#  Network-Assisted Full-Duplex Cell-Free Massive MIMO: Spectral and Energy
Efficiencies

Mohammadali Mohammadi, Tung T. Vu,
Hien Quoc Ngo, and Michail Matthaiou M. Mohammadi, T. T. Vu, H. Q. Ngo, and M.
Matthaiou are with the Centre for Wireless Innovation (CWI), Queen’s
University Belfast, U.K. Email:{m.mohammadi, hien.ngo<EMAIL_ADDRESS>(_Corresponding author: Mohammadali Mohammadi._) T. T. Vu is also with the
Department of Electrical Engineering (ISY), Linköping University, SE-581 83
Linköping, Sweden. Email<EMAIL_ADDRESS>
###### Abstract

We consider network-assisted full-duplex (NAFD) cell-free massive multiple-
input multiple-output (CF-mMIMO) systems, where full-duplex (FD) transmission
is virtually realized via half-duplex (HD) hardware devices. The HD access
points (APs) operating in uplink (UL) mode and those operating in downlink
(DL) mode simultaneously serve DL and UL user equipments (UEs) in the same
frequency bands. We comprehensively analyze the performance of NAFD CF-mMIMO
from both a spectral efficiency (SE) and energy efficiency (EE) perspectives.
Specifically, we propose a joint optimization approach that designs the AP
mode assignment, power control, and large-scale fading (LSFD) weights to
improve the sum SE and EE of NAFD CF-mMIMO systems. We formulate two mixed-
integer nonconvex optimization problems of maximizing the sum SE and EE, under
realistic power consumption models, and the constraints on minimum individual
SE requirements, maximum transmit power at each DL AP and UL UE. The
challenging formulated problems are transformed into tractable forms and two
novel algorithms are proposed to solve them using successive convex
approximation techniques. More importantly, our approach can be applied to
jointly optimize power control and LSFD weights for maximizing the sum SE and
EE of HD and FD CF-mMIMO systems, which, to date, has not been studied.
Numerical results show that: (a) our joint optimization approach significantly
outperforms the heuristic approaches in terms of both sum SE and EE; (b) in
CF-mMIMO systems, the NAFD scheme can provide approximately $30\%$ SE gains,
while achieving a remarkable EE gain of up to $200\%$ compared with the HD and
FD schemes.

††footnotetext: The work of M. Mohammadi and M. Matthaiou was supported by a
research grant from the Department for the Economy Northern Ireland under the
US-Ireland R&D Partnership Programme. The work of T. T. Vu and H. Q. Ngo was
supported by the U.K. Research and Innovation Future Leaders Fellowships under
Grant MR/S017666/1. The work of T. T. Vu was also supported in part by ELLIIT
and the KAW Foundation. Parts of this paper were presented at IEEE SPAWC 2022
[1].

###### Index Terms:

Access point mode assignment, cell-free massive multiple-input multiple-
output, energy efficiency, full-duplex, half-duplex, large-scale fading
decoding weight, network-assisted full-duplex, power control, spectral
efficiency.

##  I Introduction

Cell-free massive multiple-input multiple-output (CF-mMIMO) networks and full-
duplex (FD) communications are two technological platforms to address the
explosive growth of data demands driven by smartphones, tablets, and other
media-hungry devices and are expected to play an important part in fifth-
generation (5G) networks and beyond [Matthaiou:COMMag:2021]. CF-mMIMO
represents a scalable and practical implementation of the distributed antenna
[3] and network MIMO systems concepts [4], wherein a large number of access
points (APs) are distributed over a coverage area and coordinated by several
central processing units (CPUs) to coherently serve many user equipments (UEs)
in the same time-frequency resources [5]. The application of FD APs in CF-
mMIMO enables simultaneous transmission towards downlink (DL) UEs and
reception from uplink (UL) UEs on the same frequency, thus, holds the promise
of achieving higher spectral and energy efficiency (SE-EE) [6].

Although the potential gains of FD can be easily foreseen, the key challenge
in achieving FD communication is the high transceiver complexity, required to
mitigate the self-interference (SI) caused by the signal leakage from the
transceiver output to the input [7, 8]. While recent advances in active and
passive SI suppression (SIS) techniques [9, 10] have made the FD transceivers
feasible in practice, SIS at the FD AP imposes huge infrastructure costs to
the operators as a significant additional processing burden is required.
Furthermore, the use of FD transceivers in CF-mMIMO networks gives rise to an
additional source of interference, termed as cross-link interference (CLI),
i.e., the interference received by the receiving antennas of one AP from the
transmitting antennas of another AP, as well as the interference received by a
DL UE from other UL UEs. Both SI and CLI subsequently impact the signal
processing and resource allocation strategies, making them nontrivial tasks in
FD CF-mMIMO compared to the traditional HD CF-mMIMO. More importantly, CLI
results in higher power consumption to achieve the UEs’ SE requirements while
SIS entails power-hungry hardware at the FD APs [11]. Therefore, the
fundamental challenge behind the FD CF-mMIMO realization is how to improve the
SE while maintaining the EE at the optimum level.

The aforementioned challenges have spurred research at the system level design
for FD CF-mMIMO systems. More specifically, few efforts have been devoted
towards developing functionalities, such as power control, scheduling, and AP
selection, to improve the SE and/or EE [6, 12]. Nevertheless, SI and CLI are
still major performance bottlenecks, which are significantly detrimental from
the power consumption and EE perspective. To deal with the negative impact of
CLI, one attractive approach is to revisit and redesign the network
architecture. To this end, a novel paradigm for the physical layer of the CF-
mMIMO networks, called _network-assisted full duplexing (NAFD)_ , has been
embodied in [13]. In NAFD CF-mMIMO, the APs can operate in FD (all APs perform
DL and UL at the same time over the same frequency), hybrid-duplex (both FD
and HD APs exist in the network), or flexible-duplex (APs operate in HD mode),
thus, NAFD unifies all duplex modes in the network [13, 14]. The transmission
mode of each AP (UL reception and/or DL transmission) can be dynamically
designed based on the network conditions and requirements, accordingly both UL
and DL services could be supported simultaneously and flexibly. When NAFD is
forced to be established via pure flexible-duplex, SI and related challenges
are completely removed. Moreover, the amount of CLI becomes fairly less than
that in the FD counterpart. This is due to the fact that only a part of the
APs contribute to DL transmissions, whilst the DL transmissions and remaining
APs are dedicated to serving the UL transmissions. Furthermore, AP mode
assignment offers a flexible degree-of-freedom to better manage the CLI
compared to FD. To reduce the CLI and accordingly enhance the SE-EE of the
NAFD CF-mMIMO, joint design of AP mode assignment, UL/DL power control
coefficients, and large-scale fading decoding (LSFD) weight design for the UL
APs is a promising, but nontrivial direction, which is the main motivation of
this paper.

###  I-A Review of Related Literature

The emulation of the FD operation by spatially separating HD base stations
(BSs) in cellular networks, for serving UL and DL UEs, was first introduced in
[15]. This scheme, which is known as COMPflex in the literature, provides
advantages over using a FD BS, in terms of SE and EE. In [16], a spatial
domain duplex, called a bidirectional dynamic network, was proposed for large-
scale distributed antenna systems (DAS). In bidirectional dynamic networks,
different remote antenna units (RAUs) are used to serve the UL and DL UEs at
the same time and over the same frequency band. However, both COMPflex and
bidirectional dynamic networks are impractical due to the immense fronthaul
signaling and huge computational complexity, which poses performance
limitations and system scalability issues.

Inspired by the idea of COMPflex and by leveraging the join processing virtue
of CF-mMIMO, NAFD CF-mMIMO networks were proposed in [13]. Recently, NAFD CF-
mMIMO systems have been investigated from various aspects and/or under
different setups [13, 17, 14, 18, 19, 20, 21]. In particular, the SE of the
NAFD CF-mMIMO network has been studied in [13] for zero forcing (ZF) and
regularized ZF (RZF) precoders in DL and minimum-mean-square-error (MMSE)
receiver in UL. An optimization framework for the transceiver design for
large-scale DAS with NAFD was developed in [17], by maximizing the sum UL and
DL SE subject to quality-of-service (QoS) constraints and backhaul
constraints. The authors in [14] proposed to utilize a beamforming training
scheme to perform interference cancellation at RAUs and coherent decoding at
DL UEs. Moreover, closed-form expressions for the UL achievable rates with
maximum ratio combining (MRC) and ZF receivers and for the DL achievable rates
with maximum ratio transmission (MRT) and ZF beamforming were derived. In
[18], an optimization framework was established for joint user selection and
transceiver design in cell-free with NAFD, where the QoS constraint and
fronthaul compression are considered.

Nevertheless, a main underlying assumption in all aforementioned literature
[13, 17, 14, 18] is the fixed mode assignment in NAFD networks, i.e., the UL
or DL transmission mode of the RAUs/APs has been already given. This
assumption, however, renders the flexible adjustment of UL/DL traffic and
efficient resource usage inapplicable. In [19, 20, 21], the problem of RAU
mode selection was addressed from different perspectives. In [19], RAUs’ mode
selection for a CF-mMIMO with NAFD was investigated, to maximize the SE of DL
and UL UEs, where power budget constraints are considered. This work assumes
all the antennas of the same RAU working in the same mode and an RAU is either
UL or DL only. In [20] and [21], the authors extended their work in [19], to
consider hybrid-duplex mode in NAFD CF-mMIMO, where antenna mode assignment at
each multi-antenna RAU is investigated. More specifically, a SE maximization
problem under power and QoS constraints was studied in [20] and a two-stage
strategy of antenna mode selection and transceiver design has been proposed.
The secrecy SE maximization problem in NAFD CF-mMIMO systems, under the
constraints of the operation mode of antennas, UL receivers, and transmit
power of APs/UEs was studied in [21], where artificial noise is applied to
interfere with the eavesdropper’s reception.

###  I-B Research Gap and Main Contributions

The ongoing research efforts have mainly focused on two main directions: 1) SE
analysis of the NAFD CF-mMIMO under fixed mode assignment at the RAUs (APs),
and 2) Enhancing the SE by adopting the AP mode assignment and transceiver
design. However, the main drawback of these studies [13, 17, 14, 18, 19, 20,
21] is that they use the instantaneous channel state information (CSI) for
system level designs rather than the statistical CSI. Therefore, all resource
allocation and mode assignment designs must be recomputed quickly once the
small-scale fading coefficients are changed. Moreover, these designs must be
done at the CPUs, thus, the APs have to send all channel estimates to the CPUs
which will cause very large overhead, especially in CF-mMIMO where the numbers
of APs and UEs are very large. Finally, they require instantaneous CSI
knowledge at the UEs which causes huge resources and overhead in the systems
with many UEs [5]. Another major issue with the aforementioned optimization
frameworks is the sub-optimality, as the original problems are decoupled into
sub-problems, which are alternately solved via iterative algorithms.
Furthermore, all the pioneering research in the area of NAFD CF-mMIMO has
focused on optimization schemes to maximize the SE, while the trade-off
between SE and EE has remained an open research problem. From a green
perspective, energy consumption has become a critical concern for planning
5G/6G networks because mobile communication networks may contribute
significantly towards the global carbon footprint. To ensure sustainability,
5G networks should operate at low energy consumption levels while still
achieving large SE [22].

TABLE I:  List of Notation $K_{d}(K_{u})$ | Number of DL (UL) UEs
---|---
$M$ | Number of APs
$N$ | Number of antennas per-AP
${\bf g}_{mk}^{\mathtt{dl}}$ (${\bf g}_{m\ell}^{\mathtt{ul}}$) | Channel vector between DL UE $k$ (UL UE $\ell$) and AP $m$
$\beta_{mk}^{\mathtt{dl}}$ ($\beta_{mq}^{\mathtt{ul}}$) | Large-scale fading coefficient
$h_{k\ell}$ | Channel gain between the UL UE $\ell$ to the DL UE $k$
${\bf Z}_{mi}$ | Channel matrix from AP $m$ to AP $i$
$\eta_{mk}$ | Power control coefficient at the AP $m$
${\varsigma}_{\ell}$ | Transmit power control coefficient at UL UE $\ell$
$\rho_{t}$ | Normalized transmit power of each pilot symbol
$\rho_{d}$ | Maximum normalized transmit power at each AP
$\rho_{u}$ | Maximum normalized transmit power at each UL UE
$\tau_{c}$ | Length of coherence block
$\tau_{t}$ | Length of pilot sequences
$a_{m}$, $b_{m}$ | AP mode assignment binary variables
$\alpha_{m\ell}$ | LSFD weight at AP $m$ corresponding to UL UE $\ell$
$\mathcal{S}_{\mathtt{ul}}^{o}$ ($\mathcal{S}_{\mathtt{dl}}^{o}$) | Minimum SE required by the $\ell$-th UL UE ($k$-th DL UE)
$P_{\mathtt{cdl},m}$ ($P_{\mathtt{cul},m}$) | Internal power consumption at AP $m$ for DL (UL) transmissions
$\zeta_{m}$ ($\chi$) | Power amplifier efficiency at AP $m$ (UE)
$P_{\mathtt{bt},m}$ | Traffic-dependent backhaul power
$P_{\mathtt{fdl},m}$ ($P_{\mathtt{ful},m}$) | Fixed power consumption for each DL (UL) backhaul
$P_{\mathtt{D},\ell}$ $(P_{\mathtt{U},k})$ | Fixed power consumption for DL (UL) UE

The research on NAFD CF-mMIMO is still in its infancy stage and several key
challenges must be considered. While it is well-known that HD/FD CF-mMIMO is
energy efficient [23, 6], there is still no previous work on
characterizing/optimizing the SE and EE of NAFD CF-mMIMO, relying on
statistical CSI. To the authors’ best knowledge, the SE and EE maximization,
that takes into account the effects of imperfect CSI, hardware power
consumption, QoS requirement of all UEs, per-AP and UE power control, AP mode
assignment, and LSFD weights has not been studied for NAFD CF-mMIMO. The
closest work to our research is [24], wherein the performance of a NAFD CF-
mMIMO system under dynamic time division duplex (TDD) was analyzed, where the
transmission mode at each AP is scheduled so that the sum UL-DL SE is
maximized. However, in [24], the AP modes are scheduled by a greedy algorithm,
which is not optimal. Moreover, the SE requirements for UL and DL UEs were
ignored, and the impact of power control as well as LSFD for UL reception were
not investigated.

Motivated by filling the above-mentioned knowledge gap in the literature, we
consider an NAFD CF-mMIMO system with TDD operation, where HD multi-antenna
APs simultaneously serve multiple UL and DL UEs on the same time-frequency
resources. Then, the SE and EE of the NAFD CF-mMIMO system are investigated
comprehensively. The main contributions of this paper are summarized as
follows:

  * •

We propose a joint optimization approach of AP mode assignment, power control,
and LSFD weights to improve the SE and EE of NAFD CF-mMIMO systems. The
proposed approach is general and can be applied to both the traditional HD and
FD CF-mMIMO systems.

  * •

We formulate two novel optimization problems for SE and EE maximization of the
NAFD CF-mMIMO system. The formulated problems are under realistic power
consumption models (including power consumption for hardware and backhaul
links), individual SE requirements of both UL and DL UEs, per-AP and per-UL UE
power constraints.

  * •

Two new algorithms are then proposed to solve the challenging formulated
mixed-integer non-convex problems. In particular, we transform the formulated
problems into more tractable problems with continuous variables only. Then, we
solve the problems using successive convex approximation techniques.

  * •

Numerical results show that our joint optimization approach significantly
outperforms the heuristic approaches. The simulation results also confirm that
the NAFD scheme remarkably improves the performance of the CF-mMIMO in terms
of both the SE and EE over the traditional HD and FD schemes.

###  I-C Paper Organization and Notation

The remainder of this paper is organized as follows. In Section II, we
describe the NAFD CF-mMIMO model, derive the UL/DL SE expressions and present
the power consumption model. The formulation of the SE and EE optimization
problem and the derivation of their solutions are provided in Section III and
IV, respectively. In Section V, several benchmarks are presented. Numerical
results and discussions are provided in Section VI, while Section VII
concludes the paper.

Notation: We use bold upper case letters to denote matrices, and lower case
letters to denote vectors. The superscripts $(\cdot)^{*}$, $(\cdot)^{T}$ and
$(\cdot)^{\dagger}$ stand for the conjugate, transpose, and conjugate-
transpose (Hermitian), respectively; $\mathbf{I}_{N}$ denotes the $N\times N$
identity matrix. The zero mean circular symmetric complex Gaussian
distribution having variance $\sigma^{2}$ is denoted by
$\mathcal{CN}(0,\sigma^{2})$. Finally, $\mathbb{E}\\{\cdot\\}$ denotes the
statistical expectation. Table I lists some of the important notations used in
this article.

##  II System model

We consider a NAFD CF-mMIMO system under TDD operation, where $M$ APs serve
$K_{u}$ UL UEs and $K_{d}$ DL UEs. Each AP is connected to the CPU via a high-
capacity backhaul link. Each UE is equipped with one single antenna, while
each AP is equipped with $N$ antennas. All APs and UEs are HD devices. As
shown in Fig. 1, the assigned UL and DL APs perform simultaneous DL and UL
transmissions over the same frequency band. Each coherence block includes two
phases: UL training for channel estimation and UL-and-DL payload data
transmission.

###  II-A Uplink Training for Channel Estimation

The channel vector between the $k$-th DL UE ($\ell$-th UL UE) and the $m$-th
AP is denoted by ${\bf g}_{mk}^{\mathtt{dl}}\in\mathbb{C}^{N\times 1}$ (${\bf
g}_{m\ell}^{\mathtt{ul}}\in\mathbb{C}^{N\times 1}$), $\forall
k\in\operatorname{\mathcal{K}}_{d}\triangleq\\{1,\dots,K_{d}\\},\ell\in\operatorname{\mathcal{K}}_{u}\triangleq\\{1,\dots,K_{u}\\},m\in\operatorname{\mathcal{M}}\triangleq\\{1,\dots,M\\}$.
It is modeled as ${\bf
g}_{mk}^{\mathtt{dl}}=\sqrt{\beta_{mk}^{\mathtt{dl}}}\tilde{{\bf
g}}_{mk}^{\mathtt{dl}},~{}({\bf
g}_{m\ell}^{\mathtt{ul}}=\sqrt{\beta_{mq}^{\mathtt{ul}}}\tilde{{\bf
g}}_{m\ell}^{\mathtt{ul}})$, where $\beta_{mk}^{\mathtt{dl}}$
($\beta_{mq}^{\mathtt{ul}}$) is the large-scale fading coefficient and
$\tilde{{\bf g}}_{mk}^{\mathtt{dl}}\in\mathbb{C}^{N\times 1}$ ($\tilde{{\bf
g}}_{m\ell}^{\mathtt{ul}}\in\mathbb{C}^{N\times 1}$) is the small-scale fading
vector whose elements are independent and identically distributed (i.i.d.)
$\mathcal{CN}(0,1)$ random variables (RVs). Moreover, the channel gain between
the UL UE $\ell$ to the DL UE $k$ is denoted by $h_{k\ell}$. It can be modeled
as $h_{k\ell}=(\beta_{k\ell}^{\mathtt{du}})^{1/2}\tilde{h}_{k\ell}$, where
$\beta_{k\ell}^{\mathtt{du}}$ is the large-scale fading coefficient and
$\tilde{h}_{k\ell}$ is a $\mathcal{CN}(0,1)$ RV. Finally, the interference
links among the APs are modeled as Rayleigh fading channels. Let ${\bf
Z}_{mi}\in\mathbb{C}^{N\times N}$, $i\neq m$, be the channel matrix from AP
$m$ to AP $i$, $\forall m,i\in\operatorname{\mathcal{M}}$, whose elements are
i.i.d. $\mathcal{CN}(0,\beta_{mi})$ RVs. Here, we set
$\operatorname{\mathbf{Z}}_{mm}=0,\forall m$.

Figure 1:  Illustration of a NAFD CF-mMIMO system with the assigned UL and DL
APs along with the received desired and interference signals at a typical DL
UE and UL AP.

