# Enabling Cell-Free Massive MIMO Systems with Wireless Millimeter Wave
Fronthaul

Umut Demirhan and Ahmed Alkhateeb The authors are with the School of
Electrical, Computer and Energy Engineering, Arizona State University, Tempe,
AZ, 85287 USA. (Email: udemirhan, alkhateeb@asu.edu).

###### Abstract

Cell-free massive MIMO systems have promising data rate and uniform coverage
gains. These systems, however, typically rely on optical fiber based fronthaul
for the communication between the central processing unit (CPU) and the
distributed access points (APs), which increases the infrastructure cost,
leads to high installation time, and limits the deployment flexibility and
adaptability. To address these challenges, this paper proposes two
architectures for cell-free massive MIMO systems based on wireless fronthaul
that is operating at a higher-band compared to the access links: (i) A
wireless-only fronthaul architecture where the CPU has a wireless fronthaul
link to each AP, and (ii) a mixed-fronthaul architecture where the CPU has a
wireless link to each cluster of APs that are connected together via optical
fibers. These dual-band architectures ensure high-data rate fronthaul and
provide high capability to synchronize the distributed APs. Further, the
wireless fronthaul reduces the infrastructure cost and installation time, and
enhances the flexibility, adaptability, and scalability of the network
deployment. To investigate the achievable data rates with the proposed
architectures, we formulate the end-to-end data rate optimization problem
accounting for the various practical aspects of the fronthaul and access
links. Then, we develop a low-complexity yet efficient joint beamforming and
resource allocation solution for the proposed architectures based on user-
centric AP grouping. With this solution, we show that the proposed
architectures can achieve comparable data rates to those obtained with optical
fiber-based fronthaul under realistic assumptions on the fronthaul bandwidth,
hardware constraints, and deployment scenarios. This highlights a promising
path for realizing the cell-free massive MIMO gains in practice while reducing
the infrastructure and deployment overhead.

##  I Introduction

Can we realize the cell-free massive MIMO gains with wireless fronthaul?  This
paper attempts to answer this important question. Cell-free massive MIMO
systems are promising uniform coverage and high data rate gains, even for
scenarios with very dense users [1, 2]. A critical challenge, however, with
the current cell-free massive MIMO systems is its reliance on optical fiber-
based fronthaul for distributing data and synchronization signals from the
central processing unit (CPU) to the access points (APs). This highly
increases its infrastructure cost and installation time, and limits the
flexibility and scalability of the cell-free massive MIMO deployment. To
address these challenges, the paper proposes an alternative cell-free massive
MIMO architecture with higher-band fronthaul, e.g., a millimeter wave (mmWave)
or terahertz (THz) fronthaul for a sub-6GHz cell-free massive MIMO system. The
use of higher frequency band fronthaul has two key advantages: (i) The large
bandwidth available at the higher frequency bands provides a high data rate
fronthaul and (ii) the higher-band fronthaul signals (lower wavelength) have
high capability to synchronize the sub-6GHz AP transceivers. For this proposed
architecture, the paper investigates whether it can achieve comparable achieve
rates to those obtained by classical fiber-fronthaul based cell-free massive
MIMO systems.

###  I-A Related Works

Motivated by its potential to increase the data rate and manage the multi-user
interference, the idea of having distributed antennas simultaneously serving
the same users have been previously investigated in the network multiple-input
multiple-output (MIMO) [3], distributed MIMO [4], and coordinated multipoint
with joint transmission (CoMP-JT) [5, 6]. These earlier approaches did not
scale well with the increasing number of users and large number of access
points (APs), due to the channel estimation and feedback overhead. To overcome
this limitation, in [1, 7, 8, 9, 10], the authors investigated cell-free
massive MIMO, where reciprocity of the time-division duplexing (TDD) is
leveraged to estimate the downlink channels directly from the joint uplink
pilot transmissions. In the proposed cell-free optimization framework, the
imperfection of channel state information (CSI) was taken into account, and
only the long-term channel coefficients were utilized for the power
allocation. Despite the interesting data rate and coverage gains, the
realization of cell-free massive MIMO networks suffers from the limitations
and high cost of the CPU-AP fiber links. To reduce cost, cheaper wire-link
alternatives, with limited capacity, were considered. For example, [11]
investigated cell-free performance with quantized fronthaul transmissions.
However, the approaches in [11, 12, 13, 14] did not eliminate the wired
connection requirement, which is still associated with high infrastructure
cost, high installation time, and limited deployment flexibility.

Another relevant line of work is presented in the different network
architectures that adopted wireless fronthaul/backhaul such as small-cells
[15, 16], ultra dense networks [17], heterogeneous networks [18] and the cloud
radio access networks (C-RAN) [19, 20, 21]. In [15, 16, 17, 18], wireless
fronthaul was adopted to show the potential advantages of the mmWave backhaul.
This was however, limited to scenarios with no cooperation between
basestations. In C-RAN architectures, multiple (normally a few) base stations
coordinate to serve the users. For example, in [19], the time-frequency
resources of an OFDMA based C-RAN system consisted of a few base stations were
optimized to maximize the weighted sum-rate of large number of users. The
solution in [19] assumed the fronthaul transmission to a single base station
at a time. To overcome this limitation, the work in [20] proposed a multicast
beamforming for the downlink ultra dense C-RAN, and designed user-centric
clusters of the base stations for the transmissions. The work in [19, 20],
however, was limited to architectures with a few number of base stations. This
is partially due to the large channel estimation/feedback overhead associated
with the solutions in [19, 20], which limit them from scaling to large number
of antennas. This motivates the research for new approaches to enable the
potential gains of cell-free massive MIMO systems while relaxing the
requirement of wired fronthaul.

###  I-B Contributions

In this work, we propose a new architecture for cell-free massive MIMO systems
where the fronthaul is implemented using high-frequency (e.g., mmWave)
wireless links to serve access links at lower frequency (e.g., at a sub-6GHz
band). The proposed architecture has two key motivations: (i) The larger
available bandwidth at the high frequency band ensures high data rates for the
fronthaul and (ii) the small wavelength of the high-frequency signals provides
high synchronization accuracy for the access points that are operating at a
lower frequency band. Further, the wireless fronthaul enables a modular,
flexible and scalable architecture with low infrastructure cost and low
installation time. With all these potential gains, an important question is
whether this architecture is capable of achieving comparable data rate gains
to those achieved with optical fiber-based cell-free massive MIMO systems? To
answer this question, we develop an efficient communication scheme, analyze
its performance, and draw important insights about the proposed architecture
and data rate optimization approaches. The main contributions of the paper can
be summarized as follows:

  * •

Proposing an efficient architecture for cell-free massive MIMO systems based
on higher-frequency wireless fronthaul. The proposed architecture has the
potential of ensuring higher data rate fronthaul and high synchronization
accuracy for the distributed APs. Further, it requires low infrastructure cost
and low installation time and provides interesting flexibility and
adaptability gains for cell-free massive MIMO systems.

  * •

Developing an efficient communication model for the wireless fronthaul based
cell free massive MIMO system. This model assigns a group of APs to each user
and optimizes the multicast beamforming at the central processing unit to
simultaneously serve the AP group of each user. The adopted system model
accounts for the practical constraints on the higher-frequency band (mmWave)
beamforming architectures [22].

  * •

Formulating the end-to-end data rate maximization problem for the proposed
architecture and communication model. The optimization problem takes the
wireless fronthaul, access channels, and AP grouping into account. In
particular, it aims to determine the user-centric AP group selection,
fronthaul beamforming vectors, group time-division multiple-access (TDMA)
schedule, and AP power coefficients to maximize the end-to-end data rates.

  * •

Developing near-optimal end-to-end data rate maximization solution for the
proposed architecture. The solution adopts an iterative group selection
algorithm, which is coupled with the fronthaul and access channel data rate
maximization sub-problems. Specifically, in each iteration, the group size and
AP selection are determined based on the channel estimates and then the
fronthaul/access rates are optimized for the given groups.

  * •

Extending the architecture and developed solutions to the mixed wireless/wired
fronthaul case. A mixed-fronthaul architecture with wire-connected AP clusters
(for example through a radio stripe [2]) is proposed. In this architecture,
only the leader AP in each wire-connected cluster has a wireless fronthaul
link with the CPU. The proposed data rate maximization solutions are
generalized to account for the mixed-fronthaul.

The proposed solutions are extensively evaluated using numerical simulations
which draw important insights into the performance of the proposed cell-free
architectures. Based on the results, the high-frequency wireless fronthaul can
provide sufficient data-rates for the cell-free massive MIMO by taking
advantage of the larger bandwidth availability. With the fully wireless
architecture, it is possible to achieve data rates with only $10-20\%$
degradation compared to the fiber fronthaul based cell-free massive MIMO
architectures. Further, the mixed-fronthaul architecture significantly reduces
the bandwidth requirements and improves the data-rates. In our simulations,
the mixed-fronthaul architecture enabled data rates very similar to the fiber
fronthaul based solutions with reasonable fronthaul bandwidth requirements.

