# Beamforming Design for IRS-aided Decode-and-Forward Relay Wireless Network

Xuehui Wang, Feng Shu,  _Member_ ,  _IEEE_ , Weiping Shi, Xiaopeng Liang,
Rongen Dong, Jun Li,  _Senior Member_ ,  _IEEE_ , and Jiangzhou Wang,
_Fellow_ ,  _IEEE_ This work was supported in part by the National Natural
Science Foundation of China (Nos. 62071234 and 61771244), and the Scientific
Research Fund Project of Hainan University under Grant KYQD(ZR)-21008 and
KYQD(ZR)-21007). _(Corresponding authors: Feng Shu and Xiaopeng Liang)_.Xuehui
Wang, Feng Shu, Xiaopeng Liang, and Rongen Dong are with the School of
Information and Communication Engineering, Hainan University, Haikou, 570228,
China.Weiping Shi, Jun Li and Feng Shu are with the School of Electronic and
Optical Engineering, Nanjing University of Science and Technology, 210094,
China.Jiangzhou Wang is with the School of Engineering, University of Kent,
Canterbury CT2 7NT, U.K. Email: (e-mail: j.z.wang@kent.ac.uk).

###### Abstract

As a low-cost and low-power-consumption passive reflector, intelligent
reflecting surface (IRS) can make a significant rate improvement by building a
programmable wireless environment. To improve the rate performance and
coverage range of wireless networks, an IRS-aided decode-and-forward (DF)
relay network is proposed with multiple antennas at relay station (RS). To
achieve a high rate, an alternately iterative structure (AIS) of maximizing
receive power (Max-RP) at RS is proposed to jointly optimize the beamforming
vectors at RS and phase shifts at IRS. Considering its high-complexity, two
low-complexity Max-RP schemes of null-space projection (NSP) plus maximum
ratio combining (MRC) and IRS element selection (IRSES) plus MRC are presented
to reduce this complexity, respectively. For the former, NSP is used to
separate the reflected signal from IRS and the direct transmitted signal from
source and MRC is adopted to combine the two signals at RS. For the latter,
the basic concept of IRSES is as follows: IRS is partitioned into $M$ subsets
of elements and adjusting the phases of all elements per subset make all
reflected signals and the direct signal from source phase alignment (PA) at
the corresponding antenna of relay. Simulation results show that the proposed
three methods perform much better than the existing network with single-
antenna relay in terms of rate performance. In particular, a 85% rate gain
over existing scheme is achieved in the high signal-to-noise ratio region.
Moreover, it is verified that the positions of RS and IRS have a substantial
impact on rate performance, and there exists an optimal positions of RS and
IRS.

###### Index Terms:

Intelligent reflecting surface, decode-and-forward relay, beamforming vectors,
phase shifts, null-space projection, maximum ratio combining.

##  I Introduction

With the explosive growth of communication device nodes in wireless sensor
network (WSN), there are more stringent requirements in terms of energy
efficiency and extended coverage [1]. Relay has a strong ability to process
signal for extended communication coverage and improved rate performance [2],
[3]. However, the conventional relay is an active device, which is costly and
requires additional high energy consumption to process the signal [4], it is
crucial to develop a energy-efficient and green communication network.

Compared with the conventional relay, since intelligent reflecting surface
(IRS) does not require any radio frequency chain and baseband circuit and is
made up of low-cost and passive reflecting units, its lower-energy-consumption
and passive property is very attractive [5],[6]. Thus, IRS can be regarded as
a green reflect-to-forward relay. It is anticipated that IRS will be
potentially applied to the diverse future wireless networks such as WiFi 7,
mobile communications like sixth generation (6G), space communication, marine
communication, and emergency communication [7]. For example, for mobile
communications, IRS will be used to significantly enhance the coverage of
blind areas, especially cell edges. Due to the ability of intelligently
adjusting the amplitude and phase shift of the incident signal by a smart
controller [8], IRS may form helpful controlled multipaths, ingeniously
improve the propagation environment and provide new degrees of freedom, which
can be used to enhance the wireless network performance. Recently, IRS has
attracted much attention from both academia and industry [9]. IRS may be
adopted to aid many wireless communication directions as follows: directional
modulation [10], [11], spatial modulation [12, 13], multicast transmission
[14, 15], covert wireless communication [16, 17], unmanned aerial vehicle
(UAV) communication [18, 19], and simultaneous wireless information and power
transfer (SWIPT) [20, 21].

A combination of a relay and an IRS was shown to improve energy efficiency
[22], spectral efficiency [23] and rate performance [24, 25, 26]. In [22], the
authors proposed a multi-user mobile communication network model with the help
of a relay and an IRS, where two sub-optimal methods based on alternating
optimization (AO), called singular value decomposition (SVD) plus uplink-
downlink duality, and SVD plus zero-forcing methods were presented to
implement an enhanced performance from base station (BS) to multi-users in
terms of energy efficiency. To increase spectral efficiency, in [23] a novel
network was proposed, where some IRS elements act as active relays to amplify
incident signals, and the remaining elements reflect signals. A spectral
efficiency maximization problem was formulated and solved by the alternating
optimization method. Aiming at improving rate performance, the authors
proposed two kinds of IRS plus decode-and-forward (DF) relay [24]: distributed
and centralized. Subsequently, a sequential optimization algorithm was
presented to address the power allocation and optimization of IRS phases, and
it was demonstrated that IRS and DF relay can work in a synergistic manner to
enhance the achievable rate. In [25], a network consisting of two side-by-side
intelligent surfaces connected via a full-duplex relay was proposed to achieve
the promising rate gains with much smaller number of reflecting elements.
Moreover, a hybrid network consisting of an IRS and a single-antenna DF relay
was proposed to save a massive IRS elements compared with only IRS in [26]. In
[27], a communication system with multiple antennas at both transmitter and
receiver is considered.

From the above literature, it can be concluded that IRS has the advantages of
low cost and low energy consumption, and relay has a strong signal processing
ability such as amplify-and-forward (AF) and DF. It is interesting to combine
relay and IRS while keeping the advantages of relay and IRS. The hybrid
network combines both advantages of IRS and relay to strike a good balance
among circuit cost, energy efficiency and rate performance.

The application scenarios in the above literature are common, a hybrid network
combining IRS and multi-antenna relay is proposed to employ to WSN scenario,
which consists of a data collecting center and many other non-center nodes.
The data collecting center collects all data from other nodes, then send them
to internet with the help of relay and IRS. Furthermore, when data collecting
center-relay or relay-destination direct link has shadow fading, or in extreme
cases is completely blocked. The proposed hybrid network is still appropriate
for the practical scenario, where an IRS is placed on a high big building and
can see source, relay and destination. In other words, the signal from source
can still be transmitted to relay and destination with the help of the IRS.
This means that adding IRS can create new reflective paths among source, relay
and destination.

In this case, in order to improve the data rate of the proposed hybrid network
or dramatically extend its coverage range, three efficient beamforming methods
are proposed to achieve this goal. Our main contributions are summarized as
follows:

  1. 1.

To make a dramatic rate improvement, an IRS-aided multi-antenna relay network
model is proposed. We focus on the design of beamforming in the first time
slot due to the fact that the system model can be converted into a typical
IRS-aided three-point model. A maximizing receive power (Max-RP) using
alternately iterative structure (AIS) is proposed to maximize RP by
alternately optimizing the beamforming vector at relay station (RS) and phase
shifts at IRS. Due to the rule of Max-RP, the closed-form expressions of the
beamforming vector at RS and phase shifts at IRS are derived. From the
simulation results, the proposed Max-RP method using AIS can harvest up to 86%
rate gain over existing network of an IRS plus a single-antenna relay in [26]
in the high signal-to-noise ratio (SNR) region. The total computational
complexity is
$\mathcal{O}\\{L_{2}({N^{4}}+8M{N^{3}}+5{N^{3}}+24M{N^{2}}-2{N^{2}})\\}$, and
the highest order is $N^{4}$ float-point operations (FLOPs), which is high.

