# Self-Sustainable Reconfigurable Intelligent Surface Aided Simultaneous
Terahertz Information and Power Transfer (STIPT)

Yijin Pan, Kezhi Wang, Cunhua Pan, Huiling Zhu and Jiangzhou Wang Y. Pan is
with the National Mobile Communications Research Laboratory, Southeast
University, Nanjing 211111, China, and also with School of Engineering and
Digital Arts, University of Kent, UK.K. Wang is with the Department of
Computer and Information Sciences, Northumbria University, UK.C. Pan is with
the School of Electronic Engineering and Computer Science, Queen Mary,
University of London, UK.J. Wang and H. Zhu are with the School of Engineering
and Digital Arts, University of Kent, UK.

###### Abstract

This paper proposes a new simultaneous terahertz (THz) information and power
transfer (STIPT) system, which is assisted by reconfigurable intelligent
surface (RIS) for both the information data and power transmission. We aim to
maximize the information users’ (IUs’) sum data rate while guaranteeing the
energy users’ (EUs’) and RIS’s power harvesting requirements. To solve the
formulated non-convex problem, the block coordinate descent (BCD) based
algorithm is adopted to alternately optimize the transmit precoding of IUs,
RIS’s reflecting coefficients, and RIS’s coordinate. The Penalty Constrained
Convex Approximation (PCCA) Algorithm is proposed to solve the intractable
optimization problem of the RIS’s coordinate, where the solution’s feasibility
is guaranteed by the introduced penalties. Simulation results confirm that the
proposed BCD algorithm can significantly enhance the performance of STIPT by
employing RIS.

###### Index Terms:

Simultaneous terahertz information and power transfer (STIPT), intelligent
reflecting surface (IRS), reconfigurable intelligent surface (RIS), terahertz
(THz) communications.

##  I Introduction

The current mobile network is experiencing an unprecedented evolution with the
increasing number of attractive mobile applications, which results in
expectations for extremely high data rates for realizing a variety of
multimedia services[1]. Thanks to the abundant spectrum resources in the
terahertz (THz) band, the THz communication is able to realize high
transmission rates from hundreds of Gbps to several Tbps. As a result, THz
transmission is envisioned as an emerging solution to meet the ultra-high-
speed data rate demands of the enhanced mobile broadband (eMBB) services, such
as virtual reality and high-definition data streams[2].

At the same time, the wireless power harvesting (WPH) technology provides an
attractive approach for the Internet of things (IoT) devices with limited
battery life to reap power from external radio frequency sources [3]. The WPH
technique promises the potentials for replacing batteries in low power
consumption devices or increasing their battery lifespans [4]. With these
properties, numerous devices and sensors are being manufactured with the WPH
power supply mode, including the nanoscale IoT devices[5]. Nevertheless, the
traditional low-frequency radio waves may no longer be suitable for WPH [6].
This is because the low-frequency radio wave normally requires a large antenna
aperture to capture a large portion of radiated electromagnetic wave due to
the relatively large wavelength[7], which tends to exceed the size limit of
the IoT devices and sensors [8], especially for the nanoscale IoT devices. To
tackle this issue, a possible solution is to increase the frequency of the
radio so that the size of antennas can be miniaturized and the transmission
beam directivity can be improved[6].

Fortunately, as a key component of the 6G communication system, the THz band
bridges the gap between mmWave and optical band. Specifically, THz band ranges
from 100 GHz to 10 THz such that the wavelength of THz can greatly reduce the
required antenna aperture [9]. This makes utilizing the THz electronics for
wireless power harvesting a very attractive approach. Also, emerging rectennas
have been proposed and manufactured which make the energy harvesting in THz
bands becomes possible[10, 11]. As a result, combining the benefits of the THz
transmission in providing high-speed data rates with WPH for IoT devices will
be a promising research direction, leading to a new term of Simultaneous
Terahertz Information and Power Transfer (STIPT) network.

###  I-A Related Works

In the THz communication system, due to the ultra-high radio frequency, the
THz links are easily blocked by obstacles in transmission paths. This feature
of the THz band greatly limits the transmission distance, which necessitate
establishing efficient STIPT communications. To address this issue, it has
been proposed to utilize the reconfigurable intelligent surface (RIS), also
known as intelligent reflective surface (IRS) to help compensate for the
blocked communication links[12]. By adjusting the reflecting coefficients of
RIS, the propagation channel condition can be significantly improved to
enhance the system performance[13].

Recently, extensive efforts have been devoted to numerous applications of the
RIS-assisted transmissions[14]. A comprehensive survey about these
applications of the RIS has been given in [15], and the potential benefits
that can be brought by the RIS has been explained in various aspects in [16].
In [17], the RIS has been exploited to enhance the cell-edge performance in
multicell MIMO communication systems. Also, the RIS has been utilized to
enhance latency performance of the mobile edge computing (MEC) system in [18].
Furthermore, the RIS was also utilized to enhance the physical layer security
by improving the secrecy rate [19, 20, 21, 22, 23] and reducing the transmit
power [24]. The RIS-enhanced orthogonal frequency division multiplexing (OFDM)
system and its corresponding transmission protocol were investigated in [25].

As for the simultaneous wireless information and power transfer (SWIPT)
system, there are already a few contributions on utilizing the RIS to enhance
the performance of SWIPT systems[26]. An RIS-aided MIMO broadcasting SWIPT
system was investigated in [27], where the transmit precoding matrices and
passive phase shift matrix of the RIS were jointly optimized to maximize the
weighted sum rate of information users while guaranteeing the users’ energy
harvesting requirement. The contributions in [28] and [29] investigated the
RIS-aided secure transmission system, where the obtained secure rate and
energy efficiency were enhanced by optimizing the reflecting coefficients of
RIS, respectively. In [30], RIS was leveraged to enhance the performance of
non-orthogonal multiple access (NOMA) and the wireless power transfer (WPT)
efficiency of SWIPT. The energy harvesting efficiency of the RIS-assisted MIMO
broadcasting SWIPT system was investigated in [31], where the total transmit
power required at the AP was minimized while satisfying the QoS constraints of
the information users and the energy users. A RIS-assisted wireless power
transfer OFDM-based MEC system was investigated in [32], where the total
energy consumption was minimized by optimizing the power allocation and
computation resource allocation.

It is worth pointing out that the above works only considered the
micro/millimeter wave communications. Compared with traditional
micro/millimeter transmission system, the THz band transmissions suffer from
high molecular absorption and propagation loss[33]. The extremely short
wavelengths of THz signals makes the obstacles in the path tend to absorb THz
signals rather than reflect them[34]. Therefore, the RIS is envisioned to be a
necessity for the future THz communications to bypass blockages[35]. Most
recently, specific efforts have been devoted to the use of RIS to enhance the
THz transmission performance[36, 37]. For instance, the coverage analysis in
[38] has highlighted the impact of the molecular absorption loss on the path
loss of the RIS-assisted THz channel. As the THz band appears to be frequency-
selective with many path loss peaks, the sum rate of an RIS-assisted THz
system was maximized in [39], where a whole band is divided into several sub-
bands. The passive reflecting phase shifters of RIS was investigated in [40]
to enhance the secrecy rate of THz communication. An RIS-aided multi-user THz
MIMO system with orthogonal frequency division multiple access was
investigated in [41], where the weighted sum rate was maximized by jointly
optimizing the hybrid beamforming and reflecting matrix of the RIS. As the RIS
is able to control the propagation direction of THz waves for mitigating the
blockage issue, the RIS was utilized in [42] to assist the UAV THz
transmission, where the joint passive beamforming design and trajectory
optimization were investigated. In [42], as the available spectrum in the THz
band varies with the link distance, the unique channel fading characteristics
of the THz channel was exploited to optimize the UAV trajectory.

###  I-B Motivations and Contributions

Most of the above-mentioned RIS-aided transmission approaches are not
specifically designed for STIPT systems, as integrating RIS into the STIPT
face many challenges.

First of all, in the THz band, many path loss peaks appear and therefore the
total band is divided into many sub-bands [43]. According to [39], the
locations of these path loss peaks vary with the carrier frequencies and the
transmission distance. Consequently, it becomes challenging to efficiently
utilize the frequency-selective THz band, and the precoding for different
users needs to be designed carefully on multiple sub-bands. More importantly,
the link distance is dependent on the location of the RIS [44], but the impact
of the RIS’s location is generally ignored in the current RIS applications. In
a typical STIPT application scenario, the energy users are normally IoT
devices or sensors, and the information users are deployed for the monitor
tasks such as transmitting high-definition video/figures that entail ultra-
high data rate. The locations of energy users and information users can be
available at the network AP. Consequently, in this scenario where the
positions of users are relatively fixed, optimization of the RIS’s location
has the potential to effectively compensate for the link-distance dependent
fading in the THz band. In fact, optimizing the location of RIS with the power
constraints is challenging due to multiple periodic cosine components in the
channel expression, and currently, no efficient solutions have been reported
in the literature, to the best of our knowledge.

Secondly, due to the high radio frequencies in the THz band, the number of RIS
elements can be significantly increased within a limited area in order to
provide a better reflecting performance. According to [45], the RIS power
consumption depends on the type and the number of reflecting elements.
Therefore, the energy consumption of RIS cannot be ignored in this case, and
it is proportional to the number of RIS elements[46]. In [47], a part of the
RIS’s elements are selected to harvest the received energy and the remaining
elements help the secure information transmission, but this approach was
designed for the micro/millimeter transmissions. However, in the current RIS-
assisted SWIPT schemes, the power consumption of the RIS is generally ignored,
and it is also unclear how to power the RIS in the THz transmission system.

As a result, it is imperative to jointly consider the impact of the RIS’s
power consumption, reflecting coefficients, and its location on the STIPT’s
system. The above mentioned key fundamental issues need to be resolved, and
how to obtain an efficient STIPT communication system is still unknown.

Against the above background, in this paper, we consider the downlink
transmission of the STIPT network, where the information users (IUs) and
energy users (EUs) are jointly served by the RIS-assisted THz links. In our
system, the RIS is also equipped with the WPH module to harvest energy from
the received THz radios to maintain its circuit power consumption. The
precoding for IUs, the RIS’s reflecting coefficients, and the RIS’s coordinate
are jointly optimized to maximize the IUs’ achievable rates while satisfying
the EU’s and RIS’s power harvesting requirements. Overall, our contributions
can be summarized as follows:

  * •

We propose an RIS-aided STIPT system to simultaneously transmit information
and power for IUs and EUs, respectively. The RIS-assisted THz channel is
modelled as a function of the RIS reflecting coefficients and the coordinate
of RIS. The RIS can harvest power from the received radio, where the harvested
power can be adjusted by the amplitude of its reflecting coefficients.

