See title_page.pdf

See title_page_1.pdf

## Acknowledgments

After an intensive year, I put the finishing touches on this Master’s thesis.
I have been able to discover new elements in various areas of the beautiful
world of telecommunication. For this, I would like to sincerely thank a number
of people who have supported me during this long journey.

First of all, I want to thank my supervisor, Prof. Dr. Ir. Heidi Steendam, for
giving me the opportunity to work on this topic for my thesis. Her trust,
guidance, and support helped me to successfully complete it. I was fortunate
for her supervision, especially her patience, understanding, and availability
for discussing how to write scientific papers.

In addition, I express my gratitude to many friends and colleagues for our
fruitful discussions and friendship. Among them, my special appreciation goes
to my best friend Ata for his comments and invaluable knowledge, which reaped
us significant research outputs. I am so lucky and honored to have amazing
friends, including Pandidan, Sirus, Khasmakhu, Mamadu, Hani, Arakani,
Mahboubeh, Keyvan, and Dennis, who gave me crucial feedback. Tinus, Wim,
Jacques, and Junior count among those amazing friends whose help and concern I
will always remember.

Finally, I would like to express my most profound gratefulness to my beloved
parents, Afsaneh and Nouri, who are always sources of strength and
perseverance. I also want to thank my spiritual mother, Nancy, for being part
of my life, giving me guidance and offering me help far from my home country.
Lastly, I would like to thank my partner, Mary Grace, for her unconditional
trust, care, admiration, and love.

Jalal Jalali

June 2020

## Permission to Use Content

The author grants permission for this Master’s thesis to be used for
consultation and for parts of it to be copied for personal use. For all other
uses, however, the copyright must be respected, particularly with regard to
the obligation to explicitly credit the source when quoting the results.

Jalal Jalali

June 2020

##

Resource Allocation for SWIPT in

Multi-Service Wireless Networks

by

Jalal JALALI

A Thesis Presented in Partial Fulfillment of the Requirements for the Degree
of

Master of Science in Electrical Engineering

Academic year 2019-2020

Supervisor: Prof. Dr. Ir. Heidi STEENDAM

Faculty of Engineering and Architecture

Ghent University

Department of Telecommunications and Information Processing

Chairman: Prof. Dr. Ir. Joris WALRAEVENS

Summary

The novel resource allocation for simultaneous wireless information and power
transfer (SWIPT) is presented as a means of not only helping to communicate
and access information with increasing efficiency in the next generation of
mobile data networks, but also contributing to minimizing a network’s overall
power consumption by providing a green energy source. First, a unique
architecture is proposed that harvests energy from an access point (AP)
without the receiver needing a splitter. In the proposed system model, a
portion of the spectrum is used for information decoding (ID) while the
remaining portion is exploited for energy harvesting (EH) in an orthogonal
frequency division multiple access (OFDMA) network. To investigate the
performance gain, an optimization problem is formulated that maximizes the
harvested energy of a multi-user single-cell OFDMA downlink (DL) network with
SWIPT and also satisfies a minimum data-rate requirement for all users. A
locally optimal solution for the underlying problem, which is essentially non-
convex due to the coupling of the integer variable, is obtained by using
optimization tools. Second, the proposed system model is improved in order to
investigate the resource allocation problem of needing to maximize throughput
based on the separated receiver architecture in an OFDMA multi-user multi-cell
system that uses SWIPT. The resulting problem, which jointly optimizes the
subcarrier assignment and power allocation, is a mixed-integer non-linear
problem (MINLP) that is difficult to solve. Third, a state-of-the-art
harvesting technique at the receiver that is based on receiver antenna
selection with a co-located architecture is explored to optimize the energy
efficiency (EE) of a SWIPT-enabled multi-cell multi-user OFDMA network. This
is referred to as a “generalized antenna-switching technique”. Extensive
simulation demonstrates the superiority of the proposed methodologies and
presents interesting results.

Keywords

D.C. programming, energy efficiency, energy harvesting, green communication,
majorization minimization, MINLP, multi-user communication, non-convex
optimization, OFDMA, resource allocation, SWIPT, throughput.

###### Contents

  1. Abbreviations
  2. Operators and Symbols
  3. 1 Introduction
    1. 1.1 The Architecture of an RF Energy Harvesting Network
    2. 1.2 Energy Harvesting Modes
      1. 1.2.1 WPT
      2. 1.2.2 WPCN
      3. 1.2.3 WPBC
      4. 1.2.4 SWIPT
    3. 1.3 A SWIPT Literature Review
      1. 1.3.1 Multi-Carrier SWIPT Systems
      2. 1.3.2 SWIPT in Cognitive Radio Networks
      3. 1.3.3 Cooperative Relaying in SWIPT Networks
      4. 1.3.4 Multiple Antenna Communication in SWIPT Systems
    4. 1.4 Thesis Overview and Contributions
  4. 2 Optimization Techniques
    1. 2.1 Convex Analysis
      1. 2.1.1 Definitions
    2. 2.2 Duality Theorem
      1. 2.2.1 Weak Duality and Duality Gap
      2. 2.2.2 Strong Duality and Slater condition
    3. 2.3 Abstract Lagrangian Duality
    4. 2.4 Complementary Slackness and KKT Optimality Conditions
    5. 2.5 Interior-Point Methods
    6. 2.6 MM Approach and D.C. Programming
    7. 2.7 Optimization Packages
  5. 3 SWIPT in Single Small-Cell Networks
    1. 3.1 System Model
    2. 3.2 Optimization Problem Formulation
    3. 3.3 Solution to the Optimization Problem
      1. 3.3.1 Joint Power Allocation and Subcarrier Assignment Algorithm
    4. 3.4 Complexity Analysis
    5. 3.5 Simulation Results
      1. 3.5.1 Total Harvested Energy versus the Maximum Transmit Power
      2. 3.5.2 Average Harvested Energy versus Distance
      3. 3.5.3 Average Harvested Energy versus Number of Iterations
      4. 3.5.4 Harvested Energy versus Maximum Transmit Power at Different Target Data-rates
    6. 3.6 Summary
  6. 4 SWIPT in Multi-Cell Networks
    1. 4.1 System Model
    2. 4.2 Optimization Problem Formulation
    3. 4.3 Solution to the Optimization Problem
    4. 4.4 Computational Complexity
    5. 4.5 Low Complexity Algorithm Design (Lower Bound)
      1. 4.5.1 Low Complexity Power Control and Subcarrier Assignment
    6. 4.6 Simulation Results
      1. 4.6.1 Average Sum Data-rate versus Minimum Data-rate Requirement
      2. 4.6.2 Average Sum Data-rate versus Maximum Transmit Power
      3. 4.6.3 Average Sum Data-rate Versus Number of Iterations
      4. 4.6.4 Average Sum Data-rate versus Number of Cells
    7. 4.7 Summary
  7. 5 Generalized Antenna Switching Technique in SWIPT
    1. 5.1 System Model
    2. 5.2 Optimization Problem Formulation
    3. 5.3 Solution to the Optimization Problem
      1. 5.3.1 A. Joint SBS-subcarrier Assignment and Power Allocation
      2. 5.3.2 B. Antenna selection
    4. 5.4 Complexity Analysis
    5. 5.5 Simulation Results
      1. 5.5.1 Convergence Speed
      2. 5.5.2 Energy Efficiency versus Maximum Allowed Transmit Power
      3. 5.5.3 Energy Efficiency versus Minimum Data-rate Requirement
      4. 5.5.4 Energy Efficiency versus Distance
      5. 5.5.5 Average System Throughput versus Maximum Allowed Transmit Power
      6. 5.5.6 Average Harvested Power versus Maximum Allowed Transmit Power
    6. 5.6 Summary
  8. 6 Conclusion and Future Work
    1. 6.1 Conclusion
    2. 6.2 Future Work

###### List of Figures

  1. 1.1 The block diagram of an energy harvesting device [10].
  2. (a) General network architecture for energy harvesting
  3. (b) RF energy harvester
  4. 1.2 The WPT system model.
  5. 1.3 The WPCN system model.
  6. 1.4 The WPBC system model, where the backscatter modulation on a tag is used to reflect and modulate the incoming RF signal in order to communicate with a reader (e.g., a wireless router).
  7. 1.5 The separated SWIPT system model.
  8. 1.6 The co-located SWIPT system model.
  9. 1.7 Integrated receiver architecture designs for SWIPT.
  10. (a) Separated receiver architecture.
  11. (b) Time switching (TS) approach to realize co-located SWIPT architecture.
  12. (c) Power splitting (PS) approach to realize co-located SWIPT architecture.
  13. (d) Antenna switching (AS) approach to realize co-located SWIPT architecture.
  14. 3.1 SWIPT in a DL of a co-located multi-user small single-cell OFDMA network.
  15. 3.2 Total harvested energy versus the maximum transmit power
  16. 3.3 Average harvested energy versus distance.
  17. 3.4 Average harvested energy versus number of iterations.
  18. 3.5 Harvested energy versus sum transmit power at different target data-rates.
  19. 4.1 SWIPT in a DL of a multi-user multi-cell OFDMA network with a separated receiver architecture.
  20. 4.2 SWIPT in a DL of an OFDMA network consisting of $J=2$ small cells, where there is one user of each type in each cell, i.e., $\mathcal{K}^{ID}_{1}=\mathcal{K}^{ID}_{2}=\mathcal{K}^{EH}_{1}=\mathcal{K}^{EH}_{2}$=1.
  21. 4.3 Average sum data-rate versus minimum data-rate requirement.
  22. 4.4 Average sum data-rate versus maximum transmit power.
  23. 4.5 Average sum data-rate versus number of iteration under different initialization of the power.
  24. 4.6 Average sum data-rate versus number of iteration under different number of subcarriers.
  25. 4.7 Average sum data-rate versus number of cells.
  26. 5.1 Antenna selection architecture.
  27. 5.2 SWIPT in a DL of a multi-user multi-cell OFDMA network with generalized AS receivers.
  28. 5.3 SWIPT in a DL of an OFDMA network consisting of $J=2$ small cells, where there is one user in the intersection of the two cells, i.e., $\mathcal{K}_{1}=$1\. The user has for antennas; two of which are employed four ID and the rest for EH. Also, the user is served in the first cell and receives interference from the second cell.
  29. 5.4 Convergence speed.
  30. 5.5 Energy efficiency versus maximum allowed transmit power.
  31. 5.6 Energy efficiency versus minimum data-rate requirement.
  32. 5.7 Energy efficiency versus distance.
  33. 5.8 Average system throughput versus maximum allowed transmit power.
  34. 5.9 Average harvested power versus maximum allowed transmit power.

## Abbreviations

5G | Fifth Generation
---|---
6G | Sixth Generation
ABCS | Ambient Backscatter Communication System
AF | Amplify-and-Forward
AP | Access Point
AS | Antenna Switching
AWGN | Additive White Gaussian Noise
BS | Base Station
CRN | Cognitive Radio Network
CSI | Channel State Information
D.C. | Difference of Two Convex Functions
D2D | Device-to-Device
DC | Direct Current
DF | Decode-and-Forward
DL | Downlink
EE | Energy Efficiency
EH | Energy Harvesting
GI | Guard Interval
ICT | Information and Communication Technology
ID | Information Decoding
IFFT | Inverse Fast Fourier Transform
IoT | Internet of Things
ISI | Inter-Symbol Interference
KKT | Karush-Kuhn-Tucker
LoS | Line of Sight
LTE | Long Term Evolution
MEC | Mobile Edge Computing

MILP | Mixed Integer Linear Programming
---|---
MIMO | Multiple-Input Multiple-Output
MINLP | Mixed Integer Non-Linear Programming
MISO | Multiple-Input Single-Output
MM | Majorization Minimization
NOMA | Non-Orthogonal Multiple Access
OFDM | Orthogonal Frequency Division Multiplexing
OFDMA | Orthogonal Frequency Division Multiple Access
PS | Power Switching
QoS | Quality of Service
RF | Radio Frequency
RFID | RF Identification
SBS | Small Base Station
SE | Spectral Efficiency
SISO | Single-Input Single-Output
SNIR | Signal-to-Interference plus Noise Ratio
SNR | Signal-to-Noise Ratio
SWIPT | Simultaneous Wireless Information and Power Transfer
SDMA | Spatial Division Multiple Access
TDMA | Time Division Multiple Access
TS | Time Switching
UAV | Unmanned Aircraft Vehicles
UE | User Equipment
UL | Uplink
WDT | Wireless Data Transfer
WPBC | Wirelessly Powered Backscatter Communication
WPCN | Wirelessly Powered Communication Network
WPT | Wireless Power Transfer
WSN | Wireless Sensor Network

## Operators and Symbols

Operators

$|\mathcal{A}|$ | Cardinality of set $\mathcal{A}$
---|---
$|x|$ | Absolute value norm of $x$
$[x]^{+}$ | max$\\{0,x\\}$
$[x]^{-1}$ | Matrix inverse
$[x]^{T}$ | Matrix transpose
$(\text{·})^{(t)}$ | Value of a variable under the $t^{th}$ iteration
$f(\text{·})\mathrel{\mathop{\mathchar 58\relax}}\mathcal{A}\rightarrow\mathcal{B}$ |  $f$ is a function from set $\mathcal{A}$ into $\mathcal{B}$
$\inf f$ | Infimum of function $f$
$\sup f$ | Supremum of function $f$
$\min f$ | Minimum of function $f$
$\max f$ | Maximum of function $f$
$\nabla f$ | Gradient of function $f$
$\mathcal{L}(\text{·})$ | Lagrangian function
$\mathcal{D}(\text{·})$ | Dual function
$\sum$ | Summation operator
$\mathcal{CN}(\mu,\sigma^{2})$ | Circularly symmetric Gaussian distribution with mean $\mu$ and variance $\sigma^{2}$

Symbols

$p^{*}$ | Primal optimal value
---|---
$d^{*}$ | Dual optimal value
$\mathscr{B}$ | Bandwidth
$\mathcal{N}$ | Number or subcarriers
$\mathcal{K}$ | Number or users in the network
$\mathcal{J}$ | Number or cells in the network
$h$ | Downlink channel gain for the wireless information transfer
$g$ | Downlink channel gain for the wireless information transfer
$a$ | Binary subcarrier indicators
$x$ | Binary antenna selection indicators
EH | Amount of the harvested energy
$R$ | Amount of the data-rate
$\eta$ | Amount of the energy efficiency
$\epsilon$ | Power conversion efficiency of an energy harvesting device
$\kappa$ | Power amplifier efficiency
$\sigma^{2}$ | Noise power
$p_{max}$ | Maximum transmit power
$R_{min}$ | Minimum data-rate requirement
EHmin | Minimum harvested energy requirement

[0.55] “Invention is the most important product of man’s creative brain. The
ultimate purpose is the complete mastery of mind over the material world, the
harnessing of human nature to human needs.” – Nikola Tesla

##  Chapter 1 Introduction

Wireless communication is a broad and dynamic field that is enjoying the rapid
development of new technologies needed to cope with the massive growth in the
number of wireless communication devices and many practical applications. In
the past few decades, wireless networks and communication devices have become
an indispensable part of modern life. The next generation of wireless networks
(called “fifth generation” (5G) by the standardization community) is designed
to communicate and access information more efficiently. 5G is being deployed
to provide high data-rates for mobile users, massive internet of things (IoT)
applications, device-to-device (D2D) communications, and low latency or
extreme real-time tactile communications with high availability based on real-
time immediate interactions. High data-rate access is extremely important, but
the huge amount of power consumed by modern communication applications and
networks is a major factor in global warming. This has inspired the notion of
green and sustainable radio communication. New wireless ecosystems and data
networks supporting higher system throughput with greater energy efficiency
are needed. At the same time, they must provide a variety of services such as
wireless power transfer, mobile node positioning, cooperative sensing of the
surrounding environment, and distributed processing of wireless audio and
video signals.

