bm{\eta^{u}},\textbf{D},\mathbf{\Gamma}}{\max~{}}$
$\displaystyle\ \
\sum\limits_{t=1}^{T}\left(w\text{log}_{2}(1+\Gamma_{t})-\alpha||{\textbf{D}_{t}}||_{1}\right),$
(7a) subject to : $\displaystyle\ \ \Gamma_{t}\leq\Gamma^{*}_{t},\
t\in\mathcal{T},$ (7b)

where $\mathbf{\Gamma}=[\Gamma_{1},\Gamma_{2},...\Gamma_{T}]^{\intercal}$.
Taking the partial Lagrangian of objective function (7a) by considering
constraint (7b) with respect to $\mathbf{\Gamma}$, we have:

$\displaystyle\begin{split}L(\bm{\eta^{u}},\textbf{D},\mathbf{\Gamma},\bm{\lambda})=&\sum\limits_{t=1}^{T}\left(w\text{log}_{2}(1+\Gamma_{t})-\alpha||{\textbf{D}_{t}}||_{1}\right)\\\
&-\sum\limits_{t=1}^{T}\lambda_{t}\left(\Gamma_{t}-\Gamma^{*}_{t}\right),\end{split}$
(8)

where $\bm{\lambda}=[\lambda_{1},\lambda_{2},...\lambda_{T}]^{\intercal}$.

After computing the stationary point of (8) with respect to $\mathbf{\Gamma}$,
we get the optimal value of
$\lambda^{*}_{t}=\frac{w^{\prime}}{1+{\Gamma^{*}_{t}}}$. where
$w^{\prime}=w/{\ln 2}$ with the optimal value of
$\mathbf{\Gamma}=\mathbf{\Gamma^{*}}$, where
$\mathbf{\Gamma^{*}}=[\Gamma^{*}_{1},\Gamma^{*}_{2},...\Gamma^{*}_{T}]^{\intercal}$.
Substituting this $\lambda^{*}_{t}$ in (8) and using the Lagrangian dual
transformation [17], the problem in (6) can be reformulated as :

$\displaystyle\begin{split}&\underset{\bm{\eta^{u}},\textbf{D},\mathbf{\Gamma}}{\max~{}}\sum\limits_{t=1}^{T}\left(w\text{log}_{2}(1+\Gamma_{t})-\alpha||{\textbf{D}_{t}}||_{1}\right)-\sum\limits_{t=1}^{T}\left(\Gamma_{t}-\frac{w^{\prime}(1+{\Gamma}_{t})A^{2}p_{u}\eta_{t}^{u}(\sum\limits_{m\in\mathcal{M}}d_{mt}\gamma_{mt})^{2}}{A^{2}p_{u}\eta_{t}^{u}(\sum\limits_{m\in\mathcal{M}}d_{mt}\gamma_{mt})^{2}+{I}_{t}}\right),\\\
\end{split}$ (9a)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\text{subject
to :}\ \text{(\ref{eq : 14_1})},\text{(\ref{eq : 14_2})},\text{(\ref{eq :
14_3})},\text{(\ref{eq : 14_4})}.$ (9b)

It can be observed that after substituting the optimal value of
$\mathbf{\Gamma}$, as in (2), in equation (9a), the optimization problem in
(9) reduces exactly to the one in (6). If we look at optimization problem (9),
we still have some non-linearities and non-convexities due to a ratio term and
a product of two variables in the objective function. First, we attempt to
handle the ratio term in the objective function using the quadratic
transformation approach [17]. The quadratic transformation is performed on the
objective function (9a) such that it becomes:

$\displaystyle\begin{split}\underset{\bm{\eta^{u}},\textbf{D},\mathbf{\Gamma},\textbf{u}}{\max~{}}&\
\
\sum\limits_{t=1}^{T}\left(w\text{log}_{2}(1+\Gamma_{t})-\alpha||{\textbf{D}_{t}}||_{1}\right)-\sum\limits_{t=1}^{T}\left(\Gamma_{t}-2u_{t}\sqrt{w^{\prime}(1+{\Gamma}_{t})A^{2}p_{u}\eta_{t}^{u}(\sum\limits_{m\in\mathcal{M}}d_{mt}\gamma_{mt})^{2}}\right)\\\
&+\sum\limits_{t=1}^{T}u_{t}^{2}\left({A^{2}p_{u}\eta_{t}^{u}(\sum\limits_{m\in\mathcal{M}}d_{mt}\gamma_{mt})^{2}+{I}_{t}}\right),\end{split}$
(10)

where $\textbf{u}=[u_{1},u_{2},...u_{T}]^{\intercal}$.

