# Vector Quantization Methods for Access Point Placement in Cell-Free Massive
MIMO Systems

Govind R. Gopal, and Bhaskar D. Rao Govind R. Gopal and Bhaskar D. Rao are
with the Department of Electrical and Computer Engineering, University of
California San Diego, La Jolla, CA 92093, USA (e-mail<EMAIL_ADDRESS>brao@ucsd.edu). The work of Govind R. Gopal was supported in part by the
Center for Wireless Communications (CWC), University of California San Diego
and in part by Qualcomm Inc. through the Faculty-Mentor-Advisor program. The
work of Bhaskar D. Rao was supported by the National Science Foundation (NSF)
under Grant CCF-2124929 and Grant CCF-2225617.

###### Abstract

We examine the problem of uplink cell-free access point (AP) placement in the
context of optimal throughput. In this regard, we formulate two main placement
problems, namely the sum rate and minimum rate maximization problems, and
discuss the challenges associated with solving the underlying optimization
problem with the help of some simple scenarios. As a practical solution to the
AP placement problem, we suggest a vector quantization (VQ) approach. The
suitability of the VQ approach to cell-free AP placement is investigated by
examining three VQ-based solutions. First, the standard VQ approach, that is
the Lloyd algorithm (using the squared error distortion function) is
described. Second, the tree-structured VQ (TSVQ), which performs successive
partitioning of the distribution space is applied. Third, a probability
density function optimized VQ (PDFVQ) procedure is outlined, which enables
efficient, low complexity, and scalable placement, and is aimed at a massive
distributed multiple-input-multiple-output (MIMO) scenario. While the VQ-based
solutions do not solve the cell-free AP placement problems explicitly,
numerical experiments show that their sum and minimum rate performances are
good enough, and offer a good starting point for gradient-based optimization
methods. Among the VQ solutions, PDFVQ, with advantages over the other VQ
methods, offers a good trade-off between sum and minimum rates.

###### Index Terms:

Base station placement, Beyond 5G, Lloyd algorithm, scalability, throughput
optimization.

##  I Introduction

The concept of massive multiple-input-multiple-output (MIMO) [1] has emerged
in recent decades as a strong solution for 5G and Beyond wireless
communication systems [2, 3, 4]. By having a large number of antennas, such
systems enable higher spectral and energy efficiencies, and reduced
interference due to increased diversity [5, 6]. Distributed MIMO, comprising
distributed antenna systems (DASs), offer higher average rates over their
colocated counterparts [7, 8, 9, 10]. Distributed massive MIMO can be
predominantly split into non-cooperative and cooperative systems. Small-cell
systems comprise non-cooperative systems while cell-free systems constitute
the latter.

Cell-free massive MIMO, where there are a very large number of antennas or
access points (APs) serving users that are not divided into cells, have become
popular of late as they help in mitigating interference among users and in
enhanced spectral efficiency over small-cell systems [11, 12, 13, 14]. The
concept of “cell-free” itself arises from network MIMO [15]. However, it is to
be noted that these benefits come at the cost of large backhaul requirements
necessary for the information exchange between APs [16, 17] and the network
controller (NC) or the central processing unit, where all or most processing
is performed. The computational and processing requirements of a cell-free
system are thus high, and are expected to grow with the anticipated network
densification in Beyond 5G/6G wireless systems [18]. Hence, the conventional
small-cell architecture is still very much in use in current and near future
5G deployments [19, 20]. Nevertheless, a multitude of problems has been
explored based on the cell-free network. Prior works investigate topics such
as power optimization and energy efficiency [21, 22, 23, 24], rate
maximization [25, 26], clustering (user- and cell-centric) [27, 28, 29, 30,
31], limited fronthaul [32], pilot assignment [33], reconfigurable intelligent
surfaces [34], and federated learning [35].

In this work, our focus is on AP placement, with the prevailing question: _How
do we place APs optimally for a given user distribution?_ Here, the optimality
is in terms of throughput in cell-free systems. AP placement finds application
in scenarios such as stadiums, where the distribution of crowds changes
according to the ongoing game, or in offices where the density of employees
changes by the hour. Further, deployment of APs using unmanned aerial vehicles
(UAVs) can be advantageous in areas affected by natural (or man-made)
calamities where existing infrastructure has been destroyed [36]. Past
literature has expressed interest in AP placement, however, most of the focus
has been on single-cell or small-cell systems (e.g., [37, 38, 39, 40]). For
instance, [37] maximizes cell averaged ergodic capacity of a DAS by using
vector quantization (VQ)-based codebook design (the Lloyd algorithm) to place
antennas. In [38], antennas are placed circularly to optimize the average per-
user rate of uniformly distributed users. In [39], a 10-fold improvement in
capacity is shown in a distributed system over a colocated in a simulated
indoor environment. Further, aligning AP positions with the user density
generated around a 40% increase in capacity over uniformly distributed APs.
The authors of [40] and their related works study the deployment of
heterogeneous wireless sensor networks (placement of both APs and fusion
centers) as a source coding problem for optimal power control and with limited
communication range. Most optimization problems have mainly considered
maximizing the signal-to-noise ratio (SNR) alone, but certain works have
incorporated various forms of interference (e.g., two-cell leakage
interference in [41] and signal-to-interference-plus-noise ratio (SINR) in
[42]). Placement of UAVs as APs (UAV-APs) has also been studied (e.g., [43,
44, 45, 46, 47, 48]). [43] calculates the positions and altitudes of UAV-APs
to maximize ground user coverage as well as minimize interference to users.
The work of [44] on the other hand places UAV-APs heuristically in a spiral
fashion to guarantee user coverage. The authors of [45] find the optimal
placement and coverage radius of UAV-APs that serve users with time-varying
positions in Wi-Fi (IEEE 802.11) networks. In [46], the 3-D placement of UAV-
APs with directional antennas to minimize average user transmit powers is
investigated. A hybrid UAV-terrestrial AP network is considered in [47] where
position-flexible UAV-APs are deployed in an area occupied by fixed
terrestrial APs to maximize throughput while minimizing inter-cell
interference (ICI) by using a Lloyd-type algorithm. Finally, [48] provides an
extensive tutorial on UAVs as APs and as users (cellular connected UAVs).

The problem of AP placement in cell-free systems is fairly novel and much
prior research has not been conducted. As an example, [49] investigates the
deployment of UAVs in a cell-free network to maximize minimum SINR using a
gradient approach while considering pilot contamination and that not every AP
communicates with all users. The authors of [50] consider the placement of APs
in a distributed massive MIMO system as a combinatorial problem to minimize
transmit powers while considering antenna radiation patterns and different
channel models. While a cell-free system is not explicitly defined, the system
model considered mimics such a system. A graph-based approach is found to
yield significant power savings while ensuring placement with good coverage.
In [51], 3-D placement of UAV-APs is considered to maximize the downlink sum
rate and uses an alternating optimization method. A prior work by our group
[52] examines both sum rate and minimum rate maximization problems. This work
solves the two problems using compressed sensing techniques by dividing the
geographical area into regular grids. However, the approximations used do not
solve the two placement problems optimally.

None of the abovementioned works have investigated VQ approaches to solve the
cell-free AP placement problem. In our prior work [53], we preliminarily
investigated the utilization of the standard VQ approach, namely the Lloyd
algorithm with the squared Euclidean distance as the distortion function, to
place APs in a cell-free network. The VQ technique considers a single user
that communicates to its nearest AP with the objective function that utilizes
a distortion function averaged over the random position of this user. It does
not match the cell-free model where all users communicate to all APs. In spite
of these limitations, VQ-based solutions have some features that make them
worthy of consideration. By design, VQ offers not only a distributed solution
but can encourage cooperation (as is expected in the cell-free model) by
placing APs closer to one another in areas of higher user density.
Additionally, VQ solutions provide good initial points for gradient and
learning-based methods to solve specific throughput problems. Hence, in this
work, we explore and compare multiple VQ techniques, each with its own
benefits, that can solve for AP locations in a cell-free network, with
throughput as the performance measure.

### Contributions

To the best of our knowledge, analysis of the cell-free AP placement problem
in the context of throughput optimality and the suitability of VQ techniques
to solve the same have not been addressed in past literature. Hence, in this
work, our contributions are as follows.