In each coherence block of length $\tau_{c}$, all UEs are assumed to transmit
their pairwisely orthogonal pilot sequences of length $\tau_{t}$ to all the
APs, which requires $\tau_{t}\geq K_{d}+K_{u}$. At AP $m$,
$\operatorname{\mathbf{g}}^{\mathtt{dl}}_{mk}$ and
$\operatorname{\mathbf{g}}^{\mathtt{ul}}_{m\ell}$ are estimated by using the
received pilot signals and the MMSE estimation technique. By following
[ngo17TWC], the MMSE estimates $\hat{{\bf g}}_{mk}^{\mathtt{dl}}$ and
$\hat{{\bf g}}_{m\ell}^{\mathtt{ul}}$ of ${\bf g}_{mk}^{\mathtt{dl}}$ and
${\bf g}_{m\ell}^{\mathtt{ul}}$ are $\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\sim\mathcal{CN}(\bm{0},\gamma_{mk}^{\mathtt{dl}}\mathbf{I}_{N})$,
and $\hat{{\bf
g}}_{m\ell}^{\mathtt{ul}}\sim\mathcal{CN}(\bm{0},\gamma_{m\ell}^{\mathtt{ul}}\mathbf{I}_{N})$,
respectively, where
$\gamma_{mk}^{\mathtt{dl}}\triangleq\frac{{\tau_{t}\rho_{t}}(\beta_{mk}^{\mathtt{dl}})^{2}}{\tau_{t}\rho_{t}\beta_{mk}^{\mathtt{dl}}+1},\gamma_{m\ell}^{\mathtt{ul}}\triangleq\frac{{\tau_{t}\rho_{t}}(\beta_{mq}^{\mathtt{ul}})^{2}}{\tau_{t}\rho_{t}\beta_{mq}^{\mathtt{ul}}+1}$,
with $\rho_{t}$ being the normalized transmit power of each pilot symbol.

###  II-B Downlink-and-Uplink Payload Data Transmission

In this phase, the APs are able to switch between the UL and DL modes. The
decision of which mode is assigned to each AP is optimized to achieve the
highest sum SE or total EE of the network as will be discussed in Sections III
and IV, respectively. Note that the AP mode selection is performed on the
large-scale fading timescale which changes very slowly with time. The binary
variables to indicate the mode assignment for each AP $m$ are defined as

$\displaystyle a_{m}\triangleq\begin{cases}1,&\text{if AP $m$ operates in the
DL mode,}\\\ 0,&\mbox{otherwise}.\end{cases},\forall m$ (1) $\displaystyle
b_{m}\triangleq\begin{cases}1,&\text{if AP $m$ operates in the UL mode,}\\\
0,&\mbox{otherwise}.\end{cases},\forall m$ (2)

Here, we have

$\displaystyle a_{m}+b_{m}=1,\forall m,$ (3)

to guarantee that AP $m$ only operates in either the DL or UL mode.

####  II-B1 Downlink payload data transmission

By using the local channel estimates, the APs perform maximum-ratio (MR)
processing (a.k.a. conjugate beamforming) for the signals transmitted to
$K_{d}$ DL UEs. Our choice of MR beamforming is inspired by the fact that it
has low computational complexity and can be implemented in a distributed
manner. Thus, most of the processing is done locally at the APs and there is
no need to exchange CSI between the APs and CPU [5, 25]. Moreover, from a
green perspective, MR processing has much less power consumption as compared
to the ZF and MMSE processing schemes [26].

Let $s_{k}^{\mathtt{dl}}$ denote the intended symbol for DL UE $k$. We assume
that $s_{k}^{\mathtt{dl}}$ is a RV with zero mean and unit variance. The
transmitted signal ${\bf x}_{m}^{d}\in\mathbb{C}^{N\times 1}$ from AP $m$ in
the DL mode is generated by first scaling each symbol $s_{k}^{\mathtt{dl}}$
with the power control coefficient

$\displaystyle\theta_{mk}\geq 0,\forall m,k,$ (4)

and then multiplying them with the MR precoding vector $\left(\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\right)^{*}$ as

$\displaystyle{\bf
x}_{m}^{\mathtt{dl}}=\sqrt{\rho_{d}}\sum_{k\in\mathcal{K}_{d}}\theta_{mk}\left(\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\right)^{*}s_{k}^{\mathtt{dl}},$ (5)

where $\rho_{d}$ is the maximum normalized transmit power at each AP. Here, we
enforce

$\displaystyle(\theta_{mk}=0,\forall k,\,\,\text{if}\,\,a_{m}=0),\forall m,$
(6)

to ensure that if AP $m$ does not operate in the DL mode, all the transmit
powers $\rho_{d}\theta_{mk}^{2},\forall k$, at AP $m$ are zero. Note that AP
$m$ is required to meet the average normalized power constraint, i.e.,
$\mathbb{E}\left\\{\|{\bf x}_{m}^{\mathtt{dl}}\|^{2}\right\\}\leq\rho_{d}$,
which can also be expressed as the following per-AP power constraint [5]

$\displaystyle\sum_{k\in\mathcal{K}_{d}}\gamma_{mk}^{\mathtt{dl}}\theta_{mk}^{2}\leq\frac{1}{N},\forall
m.$ (7)

The received signal at DL UE $k$ is written as

$\displaystyle~{}y_{k}^{\mathtt{dl}}$
$\displaystyle=\sqrt{\rho_{d}}\sum_{m\in\mathcal{M}}\theta_{mk}\left({\bf
g}_{mk}^{\mathtt{dl}}\right)^{T}\left(\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\right)^{*}s_{k}^{\mathtt{dl}}$ $\displaystyle\hskip
20.00003pt+\sqrt{\rho_{d}}\sum_{m\in\mathcal{M}}\sum_{k^{\prime}\in\mathcal{K}_{d}\setminus
k}\theta_{mk^{\prime}}\left({\bf
g}_{mk}^{\mathtt{dl}}\right)^{T}\left(\hat{{\bf
g}}_{mk^{\prime}}^{\mathtt{dl}}\right)^{*}s_{k^{\prime}}^{\mathtt{dl}}$
$\displaystyle\hskip
20.00003pt+\sum_{\ell\in\mathcal{K}_{u}}h_{k\ell}x_{\ell}^{\mathtt{ul}}+w_{k}^{\mathtt{dl}},$
(8)

where $w_{k}^{\mathtt{dl}}\sim\mathcal{CN}(0,1)$ is the AWGN at DL UE $k$. We
notice that the third term in (II-B1), is the CLI caused by the UL UEs due to
concurrent transmissions of DL and UL UEs over the same frequency band, while
$x_{\ell}^{\mathtt{ul}}$ denotes the transmit signal from the $\ell$-th UL UE.

####  II-B2 Uplink payload data transmission

The transmitted signal from UL UE $\ell$ is represented by
$x_{\ell}^{\mathtt{ul}}=\sqrt{\rho_{u}{\varsigma}_{\ell}}s_{\ell}^{\mathtt{ul}}$,
where $s_{\ell}^{\mathtt{ul}}$, with
$\mathbb{E}\left\\{|s_{\ell}^{\mathtt{ul}}|^{2}\right\\}=1$, and $\rho_{u}$
denote respectively the transmitted symbol by the $\ell$-th UL UE and the
maximum normalized transmit power at each UL UE, and ${\varsigma}_{\ell}$ is
the transmit power control coefficient at UL UE $\ell$ with

$\displaystyle 0\leq{\varsigma}_{\ell}\leq 1,\forall\ell.$ (9)

The UL APs with $b_{m}=1,\forall m$, receive a transmit signal from all UL
UEs. The received signal ${\bf y}_{m}^{\mathtt{ul}}\in\mathbb{C}^{N\times 1}$
at AP $m$ in the UL mode can be written as

$\displaystyle{\bf y}_{m}^{\mathtt{ul}}$
$\displaystyle=\sqrt{\rho_{u}}\sum_{\ell\in\mathcal{K}_{u}}\sqrt{b_{m}{\varsigma}_{\ell}}{\bf
g}_{m\ell}^{\mathtt{ul}}s_{\ell}^{\mathtt{ul}}$ $\displaystyle\hskip
20.00003pt+\sqrt{\rho_{d}}\sum_{i\in\mathcal{M}\setminus
m}\sum_{k\in\mathcal{K}_{d}}\sqrt{b_{m}}\theta_{ik}{\bf Z}_{mi}(\hat{{\bf
g}}_{ik}^{\mathtt{dl}})^{*}s_{k}^{\mathtt{dl}}$ $\displaystyle\hskip
20.00003pt+\sqrt{b_{m}}{\bf w}_{m}^{\mathtt{ul}},$ (10)

where ${\bf w}_{m}^{\mathtt{ul}}$ is the $\mathcal{CN}(0,1)$ AWGN vector. We
recall that (II-B2) captures the fact that if AP $m$ does not operate in the
UL mode, i.e., $b_{m}=0$, it does not receive any signal, i.e, ${\bf
y}_{m}^{\mathtt{ul}}=\bm{0}$.

Then, AP $m$ performs MRC processing (i.e., matched filter) by applying the
Hermitian of the (locally obtained) channel estimation vector $\hat{{\bf
g}}_{m\ell}^{\mathtt{ul}}$ to the received signal in (II-B2). The resulting
$(\hat{{\bf g}}_{m\ell}^{\mathtt{ul}})^{\dagger}{\bf y}_{m}^{\mathtt{ul}}$ is
then forwarded to the CPU for signal detection. In order to improve the
achievable UL SE, the forwarded signal is further multiplied by the LSFD
weight, $\alpha_{m\ell},\forall m,k$. The aggregated received signal for UL UE
$\ell,\forall\ell,$ at the CPU can be written as [27]

$\displaystyle r_{\ell}^{\mathtt{ul}}=\sum_{m=1}^{M}\alpha_{m\ell}(\hat{{\bf
g}}_{m\ell}^{\mathtt{ul}})^{\dagger}{\bf y}_{m}^{\mathtt{ul}}.$ (11)

Finally, $s_{\ell}^{\mathtt{ul}}$ is detected from $r_{\ell}^{\mathtt{ul}}$.
Without loss of generality, we assume that

$\displaystyle|\alpha_{m\ell}|^{2}\leq 1,\quad\forall\ell,m.$ (12)

####  II-B3 Downlink SE

In order to detect $s_{k}^{\mathtt{dl}}$ from the received signal (II-B1), the
$k$-th DL UE is assumed to rely on the stochastic channel state information.
To this end, by applying the use-and-then-forget capacity-bounding technique
[5], a closed-form expression for the achievable DL SE (in bits/s/Hz) can be
obtained as

$\displaystyle~{}\mathcal{S}_{\mathtt{dl},k}^{\mathsf{NAFD}}({\bf
a},\bm{\theta},{\bm{\varsigma}})=\frac{\tau_{c}-\tau_{t}}{\tau_{c}}\log_{2}\left(1+\frac{\Xi_{k}^{2}}{\Omega_{k}}\right),$
(13)

where ${\bf a}\triangleq\\{a_{m}\\}$,
$\bm{\theta}\triangleq\\{\theta_{mk}\\}$,
${\bm{\varsigma}}\triangleq\\{{\varsigma}_{\ell}\\},\forall m,k,\ell$, and

$\displaystyle\Xi_{k}(\bm{\theta})\triangleq
N\sqrt{\rho_{d}}\sum_{{m\in\mathcal{M}}}\theta_{mk}\gamma_{mk}^{\mathtt{dl}},$
$\displaystyle\Omega_{k}(\bm{\theta},{\bm{\varsigma}})\triangleq\rho_{d}N\\!\sum_{k^{\prime}\in\mathcal{K}_{d}}\\!\sum_{m\in\mathcal{M}}\\!\\!\\!\theta_{mk^{\prime}}^{2}\beta_{mk}^{\mathtt{dl}}\gamma_{mk^{\prime}}^{\mathtt{dl}}+\rho_{u}\\!\\!\sum_{\ell\in\mathcal{K}_{u}}\\!\\!{\varsigma_{\ell}}\beta_{k\ell}^{\mathtt{du}}\\!+\\!1.$

The detailed derivation of (13) is provided in Appendix A.

####  II-B4 Uplink SE

The CPU detects the desired signal $s_{\ell}^{\mathtt{ul}}$ from ${\bf
r}_{\ell}^{\mathtt{ul}}$ in (11). Since the CPU does not know the
instantaneous CSI, it can efficiently use statistical knowledge of the
channels when performing the detection. Using again the use-and-then-forget
capacity-bounding technique [5], we obtain the achievable UL SE (in bits/s/Hz)
of the UL UE $\ell$ as

$\displaystyle\mathcal{S}_{\mathtt{ul},\ell}^{\mathsf{NAFD}}({\bf
b},\bm{\varsigma},\bm{\theta},\bm{\alpha})=\\!\frac{\tau_{c}\\!-\\!\tau_{t}}{\tau_{c}}\log_{2}(1+\text{SINR}_{\mathtt{ul},\ell}^{\mathsf{NAFD}}({\bf
b},\bm{\varsigma},\bm{\theta},\bm{\alpha})),$ (14)

where $\text{SINR}_{\mathtt{ul},\ell}^{\mathsf{NAFD}}({\bf
b},\bm{\varsigma},\bm{\theta},\bm{\alpha})$ is given in (15) at the top of the
next page,

$\displaystyle~{}\text{SINR}_{\mathtt{ul},\ell}^{\mathsf{NAFD}}({\bf
b},\bm{\varsigma},\bm{\theta},\bm{\alpha})=\frac{N\rho_{u}\left(\sum\limits_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\sqrt{b_{m}\varsigma_{\ell}}\alpha_{m\ell}\gamma_{m\ell}^{\mathtt{ul}}\right)^{2}}{\rho_{u}\sum\limits_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\sum\limits_{q\in\mathcal{K}_{u}}\\!\\!b_{m}\varsigma_{q}\alpha_{m\ell}^{2}\beta_{mq}^{\mathtt{ul}}\gamma_{m\ell}^{\mathtt{ul}}+\rho_{d}N\\!\sum\limits_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\sum\limits_{\begin{subarray}{c}i\in\mathcal{M}\end{subarray}}\sum\limits_{k\in\mathcal{K}_{d}}b_{m}\theta_{ik}^{2}\alpha_{m\ell}^{2}\gamma_{m\ell}^{\mathtt{ul}}\beta_{mi}\gamma_{ik}^{\mathtt{dl}}+\sum\limits_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}b_{m}\alpha_{m\ell}^{2}\gamma_{m\ell}^{\mathtt{ul}}}.$
(15)

${\bf b}\triangleq\\{b_{m}\\}$, and
$\bm{\alpha}\triangleq\\{\alpha_{m\ell}\\},\forall m,k,\ell$. Note that, if
$a_{m}=b_{m}=\alpha_{mk}=1,\forall m,k,$ (14) reduces to the UL SINR of the FD
CF-mMIMO system given in [28, Eq. (27)].

###  II-C Power Consumption Model

Let $P_{\ell}$ and $P_{\mathtt{U},\ell}$ be the power consumption for
transmitting signals and the required power consumption to run circuit
components for the UL transmission at UL UE $\ell$. Moreover, denote by
$P_{\mathtt{D},k}$ the power consumption to run circuit components for the DL
transmission at DL UE $k$; $P_{\mathtt{bh},m}$ is the power consumed by the
backhaul link between the CPU and AP $m$. Therefore, the total power
consumption over the considered NAFD CF-mMIMO system is modeled as [26, 23]

$\displaystyle~{}P_{\mathtt{total}}^{\mathsf{NAFD}}$
$\displaystyle=\sum_{\ell\in\mathcal{K}_{u}}(P_{\ell}+P_{\mathtt{U},\ell})+\sum_{m\in\mathcal{M}}P_{m}^{\mathsf{NAFD}}$
$\displaystyle\hskip
20.00003pt+\sum_{k\in\operatorname{\mathcal{K}}_{d}}P_{\mathtt{D},k}+\sum_{m\in\mathcal{M}}P_{\mathtt{bh},m}^{\mathsf{NAFD}},$
(16)

where $P_{m}^{\mathsf{NAFD}}$ denotes the power consumption at AP $m$ that
includes the power consumption of the transceiver chains and the power
consumed for the DL or UL transmission. The power consumption
$P_{m}^{\mathsf{NAFD}}$ can be modeled as [26, 23]

$\displaystyle P_{m}^{\mathsf{NAFD}}(\bm{a},\bm{b},\bm{\theta})=$
$\displaystyle\begin{cases}\frac{1}{\zeta_{m}}\rho_{d}\sigma_{n}^{2}\left(N\sum_{k\in\mathcal{K}_{d}}\gamma_{mk}^{\mathtt{dl}}\theta_{mk}^{2}\right)+NP_{\mathtt{cdl},m},\\\
&\text{if $a_{m}=1$ }\\\ NP_{\mathtt{cul},m},&\mbox{if
$b_{m}=1$},\end{cases},\forall m$
$\displaystyle\overset{\eqref{a}-\eqref{sumab}}{=}a_{m}\left[\frac{1}{\zeta_{m}}\rho_{d}\sigma_{n}^{2}\left(N\sum_{k\in\mathcal{K}_{d}}\gamma_{mk}^{\mathtt{dl}}\theta_{mk}^{2}\right)+NP_{\mathtt{cdl},m}\right]$
$\displaystyle\hskip 40.00006pt+b_{m}NP_{\mathtt{cul},m}$ (17)
$\displaystyle\overset{\eqref{etaa:relation}}{=}\frac{1}{\zeta_{m}}\rho_{d}\sigma_{n}^{2}\left(N\sum_{k\in\mathcal{K}_{d}}\gamma_{mk}^{\mathtt{dl}}\theta_{mk}^{2}\right)+a_{m}NP_{\mathtt{cdl},m}$
$\displaystyle\hskip 40.00006pt+b_{m}NP_{\mathtt{cul},m},$ (18)

where $0<\zeta_{m}\leq 1$ is the power amplifier efficiency at the $m$-th AP,
$\sigma_{n}^{2}$ is the noise power; $P_{\mathtt{cdl},m}$ and
$P_{\mathtt{cul},m}$ are the internal power required to run the circuit
components (e.g., converters, mixers, and filters) related to each antenna of
AP $m$ for the DL and UL transmissions, respectively. The power consumption at
UL UE $\ell$ is given by

$\displaystyle~{}P_{\ell}=\frac{1}{\chi}\rho_{u}\sigma_{n}^{2}{\varsigma}_{\ell},$
(19)

where $\chi$ is the power amplifier efficiency at UL UEs.

