# A Unified Framework for IRS Enabled Wireless Powered Sensor Networks

Zheng Chu, _Member, IEEE_ , Zhengyu Zhu, _Member, IEEE_ , Miao Zhang, _Member,
IEEE_ , Fuhui Zhou, _Senior Member, IEEE_ , Li Zhen, _Member, IEEE_ , Xueqian
Fu, _Member, IEEE_ , and Naofal Al-Dhahir, _Fellow, IEEE_

###### Abstract

This paper unveils the importance of intelligent reflecting surface (IRS) in a
wireless powered sensor network (WPSN). Specifically, a multi-antenna power
station (PS) employs energy beamforming to provide wireless charging for
multiple Internet of Things (IoT) devices, which utilize the harvested energy
to deliver their own messages to an access point (AP). Meanwhile, an IRS is
deployed to enhance the performances of wireless energy transfer (WET) and
wireless information transfer (WIT) by intelligently adjusting the phase shift
of each reflecting element. To evaluate the performance of this IRS assisted
WPSN, we are interested in maximizing its system sum throughput to jointly
optimize the energy beamforming of the PS, the transmission time allocation,
as well as the phase shifts of the WET and WIT phases. The formulated problem
is not jointly convex due to the multiple coupled variables. To deal with its
non-convexity, we first independently find the phase shifts of the WIT phase
in closed-form. We further propose an alternating optimization (AO) algorithm
to iteratively solve the sum throughput maximization problem. To be specific,
a semidefinite programming (SDP) relaxation approach is adopted to design the
energy beamforming and the time allocation for given phase shifts of WET
phase, which is then optimized for given energy beamforming and time
allocation. Moreover, we propose an AO low-complexity scheme to significantly
reduce the computational complexity incurred by the SDP relaxation, where the
optimal closed-form energy beamforming, time allocation, and phase shifts of
the WET phase are derived. Finally, numerical results are demonstrated to
validate the effectiveness of the proposed algorithm, and highlight the
beneficial role of the IRS in comparison to the benchmark schemes.

###### Index Terms:

Intelligent reflecting surface (IRS), wireless powered sensor network (WPSN),
internet of Things (IoT), phase shift, alternating optimization (AO) algorithm

##  I Introduction

In recent years, demands for massive connectivity and high data rate have
experienced an explosive growth in next generation wireless networks, which
has drawn increasing attention of academics and industries. Internet of things
(IoT) is considered as an important portion of the fifth-generation (5G)
network and beyond, which significantly enhances high-data access rate for
massive wireless devices (WDs) [1, 2]. In a generic IoT system, a number of
sensor nodes connect with an access point (AP) forming a wireless sensor
network (WSN), which has been widely applied in various practical scenarios,
including external environment monitoring, event detection for emergency
services, wireless surveillance for public safety, healthcare diagnosis, etc.
[3]. The sensors are typically low energy consumption devices, which depend on
a finite-capacity battery such that they may not have sufficient energy to
support their own computation and communication operations [4]. The sensors
are generally equipped with traditional batteries, which limits their
potential practical applications, since traditional batteries need to be
regularly maintained or even replaced to prolong their own operation lifetime,
which is often costly due to the massive number of sensors in use. In
addition, the sensors are often deployed in extreme environments,
infrastructures, or human bodies such that the battery maintenance or
replacement is very challenging. Thus, limited battery lifetime of sensors is
still a key challenge in designing future wireless networks [5].

Radio frequency (RF) wireless energy transfer (WET), as one of the promising
solutions to address the energy constraint issue, enables novel electric
energy transmission from a dedicated energy source to the WDs without any
wired connection [6]. As one important solution of RF WET, wireless powered
communication networks (WPCNs) collect energy from dedicated energy sources,
each of which has a stable energy supply, supporting wireless information
transfer (WIT). A WPCN is expected to improve the system throughput in
comparison to the conventional battery-powered communications. In WPCNs, a
classic protocol named “ _harvest-then-transmit_ ” was proposed in [7], where
the WDs first collect energy from the RF signals broadcast by an AP, and then
transmit their independent information to the AP utilizing the harvested
energy. A group of power stations (PSs) constitute of a dedicated WET network
which is deployed to support WIT in the vicinity [8, 9]. The WDs can gain
energy benefits from these PSs to prolong their own battery life via wireless
charging. The WPCN improves the energy efficiency in wireless networks which
reduces its operational cost. Thus, the WPCN is more suitable for the low
energy consumption use cases, i.e., wireless powered sensor networks (WPSNs),
which includes many sensor nodes to reduce maintenance cost as well as enhance
its deployment flexibility. In addition, the WPSN is generally used to support
low-power devices, i.e., radio frequency identification (RFID) IoT sensor
nodes and tags [5].

Moreover, 5G and beyond networks have been evolving towards machine-centric
driven by a vast range of quality of service (QoS) requirements, e.g., ultra-
high spectral efficiency and throughput. Several advanced techniques can be
used to enhance spectral efficiency and throughput, e.g., massive multiple-
input multiple-output (massive MIMO) with millimetre wave (mmWave), relaying,
and ultra-dense networks (UDNs) [10, 11, 12]. However, these techniques
typically require a large amount of RF chains over a high frequency band and
incur high energy consumption and hardware cost, which results in more energy
consumption [13, 14]. This has given rise to a novel and promising paradigm,
named _smart radio environment_ , which is a holographic wireless mode with
low cost, size, weight, and power consumption features in hardware
architecture [15]. _Smart radio environment_ offers the seamless wireless
connectivity and the capability of transmitting and processing data via
recycling the existing radio waves instead of generating new ones, which is a
transformative way to convert the traditional wireless environment into a
programmable intelligent entity [13, 14, 15].

Intelligent reflecting surface (IRS), as an enabler of _smart radio
environments_ , is one of the promising techniques for enhancing the
throughput in future sixth generation (6G) wireless communication networks
[16]. The IRS is made of a massive number of reconfigurable reflecting
elements, and coordinated by a software-oriented IRS controller [13]. These
reflecting elements typically have small, low-cost, and low-energy consumption
features, which can efficiently reflect the intended signal without a
dedicated RF processing, en/de-coding, or re-transmission [17, 18]. The IRS
exploits a novel wireless communication paradigm to achieve three-dimensional
(3D) passive beamforming gains to align the reflected signals with the direct
signal by intelligently varying the phase shift at each reflecting element.

###  I-A State of the Art

There are several existing studies in the literature that focus on the IRS
assisted wireless communication networks, such as [17, 18, 19, 20, 21, 22,
23]. In [17, 18], an IRS assisted multiple-input single-output (MISO) system
is proposed, where the desired signal is passively received and reflected by
the IRS to align the reflecting link with the direct link between the access
point (AP) and the users via intelligently adjusting the phase shift of each
reflecting element. The total transmit power was minimized subject to the
individual signal-to-interference-plus-noise ratio (SINR) constraint to
jointly design the active transmit and passive reflecting beamforming. The
energy efficiency was adopted as a performance metric in the IRS aided MISO
downlink system, which is maximized such that the power allocation and the
phase shifts of the IRS are optimally designed by an alternating optimization
(AO) algorithm, gradient descent search, and sequential fractional programming
[19]. In [20], a practical phase shift model was proposed to capture the
phase-dependent amplitude variation in the element-wise reflection design.
Furthermore, the potential of the IRS enabled wireless security was
demonstrated in [21, 22]. These works focus on the IRS-assisted MISO secure
system, where an IRS is deployed to reduce power consumption and enhance
achievable secrecy performance via alternately designing the secure transmit
and reflecting beamforming vectors. In [23], a multi-cell multiple-input
multiple-output (MIMO) system was investigated with the assistance of an IRS
in enhancing the downlink transmission for cell-edge users to mitigate the
inter-cell interference. Moreover, the IRS enabled simultaneous wireless
information and power transfer (SWIPT) was investigated in [24, 25, 26]. In
[24], an IRS was introduced in a SWIPT system to achieve high passive
beamforming gains. This enhances the WET efficiency and the rate-energy trade-
off of the IRS enabled SWIPT system via maximizing weighted harvested energy
of energy harvesting users subject to the individual SINR constraint for
information decoding users. Extension to the multiple IRSs scenario, which are
deployed to support the information/energy transfer from a AP to information
decoding/energy harvesting users [25]. The total transmit power is minimized
with the QoS constraints for all users, which demonstrates the impact of these
IRSs on energy efficiency of the SWIPT system. In [26], an IRS assisted MIMO
SWIPT system was studied to maximize the weighted sum rate (WSR) of the
information decoding users, while guaranteeing the energy harvesting
requirement of the energy harvesting users. The classic block coordinate
descent (BCD) algorithm was applied to decompose the formulated problem into
several sub-problems and alternately design the transmit precoding and phase
shift matrices.

The above-mentioned works focus on the IRS aided downlink transmission. On the
other hand, the IRS aided multiple access channel (MAC) has drawn increased
research attention recently, such as the IRS enabled MAC [27], the IRS
assisted WPCN user cooperation (UC) system [28]. In [27], the capacity region
and deployment mechanism of the IRS enabled MAC were investigated for a two-
user case with the help of distributed and centralized IRSs. Additionally, an
IRS-assisted WPCN with UC was considered in [28], where the IRS is deployed to
help a WPCN aided UC system to enhance the common throughput performance in
downlink WET and uplink WIT phases. Very recently, the IRS was introduced in
mobile edge computing (MEC) systems which coordinates the computation and
communication capabilities of wireless networks via local computing on the WDs
or offloading to the MEC server for processing their own computational tasks
[29, 30]. In the IRS assisted MEC system, the WDs offload a portion of their
computational tasks to the AP equipped with a MEC server with the assistance
of an IRS [29]. The computational latency is minimized to satisfy the
constraints of the edge computing capability, which alternately design the
computing and communication configurations. In [30], a further extension was
investigated in terms of the IRS enabled WPCN MEC network, where the IRS
assists the WDs via a reflecting link for WET and computational offloading to
reduce the total energy consumption of the WET at the AP and of the edge
computing based on orthogonal frequency-division multiplexing (OFDM)
configurations.

Although the existing state-of-the-art made variety of research contributions
in the downlink/uplink IRS aided wireless systems, the IRS aided SWIPT systems
and the IRS assisted MEC systems, there still exists a major research gap on
investigation of the IRS’s benefits in a WPSN. We introduce the IRS to enhance
energy harvesting and data transmission capabilities of the WPSN due to its
self-sustainability, which is a promising paradigm and has not yet been
investigated in the literature. Also, the IRS is deployed to properly
coordinate the energy/information RF signals which can circumvent the high RF
signal attenuation due to long distances or construct an effective energy
harvesting/charging zone. This provides an coverage enhancement for downlink
wireless charging and uplink connectivity so as to improve the throughput
performance for the WPSN. In addition, it is imperative that the energy source
is equipped with multiple antennas which realizes energy beamforming to
perform WET. This is due to the fact that the energy beamforming can bring a
significant WET efficiency in comparison to the traditional omnidirectional
multi-antenna transmission. To the best of the authors’ knowledge, there has
been few of work that modelled and investigated the IRS assisted WPSN with
multi-antenna energy source, which motivates our work in this paper.

