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# Sum-Rate Maximization in Active RIS-Assisted Multi-Antenna WPCN

Jie Jiang  Bin Lyu  Pengcheng Chen  and Zhen Yang  School of Communications
and Information Engineering, Nanjing University of Posts and
Telecommunications, Nanjing 210003, China

###### Abstract

In this paper, we propose an active reconfigurable intelligent surface (RIS)
enabled hybrid relaying scheme for a multi-antenna wireless powered
communication network (WPCN), where the active RIS is employed to assist both
wireless energy transfer (WET) from the power station (PS) to energy-
constrained users and wireless information transmission (WIT) from users to
the receiving station (RS). For further performance enhancement, we propose to
employ both transmit beamforming at the PS and receive beamforming at the RS.
We formulate a sum-rate maximization problem by jointly optimizing the RIS
phase shifts and amplitude reflection coefficients for both the WET and the
WIT, transmit and receive beamforming vectors, and network resource
allocation. To solve this non-convex problem, we propose an efficient
alternating optimization algorithm with linear minimum mean squared error
criterion, semi-definite relaxation (SDR) and successive convex approximation
techniques. Specifically, the tightness of applying the SDR is proved.
Simulation results demonstrate that our proposed scheme with 10 reflecting
elements (REs) and 4 antennas can achieve 17.78% and 415.48% performance gains
compared to the single-antenna scheme with 10 REs and passive RIS scheme with
100 REs, respectively.

###### keywords:

Wireless powered communication network, active reconfigurable intelligent
surface, beamforming, sum-rate maximization.

##  1 Introduction

With the development of the Internet-of-Things (IoT), an intelligent society
with ubiquitous interconnections and deep coverage will be truly realized.
However, wireless devices (WDs) in IoT networks are generally energy-
constrained and suffer from limited lifetime [1], which fundamentally limits
the performance of communication networks. Traditional ways of changing or
recharging the batteries manually are impossible and unacceptable, especially
when the number of WDs is numerous. Therefore, how to tackle this issue is a
critical problem in the widespread development of IoT. Wireless powered
communication has been proposed as a prospective technology for enhancing the
energy sustainability of WDs, which can be classified into two directions
based on application scenarios [2, 3, 4]. The first one focuses on
investigating simultaneous wireless information and power transfer (SWIPT),
where the base station (BS) simultaneously transfers energy and information
signals to energy receivers and information receivers via the common radio
frequency (RF) signals in the downlink (DL), resulting in a pivotal tradeoff
between the achievable rate and harvested energy [5]. In contrast to the
SWIPT, wireless powered communication network (WPCN) has been proposed as a
novel type of wireless network diagram to improve the lifetime of WDs and
enhance the deployment flexibility of IoT. In a WPCN, energy-constrained WDs
first harvest energy in the DL and then use the harvested energy to transmit
independent information in the uplink (UL) based on the widely used harvest-
then-transmit (HTT) protocol [6].

WPCN has been widely investigated in the literature [6, 7, 8, 9], which
promotes the development of WPCN. However, WPCN generally suffers from the
“doubly near-far” phenomenon if the power station (PS) and the receiving
station (RS) are co-located at the hybrid access point (HAP) [6, 8].
Specifically, a WD located far away from the HAP harvesting less energy in the
DL has to transmit information with more power in the UL, which results in an
unfair time and resource allocation among the WDs. To deal with the issue, a
promising way is to deploy the PS and RS separately [10, 11]. In [10],
multiple users harvest energy from a dedicated PS and then communicate with an
information RS following the HTT protocol. In this scenario, a user physically
close to the PS is naturally far away from the RS, and vice versa. Considering
a similar scenario, a user-centric energy-efficient (EE) problem in WPCN is
investigated in [11]. However, the performance of WPCN is still limited due to
the low efficiencies caused by the severe path-loss, which seriously affects
its practical applications.

Recently, reconfigurable intelligent surface (RIS), with the unprecedented
ability to reshape the wireless transmission environment, has drawn widespread
attentions from academia and industry [12, 13, 14, 15, 16, 17, 18]. RIS is
comprised of a large number of programmable reflecting elements (REs), which
can alter the phase shifts and amplitudes of incident signals. As such, RIS
can adaptively modify the impinging radio waves towards the appropriate
direction [13]. According to the reflection patterns, RIS can be classified as
passive RIS and active RIS. Without the property of power amplification, the
independent diffusive scatterer-based (IDS) model accounts for the basic
properties of passive RIS, which has been widely adopted in RIS-assisted
wireless communications [18]. The passive RIS is only equipped with the phase-
shift controller, while the active RIS contains both the phase-shift
controller and the active reflection-type amplifier. Hence, the active RIS can
alter both the phase shifts and amplitudes of the incident signals. It is
worth noting that the active RIS with its novel hardware structure and signal
model has been proposed in [19, 20]. Different from the full-duplex amplify-
and-forward (FD-AF) relay that requires power-consuming RF chains, the active
RIS can directly reflect and amplify the incident signals in the EM level in a
FD manner without reception. In this way, the active RIS exhibits promising
qualities, such as a low power consumption, light weight, conformal geometry
and high flexibility for practical deployment.

Currently, the passive RIS has been widely applied in WPCNs for performance
enhancement [21, 22, 23, 24]. In [21], the passive RIS is employed between the
HAP and users to improve both DL WET and UL WIT efficiencies in single-input-
single-output (SISO) WPCN. To achieve further performance improvement, the
multi-antenna technique is employed in [22], where the HAP with multi-antenna
transmits energy signals to users in the DL and receives information from
users in the UL by employing transmit bemforming and receive beamforming,
respectively. In [23], the fully dynamic RIS beamforming scheme is proposed
for WPCN, for which the phase shift vectors are independently designed over
different time slots. However, in the above works [21, 22, 23], the PS and RS
are also co-located at the HAP, which results in performance unfair among
users. To address this issue, the authors in [24] consider the scenario where
the PS and RS are separately deployed, for which the locations of RIS and
users can be carefully considered to achieve a fair performance among users.
However, as mentioned above, the passive RIS can only reflect incident signals
without amplification, which leads to the limited performance enhancement due
to the double-fading effect suffered by the reflecting links. Thus, the
energy-constrained users still need to consume much time for harvesting energy
in the DL and have less time for information transmission.

Inspired by the amplification characteristic of the active RIS, the active RIS
is confirmed to be superior to the passive RIS in terms of performance
enhancement, and thus has been considered a promising technique for IoT
networks [19, 20, 25, 26, 27, 28, 29]. The authors in [19] compare the
capacity improvement achieved by the active RIS to the passive RIS, which
demonstrats that the active RIS can fundamentally mitigate the double-fading
effect. In [20], an active RIS is applied in single input multiple output
(SIMO) systems, for which the joint optimization of phase shifts matrix and
receive beamforming is considered to obtain the maximum achievable rate. In
[25], the placement of the active RIS is optimized to enhance SISO systems’
performance. The authors in [26] propose to use the active RIS to achieve
secure transmission, which not only establishes the reliable link from the
transmitter to the receiver but also prevents the confidential information
intercepted by the eavesdropper. The active RIS-aided multiuser MISO PS-SWIPT
is studied in [27] to minimize the base station transmit power, which shows
significant improvements compared to the passive RIS-aided system. Similarly,
in [28], an active RIS is employed to assist SWIPT to boost the efficiency of
both WET and WIT, while the conclusions and approaches are inapplicable to the
WPCN system because of thire different system models.

Although the active RIS has received a lot of interests for wireless
communication networks, the applications of active RIS in WPCN is still at the
very early stage and has not been well studied in the literature. To the best
of our knowledge, there exists only one paper investigating the usage of
active RIS in WPCN [29]. Specifically, the authors in [29] investigate the
weight sum-rate maximization problem in the active RIS assisted single-antenna
WPCN, where the PS and RS is co-located at the HAP. As a result, similar to
[21, 22, 23], a part of WDs in [29] still suffer from the “doubly near-far”
phenomenon, which is not suitable for practical applications with high
requirement of performance fairness. Moreover, the authors consider a certain
simplified communication scenario where both the HAP and the WDs have single
antenna each. The transmit beamforming and receive beamforming cannot be
exploited, which is also a key technology for performance enhancement.

Motivated by the observations above, we propose an active RIS enabled hybrid
relaying scheme for the multi-antenna WPCN, where the active RIS is employed
to facilitate both the WET from the PS to energy-constrained users and the WIT
from users to the RS, which is shown in Figure 1. Compared with the existing
works which used the passive RIS in WPCN [21, 22, 23, 24], our proposed active
RIS scheme can amplify the energy signals and information signals to achieve a
satisfying system performance. Different from the single-antenna scenario
considered in [29], we propose to employ multi-antenna at both the PS and RS,
which can construct the transmit beamforming and receive beamforming for
further performance enhancement. Specifically, the transmit beamforming at the
PS can be used to enhance the WPT efficiency from the PS to users. In the
meanwhile, the receive beamforming at the RS can be used to exploit the
antenna gain and eliminate the noise caused by the active RIS. In the
considered system setup, we aim to maximize the sum-rate problem by jointly
optimizing the transmit beamforming at the PS, the receive beamforming at the
RS, the reflecting coefficients at the RIS, and network resource allocation.
It should be noted that compared to [21, 22, 23, 24, 29], our formulated
problem is much more challenging to solve and the proposed algorithms in [21,
22, 23, 24, 29] cannot be used to solve our formulated problem. Thus, we
propose an efficient algorithm to solve the formulated problem. The main
contributions of this paper are summarized as follows:

  * •

We propose an active RIS assisted multi-antenna WPCN for performance
enhancement, where the active RIS is served as a hybrid relay to achieve two
purposes, i.e., the first one is to assist the WET from the PS to users, and
the second one is to aid the WIT from users to the RS. To further improve
system performance, both transmit beamforming and receive beamforming
techniques are respectively considered at the PS and RS.