  2. 2.

To reduce the high computational complexity of the above method, a low-
complexity null-space projection (NSP)-based Max-RP plus maximum ratio
combining (MRC) is proposed in the first time slot. The two independent
receive beamforming vectors are used at RS. They utilize NSP to separate the
signal from source and the reflected signal from IRS and maximize the receive
power of the corresponding signal. Finally, MRC is adopted to combine the two
signals to improve the SNR of receive signal at RS. Our simulation results
show that the rate of the proposed NSP-based Max-RP plus MRC method is 84%
higher than that of the existing system with an IRS and a single-antenna relay
in [26] in the high SNR region. Its computational complexity is
$\mathcal{O}\\{N^{3}+M^{3}+{M^{2}}N+M{N^{2}}\\}$, which is much lower than the
Max-RP method using AIS.

  3. 3.

To further reduce the computational complexity, the IRS element selection
(IRSES)-based Max-RP plus MRC method is proposed. Here, in the first time
slot, according to the number $M$ of RS antennas, IRS elements are divided
into $M$ subsets with each subset having the same number of elements. Each
subset is mapped into one antenna at RS. In other words, there is one-to-one
mapping relationship between subsets at IRS and antennas at RS. Adjusting the
phases of all elements per subset aligns the phases of all reflected signals
and the direct signal from source at the corresponding antenna of relay.
Finally, MRC is adopted to combine all received signals at RS. From the
simulation results, the proposed Max-RP based on IRSES plus MRC method
achieves a substantial rate improvement over existing system in [26]. The
complexity is $\mathcal{O}\\{15MK+8M+10K+L_{5}(18MN+2M+3N)\\}$, which is
extremely lower than the above proposed two methods.

The remainder of this paper is organized as follows. In Section II, we
describe an IRS-aided DF relay network. In Section III, we demonstrate three
methods for a better rate performance of the proposed network. We present our
simulation results in Section IV, and draw conclusions in Section V.

_Notation_ : Scalars, vectors and matrices are respectively represented by
letters of lower case, bold lower case, and bold upper case. $(\cdot)^{*}$,
$(\cdot)^{H}$, $(\cdot)^{\dagger}$ stand for matrix conjugate, conjugate
transpose, and Moore-Penrose pseudo inverse, respectively.
$\mathbb{E}\\{\cdot\\}$, $\|\cdot\|$, $\text{tr}(\cdot)$, and
$\text{arg}(\cdot)$ denote expectation operation, 2-norm, the trace of a
matrix, and the phase of a complex number, respectively. The sign
$\textbf{I}_{N}$ is the $N\times N$ identity matrix.

##  II System Model

Figure 1: System model for an IRS-aided DF relay network.

As shown in Fig. 1, the IRS-aided DF relay network consists of a half-duplex
DF relay station (RS) with $M$ transmit antennas, an IRS with $N$ reflecting
elements, a data collector (S) and a base station (D) equipped with a single
antenna each. As a data collecting center node, S collects all data from other
sensor nodes, and transmits them towards D with the help of IRS and RS. Here,
the network is operated in a time division half-duplex scenario. The distance
from S to D are assumed so far that there is no direct link between them [26].
Due to path loss, the power of signals reflected by the IRS twice or more are
such weak that they can be ignored. Moreover, all channel state informations
(CSIs) are assumed to be perfectly known to S, RS, IRS and D. In the first
time slot, the received signal at RS is written by

$\textbf{y}_{r}=\sqrt{P_{s}}(\textbf{h}_{sr}+\textbf{H}_{ir}{\boldsymbol{\Theta}}_{1}\textbf{h}_{si})s+\textbf{n}_{r},$
(1)

where $s$ and ${P_{s}}$ are the transmit signal and power from S,
respectively. $\mathbb{E}\\{s^{H}s\\}=1$. Without loss of generality, we
assume a Rayleigh fading environment. Let
$\textbf{h}_{sr}$$\in\mathbb{C}^{M\times 1}$,
$\textbf{H}_{ir}\in\mathbb{C}^{M\times N}$ and
$\textbf{h}_{si}\in\mathbb{C}^{N\times 1}$ represent the channels from S to
RS, from IRS to RS, and from S to IRS. ${\boldsymbol{\Theta}}_{1}$ is the
diagonal reflection-coefficient matrix of IRS, which is denoted as
${\boldsymbol{\Theta}}_{1}$ =
$\text{diag}(e^{j\theta_{11}},\cdots,e^{j\theta_{1N}})$,
${\boldsymbol{\theta}}_{1}={[e^{j\theta_{11}}},\cdots,{e^{j\theta_{1N}}]^{T}}$,
$\theta_{1i}\in(0,{2\pi}]$ is the phase shift of the $i$th element.
$\textbf{n}_{r}\in\mathbb{C}^{M\times 1}$ is the additive white Gaussian noise
(AWGN) with distribution $\textbf{n}_{r}\sim\mathcal{CN}(0,\sigma_{r}^{2}{\bf
I}_{N_{r}})$. Then the received signal at RS after receive beamforming can be
expressed as

$y_{r}=\sqrt{P_{s}}\textbf{u}_{r}^{H}(\textbf{h}_{sr}+\textbf{H}_{ir}{\boldsymbol{\Theta}}_{1}\textbf{h}_{si})s+\textbf{u}_{r}^{H}\textbf{n}_{r},$
(2)

where $\textbf{u}_{r}^{H}\in\mathbb{C}^{1\times M}$ is the receive beamforming
vector, $\|\textbf{u}_{r}^{H}\|^{2}=1$. The achievable rate at RS is given by

$R_{r}=\log_{2}\left(1+\frac{P_{R}}{\sigma_{r}^{2}}\right),$ (3)

where

$P_{R}=P_{s}\text{tr}\left\\{{{{\textbf{u}_{r}^{H}}(\textbf{h}_{sr}+\textbf{H}_{ir}{\boldsymbol{\Theta}}_{1}\textbf{h}_{si})(\textbf{h}_{sr}+\textbf{H}_{ir}{\boldsymbol{\Theta}}_{1}\textbf{h}_{si})^{H}\textbf{u}_{r}}}\right\\}.$
(4)

In the second time slot, it is assumed that the original signal $s$ can be
correctly decoded by RS. The transmit signal from RS is
$\textbf{x}_{r}=\textbf{u}_{t}s$, where $\textbf{u}_{t}\in\mathbb{C}^{M\times
1}$ is the transmit beamforming vector, $\|\textbf{u}_{t}\|^{2}=1$. The
received signal at D is as follows

$y_{d}=\sqrt{P_{r}}(\textbf{h}_{rd}^{H}+\textbf{h}_{id}^{H}{\boldsymbol{\Theta}}_{2}\textbf{H}_{ri}^{H})\textbf{u}_{t}s+n_{d},$
(5)

where ${P_{r}}$ is the transmit power from RS,
$\textbf{h}_{rd}^{H}\in{\mathbb{C}^{1\times M}}$,
$\textbf{h}_{id}^{H}\in{\mathbb{C}^{1\times N}}$ and
$\textbf{H}_{ri}^{H}\in{\mathbb{C}^{N\times M}}$ are the channels from RS to
D, from IRS to D, and from RS to IRS, respectively. The diagonal reflection-
coefficient matrix of IRS is represented as
${\boldsymbol{\Theta}}_{2}=\text{diag}\left(e^{j{\theta_{21}}},\cdots,e^{j{\theta_{2N}}}\right)$,
${\boldsymbol{\theta}}_{2}={[e^{j\theta_{21}}},\cdots,{e^{j\theta_{2N}}]^{T}}$,
${\theta_{2i}}\in\left(0\right.,\left.{2\pi}\right]$ is the phase shift of the
$i$th element. ${n_{d}}$ is the AWGN with distribution
$n_{d}\sim\mathcal{CN}\left(0,{\sigma_{d}^{2}}\right)$. The achievable rate at
D can be expressed as

