# Energy-Efficient Resource Allocation in Massive MIMO-NOMA Networks with
Wireless Power Transfer: A Distributed ADMM Approach

Zhongyu Wang,  _Student Member, IEEE_ , Zhipeng Lin,  _Member, IEEE_ , Tiejun
Lv,  _Senior Member, IEEE_ , and Wei Ni,  _Senior Member, IEEE_ Manuscript
received January 18, 2020; revised June 11, 2020; accepted March 19, 2021.
_(Corresponding author: Tiejun Lv.)_ Z. Wang, Z. Lin and T. Lv are with the
School of Information and Communication Engineering, Beijing University of
Posts and Telecommunications (BUPT), Beijing 100876, China (e-mail:
{zhongyuwang, linlzp, lvtiejun}@bupt.edu.cn). W. Ni is with the Commonwealth
Scientific and Industrial Research Organisation (CSIRO), Sydney 2122,
Australia (e-mail: Wei.Ni@data61.csiro.au).

###### Abstract

In multicell massive multiple-input multiple-output (MIMO) non-orthogonal
multiple access (NOMA) networks, base stations (BSs) with multiple antennas
deliver their radio frequency energy in the downlink, and Internet-of-Things
(IoT) devices use their harvested energy to support uplink data transmission.
This paper investigates the energy efficiency (EE) problem for multicell
massive MIMO NOMA networks with wireless power transfer (WPT). To maximize the
EE of the network, we propose a novel joint power, time, antenna selection,
and subcarrier resource allocation scheme, which can properly allocate the
time for energy harvesting and data transmission. Both perfect and imperfect
channel state information (CSI) are considered, and their corresponding EE
performance is analyzed. Under quality-of-service (QoS) requirements, an EE
maximization problem is formulated, which is non-trivial due to non-convexity.
We first adopt nonlinear fraction programming methods to convert the problem
to be convex, and then, develop a distributed alternating direction method of
multipliers (ADMM)-based approach to solve the problem. Simulation results
demonstrate that compared to alternative methods, the proposed algorithm can
converge quickly within fewer iterations, and can achieve better EE
performance.

###### Index Terms:

Energy efficiency, massive MIMO NOMA, Wireless power transfer, antenna
selection, channel state information, alternating direction method of
multipliers.

##  I Introduction

###  I-A Background and Motivation

Explosive developments of the Internet-of-Things (IoT) stimulate a huge demand
for massive connectivity capability, which enables a large number of IoT
devices to access wireless networks [1]. The accessed IoT devices would
rapidly drain their batteries if they are used in high power-consumption
applications [2]. Therefore, it is important to ensure that massive energy-
constrained IoT devices can communicate with each other in a spectrally- and
energy-efficient manner while meeting quality of service (QoS) requirements
[3]. Wireless power transfer (WPT) uses uninterrupted power supply methods to
remotely charge IoT devices with small rechargeable batteries [4]. Such
technique enables battery-powered IoT devices to extract the radio frequency
(RF) signal energy to prolong their lifecycles [5]. As a result, continuous
power supply requirements of IoT devices can be met [6].

Massive MIMO has been regarded as an important technology to promote the
deployment of the fifth-generation and beyond (B5G) networks [5]. Utilizing
beams for spatial multiplexing, massive MIMO can effectively concentrate
received signal power with very narrow beams and achieve very dense deployment
for IoT networks. Massive MIMO is also an enable technique for for WPT
systems, because it can provide system-level energy efficiency (EE)[7, 8]. In
[9], user scheduling, power allocation, and rate adaptation scheme were
jointly optimized to improve the EE of orthogonal frequency division multiple
access (OFDMA) networks. Nevertheless, aforementioned transmission schemes [7,
8, 9] are carried out in an orthogonal multiple access (OMA) manner. OMA is
restrictive in terms of spectral efficiency (SE) for massive IoT devices
accessed networks, because each frequency subchannel can only be used by a
single user at a time.

To effectively use communication spectrums and improve system connectivity,
non-orthogonal multiple access (NOMA) has been proposed, which can guarantee
multiple devices access the same subchannel simultaneously in the power domain
[10]. [11] proposed a mobile user association scheme for small-cell networks.
The scheme used NOMA and successive interference cancellation (SIC) to achieve
the system utility maximization and total transmit power minimization. NOMA
can also enlarge the capacity region of OMA via the power multiplexing and SIC
at transceivers [12]. Combining NOMA with MIMO (referred to as “MIMO-NOMA”) is
able to significantly enhance the SE, channel capacity and EE [13, 14]. Under
the constraint of QoS, an EE maximization problem was studied in an uplink
NOMA-based millimeter-wave massive MIMO system [15]. By installing large scale
antennas at a base station (BS), the transmit power can be concentrated in
some areas, consequently improving the throughput and channel rate of massive
MIMO networks [16]. Nevertheless, employing massive antenna arrays would
dramatically increase energy consumption caused by of IoT networks due to the
deployment of massive antennas [17]. In this sense, it is important to find an
effective method, which can properly select the antenna number to optimize the
EE of the considered multicell WPT-enabled massive MIMO-NOMA networks.

The architecture of WPT-based massive MIMO-NOMA networks is envisioned to be a
prospective approach to enhance the SE and EE of the IoT networks. As the B5G
wireless communications have various radio resources (e.g., time, frequency,
spatial, power and code domains, etc.) [18], it is critical for massive MIMO-
NOMA IoT to achieve an energy-efficient resource allocation with incessantly
energy supply. It is also vital to find efficient resource allocation
strategies of radio resource for practical resource-constrained massive MIMO-
NOMA IoT environments with imperfect channel state information (CSI). This is
because perfect CSI is hard to be acquired. The accuracy of the obtained CSI
is dependent on channel feedback and quantization errors [19]. As a result, we
need to exploit NOMA for WPT to implement the massive connectivity for IoT
networks, and properly allocate resources in the distributed massive MIMO-NOMA
networks with multiple antenna selection (AS).

###  I-B Contribution

This paper considers the multicell massive MIMO-NOMA network, where the WPT-
powered IoT devices transmit data after they harvest sufficient energy from
the BSs. We divide the time block into two parts, i.e., the WPT time and the
wireless information transmission (WIT) time. The AS technique, which can
improve the throughput and EE performance of the system, is applied to avoid
excessive cost (such as hardware cost, device power and software computation).
To maximize the EE of the massive MIMO-NOMA network, we propose a distributed
alternating-direction-method-of-multipliers (ADMM)-based resource allocation
algorithm. The main contributions are summarized as follows.

  * •

We propose a WPT-based distributed massive MIMO-NOMA model, in which multi-
antenna BSs and massive rechargeable IoT devices are deployed, to meet the
massive connectivity requirement and achieve the high EE. In the power domain,
we employ NOMA technique to achieve SIC and improve the EE. In the time
domain, we design a two-time-slot division protocol, which can effectively
divide the WPT and WIT time.

  * •

We derive the signal-to-interference-plus-noise ratio (SINR) and EE with the
multiple AS under the perfect and imperfect CSI. A new scheme, which jointly
optimizes the power control, time division, AS and subcarrier assignment, is
proposed, and its corresponding EE optimization problem is formulated for the
distributed multicell massive MIMO-NOMA networks.

  * •

We apply nonlinear optimization to convert the original non-convex fractional
programming problem to be convex. A novel distributed ADMM-based energy-
efficient resource allocation algorithm is also proposed to divide the global
consensus problem into several sequential optimization phases. In this way,
the optimal solution of the problem can be obtained.

  * •

Extensive simulations under different parameters show substantial EE
improvements of the proposed distributed ADMM-based resource allocation
algorithm in the massive MIMO-NOMA networks. We also demonstrate that the
proposed distributed ADMM-based algorithm converges well and outperforms the
existing benchmarks in terms of overall EE.

###  I-C Organization

The rest of this paper is organized as follows. Section II is literature
review. The system model and the communication process are introduced in
Section III. Section IV formulates the EE optimization problem. In Section V,
a distributed resource allocation algorithm via ADMM is proposed to address
the EE maximization. Section VI shows the simulation results. Finally, the
conclusions and future works are drawn in Section VII.

_Notation:_ $a$, $\mathbf{a}$, $\mathcal{A}$ and $\mathbf{A}$ stand for a
scalar, a column vector, a set and a matrix, respectively; $\mathbf{I}$
represents the identity matrix; $\mathbf{A}^{H}$ represents the Hermitian
transpose of $\mathbf{A}$; $\|\cdot\|$ and $|\cdot|$ denote the L-norm and
norm of a vector; $E\left[\cdot\right]$ denotes the expectation;
$\mathcal{C}\mathcal{N}(a,b)$ denotes the distribution of a circularly
symmetric complex Gaussian (CSCG) vector with mean $a$ and variance $b$;
$\mathcal{O}(\cdot)$ denotes the computational complexity.

##  II Related Work

In recent years, WPT has widely attracted attention in academia and industry,
as it can extend the lifetime of energy-limited IoT devices [20]. [21]
proposed a simultaneous wireless information and power transfer scheme to
maximize the minimum achievable rate of all IoT devices by using power-
splitting techniques in large-scale antenna array systems. [19] maximized the
throughput of a WPT-powered massive MIMO system. [22] studied the downlink and
uplink EE tradeoff in a time-division duplexing (TDD) OFDMA system, which has
a single access point and multiple WPT-enabled IoT devices. However, the
optimization schemes in [20, 21, 19, 22] only considered OMA transmissions,
and their achieved channel capacity and SE were restrictive. Due to dense
deployment of IoT devices and the massive connectivity requirement, providing
large throughput and high SE is significantly important for future IoT
networks.

As a key technology of 5G/B5G communications, NOMA is capable of achieving the
massive connectivity in ultra-dense networks, and improving the SE and EE by
allowing the user devices to share the same subchannel simultaneously [23, 24,
25, 26, 27]. [24] showed that employing NOMA in massive MIMO networks can
markedly improve the SE, compared with those using OFDMA. In [25], power
allocation was optimized to maximize the EE of single-carrier multiuser NOMA
systems. By applying SIC and superposition coding at the WPT-enabled NOMA
receiver [26], the secure rate and EE were enhanced, which can guarantee the
information security and green communication. By applying NOMA to the
beamspace MIMO, [27] enlarged the accessed device number on the same frequency
band simultaneously, and attained a high SE.

Traditional design of massive MIMO-NOMA networks focused on improving the
system throughput or SE [6]. [28] investigated a new multi-antenna non-
orthogonal beamspace framework, which achieved the massive connectivity for
cellular massive MIMO IoT with limited spectrum resources. In [29], a joint
optimization strategy of transmission power and transfer time, was proposed
for massive antenna array systems to accomplish a long-distance and delay-
guaranteed resource allocation. Deploying multiple transmission antennas in
massive MIMO-NOMA networks could potentially scale up the system SE, but this
method resulted in expensive RF chain hardware cost, large energy consumption,
and high signal processing complexity at transmitters [30]. The AS techniques
provide a practical means to address such problem while guaranteeing the
diversity and throughput gains of massive MIMO. There are only a few studies
focusing on using AS for NOMA systems [31]. Regarding to the considered
multicell WPT-powered distributed massive MIMO-NOMA networks with AS, no
existing works have been presented to improve the system EE performance.

Most current works assumed that CSI is perfectly known at the transmitters
when allocating the radio resources [32, 7, 23, 24, 25]. As channel estimation
errors, inaccurate feedback and time-varying channel characteristic are
inevitable, perfect CSI can hardly be obtained by the transmitters [19]. In
[33], a joint optimization problem of user scheduling, power allocation and
SIC decoding was addressed under the conditions of imperfect CSI and QoS
requirements. In [34], user scheduling and power allocation were alternately
optimized to converge to the maximal EE under imperfect CSI, and it showed
that the proposed algorithm can outperform OMA schemes. Accurate CSI knowledge
is essential in WPT enabled massive MIMO-NOMA networks, as it can contribute
to both higher energy conversion efficiency and faster channel transmission
rate.

To overcome the difficulties in massive connectivity and high EE in current
resource-limited IoT environments, an energy-efficient resource allocation
optimization problem is investigated in the distributed massive MIMO-NOMA
networks deployed multiple rechargeable IoT devices. Different from the works
under perfect CSI assumption, we take the imperfect CSI into consideration.
The effect of AS on WPT-based massive MIMO-NOMA networks has also not yet been
studied. In light of these observations, the transmit power, time, AS and
subcarrier allocation are jointly investigated for the considered network.
Different from conventional greedy and centralized algorithms using the
Lagrange multiplier method [2, 9, 10, 35], we invoke the distributed ADMM
approach [36, 37] to iteratively solve the primal and dual subproblems of the
convex optimization. As a result, the maximal EE performance can be achieved
in a distributed and computationally efficient manner.