Notation: We use the following notation throughout this paper: ${\mathbf{A}}$
is a matrix, ${\mathbf{a}}$ is a vector, $a$ is a scalar, $\mathcal{A}$ is a
set. $|{\mathbf{A}}|$ is the determinant of ${\mathbf{A}}$, whereas
${\mathbf{A}}^{T}$, ${\mathbf{A}}^{H}$, ${\mathbf{A}}^{*}$,
${\mathbf{A}}^{-1}$, ${\mathbf{A}}^{\dagger}$ are its transpose, Hermitian
(conjugate transpose), conjugate, inverse, and pseudo-inverse respectively.
$\lvert\mathcal{A}\rvert$ is the cardinality of $\mathcal{A}$.
$\left\lVert{\mathbf{a}}\right\rVert$ is the $l_{2}$ norm of ${\mathbf{a}}$.
${\mathbf{I}}$ is the identity matrix.
$\mathcal{CN}({\mathbf{m}},{\mathbf{R}})$ is a complex Gaussian random vector
with mean ${\mathbf{m}}$ and covariance ${\mathbf{R}}$.
${\mathbb{E}}\left[\cdot\right]$ is used to denote expectation.
$\mathbbm{1}_{\\{\mathrm{condition}\\}}$ is the indicator function of the
condition given in the subscript.

##  II Wireless-Fronthauled Dual-Band Cell-Free Massive MIMO: The Key Idea

Cell-free massive MIMO is a promising enabler for the high data rate and
coverage requirements in future wireless communications systems. Current cell-
free massive MIMO architectures, however, assume that all the antennas are
connected to the central processing unit with wired fronthaul, that is
typically optical fiber [23]. This complicates the deployment process of these
systems and increases its cost and installation time. To overcome these
challenges, we propose an alternative architecture for cell-free massive MIMO
systems that relies on a wireless fronthaul operating at a much higher
frequency band compared to the access band, as depicted in Fig. 1. For
example, a mmWave fronthaul channel with sub-6GHz access channels or a
terahertz fronthaul supporting mmWave access. Thanks to the wireless
fronthaul, the proposed architecture is modular, flexible, scalable, with low
installation cost and time. The use of a much higher frequency band in the
fronthaul compared to the access band has the following advantages (for the
rest of the paper, we will assume a mmWave fronthaul and a sub-6GHz access)

  * •

High Data-Rate: The availability of the large bandwidth at the mmWave band
enables the central processing unit (which is equipped with a mmWave
transceiver) to support high data-rate fronthaul. This is crucial for
maintaining the high data rate gains of cell-free massive MIMO systems, as
will be shown in Section VII.

  * •

High Synchronization Accuracy: Using a mmWave fronthaul (where signals have a
small wavelength) to synchronize the sub-6GHz APs (where signals have a much
higher wavelength) has the potential of ensuring precise clock synchronization
among the APs, which is essential for the operation of the cell-free massive
MIMO systems. Specifically, the synchronization of the APs can be achieved
over the mmWave fronthaul with the aid of the CPU via master-slave type
algorithms (e.g., [24]) or via network centric solutions (e.g., [25]). The APs
use two clock signals, one for the mmWave fronthaul and the other for the
sub-6GHz access channel. The APs can synchronize their mmWave clocks and time
by referring to the CPU. These signals can then be utilized for more accurate
sub-6GHz synchronization by converting the mmWave (smaller wavelength) signal
to a large wavelength clock111For instance, if there are two mmWave clocks
that are synchronized with a $\Delta f$ frequency offset between them, the
sub-6GHz clocks obtained from the these (or from the same source) can
translate into a smaller frequency difference, possibly at the ratio of the
carrier frequencies (e.g., for $30$ GHz and $3$ GHz carrier frequencies, it
could be possible to obtain the clocks with frequency difference $\frac{\Delta
f}{10}$). A similar gain is expected if a THz-based fronthaul is leveraged to
synchronize a mmWave access channel. .

Figure 1: An illustration of the proposed architecture where a mmWave CPU
provides fronthaul to the different sets of user-centric AP groups via
different beamforming vectors. The wireless APs jointly serve their users
through a cell-free massive MIMO access channel at sub-6GHz.

Given these interesting gains of having a wireless mmWave fronthaul based
cell-free massive MIMO architecture, such as the one shown in Fig. 1, an
important question: Can this wireless fronthaul based architecture achieve the
same data rates of the fiber fronthaul based cell-free massive MIMO
architectures? In this paper, we try to answer this question. Towards this
objective, we first describe the system model of the proposed mmWave fronthaul
based cell-free massive MIMO architecture in Section III and formulate the
end-to-end (from the CPU to the users) achievable rate optimization problem in
Section IV. Then, we develop an efficient transmission strategy for the
proposed architecture in Section V and discuss some important extensions of
the proposed architecture in Section VI. Finally, the achievable rate of the
proposed architecture and comparisons with the fiber-based solutions are
provided in Section VII.

##  III System Model

We consider the distributed massive MIMO system in Fig. 1, where a CPU
communicates with $M$ wireless-APs over a high-frequency (e.g., mmWave)
wireless fronthaul and the wireless-APs serve $K$ user equipment (UEs) over a
low-frequency (e.g., sub-6GHz) channel. We will use
$\mathcal{M}=\\{1,\ldots,M\\}$ and $\mathcal{K}=\\{1,\ldots,K\\}$ to denote
the sets of $M$ APs and $K$ UEs. For the rest of the paper, we will refer to
the downlink channel from the CPU to the APs as the fronthaul channel and the
downlink channel from APs to the UEs as the access channel. Further, for the
ease of exposition and without loss of generality, we will assume that the
fronthaul channel is operating over a mmWave band, with a bandwidth
$B^{\mathrm{fh}}$, and the access channel is adopting a sub-6GHz band, with a
bandwidth $B^{\mathrm{fh}}$. The algorithms and results of the paper, however,
can be applied to other dual-band architectures, such as terahertz fronthaul
with a mmWave access channel.

To beamform the signal to the APs, the CPU employs an antenna array of $N$
elements, while the APs and UEs are, for simplicity, assumed to have single
antennas. We consider a user-centric grouping approach where the message of UE
$k$ is jointly transmitted by a subset of APs,
$\mathcal{G}_{k}\subseteq\mathcal{M}$. We denote the set of the user-centric
groups as
$\boldsymbol{\mathcal{G}}=\\{\mathcal{G}_{1},\ldots,\mathcal{G}_{K}\\}$. Note
that different groups may include the same APs since multiple APs are utilized
in the transmission to every UE. For that, we will also define
$\mathcal{U}_{m}\subseteq\mathcal{M}$ for $m\in\mathcal{M}$ as the set of
users that are being served by the AP $m$. For the distances between the CPU
and $m$-th AP and between the $m$-th AP and $k$-th user, we will use
$d^{\mathrm{fh}}_{m}$ and $d^{\mathrm{ac}}_{mk}$ to denote them. It is
important to mention here that we do not assume any knowledge about the
positions or distances between the CPU, APs, or the users.

At the Fronthaul Link: The mmWave CPU adopts TDMA to serve the $K$ user-
centric groups. In each TDMA slot, the CPU beamforms the signal towards the
APs that serve one user. The duration of the TDMA slot allocated for serving
APs of UE $k$ is denoted by $t_{k}$. Let ${\mathbf{h}}_{m}\in\mathbb{C}^{N}$
denote the channel between the CPU and the $m$th AP. If the message intended
for user $k$ is represented by $q^{\mathrm{fh}}_{k}\in\mathbb{C}$, with
${\mathbb{E}}[\left\lvert q_{k}^{\mathrm{fh}}\right\rvert^{2}]=1$, then the
received signal at the AP $m$ can be written as

$y^{\mathrm{fh}}_{m}=\sqrt{\rho^{\mathrm{fh}}}{\mathbf{h}}_{m}^{H}{\mathbf{f}}_{k}q^{\mathrm{fh}}_{k}+w^{\mathrm{fh}}_{m},$
(1)

where $\rho^{\mathrm{fh}}$ is the normalized fronthaul transmission power and
$w^{\mathrm{fh}}_{m}\sim\mathcal{CN}(0,1)$ is the receive noise at the $m$th
AP. The vector ${\mathbf{f}}_{k}\in\mathbb{C}^{N}$ is the CPU beamforming
vector intended to focus the signal to the APs that serve user $k$. To satisfy
the practical mmWave hardware constraints, we assume that the CPU adopts
analog-only beamforming implemented by a network of quantized phase shifters
[22]. This means that the beamforming vector ${\mathbf{f}}_{k}$ can only be
selected from a certain set of vectors, that we define by the codebook
$\boldsymbol{\mathcal{F}}$. If each phase shifter has $q$ bits, i.e., $2^{q}$
possible phase shift values defined by the set
$\mathcal{Q}\in\\{0,\frac{\pi}{2^{q}},\frac{2\pi}{2^{q}},...,\frac{(2^{q}-1)\pi}{{2^{q}}}\\}$,
then we can write

$\boldsymbol{\mathcal{F}}=\left\\{\frac{1}{\sqrt{N}}[e^{j\phi_{1}},\ldots,e^{j\phi_{N}}]^{T}:\phi_{n}\in\mathcal{Q},\forall
n\in\\{1,\ldots,N\\}\right\\}\textrm{ and
}{\mathbf{f}}_{k}\in\boldsymbol{\mathcal{F}}.$ (2)

We assume that the channels between the CPU and APs are available at the CPU.
This is motivated by the stationarity of the CPU and APs and the channel
reciprocity assumption.