Let $B$ be the system bandwidth. The backhaul rate between AP $m$ and the CPU
is

$\displaystyle~{}R_{m}^{\mathsf{NAFD}}({\bf x})$
$\displaystyle=B\Big{(}a_{m}\sum_{\ell\in\mathcal{K}_{u}}\mathcal{S}_{\mathtt{ul},\ell}^{\mathsf{NAFD}}({\bf
b},\bm{\varsigma},\bm{\theta},\bm{\alpha})$ $\displaystyle\hskip
40.00006pt+b_{m}\sum_{k\in\mathcal{K}_{d}}\mathcal{S}_{\mathtt{dl},k}^{\mathsf{NAFD}}({\bf
a},\bm{\theta},{\bm{\varsigma}})\Big{)},$ (20)

where ${\bf x}=\\{{\bf a},{\bf b},\bm{\varsigma},\bm{\theta},\bm{\alpha}\\}$.
Denote by $P_{\mathtt{fdl},m}$ (resp. $P_{\mathtt{ful},m}$) the fixed power
consumption for the DL (resp. UL) transmission of each backhaul, which is
traffic-independent and may depend on the distances between the APs and the
CPU and the system topology. Then, the power consumption of the backhaul
signal load to each AP $m$ is proportional to the backhaul rate as [23, 29]

$\displaystyle~{}P_{\mathtt{bh},m}^{\mathsf{NAFD}}=a_{m}P_{\mathtt{fdl},m}+b_{m}P_{\mathtt{ful},m}+R_{m}^{\mathsf{NAFD}}({\bf
x})P_{\mathtt{bt},m},$ (21)

where $P_{\mathtt{bt},m}$ is the traffic-dependent backhaul power (in Watt per
bit/s). By substituting (II-C), (19), and (21) into (II-C), we have

$\displaystyle~{}P_{\mathtt{total}}^{\mathsf{NAFD}}(\operatorname{\mathbf{x}})$
$\displaystyle=\sum_{m\in\mathcal{M}}\frac{N\rho_{d}\sigma_{n}^{2}}{\zeta_{m}}\left(\sum_{k\in\mathcal{K}_{d}}\gamma_{mk}^{\mathtt{dl}}\theta_{mk}^{2}\right)$
$\displaystyle\hskip
20.00003pt+\sum_{\ell\in\mathcal{K}_{u}}\frac{\rho_{u}\sigma_{n}^{2}}{\chi}{\varsigma}_{\ell}+P_{\mathtt{U,fixed}}+P_{bh}^{\mathsf{NAFD}}$
$\displaystyle\hskip
20.00003pt+\sum_{m\in\mathcal{M}}a_{m}(NP_{\mathtt{cdl},m}+P_{\mathtt{fdl},m})$
$\displaystyle\hskip
20.00003pt+\sum_{m\in\mathcal{M}}b_{m}(NP_{\mathtt{cul},m}+P_{\mathtt{ful},m}),$
(22)

where
$P_{\mathtt{U,fixed}}\triangleq\sum_{k\in\operatorname{\mathcal{K}}_{u}}P_{\mathtt{U},\ell}+\sum_{k\in\operatorname{\mathcal{K}}_{d}}P_{\mathtt{D},k}$
and

$\displaystyle P_{bh}^{\mathsf{NAFD}}$ $\displaystyle\triangleq
B\sum_{m\in\mathcal{M}}\Big{(}b_{m}\sum_{\ell\in\mathcal{K}_{u}}\mathcal{S}_{\mathtt{ul},\ell}^{\mathsf{NAFD}}({\bf
b},\bm{\varsigma},\bm{\theta},\bm{\alpha})$ $\displaystyle\hskip
20.00003pt+a_{m}\sum_{k\in\mathcal{K}_{d}}\mathcal{S}_{\mathtt{dl},k}^{\mathsf{NAFD}}({\bf
a},\bm{\theta},{\bm{\varsigma}})\Big{)}P_{\mathtt{bt},m},$ (23)

is the total rate-dependent power consumption in backhaul links.

##  III Spectral Efficiency Maximization

###  III-A Problem Formulation

In this subsection, we seek to optimize the UL and DL mode assignment vectors
$(\operatorname{\mathbf{a}},\operatorname{\mathbf{b}})$, power control
coefficients $(\operatorname{\bm{\theta}},\operatorname{\bm{\varsigma}})$, and
LSFD weight $\operatorname{\bm{\alpha}}$, to maximize the total SE, under the
constraints on per-UE SE, transmit power at each AP and UL UE. More precisely,
we formulate an optimization problem as

$\displaystyle\underset{{\bf x}}{\max}\,\,$
$\displaystyle\operatorname{\mathcal{S}}^{\mathsf{NAFD}}(\operatorname{\mathbf{x}})$
(24a) $\displaystyle\mathrm{s.t.}\,\,$
$\displaystyle\eqref{a}-\eqref{sumab},\eqref{theta},\eqref{etaa:relation},\eqref{DL:power:cons},\eqref{UL:power:cons},\eqref{eq:alphml}$
$\displaystyle\mathcal{S}_{\mathtt{ul},\ell}^{\mathsf{NAFD}}({\bf
b},\bm{\varsigma},{\bm{\theta}},\bm{\alpha})\geq\mathcal{S}_{\mathtt{ul}}^{o},~{}\forall\ell$
(24b) $\displaystyle\mathcal{S}_{\mathtt{dl},k}^{\mathsf{NAFD}}({\bf
a},\bm{\theta},{\bm{\varsigma}})\geq\mathcal{S}_{\mathtt{dl}}^{o},~{}\forall
k,$ (24c)

where ${\bf x}$ has been given after (II-C), $\mathcal{S}_{\mathtt{ul}}^{o}$
and $\mathcal{S}_{\mathtt{dl}}^{o}$ are the minimum SE required by the
$\ell$-th UL UE and $k$-th DL UE, respectively, to guarantee the QoS in the
network. Moreover, the total SE of a NAFD CF-mMIMO system is defined as

$\displaystyle\operatorname{\mathcal{S}}^{\mathsf{NAFD}}(\operatorname{\mathbf{x}})\triangleq\sum_{\ell\in\mathcal{K}_{u}}\mathcal{S}_{\mathtt{ul},\ell}^{\mathsf{NAFD}}({\bf
b},\bm{\varsigma},\bm{\theta},\bm{\alpha})+\sum_{k\in\mathcal{K}_{d}}\mathcal{S}_{\mathtt{dl},k}^{\mathsf{NAFD}}({\bf
a},\bm{\theta},{\bm{\varsigma}}).$ (25)

For the sake of algorithmic design in later sections, we first transform
problem (24) into a more tractable form as follows:

$\displaystyle\underset{{\bf x},{\bf q}_{\mathtt{ul}},{\bf
q}_{\mathtt{dl}}}{\min}\,\,$
$\displaystyle-\sum_{\ell\in\mathcal{K}_{u}}q_{\mathtt{ul},\ell}-\sum_{k\in\mathcal{K}_{d}}q_{\mathtt{dl},k}$
(26a) $\displaystyle\mathrm{s.t.}\,\,$
$\displaystyle\eqref{a}-\eqref{sumab},\eqref{theta},\eqref{etaa:relation},\eqref{DL:power:cons},\eqref{UL:power:cons},\eqref{eq:alphml}$
$\displaystyle{\mathcal{S}}_{\mathtt{ul},\ell}^{\mathsf{NAFD}}({\bf
b},{\bm{\theta}},\bm{\varsigma},\operatorname{\bm{\alpha}})\geq
q_{\mathtt{ul},\ell},\forall\ell$ (26b) $\displaystyle
q_{\mathtt{ul},\ell}\geq\mathcal{S}_{\mathtt{ul}}^{o},\forall\ell$ (26c)
$\displaystyle\mathcal{S}_{\mathtt{dl},k}^{\mathsf{NAFD}}({\bf
a},\bm{\theta},{\bm{\varsigma}})\geq q_{\mathtt{dl},k},\forall k$ (26d)
$\displaystyle q_{\mathtt{dl},k}\geq\mathcal{S}_{\mathtt{dl}}^{o},\forall k,$
(26e)

where
$\operatorname{\mathbf{q}}_{\mathtt{ul}}\triangleq\\{q_{\mathtt{ul},\ell}\\},\operatorname{\mathbf{q}}_{\mathtt{dl}}\triangleq\\{q_{\mathtt{dl},k}\\}$
are auxiliary variables. The problem (26) is a mixed-integer nonconvex
optimization problem due to the binary variables involved. Moreover, there is
a strong coupling between the continuous variables
($\bm{\theta},\bm{\varsigma},\bm{\alpha}$) and binary variables (${\bf a},{\bf
b}$), which makes problem (26) even more complicated. In what follows, we
first transform problem (26) into a more tractable form by exploiting the
special relationship between continuous and binary variables, and use
successive convex approximation techniques to solve the transformed problem
efficiently.

###  III-B Solution

Let us introduce the additional nonnegative variables
${\operatorname{\bm{\omega}}}\triangleq\\{{\omega}_{mq}\\},\bar{\operatorname{\bm{\omega}}}\triangleq\\{\bar{\omega}_{mq}\\},\tilde{\operatorname{\bm{\omega}}}\triangleq\\{\tilde{\omega}_{m\ell}\\},\hat{\operatorname{\bm{\omega}}}\triangleq\\{\hat{\omega}_{m\ell
k}\\},\tilde{\operatorname{\bm{\alpha}}}\triangleq\\{\tilde{\alpha}_{mk}\\},\hat{\operatorname{\bm{\alpha}}}\triangleq\\{\hat{\alpha}_{mk}\\},\bar{\operatorname{\bm{\eta}}}\triangleq\\{\bar{\eta}_{j\ell}\\},\hat{\operatorname{\bm{\eta}}}\triangleq\\{\hat{\eta}_{mjk\ell}\\}$,
where

$\displaystyle\omega_{m\ell}^{2}\leq b_{m}\varsigma_{\ell},\forall m,\ell$
(27) $\displaystyle\bar{\omega}_{mq}^{2}\geq b_{m}\varsigma_{q},\forall m,q$
(28)
$\displaystyle\omega_{m\ell}\alpha_{m\ell}\geq\tilde{\omega}_{m\ell},\forall
m,\ell$ (29)
$\displaystyle\bar{\omega}_{mq}\alpha_{m\ell}\leq\hat{\omega}_{m\ell
q},\forall m,\ell,q$ (30)
$\displaystyle\alpha_{m\ell}^{2}\leq\tilde{\alpha}_{m\ell},\forall m,\ell$
(31) $\displaystyle
b_{m}\tilde{\alpha}_{m\ell}\leq\hat{\alpha}_{m\ell},\forall m,\ell$ (32)
$\displaystyle\theta_{ik}^{2}\leq\bar{\eta}_{ik},\forall i,k$ (33)
$\displaystyle\hat{\alpha}_{m\ell}\bar{\eta}_{ik}\leq\hat{\eta}_{mi\ell
k},\forall m,i,\ell,k,$ (34)

which imply that

$\displaystyle\sqrt{b_{m}\varsigma_{\ell}}\alpha_{m\ell}\geq\tilde{\omega}_{m\ell},\forall
m,\ell$ (35) $\displaystyle
b_{m}\varsigma_{q}\alpha_{m\ell}^{2}\leq\hat{\omega}_{m\ell q}^{2},\forall
m,\ell,q$ (36) $\displaystyle
b_{m}\alpha_{m\ell}^{2}\leq\hat{\alpha}_{m\ell},\forall m,\ell$ (37)
$\displaystyle b_{m}\alpha_{m\ell}^{2}\theta_{ik}^{2}\leq\hat{\eta}_{mi\ell
k},\forall m,i,\ell,k.$ (38)

From (14), (35)–(38), we have

$\displaystyle\mathcal{S}_{\mathtt{ul},\ell}^{\mathsf{NAFD}}({\bf
b},\bm{\varsigma},\bm{\theta},\bm{\alpha})$
$\displaystyle\geq\widetilde{\mathcal{S}}_{\mathtt{ul},\ell}^{\mathsf{NAFD}}(\tilde{\operatorname{\bm{\omega}}},\hat{\operatorname{\bm{\omega}}},\hat{\operatorname{\bm{\eta}}},\hat{\operatorname{\bm{\alpha}}})$
$\displaystyle\triangleq\frac{\tau_{c}-\tau_{t}}{\tau_{c}}\log_{2}\bigg{(}1+\frac{\Psi_{\ell}^{2}(\tilde{\operatorname{\bm{\omega}}})}{\Phi_{\ell}(\hat{\bm{\omega}},\hat{\bm{\eta}},\hat{\bm{\alpha}})}\bigg{)},\forall\ell,$
(39)

where

$\displaystyle\Psi_{\ell}(\tilde{\operatorname{\bm{\omega}}})\triangleq\sqrt{N\rho_{u}}\sum_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\tilde{\omega}_{m\ell}\gamma_{m\ell}^{\mathtt{ul}},$
$\displaystyle\Phi_{\ell}(\hat{\bm{\omega}},\hat{\bm{\eta}},\hat{\bm{\alpha}})\triangleq\rho_{u}\\!\sum_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\\!\sum_{q\in\mathcal{K}_{u}}\hat{\omega}_{m\ell
q}^{2}\beta_{mq}^{\mathtt{ul}}\gamma_{m\ell}^{\mathtt{ul}}$
$\displaystyle\hskip
70.0001pt+\rho_{d}N\\!\sum_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\\!\sum_{\begin{subarray}{c}i\in\mathcal{M}\end{subarray}}\\!\sum_{k\in\mathcal{K}_{d}}\hat{\eta}_{mi\ell
k}\beta_{mi}\gamma_{m\ell}^{\mathtt{ul}}\gamma_{ik}^{\mathtt{dl}}$
$\displaystyle\hskip
70.0001pt+\sum_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\hat{\alpha}_{m\ell}\gamma_{m\ell}^{\mathtt{ul}}.$
(40)

Then, constraint (26b) can be replaced by

$\displaystyle\widetilde{\mathcal{S}}_{\mathtt{ul},\ell}^{\mathsf{NAFD}}(\tilde{\operatorname{\bm{\omega}}},\hat{\operatorname{\bm{\omega}}},\hat{\operatorname{\bm{\eta}}},\hat{\operatorname{\bm{\alpha}}})\geq
q_{\mathtt{ul},\ell},\forall\ell.$ (41)

By invoking (7), we replace constraint (6) by

$\displaystyle N\gamma_{mk}^{\mathtt{dl}}\theta_{mk}^{2}\leq
a_{m},\quad\forall m,k.$ (42)

To handle the binary constraints (1) and (2), we observe that for any real
number $x$, we have $x\in\\{0,1\\}\Leftrightarrow
x-x^{2}=0\Leftrightarrow(x\in[0,1]\,~{}\&\,~{}x-x^{2}\leq 0)$ [30]. Thus, (1)
and (2) can be replaced by the following equivalent constraint:

$\displaystyle C({\bf a},{\bf
b})\triangleq\sum_{m\in\mathcal{M}}(a_{m}\\!-\\!a_{m}^{2})+\sum_{m\in\mathcal{M}}(b_{m}\\!-\\!b_{m}^{2})\leq
0$ (43) $\displaystyle 0\leq a_{m}\leq 1,~{}0\leq b_{m}\leq 1,\forall m.$ (44)

Now, problem (26) can be written in a more tractable form as

$\displaystyle\underset{\widetilde{{\bf x}}\in\mathcal{F}}{\min}\,\,$
$\displaystyle-\sum_{\ell\in\mathcal{K}_{u}}q_{\mathtt{ul},\ell}-\sum_{k\in\mathcal{K}_{d}}q_{\mathtt{dl},k},$
(45)

where
$\widetilde{\operatorname{\mathbf{x}}}\triangleq\\{\operatorname{\mathbf{x}},\operatorname{\mathbf{q}}_{\mathtt{ul}},\operatorname{\mathbf{q}}_{\mathtt{dl}},\operatorname{\bm{\omega}},\bar{\operatorname{\bm{\omega}}},\tilde{\operatorname{\bm{\omega}}},\hat{\operatorname{\bm{\omega}}},\tilde{\operatorname{\bm{\alpha}}},\hat{\operatorname{\bm{\alpha}}},\bar{\operatorname{\bm{\eta}}},\hat{\operatorname{\bm{\eta}}}\\}$,
$\mathcal{F}\\!\triangleq\\!\\{\eqref{sumab},\eqref{theta},\eqref{DL:power:cons},\eqref{UL:power:cons},\eqref{eq:alphml},\eqref{UL:QoS:cons:2}-\eqref{DL:QoS:cons:2},\eqref{omega}-\eqref{etahat},\eqref{UL:QoS:cons:1:equi}-\eqref{abrelax}\\}$
is a feasible set. To this end, we consider the following problem

$\displaystyle\underset{\widetilde{{\bf
x}}\in\widetilde{\mathcal{F}}}{\min}\,\,$
$\displaystyle\mathcal{L}_{\mathtt{SE}}(\widetilde{{\bf x}}),$ (46)

where $\mathcal{L}_{\mathtt{SE}}(\widetilde{{\bf
x}})\triangleq-\sum_{\ell\in\mathcal{K}_{u}}q_{\mathtt{ul},\ell}-\sum_{k\in\mathcal{K}_{d}}q_{\mathtt{dl},k}+\lambda
C({\bf a},{\bf b})$ is the Lagrangian of (45) and $\lambda$ is the Lagrangian
multiplier corresponding to constraint (43). Here,
$\widetilde{\mathcal{F}}\triangleq\mathcal{F}\setminus\\{\eqref{C}\\}$.

######  Proposition 1.

The values $C_{\lambda}$ of $C$ at the solution of (46) corresponding to
$\lambda$ converge to $0$ as $\lambda\rightarrow+\infty$. Also, problem (45)
has strong duality, i.e.,

$\underset{\widetilde{{\bf
x}}\in\mathcal{F}}{\min}\,\,-\sum_{\ell\in\mathcal{K}_{u}}q_{\mathtt{ul},\ell}-\sum_{k\in\mathcal{K}_{d}}q_{\mathtt{dl},k}=\underset{\lambda\geq
0}{\sup}\,\,\underset{\widetilde{{\bf
x}}\in\widetilde{\mathcal{F}}}{\min}\,\,\mathcal{L}_{\mathtt{SE}}(\widetilde{{\bf
x}}).$ (47)

Then, problem (45) is equivalent to problem (46) at the optimal solution
$\lambda^{*}\geq 0$ of the sup-min problem in (47).

###### Proof.

The proof has a similar procedure as the proof of [30, Proposition 1], and
hence, omitted. ∎

Note that it is theoretically required to have $C_{\lambda}=0$ in order to
obtain the optimal solution to problem (45). According to Proposition 1,
$C_{\lambda}$ converges to $0$ as $\lambda\to+\infty$. For practical
implementation, it is acceptable for $C_{\lambda}$ to be sufficiently small
with a sufficiently large value of $\lambda$. In our numerical experiments,
for $\varepsilon=5\times 10^{-5}$, we see that $\lambda=1$ is enough to ensure
that $C_{\lambda}/(MK)\leq\varepsilon$. This way of selecting $\lambda$ has
been widely used in the literature, e.g., see [30] and references therein.