  * •

We investigate the sum-rate maximization problem by jointly optimizing
transmit beamforming vector at the PS, receive beamforming vectors at the RS,
phase shifts and amplitude reflection coefficients at the RIS for both the WET
and the WIT, and network resource allocation. To deal with the non-convexity
of the formulated problem, we propose an efficient alternating optimization
(AO) algorithm. Specifically, the original problem can be divided into four
sub-problems, which are solved sequentially in an alternating manner until
convergence is achieved.

  * •

For designing the receive beamforming, we apply the linear minimum mean
squared error (MMSE) criterion and obtain the closed-form expression. For the
optimization of the transmit beamforming and RIS reflecting coefficient
matrices for the WET, the semidefinite relaxation (SDR) technique is adopted
and the tightness of applying the SDR is proved. For the optimization of RIS
reflecting coefficients for the WIT, we obtain the optimal phase shifts in a
closed-form and propose a successive convex approximation (SCA) algorithm to
determine the optimal amplitude reflection coefficients. In addition, the
convergence of proposed problem is analyzed and confirmed via numerical
simulations.

  * •

Finally, numerical results are provided to evaluate the performance of
proposed scheme, which indicates that compared to the single-antenna scheme
with 10 REs and the passive-RIS scheme with 100 REs, the proposed scheme with
4 antennas and 10 REs can achieve 17.78% and 415.48% sum-rate gain,
respectively.

The rest of this paper is organized as follows. Section II describes the
system model of the active RIS-assisted multi-antenna WPCN. The sum-rate
maximization problem is formulated in Section III and solved in Section IV,
respectively. In Section V, performance is evaluated by numerical results.
Finally, this paper is concluded in Section VI.

Notations: In this paper, vectors and matrices are denoted by boldface
lowercase and uppercase letters, respectively. The operators
$(\,\cdot\,)^{T}$, $(\,\cdot\,)^{H}$, $\left|\,\cdot\,\right|$ and
$\left\|\,\cdot\,\right\|$ denote the transpose, conjugate transpose, absolute
value and the Euclidean norm, respectively. $\text{Tr}(\,\cdot\,)$ and
$\text{rank}(\,\cdot\,)$ denote the trace and rank of a matrix, respectively.
$\bm{X}\succeq 0$ represents that $\bm{X}$ is a positive semidefinite matrix.
$\mathbb{E}[\,\cdot\,]$ stands for the statistical expectation. $\bm{I}_{N}$
denotes the $N$-dimensional identity matrix. $\bm{0}$ denotes the zero
matrix/vector with appropriate size. $\mathbb{C}^{N\times M}$ denotes the set
of all $N\times M$ complex-valued matrices. $\mathbb{H}^{M}$ denotes the set
of all $M\times M$ Hermitian matrices. $\mathbb{R}^{N\times 1}$ represents the
set of all $N\times 1$ real-valued vectors.
${\mathcal{C}\mathcal{N}}\left(\mu,\sigma^{2}\right)$ denotes the distribution
of a circularly symmetric complex Gaussian random vector with mean $\mu$ and
variance $\sigma^{2}$. $\text{arg}(\,\cdot\,)$ denotes the phase extraction
operation. $\text{diag}(\bm{x})$ denotes a diagonal matrix whose diagonal
elements are from vector $\bm{x}$. $\mathrm{Im}(\,\cdot\,)$ and
$\mathrm{Re}(\,\cdot\,)$ respectively denotes the imaginary part and real part
of a complex number.

##  2 System Model

Figure 1:  System model for an active RIS-assisted multi-antenna WPCN.

As shown in Figure 1, we consider an active RIS-aided multi-antenna WPCN,
which consists of a PS with $M$ antennas, a RS with $L$ antennas, an active
RIS, and $K$ energy-constrained users each with single antenna. We assume that
each user is equipped with an energy harvesting (EH) circuit for harvesting
energy, where a rectifier is used to convert the received RF signals to direct
current (DC) signals. Then, the net energy harvested from the DC signals can
be stored in the rechargeable battery. The active RIS consists of $N$ active
REs, which can steer the reflected signals in a specific direction and also
amplify them by the active loads (negative resistance) [20]. In contrast to
the passive RE, each active RE is equipped with an additionally integrated
active reflection-type amplifier supported by a power supply. By appropriately
setting the effective resistance, it is reasonable to assume that the
reflection amplitude and phase of each element is independently [19, 20, 25,
26, 27, 28, 29]. To power the operations of active RIS and users, the PS is
equipped with a stable energy source. In addition, the PS has the capability
for performing computational tasks [21]. In particular, the users first
harvest energy from the RF signals transmitted by the PS and then use the
harvested energy to deliver information to the RS. To allievate the severe
path-loss suffered by the reflecting links, the active RIS is employed to
improve the WET efficiency from the PS to users and the WIT efficiency from
the users to the RS.

The channels are assumed to follow a quasi-static flat-fading model. That is,
all channel coefficients are constant throughout each transmission block but
vary from block to block [30]. The downlink baseband equivalent channels of
PS-to-RIS, RIS-to-$U_{k}$, and PS-to-$U_{k}$ links are denoted by
$\bm{H}_{r}\in\mathbb{C}^{N\times M}$, $\bm{h}_{u,k}^{H}\in\mathbb{C}^{1\times
N}$, and $\bm{h}_{d,k}^{H}\in\mathbb{C}^{1\times M}$, respectively, where
$U_{k}$ denotes the $k$-th user. Similarly, the uplink baseband equivalent
channels of $U_{k}$-to-RIS, RIS-to-RS, and $U_{k}$-to-RS links are
respectively denoted by $\bm{g}_{u,k}\in\mathbb{C}^{N\times 1}$,
$\bm{G}_{r}\in\mathbb{C}^{L\times N}$, and $\bm{g}_{d,k}\in\mathbb{C}^{L\times
1}$, respectively. Since there have been many efficient channel estimation
techniques proposed for RIS systems [31, 32, 33, 34], we assume the perfect
channel state information can be available in advance, which is a common
assumption considered in [21, 22, 35] and a prerequisite for investigating the
upper-bound of system performance. It should be noted that the channel
estimation error is generally inevitable [31]. However, the effect of channel
estimation error on system performance degradation is out the scope of this
paper.

According to the HTT protocol [6], the normalized transmission block of
interest is divided into $K+1$ time slots. The first time slot with duration
of $\tau_{0}\in[0,1]$ is a dedicated slot for WET, in which all users harvest
energy from the PS with the assistance of active RIS. The remaining $K$ time
slots denoted by $\bm{\tau}=[\tau_{1},\ldots,\tau_{K}]$, are used for UL WIT
via the time division multiple access (TDMA) scheme. Specifically, during
$\tau_{k}$, $k=1,\cdots,K$, $U_{k}$ delivers its information to the RS.
Without loss of generality, the whole operation time period is set to a
normalized transmission block. The network time scheduling constraint is thus
given by

$\tau_{0}+\sum_{k=1}^{K}\tau_{k}\leq 1,~{}\forall k.$ (1)

###  2.1 Wireless Energy Transfer Phase

In the WET phase, the PS transmits energy signals to all users with the
assistance of the active RIS. Denote the transmitted signal as
$\bm{s}=\bm{w}_{0}s$, where $s$ is the pseudo-random baseband signal
transmitted by the PS, and $\bm{w}_{0}\in\mathbb{C}^{M\times 1}$ is the
transmit beamforming vector. The energy constraint at the PS is expressed as

$\mathbb{E}\left[|\bm{s}|^{2}\right]=\text{Tr}\left(\bm{w}_{0}\bm{w}_{0}^{H}\right)\leq
P_{0},$ (2)

where $P_{0}$ denotes the maximum transmit power at the PS.

The reflecting coefficient matrix of the active RIS in the WET phase is
denoted by
$\bm{\Phi}_{0}=\text{diag}\left\\{\phi_{0,1},\ldots,\phi_{0,N}\right\\}\in\mathbb{C}^{N\times
N}$ with $\phi_{0,n}=a_{0,n}e^{j\theta_{0,n}}$, $n=\,1,\ldots,\,N$, where
$a_{0,n}$ and $\theta_{0,n}$ represent the amplitude reflection coefficient
and phase shift of the $n$-th RE, respectively. Without loss of generality, we
suppose each active RE has the following constraints

$a_{0,n}\leq{a_{n,max}},\,\,0\leq{\theta_{0,n}}\leq 2\pi,~{}\forall n,$ (3)

where $a_{n,max}$ is the maximum amplitude reflection coefficient of $n$-th
RE. It is worth noting that $a_{n,max}$ can be greater than 1 [25], which is a
main characteristic distinguishing the active RIS from the passive RIS since
the active load can amplify the reflected signals.