At the Access Link: Each AP decodes the CPU signal received by its mmWave
receiver and prepares it for the transmission over the sub-6GHz access
channel. Since each AP $m$ is assumed to contribute in serving a set of users
$\mathcal{U}_{m}$, the transmitted signal from the $m$th AP, $x_{m}$, can be
written as

$x_{m}=\sqrt{\rho^{\mathrm{ac}}}\sum_{k\in\mathcal{U}_{m}}\sqrt{p_{mk}}f^{\mathrm{ac}}_{mk}q^{\mathrm{ac}}_{k},$
(3)

where $\rho^{\mathrm{ac}}$ is the shared APs coefficient for the transmit
power coefficient of the APs, $f_{mk}^{\mathrm{ac}}$ and $p_{mk}$ denote the
beamforming and power control coefficients of the $m$-th AP for the $k$-th UE,
and $q^{\mathrm{ac}}_{k}$ represents the intended message for the $k$-th UE
which satisfies ${\mathbb{E}}\left[\left\lvert
q^{\mathrm{ac}}_{k}\right\rvert^{2}\right]=1$. After all the APs prepare their
messages, they transmit them to their users. Note that all the APs are assumed
to maintain sufficient clock synchronization at the sub-6GHz access channel,
which is further facilitated by the adoption of mmWave-based synchronization,
as briefly highlighted in Section II . Now, if $g_{mk}$ denotes the access
channel coefficient between the $m$-th AP and $k$-th UE, the received signal
at the UE $k$ can be expressed as

$y^{\mathrm{ac}}_{k}=\sum_{m=1}^{M}x_{m}g_{mk}+w_{k},$ (4)

where $w_{k}\sim\mathcal{CN}(0,1)$ is the receive noise at user $k$.

Channel Models: Since the fronthaul link is operating over a mmWave band, and
given that the fixed-access deployment, we adopt a LOS channel between the CPU
and the APs [22, 26]. More specifically, we consider

${\mathbf{h}}_{m}=\sqrt{N\beta^{\mathrm{fh}}_{m}}\alpha^{\mathrm{fh}}_{m}{\mathbf{a}}(\theta_{m}),$
(5)

where $\beta_{m}^{\mathrm{fh}}$ represents the large-scale fading effects,
$\alpha_{m}^{\mathrm{fh}}$ the additional phase-shift due to the LOS path, and
${\mathbf{a}}(\cdot)\in\mathbb{C}^{L}$ is the array response vector function
that takes the azimuth angle of departure from the CPU to AP $m$, denoted by
$\theta_{m}$, as the input. For the access channel, we define the channel
coefficient between AP $m$ and UE $k$ as

$g_{mk}=\sqrt{\beta^{\mathrm{ac}}_{mk}}\alpha^{\mathrm{ac}}_{mk},$ (6)

where $\beta^{\mathrm{ac}}_{mk}$ and $\alpha^{\mathrm{ac}}_{mk}$ represent the
large- and small-scale fading coefficients. The small fading coefficients are
taken as i.i.d. complex Gaussian random variables, i.e.,
$\alpha^{\mathrm{ac}}_{mk}\sim\mathcal{CN}(0,1)$.

##  IV Problem Formulation: End-to-End Rate Optimization

In this section, we formulate the optimization problem of the end-to-end
achievable rate of the proposed wireless fronthaul based cell-free massive
MIMO system. We first present the achievable rates of the access and fronthaul
channels. Then, we formulate the end-to-end rate optimization problem. In the
following, we omit the superscripts denoting fronthaul and access channel
variables as they are easily distinguished from the context.

###  IV-A Achievable Rate of the Fronthaul

Considering the system model in Section III, the achievable rate of AP $m$,
that is part of the $k$th user group $\mathcal{G}_{k}$, can be written as

$R^{\mathrm{fh}}_{m}({\mathbf{f}}_{k})=\log\bigg{(}1+{\rho^{\mathrm{fh}}\left\lvert{\mathbf{h}}_{m}^{H}{\mathbf{f}}_{k}\right\rvert^{2}}\bigg{)},$
(7)

where ${\mathbf{f}}_{k}$ is the beamforming vector used by the CPU to serve
the set of APs in the group $\mathcal{G}_{k}$. Now, note that the message of
the $k$th user is simultaneously transmitted to all the APs in
$\mathcal{G}_{k}$, and that all the APs in the group should finish receiving
the message before they start transmitting it to the user. To incorporate
that, we define the effective rate of group $k$ as the minimum rate of the APs
in the $\mathcal{G}_{k}$’s group. Mathematically, we write

$R^{\mathrm{fh}}(\mathcal{G}_{k},{\mathbf{f}}_{k})=\min_{m\in\mathcal{G}_{k}}\big{\\{}R^{\mathrm{fh}}_{m}({\mathbf{f}}_{k})\big{\\}}.$
(8)

Moreover, due to the TDMA schedule of the AP groups, the group rates will be
further scaled by the TDMA time fractions. Let $t_{k}$ denote the fraction of
TDMA time allocated to group $\mathcal{G}_{k}$. Then, the time-scaled
fronthaul rate of group $k$ can be written as
$t_{k}R^{\mathrm{fh}}(\mathcal{G}_{k},{\mathbf{f}}_{k})$. We note that these
time fractions satisfy the constraints $0<t_{k}<1$ and
$\sum_{k=1}^{K}t_{k}=1$.

###  IV-B Achievable Rate of the Cell-Free Massive MIMO Access Channel

The access channel adopts the main assumptions of the cell-free massive MIMO
architectures in [1]. Specifically, we assume the following: (i) The APs
synchronously serve the UEs without any cell boundaries, i.e., each AP can
serve any UE. (ii) Time-division duplexing (TDD) is adopted for the
transmissions, which facilitates the estimation of the downlink access channel
coefficients through the uplink pilots. (iii) Only the large-scale fading
coefficients are available at the CPU for joint power allocation. Further, we
assume that the APs adopt conjugate beamforming for the downlink transmission
to the users. More specifically, the uplink pilots are used to estimate the
uplink channels (which are also used to construct the downlink channels
leveraging channel reciprocity). Then, the information about the large-scale
fading coefficients are frequently transmitted to the CPU. The CPU uses this
large-scale fading information to determine the access channel power
coefficients. The APs adopt this power allocation while jointly serving their
users. Next, we first describe the adopted channel estimation procedure in
detail before formulating the achievable rates of the access link.

Channel Estimation: In the adopted protocol, the UEs transmit orthogonal pilot
sequences of length $L_{p}$,
$\boldsymbol{\psi}_{1},\ldots,\boldsymbol{\psi}_{K}\in\mathbb{C}^{L_{p}}$,
simultaneously to be received by all the APs. If $\rho_{t}$ denote the power
level selected for the pilot transmissions, then the received signal at AP $m$
can be written as

${\mathbf{y}}^{t}_{m}=\sqrt{\rho_{t}L_{p}}\sum_{k=1}^{K}g_{mk}\boldsymbol{\psi}_{k}+{\mathbf{n}}_{m},$
(9)

where ${\mathbf{n}}_{m}\sim\mathcal{CN}(0,{\mathbf{I}}_{L_{p}})$ is the
receive noise. With this received signal, the MMSE estimator for $g_{mk}$ can
be given by

$\hat{g}_{mk}=\frac{\sqrt{\rho_{t}L_{p}}\beta_{mk}}{1+\rho_{t}L_{p}\beta_{mk}}\boldsymbol{\psi}_{k}^{H}{\mathbf{y}}^{t}_{m}.$
(10)

We define the channel estimation error $\tilde{g}_{mk}=g_{mk}-\hat{g}_{mk}$,
and note that the estimation $\hat{g}_{mk}$ and the error $\tilde{g}_{mk}$ are
uncorrelated thanks to the estimator. Therefore, the distributions of the
estimated channel coefficients and the error can be written as

$\hat{g}_{mk}\sim\mathcal{CN}(0,\hat{\beta}_{mk}),\quad\tilde{g}_{mk}\sim\mathcal{CN}(0,\beta_{mk}-\hat{\beta}_{mk}),$
(11)

with the variance of the estimator is defined as

$\hat{\beta}_{mk}=\frac{\rho_{t}L_{p}\beta_{mk}^{2}}{1+\rho_{t}L_{p}\beta_{mk}}.$
(12)

Achievable Rate: With the adopted conjugate beamforming, the coefficients
$f^{\mathrm{ac}}_{mn}$ in the received signal equations (3)-(4) can be
replaced by $\hat{g}_{mn}^{*}$. Further, with the described changes on the
signal model, the capacity lower bound for the UEs given in [8] becomes valid,
and we can express the achievable rate of user $k$ as

$R^{\mathrm{ac}}_{k}=\log_{2}\bigg{(}1+\frac{\rho^{\mathrm{ac}}\big{(}\sum_{m=1}^{M}\sqrt{p_{mk}}\hat{\beta}_{mk}\big{)}^{2}}{\rho^{\mathrm{ac}}\sum_{m=1}^{M}\beta_{mk}\sum_{k^{\prime}=1}^{K}p_{mk^{\prime}}\hat{\beta}_{mk^{\prime}}+1}\bigg{)}.$
(13)

where we note here that the power coefficient $p_{mk}$ for the AP $m$ and UE
$k$ is set to $0$ if the AP is not in the group of that UE, i.e., if
$m\notin\mathcal{G}_{k}$. These power coefficients for each AP $m$ also
satisfy

$\sum_{k=1}^{K}p_{mk}\hat{\beta}_{mk}\leq 1,\quad\forall m\in\mathcal{M}.$
(14)

which captures the total power constraint of AP $m$.