  * •

We formulate the two main throughput optimal cell-free AP placement problems,
namely the sum rate and minimum rate maximization problems. Starting from the
simpler sum SNR problem, analysis of the sum rate problem is conducted and
simple examples are shown to describe the possible solutions and to highlight
the challenges associated with the general AP placement problem. The minimum
rate problem is briefly discussed, also with some examples.

  * •

Three VQ-based techniques to place APs are proposed and explored, namely
standard VQ, which is the Lloyd algorithm, tree-structured VQ (TSVQ), and
probability density function optimized VQ (PDFVQ). While the Lloyd algorithm
provides a well-established method to place APs, there are the disadvantage of
complexity and scalability. TSVQ, through successive partitioning of the user
area, places APs in such a way so as to foster cooperation. PDFVQ on the other
hand, allows an efficient, less computationally intensive, and easily
adaptable AP placement solution by using bit allocation, transform coding, and
scalar quantization, and is especially suitable for a scaled network with a
large number of APs.

The remainder of this manuscript is organized as follows. Section II outlines
the cell-free system model used, followed by Section III which explores the
two principal throughput optimization problems for AP placement, namely the
sum rate and minimum rate maximizations. VQ-based techniques are described in
Section IV. In Section V, we state the simulation methodology and results.
Finally, concluding remarks are provided in Section VI. Throughout this paper,
we will use bold symbols to denote vectors, $\mathbb{E}\\{\cdot\\}$ is the
expectation operator, $||\cdot||$ represents the $\ell_{2}$-norm of a vector,
and all logarithms are to the base 2.

##  II System Model

The system model outlined in [52], [53], and [54] is used. $K$ single-antenna
users are distributed over a geographical area with a probability density
function $f_{\mathbf{P}}(\mathbf{p})$, with $\mathbf{p}\in\mathbb{R}^{2}$ as
the random vector denoting the user position. $M$ single-antenna APs serve
these users, where $\mathbf{q}\in\mathbb{R}^{2}$ is the AP location. With
$m=1,2,\ldots,M$ and $k=1,2,\ldots,K$, a narrowband fading channel is
considered with

$g_{mk}=\sqrt{\beta_{mk}}h_{mk},$ (1)

where $\beta_{mk}$ and $h_{mk}\sim\mathcal{CN}(0,1)$ are the large- and small-
scale fading coefficients, respectively, independent of each other and over
coherent intervals. A general expression for $\beta_{mk}$ is

$\beta_{mk}=\frac{cz_{mk}}{\left|\left|\mathbf{p}-\mathbf{q}_{m}\right|\right|^{\gamma}},$
(2)

where $c$ is a constant, $z_{mk}$ is the shadow fading coefficient, and
$\gamma$ is the pathloss exponent. All APs are connected via error-free
backhaul links to the NC. In the cell-free uplink regime, all users are served
by all APs at the same time. The uplink received signal at AP $m$ is

$y_{m}=\sum\limits_{k=1}^{K}\sqrt{\rho_{r}}g_{mk}s_{k}+w_{m},$ (3)

where for user $k$, $\rho_{r}$ is the transmit power, $s_{k}$ is the data
symbol with $\mathbb{E}\\{|s_{k}|^{2}\\}=1$, and $w_{m}\sim\mathcal{CN}(0,1)$
is the additive noise. The received signal vector at the NC from all $M$ APs
using (3) can be written as

$\mathbf{y}=\sum\limits_{k=1}^{K}\sqrt{\rho_{r}}\mathbf{g}_{k}s_{k}+\mathbf{w},$
(4)

where $\mathbf{y}=[y_{1},y_{2},\ldots,y_{M}]^{T}$,
$\mathbf{g}_{k}=[g_{1k},g_{2k},\ldots,g_{Mk}]^{T}$, and
$\mathbf{w}=[w_{1},w_{2},\ldots,w_{M}]^{T}$. When a combiner $\mathbf{v}_{k}$
is used to estimate data symbols of user $k$ as
$\hat{s}_{k}=\mathbf{v}_{k}^{H}\mathbf{y}$, the per-user achievable rate is
$R_{k}=\mathbb{E}\\{\log(1+\phi_{k}^{\mathbf{v}_{k}})\\}$, where the
expectation is over all the small-scale and shadow fading coefficients, and
the SINR [55] is

$\phi_{k}^{\mathbf{v}_{k}}=\frac{\rho_{r}\mathbf{v}_{k}^{H}\mathbf{g}_{k}\mathbf{g}_{k}^{H}\mathbf{v}_{k}}{\mathbf{v}_{k}^{H}\mathbf{v}_{k}+\sum\limits_{\begin{subarray}{c}k^{\prime}=1\\\
k^{\prime}\neq
k\end{subarray}}^{K}\rho_{r}\mathbf{v}_{k}^{H}\mathbf{g}_{k}\mathbf{g}_{k}^{H}\mathbf{v}_{k}}.$
(5)

One such combiner is the zero forcing (ZF) detector, and results in the
processed signal at the NC as
$\mathbf{r}=\left(\mathbf{G}^{H}\mathbf{G}\right)^{-1}\mathbf{G}^{H}\mathbf{y}$,
where $\mathbf{G}=[g_{mk}]$ is a $M\times K$ matrix consisting of the channel
coefficients [52]. The achievable per-user SNR in this case is

$\psi_{k}^{\text{ZF}}=\frac{\rho_{r}}{\left[(\mathbf{G}^{H}\mathbf{G})^{-1}\right]_{kk}}.$
(6)

Using an asymptotic approximation for the SNR as outlined in [38, 52], the
per-user SNR can also be written as

$\frac{1}{M}\psi_{k}^{\text{ZF}}\xrightarrow[M\rightarrow\infty]{\text{a.s.}}\rho_{r}\overline{\beta}_{k},$
(7)

where

$\overline{\beta}_{k}\triangleq\lim_{M\rightarrow\infty}\frac{1}{M}\sum_{m}\beta_{mk}.$
(8)

##  III Throughput Formulations for the Cell-Free AP Placement Problem

There are two main formulations for cell-free AP placement in terms of
throughput optimality:

  * •

Sum rate maximization, which involves the sum of the rates of all users, as
follows

$\underset{\mathbf{q}_{1},\mathbf{q}_{2},\ldots,\mathbf{q}_{M}}{\operatorname{arg}\,\operatorname{max}}\;\sum\limits_{k=1}^{K}\log(1+\phi_{k}^{\mathbf{v}_{k}}).$
(9)

Note that this problem is identical to the average rate maximization problem
by assuming a sample mean and taking the average over the user distribution
$f_{\mathbf{P}}(\mathbf{p})$.

  * •

Minimum rate maximization, where the minimum of the rates among all of the
users is maximized

$\underset{\mathbf{q}_{1},\mathbf{q}_{2},\ldots,\mathbf{q}_{M}}{\operatorname{arg}\,\operatorname{max}}\;\min\limits_{k}\
\log(1+\phi_{k}^{\mathbf{v}_{k}}).$ (10)

The notion of fairness, which is important in a cell-free system since all
users are served by all APs, is enforced by the minimum rate problem in (10)
as opposed to the sum rate problem in (9). In practice, the 95%-likely rate,
which represents the best rate among the worst 5% of the users, is used as a
measure to evaluate the network minimum rate performance. Hence, the max-min
rate is adjusted to the 95%-likely rate, which is more robust. This is
subsequently addressed in Section IV-E.

In the ensuing sections, we discuss the above formulations. Our analysis of
the sum rate maximization problem is preceded by the simpler sum SNR
maximization problem (preliminarily discussed in [53]).

