# Intelligent Reflecting Surface Aided Full-Duplex Communication: Passive
Beamforming and Deployment Design

Yunlong Cai, Ming-Min Zhao, Kaidi Xu, and Rui Zhang  Y. Cai, M. Zhao, and K.
Xu are with the College of Information Science and Electronic Engineering,
Zhejiang University, China (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>xukaidi13@126.com). R. Zhang is with the Department of Electrical and Computer
Engineering, National University of Singapore, Singapore (e-mail:
elezhang@nus.edu.sg).

###### Abstract

This paper investigates the passive beamforming and deployment design for an
intelligent reflecting surface (IRS) aided full-duplex (FD) wireless system,
where an FD access point (AP) communicates with an uplink (UL) user and a
downlink (DL) user simultaneously over the same time-frequency dimension with
the help of IRS. Under this setup, we consider three deployment cases: 1) two
distributed IRSs placed near the UL user and DL user, respectively; 2) one
centralized IRS placed near the DL user; 3) one centralized IRS placed near
the UL user. In each case, we aim to minimize the weighted sum transmit power
consumption of the AP and UL user by jointly optimizing their transmit power
and the passive reflection coefficients at the IRS (or IRSs), subject to the
UL and DL users’ rate constraints and the uni-modulus constraints on the IRS
reflection coefficients. First, we analyze the minimum transmit power required
in the IRS-aided FD system under each deployment scheme, and compare it with
that of the corresponding half-duplex (HD) system. We show that the FD system
outperforms its HD counterpart for all IRS deployment schemes, while the
distributed deployment further outperforms the other two centralized
deployment schemes. Next, we transform the challenging power minimization
problem into an equivalent but more tractable form and propose an efficient
algorithm to solve it based on the block coordinate descent (BCD) method.
Finally, numerical results are presented to validate our analysis as well as
the efficacy of the proposed passive beamforming design.

###### Index Terms:

Intelligent reflecting surface, full-duplex, passive beamforming, deployment,
power minimization.

##  I Introduction

Recently, intelligent reflecting surface (IRS) and its various equivalents
have emerged as a promising technology to enhance the spectral efficiency of
wireless communication systems with low hardware cost and energy consumption
[1, 2, 3, 4, 5]. Specifically, IRS is generally equipped with a planar surface
composed of a large number of passive reflecting elements, each of which can
induce an independent phase shift and/or amplitude change of the incident
signal in real time. Based on the channel state information (CSI) and with the
aid of a smart controller, IRS is able to modify/reconfigure the signal
propagation by dynamically adjusting its reflection coefficients such that the
desired and interfering signals can be added constructively and destructively
at the receivers, respectively, to boost the desired signal power and/or
suppress the co-channel interference (CCI), thus achieving communication
performance improvement. Additionally, since such passive reflecting elements
do not require any active transmit radio frequency (RF) chains (which
constitute high-cost amplifiers, filters, mixers, attenuators and detectors,
etc.), their energy and hardware costs are much lower than those of the active
components in traditional base stations (BSs), access points (APs), and
relays. As a result, IRSs can be flexibly deployed in wireless networks and
seamlessly integrated into the existing cellular or WiFi systems, with
controllable interference to each other as they usually have much smaller
signal coverage than active BSs/APs/relays. Due to the above advantages, IRS
has been extensively studied under various setups (see, e.g., [6, 7, 8, 9, 10,
11]), where the effectiveness of IRS in enhancing these systems’ performance
was demonstrated.

To boost the spectral efficiency of wireless systems, the full-duplex (FD)
communication is another promising technique [12, 13, 14]. Compared with the
traditional half-duplex (HD) system, it better utilizes the spectrum by
enabling signal transmission and reception over the same time-frequency
dimension and thus can double the spectral efficiency theoretically. Although
the FD system offers promising spectral efficiency gains, it also brings
challenges in practice, such as the self-interference (SI) and CCI caused by
the simultaneous downlink (DL) and uplink (UL) transmissions. These
interferences, if left unattended, can encroach the gain offered by FD and
even degrade the system performance as compared to traditional HD.
Fortunately, many efficient SI cancellation techniques have been proposed in
the literature [14, 15, 16], such as passive suppression, analog and digital
cancellations, etc., and it has been reported that the SI can be suppressed up
to the background noise floor at an FD node [17], which leads to implementable
FD systems. However, the CCI inherited from the FD operation can also cause
significant spectral efficiency degradation, despite the fact that the SI can
be effectively suppressed. In practice, the interference from the UL users to
the DL users (UL-to-DL interference) can be detrimental if the UL and DL users
are located close to each other. To avoid the performance degradation due to
CCI, various methods such as DL/UL user pairing and transmission scheduling
have been proposed in the literature (see e.g., [18, 19] and the references
therein).

Motivated by the above, in this paper, we combine the two spectral-efficient
wireless techniques, i.e., IRS and FD into a new system, which can leverage
IRS’s interference cancellation and signal enhancement capabilities to further
improve the FD communication performance.

###  I-A Prior Works

IRS has been studied recently in various wireless systems, such as IRS-aided
orthogonal frequency division multiplexing (OFDM) [10], mmWave communication
[20, 21], physical layer security [22, 23], wireless power transfer [24, 25],
etc. To the best of our knowledge, there are only few works that have studied
IRS-aided FD systems [26, 27, 28]. Specifically, in [26], the authors studied
an IRS-aided FD multiple-input multiple-output (MIMO) two-way communication
system and the sum-rate maximization problem was considered by jointly
optimizing the active precoders at the sources and the IRS reflection
coefficients through the Arimoto-Blahut algorithm. Under the same system
model, the authors in [27] further proposed an alternating optimization
algorithm to solve the sum-rate maximization problem with lower computational
complexity. In [28], an IRS-assisted FD cognitive radio system was
investigated, where the IRS is employed to enhance the performance of the
secondary network and in the meantime mitigate the interference caused to the
primary users. Note that these existing works on IRS-aided FD systems mainly
focused on joint active and passive beamforming design, while the fundamental
advantages of IRS-aided FD versus HD system are not fully characterized yet.
Moreover, with a given number of IRS reflecting elements, there are various
IRS deployment strategies for an IRS-aided FD system, e.g., placing multiple
IRSs in the network for far-apart users separately, or forming them as one
large IRS and placing it near a cluster of nearby users, which are referred to
as distributed and centralized deployment, respectively. The IRS deployment
problem has been investigated in the point-to-point channel [29] and the UL
(DL) multiple access (broadcast) channel [30]; while it is yet unclear which
of the above two IRS deployment strategies (centralized or distributed)
achieves better performance in IRS-aided FD systems.

###  I-B Main Contributions

In this paper, we consider an IRS-aided FD system where an FD AP (equipped
with one transmit antenna and one receive antenna) communicates with a single-
antenna UL user and a single-antenna DL user simultaneously over the same
time-frequency dimension with the help of a given number of IRS reflecting
elements. For simplicity, we assume that the IRS (or IRSs if deployed in a
distributed manner) is placed in close vicinity to the users to minimize the
path loss. Under this setup, we investigate three deployment cases: 1) two
distributed IRSs placed near the UL user and DL user, respectively; 2) one
centralized IRS placed near the DL user; and 3) one centralized IRS placed
near the UL user. In each case, we aim to minimize the weighted sum transmit
power consumption of the AP and UL user by jointly optimizing their transmit
power and the passive reflection coefficients at the IRS (or IRSs), subject to
the UL and DL users’ rate constraints and the uni-modulus constraints on the
IRS reflection coefficients. To focus on the deployment strategy design and
passive beamforming optimization, we assume for simplicity the availability of
CSI for all channels involved, which can be obtained by various existing
channel estimation methods (e.g., [31, 10, 32]). The main contributions of
this paper in view of the existing literature are summarized as follows.

  * •

First, by assuming Rayleigh fading channels and with an asymptotically large
number of IRS reflecting elements, we analyze the minimum transmit power
required in the above-mentioned three IRS deployment cases. In particular, we
show that with the aid of IRS, the FD system always outperforms the HD system
regardless of the UL-to-DL interference. This is in sharp contrast with the
conventional system without IRS, where the FD operation is not always
beneficial, especially in the case that the UL-to-DL interference is severe.
Besides, we show that the minimum power consumption with the distributed
deployment (Case 1) is much lower than that with the centralized deployment
(Case 2 or 3) in the IRS-aided FD system, which unveils that the distributed
deployment generally outperforms centralized deployment in terms of power
consumption.

  * •

Second, for arbitrary channels and finite number of reflecting elements, we
propose an efficient passive beamforming design algorithm for solving the
formulated power minimization problem, which is difficult to solve due to the
non-convex objective function and uni-modulus constraints. Specifically, we
first transform the original problem into an equivalent but more tractable
form by introducing an auxiliary variable, and then propose an algorithm for
solving it by employing the block coordinate descent (BCD) method. The
computational complexity and convergence property of the proposed algorithm
are analyzed.

  * •

Finally, numerical results are presented to validate our analysis and show the
performance of the proposed algorithm. Particularly, we draw useful insights
into the impact of the distance between the UL and DL users, the number of
reflecting elements and the UL/DL target rates on the total transmit power.
Besides, performance comparison between the proposed algorithm and a low-
complexity heuristic algorithm is provided.

###  I-C Organization

The rest of this paper is organized as follows. Section II describes the
system model and formulates the optimization problem of interest. In Section
III, we analyze the minimum power consumption for FD and HD systems under
different IRS deployment cases. In Section IV, we propose a new algorithm to
optimize the passive beamforming for solving the formulated problems. The
simulation results are presented in Section V and our conclusions are drawn in
Section VI.

_Notations:_ Scalars, vectors and matrices are respectively denoted by lower
case, boldface lower case and boldface upper case letters. $\mathbf{I}$
represents an identity matrix and $\mathbf{0}$ denotes an all-zero matrix. For
a matrix $\mathbf{A}$, ${{\bf{A}}^{T}}$, $\text{conj}(\mathbf{A})$,
${{\bf{A}}^{H}}$ and $\|\mathbf{A}\|$ denote its transpose, conjugate,
conjugate transpose and Frobenius norm, respectively.
$\text{diag}(\mathbf{A})$ denotes a vector whose elements are the
corresponding ones on the main diagonal of $\mathbf{A}$. For a vector
$\mathbf{a}$, $\text{Diag}(\mathbf{a})$ denotes a diagonal matrix with each
diagonal element being the corresponding element in $\mathbf{a}$. $\mathbf{1}$
denotes an all-ones vector. $\Re\\{\cdot\\}$ ($\Im\\{\cdot\\}$) denotes the
real (imaginary) part of a variable, while $|\cdot|$ represents the absolute
value of a complex scalar. ${\mathbb{C}^{m\times n}}\;({\mathbb{R}^{m\times
n}})$ denotes the space of ${m\times n}$ complex (real) matrices. The letter
$j$ is used to represent $\sqrt{-1}$ when there is no ambiguity.
$\mathbb{Z}^{+}$ denotes a set of positive integers. The operator $\angle$
takes the phase angles of the elements in a matrix.
$\mathbf{a}\sim\mathcal{CN}(\mathbf{0},\mathbf{A})$ denotes that the random
vector $\mathbf{a}$ follows the circularly-symmetric complex Gaussian
distribution with zero mean and covariance matrix $\mathbf{A}$.

##  II System model and problem formulation

In this section, we introduce the system model and problem formulation for
three different IRS deployment cases under investigation.

###  II-A FD System

Consider an FD system consisting of an AP, a UL user and a DL user. The AP is
equipped with a single transmit antenna and a receive antenna, and operates in
the FD mode. The UL and DL users are both equipped with a single antenna, and
operate in the HD mode.111In order to focus on our study, we consider a
typical single-antenna two-user FD system; while the results in this paper can
be extended to the general setup of IRS-aided multi-antenna and/or multi-user
FD systems, which are left for future work.