Problem (46) is still difficult to solve due to the non-convex constraints
(26d) and (41). To deal with constraint (26d), we observe that

$\displaystyle\log\bigg{(}1+\frac{x^{2}}{y}\bigg{)}$
$\displaystyle\geq\log\bigg{(}1+\frac{(x^{(n)})^{2}}{y^{(n)}}\bigg{)}-\frac{(x^{(n)})^{2}}{y^{(n)}}$
$\displaystyle+2\frac{x^{(n)}x}{y^{(n)}}-\frac{(x^{(n)})^{2}(x^{2}+y)}{y^{(n)}((x^{(n)})^{2}+y^{(n)})},$
(48)

where $x>0,y>0$ [31, Eq. (40)]. Therefore,
$\mathcal{S}_{\mathtt{dl},k}^{\mathsf{NAFD}}({\bf
a},\bm{\theta},{\bm{\varsigma}})$ has a concave lower bound
$\widehat{\mathcal{S}}_{\mathtt{dl},k}^{\mathsf{NAFD}}({\bf
a},\bm{\theta},{\bm{\varsigma}})$ that is given as

$\displaystyle\widehat{\mathcal{S}}_{\mathtt{dl},k}^{\mathsf{NAFD}}(\bm{\theta},{\bm{\varsigma}})$
$\displaystyle\triangleq\frac{\tau_{c}-\tau_{t}}{\tau_{c}\log
2}\Bigg{[}\log\bigg{(}1\\!+\\!\frac{(\Xi_{k}^{(n)})^{2}}{\Omega_{k}^{(n)}}\bigg{)}\\!-\\!\frac{(\Xi_{k}^{(n)})^{2}}{\Omega_{k}^{(n)}}$
$\displaystyle\\!+\\!2\frac{\Xi_{k}^{(n)}\Xi_{k}}{\Omega_{k}^{(n)}}\\!-\\!\frac{(\Xi_{k}^{(n)})^{2}(\Xi_{k}^{2}+\Omega_{k})}{\Omega_{k}^{(n)}((\Xi_{k}^{(n)})^{2}\\!+\\!\Omega_{k}^{(n)})}\Bigg{]},$
(49)

where $\Xi_{k}$ and $\Omega_{k}$ are defined in (13). Then, constraint (26d)
is approximated by the following convex constraint

$\displaystyle\widehat{\mathcal{S}}_{\mathtt{dl},k}^{\mathsf{NAFD}}(\bm{\theta},{\bm{\varsigma}})\geq
q_{\mathtt{dl},k},\forall k.$ (50)

Similarly, to deal with constraint (41), we see that the concave lower bound
of
$\widetilde{\mathcal{S}}_{\mathtt{ul},\ell}^{\mathsf{NAFD}}(\tilde{\operatorname{\bm{\omega}}},\hat{\operatorname{\bm{\omega}}},\hat{\operatorname{\bm{\eta}}},\hat{\operatorname{\bm{\alpha}}})$
is given by

$\displaystyle\widehat{\mathcal{S}}_{\mathtt{ul},\ell}^{\mathsf{NAFD}}(\tilde{\operatorname{\bm{\omega}}},\hat{\operatorname{\bm{\omega}}},\hat{\operatorname{\bm{\eta}}},\hat{\operatorname{\bm{\alpha}}})$
$\displaystyle\triangleq\frac{\tau_{c}\\!-\\!\tau_{t}}{\tau_{c}\log
2}\Bigg{[}\log\bigg{(}1\\!+\\!\frac{(\Psi_{\ell}^{(n)})^{2}}{\Phi_{\ell}^{(n)}}\bigg{)}$
$\displaystyle-\frac{(\Psi_{\ell}^{(n)})^{2}}{\Phi_{\ell}^{(n)}}+2\frac{\Psi_{\ell}^{(n)}\Psi_{\ell}}{\Phi_{\ell}^{(n)}}-\frac{(\Psi_{\ell}^{(n)})^{2}(\Psi_{\ell}^{2}+\Phi_{\ell})}{\Phi_{\ell}^{(n)}((\Psi_{\ell}^{(n)})^{2}+\Phi_{\ell}^{(n)})}\Bigg{]}.$
(51)

Then, constraint (41) is then approximated by the following convex constraint

$\displaystyle\widehat{\mathcal{S}}_{\mathtt{ul},\ell}^{\mathsf{NAFD}}(\tilde{\operatorname{\bm{\omega}}},\hat{\operatorname{\bm{\omega}}},\hat{\operatorname{\bm{\eta}}},\hat{\operatorname{\bm{\alpha}}})\geq
q_{\mathtt{ul},\ell},\forall\ell.$ (52)

By invoking the following lower bounds [31]

$\displaystyle xy\leq 0.25[(x+y)^{2}-2(x^{(n)}-y^{(n)})(x-y)$
$\displaystyle\hskip 20.00003pt+(x^{(n)}-y^{(n)})^{2}]$ (53) $\displaystyle-
xy\leq 0.25[(x-y)^{2}-2(x^{(n)}$ $\displaystyle\hskip
20.00003pt+\\!y^{(n)})(x+y)+(x^{(n)}+y^{(n)})^{2}],$ (54)

where $\forall x\geq 0,y\geq 0$, the convex upper bound of $C({\bf a},{\bf
b})$ is given by

$\displaystyle\widetilde{C}({\bf a},{\bf
b})\triangleq\sum_{m\in\mathcal{M}}\left[a_{m}-2a_{m}^{(n)}a_{m}+(a_{m}^{(n)})^{2}\right]$
$\displaystyle\hskip
50.00008pt+\sum_{m\in\mathcal{M}}\left[b_{m}-2b_{m}^{(n)}b_{m}+(b_{m}^{(n)})^{2}\right].$
(55)

Similarly, constraints (27)–(30), (32), and (34) can be approximated by the
following convex constraints

$\displaystyle\omega_{m\ell}^{2}+0.25[(b_{m}-\varsigma_{\ell})^{2}-2(b_{m}^{(n)}+\varsigma_{\ell}^{(n)})(b_{m}+\varsigma_{\ell})$
$\displaystyle\hskip 30.00005pt+(b_{m}^{(n)}+\varsigma_{\ell}^{(n)})^{2}]\leq
0,\forall m,\ell$ (56)
$\displaystyle\\!\\!\\!\\!0.25[(b_{m}\\!+\\!\varsigma_{q})^{2}-2(b_{m}^{(n)}-\varsigma_{q}^{(n)})(b_{m}-\varsigma_{q})+(b_{m}^{(n)}-\varsigma_{q}^{(n)})^{2}]$
$\displaystyle\hskip
30.00005pt-2\bar{\omega}_{m{q}}^{(n)}\bar{\omega}_{m{q}}+(\bar{\omega}_{m{q}}^{(n)})^{2}\leq
0,\forall m,q$ (57)
$\displaystyle\\!\\!\\!\\!\tilde{\omega}_{m\ell}+0.25[(\omega_{m\ell}-\alpha_{m\ell})^{2}-2(\omega_{m\ell}^{(n)}+\alpha_{m\ell}^{(n)})(\omega_{m\ell}+\alpha_{m\ell})$
$\displaystyle\hskip
30.00005pt+(\omega_{m\ell}^{(n)}+\alpha_{m\ell}^{(n)})^{2}]\leq 0,\forall
m,\ell$ (58)
$\displaystyle\\!\\!\\!\\!0.25[(\bar{\omega}_{mq}+\alpha_{m\ell})^{2}-2(\bar{\omega}_{mq}^{(n)}-\alpha_{m\ell}^{(n)})(\bar{\omega}_{mq}-\alpha_{m\ell})$
$\displaystyle\hskip
30.00005pt+(\bar{\omega}_{mq}^{(n)}-\alpha_{m\ell}^{(n)})^{2}]-\hat{\omega}_{m\ell
q}\leq 0,\forall m,\ell,q$ (59)
$\displaystyle\\!\\!\\!\\!0.25[(b_{m}+\tilde{\alpha}_{m\ell})^{2}-2(b_{m}^{(n)}-\tilde{\alpha}_{m\ell}^{(n)})(b_{m}-\tilde{\alpha}_{m\ell})$
$\displaystyle\hskip
30.00005pt+(b_{m}^{(n)}-\tilde{\alpha}_{m\ell}^{(n)})^{2}]-\hat{\alpha}_{m\ell}\leq
0,\forall m,\ell$ (60)
$\displaystyle\\!\\!\\!\\!0.25[(\hat{\alpha}_{m\ell}+\bar{\eta}_{ik})^{2}-2(\hat{\alpha}_{m\ell}^{(n)}-\bar{\eta}_{ik}^{(n)})(\hat{\alpha}_{mk}-\bar{\eta}_{ik})$
$\displaystyle\hskip
30.00005pt+(\hat{\alpha}_{m\ell}^{(n)}-\bar{\eta}_{ik}^{(n)})^{2}]-\hat{\eta}_{mi\ell
k}\leq 0,\forall m,i,\ell,k.$ (61)

At iteration $(n+1)$, for a given point $\widetilde{{\bf x}}^{(n)}$, problem
(46) can finally be approximated by the following convex problem:

$\displaystyle\underset{\widetilde{{\bf
x}}\in\widehat{\mathcal{F}}}{\min}\,\,$
$\displaystyle\widehat{\mathcal{L}}_{\mathtt{SE}^{\mathsf{NAFD}}}(\widetilde{{\bf
x}}),$ (62)

where $\widehat{\mathcal{L}}_{\mathtt{SE}^{\mathsf{NAFD}}}(\widetilde{{\bf
x}})=-\sum_{\ell\in\mathcal{K}_{u}}q_{\mathtt{ul},\ell}-\sum_{k\in\mathcal{K}_{d}}q_{\mathtt{dl},k}+\lambda\widetilde{C}({\bf
a},{\bf b})$,
$\widehat{\mathcal{F}}\triangleq\\{\widetilde{\mathcal{F}},\eqref{DL:QoS:cons:approx},\eqref{UL:QoS:cons:approx},\eqref{omega:1:convex}-\eqref{etahat:convex}\\}\setminus\\{\eqref{DL:QoS:cons:1},\eqref{omega}\\\
-\eqref{omegahat},\eqref{alphahat},\eqref{etahat},\eqref{UL:QoS:cons:1:equi}\\}$
is a convex feasible set. In Algorithm 1, we outline the main steps to solve
problem (45). Starting from a random point $\widetilde{{\bf
x}}\in\widehat{\mathcal{F}}$, we solve (62) to obtain its optimal solution
$\widetilde{{\bf x}}^{*}$, and use $\widetilde{{\bf x}}^{*}$ as an initial
point in the next iteration. The algorithm terminates when an accuracy level
is reached. Algorithm 1 will converge to a stationary point, i.e., a Fritz
John solution, of problem (46) (hence (45) or (24)). The proof of this
convergence property uses similar steps in the proof of [30, Proposition 2],
and hence, is omitted due to lack of space.

Algorithm 1 requires solving a series of convex problems (62). For ease of
presentation, if we let $K_{d}=K_{u}=K$, problem (62) can be transformed to an
equivalent problem that involves $A_{v}\triangleq 2M+3K+9MK+MK^{2}+M^{2}K^{2}$
real-valued scalar variables, $A_{l}\triangleq 5M+4K+9MK+MK^{2}+M^{2}K^{2}$
linear constraints, $A_{q}\triangleq M+2K+7MK+MK^{2}+M^{2}K^{2}$ quadratic
constraints. Therefore, the algorithm for solving problem (46) requires a
complexity of
$\operatorname{\mathcal{O}}(\sqrt{A_{l}+A_{q}}(A_{v}+A_{l}+A_{q})A_{v}^{2})$.

Algorithm 1 Solving problem (46)

1: Initialize: $n\\!=\\!0$, $\lambda>1$, a random point $\widetilde{{\bf
x}}^{(0)}\\!\in\\!\widehat{\mathcal{F}}$.

2: repeat

3: Update $n=n+1$

4: Solve (62) to obtain its optimal solution $\widetilde{{\bf x}}^{*}$

5: Update $\widetilde{{\bf x}}^{(n)}=\widetilde{{\bf x}}^{*}$

6: until convergence

######  Remark 1 (Initial point and infeasible SE maximization problem).

Our Algorithm 1 needs a feasible point
$\widetilde{\operatorname{\mathbf{x}}}\in\widehat{\operatorname{\mathcal{F}}}$
to start. It is easy to find a point
$\widetilde{\operatorname{\mathbf{x}}}\in\widehat{\operatorname{\mathcal{F}}}\setminus\\{\eqref{UL:QoS:cons:2},\eqref{DL:QoS:cons:2}\\}$
by using a random procedure and letting the constraints in
$\widehat{\operatorname{\mathcal{F}}}\setminus\\{\eqref{UL:QoS:cons:2},\eqref{DL:QoS:cons:2}\\}$
happen with equality. However, when the minimum individual SEs required for
QoS, i.e., $\mathcal{S}_{\mathtt{ul}}^{o}$ and
$\mathcal{S}_{\mathtt{dl}}^{o}$, are large but the UEs have unfavourable links
to the APs, the QoS constraints (26c) and (26e) are not easy to satisfy. In
this case, we start with a random point
$\widetilde{\operatorname{\mathbf{x}}}\in\widehat{\operatorname{\mathcal{F}}}\setminus\\{\eqref{UL:QoS:cons:2},\eqref{DL:QoS:cons:2}\\}$
and solve the following problem (instead of problem (62)) in each iteration of
Algorithm 1

$\displaystyle\underset{\widetilde{{\bf
x}}\in\widehat{\mathcal{F}}\setminus\\{\eqref{UL:QoS:cons:2},\eqref{DL:QoS:cons:2}\\},\operatorname{\mathbf{z}}_{\mathtt{ul}},\operatorname{\mathbf{z}}_{\mathtt{dl}}}{\min}\,\,$
$\displaystyle\widehat{\mathcal{L}}_{\mathtt{SE}^{\mathsf{NAFD}}}(\widetilde{{\bf
x}})+\phi z,$ (63a)
$\displaystyle\mathrm{s.t.}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle
q_{\mathtt{ul},\ell}+z_{\mathtt{ul},\ell}\geq\mathcal{S}_{\mathtt{ul}}^{o},\forall\ell$
(63b) $\displaystyle
q_{\mathtt{dl},k}+z_{\mathtt{dl},k}\geq\mathcal{S}_{\mathtt{dl}}^{o},\forall
k$ (63c) $\displaystyle z_{\mathtt{ul},\ell}\geq 0,z_{\mathtt{dl},k}\geq
0,\forall k,\ell,$ (63d)

where
$z\triangleq\sum_{\ell\in\operatorname{\mathcal{K}}_{u}}z_{\mathtt{ul},\ell}+\sum_{k\in\operatorname{\mathcal{K}}_{d}}z_{\mathtt{dl},k}$
and $\phi>0$ is a penalty parameter. Here,
$\operatorname{\mathbf{z}}_{\mathtt{ul}}\triangleq\\{z_{\mathtt{ul},\ell}\\},\operatorname{\mathbf{z}}_{\mathtt{dl}}\triangleq\\{z_{\mathtt{dl},k}\\}$
are additional variables that makes constraints (26c), (26e) satisfied if they
are sufficiently small. Since
$\operatorname{\mathbf{z}}_{\mathtt{ul}},\operatorname{\mathbf{z}}_{\mathtt{dl}}$
are nonnegative and (63) is a minimization problem,
$\operatorname{\mathbf{z}}_{\mathtt{ul}},\operatorname{\mathbf{z}}_{\mathtt{dl}}$
are forced to approach $0$ during the iterative process of Algorithm 1. When
Algorithm 1 converges, if $z$ is smaller than a predefined error threshold,
the problem (45) or (24) is feasible with constraints (26c), (26e) satisfied,
and we take the converged point as the final solution. Otherwise, problem (45)
or (24) is considered as an infeasible problem.

##  IV Energy Efficiency Maximization

###  IV-A Problem Formulation

In this subsection, we aim at optimizing the mode assignment of the APs
$(\operatorname{\mathbf{a}},\operatorname{\mathbf{b}})$, power coefficients
$(\operatorname{\bm{\theta}},\operatorname{\bm{\varsigma}})$, and LSFD weight
$\operatorname{\bm{\alpha}}$ to maximize the total EE, under the constraints
on QoS requirements for each UE, maximum transmit power at each DL AP and each
UL UE. The total EE (in bit/Joule) is defined as the sum throughput (bit/s)
divided by the total power consumption (Watt) in the network

$\displaystyle~{}\mathtt{EE}^{\mathsf{NAFD}}(\operatorname{\mathbf{x}})=\frac{B.\mathcal{S}^{\mathsf{NAFD}}({\bf
x})}{\frac{\tau_{c}-\tau_{t}}{\tau_{c}}P_{\mathtt{total}}^{\mathsf{NAFD}}(\operatorname{\mathbf{x}})},$
(64)

where ${\bf x}$ is defined in (II-C). More precisely, the optimization problem
is formulated as follows:

$\displaystyle\underset{{\bf x}}{\max}\,\,$ $\displaystyle\hskip
10.00002pt\mathtt{EE}^{\mathsf{NAFD}}(\operatorname{\mathbf{x}})$ (65a)
$\displaystyle\mathrm{s.t.}\,\,$ $\displaystyle\hskip
10.00002pt\eqref{a}-\eqref{sumab},\eqref{theta},\eqref{etaa:relation},\eqref{DL:power:cons},\eqref{UL:power:cons},\eqref{eq:alphml},\eqref{UL:QoS:cons},\eqref{DL:QoS:cons}.$
(65b)

Problem (65a) is also a nonconvex mixed-integer problem. However, it has a
tight coupling of the AP mode assignment variables (${\bf a},{\bf b}$) and the
power consumption of backhaul signalling loads, which is not the case in the
SE maximization problem (24). On one hand, this is the issue that makes the
mathematical structure of problem (65a) significantly different from that of
problem (24). Thus, we cannot apply straightforwardly the proposed Algorithm 1
to solve problem (65a). On the other hand, this issue makes problem (65a)
technically much more challenging than problem (24) and difficult to find its
optimal solution. Therefore, instead of finding the optimal solution to the EE
problem, we aim to find its suboptimal solution.

First, we see that by the definition of $P_{bh}^{\mathsf{NAFD}}$ in (II-C), we
have

$\displaystyle P_{bh}^{\mathsf{NAFD}}<P_{bh,full}^{\mathsf{NAFD}}\triangleq
B\sum_{m\in\operatorname{\mathcal{M}}}\operatorname{\mathcal{S}}^{\mathsf{NAFD}}(\operatorname{\mathbf{x}})P_{\mathtt{bt},m},$
(66)

which is the rate-dependent power consumption when each AP shares with the CPU
full backhaul signaling loads for all the DL and UL UEs. Then,
$P_{\mathtt{total}}^{\mathsf{NAFD}}(\operatorname{\mathbf{x}})$ is always
smaller than

$\displaystyle
P_{\mathtt{total,fbh}}^{\mathsf{NAFD}}(\operatorname{\mathbf{x}})$
$\displaystyle\triangleq\sum_{m\in\mathcal{M}}\frac{N\rho_{d}\sigma_{n}^{2}}{\zeta_{m}}\left(\sum_{k\in\mathcal{K}_{d}}\gamma_{mk}^{\mathtt{dl}}\theta_{mk}^{2}\right)$
$\displaystyle\hskip
20.00003pt+\sum_{\ell\in\mathcal{K}_{u}}\frac{\rho_{u}\sigma_{n}^{2}}{\chi}{\varsigma}_{\ell}+P_{\mathtt{U,fixed}}+P_{bh,full}^{\mathsf{NAFD}}$
$\displaystyle\hskip
20.00003pt+\sum_{m\in\mathcal{M}}a_{m}(NP_{\mathtt{cdl},m}+P_{\mathtt{fdl},m})$
$\displaystyle\hskip
20.00003pt+\sum_{m\in\mathcal{M}}b_{m}(NP_{\mathtt{cul},m}+P_{\mathtt{ful},m}).$
(67)

Therefore, we have

$\displaystyle\operatorname{\mathtt{EE}}_{fullbh}^{\mathsf{NAFD}}(\operatorname{\mathbf{x}})\triangleq\frac{B.\mathcal{S}^{\mathsf{NAFD}}({\bf
x})}{\frac{\tau_{c}-\tau_{t}}{\tau_{c}}P_{\mathtt{total,fbh}}^{\mathsf{NAFD}}(\operatorname{\mathbf{x}})}<\operatorname{\mathtt{EE}}^{\mathsf{NAFD}}(\operatorname{\mathbf{x}}).$
(68)

Now, instead of solving problem (65a), we aim to solve the following problem

$\displaystyle\underset{{\bf x}}{\max}\,\,$ $\displaystyle\hskip
10.00002pt\mathtt{EE}_{fullbh}^{\mathsf{NAFD}}(\operatorname{\mathbf{x}})$
(69a) $\displaystyle\mathrm{s.t.}\,\,$ $\displaystyle\hskip
10.00002pt\eqref{a}-\eqref{sumab},\eqref{theta},\eqref{etaa:relation},\eqref{DL:power:cons},\eqref{UL:power:cons},\eqref{eq:alphml},\eqref{UL:QoS:cons},\eqref{DL:QoS:cons}.$
(69b)

Note that the solution to problem (69a) is not the optimal solution to problem
(65a) but can be sufficiently close to this solution, which is shown later in
the simulation results of Section VI. For problem (69a), we observe that