###  III-A Sum SNR Maximization

The sum rate problem stated in (9) is simplified to the sum SNR problem by
using the ZF SNR $\psi_{k}^{\text{ZF}}$ from (6) and by replacing the
summation with an expectation (the factor of $1/K$ has been neglected), where
the average is taken over the user position and the set
$\mathcal{A}=\\{z_{mk},\forall m\\}$ consisting of all shadow fading
coefficients between each user and AP. The optimization problem is written as

$\underset{\mathbf{q}_{1},\mathbf{q}_{2},\ldots,\mathbf{q}_{M}}{\operatorname{arg}\,\operatorname{max}}\;\mathbb{E}_{\mathcal{A},\mathbf{p}}\left\\{\psi_{k}^{\text{ZF}}\right\\}.$
(11)

We can simplify the above objective function using the approximation in (7)
and $\beta_{mk}$ from (2) in the following manner

$\displaystyle\mathbb{E}_{\mathcal{A},\mathbf{p}}\left\\{\psi_{k}^{\text{ZF}}\right\\}$
$\displaystyle=\mathbb{E}_{\mathcal{A},\mathbf{p}}\left\\{\sum\limits_{m=1}^{M}\beta_{mk}\right\\},$
(12)
$\displaystyle=\mathbb{E}_{\mathbf{p}}\left\\{\sum\limits_{m=1}^{M}\mathbb{E}_{\mathcal{A}}\left\\{\beta_{mk}\right\\}\right\\},$
$\displaystyle\overset{(a)}{=}\mathbb{E}_{\mathbf{p}}\left\\{\sum\limits_{m=1}^{M}\frac{c^{\prime}}{\left|\left|\mathbf{q}_{m}-\mathbf{p}\right|\right|^{\gamma}}\right\\},$
$\displaystyle\overset{(b)}{=}c^{\prime}\sum\limits_{m=1}^{M}\mathbb{E}_{\mathbf{p}}\left\\{\frac{1}{\left|\left|\mathbf{q}_{m}-\mathbf{p}\right|\right|^{\gamma}}\right\\},$

where $c^{\prime}=c\mathbb{E}_{\mathcal{A}}\\{z_{mk}\\}$, $\mathbf{p}$ is the
position of user $k$ in $(a)$, and $(b)$ uses the fact that the expectation is
a linear operator. Note that $c^{\prime}$ can be ignored as it does not effect
the optimization problem.

The following observations can be made regarding the solution of the above
objective function in (12).

  * •

A colocated solution is evident since there is no dependence between the terms
in the summation associated with each AP $m$ and hence, the optimization for
each AP can be performed separately. The suggested colocated solution may not
be a unique solution and multiple global and/or local maxima may exist for the
optimization problem. The complete characterization of the solution, however,
depends on two factors, namely the pathloss exponent $\gamma$ used as well as
the shape of the user distribution $f_{\mathbf{P}}(\mathbf{p})$ over which the
expectation is taken, e.g., uni-modal versus multi-modal density functions.

  * •

In (12), although the norm
$\left|\left|\mathbf{q}_{m}-\mathbf{p}\right|\right|$, i.e., the Euclidean
distance is strictly convex [56] in $\mathbf{q}_{m}$, its inverse is neither
concave nor convex and is also undefined at $\mathbf{q}_{m}=\mathbf{p}$ (this
can, however, be avoided by adding a small positive quantity to the
denominator, which could also account for the height of the AP). The sum of
the expectation of the inverse over the APs thus also is neither convex or
concave. Additionally, when a uni-modal distribution is assumed, we can expect
a colocated solution alone with a unique maximum. However, when a multi-modal
distribution is considered, it is expected that multiple local maxima exist
and distributed solutions may be obtained. This is explored in Section III-C.

It is worth noting that the sum SNR problem has been previously addressed in
part in [52]. In this work, while dividing the user area into regular grid
points, the sum rate maximization problem is upperbounded to a sum SNR problem
and using a compressed sensing framework, approximated to a linear program
(called the max-sum algorithm). However, as is expected from a sum SNR problem
(as discussed above), most APs are concentrated around high user density
regions (a near-colocated solution). Although a high sum rate can be achieved
with this solution, users far away from the APs are severely affected in terms
of throughput, resulting in poor minimum rate performance. Thus, this solution
is not suitable for cell-free AP placement when fairness is considered.

###  III-B Sum Rate Maximization

Returning to the sum rate maximization problem, we rewrite (9) by using the
simplifications assumed before, as follows

$\underset{\mathbf{q}_{1},\mathbf{q}_{2},\ldots,\mathbf{q}_{M}}{\operatorname{arg}\,\operatorname{max}}\;\mathbb{E}_{\mathcal{A},\mathbf{p}}\left\\{\log\left(1+\psi_{k}^{\text{ZF}}\right)\right\\},$
(13)

and the objective function, utilizing (7), can be rewritten as

$\displaystyle\mathbb{E}_{\mathcal{A},\mathbf{p}}\left\\{\log\left(1+\psi_{k}^{\text{ZF}}\right)\right\\}$
$\displaystyle=\mathbb{E}_{\mathcal{A},\mathbf{p}}\left\\{\log\left(1+\sum\limits_{m=1}^{M}\beta_{mk}\right)\right\\}.$
(14)

Similar to the SNR problem outlined before, this objective function is neither
concave nor convex. However, unlike the former, the term associated with each
AP $m$ in (14) cannot be decoupled from the terms associated with the rest of
the APs. Hence, for both uni-modal and multi-modal distributions, we can
expect only distributed solutions that maximize the sum rate.

In summary, for both the sum SNR and sum rate problems there may be multiple
local optima suggesting that the optimization problem is complex and
challenging.

###  III-C Examples for Sum SNR and Sum Rate Maximizations

Given the abovementioned complexity in solving the sum SNR and sum rate
problems, we now attempt to understand the problems and their solutions
better. For this purpose, we explore simple examples where the aforementioned
two problems are solved, and where the multiple local optima are studied to
develop intuition and insight.

####  III-C1 User distribution considered

For tractability, we consider a simple 1-D scenario where users are
distributed along a line, and the placement of four APs along the line to
maximize both sum SNR and sum rate. For this purpose, we assume a bi-modal
distribution since it is the simplest among multi-modal distributions that can
exhibit the multiple maxima as discussed in Section III-A. With the user
position denoted by $p$, the PDF of the bi-modal Gaussian considered here is

$f_{P}(p)=\pi_{1}\mathcal{N}(p|\mu_{1},\sigma^{2}_{1})+\pi_{2}\mathcal{N}(p|\mu_{2},\sigma^{2}_{2}),$
(15)

where for each Gaussian $i$, $i=1,2$, $\pi_{i}$ is the probability such that
$\pi_{1}+\pi_{2}=1$, $\mu_{i}$ is the mean, and $\sigma_{i}$ is the standard
deviation. To generate different solution structures for the two optimization
problems, we consider two distinct configurations of the user distribution:

  1. Conf. 1:

$\pi_{1}=\pi_{2}=0.5$, $\mu_{1}=-3$, $\mu_{2}=3$, and
$\sigma_{1}=\sigma_{2}=1$.

  2. Conf. 2:

$\pi_{1}=0.35$, $\pi_{2}=0.65$, $\mu_{1}=-3$, $\mu_{2}=4$, and
$\sigma_{1}=\sigma_{2}=1$.

Configuration 1 is symmetric about the origin while configuration 2 is
asymmetric.

####  III-C2 Definitions of SNR and rate

The four APs have locations $q_{1}$, $q_{2}$, $q_{3}$, and $q_{4}$, and the ZF
SNR for user $k$ can be calculated as

$\displaystyle\psi_{k}^{\text{ZF}}$
$\displaystyle=\beta_{1k}+\beta_{2k}+\beta_{3k}+\beta_{4k},$ (16)
$\displaystyle=\frac{1}{(p_{k}-q_{1})^{2}}+\frac{1}{(p_{k}-q_{2})^{2}}+\frac{1}{(p_{k}-q_{3})^{2}}+\frac{1}{(p_{k}-q_{4})^{2}},$

where, for simplicity, the transmit power is set to unity and the definition
of the large-scale fading coefficient $\beta_{mk}$ (from (2)) assumes that
shadow fading is absent, the pathloss exponent is two, and the constant is set
to one. These assumptions do not change the conclusions that are obtained in
this section. The sum SNR and sum rate quantities are then defined as
$\sum_{k=1}^{K}\psi_{k}^{\text{ZF}}$ and
$\sum_{k=1}^{K}\log(1+\psi_{k}^{\text{ZF}})$, respectively. Note that for
implementation purposes, a small quantity $\epsilon$ is added to the
denominator of $\beta_{ik}$, $i=1,2,3,4$, to prevent $\psi_{k}^{\text{ZF}}$
from approaching infinity. We then plot the sum SNR and sum rate for different
AP locations.