  * •

The optimization problem is formulated to maximize the IUs’ sum rates while
guaranteeing the power harvesting requirements of the EUs. The original non-
convex problem is first reformulated by utilizing the equivalence between the
weighted minimum mean-square error (WMMSE) and the signal-to-noise ratio
(SINR). Then, we decouple the optimization problem into three subproblems:
optimization of the precoding for IUs, RIS’s reflecting coefficients and RIS’s
coordinate.

  * •

The precoding for IUs and RIS’s reflecting coefficients are obtained by
utilizing the successive convex approximation method. To deal with the
intractable optimization problem of the RIS’s coordinate, we propose the
Penalty Constrained Convex Approximation (PCCA) Algorithm to guarantee the
solution’s feasibility and the convergence of the block coordinate descent
(BCD) algorithm.

  * •

Extensive simulation results are provided to show the performance gain
achieved by the proposed STIPT system compared with benchmarks. It is shown
that the sum rate performance of IUs is greatly affected by the RIS’s
coordinate. By utilizing the proposed BCD algorithm, the THz channel can be
optimized to fully exploit the spatial diversity so that the sum rate
performance can be significantly enhanced.

The reminder of this paper is organized as follows: Section II describes the
system model of the proposed RIS-aided STIPT system and formulates the
optimization problem. Sections III develops the detailed algorithm to solve
the formulated sum-rate maximization problem. In Section IV, the simulation
results are presented to show the performance gain and the impact of system
parameters, and Section V concludes the paper.

Notation: For a vector $\bm{x}$, $|\bm{x}|$ and $(\bm{x})^{T}$ respectively
denote its Euclidean norm and its transpose. $c$ represents the light speed.
For matrix $\bm{A}$, $\bm{A}^{*}$ and $\bm{A}^{\star}$ represent the conjugate
operator and converged solution, respectively. $\mathbb{C}^{M\times 1}$
denotes the set of $M\times 1$ complex vectors. $\text{diag}(\bm{X})$
represents the vector that is obtained from the diagonal entries of matrix
$\bm{X}$. $\bm{a}\odot\bm{b}$ represents the Hadamard (point-wise) product of
$\bm{a}$ and $\bm{b}$.

Figure 1: The RIS-assisted SWIFT system.

##  II System Model and Problem Formulation

Consider the downlink of a STIPT system that needs to serve the IoT sensors
and video transmission devices at the same time, as shown in Fig. 1. The
network access point (AP) operates in the THz band. The IoT sensors are EUs
and they need to harvest power from the THz radio, meanwhile, IUs require
high-speed data transmission for transmitting high-definition video/figures.

The AP is equipped with $N_{t}$ transmit antennas to serve $I$ IUs and $M$
EUs, and all the IUs and EUs are equipped with $N_{r}$ receive antennas. Let
$\mathcal{I}$ and $\mathcal{M}$ respectively represent the set of IUs and EUs.
Then, the set of total users is given by
$\mathcal{U}=\mathcal{I}\cup\mathcal{M}$, and the total number of all users
(including EU and IU) is denoted by $U=\left|{\cal U}\right|$. In the
following, the user $u$ can be either EU or IU.

As the THz channel is frequency-selective, the total THz band is divided into
$K$ sub-bands (SBs). Let $f_{k}$ denote the central frequency of SB $k$, and
its wavelength is given by $\lambda_{k}=\frac{c}{f_{k}}$.

Normally, the wireless transmission channel includes the line-of-sight (LOS)
link and non-line-of-sight (NLOS) links, where NLOS links consist of
reflected, scattered, and diffracted components. As the scattered and
diffracted components are shown to play insignificant roles in the received
signal power in [43, 48], similar to [49, 41], the scattered and diffracted
rays are neglected in the channel model. In addition, according to [34], the
surfaces of walls and ceilings appear “rough” for the THz signals so that they
tend to absorb and scatter the THz signals rather than reflect them. As the
RIS is specially designed to enable redirecting the incoming signal to the
desired directions, the NLOS components in this work are only contributed by
reflected paths from the RIS.

(a) LOS links from AP’s transmit antenna elements to user $u$’s first antenna
element

(b) LOS links between user $u$’s receive antenna elements and AP’s first
antenna element

Figure 2: The LOS links between AP and user $u$.

###  II-A LOS Links without RIS

Fig. 2 shows the LOS links between the AP and user $u$. As shown in Fig. 2,
the coordinate of AP’s first antenna element is denoted as
$\bm{s}^{1}_{AP}\in\mathbb{R}^{3\times 1}$, and the coordinate of user $u$’s
first antenna element is denoted as $\bm{s}^{1}_{u}\in\mathbb{R}^{3\times 1}$.
Then, the distance between the AP and the users is calculated as
$|\bm{d}_{u}|$, where $\bm{d}_{u}=\bm{s}^{1}_{u}-\bm{s}^{1}_{AP}$ is the
transmit vector for user $u$. In addition, as shown in Fig. 2, the
transmission distances between different transmit (receive) antenna elements
are different, which causes the phase difference between the channel gains in
the frequency domain. These phase differences are featured by the transmit
array vector of the AP and the receive antenna vector of the user.

Fig. 2 (a) shows the distance differences between the transmit antennas of the
AP. In Fig. 2 (a), $\bm{s}^{n_{t}}_{AP}\in\mathbb{R}^{3\times 1}$ denotes the
coordinate of AP’s $n_{t}$-th antenna element. We define directional vector
$\bm{\delta}^{n_{t}}_{AP}=\bm{s}^{n_{t}}_{AP}-\bm{s}^{1}_{AP},n_{t}=1,\cdots,N_{t}$,
and $\bm{\delta}^{1}_{AP}=[0,0,0]^{T}$. As shown in Fig. 2 (a), the phase
difference $\theta_{k,u}^{n_{t}}$ between AP’s $n_{t}$-th antenna element and
the first element is evaluated as

$\theta_{k,u}^{n_{t}}=\frac{2\pi
f_{k}}{c}\frac{(\bm{d}_{u})^{T}\bm{\delta}^{n_{t}}_{AP}}{|\bm{d}_{u}|}=\frac{2\pi}{\lambda_{k}}\frac{(\bm{d}_{u})^{T}\bm{\delta}^{n_{t}}_{AP}}{|\bm{d}_{u}|},n_{t}=1,\cdots,N_{t}.$
(1)

Then, the transmit array vector from the AP to user $u$ on SB $k$ is then
denoted by

$\bm{v}_{k,u}=[1,\cdots,\exp(-j\theta_{k,u}^{n_{t}}),\cdots,\exp(-j\theta_{k,u}^{N_{t}})]^{T}.$
(2)

Fig. 2 (b) shows the distance differences between the receive antennas of the
user $u$. In Fig. 2 (b), $\bm{s}_{u}^{n_{r}}\in\mathbb{R}^{3\times 1}$ denote
the coordinate of the $n_{r}$-th antenna element of user $u$. Define the
directional vector
$\bm{\delta}^{n_{r}}_{u}=\bm{s}_{u}^{n_{r}}-\bm{s}_{u}^{1},n_{r}=1,\cdots,N_{r},$
for the $n_{r}$-th receive antenna element. As shown in Fig. 2 (b), the phase
difference $\varrho^{n_{r}}_{k,u}$ between user’s $n_{r}$-th antenna element
and the first element is evaluated as

$\varrho^{n_{r}}_{k,u}=\frac{2\pi
f_{k}}{c}\frac{(\bm{d}_{u})^{T}\bm{\delta}^{n_{r}}_{u}}{|\bm{d}_{u}|}=\frac{2\pi}{\lambda_{k}}\frac{(\bm{d}_{u})^{T}\bm{\delta}^{n_{r}}_{u}}{|\bm{d}_{u}|},n_{r}=1,\cdots,N_{r}.$
(3)

Then, the receive array vector from user $u$ to the AP on SB $k$ is

$\bm{r}^{dir}_{k,u}=[1,\cdots,\exp(-j\varrho^{n_{r}}_{k,u}),\cdots,\exp(-j\varrho^{N_{r}}_{k,u})]^{T}.$
(4)

According to the ray tracing techniques [50], the path gain from the AP to
user $u$ on SB $k$ is evaluated as

${h}_{k,u}=\left(\frac{G_{r}G_{t}\lambda_{k}}{4\pi|\bm{d}_{u}|}\right)\exp\left({-j2\pi\frac{|\bm{d}_{u}|}{\lambda_{k}}}\right)\exp\left({-\frac{1}{2}K(f_{k})|\bm{d}_{u}|}\right),$
(5)

where $G_{r}$ and $G_{t}$ respectively represents the antenna gain of transmit
array and the antenna gain of receiving array, $|\bm{d}_{u}|$ is the distance
from IU $u$ to the AP, and $K(f_{k})$ is the overall absorption coefficient of
the transmission medium on SB $k$. Then, the LOS channel from the AP to user
$u$ on SB $k$ is denoted by

$\bm{H}_{k,u}={h}_{k,u}\bm{r}^{dir}_{k,u}\bm{v}^{H}_{k,u},u\in\mathcal{I}\cup\mathcal{M}.$
(6)

###  II-B RIS Assisted NLOS Links

The number of reflecting elements of the RIS is $N$. The coordinate of the
RIS’s first reflecting element is denoted as
$\bm{s}_{RIS}^{1}\in\mathbb{R}^{3\times 1}$.

We first consider the link from the AP to the RIS. The transmit distance
between the AP and the RIS is evaluated as $|\bm{d}_{0}|$, where the transmit
vector is $\bm{d}_{0}=\bm{s}_{RIS}^{1}-\bm{s}_{AP}^{1}$. Similar to (1), the
phase difference $\theta_{k}^{n_{t}}$ between AP’s $n_{t}$-th antenna element
and the first element is evaluated as

$\theta_{k}^{n_{t}}=\frac{2\pi}{\lambda_{k}}\frac{(\bm{d}_{0})^{T}\bm{\delta}^{n_{t}}_{AP}}{|\bm{d}_{0}|},n_{t}=1,\cdots,N_{t}.$
(7)

Then, the transmit array vector from the AP to the RIS on SB $k$ is then
denoted by

$\bm{v}_{k}=[1,\cdots,\exp(-j\theta_{k}^{n_{t}}),\cdots,\exp(-j\theta_{k}^{N_{t}})]^{T}.$
(8)

In addition, due to different transmit distance between the reflecting
elements, we define the receiving array vector $\bm{e}_{k}$ to feature the
relative phase differences between the signals received on SB $k$ at different
reflecting elements.