This paper investigates an IRS assisted WPSN, where a multi-antenna PS applies
energy beamforming to radiate wireless energy to multiple IoT devices which
then utilize the harvested energy to transmit their own information to the AP.
The IRS is deployed to improve energy harvesting and data transmission
capabilities with intelligently adjustable phase shifts. _For this novel
system model, the main contributions of this paper are summarized in the
following._

  1. 1.

We first investigate an IRS aided WPSN system, where the IRS acts as a helper
to enhance the throughput performance of the WPSN by enabling energy and
information reflections during the WET downlink and WIT uplink transmissions,
respectively.

  2. 2.

We maximize the system sum throughput to evaluate the energy harvesting and
data transmission capabilities of the IRS assisted WPSN, which jointly designs
the energy beamforming of the PS, the passive phase shifts as well as the
transmission time allocation of the WET and WIT phases. The formulated problem
is not jointly convex, due to multiple coupled variables, and cannot be solved
directly. To tackle the non-convexity, we first derive the optimal closed-form
phase shifts of the WIT phase. Next, we take into consideration an AO
algorithm to alternatively update the energy beamforming, the phase shifts of
the WET phase and the transmission time allocation. Specifically, a semi-
definite programming (SDP) relaxation scheme is applied to design the energy
beamforming as well as the time allocation for given phase shifts of the WET
phase which is also solved by the SDP relaxation for given energy beamforming
and time allocation. In addition, the rank-profile is characterized to obtain
an optimal energy beamforming, while Gaussian randomization is adopted to deal
with possible higher-rank phase shift matrix of the WET phase incurred by the
SDP relaxation.

  3. 3.

Although the SDP based AO scheme can efficiently solve the formulated problem,
it incurs a higher computational complexity and is time-consuming, especially
for a large amount of reflecting elements. Thus, it is imperative to propose
an independent scheme which can be deployed on the IRS and the IoT devices,
and significantly lowers the computational complexity incurred by the SDP
relaxation with Gaussian randomization. In light of this, the low complexity
based AO scheme is proposed. Particularly, the optimal closed-form energy
beamforming and transmission time allocation are derived by taking into
consideration eigen-decomposition, as well as the Lagrange dual method and the
KKT conditions, respectively, for given phase shifts of the WET phase. Then,
we derive the closed-form phase shifts of the WET phase by the Majorization-
Minimization (MM) algorithm for given energy beamforming and transmission time
allocation. In addition, the convergence of the proposed MM algorithm and low
complexity based AO scheme are analyzed.

The remainder of this paper is organized as follows. Section II introduces the
system model as well as the sum throughput maximization problem formulation.
Section III provides the optimal solution of the formulated problem. Numerical
results are demonstrated in Section IV to evaluate the proposed algorithm.
Finally, we conclude this paper in Section V.

###  I-B Notations

We use the upper case boldface letters for matrices and lower case boldface
letters for vectors. $\textrm{conj}(\cdot)$, $(\cdot)^{T}$ and $(\cdot)^{H}$
denote the conjugate, the transpose and conjugate transpose operations,
respectively. $\textrm{Tr}(\cdot)$ stands for trace of a matrix.
$\rho_{max}(\cdot)$ represents the maximum eigenvalue, whereas
$\bm{\nu}_{\max}(\cdot)$ denotes the eigenvector associated with the maximum
eigenvalue. $\mathbf{A}\succeq\mathbf{0}$ indicates that $\mathbf{A}$ is a
positive semidefinite matrix. $|\cdot|$ and $\|\cdot\|$ denote the absolute
value and the Euclidean norm of a vector. $\mathbf{I}$ denotes the identity
matrix with appropriate size. $\exp(\cdot)$ and $\arg$ indicate the
exponential function and the phase angle, respectively. $\mathcal{W}(\cdot)$
denotes the Lambert $\mathcal{W}$ function. $\Re\\{\cdot\\}$ represents the
real part of a complex number. $\textrm{diag}\\{\cdot\\}$ denotes a vector
that consists of the diagonal elements of a matrix or a diagonal matrix where
the diagonal elements are from a vector.

##  II System Model

Figure 1: An IRS assisted WPSN.

In this paper, we consider an IRS assisted WPSN as shown in Fig. 1. To be more
specific, a multi-antenna PS employs energy beamforming to broadcast wireless
energy to $K$ IoT devices who send their own information to an AP using the
harvested energy. Meanwhile, an IRS is deployed to enhance energy harvesting
and data transmission capabilities by generating passive energy and
information passive beamformers. It is assumed that the PS is equipped with
$N_{T}$ transmit antennas, the IRS consists of $N_{R}$ reflecting elements,
while the other devices (i.e., the AP, and all IoT devices) are single-antenna
nodes. The IRS controller is generally fixed to coordinate its reflection mode
by intelligently varying the phase shift of each reflecting element. In this
paper, all channel state information (CSI) is assumed to be perfectly known
because we are interested in an upper bound on the sum throughput performance
in the IRS assisted WPSN [7, 17]. Several channel estimation techniques have
been investigated in the literature for obtaining the CSI of the direct link
between the PS and the devices as well as the devices and the AP [7, 31].
Moreover, the IRS is equipped with a controller that coordinates its switching
between two working modes, such as reception mode for channel estimation and
reflection mode for energy/data transmission [17], which also provides a real-
time CSI feedback via this controller. Very recently, a novel channel
estimation framework was proposed based on the PARAllel FACtor (PARAFAC)
decomposition to unfold the resulting cascaded channel model [32, 33]. On the
other hand, we may take into consideration the passive pilots used for the
channel estimation of the cascaded CSI. To be specific, the reflecting
elements of the IRS passively reflect the pilot sequences transmitted from the
IoT devices to the AP/ PS to the IoT devices such that the CSI related to the
IRS can be obtained [34]. In addition, while imperfect CSI will degrade the
throughput performance due to the channel estimation errors, these channel
uncertainties have been tackled in the IRS assisted wireless networks by
taking into consideration robust resource allocations for active transmit and
passive reflecting beamformers [35, 36, 37]. In particular, these robust
schemes considered the imperfect cascaded BS-IRS-user channels, which
typically follow the bounded and statistical CSI uncertainties, and the worst-
case and outage probability robust beamforming designs are considered. We
denote
$\bm{\Theta}_{k}=\textrm{diag}\left[\beta_{k,1}\exp(j\alpha_{k,1}),...,\beta_{k,N}\exp(j\alpha_{k,N})\right],~{}|\exp(j\alpha_{k,n})|=1,~{}\forall
k\in[0,K],n\in[1,N_{R}]$ as the diagonal matrices associated with the phase
shifts of the IRS elements to reflect the $k$-th IoT device, where
$\alpha_{k,n}\in[0,2\pi)$ and $\beta_{k,n}\in[0,1]$ are the phase shift and
amplitude of the associated reflection coefficient. When $k=0$,
$\bm{\Theta}_{0}$ is known as the energy reflection phase shift matrix,
whereas $\bm{\Theta}_{k}$ is denoted as the information reflection phase
matrix when $k\in[1,K]$. Also, $\beta_{k,n},k\in[0,K],n\in[1,N_{R}]$ is
typically set to be 1 to maximize the reflected signal at the $n$-th element
of the $k$-th phase shift matrix.

###  II-A Transmission Protocol

In the IRS aided WPSN, we adopt a generic _harvest-then-transmit_ protocol,
and the whole operation time period is set to $T$. The PS first provides
wireless energy to the IoT devices during the downlink WET duration
$\tau_{0}\in[0,1]$. At the same time, the IRS collects the energy signal and
reflect it via IRS planar array. Then, these IoT devices utilize the harvested
energy to independently transmit their own information to the AP with the help
of the IRS by time division multiple access (TDMA). Thus, the uplink WIT time
duration of the $k$-th IoT device is denoted by $\tau_{k}\in[0,1]$. The time
allocation satisfies $\sum_{k=0}^{K}\tau_{k}=T$. The channel coefficients
between the PS and the $k$-th IoT device, the PS and the IRS, the IRS and the
$k$-th IoT device, the $k$-th IoT device and the AP, the $k$-th IoT device and
the IRS, as well as the IRS and the AP are denoted by
$\mathbf{g}_{d,k}\in\mathbb{C}^{N_{T}\times 1}$,
$\mathbf{G}_{0}\in\mathbb{C}^{N_{R}\times N_{T}}$,
$\mathbf{g}_{r,k}\in\mathbb{C}^{N_{R}\times 1}$,
$h_{d,k}\in\mathbb{C}^{1\times 1}$, $\mathbf{h}_{k}\in\mathbb{C}^{1\times
N_{R}}$, and $\mathbf{h}_{r}\in\mathbb{C}^{N_{R}\times 1}$,
respectively.111$\mathbf{h}_{k}=\mathbf{g}_{r,k}^{T}$. Thus, the RF harvested
energy at the $k$-th IoT device is given as222In this paper, we assume that
all IoT devices employ the linear energy harvesting. This is due to the fact
that this assumption practically holds when the harvested energy at the IoT
devices is relatively lower than its battery capacity.

$\displaystyle
E_{k}=\eta\tau_{0}\left|\left(\mathbf{g}_{r,k}^{H}\bm{\Theta}_{0}\mathbf{G}_{0}+\mathbf{g}_{d,k}^{H}\right)\mathbf{w}\right|^{2},~{}\forall
k\in[1,K],$ (1)

where $\eta\in[0,1]$ is the energy conversion efficiency; $\mathbf{w}$ denotes
the energy beamforming of the PS satisfying $\|\mathbf{w}\|^{2}\leq P_{0}$,
$P_{0}$ is the maximum transmit power available at the PS. It is assumed that
all harvested energy at each IoT device is used for the information
transmission, which achieves the maximum achievable throughput.333Practically,
each IoT device also requires constant energy consumption to support its
circuit operation. For convenience and without loss of generality, this
circuit energy consumption can be assumed to be zero in our work. Hence, the
achievable throughput of the $k$-th IoT device is given by

$\displaystyle R_{k}(\bm{\tau},\bm{\Theta})$
$\displaystyle\\!=\\!\tau_{k}\log\\!\left(\\!\\!1\\!+\\!\frac{\eta\tau_{0}\left|\left(\\!\mathbf{g}_{r,k}^{H}\bm{\Theta}_{0}\mathbf{G}_{0}\\!+\\!\mathbf{g}_{d,k}^{H}\\!\right)\mathbf{w}\right|^{2}\\!|\mathbf{h}_{k}\bm{\Theta}_{k}\mathbf{h}_{r}\\!+\\!h_{d,k}|^{2}}{\tau_{k}\sigma^{2}}\\!\right)\\!\\!,$

where $\bm{\tau}=[\tau_{0},\tau_{1},...,\tau_{K}]$,
$\bm{\Theta}=[\bm{\Theta}_{0},\bm{\Theta}_{1},...,\bm{\Theta}_{K}]$, and
$\sigma^{2}$ denotes the noise power at the AP.