In this regard, a series of procedures, specifications, requirements, and
constraints concerning the delivery of each of these services is accomplished
through gradual design. At best, this leads to ad-hoc treatment of just one of
the many services because the lack of structure causes the wireless
infrastructure to be poorly managed. In contrast, a unique shared
infrastructure provides plenty of room to holistically analyze and more
systematically design the complete wireless ecosystem and optimize a multi-
service wireless network. This systematic approach to multi-service wireless
networks is called “X-service design”. In it, all the services and
(computational and radio) resources are optimized adaptively and controlled
cooperatively.

Today’s cellular networks with approximately 6 million macro-cells worldwide
consume a peak rate of almost 12 billion watts – the majority of the world’s
wireless information and communication technology (ICT) power consumption [1].
How can we reduce this enormous consumption of power? By remembering Nikola
Tesla’s late-nineteenth-century dream of a “wirelessly powered world”. Radio
signals convey power and can potentially also be used to deliver energy to
remote devices. Tesla’s seemingly far-fetched idea has recently triggered
interest in wireless power transfer (WPT), which when coupled with information
transfer is referred to as simultaneous wireless information and power
transfer (SWIPT) [2]. WPT can be a by-product of wireless data transfer (WDT)
networks in which devices capture ambient power, and thus contribute to
minimizing a network’s overall power consumption using green energy. This
partially responds to the urgent question, although it is far from an ideal
answer. If a certain quality of power transfer must be secured, base stations
(BSs) can play an active role in power delivery and reserve part or all of
their resources for this specific objective of reducing a network’s overall
power consumption. This is one example of positioning services [3].

Let’s think for a moment of a hypothetical meeting of the minds between Claude
Shannon, the father of information theory, and Nikola Tesla. Tesla tried to
build a circuit to deliver power to a load wirelessly, Shannon wanted to use
such a circuit just for sending information [4]. Path loss, energy harvester
sensitivities that require a significant signal level, and the limits of radio
transmit power mean that WPT can be effective only over distances like those
found in ultra-dense networks [5]. Thus, network densification could be a
point of convergence for positioning, WPT, and WDT networks – with the
ultimate goal that could be described as “zero RF power” SWIPT-enabled
networks. This is where Shannon meets Tesla.

###  1.1 The Architecture of an RF Energy Harvesting Network

Wireless communication systems equipped with energy harvesting (EH) receivers
have increasingly been attracting attention [6]. A radio frequency (RF) EH
device has a sustainable power supply from a radio environment that provides
harvestable energy from RF signals for information processing and
transmission. A practical example of this is wireless sensor networks (WSNs),
in which ambient energy is converted to electrical energy via an EH device –
not only to enable a long period of operation in WSNs but also as an
alternative to replacing the battery [7]. However, traditional EH devices may
not be appropriate for many applications due to their complex mechanical
constraints such as form factor and cost, and they may not always be available
in indoor environments. Moreover, conventional EH approaches usually depend on
renewable energy sources like solar, tide, wind, thermal, geothermal, and
vibrations, which are usually unpredictable and hard to control [8]. Hence,
proactive WPT as an EH method that only needs an RF EH circuit and has low
cost and small form factor should be considered when studying how to jointly
design WPT and WDT in a SWIPT-enabled network [9, 10].

In fact, WPT presents a viable solution for facilitating sustainable
communication networks serving energy-limited communication devices in which
wireless devices can communicate through electromagnetic waves in the RF band.
In an RF energy harvesting mode, the range from 3kHz to as high as 300GHz is
designated for radio signals to carry energy as electromagnetic radiation.
Thus, by recalling that the RF signals carry both information and energy, RF
energy radiated by the transmitter(s) can be recycled at the receivers to
prolong the network’s lifetime [11].

In order to be able to harvest energy from an RF source, a general network
architecture aided with a means of harvesting modules must be present. Figure
(1.1), which was adapted directly from [12], illustrates the block diagram of
a network equipped with an RF EH device. As can be seen in figure (1.1(a)),
the application module provides the data to be processed by a low-power
microcontroller, while the low-power transceiver is employed to either
transmit the processed information or to receive data for further processing.
The next major component in figure (1.1(a)) is the RF energy harvester that
collects RF signals and converts them into electricity. The converted signal
then goes to a power management module, which either stores the electricity
obtained from the RF energy harvester in an energy storage device, such as a
rechargeable battery, thus helping the users to save excess energy for future
use (harvest-store-use mode), or uses it directly to transmit information
without saving the energy (harvest-user mode) [13].

(a) General network architecture for energy harvesting

(b) RF energy harvester

Figure 1.1: The block diagram of an energy harvesting device [10].

Figure (1.1(b)) illustrates an RF harvester device with input from an RF
antenna, an impedance matching circuit, a voltage multiplier, and a capacitor
to create the output. It is worth mentioning that an RF energy harvester
typically operates over a range of frequencies: The RF antenna that provides
input to an RF EH unit can be designed to work on either single or multiple
frequency bands, facilitating the energy harvesting from single or multiple
sources at the same time. To maximize the power transferred from the antenna
to the voltage multiplier, an impedance matching in the form of a resonator
circuit operating at the designed frequency, is utilized. Figure (1.1(b))
shows the diodes of the rectifying circuit that are the main component of the
voltage multiplier that converts RF signals into direct current (DC) voltage
levels that can be used to load an electronic circuit, where the capacitor
ensures that the energy generated is smoothly delivered to the load [14, 15,
16, 17]. Since RF signals carry both energy and information, an RF energy
harvesting device like that shown in figure (1.1), could theoretically also
simultaneously perform information decoding from the same RF signal input
using the same antenna or antenna array. This concept has been defined as
SWIPT.

###  1.2 Energy Harvesting Modes

Future generation wireless networks not only have limited spectrum resources
but also must operate with low-power batteries. Recently, energy-efficient
communication systems, or “green radios”, have been increasingly attracting
attention from the research community due to their ability to improve system
performance while simultaneously diminishing the energy consumption of the
communication devices [18]. Reducing wireless network energy consumption is
not only essential for prolonging battery lives but also crucial for the
environment.

Although ICT is to blame for more than $2$ percent of CO2 emissions worldwide,
it also presents solutions for drastically diminishing the remaining $98$
percent of CO2 emissions [19]. To significantly minimize ICTs’ carbon
footprint and environmental impact, we need new and efficient communication
techniques [20]. In recent literature, energy efficiency (EE), which measures
the number of bits communicated per unit of energy consumed (bits/joules
delivered to the receivers), has emerged as the performance metric to evaluate
a communication system’s energy consumption and guarantee green communication
[21, 22]. However, today’s galloping development of wireless communication
technologies is increasing energy consumption and carbon emissions full tilt,
further aggravating environmental concerns. According to [23], the percentage
of global carbon emissions due to ICTs is estimated to reach $5$ percent by
the end of 2020, and the situation will only be exacerbated in coming years
with the arrival of beyond 5G- and sixth generation (6G)-enabled networks.

In this regard, networks that harvest energy can definitely decrease the
carbon consumption of high data-rate wireless systems by exploiting energy
from the environment [24]. In communication devices, energy harvested from RF
signals is a random parameter that depends on the channel fading coefficients
using circuitry like that discussed in the previous section [4, 25]. Wireless
communication energy harvesting by means of an RF EH device can be explored in
a variety of ways, among which, the three most common configurations – WPT,
WPCN, WPBC, and SWIPT – will be discussed below.

####  1.2.1 WPT

As shown in figure (1.2) in this model, RF energy is transferred in the
downlink (DL) direction. A WPT-enabled network includes a transmitter
connected to the main power system, such as a base station (BS) or an access
point (AP), that supplies power to an electrical component (or a portion of a
circuit that consumes electrical power) without any wired interconnections.
The WPT-enabled network uses electromagnetic waves in the surrounding
environment that can be obtained from near-field (non-radiative) or far-field
(radiative) regions to power electrical components [26]. In general, near-
field techniques do not support the mobility of an energy harvesting device.
Therefore, it is preferable to transfer information through a far-field RF
band in which the distance is much greater than the diameter of the transmit
antenna, instead of using methods such as inductive coupling, capacitive
coupling, or magnetic resonant coupling for a near-field that corresponds to
the area of just one wavelength of the transmitting antenna [27].

WPT-enabled RF signals are anticipated to engender lots of applications and
opportunities by providing cost-efficient, predictable, dedicated, on-demand,
perpetual, and reliable energy supplies to energy-constrained wireless
networks, where no wires, contacts, or batteries are needed. Numerous research
activities are laying the groundwork for the future of wireless networking
that transcends conventional communication-centric transmission. For instance,
the authors in [25] proposed a randomly deployed power-beacon-based hybrid
cellular network that wirelessly powers mobile users as they recharge their
devices. In [28], the total outage probability of an ad-hoc network overlaid
with power-beacon was analyzed. The authors employed the stochastic geometry
method in a harvest-store-use mode to study network performance in terms of
the power and channel outage probability. In [29], multi-user multiple-input
multiple-output (MIMO) WPT was considered with a new channel learning method
that requires only one feedback bit from each EH receiver to the power
transmitter per feedback interval. The power transmitter uses the feedback
information to coordinate the transmit beamforming in subsequent training
intervals and concurrently obtains updated estimates of the MIMO channels to
different EH users by solving an optimization problem. Building on that, [30]
presented an adaptive directional WPT methodology for prolonging the lifetime
of a WSN by offering a sustainable power supply to the distributed sensor
nodes. Although today’s wireless networks were designed solely for
communication purposes, with their adoption of new technologies, they are
evolving toward 6G. Nonetheless, WPT development is still in its infancy and
has not even entered its first generation: No single standard has yet been
released on far-field WPT. The WPT framework will be particularly suitable for
future wireless networks with ubiquitous and autonomous low-power and energy-
limited devices, massive IoT connections, and D2D communications.

Figure 1.2: The WPT system model.

####  1.2.2 WPCN

Wirelessly powered communication networking (WPCN) is a new networking
paradigm in which batteries for wireless communication devices can be remotely
provisioned through microwave WPT technology. Figure (1.3) shows WPCN approach
of transferring energy in the downlink (DL) direction while information is
communicated in the uplink (UL) direction. The receiver is a low-power device
that harvests energy in the DL, and then uses the harvested energy to transfer
data in the UL. WPCN can effectively reduce cost and enhance communication
performance by eliminating the battery charge limit. In addition, WPCN enjoys
full control over its power transfer: Transmit power waveforms can be adjusted
to provide a permanent energy supply under varying physical conditions and
service requirements [31].

WPCN is intended to strike a balance between energy supply limitations and
data transmission in order to optimize communication network performance. WPCN
could be suitable for a variety of low-power applications (up to several
milliwatts) such as wireless sensor WSNs, IoT, and RF Identification (RFID)
networks. WPCN enables low-power applications to operate longer while actively
transmitting at much larger data-rates and from a greater distance than
conventional backscatter-based communications [32].

Figure 1.3: The WPCN system model.

WPCNs promise to significantly enhance performance, but building an efficient
WPCN is challenging. In [33], the authors considered a hybrid AP based on the
harvest-store-use mode with a constant power supply coordinating the wireless
energy transmissions in the DL direction to a set of distributed users that
have no other energy sources. Once the users have harvested enough energy,
they send their independent information signals to the hybrid AP in the UL
direction using time-division-multiple-access (TDMA). Because of WPCN’s
distance-dependent signal attenuation in both the DL WPT and the UL WDT, a
user who is far from the hybrid AP receives less wireless energy than a user
who is closer to the DL communication. In order to have reliable information
transmission, the distant user must transmit more power in the UL direction.
This phenomenon is known as the doubly near-far problem. [34] studied an
optimal resource allocation policy in which WPCN had simultaneous WPT in the
DL and WDT in the UL. They addressed the doubly near-far problem in which the
hybrid AP operates in full-duplex. The authors in [35] considered multiple-
input single-output (MISO) WPCN, in which the single-antenna users harvest
energy from a multi-antenna AP in DL direction and then retransmit information
to the AP in the UL direction – using the TDMA or the spatial division
multiple access (SDMA) technique. With multiple antennas, the AP can utilize
energy beamforming in the DL and employ the multiplexing-gain or receive
beamforming-gain in the UL direction [36]. In addition to all these techniques
in various directions of research, WPCNs continue to present new research
problems for future applications.

####  1.2.3 WPBC

Although WPT and WPCN offer many advantages, these EH schemes still face
critical limitations when adopted in low-power low-cost networks, such as WSN,
RFID, IoT, and D2D applications. On one hand, in WPTs, users only harvest
energy with no RF data transmission or reception, whereas in WPCNs, users may
need lots of time to harvest the RF energy needed for data transmission, which
limits the system’s performance. To overcome these deficiencies, wirelessly
powered backscatter communication (WPBC) networks, also known as ambient
backscatter communication systems (ABCSs), are proposed as an alternative that
can significantly improve network performance. This innovative technique
facilitates ubiquitous communication: Devices can interact among themselves at
unprecedented scales and in previously inaccessible locations by using
existing ambient RF signals, rather than generating their own radio waves
[37]. In WPBCs, energy is transferred to a tag in the DL direction, and
information is transferred to a reader in the UL direction, as shown in figure
(1.4). More specifically, the information on a tag (a tagged object) is
communicated to a nearby reader (e.g., an RFID reader) via backscatter
modulation (tag-to-reader UL). Since tags do not require oscillators to
generate carrier signals and do not have any dedicated power infrastructure,
backscatter communications – a passive method of communication – benefit from
much less power consumption than conventional radio communications.

Figure 1.4: The WPBC system model, where the backscatter modulation on a tag
is used to reflect and modulate the incoming RF signal in order to communicate
with a reader (e.g., a wireless router).

Quite a lot of research has been conducted on the topic of WPBC networks. In
[38], optimal resource allocation for the WPBC is investigated, where energy-
constrained users harvest energy from RF signals transmitted by a multi-
antenna source to power their future active wireless information backscatter
transmission to an AP. A theoretical trade-off in WPBC networks – between the
reliability of the backscatter communication and the harvested energy at the
tag, measured in terms of signal-to-noise ratio (SNR) at the reader – was
studied in [39]. The authors in [40] proposed a novel network architecture
that enables D2D communication to be modeled as a WPBC. In [41], the authors
aimed to integrate the WPBC and RFIDs to study perceptions of power and
spectral efficiency with respect to energy constraints. They did this by
deriving the diversity-multiplexing trade-off for RFID MIMO channels.
Notwithstanding the progress gained through implementing WPBC networks,
diverse issues such as security, reliability, the data-rate of communications,
and the development of a global standard still need to be addressed. That is
beyond the scope of this thesis.