$R_{d}=\log_{2}\left(1+\frac{P_{D}}{\sigma_{d}^{2}}\right),$ (6)

where

$P_{D}=P_{r}\text{tr}\left\\{\textbf{u}_{t}^{H}(\textbf{h}_{rd}^{H}+\textbf{h}_{id}^{H}{\boldsymbol{\Theta}}_{2}\textbf{H}_{ri}^{H})^{H}(\textbf{h}_{rd}^{H}+\textbf{h}_{id}^{H}{\boldsymbol{\Theta}}_{2}\textbf{H}_{ri}^{H})\textbf{u}_{t}\right\\}.$
(7)

The achievable rate of the proposed system is defined as follows

$R_{s}=\frac{1}{2}\text{min}\\{R_{r},R_{d}\\}.$ (8)

##  III Proposed Three High-Performance Beamforming Schemes

In this section, according to (3), maximizing rate is equivalent to Max-RP due
to the fact that the log function is a monotone increasing function of RP. In
the first time slot, three Max-RP methods: AIS-based Max-RP, NSP-based Max-RP
plus MRC, and IRSES-based Max-RP plus MRC, are proposed to optimize the phase
shift vector ${\boldsymbol{\theta}}_{1}$ of IRS and the receive beamforming
vector $\textbf{u}_{r}$ of RS. In the second time slot, RS, IRS and
destination form a typical three-point IRS-aided network, where an alternating
iteration in [28] are adopted to design the transmit beamforming vector at RS
and adjust the phases of IRS.

###  III-A Proposed AIS-based Max-RP Method

For the first time slot, the corresponding optimization problem can be casted
as

$\displaystyle\max\limits_{{\boldsymbol{\Theta}}_{1},\textbf{u}_{r}^{H}}~{}~{}~{}~{}P_{s}\text{tr}\left\\{{{{\textbf{u}_{r}^{H}}(\textbf{h}_{sr}+\textbf{H}_{ir}{\boldsymbol{\Theta}}_{1}\textbf{h}_{si})(\textbf{h}_{sr}+\textbf{H}_{ir}{\boldsymbol{\Theta}}_{1}\textbf{h}_{si})^{H}\textbf{u}_{r}}}\right\\}$
$\displaystyle~{}~{}\text{s.t.}~{}~{}~{}~{}~{}~{}\|\textbf{u}_{r}^{H}\|^{2}=1,~{}|\boldsymbol{\theta}_{1}(i)|=1,~{}\forall
i=1,\cdots,N.$ (9)

Let us define $\textbf{H}_{si}=\text{diag}\\{\textbf{h}_{si}\\}$, we have
${\boldsymbol{\Theta}}_{1}\textbf{h}_{si}=\textbf{H}_{si}{\boldsymbol{\theta}}_{1}$.
Firstly, by fixing $\textbf{u}_{r}^{H}$, (III-A) is reduced to

$\displaystyle\max\limits_{{\boldsymbol{\theta}}_{1}}~{}~{}~{}P_{s}\text{tr}({\boldsymbol{\theta}}_{1}^{H}\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si}{\boldsymbol{\theta}}_{1})+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}P_{s}\text{tr}({\boldsymbol{\theta}}_{1}^{H}\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{h}_{sr})+P_{s}\text{tr}({\boldsymbol{\theta}}_{1}\textbf{h}_{sr}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si})$
$\displaystyle~{}\text{s.t.}~{}~{}~{}~{}~{}{\boldsymbol{\theta}}_{1}^{H}{\boldsymbol{\theta}}_{1}=N.$
(10)

The Lagrangian function associated with the above optimization (III-A) is
defined as

$\displaystyle
L({\boldsymbol{\theta}}_{1},\lambda)=P_{s}\text{tr}({\boldsymbol{\theta}}_{1}^{H}\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si}{\boldsymbol{\theta}}_{1})+$
$\displaystyle~{}~{}~{}~{}P_{s}\text{tr}({\boldsymbol{\theta}}_{1}^{H}\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{h}_{sr})+P_{s}\text{tr}({\boldsymbol{\theta}}_{1}\textbf{h}_{sr}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si})+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\lambda({{\boldsymbol{\theta}}_{1}^{H}{\boldsymbol{\theta}}_{1}}-N),$
(11)

where $\lambda$ is the Lagrange multiplier, if (${\boldsymbol{\theta}}_{1}$,
$\lambda$) is the optimal solution to the above equation, the partial
derivative of the Lagrangian function with respect to
${\boldsymbol{\theta}}_{1}$ should be set to be zero as follows

$\displaystyle\frac{\partial
L({\boldsymbol{\theta}}_{1},\lambda)}{\partial{\boldsymbol{\theta}}_{1}}=P_{s}\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si}{\boldsymbol{\theta}}_{1}+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}P_{s}\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{h}_{sr}+\lambda{\boldsymbol{\theta}}_{1}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=0,$ (12)

which gives the solution

$\displaystyle\theta_{1i}=-\text{arg}\\{(\lambda{P_{s}}^{-1}\textbf{I}_{N}+\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si})^{\dagger}\cdot$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{h}_{sr})\\}_{i}.$
(13)

Inserting (III-A) back into the restriction of (III-A), we have an equation
associated with Lagrange multiplier as follows

$\det\left(\lambda{P_{s}}^{-1}\textbf{I}_{N}+\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si}\right)=b,$
(14)

where

$b=\left(\det(N^{-1}\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{h}_{sr}\textbf{h}_{sr}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si})\right)^{\frac{1}{2}}.$
(15)

Further, in accordance with the rank inequality:
$\text{rank}~{}\textbf{A}\leq\min\\{m,n\\}$, where
$\textbf{A}\in\mathbb{C}^{m\times n}$, we can get

$\text{rank}(\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{h}_{sr}\textbf{h}_{sr}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si})\leq\text{rank}(\textbf{u}_{r}^{H})=1.$
(16)

Therefore, $b=0$. Then we have the eigen decomposition as follows

$\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si}=\textbf{U}\boldsymbol{\Sigma}\textbf{U}^{H},$
(17)

where U is a unitary matrix composed of eigenvectors and
$\textbf{U}\textbf{U}^{H}$=$\textbf{U}^{H}\textbf{U}$=$\textbf{I}_{N}$.
$\boldsymbol{\Sigma}$ is a diagonal matrix composed of eigenvalues.
Substituting the above equation in (14) yields the following simplified
equation

$\displaystyle\det\left\\{\textbf{U}\left(\lambda{P_{s}}^{-1}\textbf{I}_{N}+\boldsymbol{\Sigma}\right)\textbf{U}^{H}\right\\}=\det\left\\{\textbf{U}\textbf{U}^{H}\left(\lambda{P_{s}}^{-1}\textbf{I}_{N}+\boldsymbol{\Sigma}\right)\right\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\det\left(\lambda{P_{s}}^{-1}\textbf{I}_{N}+\boldsymbol{\Sigma}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=0.$
(18)

