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# IRS Assisted MIMO Full Duplex: Rate Analysis and Beamforming Under Imperfect
CSI

Chandan Kumar Sheemar, Sourabh Solanki ,
Jorge Querol, Sumit Kumar, Symeon Chatzinotas The authors are with the
Interdisciplinary Centre for Security Reliability and Trust (SnT), University
of Luxembourg (emails:{chandankumar.sheemar, sourabh.solanki, jorge.querol,
sumit.kumar, symeon.chatzinotas}uni.lu).A preliminary version of this work
will appear in IEEE GLOBECOM 2023.

###### Abstract

Intelligent reflecting surfaces (IRS) have emerged as a promising technology
to enhance the performance of wireless communication systems. By actively
manipulating the wireless propagation environment, IRS enables efficient
signal transmission and reception. In recent years, the integration of IRS
with full-duplex (FD) communication has garnered significant attention due to
its potential to further improve spectral and energy efficiencies. IRS-
assisted FD systems combine the benefits of both IRS and FD technologies,
providing a powerful solution for the next generation of cellular systems. In
this manuscript, we present a novel approach to jointly optimize active and
passive beamforming in a multiple-input-multiple-output (MIMO) FD system
assisted by an IRS for weighted sum rate (WSR) maximization. Given the
inherent difficulty in obtaining perfect channel state information (CSI) in
practical scenarios, we consider imperfect CSI and propose a statistically
robust beamforming strategy to maximize the ergodic WSR. Additionally, we
analyze the achievable WSR for an IRS-assisted MIMO FD system under imperfect
CSI by deriving both the lower and upper bounds. To tackle the problem of
ergodic WSR maximization, we employ the concept of expected weighted minimum
mean squared error (EWMMSE), which exploits the information of the expected
error covariance matrices and ensures convergence to a local optimum. We
evaluate the effectiveness of our proposed design through extensive
simulations. The results demonstrate that our robust approach yields
significant performance improvements compared to the simplistic beamforming
approach that disregards CSI errors, while also outperforming the robust half-
duplex (HD) system considerably.

###### Index Terms:

full duplex, intelligent reflecting surfaces, robust beamforming, imperfect
CSI, ergodic weighted sum rate.

##  I Introduction

Full Duplex (FD) is a promising wireless transmission technology offering
simultaneous transmission and reception in the same frequency band, which
theoretically doubles the spectral efficiency [1, 2]. It is crucial for
sustaining the exponentially ever-increasing data rate demands as it can offer
flexible utilization of the limited wireless spectrum [3, 4]. Beyond spectral
efficiency, FD can be beneficial to improve security, enable advanced joint
communication and sensing, and reduce end-to-end delays [5, 6, 7]. However, a
major hurdle in achieving optimal FD operation is the presence of self-
interference (SI) which can reach up to $90-110$ dB higher than the desired
received signal [8, 9]. Overcoming this challenge is crucial for realizing the
full potential of FD systems.

Sophisticated self-interference cancellation (SIC) techniques hold paramount
significance in effectively reducing the power of SI to levels approaching the
noise floor thereby enabling precise reception of the desired signal. These
techniques can be categorized into two primary classifications: passive and
active SIC [8, 10]. Passive SIC involves attenuating SI by optimizing the path
loss between the transmit and receive antennas of the FD node. Through
strategic manipulation of signal propagation, passive SIC aims to effectively
diminish the power of SI. Active SIC techniques can be further divided into
two distinct domains: analog and digital techniques [11, 12, 13]. Analog SIC
primarily focuses on mitigating SI by replicating the transmitted signal with
an inverted sign prior to reaching the analog-to-digital converters (ADCs).
This approach ensures that an adequate dynamic range is preserved in the ADCs
for the desired signal. Subsequently, the residual SI that remains after the
analog-to-digital conversion stage is addressed using digital SIC techniques
in the baseband [14]. The combined utilization of active and passive SIC
techniques allows for the substantial reduction of SI near the noise floor,
resulting in a twofold increase in spectral efficiency.