###  IV-C End-to-End Achievable Rate Optimization

Based on the presented achievable rates for the fronthaul and access channels,
we now derive the end-to-end achievable rate of the system. Based on the
adopted system model in Section III, the end-to-end achievable rate of UE $k$
can be defined as the minimum of the fronthaul and the access channel rates,
which can be expressed as

$R_{k}=\min\left\\{B^{\mathrm{ac}}R^{\mathrm{ac}}_{k},\
t_{k}B^{\mathrm{fh}}R^{\mathrm{fh}}_{k}\left({\mathbf{f}}_{k}\right)\right\\}$
(15)

where the end-to-end channel rate accounts for the fronthaul and access
bandwidths (noting that the fronthaul bandwidth in our mmWave-based fronthaul
is expected to be much larger than the bandwidth of the sub-6GHz access
channel). Now, to optimize the end-to-end rate, we adopt the following
formulation of the joint max-min fair rate optimization problem

$\displaystyle\max_{\boldsymbol{\mathcal{G}},\\{{\mathbf{f}}_{k}\\},\\{t_{k}\\},\\{p_{mk}\\}}\,\min_{k}\
$ $\displaystyle\ R_{k}$ (16a) $\displaystyle\quad\mathrm{s.t.}\quad$
$\displaystyle\sum_{k\in\mathcal{K}}p_{mk}\hat{\beta}_{mk}\leq 1,\quad\forall
m\in\mathcal{M}$ (16b) $\displaystyle p_{mk}=0,\quad\forall
m\notin\mathcal{G}_{k},\ \forall k\in\mathcal{K}$ (16c) $\displaystyle
0<t_{k}<1,\quad\forall k\in\mathcal{K}$ (16d)
$\displaystyle\sum_{k\in\mathcal{K}}t_{k}=1$ (16e)
$\displaystyle{\mathbf{f}}_{k}\in\boldsymbol{\mathcal{F}},\quad\forall
k\in\mathcal{K}$ (16f)

which aims to jointly optimize the AP groups, the fronthaul beamforming
vectors, the fronthaul time allocation for different groups, and the access
channel power coefficients. The problem is non-convex and challenging,
especially due to the AP grouping and the fronthaul analog-only beamforming.
It is worth mentioning here that the AP grouping and analog beamforming can be
optimally designed via an exhaustive over all the possible groups and
candidate beam codewords, but this will require prohibitive complexity. To
reduce this complexity, we propose an iterative sub-optimal solution in the
following section.

##  V Proposed Solution

In this section, we develop a suboptimal yet efficient solution for the end-
to-end data rate maximization problem of the proposed cell-free massive
architecture. We will then show in Section VII that this developed solution
(for the wireless mmWave-fronthauled cell-free massive MIMO architecture)
achieves close performance to the upper bound which is given by the fiber-
fronthaul based cell-free massive MIMO architecture. To start, we note that
the optimization problem in (16) can be written in the equivalent form

$\displaystyle\begin{split}\max_{\boldsymbol{\mathcal{G}},\\{{\mathbf{f}}_{k}\\},\\{p_{mk}\\},\\{t_{k}\\}}\,\min\
&\left\\{B^{\mathrm{ac}}R^{\mathrm{ac}}_{k},\,t_{k}B^{\mathrm{fh}}R^{\mathrm{fh}}_{k}({\mathbf{f}}_{k})\right\\}_{k=1}^{K}\\\
\quad\mathrm{s.t.}\quad&\eqref{const:1a}-\eqref{const:1e}\end{split}$ (17)

where the objective is to maximize the minimum of all the fronthaul and access
channel rates. Before making an attempt for the solution, we highlight the
following remark.

######  Remark 1.

For the given system model and the max-min fairness problem defined in (17),
the separate optimization of the access channel and fronthaul variables depend
on the grouping variable $\boldsymbol{\mathcal{G}}$. Nevertheless, for a fixed
grouping $\boldsymbol{\mathcal{G}}$, only the objective function (but not the
constraints) retains the variables of both the fronthaul (${\mathbf{f}}_{k}$
and $t_{k}$) and access channel ($p_{mk}$), i.e., each constraint affects
either the fronthaul channel or the access channel. Therefore, for a given
grouping, a two step approach can be developed with the access channel and
fronthaul optimization steps to obtain an optimal solution.

Motivated by Remark 1, instead of the joint optimization of the fronthaul and
access rates, we first consider a given grouping structure, and design two
separate problems that maximize the fronthaul and access rates: (i) The access
channel power coefficients are optimized without any consideration of the
fronthaul, and (ii) the beamforming vectors for each group and TDMA time
fractions are determined to maximize the fronthaul rate in a fair manner. As
this process is conditioned on the grouping structure, an optimization of this
grouping is required in order to maximize the end-to-end data rates. For that,
we propose an iterative heuristic algorithm for the AP grouping where each
iteration involves solving two sub-problems for the fronthaul and access
rates. The next three subsections present the details of this proposed
approach.

###  V-A Access Channel Rate Optimization

As briefly described, for a given AP grouping selection, we first optimize the
power coefficients for the access channel, without any fronthaul limitations.
This approach allows us to allocate the power of the APs over the access
channel, in a similar way to the standard approaches in the cell-free massive
MIMO literature. To formulate the access channel optimization problem, we only
keep the access channel related terms and constraints of the original problem
defined in (17), and write

$\displaystyle\max_{\\{p_{mk}\\}}\,\min_{k}\ $ $\displaystyle\
B^{\mathrm{ac}}R^{\mathrm{ac}}_{k}$ (18a)
$\displaystyle\quad\mathrm{s.t.}\quad$
$\displaystyle\sum_{k\in\mathcal{K}}p_{mk}\hat{\beta}_{mk}\leq 1,\quad\forall
m\in\mathcal{M}$ (18b) $\displaystyle p_{mk}=0,\quad\forall
m\notin\mathcal{G}_{k},\ \forall k\in\mathcal{K},$ (18c)

where two further simplifications can be applied to the objective. First,
since $B^{\mathrm{ac}}$ is a positive constant multiplied with a function of
the variables, it can be removed. Second, the objective can be re-written in
terms of the SINR values, instead of the rates of the form
$\log(1+\textrm{SINR})$. This re-formulation does not change the optimal power
coefficients, since $\log(1+\textrm{SINR})$ is a non-decreasing function of
the SINR values. Thus, we simplify (18) and write the grouping sensitive max-
min SINR optimization for the access channel as

$\displaystyle\max_{\\{p_{mk}\\}}\,\min_{k}\ $ $\displaystyle\
\mathrm{SINR}_{k}=\frac{\rho_{d}\big{(}\sum_{m=1}^{M}\sqrt{p_{mk}}\hat{\beta}_{mk}\big{)}^{2}}{1+\rho_{d}\sum_{m=1}^{M}\beta_{mk}\sum_{k^{\prime}=1}^{K}p_{mk^{\prime}}\hat{\beta}_{mk^{\prime}}}$
(19a) $\displaystyle\quad\mathrm{s.t.}\quad$
$\displaystyle\sum_{k=1}^{K}p_{mk}\hat{\beta}_{mk}\leq 1,\quad\forall
m\in\mathcal{M}$ (19b) $\displaystyle p_{mk}=0,\quad\forall
m\notin\mathcal{G}_{k},\ \forall k\in\mathcal{K},$ (19c)

which is in the same form with the power allocation problem of the standard
cell-free massive MIMO [1] with the addition of the grouping constraint in
(19c). In fact, when all the APs transmit to all UEs, i.e.,
$\mathcal{G}_{k}=\mathcal{M}$ $\forall k\in\mathcal{K}$, (19c) does not
provide any constraints, and the presented optimization of the access channel
becomes directly equivalent to the power allocation of cell-free massive MIMO.
For the solution of (19), we first elaborate on the additional grouping
constraint (19c), which allows an AP to transmit only to the UEs whose groups
include that AP, by restricting the power allocated for the other UEs to $0$.
It is a linear equality constraint which does not affect the convexity of the
problem. Hence, for the problem (19), a similar solution to the optimal power-
allocation of the standard cell-free massive MIMO given in [1] can be applied.
To specify, as the SINR functions are quasi-concave, the problem can be
reformulated with convex constraints by introducing a variable $\gamma$, as
follows

