page1

# Enhanced Max-Min SINR for Uplink Cell-Free Massive MIMO Systems

Manijeh Bashar1, Kanapathippillai Cumanan1, Alister G. Burr1, Mérouane Debbah
2, and Hien Quoc Ngo3 , , Email: {mb1465, kanapathippillai.cumanan,
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1Department of Electronic Engineering, University of York, Heslington, York,
UK  2Large Networks and Systems Group, CentraleSupélec, Université Paris-
Saclay, France  3School of Electronics, Electrical Engineering and Computer
Science, Queen’s University Belfast, Belfast, UK

###### Abstract

In this paper, we consider the max-min signal-to-interference plus noise ratio
(SINR) problem for the uplink transmission of a cell-free Massive multiple-
input multiple-output (MIMO) system. Assuming that the central processing unit
(CPU) and the users exploit only the knowledge of the channel statistics, we
first derive a closed-form expression for uplink rate. In particular, we
enhance (or maximize) user fairness by solving the max-min optimization
problem for user rate, by power allocation and choice of receiver
coefficients, where the minimum uplink rate of the users is maximized with
available transmit power at the particular user. Based on the derived closed-
form expression for the uplink rate, we formulate the original user max-min
problem to design the optimal receiver coefficients and user power
allocations. However, this max-min SINR problem is not jointly convex in terms
of design variables and therefore we decompose this original problem into two
sub-problems, namely, receiver coefficient design and user power allocation.
By iteratively solving these sub-problems, we develop an iterative algorithm
to obtain the optimal receiver coefficient and user power allocations. In
particular, the receiver coefficients design for a fixed user power allocation
is formulated as generalized eigenvalue problem whereas a geometric
programming (GP) approach is utilized to solve the power allocation problem
for a given set of receiver coefficients. Numerical results confirm a three-
fold increase in system rate over existing schemes in the literature.

Keywords: Cell-free Massive MIMO, convex optimization, max-min SINR problem,
geometric programming, generalized eigenvalue problem.

##  I Introduction

††footnotetext: The work of K. Cumanan and A. G. Burr was supported by H2020-
MSCA-RISE-2015 under grant number 690750. The work on which this paper is
based was carried out in collaboration with COST Action CA15104 (IRACON).

Massive multiple-input multiple-output (MIMO) is one of the most promising
techniques for 5th Generation (5G) networks due to its potential for
significant rate enhancement and spectral as well as energy efficiency [1, 2,
3, 4, 5, 6]. In cell-free Massive MIMO, randomly distributed access points
(APs) jointly serve distributed users. In this paper, we propose a max-min
signal-to-interference plus noise ratio (SINR) approach for an uplink cell-
free Massive MIMO system. In [7], the authors investigate the problem of
max–min SINR in a single-cell Massive MIMO system. A similar max-min SINR
problem refereed to SINR balancing has been considered for cognitive radio
network in [8, 9, 10, 11, 12]. In [13], the same max-min SINR problem is
considered through appropriate user power allocation where a bisection search
method is utilized to determine the optimal solution. However, a novel
approach to significantly improve all users’ performance is proposed in this
paper by designing optimal receiver coefficients and user power allocation. By
employing maximal ratio combining (MRC) at the receiver, we first derive a
closed-form expression for the average uplink rate of the users. Based on
these user power allocations and receiver coefficients, we formulate the
corresponding max-min SINR problem, which is not jointly convex in terms of
the design parameters. In order to realize a solution for this non-convex
problem, we decompose the original problem into two sub-problems: receiver
coefficient design and user power allocation. An iterative algorithm is
proposed that successively solves these two sub-problems while one of the
design variables (i.e., user power allocation or receiver coefficients) is
fixed. The receiver coefficient design is formulated into a generalized
eigenvalue problem [14] whereas a geometric programming (GP) approach [15] is
exploited to solve the user power allocation problem. The performance of the
proposed scheme in terms of the user rate is significantly higher than that of
the scheme proposed in [13]. The contributions and the results of our work are
summarized as follows: 1) For the considered cell-free Massive MIMO system, we
derive the average user rate in the uplink. 2) Based on the derived user rate,
we propose a novel max-min SINR approach to significantly improve the SINR
performance in terms of the achieved user rate. The original max-min problem
formulation is not convex and therefore we decompose the original problem into
two sub-problems and propose an iterative algorithm to yield the optimal
solution. 3) The user power allocation and the receiver coefficient design
sub-problems are solved through the GP approach and the generalized eigenvalue
problem, respectively. 4) Numerical results are provided to validate the
superiority of the proposed algorithm in comparison with the scheme proposed
in [13].

Figure 1: The uplink of a cell-free Massive MIMO system with K users and M
APs. The dashed lines denote the uplink channels and the solid lines present
the backhaul links from the APs to the central processing unit (CPU).

##  II SYSTEM MODEL

We consider uplink transmission in a cell-free Massive MIMO system with $M$
randomly distributed single-antenna APs and $K$ randomly distributed single-
antenna users in the area, as shown in Fig. 1. The channel coefficient between
the $k$th user and the $m$th AP, $g_{mk}$, is modeled as [13]
$g_{mk}=\sqrt{\beta_{mk}}h_{mk},$ where $\beta_{mk}$ denotes the large-scale
fading and $h_{mk}\sim\mathcal{CN}(0,1)$ represents small-scale fading between
the $k$th user and the $m$th AP.