Besides FD technology, a promising and emerging innovation called intelligent
reflecting surface (IRS) is garnering considerable attention [15, 16]. These
surfaces provide a dynamic and adaptable wireless environment, offering full
control over the manipulation of the impinging electromagnetic field to
fulfill specific requirements [17]. Constructed from multiple cost-effective
passive meta-elements, IRSs consist of planar surfaces capable of modifying
the reflected signal without the need for a dedicated radio frequency (RF)
chain. As a result, IRSs can be deployed with significantly lower energy costs
compared to traditional active nodes. Through intelligent reconfiguration, an
IRS can fulfill various functions, including enhancing signal reception
quality by bypassing obstacles, compensating for signal fading, introducing
supplementary signal paths, improving channel statistics, and fortifying
wireless network security [18].

The recent literature on the IRSs is available in [19, 20, 21] In [19], the
recent developments in the different types of IRSs and promising candidates
for future research are discussed. In [20], the authors investigate the three-
dimensional physics-based double IRSs for the unmanned aerial vehicle-to
ground communication scenarios. A novel stochastic channel model is proposed
and the critical propagation properties of the proposed channel model, such as
the spatial cross-correlation functions, temporal auto-correlation functions,
and frequency correlation functions, with respect to different RIS reflection
phase configurations are investigated. In [21], a hybrid near-field and far-
field stochastic channel model for characterizing an IRS-assisted vehicle-to-
vehicle propagation environment is proposed. The potential of IRSs combined
with the upcoming sixth-generation (6G) [22] cellular networks can pave the
way towards smartly controlled and energy-efficient wireless systems.

###  I-A State-of-the-Art on IRS Assisted FD systems

The integration of the IRSs with FD (IRS-FD) systems, presents a highly
promising solution for meeting the escalating traffic demands of the future.
These systems offer the potential to achieve optimal utilization of available
resources, encompassing both spectrum and power allocation [23, 24, 25]. By
leveraging the intelligent capabilities of IRS and the simultaneous
transmission and reception capabilities of FD, IRS-FD systems are poised to
revolutionize wireless communication networks. This innovative approach has
garnered significant attention from researchers and experts in the field, with
studies highlighting the transformative potential of IRS-FD in addressing the
evolving requirements of modern wireless communication systems [26].

A collection of recent studies exploring IRS-FD systems can be found in [27,
28, 29, 30, 31, 32, 33, 34]. In [27], the authors investigated the performance
of the IRS-FD system by analyzing the outage and error probabilities.
Moreover, they also investigated the advantage of an IRS in partially reducing
the SI effect. In [28], the authors presented a novel beamforming design for
an FD relay assisted with one IRS to maximize the minimum achievable rate with
the $\max-\min$ optimization. In [29], the advantage of IRS in covering the
dead zones in a multi-user FD system in investigated. The problem of
minimization of the uplink (UL) and downlink (DL) power consumption under the
minimum rate constraints for an IRS-FD system is analyzed in [30]. In [31],
beamforming for point-to-point IRS-FD is investigated. In [32], the authors
studied the advantage of the IRS in improving the security of the FD systems
and presented a novel beamforming design to maximize the worst-case achievable
security rate. The authors in [33] explored a mixed time-scale beamforming
design for an IRS-assisted multi-user FD system. In addition, a deep neural
network is also developed to reduce the computational burden of the proposed
beamforming solution. Finally, in [34], the authors proposed a joint
beamforming design for weighted sum rate (WSR) maximization in a single-cell
multiple-input single-output (MISO) IRS-FD system.

Despite the fruitful results obtained in the realm of IRS-FD systems, these
studies assume perfect channel state information (CSI) availability for
optimizing the beamformers. However, in practical scenarios, this assumption
cannot be fulfilled due to the presence of inevitable errors that corrupt the
CSI. Such errors can lead to suboptimal decisions regarding the selection of
beamformers resulting in significant performance degradation. The impact of
imperfect CSI is particularly pronounced in FD systems compared to their half-
duplex (HD) counterparts [35], as errors in the CSI pertaining to the SI
channel can substantially impair system performance. The consideration of
imperfect CSI in the context of IRS-FD systems has been addressed solely in
the study presented in [32], albeit limited to a simplified point-to-point
MISO FD system. This study considered the secrecy rate as an objective
function. It is noteworthy that the problem of WSR maximization for the IRS-FD
system under imperfect CSI remains unexplored. Furthermore, the state-of-the-
art also lacked detailed analysis for the achievable WSR in IRS-FD systems
under imperfect CSI.