$\displaystyle\mathtt{EE}_{fullbh}^{\mathsf{NAFD}}$
$\displaystyle=\frac{B.\mathcal{S}^{\mathsf{NAFD}}({\bf
x})}{\frac{\tau_{c}-\tau_{t}}{\tau_{c}}P_{\mathtt{total,fbh}}^{\mathsf{NAFD}}(\operatorname{\mathbf{x}})}=\frac{1}{\frac{\tau_{c}-\tau_{t}}{\tau_{c}}\frac{P_{\mathtt{total,fbh}}^{\mathsf{NAFD}}(\operatorname{\mathbf{x}})}{B.\mathcal{S}^{\mathsf{NAFD}}({\bf
x})}}$
$\displaystyle=\frac{1}{\frac{\tau_{c}-\tau_{t}}{\tau_{c}}\left(\frac{\widetilde{P}(\operatorname{\mathbf{x}})}{B.\mathcal{S}^{\mathsf{NAFD}}({\bf
x})}+\sum_{m\in\mathcal{M}}P_{\mathtt{bt},m}\right)},$ (70)

where

$\displaystyle\widetilde{P}(\operatorname{\mathbf{x}})\triangleq$
$\displaystyle\sum_{m\in\mathcal{M}}\frac{N\rho_{d}\sigma_{n}^{2}}{\zeta_{m}}\left(\sum_{k\in\mathcal{K}_{d}}\gamma_{mk}^{\mathtt{dl}}\theta_{mk}^{2}\right)+\sum_{\ell\in\mathcal{K}_{u}}\frac{\rho_{u}\sigma_{n}^{2}}{\chi}{\varsigma}_{\ell}$
$\displaystyle+P_{\mathtt{U,fixed}}+\sum_{m\in\mathcal{M}}a_{m}(NP_{\mathtt{cdl},m}+P_{\mathtt{fdl},m})$
$\displaystyle+\sum_{m\in\mathcal{M}}b_{m}(NP_{\mathtt{cul},m}+P_{\mathtt{ful},m}).$
(71)

By invoking (IV-A), problem (69a) can be rewritten as

$\displaystyle\underset{{\bf x}}{\max}\,\,$ $\displaystyle\hskip
10.00002pt\frac{B.\mathcal{S}^{\mathsf{NAFD}}({\bf
x})}{\widetilde{P}(\operatorname{\mathbf{x}})}$ (72a)
$\displaystyle\mathrm{s.t.}\,\,$ $\displaystyle\hskip
10.00002pt\eqref{a}-\eqref{sumab},\eqref{theta},\eqref{etaa:relation},\eqref{DL:power:cons},\eqref{UL:power:cons},\eqref{eq:alphml},\eqref{UL:QoS:cons},\eqref{DL:QoS:cons}.$
(72b)

Problem (72a) is then equivalent to

$\displaystyle\underset{\operatorname{\mathbf{x}},t,\hat{p}}{\min}\,\,$
$\displaystyle-t$ (73a) $\displaystyle\mathrm{s.t.}\,\,$ $\displaystyle\hskip
10.00002pt\eqref{a}-\eqref{sumab},\eqref{theta},\eqref{etaa:relation},\eqref{DL:power:cons},\eqref{UL:power:cons},\eqref{eq:alphml},\eqref{UL:QoS:cons:1}-\eqref{DL:QoS:cons:2}$
(73b) $\displaystyle\hskip
10.00002ptt\hat{p}\leq\sum_{\ell\in\mathcal{K}_{u}}Bq_{\mathtt{ul},\ell}+\sum_{k\in\mathcal{K}_{d}}Bq_{\mathtt{dl},k}$
(73c) $\displaystyle\hskip
10.00002pt\hat{p}\geq\widetilde{P}(\operatorname{\mathbf{x}}),$ (73d)

where $t$ and $\hat{p}$ are additional variables. In the following, we
transform problem (73) into a more tractable form which is then solved by
successive convex approximation techniques.

###  IV-B Solution

Using similar steps to transform the problem of maximizing the sum SE into a
more tractable one as discussed in Section III-B, problem (73) can be
rewritten as

$\displaystyle\underset{\widehat{\operatorname{\mathbf{x}}}\in\mathcal{H}}{\min}\,\,$
$\displaystyle-t,$ (74)

where
$\mathcal{H}\\!\triangleq\\!\\{\eqref{a}-\eqref{sumab},\eqref{theta},\eqref{etaa:relation},\eqref{DL:power:cons},\eqref{UL:power:cons},\eqref{eq:alphml},\eqref{UL:QoS:cons:2}-\eqref{DL:QoS:cons:2},\eqref{omega}-\eqref{etahat},\eqref{UL:QoS:cons:1:equi}-\eqref{abrelax},\eqref{tphat},\eqref{phat}\\}$.
Now, we consider the following problem

$\displaystyle\underset{\widehat{\operatorname{\mathbf{x}}}\in\operatorname{\mathcal{H}}}{\min}\,\,$
$\displaystyle\mathcal{L}_{\mathtt{EE}^{\mathsf{NAFD}}}(\widehat{\operatorname{\mathbf{x}}}),$
(75)

where
$\mathcal{L}_{\mathtt{EE}^{\mathsf{NAFD}}}(\widehat{\operatorname{\mathbf{x}}})\triangleq-t+\lambda
C({\bf a},{\bf b})$ is the Lagrangian of (74) and $\lambda$ is the Lagrangian
multiplier corresponding to constraint (43). Here,
$\widetilde{\operatorname{\mathcal{H}}}\triangleq\operatorname{\mathcal{H}}\setminus\\{\eqref{C}\\}$.

######  Proposition 2.

The values $C_{\lambda}$ of $C$ at the solution of (75) corresponding to
$\lambda$ converge to $0$ as $\lambda\rightarrow+\infty$. Also, problem (74)
has strong duality, i.e.,

$\underset{\widehat{{\bf
x}}\in\operatorname{\mathcal{H}}}{\min}\,\,-t=\underset{\lambda\geq
0}{\sup}\,\,\underset{\widehat{{\bf
x}}\in\widetilde{\operatorname{\mathcal{H}}}}{\min}\,\,\mathcal{L}_{\mathtt{EE}^{\mathsf{NAFD}}}(\widehat{{\bf
x}}).$ (76)

Then, problem (74) is equivalent to problem (75) at the optimal solution
$\lambda^{*}\geq 0$ of the sup-min problem in (76).

Similarly, the proof of Proposition 2 follows [30], and hence, omitted.
According to Proposition 2, $C_{\lambda}$ converges to $0$ as
$\lambda\to+\infty$, and the optimal solution to problem (74) is obtained. We
recall that for practical implementation, it is acceptable for $C_{\lambda}$
to be sufficiently small with a sufficiently large value of $\lambda$. In our
numerical experiments, for $\varepsilon=5\times 10^{-5}$, we see that
$\lambda=10$ is enough to ensure that $C_{\lambda}/(MK)\leq\varepsilon$.

From (III-B), the nonconvex constraint (73c) can be approximated by the
following convex constraint

$\displaystyle
0.25[(t+\hat{p})^{2}-2(t^{(n)}-\hat{p}^{(n)})(t-\hat{p})+(t^{(n)}-\hat{p}^{(n)})^{2}]$
$\displaystyle\hskip 20.00003pt\leq
B\sum_{\ell\in\mathcal{K}_{u}}q_{\mathtt{ul},\ell}+\sum_{k\in\mathcal{K}_{d}}q_{\mathtt{dl},k}.$
(77)

To deal with the other nonconvex constraints of problem (75), similar
approximation techniques in Section VI are used. Finally, at iteration
$(n+1)$, for a given point $\widehat{{\bf x}}^{(n)}$, problem (75) can finally
be approximated by the following convex problem:

$\displaystyle\underset{\widehat{{\bf x}}\in\widehat{\mathcal{H}}}{\min}\,\,$
$\displaystyle\widehat{\mathcal{L}}_{\mathtt{EE}^{\mathsf{NAFD}}}(\widehat{{\bf
x}}),$ (78)

where $\widehat{\mathcal{L}}_{\mathtt{EE}^{\mathsf{NAFD}}}(\widetilde{{\bf
x}})=-t+\lambda\widetilde{C}({\bf a},{\bf b})$,
$\widehat{\mathcal{H}}\triangleq\\{\widehat{\operatorname{\mathcal{F}}},\eqref{tphat:convex}\\}\setminus\\{\eqref{tphat}\\}$
is a convex feasible set. In Algorithm 2, we outline the main steps to solve
problem (74). Starting from a random point $\widehat{{\bf
x}}\in\widehat{\mathcal{H}}$, we solve (78) to obtain its optimal solution
$\widehat{{\bf x}}^{*}$, and use $\widehat{{\bf x}}^{*}$ as an initial point
in the next iteration. The algorithm terminates when an accuracy level is
reached. Algorithm 2 will converge to a stationary point, i.e., a Fritz John
solution, of problem (74) (hence (73) or (69a)). The proof of this convergence
property also uses similar steps in the proof of [30, Proposition 2], and
hence, is omitted due to lack of space.

Algorithm 2 Solving problem (75)

1: Initialize: $n\\!=\\!0$, $\lambda>1$, a random point $\widehat{{\bf
x}}^{(0)}\\!\in\\!\widehat{\mathcal{H}}$.

2: repeat

3: Update $n=n+1$

4: Solve (78) to obtain its optimal solution $\widehat{{\bf x}}^{*}$

5: Update $\widehat{{\bf x}}^{(n)}=\widehat{{\bf x}}^{*}$

6: until convergence

Algorithm 2 requires solving a series of convex problems (78). Problem (78)
can be transformed to an equivalent problem that involves $A_{v}\triangleq
2M+3K+9MK+MK^{2}+M^{2}K^{2}+2$ real-valued scalar variables, $A_{l}\triangleq
5M+4K+9MK+MK^{2}+M^{2}K^{2}+1$ linear constraints, $A_{q}\triangleq
M+2K+7MK+MK^{2}+M^{2}K^{2}+1$ quadratic constraints. Therefore, the algorithm
for solving problem (46) requires a complexity of
$\operatorname{\mathcal{O}}(\sqrt{A_{l}+A_{q}}(A_{v}+A_{l}+A_{q})A_{v}^{2})$
[32].

######  Remark 2 (Initial point and infeasible EE maximization problem).

We recall that when the individual SE requirements
$\mathcal{S}_{\mathtt{ul}}^{o}$ and $\mathcal{S}_{\mathtt{dl}}^{o}$ are large
but the UEs have unfavourable links to the APs, the QoS constraints (26c) and
(26e) are difficult to satisfy. In this case, we use a similar procedure as
discussed in Remark 1 for Algorithm 2. Specifically, we start with a random
point
$\widehat{\operatorname{\mathbf{x}}}\in\widehat{\operatorname{\mathcal{H}}}\setminus\\{\eqref{UL:QoS:cons:2},\eqref{DL:QoS:cons:2}\\}$
and solve the following problem (instead of problem (78)) in each iteration of
Algorithm 2

$\displaystyle\underset{\widehat{{\bf
x}}\in\widehat{\operatorname{\mathcal{H}}}\setminus\\{\eqref{UL:QoS:cons:2},\eqref{DL:QoS:cons:2}\\},\operatorname{\mathbf{z}}_{\mathtt{ul}},\operatorname{\mathbf{z}}_{\mathtt{dl}}}{\min}\,\,$
$\displaystyle\widehat{\mathcal{L}}_{\mathtt{EE}^{\mathsf{NAFD}}}(\widehat{{\bf
x}})+\phi z,$ (79a)
$\displaystyle\mathrm{s.t.}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle\eqref{qul:s},\eqref{qdl:s},\eqref{s:cons},$ (79b)

where $z$ is defined in Remark 1. When Algorithm 2 converges, if $z$ is
smaller than a predefined error tolerance, the problem (73) or (69a) is
feasible and we take the converged point as the final solution. Otherwise,
problem (73) or (69a) is considered as an infeasible problem.

##  V Baseline Schemes

To investigate the effectiveness of our proposed optimized network-assisted
full-duplex ($\operatorname{\textbf{NAFD}}$) scheme for CF-mMIMO systems, we
introduce the following baseline schemes for comparisons in the numerical
results of Section VI.

###  V-A Network-Assisted Full-Duplex CF-mMIMO Systems: Heuristic Approaches

To show the advantages of the joint optimization of AP mode assignment, power
control, and LSFD weights in our $\operatorname{\textbf{NAFD}}$ scheme, we
consider two heuristic NAFD schemes as follows.

####  V-A1 Network-assisted full-duplex with random AP mode assignment
($\operatorname{\textbf{R-NAFD}}$)

In this scheme, we assume that the AP modes
$(\operatorname{\mathbf{a}},\operatorname{\mathbf{b}})$ are randomly assigned.
Accordingly, we optimize the power control coefficients
$(\operatorname{\bm{\eta}},\operatorname{\bm{\varsigma}})$ and LSFD weights
$\operatorname{\bm{\alpha}}$, under the same SE requirement constraints for UL
and DL UEs. The problems of sum SE maximization of the
$\operatorname{\textbf{R-NAFD}}$ scheme for the given random mode assignment
vectors $(\operatorname{\mathbf{a}},\operatorname{\mathbf{b}})$ can be
respectively expressed as

$\displaystyle\underset{\operatorname{\bm{\theta}},\operatorname{\bm{\varsigma}},\operatorname{\bm{\alpha}},{\bf
q}_{\mathtt{ul}},{\bf q}_{\mathtt{dl}}}{\min}\,\,$
$\displaystyle-\sum_{\ell\in\mathcal{K}_{u}}q_{\mathtt{ul},\ell}-\sum_{k\in\mathcal{K}_{d}}q_{\mathtt{dl},k}$
(80a) $\displaystyle\mathrm{s.t.}\,\,\,\,\,\,\,\,\,$
$\displaystyle\eqref{a}-\eqref{sumab},\eqref{theta},\eqref{etaa:relation},\eqref{DL:power:cons},\eqref{UL:power:cons},\eqref{eq:alphml},\eqref{UL:QoS:cons:1},$
$\displaystyle\eqref{UL:QoS:cons:2},\eqref{DL:QoS:cons:1},\eqref{DL:QoS:cons:2}$
(80b)

Similar to our $\operatorname{\textbf{NAFD}}$ scheme, we find the suboptimal
solution to the EE maximization problem of this scheme which is given as

$\displaystyle\underset{\operatorname{\bm{\theta}},\operatorname{\bm{\varsigma}},\operatorname{\bm{\alpha}},t,\hat{p}}{\min}\,\,$
$\displaystyle-t$ (81a) $\displaystyle\mathrm{s.t.}\,\,\,\,\,$
$\displaystyle\eqref{theta},\eqref{etaa:relation},\eqref{DL:power:cons},\eqref{UL:power:cons},\eqref{eq:alphml},\eqref{UL:QoS:cons:1}-\eqref{DL:QoS:cons:2},$
$\displaystyle\eqref{tphat},\eqref{phat}.$ (81b)

Since problems (80a) and (81a) have the same mathematical structure as
problems (26) and (73), we solve problem (80a) and problem (81a) by using
Algorithms 1 and 2 with some slight modifications.

####  V-A2 Network-assisted full-duplex CF-mMIMO with greedy AP mode
assignment, fixed power control coefficients and LSFD weights
($\operatorname{\textbf{G-NAFD}}$)

The AP mode assignment is performed by a greedy algorithm proposed in [24].
Let $\mathcal{A}_{\mathtt{ul}}$ and $\mathcal{A}_{\mathtt{dl}}$ denote the
sets containing the indices of UL APs, and DL APs, respectively. Also, let
$\operatorname{\mathcal{A}}_{s}\triangleq\operatorname{\mathcal{A}}_{\mathtt{ul}}\bigcup\operatorname{\mathcal{A}}_{\mathtt{dl}}$
be the set of assigned APs and
$\operatorname{\mathcal{A}}_{s^{\prime}}\triangleq\operatorname{\mathcal{M}}\setminus\operatorname{\mathcal{A}}_{s}$
be the set of unassigned APs. Denote by
$\mathcal{S}_{sum}(\mathcal{A}_{\mathtt{ul}},\mathcal{A}_{\mathtt{dl}})\triangleq\sum_{k=1}^{K_{d}}\mathcal{S}_{\mathtt{ul},k}(\mathcal{A}_{\mathtt{ul}},\mathcal{A}_{\mathtt{dl}})+\sum_{\ell=1}^{K_{u}}\mathcal{S}_{\mathtt{dl},\ell}(\mathcal{A}_{\mathtt{ul}},\mathcal{A}_{\mathtt{dl}})$
the sum SE that captures the dependence of the sum SE on the different choices
of $\mathcal{A}_{\mathtt{ul}}$ and $\mathcal{A}_{\mathtt{dl}}$. The greedy
algorithm for AP mode assignment is shown in Algorithm 3. The key idea of
Algorithm 3 is to take one AP out of the set of unassigned APs,
$\operatorname{\mathcal{A}}_{s^{\prime}}$, in each iteration and assign to
this AP the mode that offers the highest sum SE until
$\operatorname{\mathcal{A}}_{s^{\prime}}$ is empty. In this algorithm, the
power control coefficients and LSFD weights are fixed, i.e.,
$\theta_{mk}=\frac{a_{m}}{\sqrt{NK_{d}\gamma_{mk}^{\mathtt{dl}}}}$,
$\varsigma_{\ell}=1$, $\alpha_{m\ell}=1,\forall m,k,\ell$.

Algorithm 3 Greedy AP mode assignment for SE maximization [24]

1: Initialize: $\mathcal{A}_{\mathtt{ul}}=\mathcal{A}_{\mathtt{dl}}=\emptyset$

2: repeat

3:
$i_{\mathtt{ul}}^{\star}=\operatorname*{arg\,max}_{i\in\mathcal{A}_{s^{\prime}}}\mathcal{S}_{sum}(\mathcal{A}_{\mathtt{ul}}\bigcup\\{i\\})$

4:
$i_{\mathtt{dl}}^{\star}=\operatorname*{arg\,max}_{i\in\mathcal{A}_{s^{\prime}}}\mathcal{S}_{sum}(\mathcal{A}_{\mathtt{dl}}\bigcup\\{i\\})$

5: if
$\mathcal{S}_{sum}(\mathcal{A}_{\mathtt{ul}}\bigcup\\{i_{\mathtt{ul}}^{\star}\\})\geq\mathcal{S}_{sum}(\mathcal{A}_{\mathtt{dl}}\bigcup\\{i_{\mathtt{dl}}^{\star}\\})$
then

6: Update
$\mathcal{A}_{\mathtt{ul}}=\mathcal{A}_{\mathtt{ul}}\bigcup\\{i_{\mathtt{ul}}^{\star}\\}$

7: else

8: Update
$\mathcal{A}_{\mathtt{dl}}=\mathcal{A}_{\mathtt{dl}}\bigcup\\{i_{\mathtt{dl}}^{\star}\\}$

9: end if

10: Update
$\mathcal{A}_{s}=\mathcal{A}_{\mathtt{ul}}\bigcup\mathcal{A}_{\mathtt{dl}}$,

11: until $\mathcal{A}_{s{{}^{\prime}}}=\emptyset$

Since Algorithm 3 is proposed to only maximize the sum SE, we calculate the EE
of this $\operatorname{\textbf{G-NAFD}}$ scheme by using the AP mode solution
obtained from Algorithm 3. Specifically, we use the obtained solution
$(\mathcal{A}_{\mathtt{dl}},\mathcal{A}_{\mathtt{ul}})$ to make up the AP mode
assignment vectors $({\bf a},{\bf b})$. Then, the EE is calculated by using
(64) for given $({\bf a},{\bf b})$, in which the total power consumption is
computed by (II-C).