##  III System model

As shown in Fig. 1, we consider a distributed multicell massive MIMO-NOMA
communication system with a large number of battery-powered IoT devices. There
are $K$ cells, denoted by $\mathcal{K}=\\{1,2,\ldots,K\\}$. Each cell has a BS
at the cell center, and multiple rechargeable IoT devices randomly distributed
within the cell. The BS in the $k$-th cell (i.e., $BS_{k}$, $k\in\mathcal{K}$)
has multiple antennas, and each IoT device has a single antenna. The system
has a bunch of available subcarrier set, denoted by
$\mathcal{S}=\\{1,2,\ldots,S\\}$. The total system bandwidth is $B$, and the
subcarrier bandwidth is $B_{s}$ $(s\in\mathcal{S})$, where $B_{s}=B/S$. In the
$k$-th ($k\in\mathcal{K}$) cell, the $u$-th
($u\in\mathcal{U}=\\{1,2,\ldots,U\\}$) IoT device assigned to subcarrier $s$
is denoted as $IoT$-$D_{k,u,s}$. Every BS contains a channel estimator (CE) to
estimate the channel. Let
$\mathcal{M}=\\{{M_{1}},\ldots,{M_{k}},\ldots,{M_{K}}\\}$ be the candidate
antenna number set and
$\mathcal{N}=\\{\mathbf{N}_{1},\ldots,\mathbf{N}_{k},\ldots,\mathbf{N}_{K}\\}$
be the selected antenna vector set for all cells, where ${M_{k}}$ denotes the
total maximum antenna number of $BS_{k}$ and
$\mathbf{N}_{k,s}=\left[N_{k,1,s},N_{k,2,s},\ldots,N_{k,u,s}\right]$ is the
selected optimal antenna number vector in the $k$-th cell on subcarrier $s$.
$N_{k,u,s}$ denotes the optimal antenna number for $IoT$-$D_{k,u,s}$. Note
that in the AS process, we only select the optimal antenna at each BS. Thus,
$\mathbf{N}_{k,n}$ can be simplified as $\mathbf{N}_{k}$. In the considered
new massive MIMO-NOMA network, the harvest-then-transmit protocol is employed,
in which the BSs charge the IoT devices via WPT, and the IoT devices send
messages to the BSs by using the energy harvested from WPT process.

In Fig. 2, a two-time-slot division protocol is developed for the WPT and WIT
processes, which complies with the harvest-then-transmit protocol [32]. Since
there is no available power at IoT devices, they replenish the power from the
BSs to transmit their independent information. A complete cycle of downlink
and uplink transfer processes is treated as a time block, denoted by $T$. Each
block is partitioned into two time slots. During the first slot $\tau_{k}$,
$BS_{k}$ broadcasts wireless energy to $IoT$-$D_{k,u,s}$ via WPT technique,
and $IoT$-$D_{k,u,s}$ saves the harvesting energy in a chargeable battery.
During the second slot $(T-\tau_{k})$, all IoT devices transmit information
simultaneously by employing NOMA on subcarrier $s$.

Fig. 1: A WPT enabled massive MIMO-NOMA network with massive IoT devices. Fig.
2: Two-time-slot protocol for WPT and WIT.

In our system, $BS_{k}$ is equipped with multiple antennas, which can reduce
the EE. To save the energy and enhance the EE of the system, we allocate the
appropriate antenna number for $IoT$-$D_{k,u,s}$, denoted by $N_{k,u,s}$.
Meanwhile, the energy beamforming technique is employed for WPT to improve the
power transfer efficiency. In the rest of this section, we specify the EE for
the cases of the perfect and imperfect CSI knowledge. Notations of this paper
are collated in Table I.

Table I SUMMARY OF THE KEY NOTATIONS.
Notations Meanings $\mathcal{K}$ Set of all the cells $K$ Maximum number of
cells $k$ Index to the cells $\mathcal{U}$ Set of IoT devices in the $k$-th
cell $U$ Number of IoT devices in the $k$-th cell $u$ Index to the IoT devices
in the $k$-th cell $\mathcal{S}$ Set of subcarriers $s$ Index to the
subcarrier $S$ Maximal number of the subcarrier $T$ Time block of the downlink
WPT and uplink WIT $\tau_{k}$ Downlink WPT time slot in the $k$-th cell
$T-\tau_{k}$ Uplink WIT time slot in the $k$-th cell $BS_{k}$ BS in the $k$-th
cell $IoT$-$D_{k,u,s}$ The $u$-th device in the $k$-th cell on subcarrier $s$
$\mathbf{b}_{k,u,s}$ Energy beamforming vector $P_{k}$ Transmit power of
$BS_{k}$ $c_{k,u,s}$ Subcarrier allocation indicator for $IoT$-$D_{k,u,s}$
$\mathbf{h}_{k,u,s}$ Downlink channel between $BS_{k}$ and $IoT$-$D_{k,u,s}$
$\hat{\mathbf{h}}_{k,u,s}$ Estimated channel between $BS_{k}$ and
$IoT$-$D_{k,u,s}$ $\Gamma_{{k,u,s}}$ Co-channel interference under perfect CSI
$B_{s}$ Bandwidth of each subcarrier $\mathcal{M}$ Set of all maximal
candidate antennas $M_{k}$ Total antenna number of $BS_{k}$ in the $k$-th cell
$\mathcal{N}$ Set of the selected optimal antenna number vector
$\mathbf{N}_{k}$ Selected antenna number vector of $BS_{k}$ $N_{k,u,s}$
Optimal number of antennas for $IoT$-$D_{k,u,s}$ $\alpha_{k,u}$ Path loss
between $BS_{k}$ and $IoT$-$D_{k,u,s}$ $n_{k,u,s}^{{}^{\prime}}$ AWGN in the
downlink $n_{k,u,s}^{{}^{\prime\prime}}$ AWGN in the uplink $\eta$ Conversion
efficiency of electrical energy ${\mathbf{e}}_{k,u,s}$ Channel Estimation
error between $BS_{k}$ and $IoT$-$D_{k,u,s}$ $\Xi_{{k,u,s}}$ Channel
estimation and other IoT devices’ interferences under imperfect CSI

###  III-A Perfect Channel Condition

With perfect CSI at the energy transmitter (i.e., the BS), the downlink
received signal at $IoT$-$D_{k,u,s}$ on the subcarrier $s$ can be written as

$y_{k,u,s}^{P}=\alpha_{k,u}\mathbf{b}_{k,u,s}^{H}\mathbf{h}_{k,u,s}z_{k}+n_{k,u,s}^{{}^{\prime}}$
(1)

where $\mathbf{h}_{k,u,s}$ denotes the downlink channel gain vector between
$BS_{k}$ and $IoT$-$D_{k,u,s}$, $\alpha_{k,u}$ is the coefficient of path
loss, $\mathbf{b}_{k,u,s}$ denotes the energy beamforming vector, and $z_{k}$
is the energy signal transmitted by $BS_{k}$. $n_{k,u,s}^{{}^{\prime}}$
denotes the additive white Gaussian noise (AWGN), which obeys the CSCG
distribution, i.e.,
$n_{k,u,s}^{{}^{\prime}}\sim\mathcal{CN}\left(0,\sigma^{2}\right)$. The
transmit power of $BS_{k}$ satisfies
$E\left[\left|z_{k}\right|^{2}\right]=P_{k}$. According to the law of energy
conservation during the WPT process [17], we define the harvested energy of
$IoT$-$D_{k,u,s}$ from $BS_{k}$ as

$\displaystyle
E_{k,u,s}=\eta\tau_{k}\left(\alpha_{k,u}^{2}\left|\mathbf{b}_{{k,u,s}}^{H}\mathbf{h}_{{k,u,s}}\right|^{2}P_{k}\right)$
(2)

where $\eta\ (0\leq\eta\leq 1)$ denotes conversion efficiency from the RF
power source to electrical energy. With the optimal antenna number
$N_{k,u,s}$, the dimension of the downlink channel gain vector
$\mathbf{h}_{k,u,s}$ is $N_{k,u,s}\times 1$. The beamforming vector is
$\mathbf{b}_{k,u,s}=\frac{\mathbf{h}_{k,u,s}}{\|\mathbf{h}_{k,u,s}\|}$ and the
energy beamforming direction can be appropriately adjusted according to the
CSI knowledge at each time block.

In the WIT slot $(T-\tau_{k})$, $IoT$-$D_{k,u,s}$ uses the harvested energy to
send information to $BS_{k}$. The uplink NOMA transmission protocol is
employed to complete the message transmission. The multi-user detection and
SIC are implemented in a fixed decoding order of the received signals at the
$BS_{k}$ [38]. In the $k$-th cell, all the IoT devices are sorted by their
channel gains in descending order, i.e.,
$\left|\mathbf{h}_{k,1,s}\right|\geq\left|\mathbf{h}_{k,2,s}\right|\geq\cdots\geq\left|\mathbf{h}_{{k,u,s}}\right|\geq\cdots\geq\left|\mathbf{h}_{k,U,s}\right|$.
$BS_{k}$ decodes the messages from the IoT devices in a descending order of
channel gain. Specifically, the signal with the best channel is decoded first
while regarding the rest of IoT devices’ signals as noise, and the signal with
the worst channel is decoded finally with any interference from all other
active IoT devices canceled.

In general, for the $u$-th IoT device, $BS_{k}$ deletes the received signals
from the first IoT device to the $(u-1)$-th IoT device, and treats the signals
from the $(u+1)$-th IoT device to the $U$-th IoT device as noise. The uplink
received signal from $IoT$-$D_{k,u,s}$ at $BS_{k}$ is given by

$y_{k,u,s}^{PU}=\sqrt{\frac{E_{{k,u,s}}}{T-\tau_{k}}}\alpha_{{k,u}}\mathbf{h}_{{k,u,s}}^{{H}}{x}_{{k,u,s}}+\Gamma_{{k,u,s}}+n_{k,u,s}^{{}^{\prime\prime}}$
(3)

where $n_{k,u,s}^{{}^{\prime\prime}}$ is the AWGN in the phase of uplink WIT,
and $n_{k,u,s}^{{}^{\prime\prime}}\sim\mathcal{CN}\left(0,\sigma^{2}\right)$.
${x}_{{k,u,s}}$ is the signal from $IoT$-$D_{k,u,s}$, which is transmitted via
power-domain NOMA with the power $\frac{E_{k,u,s}}{T-\tau_{k}}$.
$\Gamma_{{k,u,s}}$ denotes both the intra-cell and inter-cell co-channel
interference signals, and it is written as

$\displaystyle{{\Gamma}_{k,u,s}}=\sum\limits_{{{u}_{1}}=u+1}^{U}{\sqrt{\frac{{{E}_{k,u_{1},s}}}{T-{{\tau}_{k}}}{{c}_{k,u_{1},s}}}}{{\alpha}_{k,{{u}_{1}}}}\mathbf{h}_{{k,u_{1},s}}^{{H}}{{{x}}_{k,u_{1},s}}$
(4) $\displaystyle+\sum\limits_{{{k}_{2}}=1,{{k}_{2}}\neq
k}^{K}{\sum\limits_{u=1}^{U}{\sqrt{\frac{{{E}_{k_{2},u,s}}}{T-{{\tau}_{{k}_{2}}}}c_{k_{2},u,s}}}{{\alpha}_{{{k}_{2}},u}}\mathbf{h}_{{k_{2},u,s}}^{{H}}{{{x}}_{k_{2},u,s}}}$

where ${{c}_{k,u_{1},s}}$ is the subcarrier allocation indicator for
$IoT$-$D_{k,u_{1},s}$. The SINR of $IoT$-$D_{k,u,s}$ on the subcarrier $s$ is
denoted by [15]

$\displaystyle{{\gamma}_{k,u,s}^{P}}=\frac{\frac{{{E}_{k,u,s}}}{T-\tau_{k}}\alpha_{k,u}^{2}{{\left|{{\mathbf{h}}_{k,u,s}}\right|}^{2}}}{I_{k,u_{1},s}+I_{k_{2},u,s}+{\sigma^{2}}}$
(5)

where $I_{k,u_{1},s}$ and $I_{k_{2},u,s}$ are the intra-cell and the inter-
cell co-channel interference, respectively, as given by

$\displaystyle I_{k,u_{1},s}$
$\displaystyle=\sum\limits_{{{u}_{1}}=u+1}^{U}{\frac{{{E}_{k,u_{1},s}}}{T-{{\tau}_{k}}}c_{k,u_{1},s}\alpha_{k,{{u}_{1}}}^{2}{{\left|{{\mathbf{h}}_{k,u_{1},s}}\right|}^{2}}},$
(6) $\displaystyle I_{k_{2},u,s}$
$\displaystyle=\sum\limits_{{{k}_{2}}=1,{{k}_{2}}\neq
k}^{K}{\sum\limits_{u=1}^{U}{\frac{{{E}_{k_{2},u,s}}}{T-{{\tau}_{{k}_{2}}}}c_{k_{2},u,s}\alpha_{{{k}_{2}},u}^{2}{{\left|{{\mathbf{h}}_{k_{2},u,s}}\right|}^{2}}}}.$
(7)