###  I-B Main Contributions

  * •

Motivated by the aforementioned considerations, we delve into the problem of
jointly optimizing the active and passive beamformers for an IRS-FD system
with multi-antenna UL and DL user, shown in Fig. 1. Recognizing the practical
challenges in attaining perfect CSI, we address the case of imperfect CSI and
adopt a statistically robust approach to design the beamformers for ergodic
WSR maximization, which pertain to the average WSR considering the CSI errors
given the channel estimates. It is noteworthy that accounting for CSI errors
in each link, alongside the presence of multi-antenna users, significantly
amplifies the complexity of beamforming design for IRS-FD systems.

  * •

The Gaussian-Kronecker model [36, 37] is adopted to characterize the CSI
errors. By leveraging the statistical distribution information of these
errors, we effectively analyze and quantify the impact of imperfect CSI on the
IRS-FD system’s performance. In our investigation, we take a rigorous
analytical approach to derive both the lower and upper bounds of the
achievable ergodic WSR, which enable us to assess the minimum and maximum
capacity of the system in the presence of imperfect CSI, respectively.

  * •

Subsequently, we formulate the problem of maximizing the ergodic WSR while
simultaneously considering constraints on the total average sum-power at the
base station and the unit-modulus phase-shift on the IRS response. To tackle
this complex problem, we leverage the relationship between the ergodic WSR and
the expected weighted minimum mean squared error (EWMMSE). This sophisticated
approach takes into account the statistical properties of CSI errors during
optimization, while considering the average mean squared error (MSE)
covariance matrices. This transformation reduces the original problem into two
distinct layers of sub-problems, which are iteratively solved. We note that
the IRS also assists in passive SIC for FD systems. However, due to the
imperfect CSI, its potential can be very limited. With the proposed robust
approach, much higher levels of SIC can be achieved, resulting in a higher
ergodic WSR for IRS-FD systems in the presence of CSI errors.

  * •

Finally, a comprehensive set of extensive simulation results is presented to
verify and substantiate the effectiveness of the proposed robust joint
beamforming design. The results corroborate the high accuracy of the derived
analytical ergodic WSR approximation. Furthermore, the proposed statistically
robust design is rigorously benchmarked against various schemes, including the
robust IRS-assisted HD system. Based on the outcomes obtained, it is concluded
that the proposed design significantly outperforms the benchmark schemes,
solidifying its superiority and efficacy under imperfect CSI.

_Paper Organization:_ The rest of the paper is organized as follows. In
Section II, we first present the system model, model the CSI errors and
formulate the problem of ergodic WSR maximization. In Sections III and IV, we
derivate the ergodic WSR and present a novel beamforming scheme that exhibits
robustness based on the WMMSE. Finally, in Sections V and VI, we present the
numerical results and draw meaningful conclusions, respectively.

_Mathematical Notations:_ Boldface lower and upper case characters denote
vectors and matrices, respectively. $\mathbb{E}\\{\cdot\\}$,
$\mbox{Tr}\\{\cdot\\}$, $\mathbf{I}$, denote expectation, trace, identity
matrix, respectively, $\widetilde{\mathbf{X}}$ denotes the transmit covariance
matrix $\widetilde{\mathbf{X}}=\mathbf{X}\mathbf{X}^{H}$. The superscripts
$(\cdot)^{T}$ and $(\cdot)^{H}$ denote transpose and conjugate-transpose
(Hermitian) operators, respectively. The $\mbox{diag}(\mathbf{x})$ denote a
diagonal matrix with vector $\mathbf{x}$ on its main diagonal and $\odot$
denotes Hadamard product. Estimate of matrix $\mathbf{X}$ is denoted with
$\hat{\mathbf{X}}$. $\mathbb{E}_{\mathbf{X}|\hat{\mathbf{X}}}$ denotes that
expectation is taken with respect to $\mathbf{X}$, given its estimate
$\hat{\mathbf{X}}$.

##  II System Model

Let $j$ and $k$ denote the multi-antenna DL and UL users served by the MIMO FD
base station (BS), respectively, and let $N_{j}$ and $M_{k}$ denote their
number of receive and transmit antennas, respectively. The FD BS is assumed to
have $M_{0}$ transmit and $N_{0}$ receive antennas.

Figure 1: An IRS-FD system with multi-antenna UL and DL users.