The quadratic transformation in equation (10) is the lower bound and concave
approximation of ratio term
$\frac{w^{\prime}(1+{\Gamma}_{t})A^{2}p_{u}\eta_{t}^{u}(\sum\limits_{m\in\mathcal{M}}d_{mt}\gamma_{mt})^{2}}{A^{2}p_{u}\eta_{t}^{u}(\sum\limits_{m\in\mathcal{M}}d_{mt}\gamma_{mt})^{2}+{I}_{t}}$
with respect to the parameter $u_{t}$. The optimal value of u can be
calculated from equation (11).

$\displaystyle
u^{*}_{t}\overset{\Delta}{=}\frac{\sqrt{w^{\prime}(1+{\Gamma}_{t})A^{2}p_{u}\eta_{t}^{u}(\sum\limits_{m\in\mathcal{M}}d_{mt}\gamma_{mt})^{2}}}{A^{2}p_{u}\eta_{t}^{u}(\sum\limits_{m\in\mathcal{M}}d_{mt}\gamma_{mt})^{2}+{I}_{t}}.$
(11)

Therefore, equation (10) is the lower bound to equation (9a), thus making the
approximation feasible.

The optimization problem becomes:

$\displaystyle\underset{\bm{\eta^{u}},\textbf{D},\mathbf{\Gamma},\textbf{u}}{\max~{}}$
$\displaystyle\ \ \text{(\ref{eq : 19})},$ (12a) subject to : $\displaystyle\
\ \text{(\ref{eq : 14_1})},\text{(\ref{eq : 14_2})},\text{(\ref{eq :
14_3})},\text{(\ref{eq : 14_4})}.$ (12b)

After substituting the optimal value of u in equation (10) and then solving
optimization problem (12) is equivalent to solving Problem (9). Notice that
Problem (12) is a convex optimization problem in different blocks. If we fix
D, $\mathbf{\Gamma}$ and u, then the objective function of problem (12a)
becomes concave with respect to $\bm{\eta^{u}}$. Then, the problem reduces to:

$\displaystyle f_{1}=\underset{\bm{\eta^{u}}}{\max~{}}$ $\displaystyle\ \
\text{(\ref{eq : 19})},$ (13a) subject to : $\displaystyle\ \ \text{(\ref{eq :
14_2})},\text{(\ref{eq : 14_3})}.$ (13b)

Similarly, upon fixing $\bm{\eta^{u}}$, $\mathbf{\Gamma}$ and u, the objective
function of (12) becomes concave with respect to D. Then, the optimization
problem becomes:

$\displaystyle f_{2}=\underset{\textbf{D}}{\max~{}}$ $\displaystyle\ \
\text{(\ref{eq : 19})},$ (14a) subject to : $\displaystyle\ \ \text{(\ref{eq :
14_1})},\text{(\ref{eq : 14_3})},\text{(\ref{eq : 14_4})}.$ (14b)

Problem 12 can be solved following the proposed iterative approach as follows.
This problem is decomposed into two distinct blocks, denoted as Problem 13 and
Problem 14. Initially, the solution for Problem 13 is obtained while keeping
D, $\mathbf{\Gamma}$, and u fixed. Subsequently, based on the solution derived
for Problem 13, the solution for Problem 14 is determined. This is repeated
until the convergence criterion is met. Algorithm 1 provides the details of
this approach.

1: Feasible Initialization: ${\bm{\eta_{k}^{u}}}^{(0)}$, $\textbf{D}^{(0)}$
and $i=0$, $\forall k$.

2: repeat

3: $i=i+1$.

4: for all t do

5: Compute $\Gamma_{t}^{(i)}$ using (2) and $u_{t}^{(i)}$ using (11).

6: end for

7: Solve optimization problem in (13) for ${\bm{\eta_{k}^{u}}}^{(i)}$.

8: for all t do

9: Again compute $\Gamma_{t}^{(i)}$ using (2) and $u_{t}^{(i)}$ using (11).

10: end for

11: Solve optimization problem in (14) for $\textbf{D}^{(i)}$.

12: Update ${\bm{\eta^{u}}}^{(i+1)}={\bm{\eta^{u}}}^{(i)}$ and
$\textbf{D}^{(i+1)}=\textbf{D}^{(i)}$.