It can be observed that
$\text{rank}(\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si})=\text{rank}(\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r})=\text{rank}(\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si})=1$,
so there is a nonzero eigenvalue and $N-1$ zero eigenvalues in
$\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si}$.
Then we construct the following equation

$\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si}(\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r})=\|\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si}\|^{2}\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r},$
(19)

where $\|\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si}\|^{2}$ is the only
nonzero eigenvalue (ie. the largest eigenvalue), so we have

$\boldsymbol{\Sigma}=\left[\begin{array}[]{cccc}\|\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si}\|^{2}&0&\cdots&0\\\
0&0&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&0\end{array}\right]_{N\times N}.$ (20)

Substituting (20) into (III-A) yields

$\left|\begin{array}[]{cccc}\lambda+\|\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si}\|^{2}{P_{s}}&0&\cdots&0\\\
0&\lambda&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&\lambda\end{array}\right|_{N\times N}=0,$ (21)

which is expanded as follows

$\lambda^{N}+\|\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si}\|^{2}{P_{s}}\lambda^{N-1}=0,$
(22)

which forms the set of candidate solutions

$\lambda\in\\{0,-\|\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si}\|^{2}{P_{s}}\\}.$
(23)

Plugging (23) into (III-A), we find that
$\lambda=-\|\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si}\|^{2}{P_{s}}$ is
not suitable for (III-A). Inserting $\lambda=0$ back into (III-A) yields the
following solution

$\theta_{1i}=-\text{arg}\\{(\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{H}_{ir}\textbf{H}_{si})^{\dagger}(\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{r}\textbf{u}_{r}^{H}\textbf{h}_{sr})\\}_{i}.$
(24)

And then by fixing $\boldsymbol{\Theta}_{1}$, the optimization problem in
(III-A) can be converted into

$\displaystyle\max\limits_{\textbf{u}_{r}^{H}}~{}~{}~{}P_{s}\text{tr}({{{\textbf{u}_{r}^{H}}\textbf{B}\textbf{u}_{r}}})$
$\displaystyle\ \text{s.t.}~{}~{}~{}~{}~{}\|\textbf{u}_{r}^{H}\|^{2}=1,$ (25)

where

$\textbf{B}=(\textbf{h}_{sr}+\textbf{H}_{ir}{\boldsymbol{\Theta}}_{1}\textbf{h}_{si})(\textbf{h}_{sr}+\textbf{H}_{ir}{\boldsymbol{\Theta}}_{1}\textbf{h}_{si})^{H},$
(26)

is a Hermitian symmetric matrix. Similar to (III-A), we have

$L(\textbf{u}_{r},\mu)=P_{s}\text{tr}(\textbf{u}_{r}^{H}\textbf{B}\textbf{u}_{r})+\mu(\|\textbf{u}_{r}^{H}\|^{2}-1),$
(27)

where $\mu$ denotes the Lagrange multiplier. Setting the partial derivative of
the Lagrange function $L(\textbf{u}_{r},\mu)$ with respect to $\textbf{u}_{r}$
equals 0, we have

$P_{s}\textbf{B}\textbf{u}_{r}+\mu\textbf{u}_{r}=0.$ (28)

Similar to (16), rank(B)$=1$, so there is a nonzero eigenvalue in B. It
indicates that $\textbf{u}_{r}$ is the eigenvector corresponding to the
nonzero eigenvalue of the matrix B. Similar to (19), we obtain the solution

$\textbf{u}_{r}=\frac{\textbf{h}_{sr}+\textbf{H}_{ir}{\boldsymbol{\Theta}}_{1}\textbf{h}_{si}}{\|\textbf{h}_{sr}+\textbf{H}_{ir}{\boldsymbol{\Theta}}_{1}\textbf{h}_{si}\|}.$
(29)

It can be observed that when one of the optimization variables
${\boldsymbol{\Theta}}_{1}$ and $\textbf{u}_{r}^{H}$ is fixed, the objective
function in (9) is convex with respect to the other variable, in each
iteration the objective function value monotonically increases with a finite
upper-bound. Therefore, it can be guaranteed that the alternately iterative
(AI) algorithm is convergent. AI between $\theta_{1i}$ and $\textbf{u}_{r}$
are performed until the condition of convergence is reached. When
$\theta_{1i}$ and $\textbf{u}_{r}$ are optimal, the $R_{r}$ in (3) is
maximized. The proposed alternate iteration algorithm for obtaining an optimal
solution $R_{r}$ is summarized in Algorithm 1.

$\bf{Algorithm~{}1}$ Proposed alternate iteration algorithm
---
1\. Initialize the receive beamforming $\textbf{u}_{r}^{1}$, set the
convergence error $\varepsilon$ and the iteration number $t$ = 1.
2\. repeat
3\. Solve (III-A) for a given $\textbf{u}_{r}^{t}$, denote the optimal
solution as ${\boldsymbol{\theta}}_{1}^{t}$ and $\theta_{1i}^{t}$.
4\. Solve (III-A) for a given ${\boldsymbol{\theta}}_{1}^{t}$, denote the
optimal solution as $\textbf{u}_{r}^{t+1}$.
5\. Update $t$ = $t$+1.
6\. until
$\left|{R_{r}^{t+1}-R_{r}^{t}}\right|\leq\varepsilon$.

According to (8), we make a comparison between $R_{r}$ and $R_{d}$, and
multiply the minimum value by 1/2 to obtain the system rate $R_{s}$. The total
complexity of proposed AIS-based Max-RP method is obtained as follows

$\displaystyle\mathcal{O}\\{L_{2}({N^{4}}+8M{N^{3}}+5{N^{3}}+24M{N^{2}}-2{N^{2}})+(3L_{1}+$
$\displaystyle~{}~{}~{}~{}~{}18L_{2})MN+(5L_{1}+2L_{2})M+(4L_{1}+3L_{2})N\\},$
(30)

where $L_{1}$ and $L_{2}$ are the AI numbers in the first time slot and the
second time slot, respectively. Obviously, the highest order of computational
complexity is $N^{4}$ float-point operations (FLOPs), which is high. So a low-
complexity NSP-based Max-RP plus MRC method is proposed to reduce the
computational complexity in the next subsection.

###  III-B Proposed NSP-based Max-RP plus MRC Method

Considering the two signals from IRS and source node overlap together and
interfere with each other, in order to explore the two-way diversity gain, two
individual receiving beamforming vectors at RS are used to separate them
respectively and MRC is used to combine them to exploit the multiple
diversities. After beamforming, the received signals at RS are rewritten by

$y_{rs}=\sqrt{P_{s}}\textbf{u}_{rs}^{H}\textbf{h}_{sr}s+\sqrt{P_{s}}\textbf{u}_{rs}^{H}\textbf{H}_{ir}{\boldsymbol{\Theta}}_{1}\textbf{h}_{si}s+\textbf{u}_{rs}^{H}\textbf{n}_{r},$
(31)
$y_{ri}=\sqrt{P_{s}}\textbf{u}_{ri}^{H}\textbf{h}_{sr}s+\sqrt{P_{s}}\textbf{u}_{ri}^{H}\textbf{H}_{ir}{\boldsymbol{\Theta}}_{1}\textbf{h}_{si}s+\textbf{u}_{ri}^{H}\textbf{n}_{r},$
(32)

where $\textbf{u}_{rs}^{H}\in\mathbb{C}^{1\times M}$ and
$\textbf{u}_{ri}^{H}\in\mathbb{C}^{1\times M}$ are defined as the beamforming
vectors for S and IRS, respectively, which extract the independent signals
from S and IRS, respectively. $\|\textbf{u}_{rs}^{H}\|^{2}=1$ and
$\|\textbf{u}_{ri}^{H}\|^{2}=1$. We assume that $\textbf{u}_{rs}^{H}$ and
$\textbf{u}_{ri}^{H}$ are in the null-space of channel $\textbf{H}_{ir}$ and
$\textbf{h}_{sr}$, respectively. It is obvious that