###  V-B Half-Duplex CF-mMIMO Systems

To show the advantages of our proposed $\operatorname{\textbf{NAFD}}$ scheme,
we compare it with the conventional HD CF-mMIMO systems
($\operatorname{\textbf{HD}}$)[5]. In this system, the DL-and-UL payload data
transmission phase is divided into two equal time fractions of length
$(\tau_{c}-\tau_{t})/2$. Each UL or DL data transmission is performed in one
time fraction. In each time fraction, all UL or DL UEs are served by all the
APs, i.e., $a_{m}=b_{m}=1,\forall m$. There is no interference from the UL UEs
to DL UEs, and from the DL APs to UL APs. There is also an additional factor
of $\frac{1}{2}$ applied in the SE expression and the total power consumption.
This factor captures the fact that each DL or UL transmission is only
performed and consumes power in half of the time fraction. In particular, the
SE expressions of DL UE $k$ is given by

$\displaystyle\mathcal{S}_{\mathtt{dl},k}^{\mathsf{HD}}(\bm{\theta},{\bm{\varsigma}})=\frac{\tau_{c}-\tau_{t}}{2\tau_{c}}\log_{2}\left(1+\text{SINR}_{\mathtt{dl},k}^{\mathsf{HD}}(\bm{\theta},{\bm{\varsigma}})\right),$
(82)

where

$\text{SINR}_{\mathtt{dl},k}^{\mathsf{HD}}(\bm{\theta},{\bm{\varsigma}})\triangleq\frac{N_{t}^{2}\rho_{d}\left(\sum_{{m\in\mathcal{M}}}\theta_{mk}\gamma_{mk}^{\mathtt{dl}}\right)^{2}}{\rho_{d}N_{t}\sum_{k^{\prime}\in\mathcal{K}_{d}}\sum_{m\in\mathcal{M}}\theta_{mk^{\prime}}^{2}\beta_{mk}^{\mathtt{dl}}\gamma_{mk^{\prime}}^{\mathtt{dl}}+1}\vspace{0em},$

while the SE of the UL UE $\ell$ is given by

$\displaystyle\mathcal{S}_{\mathtt{ul},\ell}^{\mathsf{HD}}(\bm{\varsigma},\bm{\theta},\bm{\alpha})=\\!\frac{\tau_{c}\\!-\\!\tau_{t}}{2\tau_{c}}\log_{2}\left(1+\text{SINR}_{\mathtt{ul},\ell}^{\mathsf{HD}}(\bm{\varsigma},\bm{\theta},\bm{\alpha})\right),$
(83)

where

$\displaystyle\text{SINR}_{\mathtt{dl},k}^{\mathsf{HD}}(\bm{\theta},{\bm{\varsigma}})\triangleq$
$\displaystyle\frac{N_{r}\rho_{u}\left(\sum_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\sqrt{b_{m}\varsigma_{\ell}}\alpha_{m\ell}\gamma_{m\ell}^{\mathtt{ul}}\right)^{2}}{\\!\\!\rho_{u}\\!\\!\sum_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\\!\\!\sum_{q\in\mathcal{K}_{u}}\\!\\!\\!b_{m}\varsigma_{q}\alpha_{m\ell}^{2}\beta_{mq}^{\mathtt{ul}}\gamma_{m\ell}^{\mathtt{ul}}\\!+\\!\\!\sum_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\\!b_{m}\alpha_{m\ell}^{2}\\!\gamma_{m\ell}^{\mathtt{ul}}}.\\!\vspace{0em}$
(84)

The total power consumption in the HD scheme is

$\displaystyle P_{\mathtt{total}}^{\mathsf{HD}}(\operatorname{\mathbf{x}})$
$\displaystyle\triangleq\frac{1}{2}\Bigg{[}\sum_{m\in\mathcal{M}}\frac{N\rho_{d}\sigma_{n}^{2}}{\zeta_{m}}\left(\sum_{k\in\mathcal{K}_{d}}\gamma_{mk}^{\mathtt{dl}}\theta_{mk}^{2}\right)+P_{\mathtt{U,fixed}}$
$\displaystyle\hskip
20.00003pt+\sum_{\ell\in\mathcal{K}_{u}}\frac{\rho_{u}\sigma_{n}^{2}}{\chi}{\varsigma}_{\ell}+\sum_{m\in\operatorname{\mathcal{M}}}\operatorname{\mathcal{S}}^{\mathsf{HD}}(\operatorname{\mathbf{x}})P_{\mathtt{bt},m}$
$\displaystyle\hskip
20.00003pt+\sum_{m\in\mathcal{M}}(NP_{\mathtt{cdl},m}+P_{\mathtt{fdl},m})$
$\displaystyle\hskip
20.00003pt+\sum_{m\in\mathcal{M}}(NP_{\mathtt{cul},m}+P_{\mathtt{ful},m})\Bigg{]},$
(85)

where
$\operatorname{\mathcal{S}}^{\mathsf{HD}}(\operatorname{\mathbf{x}})\triangleq\sum_{\ell\in\mathcal{K}_{u}}\mathcal{S}_{\mathtt{ul},\ell}^{\mathsf{HD}}({\bf
b},\bm{\varsigma},\bm{\theta},\bm{\alpha})+\sum_{k\in\mathcal{K}_{d}}\mathcal{S}_{\mathtt{dl},k}^{\mathsf{HD}}({\bf
a},\bm{\theta},{\bm{\varsigma}}).$

In the $\operatorname{\textbf{HD}}$ scheme, the power coefficients
($\operatorname{\bm{\theta}},\operatorname{\bm{\varsigma}}$) and LSFD weights
$\operatorname{\bm{\alpha}}$ are optimized to maximize the sum SE and total
EE. Note that the mathematical formulas of the SE expressions and the total
power consumption in (82)–(V-B) are the simplified versions of those of (13),
(14), and (IV-A) of the $\operatorname{\textbf{NAFD}}$ scheme. Therefore, the
problems of maximizing the sum SE and EE in the $\operatorname{\textbf{HD}}$
scheme are similar to problems (80a) and (81a), and hence, can be solved by
using Algorithms 1 and 2 with appropriate modifications.

###  V-C Full-Duplex CF-mMIMO Systems

We further compare our $\operatorname{\textbf{NAFD}}$ scheme with the
traditional FD CF-mMIMO systems ($\operatorname{\textbf{FD}}$) [28, 6]. In
this scheme, all the APs operate in a FD mode and serve the UL and DL UEs
simultaneously at the same frequency. Therefore, $a_{m}=b_{m}=1,\forall m$.
Each AP is equipped with $N_{t}$ transmit antennas and $N_{r}$ receive
antennas. To have a fair comparison with the
$\operatorname{\textbf{R-NAFD}},\operatorname{\textbf{G-NAFD}},\operatorname{\textbf{NAFD}},\operatorname{\textbf{HD}}$
schemes, wherein the APs operate in HD mode, the $\operatorname{\textbf{FD}}$
scheme deploys the same number of antennas as the other schemes, i.e.,
$N=N_{t}+N_{r}$, which is called a “antenna-number-preserved” condition [33,
34, 35]. Therefore, the DL SE of the $\operatorname{\textbf{FD}}$ scheme is

$\displaystyle~{}\mathcal{S}_{\mathtt{dl},k}^{\mathsf{FD}}(\bm{\theta},{\bm{\varsigma}})=\frac{\tau_{c}-\tau_{t}}{\tau_{c}}\log_{2}\left(1+\frac{(\Xi_{k}^{\mathsf{FD}})^{2}}{\Omega_{k}^{\mathsf{FD}}}\right),$
(86)

where

$\displaystyle\Xi_{k}^{\mathsf{FD}}(\bm{\theta})\triangleq
N_{t}\sqrt{\rho_{d}}\sum_{{m\in\mathcal{M}}}\theta_{mk}\gamma_{mk}^{\mathtt{dl}}$
$\displaystyle\Omega_{k}^{\mathsf{FD}}(\bm{\theta},{\bm{\varsigma}})\\!\triangleq\rho_{d}N_{t}\\!\\!\sum_{k^{\prime}\in\mathcal{K}_{d}}\\!\\!\sum_{m\in\mathcal{M}}\\!\\!\theta_{mk^{\prime}}^{2}\beta_{mk}^{\mathtt{dl}}\gamma_{mk^{\prime}}^{\mathtt{dl}}\\!+\\!\rho_{u}\\!\sum_{\ell\in\mathcal{K}_{u}}\\!\\!{\varsigma_{\ell}}\beta_{k\ell}^{\mathtt{du}}\\!+\\!1,$

while the UL SE of the $\operatorname{\textbf{FD}}$ scheme is given by

$\displaystyle\mathcal{S}_{\mathtt{ul},\ell}^{\mathsf{FD}}(\bm{\varsigma},\bm{\theta},\bm{\alpha})=\\!\frac{\tau_{c}\\!-\\!\tau_{t}}{\tau_{c}}\log_{2}(1+\text{SINR}_{\mathtt{ul},\ell}^{\mathsf{FD}}(\bm{\varsigma},\bm{\theta},\bm{\alpha})),$
(87)

where
$\text{SINR}_{\mathtt{ul},\ell}^{\mathsf{FD}}(\bm{\varsigma},\bm{\theta},\bm{\alpha})$
is given at (88) at the top of the page.

$\displaystyle~{}\text{SINR}_{\mathtt{ul},\ell}^{\mathsf{FD}}(\bm{\varsigma},\bm{\theta},\bm{\alpha})=\frac{N_{r}\rho_{u}\left(\sum_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\sqrt{\varsigma_{\ell}}\alpha_{m\ell}\gamma_{m\ell}^{\mathtt{ul}}\right)^{2}}{\rho_{u}\\!\sum_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\\!\sum_{q\in\mathcal{K}_{u}}\\!\varsigma_{q}\alpha_{m\ell}^{2}\beta_{mq}^{\mathtt{ul}}\gamma_{m\ell}^{\mathtt{ul}}+\rho_{d}N_{r}\sum_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\sum_{\begin{subarray}{c}i\in\mathcal{M}\end{subarray}}\sum_{k\in\mathcal{K}_{d}}\theta_{ik}^{2}\alpha_{m\ell}^{2}\gamma_{m\ell}^{\mathtt{ul}}\beta_{mi}\gamma_{ik}^{\mathtt{dl}}+\sum_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\alpha_{m\ell}^{2}\gamma_{m\ell}^{\mathtt{ul}}}.$
(88)

Each AP in the FD CF-mMIMO system consumes some amount of power for SIS [36,
37]. Let $P_{\text{SIS},m}$ be the power required for SIS at each receive
antenna at AP $m$. Then, the total power consumption in the FD CF-mMIMO system
is

$\displaystyle P_{\mathtt{total}}^{\mathsf{FD}}(\operatorname{\mathbf{x}})$
$\displaystyle\triangleq\\!\sum_{m\in\mathcal{M}}\\!\\!\frac{N_{t}\rho_{d}\sigma_{n}^{2}}{\zeta_{m}}\left(\sum_{k\in\mathcal{K}_{d}}\gamma_{mk}^{\mathtt{dl}}\theta_{mk}^{2}\right)+\\!\sum_{\ell\in\mathcal{K}_{u}}\\!\\!\frac{\rho_{u}\sigma_{n}^{2}}{\chi}{\varsigma}_{\ell}$
$\displaystyle+P_{\mathtt{U,fixed}}+\sum_{m\in\operatorname{\mathcal{M}}}\operatorname{\mathcal{S}}^{\mathsf{FD}}(\operatorname{\mathbf{x}})P_{\mathtt{bt},m}$
$\displaystyle+\sum_{m\in\mathcal{M}}(N_{t}P_{\mathtt{cdl},m}+P_{\mathtt{fdl},m})$
$\displaystyle+\sum_{m\in\mathcal{M}}(N_{r}P_{\mathtt{cul},m}+P_{\mathtt{ful},m})+\sum_{m\in\operatorname{\mathcal{M}}}N_{r}P_{\text{SIS},m},$
(89)

where

$\displaystyle\operatorname{\mathcal{S}}^{\mathsf{FD}}(\operatorname{\mathbf{x}})\triangleq\sum_{\ell\in\mathcal{K}_{u}}\mathcal{S}_{\mathtt{ul},\ell}^{\mathsf{FD}}({\bf
b},\bm{\varsigma},\bm{\theta},\bm{\alpha})+\sum_{k\in\mathcal{K}_{d}}\mathcal{S}_{\mathtt{dl},k}^{\mathsf{FD}}({\bf
a},\bm{\theta},{\bm{\varsigma}}).$ (90)

We note that in a FD CF-mMIMO system, the CPU needs to share with the APs the
full signaling loads of all the DL and UL UEs. Therefore, all APs contribute
to the two penultimate terms in (V-C), while in (II-C), the APs are partially
contributing to the corresponding terms.

We recall that in a $\operatorname{\textbf{FD}}$ system, the SI can be
suppressed using a dedicated hardware and the information of transmit signal.
Since the SIS process is normally imperfect, there is still a remaining level
of SI, which is called the residual SI after SIS [36]. In a FD CF-mMIMO
system, the SI at each AP can be modeled as Rayleigh fading channel [36, 38,
39]. Specifically, we denote by $\operatorname{\mathbf{Z}}_{mm}$ the residual
SI link at each AP, whose elements are i.i.d
$\operatorname{\mathcal{CN}}(0,\sigma_{\operatorname{\text{SI}}}^{2})$ RVs,
where $\sigma_{\operatorname{\text{SI}}}^{2}$ is the power of residual SI
after SIS at each AP. Note that in our $\operatorname{\textbf{NAFD}}$ scheme
where all APs operate in HD mode, there are no SI at each AP, i.e.,
$\sigma_{\operatorname{\text{SI}}}^{2}=0$. Therefore, the residual SI model is
consistent with the model of the interference matrix
$\operatorname{\mathbf{Z}}_{mm}=0,\forall m$, in Section II-A.

In the $\operatorname{\textbf{FD}}$ scheme, the power coefficients
($\operatorname{\bm{\theta}},\operatorname{\bm{\varsigma}}$) and LSFD weights
$\operatorname{\bm{\alpha}}$ are also optimized to achieve the maximum sum SE
and EE. Therefore, the problems of maximizing the sum SE and EE of the
$\operatorname{\textbf{FD}}$ scheme are similar to problems (80a) and (81a).
Here, the changes in the SE expressions, total power consumption, and the SI
matrices make no difference in the mathematical structures of the sum SE and
EE maximization problems of the $\operatorname{\textbf{FD}}$ scheme compared
to problems (80a) and (81a). Thus, the sum SE and EE maximization problems of
the $\operatorname{\textbf{FD}}$ scheme can be solved by using Algorithms 1
and 2 with some slight modifications.

######  Remark 3.

Our proposed NAFD scheme for CF-mMIMO systems cannot be compared with those in
the literature, i.e., [13, 17, 14, 18]. These works consider the SE with
instantaneous channels, while the SE and EE of our work rely only on the
statistical property of the channels.

##  VI Numerical Examples

###  VI-A Network Setup and Parameter Setting

We consider a CF-mMIMO network, where the APs and UEs are randomly distributed
in a square of $0.5\times 0.5$ km2, whose edges are wrapped around to avoid
the boundary effects. The distances between adjacent APs are at least $50$
m[40]. Unless otherwise stated, the values of the network parameters are:
$N=2$, $N_{t}=N_{r}=1$,
$\mathcal{S}_{\mathtt{dl}}^{o}=\mathcal{S}_{\mathtt{ul}}^{o}=\mathcal{S}_{QoS}$
bit/s/Hz, $K_{d}=K_{u}=K$, $\tau_{c}=200$, and $\tau_{t}=K_{d}+K_{u}$. We
further set the bandwidth $B=50$ MHz and noise figure $F=9$ dB. Thus, the
noise power $\sigma_{n}^{2}=k_{B}T_{0}BF$, where $k_{B}=1.381\times 10^{-23}$
Joules/oK is the Boltzmann constant, while $T_{0}=290^{o}$K is the noise
temperature. Let $\tilde{\rho}_{d}=1$ W, $\tilde{\rho}_{u}=0.1$ W and
$\tilde{\rho}_{t}=0.1$ W be the maximum transmit power of the APs, UL users
and UL training pilot sequences, respectively. The normalized maximum transmit
powers ${\rho}_{d}$, ${\rho}_{u}$, and ${\rho}_{t}$ are calculated by dividing
these powers by the noise power.

TABLE II: Parameters of Power Consumption Parameter | Value
---|---
Fixed power consumption/ each backhaul ($P_{\mathtt{fdl},m},P_{\mathtt{ful},m}$, $\forall m$) [26, 23] |  $0.825$ W
Internal power consumption/antenna ($P_{\mathtt{cdl},m}$ and $P_{\mathtt{cul},m},\forall m$) [23] |  $0.2$ W
Traffic-dependent backhaul power ($P_{\mathtt{bt},m},\forall m$) [26, 23] |  $0.25$ W/(Gbits/s)
Power amplifier efficiency at the APs ($\zeta_{m}$, $\forall m$) [23] | $0.4$
Power amplifier efficiency at the UEs ($\chi$) [41] | $0.3$
Fixed power consumption UL and DL UE ($P_{\mathtt{U},\ell},P_{\mathtt{D},k},\forall\ell,k$) [41] |  $0.1$ W

We model the large-scale fading coefficients $\beta_{mk}$ as [40]

$\displaystyle\beta_{mk}=10^{\frac{\text{PL}_{mk}^{d}}{10}}10^{\frac{F_{mk}}{10}},$
(91)

where $10^{\frac{\text{PL}_{mk}^{d}}{10}}$ represents the path loss, and
$10^{\frac{F_{mk}}{10}}$ represents the shadowing effect with
$F_{mk}\in\mathcal{N}(0,4^{2})$ (in dB). Here, $\text{PL}_{mk}^{d}$ (in dB) is
given by [40]

$\displaystyle\text{PL}_{mk}^{d}=-30.5-36.7\log_{10}\left(\frac{d_{mk}}{1\,\text{m}}\right),$
(92)

and the correlation among the shadowing terms from the AP $m,\forall
m\in\mathcal{M}$ to different UEs $k\in\mathcal{K}_{d}$
($\ell\in\mathcal{K}_{u}$) is expressed as:

$\displaystyle\mathbb{E}\\{F_{mk}F_{jk^{\prime}}\\}\triangleq\begin{cases}4^{2}2^{-\delta_{kk^{\prime}}/9\,\text{m}},&\text{if
$j=m$}\\\ 0,&\mbox{otherwise},\end{cases},\forall j\in\mathcal{M},$ (93)

where $\delta_{kk^{\prime}}$ is the physical distance between UEs $k$ and
$k^{\prime}$.

(a) Average sum SE versus the number of APs ($K_{d}\\!=\\!K_{u}\\!=\\!4$).

(b) Average sum SE versus the number of UEs ($M=40$).

Figure 2: Comparison among the SE achieved by the proposed
$\operatorname{\textbf{NAFD}}$ scheme and baseline schemes
($\mathcal{S}_{QoS}=0.2$ bit/s/Hz, $\sigma_{\text{SI}}^{2}/\sigma_{n}^{2}=50$
dB).

Regarding the power consumption parameters, we use the parameters of the power
consumption in [26, 23, 41], which are shown in Table II. We note that the
power consumption $P_{\text{SIS},m}$ for SIS strongly depends on the EE of the
SIS techniques, used at each AP. Therefore, for a fair comparison, in what
follows, we consider the best case of the $\operatorname{\textbf{FD}}$ scheme
with highly energy-efficient SIS techniques, where the power consumption for
SIS is sufficiently small and can be ignored in the total power consumption of
the $\operatorname{\textbf{FD}}$ scheme, i.e., $P_{\text{SIS},m}=0,\forall m$.

###  VI-B Results and Discussions

We compare our proposed optimized $\operatorname{\textbf{NAFD}}$ scheme with
the baseline schemes $\operatorname{\textbf{R-NAFD}}$,
$\operatorname{\textbf{G-NAFD}}$, $\operatorname{\textbf{HD}}$, and
$\operatorname{\textbf{FD}}$, which were discussed in Section V, in terms of
sum SE and EE. All the following average results are averaged over $200$
large-scale fading channel realizations. In each channel realization, if the
individual SE requirements are not met or the optimization problem of SE or EE
maximization of a scheme is infeasible, as discussed in Remarks 1 and 2, we
set the SE or EE of that scheme to zero. The EE values in the results are
calculated by using the solution to the problem of maximizing
$\operatorname{\mathtt{EE}}_{fullbh}^{\mathsf{NAFD}}$ (69a), which is obtained
by Algorithm 2.

####  VI-B1 Effectiveness of the $\operatorname{\textbf{NAFD}}$ scheme in
terms of SE

Figures 2(a) and 2(b) show the average SE of all the considered schemes versus
the number of APs and the different numbers of DL and UL UEs, respectively.
Numerical results lead to the following conclusions.