####  III-C3 AP location solutions considered

To evaluate and understand the sum SNR and sum rate performances of the
system, we study various AP placement scenarios. First, a colocated solution
is considered where all four APs are situated at the same location. This
location is found by sweeping the AP position denoted by $q$ across the span
of the user locations. Second, multiple semi-distributed solutions are
selected. Instead of all APs at one location, they can be allocated to each of
the Gaussians and placed at their respective means. In this scenario, we
consider three situations, namely when 2 APs each are placed at $\mu_{1}$ and
$\mu_{2}$, $3$ APs are at $\mu_{1}$ and $1$ AP is at $\mu_{2}$, and $1$ AP is
at $\mu_{1}$ and $3$ APs are at $\mu_{2}$. These three situations are termed
‘Distributed (2+2)’, ‘Distributed (3+1)’, and ‘Distributed (1+3)’,
respectively. Third, a fully distributed scenario involves starting from a
distributed (2+2) solution and moving the two APs within each Gaussian away
from each other until the maximum is achieved. Note that this fully
distributed solution represents only a local maximum. Finally, we have the
solution obtained by applying the standard Lloyd algorithm to the user
distribution.

Figure 1: Sum SNR for different AP location scenarios under user configuration
1. Figure 2: Sum SNR for different AP location scenarios under user
configuration 2.

####  III-C4 Results for sum SNR

The results of sum SNR under user configurations 1 and 2 are shown in Fig. 1
and Fig. 2, respectively. Note that for the colocated solution, the sum SNR
obtained as a function of location $q$ is plotted. For the other solutions, a
line is drawn corresponding to the sum SNRs obtained.

For configuration 1, it is observed that the peak for the colocated system
occurs at the two means $\mu_{1}$ and $\mu_{2}$. The three distributed
scenarios as well as the fully distributed scenario offer the same peak sum
SNR value as in the colocated case. It is noted that the AP locations in the
fully distributed case are the same as in distributed (2+2). Further, the
Lloyd solution (which is also a fully distributed scenario) offers a lower sum
SNR value. The sum SNR maximization for configuration 1 thus has multiple
local maxima, including both colocated and semi-distributed solutions.

For configuration 2, the Gaussian with mean $\mu_{2}$ has a higher
probability, with the result that a colocated solution with all four APs at
mean $\mu_{2}$ yields the highest sum SNR. Among the distributed solutions, it
is observed that a higher allocation of APs at $\mu_{2}$ favors a higher sum
SNR, however, with the colocated solution offering the highest. Again, both
the fully distributed and distributed (2+2) solutions have the same AP
locations and sum SNR values, with the Lloyd solution performing the worst. In
summary, the above results for sum SNR show that the AP locations that
maximize the same may be colocated or distributed depending on the user
distribution, as discussed in Section III-A.

Figure 3: Sum rate for different AP location scenarios under user
configuration 1. Figure 4: Sum rate for different AP location scenarios under
user configuration 2.

####  III-C5 Results for sum rate

The results of the sum rate metric are shown for user configurations 1 and 2
are shown in Fig. 3 and Fig. 4, respectively. In configuration 1, it is clear
that a colocated solution performs poorly and that distributed and fully
distributed solutions offer a higher sum rate. Since the distribution is
symmetric, both distributed (1+3) and distributed (3+1) solutions have the
same sum rate, which is lower than that of distributed (2+2). The Lloyd
solution performs better than the distributed solutions as no APs are
colocated, and the fully distributed solution performs the best. Note that the
fully distributed solution does not represent the optimum solution due to the
method with which the locations are generated. In configuration 2, the
colocated solution still performs the worst. However, since the Gaussian at
mean $\mu_{2}$ carries a higher proportion of users, the distributed (1+3)
solution performs better than both distributed (2+2) and distributed (3+1).
Finally, as in configuration 1, both the Lloyd and the fully distributed
solution perform better than the distributed solutions.

Figure 5: 95%-likely rate for different AP location scenarios under user
configuration 1. Figure 6: 95%-likely rate for different AP location scenarios
under user configuration 2.

####  III-C6 Results for minimum rate

We now plot the 95%-likely rates corresponding to the different AP locations
for configurations 1 and 2, in Fig. 5 and Fig. 6, respectively. Note that the
fully distributed solution in these plots maximizes the 95%-likely rates as
opposed to the sum SNR or sum rate. Even though all distributed solutions
generate the same peak sum SNR for configuration 1 as shown in Fig. 1, the
users will not achieve the same minimum rate performance in all cases. For
this user configuration, distributed (1+3) and distributed (3+1) have lower
95%-likely rates than distributed (2+2). Additionally, the Lloyd solution has
a higher minimum rate and the fully distributed solution generates the highest
95%-likely rate. Clearly, the colocated solution exhibits the worst
performance. The same observations are noted for configuration 2, with the
difference being that distributed (1+3) generates a higher 95%-likely rate
than distributed (3+1) due to the higher user proportion around mean
$\mu_{2}$.

Figure 7: Sum SNR for colocated and Lloyd solutions under uni-modal
distribution. Figure 8: Sum rate for colocated and Lloyd solutions under uni-
modal distribution.

####  III-C7 Results for a uni-modal user distribution

As a final note and to complete the discussion on the influence of the
distribution shape on the solutions, we also plot the sum SNR and sum rate
performances for a uni-modal Gaussian distribution (with zero mean and unit
variance), in Fig. 7 and Fig. 8, respectively. Like the bi-modal examples
shown prior, the colocated solution is preferred in the sum SNR case over the
distributed Lloyd solution. The opposite is true for the sum rate case. It is
also to be noted that unlike the bi-modal scenario, there is only one
colocated solution that provides the peak sum SNR.

In conclusion, all observations made in the context of the simple examples
support the complex nature of the AP placement for sum SNR and sum rate
maximization problems (Section III-A and Section III-B).

###  III-D Minimum Rate Maximization

For the sum SNR maximization case, we showed both colocated and semi-
distributed cases were favored. It was also postulated that the minimum rate
performance of these solutions would be inferior to that provided by a fully
distributed solution (like the Lloyd solution). Even in the sum rate
maximization case, we have not addressed the minimum rates of the users
explicitly and we can expect that there are users that lie far away from the
AP locations, resulting in significantly low throughput to these users, which
is not ideal from a fairness point of view. The minimum rate maximization
problem is thus of interest and has been defined above in (10). In this paper,
we do not discuss this problem as it has been addressed and solved using a
grid-based approach in [52, 54] and is called the max-min algorithm. Such a
solution is useful since the problem can be converted into a convex problem,
which can be solved easily. It is to be noted that the grid structure assumed
leads to approximate solutions and a finer grid is necessary to obtain the
optimal locations. In [53], we have further analyzed and compared the
performance of the max-min algorithm with the Lloyd algorithm.

##  IV Vector Quantization Approaches

In this section, as practical solutions to cell-free AP placement, we
investigate how VQ techniques can be applied to sum rate and minimum rate
maximizations. VQ can be implemented in a cell-free system by assuming that
each user is associated with its geographically nearest AP while adopting the
squared Euclidean distance (error) as the distortion measure of interest.
Before we delve into the various VQ techniques, we first motivate why such an
approach is useful for the placement problem.

###  IV-A Why Use a VQ Approach?

  * •

_Distributedness:_ The VQ approaches, through their formulations, are designed
to provide a distributed solution. For example, the standard Lloyd algorithm
using squared error distortion minimizes the distance of a user to its closest
AP, averaged over the entire distribution. Thus, a distributed solution is
obtained by ensuring that at least one AP is close to each user. Such a
solution also addresses the minimum rate of the system in an effective manner.

  * •

_Cooperation:_ While allowing distributedness, the VQ formulations which do
not explicitly account for cooperation, also place APs close to one another.
This clustering occurs especially in areas of high user density, thus
encouraging cooperation that is expected in a cell-free system and addresses
the system sum rate. In the case of the Lloyd algorithm, the objective
function itself allows for high AP density. In TSVQ, a practical
implementation of VQ discussed in Section IV-C below, the codepoint splitting
process in addition to the Lloyd algorithm used places APs closer to one
another and enables more cooperation.

  * •

_Initialization:_ Although it is possible to determine solutions for both
problems through gradient methods, such methods usually generate local optima
resulting in lower-than-expected performances. The VQ approaches, due to the
distributed nature of their solutions, are able to provide suitable starting
points for such gradient-based methods.