###  II-A Uplink Channel Estimation

In order to estimate the channel coefficients in the uplink, the APs employ an
minimum mean square error (MMSE) estimator. All pilot sequences used in the
channel estimation phase are collected in a matrix
${\boldsymbol{\Phi}}\in\mathbb{C}^{\tau\times K}$, where $\tau$ is the length
of the pilot sequence for each user and the $k$th column,
$\boldsymbol{\phi}_{k}$, represents the pilot sequence used for the kth user.
After performing a de-spreading operation, the MMSE estimate of the channel
coefficient between the $k$th user and the $m$th AP is given by [13]

$\displaystyle\small\\!\\!\\!\\!\hat{g}_{mk}\\!=\\!c_{mk}\\!\\!\left(\\!\\!\sqrt{\tau
p_{p}}g_{mk}\\!+\\!\sqrt{\tau p_{p}}\sum_{k^{\prime}\neq
k}^{K}g_{mk^{\prime}}\boldsymbol{\phi}_{k}^{H}\boldsymbol{\phi}_{k^{\prime}}\\!+\\!\boldsymbol{\phi}_{k}^{H}\textbf{n}_{p,m}\\!\\!\right)\\!\\!\\!,\leavevmode\nobreak\
$ (1)

where each element of $\textbf{n}_{p,m}$, $n_{p,m}\sim\mathcal{CN}(0,1)$,
denotes the noise at the $m$th antenna, $p_{p}$ represents the normalized
signal-to-noise ratio (SNR) of each pilot sequence (which we define in Section
V), and $c_{mk}$ is given by $c_{mk}=\dfrac{\sqrt{\tau p_{p}}\beta_{mk}}{\tau
p_{p}\sum_{k^{\prime}=1}^{K}\beta_{mk^{\prime}}\left|\boldsymbol{\phi}_{k}^{H}{\boldsymbol{\phi}}_{k^{\prime}}\right|^{2}+1}.$
The estimated channels in (1) are used by the APs to design the receiver
coefficients and determine power allocations at users to maintain user
fairness. In this paper, we investigate the cases of both random pilot
assignment and orthogonal pilots in cell-free Massive MIMO. Here the term
”orthogonal pilots” refers to the case where unique orthogonal pilots are
assigned to all users, while in ”random pilot assignment” each user is
randomly assigned a pilot sequence from a set of orthogonal sequences of
length $\tau$ ($<K$), following the approach of [13, 16].

###  II-B Uplink Data Transmission

In this subsection, we consider the uplink data transmission, where all users
send their signals to the APs. The transmitted signal from the $k$th user is
represented by $x_{k}=\sqrt{q_{k}}s_{k},$ where $s_{k}$
($\mathbb{E}\\{|s_{k}|^{2}\\}=1$) and $q_{k}$ denote respectively the
transmitted symbol and the transmit power at the kth user. The received signal
at the $m$th AP from all users is given by
$y_{m}=\sqrt{\rho}\sum_{k=1}^{K}g_{mk}\sqrt{q_{k}}s_{k}+n_{m},$ where
$n_{m}\sim\mathcal{CN}(0,1)$ is the noise at the $m$th AP. In addition, MRC is
employed at the APs. More precisely, the received signal at the $m$th AP,
$y_{m}$, is first multiplied with $\hat{g}_{mk}^{\ast}$. The resulting
$\hat{g}_{mk}^{\ast}y_{m}$ is then forwarded to the CPU for signal detection.
In order to improve achievable rate, the forwarded signal is further
multiplied by a receiver filter coefficient at the CPU. The aggregated
received signal at the CPU can be written as

$\displaystyle r_{k}$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{M}u_{mk}\hat{g}_{mk}^{*}y_{m}$ $\displaystyle=$
$\displaystyle\sqrt{\rho}\sum_{k^{\prime}=1}^{K}\sum_{m=1}^{M}u_{mk}\hat{g}_{mk}^{*}g_{mk^{\prime}}\sqrt{q_{k^{\prime}}}s_{k^{\prime}}+\sum_{m=1}^{M}u_{mk}\hat{g}_{mk}^{*}n_{m}.$

By collecting all the coefficients $u_{mk},\leavevmode\nobreak\
\forall\leavevmode\nobreak\ m$ corresponding to the kth user, we define
$\textbf{u}_{k}=[u_{1k},u_{2k},\cdots,u_{Mk}]^{T}$ and without loss of
generality, it is assumed that $\parallel\textbf{u}_{k}\parallel=1$. The
optimal solution of $\mathbf{u}_{k},\leavevmode\nobreak\
q_{k},\leavevmode\nobreak\ \forall\leavevmode\nobreak\ k$ for the considered
max-min SINR approach is investigated in Section IV.

##  III Performance Analysis

In this section, we derive the average user rate for the considered system
model in the previous section by following a similar approach to that in [13].
Note that the main difference between the proposed approach and the scheme in
[13] is the new set of receiver coefficients which are introduced at the CPU
to improve the achievable user rates. The benefits of the proposed approach in
terms of achieved user uplink rate is demonstrated through numerical
simulation results in Section V. In deriving the achievable rates of each
user, it is assumed that the CPU exploits only the knowledge of channel
statistics between the users and APs in detecting data from the received
signal in (II-B). Without loss of generality, the aggregated received signal
in (II-B) can be written as

$\displaystyle\small r_{k}$
$\displaystyle=\underbrace{\sqrt{\rho}\mathbb{E}\left\\{\sum_{m=1}^{M}u_{mk}\hat{g}_{mk}^{*}{g}_{mk}\sqrt{q_{k}}\right\\}}_{\text{DS}_{k}}s_{k}$
$\displaystyle\\!+\\!$
$\displaystyle\\!\underbrace{\sqrt{\rho}\left(\\!\sum_{m=1}^{M}\\!u_{mk}\hat{g}_{mk}^{*}\\!{g}_{mk}\sqrt{q_{k}}\\!-\\!\mathbb{E}\\!\left\\{\\!\sum_{m=1}^{M}\\!u_{mk}\hat{g}_{mk}^{*}{g}_{mk}\sqrt{q_{k}}\\!\right\\}\right)}_{{\text{BU}_{k}}}\\!s_{k}$
$\displaystyle+$ $\displaystyle\sum_{k^{\prime}\neq
k}^{K}\underbrace{\sqrt{\rho}\sum_{m=1}^{M}u_{mk}\hat{g}_{mk}^{*}{g}_{mk^{\prime}}\sqrt{q_{k^{\prime}}}}_{{\text{IUI}_{kk^{\prime}}}}s_{k^{\prime}}+\underbrace{\sum_{m=1}^{M}u_{mk}\hat{g}_{mk}^{*}n_{m}}_{\text{TN}_{k}},$

where $\text{DS}_{k}$ and $\text{BU}_{k}$ denote the desired signal (DS) and
beamforming uncertainty (BU) for the $k$th user, respectively, and
$\text{IUI}_{k}$ represents the inter-user-interference (IUI) caused by the
$k^{\prime}$th user. In addition, $\text{TN}_{k}$ accounts for the total noise
(TN) following the MRC detection. The average SINR of the received signal in
(III) can be defined by considering the worst-case of the uncorrelated
Gaussian noise as follows [13]:

$\displaystyle\\!\\!\\!\\!\\!{\textrm{SINR}}_{k}^{\text{UP}}\\!=\\!\dfrac{|\text{DS}_{k}|^{2}}{\\!\mathbb{E}\\{|\text{BU}_{k}|^{2}\\!\\}\\!+\\!\sum_{k^{\prime}\neq
k}^{K}\mathbb{E}\\{|\text{IUI}_{kk^{\prime}}|^{2}\\!\\}\\!+\\!\mathbb{E}\\{|\text{TN}_{k}|^{2}\\!\\}}\\!.$
(4)

$\displaystyle\begin{split}R_{k}=\log_{2}\left(1+\dfrac{\textbf{u}_{k}^{H}\left(q_{k}{\boldsymbol{\Gamma}}_{k}{\boldsymbol{\Gamma}}_{k}^{H}\right)\textbf{u}_{k}}{\textbf{u}_{k}^{H}\left(\sum_{k^{\prime}\neq
k}^{K}q_{k^{\prime}}\left|\boldsymbol{\phi}_{k}^{H}\boldsymbol{\phi}_{k^{\prime}}\right|^{2}{\boldsymbol{\Delta}}_{kk^{\prime}}{\boldsymbol{\Delta}}_{kk^{\prime}}^{H}+\sum_{k^{\prime}=1}^{K}q_{k^{\prime}}\textbf{D}_{kk^{\prime}}+\dfrac{1}{\rho}\textbf{R}_{k}\right)\textbf{u}_{k}}\right).\end{split}$
(5)

Based on the SINR definition in (4), the achievable uplink rate of the kth
user is defined in the following theorem.

######  Theorem 1.

By employing MRC detection at APs, the achievable uplink rate of the kth user
in the Cell-free Massive MIMO system with $K$ randomly distributed single-
antenna users and $M$ single-antenna APs is given by (5) (defined at the
beginning of the next page).

Note that in (5), we have
${\boldsymbol{\Gamma}}_{k}=[\gamma_{1k},\gamma_{2k},\cdots,\gamma_{Mk}]^{T}$,
$\textbf{u}_{k}=[u_{1k},u_{2k},\cdots,u_{Mk}]^{T}$,
${\boldsymbol{\Delta}}_{kk^{\prime}}=[\dfrac{\gamma_{1k}\beta_{1k^{\prime}}}{\beta_{1k}},\dfrac{\gamma_{2k}\beta_{2k^{\prime}}}{\beta_{2k}},\cdots,\dfrac{\gamma_{Mk}\beta_{Mk^{\prime}}}{\beta_{Mk}}]^{T}$,$\textbf{R}_{k}=\text{diag}\left[\gamma_{1k},\gamma_{2k},\cdots,\gamma_{Mk}\right]$,
and
$\textbf{D}_{kk^{\prime}}=\text{diag}\left[\beta_{1k^{\prime}}\gamma_{1k},\beta_{2k^{\prime}}\gamma_{2k},\cdots,\beta_{Mk^{\prime}}\gamma_{Mk}\right]$.

Proof: Please refer to Appendix A.

##  IV Proposed Max-Min SINR Scheme

In this section, we formulate the umax-min SINR problem in cell-free massive
MIMO, where the minimum uplink user rate between users is maximized while
satisfying the transmit power constraint at each user. This max-min rate
problem can be formulated as follows:

$\displaystyle P_{1}:\max_{q_{k},\textbf{u}_{k}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\min_{k=1,\cdots,K}\leavevmode\nobreak\ \leavevmode\nobreak\ R_{k}$ (6) s.t.
$\displaystyle||\textbf{u}_{k}||=1,\leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ k,\leavevmode\nobreak\ \leavevmode\nobreak\ 0\leq
q_{k}\leq p_{\text{max}}^{(k)},\leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ k,$

where $p_{max}^{(k)}$ is the maximum transmit power available at user k.
Problem $P_{1}$ is not jointly convex in terms of $\mathbf{u}_{k}$ and power
allocation $q_{k},\leavevmode\nobreak\ \forall\leavevmode\nobreak\ k$.
Therefore, this problem cannot be directly solved through existing convex
optimization software. To tackle this non-convexity issue, we divide the
original Problem $P_{1}$ into two sub-problems: receiver coefficient design
(i.e. $\mathbf{u}_{k}$) and the power allocation problem. To obtain a solution
for Problem $P_{1}$, these sub-problems are alternately solved as explained in
the following subsections.

###  IV-A Receiver Coefficients Design

In this subsection, we solve the receiver coefficient design problem to
maximize the uplink rate of each user for a given set of transmit power
allocation at all users. These coefficients (i.e., $\mathbf{u}_{k}$,
$\forall\leavevmode\nobreak\ k$) can be obtained by interdependently
maximizing the uplink SINR of each user. Hence, the optimal coefficients for
all users for a given set of transmit power allocation can be determined by
solving the following optimization problem:

$\displaystyle P_{2}:\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\max_{\mathbf{u}_{k}}\\!\frac{\mathbf{u}_{k}^{H}\left(q_{k}{\boldsymbol{\Gamma}}_{k}{\boldsymbol{\Gamma}}_{k}^{H}\right)\mathbf{u}_{k}}{\mathbf{u}_{k}^{H}\left(\\!\sum_{k^{\prime}\neq
k}^{K}q_{k^{\prime}}\left|\boldsymbol{\phi}_{k}^{H}\boldsymbol{\phi}_{k^{\prime}}\right|^{2}\\!{\boldsymbol{\Delta}}_{kk^{\prime}}\\!{\boldsymbol{\Delta}}_{kk^{\prime}}^{H}\\!+\\!\sum_{k^{\prime}=1}^{K}q_{k^{\prime}}\textbf{D}_{kk^{\prime}}\\!+\\!\frac{1}{\rho}\\!\textbf{R}_{k}\right)\\!\mathbf{u}_{k}},$
$\displaystyle\leavevmode\nobreak\ \text{s.t.}\leavevmode\nobreak\
\leavevmode\nobreak\ \parallel\mathbf{u}_{k}\parallel=1,\leavevmode\nobreak\
\leavevmode\nobreak\ \forall\leavevmode\nobreak\ \leavevmode\nobreak\ k.$ (7)