$\displaystyle\max_{\\{p_{mk}\\},\gamma}$ $\displaystyle\ \gamma$ (20a)
$\displaystyle\quad\mathrm{s.t.}\quad$
$\displaystyle\mathrm{SINR}_{k}\geq\gamma,\quad\forall k\in\mathcal{K}$ (20b)
$\displaystyle\sum_{k=1}^{K}p_{mk}\hat{\beta}_{mk}\leq 1,\quad\forall
m\in\mathcal{M}$ (20c) $\displaystyle p_{mk}=0,\quad\forall
m\notin\mathcal{G}_{k},\ \forall k\in\mathcal{K}.$ (20d)

This formulation can be optimally solved by the bisection method [27] that
iterates over $\gamma$ values. For a selected $\gamma$ value, the SINR
constraints in (20b) can be cast as second-order cone constraints, and (20)
becomes a feasibility problem that aims to find a set of power coefficients
satisfying the constraints. This feasibility problem can be solved by the
convex solvers (such as CVX [28]) as a second-order cone program (SOCP). For
the bisection method, an interval containing the optimal minimum SINR value
$\gamma^{*}\in[\gamma_{\mathrm{min}},\gamma_{\mathrm{max}}]$ is determined.
Then, the middle point of the interval
$\gamma=\frac{\gamma_{\mathrm{max}}+\gamma_{\mathrm{min}}}{2}$ is selected for
the feasibility problem. If the solution of the problem is feasible, i.e.,
there exists a power allocation solution providing the constraints, the lower
limit of the interval is updated with the middle point of the interval
($\gamma_{\mathrm{min}}=\gamma$). Otherwise, the upper limit of the interval
is updated with the same value ($\gamma_{\mathrm{max}}=\gamma$). The described
iterations are applied until the interval is smaller than a pre-determined
convergence distance.

###  V-B Fronthaul Channel Rate Optimization

In this subsection, we define an optimization problem to maximize the
fronthaul rates of the distributed AP groups in a fair manner. First, by
starting with the original formulation given in (17), we write a max-min fair
fronthaul optimization problem by eliminating the access channel related
constraints and objectives, to get

$\displaystyle\max_{\\{t_{k}\\}\\{{\mathbf{f}}_{k}\\}}\min_{k}$
$\displaystyle\ t_{k}R^{\mathrm{fh}}(\mathcal{G}_{k},{\mathbf{f}}_{k})$ (21a)
$\displaystyle\sum_{k\in\mathcal{K}}t_{k}=1$ (21b) $\displaystyle t_{k}\geq
0,\quad\forall k\in\mathcal{K}$ (21c)
$\displaystyle{\mathbf{f}}_{k}\in\boldsymbol{\mathcal{F}},\quad\forall
k\in\mathcal{K}.$ (21d)

For this problem in (21), and given the group selections
$\left\\{\mathcal{G}_{k}\right\\}$, we note that the optimal solution can be
obtained by first optimizing the rates of all the groups,
$R^{\mathrm{fh}}(\mathcal{G}_{k},{\mathbf{f}}_{k})\forall k$, and then
optimization the TDMA time allocation. This is because any set of TDMA time
fractions, $\left\\{t_{k}\right\\}$, does not affect the optimization of the
beamforming vectors. Therefore, we will first optimize the beamforming vectors
and then determine the TDMA time fractions for the groups, as described in the
remaining part of this subsection.

Beamforming Optimization The beamforming vector of each group needs to be
optimized to maximize the group rate. Since this rate is determined by the
minimum rate of the APs in the group, we can formulate the beamforming
optimization problem of any group $\mathcal{G}_{k}$ as follows

$\displaystyle\begin{split}{\mathbf{f}}_{k}^{\star}=\operatorname*{arg\,max}_{{\mathbf{f}}_{k}}\quad&\left(R^{\mathrm{fh}}(\mathcal{G}_{k},{\mathbf{f}}_{k})=\min_{m\in\mathcal{G}_{k}}\big{\\{}R^{\mathrm{fh}}_{m}({\mathbf{f}}_{k})\big{\\}}\right)\\\
\quad\mathrm{s.t.}\quad&{\mathbf{f}}_{k}\in\boldsymbol{\mathcal{F}},\end{split}$
(22)

which coincides with the well-studied multicast beamforming problem with
analog phase-shifters. The optimal solution to the problem can be obtained by
exhaustive search which has the complexity increasing exponentially with
number of antennas, i.e.,
$\mathcal{O}(\lvert\mathcal{Q}\rvert^{N})=\mathcal{O}(2^{qN})$. To reduce this
complexity, several methods have been developed in the literature [29, 30]. In
the simulation results in Section VII, we adopt the solution proposed in [29]
due to its low complexity and good performance. With the optimized beamforming
vectors, the optimal time fractions of the TDMA can be determined. Next, we
will present two approaches to solve this TDMA time fraction optimization
problem.

TDMA Optimization. Approach 1: Given the solution of the problem (22),
$\left\\{{\mathbf{f}}^{*}_{k}\right\\}$, the fronthaul rate of each group $k$
becomes a constant denoted by
$R^{\mathrm{fh}}(\mathcal{G}_{k})=R^{\mathrm{fh}}(\mathcal{G}_{k},{\mathbf{f}}^{*}_{k})$.
With these rate constants, we can simplify the fronthaul rate maximization
problem given in (21) to be

$\displaystyle\max_{\\{t_{k}\\}}\min_{k}$ $\displaystyle\
t_{k}R^{\mathrm{fh}}(\mathcal{G}_{k})\quad\mathrm{s.t.}\quad\sum_{k\in\mathcal{K}}t_{k}=1,\quad
t_{k}\geq 0,\quad\forall k\in\mathcal{K},$ (23)

where the solution can be obtained by introducing a slack variable [27],
$\eta$, and writing the equivalent formulation

$\displaystyle\max_{\\{t_{k}\\},\eta}\eta\quad\mathrm{s.t.}\quad\eta\leq
t_{k}R^{\mathrm{fh}}(\mathcal{G}_{k})\ \ \forall
k\in\mathcal{K},\quad\sum_{k\in\mathcal{K}}t_{k}=1,\quad t_{k}\geq 0\ \
\forall k\in\mathcal{K}.$ (24)

This formulation is a linear program and the solution can be attained by
finding maximum $\eta$, equivalent to the time-scaled data rate of each user,
i.e., $\eta=t_{k}R^{\mathrm{fh}}(\mathcal{G}_{k})$. Then, we have
$t_{k}=\frac{\eta}{R^{\mathrm{fh}}(\mathcal{G}_{k})}$. By the equality
constraint $\sum_{k=1}^{K}t_{k}=1$, we can write the solution for the sum
fronthaul rate, $R^{\mathrm{fh}}(\boldsymbol{\mathcal{G}})=K\eta$, by

$R^{\mathrm{fh}}(\boldsymbol{\mathcal{G}})=\frac{K}{\sum_{k=1}^{K}(R^{\mathrm{fh}}(\mathcal{G}_{k}))^{-1}}=\mathrm{HM}\\{R^{\mathrm{fh}}(\mathcal{G}_{k})\\}$
(25)

where $\mathrm{HM}\\{\cdot\\}$ denotes the harmonic mean function.

Finally, with the obtained access channel and fronthaul rates, we can take the
minimum of them to provide an efficient solution to (17) for a given grouping
structure.

The provided solution so far only aims to maximize the minimum of the rates,
which is also the objective of (17). However, the optimal solution is not
necessarily unique, which motivates the search within this set of solutions
for one that could further optimize other objectives. In particular, without
decreasing the minimum of the end-to-end rates of the given groups, we may
still be able to increase the data rates of the other groups by further
optimizing the fronthaul TDMA time allocations of these groups. To detail, the
solution given in (25) attempts to maximize the minimum of the fronthaul group
rates without accounting for the optimized access channel rates. The user
rates, however, depend on both the access and fronthaul rates. Therefore, we
may be able to further increase the data rates of the users by taking into
account the access rates as a restricting constraint. Based on that, we define
the following problem which attempts to optimize the fronthaul rate of each
group to meet the access rate of the group in a fair manner.