We consider a multi-stream approach, and the number of data streams for the UL
user $k$ and DL user $j$ are denoted as $u_{k}$ and $v_{j}$, respectively. Let
$\mathbf{U}_{k}\in\mathbb{C}^{M_{k}\times u_{k}}$ and
$\mathbf{V}_{j}\in\mathbb{C}^{M_{0}\times v_{j}}$ denote the precoders for
white unitary-variance data streams $\mathbf{s}_{k}\in\mathbb{C}^{u_{k}\times
1}$ and $\mathbf{s}_{j}\in\mathbb{C}^{v_{j}\times 1}$, respectively. We assume
that the considered FD system is aided with one IRS of size $R\times C$. Let
$\bm{\theta}=[e^{i\theta_{1}},....,e^{i\theta_{RC}}]$ denote the vector
containing the phase-shift response of its $RC$ elements, and let
$\mathbf{\Theta}=\mbox{diag}(\bm{\theta})$ denote a diagonal matrix containing
$\bm{\theta}$ on its main diagonal. Let $\mathbf{n}_{0}$ and $\mathbf{n}_{j}$
denote the noise vectors at the FD BS and DL user $j$, respectively, which are
modelled as

$\mathbf{n}_{0}=\mathcal{CN}(\bm{0},\sigma_{0}^{2}\mathbf{I}),\quad\mathbf{n}_{j}=\mathcal{CN}(\bm{0},\sigma_{j}^{2}\mathbf{I}),$
(1)

where $\sigma_{0}^{2}$ and $\sigma_{j}^{2}$ denote the noise variances at the
BS and the DL user $j$, respectively.

The channel responses from the UL user $k$ to the BS and from the BS to the DL
user $j$ are denoted with $\mathbf{H}_{k}\in\mathbb{C}^{N_{0}\times M_{k}}$
and $\mathbf{H}_{j}\in\mathbb{C}^{N_{j}\times M_{0}}$, respectively. Let
$\mathbf{H}_{0}\in\mathbb{C}^{N_{0}\times M_{0}}$ and
$\mathbf{H}_{j,k}\in\mathbb{C}^{N_{j}\times M_{k}}$ denote the SI channel
response for the FD BS and cross-interference channel response between the UL
user $k$ and the DL user $j$, respectively. The channel responses from the
transmit antenna array of the FD BS to the IRS and from the IRS to the receive
antenna array of the FD BS are denoted with
$\mathbf{H}_{\theta,0}\in\mathbb{C}^{RC\times M_{0}}$ and
$\mathbf{H}_{0,\theta}\in\mathbb{C}^{N_{0}\times RC}$, respectively. Finally,
$\mathbf{H}_{j,\theta}\in\mathbb{C}^{N_{j}\times RC}$ and
$\mathbf{H}_{\theta,k}\in\mathbb{C}^{RC\times M_{k}}$ denote the channel
responses from the IRS to the DL user $j$ and from the UL user $k$ to the IRS,
respectively.

###  II-A Imperfect CSI Modelling

In the context of the aforementioned channel matrices, it is assumed that they
have been initially estimated, potentially through techniques such as those
proposed in [38, 39, 40]. However, it should be emphasized that the problem of
CSI acquisition specifically for IRS-FD systems has not yet been thoroughly
investigated. While various approaches can be employed for CSI acquisition, it
is important to acknowledge that these estimates are prone to errors which are
inevitable in practical scenarios resulting in imperfect CSI. It is crucial to
account for the presence of imperfect CSI as it gives rise to residual SI and
interference in IRS-FD systems, thereby significantly impacting their overall
performance if not appropriately considered during the optimization process.

Let $\Delta\mathbf{H}_{k}$, $\Delta\mathbf{H}_{j}$, $\Delta\mathbf{H}_{0}$,
$\Delta\mathbf{H}_{j,k}$, $\Delta\mathbf{H}_{\theta,0}$,
$\Delta\mathbf{H}_{0,\theta}$, $\Delta\mathbf{H}_{j,\theta}$, and
$\Delta\mathbf{H}_{\theta,k}$ denote the estimation errors for the channel
responses $\mathbf{H}_{k}$, $\mathbf{H}_{j}$, $\mathbf{H}_{0}$,
$\mathbf{H}_{j,k}$, $\mathbf{H}_{\theta,0}$, $\mathbf{H}_{0,\theta}$,
$\mathbf{H}_{j,\theta}$, and $\mathbf{H}_{\theta,k}$, respectively. The true
CSI matrices can be written as a sum of the channel estimates and CSI errors
as