The received signal at $U_{k}$ during $\tau_{0}$ is given by

${y}_{u,k}=\underbrace{\bm{h}_{d,k}^{H}\bm{s}}_{\text{direct
link}}+\underbrace{\bm{h}_{u,k}^{H}\bm{\Phi}_{0}\left(\bm{H}_{r}\bm{s}+\bm{n}_{v}\right)}_{\text{RIS-
aided link}}+{n}_{u,k},~{}\forall k,$ (4)

where ${n}_{u,k}\in\mathbb{C}$ and $\bm{n}_{v}\in\mathbb{C}^{N\times 1}$
represent the additive white Gaussian noise (AWGN) at $U_{k}$ and the RIS,
respectively. Without loss of generality, we assume
${n}_{u,k}\sim\mathcal{CN}\left(0,\sigma_{u,k}^{2}\right)$ and
$\bm{n}_{v}\sim\mathcal{CN}\left(0,\sigma_{v}^{2}\bm{I}_{N}\right)$. Denote
the equivalent downlink channel as
$\bm{h}_{k}^{H}=\bm{h}_{u,k}^{H}\bm{\Phi}_{0}\bm{H}_{r}+\bm{h}_{d,k}^{H}\in\mathbb{C}^{1\times
M}$ and (4) can be rewritten as

${y}_{u,k}=\bm{h}_{k}^{H}\bm{w}_{0}s+\bm{h}_{u,k}^{H}\bm{\Phi}_{0}\bm{n}_{v}+{n}_{u,k},~{}\forall
k.$ (5)

Due to the fact that the active RIS not only amplifies the desired signal,
i.e., $\bm{s}$, but also amplifies the input noise, i.e., $\bm{n}_{v}$, it is
reasonable to consider the second term of (4) for computing the amount of
harvested energy accurately [20]. However, the noise at $U_{k}$ is generally
quite small and can be negligible. Accordingly, the harvested energy by
$U_{k}$, denoted by $E_{k}$, is given by

$E_{k}=\beta{\left|{\bm{h}_{k}^{H}}\bm{w}_{0}\right|^{2}}\tau_{0}+\beta{{\left\|{\bm{h}_{u,k}^{H}}\bm{\Phi}_{0}\right\|^{2}}{\sigma_{v}^{2}}}\tau_{0},~{}\forall
k,$ (6)

where $\beta\in\left(0,1\right]$ denotes the energy conversion efficiency of
each user. It is practical for us to consider the linear EH model here, which
is also a common assumption in the literature [6, 8, 11, 10, 22, 29].

It is worth noting that the active RIS can allocate the available reflecting
power to amplify the incident signals with active loads [20]. In the DL WET
phase, the amplification power of $\bm{s}$ and $\bm{n}_{v}$ is limited by the
RIS power budget, which is shown by the following constraint

${P_{0}}\left\|\bm{\Phi}_{0}\bm{H}_{r}\right\|^{2}+{\sigma_{v}^{2}}\left\|\bm{\Phi}_{0}\bm{I}_{N}\right\|^{2}\leq
P_{r},$ (7)

where $P_{r}$ is the maximum reflecting power for amplification at the active
RIS and substantially lower than that of an active RF amplifier [25].

###  2.2 Wireless Information Transmission Phase

In the WIT phase, the users utilize the harvested energy to transmit
information to the RS via a TDMA manner. Let $f_{k}$ denotes the information-
carrying signal of $U_{k}$ with unit power and then the transmit signal during
$\tau_{k}$ is denoted by $x_{k}=\sqrt{p_{k}}f_{k}$, where $p_{k}$ is the
transmit power at $U_{k}$. We assume that all the harvested energy at $U_{k}$
in the WET phase is used for delivering its own information. Let
$\bm{p}=[p_{1},\ldots,p_{K}]\in\mathbb{R}^{1\times K}$, which satisfies

$p_{k}\tau_{k}\leq E_{k},~{}\forall k.$ (8)

Similarly, the reflecting coefficient matrix at the active RIS for the WIT
during $\tau_{k}$ is denoted by
$\bm{\Phi}_{k}=\text{diag}\left\\{\phi_{k,1},\ldots,\phi_{k,N}\right\\}\in\mathbb{C}^{N\times
N}$, where $\phi_{k,n}=a_{k,n}e^{j\theta_{k,n}}$, $a_{k,n}$ and $\theta_{k,n}$
have the following constraints

$a_{k,n}\leq{a_{n,max}},\,\,0\leq{\theta_{k,n}}\leq 2\pi,~{}\forall
k,~{}\forall n.$ (9)

The received signal at the RS from $U_{k}$ with the assistance of active RIS
during $\tau_{k}$ is written as

$\bm{y}_{r,k}=\underbrace{\bm{g}_{d,k}^{H}x_{k}}_{\text{direct
link}}+\underbrace{\bm{G}_{r}\bm{\Phi}_{k}\left(\bm{g}_{u,k}x_{k}+\bm{n}_{v}\right)}_{\text{RIS-
aided link}}+\bm{n}_{r},~{}\forall k,$ (10)

where $\bm{n}_{r}\in\mathbb{C}^{L\times 1}$ represents the AWGN at the RS and
satisfies $\bm{n}_{r}\sim\mathcal{CN}\left(0,\sigma_{r}^{2}\bm{I}_{N}\right)$.

In the UL WIT phase, we also have the amplification power constraint at the
active RIS as follows

${p_{k}}\left\|\bm{\Phi}_{k}{\bm{g}_{u,k}}\right\|^{2}+{\sigma_{v}^{2}}\left\|\bm{\Phi}_{k}\bm{I}_{N}\right\|^{2}\leq
P_{r},~{}\forall k.$ (11)

Denote the receive beamforming vector at the RS during $\tau_{k}$ as
$\bm{w}_{k}\in\mathbb{C}^{L\times 1}$, which can be used to extract the
desired signal and suppress interference and noises. Let the equivalent uplink
channel be
$\bm{g}_{k}=\bm{g}_{d,k}+\bm{G}_{r}\bm{\Phi}_{k}\bm{g}_{u,k},~{}\forall k$.
The estimated signal at the RS during $\tau_{k}$ is expressed as

$\displaystyle\bm{u}_{k}=$ $\displaystyle\bm{w}_{k}^{H}\bm{y}_{r,k}$ (12)
$\displaystyle=$
$\displaystyle\bm{w}_{k}^{H}\bm{g}_{k}x_{k}+\bm{w}_{k}^{H}\bm{G}_{r}\bm{\Phi}_{k}\bm{n}_{v}+\bm{w}_{k}^{H}\bm{n}_{r},~{}\forall
k.$

Then, the signal-noise-ratio (SNR) at the RS during $\tau_{k}$ is written as

$\gamma_{k}=\frac{p_{k}\left|{\bm{w}_{k}^{H}}\left(\bm{G}_{r}\bm{\Phi}_{k}{\bm{g}_{u,k}}+{\bm{g}_{d,k}}\right)\right|^{2}}{\left\|\bm{w}_{k}^{H}\bm{G}_{r}\bm{\Phi}_{k}\right\|^{2}{\sigma_{v}^{2}}+{\left\|{\bm{w}_{k}}\right\|^{2}\sigma_{r}^{2}}},~{}\forall
k.$ (13)

Denote the achievable rate from $U_{k}$ to the RS as $R_{k}$, which is
formulated as

$R_{k}=\tau_{k}\log\left(1+\gamma_{k}\right),~{}\forall k.$ (14)

##  3 PROBLEM FORMULATION

In this section, we formulate the system sum-rate maximization problem by
jointly optimizing the reflecting coefficients for both the WET and the WIT at
the active RIS, the transmit beamforming at the PS, the receive beamforming at
the RS, the transmit power at each user, and the network time scheduling. The
optimization problem is formulated as

$\displaystyle(\mbox{{P1}})$
$\displaystyle\max_{\begin{subarray}{c}\tau_{0},\bm{\tau},\bm{\Phi}_{0},\bm{\Phi}_{k},\bm{w}_{0},\bm{p},{\bm{w}_{k}}\end{subarray}}\sum_{k=1}^{K}R_{k}$
(15) $\displaystyle s.t.\,\,\,\,\,\,$ (1), (2), (3), (7), (8), (9) and (11),
$\displaystyle{\tau_{0}}\geq 0,\,{\tau_{k}}\geq 0,{p_{k}}\geq 0,\,~{}\forall
k.$ (16)

where (16) indicates that the time and power variables are all nonnegative.

One can observe that the objective function in (15) is a non-concave function
due to the coupled of variables. In addition, there exists several non-convex
constraints, i.e., (2), (7), (8) and (11). It is thus challenging to solve P1
directly by standard optimization techniques. In the next section, we propose
an AO algorithm to solve it efficiently.

##  4 ALTERNATING OPTIMIZATION SOLUTION

In this section, an efficient AO algorithm is proposed to solve P1. In
particular, we decompose P1 into several subproblems and iteratively solve
them in an alternating manner. To show the procedure of AO algorithm, we
summarize a flow chart in Figure 2. Specifically, the variables are
partitioned into four blocks, $\\{\bm{w}_{k}\\}$,
$\\{\bm{w}_{0},\bm{\tau},\bm{p}\\}$,
$\\{\bm{\Phi}_{0},\tau_{0},\bm{\tau},\bm{p}\\}$, and $\\{\bm{\Phi}_{k}\\}$.
Then, the variables in each block are alternately solved by its corresponding
sub-problem with the other blocks fixed until the convergence is achieved.