13: until
$\left|\frac{f_{2}^{(i+1)}-f_{2}^{(i)}}{f_{2}^{(i)}}\right|\leq\epsilon$

Algorithm 1 Proposed Algorithm

For the convergence proof of the proposed Algorithm 1, please refer to the
appendixV.

##  IV Performance Evaluation

We performed numerical simulations on a $1000\times 1000$ square meters
geographical area with APs and user equipment UEs randomly distributed. To
simulate an infinite area network, a wrapped-around approach is used [6].
Large-scale fading coefficients are modeled using a three-slope approach for
path loss, incorporating an 8 dB standard deviation for shadow fading. The
minimum QoS parameter is taken as 0.1 bits/s/Hz for each UE. Throughout our
simulations, we keep $T=30$, $M=50\ \text{or}\ 100$, $A=4$, $\alpha=0.0005\
\text{or}\ \alpha=0.001$, $L_{c}=200$ and $L_{p}=5$, while other parameters
stay exactly the same as those outlined in [3] or as explicitly stated. All
these numerical results are averaged over $500$ simulations.

Figure 1: The variation of maximum fronthaul load on an AP, objective
function, and the sum SE, when $M=50$.

Figure 1 shows the variations in maximum front-haul load, objective function,
and the sum SE for two different regularization coefficient values, with the
number of antennas fixed at $50$. This highlights how the penalty function
affects the network’s maximum front-haul load and total sum-throughput.
Notably, the objective function value consistently remains lower than the
total SE value, exhibiting a drop attributed to the penalty function. When the
regularization coefficient is increased by two times, the maximum front-haul
load on an AP decreases by $14\%$. This reduction is due to the increased
regularization coefficient, which limits AP-UE associations. As a consequence
of the lesser number of AP-UE associations, there is a reduction in the sum SE
by nearly $1\%$. Similarly, Figure 2 illustrates variability in maximum front-
haul load, objective function, and sum SE for two different regularization
coefficient values, however with $M$ set to 100. For this scenario too, when
the regularization coefficient is increased two times, the maximum front-haul
load and total SE show a similar decreasing trend as in Fig. 1. Therefore, the
proposed approach effectively decreases the maximum front-haul load while not
significantly reducing overall sum throughput, and this is achieved by a
careful tuning of regularization coefficient.

Figure 2: The variation of maximum front-haul load on an AP, objective
function, and the sum SE, when $M=100$.

Examining Figures 1 and 2, we observe a noteworthy trend as the number of
antennas ($M$) increases from 50 to 100. The maximum front-haul load increases
by $14\%$ and $17\%$ with regularization coefficients ($\alpha$) of 0.0005 and
0.001, respectively. This growth can be attributed to the growing number of
APs in the fixed area. As the AP count increases, there are more APs in the
vicinity of each UE, resulting in a higher percentage of APs offering a strong
channel or a larger large-scale fading coefficient for each UE. This gainful
influence of numerous APs on the large-scale fading coefficient causes a
change in the influence of the penalty function, allowing more APs to
associate with each UE. While this increased association raises the maximum
front-haul load, it also results in a significant ($30\%$) boost in the
overall sum-throughput. In essence, the system responds to the increased AP
count by increasing the strength of channels across numerous APs, hence
increasing the network’s overall throughput. The penalty function is a dynamic
component of this process, optimizing AP-UE associations to optimize the
objective function while reducing negative affects via penalties.

Figure 3: The variation of the sum SE for different scenarios, when
$\alpha=0.001$. Figure 4: CDF vs per-UE SE for different scenarios, when
$\alpha=0.001$.

Figures 3 and 4 illustrate the variation of sum SE and per-UE SE respectively,
of the proposed solution in comparison to the following four scenarios: a)
where UEs transmit at full power and are served by all APs; b) fractional
power control [8]; c) where only power control is performed using the proposed
algorithm; and d) where only AP-UE association is performed using the proposed
algorithm.

Figure 3 shows that when AP-UE association and power control are carried out
simultaneously, the achievable sum SE is the largest. The proposed solution’s
sum SE surpasses that of scenarios ’a’, ’b’,’c’ and ’d’ by $16.5\%$, $4.3\%$,
$1.5\%$ and $2\%$, respectively, when $M=50$. Additionally, for case with
$M=100$, the proposed solution outperforms these scenarios by $11\%$, $3\%$,
$1.3\%$, and $1.6\%$, respectively. This demonstrates the effectiveness of the
proposed technique in optimizing the sum SE by jointly addressing the AP-UE
association and power control problems. Notably, when only AP-UE association
or only power control is conducted with the proposed method, the sum SE sees a
improvement when compared to the other two scenarios ’a’ and ’b’. This
improvement occurs while preserving the minimum QoS limitation. The proposed
approach’s versatility and effectiveness are demonstrated by its ability to
improve network performance even when improving just a single aspect of the
system, like AP-UE association or power control.