###  II-B Problem Formulation

To evaluate the performance of the IRS assisted WPSN, we are interested in
maximizing the sum throughput which jointly designs the phase shifts of the
WET and WIT phases as well as the transmission time allocation. Hence, the sum
throughput maximization problem is formulated as

$\displaystyle\max_{\mathbf{w},\bm{\tau},\bm{\Theta}}$
$\displaystyle~{}\sum_{k=1}^{K}~{}R_{k}(\bm{\tau},\bm{\Theta})$ $\displaystyle
s.t.$
$\displaystyle~{}|\exp\left(j\alpha_{k,n}\right)|\\!=\\!1,~{}n\\!\in\\![1,N_{R}],$
(2a)
$\displaystyle~{}\sum_{k=0}^{K}\tau_{k}\\!\leq\\!T,~{}\tau_{k}\\!\geq\\!0,~{}\forall
k\\!\in\\![0,K],,$ (2b) $\displaystyle~{}\|\mathbf{w}\|^{2}\leq P_{0}.$ (2c)

In problem (2), (2a) is the constraint of phase shifts, (2b) denotes the the
transmission time constraint, and (2c) is the transmit power constraint of the
PS. Problem (2) cannot be solved directly due to its non-convexity with
respect to the energy beamforming $\mathbf{w}$, the time allocation vector
$\bm{\tau}$ and the phase shift matrices $\bm{\Theta}$. In the following, we
first derive the phase shifts of the WIT phase in closed-form. Then, the SDP
based AO and low complexity based AO schemes are applied to solve problem (2)
which alternately designs the energy beamforming, the time allocation, and the
phase shifts of the WET phase.

##  III Sum Throughput Maximization

In this section, we aim to solve the sum throughput maximization problem (2).
Specifically, we first optimize the phase shift matrix
$\bm{\Theta}_{k},~{}\forall k\in[1,K]$ by maximizing its objective function.
To proceed, the following _theorem_ is presented to derive the optimal closed-
form phase shifts of the WIT phase.

###### Theorem 1

The optimal phase shifts of the WIT phases, i.e., $\bm{\Theta}_{k}$ for
$\forall k\in[1,K]$, are given as

$\displaystyle\bm{\Theta}_{k}^{*}=\textrm{diag}\left(\theta_{k,1}^{*},...,\theta_{k,N_{R}}^{*}\right),~{}\forall
k\in[1,K],~{}\forall n\in[1,N_{R}],$ (3)

where $\theta_{k,n}^{*}=\exp\left(j\alpha_{k,n}^{*}\right)$,
$\alpha_{k,n}^{*}=\arg\left(h_{d,k}\right)-\arg\left(\mathbf{b}_{k}[n]\right)$,
and $\mathbf{b}_{k}=\textrm{diag}\left(\mathbf{h}_{k}\right)\mathbf{h}_{r}$.

###### Proof:

See Appendix -A. ∎

For the optimal phase shift matrix $\bm{\Theta}_{k}^{*}$, we have the
following _proposition_

###### Proposition 1

The optimal phase shift matrix $\bm{\Theta}_{k}^{*}$ aligns the cascaded link
between the $k$-th IoT device and the AP via the IRS with the direct link
between them, which can be expressed as

$\displaystyle\mathbf{h}_{k}\bm{\Theta}_{k}^{*}\mathbf{h}_{r}=\xi_{k}h_{d,k},~{}\forall
k\in[1,K],$ (4)

where $\xi_{k}$ is positive scalar denoting the signal strength metric from
the cascaded link between the $k$-th IoT device and the AP via the IRS.

###### Proof:

See Appendix -B. ∎

By exploiting _Proposition_ 1, we have the following _remark_

###### Remark 1

By exploiting _Proposition_ 1, for given $\mathbf{w}$, $\bm{\tau}$ and
$\bm{\Theta}_{0}$, the IRS can strengthen the received signal power of the
$k$-th IoT device at the AP at most $(1+\xi_{k})^{2}$ times in comparison to
that without IRS. In addition, $\xi_{k}$ for $\forall k\in[1,K]$ is
proportional to the number of the IRS reflecting elements. Hence, a
significant improvement can be introduced in terms of the sum throughput
performance with a larger number of reflecting elements. Moreover, the IRS
deployment as well as the pathloss model of the cascaded link between the
$k$-th IoT device and the AP via the IRS also play an important role in
affecting the received signal strength [38].

To proceed, we define
$t_{k}=|\mathbf{h}_{k}\bm{\Theta}_{k}^{*}\mathbf{h}_{r}\\!+\\!h_{d,k}|^{2}$
and substitute it into problem (2) to get

$\displaystyle\max_{\mathbf{w},\bm{\tau},\bm{\Theta}_{0}}$
$\displaystyle~{}\sum_{k=1}^{K}\tau_{k}\log\left(1+\frac{\eta\tau_{0}t_{k}\left|\left(\mathbf{g}_{r,k}^{H}\bm{\Theta}_{0}\mathbf{G}_{0}\\!+\\!\mathbf{g}_{d,k}^{H}\right)\mathbf{w}\right|^{2}}{\tau_{k}\sigma^{2}}\right)$
(5a) $\displaystyle s.t.$
$\displaystyle~{}|\exp\left(j\alpha_{0,n}\right)|\\!=\\!1,~{}\forall
n\in[1,N_{R}],$ (5b)
$\displaystyle~{}\eqref{eq:Original_problem_STM_time},~{}\eqref{eq:Original_problem_STM_energy}.$
(5c)

Problem (5) is still intractable due to the non-convex nature of its objective
function (5a) and the modulus constraint (5b). In order to circumvent this
non-convex issue, we propose the AO algorithm to solve (5). Specifically, we
first investigate the SDP relaxation scheme to reformulate (5) into a convex
optimization problem to alternately design the energy beamforming, the time
allocation and the phase shifts of the WET phase. Consequently, we propose a
low complexity scheme to derive the optimal closed-form energy beamforming,
time allocation and the phase shifts of the WET phase in closed-form which
efficiently reduces the computational complexity incurred by the SDP
relaxation scheme.

###  III-A SDP Relaxation Scheme

In this section, we consider an SDP relaxation scheme to alternately design
the energy beamformer $\mathbf{w}$, the time allocation $\bm{\tau}$, and the
phase shift matrix $\bm{\Theta}_{0}$ by applying the AO algorithm [17, 24]. To
be specific, we first fix $\bm{\Theta}_{0}$ to optimize $\mathbf{w}$ and
$\bm{\tau}$, then the $\bm{\Theta}_{0}$ can be optimally designed for given
$\mathbf{w}$ and $\bm{\tau}$. To proceed, we define
$\mathbf{V}_{0}=\tau_{0}\mathbf{W}$ and $\mathbf{W}=\mathbf{w}\mathbf{w}^{H}$
and problem (5) is relaxed as

$\displaystyle\max_{\bm{\tau},\mathbf{V}_{0}\succeq\mathbf{0}}$
$\displaystyle~{}\sum_{k=1}^{K}\tau_{k}\log\left(1+\frac{\tilde{t}_{k}\textrm{Tr}\left(\mathbf{\tilde{g}}_{k}\mathbf{\tilde{g}}_{k}^{H}\mathbf{V}_{0}\right)}{\tau_{k}}\right)$
(6a) $\displaystyle s.t.$
$\displaystyle~{}\textrm{Tr}(\mathbf{V}_{0})\leq\tau_{0}P_{0},$ (6b)
$\displaystyle~{}\eqref{eq:Original_problem_STM_time},~{}\textrm{rank}(\mathbf{V}_{0})=1,$
(6c)

where $\tilde{t}_{k}=\frac{\eta t_{k}}{\sigma^{2}}$,
$\mathbf{\tilde{g}}_{k}^{H}=\mathbf{g}_{r,k}^{H}\bm{\Theta}_{0}\mathbf{G}_{0}+\mathbf{g}_{d,k}^{H}$.
The following _lemma_ is required to characterize the joint convexity of
problem (6).

###### Lemma 1

By dropping the non-convex rank-one constraint
$\textrm{rank}(\mathbf{V}_{0})=1$, (6) is jointly convex with respect to
$\bm{\tau}$ and $\mathbf{V}_{0}$.

###### Proof:

See Appendix -C. ∎

Note that the relaxed problem (6) guarantees the rank-one nature of
$\mathbf{V}_{0}$, thus, the following _lemma_ is introduced to characterize
the rank-profile of $\mathbf{V}_{0}$.

###### Lemma 2

The optimal solution to problem (6) is a rank-one matrix, i.e.,
$\textrm{rank}(\mathbf{V}_{0}^{*})=1$.

###### Proof:

See Appendix -D. ∎

Since (6) is a convex optimization problem without the non-convex rank-one
constraint, we can employ interior-point methods [39], to efficiently solve it
to find the optimal solution $(\mathbf{V}_{0}^{*},\bm{\tau}^{*})$. We further
obtain $\mathbf{W}^{*}=\frac{\mathbf{V}_{0}^{*}}{\tau_{0}^{*}}$ and
$\textrm{rank}(\mathbf{W}^{*})=1$ by applying _Lemma_ 2. Thus, the optimal
energy beamformer $\mathbf{w}^{*}$ can be recovered from $\mathbf{W}^{*}$
using eigen-decomposition.

Then, we design the phase shift matrix $\bm{\Theta}_{0}$ for given
$\mathbf{w}$ and $\bm{\tau}$. To proceed, problem (5) is relaxed as

$\displaystyle\max_{\bm{\Delta}_{0}\succeq\mathbf{0}}$
$\displaystyle~{}\sum_{k=1}^{K}\tau_{k}\log\left(1+\frac{\tau_{0}\tilde{t}_{k}\textrm{Tr}\left(\bm{\Delta}_{0}\mathbf{F}_{k}\right)}{\tau_{k}}\right)$
(7a) $\displaystyle s.t.$
$\displaystyle~{}\bm{\Delta}_{0}(n,n)=1,~{}\textrm{rank}(\bm{\Delta}_{0})=1,$
(7b)

where $\bm{\Delta}_{0}=\bm{\tilde{\theta}}_{0}\bm{\tilde{\theta}}_{0}^{H}$,
$\mathbf{F}_{k}=\mathbf{\tilde{D}}_{k}\mathbf{w}\mathbf{w}^{H}\mathbf{\tilde{D}}_{k}^{H}$,
$\bm{\tilde{\theta}}_{0}^{H}=\left[\\!\\!\begin{array}[]{cc}\bm{\theta}_{0}^{H}&1\end{array}\\!\\!\right]$,
$\bm{\theta}_{0}=[\exp(j\alpha_{0,1}),...,\exp(j\alpha_{0,N_{R}})]^{H}$, and
$\mathbf{\tilde{D}}_{k}=\left[\\!\\!\begin{array}[]{cc}\textrm{diag}\left(\mathbf{g}_{r,k}^{H}\right)\mathbf{G}_{0}\\\
\mathbf{g}_{d,k}^{H}\end{array}\\!\\!\right]$. By dropping the non-convex
rank-one constraint $\textrm{rank}(\bm{\Delta}_{0})=1$, (7) is a convex
optimization problem, which can be efficiently solved via interior-point
methods [39]. Let us denote the solution to problem (7) as
$\bm{\Delta}_{0}^{*}$. Due to the rank-one relaxation, the relaxed problem (7)
may have a higher rank solution, i.e., $\textrm{rank}(\bm{\Delta}_{0}^{*})\geq
1$. In order to recover $\tilde{\bm{\theta}}_{0}$ from $\bm{\Delta}_{0}^{*}$,
we apply a Gaussian randomization technique to construct an approximate rank-
one solution [40]. Specifically, the eigenvalue decomposition of
$\mathbf{V}_{0}^{*}$ is written as
$\bm{\Delta}_{0}^{*}=\bm{\Upsilon}\bm{\Lambda}\bm{\Upsilon}^{H}$, where
$\bm{\Upsilon}\in\mathbb{C}^{(N_{R}+1)\times(N_{R}+1)}$ is an unitary matrix,
and $\bm{\Lambda}\in\mathbb{C}^{(N_{R}+1)\times(N_{R}+1)}$ is a diagonal
matrix with eigenvalues arranged in decreasing order. To proceed, we construct
a feasible solution as
$\tilde{\bm{\theta}}_{0}^{H}=\bm{\Upsilon}\bm{\Lambda}^{\frac{1}{2}}\bm{\kappa}$,
where $\bm{\kappa}\in\mathbb{C}^{(N_{R}+1)\times 1}$ is randomly generated to
follow complex circularly symmetric uncorrelated Gaussian variables with zero-
mean and covariance matrix $\mathbf{I}_{N_{R}+1}$. According to [17], the
optimal phase vector is given by
$\bm{\theta}_{0}^{*}=[\exp(j\alpha_{0,1}^{*}),...,\exp(j\alpha_{0,N_{R}}^{*})]$,
where $\alpha_{0,n}^{*}=\arg\left(\tilde{\bm{\theta}}_{0}(n)\right)$ for
$\forall n\in[1,N_{R}]$. Thus, the optimal $\bm{\Theta}_{0}^{*}$ can be
further obtained from $\bm{\theta}_{0}^{*}$. In the proposed AO algorithm, we
replace solving problem (5) with alternately solving problems (6) and (7) in
an iterative fashion, in which the solution obtained in each iteration is used
as the initial point of the next iteration. We summarize the detailed
procedure of the proposed AO algorithm in _Algorithm_ 1.