####  1.2.4 SWIPT

Because RF signals can simultaneously carry information and energy, an
intriguing new research domain has recently developed from multiple WPT
technologies. What is called a simultaneous wireless information and power
transfer (SWIPT) is a promising technique for wireless communication systems
[8]. The SWIPT paradigm enables energy-constrained wireless user equipments
(UEs) to harvest energy and process the information simultaneously by
utilizing RF signals transmitted from a BS, mobile BS (e.g., a drone), or AP.
In this model, energy and information signals are simultaneously transferred
in the DL direction from one or multiple BSs or APs to one or multiple
receivers for simultaneous information decoding (ID) and energy harvesting
(EH). The ideal receiver for enabling SWIPT has circuitry to perform ID and EH
at the same time [2], rather than two dissimilar types of circuitry to perform
EH and ID separately at the receiver [42].

It is also worth noting that performing EH and ID operations at the same time
does not necessarily mean that these operations are carried out on the same
received signal: That is practically impossible since the information content
of the RF domain signals would be entirely destroyed by harvesting energy on
the signals. Furthermore, a single antenna receiver may not be able to create
a reliable energy supply due to its limited resources for collecting energy.
Enabling SWIPT requires using separate antennas for both the EH and ID
receivers, or splitting the received RF signal into two separate parts, one
for EH and the other for ID operations, by using a splitter.

EH and ID receivers for enabling SWIPT can be classified into two broad
architectural categories: separated or co-located receiver. In a separated
architecture, as shown in figure (1.5), the EH and ID receivers are two
distinct devices with separate antennas that experience different channels
from the transmitter. The EH receiver is a low-power device capable of
harvesting energy and the ID receiver is only able to process data. Since the
ability to harvest energy deteriorates with distance, EH receivers ought to be
closer to the BS or AP than ID receivers (which have to be spatially
separated). This explains why an inner radius and outer radius are used to
distinguish between EH and ID receivers in figure(1.5). On the other hand,
with co-located SWIPT architecture, each receiver gets identical channels from
the transmitter and is a low-power device that can perform EH and ID at the
same time, as illustrated in figure (1.6).

Figure 1.5: The separated SWIPT system model.  Figure 1.6: The co-located
SWIPT system model.

(a) Separated receiver architecture.

(b) Time switching (TS) approach to realize co-located SWIPT architecture.

(c) Power splitting (PS) approach to realize co-located SWIPT architecture.

(d) Antenna switching (AS) approach to realize co-located SWIPT architecture.

Figure 1.7: Integrated receiver architecture designs for SWIPT.

Three practical approaches to designing co-located receiver architecture for
SWIPT are time switching (TS), power splitting (PS), and antenna switching
(AS). The EH and ID receivers share the same antennas to realize the co-
located receiver architecture, as shown in figure (1.7). The receiver in the
TS approach in figure (1.7(b)) includes an EH module, an ID module, and a
switch to periodically adjust the receiving antenna for particular operations.
The receiver switches between EH and ID modes based on a pre-defined, but
optimizable, time factor or TS sequence. The TS approach necessitates careful
information/energy scheduling and accurate time perception. In the PS approach
in figure (1.7(c)), the receiver divides the received signal into two streams
of different power levels for EH and ID operations based on an optimizable PS
ratio.

Finally, figure (1.7(d)) shows how the receiver is equipped with independent
antennas for EH and ID operations in the antenna switching (AS) approach to
enable SWIPT by means of a low complexity AS algorithm. In general, an antenna
array is configured at the receiver end to take advantage of spatial
multiplexing which divides the antennas into two subsets for EH and ID
operations. One subset of antennas operates on the EH mode while the rest
executes the ID operation. It should be stressed that the AS approach is
somewhat easier and more suitable for practical SWIPT architecture designs
than the TS and PS approaches [43]. In addition, the AS approach can be
similarly adopted to optimize the separated receiver architecture as shown in
figure (1.7(a)) [44].

###  1.3 A SWIPT Literature Review

In this section, we review some of the relevant work in literature on SWIPT,
including multi-carrier SWIPT systems, SWIPT in cognitive radio networks,
cooperative relaying in SWIPT networks, and multiple antenna communication in
SWIPT systems. All the application areas associated with SWIPT apply the far-
field WPT technique for transferring power within communication systems.

####  1.3.1 Multi-Carrier SWIPT Systems

The basic idea of multi-carrier modulation is to divide the transmitted
bitstream into several smaller blocks, each containing the original bitstream,
that are sent over many different subcarriers. Under ideal propagation
conditions, the subcarriers are orthogonal. In order to alleviate the effect
of inter-symbol interference (ISI) on each subcarrier, each subcarrier should
have a bandwidth lower than the channel coherence bandwidth in a multi-carrier
network. Orthogonal frequency division multiplexing (OFDM) is one of the well-
established, discrete implementations of a multi-carrier scheme for high data-
rate wireless communications that is embraced in various standards such as
IEEE 802.11a/g/n, IEEE 802.15 ultrawideband, WiMAX, and 3GPP-Long Term
Evolution (LTE). This stimulated us to study how SWIPT can manage high date-
rate interference-free communication in multi-carrier-based systems.

In this regard, [4] investigated SWIPT over a single-user OFDM channel in
order to obtain the optimal trade-off between the achievable data-rate and the
transferred power given a specific total available power. The authors studied
a power allocation algorithm design assuming that the receiver is capable of
using one received signal to simultaneously perform EH and ID operations. A
heuristic algorithm was proposed in [45] in a study of resource allocation
policies for maximizing the harvested energy for a single user in an OFDM
SWIPT system. In [46], the EE optimization problem for OFDM-based 5G wireless
networks with SWIPT was studied, with subcarrier and power allocation jointly
optimized to maximize the system EE for single-user and multi-user cases using
the Dinkelbach iterative and the Lagrange dual methods. The authors in [47]
investigated the sum data-rate maximization problem in a multi-user OFDM
system with SWIPT capability that allows the user to either perform an ID
operation when it is active (served by the transmitter) or to be in an EH mode
when it is idle – but not simultaneously. In order to realize SWIPT in a
broadband system with transmit beamforming and PS-receiver architecture, [48]
presented a novel strategy for designing power control algorithms for both
single-user and multi-user OFDM systems: These exploit channel diversity to
concurrently enhance throughput and WPT efficiency in DL and UL directions
with variable and fixed coding rates. In [49], the weighted sum data-rate
optimization problem was studied taking into account TS and PS approaches in a
co-located architecture. In [49], the optimal design for SWIPT in DL OFDM
systems was investigated with users performing EH and ID operations on the
same signals received from a fixed AP. Studying a system model very similar to
that in [49], the authors of [50] sought to minimize the fraction of all
users’ outage when constrained by both minimum harvest energy and total
transmit power.

Two types of multiple access schemes – time division multiple access (TDMA)
and orthogonal frequency division multiple access (OFDMA) – are studied for
transmitting information in SWIPT networks in particular. TDMA permits several
users to share the same frequency channel by dividing the signal into
different non-overlapping time slots. Since TDMA occupies the entire available
bandwidth, ISI mitigation techniques are required to handle interference. On
the other hand, OFDMA technology is a promising solution for effectively
dividing the available bandwidth into orthogonal sub-channels so they can be
flexibly allocated among existing users. Although in OFDMA based networks,
“intra-cell” interference does not exist, “inter-cell” interference arising
from neighboring cells deteriorates the OFDMA networks’ overall performance.
In this respect, resource allocation is highly significant in TDMA/OFDMA-based
SWIPT cellular wireless networks: It mitigates the effect of interference with
respect to limited bandwidth, stringent power constraints, and the growing
demand for wireless communication services. In [49], both TDMA and OFDMA were
examined for information transmission. For TDMA-based information
transmission, each user applies TS so that the ID operation is used throughout
the user’s scheduled information time slot and the EH operation is applied in
all other time slots. For the OFDMA-based transmission strategy, authors
assumed that the PS approach was employed at each receiver with all
subcarriers sharing the same PS ratio at each receiver. These transmission
scenarios were employed to address the problem of maximizing the weighted sum
data-rate for all users by varying the power allocation in time and/or
frequency and also the TS/PS ratios, subject to satisfying a maximum transmit
power allowance and a minimum harvested energy constraint for each user.
Following Dinkelbach’s method, the authors in [51] investigated the secrecy EE
maximization problem by proposing power allocation and PS optimization
algorithms in an OFDMA SWIPT network with two users, where each user uses the
PS receiver approach to enable SWIPT. The resource allocation algorithm design
for EE optimization was studied in [52], in which an EE maximization problem
jointly optimizes subcarrier and power allocation as well as the PS ratios of
PS hybrid receivers in an OFDMA system with SWIPT. The underlying problem in
[52], which can be solved iteratively using the Dinkelback algorithm, was
expressed as a non-convex optimization problem that had to take into account
the maximum transmit power, the receivers’ minimum amount of EH power, and the
minimum data-rate requirements of both delay constrained services and the
network. Multi-carrier SWIPT is flourishing, with many exciting research
directions yet to be explored.

####  1.3.2 SWIPT in Cognitive Radio Networks

Radio spectrum underutilization results from conventional spectrum allocation
with a cognitive radio network (CRN) introduced to enable highly reliable and
efficient opportunistic spectrum access while increasing the number of
wireless devices. CRN allows secondary users to operate in the frequency bands
allocated to primary users, as long as they cause no harmful uncontrollable
interference to primary users [53]. OFDMA technology combined with a CRN can
help allocate radio resources to secondary users more efficiently because it
supports gained spectrum allocation by dividing the entire bandwidth into a
set of subcarriers. Two different types of cognitive radio based on spectrum-
sensing and full-coordination are recognized in the literature – spectral-
sensing to identify the channels in the radio frequency spectrum and full-
coordination to assess all the attributes that a wireless network or node can
be aware of during the communication process. A CRN is subdivided into two
main architectural designs: infrastructure-based and infrastructure-less. In
an infrastructure-based CRN, every unlicensed user transmits their data or
routing parameters through a central BS, AP, or a relay, while in an
infrastructure-less CRN, unlicensed users communicate with each other directly
using the existing communication protocols (e.g., in a peer-to-peer fashion)
that require no central entity. The combination of infrastructure-based and
infrastructure-less architectures is called a “hybrid” CRN.

In order to achieve both spectrum (spectral) efficiency (SE) and EE via
dynamic spectrum access in CRN, secondary users in a CRN can be accompanied by
the RF EH capability, as has been described very well in [54]. Focusing on the
trade-off between spectrum sensing, data transmission, and RF energy
harvesting, the study provides a detailed discussion of the dynamic channel
selection problem in a multi-channel EH-based CRN. The authors of [55]
investigated how hybrid EH cooperative spectrum sensing in heterogeneous CRNs
can enable self-sustaining green communications by reducing energy cost while
capitalizing on the idle spectrum. [56] analyzed maximizing the throughput of
cooperative CRNs with EH to obtain optimal time allocation between primary and
secondary users, and balancing the trade-off between EH and packet
transmission. Secrecy outage performance for MISO CRNs with EH in order to
maximize the secrecy and EE was studied in [57]. The same secrecy outage
performance for an underlay MIMO CRN by means of transmitting antenna
selection was investigated in [58, 59]. In both studies, EH from the primary
transmitter drives the transmitter of secondary users in order to improve both
EE and SE. In [60], a joint optimization framework to distribute power among
users at the secondary BS, allocate power for cooperative transmission, assign
a time slot for the transmission of each user, and find the PS ratio for EH
and ID receivers was explored for SWIPT-based cooperative CRNs. The authors of
[61] tried to maximize the data-rate for underlay multi-hop CRNs using TS
receivers. The problem of maximizing EH by taking into account an optimal
resource allocation policy while also satisfying constraints of minimum data-
rate, transmit power, and subcarrier was studied in [62] with respect to max-
min fairness in wideband CRN with SWIPT.

####  1.3.3 Cooperative Relaying in SWIPT Networks

Relay techniques are proposed for facilitating communication between the
source and the destination [63]. They offer a low-cost way of expanding
coverage enlargements, gaining diversity, and enhancing data-rates. However,
since the relay power supplies might be insufficient, it needs an additional,
possibly green, energy source to cooperate. Luckily, WPT techniques have the
potential to assist cooperative devices by recharging energy-limited relays to
use as tokens for future cooperation. Recent research on using SWIPT for self-
powered relaying has concentrated on two conventional cooperative relaying
systems: amplify-and-forward (AF) and decode-and-forward (DF) [64].

The past few years have witnessed new research on cooperative relaying
integrated with SWIPT networks. The authors of [65] studied a cooperative
relaying SWIPT network with a bidirectional relay capable of transferring
wireless power in the DL direction and relaying information by AF in the UL
direction. Two approaches were examined with PS and TS at the relay to
maximize the user’s data-rate. Energy harvesting and power consumption
constraints at both the relay and the user were noted. In [66], the authors
considered a cooperative PS-based SWIPT network with one source-destination
pair and multiple EH DF relays that were intended to obtain a closed-form
formulation of the outage probability achieved by the multi-relay cooperative
protocol as well as its approximation at high SNR. PS-based SWIPT in
cooperative networks with spatially random DF relays was studied in [67] in
order to characterize the outage probability and diversity gain by applying
the stochastic geometry principle. The optimization problems of PS ratios at
the relays were formulated in [68] for both DF and AF relaying protocols in a
multi-relay two-hop relay SWIPT system. Joint relay selection and resource
allocation optimization for EE maximization was studied in [69] in order to
provide a performance comparison between DF and AF relays without external
power supplies in PS-enabled SWIPT. The author of [70] investigated whether a
joint resource allocation would maximize the achievable data-rate in PS-based
and AS-based SWIPT for a two-hop cooperative transmission in which a half-
duplex multi-antenna relay adopts DF relaying strategy. Relay selection
strategies for both PS- and TS-based SWIPT in a cooperative AF relaying
network were studied in [71]. The authors tried to maximize either the amount
of EH at the user’s end under the constraint of the minimum available data-
rate in a PS-based SWIPT network or the overall user data-rate while
guaranteeing a minimum EH in a TS-based SWIPT network. A novel relay selection
and resource allocation in a two-hop relay-assisted multi-user PS-based SWIPT
OFDMA network were analyzed in [72] to optimize the subcarrier assignment,
power allocation, and users’ PS ratios as well as relays to ensure
maximization of the system’s data-rate while also satisfying minimal EH and
maximal transmit power constraints.

####  1.3.4 Multiple Antenna Communication in SWIPT Systems

Nowadays, new communication technologies can incorporate multiple antennas to
increase data-rate through multiplexing, improve performance through
diversity, and use WPT techniques to generate sufficient power for reliable
communication. These approaches require centralized or distributed antenna
array deployment at the transmitter and/or receivers to achieve substantial
array/capacity gains over single-input single-output (SISO) systems through
spatial beamforming/multiplexing [73]. Multiple-input single-output (MISO) and
MIMO are regarded as proper ways to boost the reliability and capacity of
SWIPT-enabled communication networks. Most work that has tried to integrate
SWIPT in multiple antenna wireless networks assumes that two defined user
groups need to be served, one for receiving information and the other for
receiving power to recharge their power sources. However, there are also cases
of co-located receiver architecture. The following paragraphs present
interesting research studies conducted on SWIPT in MISO and MIMO systems to
help the reader understand the challenges of multiple antenna configurations
in SWIPT.