TDMA Optimization. Approach 2: For a fair allocation, the time-scaled rate of
the groups should be equal, unless any of the constraints are met. Therefore,
the time allocated to each group should be inversely proportional to their
rates. To achieve this, we utilize the weighted logarithm function for the
objective, i.e., $\sum_{k}w_{k}\log(t_{k})$, which allows the resources to be
allocated fairly, proportional to the weights, $w_{k}$. We select the weights
as $w_{k}=\frac{1}{R^{\mathrm{fh}}(\mathcal{G}_{k})}$, to allocate the time
fractions inversely proportional to the fronthaul rates of the groups,
resulting in equal fronthaul rates. In addition, we upper-bound the fronthaul
rate of a user by the access channel rate of that user. With this upper-bound,
the total time fractions do not necessarily meet the summation equality,
$\sum_{k\in\mathcal{K}}t_{k}=1$. Hence, we relax the condition by
$\sum_{k\in\mathcal{K}}t_{k}\leq 1$, and write the problem for the time
allocation of the TDMA, that maximize the end-to-end rate in a fair manner, as
follows

$\displaystyle\max_{\\{t_{k}\\}}$
$\displaystyle\,\sum_{k\in\mathcal{K}}\frac{1}{R^{\mathrm{fh}}(\mathcal{G}_{k})}\log
t_{k}$ (26a) $\displaystyle\sum_{k\in\mathcal{K}}t_{k}\leq 1$ (26b)
$\displaystyle
t_{k}\leq\frac{B^{\mathrm{ac}}R^{\mathrm{ac}}_{k}}{B^{\mathrm{fh}}R^{\mathrm{fh}}(\mathcal{G}_{k})},\quad\forall
k\in\mathcal{K}$ (26c) $\displaystyle t_{k}\geq 0,\quad\forall
k\in\mathcal{K}$ (26d)

where the problem is concave, and the solution can be derived with Karush-
Kuhn-Tucker conditions. The resulting solution takes a similar form with the
water-filling solution with upper-bounds. Specifically, the solution can be
obtained by finding the maximum individual fronthaul rate (water level),
$\eta>0$, that satisfy the following inequality

$\sum_{k\in\mathcal{K}}t^{*}_{k}(\eta)\leq 1$ (27)

where

$t_{k}^{*}(\eta)=\min\left\\{\frac{B^{\mathrm{ac}}R^{\mathrm{ac}}_{k}}{B^{\mathrm{fh}}R^{\mathrm{fh}}(\mathcal{G}_{k})},\frac{\eta}{R^{\mathrm{fh}}(\mathcal{G}_{k})}\right\\}.$
(28)

For the optimal $\eta$ value, $\eta^{*}$, we can obtain the optimal time
fractions $t_{k}^{*}=t_{k}^{*}(\eta^{*})$. The given solution allocates the
fronthaul rates equally among UEs, until the satisfaction of individual access
channel rates or the use of the total time. In the special case of all the
fronthaul rates being smaller than the access channel rates, it allocates the
rates equally, coinciding with (25).

###  V-C End-to-End Optimization through Iterative Group Selection

With the proposed fixed-group end-to-end data rate maximization, the efficient
selection of the groups becomes crucial in achieving high data rates. However,
the group selection problem is a combinatorial problem that has high
complexity given the large number of APs and UEs. In particular, there are
$2^{M}$ possible selection of groups for each UE, leading to a total of
$2^{MK}$ distinct selections for the $K$ users. Moreover, each selection of
the groups needs to be utilized with the proposed end-to-end data rate
optimization which results in prohibitive optimization complexity for
practical systems. Hence, an efficient design for the group selection is
required. To this end, we propose a low-complexity yet efficient solution that
is motivated by understanding the end-to-end achievable rate optimization
problem and the proposed cell-free massive MIMO architecture. Next, we present
the proposed approach which has two key points, namely the group selection and
the group size optimization.

Group Selection: We adopt the following group selection approach: The group of
each user is selected as the $G$ APs with the maximum channel gains. Here, it
is important to clarify two points: (i) The group size $G$ is assumed to be
fixed for the sake of simplicity and low-complexity solution. Note that the
impact of this constraint is expected to be marginal given the high density of
the APs. (ii) The selection of the APs with the maximum channel gains will
likely lead to a set of APs that are close to each other, which will lead to
efficient beamforming design via more focused beams. Next, we mathematically
describe the proposed approach. Let
$\\{\beta^{(o)}_{1k},\ldots,\beta^{(o)}_{Mk}\\}$ define the ordered set of
channel coefficients between user $k$ and the $M$ APs. This set adopts a
descending order, i.e., the channel coefficients satisfy
$\beta^{(o)}_{mk}\geq\beta^{(o)}_{m^{\prime}k}$ for any $m\leq m^{\prime}$. In
addition, to formalize the mapping between the original channel coefficients
sets $\left\\{\beta_{1k},...,\beta_{Mk}\right\\}$ and the ordered channel
coefficients set $\\{\beta^{(o)}_{1k},\ldots,\beta^{(o)}_{Mk}\\}$, we define
the permutation $\varsigma_{k}(.)$ such that
$\beta^{(o)}_{m^{\prime}k}=\beta_{mk}$ if $\varsigma_{k}(m^{\prime})=m$, i.e.,
if the $m$th AP has the $m^{\prime}$th highest channel gain. Using these
definitions, for a given group size $G$, the group of UE $k$ can be determined
as

$\mathcal{G}_{k}=\bigcup_{m^{\prime}=1}^{G}\varsigma_{k}(m^{\prime})$ (29)

which simply selects the $G$ APs with the best channel gains. With this
approach, the group selection is reduced to the selection of the parameter
$G$, and it needs to be determined carefully.

Group Size Optimization: The number of APs per group, $G$, needs to be
optimized to maximize the end-to-end rate. Here, it is interesting to
highlight the following trade-off: A small group size may lead to more
optimized CPU-APs beamforming design and hence high fronthaul rates. At the
same time, it may also result in low APs-user beamforming gain and achievable
access channel rates. Therefore, to select $G$, one possible approach is the
trial of different group size values from a pre-determined interval (for a
given AP structure). Another approach is to start from a certain group size
$G$ and then increase/decrease the group size depending on the relation
between the access and fronthaul rates. More specifically, if the sum
fronthaul rate is larger than the sum access channel rate, i.e.,
$R^{\mathrm{fh}}\geq R^{\mathrm{acc}}$, the group size is increased.
Otherwise, if $R^{\mathrm{fh}}<R^{\mathrm{acc}}$, the group size is decreased.
Through the iterations of these updates, the group value can be locally
optimized222Although the iterative algorithm is applied through the sum data
rates, it still achieves some fairness between the users since the group size
is fixed..

##  VI Mixed-Fronthaul Cell-Free Massive MIMO Architecture

In the previous sections, we proposed and designed a cell-free massive MIMO
architecture with wireless higher-band fronthaul. While this architecture is
flexible, scalable, and has the potential of reducing the installation cost
and time of cell-free massive MIMO systems, it has a few drawbacks: (i) Adding
a dual-band relay to each remote wireless AP adds an extra cost to the system
and (ii) separately powering the distributed APs may require additional
infrastructure cost. To address these points while maintaining the same
promising gains, we propose a modified architecture with partially connected
APs. In this section, we first describe the proposed architecture in Section
VI-A and then briefly present its rate optimization approach in Section VI-B.
The performance evaluation of this architecture is then detailed in Section
VII-C.

###  VI-A Description of the Proposed Mixed-Fronthul Architecture

Figure 2: An illustration of the proposed mixed-fronthaul architecture. In
this architecture, each subset of APs are connected together via an optical
fiber forming a cluster. Each cluster has one leading AP that is responsible
for the wireless fronthaul communication with the central processing unit. For
the access link, each user is served by a group of AP clusters. This
architecture has the potential of reducing the fronthaul cost/complexity while
maintaining the data rate/coverage gains as illustrated in Section VII.

To reduce the cost of the distributed APs while maintaining the potential
deployment/operational gains of using wireless fronthaul, we propose the
alternative cell-free massive MIMO architecture depicted in Fig. 2. In this
architecture, every cluster/set of distributed APs is connected with wired
connections, e.g., an optical fiber. Further, each set of APs includes one
leading AP that has a wireless-fronthaul to the central unit. This leading AP
will be responsible for transmitting/receiving the cluster data to/from the
central unit. It is worth noting here that the connection between the APs in
each cluster could be realized using radio stripes [2]. With this
implementation, the proposed mixed-fronthaul cell-free massive MIMO
architectures combines the gains of the radio stripes and the installation
cost/flexibility gains of the higher-band wireless fronthaul, shaping a
promising and practical solution for cell-free massive MIMO systems.

From an operation perspective, the proposed mixed-fronthaul architecture
reduces the number of APs that are simultaneously using the wireless
fronthaul, which relaxes the fronthaul requirements in terms of the equipment
cost, the required bandwidth, and the beamforming design. Therefore, this
mixed-fronthaul architecture has the potential of further improving the
wireless fronthaul rates since better beams can be utilized by the central
unit in serving the clusters leading APs. In terms of the end-to-end
communication model, the main difference between the mixed-fronthaul
architecture and the original proposed architecture in Section II is that all
the APs in one cluster communicate with the central unit through one leading
AP. Therefore, and to simplify the end-to-end rate optimization, we assume
that all the APs in one cluster will serve the same user. If the number of APs
in each cluster is large, however, it will be more important to relax this
constraint, i.e., to allow any user to be served by only a subset of the
cluster APs. In the next subsection, we elaborate more on the proposed rate-
optimization approach.