###### Algorithm 1

The SDP based AO algorithm to solve problem (5).

  1. 1.

Initialization: IRS phase shift matrix $\bm{\Theta}_{0}^{(0)}$ with iteration
number $m$.

  2. 2.

Repeat: Iteration number $m$

    1. (a)

Fix $\bm{\Theta}_{0}=\bm{\Theta}_{0}^{(m)}$ to obtain the optimal solution of
the energy beamforming $\mathbf{w}^{(m+1)}$ and the time allocation
$\bm{\tau}^{(m+1)}$ by solving problem (6).

    2. (b)

Fix $\mathbf{w}=\mathbf{w}^{(m+1)}$ and $\bm{\tau}=\bm{\tau}^{(m+1)}$ to
obtain the optimal solution of the phase shift matrix
$\bm{\theta}_{0}^{(m+1)}$ by solving problem (7).

    3. (c)

Set
$\bm{\Theta}_{0}^{(m+1)}=\textrm{diag}\left(\bm{\theta}_{0}^{(m+1)}\right)$.

    4. (d)

Update: $m=m+1$

  3. 3.

Until convergence.

Now, the following _proposition_ is required to characterize the convergence
of _Algorithm_ 1

###### Proposition 2

The objective value of problem (6) produces a non-descending trend over
iterations by employing _Algorithm_ 1, and its convergence is guaranteed.

###### Proof:

See Appendix -E. ∎

Note that the maximum values of problems(6) and (7) serve as upper bounds for
the reformulated counterparts of problem (5) with respect to $\mathbf{w}$ for
given $\bm{\tilde{\theta}}_{0}$ and with respect to $\bm{\tilde{\theta}}_{0}$
for given $\mathbf{w}$, respectively [41]. We solve problems (6) and (7) in
Steps (2-a) and (2-b) of _Algorithm_ 1 in an iterative fashion such that these
upper bounds can be monotonically tightened, and the objective values of each
iteration $(\mathbf{w}^{(m)},\bm{\tau}^{(m)},\bm{\Theta}_{0}^{(m)})$ yield a
non-decreasing sequence that converges to a stationary value in polynomial
time. Any limit point of the sequence
$(\mathbf{w}^{(m)},\bm{\tau}^{(m)},\bm{\Theta}_{0}^{(m)})$ is a stationary
point of problem (5) [42, 41]. Also, provided that problem (5) is always
feasible, _Algorithm_ 1 yields a feasible solution [43].

###  III-B Low Complexity Scheme

In Section III-A, we proposed the SDP based AO scheme to solve problem (5),
which alternately designs the energy beamforming, the time allocation, and the
phase shifts of the WET phase. However, the AO base SDP scheme may incur a
very high computational complexity due to the SDP relaxation with Gaussian
randomization. Thus, it is imperative to develop an independent scheme which
can be deployed on the IRS and the IoT devices. In this subsection, we propose
the low complexity based AO scheme which derives an optimal closed-form
solution to alternately solve problem (5) to significantly reduce the
computation complexity compared with the SDP based AO scheme.

####  III-B1 Optimization of $\mathbf{w}$ and $\tau$ for given $\Theta_{0}$

First, let $\mathbf{W}=\mathbf{w}\mathbf{w}^{H}$, we rewrite problem (5) for
given $\bm{\Theta}_{0}$ as (8) 444Here, problem (5) is rewritten as (8)
instead of (6). This is due to the fact that it is more convenient to derive
closed-form solutions of the transmission time allocation and the energy
beamforming. Note that both problems (6) and (8) have the same solution.

$\displaystyle\max_{\mathbf{W}\succeq\mathbf{0},\bm{\tau}}$
$\displaystyle~{}\sum_{k=1}^{K}\tau_{k}\log\left(1+\frac{\tau_{0}\tilde{t}_{k}\textrm{Tr}(\mathbf{\tilde{g}}_{k}\mathbf{\tilde{g}}_{k}^{H}\mathbf{W})}{\tau_{k}}\right)$
(8a) $\displaystyle s.t.$ $\displaystyle~{}\textrm{Tr}(\mathbf{W})\leq
P_{0},~{}\eqref{eq:Original_problem_STM_time}.$ (8b)

Note that $\textrm{rank}(\mathbf{W})=1$ is satisfied in problem (8) due to the
SDP relaxation (i.e., $\mathbf{W}=\mathbf{w}\mathbf{w}^{H}$). Problem (8) is
not jointly convex with respect to $\mathbf{W}$ and $\bm{\tau}$ due to the
non-convex property of (8a) which cannot be directly solved. In order to
circumvent this issue, we consider the following Lagrange dual function for
given $\mathbf{W}$ to derive a closed-form solution for $\bm{\tau}$.

$\displaystyle\mathcal{L}(\bm{\tau},\mu)$
$\displaystyle~{}=\sum_{k=1}^{K}\tau_{k}\log\left(1+\frac{\tau_{0}\tilde{t}_{k}\textrm{Tr}(\mathbf{\tilde{g}}_{k}\mathbf{\tilde{g}}_{k}^{H}\mathbf{W})}{\tau_{k}}\right)$
$\displaystyle~{}~{}~{}~{}-\mu\left(\tau_{0}+\sum_{k=1}^{K}\tau_{k}-T\right),$
(9)

where $\mu$ is the non-negative dual multiplier associated with the constraint
(2b). Also, its dual problem is given by

$\displaystyle\min_{\bm{\tau}\in\mathcal{S}}~{}\mathcal{L}(\bm{\tau},\mu),$
(10)

where $\mathcal{S}$ is the feasible set of any $\tau_{k}$ for $\forall
k\in[0,K]$, which has been shown in the constraint (2b). Note that (8) can be
reformulated into a convex optimization problem and guarantees Slater’s
condition, since $\bm{\tau}\in\mathcal{S}$, $\bm{\tau}\succeq\mathbf{0}$, and
$\sum_{k=1}^{K}\tau_{k}<T$. Hence, strong duality holds such that the optimal
solution to (8) satisfies the Karush-Kuhn-Tucker (KKT) conditions [39], which
is given by

$\displaystyle~{}\mu\left(\tau_{0}+\sum_{k=1}^{K}\tau_{k}-T\right)=0,$ (11a)
$\displaystyle~{}\frac{\partial\mathcal{L}}{\partial\tau_{k}}=0,~{}\forall
k\in[1,K].$ (11b)

From (11a), we have the optimal dual variable satisfies $\mu^{*}>0$, since the
optimal solution to problem (8) (i.e., $\bm{\tau}^{*}$) guarantees the
equality $\sum_{k=0}^{K}\tau_{k}=T$. According to (11b), we take into
consideration the first-order derivative of (III-B1) in terms of $\tau_{k}$
for $\forall k\in[1,K]$ and set it to zero, which is given by

$\displaystyle\\!\\!\\!\\!\\!\log\left(\\!1\\!+\\!\frac{\tau_{0}\tilde{t}_{k}\textrm{Tr}(\mathbf{\tilde{g}}_{k}\mathbf{\tilde{g}}_{k}^{H}\mathbf{W})}{\tau_{k}}\\!\right)\\!-\\!\frac{\tau_{0}\tilde{t}_{k}\textrm{Tr}(\mathbf{\tilde{g}}_{k}\mathbf{\tilde{g}}_{k}^{H}\mathbf{W})}{\tau_{k}+\tau_{0}\tilde{t}_{k}\textrm{Tr}(\mathbf{\tilde{g}}_{k}\mathbf{\tilde{g}}_{k}^{H}\mathbf{W})}\\!=\\!\mu.\\!\\!\\!\\!$
(12)

The left hand side (LHS) of (12) can be written in the form of
$f(x)=\log(1+x)-\frac{x}{1+x}$, which is a monotonically increasing function
with respect to $x$. To guarantee the $K$ equations in (12), we further have

$\displaystyle\frac{\tau_{0}\tilde{t}_{1}\textrm{Tr}(\mathbf{\tilde{g}}_{1}\mathbf{\tilde{g}}_{1}^{H}\mathbf{W})}{\tau_{1}}\\!=\\!...\\!=\\!\frac{\tau_{0}\tilde{t}_{K}\textrm{Tr}(\mathbf{\tilde{g}}_{K}\mathbf{\tilde{g}}_{K}^{H}\mathbf{W})}{\tau_{K}}.$
(13)

Let
$\frac{1}{\lambda}=\frac{\tau_{0}\tilde{t}_{k}\textrm{Tr}(\mathbf{\tilde{g}}_{k}\mathbf{\tilde{g}}_{k}^{H}\mathbf{W})}{\tau_{k}}$,
we have

$\displaystyle\tau_{k}=\lambda\tau_{0}\tilde{t}_{k}\textrm{Tr}(\mathbf{\tilde{g}}_{k}\mathbf{\tilde{g}}_{k}^{H}\mathbf{W}),$
(14)

Substitute (14) into the constraint (2b), the following derivations are
required to obtain the following expression for $\lambda$

$\displaystyle\lambda=\frac{T-\tau_{0}}{\sum_{k=1}^{K}\tau_{0}\tilde{t}_{k}\textrm{Tr}(\mathbf{\tilde{g}}_{k}\mathbf{\tilde{g}}_{k}^{H}\mathbf{W})}.$
(15)

Substitute (15) into (14), the optimal solution of $\tau_{k}$ for $\forall
k\in[1,K]$ is given by

$\displaystyle\tau_{k}^{*}=\frac{(T-\tau_{0})\tau_{0}\tilde{t}_{k}\textrm{Tr}(\mathbf{\tilde{g}}_{k}\mathbf{\tilde{g}}_{k}^{H}\mathbf{W})}{\sum_{k=1}^{K}\tau_{0}\tilde{t}_{k}\textrm{Tr}(\mathbf{\tilde{g}}_{k}\mathbf{\tilde{g}}_{k}^{H}\mathbf{W})}.$
(16)

We substitute (16) into problem (8) and let
$\mathbf{\tilde{G}}=\left[\sqrt{\tilde{t}_{1}}\mathbf{\tilde{g}}_{1},...,\sqrt{\tilde{t}_{K}}\mathbf{\tilde{g}}_{K}\right]$,
(8) is written as

$\displaystyle\max_{\mathbf{W}\succeq\mathbf{0},\tau_{0}}$
$\displaystyle~{}(T-\tau_{0})\log\left(1+\frac{\tau_{0}\textrm{Tr}\left(\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\mathbf{W}\right)}{T-\tau_{0}}\right)$
(17a) $\displaystyle s.t.$ $\displaystyle~{}\textrm{Tr}(\mathbf{W})\leq
P_{0},~{}0\leq\tau_{0}\leq T.$ (17b)

Problem (17) is still not jointly convex in terms of $\mathbf{W}$ and
$\tau_{0}$. In the following, we derive the optimal closed-form solution
$\mathbf{W}$ for given $\tau_{0}$ using the following _theorem_.