In [74], the capacity region of a MISO broadcast channel featuring SWIPT was
evaluated. The approach used was based on solving a sequence of weighted sum
data-rate maximization problems subject to a maximum transmit power constraint
for the AP, and a set of minimum harvested energy constraints for individual
EH receivers. In this study a multi-antenna AP simultaneously delivers
information and energy via RF signals to multiple single-antenna receivers in
a separated receiver architecture. The authors of [75] studied joint power
allocation and PS ratio optimization for a multi-user MISO SWIPT network with
the aim of maximizing the users’ minimum signal-to-interference plus noise
ratio (SINR) under the maximum transmit power and the minimum energy
harvesting constraints.

In [76], a multi-user MISO full-duplex system with PS-based SWIPT was proposed
for a resource allocation policy design that jointly optimizes the PS ratios,
the beamforming matrix, and the transmit power – subject to satisfying maximal
SINR and harvested power constraints. A novel beamforming design to minimize
transmission power in a multi-user MISO SWIPT system was proposed in [77] for
TS and PS receiver architectures. Joint transceiver optimization of
beamforming vectors and TS ratios for MISO SWIPT systems was examined in [78].
In [79], the joint design of the beamforming vector and the artificial noise
covariance matrix for a MISO SWIPT with multiple eavesdroppers was studied by
analyzing a proportional secrecy EE maximization problem.

Two different scenarios for the MIMO broadcast system were investigated in
[80], namely, separate and co-located receivers architecture in which all the
transmitters and receivers were equipped with multiple antennas. In the first
scenario (separated receivers), the best transmission strategy for MIMO SWIPT
systems was designed to attain different trade-offs between maximum
information data-rate and energy transfer in the boundary of a so-called rate-
energy region. For the other scenario (co-located receivers), TS and PS
approaches were applied to determine an outer bound for the achievable rate-
energy region. The authors of [81] similarly described the trade-off between
maximum energy transfer versus maximum information data-rate under the
nonlinear EH model. A novel power allocation algorithm for SWIPT-enabled
multi-user MIMO DL systems with separated receiver architecture was proposed
in [82], based on the block diagonalization precoding technique that
completely suppresses multi-user interference while maximizing a network’s
data-rate.

In [83], the problem of antenna selection (at the transmitter) and transmit
covariance matrix design was studied in the effort to jointly maximize the
data-rate of a MIMO broadcast system with SWIPT, given the multidimensional
trade-off between the minimum data-rate requirement of ID users and the
minimum energy harvesting of EH users in a separated architecture. A general
multi-objective optimization problem, in which data-rate and harvested power
are simultaneously optimized for each user, is proposed in [84] for a multi-
user MIMO network implementing SWIPT. A secrecy data-rate maximization problem
for SWIPT in cognitive MIMO networks was studied in [85]: An interference
power constraint was imposed to protect the primary user while the secondary
EH receiver had a minimum energy harvesting constraint. Unlike the MIMO SWIPT
systems, the weighted minimum mean squared error criterion – rather than the
data-rate of the network – was investigated in [86] for a separated SWIPT
architecture. The secure transmission issue for PS-enabled SWIPT in a multi-
user MIMO system with multiple external eavesdroppers is presented in [87].
Robust beamforming with imperfect channel state information (CSI) was designed
with the maximum transmission power optimized subject to a minimum achievable
secrecy data-rate and EH. EE maximization in SWIPT-based MIMO broadcast
channels for IoT applying the TS approach was studied in [88] as a way of
obtaining resource allocation policies that consider per-user minimum
harvested energy constraints.

###  1.4 Thesis Overview and Contributions

Although studies have thus far concentrated on solving problems in wireless
communication networks empowered by WPT technologies, various problems in the
field remain to be answered. In this thesis, we attempt to fill in several
crucial gaps in the literature by designing, analyzing, and optimizing
resource allocation problems with different objectives for SWIPT in multi-
service energy-constrained wireless networks. The overview and specific
contributions of the main chapters are:

Chapter 2 Optimization Techniques

In our study, we refer to several basic optimization techniques. Chapter 2
discusses them and provides some approaches to optimization.

Chapter 3 SWIPT in Single Small-Cell Networks

In chapter 3, we provide a novel architecture for harvesting energy from an AP
without needing a splitter at the receiver. We propose a new system model in
which a designated portion of the spectrum is used for ID operation while
another portion is exploited for EH operation, and investigate how much
performance is gained. The main objective of this chapter is to design a
resource allocation policy that maximizes the harvested energy of a multi-user
DL OFDMA network with SWIPT while satisfying a minimum data-rate requirement
for all users. We then use optimization tools to obtain a locally optimal
solution for the underlying problem, which is essentially non-convex.

The results of this chapter with slight improvements to the system model, as
presented in the extended abstract, will be submitted to IEEE Communications
Letters:

[2.5em]0em ——, “Optimal Resource Allocation for MC-NOMA in SWIPT-enabled
Networks,” to be submitted to IEEE Communications Letters.

Chapter 4 SWIPT in Multi-Cell Networks

In chapter 4, we study allocating resources to maximize the data-rate based on
the separated receiver architecture in a SWIPT OFDMA multi-user multi-cell
system. We distribute the users between the inner and outer cells. Nearby
users can harvest energy from the AP; users at a distance get their wireless
information from the nearest AP. Although it seems that the AP supports EH
users by consuming some power, in this chapter, we discuss whether it is
justifiable in light of the fact that this might cause the performance gain of
a multi-user multi-cell OFDMA network to deteriorate. The resulting problem,
which jointly optimizes the subcarrier assignment and the power allocation, is
an intractable mixed-integer non-linear problem. Because of that, we use a
minorization maximization approach based on the difference of convex functions
programming with a surrogate function to approximate the non-convex objective
function.

Chapter 5 Antenna Selection Technique in SWIPT

Finally, in chapter 5, we describe a new harvesting technique at the receiver
that is based on the receiver antenna selection for a multi-user multi-cell
SWIPT OFDMA system with a co-located architecture. This we call a “generalized
antenna switching technique”. As a performance metric, we optimize EE, which
is highly appreciated for investigating resource allocation in the next
generation of wireless communication. The underlying problem in this chapter
is non-convex because we incorporate both interference and integer variables.
We relax the integer variable, and then apply the big-M formulation to make
sure that relaxed variables take binary values. After that, we use the
minorization maximization approach employing a first-order Taylor
approximation. As a last step to convexify the EE optimization problem, we
apply the Dinkelback algorithm to transform the objective function into a non-
fractional function. The simulation result concludes the amount of performance
gain that can be obtained through generalized antenna switching architecture.

The results of this chapter with an improvement in the system model will be
submitted to IEEE JSTSP (Special Issue on Signal Processing Advances in
Wireless Transmission of Information and Power). A very simplified version of
the system model without SWIPT has already been accepted by the VTC 2020
conference:

[2.5em]0em Jalal Jalali, Ata Khalili, and Heidi Steendam, “Antenna Selection
and Resource Allocation in Downlink MISO OFDMA Femtocell Networks,” accepted
for presentation at IEEE VTC 2020.

[2.5em]0em ——, “Simultaneous Wireless Information and Power Transfer via a
Joint Resource Allocation and Generalized Antenna Switching Strategy,” to be
submitted to IEEE JSTSP.

Chapter 6 Conclusion and Future Work

Chapter 6 summarizes the results and provides suggestions for future research.

##  Chapter 2 Optimization Techniques

In this chapter, we introduce some basic properties of convex functions that
can be useful for better understanding this thesis [89].

###  2.1 Convex Analysis

####  2.1.1 Definitions

Let $f(\text{·})\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a convex function. Then,
$f(\text{·})$ is convex, if for each $\lambda\in[0,1]$, we have

$f(\lambda\mathbf{x}_{1}+(1-\lambda)\mathbf{x}_{2})\leq\lambda
f(\mathbf{x}_{1})+(1-\lambda)f(\mathbf{x}_{2}),$ (2.1)

for all $\mathbf{x}_{1}$, $\mathbf{x}_{2}\in\mathbb{R}^{n}$. Geometrically
speaking, the above inequality states that the line segment between
$(x_{1},f(x_{1}))$ and $(x_{2},f(x_{2}))$, which is the chord from $x_{1}$ to
$x_{2}$, lies on top of the graph of $f(\text{·})$. A function $f(\text{·})$
is said to be strictly convex if (2.1) holds with strict inequality. Moreover,
supposing that $f(\text{·})$ is differentiable, i.e., its gradient exists,
then each convex function satisfies the following inequality

$f(\mathbf{{x}})\geq
f(\tilde{\mathbf{{x}}})+\nabla_{\mathbf{x}}f(\tilde{\mathbf{x}})^{T}(\mathbf{x}-\tilde{\mathbf{x}}),$
(2.2)

where $\nabla_{\mathbf{x}}$ is the gradient vector with respect to
$\mathbf{x}$ at $\tilde{\mathbf{x}}$, and ${\square}^{T}$ is the transpose
operation on ${\square}$. The inequality (2.2) asserts that for a convex
function, a global underestimator of the function can be easily derived via
its first-order Taylor approximation. Consequently, the first-order Taylor
approximation of a convex function is always a global underestimator of the
function. This inequality additionally confirms that global information of a
convex function can be obtained through its local information, i.e., its value
and derivative at a point.

###  2.2 Duality Theorem

We consider the following optimization problem, also known as the primal
problem, written in its general form as

$\displaystyle\min_{\mathbf{x}\in\mathcal{X}}\leavevmode\nobreak\
f(\mathbf{x})$ (2.3) $\displaystyle s.t.\mathrel{\mathop{\mathchar
58\relax}}\leavevmode\nobreak\ $ $\displaystyle g_{i}(\mathbf{x})\leq
0,\leavevmode\nobreak\ \forall i=1,...,I,$ $\displaystyle
h_{l}(\mathbf{x})=0,\leavevmode\nobreak\ \forall l=1,...,L,$

where $f(\text{·})\mathrel{\mathop{\mathchar 58\relax}}\leavevmode\nobreak\
\mathbb{R}^{n}\rightarrow\mathbb{R}$ is the objective function, and
$\mathbf{x}\in\mathbb{R}^{n}$ is the vector of optimization variables inside
the feasible set $\mathcal{X}$. This optimization problem has $I$ inequality
constraints and $L$ equality constraints. Furthermore, we also refer to
$p^{*}$ as the optimal value of the optimization problem in (2.3).

The Lagrangian duality of the objective function of (2.3) is given by

$\mathcal{L}(\mathbf{x},\boldsymbol{\mu},\boldsymbol{\nu})=f(\mathbf{x})+\sum_{i=1}^{I}\mu_{i}g_{i}(\mathbf{x})+\sum_{l=1}^{L}\nu_{l}h_{l}(\mathbf{x}),$
(2.4)

where $\boldsymbol{\mu}$ and $\boldsymbol{\nu}$ are called the vector of
Lagrangian multiplier or the dual variables with respect to inequality and
equality constraints associated with the problem (2.3) that have $\mu_{i}$’s
and $\nu_{l}$’s as the elements of the corresponding vectors. The essential
purpose of Lagrangian duality is to get somehow rid of the constraints in
(2.3) by adding a weighted sum of the constraint functions to the objective
function. We can now define the corresponding Lagrange dual function (or just
dual function), which is formally stated as

$\mathcal{D}(\boldsymbol{\mu},\boldsymbol{\nu})=\inf_{\mathbf{x}}\mathcal{L}(\mathbf{x},\boldsymbol{\mu},\boldsymbol{\nu}).$
(2.5)

Note that even though the primal problem could be non-convex, the dual problem
is always a convex optimization problem since the dual function is a point-
wise infimum. This infimum can be seen as the greatest lower bound of a family
of affine functions with respect to $\boldsymbol{\mu}$ and $\boldsymbol{\nu}$.

The Lagrange dual function in (2.5) gives us a lower bound on the optimal
value $p^{*}$ of the primal problem (2.3). In order to find the best lower
bound for the primal problem, the following optimization problem can be
defined from the Lagrange dual function

$\max_{\boldsymbol{\mu},\boldsymbol{\nu}}\leavevmode\nobreak\
\mathcal{D}(\boldsymbol{\mu},\boldsymbol{\nu}).$ (2.6)

This problem is known as the Lagrange dual problem corresponding to the primal
problem. Moreover, if $\boldsymbol{\mu}^{*}$ and $\boldsymbol{\nu}^{*}$ are
the optimal values for the Lagrange dual problem in (2.6), they are
traditionally called dual optimal or optimal Lagrange. It should also be noted
since the objective to be maximized is concave in (2.6), the Lagrange dual
problem is a convex optimization problem no matter the primal problem in (2.3)
is convex or not.

####  2.2.1 Weak Duality and Duality Gap

Let $\mathbf{x}^{*}$ be a feasible solution for the primal problem, i.e.,
$p^{*}$ and $(\boldsymbol{\mu}^{*},\boldsymbol{\nu}^{*})$ are a feasible
solution to the dual problem, that is, $d^{*}$. According to weak duality, we
have the following inequality for a general (possibly non-convex) problem

$d^{*}\leq p^{*}.$ (2.7)

It must be noted that the weak duality inequality also holds when $d^{*}$ and
$p^{*}$ are infinite. On the other hand, the difference between the primal
optimal value and dual primal value, i.e., $p^{*}-d^{*}$ is called the optimal
duality gap. It should be stated that the optimal duality gap is always non-
negative. Since the dual problem is always convex, and often can be solved
efficiently to determine $d^{*}$, the inequality in (2.7) is quite useful in
finding a lower bound on the optimal value of a problem that is difficult to
solve.

####  2.2.2 Strong Duality and Slater condition

If the duality gap is zero, i.e., $p^{*}=d^{*}$, the strong duality holds. The
strong duality indicates that the best bound that can be achieved from the
Lagrange dual function is tight. Moreover, in strong duality, since the gap
between primal and dual is zero, solving the dual problem is equivalent to
solving the primal problem.

A sufficient condition for strong duality to hold for a convex optimization
problem is the Slater condition or Slater’s condition. In particular, if the
Slater condition holds for the primal problem, then the duality gap is zero,
which implies strong duality for convex problems. And if the dual optimal
value is finite, then it is attained, i.e., a dual feasible
$(\boldsymbol{\mu}^{*},\boldsymbol{\nu}^{*})$ exists that satisfies
${\mathcal{D}(\boldsymbol{\mu}^{*},\boldsymbol{\nu}^{*})=d^{*}=p^{*}}$. In
general, there exist so many results that establish conditions on the
optimization problem, that yield strong duality. These conditions are coined
constraint qualifications, where the Slater condition is only a simple
specific example of many.