Algorithm 1 Proposed Algorithm to Solve $P_{1}$

1. Initialize $\textbf{q}^{(0)}=[q_{1}^{(0)},q_{2}^{(0)},\cdots,q_{K}^{(0)}]$, $i=1$

2. Repeat

3. $i=i+1$

4. Set $\textbf{q}^{(i)}=\textbf{q}^{(i-1)}$ and find the optimal receiver coefficients $\textbf{U}^{(i)}=[\textbf{u}^{(i)}_{1},\textbf{u}^{(i)}_{2},\cdots,\textbf{u}^{(i)}_{K}]$ through solving the generalized eigenvalue Problem $P_{2}$ in (IV-A)

5. Compute $\textbf{q}^{(i+1)}$ through solving Problem $P_{4}$ in (9).

6. Go back to Step 3 and repeat until required accuracy.

Problem $P_{2}$ is a generalized eigenvalue problem [14], where the optimal
solutions can be obtained by determining the generalized eigenvalue of the
matrix pair
$\mathbf{A}_{k}=q_{k}{\boldsymbol{\Gamma}}_{k}{\boldsymbol{\Gamma}}_{k}^{H}$
and $\mathbf{B}_{k}=\\!\sum_{k^{\prime}\neq
k}^{K}q_{k^{\prime}}|\boldsymbol{\phi}_{k}^{H}\boldsymbol{\phi}_{k^{\prime}}|^{2}\\!{\boldsymbol{\Delta}}_{kk^{\prime}}\\!{\boldsymbol{\Delta}}_{kk^{\prime}}^{H}\\!+\\!\sum_{k^{\prime}=1}^{K}q_{k^{\prime}}\textbf{D}_{kk^{\prime}}\\!+\\!\frac{1}{\rho}\\!\textbf{R}_{k}\\!$
corresponding to the maximum generalized eigenvalue.

###  IV-B Power Allocation

In this subsection, we solve the power allocation problem for a set of fixed
receiver coefficients. The power allocation problem can be formulated into the
following max-min problem:

$\displaystyle P_{3}:\max_{q_{k}}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \min_{k=1,\cdots,K}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{SINR}_{k}$ (8) s.t. $\displaystyle 0\leq q_{k}\leq
p_{max}^{(k)},\leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ k,$

Without loss of generality, Problem $P_{3}$ can be rewritten by introducing a
new slack variable as

$\displaystyle P_{4}:\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \max_{t,q_{k}}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ t$ (9) $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{s.t.}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ 0$ $\displaystyle\leq q_{k}\leq
p_{max}^{(k)},\leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ k,\leavevmode\nobreak\ \leavevmode\nobreak\
\text{SINR}_{k}\geq t,\leavevmode\nobreak\ \forall\leavevmode\nobreak\ k.$

Proposition 1: Problem $P_{4}$ can be formulated into a GP.
Proof: Please refer to Appendix B.
Therefore, this problem can be efficiently solved through existing convex
optimization software. Based on these two sub-problems, an iterative algorithm
is developed by alternately solving each sub-problem in each iteration. The
proposed algorithm is summarized in Algorithm 1.

##  V Convergence analysis

In this section, the convergence analysis of the proposed Algorithm 1 is
provided. Two sub-problems are alternately solved to determine the solution to
Problem $P_{1}$. At each iteration, one of the design parameters is determined
by solving the corresponding sub-problem while other design variable is fixed.
Note that each sub-problem provides an optimal solution for the other given
design variable. At the $i$th iteration, the receiver filter coefficients
$\textbf{u}_{k}^{(i)},\leavevmode\nobreak\ \forall k$ are determined for a
given power allocation $\textbf{q}^{(i)}$ and similarly, the power allocation
$\textbf{q}^{(i+1)}$ is updated for a given set of receiver filter
coefficients $\textbf{u}_{k}^{(i)},\leavevmode\nobreak\ \forall k$. The
optimal power allocation $\textbf{q}^{(i+1)}$ obtained for a given
$\textbf{u}_{k}^{(i)}$ achieves an uplink rate greater than or equal to that
of the previous iteration. In addition, the power allocation
$\textbf{q}^{(i)}$ is also a feasible solution in determining
$\textbf{q}^{(i+1)}$ as the receiver filter coefficients
$\textbf{u}_{k}^{(i+1)},\leavevmode\nobreak\ \forall k$ are determined for a
given $\textbf{q}^{(i)}$. This reveals that the achieved uplink rate
monotonically increases with each iteration, which can be also observed from
the simulation results presented in Fig. 4. As the achievable uplink max-min
rate is upper bounded by a certain value for a given set of per-user power
constraints, the proposed algorithm converges to a particular solution.
Fortunately, the proposed Algorithm 1 converges to the optimal solution, as we
will prove by establishing the uplink-downlink duality in the following
section.

##  VI Numerical Results and Discussion

In this section, we provide numerical simulation results to validate the
performance of the proposed max-min SINR approach with different parameters. A
cell-free Massive MIMO system with $M$ APs and $K$ single antenna users is
considered in a $D\times D$ simulation area, where both APs and users are
randomly distributed. In the following subsections, we define the simulation
parameters and then present the corresponding simulation results.

Figure 2: The cumulative distribution of the per-user uplink rate, with
orthogonal and random pilots for $M=60$, $K=20$ and $D=1$ km.