###### Theorem 2

The optimal closed-form solution to problem (17), i.e., $\mathbf{W}$, is given
by
$\mathbf{W}^{*}=P_{0}\bm{\nu}_{\max}\left(\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\right)\bm{\nu}_{\max}\left(\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\right)^{H}$.

###### Proof:

See Appendix -F. ∎

Due to the rank-one condition $\textrm{rank}(\mathbf{W})=1$, the optimal
energy beamformer is easily derived as
$\mathbf{w}^{*}=\sqrt{P_{0}}\bm{\nu}_{\max}\left(\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\right)$.
By applying _Theorem_ 2, problem (17) can be rewritten with respect to
$\tau_{0}$ as

$\displaystyle\max_{\tau_{0}}$
$\displaystyle~{}(T-\tau_{0})\log\left(1+\frac{\tau_{0}P_{0}\rho_{\max}\left(\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\right)}{T-\tau_{0}}\right)$
$\displaystyle s.t.$ $\displaystyle~{}0\leq\tau_{0}\leq T.$ (18)

Problem (III-B1) is a single-value optimization problem, where the optimal
time allocation $\tau_{0}$ can be easily attained by using one-dimensional
line search, e.g., golden search [44]. However, this numerical search will
introduce a very high computational complexity. In order to circumvent this
issue, we propose a low complexity method to derive the optimal closed-form
$\tau_{0}$ without such an exhaustive search. To proceed, the following
_theorem_ is required

###### Theorem 3

The optimal time allocation $\tau_{0}$ can be derived in closed-form as

$\displaystyle\tau_{0}^{*}\\!=\\!\frac{\exp\left[\mathcal{W}\left(\frac{P_{0}\rho_{\max}\left(\\!\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\\!\right)\\!-\\!1}{\exp(1)}\right)\\!+\\!1\right]\\!-\\!1}{P_{0}\rho_{\max}\left(\\!\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\\!\right)\\!-\\!1\\!+\\!\exp\left[\mathcal{W}\left(\frac{P_{0}\rho_{\max}\left(\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\right)\\!-\\!1}{\exp(1)}\right)\\!+\\!1\right]}T.$
(19)

###### Proof:

See Appendix -G. ∎

####  III-B2 Optimization of $\Theta_{0}$ for given $\mathbf{w}$ and $\tau$

Here, we first rewrite problem (5) with respect to $\bm{\Theta}_{0}$ as

$\displaystyle\max_{\bm{\Theta}_{0}}$
$\displaystyle~{}(1\\!-\\!\tau_{0})\log\left(\\!1\\!+\\!\frac{\tau_{0}\sum_{k=1}^{K}\tilde{t}_{k}\left|\left(\mathbf{g}_{r,k}^{H}\bm{\Theta}_{0}\mathbf{G}_{0}\\!+\\!\mathbf{g}_{d,k}^{H}\right)\mathbf{w}\right|^{2}}{1\\!-\\!\tau_{0}}\right)$
$\displaystyle s.t.$
$\displaystyle~{}|\exp\left(j\alpha_{0,n}\right)|\\!=\\!1,~{}\forall
n\in[1,N_{R}],$ (20)

Problem (III-B2) is not convex due to its non-convex unit-modulus constraint.
Thus, we propose a novel method to derive the optimal closed-form phase shifts
of the WET phase. It is easily verified that maximizing problem (III-B2) is
equivalent to the maximization of the following problem

$\displaystyle\max_{\bm{\theta}_{0}}$
$\displaystyle~{}\sum_{k=1}^{K}\tilde{t}_{k}\left|\bm{\theta}_{0}^{H}\mathbf{b}_{k}+d_{k}\right|^{2},$
(21a) $\displaystyle s.t.$
$\displaystyle~{}|\bm{\theta}_{0}(n)|\\!=\\!1,~{}\forall n\in[1,N_{R}],$ (21b)

where
$\mathbf{b}_{k}=\textrm{diag}\left(\mathbf{g}_{r,k}^{H}\right)\mathbf{G}_{0}\mathbf{w}$
and $d_{k}=\mathbf{g}_{d,k}^{H}\mathbf{w}$. By applying a few mathematical
manipulations to (21a), we have

$\displaystyle\sum_{k=1}^{K}\tilde{t}_{k}\left|\bm{\theta}_{0}^{H}\mathbf{b}_{k}+d_{k}\right|^{2}$
$\displaystyle~{}=\bm{\theta}_{0}^{H}\bm{\Phi}_{1}\bm{\theta}_{0}+2\Re\left\\{\bm{\theta}_{0}^{H}\bm{\gamma}\right\\}+e_{1},$
(22)

where
$\bm{\Phi}_{1}=\sum_{k=1}^{K}\tilde{t}_{k}\mathbf{b}_{k}\mathbf{b}_{k}^{H}$,
$\bm{\gamma}=\sum_{k=1}^{K}\tilde{t}_{k}\textrm{conj}(d_{k})\mathbf{b}_{k}$,
and $e_{1}=\sum_{k=1}^{K}\tilde{t}_{k}d_{k}\textrm{conj}(d_{k})$. Accordingly,
problem (21) is equivalent to

$\displaystyle\min_{\bm{\theta}_{0}}$
$\displaystyle~{}\bm{\theta}_{0}^{H}\bm{\Phi}\bm{\theta}_{0}-2\Re\left\\{\bm{\theta}_{0}^{H}\bm{\gamma}\right\\}+e,$
(23a) $\displaystyle s.t.$ $\displaystyle~{}|\bm{\theta}_{0}(n)|=1,~{}\forall
n\in[1,N_{R}],$ (23b)

where $\bm{\Phi}=-\bm{\Phi}_{1}$, and $e=-e_{1}$.

Problem (23) is still intractable due to its unit modulus equality constraint
(23b). In order to tackle this issue, the Majorization-Minimization (MM)
algorithm is adopted, and a sequence of tractable sub-problems are considered
to iteratively solve problem (23) by approximating its objective function and
constraint set [42]. Now, consider the following problem:

$\displaystyle\min_{\mathbf{x}}$
$\displaystyle~{}f_{0}(\mathbf{x}),~{}s.t.~{}f_{i}(\mathbf{x})\leq
0,~{}i=1,...,L.$ (24)

We approximate both the objective function and feasible constraints set of
problem (24) at each iteration.555Here we assume that $f_{i}$ is differential
[42]. Thus, the following convex sub-problem can be solved at the $m$-th
iteration.

$\displaystyle\min_{\mathbf{x}}$
$\displaystyle~{}g_{0}(\mathbf{x}|\mathbf{x}^{(m)}),~{}s.t.~{}g_{i}(\mathbf{x}|\mathbf{x}^{(m)})\leq
0,i=1,...,L,$ (25)

where $g_{i}(\mathbf{*}|\mathbf{x}^{(m)})$, $\forall m=0,...,L$ denotes a
convex function which guarantees the following conditions:

$\displaystyle g_{i}(\mathbf{x}^{(m)}|\mathbf{x}^{(m)})$
$\displaystyle=f_{i}(\mathbf{x}^{(m)}),$ $\displaystyle
g_{i}(\mathbf{x}|\mathbf{x}^{(m)})$ $\displaystyle\geq f_{i}(\mathbf{x}),$
(26) $\displaystyle\nabla g_{i}(\mathbf{x}^{(m)}|\mathbf{x}^{(m)})$
$\displaystyle=\nabla f_{i}(\mathbf{x}^{(m)}).$

The sequence $\mathbf{x}^{(m)}$ results in a monotonically decreasing
$f_{0}(\mathbf{x})$ which converges to a KKT point [42]. In other words, the
sub-problem (25) is introduced from the upper bound of the objective function
in (24) via a convex surrogate function, and the feasible set in (24) is
approximated via linearization [42].

###### Proposition 3

[45, 42] The objective function (23a) is approximated in the following for any
given $\bm{\theta^{(m)}}$ at the $m$-th iteration and for any feasible
$\bm{\theta}$.

$\displaystyle~{}f(\bm{\theta}_{0})\triangleq~{}\bm{\theta}_{0}\bm{\Phi}\bm{\theta}_{0}^{H}\\!-\\!2\Re\\{\bm{\theta}_{0}\bm{\gamma}\\}\\!+\\!e$
$\displaystyle\leq\bm{\theta}_{0}\bm{\Upsilon}\bm{\theta}_{0}^{H}\\!-\\!2\Re\left\\{\bm{\theta}_{0}\left[(\bm{\Upsilon}-\bm{\Phi})\tilde{\bm{\theta}}_{0}^{H}\\!+\\!\bm{\gamma}\right]\right\\}\\!+\\!\tilde{\bm{\theta}}_{0}(\bm{\Upsilon}\\!-\\!\bm{\Phi})\tilde{\bm{\theta}}_{0}^{H}\\!+\\!e$
$\displaystyle\\!=\\!\rho_{\max}(\bm{\Phi})\|\bm{\theta}_{0}\|^{2}\\!-\\!2\mathcal{R}\left\\{\\!\bm{\theta}_{0}\left[\left(\\!\rho_{\max}(\bm{\Phi})\mathbf{I}_{N_{R}\times
N_{R}}\\!-\\!\bm{\Phi}\\!\right)\tilde{\bm{\theta}}_{0}^{H}\\!+\\!\bm{\gamma}\right]\\!\right\\}$
$\displaystyle~{}~{}~{}~{}+\tilde{e}~{}\triangleq~{}g(\bm{\theta}|\bm{\theta}^{(m)}),$
(27)

where
$\tilde{e}=\tilde{\bm{\theta}}_{0}\left[\rho_{\max}(\bm{\Phi})\mathbf{I}_{N_{R}\times
N_{R}}-\bm{\Phi}\right]\tilde{\bm{\theta}}_{0}^{H}+e$, and
$\bm{\Upsilon}=\rho_{\max}(\bm{\Phi})\mathbf{I}_{N_{R}\times N_{R}}$. Also,
$\tilde{\bm{\theta}}_{0}$ denotes the approximated solution to
$\bm{\theta}_{0}$ which is achieved in the previous iteration via the
alternating algorithm.