Figure 2: A flow chart of the proposed algorithm.

###  4.1 Linear MMSE-based Receive Beamforming Optimization

With the other variables fixed, we first design the receive beamforming
vectors $\\{\bm{w}_{k}\\}$. To cope with the interference caused by
$\bm{n}_{v}$ and $\bm{n}_{r}$ in (13), we apply the linear MMSE criterion
here. Based on this criterion, the MMSE-based receive beamforming is given by

$\bm{w}_{k}^{*}=\left(\bm{g}_{k}\bm{g}_{k}^{H}+\frac{\sigma_{v}^{2}}{p_{k}}\bm{G}_{r}\bm{\Phi}_{k}\bm{\Phi}_{k}^{H}\bm{G}_{r}^{H}+\frac{\sigma_{r}^{2}}{p_{k}}I_{L}\right)^{-1}\bm{g}_{k},~{}\forall
k.$ (17)

###  4.2 SDP-based Transmit Beamforming Optimization

We then proceed to optimize $\\{\bm{w}_{0},\bm{\tau},\bm{p}\\}$ with the other
variables $\\{\bm{w}_{k},\bm{\Phi}_{0},\tau_{0},\bm{\Phi}_{k}\\}$ fixed.
Letting $e_{k}={p_{k}}{\tau_{k}}$ and $\bm{e}=[e_{1},\ldots,e_{K}]$ and
applying the obtained results in (17), P1 can be simplified as follows

$\displaystyle(\mbox{{P2}})\max_{\begin{subarray}{c}{\bm{w}_{0}},{\bm{\tau}},{\bm{e}}\end{subarray}}$
$\displaystyle\sum_{k=1}^{K}\tau_{k}\log\left(1+{\epsilon_{k}}\frac{e_{k}}{\tau_{k}}\right)$
(18) $\displaystyle s.t.\,\,\,\,\,\,$ $\displaystyle{e_{k}}\leq
E_{k},~{}\forall k,$ (19) $\displaystyle{\tau_{k}}\geq 0,\,{e_{k}}\geq
0,~{}\forall k,$ (20) (1), (2) and (11),

where
$\epsilon_{k}=\frac{\left|{(\bm{w}_{k}^{*})^{H}}\left({\bm{G}_{r}}{\bm{\Phi}_{k}}{\bm{g}_{u,k}}+{\bm{g}_{d,k}}\right)\right|^{2}}{\left\|(\bm{w}_{k}^{*})^{H}{\bm{G}_{r}}\bm{\Phi}_{k}\right\|^{2}{\sigma_{v}^{2}}+{\left\|{\bm{w}_{k}^{*}}\right\|^{2}\sigma_{r}^{2}}}$,
$\forall k$. It can be found that P2 is highly non-convex. To solve it
efficiently, the SDR technique [36] is employed. Define
$\bm{H}_{k}=\bm{h}_{k}\bm{h}_{k}^{H}$, and
$\bm{G}_{u,k}=\text{diag}\left(\left|\bm{g}_{u,k,1}\right|^{2},\left|\bm{g}_{u,k,2}\right|^{2},\ldots,\left|\bm{g}_{u,k,N}\right|^{2}\right)$,
$\forall k$. Let $\bm{W}_{0}=\bm{w}_{0}\bm{w}_{0}^{H}$, which satisfies
${\bm{W}_{0}}\succeq 0$ and $\text{rank}(\bm{W}_{0})=1$. Then, (19) is
rewritten as

${e_{k}}\leq\beta{\tau_{0}}[{\rm
Tr}\left(\bm{H}_{k}\bm{W}_{0}\right)+\left\|\bm{h}_{u,k}^{H}\bm{\Phi}_{0}\right\|^{2}{\sigma_{v}^{2}}],\forall
k.$ (21)

Denote the RIS reflecting coefficient vector for the WET as
$\bm{\varphi}_{k}=\left[{\phi_{k,1}},{\phi_{k,2}},\ldots,{\phi_{k,N}}\right]^{T}$,
$\forall k$, (11) can be recast as

${e_{k}}{\bm{\varphi}_{k}^{H}}{G_{u,k}{\bm{\varphi}_{k}}}+{\tau_{k}}{\sigma_{v}^{2}}{\bm{\varphi}_{k}^{H}}{\bm{\varphi}_{k}}\leq{\tau_{k}}P_{r},~{}\forall
k.$ (22)

Then, P2 can be equivalently transformed into

$\displaystyle(\mbox{{P2-1}})\max_{\bm{W}_{0},\bm{\tau},\bm{e}}\,\,\,\,\,\,$
$\displaystyle\sum_{k=1}^{K}\tau_{k}\log\left(1+{\epsilon_{k}}\frac{e_{k}}{\tau_{k}}\right)$
(23) $\displaystyle s.t.\,\,\,\,\,\,$ $\displaystyle{\rm
Tr}\left(\bm{W}_{0}\right)\leq P_{0},$ (24) $\displaystyle{\bm{W}_{0}}\succeq
0,$ (25) $\displaystyle\text{rank}(\bm{W}_{0})=1,$ (26)
$\displaystyle\mbox{\eqref{time}, \eqref{positive_2}, \eqref{E_k3} and
\eqref{Pr_WIT2}}.$

Since the rank-one constraint in (26) is non-convex, we employ the SDR
technique to relax it. Thus, P2-1 becomes to be a convex semidefinite program
(SDP) and can be solved with the interior-point method [37].

######  Proposition 4.1.

The optimal transmit beamforming matrix obtained by solving the relaxed
version of P2-1, denoted by $\bm{W}_{0}^{*}$, is rank-one.

######  Proof 4.2.

Please refer to Appendix A.

According to Proposition 4.1, the tightness of SDR is guaranteed. Hence, we
can employ Cholesky decomposition to obtain the optimal energy beamforming
vector $\bm{w}_{0}^{*}$.

###  4.3 SDP-based RIS Reflecting Coefficients for the WET and Resource
Allocation Optimization

In this sub-section, we focus on optimizing the reflecting beamforming at the
RIS in the WET phase, the transmit power at each user, and the network time
scheduling. Since $\tau_{0}$ and $\bm{\Phi}_{0}$ are coupled, we first
optimize $\\{\bm{\Phi}_{0},\bm{\tau},\bm{p}\\}$ with $\tau_{0}$ given. Define
${\bm{\Psi}_{0}=\bm{\tilde{\varphi}_{0}}\bm{\tilde{\varphi}_{0}}^{H}}$ with
${\bm{\Psi}_{0}}\succeq 0$ and $\text{rank}(\bm{\Psi}_{0})=1$, where
${\bm{\tilde{\varphi}_{0}}=[{\bm{\varphi_{0}}^{H}},1]^{H}}$ and
$\bm{\varphi}_{0}=\left[{\phi_{0,1}},\ldots,{\phi_{0,N}}\right]^{T}$. Let
$\bm{H}_{u,k}=\text{diag}\left\\{\bm{h}_{u,k,1},\ldots,\bm{h}_{u,k,N}\right\\}$
and
$\bm{Q}_{u,k}=\text{diag}\left\\{\left|\bm{h}_{u,k,1}\right|^{2},\ldots,\left|\bm{h}_{u,k,N}\right|^{2},1\right\\}$.
Then, (7) and (19) are respectively reformulated as

${P_{0}}{\rm Tr}({\tilde{\bm{H}}_{r}}\bm{\Psi}_{0})+{\sigma_{v}^{2}}{\rm
Tr}(\bm{\Psi}_{0})\leq P_{r},$ (27) ${e_{k}}\leq\beta{\tau_{0}}{\rm
Tr}[(\bm{V}+{\sigma_{v}^{2}}\bm{Q}_{u,k})\bm{\Psi}_{0}]-\beta{\tau_{0}}{\sigma_{v}^{2}},~{}\forall
k,$ (28)

where

$\bm{V}=\left[\begin{matrix}\bm{H}_{u,k}^{H}\bm{H}_{r}\bm{W}_{0}\bm{H}_{r}^{H}\bm{H}_{u,k}&\bm{H}_{u,k}^{H}\bm{H}_{r}\bm{W}_{0}\bm{h}_{d,k}\\\
\bm{h}_{d,k}^{H}\bm{W}_{0}\bm{H}_{r}^{H}\bm{H}_{u,k}&\bm{h}_{d,k}^{H}\bm{W}_{0}\bm{h}_{d,k}\end{matrix}\right],$

and

$\tilde{\bm{H}}_{r}={\left[\begin{matrix}\bm{H}_{r}\bm{H}_{r}^{H}&\bm{0}\\\
\bm{0}&\bm{0}\end{matrix}\right]}.$

With the obtained solutions in Sections 4.1 and 4.2, P1 can be equivalently
written as

$\displaystyle(\mbox{{P2-2}})\max_{\begin{subarray}{c}{\bm{\Psi}_{0}},\bm{\tau},\bm{e}\end{subarray}}$
$\displaystyle\sum_{k=1}^{K}\tau_{k}\log\left(1+{\epsilon_{k}}\frac{e_{k}}{\tau_{k}}\right)$
(29) $\displaystyle s.t.\,\,\,\,\,\,$ $\displaystyle{\bm{\Psi}_{0}}\succeq
0,\,\text{rank}(\bm{\Psi}_{0})=1,$ (30)
$\displaystyle{[\bm{\Psi}_{0}]_{n,n}}\leq{a_{max}^{2}},~{}\forall n,$ (31)
$\displaystyle{[\bm{\Psi}_{0}]_{N+1,N+1}}=1,\,\,\,\,\,$ (32) (1), (20), (22),
(27) and (28).