  * •

The optimized $\operatorname{\textbf{NAFD}}$ scheme outperforms the heuristic
$\operatorname{\textbf{R-NAFD}}$ and $\operatorname{\textbf{G-NAFD}}$ schemes.
More specifically, it provides performance gains of up to $12\%$ and $150\%$
over $\operatorname{\textbf{R-NAFD}}$ and $\operatorname{\textbf{G-NAFD}}$,
respectively, which highlights the advantage of our joint optimization
solution over heuristic ones. On the other hand, the remarkable performance
gap between the $\operatorname{\textbf{R-NAFD}}$ and
$\operatorname{\textbf{G-NAFD}}$ verifies the effectiveness of the joint power
control and LSFD weight design in NAFD CF-mMIMO systems. Interestingly,
$\operatorname{\textbf{R-NAFD}}$ scheme offers an acceptable SE performance
compared with $\operatorname{\textbf{NAFD}}$, which balances the trade-off
between performance and complexity. Therefore, it can be deployed instead of
the $\operatorname{\textbf{NAFD}}$ scheme if complexity is an issue.

  * •

In the comparison between the proposed $\operatorname{\textbf{NAFD}}$ scheme
and the conventional $\operatorname{\textbf{HD}}$,
$\operatorname{\textbf{FD}}$ schemes, the $\operatorname{\textbf{NAFD}}$
scheme achieves the best SE performance, while the
$\operatorname{\textbf{HD}}$ scheme offers the worst SE performance. This is
reasonable because, in $\operatorname{\textbf{NAFD}}$ and
$\operatorname{\textbf{FD}}$ schemes, the UL and DL data transmissions are
performed simultaneously, thus, the pre-log factor $\frac{1}{2}$, that comes
up in the SE expression of the $\operatorname{\textbf{HD}}$ scheme, is
eliminated. The $\operatorname{\textbf{NAFD}}$ scheme has a smaller number of
APs to serve DL or UL UEs than the $\operatorname{\textbf{FD}}$ scheme, which
could lead to lower power for both desired signals and CLI. Nevertheless, with
optimizing AP mode assignment, $\operatorname{\textbf{NAFD}}$ is more
efficient than $\operatorname{\textbf{FD}}$ in managing the power resource for
sufficiently high power of desired signals and lower power of CLI. Moreover,
in the $\operatorname{\textbf{NAFD}}$ scheme, the detrimental impact of
residual SI, which exists in the $\operatorname{\textbf{FD}}$ scheme, is
completely removed.

  * •

The SE gain of $\operatorname{\textbf{NAFD}}$ over
$\operatorname{\textbf{HD}}$ scheme increases when the number of APs
increases. This is because the $\operatorname{\textbf{NAFD}}$ scheme has more
degrees-of-freedom in terms of AP mode assignment to manage the CLI. However,
this gain is saturated at around $30\%$. This is because of the AP-to-AP
interference. This interference is the fundamental limit of both the
$\operatorname{\textbf{NAFD}}$ and $\operatorname{\textbf{FD}}$ schemes and
increases when the number of APs increases. We also note that, due to the
presence of residual SI and CLI, the QoS requirement, both the
$\operatorname{\textbf{NAFD}}$ and $\operatorname{\textbf{FD}}$ (under
“antenna-number-preserved” condition) schemes cannot achieve the promised
double gain over the $\operatorname{\textbf{HD}}$ scheme. When the individual
SE constraints are applied, the system needs to spend more power on the UEs
with unfavorable links (i.e., lower SEs) to guarantee the SE requirements of
these UEs larger than the minimum SE threshold
$\operatorname{\mathcal{S}}_{\mathtt{ul},\ell}^{o}$ or
$\operatorname{\mathcal{S}}_{\mathtt{dl},k}^{o}$. Therefore, there is stronger
interference for the UEs with favorable links, and the SE of these UEs are
sacrificed to compensate for the SE of the UEs with unfavorable links.

  * •

The SEs of all the considered schemes, except
$\operatorname{\textbf{G-NAFD}}$, are monotonically improved when the number
of UEs increases. Note that increasing the number of UEs causes stronger CLI.
This result implies that our proposed joint optimization approach can
effectively manage the CLI in $\operatorname{\textbf{NAFD}}$,
$\operatorname{\textbf{R-NAFD}}$, $\operatorname{\textbf{FD}}$ and
$\operatorname{\textbf{HD}}$ schemes. In contrast, the heuristic
$\operatorname{\textbf{G-NAFD}}$ scheme fails to deal with the interference.
In our simulation results, it often violates the individual SE requirements
and the probability of its SE set to be zero is high.

  * •

The gap between the $\operatorname{\textbf{NAFD}}$ and conventional schemes
($\operatorname{\textbf{FD}}$ and $\operatorname{\textbf{HD}}$) is diminished
when the number of UEs increases. Moreover, $\operatorname{\textbf{FD}}$
outperforms $\operatorname{\textbf{NAFD}}$ when $K_{u}=K_{d}>6$. These results
are reasonable because the number of APs for UL or DL transmission in the
$\operatorname{\textbf{NAFD}}$ scheme is smaller than that in the
$\operatorname{\textbf{FD}}$ and $\operatorname{\textbf{HD}}$ schemes. When
the number of UEs increases to a sufficient value, the degrees-of-freedom of
the $\operatorname{\textbf{FD}}$ and $\operatorname{\textbf{HD}}$ schemes to
manage CLI become more than those of the $\operatorname{\textbf{NAFD}}$ and
dominate the SE improvement achieved by optimizing the AP mode assignment in
$\operatorname{\textbf{NAFD}}$.

(a) $K_{d}=K_{u}=4$.

(b) $M=40$.

Figure 3: Comparison among the EE of the proposed
$\operatorname{\textbf{NAFD}}$ scheme and baseline schemes
($\mathcal{S}_{QoS}=0.2$ bit/s/Hz, $\sigma_{\text{SI}}^{2}/\sigma_{n}^{2}=50$
dB). Figure 4: Impact of the number of AP antennas on the SE
($NM=80,K_{d}=K_{u}=4,\mathcal{S}_{QoS}=0.2$ bit/s/Hz,
$\sigma_{\text{SI}}^{2}/\sigma_{n}^{2}=50$ dB). Figure 5: Impact of the number
of AP antennas on the EE ($NM=80,K_{d}=K_{u}=4,\mathcal{S}_{QoS}=0.2$
bit/s/Hz, $\sigma_{\text{SI}}^{2}/\sigma_{n}^{2}=50$ dB).

####  VI-B2 Effectiveness of the $\operatorname{\textbf{NAFD}}$ scheme in
terms of EE

Figures 3(a) and 3(b) show the average EE of the
$\operatorname{\textbf{NAFD}}$ and baseline schemes as a function of the
number of APs and number of UEs, respectively. From these figures, we have the
following observations.

  * •

The $\operatorname{\textbf{NAFD}}$ scheme provides a better EE performance
than both the heuristic $\operatorname{\textbf{R-NAFD}}$ and
$\operatorname{\textbf{G-NAFD}}$ schemes. This result highlights the
effectiveness of our joint optimization solution over heuristic solutions. The
gap between $\operatorname{\textbf{NAFD}}$ and
$\operatorname{\textbf{G-NAFD}}$ is always noticeable, while that between
$\operatorname{\textbf{NAFD}}$ and $\operatorname{\textbf{R-NAFD}}$ is only
remarkable when $K$ is large enough and $M/K$ is small enough. This indicates
that the $\operatorname{\textbf{R-NAFD}}$ scheme can be utilized for the
systems that have a large value of $M/K$ with a small number of UEs and
requires a lower computational complexity solution with an acceptable loss in
the EE performance.

  * •

When the number of UEs increases, the EE performance of
$\operatorname{\textbf{G-NAFD}}$ is degraded, offering the worse performance
for $K\geq 4$ among all schemes. However, the EE achieved by the
$\operatorname{\textbf{R-NAFD}}$ is improved by increasing $K$ and then
sharply degraded when $K$ increases beyond $6$. This is because the CLI is
stronger and increases the probability of infeasible cases of these schemes.

  * •

The $\operatorname{\textbf{NAFD}}$ scheme achieves a noticeable EE improvement
(i.e., up to $200\%$) compared with the $\operatorname{\textbf{FD}}$ and
$\operatorname{\textbf{HD}}$ schemes. This result is intuitive because the
$\operatorname{\textbf{NAFD}}$ scheme can provide a better SE compared with
the $\operatorname{\textbf{FD}}$ and $\operatorname{\textbf{HD}}$ schemes.
Moreover, the $\operatorname{\textbf{NAFD}}$ scheme has a lower number of APs
serving DL or UL transmission than the $\operatorname{\textbf{R-NAFD}}$ and
$\operatorname{\textbf{G-NAFD}}$ schemes, which results in a lower power
consumption to the circuit components of the APs (i.e., $P_{\mathtt{cdl},m}$
and $P_{\mathtt{cul},m}$) as well as more power for backhaul links (i.e.,
$P_{\mathtt{fdl},m}$, $P_{\mathtt{ful},m}$).

  * •

The $\operatorname{\textbf{HD}}$ scheme slightly outperforms the
$\operatorname{\textbf{FD}}$ scheme in terms of EE. This is due to the fact
that in the $\operatorname{\textbf{HD}}$ scheme, the APs (UEs) consume power
in DL (UL) only during half of each time slot, which is reflected by the
factor of $\frac{1}{2}$ in (V-B). This advantage of the
$\operatorname{\textbf{HD}}$ scheme outperforms the disadvantage of having a
lower SE than the $\operatorname{\textbf{FD}}$ scheme.

####  VI-B3  Impact of the number of AP antennas on SE and EE

Figures 4 and 5 illustrate the average sum SE and EE of NAFD, traditional HD,
and FD schemes under different numbers of antennas per AP, respectively. Here,
the total number of AP antennas is kept fixed, i.e., $NM=80$. It can be seen
from Figure 1 that by increasing the number of antennas at each AP, the sum SE
of the NAFD is monotonously increased, while in the cases of FD and HD
networks, it does not change much for $N\geq 10$. Therefore, by increasing the
number of antennas at each AP, the benefits of the optimized NAFD network are
more pronounced.

From Fig. 5, we can see that there is an optimal number of antennas per AP for
a maximum EE, e.g., $N=5$ in the considered setting. In principle, this is
reasonable because when the number of AP antennas increases, the number of APs
decreases. On one hand, there are possibly UEs that are now far away from the
APs and have low SEs, which slows down the increase in the sum SE. On the
other hand, the APs need to use more power to serve these far UEs to guarantee
their quality-of-service SEs, which leads to a lower EE. In the regime of
small values of $N$, i.e., $N\leq 5$, the number of APs, i.e., $M\geq 16$, is
still large and the APs create proper coverage for all the UEs. Thus,
increasing the number of APs leads to higher achievable UE rates without
increasing the power consumption too much, and hence, increases the EE. In the
regime of large values of $N$, e.g., $N\geq 8$, the number of APs, i.e.,
$M\leq 10$, is small and the coverage in the network is poor. The increase in
power consumption to achieve QoS SEs dominates the increase in the achievable
UE rates, which leads to a significant decrease in the EE. Also, in this
regime, the EE slightly increases when $N$ increases. This is because the
decrease in the number of APs now leads to a reduction in the total fixed
power consumption and the total power consumption in backhaul links, while the
increase in the achievable SE of UEs is not big enough.

####  VI-B4 Impact of residual SI on the sum SE of the
$\operatorname{\textbf{FD}}$ scheme

Figure 6 depicts the average SEs of CF-mMIMO systems with the proposed
$\operatorname{\textbf{NAFD}}$ and traditional $\operatorname{\textbf{HD}}$
and $\operatorname{\textbf{FD}}$ schemes. We recall that the SI has no effect
on the performance of the $\operatorname{\textbf{NAFD}}$ and
$\operatorname{\textbf{HD}}$ schemes. The first observation is that
$\operatorname{\textbf{FD}}$ scheme outperforms $\operatorname{\textbf{HD}}$
counterpart at a sufficiently small level of residual SI.
$\operatorname{\textbf{FD}}$ scheme provides more than $18\%$ SE performance
gain over the $\operatorname{\textbf{HD}}$. However, its performance is
dramatically degraded when $\sigma_{\text{SI}}^{2}/\sigma_{n}^{2}$ increases.
Another interesting observation is that the proposed
$\operatorname{\textbf{NAFD}}$ scheme provides the best performance
irrespective of the residual SI strength and significantly better performance
than $\operatorname{\textbf{FD}}$ for
$\sigma_{\text{SI}}^{2}/\sigma_{n}^{2}>90$ dB. This result further confirms
that the $\operatorname{\textbf{NAFD}}$ scheme is well-suited for CF-mMIMO
networks.

Figure 6: Average SE versus different residual SI power levels
($M=40,K_{d}=K_{u}=4,\mathcal{S}_{QoS}=0.2$ bit/s/Hz). Figure 7: Impact of
individual SE requirements on EE ($M\\!=\\!40,K_{d}\\!=\\!K_{u}\\!=\\!4$
bit/s/Hz, $\sigma_{\text{SI}}^{2}/\sigma_{n}^{2}=50$ dB).

The SE of the NAFD is relatively greater than that of the FD scheme,
especially when the number of APs and/or the number of antennas per each AP is
increased. The EE gain achieved by the NAFD over the FD is much greater than
the SE gain. This is because the power consumption of the NAFD is remarkably
less than that of the FD. In a FD system, all APs contribute to UL and DL
transmissions, while in a NAFD system, a part of the APs are assigned for DL
transmission and the other part of the APs are used for UL reception. Thus,
the power consumption of the transceiver chains and the power consumed for UL
and DL transmission is significantly reduced.

####  VI-B5 Impact of individual SE requirements on EE

Figure 7 shows the EE of CF-mMIMO systems with $\operatorname{\textbf{NAFD}}$,
$\operatorname{\textbf{HD}}$, and $\operatorname{\textbf{FD}}$ schemes versus
the individual SE requirement $\mathcal{S}_{QoS}$. It can be seen that the EE
of all schemes decreases as $\mathcal{S}_{QoS}$ increases. This observation
can be interpreted as follows. To achieve higher SE requirements, more power
is consumed in the system, and then the CLI becomes more severe, which finally
leads to lower EE values. Moreover, we can observe that both
$\operatorname{\textbf{HD}}$ and $\operatorname{\textbf{FD}}$ schemes fail to
satisfy the SE requirement for $\mathcal{S}_{QoS}\geq 1.8$ bits/s/Hz, while
the $\operatorname{\textbf{NAFD}}$ scheme still meets the individual SE
requirements of the system. This result shows that the
$\operatorname{\textbf{NAFD}}$ scheme can be much more energy-efficient than
the $\operatorname{\textbf{HD}}$ and $\operatorname{\textbf{FD}}$ schemes
while achieving a high SE target.

####  VI-B6 Quality of the proposed EE maximization solution

We investigate the quality of the proposed solution for the EE maximization by
demonstrating the gap between $\operatorname{\mathtt{EE}}^{\mathsf{NAFD}}$ and
$\operatorname{\mathtt{EE}}_{fullbh}^{\mathsf{NAFD}}$ in Fig. 8. It is clear
that the gap is small for different numbers of APs $M$. This is because the
only difference between maximizing
$\operatorname{\mathtt{EE}}^{\mathsf{NAFD}}$ and
$\operatorname{\mathtt{EE}}_{fullbh}^{\mathsf{NAFD}}$ is to design the AP mode
assignment for optimizing the rate-dependent power consumption
$P_{bh}^{\mathsf{NAFD}}$ in backhaul links. However, $P_{bh}^{\mathsf{NAFD}}$
contributes only a small portion to the total power consumption
$P_{\mathtt{total}}^{\mathsf{NAFD}}$, which is shown in Table III.
Specifically, the maximum contribution of $P_{bh}^{\mathsf{NAFD}}$ in
$P_{\mathtt{total}}^{\mathsf{NAFD}}$ is $7.95\%$, and even the maximum
contribution of $P_{bh,full}^{\mathsf{NAFD}}$ in
$P_{\mathtt{total,fbh}}^{\mathsf{NAFD}}$ is only $14.18\%$. Therefore,
optimizing the AP mode assignment for further reducing rate-dependent power
consumption $P_{bh}^{\mathsf{NAFD}}$ brings no benefit to reduce
$P_{\mathtt{total}}^{\mathsf{NAFD}}$ much. Hence, the proposed solution for
the $\operatorname{\mathtt{EE}}_{fullbh}^{\mathsf{NAFD}}$ maximization problem
in (69a) is sufficiently close to the solution of the
$\operatorname{\mathtt{EE}}$ maximization problem (65a).

Figure 8:  Comparison of $\operatorname{\mathtt{EE}}^{\mathsf{NAFD}}$ and $\operatorname{\mathtt{EE}}_{fullbh}^{\mathsf{NAFD}}$ ($K_{d}=K_{u}=4,\mathcal{S}_{QoS}=0.2$ bit/s/Hz, $\sigma_{\text{SI}}^{2}/\sigma_{n}^{2}=50$ dB). TABLE III: Average $\frac{P_{bh}^{\mathsf{NAFD}}}{P_{\mathtt{total}}^{\mathsf{NAFD}}}$ and $\frac{P_{bh,full}^{\mathsf{NAFD}}}{P_{\mathtt{total,fbh}}^{\mathsf{NAFD}}}$ $M$ | $20$ | $30$ | $40$ | $50$
---|---|---|---|---
$\frac{P_{bh}^{\mathsf{NAFD}}}{P_{\mathtt{total}}^{\mathsf{NAFD}}}$ | $6.71\%$ | $6.91\%$ | $7.54\%$ | $7.95\%$
$\frac{P_{bh,full}^{\mathsf{NAFD}}}{P_{\mathtt{total,fbh}}^{\mathsf{NAFD}}}$ | $10.55\%$ | $11.42\%$ | $12.97\%$ | $14.18\%$

##  VII Conclusion

We have investigated the sum SE and EE performance of NAFD CF-mMIMO systems.
We proposed a large-scale-fading-based joint optimization approach of
designing the AP mode assignment, UL and DL power control, and LSFD weights to
maximize the sum SE and EE under a realistic power consumption model,
individual QoS SE requirements and transmit power constraints. The proposed
approach was then applied to maximize the sum SE and EE of CF-mMIMO systems
with heuristic NAFD approaches as well as traditional HD and FD approaches. We
showed that our jointly optimized NAFD approach provides significant SE and EE
gains over the heuristic NAFD approaches. Our results also confirm that in a
CF-mMIMO system, the NAFD scheme can achieve a noticeable SE gain, while
improving remarkably the EE compared with the HD and FD schemes. Insights from
the simulation results demonstrated that the ratio between the number of APs
and the number of UEs is a dominating factor of the system performance. If the
ratio is large, the NAFD scheme with random AP mode assignment offers an
acceptable performance; accordingly, balances the performance and complexity
of the NAFD scheme. Finally, finding lower-complexity resource allocation
approaches, such as machine learning-based algorithms, that can achieve
acceptable SE and EE performance in NAFD CF-mMIMO is a timely research topic
for future research.