_Proposition_ 3 constructs a surrogate function of (23a), and it is easily
verified that $g(\bm{\theta}|\bm{\theta}^{(m)})$ in (3) guarantees the
conditions in (III-B2). By exploiting _Proposition_ 3, problem (23) is
reformulated as

$\displaystyle\min_{\bm{\theta}_{0}}$
$\displaystyle~{}\rho_{\max}\left(\bm{\Phi}\right)\|\bm{\theta}_{0}\|^{2}-2\Re\left\\{\bm{\theta}_{0}^{H}\tilde{\bm{\gamma}}\right\\},$
$\displaystyle s.t.$ $\displaystyle~{}|\bm{\theta}_{0}(n)|=1,~{}\forall
n\in[1,N_{R}],$ (28)

where $\tilde{\bm{\gamma}}=\left(\rho_{\max}(\bm{\Phi})\mathbf{I}_{N_{R}\times
N_{R}}-\bm{\Phi}\right)\tilde{\bm{\theta}}_{0}^{H}+\bm{\gamma}$. Also, it is
easy to obtain $\|\bm{\theta}_{0}\|^{2}=N_{R}$ due to
$|\bm{\theta}_{0}(n)|=1,~{}\forall n\in[1,N_{R}]$. The term
$\Re\left\\{\bm{\theta}_{0}\tilde{\bm{\gamma}}\right\\}$ can be maximized when
the phases of $\bm{\theta}_{0}(n)$ and $\tilde{\bm{\gamma}}(n)$ are identical.
Hence, the optimal solution to (III-B2) is derived as

$\displaystyle\bm{\theta}_{0}^{*}=\left[\\!\\!\begin{array}[]{ccc}\exp\left(j\arg[\tilde{\bm{\gamma}}(1)]\right)\\!\\!&,...,&\\!\\!\exp\left(j\arg[\tilde{\bm{\gamma}}(N)]\right)\end{array}\\!\\!\right].$
(30)

Although problem (21) can be easily relaxed as a SDP, the optimal closed-form
phase shift in (30) provides a more efficient way for practical implementation
and reduction of the computational complexity introduced by the SDP,
especially for a large number of reflecting elements $N_{R}$.

###### Algorithm 2

The low complexity based AO algorithm to solve problem (5).

  1. 1.

Initialization: IRS phase shift matrix $\bm{\Theta}_{0}^{(0)}$ with iteration
number $m$.

  2. 2.

Repeat: Iteration number $m$

    1. (a)

Fix $\bm{\Theta}_{0}=\bm{\Theta}_{0}^{(m)}$ to optimize energy beamforming
$\mathbf{w}^{(m+1)}$ and time allocation
$\bm{\tau}^{(m+1)}=\left[\tau_{0}^{(m+1)},...,\tau_{K}^{(m+1)}\right]$.

      1. i.

Obtain $\mathbf{w}^{(m+1)}$ via _Theorem_ 2.

      2. ii.

Obtain $\tau_{0}^{(m+1)}$ via _Theorem_ 3.

      3. iii.

Obtain $\tau_{k}^{(m+1)}$ for $\forall k\in[1,K]$ via (16).

    2. (b)

Fix $\mathbf{w}=\mathbf{w}^{(m+1)}$ and $\bm{\tau}=\bm{\tau}^{(m+1)}$ to
optimize phase shift vector $\bm{\theta}_{0}^{(m+1)}$ which can be obtained
via (30).

    3. (c)

Set
$\bm{\Theta}_{0}^{(m+1)}=\textrm{diag}\left(\bm{\theta}_{0}^{(m+1)}\right)$.

    4. (d)

Update $m=m+1$.

  3. 3.

Until convergence.

The objective value of problem (8) produces a non-decreasing trend over
iterations by applying _Algorithm_ 2, and its convergence is guaranteed. The
proof of this statement is similar to that of _Proposition_ 2, and is omitted
here due to space limitation.

##  IV Numerical Results

Figure 2: System deployment.

In this section, numerical results are presented to illustrate the theoretical
results of the proposed algorithms as derived in Section III. In our
simulations, we consider the system deployment shown in Fig. 2. Unless
otherwise stated, the PS, the AP, and the IRS are located at $(-10,0,0)$,
$(10,0,0)$ and $(-2,6,0)$. All IoT sensor nodes are assigned at
$(0,0,\frac{n\times l}{2})$ if $n=1,...,(2k+1)$, and
$\left(0,0,-\frac{(n-1)\times l}{2}\right)$, if $n=2,...,2k$, where $l$ is the
interval between two neighbouring IoT devices. The channel coefficient is
composed of distance-dependent path loss model and small-scale fading. The
path loss model is set to $P_{L}=Ad^{-\varepsilon}$, where
$A=-30~{}\textrm{dB}$, $\varepsilon$ is the path loss exponent, and $d$
represents the distance between any two nodes, i.e., the PS and the IRS, the
PS and the $k$-th IoT devices, the IRS and the $k$-th IoT device, the IRS and
the AP, as well as the $k$-th IoT device and the AP. The channel coefficients
between the PS and the IRS, the IRS and the $k$-th IoT device, the $k$-th IoT
device and the IRS, as well as the IRS and the AP are modelled as
$\mathbf{G}_{0}=\sqrt{\frac{K_{1}}{K_{1}+1}}\mathbf{G}_{0}^{\textrm{LOS}}+\sqrt{\frac{1}{K_{1}+1}}\mathbf{G}_{0}^{\textrm{NLOS}}$,
$\mathbf{g}_{r,k}=\sqrt{\frac{K_{1}}{K_{1}+1}}\mathbf{g}_{r,k}^{\textrm{LOS}}+\sqrt{\frac{1}{K_{1}+1}}\mathbf{g}_{r,k}^{\textrm{NLOS}}$,
$\mathbf{h}_{k}=\mathbf{g}_{r,k}^{T}$, and
$\mathbf{h}_{r}=\sqrt{\frac{K_{1}}{K_{1}+1}}\mathbf{h}_{r}^{\textrm{LOS}}+\sqrt{\frac{1}{K_{1}+1}}\mathbf{h}_{r}^{\textrm{NLOS}}$,
where $\mathbf{G}_{0}^{\textrm{LOS}}$, $\mathbf{g}_{r,k}^{\textrm{LOS}}$, and
$\mathbf{h}_{r}^{\textrm{LOS}}$ denote the line-of-sight (LOS) deterministic
components of the corresponding channel coefficients;
$\mathbf{G}_{0}^{\textrm{NLOS}}$, $\mathbf{g}_{r,k}^{\textrm{NLOS}}$, and
$\mathbf{h}_{r}^{\textrm{NLOS}}$ are the non-line-of-sight (NLOS) components
of the corresponding channel coefficients which follow the Rayleigh fading. In
addition, $K_{1}$ is the Rician factor which is set to $5~{}\textrm{dB}$ for
convenience and without loss of generality. The remaining small-scale channel
coefficients are generated as circularly symmetric Gaussian random variables
with zero mean and unit variance. Other parameters of the simulations are: the
total transmission time period $T=1$, the number of transmit antenna at the PS
$N_{T}=6$, number of the IRS reflecting elements $N_{R}=30$, number of IoT
devices $K=5$, transmit power at the PS $P_{0}=25~{}\textrm{dBm}$, noise power
at the AP $\sigma^{2}=-90~{}\textrm{dBm}$, and energy conversion efficiency
$\eta=0.8$, unless otherwise specified.

Figure 3: Convergence of _Algorithm_ 1 and _Algorithm_ 2.

Figure 4: Sum throughput versus transmit power at the PS ($P_{0}$).

We first plot the convergence performance of _Algorithm_ 1 and _Algorithm_ 2
in Fig. 4. From this figure, it is seen that the sum throughput increases with
iterations and then achieves a converged value within five iterations, which
confirms the convergence of the proposed AO algorithms. Also, the proposed low
complexity scheme performance matches with the SDP relaxation scheme when the
algorithms converge, which confirms the theoretical derivations as shown in
Section III-B.

To highlight our proposed schemes, we also evaluate the performance of the
following benchmark schemes under the same parameters for comparison.

  * •

Random phase shift (RPS): the phase shifts are uniformly and randomly
distributed in $[0,2\pi)$.

  * •

Fixed time allocation (FTA) ($\tau_{0}=0.5$). [46].

  * •

Equal time allocation (ETA) ($\tau_{k}=\frac{1}{K+1}$, $\forall k\in[0,K]$)
[46].

  * •

Without IRS [7].

In Fig. 4, the sum throughput versus the transmit power at the PS $P_{0}$ is
plotted for each scheme. From this figure, one can observe that the sum
throughput increases with $P_{0}$, because more energy is collected by the IoT
devices with a higher PS’s transmit power which yields a larger sum
throughput. In addition, the performances of the proposed SDP based AO and low
complexity based AO schemes are identical and significantly outperform the
benchmark schemes and the gap between the proposed schemes and the benchmark
schemes becomes larger. This highlights the beneficial role of the IRS in the
WPSN.

Figure 5: Sum throughput versus number of reflecting elements at the IRS
$N_{R}$.

Figure 6: Sum throughput versus number of transmit antenna at the PS $N_{T}$.

Next, we evaluate the sum throughput versus the number of reflecting elements
at the IRS $N_{R}$ in Fig. 6. It can observed from this figure that large-size
reflecting arrays can achieve higher sum throughput than all benchmark
schemes. In benchmark schemes, FTA achieves a slower increase in terms of sum
throughput, since it employs a given time allocation instead of optimal time
allocation, while RPS and ETA have an inconspicuous increase due to random
phase shift distributed and equal time allocation. The scheme without IRS
remains constant with $N_{R}$ since the IRS does not involve energy and
information reflections during the WET and WIT phases.

Figure 7: Sum throughput versus number of IoT devices $K$.

Figure 8: Sum throughput versus pathloss exponent between the PS and the IRS.

Then, we demonstrate the impact of the number of transmit antennas at the PS
$N_{T}$ and IoT devices $K$, on the sum throughput in Fig. 6 and Fig. 8,
respectively. Fig. 6 demonstrates that the proposed schemes outperform all
benchmark schemes in terms of sum throughput which increases with $N_{T}$.
This is due to the fact that a larger number of transmit antennas at the PS
brings more power via energy beamforming to improve energy/information
reflection efficiency. In addition, Fig. 8 shows that a large number of IoT
devices $K$ improve the throughput performance for each scheme. To be
specific, the sum throughput gradually increases with respect to $K$ and until
about $K\geq 8$. This is due to the fact that as $K$ increases, more and more
IoT devices are located farther away from the PS, the IRS, and the AP (refer
to the system deployment in Fig. 2) which degrades the energy and information
reflection efficiencies such that the sum throughput gain becomes smaller.