Generally, IRS is deployed near the users for enhancing their performance.
Assuming a total of $N\in\mathbb{Z}^{+}$ IRS reflecting elements utilized in
the system, for this case depicted in Fig. 1 (a), there are two IRSs, namely
IRS 1 and IRS 2, which are equipped with $\rho N\in\mathbb{Z}^{+}$ and
$(1-\rho)N\in\mathbb{Z}^{+}$ reflecting elements and deployed near the UL/DL
user, respectively, where $0<\rho<1$ denotes a preset ratio. Since IRS 1 and
IRS 2 are sufficiently far from the DL user and the UL user, respectively, the
link that IRS 1 reflects the signals from the AP to the DL user as well as
that IRS 2 reflects the signals from the UL user to the AP can be neglected,
as compared to the other links shown in the figure. The received signal at the
AP is expressed as

$y_{A}=h_{AU}\sqrt{p_{U}}s_{U}+\mathbf{f}^{H}_{AI}\mathbf{\Theta}_{U}\mathbf{f}_{IU}\sqrt{p_{U}}s_{U}+n_{A},$
(1)

where $s_{U}$ denotes the transmit signal of the UL user, $p_{U}>0$ denotes
the transmit power of the UL user, and $n_{A}\in\mathbb{C}$ denotes the
independent and identically distributed (i.i.d.) complex Gaussian noise at the
AP with zero mean and variance $\sigma^{2}_{A}$.
$\mathbf{f}_{IU}\in\mathbb{C}^{\rho N\times 1}$ and
$\mathbf{f}_{AI}\in\mathbb{C}^{\rho N\times 1}$ denote respectively the
channel vector between the UL user and IRS 1, and that between IRS 1 and the
AP. $h_{AU}\in\mathbb{C}$ denotes the channel coefficient between the UL user
and the AP. $\mathbf{\Theta}_{U}\in\mathbb{C}^{\rho N\times\rho N}$ denotes
the passive beamforming matrix at IRS 1 placed near the UL user, which is a
diagonal matrix due to no signal coupling/joint processing over its passive
reflecting elements. Since the CSI of the SI link can be obtained at the AP,
based on certain interference cancellation techniques [33, 34], we assume that
the SI at the AP can be eliminated completely for the sake of exposition.

On the other hand, the received signal at the DL user is given by

$y_{D}=h_{DA}\sqrt{p_{A}}s_{D}+\mathbf{g}^{H}_{DI}\mathbf{\Theta}_{D}\mathbf{g}_{IA}\sqrt{p_{A}}s_{D}+\underbrace{g\sqrt{p_{U}}s_{U}+\mathbf{f}^{H}_{DI}\mathbf{\Theta}_{U}\mathbf{f}_{IU}\sqrt{p_{U}}s_{U}+\mathbf{g}^{H}_{DI}\mathbf{\Theta}_{D}\mathbf{g}_{IU}\sqrt{p_{U}}s_{U}}_{\textrm{interference
from the UL user}}+n_{D},$ (2)

where $s_{D}$ and $p_{A}>0$ denote the transmit signal for the DL user and the
transmit power at the AP, respectively.
$\mathbf{g}_{IA}\in\mathbb{C}^{(1-\rho)N\times 1}$,
$\mathbf{g}_{DI}\in\mathbb{C}^{(1-\rho)N\times 1}$,
$\mathbf{f}_{DI}\in\mathbb{C}^{\rho N\times 1}$ and
$\mathbf{g}_{IU}\in\mathbb{C}^{(1-\rho)N\times 1}$ denote the channel vectors
between the AP and IRS 2, between IRS 2 and the DL user, between IRS 1 and the
DL user, and between the UL user and IRS 2, respectively.
$h_{DA}\in\mathbb{C}$ denotes the channel coefficient between the AP and the
DL user. $\mathbf{\Theta}_{D}\in\mathbb{C}^{(1-\rho)N\times(1-\rho)N}$ denotes
the diagonal beamforming matrix at IRS 2 which is placed near the DL user. $g$
denotes the channel coefficient between the UL user and the DL user.
$n_{D}\in\mathbb{C}$ denotes the i.i.d. complex Gaussian noise at the DL user
with zero mean and variance $\sigma^{2}_{D}$. The link between the two IRSs is
also neglected due to their large distance and thus the high path loss.

Figure 1:  (a) Case 1: IRSs placed near both users; (b) Case 2: IRS placed
near the downlink user only; (c) Case 3: IRS placed near the uplink user only.

Let us define the effective channel power gains, namely,
$\lambda_{D}(\bm{\theta}_{D})\triangleq|h_{DA}+\mathbf{g}^{H}_{DA}\bm{\theta}_{D}|^{2}$,
where
$\mathbf{g}_{DA}\triangleq\textrm{Diag}(\textrm{conj}(\mathbf{g}_{IA}))\mathbf{g}_{DI}$
and $\bm{\theta}_{D}\triangleq\textrm{diag}(\bm{\Theta}_{D})$,
$\lambda_{U}(\bm{\theta}_{U})\triangleq|h_{AU}+\mathbf{f}^{H}_{AU}\bm{\theta}_{U}|^{2}$,
where
$\mathbf{f}_{AU}\triangleq\textrm{Diag}(\textrm{conj}(\mathbf{f}_{IU}))\mathbf{f}_{AI}$
and $\bm{\theta}_{U}\triangleq\textrm{diag}(\bm{\Theta}_{U})$, and
$\lambda_{DU}(\bm{\theta}_{U},\bm{\theta}_{D})\triangleq|g+\mathbf{d}^{H}\bm{\theta}_{DU}|^{2}$,
where
$\bm{\theta}_{DU}\triangleq[\bm{\theta}^{H}_{U},\bm{\theta}^{H}_{D}]^{H}$,
$\mathbf{d}\triangleq[\mathbf{f}^{H}_{DU},\mathbf{g}^{H}_{DU}]^{H}$,
$\mathbf{f}_{DU}\triangleq\textrm{Diag}(\textrm{conj}(\mathbf{f}_{IU}))\mathbf{f}_{DI}$,
and
$\mathbf{g}_{DU}\triangleq\textrm{Diag}(\textrm{conj}(\mathbf{g}_{IU}))\mathbf{g}_{DI}$.
We aim to minimize the weighted sum transmit power consumption of the UL user
and AP by jointly optimizing their corresponding transmit power levels $p_{U}$
and $p_{A}$, and the passive beamforming vectors $\bm{\theta}_{U}$ and
$\bm{\theta}_{D}$ at the two IRSs, subject to the UL and DL users’ rate
constraints and the uni-modulus constraints on the elements of the IRS passive
beamforming vector. Accordingly, the optimization problem is formulated as

$\displaystyle\min_{p_{U},p_{A},\bm{\theta}_{U},\bm{\theta}_{D}}\quad$
$\displaystyle\kappa_{U}p_{U}+\kappa_{A}p_{A}$ (3a) s.t.
$\displaystyle\log\left(1+\frac{p_{U}\lambda_{U}(\bm{\theta}_{U})}{\sigma^{2}_{A}}\right)\geq\gamma_{A},$
(3b)
$\displaystyle\log\left(1+\frac{p_{A}\lambda_{D}(\bm{\theta}_{D})}{p_{U}\lambda_{DU}(\bm{\theta}_{U},\bm{\theta}_{D})+\sigma^{2}_{D}}\right)\geq\gamma_{D},$
(3c)
$\displaystyle|\bm{\theta}_{U}(n_{1})|=1,\,|\bm{\theta}_{D}(n_{2})|=1,\,\forall
n_{1}\in\\{1,\ldots,\rho N\\},\,\forall n_{2}\in\\{1,\ldots,(1-\rho)N\\},$
(3d)

where we have assumed base-2 for the logarithm function, $\kappa_{U}>0$ and
$\kappa_{A}>0$ represent the power weights corresponding to the UL user and
AP, respectively. Constraints (3b) and (3c) correspond to the rate
requirements for the UL and DL transmissions, respectively, where $\gamma_{A}$
and $\gamma_{D}$ denote the target rates in bits per second per Hertz
(bits/s/Hz). (3d) denotes the uni-modulus constraints on all elements of the
IRS passive beamforming vector.

Next, we consider two special cases of the general IRS deployment introduced
in the above. First, for the case of placing all reflecting elements to IRS 2
near the DL user (i.e., with $\rho=0$) as shown in Fig. 1 (b), let
$\bm{\theta}_{DU}\in\mathbb{C}^{N\times 1}$ denote the entire passive
beamforming vector at IRS 2. By defining
$\tilde{\mathbf{g}}_{IA}\in\mathbb{C}^{N\times 1}$,
$\tilde{\mathbf{g}}_{DI}\in\mathbb{C}^{N\times 1}$ and
$\tilde{\mathbf{g}}_{IU}\in\mathbb{C}^{N\times 1}$ as the channel vectors
between the AP and IRS 2, between IRS 2 and the DL user, and between the UL
user and IRS 2, respectively, the optimization problem in this case can be
formulated by simplifying (3) as

$\displaystyle\min_{p_{U},p_{A},\bm{\theta}_{DU}}\quad$
$\displaystyle\kappa_{U}p_{U}+\kappa_{A}p_{A}$ (4a) s.t.
$\displaystyle\log\left(1+\frac{p_{U}|h_{AU}|^{2}}{\sigma^{2}_{A}}\right)\geq\gamma_{A},$
(4b)
$\displaystyle\log\left(1+\frac{p_{A}\omega_{D}(\bm{\theta}_{DU})}{p_{U}\omega_{DU}(\bm{\theta}_{DU})+\sigma^{2}_{D}}\right)\geq\gamma_{D},$
(4c) $\displaystyle|\bm{\theta}_{DU}(n)|=1,\forall n\in\\{1,\ldots,N\\}.$ (4d)

Here we have
$\omega_{D}(\bm{\theta}_{DU})\triangleq|h_{DA}+\tilde{\mathbf{g}}^{H}_{DA}\bm{\theta}_{DU}|^{2}$,
where
$\tilde{\mathbf{g}}_{DA}\triangleq\textrm{Diag}(\textrm{conj}(\tilde{\mathbf{g}}_{IA}))\tilde{\mathbf{g}}_{DI}$,
and
$\omega_{DU}(\bm{\theta}_{DU})\triangleq|g+\tilde{\mathbf{g}}^{H}_{DU}\bm{\theta}_{DU}|^{2}$,
where
$\tilde{\mathbf{g}}_{DU}\triangleq\textrm{Diag}(\textrm{conj}(\tilde{\mathbf{g}}_{IU}))\tilde{\mathbf{g}}_{DI}$.

Second, for the other case of placing all reflecting elements to IRS 1 near
the UL user (i.e., with $\rho=1$) as depicted in Fig. 1 (c), let us define
$\tilde{\mathbf{f}}_{IU}\in\mathbb{C}^{N\times 1}$ and
$\tilde{\mathbf{f}}_{AI}\in\mathbb{C}^{N\times 1}$ as the channel vectors
between the UL user and IRS 1 and between IRS 1 and the AP, respectively, and
$\tilde{\mathbf{f}}_{DI}\in\mathbb{C}^{N\times 1}$ as the channel vector
between IRS 1 and the DL user. Similarly, the optimization problem in this
case can be formulated as

$\displaystyle\min_{p_{U},p_{A},\bm{\theta}_{DU}}\quad$
$\displaystyle\kappa_{U}p_{U}+\kappa_{A}p_{A}$ (5a) s.t.
$\displaystyle\log\left(1+\frac{p_{U}\xi_{U}(\bm{\theta}_{DU})}{\sigma^{2}_{A}}\right)\geq\gamma_{A},$
(5b)
$\displaystyle\log\left(1+\frac{p_{A}|h_{DA}|^{2}}{p_{U}\xi_{DU}(\bm{\theta}_{DU})+\sigma^{2}_{D}}\right)\geq\gamma_{D},$
(5c) $\displaystyle|\bm{\theta}_{DU}(n)|=1,\forall n\in\\{1,\ldots,N\\}.$ (5d)

Here we define
$\xi_{U}(\bm{\theta}_{DU})\triangleq|h_{AU}+\tilde{\mathbf{f}}^{H}_{AU}\bm{\theta}_{DU}|^{2}$,
where
$\tilde{\mathbf{f}}_{AU}\triangleq\textrm{Diag}(\textrm{conj}(\tilde{\mathbf{f}}_{IU}))\tilde{\mathbf{f}}_{AI}$,
and
$\xi_{DU}(\bm{\theta}_{DU})\triangleq|g+\tilde{\mathbf{f}}^{H}_{DU}\bm{\theta}_{DU}|^{2}$,
where
$\tilde{\mathbf{f}}_{DU}\triangleq\textrm{Diag}(\textrm{conj}(\tilde{\mathbf{f}}_{DI}))\tilde{\mathbf{f}}_{IU}$.