###  2.3 Abstract Lagrangian Duality

In order to study duality in optimization models, two approaches exist
historically, and the duality results are manifested as referred to as: i)
Classical Lagrangian and ii) Abstract Lagrangian. Among these two forms, the
classical Lagrangian form is more extensively used in the literature. What we
have discussed so far is indeed the classical Lagrangian form of duality. As
seen, classical Lagrangian typically starts from a primal problem while the
Lagrangian and the Dual Lagrangian problems are established subsequently.
However, at a more abstract level, an abstract Lagrangian function is used to
derive the primal and dual optimization problems. Here, we briefly discuss an
abstract version of Lagrangian duality that is elaborated in more significant
details in [90]. In this version, through a certain real-valued abstract
Lagrangian function, the primal and dual costs are taken into account, such
that

$\displaystyle(\textrm{Primal problem})$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \min_{\mathbf{x}\in\mathcal{X}}\leavevmode\nobreak\
\mathcal{F}(\mathbf{x})\leavevmode\nobreak\ \leavevmode\nobreak\
\textrm{where}\leavevmode\nobreak\
\mathcal{F}(\mathbf{x})=\leavevmode\nobreak\ \sup_{\mathbf{y}\in
Y}\leavevmode\nobreak\ \mathcal{L}(\mathbf{x},\mathbf{y}),$
$\displaystyle(\textrm{Dual problem})\leavevmode\nobreak\ $
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\max_{\mathbf{y}\in\mathcal{Y}}\leavevmode\nobreak\
\mathcal{G}(\mathbf{y})\leavevmode\nobreak\ \leavevmode\nobreak\
\textrm{where}\leavevmode\nobreak\
\mathcal{G}(\mathbf{y})=\leavevmode\nobreak\
\inf_{\mathbf{x}\in\mathcal{X}}\leavevmode\nobreak\
\mathcal{L}(\mathbf{x},\mathbf{y}),$

where $\mathcal{L}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{X}\times\mathcal{Y}\longrightarrow R$ is the abstract
Lagrangian function pertaining to $\mathcal{X}$ and $\mathcal{Y}$ as
appropriate domains defined in some primal and dual spaces, respectively.
Moreover, the supremum can be seen as the least upper bound of a family of
affine functions with respect to $\mathbf{x}$ and $\mathbf{y}$. This approach
to duality is based on conjugate duality, where a convexity assumption is
always made [91]. This approach also puts a strong emphasis on the minimax and
saddle point theorems, which are given below.

$\bullet$ Minimax Theorem: This theorem provides the condition that guarantees
the strong max-min property or the saddle point as follows

$\sup_{\mathbf{y}\in\mathcal{Y}}\leavevmode\nobreak\
\inf_{\mathbf{x}\in\mathcal{X}}\leavevmode\nobreak\
\mathcal{H}(\mathbf{x},\mathbf{y})=\inf_{\mathbf{x}\in\mathcal{X}}\leavevmode\nobreak\
\sup_{\mathbf{y}\in\mathcal{Y}}\leavevmode\nobreak\
\mathcal{H}(\mathbf{x},\mathbf{y}).$ (2.8)

It should be noted that the above equality, strong max-min property, holds
only in special cases. This is, in particular, true, when for example,
$\mathcal{H}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{X}\times\mathcal{Y}\longrightarrow R$ is the Lagrangian of
a problem where the strong duality holds.

$\bullet$ Saddle Point Theorem: Under suitable conditions, there exists a
saddle point for $\mathcal{S}(\text{·})$ referred to as a pair
$(\mathbf{x}^{*},\mathbf{y}^{*})\in\mathcal{X}\times\mathcal{Y}$ such that for
all $(\mathbf{x},\mathbf{y})\in\mathcal{X}\times\mathcal{Y}$,

$\mathcal{S}(\mathbf{x}^{*},\mathbf{y})\leq\mathcal{S}(\mathbf{x}^{*},\mathbf{y}^{*})\leq\mathcal{S}(\mathbf{x},\mathbf{y}^{*}).$
(2.9)

In (2.9), $\mathcal{S}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{X}\times\mathcal{Y}\longrightarrow R$ is the Lagrangian of
a problem where the strong duality holds. In other words,
$\mathcal{S}(\textbf{x}^{*},\textbf{y}^{*})=\sup_{\textbf{y}\in\mathcal{Y}}\leavevmode\nobreak\
\mathcal{S}(\textbf{x},\textbf{y}^{*})$, and
$\mathcal{S}(\textbf{x}^{*},\textbf{y}^{*})=\inf_{\textbf{x}\in\mathcal{X}}\leavevmode\nobreak\
\mathcal{S}(\textbf{x}^{*},\textbf{y})$. This indicates that the strong max-
min property (2.8) holds with the common value of
$\mathcal{S}(\textbf{x}^{*},\textbf{y}^{*})$.

###  2.4 Complementary Slackness and KKT Optimality Conditions

Suppose that both the primal and dual optimal values exist and are equal. This
means the strong duality holds. We further assume that $\mathbf{x^{*}}$ and
$(\boldsymbol{\mu}^{*},\boldsymbol{\nu}^{*})$ to be a primal optimal and a
dual optimal point, respectively. Therefore, we have

$\displaystyle
f(\mathbf{x^{*}})=\mathcal{D}(\boldsymbol{\mu}^{*},\boldsymbol{\nu}^{*})\leq
f(\mathbf{x^{*}})+\sum_{i=1}^{I}\mu^{*}_{i}g_{i}(\mathbf{x^{*}})+\sum_{l=1}^{L}\nu^{*}_{l}h_{l}(\mathbf{x^{*}})\leq
f(\mathbf{x^{*}}).$ (2.10)

The first inequality in (2.10) holds since the infimum of the Lagrangian over
$\mathbf{x}$ is less than or equal to its value at
$\mathbf{x}=\mathbf{x^{*}}$. However, the last inequality follows from
${\mu_{i}^{*}\geq 0,\leavevmode\nobreak\ g_{i}(\mathbf{x^{*}})\leq
0,\leavevmode\nobreak\ \forall i=1,...,I}$, and ${h_{l}(\mathbf{x^{*}})\leq
0,\leavevmode\nobreak\ \forall l=1,...,L}$. An important conclusion that one
can make from (2.10) is that

$\mu^{*}_{i}g_{i}(\mathbf{x^{*}})=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \forall i=1,...,I.$ (2.11)

This condition is called the complementary slackness. It confirms that one can
go from the optimal primal solution to the optimal dual solution, and vice
versa, if the strong duality holds. Moreover, the complementary slackness
verifies that a solution is optimal, by checking if there is a dual solution.

Now, we introduce the Karush-Kuhn-Tucker (KKT) conditions assuming that all
the functions both in the objective and the constrains in (2.3) are
differentiable. Just same as was assumed in (2.10), let’s also suppose the
primal and dual variables at the optimum point, for which strong duality
obtains, are $\mathbf{x}^{*}$ and
$(\boldsymbol{\mu}^{*},\boldsymbol{\nu}^{*})$, respectively. The KKT
conditions have the following properties

$\displaystyle g_{i}(\mathbf{x}^{*})$ $\displaystyle\leq
0,\leavevmode\nobreak\ \forall i=1,...,I,$ (2.12a) $\displaystyle
h_{l}(\mathbf{x}^{*})$ $\displaystyle=0,\leavevmode\nobreak\ \forall
l=1,...,L,$ (2.12b) $\displaystyle\mu_{i}^{*}$ $\displaystyle\geq
0,\leavevmode\nobreak\ \forall i=1,...,I,$ (2.12c)
$\displaystyle\mu_{i}^{*}g_{i}(\mathbf{x}^{*})$
$\displaystyle=0,\leavevmode\nobreak\ \forall i=1,...,I,$ (2.12d)
$\displaystyle\nabla_{\mathbf{x}}f(\mathbf{x}^{*})+\sum_{i=1}^{I}\mu_{i}^{*}\nabla_{\mathbf{x}}g_{i}(\mathbf{x}^{*})+\sum_{l=1}^{L}\nu_{l}^{*}\nabla_{\mathbf{x}}h_{l}(\mathbf{x}^{*})$
$\displaystyle=0,$ (2.12e)

where $\mu_{i}^{*}$ and $\nu_{l}^{*}$ are the elements of Lagrangian vectors
$\boldsymbol{\mu}^{*}$ and $\boldsymbol{\nu}^{*}$, respectively. Also,
$\nabla_{\textbf{x}}$ denotes the gradient of a function with respect to x in
(2.12e). Note that the KKT conditions are necessary and sufficient conditions
for the optimality of the convex optimization problem with differentiable
objective and constraint functions. However, if the problem is non-convex, the
KKT conditions would only provide the necessary conditions for optimality
given that the objective and constraints are differentiable.

###  2.5 Interior-Point Methods

The literature on interior-point methods is very extensive, and research is
still flourishing. This paragraph can only serve as a very condensed
introduction. Interior-point methods can be seen as a branch in the
classification of convex optimization algorithms that solve linear and
nonlinear convex optimization problems. KKT conditions, another branch of
optimization algorithms in this classification, obtain a collection of linear
equations that can be solved analytically, for example a quadratic
optimization problem with linear equality constraints. Interior-point methods,
on the other hand, solve an optimization problem with linear equality and
inequality constraints by the relaxation of a problem with only a set of
linear equality constraints. The motivation for calling such methods an
“interior-point method” lies in the fact that these methods begin their search
for an optimal solution in the interior of the feasible region, and travel on
a path towards the boundary, converging at the optimum.

###  2.6 MM Approach and D.C. Programming

MM algorithms are an appropriate tool to reduce a given optimization problem
into a series of simpler problems. In this sense, an MM algorithm is not an
algorithm, but rather an appropriate principal way of designing optimization
algorithms for high dimensional settings, where the classical methods of
optimization do not work well. MM algorithms are not new. The celebrated
Expectation Maximization algorithm is a particular case of MM algorithms that
is extensively used in electrical engineering applications and in other
fields. The reason for selecting the MM acronym is two-fold. An MM algorithm
operates on a more straightforward and simpler surrogate function that
majorizes/minorizes (the first M of MM) the objective function in a
minimization/maximization (the second M of MM) optimization problem. Thus, the
MM stands for either Majorization Minimization or Minorization Maximization,
depending on the application. In the next few paragraphs, we consider a
majorization minimization problem to explain how the algorithm works.

Consider the following optimization problem

$\displaystyle\min_{\textbf{x}\in\mathcal{X}}\leavevmode\nobreak\ $
$\displaystyle f(\textbf{x}),$ (2.13)

where x is the optimization variable vector belonging to the feasible set
${\mathcal{X}}$. In order to majorize the function $f(\textbf{x})$ at
$\textbf{x}^{n}$, there exists a surrogate function
$g(\textbf{x}|\textbf{x}^{n})$ that satisfies two conditions

$\displaystyle f(\textbf{x}^{n})$
$\displaystyle=g(\textbf{x}^{n}|\textbf{x}^{n}),$ (2.14) $\displaystyle
f(\textbf{x})$ $\displaystyle\leq
g(\textbf{x}|\textbf{x}^{n}),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \textbf{x}\neq\textbf{x}^{n}.$ (2.15)

The first condition (2.14) is called the tangency condition at the current
iteration step. This condition grantees $g(\textbf{x}^{n}|\textbf{x}^{n})$ is
tangent to $f(\textbf{x})$ at $\textbf{x}^{n}$. The second condition, on the
other hand, (2.15) makes sure the $g(\textbf{x}|\textbf{x}^{n})$ is dominant
in a sense that it always lies above the surface of $f(\textbf{x})$ except at
$\textbf{x}^{n}$. Besides, if a function $g(\textbf{x}|\textbf{x}^{n})$
majorizes the function $f(\textbf{x})$ at $\textbf{x}^{n}$, it can be easily
perceived that -$g(\textbf{x}|\textbf{x}^{n})$ minorizes -$f(\textbf{x})$.

Another very important result of the MM algorithms is the descent property.
Starting from $\textbf{x}^{0}\in\mathcal{X}$ as an initial point for the
feasible set ${\mathcal{X}}$, an MM algorithm generates a sequence of feasible
point $\textbf{x}^{n}$. At point $\textbf{x}^{n}$ in the majorization step, a
continuous surrogate function is constructed that satisfies the domination
condition in (2.15)

$g(\textbf{x}|\textbf{x}^{n})\geq
f(\textbf{x})+g(\textbf{x}^{n}|\textbf{x}^{n})-f(\textbf{x}^{n}),\leavevmode\nobreak\
\textbf{x}\neq\textbf{x}^{n}.$ (2.16)

Hence, in the minimization step, the following update rule can be applied

$\textbf{x}^{n+1}\in\min_{\textbf{x}\in\mathcal{X}}g(\textbf{x}|\textbf{x}^{n}).$
(2.17)

It is easy to show that the generated sequence $f(\textbf{x}^{n})$ is non-
increasing. Thus, we have

$f(\textbf{x}^{n+1})\leq
g(\textbf{x}^{n+1}|\textbf{x}^{n})-g(\textbf{x}^{n}|\textbf{x}^{n})+f(\textbf{x}^{n})\leq
g(\textbf{x}^{n}|\textbf{x}^{n})-g(\textbf{x}^{n}|\textbf{x}^{n})+f(\textbf{x}^{n})=f(\textbf{x}^{n}),$
(2.18)

where the first inequality comes from (2.16), and the second inequality is the
direct consequence of (2.17). The property in (2.18), the descent property,
gives a remarkable numerical stability to MM algorithms. Hence, instead of
minimizing the cost function $f(\textbf{x})$ directly, the MM algorithms
stably optimize a sequence of tractable approximate surrogate objective
functions $g(\textbf{x}|\textbf{x}^{n})$ that minorize $f(\textbf{x})$ as
tightly as possible.

The MM algorithms can easily be connected to other algorithmic frameworks [92,
93, 94]. One of the application areas of the MM algorithms is in Difference of
Convex functions (D.C.) programming problems. The general form of D.C.
functions is

$\displaystyle\min_{\textbf{x}}\leavevmode\nobreak\
f_{0}(\textbf{x})-h_{0}(\textbf{x})$ (2.19) s.t.: $\displaystyle
f_{i}(\textbf{x})-h_{i}(\textbf{x})\leq 0,\leavevmode\nobreak\ \forall
i=1,...,m,$ (2.20)

where $f_{i}$’s and $h_{i}$’s are all convex functions. We further assume that
$f_{i}$’s and $h_{i}$’s are twice differentiable, and are strictly convex
without loss of generality according to (2.1). Among various algorithms having
desirable properties for the solution of D.C. problems, the MM scheme, which
solves a sequence of convex problems acquired by linearizing non-convex parts
in the objective function as well as the constraints, is preferred.
Accordingly, an approximate solution can be found that iteratively solves
(2.19) through defining the following convex subproblem

$\displaystyle\min_{\textbf{x}}\leavevmode\nobreak\
g_{0}(\textbf{x}|\textbf{x}^{n})$ (2.21) s.t.: $\displaystyle
g_{i}(\textbf{x}|\textbf{x}^{n})\leq 0,\leavevmode\nobreak\ \forall
i=1,...,m,$ (2.22)

where

$g_{i}(\textbf{x}|\textbf{x}^{n})=f_{i}(\textbf{x})-\Big{(}h_{i}(\textbf{x}^{n})+\nabla_{\textbf{x}}h_{i}(\textbf{x}^{n})^{T}(\textbf{x}-\textbf{x}^{n})\Big{)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \forall i\in\\{0,...,m\\}.$ (2.23)

The aforementioned approximation satisfies the MM principle and is a tight
upper bound of $f_{i}-h_{i}$ with equality achieved at
$\textbf{x}=\textbf{x}^{n}$. This technique is used several times throughout
the thesis. Moreover, the solution methodology for the MM algorithm is
summarized in Algorithm 1. A valid question to be asked at this point would be
how good the convergence behaviors of the MM algorithms are. For the answer,
the interested reader is referred to [95, 96, 97].