$\textbf{u}_{rs}^{H}\textbf{H}_{ir}=\textbf{0}^{T},\
\textbf{u}_{ri}^{H}\textbf{h}_{sr}=0,$ (33)

which mean that the signals from S and IRS do not interfere with each other at
RS. Correspondingly, (31) and (32) are reduced to

$y_{rs}=\sqrt{P_{s}}\textbf{u}_{rs}^{H}\textbf{h}_{sr}s+\textbf{u}_{rs}^{H}\textbf{n}_{r},$
(34)
$y_{ri}=\sqrt{P_{s}}\textbf{u}_{ri}^{H}\textbf{H}_{ir}{\boldsymbol{\Theta}}_{1}\textbf{h}_{si}s+\textbf{u}_{ri}^{H}\textbf{n}_{r}.$
(35)

Maximizing $R_{r}$ is equal to maximizing the power of $y_{rs}$ and $y_{ri}$,
respectively. The corresponding optimization problem of the power of $y_{rs}$
is given by

$\displaystyle\max\limits_{\textbf{u}_{rs}^{H}}~{}~{}~{}~{}P_{s}\text{tr}(\textbf{u}_{rs}^{H}\textbf{h}_{sr}\textbf{h}_{sr}^{H}\textbf{u}_{rs})$
$\displaystyle~{}\text{s.t.}~{}~{}~{}~{}~{}~{}\|\textbf{u}_{rs}^{H}\|^{2}=1,~{}\textbf{u}_{rs}^{H}\textbf{H}_{ir}=\textbf{0}^{T}.$
(36)

According to the second constraint of the above optimization problem, let us
define $\textbf{u}_{rs}$ as follows

$\textbf{u}_{rs}=(\textbf{I}_{M}-\textbf{H}_{ir}(\textbf{H}_{ir}^{H}\textbf{H}_{ir})^{\dagger}\textbf{H}_{ir}^{H})\textbf{v}_{rs}=\textbf{P}\textbf{v}_{rs},$
(37)

where
$\textbf{P}=\textbf{I}_{M}-\textbf{H}_{ir}(\textbf{H}_{ir}^{H}\textbf{H}_{ir})^{\dagger}\textbf{H}_{ir}^{H}$,
$\textbf{v}_{rs}$ is a new variable and $\|\textbf{v}_{rs}\|=1$. Inserting the
above equation back into (III-B), we obtain the simplified optimization
problem as follows

$\displaystyle\max\limits_{\textbf{v}_{rs}^{H}}~{}~{}~{}~{}P_{s}\textbf{v}_{rs}^{H}\textbf{P}^{H}\textbf{h}_{sr}\textbf{h}_{sr}^{H}\textbf{P}\textbf{v}_{rs}$
$\displaystyle~{}\text{s.t.}~{}~{}~{}~{}~{}~{}\|\textbf{v}_{rs}\|^{2}=1.$ (38)

Similar to (III-A), the solution $\textbf{v}_{rs}$ is given by

$\textbf{v}_{rs}=\frac{\textbf{P}^{H}\textbf{h}_{sr}}{\|\textbf{P}^{H}\textbf{h}_{sr}\|},$
(39)

which yields

$\textbf{u}_{rs}=\frac{\textbf{P}\textbf{v}_{rs}}{\|\textbf{P}\textbf{v}_{rs}\|}=\frac{(\textbf{I}_{M}-\textbf{H}_{ir}(\textbf{H}_{ir}^{H}\textbf{H}_{ir})^{\dagger}\textbf{H}_{ir}^{H})^{2}\textbf{h}_{sr}}{\|(\textbf{I}_{M}-\textbf{H}_{ir}(\textbf{H}_{ir}^{H}\textbf{H}_{ir})^{\dagger}\textbf{H}_{ir}^{H})^{2}\textbf{h}_{sr}\|}.$
(40)

In the same manner, the optimization problem of the power of $y_{ri}$ is also
written by

$\displaystyle\max\limits_{\textbf{u}_{ri}^{H},{\boldsymbol{\theta}}_{1}}~{}~{}~{}~{}P_{s}\text{tr}(\textbf{u}_{ri}^{H}\textbf{H}_{ir}\textbf{H}_{si}{\boldsymbol{\theta}}_{1}{\boldsymbol{\theta}}_{1}^{H}\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{ri})$
$\displaystyle~{}\text{s.t.}~{}~{}~{}~{}~{}~{}\|\textbf{u}_{ri}^{H}\|^{2}=1,~{}\textbf{u}_{ri}^{H}\textbf{h}_{sr}=0,~{}{\boldsymbol{\theta}}_{1}^{H}{\boldsymbol{\theta}}_{1}=N.$
(41)

For a given $\textbf{u}_{ri}$, the solution $\theta_{1i}$ can be expressed as

$\theta_{1i}=-\text{arg}\\{\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{ri}\\}_{i}.$
(42)

For a given ${\boldsymbol{\theta}}_{1}$, the solution $\textbf{u}_{ri}$ can be
represented as

$\textbf{u}_{ri}=\frac{(\textbf{I}_{M}-\textbf{h}_{sr}(\textbf{h}_{sr}^{H}\textbf{h}_{sr})^{\dagger}\textbf{h}_{sr}^{H})^{2}\textbf{H}_{ir}\textbf{H}_{si}{\boldsymbol{\theta}}_{1}}{\|(\textbf{I}_{M}-\textbf{h}_{sr}(\textbf{h}_{sr}^{H}\textbf{h}_{sr})^{\dagger}\textbf{h}_{sr}^{H})^{2}\textbf{H}_{ir}\textbf{H}_{si}{\boldsymbol{\theta}}_{1}\|}.$
(43)

Similar to Algorithm 1, $\textbf{u}_{ri}$ and ${\boldsymbol{\theta}}_{1}$ are
performed via AI procedure to obtain the maximum power of $y_{ri}$. Then the
joint received signal at RS by applying MRC can be expressed as

$\displaystyle
y_{r}=\frac{(\textbf{u}_{rs}^{H}\textbf{h}_{sr})^{H}y_{rs}+(\textbf{u}_{ri}^{H}\textbf{H}_{ir}\textbf{H}_{si}\boldsymbol{\theta_{1}})^{H}y_{ri}}{\|\textbf{u}_{rs}^{H}\textbf{h}_{sr}+\textbf{u}_{ri}^{H}\textbf{H}_{ir}\textbf{H}_{si}\boldsymbol{\theta_{1}}\|}$
$\displaystyle=\sqrt{P_{s}}\left(\frac{\textbf{h}_{sr}^{H}\textbf{u}_{rs}\textbf{u}_{rs}^{H}\textbf{h}_{sr}+\boldsymbol{\theta}_{1}^{H}\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{ri}\textbf{u}_{ri}^{H}\textbf{H}_{ir}\textbf{H}_{si}\boldsymbol{\theta_{1}}}{\|\textbf{u}_{rs}^{H}\textbf{h}_{sr}+\textbf{u}_{ri}^{H}\textbf{H}_{ir}\textbf{H}_{si}\boldsymbol{\theta_{1}}\|}\right)s+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left(\frac{\textbf{h}_{sr}^{H}\textbf{u}_{rs}\textbf{u}_{rs}^{H}+\boldsymbol{\theta}_{1}^{H}\textbf{H}_{si}^{H}\textbf{H}_{ir}^{H}\textbf{u}_{ri}\textbf{u}_{ri}^{H}}{\|\textbf{u}_{rs}^{H}\textbf{h}_{sr}+\textbf{u}_{ri}^{H}\textbf{H}_{ir}\textbf{H}_{si}\boldsymbol{\theta_{1}}\|}\right)\textbf{n}_{r}.$
(44)