$\displaystyle\mathbf{H}_{k}=\hat{\mathbf{H}}_{k}+\Delta\mathbf{H}_{k},\quad\quad\quad\;\mathbf{H}_{j}=\hat{\mathbf{H}}_{j}+\Delta\mathbf{H}_{j},$
(2)
$\displaystyle\mathbf{H}_{0}=\hat{\mathbf{H}}_{0}+\Delta\mathbf{H}_{0},\quad\quad\quad\;\hskip
1.99168pt\mathbf{H}_{j,k}=\hat{\mathbf{H}}_{j,k}+\Delta\mathbf{H}_{j,k},$
$\displaystyle\mathbf{H}_{\theta,0}=\hat{\mathbf{H}}_{\theta,0}+\Delta\mathbf{H}_{\theta,0},\;\quad\;\mathbf{H}_{0,\theta}=\hat{\mathbf{H}}_{0,\theta}+\Delta\mathbf{H}_{0,\theta}$
$\displaystyle\mathbf{H}_{j,\theta}=\hat{\mathbf{H}}_{j,\theta}+\Delta\mathbf{H}_{j,\theta},\quad\;\;\;\mathbf{H}_{\theta,k}=\hat{\mathbf{H}}_{\theta,k}+\Delta\mathbf{H}_{\theta,k},$

where the channel matrices of the form $\hat{\mathbf{X}}$ denote the channel
estimates. To model the estimation errors, we adopt the Gaussian Kronecker
model [36], which dictates that

$\begin{cases}\Delta\mathbf{H}_{k}=\mathcal{CN}\Big{(}\bm{0},\mathbf{J}_{k}\otimes\mathbf{K}_{k}\Big{)},\\\
\Delta\mathbf{H}_{\theta,0}=\mathcal{CN}\Big{(}\bm{0},\mathbf{J}_{\theta,0}\otimes\mathbf{K}_{\theta,0}\Big{)},\\\
\Delta\mathbf{H}_{0}=\mathcal{CN}\Big{(}\bm{0},\mathbf{J}_{0}\otimes\mathbf{K}_{0}\Big{)},\\\
\Delta\mathbf{H}_{j,k}=\mathcal{CN}\Big{(}\bm{0},\mathbf{J}_{j,k}\otimes\mathbf{K}_{j,k}\Big{)},\\\
\Delta\mathbf{H}_{j}=\mathcal{CN}\Big{(}\bm{0},\mathbf{J}_{j}\otimes\mathbf{K}_{j}\Big{)},\\\
\Delta\mathbf{H}_{0,\theta}=\mathcal{CN}\Big{(}\bm{0},\mathbf{J}_{0}\otimes\mathbf{K}_{\theta}\Big{)},\\\
\Delta\mathbf{H}_{j,\theta}=\mathcal{CN}\Big{(}\bm{0},\mathbf{J}_{j,\theta}\otimes\mathbf{K}_{j,\theta}\Big{)},\\\
\Delta\mathbf{H}_{\theta,k}=\mathcal{CN}\Big{(}\bm{0},\mathbf{J}_{\theta,k}\otimes\mathbf{K}_{\theta,k}\Big{)},\\\
\end{cases}$ (3)

where the matrices $\mathbf{J}$ and $\mathbf{K}$ are the covariance matrices
seen from the transmitter and receiver [36, 37], respectively. The estimation
errors are uncorrelated with the estimated channel matrices [41], and hence we
have