###  III-B Imperfect Channel Condition

In practical massive MIMO-NOMA communication networks, the errors in channel
estimation, feedback and quantization result in imperfect CSI at the
transmitters, degrading the system EE. This paper investigates a quasi-static
Rayleigh channel, where the channel between $BS_{k}$ and $IoT$-$D_{k,u,s}$
remains unchanged for each time block $T$, and varies independently between
different time blocks [39]. In each block, $BS_{k}$ uses the CE to estimate
the channel by minimum mean square error (MMSE) method. The estimated channel
is denoted as

$\displaystyle\hat{\mathbf{h}}_{k,u,s}=\mathbf{h}_{k,u,s}-{\mathbf{e}}_{k,u,s}$
(8)

where
${\mathbf{e}}_{k,u,s}\sim\mathcal{CN}(0,\sigma_{{{e}}_{k,u,s}}^{2}\mathbf{I}_{{\mathbf{e}}_{k,u,s}})$
denotes the channel estimation error, and $\mathbf{I}_{{\mathbf{e}}_{k,u,s}}$
stands for the identity matrix. To efficiently harvest energy, the energy
beamforming policy is designed as $\hat{\mathbf{b}}_{k,u,s}$, using the
maximal ratio transmission (MRT) linear precoding technique. Considering the
beamforming strategy and practical (imperfect) CSI, the system adjusts the
energy transfer direction, so that the IoT devices harvest as much energy as
possible. The received energy signal of $IoT$-$D_{k,u,s}$ in the WPT phase is
written as

$\displaystyle
y_{k,u,s}^{imP}=\alpha_{k,u}\hat{\mathbf{b}}_{k,u,s}^{H}{\mathbf{h}}_{k,u,s}z_{k}+n_{k,u,s}^{{}^{\prime}}$
(9)

where
$\hat{\mathbf{b}}_{k,u,s}=\frac{\hat{\mathbf{h}}_{k,u,s}}{\|\hat{\mathbf{h}}_{k,u,s}\|}$.
The same as the perfect CSI case, the energy harvested by $BS_{k}$ under
imperfect CSI knowledge is denoted by

$\displaystyle
E_{k,u,s}=\eta\tau_{k}\left(\alpha_{k,u}^{2}\left|\hat{\mathbf{b}}_{{k,u,s}}^{H}{\mathbf{h}}_{{k,u,s}}\right|^{2}P_{k}\right)$
(10) $\displaystyle\ \ \ \ \ \ \ \
=\eta\tau_{k}\alpha_{k,u}^{2}\varpi_{k,u,s}P_{k}$

where $\varpi_{k,u,s}$ is the degradation coefficient resulting from imperfect
CSI, and
$\varpi_{k,u,s}=\frac{\sigma^{2}_{e_{k,u,s}}}{1+\sigma^{2}_{e_{k,u,s}}}+\frac{\left\|\hat{\mathbf{h}}_{k,u,s}\right\|^{2}}{\left({1+\sigma^{2}_{e_{k,u,s}}}\right)^{2}}$
[17].

In the phase of WPT, the received signal from $IoT$-$D_{k,u,s}$ to $BS_{k}$
can be expressed as

$\displaystyle
y_{k,u,s}^{imPU}\text{=}\sqrt{\frac{E_{{k,u,s}}}{T-\tau_{k}}}\alpha_{{k,u}}\hat{\mathbf{h}}_{{k,u,s}}^{H}{x}_{{k,u,s}}\text{+}\Xi_{{k,u,s}}\text{+}n_{k,u,s}^{{}^{\prime\prime}}$
(11)

where $\Xi_{{k,u,s}}$ denotes the interferences caused by the channel
estimation and other IoT devices, and can be given by

$\displaystyle{{\Xi}_{k,u,s}}$
$\displaystyle={\sqrt{\frac{{{E}_{k,u,s}}}{T-\tau_{k}}}}{{\alpha}_{k,{{u}}}}\hat{\mathbf{e}}_{k,u,s}^{H}{{{x}}_{k,u,s}}+$
(12)
$\displaystyle\sum\limits_{{{u}_{1}}=u+1}^{U}{\sqrt{\frac{{{E}_{k,u_{1},s}}}{T-\tau_{k}}{{c}_{k,u_{1},s}}}}{{\alpha}_{k,{{u}_{1}}}}\hat{\mathbf{h}}_{k,u_{1},s}^{H}{{{x}}_{k,u_{1},s}}+$
$\displaystyle\sum\limits_{{{k}_{2}}=1,{{k}_{2}}\neq
k}^{K}{\sum\limits_{u=1}^{U}{\sqrt{\frac{{{E}_{k_{2},u,s}}}{T-{{\tau}_{{{k}_{2}}}}}c_{k_{2},u,s}}}{{\alpha}_{{{k}_{2}},u}}\hat{\mathbf{h}}_{k_{2},u,s}^{H}{{{x}}_{k_{2},u,s}}}.$

To effectively complete accurate signal detection and decoding in the uplink,
it is important to acquire CSI knowledge derived from the known training
pilots which are transmitted in regular time intervals. Once the BS receives
the superimposed signals of the IoT devices, several steps of signal
processing will be employed to retrieve each IoT device’s desired signal.
Generally, the message of IoT device with the highest channel gain is first
decoded to guarantee that $BS_{k}$ detects this IoT device’s message while
treating other IoT devices’ messages with low channel gain as noise. For an
ideal SIC, a replica of strong interference signal successfully detected by
the BS is first generated before the BS subtracting it from the received
signals, which are beneficial to cancel the effects from subsequent signal
detection of IoT devices under weak channel conditions. Nonetheless, affected
by the channel noises and estimation errors, it is prone to generate the
errors for the IoT devices’ signal detection under strong channel conditions.
Therefore, error propagation while executing SIC is inevitable for the weak
IoT devices. To alleviate the interference between IoT devices, the symbol
level SIC and code word level SIC [40] are jointly adopted to mitigate the
error propagation under imperfect channel estimation.

For the imperfect case, we assume that
$\left|\hat{\mathbf{h}}_{k,1,s}\right|\geq\cdots\geq\left|\hat{\mathbf{h}}_{{k,u,s}}\right|\geq\cdots\geq\left|\hat{\mathbf{h}}_{k,U,s}\right|$.
As the signals of IoT devices with the strong channel gain are detected and
decoded, the SIC can be effectively performed to cancel the inter-IoT-device
interference. The SINR of $IoT$-$D_{k,u,s}$ under imperfect CSI is written as
[15, 34]

$\displaystyle{{\gamma}_{k,u,s}^{imP}}=\frac{\frac{{{E}_{k,u,s}}}{T-\tau_{k}}\alpha_{k,u}^{2}{{\left|{{\hat{\mathbf{h}}}_{k,u,s}}\right|}^{2}}}{I_{k,u,s}^{e}+I_{k,u_{1},s}^{{}^{\prime}}+I_{k_{2},u,s}^{{}^{\prime}}+{\sigma^{2}}}$
(13)

where $I_{k,u,s}^{e}$ is the estimation noise since self-interference caused
by the channel estimation error can be regarded as a noise.
$I_{k,u_{1},s}^{{}^{\prime}}$ is intra-cell co-channel interference.
$I_{k_{2},u,s}^{{}^{\prime}}$ is inter-cell co-channel interference. They can
be expressed as

$\displaystyle I_{k,u,s}^{e}$
$\displaystyle={\frac{{{E}_{k,u,s}}}{T-\tau_{k}}\alpha_{k,{{u}}}^{2}\sigma^{2}_{e_{k,u,s}}},$
(14) $\displaystyle I_{k,u_{1},s}^{{}^{\prime}}$
$\displaystyle=\sum\limits_{{{u}_{1}}=u+1}^{U}{\frac{{{E}_{k,u_{1},s}}}{T-{{\tau}_{k}}}c_{k,u_{1},s}\alpha_{k,{{u}_{1}}}^{2}{{\left|{{\hat{\mathbf{h}}}_{k,u_{1},s}}\right|}^{2}}},$
(15) $\displaystyle I_{k_{2},u,s}^{{}^{\prime}}$
$\displaystyle=\sum\limits_{{{k}_{2}}=1,{{k}_{2}}\neq
k}^{K}{\sum\limits_{u=1}^{U}{\frac{{{E}_{k_{2},u,s}}}{T-\tau_{{k}_{2}}}c_{k_{2},u,s}\alpha_{{{k}_{2}},u}^{2}{{\left|{{\hat{\mathbf{h}}}_{k_{2},u,s}}\right|}^{2}}}}.$
(16)

###  III-C Definition of the EE

According to the channel hardening characteristic of the massive MIMO-NOMA
networks and the Shannon capacity formula [41], the data rate from
$IoT$-$D_{k,u,s}$ to $BS_{k}$ at the $s$-th subcarrier is given by

$\displaystyle
R_{k,u,s}=B_{s}\log_{2}\left[1+\left(1+\ln\frac{M_{k}}{N_{k,u,s}}\right)\gamma_{k,u,s}{N_{k,u,s}}\right]$
(17)

where $\gamma_{k,u,s}=\gamma_{k,u,s}^{P}$ in the case of perfect CSI, and
$\gamma_{k,u,s}=\gamma_{k,u,s}^{imP}$ in the case of imperfect CSI. The
validity of (17) under imperfect CSI is proved in Appendix A.

The total throughput of the considered system is denoted as

$\displaystyle
R_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)\text{=}\sum_{k=1}^{K}\sum_{u=1}^{U}\sum_{s=1}^{S}c_{k,u,s}(T-\tau_{k})R_{k,u,s}$
(18)

where $\boldsymbol{P}=[P_{1},P_{2},\ldots,P_{K}]$,
$\boldsymbol{\tau}=[\tau_{1},\tau_{2},\ldots,\tau_{K}]$,
$\boldsymbol{N}=[N_{k,u,s}]$ and $\boldsymbol{C}=[c_{k,u,s}](\forall
k\in\mathcal{K},u\in\mathcal{U},s\in\mathcal{S})$ are the vectors of transmit
power, time, selected antenna and subcarrier allocation policies,
respectively. The total consumed energy is given by

$\displaystyle{E_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)=}$
(19)
$\displaystyle\sum_{k=1}^{K}\left(\left(P_{\text{bs}}\max_{u}\\{N_{k,u,s}\\}\text{+}UP_{\text{user}}\right)T\text{+}\sum_{s=1}^{S}P_{k}c_{k,u,s}\tau_{k}\right)$

where $P_{\text{bs}}$ and $P_{\text{user}}$ are the power consumptions of each
antenna at $BS_{k}$ and each IoT device, respectively.
$P_{\text{bs}}=P_{\text{DAC}}+P_{\text{mix}}+P_{\text{filt}}$ and
$P_{\text{user}}=P_{\text{syn}}+P_{\text{LNA}}+P_{\text{mix}}+P_{\text{IFA}}+P_{\text{filr}}+P_{\text{ADC}}$
[41, 42], where $P_{\mathrm{DAC}}$ is the power consumption of the digital-to-
analog converter; $P_{\mathrm{mix}}$ is the power consumption of the mixer;
$P_{\mathrm{LNA}}$ is the power consumption of the low-noise amplifier;
$P_{\mathrm{IFA}}$ is the power consumption of the intermediate frequency
amplifier; $P_{\mathrm{filt}}$ is the power consumption of active filters at
the transmitter; $P_{\mathrm{filr}}$ is the power consumption of active
filters at the receiver; $P_{\mathrm{ADC}}$ is the power consumption of the
analog-to-digital converter; and $P_{\text{syn }}$ is the power consumption of
the frequency synthesizer.