###  II-B HD System

As for the corresponding HD system, the received signal vectors at the AP and
DL user for the three deployment schemes are similar to that of the FD system,
but there is no interference term from the UL user due to the orthogonal UL
and DL transmissions. The detailed expressions of the received signals are
thus omitted for brevity. As a result, the optimization problem of Case 1 in
the HD system is expressed as

$\displaystyle\min_{p_{U},p_{A},\bm{\theta}_{U},\bm{\theta}_{D}}\quad$
$\displaystyle\kappa_{U}p_{U}+\kappa_{A}p_{A}$ (6a) s.t.
$\displaystyle\frac{1}{2}\log\left(1+\frac{p_{U}\lambda_{U}(\bm{\theta}_{U})}{\frac{1}{2}\sigma^{2}_{A}}\right)\geq\gamma_{A},$
(6b)
$\displaystyle\frac{1}{2}\log\left(1+\frac{p_{A}\lambda_{D}(\bm{\theta}_{D})}{\frac{1}{2}\sigma^{2}_{D}}\right)\geq\gamma_{D},$
(6c) $\displaystyle\eqref{constantmodulus2223},$

where (6b) and (6c) denote the user rate constraints, and the factor
$\frac{1}{2}$ is due to the fact that the uplink and downlink transmissions
are allocated with either half of the bandwidth or half of the time as
compared to the FD system. The optimization problems for Cases 2 and 3 in the
HD system can be similarly formulated, thus they are omitted here.

In the following two sections, we first analyze the minimum power consumption
required in different IRS deployment cases with given IRS passive beamforming
vectors. Then, we propose an efficient algorithm to optimize the passive
beamforming and thereby solve the above problems.

##  III Performance Analysis

In this section, we analyze the minimum transmit power required in the IRS-
aided FD system under each IRS deployment scheme, and compare it with that of
the corresponding HD system. We show that the FD system outperforms its HD
counterpart for all IRS deployment schemes, while the distributed deployment
further outperforms the other two centralized deployment schemes in the IRS-
aided FD system.

###  III-A Comparison of FD versus HD Systems

In this subsection, we compare the power consumption of IRS-aided FD system
with its HD counterpart under the three considered IRS deployment cases.

####  III-A1 Case 1

First, consider the case of placing IRSs near both users as shown in Fig. 1
(a). For the FD system, since the user rates are monotonically increasing with
$p_{U}$ and $p_{A}$, their optimal solutions in (3) should guarantee that the
inequality constraints (3b) and (3c) are met with equality. Thus, the optimal
transmit power $p_{U}^{*}$ and $p_{A}^{*}$ are given by

$p_{U}^{*}=\frac{(2^{\gamma_{A}}-1)\sigma^{2}_{A}}{\lambda_{U}(\bm{\theta}_{U})},$
(7)
$p_{A}^{*}=\frac{(2^{\gamma_{D}}-1)\sigma^{2}_{A}(2^{\gamma_{A}}-1)\lambda_{DU}(\bm{\theta}_{U},\bm{\theta}_{D})}{\lambda_{D}(\bm{\theta}_{D})\lambda_{U}(\bm{\theta}_{U})}+\frac{(2^{\gamma_{D}}-1)\sigma^{2}_{D}}{\lambda_{D}(\bm{\theta}_{D})}.$
(8)

By defining $\bm{\theta}^{*}_{U}$ and $\bm{\theta}^{*}_{D}$ as the optimal IRS
passive beamforming/phase-shift vectors for Case 1, the minimum power
consumption of the FD system can be expressed as

$L_{1}(\bm{\theta}^{*}_{U},\bm{\theta}^{*}_{D})\triangleq\kappa_{U}p^{*}_{U}+\kappa_{A}p^{*}_{A}=\frac{\bar{\gamma}_{1}\lambda_{DU}(\bm{\theta}^{*}_{U},\bm{\theta}^{*}_{D})}{\lambda_{D}(\bm{\theta}^{*}_{D})\lambda_{U}(\bm{\theta}^{*}_{U})}+\frac{\bar{\gamma}_{2}}{\lambda_{D}(\bm{\theta}^{*}_{D})}+\frac{\bar{\gamma}_{3}}{\lambda_{U}(\bm{\theta}^{*}_{U})},$
(9)

where
$\bar{\gamma}_{1}\triangleq\kappa_{A}(2^{\gamma_{D}}-1)\sigma^{2}_{A}(2^{\gamma_{A}}-1)$,
$\bar{\gamma}_{2}\triangleq\kappa_{A}(2^{\gamma_{D}}-1)\sigma^{2}_{D}$ and
$\bar{\gamma}_{3}\triangleq\kappa_{U}(2^{\gamma_{A}}-1)\sigma^{2}_{A}$.

By recalling problem (6), similarly, the minimum power consumption of the HD
system in Case 1 can be expressed as

$\tilde{L}_{1}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})=\frac{\kappa_{U}(2^{2\gamma_{A}}-1)\sigma^{2}_{A}}{2\lambda_{U}(\bm{\theta}^{**}_{U})}+\frac{\kappa_{A}(2^{2\gamma_{D}}-1)\sigma^{2}_{D}}{2\lambda_{D}(\bm{\theta}^{**}_{D})},$
(10)

where
$\bm{\theta}^{**}_{U}\triangleq\arg\max_{\bm{\theta}_{U}}\lambda_{U}(\bm{\theta}_{U})$
and
$\bm{\theta}^{**}_{D}\triangleq\arg\max_{\bm{\theta}_{D}}\lambda_{D}(\bm{\theta}_{D})$
denote the channel gain maximization (CGM) based beamforming vectors at both
IRSs, which are optimal for the HD system.
$\lambda_{U}(\bm{\theta}^{**}_{U})\triangleq|h_{AU}+\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|^{2}$
and
$\lambda_{D}(\bm{\theta}^{**}_{D})\triangleq|h_{DA}+\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|^{2}$
denote the maximum channel gains for the link between the UL user and the AP
and that between the AP and the DL user, respectively. It is readily seen that

$\bm{\theta}^{**}_{U}=\exp(j(\angle
h_{AU}\mathbf{1}-\angle\mathbf{f}_{AU})),\,\,\bm{\theta}^{**}_{D}=\exp(j(\angle
h_{DA}\mathbf{1}-\angle\mathbf{g}_{DA})).$ (11)

Based on (9) and (10), we have the following theorem.

######  Theorem 1.

The minimum power consumption of the FD system is lower than that of the HD
system in Case 1, i.e.,
$\tilde{L}_{1}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})\geq
L_{1}(\bm{\theta}^{*}_{U},\bm{\theta}^{*}_{D})$, if the following condition is
satisfied:

$\begin{split}\frac{\kappa_{A}|g|^{2}}{N^{2}}\leq
U(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})&\triangleq\frac{\kappa_{U}(|h_{DA}|^{2}+|\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|^{2}-2|h_{DA}||\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|)(2^{\gamma_{A}}-1)}{6(2^{\gamma_{D}}-1)N^{2}}\\\
&\quad+\frac{\kappa_{A}(|h_{AU}|^{2}+|\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|^{2}-2|h_{AU}||\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|)(2^{\gamma_{D}}-1)\sigma^{2}_{D}}{6(2^{\gamma_{A}}-1)\sigma^{2}_{A}N^{2}}\\\
&\quad-\frac{\kappa_{A}(|\mathbf{f}^{H}_{DU}\bm{\theta}^{**}_{U}|^{2}+|\mathbf{g}^{H}_{DU}\bm{\theta}^{**}_{D}|^{2})}{N^{2}}.\end{split}$
(12)

Moreover, assuming Rayleigh fading channels for all IRS-related links, i.e.,
$\mathbf{f}_{IU}\sim\mathcal{CN}(\mathbf{0},\varrho^{2}_{f_{IU}}\mathbf{I})$,
$\mathbf{f}_{AI}\sim\mathcal{CN}(\mathbf{0},\varrho^{2}_{f_{AI}}\mathbf{I})$,
$\mathbf{g}_{IA}\sim\mathcal{CN}(\mathbf{0},\varrho^{2}_{g_{IA}}\mathbf{I})$,
$\mathbf{g}_{DI}\sim\mathcal{CN}(\mathbf{0},\varrho^{2}_{g_{DI}}\mathbf{I})$,
$\mathbf{f}_{DI}\sim\mathcal{CN}(\mathbf{0},\varrho^{2}_{f_{DI}}\mathbf{I})$
and
$\mathbf{g}_{IU}\sim\mathcal{CN}(\mathbf{0},\varrho^{2}_{g_{IU}}\mathbf{I})$,
it holds that when $N$ becomes asymptotically large,

$U(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})\rightarrow\frac{\kappa_{U}(2^{\gamma_{A}}-1)(1-\rho)^{2}\pi^{2}\varrho^{2}_{f_{IU}}\varrho^{2}_{f_{AI}}}{96(2^{\gamma_{D}}-1)}+\frac{\bar{\gamma}_{2}\pi^{2}\varrho^{2}_{g_{IA}}\varrho^{2}_{g_{DI}}\rho^{2}}{96(2^{\gamma_{A}}-1)\sigma^{2}_{A}}>0.$
(13)

###### Proof.

Please refer to Appendix A. ∎

Since in (12) $\frac{\kappa_{A}|g|^{2}}{N^{2}}\rightarrow 0$ as
$N\rightarrow\infty$, the above condition is always satisfied for sufficiently
large $N$, which implies that with the help of large IRSs, in Case 1 the power
consumption of the FD system is asymptotically lower than that of the HD
system, regardless of the interference channel gain $|g|^{2}$. Intuitively,
this can be explained by the fact that with sufficiently large IRSs, the
interference between the UL user and the DL user can be effectively
suppressed, such that FD is ensured to be more spectral efficient than HD.

######  Remark 1.

_In contrast, it can be similarly shown that without using IRSs, to guarantee
that the power consumption of the FD system is lower than that of the HD
system, the following condition must be met:_

$\kappa_{A}|g|^{2}\leq\frac{\kappa_{U}(2^{\gamma_{A}}-1)|h_{DA}|^{2}}{2(2^{\gamma_{D}}-1)}+\frac{\kappa_{A}(2^{\gamma_{D}}-1)|h_{AU}|^{2}\sigma^{2}_{D}}{2(2^{\gamma_{A}}-1)\sigma^{2}_{A}},$
(14)

_which, however, does not always hold, especially when the UL-to-DL
interference is severe, i.e., when $|g|^{2}$ is large._

####  III-A2 Case 2

Next, we consider the case of placing all IRS elements to IRS 2 near the DL
user. Based on problem (4), the optimal transmit power $p_{U}^{*}$ and
$p_{A}^{*}$ for Case 2 are given by

$p_{U}^{*}=\frac{(2^{\gamma_{A}}-1)\sigma^{2}_{A}}{|h_{AU}|^{2}},$ (15)
$p_{A}^{*}=\frac{(2^{\gamma_{D}}-1)\sigma^{2}_{A}(2^{\gamma_{A}}-1)\omega_{DU}(\bm{\theta}_{DU})}{|h_{AU}|^{2}\omega_{D}(\bm{\theta}_{DU})}+\frac{(2^{\gamma_{D}}-1)\sigma^{2}_{D}}{\omega_{D}(\bm{\theta}_{DU})}.$
(16)

By defining $\bm{\theta}^{*}_{DU}$ as the optimal IRS phase-shift vector for
Case 2, the minimum power consumption of the FD system can be expressed as

$L_{2}(\bm{\theta}^{*}_{DU})\triangleq\kappa_{U}p^{*}_{U}+\kappa_{A}p^{*}_{A}=\frac{\bar{\gamma}_{1}\omega_{DU}(\bm{\theta}^{*}_{DU})}{|h_{AU}|^{2}\omega_{D}(\bm{\theta}^{*}_{DU})}+\frac{\bar{\gamma}_{2}}{\omega_{D}(\bm{\theta}^{*}_{DU})}+\frac{\bar{\gamma}_{3}}{|h_{AU}|^{2}}.$
(17)

Moreover, similar to (6b) and (6c), the user rate constraints for the
corresponding HD system in this case can be expressed as

$\displaystyle\frac{1}{2}\log\left(1+\frac{p_{U}|h_{AU}|^{2}}{\frac{1}{2}\sigma^{2}_{A}}\right)\geq\gamma_{A},$
(18a)
$\displaystyle\frac{1}{2}\log\left(1+\frac{p_{A}\omega_{D}(\bm{\theta}_{DU})}{\frac{1}{2}\sigma^{2}_{D}}\right)\geq\gamma_{D}.$
(18b)

Then, the minimum power consumption of the HD system in Case 2 can be
expressed as

$\tilde{L}_{2}(\bm{\theta}^{**}_{DU})=\frac{\kappa_{U}(2^{2\gamma_{A}}-1)\sigma^{2}_{A}}{2|h_{AU}|^{2}}+\frac{\kappa_{A}(2^{2\gamma_{D}}-1)\sigma^{2}_{D}}{2\omega_{D}(\bm{\theta}^{**}_{DU})},$
(19)

where
$\bm{\theta}^{**}_{DU}\triangleq\arg\max_{\bm{\theta}_{DU}}\omega_{D}(\bm{\theta}_{DU})$
denotes the CGM beamforming vector in this case, which is optimal for the HD
system. Similarly, we have

$\bm{\theta}^{**}_{DU}=\exp(j(\angle
h_{DA}\mathbf{1}-\angle\tilde{\mathbf{g}}_{DA})).$ (20)

The CGM beamforming leads to the channel gains
$\omega_{D}(\bm{\theta}^{**}_{DU})\triangleq|h_{DA}+\tilde{\mathbf{g}}^{H}_{DA}\bm{\theta}^{**}_{DU}|^{2}$
and $\omega_{DU}(\bm{\theta}^{**}_{DU})$
$\triangleq|g+\tilde{\mathbf{g}}^{H}_{DU}\bm{\theta}^{**}_{DU}|^{2}$ for the
link between the AP and the DL user and that between the UL user and the DL
user, respectively. By comparing $L_{2}(\bm{\theta}^{*}_{DU})$ and
$\tilde{L}_{2}(\bm{\theta}^{**}_{DU})$, we obtain the following theorem.