Indeed, our proposed technique aims to enhance the sum SE. Figure 4 shows how
our strategy has a significant influence on the $90\%-likely$ per-user SE. Our
proposed scheme outperforms scenarios ’a’, ’b’, ’c’, and ’d’ in terms of
$90\%-likely$ per-user SE by $175\%$, $22\%$, $4.7\%$, and $11.7\%$,
respectively, at $M=50$. Similarly, for $M=100$, our approach outperforms the
same cases by $52.3\%$, $6.7\%$, $2.1\%$, and $3.2\%$ in terms of
$90\%-likely$ per-user SE. The findings illustrate an additional benefit of
the method we propose, demonstrating its massive contribution to improve per-
user throughput while also increasing the total throughput.

Figure 5: The variation of SE with and without discretization of elements of
D.

In Figure 5, we provide a numerical justification for relaxation proposed in
(5). It shows the variation of the SE with and without the discretization of
matrix D. The figure shows that for $m=50$, the variation in the SE is
negligible with and without the discretization of elements of matrix D. For
the case when $m=100$, the difference in SE is mere $0.2\%$. The negligible SE
variation can be attributed to the fact that penalty parameter does not force
elements of matrix D too much away from zero or one, thus our proposed
solution does not require additional algorithm for AP-UE association.

Figure 6: The convergence graph of Algorithm 1, when $\alpha=0.001$, $T=30$
and $M=50$.

In Figure 6, we provide a numerical evidence for the convergence Algorithm 1.
The figure shows that as the number of iterations increases, so does the value
of the objective function, leading up in a non-decreasing function. However,
the rate of rise gradually decreases, ensuring that the rate of increase of
objective function finally becomes lower than the defined tolerance of
$\epsilon=5\times 10^{-3}$. This observation guarantees the convergence of
Algorithm 1.

##  V Conclusion

In this work, we address the challenge of joint AP-UE association and power
control in uplink distributed mMIMO systems by introducing an l1-penalty based
joint mixed-integer non-convex optimization problem. Our technique strives to
maximize system throughput while adhering to minimum quality of service (QoS)
requirements. For tackling this optimization problem, we reformulate the
original problem and develop an iterative technique that guarantees
convergence. Through numerical simulations, we demonstrate that optimizing AP-
UE association and power control factor jointly results in a significant
increase in the system’s sum-throughput when compared to the cases where just
one parameter is optimized. Another interesting observation is the significant
decrease in the maximum front-haul load at the expense of marginal drop in the
sum-throughput. A highlight of our proposed scheme is the significant
enhancement in the $90\%-likely$ per-UE SE. It demonstrates the significance
and efficacy of the proposed scheme when compared to other schemes in
maximizing system performance and providing a better user experience in uplink
distributed mMIMO systems. Overall, our findings underline the need of
optimizing AP-UE association and power regulation simultaneously, emphasizing
the potential for significant throughput gains and per-user SE improvements in
uplink distributed mMIMO systems.

Convergence Analysis : For the convergence analysis of the Algorithm 1, the
primary goal is to prove that both optimization problem defined in (13) and
(14) converges to the their respective optimal solutions. Thereafter, we prove
that in each iteration of Algorithm 1, the optimization function in (12a) is
non-decreasing with respect to $\mathbf{\eta^{u}}$ and D.

The feasible set of smooth concave objective function $\mathbb{F}_{1}$ (13a),
articulated by Equation (13b), is closed and convex. As a result,
$\mathbb{F}_{1}$ is inherently continuous and differentiable over its feasible
set, implying the existence of an optimal solution for $\mathbb{F}_{1}$. The
feasible set, defined by Equation (14b), of non-smooth concave objective
function $\mathbb{F}_{2}$ (14a), is closed and convex. However,
$\mathbb{F}_{2}$ due to the l1-penalty lacks continuous differentiability over
its feasible set. Despite this, $\mathbb{F}_{2}$ can be separated into smooth
and non-smooth part. The convergence of $\mathbb{F}_{2}$ is guaranteed if
gradient of the smooth part of $\mathbb{F}_{2}$ is Lipschitz continuous over
its feasible set [18]. Thus the second derivative of the $\mathbb{F}_{2}$
without the non-smooth part with respect to the AP-UE association variable,
$d_{mt}$ is given by :