$\begin{cases}\mathbf{H}_{k}=\mathcal{CN}\Big{(}\hat{\mathbf{H}}_{k},\mathbf{J}_{k}\otimes\mathbf{K}_{k}\Big{)},\\\
\mathbf{H}_{\theta,0}=\mathcal{CN}\Big{(}\hat{\mathbf{H}}_{\theta,0},\mathbf{J}_{\theta,0}\otimes\mathbf{K}_{\theta,0}\Big{)},\\\
\mathbf{H}_{0}=\mathcal{CN}\Big{(}\hat{\mathbf{H}}_{0},\mathbf{J}_{0}\otimes\mathbf{K}_{0}\Big{)},\\\
\mathbf{H}_{j,k}=\mathcal{CN}\Big{(}\hat{\mathbf{H}}_{j,k},\mathbf{J}_{j,k}\otimes\mathbf{K}_{j,k}\Big{)},\\\
\mathbf{H}_{j}=\mathcal{CN}\Big{(}\hat{\mathbf{H}}_{j},\mathbf{J}_{j}\otimes\mathbf{K}_{j}\Big{)},\\\
\mathbf{H}_{0,\theta}=\mathcal{CN}\Big{(}\hat{\mathbf{H}}_{0,\theta},\mathbf{J}_{0,\theta}\otimes\mathbf{K}_{0,\theta}\Big{)},\\\
\mathbf{H}_{j,\theta}=\mathcal{CN}\Big{(}\hat{\mathbf{H}}_{j,\theta},\mathbf{J}_{j,\theta}\otimes\mathbf{K}_{j,\theta}\Big{)},\\\
\mathbf{H}_{\theta,k}=\mathcal{CN}\Big{(}\hat{\mathbf{H}}_{\theta,k},\mathbf{J}_{\theta,k}\otimes\mathbf{K}_{\theta,k}\Big{)}.\\\
\end{cases}$ (4)

Let
$\overline{\mathbf{H}}_{0,k},\overline{\mathbf{H}}_{0,0},\overline{\mathbf{H}}_{j},\overline{\mathbf{H}}_{j,k}$
denote the effective channel responses affected by the estimation errors
defined as

$\overline{\mathbf{H}}_{k}=\Big{(}\hat{\mathbf{H}}_{k}+\Delta\mathbf{H}_{k}\Big{)}+\Big{(}\hat{\mathbf{H}}_{0,\theta}+\Delta\mathbf{H}_{0,\theta}\Big{)}\mathbf{\Theta}\Big{(}\hat{\mathbf{H}}_{\theta,k}+\Delta\mathbf{H}_{\theta,k}\Big{)},$
(5a)
$\overline{\mathbf{H}}_{0}=\Big{(}\hat{\mathbf{H}}_{0}+\Delta\mathbf{H}_{0}\Big{)}+\Big{(}\hat{\mathbf{H}}_{0,\theta}+\Delta\mathbf{H}_{0,\theta}\Big{)}\mathbf{\Theta}\Big{(}\hat{\mathbf{H}}_{\theta,0}+\Delta\mathbf{H}_{\theta,0}\Big{)},$
(5b)
$\overline{\mathbf{H}}_{j}=\Big{(}\hat{\mathbf{H}}_{j}+\Delta\mathbf{H}_{j}\Big{)}+\Big{(}\hat{\mathbf{H}}_{j,\theta}+\Delta\mathbf{H}_{j,\theta}\Big{)}\mathbf{\Theta}\Big{(}\hat{\mathbf{H}}_{\theta,0}+\Delta\mathbf{H}_{\theta,0}\Big{)},$
(5c)
$\overline{\mathbf{H}}_{j,k}=\Big{(}\hat{\mathbf{H}}_{j,k}+\Delta\mathbf{H}_{j,k}\Big{)}+\Big{(}\hat{\mathbf{H}}_{j,\theta}+\Delta\mathbf{H}_{j,\theta}\Big{)}\mathbf{\Theta}\Big{(}\hat{\mathbf{H}}_{\theta,k}+\Delta\mathbf{H}_{\theta,k}\Big{)}.$
(5d)

Further, let $\mathbf{y}_{k}$ and $\mathbf{y}_{j}$ denote the signals received
by the FD BS, from UL user $k$, and by the DL user $j$, respectively. By using
(5), they can be written as

$\displaystyle\mathbf{y}_{k}=$
$\displaystyle\overline{\mathbf{H}}_{k}\mathbf{U}_{k}\mathbf{s}_{k}+\overline{\mathbf{H}}_{0}\mathbf{V}_{j}\mathbf{s}_{j}+\mathbf{n}_{0},$
(6a) $\displaystyle\mathbf{y}_{j}=$
$\displaystyle\overline{\mathbf{H}}_{j}\mathbf{V}_{j}\mathbf{s}_{j}+\overline{\mathbf{H}}_{j,k}\mathbf{U}_{k}\mathbf{s}_{k}+\mathbf{n}_{j}.$
(6b)