Furthermore, the impact of the reflection’s path loss model on the sum
throughput is evaluated in Fig. 8 and Fig. 10, which investigate the energy
reflection performance in the reflecting link from the PS to the IRS as well
as the IRS to the IoT devices, respectively. Fig. 8 depicts the sum throughput
versus the path loss exponent between the PS and the IRS, i.e., $\varepsilon$
is denoted by $\varepsilon_{\textrm{PS2IRS}}$. It is apparent from this figure
that the sum throughput exhibits a declining trend with
$\varepsilon_{\textrm{PS2IRS}}$ for the IRS assisted schemes, since a larger-
scale fading between the PS and the IRS will result in a weaker energy signal
reflected from the IRS, diminishing its benefits. Moreover, the proposed
schemes have a significantly better performance than all benchmark schemes in
terms of sum throughput. Specifically, the FTA can achieve a very close
performance to the proposed schemes for a larger region of the path loss
exponent $\varepsilon_{\textrm{PS2IRS}}$, which demonstrates that the optimal
time allocation $\tau_{0}$ of the proposed schemes approaches $0.5$ at this
point. Since the IRS is not deployed in the traditional WPSN, its sum
throughput remains constant with $\varepsilon_{\textrm{PS2IRS}}$. Fig. 10
depicts the sum throughput versus the path loss exponent between the IRS and
the IoT devices, i.e., $\varepsilon$ is denoted by
$\varepsilon_{\textrm{IRS2D}}$. Interestingly, similar trends and arguments
from Fig. 8 can be also observed here in terms of the sum throughput, which
has been omitted due to space limitation. Fig. 10 depicts the sum throughput
versus the path loss exponent between the IRS and the AP, i.e., $\varepsilon$
is denoted by $\varepsilon_{\textrm{IRS2AP}}$, which studies the information
reflection of the reflecting link from the IRS to the AP. A slight throughput
decreases is observed from this figure for the proposed schemes, which
outperform their counterparts, i.e., RPS, FTA and ETA. This demonstrates that
the information reflection of the IRS has a slight impact on the sum
throughput due to large-scale fading and attenuation of the information signal
transmitted by the energy harvested IoT devices.

Figure 9: Sum throughput versus pathloss exponent between the IRS and the IoT
devices.

Figure 10: Sum throughput versus pathloss exponent between the IRS and the AP.

Figure 11: Sum throughput versus x-coordinate of the IRS.

Figure 12: Sum throughput versus x-coordinate of the IoT devices.

Finally, we evaluate the impact of the x-coordinate of the IRS (denoted by
$X_{IRS}$) and the IoT devices (denoted by $X_{U}$) on the sum throughput in
Fig. 12 and 12, respectively. In Fig. 12, we plot the sum throughput versus
the x-coordinate of the IRS, i.e., $X_{IRS}$. It is apparent from this figure
that the sum throughput first increases and then decreases with $X_{IRS}$.
This illustrates that the optimal deployment of the IRS effectively improves
the energy collection at the IoT devices so as to maximize the information
reception at the AP. Fig. 12 shows the sum throughput versus x-coordinate of
the IoT devices, i.e., $X_{U}$. The PS and AP are located $(-5,0,0)$ and
$(5,0,0)$ for convenience and without loss of generality. As seen in this
figure, the sum throughput first decreases and then increases for each scheme.
This illustrates that the IoT devices should be deployed either in the
proximity to the AP or to the PS, resulting in a higher sum throughput.

##  V Conclusion

In this paper, we maximize the sum throughput of an IRS assisted WPSN by
optimally designing the energy beamforming, the phase shifts of the WET and
WIT phases and the transmission time allocation. We first derive the optimal
phase shifts of the WIT phase in closed-form. Then, the AO algorithm was
applied to alternately solve the sum throughput maximization problem.
Specifically, the SDP relaxation scheme was adopted to alternately design the
energy beamforming and the phase shifts of the WET phase, respectively. In
addition, a low complexity scheme was proposed to derive the optimal energy
beamformer, the phase shifts of WET phase and the time allocation in closed-
form. Numerical results were presented to quantify the sum throughput gains of
the proposed algorithm in comparison to the several benchmarks. For future
work, it is interesting to investigate robust designs for the IRS assisted
WPSN, where the imperfect cascaded CSIs of the PS-IRS-IoT device and IoT
device-IRS-AP channels are unknown. These cascaded CSI uncertainties follow
the bounded and statistical models, and the worst-case and outage probability
robust beamforming designs may be adopted in the IRS assisted WPSN. However,
these robust designs may demonstrate that a low complexity scheme is not
feasible since it may not be possible to derive closed-form solutions of the
phase shifts of the IRS, the transmission time allocation, and the energy
beamforming.

###  -A Proof of _Theorem_ 1

To prove _Theorem_ 1, we first need to solve an equivalent sub-problem. To
proceed, it is easily verified that the throughput at the AP from the $k$-th
device $R_{k}$ is monotonically increasing with respect to
$\left|\mathbf{h}_{k}\bm{\Theta}_{k}\mathbf{h}_{r}+h_{d,k}\right|^{2}$. Thus,
solving problem (2) with respect to $\bm{\Theta}_{k}$ is equivalent to
maximizing
$\left|\mathbf{h}_{k}\bm{\Theta}_{k}\mathbf{h}_{r}+h_{d,k}\right|^{2}$, for
$\forall k\in[1,K]$, each of which only relies on $\bm{\Theta}_{k}$ with
$\left|\exp\left(j\alpha_{k,n}\right)\right|=1$ and $\alpha_{k,n}\in[0,2\pi]$.
Thus, we solve the following sub-problem with respect to $\bm{\Theta}_{k}$,
$\forall k\in[1,K]$, instead of solving (2)

$\displaystyle\max_{\bm{\Theta}_{k}}$
$\displaystyle~{}\left|\mathbf{h}_{k}\bm{\Theta}_{k}\mathbf{h}_{r}+h_{d,k}\right|^{2}$
$\displaystyle s.t.$
$\displaystyle~{}\left|\exp\left(j\alpha_{k,n}\right)\right|=1,~{}\forall
k\in[1,K],~{}\forall n\in[1,N_{R}].$ (31)

In order to solve (-A), its objective function can be equivalently modified as

$\displaystyle|\mathbf{h}_{k}\bm{\Theta}_{k}\mathbf{h}_{r}+h_{d,k}|^{2}$
$\displaystyle~{}=|\bm{\theta}_{k}\mathbf{b}_{k}+h_{d,k}|^{2},~{}\forall
k\in[1,K],$ (32)

where $\mathbf{b}_{k}=\textrm{diag}(\mathbf{h}_{k})\mathbf{h}_{r}$,
$\bm{\theta}_{k}=\left[\theta_{k,1},...,\theta_{k,N_{R}}\right]=\left[\exp(j\alpha_{k,1}),...,\exp(j\alpha_{K,N_{R}})\right]$,
$|\bm{\theta}_{k}(n)|=1$, $\forall k\in[1,K],~{}n\in[1,N_{R}]$. To proceed, we
apply the following triangle inequality [17, 28]

$\displaystyle\\!\\!\\!\\!\\!|\bm{\theta}_{k}\mathbf{b}_{k}\\!+\\!h_{d,k}|\\!\leq\\!\sum_{n=1}^{N}|\theta_{k,n}\mathbf{b}_{k}(n)|\\!+\\!|h_{d,k}|\\!=\\!\sum_{n=1}^{N}|\mathbf{b}_{k}(n)|\\!+\\!|h_{d,k}|,$
(33)

where $\mathbf{b}_{k}(n)$ is the $n$-th element of $\mathbf{b}_{k}$ and the
equality holds with $|\theta_{k,n}|=1$ for $n\in[1,N_{R}]$. We obtain the
upper bound in (33) via
$\alpha_{k,n}=\arg(h_{d,k})-\arg\left(\mathbf{b}_{k}(n)\right)$ and
$\theta_{k,n}=\exp(j\alpha_{k,n})$, where $\arg(\cdot)$ is the the phase
operator. The optimal solution to problem (-A) is denoted by
$\bm{\theta}_{k}^{*}$, and equivalently the optimal phase shift matrix
$\bm{\Theta}_{k}^{*}$ is found from $\bm{\theta}_{k}^{*}$.

###  -B Proof of _Proposition_ 1

By applying a few basic mathematical manipulations, the term
$\left|\mathbf{h}_{k}\bm{\Theta}_{k}\mathbf{h}_{r}+h_{d,k}\right|^{2}$ is
further equivalent to

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\left|\mathbf{h}_{k}\bm{\Theta}_{k}\mathbf{h}_{r}\\!+\\!h_{d,k}\right|^{2}\\!=\\!\left|\mathbf{h}_{k}\bm{\Theta}_{k}\mathbf{h}_{r}\right|^{2}\\!+\\!\left|h_{d,k}\right|^{2}$
$\displaystyle~{}~{}\\!+\\!2\left|\mathbf{h}_{k}\bm{\Theta}_{k}\mathbf{h}_{r}\right|\left|h_{d,k}\right|\cos\left[\arg(h_{d,k})\\!-\\!\arg(\mathbf{h}_{k}\bm{\Theta}_{k}\mathbf{h}_{r})\right].$
(34)

From (-B), it is easily verified that
$\left|\mathbf{h}_{k}\bm{\Theta}_{k}\mathbf{h}_{r}+h_{d,k}\right|^{2}$
achieves its maximum value if

$\cos\left[\arg(h_{d,k})-\arg(\mathbf{h}_{k}\bm{\Theta}_{k}\mathbf{h}_{r})\right]=1.$

This means that the phases of the direct and cascaded links between the
$k$-the IoT device and the AP are identical, i.e.,
$\arg(h_{d,k})=\arg(\mathbf{h}_{k}\bm{\Theta}_{k}\mathbf{h}_{r})$, thus, (4)
holds. This completes the proof of _Proposition_ 1.

###  -C Proof of _Lemma_ 1

First, the objective function (6a) is the sum of logarithm functions, i.e.,
$\tau_{k}\log\left(1+\frac{\tilde{t}_{k}\textrm{Tr}\left(\mathbf{\tilde{g}}_{k}\mathbf{\tilde{g}}_{k}^{H}\mathbf{V}_{0}\right)}{\tau_{k}}\right)$,
each of which preserves its concavity with respect to $\bm{\tau}$ and
$\mathbf{V}_{0}$ since it is the perspective function of the concave function
$\log\left(1+\tilde{t}_{k}\textrm{Tr}\left(\mathbf{\tilde{g}}_{k}\mathbf{\tilde{g}}_{k}^{H}\mathbf{V}_{0}\right)\right)$.
In addition, all the constraints (6b) and (2b) are affine with $\bm{\tau}$ and
$\mathbf{V}_{0}$. We have completed the proof of _Lemma_ 1.

###  -D Proof of _Lemma_ 2

For given $\bm{\tau}$, we consider the following problem

$\displaystyle\max_{\mathbf{V}_{0}\succeq\mathbf{0}}$
$\displaystyle~{}\sum_{k=1}^{K}\tau_{k}\log\left(1+\frac{\tilde{t}_{k}\textrm{Tr}\left(\mathbf{\tilde{g}}_{k}\mathbf{\tilde{g}}_{k}^{H}\mathbf{V}_{0}\right)}{\tau_{k}}\right)$
(35a) $\displaystyle s.t.$
$\displaystyle~{}\textrm{Tr}(\mathbf{V}_{0})\leq\tau_{0}P_{0},$ (35b)

It is observed that (35a) is a logarithmic function which is concave but non-
linear. To linearize it, we take into consideration a successive convex
approximation (SCA) to transform (35a) into a series of linear programming
(LP) as follows

$\displaystyle\mathbf{V}_{0}^{(m+1)}=\arg\max_{\mathbf{V}_{0}\succeq\mathbf{0}}$
$\displaystyle~{}\sum_{k=1}^{K}\frac{\tilde{t}_{k}\textrm{Tr}\left(\mathbf{\tilde{g}}_{k}\mathbf{\tilde{g}}_{k}^{H}\mathbf{V}_{0}\right)}{1+\frac{\tilde{t}_{k}\textrm{Tr}\left(\mathbf{\tilde{g}}_{k}\mathbf{\tilde{g}}_{k}^{H}\mathbf{V}_{0}^{(m)}\right)}{\tau_{k}}}$
$\displaystyle s.t.$
$\displaystyle~{}\textrm{Tr}(\mathbf{V}_{0})\leq\tau_{0}P_{0}$ (36)

where $\mathbf{V}_{0}^{(m)}$ denotes the optimal solution to (-D) at the
$m$-th iteration. By exploiting [47], it is easily verified that
$\mathbf{V}_{0}^{(m+1)}$ can be obtained by iteratively solving (-D) which
yields a rank-one solution at each iteration. Thus, the optimal solution to
problem (6), denoted by $\mathbf{V}_{0}^{*}$, always returns a rank-one
matrix. This completes the proof of _Lemma_ 2.