Generally, the system EE can be defined by information bit number reliably
transmitted to all BSs per unit of the consumed energy. Thus, the EE of the
considered massive MIMO-NOMA networks can be calculated as

$\displaystyle{\eta_{\text{EE}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)}=\frac{R_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)}{E_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)}.$
(20)

##  IV Problem Formulation and Transformation

###  IV-A Problem Formulation

In this section, an energy-efficient scheme of power, time, antenna and
subcarrier allocation is investigated, which can achieve an optimal EE for the
massive MIMO-NOMA networks. The EE maximization problem of the considered
system is formulated as

$\displaystyle\ \ \ \ \mathbf{P1}:\ \ \ \ \
\underset{\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}}{\mathop{\max}}\
\
\eta_{\text{EE}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)$
(21a) $\displaystyle s.t.\mathbf{C1}:\ \ \
0\leq{{P}_{k}}\leq{{P}_{\text{bs},\max}},\ \ \forall k\in\mathcal{K},$ (21b)
$\displaystyle\ \ \ \ \mathbf{C2}:\ \ \ 0\leq\tau_{k}\leq T,\ \ \forall
k\in\mathcal{K},$ (21c) $\displaystyle\ \ \ \ \mathbf{C3}:\ \ \
0\leq\frac{{{E}_{k,u,s}}}{T-\tau_{k}}\leq{{P}_{\text{user},\max}},\forall
k\in\mathcal{K},u\in\mathcal{U},s\in\mathcal{S},$ (21d) $\displaystyle\ \ \ \
\mathbf{C4}:\ \ \ {{{R}_{k,u,s}}}\geq{{R}_{\min}},\forall
k\in\mathcal{K},u\in\mathcal{U},s\in\mathcal{S},$ (21e) $\displaystyle\ \ \ \
\mathbf{C5}:\ \ \ N_{k,u,s}\in\\{1,2,...,{M}_{k}\\},\forall
k\in\mathcal{K},u\in\mathcal{U},s\in\mathcal{S},$ (21f) $\displaystyle\ \ \ \
\mathbf{C6}:\ \ \ {c_{k,u,s}}\in\\{0,1\\},\sum_{u=1}^{U}c_{k,u,s}\leq
U,\forall k\in\mathcal{K},u\in\mathcal{U},$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ s\in\mathcal{S},$ (21g)

where $\mathbf{C1}$ specifies the maximal transmit power
${P}_{\text{bs},\max}$ of $BS_{k}$; $\mathbf{C2}$ represents the range of the
energy transfer time; $\mathbf{C3}$ ensures that the transmit power of
$IoT$-$D_{k,u,s}$ should be non-negative and no larger than
${{P}_{\text{user},\max}}$; $\mathbf{C4}$ ensures that the channel rate of
$IoT$-$D_{k,u,s}$ should be no lower than the minimal rate $R_{min}$;
$\mathbf{C5}$ controls the number of active antennas allocated for each IoT
device to provide the fairness among IoT devices and save the RF cost; and
$\mathbf{C6}$ ensures subcarrier allocation, and one subcarrier can be
multiplexed by at most $U$ IoT devices.

###  IV-B Problem Transformation

Problem $\mathbf{P1}$ is a fractional programming problem, which is non-
convex. To solve $\mathbf{P1}$, we first prove the concavity of
$R_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)$
(see Appendix B), and then use the Dinkelbach method [43] to convert it to a
linear problem. We provide the following theorem that presents the sufficient
and necessary conditions for the optimal solution to $\mathbf{P1}$.

######  Theorem 1.

The maximum EE can be achieved, if and only if

$\displaystyle\underset{\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}}{\mathop{\max}}\
\
\\{R_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)-\eta_{\text{EE}}^{*}E_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)\\}$
(22)
$\displaystyle=R_{\text{tot}}\left(\boldsymbol{P}^{*},\boldsymbol{\tau}^{*},\boldsymbol{N}^{*},\boldsymbol{C}^{*}\right)-\eta_{\text{EE}}^{*}E_{\text{tot}}\left(\boldsymbol{P}^{*},\boldsymbol{\tau}^{*},\boldsymbol{N}^{*},\boldsymbol{C}^{*}\right)=0,$

where $\eta_{\text{EE}}^{*}$ is the global optimal EE, as given by

$\eta_{\text{EE}}^{*}=\frac{R_{\text{tot}}\left(\boldsymbol{P}^{*},\boldsymbol{\tau}^{*},\boldsymbol{N}^{*},\boldsymbol{C}^{*}\right)}{E_{\text{tot}}\left(\boldsymbol{P}^{*},\boldsymbol{\tau}^{*},\boldsymbol{N}^{*},\boldsymbol{C}^{*}\right)}.$
(23)

Herein, $\boldsymbol{P}^{*}$, $\boldsymbol{\tau}^{*}$, $\boldsymbol{N}^{*}$
and $\boldsymbol{C}^{*}$ denote the optimal power, WPT time, AS and subcarrier
allocation policies, respectively.

###### Proof.

See Appendix C. ∎

According to Theorem 1, $\mathbf{P1}$ can be converted to a subtractive linear
form, which is given by

$\displaystyle\mathbf{P2}:\underset{\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}}{\mathop{\max}}\\{R_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)-\eta_{\text{EE}}E_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)\\}$
(24a) $\displaystyle s.t.\ \ \ \ \mathbf{C1,C2,C3,C4,C5,C6}.$ (24b)

We further deal with the subtractive problem $\mathbf{P2}$, which, however, is
a non-convex integer programming problem. Solving this problem wound suffer a
prohibitively high computational complexity. To reduce the computation
complexity, we solve it in the dual domain and obtain an equivalent solution
(for e.g., when using the simplex method, if the original problem has many
constraints and few variables, solving it in dual domain is feasible and
efficient). Although there is a non-zero duality gap between the primal
problem and dual problem, [44] has proved that it is negligible for such non-
convex problem in the time-sharing multi-carrier system. Since the number of
subcarriers is large, e.g., from 32 to 256, the solution is asymptotically
optimal. According to [44], we relax $N_{k,u,s}$ and $c_{k,u,s}$ in
$\mathbf{C5}$ and $\mathbf{C6}$ to be continuous values [45, 17]. We define
two new auxiliary continuous variables
$N_{k,u,s}^{\dagger}\in\left[1,M_{k}\right]$ and
$c_{k,u,s}^{{\dagger}}\in\left[0,1\right]$ to replace the original variables
$N_{k,u,s}$ and $c_{k,u,s}$. $\mathbf{P2}$ can be transformed into
$\mathbf{P3}$ as follows.

$\displaystyle\mathbf{P3}:\underset{\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N}^{\dagger},\boldsymbol{C}^{\dagger}}{\max}\\{R_{\text{tot}}(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N}^{\dagger},\boldsymbol{C}^{\dagger})-\eta_{\text{EE}}E_{\text{tot}}(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N}^{\dagger},\boldsymbol{C}^{\dagger})\\}$
(25a) $\displaystyle\ \ s.t.\ \ \ \ \ \ \ \ \
\mathbf{C1},\mathbf{C2},\mathbf{C3},\mathbf{C4},$ (25b) $\displaystyle\ \ \ \
\ \ \ \ \ \ \ \ \ \ \
\overset{\scriptscriptstyle\smile}{\mathbf{C5}}:N_{k,u,s}^{\dagger}\in\left[1,M_{k}\right],$
(25c) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\overset{\scriptscriptstyle\smile}{\mathbf{C6}}:c_{k,u,s}^{{\dagger}}\in\left[0,1\right],\sum_{u=1}^{U}c_{k,u,s}^{{\dagger}}\in\left[0,U\right].$
(25d)

By solving the optimal solution in the dual domain, the asymptotically optimal
solution for $\mathbf{P3}$ can be obtained according to the principle of
strong duality theory [46]. The sub-optimality of AS can be given as
$N_{k,u,s}=\left\lfloor N_{k,u,s}^{\dagger}\right\rfloor$.

As a result, we design a new iterative algorithm with convergence to obtain
$\eta_{\text{EE}}^{*}$, which is given in Alg. 1. The allocation strategy of
power, time, AS and subcarrier allocation can be optimized by solving
$\mathbf{P3}$. As shown in Theorem 1, given a radio resource allocation
strategy, e.g.,
$\\{\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\\}$, we can
find the corresponding solution for $\eta_{\text{EE}}$. By running such
iterative process, the optimal solution for resource allocation and
$\eta_{\text{EE}}$ can be obtained. We proceed to adopt ADMM method to
decentralize and speed up the algorithm in next section.

Algorithm 1 An iterative algorithm for obtaining $\eta_{EE}^{*}$ by using
Dinkelbach’s method.

1: Initialization: Initialize the maximum iteration index, $\Upsilon_{\max}$,
the maximum tolerance $\varepsilon$, and the inner loop iteration index
$\Upsilon=1$.

2: while (not converge) or $\Upsilon$ = $\Upsilon_{\max}$ do

3: Solve the transformed $\mathbf{P3}$ in (25) for given $\eta_{EE}$ and get
the resource allocation policies
$\\{\boldsymbol{P}^{\prime},\boldsymbol{\tau}^{\prime},\boldsymbol{N}^{\prime},\boldsymbol{C}^{\prime}\\}$.

4: if
$|{R_{\text{tot}}(\boldsymbol{P}^{\prime},\boldsymbol{\tau}^{\prime},\boldsymbol{N}^{\prime},\boldsymbol{C}^{\prime})}-{\eta_{EE}}{E_{\text{tot}}(\boldsymbol{P}^{\prime},\boldsymbol{\tau}^{\prime},\boldsymbol{N}^{\prime},\boldsymbol{C}^{\prime})}|\leq\varepsilon$
then

5: Convergence = true;

6: return
$\\{\boldsymbol{P}^{*},\boldsymbol{\tau}^{*},\boldsymbol{N}^{*},\boldsymbol{C}^{*}\\}=\\{\boldsymbol{P}^{\prime},\boldsymbol{\tau}^{\prime},\boldsymbol{N}^{\prime},\boldsymbol{C}^{\prime}\\}$
and obtain the optimal ratio $\eta_{EE}^{*}$ by $\mathbf{Theorem1}$;

7: else

8: Convergence = false;

9: return obtain a ratio of the total throughput to energy consumption
$\eta_{EE}=\frac{R_{\text{tot}}(\boldsymbol{P}^{\prime},\boldsymbol{\tau}^{\prime},\boldsymbol{N}^{\prime},\boldsymbol{C}^{\prime})}{E_{\text{tot}}(\boldsymbol{P}^{\prime},\boldsymbol{\tau}^{\prime},\boldsymbol{N}^{\prime},\boldsymbol{C}^{\prime})}$
and $\Upsilon$ = $\Upsilon+1$;

10: end if

11: end while

##  V Energy-Efficient Resource Allocation Via ADMM

In this section, we propose a novel distributed ADMM-based resource allocation
algorithm to divide the convex problem into several sequential optimization
phases, which improves the efficiency of solving the problem and the quality
of optimal solution. To solve non-convex optimization problems, traditional
methods first relax and convert it to a convex one, and then use the greedy
and centralized algorithms based on Lagrange multiplier method [2, 9, 10] to
solve the converted problem. These methods have relatively high iteration
numbers due to the low efficiency of the algorithm execution, even only
reaching a local optimum.

We first briefly review ADMM. Next, the applicability of distributed ADMM for
this problem is illustrated. To efficiently solve $\mathbf{P3}$, we propose a
distributed ADMM-based resource allocation algorithm for the considered
network.

###  V-A Introduction to ADMM

ADMM is an efficient algorithm, which can be applicable to distributed convex
optimization [47, 48]. An important property of ADMM is that it can quickly
converge with a high accuracy of the optimal solution. ADMM has been
successfully applied in diverse optimization problems, e.g., statistical
learning, multi-period portfolio optimization, network scheduling [36], [47].
The downside of the standard ADMM algorithm is that the problem to be solved
can only be partitioned into two subproblems. Standard ADMM cannot be
implemented in a distributed manner for the massive MIMO-NOMA network. For
this reason, distributed ADMM is regarded as an effective approach, which can
combine the benefits of dual decomposition and augmented Lagrangian for
multiple constraint optimization problems [36, 37]. By using the distributed
ADMM, a global consensus problem is divided into some local subproblems. The
solutions to local subproblems are further coordinated to obtain a consensus
solution to the original problem.

###  V-B Transformation of $\boldsymbol{P3}$

We define several local variables, $\tilde{\tau}_{k}=\tau_{k}$,
$\tilde{n}_{k,u,s}=N_{k,u,s}^{\dagger}$, and
$\tilde{c}_{k,u,s}=c_{k,u,s}^{\dagger}$. By substituting the local variables
into $\mathbf{P3}$, $\mathbf{P3}$ is equivalent to the following form