######  Theorem 2.

The minimum power consumption of the FD system is lower than that of the HD
system in Case 2, if the following condition is satisfied:

$\begin{split}\frac{\kappa_{A}|g|^{2}}{N^{2}}\leq
U(\bm{\theta}^{**}_{DU})&\triangleq\frac{\kappa_{U}(2^{\gamma_{A}}-1)(|h_{DA}|^{2}+|\tilde{\mathbf{g}}^{H}_{DA}\bm{\theta}^{**}_{DU}|^{2}-2|h_{DA}||\tilde{\mathbf{g}}^{H}_{DA}\bm{\theta}^{**}_{DU}|)}{4(2^{\gamma_{D}}-1)N^{2}}\\\
&\quad+\frac{\kappa_{A}(2^{\gamma_{D}}-1)|h_{AU}|^{2}\sigma^{2}_{D}}{4(2^{\gamma_{A}}-1)\sigma^{2}_{A}N^{2}}-\frac{\kappa_{A}|\tilde{\mathbf{g}}^{H}_{DU}\bm{\theta}^{**}_{DU}|^{2}}{N^{2}}.\end{split}$
(21)

Assuming Rayleigh fading channels for all IRS-related links, i.e.,
$\tilde{\mathbf{g}}_{IA}\sim\mathcal{CN}(\mathbf{0},\varrho^{2}_{\tilde{g}_{IA}}\mathbf{I})$,
$\tilde{\mathbf{g}}_{DI}\sim\mathcal{CN}(\mathbf{0},\varrho^{2}_{\tilde{g}_{DI}}\mathbf{I})$
and
$\tilde{\mathbf{g}}_{IU}\sim\mathcal{CN}(\mathbf{0},\varrho^{2}_{\tilde{g}_{IU}}\mathbf{I})$,
it follows that when $N$ becomes asymptotically large,

$U(\bm{\theta}^{**}_{DU})\rightarrow\frac{\kappa_{U}(2^{\gamma_{A}}-1)(16|h_{DA}|^{2}+\pi^{2}\varrho^{2}_{\tilde{g}_{IA}}\varrho^{2}_{\tilde{g}_{DI}})}{64(2^{\gamma_{D}}-1)}>0.$
(22)

###### Proof.

Please refer to Appendix B. ∎

From the above, we see that with sufficiently large $N$, in Case 2 the FD
system always outperforms the HD system, regardless of the interference
channel gain $|g|^{2}$.

####  III-A3 Case 3

Last, for the case of placing all reflecting elements to IRS 1 near the UL
user, the minimum power consumption of the FD system can be expressed as

$L_{3}(\tilde{\bm{\theta}}^{*}_{DU})\triangleq\frac{\bar{\gamma}_{1}\xi_{DU}(\tilde{\bm{\theta}}^{*}_{DU})}{|h_{DA}|^{2}\xi_{U}(\tilde{\bm{\theta}}^{*}_{DU})}+\frac{\bar{\gamma}_{3}}{\xi_{U}(\tilde{\bm{\theta}}^{*}_{DU})}+\frac{\bar{\gamma}_{2}}{|h_{DA}|^{2}},$
(23)

where $\tilde{\bm{\theta}}^{*}_{DU}$ denotes the optimal IRS phase-shift
vector for Case 3. By following the same approach for Case 2, we can obtain
the same conclusion for the FD system in Case 3 when $N$ becomes sufficiently
large. Hence, we omit the details for brevity.

###  III-B Comparison of Different IRS Deployment Cases

In the following, we focus on the FD system and compare the minimum power
consumption among Case 1, Case 2 and Case 3 to find which deployment strategy
performs the best. In particular, we derive an asymptotic result when the
number of reflecting elements $N$ becomes large.

With the minimum power consumption in Case 1 expressed in (9), we provide the
following theorem.

######  Theorem 3.

Assuming Rayleigh fading channels for all IRS-related links, the upper bound
of the minimum power consumption of the FD system in Case 1 quadratically
decreases to $0$ with the increasing of $N$.

###### Proof.

Please refer to Appendix C. ∎

Besides, we can rewrite the expressions of the minimum power consumption of
the FD system in (17) and (23) for Case 2 and Case 3 as follows:

$L_{2}(\bm{\theta}^{*}_{DU})=\frac{\bar{\gamma}_{1}\omega_{DU}(\bm{\theta}^{*}_{DU})}{|h_{AU}|^{2}\omega_{D}(\bm{\theta}^{*}_{DU})}+\frac{\bar{\gamma}_{2}}{\omega_{D}(\bm{\theta}^{*}_{DU})}+\frac{\bar{\gamma}_{3}}{|h_{AU}|^{2}}>\frac{\bar{\gamma}_{3}}{|h_{AU}|^{2}}>0,$
(24)
$L_{3}(\tilde{\bm{\theta}}^{*}_{DU})=\frac{\bar{\gamma}_{1}\xi_{DU}(\tilde{\bm{\theta}}^{*}_{DU})}{|h_{DA}|^{2}\xi_{U}(\tilde{\bm{\theta}}^{*}_{DU})}+\frac{\bar{\gamma}_{3}}{\xi_{U}(\tilde{\bm{\theta}}^{*}_{DU})}+\frac{\bar{\gamma}_{2}}{|h_{DA}|^{2}}>\frac{\bar{\gamma}_{2}}{|h_{DA}|^{2}}>0.$
(25)

It is observed from (24) and (25) that $L_{2}(\bm{\theta}^{*}_{DU})$ and
$L_{3}(\tilde{\bm{\theta}}^{*}_{DU})$ are lower-bounded by
$\frac{\bar{\gamma}_{3}}{|h_{AU}|^{2}}$ and
$\frac{\bar{\gamma}_{2}}{|h_{DA}|^{2}}$, respectively. Based on (24), (25) and
Theorem 3, we can conclude that with sufficiently large $N$, the minimum power
consumption in Case 1 is much lower than that in either Case 2 or 3, which
means that the distributed deployment generally outperforms the centralized
deployment in terms of power consumption. The reason is that, when
$N\rightarrow\infty$, in Case 1 both users have asymptotically large channel
gains and the DL user has diminishing interference, whereas Case 2 and Case 3
cannot achieve all the above at the same time.

##  IV  passive beamforming Optimization

In this section, we propose a new BCD-based algorithm to optimize the passive
beamforming vector at the IRSs, for a general channel model and an arbitrary
value of $N$. Subsequently, we discuss the convergence and computational
complexity of the proposed algorithm.

###  IV-A BCD-Based Passive Beamforming Design

Let us first focus on Case 1 to introduce the proposed algorithm in general
(as Cases 2 and 3 are special cases of Case 1). By substituting the optimal
transmit powers in (7) and (8) into the original optimization problem (3), we
have the following equivalent problem:

$\begin{split}\min_{\bm{\phi}_{U},\bm{\phi}_{D}}&\quad\frac{\bar{\gamma}_{1}|g+\mathbf{f}^{H}_{DU}\bm{\psi}_{U}+\mathbf{g}^{H}_{DU}\bm{\psi}_{D}|^{2}}{|h_{DA}+\mathbf{g}^{H}_{DA}\bm{\psi}_{D}|^{2}|h_{AU}+\mathbf{f}^{H}_{AU}\bm{\psi}_{U}|^{2}}+\frac{\bar{\gamma}_{2}}{|h_{DA}+\mathbf{g}^{H}_{DA}\bm{\psi}_{D}|^{2}}+\frac{\bar{\gamma}_{3}}{|h_{AU}+\mathbf{f}^{H}_{AU}\bm{\psi}_{U}|^{2}},\end{split}$
(26)

where $\bm{\psi}_{U}\triangleq\exp(j\bm{\phi}_{U})$,
$\bm{\psi}_{D}\triangleq\exp(j\bm{\phi}_{D})$,
$\bm{\phi}_{U}\triangleq\angle\bm{\theta}_{U}$ and
$\bm{\phi}_{D}\triangleq\angle\bm{\theta}_{D}$. Note that in (26), the phase-
shift values (i.e., $\bm{\phi}_{U}$ and $\bm{\phi}_{D}$) of the IRS reflection
coefficients are regarded as the optimization variables and
$\\{\bm{\psi}_{U},\bm{\psi}_{D}\\}$ are treated as functions of these phase-
shift values, thus the original uni-modular constraint (3d) in problem (3) can
be safely ignored.

Problem (26) is still difficult to solve due to the fractional coupling terms
in the objective function. In this paper, we first transform (26) into an
equivalent but more tractable form and then develop a BCD-based algorithm to
solve the equivalent problem. By reducing the fractions of the objective
function in (26) to a common denominator and taking the reciprocal, we obtain
an equivalent expression of problem (26) as

$\max_{\bm{\phi}_{U},\bm{\phi}_{D}}\quad\frac{|h_{DA}+\mathbf{g}^{H}_{DA}\bm{\psi}_{D}|^{2}|h_{AU}+\mathbf{f}^{H}_{AU}\bm{\psi}_{U}|^{2}}{\bar{\gamma}_{1}|g+\mathbf{f}^{H}_{DU}\bm{\psi}_{U}+\mathbf{g}^{H}_{DU}\bm{\psi}_{D}|^{2}+\bar{\gamma}_{2}|h_{AU}+\mathbf{f}^{H}_{AU}\bm{\psi}_{U}|^{2}+\bar{\gamma}_{3}|h_{DA}+\mathbf{g}^{H}_{DA}\bm{\psi}_{D}|^{2}}.$
(27)

Then, we can state the following theorem.

######  Theorem 4.

By introducing an auxiliary variable $v\in\mathbb{R}$, problem (27) can be
equivalently formulated as the following problem in the sense that both
problems share the same global optimal solutions for $\bm{\phi}_{U}$ and
$\bm{\phi}_{D}$:

$\begin{split}\min_{v,\bm{\phi}_{U},\bm{\phi}_{D}}&\quad
v^{2}(\bar{\gamma}_{1}|g+\mathbf{f}^{H}_{DU}\bm{\psi}_{U}+\mathbf{g}^{H}_{DU}\bm{\psi}_{D}|^{2}+\bar{\gamma}_{2}|h_{AU}+\mathbf{f}^{H}_{AU}\bm{\psi}_{U}|^{2}+\bar{\gamma}_{3}|h_{DA}+\mathbf{g}^{H}_{DA}\bm{\psi}_{D}|^{2})\\\
&\quad-2\Re(\emph{{conj}}(v)(h_{DA}+\mathbf{g}^{H}_{DA}\bm{\psi}_{D})(h_{AU}+\mathbf{f}^{H}_{AU}\bm{\psi}_{U})).\end{split}$
(28)

###### Proof.

Please refer to Appendix D. ∎

Problem (28) can be solved based on the BCD method, where we partition the
variables into multiple blocks which are updated sequentially in each
iteration. To this end, the variables are updated as follows: 1) update $v$ by
fixing the other variables; 2) update $\bm{\phi}_{U}(n),\forall n,$
sequentially with the other variables fixed; 3) update
$\bm{\phi}_{D}(n),\forall n,$ sequentially with the other variables fixed. The
detailed updating procedure is presented as follows.