##  Appendix A  Downlink SE Derivation

According to (II-B1), in order to detect $s_{k}^{\mathtt{dl}}$, the $k$-th DL
UE need to have access to the effective channel
$\sqrt{\rho_{d}}\sum_{m\in\mathcal{M}}\theta_{mk}\left({\bf
g}_{mk}^{\mathtt{dl}}\right)^{T}\left(\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\right)^{*}$. However, since there is no pilot in DL,
this CSI is not available at DL UE $k$. To deal with this challenge, UE $k$
will rely on the stochastic CSI to detect $s_{k}^{\mathtt{dl}}$. The received
signal in (II-B1) can be rewritten as

$\displaystyle~{}y_{k}^{\mathtt{dl}}$
$\displaystyle=\mathbb{DS}_{k}s_{k}^{\mathtt{dl}}+\mathbb{BU}_{k}s_{k}^{\mathtt{dl}}+\sum_{k^{\prime}\in\mathcal{K}_{d}\setminus
k}\mathbb{DI}_{kk^{\prime}}s_{k^{\prime}}^{\mathtt{dl}}$ $\displaystyle\hskip
20.00003pt+\sum_{\ell\in\mathcal{K}_{u}}\mathbb{UI}_{\ell}s_{\ell}^{\mathtt{ul}}+w_{k}^{\mathtt{dl}},$
(94)

where

$\displaystyle\mathbb{DS}_{k}$
$\displaystyle\triangleq\sqrt{\rho_{d}}\mathbb{E}\bigg{\\{}\sum_{m\in\mathcal{M}}\eta_{mk}^{1/2}\left({\bf
g}_{mk}^{\mathtt{dl}}\right)^{T}\left(\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\right)^{*}\bigg{\\}},$ $\displaystyle\mathbb{BU}_{k}$
$\displaystyle\triangleq\sqrt{\rho_{d}}\Bigg{(}\sum_{m\in\mathcal{M}}\eta_{mk}^{1/2}\left({\bf
g}_{mk}^{\mathtt{dl}}\right)^{T}\left(\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\right)^{*}$ $\displaystyle\hskip
40.00006pt-\mathbb{E}\left\\{\sum_{m\in\mathcal{M}}\eta_{mk}^{1/2}\left({\bf
g}_{mk}^{\mathtt{dl}}\right)^{T}\left(\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\right)^{*}\right\\}\Bigg{)}$
$\displaystyle\mathbb{DI}_{kk^{\prime}}$
$\displaystyle\triangleq\sqrt{\rho_{d}}\sum_{m\in\mathcal{M}}\eta_{mk^{\prime}}^{1/2}\left({\bf
g}_{mk}^{\mathtt{dl}}\right)^{T}\left(\hat{{\bf
g}}_{mk^{\prime}}^{\mathtt{dl}}\right)^{*},$ $\displaystyle\mathbb{UI}_{\ell}$
$\displaystyle\triangleq h_{k\ell}\sqrt{\rho_{u}\tilde{\varsigma}_{\ell}},$
(95)

represent the strength of the desired DL signal, the beamforming gain
uncertainty, cross-link interference caused by the $k^{\prime}$-th DL UE, and
cross-link interference caused by the $\ell$-th UL UE, respectively. The sum
of the last four terms in (A) is treated as the effective noise, which is
uncorrelated with the first term, i.e., the desired signal [5]. Utilizing the
use-and-then-forget capacity bounding technique in [5], the corresponding SE
of the DL UE $k$ is given by

$\displaystyle\mathcal{S}_{\mathtt{dl},k}=\frac{\tau_{c}-\tau_{t}}{\tau_{c}}\log_{2}\Big{(}1+$
$\displaystyle\vspace{0em}\frac{|\mathbb{DS}_{k}|^{2}}{\mathbb{E}\left\\{\\!|\mathbb{BU}_{k}|^{2}\\!\right\\}\\!\\!+\\!\sum_{k^{\prime}\in\mathcal{K}_{d}\setminus
k}\\!\\!\mathbb{E}\left\\{\\!|\mathbb{DI}_{kk^{\prime}}|^{2}\\!\right\\}+\\!\sum_{\ell\in\mathcal{K}_{u}}\\!\\!\mathbb{E}\left\\{\\!|\mathbb{UI}_{\ell}|^{2}\\!\right\\}\\!+\\!1}\Big{)}.$
(96)

Therefore, we need to compute $\mathbb{DS}_{k}$,
$\mathbb{E}\left\\{|\mathbb{BU}_{k}|^{2}\right\\}$,
$\mathbb{E}\left\\{|\mathbb{DI}_{kk^{\prime}}|^{2}\right\\}$, and
$\mathbb{E}\left\\{|\mathbb{UI}_{\ell}|^{2}\right\\}$. To compute
$\mathbb{DS}_{k}$, we have

$\displaystyle~{}\mathbb{DS}_{k}$
$\displaystyle=\sqrt{\rho_{d}}\mathbb{E}\bigg{\\{}\sum_{m\in\mathcal{M}}\eta_{mk}^{1/2}\left(\hat{{\bf
g}}_{mk}^{\mathtt{dl}}+\tilde{{\bf
g}}_{mk}^{\mathtt{dl}}\right)^{T}\left(\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\right)^{*}\bigg{\\}}$
$\displaystyle=\sqrt{\rho_{d}}\sum_{m\in\mathcal{M}}\eta_{mk}^{1/2}\mathbb{E}\bigg{\\{}\|\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\|^{2}\bigg{\\}}$
$\displaystyle=\sqrt{\rho_{d}}\sum_{m\in\mathcal{M}}N\eta_{mk}^{1/2}\gamma_{mk}^{\mathtt{dl}},$
(97)

where we have used the fact that $\hat{{\bf g}}_{mk}^{\mathtt{dl}}$ and
$\tilde{{\bf g}}_{mk}^{\mathtt{dl}}$ are zero mean and independent.

Noticing that the variance of a sum of independent RVs is equal to the sum of
the variances, we can derive
$\mathbb{E}\left\\{|\mathbb{BU}_{k}|^{2}\right\\}$ as

$\displaystyle~{}\mathbb{E}\left\\{|\mathbb{BU}_{k}|^{2}\right\\}$
$\displaystyle={\rho_{d}}\sum_{m\in\mathcal{M}}\eta_{mk}\mathbb{E}\bigg{\\{}\bigg{|}\left({\bf
g}_{mk}^{\mathtt{dl}}\right)^{T}\left(\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\right)^{*}$ $\displaystyle\hskip
50.00008pt-\mathbb{E}\Big{\\{}\left({\bf
g}_{mk}^{\mathtt{dl}}\right)^{T}\left(\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\right)^{*}\Big{\\}}\bigg{|}^{2}\bigg{\\}}$
$\displaystyle={\rho_{d}}\sum_{m\in\mathcal{M}}\eta_{mk}\bigg{(}\mathbb{E}\bigg{\\{}\bigg{|}\left({\bf
g}_{mk}^{\mathtt{dl}}\right)^{T}\left(\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\right)^{*}\bigg{|}^{2}\bigg{\\}}$ $\displaystyle\hskip
50.00008pt-\bigg{|}\mathbb{E}\Big{\\{}\left({\bf
g}_{mk}^{\mathtt{dl}}\right)^{T}\left(\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\right)^{*}\Big{\\}}\bigg{|}^{2}\bigg{)}$
$\displaystyle={\rho_{d}}\sum_{m\in\mathcal{M}}\eta_{mk}\bigg{(}\mathbb{E}\Big{\\{}\Big{|}\left(\tilde{{\bf
g}}_{mk}^{\mathtt{dl}}\right)^{T}\left(\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\right)^{*}\Big{|}^{2}\Big{\\}}$ $\displaystyle\hskip
50.00008pt+\mathbb{E}\Big{\\{}\big{\|}\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\big{\|}^{4}\Big{\\}}-N^{2}(\gamma_{mk}^{\mathtt{dl}})^{2}\bigg{)},$
$\displaystyle={\rho_{d}}N\sum_{m\in\mathcal{M}}\eta_{mk}\gamma_{mk}^{\mathtt{dl}}\beta_{mk}^{\mathtt{dl}},$
(98)

where the final result follows from the fact that
$\mathbb{E}\big{\\{}\big{|}\left(\tilde{{\bf
g}}_{mk}^{\mathtt{dl}}\right)^{T}\left(\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\right)^{*}\big{|}^{2}\big{\\}}=\mathbb{E}\big{\\{}\big{|}\tilde{{\bf
g}}_{mk}^{\mathtt{dl}}|^{2}\big{\\}}\mathbb{E}\big{\\{}|\hat{{\bf
g}}_{mk}^{\mathtt{dl}}|^{2}\big{\\}}=N\gamma_{mk}^{\mathtt{dl}}(\beta_{mk}^{\mathtt{dl}}-\gamma_{mk}^{\mathtt{dl}})$
and $\mathbb{E}\big{\\{}\big{\|}\hat{{\bf
g}}_{mk}^{\mathtt{dl}}\big{\|}^{4}\big{\\}}=N(N+1)(\gamma_{mk}^{\mathtt{dl}})^{2}$.

Following similar steps, we can compute
$\mathbb{E}\left\\{|\mathbb{UI}_{\ell}|^{2}\right\\}$ and
$\mathbb{E}\left\\{|\mathbb{UI}_{\ell}|^{2}\right\\}$ as

$\displaystyle~{}\mathbb{E}\left\\{|\mathbb{DI}_{kk^{\prime}}|^{2}\right\\}$
$\displaystyle={\rho_{d}}N\sum_{m\in\mathcal{M}}\eta_{mk}\gamma_{mk}^{\mathtt{dl}}\beta_{mk}^{\mathtt{dl}},$
$\displaystyle\mathbb{E}\left\\{|\mathbb{UI}_{\ell}|^{2}\right\\}$
$\displaystyle={\rho_{u}\tilde{\varsigma}_{\ell}}\beta_{k\ell}^{\mathtt{du}}.$
(99)

To this end, by substituting (A), (A), and (A) into (A), the desired result in
(13), is obtained.

## References

  * [1] M. Mohammadi, T. T. Vu, B. Naderi Beni, H. Q. Ngo, and M. Matthaiou, “Virtually full-duplex cell-free massive MIMO with access point mode assignment,” in _Proc. IEEE Int. Workshop Signal Process. Adv. Wireless Commun. (SPAWC)_ , Jul. 2022, pp. 1-5.
  * [2] M. Matthaiou, _et al,_ “The road to 6G: Ten physical layer challenges for communications engineers,” _IEEE Commun. Mag.,_ vol. 59, no. 1, pp. 64-69, Jan. 2021.
  * [3] H. Zhu, “Performance comparison between distributed antenna and microcellular systems,” _IEEE J. Sel. Areas Commun.,_ vol. 29, no. 6, pp. 1151-1163, June 2011.
  * [4] S. Venkatesan, A. Lozano, and R. Valenzuela, “Network MIMO: Overcoming intercell interference in indoor wireless systems,” in _Proc. IEEE Asilomar Conf. Signals Syst. Comput.,_ Nov. 2007, pp. 83-87.
  * [5] H. Q. Ngo, A. Ashikhmin, H. Yang, E. G. Larsson, and T. L. Marzetta, “Cell-free massive MIMO versus small cells,” _IEEE Trans. Wireless Commun.,_ vol. 16, no. 3, pp. 1834-1850, Mar. 2017.
  * [6] H. V. Nguyen _et al._ , “On the spectral and energy efficiencies of full-duplex cell-free massive MIMO,” _IEEE J. Sel. Areas Commun._ , vol. 38, no. 8, pp. 1698-1718, Aug. 2020.
  * [7] M. Duarte, “Full-duplex wireless: Design, implementation and characterization,” Ph.D. dissertation, Dept. Elect. and Computer Eng., Rice University, Houston, TX, 2012.
  * [8] A. Sabharwal, P. Schniter, D. Guo, D. W. Bliss, S. Rangarajan, and R. Wichman, “In-band full-duplex wireless: Challenges and opportunities,” _IEEE J. Sel. Areas Commun._ , vol. 32, no. 9, pp. 1637-1652, Sep. 2014.
  * [9] D. Korpi, L. Anttila, V. Syrjal ¨ a, and M. Valkama, “Widely linear digital self-interference cancellation in direct-conversion full-duplex transceiver,” _IEEE J. Sel. Areas Commun._ , vol. 32, pp. 1674-1687, Sep. 2014.
  * [10] S. Hong _et al._ , “Applications of self-interference cancellation in 5G and beyond,” _IEEE Commun. Mag._ , vol. 52, no. 2, pp. 114-121, Feb. 2014.
  * [11] Z. Zhang, X. Chai, K. Long, A. V. Vasilakos, and L. Hanzo, “Full duplex techniques for 5G networks: self-interference cancellation, protocol design, and relay selection,” _IEEE Commun. Mag.,_ vol. 53, no. 5, pp. 128-137, May 2015.
  * [12] S. Datta, D. N. Amudala, E. Sharma, R. Budhiraja, and S. S. Panwar, “Full-duplex cell-free massive MIMO systems: Analysis and decentralized optimization,” _IEEE Open J. Commun. Society,_ vol. 3, pp. 31-50, Dec. 2021.
  * [13] D. Wang, M. Wang, P. Zhu, J. Li, J. Wang, and X. You, “Performance of network-assisted full-duplex for cell-free massive MIMO,” _IEEE Trans. Commun._ , vol. 68, no. 3, pp. 1464-1478, Mar. 2020.
  * [14] J. Li, Q. Lv, P. Zhu, D. Wang, J. Wang, and X. You, “Network-assisted full-duplex distributed massive MIMO systems with beamforming training based CSI estimation,” _IEEE Trans. Wireless Commun.,_ vol. 20, no. 4, pp. 2190-2204, Apr. 2021.
  * [15] H. Thomsen, P. Popovski, E. d. Carvalho, N. K. Pratas, D. M. Kim, and F. Boccardi, “CoMPflex: CoMP for in-band wireless full duplex,” _IEEE Wireless Commun. Lett._ , vol. 5, no. 2, pp. 144–147, Apr. 2016
  * [16] Y. Xin, R. Zhang, D. Wang, J. Li, L. Yang, and X. You, “Antenna clustering for bidirectional dynamic network with large-scale distributed antenna systems,” _IEEE Access_ , vol. 5, pp. 4037-4047, 2017.
  * [17] X. Xia, P. Zhu, J. Li, D. Wang, Y. Xin, and X. You, “Joint sparse beamforming and power control for a large-scale DAS with networkassisted full duplex,” IEEE Trans. Veh. Technol., vol. 69, no. 7, pp. 7569-7582, July 2020.
  * [18] X. Xia _et al._ , “Joint user selection and transceiver design for cell-free with network-assisted full duplexing,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 12, pp. 7856-7870, Dec. 2021.
  * [19] Y. Zhu, J. Li, P. Zhu, H. Wu, D. Wang, and X. You, “Optimization of duplex mode selection for network-assisted full-duplex cell-free massive MIMO systems,” _IEEE Commun. Lett._ , vol. 25, no. 11, pp. 3649-3653, Nov. 2021.
  * [20] X. Xia, P. Zhu, J. Li, H. Wu, D. Wang, and Y. Xin, “Joint optimization of spectral efficiency for cell-free massive MIMO with networkassisted full duplexing,” _Science China Inf. Sciences_ , vol. 64, no. 8, pp. 1-16, 2021.
  * [21] X. Xia et al., “Joint uplink power control, downlink beamforming, and mode selection for secrecy cell-free massive MIMO with network-assisted full duplexing,” _IEEE Sys. J._ , pp. 1-12, Jul. 2022.
  * [22] M. Shafi _et al_., “5G: A tutorial overview of standards, trials, challenges, deployment, and practice,” _IEEE J. Sel. Areas Commun._ , vol. 35, no. 6, pp. 1201-1221, June 2017.
  * [23] H. Q. Ngo, L.-N. Tran, T. Q. Duong, M. Matthaiou, and E. G. Larsson,“On the total energy efficiency of cell-free massive MIMO,” _IEEE Trans. Green Commun. Netw._ , vol. 2, no. 1, pp. 25-39, Mar. 2018.
  * [24] A. Chowdhury, R. Chopra, and C. R. Murthy, “Can dynamic TDD enabled half-duplex cell-free massive MIMO outperform full-duplex cellular massive MIMO?” _IEEE Trans. Commun._ , vol. 70, no. 7, pp. 4867-4883, May 2022.
  * [25] J. Zheng, J. Zhang, E. Björnson, and B. Ai, “Impact of channel aging on cell-free massive MIMO over spatially correlated channels,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 10, pp. 6451-6466, Oct. 2021.
  * [26] E. Björnson, L. Sanguinetti, J. Hoydis, and M. Debbah, “Optimal design of energy-efficient multi-user MIMO systems: Is massive MIMO the answer?” _IEEE Trans. Wireless Commun._ , vol. 14, no. 6, pp. 3059-3075, June 2015.
  * [27] M. Bashar, K. Cumanan, A. G. Burr, M. Debbah, and H. Q. Ngo, “On the uplink max–min SINR of cell-free massive MIMO systems,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 4, pp. 2021-2036, Apr. 2019.
  * [28] T. T. Vu, D. T. Ngo, H. Q. Ngo, and T. Le-Ngoc, “Full-duplex cell-free massive MIMO,” in _Proc. IEEE Int. Conf. Commun. (ICC)_ , May 2019, pp. 1-6.
  * [29] M. Bashar et al., “Uplink spectral and energy efficiency of cell-free massive MIMO with optimal uniform quantization,” _IEEE Trans. Commun._ , vol. 69, no. 1, pp. 223-245, Oct. 2021.
  * [30] T. T. Vu, D. T. Ngo, M. N. Dao, S. Durrani, and R. H. Middleton, “Spectral and energy efficiency maximization for content-centric CRANs with edge caching,” _IEEE Trans. Commun._ , vol. 66, no. 12, pp. 6628-6642, Dec. 2018.
  * [31] T. T. Vu, D. T. Ngo, N. H. Tran, H. Q. Ngo, M. N. Dao, and R. H. Middleton, “Cell-free massive MIMO for wireless federated learning,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 10, pp. 6377-6392, Oct. 2020.
  * [32] H. H. M. Tam, H. D. Tuan, D. T. Ngo, T. Q. Duong, and H. V. Poor, “Joint load balancing and interference management for small-cell heterogeneous networks with limited backhaul capacity,” _IEEE Trans. Wireless Commun._ , vol. 16, no. 2, pp. 872-884, Feb. 2017.
  * [33] E. Aryafar, M. A. Khojastepour, K. Sundaresan, S. Rangarajan, and M. Chiang, “MIDU: Enabling MIMO full duplex,” in _Proc. IEEE Annual Int. Conf. Mobile Comput. Netw. (MOBICOM)_ , Aug. 2012, pp. 257-268.
  * [34] H. A. Suraweera, I. Krikidis, G. Zheng, C. Yuen, and P. J. Smith, “Low-complexity end-to-end performance optimization in MIMO full-duplex relay systems,” _IEEE Trans. Wireless Commun._ , vol. 13, no. 2, pp. 913-927, Feb. 2014.
  * [35] M. Mohammadi, H. A. Suraweera, and C. Tellambura, “Uplink/downlink rate analysis and impact of power allocation for full-duplex cloud-RANs,” _IEEE Trans. Wireless Commun._ , vol. 17, no. 9, pp. 5774-5788, Sept. 2018.
  * [36] T. Riihonen, S. Werner, and R. Wichman, “Mitigation of loopback self-interference in full-duplex MIMO relays,” _IEEE Trans. Signal Process._ , vol. 59, no. 12, pp. 5983-5993, Dec. 2011
  * [37] Y. Zhang, M. Xiao, S. Han, M. Skoglund, and W. Meng, “On precoding and energy efficiency of full-duplex millimeter-wave relays,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 3, pp. 1943-1956, Mar. 2019.
  * [38] H. Q. Ngo, H. A. Suraweera, M. Matthaiou, and E. G. Larsson, “Multipair full-duplex relaying with massive arrays and linear processing,” _IEEE J. Sel. Areas Commun._ , vol. 32, no. 9, pp. 1721-1737, Sep. 2014.
  * [39] D. Nguyen, L.-N. Tran, P. Pirinen, and M. Latva-aho, “Precoding for full-duplex multiuser MIMO systems: Spectral and energy efficiency maximization,” _IEEE Trans. Signal Process._ , vol. 61, no. 16, pp. 4038- 4050, Jun. 2013.
  * [40] E. Bjornson and L. Sanguinetti, “Making cell-free massive MIMO ¨ competitive with MMSE processing and centralized implementation,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 1, pp. 77-90, Jan. 2020.
  * [41] M. Bashar, K. Cumanan, A. G. Burr, H. Q. Ngo, E. G. Larsson, and P. Xiao, “Energy efficiency of the cell-free massive MIMO uplink with optimal uniform quantization,” _IEEE Trans. Green Commun. Netw._ , vol. 3, no. 4, pp. 971-987, Dec. 2019.