$\displaystyle\mathbf{P4}:\ \ \ \ \
\underset{\boldsymbol{P},\tilde{\boldsymbol{\tau}},{\tilde{\boldsymbol{N}}},{\tilde{\boldsymbol{C}}}}{\max}\
\Lambda\left(\boldsymbol{P},\tilde{\boldsymbol{\tau}},{\tilde{\boldsymbol{N}}},{\tilde{\boldsymbol{C}}}\right)$
(26a) $\displaystyle s.t.\ \ \ \ \ \ \ \tilde{\mathbf{C}}\mathbf{1}:\ \ \
0\leq P_{k}\leq P_{\text{bs},\max},$ (26b) $\displaystyle\ \ \ \ \ \ \ \ \ \ \
\tilde{\mathbf{C}}\mathbf{2}:\ \ \ 0\leq\tilde{\tau}_{k}\leq T,$ (26c)
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \tilde{\mathbf{C}}\mathbf{3}:\ \ \
0\leq\frac{\tilde{E}_{k,u,s}}{T-\tilde{\tau}_{k}}\leq P_{\text{user},\max},$
(26d) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \tilde{\mathbf{C}}\mathbf{4}:\ \ \
\tilde{R}_{k,u,s}\geq R_{\min},$ (26e) $\displaystyle\ \ \ \ \ \ \ \ \ \ \
\tilde{\mathbf{C}}\mathbf{5}:\quad 1\leq\tilde{n}_{k,u,s}\leq M_{k},$ (26f)
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \tilde{\mathbf{C}}\mathbf{6}:\ \ \
0\leqslant\tilde{c}_{k,u,s}\leqslant
1,\sum\nolimits_{{u=1}}^{{U}}{{\tilde{c}_{k,u,s}}}\leq U,$ (26g)

where
$\Lambda\left(\boldsymbol{P},\tilde{\boldsymbol{\tau}},{\tilde{\boldsymbol{N}}},{\tilde{\boldsymbol{C}}}\right)$
is the EE objective function after transformation, i.e.,
$\Lambda\left(\boldsymbol{P},\tilde{\boldsymbol{\tau}},{\tilde{\boldsymbol{N}}},{\tilde{\boldsymbol{C}}}\right)=R_{\text{tot}}\left(\boldsymbol{P},\tilde{\boldsymbol{\tau}},{\tilde{\boldsymbol{N}}},{\tilde{\boldsymbol{C}}}\right)-\eta_{\text{EE}}E_{\text{tot}}\left(\boldsymbol{P},\tilde{\boldsymbol{\tau}},{\tilde{\boldsymbol{N}}},{\tilde{\boldsymbol{C}}}\right)$.
With mathematical manipulations, the primal non-convex optimization problem
$\mathbf{P1}$ can be converted to a convex problem $\mathbf{P4}$. According to
the property of the perspective function [49], $\mathbf{P4}$ is convex with
respect to the variables $P_{k}$, $\tilde{\tau}_{k}$ and $\tilde{n}_{k,u,s}$.
The convexity of $\mathbf{P4}$ is proved in Appendix D. As the transmit power
$P_{k}$ concerning different $BS_{k}$ is correlated to some extent, the
coupling among all BSs urgently needs to be decoupled to solve $\mathbf{P4}$
by applying distributed ADMM. We denote the local copy of $P_{k}$ as
$\tilde{P}_{k}$, which is the perception of the global variables. Through
introducing the local variables $\tilde{P}_{k}$, $\tilde{\tau}_{k}$,
$\tilde{n}_{k,u,s}$ and $\tilde{c}_{k,u,s}$, a feasible local variable set of
$BS_{k}$ is given by

$\mathbf{X_{k}}=\\{\tilde{P}_{k},\tilde{\tau}_{k},\tilde{n}_{k,u,s},\tilde{c}_{k,u,s}|\tilde{\mathbf{C2}},\tilde{\mathbf{C3}},\tilde{\mathbf{C4}},\tilde{\mathbf{C5}},\tilde{\mathbf{C6}}\\}.$
(27)

Furthermore, we define an associated local cost function with local variables
$\tilde{P}_{k}$, $\tilde{\tau}_{k}$, $\tilde{n}_{k,u,s}$ and
$\tilde{c}_{k,u,s}$, as given by

$\displaystyle\ \ \
{{g}_{k}}({{{\tilde{P}}}_{k}},{{{\tilde{\tau}}}_{k}},\tilde{n}_{k,u,s},\tilde{c}_{k,u,s})=$
(28) $\displaystyle\left\\{\begin{aligned}
&\sum\limits_{u=1}^{U}{\sum\limits_{s=1}^{S}{\tilde{c}_{k,u,s}{{{\tilde{R}}}_{k,u,s}}}}-\eta_{k}^{*}\left(({{P}_{\text{bs}}}\underset{u}{\mathop{\max}}\,\\{\tilde{n}_{k,u,s}\\}\text{+}U{{P}_{\text{user}}})T\right.\\\
&+{{{\tilde{P}}}_{k}}\sum\limits_{s=1}^{S}\left.{\tilde{c}_{k,u,s}\,{{{\tilde{\tau}}}_{k}}}\right),\text{if}\
\tilde{{P}_{k}},{{{\tilde{\tau}}}_{k}},\tilde{n}_{k,u,s},\tilde{c}_{k,u,s}\in{{\mathbf{X}}_{\mathbf{k}}},\\\
&\infty,\ \ \ \text{otherwise.}\\\ \end{aligned}\right.$

As a consequence, the global consensus problem $\mathbf{P4}$ can be
transformed into $\mathbf{P4}$, as shown

$\displaystyle\mathbf{P5}:\ \ \ \ \ \ \
\underset{\tilde{\boldsymbol{P}},\tilde{\boldsymbol{\tau}},\tilde{\boldsymbol{N}},{\tilde{\boldsymbol{C}}}}{\min}\
\sum_{k=1}^{K}g_{k}\left(\tilde{P}_{k},\tilde{\tau}_{k},\tilde{n}_{k,u,s},\tilde{c}_{k,u,s}\right)$
(29a) $\displaystyle s.t.\ \ \ \ \ \ \ \ \
\widehat{\mathbf{C}}\mathbf{1}:\tilde{P}_{k}=P_{k},\ \ 1\leq k\leq K.$ (29b)

###  V-C Energy-efficient Solution via ADMM

The proposed algorithm, Alg. 1, is extended to achieve the efficient resource
allocation of multicell massive MIMO-NOMA networks in a fully distributed
fashion. Inspired by [36], $\mathbf{P5}$ is a global consensus problem, whose
solution can be obtained by using ADMM approaches in a distributed manner. The
mechanism of ADMM starts with constructing an augmented Lagrangian function
concerning the consensus constraints [36]. Not only does the augmented
Lagrangian include a set of consensus constraints weighted by the Lagrangian
multipliers (i.e., the traditional Lagrangian), but also adds a regularized
quadratic term: a squared $\text{L}_{2}$ norm of the consensus constraints. We
use $\lambda_{k}$ to denote the Lagrangian multiplier corresponding to the
$k$-th consensus constraint of (29), the augmented Lagrangian of (29) is given
by

$\displaystyle\ \ \ \
{{\mathcal{L}}_{\rho}}\left(\\{{{{\tilde{P}}}_{k}},{{{\tilde{\tau}}}_{k}},\tilde{n}_{k,u,s},\tilde{c}_{k,u,s}\\},\\{{{P}_{k}}\\},\text{
}\\!\\!\\{\\!\\!\text{ }{{\lambda}_{k}}\text{ }\\!\\!\\}\\!\\!\text{ }\right)$
(30)
$\displaystyle=\sum\limits_{k=1}^{K}{{{g}_{k}}\left({{{\tilde{P}}}_{k}},{{{\tilde{\tau}}}_{k}},\tilde{n}_{k,u,s},\tilde{c}_{k,u,s}\right)}+\sum\limits_{k=1}^{K}{{{\lambda}_{k}}\left({{{\tilde{P}}}_{k}}-{{P}_{k}}\right)}$
$\displaystyle\ \
+\frac{\rho}{2}\sum\limits_{k=1}^{K}{{{\left({{{\tilde{P}}}_{k}}-{{P}_{k}}\right)}^{2}}}$

where $\rho\in\mathbb{R}^{+}$ denotes a positive penalty parameter which has
the function of adjusting the convergence rate of ADMM approach. $\rho$ also
provides strict convexity with respect to global and local variables in (30),
and thus problem (29) is solvable [50]. As the structure of multicell massive
MIMO-NOMA is interconnected, the augmented Lagrangian is separable among the
BSs. Then, we execute the central parts of ADMM, which are used to update the
global variables (i.e., $\boldsymbol{P}$), the local variables
(i.e.,$\\{\tilde{\boldsymbol{P}},\tilde{\boldsymbol{\tau}},{\tilde{\boldsymbol{N}}},{\tilde{\boldsymbol{C}}}\\}$),
and the Lagrange multipliers (i.e., $\boldsymbol{\lambda}$) by using the
Gauss-Seidel method. In each iteration of executing ADMM, all BSs share the
knowledge of interrelated inter-cell interference temperatures and consensus
EE through updating the global variables. Then, each BS independently solves
its own subproblem and updates Lagrange multipliers to synchronize involved
variables. The sequential optimization steps of the updated procedures are
given by

$\displaystyle\\{P_{k}\\}^{[t+1]}:=$
$\displaystyle\arg\underset{\boldsymbol{P}}{\min}\left\\{\sum_{k=1}^{K}\lambda_{k}^{[t]}\left(\tilde{P}_{k}^{[t]}-P_{k}\right)+\frac{\rho}{2}\sum_{k=1}^{K}\left(\tilde{P}_{k}^{[t]}-P_{k}\right)^{2}\right\\},$
(31)
$\displaystyle\left\\{\tilde{P}_{k},\tilde{\tau}_{k},\tilde{n}_{k,u,s},\tilde{c}_{k,u,s}\right\\}^{[t+1]}:=$
$\displaystyle\arg\underset{\tilde{\boldsymbol{P}},\tilde{\boldsymbol{\tau}},{\tilde{\boldsymbol{N}}},{\tilde{\boldsymbol{C}}}}{\min}\left\\{g_{k}\left(\tilde{P}_{k},\tilde{\tau}_{k},\tilde{n}_{k,u,s},\tilde{c}_{k,u,s}\right)+\lambda_{k}\left(\tilde{P}_{k}-P_{k}^{[t+1]}\right)\right.$
$\displaystyle\left.+\frac{\rho}{2}\left(\tilde{P}_{k}-P_{k}^{[t+1]}\right)^{2}\right\\},$
(32)
$\displaystyle\left\\{\lambda_{k}\right\\}^{[t+1]}:=\left\\{\lambda_{k}\right\\}^{[t]}+\rho\left(\tilde{P}_{k}^{[t+1]}-P_{k}^{[t+1]}\right)$
(33)

where the superscript $t$ indicates the $t$-th iteration.

###  V-D Distributed ADMM-based Resource Allocation Algorithm

As illustrated in Alg. 2, a distributed ADMM-based resource allocation
algorithm is summarized. The algorithm consists of three steps: (a) update of
global variables, (b) update of local variables, and (c) update of Lagrange
multipliers. Generally, in a relatively long period (or one iteration
instead), all the BSs first update the global variables to keep a consistent
interference level and the consistent EE. After that, each BS independently
addresses the self subproblems, such as AS and subcarrier allocation. Then, by
collecting and using the local CSI of IoT devices, Lagrange multipliers are
updated at the BSs. Once a local convergence is reached, the BSs exchange the
values of power with each other to attain the optimal solution and make proper
adjustments for local resource allocation in the next round. Through such an
interactive and iterative process, the signaling overhead can decrease. No CSI
needs to be exchanged between the BSs and IoT devices. Moreover, a large
problem is decoupled into several distributed small problems to be
simultaneously solved, hence remarkably improving the efficiency of solving
the optimization problem. Only the EE of each BS needs to be exchanged to run
Alg. 1. For Alg. 2, we have the following key steps to elaborate.

Algorithm 2 A distributed ADMM-based resource allocation algorithm.

1: Initialization: a) Initialize the parameters: $K$, $U$, $M_{k}$,
$\alpha_{k,u}$, $P_{\text{bs,max}}$, $P_{\text{user,max}}$, $R_{\text{min}}$,
$P_{\text{bs}}$, $P_{\text{user}}$, $\eta_{\text{EE}}$; b) Initialize the
transmit power $P_{k}$, the Langrange multiplier $\lambda_{k}$ and a small
enough stop criterion threshold $\epsilon$; c) Collect the CSI of IoT devices
in the coverage range of $BS_{k}$ by MMSE method to estimate their channels.

2: for t=1,2,…,G do

3: Initialize $P_{k}^{[\text{t}]}$ and $\lambda_{k}^{\text{[t]}}$ of $BS_{k}$.

4: $BS_{k}$ updates $P_{k}^{\text{[t+1]}}$ according to (31) through an
average consensus algorithm [51].

5: $BS_{k}$ receives the results of $P_{k}^{\text{[t+1]}}$ from each $BS_{k}$
and updates
$\left\\{\tilde{P}_{k},\tilde{\tau}_{k},\tilde{n}_{k,u,s},\tilde{c}_{k,u,s}\right\\}^{[t+1]}$
by taking (32).

6: Update $\lambda_{k}^{\text{[t+1]}}$ via taking the iterative projection
method in (33).

7: if $\epsilon$ in (34) is sufficiently small and satisfied then

8: Go to Step 11;

9: end if

10: end for

11: Output: The optimal resource allocation policy
$\left\\{\tilde{P}_{k},\tilde{\tau}_{k},\tilde{n}_{k,u,s},\tilde{c}_{k,u,s}\right\\}^{[t+1]}$.

1) Update of global variables $\\{P_{k}\\}_{k\in\mathcal{K}}$

The iteration process of distributed ADMM-based algorithm begins with the
update of the global variables, which can be handled by fixing the local
variables when solving the problem $\underset{\boldsymbol{P}}{\min}\
{{\mathcal{L}}_{\rho}}\left(\\{{{{\tilde{P}}}_{k}},{{{\tilde{\tau}}}_{k}},\tilde{n}_{k,u,s},\tilde{c}_{k,u,s}\\},\\{{{P}_{k}}\\},\text{
}\\!\\!\\{\\!\\!\text{ }{{\lambda}_{k}}\text{ }\\!\\!\\}\\!\\!\text{
}\right)$. Note that in (31), $\tilde{P}_{k}^{[t]}$ is the value obtained
after the $t$-th iteration and is exchanged by all BSs.