$\displaystyle\frac{\partial^{2}\mathbb{F}_{1}}{\partial
d_{mt}^{2}}=\sum\limits_{t=1}^{T}\left(2w^{\prime}(1+\mathbf{\Gamma})A^{2}p_{u}\eta_{t}^{u}\gamma^{2}_{mt}-\frac{P(\sum\limits_{t^{\prime}=1}^{T}A^{2}p_{u}\eta_{t^{\prime}}^{u}\left|\Psi^{H}_{t}\Psi_{t^{\prime}}\right|^{2}(\gamma_{mt}\frac{\beta_{mt^{\prime}}}{\beta_{mt}})^{2})}{Q}-\frac{PQ^{\prime}}{Q}+\frac{PQ^{\prime
2}}{Q^{2}}\right),$ (15)

where $P$, $Q$ and $Q^{\prime}$ are given by, respectively:

$\displaystyle
P=w^{\prime}(1+\mathbf{\Gamma})A^{2}p_{u}\eta_{t}^{u}(\sum\limits_{m\in\mathcal{M}}d_{mt}\gamma_{mt})^{2},$
(16) $\displaystyle
Q=\sum\limits_{t^{\prime}=1}^{T}A^{2}p_{u}\eta_{t^{\prime}}^{u}\left|\Psi^{H}_{t}\Psi_{t^{\prime}}\right|^{2}(\sum\limits_{m\in\mathcal{M}}d_{mt}\gamma_{mt}\frac{\beta_{mt^{\prime}}}{\beta_{mt}})^{2}+\sum\limits_{t^{\prime}=1}^{T}Ap_{u}\eta_{t^{\prime}}^{u}\sum\limits_{m\in\mathcal{M}}d_{mt}\gamma_{mt}\beta_{mt^{\prime}}+\sum\limits_{m\in\mathcal{M}}Ad_{mt}\gamma_{mt},$
(17) $\displaystyle
Q^{\prime}=\sum\limits_{t^{\prime}=1}^{T}A^{2}p_{u}\eta_{t^{\prime}}^{u}\left|\Psi^{H}_{t}\Psi_{t^{\prime}}\right|^{2}(\sum\limits_{m\in\mathcal{M}}d_{mt}\gamma_{mt}\frac{\beta_{mt^{\prime}}}{\beta_{mt}})\gamma_{mt}\frac{\beta_{mt^{\prime}}}{\beta_{mt}}+\sum\limits_{t^{\prime}=1}^{T}Ap_{u}\eta_{t^{\prime}}^{u}\gamma_{mt}\beta_{mt^{\prime}}+A\gamma_{mt}$
(18)

Notably, all the terms in the above double partial derivative (15) are bounded
from above, thus the gradient of the $\mathbb{F}_{2}$ is Lipschitz continuous
over its feasible set.

To prove the non-decreasing nature of the optimization function
$f(\mathbf{\eta^{u}},\textbf{D})$ defined in (12a), let
${\mathbf{\eta^{u}}}^{*}$ be the optimal value of function $f$, when D is
fixed. Then, for fixed D, the inequality
$f({\mathbf{\eta^{u}}}^{*},{\textbf{D}}^{(i)})\geq
f({\mathbf{\eta^{u}}}^{(i)},{\textbf{D}}^{(i)})$ always holds due to the
concavity of the function $f$. For the optimal $\textbf{D}^{*}$, when
$\mathbf{\eta^{u}}$ is fixed at ${\mathbf{\eta^{u}}}^{(i+1)}$, the inequality
$f({\mathbf{\eta^{u}}}^{(i+1)},{\textbf{D}}^{*})\geq
f({\mathbf{\eta^{u}}}^{(i+1)},{\textbf{D}}^{(i)})$ always holds as $f$ is
concave for fixed D. It can be inferred that
$f({\mathbf{\eta^{u}}}^{(i+1)},{\textbf{D}}^{(i+1)})\geq
f({\mathbf{\eta^{u}}}^{(i)},{\textbf{D}}^{(i)})$, indicating $f$ is non-
decreasing in each iteration. It makes the optimization function monotonically
increasing in each iteration and also the optimization function is bounded
from above, thereby guaranteeing the convergence of the proposed algorithm.

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