Similar to (3), the directional vector is defined as
$\bm{\delta}^{n}_{RIS}=\bm{s}_{RIS}^{n}-\bm{s}_{RIS}^{1}$, $n=1,\cdots,N$, and
the phase difference between RIS’s $n$-th reflecting element and the first
reflecting element is

$\vartheta_{k}^{n}=\frac{2\pi}{\lambda_{k}}\frac{(\bm{d}_{0})^{T}\bm{\delta}^{n}_{RIS}}{|\bm{d}_{0}|},n=1,\cdots,N.$
(9)

Then, the receive array vector is given by

$\bm{e}_{k}=\left[1,\cdots,\exp(-j\vartheta_{k}^{n}),\cdots,\exp(-j\vartheta_{k}^{N})\right]^{T}.$
(10)

As the path-loss gain from the AP to RIS on SB $k$ is evaluated as

$H_{k}=\left(\frac{G_{t}\lambda_{k}}{4\pi|\bm{d}_{0}|}\right)\exp\left({-j2\pi\frac{|\bm{d}_{0}|}{\lambda_{k}}}\right)\exp\left({-\frac{1}{2}K(f_{k})|\bm{d}_{0}|}\right),$
(11)

then the LOS channel from the AP to RIS on SB $k$ is denoted by

$\bm{H}_{k}=H_{k}\bm{e}_{k}\bm{v}_{k}^{H}.$ (12)

Then, we consider the links from the RIS to the users. The transmit distance
between the RIS and the user $u$ is $|\bm{d}_{0,u}|$, where
$\bm{d}_{0,u}=\bm{s}_{u}^{1}-\bm{s}_{RIS}^{1}$. Similarly, the phase
difference between the $n$-th reflecting element and the first reflecting
element is

$\vartheta^{n}_{k,u}=\frac{2\pi}{\lambda_{k}}\frac{(\bm{d}_{0,u})^{T}\bm{\delta}^{n}_{RIS}}{|\bm{d}_{0,u}|},n=1,\cdots,N,$
(13)

and the transmit array vector from the RIS to user $u$ on SB $k$ is expressed
as

$\bm{e}_{k,u}=\left[1,\cdots,\exp(-j\vartheta^{n}_{k,u}),\cdots,\exp(-j\vartheta^{N}_{k,u})\right]^{T}.$
(14)

As the power consumption of the RIS cannot be ignored, we assume that the WPH
module is equipped in the RIS so that the RIS can also harvest energy from the
radios sent by AP. As a result, the reflecting coefficient is denoted by
$\Phi_{n}=\beta_{n}\exp(j\phi_{n})$, where $\beta_{n}$ and $\phi_{n}$
respectively represent the amplitude and the phase shift of the $n$-th
reflecting element. Then, the phase shift matrix of the RIS is denoted by

$\bm{\Phi}=\text{diag}\\{\beta_{n}\exp(j\phi_{n}),n=1,\cdots,N\\}.$ (15)

We then have the following constraints for the reflecting coefficients as

$C1:|\beta_{n}\exp(j\phi_{n})|\leq 1,n=1,\cdots,N.$ (16)

In addition, the phase difference at SB $k$ between user $u$’s first receive
antenna element and the $n_{r}$-th element is

$\zeta^{n_{r}}_{k,u}=\frac{2\pi}{\lambda_{k}}\frac{(\bm{d}_{0,u})^{T}\bm{\delta}^{n_{r}}_{u}}{|\bm{d}_{0,u}|},n_{r}=1,\cdots,N_{r}.$
(17)

Then, the receive array vector from user $u$ to RIS on SB $k$ is expressed as

$\bm{r}_{k,u}=\left[1,\cdots,\exp(-j\zeta^{n_{r}}_{k,u}),\cdots,\exp(-j\zeta^{N_{r}}_{k,u})\right]^{T}.$
(18)

The cascaded channel gain of the AP-RIS-user $u$ link on SB $k$ can be
expressed as [51]

${g}_{k,u}=\left(G_{t}G_{r}\frac{\lambda_{k}}{8\sqrt{\pi^{3}}|\bm{d}_{0,u}||\bm{d}_{0}|}\right)\exp\left({-j2\pi\frac{|\bm{d}_{0,u}|+|\bm{d}_{0}|}{\lambda_{k}}}\right)\exp\left({-\frac{1}{2}K(f_{k})(|\bm{d}_{0,u}|+|\bm{d}_{0}|)}\right),$
(19)

where $|\bm{d}_{0,u}|$ and $|\bm{d}_{0}|$ represent the distance from the RIS
to the user $u$ and the AP, respectively. Overall, the AP-RIS-user $u$ link on
SB $k$ is given by

$\bm{G}_{k,u}=g_{k,u}\bm{r}_{k,u}\bm{e}_{k,u}^{H}\bm{\Phi}\bm{e}_{k}\bm{v}_{k}^{H},u\in\mathcal{I}\cup\mathcal{M}.$
(20)

###  II-C Information Transfer

The signal vector transmitted from the AP to IU $i$ on SB $k$ is
$\bm{s}_{k,i}\in\mathbb{C}^{d\times 1}$. Suppose that the data symbol
$\bm{s}_{k,i}$ satisfies $\mathbb{E}[\bm{s}_{k,i}\bm{s}^{*}_{k,i}]=\bm{I}_{d}$
and $\mathbb{E}[\bm{s}_{k,i}\bm{s}^{*}_{k,j}]=\bm{0}$ for $i\neq j$. Let
$\bm{F}_{k,i}\in\mathbb{C}^{N_{t}\times d}$ denote the precoding matrix used
by the AP for IU $i$ on SB $k$. Then, the transmitted signal
$\bm{x}_{k}\in\mathbb{C}^{N_{t}\times 1}$ from the AP on SB $k$ is

$\bm{x}_{k}=\sum_{i=1}^{I}\bm{F}_{k,i}\bm{s}_{k,i}.$ (21)

With the aid of the RIS, the received signal at the IU $i$ on SB $k$ is

$\bm{y}_{k,i}=(\bm{H}_{k,i}+\bm{G}_{k,i})\bm{x}_{k}+\bm{n}_{k,i}=\bm{Z}_{k,i}\bm{x}_{k}+\bm{n}_{k,i},$
(22)

where $\bm{Z}_{k,i}=\bm{H}_{k,i}+\bm{G}_{k,i}$, and $\bm{n}_{k,i}$ is the
additive Gaussian noise.

Then, the achievable data rate of IU $i$ on SB $k$ is given by

$R_{k,i}=\log\left|\bm{I}_{N_{r}}+\bm{F}^{H}_{k,i}\bm{Z}^{H}_{k,i}\bm{Z}_{k,i}\bm{F}_{k,i}\bm{J}_{k,i}^{-1}\right|,$
(23)

where $\bm{J}_{k,i}=\sum_{u\neq
i}^{I}\bm{Z}_{k,i}\bm{F}_{k,u}\bm{F}^{H}_{k,u}\bm{Z}^{H}_{k,i}+\sigma_{k,i}^{2}\bm{I}_{N_{r}}$,
and $\sigma_{k,i}^{2}$ is the noise power.

Furthermore, as the transmit power is limited, we have the following
constraints for the precoding matrices:

$C2:\sum_{k=1}^{K}\sum_{i=1}^{I}\|\bm{F}_{k,i}\|^{2}_{F}\leq P_{T}^{max}.$
(24)

###  II-D Energy Harvesting

As the RIS also harvests energy from the AP, so that the reflecting
coefficients can be adjusted to satisfy the energy harvesting requirement.
That is to say, a part of the AP’s energy is reflected by the RIS, and the
remaining part is fed into the RIS’s WPH unit for harvesting. The power
received by RIS on SB $k$ is

$q^{in}_{k}=\sum_{i=1}^{I}\text{tr}\left(\bm{F}^{H}_{k,i}\bm{H}_{k}^{H}\bm{H}_{k}\bm{F}_{k,i}\right).$
(25)

The reflected power by RIS on SB $k$ is

$q^{out}_{k}=\sum_{i=1}^{I}\text{tr}\left(\bm{F}^{H}_{k,i}\bm{H}^{H}_{k}{\bm{\Phi}^{H}\bm{\Phi}}\bm{H}_{k}\bm{F}_{k,i}\right).$
(26)

Then, the harvested power by RIS is calculated by

$P_{RIS}=\sum_{k=1}^{K}\eta_{k}(q^{in}_{k}-q^{out}_{k}).$ (27)

where $\eta_{k}$ denotes the power harvesting efficiency on SB $k$, since the
RF-DC conversion efficiency is dependent on the carrier’s frequency. Let
$P^{I}$ denote the required power for RIS, then we have

$C3:\sum_{k=1}^{K}\sum_{i=1}^{I}\eta_{k}\text{tr}\left(\bm{F}^{H}_{k,i}\bm{H}^{H}_{k}(\bm{I}-\bm{\Phi}^{H}\bm{\Phi})\bm{H}_{k}\bm{F}_{k,i}\right)\geq
P^{I}.$ (28)

Similarly, the power harvested by EU $m$ should satisfy the following
constraint:

$C4:\sum_{k=1}^{K}\sum_{i=1}^{I}\eta_{k}\text{tr}\left(\bm{Z}_{k,m}\bm{F}_{k,i}\bm{F}^{H}_{k,i}\bm{Z}^{H}_{k,m}\right)\geq
P^{U}_{m},m\in\mathcal{M},$ (29)

where $P^{U}_{m}$ is the required power of EU $m$, and $\bm{Z}_{k,m}$ is the
composite channel gain between the AP and the EU $m$ on SB $k$.

###  II-E Problem Formulation

For simplicity, we define $\bm{\beta}=[\beta_{1},\cdots,\beta_{N}]$ as the
amplitude vector of the reflecting coefficients, and define
$\bm{\phi}=[\phi_{1},\cdots,\phi_{N}]$ as the phase shifts of the reflecting
coefficients. For ease of presentation, in the following, we utilize the
notation $\bm{L}$ to represent the coordinate of RIS, which is optimized in
the following section. In the system model, we take the coordinate of RIS’s
first reflecting element, i.e., $\bm{s}_{RIS}^{1}$, as the reference
coordinate in the system model. That is to say, $\bm{L}$ is equivalent to
$\bm{s}_{RIS}^{1}$ in the following.

It is observed that the sum rate of IUs and the harvested power for EUs are
dependent on the coordinate of RIS $\bm{L}$, the transmit precoding matrices
and the reflecting coefficients of the RIS. Then, we can formulate the problem
as:

$\begin{split}\underset{\bm{\beta},\bm{\phi},\bm{L},\bm{F}_{k,i}}{\text{max}}&\quad\quad
R_{s}=\sum_{k=1}^{K}\sum_{i=1}^{I}R_{k,i}\\\ \text{s.t.}&\quad
C1-C4.\end{split}$ (30)

It is observed that Problem (30) is non-convex and difficult to solve due to
the following reasons. First of all, the optimization variables are coupled
together and the objective function is intractable. Moreover, according to the
RIS-assisted channel model in Section II-B, there is a complicated
relationship between the position of RIS, the RIS’s reflecting coefficients
and the channel gain. Therefore, effective reformulation and simplification
are required to tackle the above optimization problem.