2) Update of local variables
$\\{\tilde{P}_{k},\tilde{\tau}_{k},\tilde{n}_{k,u,s},\tilde{c}_{k,u,s}\\}_{k\in\mathcal{K}}$

For Step 5, the problems of finding the local selected antenna number and the
resource allocation strategies are separable among different BSs. Thus, the
update of
$\\{\tilde{P}_{k},\tilde{\tau}_{k},\tilde{n}_{k,u,s},\tilde{c}_{k,u,s}\\}_{k\in\mathcal{K}}$
is decomposed into $K$ subproblems. Each subproblem can be handled locally by
each BS. The optimization subproblem at the $(t+1)$-th iteration in (32) can
be tackled in parallel by the BS. Specifically, $BS_{k}$ is able to locally
solve the convex subproblem of (32) only using its local CSI
${\mathbf{h}}_{{k,u,s}}$ and $\\{P_{k}\\}^{[t+1]}$ from the update of global
variables.

3) Update of Lagrange Multiplier $\\{\lambda_{k}\\}_{k\in\mathcal{K}}$

The last step of ADMM is to update the Lagrange multiplier for Step 6 by using
(33). As the current iterations of the local variables and global variables
are available to all BSs, updating the Lagrange multiplier in this step does
not require additional information exchange, and therefore will not produce
additional signaling overhead.

updating the Lagrange multiplier does not require additional information
exchange, and thus incurs no signaling overhead.

4) Stop criteria and convergence

The iteration stops if

$\displaystyle\left\|\mathbf{r}_{k}^{[t+1]}\right\|_{2}^{2}=\left\|\tilde{\mathbf{P}}_{k}^{[t+1]}-\mathbf{P}_{k}^{[t+1]}\right\|_{2}^{2}\leq\epsilon,\forall{k}\in\mathcal{K}$
(34)

where $\mathbf{r}_{k}$ denotes the residual between the local and global
variables. For Step 9, the proposed distributed ADMM-based energy-efficient
algorithm guarantees the residual convergence, objective convergence and dual
variable convergence if $t\rightarrow\infty$. As the objective function of
$\mathbf{P5}$ is a closed convex form, the Lagrangian ${{\mathcal{L}}_{\rho}}$
has a saddle point.

After obtaining the optimal solution, the antenna number is approximated as
$N_{k,u,s}=\left\lfloor N_{k,u,s}^{\dagger}\right\rfloor$ by considering that
the number of selected antennas is an integer. The complexity of Alg. 2 during
each iteration loop is influenced and charged by the complexity of solving
subproblem (32) of $\mathbf{P5}$ at each BS. The polynomial time-complexity of
addressing this problem is related to the number of IoT devices, i.e.,
$\mathcal{O}(U)$. The non-ergodic convergence rate of Alg. 2 is
$\mathcal{O}(1/t)$, where $t$ is the index of iterations. In $t$-th iteration
of the loop, (31), (32) and (33) are updated sequentially with the
computational complexity of $\mathcal{O}(KUS\text{log}_{2}({U^{2}/2+2KU}))$.
Therefore, the whole computational complexity of Alg. 2 is
$\mathcal{O}(KUS\text{log}_{2}({U^{2}/2+2KU})t)$.

##  VI Simulation Results

In this section, the EE performance of the proposed distributed ADMM-based
energy-efficient resource allocation algorithm (denoted as “PA”) is presented.
We consider multiple cells, and the radius of each cell is set as 500 m. The
system bandwidth is normalized with $B=1$ Hz and the bandwidth of each
subcarrier is $B_{s}=B/S$ Hz. In the massive MIMO-NOMA networks, the IoT
devices are randomly and uniformly distributed within each cell. Both path
loss and Rayleigh fading are considered. The simulation results are based on
averaging 10,000 iterations. According to [10] and [42], some simulation
parameters are listed in Table II.

Table II SIMULATION PARAMETERS.
Parameter Value Parameter Value $K$ 6 $\eta$ 0.8 $U$ 15 $\epsilon$ $10^{-7}$
$S$ 20 $\sim$ 40 $B$ $1$ Hz $P_{\text{bs,max}}$ 46 dBm $P_{\text{DAC}}$ 10 mW
$P_{\text{user,max}}$ 23 dBm $P_{\text{ADC}}$ 10 mW $R_{\text{min}}$ 0.1
bit/s/Hz $P_{\text{filr}}$ 2.5 mW ${\alpha_{k,u}}$ $1/75$ $P_{\text{filt}}$
2.5 mW $P_{\text{mix}}$ 30.3 mW $P_{\text{syn}}$ 50 mW $P_{\text{LNA}}$ 20 mW
$P_{\text{IFA}}$ 3 mW

To show the effects of AS, channel information and transmission conditions,
the performance of AS under imperfect CSI through NOMA mode is presented in
Fig. 3 by plotting the EE versus the communication distance. We use “OTS1” and
“OTS2” to denote the OMA (i.e., OFDMA) transmission scheme with AS and perfect
CSI [45], and the OMA transmission scheme without AS [29], respectively. In
Fig. 3, we see that with the increase of the distance between $BS_{k}$ and
$IoT$-$D_{k,u,s}$, the system EE decreases. The reason is that as the distance
increases, $BS_{k}$ needs to transmit more power to provide the IoT devices to
guarantee QoS. Fig. 3 also shows that the EE under perfect CSI is higher than
that under imperfect CSI, due to the impact of CSI and feedback errors. The EE
performance with AS is superior to the case with no AS. We also compare the EE
performance between NOMA and OMA transmissions under the consideration of AS
and CSI. We can see that the proposed distributed ADMM algorithm via NOMA
outperforms “OTS1” and “OTS2” in terms of EE.

In Fig. 4, we compare the EE of “PA” with that of “OTS1” and “OTS2” under
perfect and imperfect CSI. We can observe that the EE of “PA” under perfect
CSI is superior to all other cases thanks to its high-efficient AS scheme and
perfect channel conditions. Considering the estimation error in the practical
channel environment, the EE of “PA” under imperfect CSI is lower than that
under perfect CSI. The EE of OMA-based “OTS2” with no AS and imperfect CSI has
the worst performance. This is because it does not adopt the AS technique or
NOMA transmission. In terms of transmission mode, the EE of “PA” based on NOMA
outperforms the OMA-based “OTS1” and “OTS2” whether the CSI is perfect or not.
This is because power domain NOMA technique can improve the accessing number
of IoT devices and increase the SE and EE by employing the multiple access on
the same time-frequency resource block.

Fig. 3: EE comparison between PA and reference schemes with different AS, CSI
and transmission conditions.

Fig. 4: Comparison between NOMA-based PA and OMA-based benchmark schemes under
perfect and imperfect CSI.

Fig. 5 illustrates the influence of imperfect CSI on the system EE under
different channel estimation parameters. We set three values for the variance
of channel estimation error, i.e., $\sigma_{e_{k,u,s}}^{2}=0.7$,
$\sigma_{e_{k,u,s}}^{2}=0.5$, and $\sigma_{e_{k,u,s}}^{2}=0.3$. The results of
the proposed distributed resource allocation algorithm under perfect CSI are
presented for comparison. We can observe in Fig. 5 that, as the distance
between $BS_{k}$ and $IoT$-$D_{k,u,s}$ increases, the system EE of the
proposed distributed algorithm declines moderately for all the cases. We can
also see that Alg. 2 can obtain the optimal EE when the channel is perfect.
Fig. 5 shows the performance difference between the perfect and imperfect CSI
cases. When the transmission distance is 100 m, the system EE under perfect
CSI is approximately 3.5 times higher than that under imperfect CSI with
$\sigma_{e_{k,u,s}}^{2}=0.7$. When the transmission distance increases to 200
m, the EE under perfect CSI is nearly 1.5 times larger than that with
$\sigma_{e_{k,u,s}}^{2}=0.7$. As the channel estimation variance enlarges, the
gap between different variances becomes large. For instance, the EE with
$\sigma_{e_{k,u,s}}^{2}=0.3$ is higher than the other two cases, i.e., the EE
with $\sigma_{e_{k,u,s}}^{2}=0.5$ and $\sigma_{e_{k,u,s}}^{2}=0.7$. All the
observations in Fig. 5 show that the system EE depends on the accuracy of the
obtained CSI.

Fig. 6 shows the EE versus the number of antenna under perfect and imperfect
CSI. We can see that, the EE first increases and then decreases with the
increasing antenna number. For example, the EE curve under imperfect CSI
($\sigma_{e_{k,u,s}}^{2}=0.6$) grows approximately from $0.6\times 10^{-3}$
bits/J/Hz with $M_{k}=10$, to $2.2\times 10^{-3}$ bits/J/Hz with $M_{k}=30$.
After reaching the peak at $2.2\times 10^{-3}$ bits/J/Hz, the system EE of
“PA” shows a downward trend. The reason is that an appropriate antenna number
can effectively improve the system EE, but excessive antennas would result in
a rapidly increasing power consumption and reduce the EE. Moreover, AS can
alleviate hardware complexity of multi-antenna systems. Therefore, the optimal
antenna number needs to be reasonably determined.

Fig. 7 shows the EE with different antenna numbers and CSI estimation errors.
As shown in Fig. 7, there is an optimal transmit power for a given antenna
number and CSI to achieve the maximum EE. As the transmit power increases, the
system EE ascends slightly, then arrives at its maximum and lastly descends
moderately under either perfect or imperfect CSI. This shows that proper
allocation of transmit powers can improve the system EE. Fig. 7 also shows
that, under different antenna numbers, the optimal EE is different. When the
antenna number $N_{k,u,s}$ = 30, the EE optimum is $3.2\times 10^{-3}$
bits/J/Hz, which is higher than other three cases, i.e., $N_{k,u,s}$ = 20,
$N_{k,u,s}$ = 25, and $N_{k,u,s}$ = 40. We also compare the system EE under
perfect and imperfect CSI. The results demonstrate that the EE under perfect
CSI is superior to that under the imperfect CSI, which is consistent with
Figs. 3–6.

Fig. 5: EE comparison of PA under perfect and imperfect CSI.

Fig. 6: EE versus number of antenna under perfect and imperfect CSI.

Fig. 7: Impact of transmit power on EE with different number of antenna and
CSI estimation errors.

Fig. 8: EE versus transmit power under different subcarrier schemes.

Fig. 8 shows the EE performance versus transmit power under the different
subcarrier processing mode. For simplicity, we use “MC-NOMA without SCA” to
denote the benchmark algorithm in the multi-carrier non-orthogonal multiple
access system without considering the subcarrier allocation (SCA) [35]. Under
imperfect CSI, we present the EE results of “PA” and “MC-NOMA without SCA”. We
can observe that the EE of “PA” has a better performance than that of “MC-NOMA
without SCA” whether the number of subcarriers equals 20 or 30. Moreover, the
figure also shows that using more subcarriers reduces system EE. When the
number of subcarriers is 20, the EE of “PA” is higher than the case where 30
subcarriers are considered. This is because more subcarriers narrow the width
of the subcarrier bandwidths. As a result, it requires more computations when
allocating the subcarriers to the IoT devices, reducing the system EE.

Considering the feature of massive MIMO, Fig. 9 shows the EE performance
comparison between MIMO and massive MIMO. We use the proposed AS-based
distributed algorithm to solve the EE maximization. The base station with
256/128/64 antenna arrays serves 15 single-antenna IoT devices. We also
compare the EE of the traditional MIMO network when the number of antennas is
2, 4, and 8, with the number of massive MIMO is 64, 128, and 256. We see that
the EE with the massive MIMO-NOMA technique outperforms than that with the
traditional MIMO-NOMA technique whether in Fig. 9(a) or (b). As the total
antenna number $M_{k}$ increases, the increasing rate of system EE slows down.
This can be seen from the massive MIMO case corresponding to
$M_{k}=64,128,256$. As we can observe, the EE with massive MIMO-NOMA technique
under the case of $M_{k}=256$ approaches the EE under the case of
$M_{k}\rightarrow\infty$. These results verify that massive MIMO is superior
to traditional MIMO due to its multi-antenna advantages, and infinite EE
cannot be achieved by taking the total number of antennas to infinity. Fig. 9
also verifies that multi-antenna selection in massive MIMO-NOMA systems can
obtain a high EE.