In Step 1, we optimize $v$ by fixing the other variables. The subproblem for
$v$ is given by

$\begin{split}\min_{v}\quad v^{2}a-2\Re(\textrm{conj}(v)b),\end{split}$ (29)

where
$a=\bar{\gamma}_{1}|g+\mathbf{f}^{H}_{DU}\bm{\psi}_{U}+\mathbf{g}^{H}_{DU}\bm{\psi}_{D}|^{2}+\bar{\gamma}_{2}|h_{AU}+\mathbf{f}^{H}_{AU}\bm{\psi}_{U}|^{2}+\bar{\gamma}_{3}|h_{DA}+\mathbf{g}^{H}_{DA}\bm{\psi}_{D}|^{2}$
and
$b=(h_{DA}+\mathbf{g}^{H}_{DA}\bm{\psi}_{D})(h_{AU}+\mathbf{f}^{H}_{AU}\bm{\psi}_{U})$.
We can obtain the optimal solution of this subproblem as $v^{*}=\frac{b}{a}$
by checking the first order optimality condition.

In Step 2, we optimize $\bm{\phi}_{U}(n),\forall n$, sequentially by fixing
the other variables. The corresponding subproblem for $\bm{\phi}_{U}(n)$ is
given by

$\begin{split}\min_{\bm{\phi}_{U}(n)}&\quad
v^{2}(\bar{\gamma}_{1}|g+\mathbf{f}^{H}_{DU}\bm{\psi}_{U}+\mathbf{g}^{H}_{DU}\bm{\psi}_{D}|^{2}+\bar{\gamma}_{2}|h_{AU}+\mathbf{f}^{H}_{AU}\bm{\psi}_{U}|^{2})\\\
&\quad-2\Re(\textrm{conj}(v)(h_{DA}+\mathbf{g}^{H}_{DA}\bm{\psi}_{D})(h_{AU}+\mathbf{f}^{H}_{AU}\bm{\psi}_{U})).\end{split}$
(30)

By appropriately rearranging the objective function of problem (30), we have
the following equivalent problem:

$\begin{split}\min_{\bm{\phi}_{U}(n)}&\quad\bm{\psi}_{U}^{H}\mathbf{A}_{\psi_{U}}\bm{\psi}_{U}-2\Re(\bm{\psi}_{U}^{H}\mathbf{b}_{\psi_{U}}),\end{split}$
(31)

where
$\mathbf{A}_{\psi_{U}}\triangleq|v|^{2}(\bar{\gamma}_{1}\mathbf{f}_{DU}\mathbf{f}_{DU}^{H}+\bar{\gamma}_{2}\mathbf{f}_{AU}\mathbf{f}_{AU}^{H})$
and $\mathbf{b}_{\psi_{U}}\triangleq
v(\textrm{conj}(h_{DA})+\bm{\phi}_{D}^{H}\mathbf{g}_{DA})\mathbf{f}_{AU}-|v|^{2}(\bar{\gamma}_{1}\mathbf{f}_{DU}(g+\mathbf{g}_{DU}^{H}\bm{\phi}_{D})+\bar{\gamma}_{2}\mathbf{f}_{AU}h_{AU})$.
It is readily seen that the objective function of problem (31) is a quadratic
function with respect to $\bm{\psi}_{U}(n)$. Hence, by omitting the irrelevant
constant terms, we can rewrite problem (31) as follows:

$\begin{split}\min_{\bm{\phi}_{U}(n)}&\quad\bar{a}_{\psi_{U,n}}|\bm{\psi}_{U}(n)|^{2}-2\Re(\textrm{conj}(\bar{b}_{\psi_{U,n}})\bm{\psi}_{U}(n)),\end{split}$
(32)

where $\bar{a}_{\psi_{U,n}}$ is a known coefficient that is not related to
$\bm{\phi}_{U}(n)$,
$\bar{b}_{\psi_{U,n}}=\mathbf{A}_{\psi_{U}}(n,n)\bm{\psi}_{U}(n)-\mathbf{A}_{\psi_{U}}(n,:)\bm{\psi}_{U}+\mathbf{b}_{\psi_{U}}(n),$
and $\mathbf{A}_{\psi_{U}}(n,:)$ denotes the $n$th row vector of matrix
$\mathbf{A}_{\psi_{U}}$. Obviously, the optimal solution of problem (32) is
given by $\bm{\phi}^{*}_{U}(n)=\angle\bar{b}_{\psi_{U,n}}$.

In Step 3, we optimize $\bm{\phi}_{D}(n),\forall n$, sequentially by fixing
the other variables. The corresponding subproblem is given by

$\begin{split}\min_{\bm{\phi}_{D}(n)}&\quad
v^{2}(\bar{\gamma}_{1}|g+\mathbf{f}^{H}_{DU}\bm{\psi}_{U}+\mathbf{g}^{H}_{DU}\bm{\psi}_{D}|^{2}+\bar{\gamma}_{3}|h_{DA}+\mathbf{g}^{H}_{DA}\bm{\psi}_{D}|^{2})\\\
&\quad-2\Re(\textrm{conj}(v(h_{DA})+\mathbf{g}^{H}_{DA}\bm{\psi}_{D})(h_{AU}+\mathbf{f}^{H}_{AU}\bm{\psi}_{U})).\end{split}$
(33)

This subproblem can be similarly solved as problem (30), thus the details are
omitted for brevity.

In summary, we can solve problem (26) by iterating over the abovementioned
three steps and the overall BCD-based algorithm is summarized in Algorithm 1.

Algorithm 1 Proposed BCD-based passive beamforming design algorithm

  * 1.

Define the tolerance of accuracy $\delta$. Initialize the algorithm with a
feasible point. Set the iteration number $i=0$ and the maximum iteration
number $N_{\max}$.

  * 2.

Repeat

    * –

Update $v$, $\bm{\phi}_{U}(n),\forall n$ and $\bm{\phi}_{D}(n),\forall n$
sequentially according to Steps 1-3, respectively.

    * –

Update the iteration number : $i\leftarrow i+1$.

  * 3.

Until the fractional decrease of (28) is less than $\delta$ or the maximum
number of iterations is reached, i.e., $i>{N_{\max}}$.

Besides, the passive beamforming optimization problems for Case 2 and Case 3
can be written as

$\begin{split}\min_{\bm{\phi}_{DU}}&\quad\frac{\bar{\gamma}_{1}|g+\tilde{\mathbf{g}}^{H}_{DU}\bm{\psi}_{DU}|^{2}}{|h_{AU}|^{2}|h_{DA}+\tilde{\mathbf{g}}^{H}_{DA}\bm{\psi}_{DU}|^{2}}+\frac{\bar{\gamma}_{2}}{|h_{DA}+\tilde{\mathbf{g}}^{H}_{DA}\bm{\psi}_{DU}|^{2}},\end{split}$
(34)
$\begin{split}\min_{\tilde{\bm{\phi}}_{DU}}&\quad\frac{\bar{\gamma}_{1}|g+\tilde{\mathbf{f}}^{H}_{DU}\tilde{\bm{\psi}}_{DU}|^{2}}{|h_{DA}|^{2}|h_{AU}+\tilde{\mathbf{f}}^{H}_{AU}\tilde{\bm{\psi}}_{DU}|^{2}}+\frac{\bar{\gamma}_{3}}{|h_{AU}+\tilde{\mathbf{f}}^{H}_{AU}\tilde{\bm{\psi}}_{DU}|^{2}},\end{split}$
(35)

respectively, where $\bm{\psi}_{DU}\triangleq\exp(j\bm{\phi}_{DU})$,
$\bm{\phi}_{DU}\triangleq\angle\bm{\theta}_{DU}$,
$\tilde{\bm{\psi}}_{DU}\triangleq\exp(j\tilde{\bm{\phi}}_{DU})$ and
$\tilde{\bm{\phi}}_{DU}\triangleq\angle\tilde{\bm{\theta}}_{DU}$. Since
Algorithm 1 can be easily modified to solve problems (34) and (35) as well,
the details are omitted here.

###  IV-B Complexity and Convergence of Algorithm 1

The complexity of Algorithm 1 is dominated by the matrix multiplication
operations required for updating $\bm{\phi}_{U}$ and $\bm{\phi}_{D}$. Thus, by
omitting the lower order terms, the overall complexity of Algorithm 1 is given
by $\mathcal{O}(IN^{2})$, where $I$ is the iteration number required by
Algorithm 1.

As for the convergence property of Algorithm 1, it is seen that each
subproblem in the proposed BCD-based algorithm is globally and uniquely
solved. Therefore, according to Proposition 2.7.1 (Convergence of Block
Coordinate Descent) in [35], Algorithm 1 is guaranteed to converge to a
Karush–Kuhn–Tucker (KKT) point of problem (28), i.e., a solution satisfying
the KKT conditions of the problem. Furthermore, according to [36], problem
(28) and problem (27) share the same KKT points. Therefore, Algorithm 1 is
guaranteed to converge to a KKT point of problem (27), which is also a KKT
point of problem (26).

##  V  Simulation results

In this section, we validate our analysis and evaluate the performance of the
proposed algorithm based on simulations. The path loss is modeled as
$L=C_{0}(d_{\textrm{link}}/D_{0})^{-\alpha}$, where $C_{0}$ is the path loss
at the reference distance of $D_{0}=1$ m, $\alpha$ is the path-loss exponent
and $d_{\textrm{link}}$ represents the link distance. We denote the path-loss
exponents of the AP-user (both UL and DL), AP-IRS, IRS-user and user-user
links as $\alpha_{DA}$, $\alpha_{IA}$ $\alpha_{DI}$ and $\alpha_{DU}$,
respectively, and set $\alpha_{DA}=3.6$ and
$\alpha_{IA}=\alpha_{DI}=\alpha_{DU}=2.2$. In the simulations, we construct a
3-D coordinate system where the AP is located on the $x$-axis and the IRSs are
placed in the planes parallel to the $y-z$ plane, as illustrated in Fig. 2.
The distance between the UL user and the DL user is represented by $d$. The
reference antenna at AP is located at $(2+\frac{d}{2}\;\textrm{m},0,0)$, the
UL user is located at $(2\;\textrm{m},150\;\textrm{m},0)$, and the DL user is
located at $(d+2\;\textrm{m},150\;\textrm{m},0)$. For Case 1, the reference
reflecting elements at the two IRSs are placed at
$(0,150\;\textrm{m},3\;\textrm{m})$ and
$(d+4\;\textrm{m},150\;\textrm{m},3\;\textrm{m})$, respectively. For Case 2
and Case 3, the reference reflecting element at the combined IRS is placed at
$(d+4\;\textrm{m},150\;\textrm{m},3\;\textrm{m})$ and
$(0,150\;\textrm{m},3\;\textrm{m})$, respectively.

Figure 2: Simulation setup. (a) Case 1: Place IRSs on both user sides; (b)
Case 2: IRS is placed near the DL user only; (c) Case 3: IRS is placed near
the UL user only.

Besides, we assume the Rician fading channel model for all links in our
simulations since both line-of-sight (LoS) and non-LoS (NLoS) components may
exist in practical channels. Accordingly, the AP-IRS and IRS-user channels are
modeled as

$\mathbf{f}=\sqrt{\frac{\beta}{1+\beta}}\mathbf{f}^{\textrm{LoS}}+\sqrt{\frac{1}{1+\beta}}\mathbf{f}^{\textrm{NLoS}},$
(36)

where $\beta$ is the Rician factor, $\mathbf{f}^{\textrm{LoS}}$ and
$\mathbf{f}^{\textrm{NLoS}}$ denote the deterministic LoS and NLoS components,
respectively. Here we denote the Rician factors of the AP-IRS and IRS-user
links as $\beta_{IA}$ and $\beta_{DI}$, respectively, and we set
$\beta_{IA}=9$ dB and $\beta_{DI}=6$ dB in our simulations. $g$ is modeled
similarly, with the Rician factor equal to $4$ dB. Without loss of generality,
we set $\rho=0.5$ and $\kappa_{U}=\kappa_{A}=1$ in all of our simulations.

In the following, we first show the advantage of using IRS over without using
IRS by comparing the minimum transmit power required in the FD and HD systems
under different IRS deployment strategies. Then, we compare the performance of
the three IRS deployment strategies based on the FD system.

###  V-A FD versus HD Systems

In this subsection, we aim to verify our analytical results in Section III-A
by simulations. Thus, the IRS reflection coefficients for the considered three
deployment strategies are set based on the CGM beamforming given in Section
III to maximize the channel power gains of the IRS-related links. Unless
otherwise specified, we set $d=80$ m and $\gamma_{A}=\gamma_{D}=4$ bits/s/Hz.

Figure 3: Minimum power consumption in Case 1: (a) Transmit power versus the
distance between the UL and DL users; (b) Transmit power versus the total
number of IRS reflecting elements.