Similarly, after the relaxation of the rank-one constraint in (30), P2-2 is
also an SDP and can be solved by the interior-point method. Recall that the
tightness of optimizing $\bm{W}_{0}$ by SDR can be guaranteed, we can also
prove that the obtained solution $\bm{\Psi}_{0}$ is rank-one. Then,
$\bm{\tilde{\varphi}}_{0}$ can be recovered by implementing Cholesky
decomposition of $\bm{\Psi}_{0}$, and the optimal reflection coefficient
vector for the WET $\bm{\varphi_{0}}^{*}$ can be obtanied by linear operation
from $\bm{\tilde{\varphi}}_{0}$. Subsequently, the optimal RIS reflecting
coefficient matrix $\bm{\Phi}_{0}^{*}$ can be obtained by
$\bm{\Phi}_{0}^{*}\,=\,\text{diag}((\bm{{\varphi}}_{0}^{H})^{*})$.

Finally, we continue to update the optimal energy transmission time
$\tau_{0}\in[0,1]$ by the one-dimensional search method. Thus, the maximum
sum-rate of this sub-problem is achieved with the optimal solution
$\\{\bm{\Phi}_{0}^{*},\tau_{0}^{*},\bm{\tau}^{*},\bm{p}^{*}\\}$. The procedure
is summarized in Algorithm 1.

Algorithm 1 SDP-based RIS reflecting coefficients for the WET and resource
allocation optimization

0: $\bm{w}_{0},\\{\bm{w}_{k}\\},\\{\bm{\Phi}_{k}\\},~{}\forall k$

0: $\bm{\Phi}_{0}^{*},\tau_{0}^{*},\bm{\tau}^{*},\bm{p}^{*}$

1: Initialization: The maximum objective function value $R_{\max}=0$ and the
step size $\delta$.

2: for $\tau_{0}=0:\delta:1$ do

3: Given $\bm{w}_{0},\\{\bm{w}_{k}\\},\\{\bm{\Phi}_{k}\\}$, we obtain
${\tau_{0}^{\prime}}$, ${\bm{\Psi}_{0}^{\prime}}$, ${\tau_{k}^{\prime}}$,
${e_{k}^{\prime}}$, $~{}\forall k$ by solving P2-2.

4: Calculate
$R\,=\,\sum_{k=1}^{K}\tau_{k}^{\prime}\log\left(1+{\epsilon_{k}}\frac{e_{k}^{\prime}}{\tau_{k}^{\prime}}\right)$.

5: if $R>R_{\max}$ then

6: Update $R_{max}\leftarrow R$.

7: Update
$\tau_{0}\leftarrow\tau_{0}^{\prime},\bm{\Psi}_{0}\leftarrow\bm{\Psi}_{0}^{\prime},\tau_{k}\leftarrow\tau_{k}^{\prime},e_{k}\leftarrow
e_{k}^{\prime}$.

8: end if

9: end for

10: Obtain $\bm{\tilde{\varphi}_{0}}$ from $\bm{\Psi}_{0}$ by Cholesky
decomposition.

11: Obtain $\bm{{\varphi}}_{0}$ from $\bm{\tilde{\varphi}}_{0}$.

12: Set $\bm{\Phi}_{0}^{*}\,=\,\text{diag}((\bm{{\varphi}}_{0}^{H})^{*})$.

13: Calculate $p_{k}^{*}\,=\,e_{k}^{*}/\tau_{k}^{*}$.

14: return $\bm{\Phi}_{0}^{*},\tau_{0}^{*},\bm{\tau}^{*},\bm{p}^{*}$.

###  4.4 SCA-based RIS Reflecting Coefficients Optimization for the WIT

In this sub-section, we investigate the optimization of RIS reflecting
coefficient matrix $\bm{\Phi}_{k}$ in the WIT phase, which is given by

$\displaystyle(\mbox{{P3}})$
$\displaystyle\max_{\begin{subarray}{c}\bm{\Phi}_{k}\end{subarray}}\sum_{k=1}^{K}R_{k}$
(33) $\displaystyle s.t.\,\,\,\,\,\,$ (9) and (11).

Note that P3 is still a non-convex optimization problem as the active RIS
introduces additional noise term in the denominator of the objective function,
which results in a quadratic fractional programming problem. In fact, the RIS
reflecting coefficients include amplitude reflection coefficients and phase
shifts. To simplify this problem, we derive the optimal phase shifts in the
closed-form and then exploit an SCA algorithm to obtain the near-optimal
amplitude reflection coefficients according to [20].

It is worth noting that P3 can be decomposed into $K$ independent subproblems,
each of which maximizes the SNR of $U_{k}$ at the RS during $\tau_{k}$ with
respect to the RIS reflection coefficient vector $\bm{\varphi}_{k}$.
Specifically, with $\bm{w}_{k}^{*}$ obtained in (17) and introducing some new
auxiliary variables, the $k$-th SNR maximization problem can be formulated as

$\gamma_{k}=\frac{{p_{k}}\left|{\bm{b}_{k}^{H}}{\bm{\varphi}_{k}}+{g_{d,k}}\right|^{2}}{{{\bm{\varphi}_{k}^{H}}{\bm{Q}_{r}}{\bm{\varphi}_{k}}}{\sigma_{v}^{2}}+{\sigma_{r}^{2}}},$
(34)

where $g_{d,k}=\bm{w}_{k}^{H}\bm{g}_{d,k}$,
$\bm{g}_{r,k}^{H}=\bm{w}_{k}^{H}\bm{G}_{r}$,
$\bm{b}_{k}^{H}=\bm{g}_{r,k}^{H}\text{diag}\left(\bm{g}_{u,k}\right)$,
$\bm{Q}_{r}=\text{diag}\left\\{\left|\bm{g}_{r,k,1}\right|^{2},\ldots,\left|\bm{g}_{r,k,N}\right|^{2}\right\\}$,
$\forall k$. Let $\bm{F}_{k}={p_{k}}{\bm{G}_{u,k}}+{\sigma_{v}^{2}}{I_{N}}$,
$\forall k$. The subproblem can be expressed as

$\displaystyle(\mbox{{P4}})\max_{{\bm{\varphi}_{k}}}\,\,\,$
$\displaystyle\gamma_{k}$ (35) $\displaystyle s.t.\,\,\,\,\,\,$
$\displaystyle{\bm{\varphi}_{k}^{H}}{\bm{F}_{k}}{\bm{\varphi}_{k}}\leq
P_{r},~{}\forall k,$ (36) (9).

To solve P4, we decompose the optimization of RIS reflecting coefficient
vector $\bm{\varphi}_{k}$ into two sub-problems for the amplitude reflection
coefficient design and the optimal phase shift design, respectively. Let
$\bm{\varphi}_{k}=\bm{\varTheta}_{k}\bar{\bm{\varphi}}_{k}$, where
$\bm{\varTheta}_{k}=\text{diag}\left\\{e^{j\theta_{k,1}},\ldots,e^{j\theta_{k,N}}\right\\}\in\mathbb{C}^{N\times
N}$ and
$\bar{\bm{\varphi}}_{k}=\left[a_{k,1},\ldots,a_{k,N}\right]^{T}\in\mathbb{R}^{N\times
1}$.

####  4.4.1 Optimization of phase shifts for the WIT

The optimal design of phase shifts is given in the following proposition.

######  Proposition 4.3.

The optimal RIS phase shift of the $n$-th RE for the WIT during $\tau_{k}$ is
derived as

$\displaystyle\theta_{k,n}^{*}=arg(g_{d,k})-arg(\bm{g}_{u,k,n})+arg(\bm{g}_{r,k,n}),$
(37)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall
k,\forall n,$

where $\bm{g}_{u,k,n}$ and $\bm{g}_{r,k,n}$ denote the $n$-th element of the
vector $\bm{g}_{u,k}$ and $\bm{g}_{r,k}$, respectively.

######  Proof 4.4.

Please refer to Appendix B.