In Fig. 10, we evaluate the EE by jointly varying the transmit power and WPT
time. As shown in Fig. 10, the optimal WPT time slot $\tau_{k}$ can be
obtained for a given transmit power. For different numbers of selected
antennas, Fig. 10 presents the EE of four cases, Case 1: $N_{k,u,s}$ = 20;
Case 2: $N_{k,u,s}$ = 25; Case 3: $N_{k,u,s}$ = 30; and Case 4: $N_{k,u,s}$ =
40. We observe that, the EE firstly increases, then climbs to the maximum and
decreases with the increasing WPT time. When the WPT time slot is between $0$
and $1$ (e.g., 0.5), the EE first ascends, then arrives at the maximum and
finally descends. From the four subfigures of Fig. 10, we see that the system
EE varies with the different antenna number. For example, the maximal EE with
$N_{k,u,s}=30$ antennas is $6\times 10^{-4}$ bits/J/Hz, higher than the other
three cases. Therefore, on top of AS, the joint optimization of the power
$P_{k}$ and the WPT time $\tau_{k}$ is significant to obtain the maximum EE.
Fig. 10 confirms the validity of the proposed energy-efficient resource
allocation algorithm in which the power and time are jointly optimized.

Fig. 9: EE comparison between massive MIMO-NOMA and traditional MIMO-NOMA.
Fig. 9(a) presents the EE versus transmission distance under different antenna
number; Fig. 9(b) presents the EE versus transmit power under different
antenna number.

Fig. 10: EE versus transmit power and first time slot.

To show the convergence performance of “PA”, we compare it with a Dinkelbach-
based centralized algorithm [52], and an OMA-based scheme using the Lagrange
multipliers [53], in Fig. 11. We denote the benchmark algorithms as
“Dinkelbach-based CA” [54] and “LA OMA” [53]. The convergence of “PA” and the
impact of the convergence parameter $\rho$ are presented in Fig. 11. It shows
that the optimal EE with a small number of iterations can be achieved by
adopting ADMM. When $\rho$ = 0.068, the optimal EE can be obtained within nine
iterations. When $\rho$ = 0.088, the optimal EE can be obtained within seven
iterations. These results illustrate that the convergence rate is dependent on
the value of $\rho$. We also analyse the convergence difference of among
“Dinkelbach-based CA”, “LA OMA” and “PA” to show the performance loss in terms
of the EE and convergence rate. Comparing “PA” with “Dinkelbach-based CA” and
“LA OMA”, we find that the EE first increases as the iteration index grows,
and then the system EE stays unchanged. It can also be found that “PA” and
“Dinkelbach-based CA” have very similar EE performances. Fig. 11 shows that
the EE performance difference is negligible, and the distributed method has
fast convergence. This is due to the vibration effect of the Dinkelbach method
in the vicinity of the optimal solution. We remark that the Dinkelbach method
is merely on the basis of closed-form expressions which require low
computational complexity per iteration but more iterations. The reason for
“PA” with a higher convergence rate is that “PA” splits the target function of
the primal variable to simplify the optimization problem and the augmented
Lagrangian function has a saddle point contributing a faster convergence.

Fig. 11: Convergence of the proposed algorithm and reference algorithms.

##  VII Conclusion

This paper has investigated the energy-efficient resource allocation in the
WPT-based massive MIMO-NOMA networks. The joint transmit power, WPT time,
antenna selection and subcarrier allocation scheme has been proposed to solve
the system EE maximization problem. We have invoked the nonlinear fraction
programming approach to convert the original non-convex problem to be convex,
and solve the problem by developing a novel distributed ADMM-based energy-
efficient resource allocation algorithm. We have considered both the perfect
and imperfect CSI, and analysed the impact of channel estimation error on the
EE performance. Simulation results have shown the convergence and
effectiveness of the proposed algorithm, as well as the superior EE
performance over the benchmark schemes.

There are many promising directions worth researching in the future. For
instance, the IRS-assisted wireless communication is a young-born paradigm
that has attracted researchers’ attention. We can consider IRS-enabled massive
MIMO-NOMA to improve the EE of the RF-based WPT networks. UAV-assisted massive
MIMO-NOMA is also an interesting topic, as it can effectively support the
coverage and throughput of wireless communication by leveraging the strength
of line-of-sight connections. Moreover, it is worth extending our work to more
practical and complicated scenarios, such as the networks with multiple
dynamic IoT devices using different energy harvesting techniques.

##  Appendix A Derivation of (17)

In the considered WPT enabled massive MIMO-NOMA network, we take the $k$-th
cell as a example, where an IoT device transmits data packets on the $s$-th
subcarrier. For simplicity, we omit the subscript $k$ and $s$. According to
[55], we have ${{x}_{1}}>{{x}_{2}}>\cdots>{{x}_{u}}\cdots>{{x}_{U}}$ as the
ordered random signal variables from the IoT devices in the $k$-th cell. With
massive antennas at each BS, the selected optimal antenna number $N$ satisfies
the condition of $1\leq N\leq M$. When $M\rightarrow\infty$, the distribution
of trimmed sum $\sum\limits_{u=1}^{N}{{{x}_{u}}}$ of IoT devices is
asymptotically normal. When ${{x}_{u}}$ is a chi-square random variable signal
propagated through ${\hat{h}}_{u}$ with the degree of freedom 2, the mean and
variance can be found in [56]. The channel coefficient sum between the $u$-th
IoT device and the $BS_{k}$ can be denoted by

$\sum\limits_{u=1}^{N}{{{\left|{{{\hat{h}}}_{u}}\right|}^{2}}}\sim\mathcal{N}\left(N\left(1+\ln\frac{M}{N}\right),N\left(2-\frac{N}{M}\right)\right)$
(35)

where ${{\hat{h}}_{u}}$ denotes the subchannel of channel matrix
$\mathbf{\hat{H}}$ between the IoT devices and the BS. According to the [57],
in our proposed WPT enabled massive MIMO-NOMA network, the mutual information
under the antenna selection scheme is expressed by

${{I}_{sel}}\text{=}{{\log}_{2}}\left|1\text{+}\gamma\sum\limits_{u=1}^{N}{{{\left|{{{\hat{h}}}_{u}}\right|}^{\text{2}}}}\right|\text{=}{{\log}_{2}}\left|1\text{+(1+ln}\frac{M}{N}\text{)}\gamma
N\right|+{{\log}_{2}}\left|\xi\right|$ (36)

where $\gamma$ is the SNR measured at each receive antenna.
$\left|\cdot\right|$ takes the absolute value, and the introduced new variable
$\xi$ is given by

$\xi=1+\frac{\gamma\left(\sum\limits_{u=1}^{N}{{{\left|{{{\hat{h}}}_{u}}\right|}^{2}}}-N\left(1+\ln\frac{M}{N}\right)\right)}{1+\left(1+\ln\frac{M}{N}\right)\gamma
N}.$ (37)

According to (35), $\xi$ obeys the distribution as follows

$\xi\sim\mathcal{N}\left(1,\frac{{{\gamma}^{2}}N\left(2-\frac{N}{M}\right)}{{{\left(1+\left(1+\ln\frac{M}{N}\right)\gamma
N\right)}^{2}}}\right).$ (38)

As the property of folded normal distribution of variable
$y\text{=}\left|\xi\right|$, the mutual information of WPT enabled massive
MIMO-NOMA networks can be further derived as

$\displaystyle{{I}_{sel}}\text{=}{{\log}_{2}}\left(1\text{+}(1\text{+}\ln\frac{M}{N})\gamma
N\right)\text{+}(y-1){{\log}_{2}}e\text{+}O\left({{(y-1)}^{2}}\right).$ (39)

As $O\left({{(y-1)}^{2}}\right)$ is asymptotically zero [57], ${{I}_{sel}}$
obeys the distribution as follows

$\displaystyle{{I}_{sel}}\sim\mathcal{F}\mathcal{N}\left({{\log}_{2}}[1+(1+\ln\frac{M}{N})\gamma
N]\right.,\left.\frac{{{\left({{\log}_{2}}e\right)}^{2}}{{\gamma}^{2}}N\left(2-\frac{N}{M}\right)}{{{\left(1+\left(1+\ln\frac{M}{N}\right)\gamma
N\right)}^{2}}}\right).$ (40)

The proof of data rate in (17) under imperfect CSI is completed. ∎

##  Appendix B The Concavity Proof of
$R_{tot}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)$
in (18)

To prove the joint concavity of
$R_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)$
in (18) concerning the variables
$\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{L}$ and $\boldsymbol{C}$, we
first take successive convex approximation methods to relax the discrete
variables $N_{k,u,s}$ and $c_{k,u,s}$ into continuous slack variables.
Further, the Hessian matrix of $R_{tot}$ can be denoted by

$\displaystyle{{\mathbf{H}}_{R}}=$ (41)
$\displaystyle\left[\begin{matrix}\frac{{{\partial}^{2}}{{R}_{\text{tot}}}}{\partial
P_{k}^{2}}&\frac{{{\partial}^{2}}{{R}_{\text{tot}}}}{\partial{{P}_{k}}\partial{{\tau}_{k}}}&\frac{{{\partial}^{2}}{{R}_{\text{tot}}}}{\partial{{P}_{k}}\partial
N_{k,u,s}}&\frac{{{\partial}^{2}}{{R}_{\text{tot}}}}{\partial{{P}_{k}}\partial{c_{k,u,s}}}\\\
\frac{{{\partial}^{2}}{{R}_{\text{tot}}}}{\partial{{\tau}_{k}}\partial{{P}_{k}}}&\frac{{{\partial}^{2}}{{R}_{\text{tot}}}}{\partial\tau_{k}^{2}}&\frac{{{\partial}^{2}}{{R}_{\text{tot}}}}{\partial{{\tau}_{k}}\partial
N_{k,u,s}}&\frac{{{\partial}^{2}}{{R}_{\text{tot}}}}{\partial{{\tau}_{k}}\partial{c_{k,u,s}}}\\\
\frac{{{\partial}^{2}}{{R}_{\text{tot}}}}{\partial
N_{k,u,s}\partial{{P}_{k}}}&\frac{{{\partial}^{2}}{{R}_{\text{tot}}}}{\partial
N_{k,u,s}\partial{{\tau}_{k}}}&\frac{{{\partial}^{2}}{{R}_{\text{tot}}}}{\partial
N_{k,u,s}^{2}}&\frac{{{\partial}^{2}}{{R}_{\text{tot}}}}{\partial
N_{k,u,s}\partial{c_{k,u,s}}}\\\
\frac{{{\partial}^{2}}{{R}_{\text{tot}}}}{\partial{c_{k,u,s}}\partial{{P}_{k}}}&\frac{{{\partial}^{2}}{{R}_{\text{tot}}}}{\partial{c_{k,u,s}}\partial{{\tau}_{k}}}&\frac{{{\partial}^{2}}{{R}_{\text{tot}}}}{\partial{c_{k,u,s}}\partial
N_{k,u,s}}&\frac{{{\partial}^{2}}{{R}_{\text{tot}}}}{\partial
c_{k,u,s}^{2}}\\\ \end{matrix}\right].$

More specifically, we calculate the partial derivatives of each element of the
Hessian matrix and we have $\frac{{{\partial}^{2}}{{R}_{\text{tot}}}}{\partial
P_{k}^{2}}\text{=-}\frac{f_{11}^{1}f_{11}^{3}\left(2{{\left(f_{11}^{2}\right)}^{2}}{{P}_{k}}+2f_{11}^{1}f_{11}^{2}{{P}_{k}}+2f_{11}^{2}f_{11}^{3}+f_{11}^{1}f_{11}^{3}\right)}{{{\left(\left[{{\left(f_{11}^{2}\right)}^{2}}+f_{11}^{1}f_{11}^{2}\right]P_{k}^{2}+\left(2f_{11}^{2}f_{11}^{3}+f_{11}^{1}f_{11}^{3}\right){{P}_{k}}+{{\left(f_{11}^{3}\right)}^{2}}\right)}^{2}}}<0$,
where
$f_{11}^{1}\text{=}KUS{{B}_{s}}\eta{{\tau}_{k}}\alpha_{k,u}^{4}{{\left|{{\mathbf{h}}_{k,u,s}}\right|}^{4}}$,
$f_{11}^{2}\text{=}\left(U-u+1\right)\eta{{\tau}_{k}}\cdot$
$\alpha_{k,{{u}_{1}}}^{4}{{\left|{{\mathbf{h}}_{k,u_{1},s}}\right|}^{4}}{{c}_{k,u_{1},s}}$,
$f_{11}^{3}\text{=}\left(T\text{-}{{\tau}_{k}}\right)\left({{I}_{k_{2},u,s}}+{{\sigma}^{2}}\right)$.
Similarly, we calculate the second partial derivatives of the other elements
of the upper triangular Hessian matrix, and obtain that the sign of each
element is less than 0 according to the property of symmetric matrix. We take
any two independent variables ${{x}_{1}}$ and ${{x}_{2}}$
(${{x}_{1}},{{x}_{2}}\in\\{\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\\}$),
and the difference between ${{x}_{1}}$ and ${{x}_{2}}$ is denoted by
$\Delta\mathbf{x}$, then the inequality
$\Delta{{\mathbf{x}}^{T}}\mathbf{H}_{R}\Delta\mathbf{x}\leq 0$ holds. Thus,
${{\mathbf{H}}_{R}}$ is semi-negative definite, and the concavity of
${{R}_{\text{tot}}}$ is proved.