##  III Solution Analysis

The original Problem (30) is non-convex and challenging to solve, we first
reformulate the problem by leveraging the equivalence between the minimum
mean-square error (MMSE) and the signal-to-noise ratio (SINR). At IU $i$, the
receive decoding matrix $\bm{U}_{k,i}\in\mathbb{C}^{N_{r}\times d}$ is applied
to the received signal on SB $k$ so that
$\hat{\bm{s}}_{k,i}=\bm{U}_{k,i}^{H}\bm{y}_{k,i}$. Then, the received mean
square error (MSE) of IU $i$ on SB $k$ is given by

$\displaystyle\bm{E}_{k,i}$
$\displaystyle=\mathbb{E}_{\bm{s},\bm{n}}\left[(\hat{\bm{s}}_{k,i}-\bm{s}_{k,i})(\hat{\bm{s}}_{k,i}-\bm{s}_{k,i})^{H}\right]$
$\displaystyle=(\bm{U}_{k,i}^{H}\bm{Z}_{k,i}\bm{F}_{k,i}-\bm{I}_{d})^{2}+\sum_{u\neq
i}^{I}\bm{U}_{k,i}^{H}\bm{Z}_{k,i}\bm{F}_{k,u}\bm{F}^{H}_{k,u}\bm{Z}^{H}_{k,i}\bm{U}_{k,i}+\bm{U}_{k,i}^{2}\sigma^{2}_{k,i}.$
(31)

The optimal MMSE decoding matrix $\\{\bm{U}_{k,i}\\}$ is given by

$\bm{U}_{k,i}=(\bm{Z}_{k,i}\bm{F}_{k,i}\bm{F}^{H}_{k,i}\bm{Z}^{H}_{k,i}+\bm{J}_{k,i})^{-1}\bm{Z}_{k,i}\bm{F}_{k,i}.$
(32)

Then, substituting (32) into (III), we have

$\bm{E}_{k,i}^{\text{min}}=\bm{I}_{d}-\bm{F}^{H}_{k,i}\bm{Z}^{H}_{k,i}(\bm{Z}_{k,i}\bm{F}_{k,i}\bm{F}^{H}_{k,i}\bm{Z}^{H}_{k,i}+\bm{J}_{k,i})^{-1}\bm{Z}_{k,i}\bm{F}_{k,i}.$
(33)

According to the relationship between the $\bm{E}_{k,i}^{\text{min}}$ and the
SINR shown in [52], the original Problem (30) can be reformulated as Problem
(34) by introducing a set of auxiliary variables $\\{\bm{W}_{k,i}\\}$ together
with the receiving matrices $\\{\bm{U}_{k,i}\\}$.

$\begin{split}\underset{\bm{\beta},\bm{\phi},\bm{L},\bm{F}_{k,i},\bm{W}_{k,i},\bm{U}_{k,i}}{\text{min}}&\quad
O^{tot}=\sum_{k=1}^{K}\sum_{i=1}^{I}\left(\text{tr}\left(\bm{W}_{k,i}\bm{E}_{k,i}\right)-\log|\bm{W}_{k,i}|\right)\\\
\text{s.t.}&\quad C1-C4.\end{split}$ (34)

Although Problem (34) has more optimization variables, the objective function
of Problem (34) is more tractable. Consequently, Problem (34) can be solved by
employing the BCD algorithm, where the optimization variables can be
iteratively obtained while keeping the others fixed. That is to say, we
decouple the optimization problem into three subproblems: optimization of the
precoding for IUs, RIS’s reflecting coefficients and RIS’s coordinate. Note
that the receiving matrices $\\{\bm{U}_{k,i}\\}$ and the auxiliary matrices
$\bm{W}_{k,i}$ can be directly solved according to the above analysis. Then,
the optimal decoding matrix $\\{\bm{U}_{k,i}\\}$ is given by (32), and the
optimal $\bm{W}_{k,i}^{*}$ is given by

$\bm{W}^{*}_{k,i}=(\bm{E}_{k,i}^{\text{min}})^{-1}.$ (35)

###  III-A Precoding Matrices Design

Given the coordinate of RIS $\bm{L}$, RIS’s reflecting coefficients, auxiliary
matrices $\bm{W}_{k,i}$ and $\\{\bm{U}_{k,i}\\}$, the precoding matrices are
optimized in this section. By substituting the MSE $\bm{E}_{k,i}$ in (III)
into (34) and discarding the constant terms, the precoding matrices
$\bm{F}_{k,i}$ are determined by the following problem

$\begin{split}\underset{\bm{F}_{k,i}}{\text{min}}&\sum_{k=1}^{K}\sum_{i=1}^{I}\text{tr}\left(\bm{F}^{H}_{k,i}\bm{\bar{W}}_{k}\bm{F}_{k,i}\right)-\sum_{k=1}^{K}\sum_{i=1}^{I}2\Re\left[\text{tr}(\bm{\bar{Z}}_{k,i}\bm{F}_{k,i})\right]\\\
\text{s.t.}&\quad C2-C4,\end{split}$ (36)

where
$\bm{\bar{W}}_{k}=\sum_{i=1}^{I}\bm{Z}^{H}_{k,i}\bm{U}_{k,i}\bm{W}_{k,i}\bm{U}^{H}_{k,i}\bm{Z}_{k,i}$
and $\bm{\bar{Z}}_{k,i}=\bm{W}_{k,i}\bm{U}^{H}_{k,i}\bm{Z}_{k,i}$. Although
the objective function of Problem (36) is convex, the energy harvesting
constraints $C3$ and $C4$ are non-convex. Note that
$\bm{I}-\bm{\Phi}^{H}_{n}\bm{\Phi}_{n}$ is positive definite. As a result, we
adopt the successive convex approximation method by leveraging the first-order
Taylor expansions with the given precoding matrix $\bm{\bar{F}}_{k,i}$ as

$\displaystyle\text{tr}(\bm{F}^{H}_{k,i}\bm{B}_{k}\bm{{F}}_{k,i})\geq
2\Re\\{\text{tr}(\bm{\bar{F}}^{H}_{k,i}\bm{B}_{k}\bm{F}_{k,i})\\}-\text{tr}(\bm{\bar{F}}^{H}_{k,i}\bm{B}_{k}\bm{\bar{F}}_{k,i}),$
(37) $\displaystyle\text{tr}(\bm{{F}}^{H}_{k,i}\bm{C}_{k,m}\bm{{F}}_{k,i})\geq
2\Re\\{\text{tr}(\bm{\bar{F}}^{H}_{k,i}\bm{C}_{k,m}\bm{F}_{k,i})\\}-\text{tr}(\bm{\bar{F}}^{H}_{k,i}\bm{C}_{k,m}\bm{\bar{F}}_{k,i}),$
(38)

where
$\bm{B}_{k}=\eta_{k}\bm{H}^{H}_{k}(\bm{I}-\bm{\Phi}^{H}\bm{\Phi})\bm{H}_{k}$
and $\bm{C}_{k,m}=\eta_{k}\bm{Z}^{H}_{k,m}\bm{Z}_{k,m}$. Then, $C3$ and $C4$
can be respectively reformulated as

$\displaystyle\sum_{k=1}^{K}\sum_{i=1}^{I}2\Re\\{\text{tr}(\bm{\bar{F}}^{H}_{k,i}\bm{B}_{k}\bm{F}_{k,i})\\}\geq\sum_{k=1}^{K}\sum_{i=1}^{I}\text{tr}(\bm{\bar{F}}^{H}_{k,i}\bm{B}_{k}\bm{\bar{F}}_{k,i})+P_{RIS},$
(39)
$\displaystyle\sum_{k=1}^{K}\sum_{i=1}^{I}2\Re\\{\text{tr}(\bm{\bar{F}}^{H}_{k,i}\bm{C}_{k,m}\bm{F}_{k,i})\\}\geq\sum_{k=1}^{K}\sum_{i=1}^{I}\text{tr}(\bm{\bar{F}}^{H}_{k,i}\bm{C}_{k,m}\bm{\bar{F}}_{k,i})+P^{UE}_{m}.$
(40)

Then, by substituting $C3$ and $C4$ with (39) and (40) respectively, Problem
(36) can be transformed into a series of convex problems, which can be solved
by standard tools, such as the CVX.

###  III-B RIS Reflecting Coefficient Optimization

Given $\bm{W}_{k,i}$, $\\{\bm{U}_{k,i}\\}$, $\\{\bm{F}_{k,i}\\}$ and the RIS’s
coordinate $\bm{L}$, we consider the optimization of RIS’s reflecting
coefficients $\varphi_{n}=\beta_{n}\exp(j\phi_{n})$, where the reflecting
matrix is $\bm{\Phi}=\text{diag}\\{[\varphi_{n}]_{n=1}^{N}\\}$.

According to the MSE given in (III), we have

$\text{tr}(\bm{W}_{k,i}\bm{E}_{k,i})=\text{tr}(\bm{W}_{k,i}\bm{U}_{k,i}^{H}\bm{Z}_{k,i}\bm{F}^{s}_{k}\bm{Z}_{k,i}^{H}\bm{U}_{k,i})-2\Re{\left[\text{tr}(\bm{Z}_{k,i}\bm{F}_{k,i}\bm{W}_{k,i}\bm{U}^{H}_{k,i})\right]}+\text{const},$
(41)

where $\bm{F}^{s}_{k}=\sum_{u=1}^{I}\bm{F}_{k,u}\bm{F}^{H}_{k,u}$. The term
“const” denotes the constant that is irrelevant with the reflecting
coefficients $\varphi_{n}$. As $\bm{Z}_{k,i}=\bm{H}_{k,i}+\bm{G}_{k,i}$, by
removing the irrelevant terms in (41), the reflecting coefficient optimization
problem is formulated as

$\begin{split}\underset{\bm{\varphi}}{\text{min}}&\sum_{k=1}^{K}\sum_{i=1}^{I}O_{k,i}(\bm{\varphi})\\\
\text{s.t.}&\quad C1,C3,C4,\end{split}$ (42)

where $\bm{\varphi}=[\varphi_{1},\cdots,\varphi_{N}]^{T}$. In the objective
function of (42), for simplicity, we define
$\bm{\bar{U}}_{k,i}=\bm{U}_{k,i}\bm{W}_{k,i}\bm{U}^{H}_{k,i}$,
$\bm{\bar{F}}_{k,i}=\bm{F}_{k,i}\bm{W}_{k,i}\bm{U}_{k,i}^{H}$, and then we
have

$O_{k,i}(\bm{\varphi})=2\Re[\text{tr}(\bm{G}_{k,i}\bm{F}^{s}_{k}\bm{H}^{H}_{k,i}\bm{\bar{U}}_{k,i})]+\text{tr}(\bm{G}_{k,i}\bm{F}^{s}_{k}\bm{G}^{H}_{k,i}\bm{\bar{U}}_{k,i})-2\Re[\text{tr}(\bm{G}_{k,i}\bm{\bar{F}}_{k,i})].$
(43)

Note that $\bm{\bar{U}}_{k,i}$ is hermitian, but it is still difficult to
solve Problem (42) with this formulation.