###  IV-B Standard VQ

The standard and simplest VQ technique is the Lloyd algorithm [57] that
utilizes the squared Euclidean distance between APs and users as the
distortion measure. This squared error (SE) distortion between a user at
$\mathbf{p}$ and an AP at $\mathbf{q}_{m}$ is denoted as follows

$d_{\text{SE}}(\mathbf{p},\mathbf{q}_{m})=\left|\left|\mathbf{p}-\mathbf{q}_{m}\right|\right|^{2}.$
(17)

Details of how the Lloyd algorithm can be applied to AP placement have been
elucidated in [42]. We provide the algorithm (Algorithm 1) below.

Algorithm 1 Lloyd Algorithm

1:Initialize random AP locations
$\mathbf{q}_{1}^{(0)},\mathbf{q}_{2}^{(0)},\ldots,\mathbf{q}_{M}^{(0)}$.

2:Use the NNC to determine the cells
$\mathcal{C}_{1}^{(i+1)},\mathcal{C}_{2}^{(i+1)},\ldots,\mathcal{C}_{M}^{(i+1)}$
such that
$\mathcal{C}_{m}^{(i+1)}\\!=\\!\left\\{\\!\mathbf{p}_{k}\\!:\\!d_{\text{SE}}\\!\left(\\!\mathbf{p}_{k},\mathbf{q}_{m}^{(i)}\\!\right)\\!\leq\\!d_{\text{SE}}\\!\left(\\!\mathbf{p}_{k},\mathbf{q}_{l}^{(i)}\\!\right)\\!,\forall
l\\!\neq m\\!\right\\}.$

3:Use the CC to determine the AP locations
$\mathbf{q}_{1}^{(i+1)},\mathbf{q}_{2}^{(i+1)},\ldots,\mathbf{q}_{M}^{(i+1)}$
such that
$\mathbf{q}_{m}^{(i+1)}=\frac{1}{\left|\mathcal{C}_{m}^{(i+1)}\right|}\sum\limits_{\mathbf{p}_{k}\in\mathcal{C}_{m}^{(i+1)}}\mathbf{p}_{k}.$

4:Repeat from step 2 until convergence (MSE falls below a threshold).

It is important to note that while the Lloyd algorithm provides distributed AP
locations and a user is expected to be close to at least one AP, the minimum
rate metric is not explicitly contained in its objective function.
Additionally, the assumed cooperation between the APs enabling higher sum rate
is also not explicitly included in the optimization. However, the Lloyd
algorithm indirectly fosters cooperation since the AP density is proportional
to a power of the user density under a high resolution approximation [57], as
follows

$g_{\mathbf{P}}(\mathbf{p})=\frac{f_{\mathbf{P}}^{\frac{1}{2}}(\mathbf{p})}{\int
f_{\mathbf{P}}^{\frac{1}{2}}(\mathbf{p}^{\prime})\mathrm{d}\mathbf{p}^{\prime}}.$
(18)

Therefore, APs in the high user density areas can cooperate with one another.

###  IV-C Tree-Structured VQ

Tree-structured VQ (TSVQ) [57] is an alternate VQ approach where the codebook
search time is reduced compared to standard VQ. In this technique, the input
training set is partitioned into a hierarchy of Voronoi regions, which allows
a tree to be generated for encoding. Thus, TSVQ differs from the standard VQ
discussed above in that the final required codebook is generated by the
successive splitting of intermediate codebooks, starting with a single
centroid. The Lloyd algorithm (Algorithm 1) is applied to each stage of the
hierarchy and its application is confined to the partitioned training set of
the previous stage.

_Why use TSVQ?_ Designed primarily for relatively fast codebook search
properties, TSVQ has the following benefits.

  * •

_Initialization:_ An advantage of TSVQ is that initialization of the AP
locations is not required, since it starts with a single centroid. Compared to
the standard VQ, this avoids the random initializations (considered sub-par)
and the calculations needed for advanced initialization methods such as
k-means++ [58].

  * •

_Cooperation:_ Due to the successive splitting of the intermediate codepoints,
we can expect pairs of APs to be closer to each other compared to standard VQ.
This enables cooperation among APs, increasing the system sum rate.

  * •

_Flexibility:_ When the number of APs is changed for the same user
distribution, codepoints can either be merged (when the number of APs is
reduced) or split (when increased). The choice of codepoints for merging or
splitting will be on the basis of the rate performance associated with the
codepoints.

It should be noted that in general, TSVQ does not find the closest AP to each
user and there is a small decrease expected in the performance of TSVQ (in
terms of mean squared error) when compared to standard VQ.

_TSVQ Algorithm_. In our implementation of TSVQ for cell-free AP placement, we
limit ourselves to balanced binary trees, i.e., at each stage, every
intermediate codepoint is split into two codepoints, to favor the lowest
complexity. The TSVQ algorithm is outlined in Algorithm 2. The stages are
indexed by $j$ (root node is stage 0) and the set of codepoints is represented
by $\mathcal{P}_{j}$ for stage $j$. The codepoints are indexed by $i$, and the
number of codepoints increases at each stage. The codepoints generated after
the algorithm converges are the required AP positions. $\mathcal{R}$ indicates
the set of all users while $\mathcal{R}_{i}$ denotes the set of users
associated with codepoint $i$. The splitting of codepoints is performed by
generating two perturbations of the original codepoint.

Algorithm 2 Tree-Structured Vector Quantization Algorithm

1:Initialize $\mathcal{P}_{0}$ with the codepoint of all users in
$\mathcal{R}$.

2:Split the codepoint(s) in $\mathcal{P}_{j}$ into two.

3:Apply Lloyd algorithm (Algorithm 1) to each split pair for training data
$\mathcal{R}_{i}$ at stage $j$.

4:Partition $\mathcal{R}_{i}$ at stage $j$ into two sets corresponding to new
codepoints.

5:Update $\mathcal{P}_{j}$ with new codepoints.

6:Repeat from step 2 for next stage $j+1$ until $M$ codepoints are generated.

###  IV-D PDF Optimized VQ

So far, we have considered a full-scale version of VQ (the Lloyd algorithm)
and TSVQ, which involves hierarchical codebook generation. Both flavors of VQ,
however, come with disadvantages such as complexity, scalability, and
learnability which will be elaborated shortly. In this section, we outline the
probability density function optimized VQ (PDFVQ) procedure, first developed
in [59], for use in cell-free AP placement that can address these
shortcomings. Here, an efficient quantizer is generated by first estimating
the user distribution PDF using the expectation-maximization (EM) algorithm
[60] and assuming a Gaussian mixture model (GMM). Then, by leveraging both bit
allocation and transform coding, closed-form expressions are defined to
allocate bits to each cluster and along every dimension (x- and y-coordinates
of the 2-D user density) so that scalar quantizers can be used to generate the
required AP codebook.

_Why use PDFVQ?_ Although designed for high-dimensional source inputs, this
procedure addresses the shortcomings of the two prior VQ approaches in the
following manner.

  * •

_Complexity:_ In the two prior VQ approaches, the computational complexity and
memory required are high. In TSVQ, the Lloyd algorithm is applied a number of
times depending on the codebook splits performed to obtain the required number
of APs. The overall complexity of the Lloyd algorithm is $\mathcal{O}(2KMI)$,
where the factor ‘$2$’ is due to the 2-D user density and $I$ denotes the
number of iterations required for convergence. It is also observed that the
number of executed iterations $I$ increases with the number of APs $M$. In
addition to the simple expressions for AP allocations for each cluster and
dimension of the 2-D user density, through the use of scalar quantizers for
each dimension, PDFVQ enjoys lower complexity as the quantizers work on both a
lower number of users and APs, consequently performing a lower number of
iterations. Further, the scalar quantizers can also be implemented by means of
look-up tables [61] that determine the codebook. Thus, there exists a simple
closed-form mapping from user density to the codepoints (AP locations). The
memory requirements in PDFVQ are also small as a result of transform coding.

  * •

_Scalability:_ The two prior VQ solutions are not easily scalable with bit
rate, that is, the number of APs. For the Lloyd algorithm alone, the whole
procedure must be repeated. In the case of TSVQ, additional successive
splitting steps must be performed, with the number of times that the Lloyd
algorithm is implemented doubling at every step. In PDFVQ, when the number of
bits is changed, then the PDF estimation phase remains unchanged and the bit
allocation and codebook generation are readily obtained as closed-form
expressions exist. Thus, PDFVQ facilitates scalability, both with respect to
dimensions (of the source distribution) as well as the number of bits.