The signals from S and IRS received by RS are independent, thus the achievable
rate at RS can be calculated as

$R_{r}=\log_{2}\left(1+\frac{(\|\textbf{u}_{rs}^{H}\textbf{h}_{sr}\|^{4}+\|\textbf{u}_{ri}^{H}\textbf{H}_{ir}\textbf{H}_{si}\boldsymbol{\theta_{1}}\|^{4})P_{s}}{\|\textbf{u}_{rs}^{H}\textbf{h}_{sr}+\textbf{u}_{ri}^{H}\textbf{H}_{ir}\textbf{H}_{si}\boldsymbol{\theta_{1}}\|^{2}\sigma_{r}^{2}}\right).$
(45)

Compare $R_{r}$ and $R_{d}$ to obtain the system rate $R_{s}$. The complexity
of the proposed NSP-based Max-RP plus MRC method is written by

$\displaystyle\mathcal{O}\\{{N^{3}}+2(1+L_{3}){M^{3}}+2(1+L_{3}){M^{2}}N+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(4+3L_{3})M{N^{2}}+(4+3L_{3}){M^{2}}-L_{3}{N^{2}}-$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1-5L_{3}-18L_{4})MN-(1-4L_{3}-2L_{4})M+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1+2L_{3}+3L_{4})N\\},$
(46)

where $L_{3}$ and $L_{4}$ respectively denote the AI numbers in the first time
slot and the second time slot. The highest order of complexity is $N^{3}$
FLOPs, which is lower than AIS-based Max-RP Method and still very high. To
further reduce the computational complexity, an IRSES-based Max-RP plus MRC
method is proposed in the following subsection.

###  III-C Proposed IRSES-based Max-RP plus MRC Method

As shown in Fig. 2, according to the number $M$ of RS antennas, the $N$
elements in IRS are randomly divided into $M$ suarrays evenly, each of which
is mapped into one antenna at RS with $\frac{N}{M}$ = $K$ elements. Fig. 3 is
the block diagram for IRSES-based Max-RP plus MRC method at RS, the details
are described as below.

Figure 2: System model for IRSES-based Max-RP plus MRC method in the first
time slot.
Figure 3: Block diagram for IRSES-based Max-RP plus MRC method.

A subarray denoted as ${{\cal E}_{I,m}}$ is randomly chosen to reflect the
signal from S to antenna $m$ of RS, ${{\cal E}_{I,m}}$ =
$\left\\{{{1_{m}},{2_{m}},\cdots,{K_{m}}}\right\\}$. Let ${\cal E}_{I}$=
${\cal E}_{I,1}\cup{\cal E}_{I,2}\cdots\cup{\cal E}_{I,M}$ be the subarray
set. ${\cal E}_{I,1},{\cal E}_{I,2},\cdots,{\cal E}_{I,M}$ is a partition of
set ${\cal E}_{I}$, and ${\cal E}_{I,m}\cap{\cal E}_{I,n}$= $\Phi$, where
$m,n$ $\in\left\\{{1,2,\cdots,M}\right\\}$. The remaining subsets are denoted
as ${\cal E}_{I,-m}$, and ${\cal E}_{I}$= ${\cal E}_{I,m}$ $\cup$ ${\cal
E}_{I,-m}$. The received signal of antenna $m$ at RS can be written by

$\displaystyle
y_{r,m}=\sqrt{P_{s}}(h_{sr,m}+\textbf{h}_{ir,m}\boldsymbol{\Theta}_{1,m}\textbf{h}_{si,m})s+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sqrt{P_{s}}\textbf{h}_{ir,-m}\boldsymbol{\Theta}_{1,-m}\textbf{h}_{si,-m}s+n_{r,m},$
(47)

where $h_{sr,m}\in{\mathbb{C}^{1\times 1}}$ is the $m$th element of
$\textbf{h}_{sr}$, $\textbf{h}_{ir,m}\in{\mathbb{C}^{1\times K}}$ denotes the
channel from the selected subset ${\cal E}_{I,m}$ to antenna $m$ at RS, and
includes $K$ elements which are determined by ${\cal E}_{I,m}$ in the $m$th
row of $\textbf{H}_{ir}$. $\textbf{h}_{si,m}\in{\mathbb{C}^{K\times 1}}$
represents the channel from S to ${\cal E}_{I,m}$ and also consists of $K$
elements decided by ${\cal E}_{I,m}$ in $K$ rows of $\textbf{h}_{si}$. The
diagonal reflection-coefficient matrix of elements in ${\cal E}_{I,m}$ is
denoted as
$\boldsymbol{\Theta}_{1,m}=\text{diag}({e^{j\theta_{1,m}(1)}},\cdots,{e^{j\theta_{1,m}(K)}})$.
$\textbf{h}_{ir,-m}\in{\mathbb{C}^{1\times(N-K)}}$ is the channel from the
unselected subset ${\cal E}_{I,-m}$ to antenna $m$ at RS,
$\textbf{h}_{si,-m}\in{\mathbb{C}^{(N-K)\times 1}}$ is the channel from S to
${\cal E}_{I,-m}$. $\boldsymbol{\Theta}_{1,-m}$ is the diagonal reflection-
coefficient matrix of elements in ${\cal E}_{I,-m}$, which can be expressed as
$\boldsymbol{\Theta}_{1,-m}=\text{diag}({e^{j\theta_{1,-m}(1)}},\cdots,{e^{j\theta_{1,-m}(N-K)}})$.
$n_{r,m}$ is the AWGN with the distribution of
$n_{r,m}\sim\mathcal{CN}\left(0,{\sigma_{r,m}^{2}}\right)$. While according to
law of large numbers, the signals reflected by the unselected subset ${\cal
E}_{I,-m}$ overlap to zero at antenna $m$. (III-C) is simplified and expanded
as follows

$\displaystyle y_{r,m}$
$\displaystyle=\sqrt{P_{s}}(h_{sr,m}+\textbf{h}_{ir,m}\boldsymbol{\Theta}_{1,m}\textbf{h}_{si,m})s+n_{r,m}$
$\displaystyle=\sqrt{P_{s}}\left(h_{sr,m}+\sum\limits_{i=1}^{K}\textbf{h}_{ir,m}(i)\boldsymbol{\Theta}_{1,m}(i,i)\textbf{h}_{si,m}(i)\right)s+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}n_{r,m},$ (48)

where $\textbf{h}_{ir,m}(i)$ and $\textbf{h}_{si,m}(i)$ are the $i$th element
of $\textbf{h}_{ir,m}$ and $\textbf{h}_{si,m}$, respectively.
$\boldsymbol{\Theta}_{1,m}(i,i)$ is the $i$th element in diagonal of
$\boldsymbol{\Theta}_{1,m}$. The phases of $K$ elements in ${\cal E}_{I,m}$
are adjusted to make the phases of all reflected signals and the direct signal
from S aligned at antenna $m$ of RS. So as to maximize the power of the signal
received by antenna $m$ for obtaining better rate performance. Thus, the
corresponding phase shift of the $i$th element is calculated as

$\theta_{1,m}(i)=\text{arg}(h_{sr,m})-\text{arg}(\textbf{h}_{ir,m}(i))-\text{arg}(\textbf{h}_{si,m}(i)).$
(49)

Finally, MRC is adopted to combine all received signals at RS, the received
signal can be expressed as follows