To obtain a more tractable problem formulation, we define
$\bm{u}_{k,i}=\bm{e}_{k,i}^{H}\odot\bm{e}_{k}^{T}$. Then, the RIS assisted
channel gain is represented as

$\bm{G}_{k,i}=g_{k,i}(\bm{u}_{k,i}\bm{\varphi})\bm{r}_{k,i}\bm{v}_{k}^{H}.$
(44)

Substituting (44) into (43), we have

$O_{k,i}(\bm{\varphi})=2\Re\\{g_{k,i}\xi_{k,i}\bm{u}_{k,i}\bm{\varphi}\\}+A_{k,i}(g_{k,i})^{2}(\bm{u}_{k,i}\bm{\varphi})^{2},$
(45)

where
$A_{k,i}=\text{tr}(\bm{r}_{k,i}\bm{v}_{k}^{H}\bm{F}^{s}_{k}\bm{v}_{k}\bm{r}^{H}_{k,i}\bm{\bar{U}}_{k,i})$,
and
$\xi_{k,i}=(\text{tr}(\bm{r}_{k,i}\bm{v}_{k}^{H}\bm{F}^{s}_{k}\bm{H}^{H}_{k,i}\bm{\bar{U}}_{k,i})-\text{tr}(\bm{r}_{k,i}\bm{v}_{k}^{H}\bm{\bar{F}}_{k,i}))$.

Similarly, according to (12), constraint $C3$ can be reformulated as

$\text{tr}\left(\bm{F}^{H}_{k,i}\bm{H}^{H}_{k}(\bm{I}-\bm{\Phi}^{H}\bm{\Phi})\bm{H}_{k}\bm{F}_{k,i}\right)=(H_{k})^{2}\text{tr}\left(\bm{F}^{H}_{k,i}\bm{v}_{k}\bm{v}_{k}^{H}\bm{F}_{k,i}\right)(N-\bm{\varphi}^{H}\bm{\varphi}),$
(46)

where $\bm{e}_{k}^{H}\bm{e}_{k}=N$ is utilized.

Similarly, for constraint $C4$, by substituting
$\bm{Z}_{k,m}=\bm{H}_{k,m}+\bm{G}_{k,m}$ and (44) into $C4$, we have the
following reformulation as

$\sum_{i=1}^{I}\text{tr}\left(\bm{Z}_{k,m}\bm{F}_{k,i}\bm{F}^{H}_{k,i}\bm{Z}^{H}_{k,m}\right)=(g_{k,m})^{2}\Lambda_{k,m}(\bm{u}_{k,m}\bm{\varphi})^{2}+2\Re\\{g_{k,m}w_{k,m}\bm{u}_{k,m}\bm{\varphi}\\}+Q_{k,m},$
(47)

where
$\Lambda_{k,m}=\text{tr}\left(\bm{r}_{k,i}\bm{v}_{k}^{H}\bm{F}^{s}_{k}\bm{v}_{k}\bm{r}_{k,i}^{H}\right)$,
$w_{k,m}=\text{tr}\left(\bm{r}_{k,i}\bm{v}_{k}^{H}\bm{F}^{s}_{k}\bm{H}^{H}_{k,m}\right)$,
and
$Q_{k,m}=\text{tr}\left(\bm{H}_{k,m}\bm{F}^{s}_{k}\bm{H}^{H}_{k,m}\right)$.

According to (44), (46) and (47), the RIS’s reflecting coefficient problem can
be reformulated as

$\displaystyle\underset{\bm{\varphi}}{\text{min}}$
$\displaystyle\quad\quad\bm{\varphi}^{H}\bm{A}\bm{\varphi}+\Re\\{\bm{\xi}\bm{\varphi}\\}$
(48a) s.t.
$\displaystyle\quad\bm{\varphi}^{H}\bm{\Lambda}_{m}\bm{\varphi}+\Re\\{\bm{\omega}_{m}\bm{\varphi}\\}\geq\tilde{P}^{U}_{m}.$
(48b) $\displaystyle\quad(N-\bm{\varphi}^{H}\bm{\varphi})C_{RIS}\geq P^{I}$
(48c) $\displaystyle\quad|\bm{\varphi}|\leq\bm{1}.$ (48d)

where

$\displaystyle\bm{A}=\sum_{k=1}^{K}\sum_{i=1}^{I}A_{k,i}(g_{k,i})^{2}\bm{u}^{H}_{k,i}\bm{u}_{k,i},\
\bm{\xi}=\sum_{k=1}^{K}\sum_{i=1}^{I}2g_{k,i}\xi_{k,i}\bm{u}_{k,i},$
$\displaystyle\bm{\Lambda}_{m}=\sum_{k=1}^{K}\eta_{k}(g_{k,m})^{2}\Lambda_{k,m}\bm{u}^{H}_{k,m}\bm{u}_{k,m},\
\bm{\omega}_{m}=2\sum_{k=1}^{K}\eta_{k}g_{k,m}w_{k,m}\bm{u}_{k,m},$
$\displaystyle\tilde{P}^{U}_{m}={P}^{U}_{m}-\sum_{k=1}^{K}Q_{k,m},\text{and}\
C_{RIS}=\sum_{k=1}^{K}\sum_{i=1}^{I}\eta_{k}(H_{k})^{2}\text{tr}\left(\bm{F}^{H}_{k,i}\bm{v}_{k}\bm{v}_{k}^{H}\bm{F}_{k,i}\right).$

However, it is observed that constraint (48b) is non-convex. Note that
$\bm{\Lambda}_{m}$ is positive-definite so that we adopt the first-order
Taylor expansion for convex approximation. At given $\bm{\bar{\varphi}}$, we
have

$\bm{\varphi}^{H}\bm{\Lambda}_{m}\bm{\varphi}\geq
2\Re\\{\bm{\varphi}^{H}\bm{\Lambda}_{m}\bm{\bar{\varphi}}\\}-\bm{\bar{\varphi}}^{H}\bm{\Lambda}_{m}\bm{\bar{\varphi}}.$
(49)

By utilizing (49) to simplify the (48b), Problem (48) can be transformed into
a series of simple convex problems, which can be easily solved by CVX.

###  III-C Optimization of RIS’s Coordinate

We consider the optimization of RIS coordinate with given $\bm{W}_{k,i}$ and
$\\{\bm{U}_{k,i}\\}$, $\\{\bm{F}_{k,i}\\}$ and the phase shift matrix
$\bm{\Phi}$. In this case, based on the formulations given in (45), (46) and
(47), the original Problem (34) with respect to the RIS coordinate is
formulated as

$\displaystyle\underset{\bm{L}}{\text{min}}$ $\displaystyle\quad
O^{tot}(\bm{L})=\sum_{k=1}^{K}\sum_{i=1}^{I}{g}_{k,i}(\bm{L})^{2}E_{k,i}(\bm{L})+\Re\\{{g}_{k,i}(\bm{L})F_{k,i}(\bm{L})\\}+\text{Cst}(\bm{W}_{k,i},\bm{U}_{k,i},\bm{F}_{k,i})$
(50a) s.t.
$\displaystyle\quad\sum_{k=1}^{K}({g}_{k,i}(\bm{L})^{2}\lambda_{k,m}(\bm{L})+\Re\\{g_{k,m}(\bm{L})\chi_{k,m}(\bm{L})\\}+\eta_{k}Q_{k,m})\geq
P^{U}_{m},$ (50b)
$\displaystyle\quad\sum_{k=1}^{K}H_{k}(\bm{L})^{2}D_{k}(\bm{L})\geq P^{I},$
(50c)

where $\text{Cst}(\bm{W}_{k,i},\bm{U}_{k,i},\bm{F}_{k,i})$ is the constant
term, and

$\displaystyle
E_{k,i}(\bm{L})=A_{k,i}(\bm{L})(\bm{u}_{k,i}(\bm{L})\bm{\varphi})^{2},$ (51)
$\displaystyle
F_{k,i}(\bm{L})=2\xi_{k,i}(\bm{L})\bm{u}_{k,i}(\bm{L})\bm{\varphi},$ (52)
$\displaystyle\lambda_{k,m}(\bm{L})=\eta_{k}\Lambda_{k,m}(\bm{L})(\bm{u}_{k,m}(\bm{L})\bm{\varphi})^{2},$
(53)
$\displaystyle\chi_{k,m}(\bm{L})=2\eta_{k}w_{k,m}(\bm{L})\bm{u}_{k,m}(\bm{L})\bm{\varphi},$
(54) $\displaystyle
D_{k}(\bm{L})=\sum_{i=1}^{I}\eta_{k}\text{tr}\left(\bm{F}^{H}_{k,i}\bm{v}_{k}(\bm{L})\bm{v}_{k}^{H}(\bm{L})\bm{F}_{k,i}\right)(N-\bm{\varphi}^{H}\bm{\varphi}).$
(55)

According to the channel model, many periodic cosine components with respect
to the SB’s index and UE’s index are involved in $E_{k,i}(\bm{L})$,
$F_{k,i}(\bm{L})$, $\lambda_{k,m}(\bm{L})$, $\chi_{k,m}(\bm{L})$ and
$D_{k}(\bm{L})$. However, they are all dependent on the RIS’s coordinate
$\bm{L}=[X,Y,Z]$. Their complex expressions make it very difficult to directly
optimize the objective function given in (50a). Therefore, we seek to find a
tractable formulation of the coordinate optimization problem by regarding
these intractable terms as constants.

First of all, for ease of exposition, we define auxiliary variables $r_{u}$
and $d_{0}$, which are dependent on the RIS’s coordinate $\bm{L}$ by

$d_{0}=|\bm{L}-\bm{s}_{AP}^{1}|,r_{u}=|\bm{L}-\bm{s}_{u}^{1}|,u\in\mathcal{I}\cup\mathcal{M}.$
(56)

Then, we define the function $f_{k,u}(r_{u},d_{0})$ with respect to
$(r_{u},d_{0})$ as

$f_{k,u}(r_{u},d_{0})=\frac{\mu_{k}}{r_{u}d_{0}}\exp\left({-K_{k}(r_{u}+d_{0})}\right),u\in\mathcal{I}\cup\mathcal{M},$
(57)

where $\mu_{k}=\frac{G_{t}G_{r}\lambda_{k}}{8\sqrt{\pi^{3}}}$, and
$K_{k}=\frac{K(f_{k})}{2}$.