Algorithm 1 The MM Approach

1: $\mathbf{Initialize}$ [1em]0em iteration index $n=0$ with the maximum
number of iteration $N_{\max}$ and find a feasible point $\mathbf{x}^{0}$.

2: repeat

3: [1em]0em Find $\mathbf{x}^{n}$ by solving the optimization problem (2.17)
and store as $\mathbf{x}$.

4: [1em]0em Set $n=n+1$ and $\mathbf{x}^{n}=\mathbf{x}$.

5: until some convergence criterion is met or $n=N_{\max}$

6: return optimal $\mathbf{x}$

###  2.7 Optimization Packages

Many optimization tools and packages exist for solving any given optimization
problem. This paragraph introduces a few of the most popular optimization
packages. The GLPK is a package designated for solving large-scale linear
programming (LP), mixed integer programming (MIP), and other similar problems.
The Gurobi optimizer is a well-known commercial optimization solver for LP,
quadratic programming (QP), and MIPs (including mixed-integer linear
programming (MILP), mixed-integer quadratic programming (MIQP), and mixed-
integer quadratically constrained programming (MIQCP)). The Mosek solver is
another widely used optimization package that solves LP, QP, MIP, second-order
cone programming (SOCP), and semi-definite programming (SDP). The SeDuMi and
SDPT3 are two other leading solvers for SDPs. The list of packages goes on and
on, with new packages always being added and the well-known ones continuously
being updated to respond to the ever-rising demand for higher performance
speed to solve problems with unimaginably large dimensions. The purpose of the
thesis is not to develop such optimization solvers nor to improve them, but
rather to use the existing ones to solve the problems at hand. A whole
different research domain investigates and designs optimization solvers for
specific needs.

##  Chapter 3 SWIPT in Single Small-Cell Networks

As explained in the introductory chapter, wireless power transfer (WPT)
provides wireless devices with continuous and stable energy. Radio frequency
(RF)-enabled simultaneous wireless information and power transfer (SWIPT) has
the capability of using an innovative way to simultaneously transfer
information over a single radio waveform as power. Consequently, SWIPT-based
networks contribute exceptional benefit to users by conveniently utilizing
radio signals to transfer both energy and information. SWIPT is attracting
ever more attention due to its ability to provide green communication services
more efficiently by broadcasting information and power on orthogonal and non-
orthogonal resources. Furthermore, resource allocation for SWIPT-enabled
networks carefully takes into account the performance of both information and
power transfer, unlike traditional wireless systems. In this sense, resource
allocation in SWIPT systems can improve network performance and make maximum
use of network resources while satisfying the quality of service requirement
by flexibly allocating and dynamically adjusting the network’s available
resources.

A great deal of research has been conducted on resource allocation for SWIPT
in various types of wireless communication networks. In [48] for instance, the
focus was on resource allocation based on the orthogonal frequency division
multiplexing (OFDM) for different system configurations in a SWIPT-enabled
mobile network that maximizes throughput by using separated receiver
architecture. The optimal design for maximizing the weighted sum data-rate
over all users for SWIPT in downlink (DL) OFDM systems was explored in [49].
In it, users harvest energy and decode information using the same receiving
signals that carry both energy and information from a fixed access point (AP).
Each user also applies either power splitting (PS) or time switching (TS)
receiver architecture to coordinate the energy harvesting (EH) and information
decoding (ID) operations. The authors in [98] investigated resource allocation
for maximizing the effective capacity and effective energy efficiency (EE) of
a DL multi-user OFDM system by considering both PS and TS architectures for
receivers using SWIPT. In [52], the ideal resource allocation algorithm design
to maximize EE of data transmission was studied by considering PS hybrid
receivers in an orthogonal frequency division multiple access (OFDMA) network.
Most of the previous works had investigated a SWIPT system based on the
splitter using PS or TS architecture to separate the received signal for
either EH or ID operation. However, [45] considered a SWIPT-assisted joint
subcarrier and power allocation based on an OFDM system using a heuristic
solution methodology in which neither time nor power splitter was used to
maximize EH. In this chapter, we aim to fill the knowledge gap in existing
literature by improving the paradigm of resource allocation policy in SWIPT
networks without the need for a splitter in a co-located architecture.

We study a simple scenario often found in the literature that captures the
essential characteristics of SWIPT-enabled networks in this chapter: the
optimal design for resource allocation in a multi-user OFDMA network with
SWIPT using the same signals received from a fixed AP to perform both EH and
ID operations. The complexity of the receiver is significantly reduced because
there is no need for a splitter to perform appropriately: The receiver has no
time or power splitter. The resource allocator only needs to know which group
of subcarriers is allocated for EH and which for ID operation. This
information is derived from the channel state information (CSI) and the
relevant algorithm. We do not address the effect of interference in this
chapter; instead we investigate the problem of resource allocation design in a
single small-cell network, which allows us to address the problem of
maximizing the energy harvested by all users. More specifically, the problem
of joint power allocation and subcarrier assignment is investigated in order
to maximize harvested energy while fulfilling the minimum data-rate
requirement for each user. Since each subcarrier can be configured
independently in OFDMA systems, information and power are transferred
separately on different subcarriers with different waveforms. We thus try to
determine a resource allocation policy that includes joint power allocation
and subcarrier assignment algorithms based on an OFDMA network for a DL of a
single small-cell multi-user system by means of SWIPT. The underlying
optimization problem calls for mixed-integer linear programming (MILP). To
tackle the problem, we employ the majorization minimization (MM) optimization
approach to obtain a close-to-optimal resource allocation policy. Simulation
results demonstrate that our proposed algorithm achieves excellent performance
as compared to other possible solutions presented in the literature.

###  3.1 System Model

We consider a DL of an OFDMA network in a single small-cell scenario
consisting of an indoor AP and $K$ co-located users as shown in figure (3.1).
In particular, users receive the intended signal from the AP for EH and ID
operations, simultaneously. Furthermore, the AP and all the receivers are
equipped with a single antenna in this configuration. We additionally assume
that the entire frequency band of $\mathscr{B}$ is partitioned into $N$
subcarriers, each having a bandwidth of $\mathscr{W}$. It also needs to be
stated that a portion of the spectrum is used for ID while the remaining
portion is exploited for EH proclaiming a demand for the use of two separate
filters at receivers [99]. Moreover, the set of users and subcarriers are
denoted by $\mathcal{K}=\\{1,2,...,K\\}$ and
$\mathcal{N}=\\{1,2,...,Z,Z+1,...,N\\}$, respectively. Let
$\mathcal{N}_{i}=\\{1,2,,...,Z\\}$ denote the set of subcarriers for ID,
whereas the remaining subcarriers
$\mathcal{N}_{e}=\mathcal{N}-\mathcal{N}_{i}=\\{Z+1,Z+2,...,N\\}$ is used to
indicate the set of subcarriers for EH. Note that the optimal value of $Z$,
i.e. the cardinally of the set $\mathcal{N}_{i}$, can be obtained [45].
However, we choose not to adapt the problem of finding the optimal $Z$ in our
proposed problem design for the sake of avoiding repetition.

Figure 3.1: SWIPT in a DL of a co-located multi-user small single-cell OFDMA
network.

Moreover, it is further assumed that all the subcarriers are considered to be
perfectly orthogonal to one another, and no inter-subcarrier interference
exists. Hence, the subcarrier assignment variable is given by

$a_{n,k}=\begin{cases}1,&\text{if subcarrier $n$ is assigned to user {k}},\\\
0,&\text{otherwise}.\end{cases}$

Let $h_{n,k}$ denote the DL channel coefficient from the AP to the $k^{th}$
user over the subcarrier $n$. We assume that the perfect CSI is available at a
centralized resource allocator to design resource allocation policy.
Specifically, it is presumed that the AP broadcasts orthogonal preambles,
pilot signals, in the DL to the users. Then, through a feedback channel, each
user estimates the CSI and transfers this information back to the AP.
Afterward, the corresponding AP listens to the sounding reference signals
communicated by the users and sends the CSI to the centralized controller for
resource allocation design. Now, by denoting $p_{n,k}$ as the DL transmit
power of AP to the $k^{th}$ user over the subcarrier $n$, the DL signal-to-
noise ratio (SNR) of user $k$ in subcarrier $n$ is defined as

$\gamma_{n,k}=\frac{a_{n,k}p_{n,k}|h_{n,k}|^{2}}{\sigma^{2}_{n,k}},$ (3.1)

where $\sigma^{2}_{n,k}$ denotes the additive noise power. More specifically,
at receiver $k$, the received signal on each subcarrier is corrupted by noise
$n_{n,k}$. This noise is modeled as an additive white Gaussian noise (AWGN)
random variable with zero mean and variance $\sigma^{2}_{n,k}$, denoted by a
circularly symmetric Gaussian distribution referred to as
$n_{n,k}\sim\mathcal{CN}(0,\sigma^{2}_{n,k})$. However, for the sake of
simplicity, we consider $\sigma^{2}_{n,k}$ = $\sigma^{2}$ throughout this
chapter meaning that the variance of the noise is the same over all
subcarriers for all users. According to the Shannon capacity formula, the
data-rate of the $k^{th}$ user over the subcarrier $n$ can be expressed as

$R_{n,k}=\log_{2}\bigg{(}1+\gamma_{n,k}\bigg{)}.$ (3.2)

For facilitating the presentation, we denote $\textbf{p}\in\mathbb{R}^{1\times
KN}$ and $\textbf{a}\in\mathbb{Z}^{1\times KN}$ as vectors of optimization
problem for power allocation and subcarrier assignment, respectively.
Consequently, the data-rate of the $k^{th}$ user in DL is given as

$R_{k}(\textbf{a},\textbf{p})=\sum_{n\in\mathcal{N}_{i}}R_{n,k}.$ (3.3)

Furthermore, to guarantee the quality of service (QoS) of users, a minimum
data-rate denoted by $R_{min}$, should be provided for each user. That is

$R_{k}(\textbf{a},\textbf{p})\geq R_{min},\leavevmode\nobreak\ \forall
k\in\mathcal{K}.$ (3.4)

Moreover, the amount of the harvested energy can be stated as

$\textrm{EH}(\textbf{a},\textbf{p})=\sum_{k\in\mathcal{K}}\epsilon_{k}\bigg{(}\sum_{n\in\mathcal{N}_{e}}a_{n,k}p_{n,k}|h_{n,k}|^{2}\bigg{)},$
(3.5)

where $\epsilon_{k}$ is the power efficiency of the $k^{th}$ user capable of
energy harvesting, which takes its value from the interval $0<\epsilon_{k}<1$.
It should be noted that $\sigma^{2}$ also contributes to the EH formula in
(3.5). However, since its value is very small, it is neglected in the
representation of the EH formula.

###  3.2 Optimization Problem Formulation

The main objective persuaded in this chapter is to assign subcarrier(s) and
set the transmit power(s), for each user, such that the total harvested energy
in (3.5) is maximized. Thus, we can formulate the optimization problem as

$\displaystyle\max_{\textbf{a},\textbf{p}}\leavevmode\nobreak\
\textrm{EH}(\textbf{a},\textbf{p})$ (3.6a) $\displaystyle
s.t.\mathrel{\mathop{\mathchar 58\relax}}$ $\displaystyle\leavevmode\nobreak\
C_{1}\mathrel{\mathop{\mathchar 58\relax}}\sum_{k\in\mathcal{K}}a_{n,k}\leq
1,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \forall
n\in\mathcal{N}_{i},$ (3.6b) $\displaystyle\leavevmode\nobreak\
C_{2}\mathrel{\mathop{\mathchar 58\relax}}\sum_{k\in\mathcal{K}}a_{n,k}\leq
1,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \forall
n\in\mathcal{N}_{e},$ (3.6c) $\displaystyle\leavevmode\nobreak\
C_{3}\mathrel{\mathop{\mathchar
58\relax}}\sum_{k\in\mathcal{K}}\leavevmode\nobreak\
\sum_{n\in\mathcal{N}}a_{n,k}p_{n,k}\leq p_{max},$ (3.6d)
$\displaystyle\leavevmode\nobreak\ C_{4}\mathrel{\mathop{\mathchar
58\relax}}R_{k}(\textbf{a},\textbf{p})\geq R_{min},\leavevmode\nobreak\
\forall k\in\mathcal{K},$ (3.6e) $\displaystyle\leavevmode\nobreak\
C_{5}\mathrel{\mathop{\mathchar
58\relax}}a_{n,k}\in\\{0,1\\},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall k\in\mathcal{K},\leavevmode\nobreak\ \forall
n\in\mathcal{N}.$ (3.6f)

In this optimization problem, $C_{1}$ and $C_{2}$ indicate that each
subcarrier can be assigned to at most one user for ID and EH operation,
respectively. $C_{3}$ states that power constraint for the AP with a maximum
transmit power allowance of $p_{max}$. The forth constraint, $C_{4}$,
guarantees the QoS for each user. Finally, by keeping in mind that the
subcarrier assignment variable is binary, the last constraint, $C_{5}$, makes
sure that different subcarriers take their values from a binary set. This
means whether a given subcarrier is going to be selected to maximize the
energy harvesting, i.e., the objective of the optimization problem at hand.

One can readily conclude that the optimization problem in (3.6) is a non-
convex mixed-integer linear programming (MILP) problem [100] due to the binary
constraint for the subcarrier assignment in $C_{5}$ and the non-linearity of
the QoS constraint. In general, it is impossible to find an optimal solution
for a non-convex MILP in a polynomial-time. However, in the next section, we
exploit an approach to find a locally optimal solution for the considered
system. Furthermore, we propose a suboptimal resource allocation algorithm,
which has a polynomial-time computational complexity to strike a balance
between complexity and system performance.

###  3.3 Solution to the Optimization Problem

In this section, we try to solve the problem in (3.6) using the MM approach by
constructing a sequence of surrogate functions to approximate the non-convex
problem. The MM procedure in our setting, i.e., the problem (3.6), consists of
two major steps. In the first minorization step, a surrogate function is found
that locally approximates the transformed objective function with their
difference maximized at the current point. That is to say, the transformed
objective function is lower-bounded by the surrogate function up to a constant
value. Next, the surrogate function is maximized in the maximization step. By
inspiring this approach, we make a convex approximation which can be globally
and efficiently solved using optimization packages incorporating the MM
algorithm. Nonetheless, finding a proper surrogate function that yields a low-
complexity algorithm is not a straightforward task. On the one hand, a
surrogate function that attempts to mimic the form of the objective function
and obtains a fast convergence speed is preferable. On the other hand, the
surrogate function should be easy to maximize in such a way that the
computational cost per iteration remains low. Realizing the appropriate trade-
off between the above-mentioned conflicting goals necessitates experiences in
applying inequalities to particular problems, as becomes evident in the next
subsection. For more details please refer to [92, 101, 102, 103, 104] and the
references therein.