First, consider the performance comparison between the FD and HD systems in
Case 1. Fig. 3 shows the minimum total transmit power consumption (for both
the AP and UL user) versus the distance between the UL and DL users, $d$, as
well as the total number of IRS reflecting elements, $N$. From Fig. 3 (a), we
can see that the power consumption of the FD system decreases with the
increasing of $d$ while the power consumption of the HD system gradually
increases with the increasing of $d$. This is because the UL-to-DL
interference in general weakens as $d$ increases, which is beneficial for
improving the performance of the FD system; on the other hand, larger $d$
implies longer distance between the AP and users and more severe path losses
of the AP-user links, therefore although there is no UL-to-DL interference in
the HD system, its performance could be impaired as $d$ increases. Moreover,
it is observed that the transmit powers of all considered systems decrease
with the increasing of $N$, which is expected since larger $N$ leads to higher
passive beamforming gain. Fig. 3 (b) illustrates the relationship between the
transmit power and the number of reflecting elements at the IRSs for both FD
and HD systems. From the results, we can see that the transmit power of the FD
system is larger than that of the HD system when there is no IRS or $N$ is
small; however the transmit power of the FD system becomes much smaller than
that of the HD system with the increasing of $N$, which verifies the results
in Theorem 1.

Figure 4: Minimum power consumption in Case 2: (a) Transmit power versus the
distance between the UL and DL users; (b) Transmit power versus the total
number of IRS reflecting elements. Figure 5: Minimum power consumption in Case
3: (a) Transmit power versus the distance between the UL and DL users; (b)
Transmit power versus the total number of IRS reflecting elements.

Then, we compare the transmit power consumption in the FD and HD systems for
Case 2 and Case 3. Fig. 4 (a) and (b) illustrate the transmit power
performance versus the distance $d$ and the number of IRS reflecting elements
in Case 2, respectively, and the same comparison is shown in Fig. 5 (a) and
(b) for Case 3. Similar to Case 1, from Fig. 4 (a) and Fig. 5 (a), the
transmit power of the FD system decreases while that of the HD system slightly
increases with $d$ in both Case 2 and Case 3. The reason for this phenomenon
is the same as the aforementioned one for Case 1. In Fig. 4 (b) and Fig. 5
(b), the trend of the curves regarding the HD system with IRS does not
decrease much with the increasing number of reflecting elements, which is
different from its counterpart in Case 1. This can be explained by the fact
that with only one large IRS deployed near the DL (or UL) user, the power
consumption required for achieving the rate requirement of the UL (DL) user
cannot be reduced, thus the total transmit power consumption in Case 2 (Case
3) is dominated by the power required by the UL (DL) transmission which
remains unchanged under different values of $N$. From the above results, we
can see that the FD and HD systems with IRS achieve much lower transmit power
consumption as compared to those without IRS. Furthermore, the transmit power
of the FD system becomes much smaller than that of the HD system with the
increasing of $N$. The results thus verify the discussions in Section III-A
for Cases 2 and 3.

###  V-B Comparison of Different IRS Deployment Strategies in FD System

In this subsection, we evaluate the performance of our proposed BCD-based
passive beamforming algorithm (i.e., Algorithm 1) in the FD system and compare
the performance of different IRS deployment strategies. Unless otherwise
specified, we assume $\gamma_{A}=\gamma_{D}=4$ bits/s/Hz, $N=2000$ and $d=40$
m for all cases.

Figure 6: Convergence behavior of the proposed BCD-based passive beamforming
algorithm.

We first investigate the convergence behavior of our proposed BCD-based
algorithm. Fig. 6 shows the achieved objective value versus the number of
iterations in Case 1. The phase shifts are randomly initialized within
$[0,2\pi)$. It is observed that the transmit power achieved by the proposed
algorithm decreases monotonically with the number of iterations and the
proposed algorithm can converge within 60 iterations. Moreover, the proposed
algorithm has a low complexity due to the closed-form solution (multiplication
operation only) of each step shown in Algorithm 1.

Next, we study in Fig. 7 the effect of the rate requirement of the UL user,
i.e., $\gamma_{A}$, on the transmit power consumption, under the three IRS
deployment strategies. We can see from this figure that the transmit power of
Case 1 is significantly lower than that of Case 2 or Case 3, which shows the
advantage of distributed deployment over centralized deployment under the
considered FD system. Besides, we observe that the transmit power in Case 3
increases slower than that in Case 2 with the increase of $\gamma_{A}$. This
is because in Case 3, the IRS is placed near the UL user and thus the rate
requirement of the UL user can be met easier. Moreover, we can see that the
proposed BCD-based passive beamforming algorithm achieves better performance
compared to the CGM passive beamforming algorithm, especially when
$\gamma_{A}$ is large. This is mainly because the UL-to-DL interference
becomes the performance bottleneck in the high-$\gamma_{A}$ regime and simply
employing the CGM based beamforming cannot effectively suppress this
interference.

Figure 7:  Transmit power versus $\gamma_{A}$ for the proposed BCD-based
algorithm and the CGM algorithm: (a) Case 1; (b) Case 2 and Case 3.

Similarly, we explore in Fig. 8 the effect of the rate requirement of the DL
user, i.e., $\gamma_{D}$, on the transmit power consumption. We can see that
Case 1 also outperforms Case 2 and Case 3 under this setup. Besides, the
transmit power in Case 2 increases slower than that in Case 3 as $\gamma_{D}$
increases since the rate requirement of the DL user can be more easily
satisfied in this case.

Figure 8: Transmit power versus $\gamma_{D}$ for the proposed BCD-based
algorithm and the CGM algorithm: (a) Case 1; (b) Case 2 and Case 3. Figure 9:
Transmit power versus the number of IRS reflecting elements for the proposed
BCD-based algorithm and the CGM algorithm: (a) Case 1; (b) Case 2 and Case 3.

Finally, in Fig. 9, we investigate the effect of the number of IRS reflecting
elements on the transmit power performance for the three IRS deployment
strategies. It can be seen that the performance of Case 1 is the best among
all considered. Besides, the distributed deployment strategy in Case 1 also
provides the largest performance improvement when $N$ is increased from $400$
to $2000$, e.g., the transmit power when $N=2000$ is about $20$ dBm lower than
that when $N=400$ in Case 1, but this gain is only around $10$ dBm in Cases 2
and 3. These results coincide with the analysis in Section III-B. Furthermore,
we can see that the proposed BCD-based algorithm still outperforms the CGM
algorithm and the gain increases with the number of reflecting elements. This
is because larger $N$ offers more flexibility when designing the reflection
coefficients and a more significant tradeoff between channel gain and UL-to-DL
interference can be achieved by the proposed BCD-based algorithm, while only
channel gain maximization is considered in the CGM algorithm.

##  VI Conclusions and Future Work

In this paper, we have studied three deployment cases for an IRS-aided FD
wireless system. In each case, the weighted sum transmit power consumption of
the AP and UL user is minimized by jointly optimizing their transmit power and
the passive reflection coefficients at the IRS (or IRSs), subject to the UL
and DL users’ rate constraints and the uni-modulus constraints on the IRS
reflection coefficients. We have analyzed the minimum transmit power in the
IRS-aided FD system under the three deployment schemes, as compared with that
of the corresponding HD system. Specifically, we have showed that the FD
system outperforms its HD counterpart for all IRS deployment schemes, while
the distributed deployment further outperforms the other two centralized
deployment schemes. Moreover, we have developed an efficient BCD-based
algorithm for passive beamforming design in a general system setup. Finally,
numerical results have been presented to verify our analysis and the efficacy
of the proposed passive beamforming design algorithm.

In the following, we briefly discuss some important issues/aspects of this
work that are not addressed yet, so as to motivate future research.

  * •

For simplicity, we only considered the single UL/DL user case and the two
users are assumed to be equipped with a single antenna each. Besides, the FD
AP is also assumed to have a single transmit/receive antenna. Generally, there
may exist multiple UL/DL users, and the AP as well as users are usually
equipped with multiple antennas for high-rate communications, therefore the
joint design of active beamforming at the AP/users and passive beamforming at
the IRS is an important problem to investigate in the future.

  * •

In this paper, we considered continuous phase shifts at the reflecting
elements of the IRS, which may be practically difficult to implement due to
the hardware limitation. Besides, the reflection amplitudes at the IRS were
assumed to be 1 for simplicity and the potential joint optimization of
reflection amplitudes and phase shifts [37] was not exploited. Thus,
beamforming optimization with discrete phase shifts at the IRS and joint
reflection amplitude and phase-shift optimization are interesting topics,
which are worthy of further investigation.

  * •

In practice, the SI due to the FD operation cannot be perfectly canceled in
general and CSI errors are inevitable in practice due to limited channel
training resources. Therefore, beamforming optimization under more practice
channel model is also an important problem to investigate in future work.

##  Appendix A Proof for Theorem 1

By comparing $L_{1}(\bm{\theta}^{*}_{U},\bm{\theta}^{*}_{D})$ and
$\tilde{L}_{1}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})$, we obtain

$\begin{split}&\tilde{L}_{1}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})-L_{1}(\bm{\theta}^{*}_{U},\bm{\theta}^{*}_{D})\\\
&=\frac{\kappa_{U}(2^{\gamma_{A}}+1)(2^{\gamma_{A}}-1)\sigma^{2}_{A}}{2\lambda_{U}(\bm{\theta}^{**}_{U})}+\frac{\kappa_{A}(2^{\gamma_{D}}+1)(2^{\gamma_{D}}-1)\sigma^{2}_{D}}{2\lambda_{D}(\bm{\theta}^{**}_{D})}\\\
&\quad-\left(\frac{\bar{\gamma}_{3}}{\lambda_{U}(\bm{\theta}^{*}_{U})}+\frac{\bar{\gamma}_{2}}{\lambda_{D}(\bm{\theta}^{*}_{D})}+\frac{\bar{\gamma}_{1}\lambda_{DU}(\bm{\theta}^{*}_{U},\bm{\theta}^{*}_{D})}{\lambda_{D}(\bm{\theta}^{*}_{D})\lambda_{U}(\bm{\theta}^{*}_{U})}\right)\\\
&\geq\tilde{L}_{1}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})-L_{1}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})\\\
&=\frac{\kappa_{U}(2^{\gamma_{A}}-1)^{2}\sigma^{2}_{A}}{2\lambda_{U}(\bm{\theta}^{**}_{U})}+\frac{\kappa_{A}(2^{\gamma_{D}}-1)^{2}\sigma^{2}_{D}}{2\lambda_{D}(\bm{\theta}^{**}_{D})}-\frac{\bar{\gamma}_{1}\lambda_{DU}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})}{\lambda_{D}(\bm{\theta}^{**}_{D})\lambda_{U}(\bm{\theta}^{**}_{U})},\end{split}$
(37)

where
$\lambda_{DU}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})\triangleq|g+\mathbf{f}^{H}_{DU}\bm{\theta}^{**}_{U}+\mathbf{g}^{H}_{DU}\bm{\theta}^{**}_{D}|^{2}$
denotes the channel power gain between the UL user and the DL user with the
CGM beamforming and the inequality holds since the CGM beamforming vectors
$\bm{\theta}^{**}_{U}$ and $\bm{\theta}^{**}_{D}$ shown in (11) are suboptimal
solutions of problem (3) in general and
$L_{1}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})\geq
L_{1}(\bm{\theta}^{*}_{U},\bm{\theta}^{*}_{D})$. Note that in light of the
Cauchy-Schwarz inequality222Our result also holds by the fact that for two
complex numbers $x$ and $y$, the following inequality holds
$||x|-|y||\leq|x+y|\leq|x|+|y|$., we can obtain

$\begin{split}\lambda_{DU}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})\leq(|g|+|\mathbf{f}^{H}_{DU}\bm{\theta}^{**}_{U}|+|\mathbf{g}^{H}_{DU}\bm{\theta}^{**}_{D}|)^{2}\leq
3(|g|^{2}+|\mathbf{f}^{H}_{DU}\bm{\theta}^{**}_{U}|^{2}+|\mathbf{g}^{H}_{DU}\bm{\theta}^{**}_{D}|^{2}).\end{split}$
(38)

Thus, we further have

$\begin{split}&\tilde{L}_{1}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})-L_{1}(\bm{\theta}^{*}_{U},\bm{\theta}^{*}_{D})\\\
&\geq\frac{\kappa_{U}(2^{\gamma_{A}}-1)^{2}\sigma^{2}_{A}}{2\lambda_{U}(\bm{\theta}^{**}_{U})}+\frac{\kappa_{A}(2^{\gamma_{D}}-1)^{2}\sigma^{2}_{D}}{2\lambda_{D}(\bm{\theta}^{**}_{D})}-\frac{3\bar{\gamma}_{1}(|g|^{2}+|\mathbf{f}^{H}_{DU}\bm{\theta}^{**}_{U}|^{2}+|\mathbf{g}^{H}_{DU}\bm{\theta}^{**}_{D}|^{2})}{\lambda_{D}(\bm{\theta}^{**}_{D})\lambda_{U}(\bm{\theta}^{**}_{U})}.\end{split}$
(39)