The concavity proof of (18) is completed. ∎

##  Appendix C Derivation of Theorem 1

###  C-A Sufficiency

The proof of Theorem 1 can be established in the same way as in [43]. For
simplifying notations, we define $\Delta$, $\Theta$, $\Re$ and $\Im$ as the
set of feasible solutions for $\mathbf{P1}$ in (21), and
${\eta_{\text{EE}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)=}{R_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)}/{E_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)}$.
Without loss of generality, $\eta_{\text{EE}}^{*}$, $P_{k}^{*}\in\Delta$,
$\tau_{k}^{*}\in\Theta$, $N_{k,u,s}^{*}\in\Re$ and $c_{k,u,s}^{*}\in\Im$ are
defined as the optimal EE, power allocation, time partition scheme, AS and
subcarrier allocation policies of the primal problem $\mathbf{P1}$,
respectively. We can obtain the optimal EE as

$\displaystyle\eta_{\text{EE}}^{*}=\frac{R_{\text{tot}}\left(\boldsymbol{P}^{*},\boldsymbol{\tau}^{*},\boldsymbol{N}^{*},\boldsymbol{C}^{*}\right)}{E_{\text{tot}}\left(\boldsymbol{P}^{*},\boldsymbol{\tau}^{*},\boldsymbol{N}^{*},\boldsymbol{C}^{*}\right)}\geq\frac{R_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)}{E_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)},$
(42) $\displaystyle\ \ \ \ \ \forall\
P_{k}\in\Delta,\tau_{k}\in\Theta,N_{k,u,s}\in\Re,c_{k,u,s}\in\Im.$

Since the denominator of (42) is positive, according to the transformation of
the equation and inequality, we have

$\left\\{\begin{aligned}
&{{R}_{\text{tot}}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)-\eta_{EE}^{*}{{E}_{\text{tot}}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)\leq
0,\\\
&{{R}_{\text{tot}}}\left(\boldsymbol{P}^{*},\boldsymbol{\tau}^{*},\boldsymbol{N}^{*},\boldsymbol{C}^{*}\right)-\eta_{EE}^{*}{{E}_{\text{tot}}}\left(\boldsymbol{P}^{*},\boldsymbol{\tau}^{*},\boldsymbol{N}^{*},\boldsymbol{C}^{*}\right)=0.\\\
\end{aligned}\right.$ (43)

We conclude that
$\underset{\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}}{\mathop{\max}}R_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)-\eta_{\text{EE}}$
$\cdot
E_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)$
$=0$ is achievable and feasible at the optimal power, time, AS and subcarrier
allocation $\boldsymbol{P}^{*}$, $\boldsymbol{\tau}^{*}$,
$\boldsymbol{N}^{*}$, $\boldsymbol{C}^{*}$.

###  C-B Necessity

The converse implication of Theorem 1 can be proved as follows.
$\boldsymbol{P}^{\prime}$, $\boldsymbol{\tau}^{\prime}$,
$\boldsymbol{N}^{\prime}$ and $\boldsymbol{C}^{\prime}$ are assumed as the
optimal power, WPT time, AS and subcarrier allocation policies of the
equivalent objective function so as to satisfy the following condition

$\begin{aligned}
&{{R}_{\text{tot}}}\left(\boldsymbol{P}^{\prime},\boldsymbol{\tau}^{\prime},\boldsymbol{N}^{\prime},\boldsymbol{C}^{\prime}\right)-\eta_{EE}^{*}{{E}_{\text{tot}}}\left(\boldsymbol{P}^{\prime},\boldsymbol{\tau}^{\prime},\boldsymbol{N}^{\prime},\boldsymbol{C}^{\prime}\right)=0\\\
\end{aligned}.$ (44)

Given any feasible power, time, subcarrier allocation and AS policies,
$P_{k}\in\Delta$, $\tau_{k}\in\Theta$, $c_{k,u,s}\in\Im$ and
$N_{k,u,s}\in\Re$, we have

$\displaystyle\ \ \ \
{{R}_{\text{tot}}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)-\eta_{EE}^{*}{{E}_{\text{tot}}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)$
(45)
$\displaystyle\leq{{R}_{\text{tot}}}\left(\boldsymbol{P}^{\prime},\boldsymbol{\tau}^{\prime},\boldsymbol{N}^{\prime},\boldsymbol{C}^{\prime}\right)-\eta_{EE}^{*}{{E}_{\text{tot}}}\left(\boldsymbol{P}^{\prime},\boldsymbol{\tau}^{\prime},\boldsymbol{N}^{\prime},\boldsymbol{C}^{\prime}\right)=0.$

The inequality (45) yields

$\displaystyle\frac{R_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)}{E_{\text{tot}}\left(\boldsymbol{P},\boldsymbol{\tau},\boldsymbol{N},\boldsymbol{C}\right)}\leq\eta_{\text{EE}}^{*},$
$\displaystyle\forall\
P_{k}\in\Delta,\tau_{k}\in\Theta,N_{k,u,s}\in\Re,c_{k,u,s}\in\Im$ (46)
$\displaystyle\frac{R_{\text{tot}}\left(\boldsymbol{P}^{\prime},\boldsymbol{\tau}^{\prime},\boldsymbol{N}^{\prime},\boldsymbol{C}^{\prime}\right)}{E_{\text{tot}}\left(\boldsymbol{P}^{\prime},\boldsymbol{\tau}^{\prime},\boldsymbol{N}^{\prime},\boldsymbol{C}^{\prime}\right)}=\eta_{\text{EE}}^{*}.$
(47)

This indicates that the optimal solutions $\boldsymbol{P}^{\prime}$,
$\boldsymbol{\tau}^{\prime}$, $\boldsymbol{N}^{\prime}$ and
$\boldsymbol{C}^{\prime}$ for the objective function corresponds to the
optimal resource allocation policies for the original objective $\mathbf{P1}$.

The proof of Theorem 1 is completed. ∎

##  Appendix D The Convexity Proof of P4

In light of the definition of Hessian matrix, if $f$ is a real-valued function
$f(x_{1},x_{2},...,x_{n})$ and differentiable with respect to the independent
variable $x_{i},i\in[1,n]$, the Hessian matrix of $f$ is defined as
$\mathbf{H}(f)_{i,j}(x)=D_{i}D_{j}f(x)$, where $D_{i}$ (and $D_{j}$) denotes
the differential operator with respect to the $i$-th ($j$-th) independent
variable. The Hessian matrix of
$\Lambda({{{P}_{k}}},{\tilde{{\tau}}_{k}},{{\tilde{n}}_{k,u,s}},{\tilde{c}_{k,u,s}})$
of $\mathbf{P4}$ in (26) can be given by

$\mathbf{H}=\left[\begin{matrix}\frac{{{\partial}^{2}}{\Lambda}}{\partial
P_{k}^{2}}&\frac{{{\partial}^{2}}{\Lambda}}{\partial{{P}_{k}}\partial{{{\tilde{\tau}}}_{k}}}&\frac{{{\partial}^{2}}{\Lambda}}{\partial{{P}_{k}}\partial\tilde{n}_{k,u,s}}\\\
\frac{{{\partial}^{2}}{\Lambda}}{\partial{{{\tilde{\tau}}}_{k}}\partial{{P}_{k}}}&\frac{{{\partial}^{2}}{\Lambda}}{\partial\tilde{\tau}_{k}^{2}}&\frac{{{\partial}^{2}}{\Lambda}}{\partial{{{\tilde{\tau}}}_{k}}\partial\tilde{n}_{k,u,s}}\\\
\frac{{{\partial}^{2}}{\Lambda}}{\partial\tilde{n}_{k,u,s}\partial{{P}_{k}}}&\frac{{{\partial}^{2}}{\Lambda}}{\partial\tilde{n}_{k,u,s}\partial{{{\tilde{\tau}}}_{k}}}&\frac{{{\partial}^{2}}{\Lambda}}{\partial\tilde{n}_{k,u,s}^{2}}\\\
\end{matrix}\right].$ (48)

In (48), we let $\Lambda$ =
$\Lambda({{{P}_{k}}},{\tilde{{\tau}}_{k}},{{\tilde{n}}_{k,u,s}},{\tilde{c}_{k,u,s}})$
for notational simplicity. Further, we divide $\mathbf{H}$ into the sub-
matrices according to the order of the matrix, as given by

$\mathbf{H_{1}}=\left[\begin{matrix}\frac{{{\partial}^{2}}{\Lambda}}{\partial
P_{k}^{2}}\end{matrix}\right],\mathbf{H_{2}}=\left[\begin{matrix}\frac{{{\partial}^{2}}{\Lambda}}{\partial
P_{k}^{2}}&\frac{{{\partial}^{2}}{\Lambda}}{\partial{{P}_{k}}\partial{{{\tilde{\tau}}}_{k}}}\\\
\frac{{{\partial}^{2}}{\Lambda}}{\partial{{{\tilde{\tau}}}_{k}}\partial{{P}_{k}}}&\frac{{{\partial}^{2}}{\Lambda}}{\partial\tilde{\tau}_{k}^{2}}\\\
\end{matrix}\right],\mathbf{H_{3}}=\mathbf{H}.$ (49)

As the determinants of sub-matrices affect the positive definition of the
original matrix, by evaluating the determinants of the sub-matrices, we have

$|\mathbf{H_{1}}|\leq 0,|\mathbf{H_{2}}|\geq 0,|\mathbf{H_{3}}|\leq 0.$ (50)

Thus, $\mathbf{H}$ is semi-negative definite. We can conclude that
$\Lambda({{{P}_{k}}},{\tilde{{\tau}}_{k}},$
${{\tilde{n}}_{k,u,s}},{\tilde{c}_{k,u,s}})$ of $\mathbf{P4}$ is a jointly
convex function of variables ${P}_{k},{\tilde{{\tau}}_{k}}$, and
${{\tilde{n}}_{k,u,s}}$.

The convexity proof of $\mathbf{P4}$ is completed. ∎

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|  Zhongyu Wang (S’19) received the M.S. degree in computer science and
technology from School of Information Science and Engineering, Yanshan
University, Qinhuangdao, China. She is currently working toward the Ph.D.
degree in information and communication engineering with the School of
Information and Communication Engineering, Beijing University of Posts and
Telecommunications, Beijing, China. Her current research interests include
ultra-reliable low latency communication, non-orthogonal multiple access,
massive MIMO, energy harvesting technology, and UAV communication.
---|---
|  Zhipeng Lin (M’20) is currently working toward the dual Ph.D. degrees in
communication and information engineering with the School of Information and
Communication Engineering, Beijing University of Posts and Telecommunications,
Beijing, China, and the School of Electrical and Data Engineering, University
of Technology of Sydney, NSW, Australia. His current research interests
include millimeter-wave communication, massive MIMO, hybrid beamforming,
wireless localization, and tensor processing.
---|---
|  Tiejun Lv (M’08-SM’12) received the M.S. and Ph.D. degrees in electronic
engineering from the University of Electronic Science and Technology of China
(UESTC), Chengdu, China, in 1997 and 2000, respectively. From January 2001 to
January 2003, he was a Postdoctoral Fellow with Tsinghua University, Beijing,
China. In 2005, he was promoted to a Full Professor with the School of
Information and Communication Engineering, Beijing University of Posts and
Telecommunications (BUPT). From September 2008 to March 2009, he was a
Visiting Professor with the Department of Electrical Engineering, Stanford
University, Stanford, CA, USA. He is the author of 3 books, more than 90
published IEEE journal papers and 190 conference papers on the physical layer
of wireless mobile communications. His current research interests include
signal processing, communications theory and networking. He was the recipient
of the Program for New Century Excellent Talents in University Award from the
Ministry of Education, China, in 2006. He received the Nature Science Award in
the Ministry of Education of China for the hierarchical cooperative
communication theory and technologies in 2015.
---|---
|  Wei Ni (M’09-SM’15) received the B.E. and Ph.D. degrees in Electronic
Engineering from Fudan University, Shanghai, China, in 2000 and 2005,
respectively. Currently, he is a Group Leader and Principal Research Scientist
at CSIRO, Sydney, Australia, and an Adjunct Professor at the University of
Technology Sydney and Honorary Professor at Macquarie University, Sydney. He
was a Postdoctoral Research Fellow at Shanghai Jiaotong University from 2005 –
2008; Deputy Project Manager at the Bell Labs, Alcatel/Alcatel-Lucent from
2005 to 2008; and Senior Researcher at Devices R&D, Nokia from 2008 to 2009.
His research interests include signal processing, optimization, learning, and
their applications to network efficiency and integrity. Dr Ni is the Chair of
IEEE Vehicular Technology Society (VTS) New South Wales (NSW) Chapter since
2020 and an Editor of IEEE Transactions on Wireless Communications since 2018.
He served first the Secretary and then Vice-Chair of IEEE NSW VTS Chapter from
2015 to 2019, Track Chair for VTC-Spring 2017, Track Co-chair for IEEE VTC-
Spring 2016, Publication Chair for BodyNet 2015, and Student Travel Grant
Chair for WPMC 2014.
---|---