####  3.3.1 Joint Power Allocation and Subcarrier Assignment Algorithm

In this subsection, we propose a locally optimal solution for the optimization
problem in (3.6). It is worth mentioning that the multiplication of two
variables in the objective function of the problem in (3.6) as well as the
constrain $C_{4}$ are the obstacles for the design of a computationally
efficient resource allocation algorithm. Since the multiplication of two
variables in (3.6d), i.e., $a_{n,k}p_{n,k}$, is non-convex, we define the
product terms as $\tilde{p}_{n,k}=a_{n,k}p_{n,k}$. In order to handle this
difficulty of the non-convexity, we adopt the big-M formulation [105] to
decouple the product terms. Therefore, the following additional constraints
are imposed accordingly as

$\displaystyle C_{6}\mathrel{\mathop{\mathchar 58\relax}}\tilde{p}_{n,k}\leq
p_{max}a_{n,k},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \forall
k\in\mathcal{K},\leavevmode\nobreak\ \forall n\in\mathcal{N},$ (3.7)
$\displaystyle C_{7}\mathrel{\mathop{\mathchar 58\relax}}\tilde{p}_{n,k}\leq
p_{n,k},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \forall
k\in\mathcal{K},\leavevmode\nobreak\ \forall n\in\mathcal{N},$ (3.8)
$\displaystyle C_{8}\mathrel{\mathop{\mathchar 58\relax}}\tilde{p}_{n,k}\geq
p_{n,k}-(1-a_{n,k})p_{max},\leavevmode\nobreak\ \forall
k\in\mathcal{K},\leavevmode\nobreak\ \forall n\in\mathcal{N},$ (3.9)
$\displaystyle C_{9}\mathrel{\mathop{\mathchar 58\relax}}\tilde{p}_{n,k}\geq
0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \forall
k\in\mathcal{K},\leavevmode\nobreak\ \forall n\in\mathcal{N},$ (3.10)

where $\tilde{\textbf{p}}\in\mathbb{R}^{1\times KN}$ is the collection of all
$\tilde{p}_{n,k}$’s. Therefore, the original optimization problem in (3.6) can
be recast in equivalent form as

$\displaystyle\max_{{\textbf{a}},\textbf{p},\tilde{\textbf{p}}}\leavevmode\nobreak\
\widehat{\overline{\textrm{EH}}}(\textbf{a},\textbf{p},\tilde{\textbf{p}})$
(3.11a) $\displaystyle s.t.\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\leavevmode\nobreak\ C_{1}-C_{2},C_{5}-C_{9},$ (3.11b)
$\displaystyle\leavevmode\nobreak\ C_{3}\mathrel{\mathop{\mathchar
58\relax}}\sum_{k\in\mathcal{K}}\leavevmode\nobreak\
\sum_{n\in\mathcal{N}}\tilde{p}_{n,k}\leq p_{max},$ (3.11c)
$\displaystyle\leavevmode\nobreak\ C_{4}\mathrel{\mathop{\mathchar
58\relax}}\sum_{n\in\mathcal{N}}\log_{2}(1+\frac{\tilde{p}_{n,k}|h_{n,k}|^{2}}{\sigma^{2}})\geq
R_{min},\leavevmode\nobreak\ \forall k\in\mathcal{K},$ (3.11d)

where, by revisiting the definition of the harvested energy in (3.5), the
objective function of (3.11) is as follows

$\widehat{\overline{\textrm{EH}}}(\textbf{a},\textbf{p},\tilde{\textbf{p}})=\sum_{k\in\mathcal{K}}\epsilon_{k}\bigg{(}\sum_{n\in\mathcal{N}_{e}}\tilde{p}_{n,k}|h_{n,k}|^{2}\bigg{)}.$
(3.12)

The optimization problem of (3.11) is still a non-convex MILP problem, which
is complicated to solve. To facilitate the solution design, we restate the
integer variable in constraint $C_{5}$ as the intersection of the following
regions [106, 107]

$\displaystyle\dot{C}_{5}\mathrel{\mathop{\mathchar 58\relax}}0\leq
a_{n,k}\leq 1,\leavevmode\nobreak\ \forall
k\in\mathcal{K},\leavevmode\nobreak\ \forall n\in\mathcal{N},$ (3.13)
$\displaystyle\ddot{C}_{5}\mathrel{\mathop{\mathchar
58\relax}}\sum_{k\in\mathcal{K}}\sum_{n\in\mathcal{N}}a_{n,k}-(a_{n,k})^{2}\leq
0.$ (3.14)

The above transformations make integer optimization variables continuous with
values between zero and one. Therefore, the original problem in (3.11) can be
rewritten as

$\displaystyle\max_{{\textbf{a}},\textbf{p},\tilde{\textbf{p}}}\leavevmode\nobreak\
\widehat{\overline{\textrm{EH}}}(\textbf{a},\textbf{p},\tilde{\textbf{p}})$
(3.15) $\displaystyle s.t.\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\leavevmode\nobreak\
C_{1}-C_{4},\dot{C}_{5},\ddot{C}_{5},C_{6}-C_{9}.$

Note that the optimization problem in (3.15) is a continuous optimization
problem with respect to all variables. However, our concern is to obtain
integer solutions for $a_{n,k}$. To this end, we add a penalty term to the
objective function. The integer variable is now relaxed to take any values
between zero to one. Thereby, the problem can be restated as follows

$\displaystyle\max_{\textbf{a},\textbf{p},\tilde{\textbf{p}}}\mathcal{L}(\textbf{a},\textbf{p},\tilde{\textbf{p}},\lambda)$
(3.16) $\displaystyle s.t.\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\leavevmode\nobreak\ C_{1}-C_{4},\dot{C}_{5},C_{6}-C_{9},$

where $\mathcal{L}(\textbf{a},\textbf{p},\tilde{\textbf{p}},\lambda)$ is the
abstract Lagrangian duality [91] associated to (3.15), and is defined as

$\displaystyle\mathcal{L}(\textbf{a},\textbf{p},\tilde{\textbf{p}},\lambda)=\widehat{\overline{\textrm{EH}}}(\textbf{a},\textbf{p},\tilde{\textbf{p}})-\lambda\bigg{(}\sum_{k\in\mathcal{K}}\sum_{n\in\mathcal{N}}a_{n,k}-(a_{n,k})^{2}\bigg{)}.$
(3.17)

In (3.17), $\lambda$ acts as a penalty factor to penalize the objective
function when $a_{n,k}$ is not an integer variable.

######  Proposition 1.

In the abstract Lagrangian of (3.16), the $\lambda$ acts as a penalty factor
to penalize the objective function when the subcarrier allocation variable
$a_{n,k}$ does not have a binary value. For sufficiently large values of
$\lambda$, the optimization problem in (3.16) is equivalent to (3.15) with
both problems yielding the same optimal results.

###### Proof.

By using the abstract Lagrangian duality, the proof of Proposition 1 is
presented. Accordingly, the primal and dual problem of (3.15) can be written
respectively as

$\displaystyle(\textrm{Primal problem})$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\
p^{*}=\max_{\textbf{a},\textbf{p},\tilde{\textbf{p}}}\leavevmode\nobreak\
\min_{\lambda}\mathcal{L}(\textbf{a},\textbf{p},\tilde{\textbf{p}},\lambda),$
(3.18) $\displaystyle(\textrm{Dual problem})\leavevmode\nobreak\ $
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
d^{*}=\min_{\lambda}\leavevmode\nobreak\
\max_{\textbf{a},\textbf{p},\tilde{\textbf{p}}}\mathcal{L}(\textbf{a},\textbf{p},\tilde{\textbf{p}},\lambda).$
(3.19)

Now, by defining
$\mathcal{G}(\lambda)\triangleq\max\limits_{\textbf{a},\textbf{p},\tilde{\textbf{p}}}\mathcal{L}(\textbf{a},\textbf{p},\tilde{\textbf{p}},\lambda)$,
the following inequality holds according to the weak duality theorem

$\displaystyle
p^{*}=\max_{\textbf{a},\textbf{p},\tilde{\textbf{p}}}\min_{\lambda}\mathcal{L}(\textbf{a},\textbf{p},\tilde{\textbf{p}},\lambda)\leq\min_{\lambda\geq
0}\mathcal{G}(\lambda)=d^{*}.$ (3.20)

Additionally, it should be pointed out that for
$\textbf{a},\textbf{p},\tilde{\textbf{p}}\leavevmode\nobreak\
s.t.\mathrel{\mathop{\mathchar
58\relax}}C_{1}-C_{4},\dot{C}_{5},\ddot{C}_{5},C_{6}-C_{9}$, two cases can be
observed.

_Case 1_ : In the first case, it is assumed that at the optimal subcarrier
allocation point, we have

$\ddot{C}_{5}\mathrel{\mathop{\mathchar 58\relax}}\leavevmode\nobreak\
\sum_{k\in\mathcal{K}}\sum_{n\in\mathcal{N}}a_{n,k}-(a_{n,k})^{2}=0.$ (3.21)

Consequently, $d^{*}$ becomes a feasible solution of (3.15). Afterward,
substituting the optimal value of $\lambda$, i.e., $\lambda^{*}$, into the
abstract Lagrangian duality problem of (3.16) results in

$d^{*}=\mathcal{G}(\lambda^{*})=\max_{\textbf{a},\textbf{p},\tilde{\textbf{p}}}\textrm{EH}(\textbf{a},\textbf{p},\tilde{\textbf{p}})=p^{*}.$
(3.22)

Furthermore, considering the optimization problem in (3.16) and assuring that
the subcarrier allocation variable, $a_{n,k}$, takes its values in the region
a$\leavevmode\nobreak\ \in\dot{C}_{5},\ddot{C}_{5}$, one can conclude that
$\mathcal{G}(\lambda)$ is a monotonically decreasing function with respect to
$\lambda$. On the other hand, it can be asserted that
$d^{*}$=$\min_{\lambda\geq 0}\mathcal{G}(\lambda)$. Therefore, we have

$\mathcal{G}(\lambda)=d^{*},\forall\leavevmode\nobreak\
\lambda\geq\lambda_{0},$ (3.23)

where $\lambda_{0}$ is a given value. This confirms that for any value of
$\lambda\geq\lambda^{*}$, the solution of (3.16) can return the optimal
solution of (3.15).

_Case 2_ : In the second case, it is assumed that the subcarrier allocation
variable, $a_{n,k}$, takes some value from the interval between zero and one.
This is equivalent to satisfying the following inequality

$\ddot{C}_{5}\mathrel{\mathop{\mathchar
58\relax}}\sum_{k\in\mathcal{K}}\sum_{n\in\mathcal{N}}a_{n,k}-(a_{n,k})^{2}>0.$
(3.24)

In this regard, one may conclude $\mathcal{G}(\lambda^{*})$ tends to $-\infty$
at the optimal point by referring to (3.16) and (3.20). Nevertheless, this
cannot happen. Since the primal solution must always be greater than zero,
this would be a contradiction as it states that $\mathcal{G}(\lambda^{*})$
would be limited from below by the solution of (3.15). Hence,
$\sum_{k\in\mathcal{K}}\sum_{n\in\mathcal{N}}a_{n,k}-(a_{n,k})^{2}=0$.
Therefore, the solution of (3.16) yields the optimal solution of (3.15). ∎

Now, we can express the optimization problem in (3.15) in terms of difference
of convex functions (D.C.) as follows

$\displaystyle\max_{\textbf{a},\textbf{p},\tilde{\textbf{p}}}\leavevmode\nobreak\
\widehat{\overline{\textrm{EH}}}(\textbf{a},\textbf{p},\tilde{\textbf{p}})-\lambda\big{(}\mathcal{V}(\textbf{a})-\mathcal{W}(\textbf{a})\big{)}$
(3.25) $\displaystyle s.t.\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\leavevmode\nobreak\ C_{1}-C_{4},\dot{C}_{5},C_{6}-C_{9},$

where

$\displaystyle\mathcal{V}(\textbf{a})$
$\displaystyle=\sum_{k\in\mathcal{K}}\sum_{n\in\mathcal{N}}a_{n,k},$ (3.26)
$\displaystyle\mathcal{W}(\textbf{a})$
$\displaystyle=\sum_{k\in\mathcal{K}}\sum_{n\in\mathcal{N}}(a_{n,k})^{2},$
(3.27)

are all convex functions, and $\lambda$ is the penalty factors to penalize the
objective function when $a_{n,k}$ is not integer values. Although all terms in
the objective function of (3.25) are convex, the subtraction of two convex
functions are not necessarily convex [94, 107, 108]. To make a convex
approximation for the objective function, we adopt the MM algorithm by
constructing a surrogate function via a first-order Taylor approximation as

$\displaystyle\mathcal{W}(\textbf{a})\simeq\mathcal{W}(\textbf{a}^{t-1})+\nabla_{\textbf{a}}\mathcal{W}(\textbf{a}^{t-1})^{T}.(\textbf{a}-\textbf{a}^{t-1})\triangleq\tilde{\mathcal{W}}(\textbf{a}),$
(3.28)

where $t$ denotes the iteration number, the $\textbf{a}^{t-1}$ is the solution
of the problem at $(t-1)^{th}$ iteration, and $\nabla_{\square}$ represents
the gradient with respect to ${\square}$. Approximation (3.28), satisfies the
MM criteria and is a tight upper bound of $\mathcal{W}(\textbf{a})$ [92, 108].
Therefore, using the MM approach while constructing a sequence of surrogate
functions at the $t^{th}$ iteration, we can solve the following convex problem
instead of dealing with the non-convex optimization problem in (3.25). Thus,
we have

$\displaystyle\max_{\textbf{a},\textbf{p},\tilde{\textbf{p}}}\leavevmode\nobreak\
\widehat{\overline{\textrm{EH}}}(\textbf{a},\textbf{p},\tilde{\textbf{p}})-\lambda\big{(}\mathcal{V}(\textbf{a})-\tilde{\mathcal{W}}(\textbf{a})\big{)}$
(3.29) $\displaystyle s.t.\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\leavevmode\nobreak\ C_{1}-C_{4},\dot{C}_{5},C_{6}-C_{9}.$

It is easy to demonstrate that the optimization problem (3.29) is convex and
can be solved efficiently via D.C. approximation based on the interior point
methods [92]. As a consequence, the solution of (3.29) would be an
approximation to the solution of the original problem given in (3.25).
However, in D.C. programming, the iteration begins from a feasible initial
point and solves the optimization problem iteratively until it eventually
approaches a close-to-optimal solution [107, 108, 109]. Besides, it must be
noticed that the MM approach produces a sequence of improved feasible
solutions with the adopted D.C. approximation, which would ultimately converge
to a locally optimal solution
$(\textbf{a}^{*},\textbf{p}^{*},\tilde{\textbf{p}}^{*})$ using standard convex
program solvers such as CVX.

######  Proposition 2.

By incorporating D.C. approximation, the solution of the problem (3.29)
becomes a tightly lower-bounded solution from below for the original problem
(3.25) at the end of each iteration.