####  4.4.2 Optimization of amplitude reflection coefficients for the WIT

For P4, the optimal design of phase shifts shown in Proposition 4.3 holds
because the value of the amplification power in (9) and (36) and the noise
power in the denominator of (34) are independent with the phase shift of each
RE. In addition, optimizing $\theta_{k,n}$ is equivalent to maximizing the
objective function in (34) [20]. With the optimal phase shifts in Proposition
4.3, we proceed to optimize the RIS amplitude reflection coefficients for the
WIT. In particular, P4 can be simplified as

$\displaystyle(\mbox{{P4-1}})\max_{\bar{\varphi}_{k}}\,\,\,\,\,\,$
$\displaystyle\bar{\gamma_{k}}=\frac{{p_{k}}\left|{\bar{\bm{b}}_{k}^{H}}\bar{\bm{\varphi}}_{k}+\left|g_{d,k}\right|\right|^{2}}{{{\bar{\bm{\varphi}}_{k}^{H}}{\bm{Q}_{r}}{\bar{\bm{\varphi}}_{k}}}{\sigma_{v}^{2}}+{\sigma_{r}^{2}}}$
(38) $\displaystyle s.t.\,\,\,\,\,\,$
$\displaystyle{\bar{\bm{\varphi}}_{k}^{H}}{\bm{F}_{k}}{\bar{\bm{\varphi}}_{k}}\leq
P_{r},~{}\forall k,$ (39) $\displaystyle a_{k,n}\leq{a_{n,\max}},~{}\forall
k,~{}\forall n,$ (40)

where $\bar{\gamma}_{k}=\left|\gamma_{k}\right|$, and $\bar{\bm{b}}_{k}$ is
element-wise modulus of $\bm{b}_{k}$. To deal with the non-convexity of the
objective function (38), we introduce a new auxiliary variable
${n_{k}}={{\bar{\bm{\varphi}}_{k}^{H}}{\bm{Q}_{r}}{\bar{\bm{\varphi}}_{k}}}{\sigma_{v}^{2}}+{\sigma_{r}^{2}}$,
which denotes the noise power received at the RIS. Then, P4-1 can be converted
into the following equivalent form

$\displaystyle(\mbox{{P4-2}})\max_{\bar{\gamma}_{k},n_{k},\bar{\bm{\varphi}}_{k}}$
$\displaystyle\,\,\bar{\gamma}_{k}$ (41) $\displaystyle s.t.$
$\displaystyle\sqrt{p_{k}}\left(\bar{\bm{b}}_{k}^{H}\bar{\bm{\varphi}}_{k}+\left|g_{d,k}\right|\right)\geq\sqrt{n_{k}\bar{\gamma}_{k}},~{}\forall
k,$ (42)
$\displaystyle\bar{\bm{\varphi}}_{k}^{H}\bm{Q}_{r}\bar{\bm{\varphi}}_{k}\sigma_{v}^{2}+\sigma_{r}^{2}\leq
n_{k},~{}\forall k,$ (43) $\displaystyle\mbox{\eqref{Pr_WIT4} and
\eqref{RE_WIT2}}.$

However, the constraint (42) is still non-convex. To solve P4-1 efficiently,
we exploit the SCA algorithm to approximate the square root by a convex upper-
bound in each iteration. Define ${\bar{\gamma}_{k}(t)}$ and ${n_{k}(t)}$ as
the iterative optimization variables after the $t$-th step iteration. In terms
of $\left\\{\bar{\gamma}_{k}\left(t\right),n_{k}\left(t\right)\right\\}$, the
first-order Taylor polynomial is used to approximate
${\sqrt{n_{k}\bar{\gamma}_{k}}}$, which is given by

$\displaystyle\sqrt{n_{k}\bar{\gamma}_{k}}\leq$
$\displaystyle\mathcal{G}(\bar{\gamma}_{k},n_{k};t)$ (44) $\displaystyle=$
$\displaystyle\sqrt{\bar{\gamma}_{k}(t)n_{k}(t)}+\frac{1}{2}\left(\frac{n_{k}(t)}{\bar{\gamma}_{k}(t)}\right)^{\frac{1}{2}}\left[\bar{\gamma_{k}}-\bar{\gamma}_{k}(t)\right]$
$\displaystyle+\frac{1}{2}\left(\frac{\bar{\gamma}_{k}(t)}{n_{k}(t)}\right)^{\frac{1}{2}}\left[n-n_{k}(t)\right].$

Based on (44), (42) can be rewritten as

$\sqrt{p_{k}}\left(\bar{\bm{b}}_{k}^{H}\bar{\bm{\varphi}}_{k}+\left|g_{d,k}\right|\right)\geq\mathcal{G}(\bar{\gamma}_{k},n_{k};t),~{}\forall
k.$ (45)

Then, P4-2 can be reformulated as the following problem

$\displaystyle(\mbox{{P4-3}})\max_{\bar{\gamma}_{k},n_{k},\bar{\bm{\varphi}}_{k}}$
$\displaystyle\,\,\bar{\gamma}_{k},$ (46) $\displaystyle s.t.$ (39), (40),
(43) and (45).

As P4-3 is convex and can be solved by the inter-point method. We then discuss
the initialization of ${\bar{\gamma}_{k}(t)}$ and ${n_{k}(t)}$. First, we
propose an initial solution $\bar{\bm{\varphi}}_{k}(0)$ by solving a simple
feasible version of problem P4-1, i.e., $\bar{\bm{\varphi}}_{k}(0)$,
satisfying the constraints (39) and (40). Then, the reasonable initialization
of ${\bar{\gamma}_{k}(0)}$ and ${n_{k}(0)}$ is given by

$\displaystyle\bar{\gamma}_{k}(0)$
$\displaystyle=\frac{{p_{k}}\left|{\bar{\bm{b}}_{k}^{H}}\bar{\bm{\varphi}}_{k}(0)+\left|g_{d,k}\right|\right|^{2}}{{{\sigma_{v}^{2}}\bar{\bm{\varphi}}_{k}^{H}(0){\bm{Q}_{r}}\bar{\bm{\varphi}}_{k}}(0)+{\sigma_{r}^{2}}},$
(47) $\displaystyle n_{k}(0)$
$\displaystyle={\sigma_{v}^{2}}\bar{{\bm{\varphi}}}_{k}^{H}(0)\bm{Q}_{r}\bar{\bm{\varphi}}_{k}(0)+{\sigma_{r}^{2}}.$
(48)

With the initialization described in (47) and (48), the optimal amplitude
reflection coefficients for the WIT, denoted by $\bar{\bm{\varphi}}_{k}^{*}$,
can be obtained by iteratively solving P4-3 until the convergence is achieved.
As a result, the RIS reflecting coefficients during $\tau_{k}$ can be
calculated by

$\bm{\Phi}_{k}^{*}=\text{diag}\left(\bm{\varTheta}_{k}^{*}\bar{\bm{\varphi}}_{k}^{*}\right),~{}\forall
k,$ (49)

where
$\bm{\varTheta}_{k}^{*}=\text{diag}\\{e^{j\theta_{k,1}^{*}},\ldots,e^{j\theta_{k,N}^{*}}\\}$.

The detailed description of optimizing the RIS reflecting coefficients in P3
is summarized in the Algorithm 2.

Algorithm 2 SCA-based RIS reflecting coefficients for the WIT

0: $\\{\bm{w}_{k}\\},~{}\bm{p},~{}\forall k$

0: $\\{\bm{\Phi}_{k}^{*}\\},~{}\forall k$

1: Initialization: $\bar{\gamma}_{k}(t)$, $n_{k}(t)$,
$\bar{\bm{\varphi}}_{k}(t)$, and $t=0$.

2: Obtain $\theta_{k,n}^{*}$ in Proposition 4.3 and have
$\bm{\varTheta}_{k}^{*}$.

3: repeat

4: $t=t+1$.

5: Update $\bar{\gamma}_{k}(t)$, $n_{k}(t)$, $\bar{\bm{\varphi}}_{k}(t)$ by
solving P4-3.

6: until the convergence is achieved.

7: Obtain
$\bm{\Phi}_{k}^{*}=\text{diag}\left(\bm{\varTheta}_{k}^{*}\bar{\bm{\varphi}}_{k}^{*}\right)$,
where $\bar{\bm{\varphi}}_{k}^{*}=\bar{\bm{\varphi}}_{k}(t)$.

8: return $\\{\bm{\Phi}_{k}^{*}\\}$.

###  4.5 Algorithm Summarization and Analysis

Based on the above analysis, the algorithm for solving P1 is summarized in
Algorithm 3. Based on the optimality analysis,the objective function of P1 is
a non-decreasing function. Due to the power budget constraint (2), (7) and
(11), the optimal objective value of problem P1 is bounded. Hence, the
convergence of Algorithm 2 can be thus guaranteed, which will be also
confirmed by numerical simulations in Section 5.

####  4.5.1 Complexity Analysis

The computational complexity of our proposed AO algorithm is analyzed as
follows, which contains the linear MMSE-based receive beamforming calculation,
the SDR algorithm and the SCA algorithm in each iteration. For the linear
MMSE-based receive beamforming optimization, we derive a closed-form solution
to the (17), and the approximate worst-case computational complexity is given
by $\mathcal{O}\left(KL\max(N,L)^{2}\right)$. According to [36], for the
subproblem of SDP-based transmit beamforming optimization, the worst-case
computational complexity is
$\mathcal{O}\left(\max(K,M)^{4.5}\log(1/\epsilon)\right)$, where $\epsilon$ is
the computational accuracy of the interior-point method in CVX. Similarly, for
the SDP-based RIS reflecting coefficients for the WET and resource allocation
optimization, the worst-case computational complexity is
$\mathcal{O}\left(\mathcal{I}_{\tau}\max(N,K)^{4.5}\log(1/\epsilon)\right)$,
where $\mathcal{I}_{\tau}$ is the iteration number for updating $\tau_{0}$.
For the SCA-based RIS reflecting coefficients optimization in the WIT, the
computational complexity is less than
$\mathcal{O}\left(\mathcal{I}_{S}KN^{3.5}\log(N/\epsilon)\right)$, where
$\mathcal{I}_{S}$ is the iteration number for the SCA algorithm. Thus, the
computational complexity of the overall AO algorithm is given by

$\displaystyle\mathcal{O}\left(\mathcal{I}_{A}\left(KL\max(N,L)^{2}+\max(K,M)^{4.5}\log(1/\epsilon)\right.\right.$
(50)
$\displaystyle\left.\left.+\mathcal{I}_{\tau}\max(N,K)^{4.5}\log(1/\epsilon)+\mathcal{I}_{S}KN^{3.5}\log(N/\epsilon)\right)\right),$

where $\mathcal{I}_{A}$ denotes the number of iterations required for
convergence.