To model the channel coefficients between users and APs, the coefficient
$\beta_{mk}$ is given by
$\beta_{mk}=\text{PL}_{mk}.10^{\frac{\sigma_{sh}z_{mk}}{10}}$ where
$\text{PL}_{mk}$ is the path loss from the $k$th user to the $m$th AP, and
$10^{\frac{\sigma_{sh}\leavevmode\nobreak\ z_{mk}}{10}}$ denotes the shadow
fading with standard deviation $\sigma_{sh}$, and $z_{mk}\sim\mathcal{N}(0,1)$
[13]. The noise power is given by $P_{n}=\text{BW}k_{B}T_{0}W,$ where
$\text{BW}=20$ MHz denotes the bandwidth, $k_{B}=1.381\times 10^{-23}$
represents the Boltzmann constant, and $T_{0}=290$ (Kelvin) denotes the noise
temperature. Moreover, $W=9$dB, and denotes the noise figure [13]. It is
assumed that that $\bar{P}_{p}$ and $\bar{\rho}$ denote the pilot sequence and
the uplink data, respectively, where $P_{p}=\frac{\bar{P}_{p}}{P_{n}}$ and
$\rho=\frac{\bar{\rho}}{P_{n}}$. In simulations, we set $\bar{P}_{p}=100$mW
and $\bar{\rho}=100$mW. Similar to [13], we suppose the simulation area is
wrapped around at the edges which can simulate an area without boundaries.
Hence, the square simulation area has eight neighbors. We evaluate the rate of
the system over 300 random realizations of the locations of APs, users and
shadowing.

Figure 3: The cumulative distribution of the per-user uplink rate, with random
pilots for $M=100$, $K=40$, $\tau=20$, and $D=1$ km. Figure 4: The convergence
of the proposed max-min SINR approach for $M=60$, $K=20$, $\tau=10$, and $D=1$
km.

In this subsection, we investigate the effect of the max-min SINR problem on
the system performance. Fig. 2 compares the cumulative distribution of the
achievable uplink rates for our proposed algorithm with the power allocation
scheme in [13], for three cases of orthogonal pilots, random pilots with
$\tau=10$ and $\tau=5$ for the length of pilot sequence. In Fig. 2, $M=60$ APs
and $K=20$ users are randomly distributed through the simulation area of size
$1\times 1$ $\text{km}^{2}$. As the figure shows, the performance of the
proposed scheme is almost three times than that of the scheme in [13].

In Fig. 3, we compare the performance of the proposed max-min SINR approach
with the scheme in [13] for the case of $M=100$ APs, $K=40$ users and
$\tau=20$ as the length of the pilot sequence. Fig. 3 shows the superiority of
the proposed iterative algorithm over the power allocation scheme in [13].
Moreover, Fig. 3 demonstrates that the rate of the proposed max-min SINR
approach is more concentrated around the median. Fig. 4 investigates the
convergence of the proposed max-min SINR algorithm for a set of different
channel realizations. The figure shows that the proposed algorithm converges
after a few iterations, while the minimum rate of the users increases with the
iteration number.

##  VII Conclusions

We have considered the max-min optimization problem in cell-free Massive MIMO
systems, and propose an efficient solution that maximizes the smallest of the
uplink rate of the users. We propose to divide the original max-min problem
into two sub-problems which can be iteratively solved by exploiting
generalized eigenvalue problem and GP. The simulation results showcased the
effectiveness of the proposed scheme in terms of maximising the smallest of
the uplink rate of the users compared with existing schemes.

## Appendix A: Proof of Theorem 1

The desired signal for the user $k$ is given by

$\displaystyle\small\text{DS}_{k}\\!=\\!\mathbb{E}\left\\{\\!\sum_{m=1}^{M}\\!u_{mk}\hat{g}_{mk}^{*}g_{mk}\\!\sqrt{q_{k}}\right\\}\\!=\\!\sqrt{q_{k}}\\!\sum_{m=1}^{M}\\!u_{mk}\\!\gamma_{mk}.\leavevmode\nobreak\
$ (10)

Hence,
$\left|\text{DS}_{k}\right|^{2}=q_{k}\left(\sum_{m=1}^{M}u_{mk}\gamma_{mk}\right)^{2}.$
Moreover, the term $\mathbb{E}\\{\left|\text{BU}_{k}\right|^{2}\\}$ can be
obtained as

$\displaystyle\mathbb{E}\\!\\!\\!\\!\\!\\!$
$\displaystyle\left\\{\left|\text{BU}_{k}\right|^{2}\right\\}=\rho\mathbb{E}\left\\{\left|\sum_{m=1}^{M}u_{mk}\hat{g}_{mk}^{*}{g}_{mk}\sqrt{q_{k}}\right.\right.$
$\displaystyle-$
$\displaystyle\left.\left.\rho\mathbb{E}\left\\{\sum_{m=1}^{M}u_{mk}\hat{g}_{mk}^{*}{g}_{mk}\sqrt{q_{k}}\right\\}\right|^{2}\right\\}=\rho
q_{k}\sum_{m=1}^{M}u_{mk}^{2}\gamma_{mk}\beta_{mk},$

where the last equality comes from the analysis in [13, Appendix A]] and using
the following fact;
$\gamma_{mk}=\mathbb{E}\\{\left|\hat{g}_{mk}\right|^{2}\\}=\sqrt{\tau
p_{p}}\beta_{mk}c_{mk}$. The term
$\mathbb{E}\\{\left|\text{IUI}_{kk^{\prime}}\right|^{2}\\}$ is obtained as

$\displaystyle\mathbb{E}$ $\displaystyle\\{|$
$\displaystyle\text{IUI}_{kk^{\prime}}|^{2}\\}=\rho\mathbb{E}\left\\{\left|\sum_{m=1}^{M}u_{mk}\hat{g}_{mk}^{*}{g}_{mk^{\prime}}\sqrt{q_{k^{\prime}}}\right|^{2}\right\\}$
$\displaystyle=$
$\displaystyle\rho\mathbb{E}\left\\{\left|\sum_{m=1}^{M}c_{mk}u_{mk}{g}_{mk^{\prime}}\sqrt{q_{k^{\prime}}}\right.\right.$
$\displaystyle\left.\left.\left(\sqrt{\tau
p_{p}}\sum_{i=1}^{K}g_{mi}\boldsymbol{\phi}_{k}^{H}\boldsymbol{\phi}_{i}\\!+\\!\boldsymbol{\phi}_{k}^{H}\textbf{n}_{p,m}\right)^{*}\right|^{2}\right\\}$
$\displaystyle=$
$\displaystyle\rho\underbrace{q_{k^{\prime}}\mathbb{E}\left\\{\left|\sum_{m=1}^{M}c_{mk}u_{mk}g_{mk^{\prime}}\tilde{n}_{mk}^{*}\right|^{2}\right\\}}_{A}$
$\displaystyle\\!+\\!$ $\displaystyle\rho\underbrace{\tau
p_{p}\mathbb{E}\\!\left\\{\\!q_{k^{\prime}}\left|\\!\sum_{m=1}^{M}\\!c_{mk}u_{mk}{g}_{mk^{\prime}}\left(\sum_{i=1}^{K}g_{mi}\\!\boldsymbol{\phi}_{k}^{H}\boldsymbol{\phi}_{i}\\!\right)^{*}\\!\right|^{2}\right\\}}_{B},$