###  II-B Problem Formulation

In light of the corrupted CSI across all nodes, designing the beamformer and
IRS phase response based solely on the knowledge of estimates
$\hat{\mathbf{H}}_{k},\hat{\mathbf{H}}_{j},\hat{\mathbf{H}}_{0},\hat{\mathbf{H}}_{j,k},\hat{\mathbf{H}}_{\theta,0},\hat{\mathbf{H}}_{0,\theta},\hat{\mathbf{H}}_{j,\theta},\hat{\mathbf{H}}_{\theta,k}$
can result in substantial performance degradation attributable to the inherent
mismatch.

In this endeavor, our objective is to maximize the ergodic WSR of the MIMO
IRS-FD system considering the presence of imperfect CSI along with the
statistics of CSI errors and channel estimates. Let
$\mathcal{R}=\mathcal{R}_{k}+\mathcal{R}_{j}$ represent the WSR of the system
under perfect CSI, where $\mathcal{R}_{k}$ and $\mathcal{R}_{j}$ denote the
weighted rates of users $k$ and $j$, respectively. The ergodic WSR, which
captures the average WSR considering the CSI errors, can be expressed as
$\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}}[\mathcal{R}]$. However, solving this
problem directly poses challenges. To address this, the Jensen’s inequality is
employed, enabling the relocation of the expectation operator onto the
arguments of the logarithm, yielding
$\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}}[\mathcal{R}]\geq\mathcal{R}(\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}})$
[42], where $\mathcal{R}(\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}})$ emphasize
that the rate is evaluated with the expectation taken with respect to the
channel, given the estimates. Formally, the maximization problem for the
ergodic WSR $\mathcal{R}(\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}})$ can be
formulated as follows:

$\underset{\begin{subarray}{c}\mathbf{V}_{j},\mathbf{U}_{k}\end{subarray},\mathbf{\Theta}}{\max}\quad\mathcal{R}_{k}(\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}})+\mathcal{R}_{j}(\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}})$
(7a)
$\text{s.t.}\quad\mbox{Tr}\Big{(}\mathbf{U}_{k}\mathbf{U}_{k}^{H}\Big{)}\preceq\alpha_{k},$
(7b)
$\quad\quad\mbox{Tr}\Big{(}\mathbf{V}_{j}\mathbf{V}_{j}^{H}\Big{)}\leq\alpha_{0},$
(7c) $\quad\quad\quad\Big{|}\bm{\theta}(i)\Big{|}=1,\quad\forall i.$ (7d)

The statistical expectation is taken with respect to the CSI, with the
distribution given in (4). As the precise CSI remains unknown, the constraints
(7b)-(7c) denote the average power constraint in UL and DL, given the channel
estimates $\hat{\mathbf{H}}$, and (7d) denotes the unit-modulus constraint
imposed on the IRS phase-response.

##  III Ergodic WSR Analysis with Imperfect CSI

In the subsequent analysis, we derive the expression for the ergodic WSR
$\mathcal{R}(\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}})$ considering the
presence of imperfect CSI in the MIMO IRS-FD system. We define
$\widetilde{\mathbf{U}}_{k}=\mathbf{U}_{k}\mathbf{U}_{k}^{H}$ and
$\widetilde{\mathbf{V}}_{j}=\mathbf{V}_{j}\mathbf{V}_{j}^{H}$ as the transmit
covariance matrices for UL user $k$ and DL user $j$, respectively. When
imperfect CSI is taken into account, the received signal plus interference and
noise covariance matrices, denoted as $\mathbf{R}_{k}$ and $\mathbf{R}_{j}$,
encompassing both the CSI errors and the IRS phase response $\mathbf{\Theta}$,
can be expressed as follows:

$\displaystyle\mathbf{R}_{k}=$
$\displaystyle\overline{\mathbf{H}}_{k}\widetilde{\mathbf{U}}_{k}\overline{\mathbf{H}}_{k}^{H}+\overline{\mathbf{H}}_{0}\widetilde{\mathbf{V}}_{j}\overline{\mathbf{H}}_{0}^{H}+\sigma_{0}^{2}\mathbf{I}.$
(8a) $\displaystyle\mathbf{R}_{j}=$
$\displaystyle\overline{\mathbf{H}}_{j}\widetilde{\mathbf{V}}_{j}\overline{\mathbf{H}}_{j}^{H}+\overline{\mathbf{H}}_{j,k}\widetilde{\mathbf{U}}_{k}\overline{\mathbf{H}}_{j,k}^{H}+\sigma_{j}^{2}\mathbf{I}.$
(8b)