  * •

_Learnability:_ It is expected that the user density varies over time, in
which case the AP positions must be adapted to account for this change. Both
prior VQ schemes are not amenable to a learning environment as they cannot
adapt quickly to changes in the user distribution as well as number of APs,
and their computationally complex procedures must be repeated to optimize to
the new environment. In PDFVQ, when the source distribution changes, both the
density estimation and codebook generation steps must be implemented, however,
only the parameters of the density have to be learned (the EM algorithm is
easily updated using existing values). If the number of APs only is changed,
then the closed-form expressions (which are not altered) and scalar quantizers
alone are needed, resulting in computational savings, faster adaptability, and
enabling an ‘on-the-fly’ placement.

Due to the abovementioned advantages, PDFVQ is thus suitable for AP placement
both in an environment where the user distribution changes over time as well
as the large-scale massive MIMO scenario with a very high number of APs. It is
also worth noting that the PDFVQ procedure is independent of the chosen
distortion measure for the scalar quantizers, but for simplicity, we will
consider that the Lloyd algorithm is used.

_PDFVQ Algorithm_. The PDFVQ procedure from [59] applied to AP placement
(involving only two dimensions) is outlined in Algorithm 3 below. Note that in
this paper, we model the user distributions as GMMs, which avoids the need for
parameter estimation. The GMM considered is of the form

$f_{\mathbf{P}}(\mathbf{p})=\sum\limits_{l=1}^{L}p_{l}\mathcal{N}\left(\mathbf{p}|\boldsymbol{\mu}_{l},\boldsymbol{\Sigma}_{l}\right),$
(19)

where $L$ is the number of mixture components (clusters), and $p_{l}$,
$\boldsymbol{\mu}_{l}$, and $\boldsymbol{\Sigma}_{l}$ are the probability,
mean, and covariance matrix, respectively, of mixture component $l$. The total
number of bits available $b_{\text{tot}}$ is $\log_{2}M$. It is important to
note that in both step 1 and step 2, the resulting bit allocations $b_{l}$ and
$b_{l,j}$, respectively, are _not_ expected to be integers. The number of
levels computed in step 2 corresponding to the bit allocation $b_{l,j}$, that
is, $V_{l,j}=2^{b_{l,j}}$, is ultimately rounded off to the nearest integer,
and represents the number of APs. As ‘bit’ generally refers to an integer
quantity, to avoid subsequent confusion, we omit the word ’bit’ when
discussing allocations. The codebooks generated in step 3 for each cluster are
the required AP locations.

Algorithm 3 PDF Optimized Vector Quantization Algorithm

1:Determine the allocation $b_{l}$ to cluster $l$ given the total budget
$b_{\text{tot}}$ using
$2^{b_{l}}=2^{b_{\text{tot}}}\frac{\sqrt{p_{l}c_{l}}}{\sum\limits_{j=1}^{L}\sqrt{p_{l}c_{l}}},$
(20) where $c_{l}=\sqrt{\lambda_{l,1}\lambda_{l,2}}$,
$\boldsymbol{\lambda}_{l}=\textrm{diag}(\lambda_{l,1},\lambda_{l,2})$, and
$\boldsymbol{\Sigma}_{l}=\mathbf{Q}_{l}\boldsymbol{\lambda}_{l}\mathbf{Q}_{l}^{T}$
is the eigen value decomposition.

2:With each cluster $l$, compute the allocation along each dimension
$b_{l,j}$, $j=1,2$, using
$b_{l,j}=\frac{b_{l}}{2}+\frac{1}{2}\log\left[\frac{\lambda_{l,j}}{c_{l}}\right].$
(21) Compute and round off the corresponding level $V_{l,j}=2^{b_{l,j}}$.

3:Generate the codebook $\mathcal{R}_{l}$ for each cluster using
$\mathcal{R}_{l}=\left\\{\mathbf{q}|\mathbf{q}=\mathbf{Q}_{l}\mathbf{y}+\boldsymbol{\mu}_{l},\mathbf{y}\in\mathcal{T}_{l}\right\\},$
(22) where $\mathcal{T}_{l}$ is the set of vectors given by the Cartesian
product $\mathcal{T}_{l}=\mathcal{S}_{l,1}\times\mathcal{S}_{l,2}$, with
$\mathcal{S}_{l,j}$, $j=1,2$ being the optimal $V_{l,j}$-level scalar
quantizer of a univariate Gaussian with variance $\lambda_{l,j}$.

###  IV-E Gradient Approaches

As alluded to above, the VQ approaches do not explicitly solve the sum rate
and minimum rate maximization problems. They, however, provide reasonably good
starting points to apply gradient ascent for both problems. Accordingly, we
present both of the gradient calculations below, and are called the max-sum
and max-min gradient, respectively.

####  IV-E1 Max-sum gradient

To maximize sum rate, the gradient update expression with $j$ as the iteration
index is

$\mathbf{q}_{m}^{(j+1)}=\mathbf{q}_{m}^{(j)}+\delta\frac{\partial}{\partial\mathbf{q}_{m}^{(j)}}\left\\{\sum\limits_{k=1}^{K}\log\left(1+\sum\limits_{m=1}^{M}\beta_{mk}\right)\right\\},\
\forall m,$ (23)

where in the objective function from (14), we have neglected the shadow fading
$z_{mk}$ and replaced the expectation with the sum over the users. The
gradient in (23) is calculated as

$\frac{\partial}{\partial\mathbf{q}_{m}^{(j)}}\left\\{\log\left(1+\sum\limits_{m=1}^{M}\beta_{mk}\right)\right\\}\\\
=\frac{\gamma}{2}\sum\limits_{k=1}^{K}\frac{1}{1+\psi_{k}^{\text{ZF}}}\frac{(\mathbf{p}_{k}-\mathbf{q}_{m}^{(j)})}{||\mathbf{p}_{k}-\mathbf{q}_{m}^{(j)}||^{\gamma+2}}.$
(24)

####  IV-E2 Max-min gradient

For minimum rate, taking the gradient of the rate of the worst user is fragile
since the absolute value of the minimum rate can vary significantly across the
iterations of gradient ascent causing convergence issues. Additionally, we are
also interested in evaluating the performance of the cell-free system in terms
of the 95%-likely rate, which quantifies the best rate of the worst 5% of the
users. Accordingly, we consider the sum of rates corresponding to the worst 5%
of the users, represented by $\mathcal{K}_{5\%}$, as

$\mathbf{q}_{m}^{(j+1)}=\mathbf{q}_{m}^{(j)}+\delta\frac{\partial}{\partial\mathbf{q}_{m}^{(j)}}\left\\{\sum\limits_{k\in\mathcal{K}_{5\%}}\log\left(1+\sum\limits_{m=1}^{M}\beta_{mk}\right)\right\\},\
\forall m,$ (25)

which uses the same simplifications as in (23), and the gradient is

$\frac{\partial}{\partial\mathbf{q}_{m}^{(j)}}\left\\{\log\left(1+\sum\limits_{m=1}^{M}\beta_{mk}\right)\right\\}\\\
=\frac{\gamma}{2}\sum\limits_{k\in\mathcal{K}_{5\%}}\frac{1}{1+\psi_{k}^{\text{ZF}}}\frac{(\mathbf{p}_{k}-\mathbf{q}_{m}^{(j)})}{||\mathbf{p}_{k}-\mathbf{q}_{m}^{(j)}||^{\gamma+2}}.$
(26)

##  V Simulation Methodology and Results

###  V-A Simulation Parameters

In a geographical area of dimensions $2\text{ km}\times 2\text{ km}$, we
consider $M=32$ APs and $K=4$ users, since $M\gg K$ for a cell-free system.
For the purposes of placement, however, we use $2000$ users that are
distributed according to a GMM of the form in (19), with covariance
$\boldsymbol{\Sigma}_{l}=\sigma^{2}_{l}\mathbf{I}$, where $\sigma_{l}$ is the
standard deviation of mixture component $l$ and $\mathbf{I}$ is the identity
matrix. The GMM used in our simulations has parameters $L=3$,
$\boldsymbol{\mu}_{1}=[0.5,-0.5]^{T}$, $\boldsymbol{\mu}_{2}=[0,0.5]^{T}$,
$\boldsymbol{\mu}_{3}=[-0.5,0]^{T}$, $\sigma_{1}=\sigma_{2}=\sigma_{3}=100$,
$p_{1}=0.6$, and $p_{2}=p_{3}=0.2$. The pathloss model from [54, (4.34)] is
used with shadow fading ignored for simplicity and the transmit power
$\rho_{r}$ of the users is increased from $5$ to $30$ dB. The max-sum and max-
min gradient methods utilize step sizes of $\delta=10^{3}$ and $\delta=3\times
10^{4}$, respectively.