where the third equality in (Appendix A: Proof of Theorem 1) is due to the
fact that for two independent random variables $X$ and $Y$ and
$\mathbb{E}\\{X\\}=0$, we have
$\mathbb{E}\\{\left|X+Y\right|^{2}\\}=\mathbb{E}\\{\left|X\right|^{2}\\}+\mathbb{E}\\{\left|Y\right|^{2}\\}$
[13, Appendix A]. Since
$\tilde{n}_{mk}=\boldsymbol{\phi}_{k}^{H}\textbf{n}_{p,m}\sim\mathcal{CN}(0,1)$
is independent from the term $g_{mk^{\prime}}$ similar to [13, Appendix A],
the term $A$ in (Appendix A: Proof of Theorem 1) immediately is given by
$A=q_{k^{\prime}}\sum_{m=1}^{M}c_{mk}^{2}u_{mk}^{2}\beta_{mk^{\prime}}.$The
term $B$ in (Appendix A: Proof of Theorem 1) can be obtained as

$\displaystyle\small B$ $\displaystyle=$ $\displaystyle\underbrace{\tau
p_{p}q_{k^{\prime}}\mathbb{E}\left\\{\left|\sum_{m=1}^{M}c_{mk}u_{mk}\left|{g}_{mk^{\prime}}\right|^{2}\boldsymbol{\phi}_{k}^{H}{\boldsymbol{\phi}}_{k^{\prime}}\right|^{2}\right\\}}_{C}$
$\displaystyle+$ $\displaystyle\underbrace{\tau
p_{p}q_{k^{\prime}}\mathbb{E}\left\\{\left|\sum_{m=1}^{M}c_{mk}u_{mk}{g}_{mk^{\prime}}\left(\sum_{i\neq
k^{\prime}}^{K}g_{mi}\boldsymbol{\phi}_{k}^{H}\boldsymbol{\phi}_{i}\right)^{*}\right|^{2}\right\\}}_{D}.$

The first term in (Appendix A: Proof of Theorem 1) is given by

$\displaystyle\small C$ $\displaystyle=$ $\displaystyle\tau
p_{p}q_{k^{\prime}}\mathbb{E}\left\\{\left|\sum_{m=1}^{M}c_{mk}u_{mk}\left|{g}_{mk^{\prime}}\right|^{2}\boldsymbol{\phi}_{k}^{H}{\boldsymbol{\phi}}_{k^{\prime}}\right|^{2}\right\\}$
(14) $\displaystyle=$ $\displaystyle\tau
p_{p}q_{k^{\prime}}\left|\boldsymbol{\phi}_{k}^{H}{\boldsymbol{\phi}}_{k^{\prime}}\right|^{2}\sum_{m=1}^{M}c_{mk}^{2}u_{mk}^{2}\beta_{mk^{\prime}}^{2}$
$\displaystyle+$ $\displaystyle
q_{k^{\prime}}\left|\boldsymbol{\phi}_{k}^{H}{\boldsymbol{\phi}}_{k^{\prime}}\right|^{2}\left(\sum_{m=1}^{M}u_{mk}\gamma_{mk}\dfrac{\beta_{mk^{\prime}}}{\beta_{mk}}\right)^{2},$

where the last equality is derived based on the fact $\gamma_{mk}=\sqrt{\tau
p_{p}}\beta_{mk}c_{mk}$. The second term in (Appendix A: Proof of Theorem 1)
can be obtained as

$\displaystyle D$ $\displaystyle=$ $\displaystyle\tau
p_{p}q_{k^{\prime}}\mathbb{E}\left\\{\left|\sum_{m=1}^{M}c_{mk}u_{mk}{g}_{mk^{\prime}}\left(\sum_{i\neq
k^{\prime}}^{K}g_{mi}\boldsymbol{\phi}_{k}^{H}\boldsymbol{\phi}_{i}\right)^{*}\right|^{2}\right\\}$
(15) $\displaystyle=$ $\displaystyle\tau p_{p}\sum_{m=1}^{M}\sum_{i\neq
k^{\prime}}^{K}q_{k^{\prime}}c_{mk}^{2}u_{mk}^{2}\beta_{mk^{\prime}}\beta_{mi}\left|\boldsymbol{\phi}_{k}^{H}\boldsymbol{\phi}_{i}\right|^{2}.$

Hence, (Appendix A: Proof of Theorem 1) can be written as

$\displaystyle\small\mathbb{E}\\!\\!\\!\\!\\!\\!$
$\displaystyle\left\\{\left|\text{IUI}_{kk^{\prime}}\right|^{2}\right\\}=\underbrace{q_{k^{\prime}}\sum_{m=1}^{M}c_{mk}^{2}u_{mk}^{2}\beta_{mk^{\prime}}}_{C_{1}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle+$ $\displaystyle\tau
p_{p}q_{k^{\prime}}\left|\boldsymbol{\phi}_{k}^{H}{\boldsymbol{\phi}}_{k^{\prime}}\right|^{2}\sum_{m=1}^{M}c_{mk}^{2}u_{mk}^{2}\beta_{mk^{\prime}}^{2}$
$\displaystyle+$ $\displaystyle\tau
p_{p}q_{k^{\prime}}\sum_{m=1}^{M}\sum_{i\neq
k^{\prime}}^{K}c_{mk}^{2}u_{mk}^{2}\beta_{mk^{\prime}}\beta_{mi}\left|\boldsymbol{\phi}_{k}^{H}\boldsymbol{\phi}_{i}\right|^{2}$
$\displaystyle+$ $\displaystyle
q_{k^{\prime}}\left|\boldsymbol{\phi}_{k}^{H}{\boldsymbol{\phi}}_{k^{\prime}}\right|^{2}\left(\sum_{m=1}^{M}u_{mk}\gamma_{mk}\dfrac{\beta_{mk^{\prime}}}{\beta_{mk}}\right)^{2},$
(17)