####  4.5.2 Optimality Analysis

As the formulated problem P1 is extremely non-convex, it is very difficult to
obtain the globally optimal solution. To solve P1 efficiently, we propose an
efficient AO algorithm to obtain the suboptimal solutions. Firstly, we obtain
the optimal receive beamforming $\left\\{\bm{w}_{k}\right\\}$ in a closed-
form, which are the globally optimal solutions. With the obtained receive
beamforming solutions, the formulated problem can be simplified but is still
non-convex. Secondly, the SDR technique is adopted to optimize the transmit
beamforming, the RIS reflecting coefficient matrices for the WET phase, the
transmit power at each user, and the network time scheduling. In particular,
we prove the obtained solutions of P2-1 and P2-2 are rank-one. Since the
tightness of applying SDR can be guaranteed, the obtained solutions
$\\{{\bm{w}_{0},\bm{\Phi}_{0},\bm{\tau},\bm{p}}\\}$ are globally optimal [23].
Then, we use the one-dimensional search method to exploit the optimal energy
transmission time $\tau_{0}$ by setting an appropriate step size. Thirdly, for
the optimization of RIS reflecting coefficients for the WIT phase, we
decompose P4 into two sub-problems. On the one hand, the optimal phase shifts
have been derived in a closed-form which has been proved. On the other hand,
P4-1 is solved by Algorithm 2, which obtains the reflection amplitudes of
$\\{\bm{\Phi}_{k}\\}$ are near-optima of the original problem [38]. Hence,
Algorithm 3 can be used to obtain the near-optimal solutions to P1 with a high
accuracy.

Algorithm 3 AO algorithm for P1

1: Initialization:
$\bm{w}_{0},\\{\bm{w}_{k}\\},\bm{\Phi}_{0},\\{\bm{\Phi}_{k}\\},\tau_{0},\bm{\tau},\bm{p},~{}\forall
k$.

2: repeat

3: Given $\bm{\Phi}_{k}$ and $\bm{p}$, update $\\{\bm{w}_{k}\\}$ by (17).

4: Given $\bm{\Phi}_{0}$, $\tau_{0}$, $\\{\bm{\Phi}_{k}\\}$,
$\\{\bm{w}_{k}\\}$, update $\bm{w}_{0}$ with by solving P2-1.

5: Given $\bm{w}_{0}$, $\\{\bm{w}_{k}\\}$, $\\{\bm{\Phi}_{k}\\}$, update
$\bm{\Phi}_{0}$, $\tau_{0}$, $\bm{\tau}$ and $\bm{p}$ by Algorithm 1.

6: Given $\\{\bm{w}_{k}\\}$ and $\bm{p}$, update {$\bm{\Phi}_{k}\\}$ by
Algorithm 2.

7: until $\sum_{k=1}^{K}R_{k}$ converged.

8: return
$\bm{w}_{0}^{*},\\{\bm{w}_{k}^{*}\\},\bm{\Phi}_{0}^{*},\\{\bm{\Phi}_{k}^{*}\\},\tau_{0}^{*},\bm{\tau}^{*},\bm{p}^{*}$.

##  5 Numerical Results

Figure 3: Placement model of simulation setup.

In this section, numerical results are presented to evaluate the performance
of the proposed scheme. As shown in Figure 3, we consider that the simulated
network deployment is a 2-D coordinate system, where the coordinates of the
PS, the RIS, and the RS are given as (0,0), ($x_{r}$, 0), and ($x_{s}$,0),
respectively, the users are randomly deployed within a circular area centered
at ($x_{u}$, $x_{h}$) with radius 1m. We follow the channel model considered
in [29]. In particular, the large-scale path-loss is modeled as
$L=A(d/d_{0})^{-\alpha}$, where $A$ is the path-loss at the reference distance
$d_{0}=1$m and set as $A=-30$dB, $d$ denotes the distance between two nodes,
and $\alpha$ is the path-loss exponent. For the RIS related links, the path-
loss exponent is set as 2.2 since the location of RIS can be carefully
designed to avoid the severe signal blockage. While the path-loss exponents
for the RIS unrelated links are set as 3.5 due to the users’ random
deployment. We assume the direct link channels follow Rayleigh fading but the
RIS related channels follow Rician fading. Specifically, the small-scale
channel from the PS to the RIS can be expressed as

$\bm{H_{r}}=\left(\sqrt{\frac{\beta_{r}}{\beta_{r}+1}}\bar{\bm{H}}_{r}^{\text{LoS}}+\sqrt{\frac{1}{\beta_{r}+1}}\bar{\bm{H}}_{r}^{\text{NLoS}}\right)$
(51)

where $\beta_{r}$ is the Rician factor for the PS-RIS link,
$\bar{\bm{H}}_{r}^{\text{LoS}}$ denotes the deterministic line of sight (LoS)
component, and $\bar{\bm{H}}_{r}^{\text{NLoS}}$ denotes the non-LoS compotent
with circularly symmetric complex Gaussian random variables with zero mean and
unit variance. The other channels can be similarly defined. Unless otherwise
stated, other parameters are given as follows: $\beta_{r}=10$ [39],
$\rho=0.8$, $\sigma_{v}^{2}=\sigma_{r}^{2}=-90$dBm, $P_{0}=20$dBm [20],
$P_{r}=20$dBm, $a_{max}=25$dB [40], $N=10$, $K=4$, $M=4$, $L=4$, $x_{r}=10$m,
$x_{u}=10$m, $x_{s}=20$m, and $x_{h}=2$m.

For comparisons, we also evaluate the performance of the following benchmark
schemes:

  * (1)

Active RIS-aided single-antenna WPCN scheme (Active-SA).

  * (2)

Passive RIS-aided multi-antenna WPCN scheme (Passive-MA).

  * (3)

Active RIS-aided multi-antenna WPCN with uniform energy beamforming scheme
(Active-MA-UEBF).

Notice that for the multi-antenna schemes, the number of antennas is set as 4
in the PS and RS. In addition, we set the number of REs in the Passive-MA
scheme as $N=100$ to show the superiority of the proposed scheme.

Before performance comparisons, we first show the convergence performance of
the proposed AO algorithm in Figure 4. One can observe that as the number of
iterations increases, the sum-rate first increases but finally converges to a
constant after nearly 8 iterations. This demonstrates that the convergence of
the proposed scheme can be achieved quickly. The other observation is that the
effect of the parameter setting on convergence is limited.

Figure 4:  Convergence behavior of the proposed scheme under different
parameter settings.

Figure 5 shows the impact of the transmit power at the PS (i.e., $P_{0}$) on
the sum-rate when the RIS’s maximum reflecting power $P_{r}=10,\,20$ dBm and
the RIS’s maximum amplitude reflection coefficient $a_{\max}=10,\,25$ dB,
respectively. In general, the proposed scheme outperforms the Active-SA scheme
with the same parameters, which confirms that the assistance of multiple
antennas can achieve a significant performance gain by constructing the
transmit beamforming at the PS and the receive beamforming at the RS. For a
given $a_{max}=25$ dB, our proposed scheme with 10 REs can achieve 415.48%
performance gain compared to the passive RIS scheme with 100 REs when
$P_{0}=20$ dBm. Indeed, the active RIS can considerably make use of its
amplification characteristic to amplify the energy signals at low transmit
power and thereby realize a superior capability at the cost of additional
power consumption. For a given $a_{max}=10$ dB, it can be seen that the sum-
rates achieved by the proposed scheme with $P_{r}=20$ dBm and $P_{r}=10$ dBm
are almost the same, which implies that the amplification power constraints
defined in (7) and (11) are inactive since $a_{\max}$ is limited for the small
transmit power. Note that, the performance gap is significant between the
scheme with $a_{\max}=25$ dB and $a_{\max}=10$ dB because $a_{\max}$ directly
limits the amplitude reflection coefficient of the active RIS. In addition,
the sum-rate of the passive-MA scheme is generally lower than the active RIS
schemes with the same REs and the passive RIS needs to be equipped with more
REs (e.g., 100 REs) to achieve the similar performance.

Figure 5:  Sum-rate versus the transmit power at the PS.  Figure 6: Sum-rate
versus number of reflecting elements at the RIS.

In Figure 6, we evaluate the sum-rate versus the number of reflecting elements
at the RIS. It can be seen that the proposed schemes can achieve a higher
performance gain compared with the other benchmark schemes. With an increasing
number of reflecting elements, the sum-rate increases due to the fact that
more transmission links can be provided for both the WET and the WIT. In
addition, to investigate the best system performance, the maximum number of
users is set to be equal to the number of REs. Since the active RIS can
amplify the incident signals, a limited number of REs is sufficient to reach
the desired SNR. Therefore, the size of active RIS can be reduced, making it
applicable to the scenario where the space for the RIS deployment is limited.