$y_{r}=[u_{r,1},u_{r,2},\cdots,u_{r,M}][y_{r,1},y_{r,2},\cdots,y_{r,M}]^{T},$
(50)

where

$u_{r,m}=\frac{(h_{sr,m}+\textbf{h}_{ir,m}\boldsymbol{\Theta}_{1,m}\textbf{h}_{si,m})^{H}}{\|h_{sr,m}+\textbf{h}_{ir,m}\boldsymbol{\Theta}_{1,m}\textbf{h}_{si,m}\|}.$
(51)

The achievable rate at RS can be represented as

$R_{r}=\log_{2}\left(1+\frac{\sum_{m=1}^{M}\|h_{sr,m}+\textbf{h}_{ir,m}\boldsymbol{\Theta}_{1,m}\textbf{h}_{si,m}\|^{4}P_{s}}{\sum_{m=1}^{M}\|h_{sr,m}+\textbf{h}_{ir,m}\boldsymbol{\Theta}_{1,m}\textbf{h}_{si,m}\|^{2}\sigma_{r,m}^{2}}\right).$
(52)

Further, taking the minimum of $R_{r}$ and $R_{d}$ to obtain achievable rate
$R_{s}$. The complexity of the proposed IRSES-based Max-RP plus MRC method is
given by

$\mathcal{O}\\{15MK+8M+10K+L_{5}(18MN+2M+3N)\\},$ (53)

where $L_{5}$ is the AI number in the second time slot. It can be seen that
the highest order of complexity is $MN$ FLOPs, which is much lower than the
above two methods. Especially in massive IRS scenario, the gap among the
complexity of the proposed three methods is more obvious.

##  IV Simulation And Numerical Results

In this section, numerical simulations are performed to evaluate and compare
the rate performance between an IRS-aided multi-antenna DF relay network and
that with single-antenna RS in [26]. Additionally, it is assumed that IRS and
RS are deployed with the same abscissas, the positions of S, RS, IRS, and D
are given as (0,0), (50m,0), (50m,10m) and (100m,0), respectively. The
amplitude attenuation of the received signal is $(d)^{-(\frac{\alpha}{2})}$
caused by path loss, where $d$ is the distance between transmitter and
receiver, and $\alpha$ is the path-loss exponent. The antenna gains at S, RS
and D are 5 dBi, 5dBi, and 2 dBi, respectively. System parameters are set as
follows: $P_{s}$ = $P_{r}$ = 10dBW, $\alpha$ = 2.4, and $\sigma_{d}^{2}$ =
$\sigma_{r}^{2}=\sigma_{w}^{2}$. SNR is defined as
$(P_{s}+P_{r})/\sigma_{w}^{2}$.

Figure 4: Convergence of proposed methods with $M$ = 50 and $N$ = 50.
Figure 5: Computational complexity versus $N$ with $M$ = 50.
Figure 6: Achievable rate versus SNR with $M$ = 16 and $N$ = 160.
Figure 7: Achievable rate versus SNR with $M$ = 50 and $N$ = 200.
Figure 8: Achievable rate versus $N$ with $M$ = 16 and SNR = 30dB.
Figure 9: Achievable rate versus $M$ with $N$ = 200 and SNR = 30dB.

Figure 10: (a) How to move IRS and RS. (b) Achievable rate versus distance
with $M$ = 50, $N$ = 200 and SNR = 30dB.

Fig. 4 shows the achievable rate versus the number of iterations with $M$ = 50
and $N$ = 50 in the first time slot. It is obvious that the proposed AIS-based
Max-RP and NSP-based Max-RP plus MRC methods require about only three
iterations to achieve the rate ceil.

Fig. 5 plots the curves of computational complexity versus $N$ with $M$ = 50.
It demonstrates that the computational complexities of the proposed three
methods, AIS-based Max-RP, NSP-based Max-RP plus MRC, and IRSES-based Max-RP
plus MRC, increase as $N$ increases. Clearly, the first and third methods have
the highest and lowest computational complexities, respectively. The second
one is in between them.

Fig. 6 shows the curves of achievable rate versus SNR with $M$ = 16 and $N$ =
160. It is clear that the rate performance of the proposed three methods with
fixed IRS phases (i.e., ${\boldsymbol{\Theta}}$ = I) to eliminate the impact
of IRS elements on the performance is more than 5.24 bits/s/Hz, which is 25.6%
higher than that of the existing system with an IRS and a single-antenna relay
in [26]. Compared to the above methods with ${\boldsymbol{\Theta}}$ = I, the
performance of the proposed three methods with joint RS active beamforming and
IRS passive beamforming can approximately harvest up to 25.1% rate gain.
Therefore, it is verified that the rate gain results from the increased number
of antennas at RS and from the joint RS active beamforming and IRS passive
beamforming.

Fig. 7 illustrates the curves of achievable rate versus SNR with $M$ = 50 and
$N$ = 200. It can be seen that the proposed methods make a significant
performance improvement over that with single-antenna RS in [26]. For example,
when SNR equals 30dB, the proposed worst method, IRSES-based Max-RP plus MRC,
can harvest up to 78.6% rate gain over that method in [26]. The best method
AIS-based Max-RP approximately has a 80.8% rate gain over that in [26]. This
shows that as SNR increases, significant rate gains are achieved for the
proposed network with IRS plus multi-antenna RS.

Fig. 8 demonstrates the achievable rate versus the number $N$ of reflecting
elements at IRS. It is observed that the proposed three methods still
outperform the method with single-antenna RS in [26]. For small-scale and
medium-scale IRS, the proposed three methods have the following increasing
order on rate as follows: NSP-based Max-RP plus MRC, AIS-based Max-RP and
IRSES-based Max-RP plus MRC. As the number of elements at IRS goes to a large-
scale, their order are becomes as follows: IRSES-based Max-RP plus MRC, NSP-
based Max-RP plus MRC and AIS-based Max-RP.

Fig. 9 shows the curves of achievable rate versus $M$ with $N$ = 200 and SNR =
30dB. It can be seen that as the number $M$ of antennas at RS increases, the
rate performance increases. The proposed three methods: AIS-based Max-RP, NSP-
based Max-RP plus MRC and IRSES-based Max-RP plus MRC have harvested
significant rate performance gains over that in [26], IRS-only-aided network
and relay-only-aided network.

In order to observe the effect of positions of IRS and RS on rate performance,
Fig. 10 (a) shows how to move both IRS and RS, where IRS and RS move toward D
together along the direction parallel to the line segment SD with $d$ denoting
the horizontal distance from S to IRS and RS . Fig. 10 (b) plots the
corresponding curves of achievable rate versus $d$ with $M$ = 50, $N$ = 200
and SNR = 30dB. When $d$ = 30m, the proposed three methods can achieve their
largest rate peak. While $d$ $>$ 30m, the rate performance degrades gradually.
Regardless of $d$, the proposed three methods: AIS-based Max-RP, NSP-based
Max-RP plus MRC and IRSES-based Max-RP plus MRC still perform much better than
that in [26], IRS-only-aided network and relay-only-aided network in terms of
rate performance.

##  V Conclusions

In this paper, we have made an investigation on an IRS-aided DF relay network
with multi-antenna at RS. In order to improve the rate performance, three
high-performance schemes, namely AIS-based Max-RP, NSP-based Max-RP plus MRC
and IRSES-based Max-RP plus MRC, were proposed. The third method is very
attractive due to its low-complexity and excellent rate performance for small
and medium -scale IRSs. Simulation results show that the proposed three
methods can approximately harvest up to 85% rate gain over existing network
with single-antenna RS in almost all SNR regions. Thus, an IRS-aided multi-
antenna relay network will provide an enhanced network performance and
extended network coverage for the future mobile communications, WSNs, and
internet of things.