and

$\displaystyle C_{2}$ $\displaystyle=$ $\displaystyle\tau
p_{p}q_{k^{\prime}}\left|\boldsymbol{\phi}_{k}^{H}{\boldsymbol{\phi}}_{k^{\prime}}\right|^{2}\sum_{m=1}^{M}c_{mk}^{2}u_{mk}^{2}\beta_{mk^{\prime}}^{2}$
(18) $\displaystyle+$ $\displaystyle\underbrace{\tau
p_{p}q_{k^{\prime}}\sum_{m=1}^{M}\sum_{i\neq
k^{\prime}}^{M}c_{mk}^{2}u_{mk}^{2}\beta_{mk^{\prime}}\beta_{mi}\left|\boldsymbol{\phi}_{k}^{H}\boldsymbol{\phi}_{i}\right|^{2}}_{C_{3}}.$

For the last term of (18), we have

$\displaystyle C_{3}$ $\displaystyle=$ $\displaystyle\tau
p_{p}q_{k^{\prime}}\sum_{m=1}^{M}\sum_{i\neq
k^{\prime}}^{K}c_{mk}^{2}u_{mk}^{2}\beta_{mk^{\prime}}\beta_{mi}\left|\boldsymbol{\phi}_{k}^{H}\boldsymbol{\phi}_{i}\right|^{2}$
(19) $\displaystyle=$ $\displaystyle\sqrt{\tau
p_{p}}q_{k^{\prime}}\sum_{m=1}^{M}u_{mk}^{2}c_{mk}\beta_{mk^{\prime}}\beta_{mk}-q_{k^{\prime}}\sum_{m=1}^{M}u_{mk}^{2}c_{mk}^{2}\beta_{mk^{\prime}}$
$\displaystyle-$ $\displaystyle\tau
p_{p}q_{k^{\prime}}\sum_{m=1}^{M}u_{mk}^{2}c_{mk}^{2}\beta_{mk^{\prime}}\left|\boldsymbol{\phi}_{k}^{H}{\boldsymbol{\phi}}_{k^{\prime}}\right|^{2},$

where in the last step, we used equation (II-A). As a result,
$C_{1}+C_{2}=\sqrt{\tau
p_{p}}q_{k^{\prime}}\sum_{m=1}^{M}u_{mk}^{2}c_{mk}\beta_{mk^{\prime}}\beta_{mk}$.
Then finally we have

$\displaystyle\mathbb{E}\left\\{\left|\text{IUI}_{kk^{\prime}}\right|^{2}\right\\}$
$\displaystyle=\rho
q_{k^{\prime}}\left(\sum_{m=1}^{M}u_{mk}^{2}\beta_{mk^{\prime}}\gamma_{mk}\right)$
$\displaystyle+$ $\displaystyle\rho
q_{k^{\prime}}\left|\boldsymbol{\phi}_{k}^{H}{\boldsymbol{\phi}}_{k^{\prime}}\right|^{2}\left(\sum_{m=1}^{M}u_{mk}\gamma_{mk}\dfrac{\beta_{mk^{\prime}}}{\beta_{mk}}\right)^{2}.$

The total noise for the user $k$ is given by

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathbb{E}\left\\{\left|\text{TN}_{k}\right|^{2}\right\\}\\!=\\!\mathbb{E}\left\\{\left|\sum_{m=1}^{M}u_{mk}\hat{g}_{mk}^{*}n_{m}\right|^{2}\\!\right\\}\\!=\\!\sum_{m=1}^{M}u_{mk}^{2}\gamma_{mk},$
(21)

where the last equality is due to the fact that the terms $\hat{g}_{mk}$ and
$n_{m}$ are uncorrelated. Finally, by substituting (Appendix A: Proof of
Theorem 1), (Appendix A: Proof of Theorem 1), (Appendix A: Proof of Theorem 1)
and (21) into (4), SINR of $k$th user is obtained by (5), which completes the
proof of Theorem 1. $\blacksquare$

## Appendix B: Proof of Proposition 1

The standard form of GP is defined as follows [15]:

$\displaystyle P_{5}:\min\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ f_{0}(\textbf{x})$ (22) s.t. $\displaystyle
f_{i}(\textbf{x})\leq 1,\leavevmode\nobreak\ \leavevmode\nobreak\
i=1,\cdots,m,$ $\displaystyle g_{i}(\textbf{x})=1,\leavevmode\nobreak\
\leavevmode\nobreak\ i=1,\cdots,p,$

where $f_{0}$ and $f_{i}$ are posynomial and $g_{i}$ are monomial functions.
Moreover, $\textbf{x}=\\{x_{1},\cdots,x_{n}\\}$ represent the optimization
variables. The SINR constraint in (9) is not a posynomial function in its
form, however it can be rewritten into the following posynomial function:

$\displaystyle\small\dfrac{\mathbf{u}_{k}^{H}\\!\left(\\!\sum_{k^{\prime}\neq
k}^{K}\\!q_{k^{\prime}}\\!|\\!\boldsymbol{\phi}_{k}^{H}\\!\boldsymbol{\phi}_{k^{\prime}}|^{2}\\!{\boldsymbol{\Delta}}_{kk^{\prime}}\\!{\boldsymbol{\Delta}}_{kk^{\prime}}^{H}\\!+\\!\sum_{k^{\prime}=1}^{K}\\!q_{k^{\prime}}\\!\textbf{D}_{kk^{\prime}}\\!+\\!\frac{1}{p}\\!\textbf{R}_{k}\\!\right)\\!\mathbf{u}_{k}}{\mathbf{u}_{k}^{H}\\!\left(q_{k}{\boldsymbol{\Gamma}}_{k}{\boldsymbol{\Gamma}}_{k}^{H}\right)\mathbf{u}_{k}}\\!$
$\displaystyle<$ $\displaystyle\\!\dfrac{1}{t},$ $\displaystyle\forall k.$
(23)

By applying a simple transformation, (23) is equivalent to the following
inequality:

$\displaystyle q_{k}^{-1}\left(\sum_{k^{\prime}\neq
k}^{K}\\!a_{kk^{\prime}}q_{k^{\prime}}\\!+\\!\sum_{k^{\prime}=1}^{K}b_{kk^{\prime}}q_{k^{\prime}}+c_{k}\right)<\dfrac{1}{t},$
(24)

where
$a_{kk^{\prime}}=\frac{\mathbf{u}_{k}^{H}(|\\!\boldsymbol{\phi}_{k}^{H}\\!\boldsymbol{\phi}_{k^{\prime}}|^{2}\\!{\boldsymbol{\Delta}}_{kk^{\prime}}\\!{\boldsymbol{\Delta}}_{kk^{\prime}}^{H})\mathbf{u}_{k}}{\mathbf{u}_{k}^{H}\\!({\boldsymbol{\Gamma}}_{k}{\boldsymbol{\Gamma}}_{k}^{H})\mathbf{u}_{k}}$,
$b_{kk^{\prime}}=\frac{\mathbf{u}_{k}^{H}\textbf{D}_{kk^{\prime}}\mathbf{u}_{k}}{\mathbf{u}_{k}^{H}\\!({\boldsymbol{\Gamma}}_{k}{\boldsymbol{\Gamma}}_{k}^{H})\mathbf{u}_{k}}$,
and
$c_{k}=\frac{\mathbf{u}_{k}^{H}\textbf{R}_{k}\mathbf{u}_{k}}{p\mathbf{u}_{k}^{H}\\!({\boldsymbol{\Gamma}}_{k}{\boldsymbol{\Gamma}}_{k}^{H})\mathbf{u}_{k}}$.
The transformation in (24) shows that the left-hand side of (23) is a
polynomial function. Therefore, the power allocation Problem $P_{4}$ is a
standard GP (convex problem), where the objective function and constraints are
monomial and polynomials, which completes the proof of Proposition 1.
$\blacksquare$

## References

  * [1] A. Zappone, L. Sanguinetti, G. Bacci, E. A. Jorswieck, and M. Debbah, “Energy-efficient power control: a look at 5G wireless technologies,” _IEEE Trans. Signal Process._ , vol. 64, no. 7, pp. 1668–1683, Apr. 2016\.
  * [2] E. Björnson, E. A. Jorswieck, M. Debbah, and B. Ottersten, “Multiobjective signal processing optimization: the way to balance conflicting metrics in 5G systems,” _IEEE Signal Process. Mag._ , vol. 31, no. 6, pp. 14–23, Oct. 2014.
  * [3] A. G. Burr, M. Bashar, and D. Maryopi, “Cooperative access networks: Optimum fronthaul quantization in distributed Massive MIMO and cloud RAN,” in _Proc. IEEE VTC_ , Jun. 2018, pp. 1–5.
  * [4] M. Bashar, K. Cumanan, A. G. Burr, H. Q. Ngo, and M. Debbah, “Cell-free Massive MIMO with limited backhaul,” in _Proc. IEEE ICC_ , May 2018, pp. 1–7.
  * [5] M. Bashar, A. G. Burr, K. Haneda, and K. Cumanan, “Robust user scheduling with COST 2100 channel model for Massive MIMO networks,” _Accepted to publish in IET microwave antenna and propagation_ , Feb. 2018.
  * [6] Y. Lejosne, M. Bashar, D. Slock, and Y. Yuan-Wu, “From MU Massive MISO to pathwise MU Massive MIMO,” in _Proc. IEEE SPAWC_ , Jun. 2014, pp. 16–20.
  * [7] H. Sifaou, A. Kammoun, L. Sanguinetti, M. Debbah, and M. Alouini, “Max–min SINR in large-scale single-cell MU-MIMO: asymptotic analysis and low-complexity transceivers,” _IEEE Trans. Signal Process._ , vol. 65, no. 7, pp. 1841–1854, Apr. 2017.
  * [8] K. Cumanan, R. Krishna, L. Musavian, and S. Lambotharan, “Joint beamforming and user maximization techniques for techniques for cognitive radio networks based on branch and bound method,” _IEEE Trans. Wireless Commun._ , vol. 9, no. 10, pp. 3082–3092, Oct. 2010.
  * [9] K. Cumanan, L. Musavian, S. Lambotharan, and A. B. Gershman, “SINR balancing technique for downlink beamforming in cognitive radio networks,” _IEEE Signal Process. Lett._ , vol. 17, no. 2, pp. 133–136, Feb. 2010.
  * [10] K. Cumanan, R. Krishna, Z. Xiong, and S. Lambotharan, “SINR balancing technique and its comparison to semidefinite programming based QoS provision for cognitive radios,” in _Proc. IEEE VTC_ , Apr. 2009, pp. 1–5.
  * [11] K. Cumanan, Y. Rahulamathavan, S. Lambotharan, and Z. Ding, “A mixed quality of service based linear transceiver design for a multiuser MIMO network with linear transmit covariance constraints,” in _Proc. IEEE WCNC_ , Apr. 2013, pp. 3895–3899.
  * [12] K. Cumanan, J. Tang, and S. Lambotharan, “Rate balancing based linear transceiver design for multiuser MIMO system with multiple linear transmit covariance constraints,” in _Proc. IEEE ICC_ , Jun. 2011, pp. 1–5.
  * [13] H. Q. Ngo, A. Ashikhmin, H. Yang, E. G. Larsson, and T. L. Marzetta, “Cell-free Massive MIMO versus small cells,” _IEEE Trans. Wireless Commun._ , vol. 16, no. 3, pp. 1834–1850, Mar. 2017.
  * [14] G. Golub and C. V. Loan, _Matrix Computations_ , 2nd ed. Baltimore, MD: The Johns Hopkins Univ. Press, 1996.
  * [15] S. Boyd and L. Vandenberghe, _Convex Optimization_. Cambridge, UK: Cambridge University Press, 2004.
  * [16] H. Ahmadi, A. Farhang, N. Marchetti, and A. MacKenzie, “A game theoretic approach for pilot contamination avoidance in Massive MIMO,” _IEEE Wireless Commun. Lett._ , vol. 5, no. 1, pp. 12–15, Feb. 2016.