###### Proof.

In the $t^{th}$ iteration, the objective function of (3.25) is

$\widehat{\overline{\textrm{EH}}}(\textbf{a}^{t},\textbf{p}^{t},\tilde{\textbf{p}}^{t})-\lambda\big{(}\mathcal{V}(\textbf{a}^{t})-\mathcal{W}(\textbf{a}^{t})\big{)}.$

Subsequently, in the next iteration, we have

$\displaystyle\widehat{\overline{\textrm{EH}}}(\textbf{a}^{t+1},$
$\displaystyle\textbf{p}^{t+1},\tilde{\textbf{p}}^{t+1})-\lambda\big{(}\mathcal{V}(\textbf{a}^{t+1})-\mathcal{W}(\textbf{a}^{t+1})\big{)}$
$\displaystyle\geq\widehat{\overline{\textrm{EH}}}(\textbf{a}^{t+1},\textbf{p}^{t+1},\tilde{\textbf{p}}^{t+1})-\lambda\big{(}\mathcal{V}(\textbf{a}^{t+1})-\mathcal{W}(\textbf{a}^{t})\big{)}+\lambda\nabla_{\textbf{a}}\mathcal{W}(\textbf{a}^{t})^{T}.(\textbf{a}-\textbf{a}^{t})$
$\displaystyle=\max_{{\textbf{a}},{\textbf{p}},\tilde{\textbf{p}}}\widehat{\overline{\textrm{EH}}}(\textbf{a},\textbf{p},\tilde{\textbf{p}})-\lambda\big{(}\mathcal{V}(\textbf{a})-\mathcal{W}(\textbf{a}^{t})-\nabla_{\textbf{a}}\mathcal{W}(\textbf{a}^{t})^{T}.(\textbf{a}-\textbf{a}^{t})\big{)}$
$\displaystyle\geq\widehat{\overline{\textrm{EH}}}(\textbf{a},\textbf{p},\tilde{\textbf{p}})-\lambda\big{(}\mathcal{V}(\textbf{a})-\mathcal{W}(\textbf{a}^{t})-\nabla_{\textbf{a}}\mathcal{W}(\textbf{a}^{t})^{T}.(\textbf{a}^{t}-\textbf{a}^{t})\big{)}$
$\displaystyle=\widehat{\overline{\textrm{EH}}}(\textbf{a}^{t},\textbf{p}^{t},\tilde{\textbf{p}}^{t})-\lambda\big{(}\mathcal{V}(\textbf{a}^{t})-\mathcal{W}(\textbf{a}^{t})\big{)}.$

This completes the proof. ∎

One can readily verify that the objective function of (3.29) takes larger
values as the iteration continues. Hence, we adopt an iterative solution to
tighten the obtained upper bound based on the Algorithm 2.

Algorithm 2 Proposed Iterative Method via D.C. Programming Based on the MM
Approach

1: $\mathbf{Initialize}$ [1em]0em MM iteration index $t=0$ with maximum number
of MM iteration $T_{max}$, feasible set vector $\mathbf{a}^{0}$,
$\mathbf{p}^{0}$, and $\tilde{\mathbf{p}}^{{0}}$,and the penalty factor
$\lambda\gg 1$.

2: repeat

3: [1em]0em Update $\tilde{\mathcal{W}}(\textbf{a})$ based on (3.28).

4: [1em]0em Solve optimization problem of (3.29) and store the intermediate
resource allocation policy $\textbf{a}^{t}$, $\textbf{p}^{t}$, and
$\tilde{\textbf{p}}^{t}$.

5: [1em]0em Set $t=t+1$.

6: [1em]0em Set {$\mathbf{a}^{t},\mathbf{p}^{t}$,$\tilde{\mathbf{p}}^{t}$}=
{$\mathbf{a},\mathbf{p}$,$\tilde{\mathbf{p}}$}.

7: until Convergence or $t=T_{max}$

8: return {$\mathbf{a}^{*},{\mathbf{p}}^{*}$,$\tilde{\mathbf{p}}^{*}$} $=$
{$\mathbf{a}^{t},{\mathbf{p}}^{t}$,$\tilde{\mathbf{p}}^{t}$}

###  3.4 Complexity Analysis

As can be observed, the joint optimization problem in (3.29) involves $KN$
variables and $N+K+5NK$ linear constraints. Consequently, it can be concluded
that the overall computational complexity of optimization problem is
$\mathcal{O}(NK)^{2}(N+K+5NK)$. This is asymptotically equal to
$\mathcal{O}(NK)^{3}$, exhibiting a polynomial-time complexity.

###  3.5 Simulation Results

Table 3.1: Simulation Parameters Parameter | Value
---|---
Cell coverage ($d_{max}$) |  $20$ m
Reference distance ($d_{0}$) |  $5$ m
The number of user ($K$) | $4$
The number of subcarriers ($N$) | $16$
Noise power ($\sigma^{2}$) |  $-120$ dBm
The bandwidth of each subcarrier |  $180$ kHz
Path loss exponent ($\alpha$) | $2.76$
Path loss model for cellular links | $31.7+27.6\log(\frac{d}{d_{0}})$
Multi-path fading distribution | Rician fading with factor $3$ dB
Power conversion efficiency ($\epsilon$) | $30\%$
The maximum transmit power of the AP ($p_{max}$) |  $30$ dBm
The minimum data-rate requirement for the $k^{th}$ user (${R}_{\textnormal{min}}$) |  $1$ bps$/$Hz
Channel realization number | $100$

In this section, the performance gain of the proposed joint subcarrier
assignment and power allocation algorithm for SWIPT in the DL direction of a
single-cell multi-user OFDMA system is evaluated through extensive
simulations. The radius of the cell, $d_{max}$, is 20 meters, with a reference
distance, $d_{0}$, of 5 meters. Moreover, there are $K=4$ uniformly and
randomly located users between, $d_{0}$, the reference distance and maximum
coverage of the small-cell, $d_{max}$. We further assume a frequency-selective
fading channel and consider the central carrier frequency is set to 3 GHz with
the bandwidth of each subcarrier being 180 kHz. The number of subcarriers is
$N=16$, where the optimal set cardinality of subcarriers for ID and EH is
determined based on [45]. The variance of the background noise at the receiver
is equal to ${\sigma_{n,k}^{2}=\sigma^{2}=-120}$ dBm throughout the
simulations. Since a line-of-sight (LoS) signal is expected in the received
signal, the small-scale fading channel is modeled as Rician fading with Rician
factor $\rho=3$ dB. Besides, the Rician flat fading channel gains include a
distance-dependent path loss model of $31.7+10\alpha\log(\frac{d}{d_{0}})$
[dB] (where $d$ is the distance between the transmitter and the receiver) and
a log-normal shadowing component with $8$ dB standard deviation where the path
loss exponent is equal to $\alpha=2.8$ [110]. These parameters for propagation
modeling and simulations follow the suggestions in 3GPP evaluation methodology
[111]. The power conversion efficiency of all users, $\epsilon_{k}$, is
assumed to be the same and is equal to $\epsilon_{k}=\epsilon=0.3$. The target
transmission rate $R_{min}=1$ bit/second/Hz (bps/Hz) unless otherwise stated.
Moreover, we conduct Monte Carlo simulations by generating random realizations
of the channel gains to obtain the average harvested energy of the network.
Finally, the setting summarized in Table (3.1) are used unless otherwise
specified.

####  3.5.1 Total Harvested Energy versus the Maximum Transmit Power

Figure (3.2) shows the total harvested energy versus the maximum transmit
power $p_{max}$. As can be observed, the average total harvested energy grows
monotonically as the maximum transmit power increases. Besides, harvested
energy for the lower values of the maximum transmit power is low as compared
to higher values. This is due to the inability of the AP to contribute to
energy harvesting as it is forced to ensure the QoS requirements.
Consequently, for the higher values of the maximum transmit power, yet with
the same data-rate requirement, the AP can help users to harvest more energy
since fewer subcarriers are assigned to ID and more to EH. Hence, excess
subcarriers are utilized to provide more harvested energy.

Figure 3.2: Total harvested energy versus the maximum transmit power

For comparison, the performance of the proposed joint optimization algorithm,
Method A, is compared with the following benchmark algorithms as of four
Methods B-E. Method B is the proposed method in [45]. In this method, a subset
of subcarriers is assigned to EH to maximize the harvesting energy based on
the resource allocation design. On the other hand, the remaining ones are
allocated for ID with an imposed QoS requirement. Method C examines the
proposed method in [49] in which each subcarrier set is divided into two
groups. Specifically, one group is employed for EH, and the other one is
utilized for ID while considering fixed PS ratios. Method D is our proposed
method with equal power allocation. This method is based on the proposed
algorithm for the subcarrier assignment, while equal power is allocated across
subcarrier for each user. Method E is our proposed method with an equal power
allocation alongside a random subcarrier assignment. This method considers
equal power, where the subcarrier assignment is done randomly for meeting the
data-rate requirement. Once the minimum data-rate requirement is satisfied,
the remaining subcarriers are assigned to EH.

It can be seen that our proposed method outperforms the other benchmark
algorithms due to the joint optimization framework. As can be seen, Method B
performs better than Method C. This is because Method B considers the power
allocation and subcarrier assignment based on a heuristic search, while in
Method C, each subcarrier is split into two streams with a fixed PS ratio. In
particular, in Method C, the receiver does not adapt to the channel condition
of each subcarrier, and each subcarrier splits the same ratio of power for ID
and EH. This results in the degradation of the performance gain and less
harvested energy comparatively. Another interesting observation is that
considering a joint subcarrier assignment and power allocation in the
optimization problem leads to a remarkable enhancement of the performance gain
due to mitigating the harmful effect of deep fades experienced in the wireless
channel as the resources allocator carefully takes this phenomenon under
consideration. It should be noted that the reason the Method E performs worse
than the rest is that the subcarriers are assigned randomly together with an
equal power assignment to users. Although the performance can improve if the
subcarriers are assigned according to the designed resource allocation policy
as it is the case in Method D.

####  3.5.2 Average Harvested Energy versus Distance

Figure 3.3 depicts the harvested energy versus distance between transmitter
and receiver. It is observed that as the distance increases, the harvested
energy decreases. The reason for this is that by increasing the distance, the
channel strength would become weak, and subsequently, more subcarriers with
more power are needed to be assigned to meet the minimum required data-rate.
Hence, less energy would be harvested by the users since the AP first needs to
consume power to secure the quality of experience provisioning. Once the
minimum data-rate is met, the rest of the subcarriers would be exploited for
EH. It should be noted that the Methods A-E are the same as defined in the
last subsection. The explanations and superiority of Method A also stay the
same and is because of our joint optimization network resource allocation
framework.

Figure 3.3: Average harvested energy versus distance.

####  3.5.3 Average Harvested Energy versus Number of Iterations

Figure (3.4) illustrates the convergence of the proposed algorithm for
different initialization for the power allocation. It can be observed that the
average harvested energy rises up gradually as the number of iterations
increases and tends to a stationary point no matter how the power is allocated
among users over all subcarriers initially. This is because a more accurate,
locally optimal solution can be found by increasing the number of iterations.
Likewise, this is equivalent to saying that the algorithm is getting closer to
satisfying convergence requirements. In the figure, we compare three initial
selections for the power control; $\mathbf{p}^{0}(i)=\frac{p_{max}}{N}$, i.e.,
equal power is assigned to all users over all subcarriers, be it to an EH or
an ID operation, $\mathbf{p}^{0}(i)=p_{max}$, i.e., all subcarriers have the
maximum power, and $\mathbf{p}^{0}(i)=0$, no power is assigned to different
subcarriers to start with. Note that the last two power assignments do not
satisfy the constraint ${C_{3}}$ in (3.11c). Moreover, even though these
curves are obtained for different initial power allocations, they all converge
to almost the same value. However, the convergence rate differs significantly
from the different power allocation settings: while the initial setting
$\mathbf{p}^{0}(i)=0$ converges the slowest to the optimal harvested energy,
the setting $\mathbf{p}^{0}(i)=\frac{p_{max}}{N}$, which satisfies the
constraints of the optimization problem, converges the fastest, i.e., less
than nine iterations are required.

Figure 3.4: Average harvested energy versus number of iterations. Figure 3.5:
Harvested energy versus sum transmit power at different target data-rates.

####  3.5.4 Harvested Energy versus Maximum Transmit Power at Different Target
Data-rates

Figure (3.5) illustrates the harvested energy versus the maximum transmit
power, $p_{max}$, for different data-rate requirements. It can be seen that
less energy would be harvested by increasing the minimum data-rate
requirement, $R_{min}$. The reason behind it is quite evident. Considering a
fixed maximum transmit power at the AP, more subcarriers with more power would
be required to be assigned to each user when the minimum data-rate requirement
becomes larger. This means that fewer subcarriers would be assigned to EH
operation with lower allocated power. This, in turn, causes less energy to be
harvested by the users over the EH subcarriers as the minimum data-rate
requirement increases.

###  3.6 Summary

In this chapter, we investigated the problem of joint subcarrier assignment
and power control for SWIPT in a multi-user single small-cell network to
maximize the harvesting energy while respecting the minimum required data-rate
for each user. We reduced the complexity of the receiver in our proposed
algorithm as the receiver does not need a splitter to perform appropriately;
that is, neither time nor power splitter is utilized at the receiver. The only
knowledge that the resource allocator needs to know is which group of
subcarriers is allocated for EH and which group for ID operation, where this
knowledge is derived based on the CSI and the designed resource allocation
policy. The problem considered in this chapter was a non-convex MILP problem,
which is generally difficult to solve. In order to circumvent this difficulty,
we relaxed the integer variables and employed the MM approach to convexify the
problem. Afterward, the problem was solved in an iterative manner to obtain a
locally optimal solution. Simulation results demonstrated that the designed
algorithms performed better than other algorithms that have been addressed in
the literature.

##  Chapter 4 SWIPT in Multi-Cell Networks

Most of the literature on the subject of simultaneous wireless information and
power transfer (SWIPT) considers a single-cell network, in which users’ data-
rate is a function of their signal-to-noise ratio (SNR). Although the absence
of interference simplifies problem solving, it is clear that these kinds of
system models do not accurately represent real-world networks. In contrast to
chapter 3, here we examine a multi-cell network that incorporates interference
in the data-rate function. It is not always the best practice to maximize
energy harvesting when it could result in the network having a lower total
data-rate or throughput. Data-rate is conventionally defined as the number of
delivered information bits per channel use measured in bit/second/Hz (bps/Hz).
In this chapter, we use a smart resource allocation policy to maximize the
data-rate, subject to a minimum harvested energy constraint for users.

The problem of resource allocation in multi-cell networks has already been
addressed by other researchers, for instance, the authors of [112], who
studied an energy-efficient communication in the downlink (DL) of an
orthogonal frequency division multiple access (OFDMA) multi-cell network with
large numbers of single antenna cooperative base stations (BSs). They based
their work on joint BS zero-forcing beamforming with full channel state
information (CSI) while looking at the trade-off between energy efficiency,
backhaul capacity, and network capacity. The authors of [113] studied an
adaptive fairness scheduling algorithm in co-located SWIPT-enabled multi-cell
DL networks that employs an adjustable weighted sum method as the utility
function of each user to study fairness among users. With the aim of
maximizing energy harvesting efficiency, the authors of [114] investigated the
beamforming design for multi-cell multi-user networks with SWIPT in a co-
located receiver architecture. A max-min signal-to-interference plus
interference ratio (SINR) problem in the DL of a dense multi-cell SWIPT-
enabled network was explored in [115], in which the users near their serving
BS can perform both energy harvesting (EH) and information decoding (ID) using
the power splitting receiver approach, whereas far-field users can only