According to (19), the cascaded channel gain ${g}_{k,i}(\bm{L})$ can be
represented as a function with respect to $(r_{i},d_{0})$ as

${g}_{k,i}(r_{i},d_{0})=f_{k,i}(r_{i},d_{0})\exp\left({-j2\pi\frac{r_{i}+d_{0}}{\lambda_{k}}}\right),u\in\mathcal{I}\cup\mathcal{M}.$
(58)

Next, the objective function and constraints are investigated step by step. As
the solution is obtained by the iterative algorithm, we adopt a given RIS’s
coordinate obtained at the $(n)$-th iteration denoted as $\bm{L}^{(n)}$ to
help find the tractable formulation of Problem (50).

####  III-C1 Simplification of Objective

To simplify objective (50a), substituting the coordinate $\bm{L}^{(n)}$ into
(51) and (52), we can obtain the following constants as

$E_{k,i}\triangleq
E_{k,i}(\bm{L}^{(n)}),F_{k,i}\triangleq\Re\left\\{\exp\left({-j2\pi\frac{\tilde{r}_{i}+\tilde{d}_{0}}{\lambda_{k}}}\right)F_{k,i}(\bm{L}^{(n)})\right\\},i\in\mathcal{I},$

where the constants $\tilde{r}_{i}$ and $\tilde{d}_{0}$ are obtained by
leveraging the coordinate $\bm{L}^{(n)}$ in $(n)$-th iteration as

$\tilde{r}_{i}=|\bm{L}^{(n)}-\bm{s}_{i}^{1}|,\tilde{d}_{0}=|\bm{L}^{(n)}-\bm{s}_{AP}^{1}|.$

The Hessian matrix of $f_{k,i}(r_{i},d_{0})$ with respect to $(r_{i},d_{0})$
is given by

$\bm{\bigtriangledown}^{2}f_{k,i}=\frac{\mu_{k}}{r_{i}d_{0}}\exp\left({-K_{k}(r_{u}+d_{0})}\right)\left[\begin{array}[]{cc}\frac{(K_{k}r_{u}+1)^{2}+1}{r_{i}^{2}}&\frac{(K_{k}r_{u}+1)(K_{k}d_{0}+1)}{r_{i}d_{0}}\\\
\frac{(K_{k}r_{u}+1)(K_{k}d_{0}+1)}{r_{i}d_{0}}&\frac{(K_{k}d_{0}+1)^{2}+1}{d_{0}^{2}}\end{array}\right].$
(59)

Then, it is observed that $\bm{\bigtriangledown}^{2}f_{k,i}$ is positive-
definite so that $f^{2}_{k,i}(r_{i},d_{0})$ is convex with respect to
$(r_{i},d_{0})$. However, note that the calculated coefficient $F_{k,i}$ is
not necessarily positive. As a result, the first-order Taylor expansion of
$f_{k,i}(r_{i},d_{0})$ is adopted for simplification as

$f_{k,i}(r_{i},d_{0})\geq
f_{k,i}(\tilde{r}_{i},\tilde{d}_{0})+\nabla_{r_{i}}f_{k,i}(\tilde{r}_{i},\tilde{d}_{0})(r_{i}-\tilde{r}_{i})+\nabla_{d_{0}}f_{k,i}(\tilde{r}_{i},\tilde{d}_{0})(d_{0}-\tilde{d}_{0}),$
(60)

where the derivative is

$\nabla_{x}f_{k,i}(x,y)=-\mu_{k}\frac{K_{k}x+1}{x^{2}y}\exp\left({-K_{k}(x+y)}\right).$
(61)

In addition, the Hessian matrix of $f^{2}_{k,i}(r_{i},d_{0})$ with respect to
$(r_{i},d_{0})$ is

$\bm{\bigtriangledown}^{2}f^{2}_{k,i}=\frac{\mu_{k}^{2}}{r^{2}_{i}d^{2}_{0}}\exp\left({-2K_{k}(r_{u}+d_{0})}\right)\left[\begin{array}[]{cc}\frac{(2K_{k}r_{u}+2)^{2}+2}{r_{i}^{2}}&\frac{(2K_{k}r_{u}+2)(2K_{k}d_{0}+2)}{r_{i}d_{0}}\\\
\frac{(2K_{k}r_{u}+2)(2K_{k}d_{0}+2)}{r_{i}d_{0}}&\frac{(2K_{k}d_{0}+2)^{2}+2}{d_{0}^{2}}\end{array}\right].$
(62)

As $\bm{\bigtriangledown}^{2}f^{2}_{k,i}$ is positive definite,
$f_{k,i}(r_{i},d_{0})$ is convex with respect to $(r_{i},d_{0})$. Furthermore,
it is observed that the calculated coefficient $E_{k,i}$ is always positive.
Consequently, by leveraging (58) and (60), the calculated $E_{k,i}$, $F_{k,i}$
and discarding the irrelevant constants, a simplified version of the objective
(50a) with a given coordinate $\bm{L}^{(n)}$ is given as

$O_{\bm{L}^{(n)}}(r_{i},d_{0})=\sum_{k=1}^{K}\sum_{i=1}^{I}(f^{2}_{k,i}(r_{i},d_{0})E_{k,i}+F_{k,i}(\nabla_{r_{i}}f_{k,i}(\tilde{r}_{i},\tilde{d}_{0})r_{i}+\nabla_{d_{0}}f_{k,i}(\tilde{r}_{i},\tilde{d}_{0})d_{0})).$
(63)

It is observed that the objective $O_{\bm{L}^{(n)}}(r_{i},d_{0})$ is convex
with respect to $(r_{i},d_{0})$.

####  III-C2 Simplification of Constraints for EUs

Then, we deal with constraint (50b) for the EU $m$.

By substituting the coordinate $\bm{L}^{(n)}$ into (53) and (54), the
following constants can be obtained as

$\lambda_{k,m}\triangleq\lambda_{k,m}(\bm{L}^{(n)}),\chi_{k,m}\triangleq\Re\left\\{\exp\left({-j2\pi\frac{\tilde{r}_{m}+\tilde{d}_{0}}{\lambda_{k}}}\right)\chi_{k,m}(\bm{L}^{(n)})\right\\},m\in\mathcal{M},$

where the constant $\tilde{r}_{m}=|\bm{L}^{(n)}-\bm{s}_{m}^{1}|$ is calculated
with the coordinate $\bm{L}^{(n)}$ .

Then, by employing the function $f_{k,m}(r_{m},d_{0}),m\in\mathcal{M}$ in
(57), the constants $\lambda_{k,m}$ and $\chi_{k,m}$, constraint (50b) is
rewritten as

$\sum_{k=1}^{K}(f^{2}_{k,m}(r_{m},d_{0})\lambda_{k,m}+f_{k,m}(r_{u},d_{0})\chi_{k,m}+\eta_{k}Q_{k,m})\geq
P^{U}_{m},m\in\mathcal{M}.$ (64)

However, (64) is still non-convex. As $f_{k,m}(r_{u},d_{0})$ and
$f^{2}_{k,m}(r_{u},d_{0})$ are both convex functions, their first-order Taylor
expansions are

$\displaystyle f^{2}_{k,m}(r_{m},d_{0})$ $\displaystyle\geq
f^{2}_{k,m}(\tilde{r}_{m},\tilde{d}_{0})+\nabla_{r_{m}}f^{2}_{k,m}(\tilde{r}_{m},\tilde{d}_{0})(r_{m}-\tilde{r}_{m})+\nabla_{d_{0}}f^{2}_{k,m}(\tilde{r}_{m},\tilde{d}_{0})(d_{0}-\tilde{d}_{0}),$
(65) $\displaystyle f_{k,m}(r_{m},d_{0})$ $\displaystyle\geq
f_{k,i}(\tilde{r}_{m},\tilde{d}_{0})+\nabla_{r_{m}}f_{k,m}(\tilde{r}_{i},\tilde{d}_{0})(r_{m}-\tilde{r}_{m})+\nabla_{d_{0}}f_{k,m}(\tilde{r}_{m},\tilde{d}_{0})(d_{0}-\tilde{d}_{0}),$
(66)

where $\nabla_{x}f_{k,m}(x,y)$ is given in (61), and
$\nabla_{x}f^{2}_{k,m}(x,y)=2f_{k,m}(x,y)\nabla_{x}f_{k,m}(x,y)$.

Then, substituting (65) and (66) into (64), constraint (50b) for the EU $m$
can be further reformulated as

$\displaystyle
A_{m}(\bm{L}^{(n)},\tilde{r}_{m},\tilde{d}_{0})r_{m}+B_{m}(\bm{L}^{(n)},\tilde{r}_{m},\tilde{d}_{0})d_{0}+C_{m}(\bm{L}^{(n)},\tilde{r}_{m},\tilde{d}_{0})\geq
P_{m}^{U},m\in\mathcal{M},$ (67)

where
$A_{m}(\bm{L}^{(n)},\tilde{r}_{m},\tilde{d}_{0})=\sum_{k=1}^{K}\left(\lambda_{k,m}(\bm{L}^{(n)})\nabla_{r_{m}}f^{2}_{k,m}(\tilde{r}_{m},\tilde{d}_{0})+\chi_{k,m}(\bm{L}^{(n)})\nabla_{r_{m}}f_{k,m}(\tilde{r}_{m},\tilde{d}_{0})\right)$,
$\quad
B_{m}(\bm{L}^{(n)},\tilde{r}_{m},\tilde{d}_{0})=\sum_{k=1}^{K}\left(\lambda_{k,m}(\bm{L}^{(n)})\nabla_{d_{0}}f^{2}_{k,m}(\tilde{r}_{m},\tilde{d}_{0})+\chi_{k,m}(\bm{L}^{(n)})\nabla_{d_{0}}f_{k,m}(\tilde{r}_{m},\tilde{d}_{0})\right)$,
and
$\quad
C_{m}(\bm{L}^{(n)},\tilde{r}_{m},\tilde{d}_{0})=\sum_{k=1}^{K}(\lambda_{k,m}(\bm{L}^{(n)})f^{2}_{k,m}(\tilde{r}_{m},\tilde{d}_{0})+\chi_{k,m}(\bm{L}^{(n)})f_{k,m}(\tilde{r}_{m},\tilde{d}_{0}))-A_{m}(\bm{L}^{(n)},\tilde{r}_{m},\tilde{d}_{0})\tilde{r}_{m}-B_{m}(\bm{L}^{(n)},\tilde{r}_{m},\tilde{d}_{0})\tilde{d}_{0}+\sum_{k=1}^{K}\eta_{k}Q_{k,m}$.