## References

  * [1] P. Huang, Y. Hao, T. Lv, J. Xing, J. Yang, and P. T. Mathiopoulos, “Secure beamforming design in relay-assisted internet of things,” _IEEE Internet Things J._ , vol. 6, no. 4, pp. 6453–6464, Aug. 2019.
  * [2] E. Yilmaz, R. Zakhour, D. Gesbert, and R. Knopp, “Multi-pair two-way relay channel with multiple antenna relay station,” in _2010 IEEE International Conference on Communications_ , Jul. 2010, pp. 1–5.
  * [3] G. Farhadi and N. Beaulieu, “On the ergodic capacity of multi-hop wireless relaying systems,” _IEEE Trans. Commun._ , vol. 8, no. 5, pp. 2286–2291, May 2009.
  * [4] Q. Gu, D. Wu, X. Su, J. Jin, Y. Yuan, and J. Wang, “Performance comparison between reconfigurable intelligent surface and relays: Theoretical methods and a perspective from operator,” 2021. [Online]. Available: https://arxiv.org/abs/2101.12091
  * [5] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network,” _IEEE Commun. Mag._ , vol. 58, no. 1, pp. 106–112, Jan. 2020.
  * [6] M. Hua and Q. Wu, “Joint dynamic passive beamforming and resource allocation for IRS-aided full-duplex WPCN,” 2021. [Online]. Available: https://arxiv.org/abs/2108.06660
  * [7] Y. Liang, J. Chen, R. Long, Z. He, X. Lin, C. Huang, S. Liu, X. Shen, and M. D. Renzo, “Reconfigurable intelligent surfaces for smart wireless environments: channel estimation, system design and applications in 6G networks,” _Sci. China Inf. Sci._ , vol. 64, no. 10, pp. 1–21, Jul. 2021.
  * [8] H.-M. Wang, J. Bai, and L. Dong, “Intelligent reflecting surfaces assisted secure transmission without eavesdropper’s CSI,” _IEEE Signal Process. Lett._ , vol. 27, pp. 1300–1304, Jul. 2020.
  * [9] E. Basar, M. Di Renzo, J. De Rosny, M. Debbah, M.-S. Alouini, and R. Zhang, “Wireless communications through reconfigurable intelligent surfaces,” _IEEE Access_ , vol. 7, pp. 116 753–116 773, Sep. 2019.
  * [10] F. Shu, Y. Teng, J. Li, M. Huang, W. Shi, J. Li, Y. Wu, and J. Wang, “Enhanced secrecy rate maximization for directional modulation networks via IRS,” _IEEE Trans. Commun._ , vol. 69, no. 12, pp. 8388–8401, Dec. 2021.
  * [11] L. Lai, J. Hu, Y. Chen, H. Zheng, and N. Yang, “Directional modulation-enabled secure transmission with intelligent reflecting surface,” in _2020 IEEE 3rd International Conference on Information Communication and Signal Processing (ICICSP)_ , Oct. 2020, pp. 450–453.
  * [12] F. Shu, X. Jiang, W. Cai, W. Shi, M. Huang, J. Wang, and X. You, “Beamforming and transmit power design for intelligent reconfigurable surface-aided secure spatial modulation,” 2021. [Online]. Available: https://arxiv.org/abs/2106.03616
  * [13] S. Luo, P. Yang, Y. L. Che, K. Yang, K. Wu, K. C. Teh, and S. Li, “Spatial modulation for RIS-assisted uplink communication: Joint power allocation and passive beamforming design,” _IEEE Trans. Commun._ , vol. PP, no. 99, pp. 1–1, Jul. 2021.
  * [14] W. Shi, J. Li, G. Xia, Y. Wang, X. Zhou, Y. Zhang, and F. Shu, “Secure multigroup multicast communication systems via intelligent reflecting surface,” _China Commun._ , vol. 18, no. 3, pp. 39–51, Mar. 2021.
  * [15] M. Hua, Q. Wu, D. Ng, J. Zhao, and L. Yang, “Intelligent reflecting surface-aided joint processing coordinated multipoint transmission,” _IEEE Trans. Commun._ , vol. 69, no. 3, pp. 1650–1665, Mar. 2021.
  * [16] X. Zhou, S. Yan, Q. Wu, F. Shu, and D. W. K. Ng, “Intelligent reflecting surface (IRS)-aided covert wireless communications with delay constraint,” _IEEE Trans. Wireless Commun._ , vol. PP, no. 99, pp. 1–1, Jul. 2021.
  * [17] X. Lu, E. Hossain, T. Shafique, S. Feng, H. Jiang, and D. Niyato, “Intelligent reflecting surface enabled covert communications in wireless networks,” _IEEE Network_ , vol. 34, no. 5, pp. 148–155, Oct. 2020.
  * [18] Y. Pan, K. Wang, C. Pan, H. Zhu, and J. Wang, “UAV-assisted and intelligent reflecting surfaces-supported terahertz communications,” _IEEE Wireless Commun. Lett._ , vol. 10, no. 6, pp. 1256–1260, Jun. 2021.
  * [19] Z. Wei, Y. Cai, Z. Sun, D. W. K. Ng, J. Yuan, M. Zhou, and L. Sun, “Sum-rate maximization for IRS-assisted UAV OFDMA communication systems,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 4, pp. 2530–2550, Apr. 2021\.
  * [20] Q. Wu and R. Zhang, “Weighted sum power maximization for intelligent reflecting surface aided SWIPT,” _IEEE Wireless Commun. Lett._ , vol. 9, no. 5, pp. 586–590, May 2020.
  * [21] W. Shi, X. Zhou, L. Jia, Y. Wu, F. Shu, and J. Wang, “Enhanced secure wireless information and power transfer via intelligent reflecting surface,” _IEEE Commun. Lett._ , vol. 25, no. 4, pp. 1084–1088, Apr. 2021.
  * [22] M. Obeed and A. Chaaban, “Relay-reconfigurable intelligent surface cooperation for energy-efficient multiuser systems,” in _2021 IEEE International Conference on Communications Workshops (ICC Workshops)_ , Jul. 2021, pp. 1–6.
  * [23] N. T. Nguyen, Q.-D. Vu, K. Lee, and M. Juntti, “Spectral efficiency optimization for hybrid relay-reflecting intelligent surface,” in _2021 IEEE International Conference on Communications Workshops (ICC Workshops)_ , Jul. 2021, pp. 1–6.
  * [24] I. Yildirim, F. Kilinc, E. Basar, and G. C. Alexandropoulos, “Hybrid RIS-empowered reflection and decode-and-forward relaying for coverage extension,” 2020. [Online]. Available: https://arxiv.org/abs/2012.12329
  * [25] X. Ying, U. Demirhan, and A. Alkhateeb, “Relay aided intelligent reconfigurable surfaces: Achieving the potential without so many antennas,” 2020\. [Online]. Available: https://arxiv.org/abs/2006.06644
  * [26] Z. Abdullah, G. Chen, S. Lambotharan, and J. A. Chambers, “A hybrid relay and intelligent reflecting surface network and its ergodic performance analysis,” _IEEE Wireless Commun. Lett._ , vol. 9, no. 10, pp. 1653–1657, Oct. 2020.
  * [27] G. Xia, Y. Lin, X. Zhou, W. Zhang, F. Shu, and J. Wang, “Hybrid precoding design for secure generalized spatial modulation with finite-alphabet inputs,” _IEEE Trans. Commun._ , vol. 69, no. 4, pp. 2570–2584, Apr. 2021\.
  * [28] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 11, pp. 5394–5409, Nov. 2019.