Figure 7:  Sum-rate versus the number of users.

In Figure 7, we study the effect of number of user on the sum-rate. As the
number of users increases, the total amount of harvested energy by users
improves, which results in a higher sum-rate. Nonetheless, when the number of
users reaches a threshold, e.g. $K=8$, the sum-rate achieved by our proposed
scheme becomes to be saturated. This is due to the fact that the increment of
number of users reduces the energy transfer duration and the time allocated to
each user for information transmission, which makes the sum-rate converge to a
constant. Again, our proposed scheme notably outperforms the other benchmark
schemes. For example, when the number of users is $K=4$, our proposed scheme
can achieve 17.78% and 415.48% performance gain compared with the Active-SA
scheme and the Passive-MA scheme with 100 REs, respectively.

Figure 8: Sum-rate versus x-coordinate of the RIS.

In Figure 8, we plot the sum-rate versus the horizontal ordinate of the RIS.
As $x_{r}$ varies, the sum-rates of all schemes first increase but then
decrease. Compared to the scenario that the RIS is close to the RS, by
deploying the RIS near the PS, e.g., $x_{r}=1$, the sum-rate can be improved.
It is because the users can harvest more energy assisted by the active RIS.
Moreover, we can observe that the sum-rate is maximized at $x_{r}=10$, where
the reflecting link between the active RIS and each user is strongest so the
users can benefit from a larger amplification and reflection gain. However,
when the RIS is neither close to the PS nor the users, both the PS-RIS link
and the RIS-users links become weak, which results in the reduce of harvested
energy. Furthermore, since the Active-MA-UEBF scheme adopts the uniform energy
beamforming, the energy signals cannot adaptively align with the direction of
the desired channels, which results in a low WET efficiency. In addition, the
schemes with the active RIS can achieve a much better performance than the
passive RIS scheme, which demonstrates that the active RIS with the
amplification functionality can significantly mitigates the double-fading
effect. The above observation demonstrates that the location of the active RIS
should be carefully designed.

##  6 CONCLUSIONS

In this paper, we have proposed an active RIS assisted relaying scheme to
enhance the performance of multiuser multi-antenna WPCN, which is involved in
both the WET from the PS to users and the WIT from users to the RS. To further
enhance system performance, both transmit beamforming at the PS and receive
beamforming at the RS have been designed. We have formulated a system sum-rate
maximization problem by jointly optimizing the RIS reflection coefficients for
both the WET and the WIT, transmit and receive beamforming vectors, transmit
power at each user, and network time scheduling. As the formulated problem is
non-convex, we have proposed an AO algorithm with linear MMSE, SDR and SCA
techniques to solve it efficiently. Finally, numerical results have been
provided to confirm the performance superiority of the proposed scheme.

## Appendix

### A. Proof of Proposition 1

The Lagrangian function of P2-1 can be expressed as

$\displaystyle\mathcal{L}=$
$\displaystyle\sum_{k=1}^{K}\lambda_{k}\beta\text{Tr}(\bm{H}_{d,k}\bm{W}_{0})-\xi\text{Tr}(\bm{W}_{0})$
(52) $\displaystyle+\text{Tr}(\bm{\Omega}\bm{W}_{)}+\delta,~{}\forall k,$

where $\lambda_{k}\geq 0$, $\xi\geq 0$, and $\bm{\Omega}\in\mathbb{H}^{M}$ are
the Lagrange multipliers associated with constraints (21), (22), and (24),
respectively, $\delta$ denotes the term unrelated with $\bm{W}_{0}$. The
Karush-Kuhn-Tucker (KKT) conditions of P2-1 are given as follows

$\displaystyle\frac{\partial{\mathcal{L}}}{\partial{\bm{W}_{0}}}=\sum_{k=1}^{K}\lambda_{k}^{*}\beta\bm{H}_{d,k}-\xi^{*}\bm{I}_{M}+\bm{\Omega}^{*}=0,$
(53) $\displaystyle\bm{\Omega}^{*}\bm{W}_{0}^{*}=0,$ (54)

where $\lambda_{k}^{*}$, $\xi^{*}$ and $\bm{\Omega}^{*}$ are the optimal
Lagrangian multipliers for the dual problem of P2-1. It can be proved that
$\lambda_{k}^{*}>0$ and $\xi^{*}>0$ since the constraints (21) and (22)are
equalities in the optimal condition. Based on (53) and (54), it is
straightforward to obtain the following equality

$\displaystyle(\xi^{*}\bm{I}_{M}-\sum_{k=1}^{K}\lambda_{k}^{*}\beta\bm{H}_{d,k})\bm{W}_{0}^{*}=0$
(55)

According to [41],
$\text{rank}(\xi^{*}\bm{I}_{M}-\sum_{k=1}^{K}\lambda_{k}^{*}\beta\bm{H}_{d,k})\geq
M-1$ due to the fact that $\bm{H}_{d,k}$ for $\forall k$ are independently
distributed. Thus, from (55), we can obtain that $\text{rank}(\bm{W}_{0})\leq
1$. It is obvious that $\bm{W}_{0}=\bm{0}$ is not the optimal solution to
P2-1. Hence, we derive that $\text{rank}(\bm{W}_{0})=1$, which thus proves
Proposition 4.1.

### B. Proof of Proposition 2

Since $\bm{Q}_{r}$ and $\bm{F}_{k}$ are diagonal matrices, we observe that the
noise power in the denominator of (35) and the amplification power in (9) and
(36) are independent of the phase shift of each RE. Therefore, maximizing
$\gamma_{k}$ with respect to $\bm{\varTheta}_{k}$ is equivalent to the
following optimization problem

$\displaystyle(\mbox{{P4-4}})\max_{{\bm{\varTheta}_{k}}}\,\,\,\,\,\,$
$\displaystyle\left|{\bm{b}_{k}^{H}}\bm{\varTheta}_{k}{\bar{\bm{\varphi}}_{k}}+{g_{d,k}}\right|^{2}$
(56) $\displaystyle s.t.\,\,\,\,\,\,$
$\displaystyle\left|\varTheta_{k,n}\right|=1,~{}\forall k,~{}\forall n.$ (57)

We rewrite the objective function as

$\left|\bm{b}_{k}^{H}\bm{\varTheta}_{k}\bar{\bm{\varphi}}_{k}\right|^{2}+\left|{g_{d,k}}\right|^{2}+2\left|\bm{b}_{k}^{H}\bm{\varTheta}_{k}\bar{\bm{\varphi}}_{k}\right|\left|{g_{d,k}}\right|\cos\alpha,$
(58)

where
$\alpha=\arctan\frac{\mathrm{Im}(\bm{b}_{k}^{H}\bm{\varTheta}_{k}\bar{\bm{\varphi}}_{k})}{\mathrm{Re}(\bm{b}_{k}^{H}\bm{\varTheta}_{k}\bar{\bm{\varphi}}_{k})}-\arctan\frac{\mathrm{Im}({g_{d,k}})}{\mathrm{Re}({g_{d,k}})}$.
Obviously, the maximum of
$\left|{\bm{b}_{k}^{H}}\bm{\varTheta}_{k}{\bar{\bm{\varphi}}_{k}}+{g_{d,k}}\right|^{2}$
is achieved when
$arg(\bm{b}_{k}^{H}\bm{\varTheta}_{k}\bar{\bm{\varphi}}_{k})=arg(g_{d,k})\triangleq\omega$.
Let $v_{k}=[v_{k,1},v_{k,2},...,v_{k,N}]^{T}\in\mathbb{R}^{N\times 1}$ and
$\bm{\xi}_{k}=\text{diag}(\bm{b}_{k}^{H})\bar{\bm{\varphi}}_{k}$. As
${\bm{b}_{k}^{H}}\bm{\varTheta}_{k}{\bar{\bm{\varphi}}_{k}}=\bm{v}_{k}^{H}\bm{\xi}_{k}$,
P4-4 can be rewritten as

$\displaystyle(\mbox{{P4-5}})\max_{{\bm{v}_{k}}}\,\,\,\,\,\,$
$\displaystyle\left|\bm{v}_{k}^{H}\bm{\xi}_{k}\right|^{2}$ (59) $\displaystyle
s.t.\,\,\,\,\,\,$ $\displaystyle\left|\bm{v}_{k,n}\right|=1,~{}\forall
k,~{}\forall n,$ (60) $\displaystyle
arg(\bm{v}_{k}^{H}\bm{\xi}_{k})=\omega,~{}\forall k.$ (61)

Based on [12], the optimal solution to P4-5 can be expressed as
$\bm{v}_{k}^{*}=e^{j(\omega-arg(\bm{\xi}_{k}))}=e^{j(\omega-
arg(\text{diag}(\bm{b}_{k}^{H})\bar{\bm{\varphi}}_{k}))}$. Then, the optimal
RIS phase shift for the $n$-th RE is expressed as
$\theta_{k,n}=arg(g_{d,k})-arg(\bm{b}_{k,n}^{H})-arg(\bar{\bm{\varphi}}_{k,n})$.
Finally, we obtain that
$\theta_{k,n}=arg(g_{d,k})-arg(\bm{g}_{u,k,n})+arg(\bm{g}_{r,k,n})$, $\forall
k$, $\forall n$. This completes the proof of Proposition 4.3.

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