###  VI-B Proposed Rate Optimization Approach

In this subsection, we extend the rate optimization solution presented in
Section V to the modified cell-free massive MIMO architecture with mixed-
fronthaul. First, we assume that all the APs in one cluster will be serving
the same set of UEs. This is motivated by the negligible data transmission
cost within the same cluster. Hence, the group of APs serving each user $k$
will be determined by selected the set of clusters (instead of selecting
individual APs in the separate-AP architecture in Section II-Section V).
Formally, assume that there are $L$ distinct AP clusters, such that
$\mathcal{C}_{l}\subseteq\mathcal{M}$, $\forall l\in\\{1,\ldots,L\\}$ with
$\bigcup_{l=1}^{L}\mathcal{C}_{l}=\mathcal{M}$ and
$\mathcal{C}_{l}\cap\mathcal{C}_{l^{\prime}}=\emptyset$, $\forall l\neq
l^{\prime}$. Then, the APs group of user $k$, which we denote as
$\mathcal{G}^{\mathrm{cluster}}_{k}$ to emphasize that there are selected from
the available AP clusters will satisfy
$\mathcal{G}^{\mathrm{cluster}}_{k}\subseteq\\{1,\ldots,L\\}$. Further, we let
$a_{l}$ represent the leading AP of the $l$th cluster. Next, we revisit the
fronthaul/access rate optimization and the group selection methodology
presented in Section V from the lens of the mixed fronthaul architecture.

Fronthaul Rate Optimization: The initial fronthaul rate optimization was
defined in (21), which aims to optimize the time and beams of the group
$\mathcal{G}_{k}$. With the mixed architecture, the CPU only needs to transmit
the data to the leading APs of the clusters of the $k$-th UE. To simplify the
description, we define the user-centric fronthaul groups
$\mathcal{G}_{k}^{\mathrm{fh}}=\\{a_{l}:\
l\in\mathcal{G}^{\mathrm{cluster}}_{k}\\}$ of the leading APs of the clusters.
This notation allows a straightforward use of the fronthaul formulation in
(21), only by adopting $\mathcal{G}_{k}^{\mathrm{fh}}$ instead of
$\mathcal{G}_{k}$. With this fronthaul group definition, the rest of the
fronthaul optimization methodology can be carried out in the same form.

Access Rate Optimization: Similar to the fronthaul, we let
$\mathcal{G}^{\mathrm{ac}}_{k}$ denote the set of APs serving each use $k$
over the access channel. This can be described as
$\mathcal{G}^{\mathrm{ac}}_{k}=\\{\mathcal{C}_{l}:\
l\in\mathcal{G}^{\mathrm{cluster}}_{k}\\}$. With this notation, we can utilize
the original access channel formulation in (18) and the adopted rate
optimization approach, by replacing the AP groups
$\left\\{\mathcal{G}_{k}\right\\}$ with the access channel groups
$\left\\{\mathcal{G}^{\mathrm{ac}}_{k}\right\\}$.

Group Selection: For the selection of the groups, as the user groups are
defined in terms of the clusters, a metric for each cluster is needed. For
this purpose, we consider the sum of the channel gains of the APs of the
clusters, i.e., $\bar{\beta}_{lk}=\sum_{m\in\mathcal{C}_{l}}\beta_{mk}$, and
select the $G$ clusters with the maximum value as the clusters of a UE,
$\mathcal{G}^{\mathrm{cluster}}_{k}$. After determining the active clusters of
UEs, the obtained groups of clusters can be converted to the fronthaul and
access channel groups of the APs by earlier definitions. Then, the access
channel and fronthaul optimization problems can be utilized over the access
channel and fronthaul groups, completing an iteration of end-to-end data rate
maximization. The described process will evaluated in the numerical results
section to show the performance of connected APs.

##  VII Numerical Results

In this section, we evaluate the performance of the proposed cell-free massive
MIMO architectures and the developed rate optimization algorithms. Further,
these proposed solutions are compared with the classical fiber-fronthaul based
cell-free massive MIMO systems. The flow of this section is as follows. First,
we present the adopted system setup and parameters in Section VII-A. Then, we
evaluate the performance of the two proposed cell-free massive MIMO
architectures, namely with separate APs and connected APs, in Sections VII-B
and VII-C.

TABLE I: Simulation Parameters Parameter | Fronthaul | Access Channel
---|---|---
Frequency ($f_{c}^{\mathrm{fh}},f_{c}^{\mathrm{ac}}$) |  $28$ GHz |  $3.5$ GHz
Bandwidth (B) |  $2$ GHz |  $20$ MHz
Power ($\tilde{\rho}^{\mathrm{fh}},\tilde{\rho}^{\mathrm{ac}}$) |  $30$ dBm |  $10$ dBm
Noise Figure ($\sigma^{2}$) | 9dB
Antenna Spacing ($d_{A}$) | $\lambda/2$ | N/A
CPU Antennas ($N$) | 128

###  VII-A Simulation Setup

In the simulation, we considered the proposed cell-free massive MIMO
architectures in Section II (for the separate APs case) and in Section VI (for
the connected APs case). For both architectures, we adopt mmWave fronthaul
channels at $f^{\mathrm{fh}}=28$ GHz carrier frequency, and access channels at
$f^{\mathrm{ac}}=3.5$ GHz. The bandwidth allocated for the access channels is
$B^{\mathrm{ac}}=20$ MHz, and various values for the fronthaul bandwidth are
considered, as will be discussed in the results. The noise figure is set to
$9$ dB for both the fronthaul and access links. The central unit is deploying
a uniform linear array with $N=128$ elements and half wavelength antenna
spacing. The transmit power of the central unit is assumed to be
$\tilde{\rho}^{\mathrm{fh}}=30$ dBm for the fronthaul links while the APs and
users transmit powers are set to
$\tilde{\rho}^{\mathrm{ac}}=\tilde{\rho}^{t}=10$ dBm for the access links and
pilot transmissions. The noise power is determined based on the bandwidth of
the fronthaul and access channels, following $\sigma_{n}^{2}=\sigma^{2}\cdot
B\cdot k\cdot T$, with the Boltzmann constant $k$, the channel bandwidth $B$,
the temperature $T=$290\text{\,}\mathrm{K}$$, and the noise figure
$\sigma^{2}$. The normalized power levels of the different channels are then
determined by $\rho=\frac{\tilde{\rho}}{\sigma_{n}^{2}}$ with the the
corresponding parameters of the specific channel. These baseline system
parameters are summarized in Table I. The values that are selected differently
from the baseline parameters in different figures are explicitly stated.

For the large scale channel coefficients, we adopt the 3GPP Urban Micro (UMi)
street-canyon model given in [31] as follows:

$\begin{split}&\mathrm{PL}(d,f_{c})=32.4\,\mathrm{(dB)}+20\log_{10}d\,\mathrm{(m)}+21\log_{10}f_{c}\,\mathrm{(GHz)}\,\mathrm{(dB)}.\end{split}$

Specifically, the mmWave fronthaul and sub-$6$ GHz access channel large scale
coefficients can be written in terms of the path-loss function as

$\beta^{\mathrm{ac}}_{m}=\mathrm{PL}(d_{m}^{\mathrm{fh}},f_{c}^{\mathrm{fh}}),\quad\beta^{\mathrm{ac}}_{mk}=\mathrm{PL}(d_{mk}^{\mathrm{ac}},f_{c}^{\mathrm{ac}})$

The phase-shift of the fronthaul is taken as $\alpha_{m}^{\mathrm{fh}}=1$
while the small-scaling fading of the access channel is generated by
$\alpha_{mk}^{\mathrm{ac}}\sim\mathcal{CN}(0,1)$. For the phase-shift set of
analog beamforming at the CPU, we use $q=3$ bit quantizers, and uniformly
sample the angle space. Next, we evaluate the performance of the proposed
solutions.

(a) Group of 12 APs

(b) Group of 25 APs

Figure 3: The beamforming gains of the optimized beamforming vectors for the
groups of (a) 12 APs and (b) 25 APs. The given scenario adopts randomly placed
$M=100$ APs and $K=10$ UEs. The beamforming gain of a group is depends on the
group size, decreasing with larger groups.

###  VII-B Evaluating the Proposed Architecture with Separate APs (Wireless-
Only Fronthaul)

The objective of this subsection is to evaluate the performance of the
proposed cell-free massive MIMO architecture with wireless fronthaul (and
separate APs). In particular, we want to draw some insights into how this
architecture performs if compared with the classical fiber-fronthaul based
cell-free massive MIMO architecture and whether it can achieve comparable data
rates. To do that, we consider a setup where the UEs and APs are randomly
placed over an area of $100m\times 100m$ centered at $(0,0)$. Unless otherwise
stated, the following simulations assume that $K=10$ UEs and $M=100$ APs are
placed in the considered area. The central unit (CPU) is located at
${\mathbf{z}}_{0}=(x_{0},y_{0})=(-D,0)$, i.e., at a distance $D=100$m away
from the center of the square area. The simulations are averaged over $250$
realizations. Each realization drops the APs and UEs randomly adopting a
uniform probability distribution. With this setup, we study the following
important questions.