###  -E Proof of _Proposition_ 2

In order to prove _Proposition_ 2, the feasible solution and the objective
value of problem (6) are denoted by
$\left(\mathbf{\tilde{w}},\bm{\tilde{\tau}},\bm{\tilde{\Theta}}_{0}\right)$
and
$\tilde{f}\left(\mathbf{\tilde{w}},\bm{\tilde{\tau}},\bm{\tilde{\Theta}}_{0}\right)$,
respectively. From step (2-b and 2-c) of _Algorithm_ 1, there exists a
feasible solution to problem (7), i.e.,
$(\mathbf{\tilde{w}}^{(m+1)},\bm{\tilde{\tau}}^{(m+1)},\bm{\tilde{\Theta}}_{0}^{(m+1)})$,
which is also feasible to problem (6). As such,
$\left(\mathbf{\tilde{w}}^{(m+1)},\bm{\tilde{\tau}}^{(m+1)},\bm{\tilde{\Theta}}_{0}^{(m)}\right)$
in step (2-a) is the feasible solution to (6) in the $m$-th iteration. Thus,
the following relation is easily achieved

$\displaystyle~{}\tilde{f}\left(\mathbf{\tilde{w}}^{(m)},\bm{\tilde{\tau}}^{(m)},\bm{\tilde{\Theta}}_{0}^{(m)}\right)\leq\tilde{f}\left(\mathbf{\tilde{w}}^{(m+1)},\bm{\tilde{\tau}}^{(m+1)},\bm{\tilde{\Theta}}_{0}^{(m)}\right)$
$\displaystyle~{}~{}~{}~{}\leq\tilde{f}\left(\mathbf{\tilde{w}}^{(m+1)},\bm{\tilde{\tau}}^{(m+1)},\bm{\tilde{\Theta}}_{0}^{(m+1)}\right),$
(37)

where the first inequality holds due to the fact that $\mathbf{w}^{(m+1)}$ and
$\bm{\tau}^{(m+1)}$ are the optimal solution to problem (6) for given
$\bm{\Theta}_{0}^{(m)}$ in step (2-a) of _Algorithm_ 1; The second inequality
holds due to the fact that $\bm{\Theta}_{0}^{(m+1)}$ is the sub-optimal
solution to problem (7) for given $\mathbf{w}^{(m+1)}$ and
$\bm{\tau}^{(m+1)}$. Thus, (-E) has a monotonically increasing trend of
$\tilde{f}\left(\mathbf{\tilde{w}},\bm{\tilde{\tau}},\bm{\tilde{\Theta}}_{0}\right)$.
Moreover,
$\tilde{f}\left(\mathbf{\tilde{w}},\bm{\tilde{\tau}},\bm{\tilde{\Theta}}_{0}\right)$
is an upper-bounded due to the transmit power constraint (6b). This completes
the proof of _Proposition_ 2.

###  -F Proof of _Theorem_ 2

It is easily verified that (17a) is an logarithmic function which is
increasing with respect to the term
$\textrm{Tr}\left(\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\mathbf{W}\right)$,
thus, the optimal solution to problem (17) is equivalent to solving the
following problem

$\displaystyle\max_{\mathbf{W}\succeq\mathbf{0}}$
$\displaystyle~{}\textrm{Tr}\left(\mathbf{\tilde{G}}^{H}\mathbf{W}\mathbf{\tilde{G}}\right),~{}s.t.~{}\textrm{Tr}\left(\mathbf{W}\right)\leq
P_{0}.$ (38)

We take into consideration the eigen-decomposition of $\mathbf{W}$ as

$\displaystyle\mathbf{W}=\mathbf{\tilde{U}}\bm{\Delta}\mathbf{\tilde{U}}^{H},$
(39)

where $\mathbf{\tilde{U}}\in\mathbb{C}^{N_{T}\times N_{T}}$ denotes a unitary
matrix, and $\bm{\Delta}=\textrm{diag}(\varphi_{1},...,\varphi_{N_{T}})$ with
$\varphi_{1}\geq,...,\varphi_{N_{T}}$ as well as
$\sum_{m=1}^{N_{T}}\varphi_{m}\leq P_{0}$. By substituting (39) into the
objective function (38), we have

$\displaystyle\textrm{Tr}\left(\mathbf{\tilde{G}}^{H}\mathbf{W}\mathbf{\tilde{G}}\right)$
$\displaystyle~{}=\textrm{Tr}\left(\mathbf{\tilde{G}}^{H}\mathbf{\tilde{U}}\bm{\Delta}\mathbf{\tilde{U}}^{H}\mathbf{\tilde{G}}\right)=\textrm{Tr}\left(\mathbf{\bar{G}}^{H}\bm{\Delta}\mathbf{\bar{G}}\right)$
$\displaystyle~{}=\sum_{m=1}^{N_{T}}\varphi_{m}\|\mathbf{\bar{g}}_{m}\|^{2}\leq
P_{0}\|\mathbf{\bar{g}}_{1}\|^{2},$ (40)

where
$\mathbf{\bar{G}}^{H}=\mathbf{\tilde{G}}^{H}\mathbf{\tilde{U}}=\left[\mathbf{\bar{g}}_{1},...,\mathbf{\bar{g}}_{N_{T}}\right]$
with $\mathbf{\bar{g}}_{m}\in\mathbb{C}^{K\times 1}$. Note that the equality
in (-F) holds if
$\|\mathbf{\bar{g}}_{1}\|^{2}=\max_{m\in[1,N_{T}]}\|\mathbf{\bar{g}}_{m}\|^{2}$
and $[\varphi_{1},\varphi_{2},...,\varphi_{N_{T}}]=[P_{0},0,...,0]$.
Furthermore, the optimal $\bm{\Delta}$ can be derived as

$\displaystyle\bm{\Delta}=\textrm{diag}\left(P_{0},0,...,0\right).$ (41)

Let
$\mathbf{\tilde{U}}=\left[\mathbf{\tilde{u}}_{1},...,\mathbf{\tilde{u}}_{N_{T}}\right]$
with $\mathbf{\tilde{u}}_{m}\in\mathbb{C}^{N_{T}\times 1}$, we substitute (41)
into (39), the optimal solution to
$\mathbf{W}=P_{0}\mathbf{\tilde{u}}_{1}\mathbf{\tilde{u}}_{1}^{H}$. Moreover,
we define $N=\min(N_{T},K)$ and denote the singular value decomposition (SVD)
of $\mathbf{\tilde{G}}$ as

$\displaystyle\mathbf{\tilde{G}}=\mathbf{V}_{\mathbf{\tilde{G}}}\bm{\Gamma}^{\frac{1}{2}}\mathbf{U}_{\mathbf{\tilde{G}}}^{H},$
(42)

where $\mathbf{V}_{\mathbf{\tilde{G}}}\in\mathbb{C}^{N_{T}\times N}$ and
$\mathbf{U}_{\mathbf{\tilde{G}}}^{H}\in\mathbb{C}^{N\times N_{T}}$ each of
which is a unitary matrix, and
$\bm{\Gamma}=\textrm{diag}\left(\rho_{1},...,\rho_{N}\right)$ with
$\rho_{1}\geq,...,\geq\rho_{N}$. Note that
$\mathbf{U}_{\mathbf{\tilde{G}}}=[\mathbf{u}_{1},...,\mathbf{u}_{N}]$, where
$\mathbf{u}_{i}\in\mathbb{C}^{N_{T}\times 1},(\forall i\in[1,N])$ denotes the
$i$-th column of $\mathbf{U}_{\mathbf{\tilde{G}}}$.

Due to the SVD in (42), $\|\mathbf{\bar{g}}_{1}\|^{2}$ is the maximum for all
terms $\|\mathbf{\bar{g}}_{m}\|^{2}$ if and only if $\mathbf{\tilde{u}}_{1}$
is also the first column of $\mathbf{U}_{\mathbf{\tilde{G}}}$, i.e.,
$\mathbf{\tilde{u}}_{1}$ is equivalent to $\mathbf{u}_{1}$, which is also
known as the eigenvector associated with the maximum eigenvalue of
$\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}$, denoted by $\rho_{1}$. Thus, the
optimal solution to problem (38) is given by
$\mathbf{W}^{*}=P_{0}\bm{\nu}_{\max}\left(\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\right)\bm{\nu}_{\max}\left(\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\right)^{H}$,
which completes the proof of _Theorem_ 2.

###  -G Proof of _Theorem_ 3

In order to prove _Theorem_ 19, we first rewrite the objective function in
(III-B1) as

$\displaystyle
f(\tau_{0})=(T-\tau_{0})\log\left(1+\frac{\tau_{0}P_{0}\rho_{\max}\left(\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\right)}{T-\tau_{0}}\right).$
(43)

We take the first-order derivative of $f(\tau_{0})$ with respect to $\tau_{0}$
and set it to zero,

$\displaystyle~{}\frac{\partial
f(\tau_{0})}{\partial\tau_{0}}=-\log\left(1+\frac{\tau_{0}P_{0}\rho_{\max}\left(\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\right)}{T-\tau_{0}}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\frac{\frac{TP_{0}\rho_{\max}\left(\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\right)}{T-\tau_{0}}}{1+\frac{\tau_{0}P_{0}\rho_{\max}\left(\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\right)}{T-\tau_{0}}}=0.$
(44)

Let
$x=1+\frac{\tau_{0}P_{0}\rho_{\max}\left(\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\right)}{T-\tau_{0}}$,
(-G) is equivalently written as

$\displaystyle
x\log(x)-x=P_{0}\rho_{\max}\left(\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\right)-1.$
(45)

After a series of mathematical manipulations, (45) is equivalent to

$\displaystyle\eqref{eq:Tau0_relation1}$
$\displaystyle~{}\Rightarrow~{}\log\left(\frac{x}{\exp(1)}\right)\exp\left[\log\left(\frac{x}{\exp(1)}\right)\right]$
$\displaystyle=\frac{P_{0}\rho_{\max}\left(\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\right)-1}{\exp(1)}.$
(46)

By exploiting the Lambert $\mathcal{W}$ function, i.e.,
$\bar{x}\exp(\bar{x})=\bar{y}\Rightarrow\bar{x}=\mathcal{W}(\bar{y})$,

$\displaystyle\eqref{eq:Tau0_relation2}$
$\displaystyle~{}x=\exp\left[\mathcal{W}\left(\frac{P_{0}\rho_{\max}\left(\mathbf{\tilde{G}}\mathbf{\tilde{G}}^{H}\right)-1}{\exp(1)}\right)+1\right].$
(47)

From (47), the optimal time allocation can be easily obtained in (19), which
completes the proof of _Theorem_ 3.

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