####  III-C3 Simplification of Constraints for RIS

Similarly, the constraint (50c) for the RIS is simplified in the following. We
first define function $h_{k}(d_{0})$ as

$h_{k}(d_{0})=\frac{\rho_{k}}{d_{0}^{2}}\exp\left(-2K_{k}d_{0}\right),$ (68)

where $\rho_{k}=\left(\frac{\lambda_{k}}{4\pi}\right)^{2}$. By checking the
Hessian matrix of $h_{k}(d_{0})$, it can be verified that $h_{k}(d_{0})$ is
convex with respect to $d_{0}$. Also, by substituting the coordinate
$\bm{L}^{(n)}$ into (55), constant $D_{k}$ can be obtained as $D_{k}\triangleq
D_{k}(\bm{L}^{(n)})$.

However, by employing the function $h_{k}(d_{0})$ in (68), and substituting
$D_{k}$ into (50b), the following constraint is still non-convex:

$\sum_{k=1}^{K}h_{k}(d_{0})D_{k}\geq P^{I}.$ (69)

Therefore, the first-order Taylor expansion of $h_{k}(d_{0})$ is utilized for
convex approximation, which is

$\displaystyle h_{k}(d_{0})$ $\displaystyle\geq
h_{k}(\tilde{d}_{0})+\nabla_{d_{0}}^{h_{k}}(\tilde{d}_{0})(d_{0}-\tilde{d}_{0}),$
(70)

where $\nabla_{d_{0}}^{h_{k}}(\tilde{d}_{0})$ represents the first-order
derivative of $h_{k}(d_{0})$ given by

$\nabla_{d_{0}}^{h_{k}}(d_{0})=-\rho_{k}\frac{2K_{k}d_{0}+2}{d_{0}^{3}}\exp\left(-2K_{k}d_{0}\right).$

Then, substituting (70) into (50c), we have

$\displaystyle
A_{RIS}(\bm{L}^{(n)},\tilde{d}_{0})d_{0}+B_{RIS}(\bm{L}^{(n)},\tilde{d}_{0})\geq
P^{I},$ (71)

where the constants are evaluated as

$\displaystyle
A_{RIS}(\bm{L}^{(n)},\tilde{d}_{0})=\sum_{k=1}^{K}D_{k}(\bm{L}^{(n)})\nabla_{d_{0}}^{h_{k}}(\tilde{d}_{0}),$
(72) $\displaystyle
B_{RIS}(\bm{L}^{(n)},\tilde{d}_{0})=\sum_{k=1}^{K}D_{k}(\bm{L}^{(n)})(h_{k}(\tilde{d}_{0})-\tilde{d}_{0}\nabla_{d_{0}}^{h_{k}}(\tilde{d}_{0})).$
(73)

####  III-C4 Simplification of Problem (50)

Finally, given RIS’s coordinate obtained at the $(n)$-th iteration denoted as
$\bm{L}^{(n)}$, by replacing $O^{tot}(\bm{L})$ with
$O_{\bm{L}^{(n)}}(r_{i},d_{0})$ in (63), replacing (50b) and (50c) with (67)
and (71) respectively, Problem (50) is reformulated as

$\displaystyle\underset{\bm{L},r_{u},d_{0}}{\text{min}}$
$\displaystyle\quad\quad O_{\bm{L}^{(n)}}(r_{i},d_{0})$ (74a) s.t.
$\displaystyle\quad|\bm{L}-\bm{s}_{u}^{1}|\leq
r_{u},u\in\mathcal{I}\cup\mathcal{M}$ (74b)
$\displaystyle\quad|\bm{L}-\bm{s}_{AP}^{1}|\leq d_{0}.$ (74c)
$\displaystyle\quad(\ref{UEPwst}),(\ref{RISPwst}).$

Constraints (74b) and (74c) are introduced for the auxiliary variables $r_{u}$
and $d_{0}$. Then, it can be verified that Problem (74) is convex, which can
be readily solved by CVX.

However, note that Problem (74) is the simplified version of Problem (50). The
optimal solution to Problem (74) denoted by $(\bm{L}^{*},r_{u}^{*},d_{0}^{*})$
may not satisfy all the constraints of Problem (50). Consequently, we need to
add the following procedure to ensure that the obtained solution
$(\bm{L}^{*},r_{u}^{*},d_{0}^{*})$ is a feasible solution to Problem (50).

####  III-C5 Feasibility guarantee

First we define the following penalty indicators as

$\displaystyle\alpha=A_{RIS}(\bm{L}^{*},d^{*}_{0})d^{*}_{0}+B_{RIS}(\bm{L}^{*},d^{*}_{0})-P^{I},$
(75)
$\displaystyle\alpha_{m}=A_{m}(\bm{L}^{*},r^{*}_{m},d^{*}_{0})r^{*}_{m}+B_{m}(\bm{L}^{*},r^{*}_{m},d^{*}_{0})d^{*}_{0}+C_{m}(\bm{L}^{*},r^{*}_{m},d^{*}_{0})-P_{m}^{U}.$
(76)

If $\alpha<0$, the obtained coordinate $\bm{L}^{*}$ does not satisfy
constraint (50c). This implies that the distance between RIS and AP is too
large, i.e., the power harvested by the RIS does not exceed the requirement.
By checking the first-order derivative of $h_{k}(d_{0})$ with respect to
$d_{0}$ , it is verified that $\nabla_{d_{0}}^{h_{k}}(d)<0,\forall d\geq 0$.
According to (72), as $D_{k}(\bm{L}^{*})\geq 0$ for all coordinates, so that
$A_{RIS}(\bm{L}^{*},d^{*}_{0})<0$. Consequently, to ensure that (50c) is
satisfied, the RIS should be closer to the AP, i.e., $d_{0}$ should be
reduced. To this end, the required harvested power of RIS, i.e., $P^{I}$ is
modified as

${P^{I}}^{\prime}=P^{I}+\epsilon,\text{if }\alpha<0,$ (77)

where $\epsilon>0$ is the introduced penalty. 111In practise, penalties are
set as $1\%$ of the required powers for satisfactory performance, i.e.,
$\epsilon=1\%P^{I}$ and $\epsilon_{m}=1\%P_{m}^{U}$.

Also, if $\alpha_{m}<0$, the obtained coordinate $\bm{L}^{*}$ does not satisfy
constraint (50b) for EU $m$. Note that $\nabla_{x}f_{k,m}(x,y)<0$,
$\nabla_{y}f_{k,m}(x,y)<0$ and $\lambda_{k,m}(\bm{L})\geq 0,\forall\bm{L}$.
Similarly, according to (67), this implies that $d_{0}$ or $r_{m}$ should be
decreased to satisfy constraint (50b). Given that $d_{0}$ is modified by
adjusting $P^{I}$, therefore, we only adjust the $P_{m}^{U}$ to reduce
$d_{m}$, which should be modified as

${P_{m}^{U}}^{\prime}=P_{m}^{U}+\epsilon_{m},\text{if }\alpha_{m}<0,$ (78)

where $\epsilon_{m}>0$ is the penalty for EU $m$.

Overall, constraints (71) and (67) are updated by replacing ${P^{I}}$ and
${P_{m}^{U}}$ with ${P^{I}}^{\prime}$ and ${P_{m}^{U}}^{\prime}$ in (77) and
(78), respectively. The solution to Problem (74) with the modified constraints
(71) and (67) should be updated accordingly. Then, the finally obtained
solution $(\bm{L}^{*},r_{u}^{*},d_{0}^{*})$ is guaranteed to satisfy all the
constraints of Problem (50) when $\alpha\geq 0$ and $\alpha_{m}\geq 0,\forall
m$.

_Remark_ 1: Once the obtained $(\bm{L}^{*},r_{u}^{*},d_{0}^{*})$ is a feasible
solution to the original Problem (50). We denote it as the $(n+1)$-th
coordinate of the RIS, i.e., $\bm{L}^{(n+1)}=\bm{L}^{*}$ when $\alpha\geq 0$
and $\alpha_{m}\geq 0,\forall m$. Then, Problem (74) can be formulated based
on $\bm{L}^{(n+1)}$, and the feasible coordinate of RIS for the $(n+2)$-th
iteration can be obtained by leveraging the penalties for power harvesting
constraints.

_Remark_ 2: In addition, as the objective function
$O_{\bm{L}^{(n)}}(r_{i},d_{0})$ is the approximation of (50a), we need to
check the original objective value $O^{tot}(\bm{L}^{(n+1)})$ in (50a) at each
iteration. Note that the receiving matrix $\bm{U}^{n}_{k,i}$ aims to minimize
the MSE for a given channel $\bm{Z}(\bm{L}^{(n)})$, and weight
$\bm{W}^{(n)}_{k,i}$ is dependent on the current MMSE matrix. As the channel
is optimized by adopting the RIS’s new coordinate $\bm{L}^{(n+1)}$, the
receiving matrix $\bm{U}_{k,i}$ matched to this new channel and the obtained
MMSE should be updated according to (32) and (35), respectively. That is to
say, the objective value at the $(n+1)$-th iteration is evaluated as
$O^{tot}(\bm{L}^{(n+1)},\bm{W}^{n+1}_{k,i},\bm{U}^{n+1}_{k,i})$.

_Remark_ 3: Denote the initial coordinate of RIS as $\bm{L}^{(0)}$. To
guarantee the monotonicity of RIS’s position optimization, we need to find the
optimized coordinate $\bm{L}^{(n^{\prime})}$ which can reduce the objective
$O^{tot}$ compared with the initial coordinate $\bm{L}^{(0)}$. That is to
say222As $O^{tot}$ consists of two parts with different physical meanings:
weighted MSE and rate, their numerical values may have more than 3 orders of
magnitude difference. In this case, it is better to check these two parts
separately for satisfactory performance. ,

$O^{tot}(\bm{L}^{(n^{\prime})},\bm{W}^{n^{\prime}}_{k,i},\bm{U}^{n^{\prime}}_{k,i})<O^{tot}(\bm{L}^{(0)},\bm{W}^{0}_{k,i},\bm{U}^{0}_{k,i}).$
(79)

Although the obtained objective in $(n+1)$-th iteration may be larger than
that of the $(n)$-th iteration, we still update Problem (74) by utilizing
$\bm{L}^{(n+1)}$ for the next feasible solution when the coordinate
$\bm{L}^{(n^{\prime})}$ satisfying (79) have not be found.

_Remark_ 4: Note that Problem (74) may become infeasible due to the penalties
added in constraints (71) and (67). Consequently, it may exist the case that
the coordinate $\bm{L}^{(n^{\prime})}$ satisfying (79) cannot be found. In
this case, the RIS should keep the initial coordinate $\bm{L}^{(0)}$, i.e.,
$\bm{L}^{(n^{\prime})}=\bm{L}^{(0)}$.

Overall, the above analysis can be summarized as the following Penalty
Constrained Convex Approximation (PCCA) Algorithm 1 to optimize the RIS’s
coordinate.