###  V-B Performance Measures

The per-user achievable rate is used, and is defined for user $k$ as

$R_{k}=\mathbb{E}\left\\{\log_{2}\left(1+\psi_{k}^{\text{ZF}}\right)\right\\},$
(27)

with $\psi_{k}^{\text{ZF}}$ from (6) and the rate values are generated using
Monte Carlo iterations. Further, algorithms are run multiple times and the
solution that yields the best result is chosen. The maximum number of
iterations for the Lloyd algorithm is set to $50$. For comparison among the
algorithms, both the sum rate and the 95%-likely rate measures are used. The
relative performance between algorithms (say, algorithm 2 over algorithm 1)
can be calculated by using the following measure expressed as percentage

$\text{Improvement Ratio}=\frac{\mathcal{P}^{\text{Algorithm
2}}-\mathcal{P}^{\text{Algorithm 1}}}{\mathcal{P}^{\text{Algorithm 1}}}\times
100,$ (28)

where $\mathcal{P}$ is either the sum rate or 95%-likely rate.

We note here that while the sum rate and minimum rate maximization problems
are solved by the max-sum and max-min algorithms, respectively, in [52], we do
not include the results here. As described in Section III-A, the max-sum
algorithm does not solve the sum rate problem accurately and generates a
colocated-like solution with a remarkably poor 95%-likely rate performance
(reflected in the numerical simulations of [52]). Further, in [53], we compare
the grid-based max-min performance with the Lloyd algorithm and show that
their rate performances are comparable. Even though a higher 95%-likely rate
is achievable with the max-min algorithm, the simplicity of the Lloyd
algorithm outweighs the complexity associated with the increased grid
resolution needed.

###  V-C Numerical Results

Figure 9: Final AP locations of the Lloyd algorithm, TSVQ, and PDFVQ at $\rho_{r}=30$ dB. Figure 10: Sum rate as a function of $\rho_{r}$ for the Lloyd algorithm, TSVQ, and PDFVQ. Figure 11: 95%-likely rate as a function of $\rho_{r}$ for the Lloyd algorithm, TSVQ, and PDFVQ. TABLE I: Rate Improvement of TSVQ and PDFVQ Relative to the Lloyd Algorithm at $\rho_{r}=30$ dB Algorithm | Sum Rate | 95%-Likely Rate
---|---|---
TSVQ |  $3.82$% |  $-7.98$%
PDFVQ |  $2.78$% |  $-0.82$%

Experiment 1. We compare the performances of TSVQ and PDFVQ to the standard
Lloyd algorithm. Their final AP locations and the distribution of the 2000
users (as gray circles) are shown in Fig. 9, where the Lloyd algorithm results
in APs that are more distributed than TSVQ. This is expected since the TSVQ
algorithm successively splits Voronoi regions and after completion does not
necessarily associate each user to its closest AP. For PDFVQ, the allocation
procedure within each cluster results in 3.85, 2.93, and 2.93 APs along each
dimension, respectively, since the GMM clusters are symmetric along both x-
and y-coordinates. Although rounding off these values would result in 4, 3,
and 3 APs along each coordinate, the total number of APs would then be 34.
Thus, to limit to $M=32$ APs, we select 4, 2, and 2 APs along the x-coordinate
and 4, 4, and 4 APs along the y-coordinate. This combination is selected as it
provides the best result through repeated trials. The AP locations are
observed to be more regular and grid-like due to the scalar quantizers used in
the transform coding design, when compared to the standard Lloyd algorithm or
TSVQ. The sum and 95%-likely rates of the three VQ-based algorithms are
plotted in Fig. 10 and Fig. 11, respectively. The improvements in sum and
95%-likely rates, expressed as percentages, of TSVQ and PDFVQ over the Lloyd
algorithm at transmit power $\rho_{r}=30$ dB are noted in Table I. It is seen
that while the sum rate of TSVQ is greater than that of the Lloyd algorithm
(by nearly $4$%), the 95%-likely rate is worse (by nearly $8$%). For PDFVQ,
the sum rate offered is close to that of TSVQ while the 95%-likely rate
approaches but does not equal that of the Lloyd algorithm (with the difference
being less than $1$%), due to the fact that the AP positions are far away from
a small group of users in two mixture components. While for the user
distribution considered PDFVQ offers a solution providing higher sum rate and
similar 95%-likely rate to the Lloyd algorithm, it can be expected that this
performance will reduce as the variance of the mixture components is
increased.

Experiment 2. In Experiment 1, we compared the VQ approaches for a GMM
consisting of Gaussian mixture components with spherical (proportionate to the
identity matrix) covariance matrices. In order to show an example of PDFVQ
applied to a full (non-diagonal) covariance matrix, we now consider a user
distribution where the second component of the GMM considered is modified to
have a covariance matrix of
$\boldsymbol{\Sigma}_{2}=\begin{bmatrix}\sigma^{2}&2\sigma^{2}/3\\\
2\sigma^{2}/3&2\sigma^{2}\end{bmatrix}$ with $\sigma=100$. When PDFVQ is
applied to this distribution, $4$, $2$, $3$ APs along the x-coordinate and
$4$, $5$, and $3$ APs along the y-coordinate are allocated, totaling $35$ APs.
The best allocation for the desired $32$ APs is found to be $4$, $2$, and $4$,
and $4$, $4$, and $2$ APs along the x- and y-coordinates, respectively. The
positions of such APs along with those of the Lloyd algorithm and TSVQ are
shown in Fig. 12. The sum rate and 95%-likely rates corresponding to these
locations are provided in Fig. 13 and Fig. 14, respectively, and the rate
improvements are tabulated in Table II. Similar to the GMM with spherical
Gaussians in Experiment 1, we observe that PDFVQ is able to match the sum rate
performance of TSVQ. However, unlike above, the 95%-likely rate, like TSVQ, is
lower than the Lloyd algorithm by nearly $13$% since the space-filling
advantage of the Lloyd algorithm is lost. Based on the above two experiments,
it could be concluded that PDFVQ is a reasonable alternative to TSVQ that
provides a similar or superior performance.

Although it is not the case here, we also note that the clusters of the GMM
may be placed quite close to one another or may even overlap marginally. When
PDFVQ is applied, the APs associated with the overlapping clusters may be
closely placed and is inefficient. In such a scenario, starting from the
cluster pair with most overlap (the degree of which can be computed by a
measure such as overlap rate [62]), the AP allocation of the cluster with the
higher 95%-likely rate could be decreased and re-allocated to another cluster
with a poor performance. This process can be repeated for all overlapping
clusters to improve the 95%-likely rate of the network.