By multiplying
$\frac{\lambda_{D}(\bm{\theta}^{**}_{D})\lambda_{U}(\bm{\theta}^{**}_{U})}{3(2^{\gamma_{D}}-1)\sigma^{2}_{A}(2^{\gamma_{A}}-1)N^{2}}>0$
on both sides of (39), it yields

$\begin{split}&\frac{\lambda_{D}(\bm{\theta}^{**}_{D})\lambda_{U}(\bm{\theta}^{**}_{U})(\tilde{L}_{1}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})-L_{1}(\bm{\theta}^{*}_{U},\bm{\theta}^{*}_{D}))}{3(2^{\gamma_{D}}-1)\sigma^{2}_{A}(2^{\gamma_{A}}-1)N^{2}}\\\
&\geq\frac{\kappa_{U}\lambda_{D}(\bm{\theta}^{**}_{D})(2^{\gamma_{A}}-1)}{6(2^{\gamma_{D}}-1)N^{2}}+\frac{\kappa_{A}\lambda_{U}(\bm{\theta}^{**}_{U})(2^{\gamma_{D}}-1)\sigma^{2}_{D}}{6(2^{\gamma_{A}}-1)\sigma^{2}_{A}N^{2}}-\frac{\kappa_{A}(|g|^{2}+|\mathbf{f}^{H}_{DU}\bm{\theta}^{**}_{U}|^{2}+|\mathbf{g}^{H}_{DU}\bm{\theta}^{**}_{D}|^{2})}{N^{2}}\\\
&\geq\frac{\kappa_{U}(|h_{DA}|^{2}+|\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|^{2}-2|h_{DA}||\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|)(2^{\gamma_{A}}-1)}{6(2^{\gamma_{D}}-1)N^{2}}\\\
&\quad+\frac{\kappa_{A}(|h_{AU}|^{2}+|\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|^{2}-2|h_{AU}||\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|)(2^{\gamma_{D}}-1)\sigma^{2}_{D}}{6(2^{\gamma_{A}}-1)\sigma^{2}_{A}N^{2}}\\\
&\quad-\frac{\kappa_{A}(|g|^{2}+|\mathbf{f}^{H}_{DU}\bm{\theta}^{**}_{U}|^{2}+|\mathbf{g}^{H}_{DU}\bm{\theta}^{**}_{D}|^{2})}{N^{2}}=U(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})-\frac{\kappa_{A}|g|^{2}}{N^{2}},\end{split}$
(40)

where the second inequality holds due to the fact that

$\lambda_{U}(\bm{\theta}^{**}_{U})\triangleq|h_{AU}+\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|^{2}\geq(|h_{AU}|-|\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|)^{2}=|h_{AU}|^{2}+|\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|^{2}-2|h_{AU}||\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|,$
(41)
$\lambda_{D}(\bm{\theta}^{**}_{D})\triangleq|h_{DA}+\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|^{2}\geq(|h_{DA}|-|\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|)^{2}=|h_{DA}|^{2}+|\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|^{2}-2|h_{DA}||\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|.$
(42)

Based on (40), we can guarantee
$\tilde{L}_{1}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})\geq
L_{1}(\bm{\theta}^{*}_{U},\bm{\theta}^{*}_{D})$ if (12) is satisfied.

Moreover, assuming Rayleigh fading channels for all IRS-related links, due to
(11), we have
$|\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|=\sum^{(1-\rho)N}_{n=1}|g_{IA,n}||g_{DI,n}|$
and $|\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|=\sum^{\rho
N}_{n=1}|f_{IU,n}||f_{AI,n}|$, where $f_{IU,n}$ and $f_{AI,n}$ denote the
$n$-th elements of $\mathbf{f}_{IU}$ and $\mathbf{f}_{AI}$, respectively, and
$g_{IA,n}$ and $g_{DI,n}$ denote the $n$-th elements of $\mathbf{g}_{IA}$ and
$\mathbf{g}_{DI}$, respectively. Since $|f_{IU,n}|$, $|f_{AI,n}|$,
$|g_{IA,n}|$ and $|g_{DI,n}|$ are statistically independent and follow
Rayleigh distribution with mean values $\frac{\sqrt{\pi}\varrho_{f_{IU}}}{2}$,
$\frac{\sqrt{\pi}\varrho_{f_{AI}}}{2}$,
$\frac{\sqrt{\pi}\varrho_{g_{IA}}}{2}$, and
$\frac{\sqrt{\pi}\varrho_{g_{DI}}}{2}$, respectively, we have
$\mathbb{E}(|f_{IU,n}||f_{AI,n}|)=\frac{\pi\varrho_{f_{IU}}\varrho_{f_{AI}}}{4}$
and
$\mathbb{E}(|g_{IA,n}||g_{DI,n}|)=\frac{\pi\varrho_{g_{IA}}\varrho_{g_{DI}}}{4}$.
When $N\rightarrow\infty$, we obtain

$\left|\frac{\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}}{(1-\rho)N}\right|=\frac{\sum^{(1-\rho)N}_{n=1}|f_{IU,n}||f_{AI,n}|}{(1-\rho)N}\rightarrow\frac{\pi\varrho_{f_{IU}}\varrho_{f_{AI}}}{4},$
(43) $\left|\frac{\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}}{\rho
N}\right|=\frac{\sum^{\rho N}_{n=1}|g_{IA,n}||g_{DI,n}|}{\rho
N}\rightarrow\frac{\pi\varrho_{g_{IA}}\varrho_{g_{DI}}}{4}.$ (44)

Similarly, we have

$|\frac{\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}}{N}|^{2}\rightarrow\frac{\pi^{2}\varrho^{2}_{f_{IU}}\varrho^{2}_{f_{AI}}(1-\rho)^{2}}{16},\,\,|\frac{\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}}{N}|^{2}\rightarrow\frac{\pi^{2}\varrho^{2}_{g_{IA}}\varrho^{2}_{g_{DI}}\rho^{2}}{16}.$
(45)

By leveraging the Lindeberg-Lévy central limit theorem [38] and the conclusion
in [6], we have
$\mathbf{f}^{H}_{DU}\bm{\theta}^{**}_{U}\sim\mathcal{CN}(0,\rho
N\varrho^{2}_{f_{IU}}\varrho^{2}_{f_{DI}})$ and
$\mathbf{g}^{H}_{DU}\bm{\theta}^{**}_{D}\sim\mathcal{CN}(0,(1-\rho)N\varrho^{2}_{g_{IA}}\varrho^{2}_{g_{IU}})$
as $N\rightarrow\infty$, and
$\frac{|\mathbf{f}^{H}_{DU}\bm{\theta}^{**}_{U}|^{2}+|\mathbf{g}^{H}_{DU}\bm{\theta}^{**}_{D}|^{2}}{N}\rightarrow\rho\varrho^{2}_{f_{IU}}\varrho^{2}_{f_{DI}}+(1-\rho)\varrho^{2}_{g_{IA}}\varrho^{2}_{g_{IU}}$.
Thus, we obtain that when $N$ becomes asymptotically large,
$U(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})$ in this condition satisfies
(13). This thus completes the proof.

##  Appendix B Proof for Theorem 2

By comparing $L_{2}(\bm{\theta}^{*}_{DU})$ and
$\tilde{L}_{2}(\bm{\theta}^{**}_{DU})$, we obtain

$\begin{split}&\tilde{L}_{2}(\bm{\theta}^{**}_{DU})-L_{2}(\bm{\theta}^{*}_{DU})\\\
&=\frac{\kappa_{U}(2^{\gamma_{A}}+1)(2^{\gamma_{A}}-1)\sigma^{2}_{A}}{2|h_{AU}|^{2}}+\frac{\kappa_{A}(2^{\gamma_{D}}+1)(2^{\gamma_{D}}-1)\sigma^{2}_{D}}{2\omega_{D}(\bm{\theta}^{**}_{DU})}\\\
&\quad-\left(\frac{\bar{\gamma}_{3}}{|h_{AU}|^{2}}+\frac{\bar{\gamma}_{2}}{\omega_{D}(\bm{\theta}^{*}_{DU})}+\frac{\bar{\gamma}_{1}\omega_{DU}(\bm{\theta}^{*}_{DU})}{|h_{AU}|^{2}\omega_{D}(\bm{\theta}^{*}_{DU})}\right)\\\
&\geq\tilde{L}_{2}(\bm{\theta}^{**}_{DU})-L_{2}(\bm{\theta}^{**}_{DU})\\\
&=\frac{\kappa_{U}(2^{\gamma_{A}}-1)^{2}\sigma^{2}_{A}}{2|h_{AU}|^{2}}+\frac{\kappa_{A}(2^{\gamma_{D}}-1)^{2}\sigma^{2}_{D}}{2\omega_{D}(\bm{\theta}^{**}_{DU})}-\frac{\bar{\gamma}_{1}\omega_{DU}(\bm{\theta}^{**}_{DU})}{|h_{AU}|^{2}\omega_{D}(\bm{\theta}^{**}_{DU})}.\end{split}$
(46)

Note that
$\omega_{DU}(\bm{\theta}^{**}_{DU})\triangleq|g+\tilde{\mathbf{g}}^{H}_{DU}\bm{\theta}^{**}_{DU}|^{2}\leq(|g|+|\tilde{\mathbf{g}}^{H}_{DU}\bm{\theta}^{**}_{DU}|)^{2}\leq
2(|g|^{2}+|\tilde{\mathbf{g}}^{H}_{DU}\bm{\theta}^{**}_{DU}|^{2})$. Thus, we
have

$\begin{split}\tilde{L}_{2}(\bm{\theta}^{**}_{DU})-L_{2}(\bm{\theta}^{*}_{DU})&\geq\frac{\kappa_{U}(2^{\gamma_{A}}-1)^{2}\sigma^{2}_{A}}{2|h_{AU}|^{2}}+\frac{\kappa_{A}(2^{\gamma_{D}}-1)^{2}\sigma^{2}_{D}}{2\omega_{D}(\bm{\theta}^{**}_{DU})}-\frac{2\bar{\gamma}_{1}(|g|^{2}+|\tilde{\mathbf{g}}^{H}_{DU}\bm{\theta}^{**}_{DU}|^{2})}{|h_{AU}|^{2}\omega_{D}(\bm{\theta}^{**}_{DU})}.\end{split}$
(47)

By multiplying
$\frac{|h_{AU}|^{2}\omega_{D}(\bm{\theta}^{**}_{DU})}{2(2^{\gamma_{D}}-1)\sigma^{2}_{A}(2^{\gamma_{A}}-1)N^{2}}>0$
on both sides of (47) and due to the fact that
$\omega_{D}(\bm{\theta}^{**}_{DU})\triangleq|h_{DA}+\tilde{\mathbf{g}}^{H}_{DA}\bm{\theta}^{**}_{DU}|^{2}\geq(|h_{DA}|-|\tilde{\mathbf{g}}^{H}_{DA}\bm{\theta}^{**}_{DU}|)^{2}=|h_{DA}|^{2}+|\tilde{\mathbf{g}}^{H}_{DA}\bm{\theta}^{**}_{DU}|^{2}-2|h_{DA}||\tilde{\mathbf{g}}^{H}_{DA}\bm{\theta}^{**}_{DU}|$,
we obtain

$\begin{split}&\frac{|h_{AU}|^{2}\omega_{D}(\bm{\theta}^{**}_{DU})(\tilde{L}_{2}(\bm{\theta}^{**}_{DU})-L_{2}(\bm{\theta}^{*}_{DU}))}{2(2^{\gamma_{D}}-1)\sigma^{2}_{A}(2^{\gamma_{A}}-1)N^{2}}\\\
&\geq\frac{\kappa_{U}\omega_{D}(\bm{\theta}^{**}_{DU})(2^{\gamma_{A}}-1)}{4(2^{\gamma_{D}}-1)N^{2}}+\frac{\kappa_{A}|h_{AU}|^{2}(2^{\gamma_{D}}-1)\sigma^{2}_{D}}{4(2^{\gamma_{A}}-1)\sigma^{2}_{A}N^{2}}-\frac{\kappa_{A}(|g|^{2}+|\tilde{\mathbf{g}}^{H}_{DU}\bm{\theta}^{**}_{DU}|^{2})}{N^{2}}\\\
&\geq\frac{\kappa_{U}(|h_{DA}|^{2}+|\tilde{\mathbf{g}}^{H}_{DA}\bm{\theta}^{**}_{DU}|^{2}-2|h_{DA}||\tilde{\mathbf{g}}^{H}_{DA}\bm{\theta}^{**}_{DU}|)(2^{\gamma_{A}}-1)}{4(2^{\gamma_{D}}-1)N^{2}}+\frac{\kappa_{A}|h_{AU}|^{2}(2^{\gamma_{D}}-1)\sigma^{2}_{D}}{4(2^{\gamma_{A}}-1)\sigma^{2}_{A}N^{2}}\\\
&\quad-\frac{\kappa_{A}(|g|^{2}+|\tilde{\mathbf{g}}^{H}_{DU}\bm{\theta}^{**}_{DU}|^{2})}{N^{2}}=U(\bm{\theta}^{**}_{DU})-\frac{\kappa_{A}|g|^{2}}{N^{2}}.\end{split}$
(48)

Therefore, we can guarantee that the power consumption of the FD system is
lower than that of the HD system in Case 2, i.e.,
$\tilde{L}_{2}(\bm{\theta}^{**}_{DU})\geq L_{2}(\bm{\theta}^{*}_{DU})$, if the
condition (21) is satisfied.