What is the performance of the fronthaul beamforming?  To evaluate the
performance of the adopted fronthaul beamforming approach in our wireless-
fronthaul based cell-free massive MIMO architecture, we plot the achievable
beamforming gain for the considered deployment in Fig. 3. This figure shows
the beamforming gain in x-y coordinates for two different AP group sizes,
namely a group of 12 APs in Fig. 3(a) and a group of 25 APs in Fig. 3(b).
These figures provide a visual verification that the central unit focuses it
multi-cast beam towards the APs in the group of interest. In terms of
beamforming gain, it varies from 12 (i.e., around 11dB) in the case of 12 APs
group to 4 (i.e., around 6dB) in the 25 APs/group case. This highlights the
trade-off between the fronthaul rate and access rate as the group sizes
increases, as this decreases the fronthaul rate while increasing the access
rate. It is worth mentioning here that this fronthaul beamforming can be
further improved when adopting hybrid precoding approaches [32, 22]. Also,
leveraging machine learning could enable the autonomous design of these
beamforming vectors/codebooks [33, 34, 35]. This can further reduce the
depolyment overhead.

(a) Sum fronthaul and access channel rates

(b) Sum fronthaul, access channel and end-to-end rates

Figure 4: The access channel and fronthaul sum data rates of the proposed
architecture with different group size and fronthaul bandwidth values. (a)
shows the change in the fronthaul rates based on the available bandwidth an
group size. (b) includes the sum of the end-to-end data rates reflecting the
exact performance.

What are the achievable fronthaul and access rates? Now, we want to evaluate
the fronthaul and access achievable rates using the developed rate
optimization solutions in Section V. Adopting the same setup described earlier
in this section with 10 UEs and 100 APs, we plot the achievable
fronthaul/access sum-rates in Fig. 4(a) versus the AP group size. The group
for each user is determined based on the channel gain criterion described in
Section V-C, the access rate is optimized based on the solution presented in
Section V-A, and the fronthaul time allocation optimization is implemented
based on Approach 1 in V-B. As shown in Fig. 4(a), as the group size
increases, the access rate increases because of the higher access beamforming
gain and the fronthaul rate decreases because of the lower fronthaul multicast
beamforming gain. The intersection point of the fronthaul/access rates can be
considered as a good approximation for the achievable end-to-end sum-rate, as
will be discussed shortly in Fig. 4(b). As illustrated in Fig. 4(a), with high
enough fronthaul bandwidth, e.g., 2GHz, the proposed rate optimization
approach and cell-free massive MIMO architecture can achieve around $620$Mbps
sum-rate. This is very close to the $690$Mbps sum-rate achieved by the upper
bound, which is given by the classical fiber-fronthaul based cell-free massive
MIMO system. These results highlight the promising data rates of the proposed
rate optimization solutions.

Can the proposed architecture with wireless fronthaul approach the data rates
of the classical fiber-fronthaul based cell-free massive MIMO architecture?
This is a key question that we are trying to answer in this paper.
Essentially, the proposed architecture with wireless higher-band fronthaul has
clear gains in terms of the installation cost/time/flexibility, but could it
also provide comparable achievable rates to the fiber-based architecture? In
Fig. 4(a), we provided some insights into the answer of this question by
separately evaluating the achievable fronthaul and access rates. The exact
achievable rate, however, is the one with matched fronthaul and access rates.
To evaluate this exact rate, we adopt the fronthaul rate optimization
(Approach 2) in Section V-B which ensures that the fronthaul rate of each link
does not exceed its corresponding access rate. In Fig. 4(b), we plot the
achievable rates for the access and fronthaul links using Approach 2. This
rate then represents the exact end-to-end achievable rates using the proposed
architecture. For reference, we also plot the fronthaul rates using Approach
1. As shown in Fig. 4(b), with sufficient fronthaul bandwidth, the exact
achievable end-to-end data rates using the proposed architecture are very
comparable to the classical fiber-based cell-free massive MIMO architecture.
For example, with $480$MHz fronthaul bandwidth, the proposed solution achieves
around $580$Mbps sum-rate compared to $690$Mbps for the fiber-based
architecture. This difference can be further reduced by increasing the
fronthaul bandwidth. It is also good to note that the exact rate is close to
the approximation discussed in Fig. 4(a), which is given by the intersection
of the access rate and the Approach 1 fronthaul rate.

Figure 5: The access channel and fronthaul sum data rates of the proposed
architecture with different number of randomly placed APs. The figure shows
the relationship of the rates for different group size, $G$, values. With
increasing number of APs, $M$, the group size providing the maximum data rate
decreases.

Can the proposed architecture support a large number of distributed APs? The
scalability is a key objective of cell-free massive MIMO systems to be able to
support more users/larger areas and to increase the beamforming
gains/achievable rates. To draw some insights into the capability of the
proposed architecture in supporting large numbers of APs, we plot the
fronthaul/access rates vs the number of APs, $M$, in Fig. 5. As shown in this
figure, for each group size, the access rates increase with denser APs due to
the higher access beamforming gains. For the fronthaul links, adding more APs
reduces the achievable rates due to the harder multicast beamforming design
problem which leads to lower fronthaul beamforming gains. It is worth noting
here that when $M\leq G$, the fronthaul rate decreases more rapidly as every
new AP will also increase the effective group size and hence further
complicates the beamforming design problem. Overall, however, the end-to-end
data rates, which are approximately given by the intersections of the
fronthaul and access channel rates, are increasing with more APs. This is also
very comparable to the increase in data rates experienced by the fiber-based
architecture, which highlights the potential of the proposed solution.

(a) Group size of 10 clusters

(b) Group size of 9 clusters

Figure 6: The beamforming gains with the optimized beamforming vectors are
shown for the mixed-fronthaul architecture. In (a), all of the clusters are
activated, hence the beam gain is maximized towards all of the leading APs. In
(b), the bottom $9$ clusters are activated, and the beam gain is not maximized
for the leading AP of the top cluster.

###  VII-C Evaluating the Proposed Architecture with Connected APs (Mixed-
Fronthaul)

In this subsection, we evaluate the performance of the proposed modified
architecture in Section VI with connected APs. As shown in Fig. 6, we consider
a square-grid of APs, where the APs are located uniformly in an $10\times 10$
grid with equal distance between the rows and columns. For simplicity, we
assume that the 10 APs in each line in the $y$-axis form a cluster. The
leaders of the clusters (that communicate with the central unit through the
mmWave fronthaul) are selected as the closest APs to the central unit.
Further, we assume that $K=$10 UEs are randomly located in the 100m $x$ 100m
area of interest. The central unit is located at the origin, as depicted in
Fig. 6, and is equipped with a ULA of $N=128$ antenna elements.

When does the achievable data rate with the proposed architecture approach the
optimal solution? To evaluate the achievable data rates with the proposed
connected-APs cell-free massive MIMO architecture, we adopt the setup
described in the previous paragraph, which is also depicted in Fig. 6. For
this setup, we first investigate the achievable beamforming gains for
different numbers of AP clusters per group. To illustrate, we consider one UE
and plot the beamforming gains for the clusters serving this UE in two
scenarios: (i) When a group of 10 AP clusters (i.e., all the clusters) jointly
serve this UE as shown in Fig. 6(a) and (ii) when a group of the 9 bottom
clusters serve the UE, as shown in Fig. 6(b). As expected, when the number of
AP clusters per group increases, the beamforming gain decreases. Next, we
calculate the achievable fronthaul and access rates of the same setup in Fig.
7 for different numbers of AP clusters per group. As shown in this figure, the
modified architecture with connected APs has clear data rate gains compared to
the separate APs architecture studied in Fig. 4. With only $80$MHz fronthaul
bandwidth, the proposed connected-APs architecture achieves nearly the same
end-to-end rates obtained by the upper bound, which is defined by the
classical fiber-fronthaul based cell-free massive MIMO architecture. This
provides a promising solution for flexible, scalable, and efficient cell-free
massive MIMO systems.

Figure 7: For the mixed-fronthaul architecture, the fronthaul, access channel
and end-to-end sum rates are shown for different number of group size (of
clusters). In the figure, the $10\times 10$ APs rectangular grid scenario
illustrated in Fig. 6 is adopted and each cluster consists of $10$ APs.

##  VIII Conclusions

In this work, we proposed two wireless-fronthaul based architectures for cell-
free massive MIMO systems where the fronthaul link is operating at a higher
band (e.g., mmWave) compared to the access band (e.g., sub-6GHz). We
formulated the end-to-end data rate optimization problems for these
architectures based on user-centric AP grouping. Then, we developed efficient
transmission and resource allocation strategies that can achieve near-optimal
performance. Simulation results showed that the proposed architectures can
achieve data rates that approach the optimal rates obtained with optical
fiber-based fronthaul with reasonable assumptions on the number or CPU
antennas, coverage area, number of users, etc. The results also showed that
these architecture are capable of supporting large numbers of APs under
realistic fronthaul bandwidth requirements. For the future work, it is
interesting to develop more advanced fronthaul beamforming solutions using
hybrid analog/digital architectures. It is also interesting to investigate the
design of machine learning based adaptive AP grouping and beamforming
optimization approaches for the proposed wireless-fronthaul based cell-free
massive MIMO architectures.

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