Figure 12: Final AP locations of the Lloyd algorithm, TSVQ, and PDFVQ with a full covariance matrix at $\rho_{r}=30$ dB. Figure 13: Sum rate as a function of $\rho_{r}$ for the Lloyd algorithm, TSVQ, and PDFVQ with a full covariance matrix. Figure 14: 95%-likely rate as a function of $\rho_{r}$ for the Lloyd algorithm, TSVQ, and PDFVQ with a full covariance matrix. TABLE II: Rate Improvement of TSVQ and PDFVQ Relative to the Lloyd Algorithm With a Full Covariance Matrix at $\rho_{r}=30$ dB Algorithm | Sum Rate | 95%-Likely Rate
---|---|---
TSVQ |  $5.59$% |  $-13.67$%
PDFVQ |  $4.98$% |  $-12.85$%

Experiment 3. In this experiment, we use the gradient approaches outlined in
Section IV-E to improve the sum rate and 95%-likely rate performances shown in
Experiment 2 for the GMM with a full covariance matrix. First, we show that
using the max-sum gradient ascent, we can increase the sum rates of each of
the Lloyd, TSVQ, and PDFVQ performances, shown in Fig. 15. Table III lists the
percentage rate improvements of the max-sum gradient applied to each VQ
approach, over the Lloyd algorithm alone. The gradient ascent applied to both
the TSVQ and PDFVQ AP locations yields the highest sum rate (nearly $10$% over
the Lloyd algorithm) while the rate obtained when the ascent operation is
applied with the Lloyd algorithm as the starting point is nearly the same as
the sum rates of PDFVQ and TSVQ. This occurrence is due to the fact that the
gradient ascent iterations with the Lloyd AP solutions as the initial points
converge to a local optimum which is different from that obtained when the
ascent is applied to PDFVQ or TSVQ. The 95%-likely rate performances shown in
Fig. 16, when the max-sum gradient is applied to PDFVQ and TSVQ, do not change
significantly (an increase is observed at $\rho_{r}=30$ dB) and decrease when
the gradient is applied to the Lloyd AP positions. Thus, in terms of sum rate,
the PDFVQ provides the best solution out of all VQ approaches and a further
increase in sum rate (about $4$% over PDFVQ) without negatively affecting the
minimum rate performance (about $5$% increase) is achieved by using the max-
sum gradient ascent. Next, the max-min gradient ascent is applied and the
95%-likely rate is observed to increase as shown in Fig. 17. The best rate is
obtained when the ascent algorithm is applied to the Lloyd solution as opposed
to when it is applied to either TSVQ or PDFVQ where it is able to match the
performance of the Lloyd algorithm. Table IV informs us that while a $14$%
improvement in the 95%-likely rate is achieved by applying the gradient to the
Lloyd solution, the difference of the gradient applied to TSVQ or PDFVQ from
the Lloyd algorithm is only up to $4$%. The sum rates corresponding to the
95%-likely rates are plotted in Fig. 18, where as a result of the 95%-likely
rate improvement, the sum rate when the ascent is applied to the Lloyd
algorithm is the least value. In contrast, despite the increase in 95%-likely
rate, the sum rate performances when ascent is applied to TSVQ and PDFVQ are
nearly the same as TSVQ and PDFVQ itself. Hence, with its simpler design,
PDFVQ along with max-min gradient offers a good tradeoff between a 95%-likely
rate similar to and a sum rate higher (by over $5$%) than the Lloyd algorithm.
It should, however, be mentioned that if 95%-likely rate (or minimum rate) is
the sole performance measure of interest, the Lloyd algorithm alone is a
straightforward choice. The gradient approach requires the choice of an
appropriate step size for convergence while the Lloyd algorithm is known to
converge [63].

Figure 15: Sum rate as a function of $\rho_{r}$ for the Lloyd algorithm, TSVQ, and PDFVQ with a full covariance matrix along with max-sum gradient. Figure 16: 95%-likely rate as a function of $\rho_{r}$ for the Lloyd algorithm, TSVQ, and PDFVQ with a full covariance matrix along with max-sum gradient. TABLE III: Rate Improvement of the VQ Approaches with the Max-Sum Gradient Relative to the Lloyd Algorithm at $\rho_{r}=30$ dB Algorithm | Sum Rate | 95%-Likely Rate
---|---|---
Lloyd + Max-Sum Gradient |  $3.88$% |  $-5.60$%
TSVQ + Max-Sum Gradient |  $9.80$% |  $-7.52$%
PDFVQ + Max-Sum Gradient |  $9.34$% |  $-8.66$%
Figure 17: 95%-likely rate as a function of $\rho_{r}$ for the Lloyd algorithm, TSVQ, and PDFVQ with a full covariance matrix along with max-min gradient. Figure 18: Sum rate as a function of $\rho_{r}$ for the Lloyd algorithm, TSVQ, and PDFVQ with a full covariance matrix along with max-min gradient. TABLE IV: Rate Improvement of the VQ Approaches with the Max-Min Gradient Relative to the Lloyd Algorithm at $\rho_{r}=30$ dB Algorithm | 95%-Likely Rate | Sum Rate
---|---|---
Lloyd + Max-Min Gradient |  $13.93$% |  $-3.73$%
TSVQ + Max-Min Gradient |  $-3.60$% |  $5.73$%
PDFVQ + Max-Min Gradient |  $-2.48$% |  $5.30$%

Experiment 4. In this experiment, we quantify the effect of a time-varying
user density on the network performance and show the need for an easily
adaptable AP placement algorithm. We consider a simple situation where the
users are _initially_ distributed as a single-cluster GMM ($L=1$ in (19)) with
mean $\boldsymbol{\mu}=[0,0]^{T}$ and covariance
$\boldsymbol{\Sigma}=\begin{bmatrix}\sigma^{2}&\sigma^{2}/3\\\
\sigma^{2}/3&\sigma^{2}/2\end{bmatrix}$ where $\sigma=200$, and call it
density A. Assuming a total of $M=18$ APs, we calculate the PDFVQ AP locations
for this user density. Over time, we consider that the user distribution
changes to density B, where the users are more spread out along another
direction to that of density A, with covariance matrix
$\boldsymbol{\Sigma}=\begin{bmatrix}\sigma^{2}/2&\sigma^{2}/2\\\
\sigma^{2}/2&\sigma^{2}\end{bmatrix}$ and $\sigma=300$. These two user
densities and the PDFVQ AP locations determined for density A are shown in
Fig. 19. We evaluate the sum rate and 95%-likely rate performances when the AP
locations are matched to density A and consider both user densities A and B,
so that the loss due to mismatch of the AP locations to the user density B is
also shown. For completeness, we also compute the rate values when PDFVQ AP
locations are determined and matched to density B. As expected, in both the
sum rate plotted in Fig. 20 and the 95%-likely rate plotted in Fig. 21, there
is a significant loss in performance for the PDFVQ AP locations which are
matched to density A, when the users re-position to density B. The relative
losses in the sum rate and 95%-likely rate at $\rho_{r}=30$ dB are $32.67$%
and $89.54$%, respectively. To prevent this performance decrease, there is a
need to re-calculate the AP locations for the new user density. Clearly, when
the AP locations are matched via PDFVQ to the user density B, the rate
performances in Fig. 20 and Fig. 21 are improved over the diminished
performance of the mismatched case. With easy adaptability and the other
aforementioned advantages, PDFVQ offers the best method for cell-free AP
placement among the VQ techniques discussed in this paper. In future work, we
will address how the changing user densities can be learned for use in PDFVQ.

(a) Density A.

(b) Density B.

Figure 19: The two user densities A and B considered for Experiment 4 along
the PDFVQ AP locations matched to density A. Figure 20: Sum rate as a function
of $\rho_{r}$ for Experiment 4. Figure 21: 95%-likely rate as a function of
$\rho_{r}$ for Experiment 4.

##  VI Conclusion

In this paper, we have addressed access point (AP) placement in cell-free
massive multiple-input-multiple-output (MIMO) systems under a throughput
criteria. We investigated the two main optimization problems in this regard,
namely the sum rate and minimum rate maximization problems. To understand
their solution frameworks, simple examples were constructed and analyzed
exposing the difficulty in solving the problems. Therefore, as a practical
approach, the use of vector quantization (VQ)-based methods, namely the
popular Lloyd algorithm, tree-structured VQ (TSVQ), and probability density
function optimized VQ (PDFVQ), to cell-free AP placement, was investigated.
Among the three algorithms presented, although the tree-structured VQ (TSVQ)
provides better sum rate (as it fosters cooperation among APs by placing them
closer) compared to the Lloyd algorithm, it suffers from high complexity, poor
scalability, and the inability to easily adapt to new environments. PDFVQ,
which overcomes the aforementioned shortcomings, allowed a more efficient
generation of the codebook and generated a sum rate similar to and 95%-likely
rate higher than TSVQ and close to the Lloyd algorithm. Additionally, for
gradient-based maximization methods, PDFVQ is found to provide good initial
points. It was observed numerically that, over the Lloyd algorithm, an
increase of $9$% in sum rate and a difference of just $2.5$% in the 95%-likely
rate was achieved by applying max-sum and max-min gradient ascent algorithms,
respectively, with the PDFVQ AP locations as starting points. Thus, PDFVQ
offers a convenient, less computationally intensive, and easily scalable AP
placement technique for cell-free networks.

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