Assuming Rayleigh fading channels for all IRS-related links, when
$N\rightarrow\infty$, similarly by leveraging the Lindeberg-Lévy central limit
theorem [38], we have
$\left|\frac{\tilde{\mathbf{g}}^{H}_{DA}\bm{\theta}^{**}_{DU}}{N}\right|^{2}\rightarrow\frac{\pi^{2}\varrho^{2}_{\tilde{g}_{IA}}\varrho^{2}_{\tilde{g}_{DI}}}{16}$,
$\frac{|\tilde{\mathbf{g}}^{H}_{DA}\bm{\theta}^{**}_{DU}|}{N^{2}}\rightarrow
0$, and
$\left|\frac{\tilde{\mathbf{g}}^{H}_{DU}\bm{\theta}^{**}_{DU}}{N}\right|^{2}\rightarrow
0$. Thus, we obtain that when $N$ becomes asymptotically large,
$U(\bm{\theta}^{**}_{DU})$ in this condition satisfies (22). The proof is thus
completed.

##  Appendix C Proof for Theorem 3

Let us rewrite the expression of the minimum power consumption of the FD
system in Case 1 as

$L_{1}(\bm{\theta}^{*}_{U},\bm{\theta}^{*}_{D})=\frac{\bar{\gamma}_{1}\lambda_{DU}(\bm{\theta}^{*}_{U},\bm{\theta}^{*}_{D})}{\lambda_{D}(\bm{\theta}^{*}_{D})\lambda_{U}(\bm{\theta}^{*}_{U})}+\frac{\bar{\gamma}_{2}}{\lambda_{D}(\bm{\theta}^{*}_{D})}+\frac{\bar{\gamma}_{3}}{\lambda_{U}(\bm{\theta}^{*}_{U})}.$
(49)

Based on the Cauchy-Schwarz inequality, we have

$\lambda_{D}(\bm{\theta}^{**}_{D})\triangleq|h_{DA}+\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|^{2}\geq(|h_{DA}|-|\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|)^{2}=|h_{DA}|^{2}+|\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|^{2}-2|h_{DA}||\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|,$
(50)
$\lambda_{U}(\bm{\theta}^{**}_{U})\triangleq|h_{AU}+\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|^{2}\geq(|h_{AU}|-|\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|)^{2}=|h_{AU}|^{2}+|\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|^{2}-2|h_{AU}||\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|.$
(51)

Then, with the suboptimal solutions $\bm{\theta}^{**}_{U}$ and
$\bm{\theta}^{**}_{D}$, we further obtain the following inequalities for the
minimum power consumption:

$\begin{split}&L_{1}(\bm{\theta}^{*}_{U},\bm{\theta}^{*}_{D})\\\
&\leq\frac{\bar{\gamma}_{1}\lambda_{DU}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})}{\lambda_{D}(\bm{\theta}^{**}_{D})\lambda_{U}(\bm{\theta}^{**}_{U})}+\frac{\bar{\gamma}_{2}}{\lambda_{D}(\bm{\theta}^{**}_{D})}+\frac{\bar{\gamma}_{3}}{\lambda_{U}(\bm{\theta}^{**}_{U})}\\\
&\leq\frac{\bar{\gamma}_{1}\lambda_{DU}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})}{(|h_{AU}|-|\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|)^{2}(|h_{DA}|-|\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|)^{2}}+\frac{\bar{\gamma}_{2}}{(|h_{DA}|-|\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|)^{2}}+\frac{\bar{\gamma}_{3}}{(|h_{AU}|-|\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|)^{2}}\\\
&\leq\tilde{U}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})\\\
&\triangleq\frac{3\bar{\gamma}_{1}(|g|^{2}+|\mathbf{f}^{H}_{DU}\bm{\theta}^{**}_{U}|^{2}+|\mathbf{g}^{H}_{DU}\bm{\theta}^{**}_{D}|^{2})}{(|h_{AU}|^{2}+|\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|^{2}-2|h_{AU}||\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|)(|h_{DA}|^{2}+|\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|^{2}-2|h_{DA}||\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|)}\\\
&\quad+\frac{\bar{\gamma}_{2}}{|h_{DA}|^{2}+|\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|^{2}-2|h_{DA}||\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|}+\frac{\bar{\gamma}_{3}}{|h_{AU}|^{2}+|\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|^{2}-2|h_{AU}||\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|},\end{split}$
(52)

where the last inequality holds due to the fact that

$\begin{split}\lambda_{DU}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})&\triangleq|g+\mathbf{f}^{H}_{DU}\bm{\theta}^{**}_{U}+\mathbf{g}^{H}_{DU}\bm{\theta}^{**}_{D}|^{2}\leq(|g|+|\mathbf{f}^{H}_{DU}\bm{\theta}^{**}_{U}|+|\mathbf{g}^{H}_{DU}\bm{\theta}^{**}_{D}|)^{2}\\\
&\leq
3(|g|^{2}+|\mathbf{f}^{H}_{DU}\bm{\theta}^{**}_{U}|^{2}+|\mathbf{g}^{H}_{DU}\bm{\theta}^{**}_{D}|^{2}).\end{split}$
(53)

Furthermore, let us rewrite
$\tilde{U}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})$ as

$\begin{split}&\tilde{U}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})\\\
&\triangleq\frac{3\bar{\gamma}_{1}(|g|^{2}+N\frac{(|\mathbf{f}^{H}_{DU}\bm{\theta}^{**}_{U}|^{2}+|\mathbf{g}^{H}_{DU}\bm{\theta}^{**}_{D}|^{2})}{N})}{(|h_{DA}|^{2}+N^{2}\frac{|\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|^{2}}{N^{2}}-2N|h_{DA}|\frac{|\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|}{N})(|h_{AU}|^{2}+N^{2}\frac{|\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|^{2}}{N^{2}}-2N|h_{AU}|\frac{|\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|}{N})}\\\
&\quad+\frac{\bar{\gamma}_{2}}{|h_{DA}|^{2}+N^{2}\frac{|\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|^{2}}{N^{2}}-2N|h_{DA}|\frac{|\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}|}{N}}+\frac{\bar{\gamma}_{3}}{|h_{AU}|^{2}+N^{2}\frac{|\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|^{2}}{N^{2}}-2N|h_{AU}|\frac{|\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}|}{N}}.\end{split}$
(54)

Assuming Rayleigh fading channels for all IRS-related links, similarly when
$N\rightarrow\infty$ we obtain

$\begin{split}&|\frac{\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}}{N}|\rightarrow\frac{\pi\varrho_{g_{IA}}\varrho_{g_{DI}}(1-\rho)}{4},\,|\frac{\mathbf{g}^{H}_{DA}\bm{\theta}^{**}_{D}}{N}|^{2}\rightarrow\frac{\pi^{2}\varrho^{2}_{g_{IA}}\varrho^{2}_{g_{DI}}(1-\rho)^{2}}{16},\,|\frac{\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}}{N}|\rightarrow\frac{\pi\varrho_{f_{IU}}\varrho_{f_{AI}}\rho}{4},\\\
&|\frac{\mathbf{f}^{H}_{AU}\bm{\theta}^{**}_{U}}{N}|^{2}\rightarrow\frac{\pi^{2}\varrho^{2}_{f_{IU}}\varrho^{2}_{f_{AI}}\rho^{2}}{16},\,\frac{|\mathbf{f}^{H}_{DU}\bm{\theta}^{**}_{U}|^{2}+|\mathbf{g}^{H}_{DU}\bm{\theta}^{**}_{D}|^{2}}{N}\rightarrow\rho\varrho^{2}_{f_{IU}}\varrho^{2}_{f_{DI}}+(1-\rho)\varrho^{2}_{g_{DI}}\varrho^{2}_{g_{IU}}.\end{split}$
(55)

Therefore, when $N$ becomes asymptotically large, the upper bound of
$L_{3}(\bm{\theta}^{*}_{U},\bm{\theta}^{*}_{D})$ satisfies

$\begin{split}\tilde{U}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})&\rightarrow\frac{3\bar{\gamma}_{1}(|g|^{2}+N(\rho\varrho^{2}_{f_{IU}}\varrho^{2}_{f_{DI}}+(1-\rho)\varrho^{2}_{g_{DI}}\varrho^{2}_{g_{IU}}))}{(|h_{DA}|^{2}+\frac{\varpi^{2}_{g}}{16}N^{2}-\frac{|h_{DA}|\varpi_{g}}{2}N)(|h_{AU}|^{2}+\frac{\varpi^{2}_{f}}{16}N^{2}-\frac{|h_{AU}|\varpi_{f}}{2}N)}\\\
&\quad+\frac{\bar{\gamma}_{2}}{|h_{DA}|^{2}+\frac{\varpi^{2}_{g}}{16}N^{2}-\frac{|h_{DA}|\varpi_{g}}{2}N}+\frac{\bar{\gamma}_{3}}{|h_{AU}|^{2}+\frac{\varpi^{2}_{f}}{16}N^{2}-\frac{|h_{AU}|\varpi_{f}}{2}N}\rightarrow
0,\end{split}$ (56)

where $\varpi_{g}\triangleq\pi\varrho_{g_{IA}}\varrho_{g_{DI}}(1-\rho)$ and
$\varpi_{f}\triangleq\pi\varrho_{f_{IU}}\varrho_{f_{AI}}\rho$. Since the last
two terms on the right hand side (RHS) of (56) dominate in the magnitude of
$\tilde{U}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})$, we can see that
$\tilde{U}(\bm{\theta}^{**}_{U},\bm{\theta}^{**}_{D})$ decreases quadratically
with the increasing of $N$. The proof is thus completed.

##  Appendix D  Proof of equivalence between (27) and (28)

By fixing the other variables, the optimum $v$ for minimizing (28) is given by

$v=\frac{|h_{DA}+\mathbf{g}^{H}_{DA}\bm{\theta}_{D}||h_{AU}+\mathbf{f}^{H}_{AU}\bm{\theta}_{U}|}{\bar{\gamma}_{1}|g+\mathbf{f}^{H}_{DU}\bm{\theta}_{U}+\mathbf{g}^{H}_{DU}\bm{\theta}_{D}|^{2}+\bar{\gamma}_{2}|h_{AU}+\mathbf{f}^{H}_{AU}\bm{\theta}_{U}|^{2}+\bar{\gamma}_{3}|h_{DA}+\mathbf{g}^{H}_{DA}\bm{\theta}_{D}|^{2}}.$
(57)

By substituting (57) into (28), we have the following equivalent optimization
problem:

$\max_{\bm{\phi}_{U},\bm{\phi}_{D}}\quad\frac{|h_{DA}+\mathbf{g}^{H}_{DA}\bm{\psi}_{D}|^{2}|h_{AU}+\mathbf{f}^{H}_{AU}\bm{\psi}_{U}|^{2}}{\bar{\gamma}_{1}|g+\mathbf{f}^{H}_{DU}\bm{\psi}_{U}+\mathbf{g}^{H}_{DU}\bm{\psi}_{D}|^{2}+\bar{\gamma}_{2}|h_{AU}+\mathbf{f}^{H}_{AU}\bm{\psi}_{U}|^{2}+\bar{\gamma}_{3}|h_{DA}+\mathbf{g}^{H}_{DA}\bm{\psi}_{D}|^{2}}.$
(58)

It is readily seen that (27) is equivalent to (28). This thus completes the
proof.

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