Algorithm 1 Penalty Constrained Convex Approximation Algorithm (PCCA)

1: Initialize coordinate $\bm{L}^{(0)}$, and the objective
$O^{tot}(\bm{L}^{(0)},\bm{W}^{0}_{k,i},\bm{U}^{0}_{k,i})$;

2: Initialize iterative number $n=0$ and maximum number of iterations
$N_{max}$;

3: repeat

4: Obtain $(\bm{L}^{*},r_{u}^{*},d_{0}^{*})$ by solving Problem (74);

5: if $\alpha\geq 0$ and $\alpha_{m}\geq 0$ then

6: Update $\bm{L}^{n+1}=\bm{L}^{*}$;

7: Calculate $\bm{W}^{n+1}_{k,i},\bm{U}^{n+1}_{k,i}$ according to (32) and
(35), respectively.

8: Calculate $O^{tot}(\bm{L}^{n+1},\bm{W}^{n+1}_{k,i},\bm{U}^{n+1}_{k,i})$;

9: if
$O^{tot}(\bm{L}^{n+1},\bm{W}^{n+1}_{k,i},\bm{U}^{n+1}_{k,i})<O^{tot}(\bm{L}^{(0)},\bm{W}^{0}_{k,i},\bm{U}^{0}_{k,i})$
then

10: $\bm{L}^{(n^{\prime})}=\bm{L}^{n+1}$;

11: end if

12: else

13: Update $P_{m}^{U}$ for all $m,\alpha_{m}<0$ according to (76) ;

14: Update $P^{I}$ according to (75) if $\alpha<0$;

15: if Problem (74) is not feasible then

16: Set $n=N_{max}$ and $\bm{L}^{(n^{\prime})}=\bm{L}^{0}$;

17: end if

18: end if

19: until $n=N_{max}$;

19:
$\bm{L}^{(n^{\prime})}$,$\bm{U}^{n^{\prime}}_{k,i},\bm{W}^{n^{\prime}}_{k,i}$;

###  III-D BCD Algorithm to Solve Problem (34)

Based on the above analysis, a BCD based alternating optimization algorithm is
proposed for alternately optimizing the precoding matrices of the AP, the
phase shifts of the RIS and the coordinate of RIS. The detailed algorithm is
presented as in Algorithm 2.

Algorithm 2 The BCD Algorithm to Solve Problem (34)

1: Initialize feasible $\bm{L}^{0}$, $\bm{\varphi}^{0}$, and
$\bm{F}^{0}_{k,i}$.

2: Initialize $\bm{U}^{(0)}_{k,i}$ and $\bm{W}^{(0)}_{k,i}$ according to (32)
and (35), respectively.

3: Initialize maximum number of iterations $S_{max}$ and the iterative number
$s=0$.

4: repeat

5: Calculate $\bm{F}_{k,i}^{(s+1)}$ by solving the convex approximations of
Problem (36);

6: Calculate $\bm{\varphi}^{(s+1)}$ by solving the convex approximations of
Problem (48);

7: Calculate $\\{\bm{L}^{(s+1)},\bm{U}^{(s+1)}_{k,i},\bm{W}^{(s+1)}_{k,i}\\}$
according to the PCCA Algorithm;

8: until $s=S_{max}$.

The proposed BCD algorithm is guaranteed to converge. Specifically, the
generated solutions $\\{\bm{L}^{(s)},r^{(s)}_{u},d^{(s)}_{0}\\}$ by the PCCA
algorithm are always feasible to the coordinate optimization problem (50) by
updating $P_{m}^{U}$ and $P^{I}$. Furthermore, according to the steps 15-17
and steps 9-11 of the PCCA algorithm, the obtained objective value
$\\{O^{tot}(\bm{L}^{s},\bm{W}^{s}_{k,i},\bm{U}^{s}_{k,i})\\}$ is monotonically
decreasing. Then, according to Section III-A and Section III-B, Problem (36)
and Problem (48) are reformulated based on the Taylor-expansion based convex
optimization. It can be readily verified that the sequence of solutions
generated by the BCD Algorithm is always feasible for Problem (34). The
monotonic property of the BCD Algorithm can be similarly proved by using the
method in [53].

##  IV Simulation Results

Simulation results are presented in this section to evaluate the performance
of the proposed algorithm. In the simulation, THz frequency range at the AP is
$300$-$340$ GHz, the bandwidth of each sub-band is $20$ GHz, and the molecular
absorption coefficients are generated according to [44]. The AP is located
along the Y-axis with height of $2$ m. The AP’s transmit antenna is modelled
as a uniform planar array (UPA) with size of $N_{t}=5\times 10$, and
$P_{t}^{max}=10$ W. The IUs and EUs are both equipped with $2$ receive
antennas. The antenna gain is set as $G_{t}=15$ and $G_{r}=6$. The separations
between the transmit/receive antennas are set to be $0.1$ mm.

As shown in Fig. 3, there are $2$ IUs and $2$ EUs which are randomly
distributed in a square area with width $3$ m. The RIS is installed on the
X-axis, and the separations between the RIS reflecting elements are $0.1$ mm.
In Fig. 3, the number of reflecting elements is set to $100$. The initial
coordinate of the RIS in X-axis is marked by “*” and the optimized coordinate
in X-axis is marked by “$\times$”. It is observed that in the layout shown in
Fig. 3, the optimized coordinate of the RIS is updated by the proposed BCD
algorithm.

The proposed BCD algorithm given in Algorithm 2 is labelled as “PropBCD”. For
performance comparison, we consider two benchmark schemes:

  * •

The first scheme which is labelled as “BeamOpt” only optimizes the transmit
precoding matrices with the fixed RIS’s phase shift and coordinate; This
scheme can be obtained by removing step 6 and step 7 of Algorithm 2.

  * •

The scheme labelled as “FixedLoc” optimizes both the transmit precoding
matrices and the RIS’s phase shift, where the RIS’s coordinate is kept fixed.
This scheme is obtained by removing step 7 of Algorithm 2.

Figure 3: The simulation scenario of the STIPT system. Figure 4: The
convergence performance of different algorithms.

Fig. 4 shows the convergence performance of the proposed algorithm and the
benchmarks, where its simulation scenario is shown in Fig. 3. In Fig. 4, the
required harvest power by RIS and the EU are set as $0.1$ mW. It is observed
that the achieved sum rate of IUs increases with the number of iterations for
all considered cases, and all the considered schemes converge within 7
iterations. As expected, the proposed BCD algorithm achieves the best
performance. Significant rate improvement can be obtained by optimizing the
RIS’s coordinate.

Figure 5: The sum rate versus the number of reflecting elements. Figure 6: The
sum rate versus the number of IUs.

Fig. 5 shows the achieved sum rates of IUs by different schemes versus the
number of RIS reflecting elements. The simulation results are averaged by 100
random realizations, where the initial coordinates of the RIS, the IUs, and
the EUs are randomly generated. As shown in Fig. 5, the sum rates of IUs
achieved by the proposed BCD algorithm and the “FixedLoc” algorithm increase
with the number of reflecting elements. However, the proposed BCD algorithm
can achieve a higher sum rate, and the performance gap increases with the
number of reflecting elements. This implies that the proposed BCD algorithm
can fully exploit the potential benefits provided by the RIS, especially in
this STIPT system. In addition, it is observed that the sum rate of the
“BeamOpt” algorithm remains the same as the number of reflecting elements
increases since the RIS related parameters are not optimized in this scheme.

Fig. 6 shows the achieved sum rates of the IUs by different schemas versus the
number of IUs in the system. It is interesting to see that the achieved sum
rates by the “PropBCD” algorithm increases with the number of IUs. Meanwhile
the sum rate obtained by the “BeamOpt” algorithm slightly increases, and that
of the “FixedLoc” algorithm keeps fixed. This result clearly validates the
benefits provided by the RIS in the STIPT system. By utilizing the RIS, the
transmission channel can also be optimized to fully exploit the spatial
diversity so that the sum rate performance can be enhanced. In particular, in
the THz system, where the channel gain is very sensitive to the transmit
distance, optimizing the RIS’s coordinate can help provide considerable
performance gain.

Figure 7: The sum rate versus the harvested power of EUs.

Fig. 7 shows the achieved sum rates of the IUs versus the number of EUs in the
system, and the number of IUs is fixed to be 2. Then, it is observed that the
sum rate of the IUs decreases with the number of EUs for all cases, as more
power needs to be harvested for the EUs. In addition, the performance gap
between the “PropBCD” algorithm and the other benchmarks increases with the
number of EUs, which also shows the superiority of proposed algorithm.

Figure 8: The sum rate versus the harvested power of EUs. Figure 9: The sum
rate versus the harvested power of RISs

Fig. 8 shows the impact of the harvest power required by the EUs on the sum
rate performance. In Fig. 8, the required power of the RIS is fixed to $0.1$
mW, and the number of EUs is $2$. It is observed that the proposed BCD
algorithm outperforms the other two algorithms. As shown in Fig. 8, the
achieved sum rate of IUs by the BCD algorithm decreases with the required
harvested power of EUs for all considered cases. In addition, the performance
gap between the “PropBCD” algorithm and the other schemes keeps stable with
the required power. This implies that the proposed algorithm still has stable
performance advantages when the power harvesting demand from EUs increases.

Finally, Fig. 9 shows the impact of the harvest power required by the RIS on
sum rate performance. In Fig. 9, the required power of the EU is $0.1$ mW, and
the other simulation parameters are the same as those of Fig. 8. As expected,
the proposed BCD algorithm outperforms the other two algorithms, but the
performance gap slightly decreases with the required power to harvest. It
becomes more difficult to find the new coordinate of RIS to improved IU’s rate
while guaranteeing the more stringent harvested power constraints of RIS.
Also, compared with the impact of the EU’s harvested power shown in Fig. 8,
RIS’s required harvest power has a slighter impact on rate performance.

##  V Conclusions

In this paper, we have investigated a new simultaneous THz information and
power transfer system, named as STIPT, where the RIS is utilized to support
the THz transmission. In this system, the RIS can utilize the power harvesting
technology to self-sustain its power consumption. The optimization problem has
been formulated to maximize the IUs’ sum rate while guaranteeing the EU’s and
RIS’s power harvesting requirements. A BCD-based alternating optimization
algorithm has been proposed to optimize the transmit precoding for IUs, the
RIS’s reflecting coefficients and the RIS’s coordinate. Simulation results
have shown that the proposed algorithm can achieve considerable performance
gain in terms of the sum rate. With the assistance of RIS, the transmission
channel can be optimized to fully exploit the spatial diversity so that the
sum rate performance can be enhanced. As the channel gain of the THz
transmission is very sensitive to the transmission distance, optimizing the
RIS’s coordinate in the STIPT system can help provide considerable performance
gain.

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