The interference plus noise covariance matrices can be obtained as
$\mathbf{R}_{\overline{k}}=\mathbf{R}_{k}-\mathbf{S}_{k},\mathbf{R}_{\overline{j}}=\mathbf{R}_{j}-\mathbf{S}_{j}$,
with $\mathbf{S}_{k}$ and $\mathbf{S}_{j}$ denoting the useful received signal
covariance part.

######  Theorem 1.

Given the statistical distribution of the CSI errors (3) and the channel
estimates, the ergodic WSR
$\mathcal{R}(\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}})$ of an MIMO IRS-FD
system under imperfect CSI can be approximated as

$\displaystyle\mathcal{R}(\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}})$
$\displaystyle=w_{k}\mbox{ln}\Big{[}\mbox{det}\Big{(}\mathbf{I}+\mathbf{U}_{k}^{H}\Big{(}\hat{\mathbf{H}}_{k}+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}\Big{)}^{H}\mathbf{\Sigma}_{\overline{k}}^{-1}$
(9)
$\displaystyle\quad\quad\quad\quad\quad\quad\Big{(}\hat{\mathbf{H}}_{k}+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}\Big{)}\mathbf{U}_{k}\Big{)}\Big{]}$
$\displaystyle+w_{j}\mbox{ln}\Big{[}\mbox{det}\Big{(}\mathbf{I}+\mathbf{V}_{j}^{H}\Big{(}\hat{\mathbf{H}}_{j}+\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}\Big{)}^{H}\mathbf{\Sigma}_{\overline{j}}^{-1}$
$\displaystyle\quad\quad\quad\quad\quad\quad\Big{(}\hat{\mathbf{H}}_{j}+\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}\Big{)}\mathbf{V}_{j}\Big{)}\Big{]},$

where $\mathbf{\Sigma}_{\overline{k}}$ and $\mathbf{\Sigma}_{\overline{j}}$
are given as in (10), at the top of the next page and $w_{k}$ and $w_{j}$
denote the weights.

$\displaystyle\mathbf{\Sigma}_{\overline{k}}$
$\displaystyle=\mbox{Tr}\Big{(}\widetilde{\mathbf{U}}_{k}\mathbf{J}_{k}^{T}\Big{)}\mathbf{K}_{k}+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\mbox{Tr}\Big{(}\widetilde{\mathbf{U}}_{k}\mathbf{J}_{\theta,k}^{T}\Big{)}\mathbf{K}_{\theta,k}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}+\mbox{Tr}\Big{(}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}\mathbf{\Theta}^{H}\mathbf{J}_{0,\theta}^{T}\Big{)}\mathbf{K}_{0,\theta}+\mbox{Tr}\Big{(}\widetilde{\mathbf{U}}_{k}\mathbf{J}_{\theta,k}^{T}\Big{)}$
(10a)
$\displaystyle\times\mbox{Tr}\Big{(}\mathbf{\Theta}\mathbf{K}_{\theta,k}\mathbf{\Theta}^{H}\mathbf{J}_{0,\theta}^{T}\Big{)}\mathbf{K}_{0,\theta}+\hat{\mathbf{H}}_{0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{0}^{H}+\hat{\mathbf{H}}_{0}\widetilde{\mathbf{V}}_{j}\mathbf{H}_{\theta,0}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}+\mbox{Tr}\Big{(}\widetilde{\mathbf{V}}_{j}\mathbf{J}_{0}^{T}\Big{)}\mathbf{K}_{0}^{T}+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{0}^{H}$
$\displaystyle+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\mathbf{H}_{\theta,0}\widetilde{\mathbf{V}}_{j}\mathbf{H}_{\theta,0}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\mbox{Tr}\Big{(}\widetilde{\mathbf{V}}_{j}\mathbf{J}_{\theta,0}^{T}\Big{)}\mathbf{K}_{\theta,0}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}+\mbox{Tr}\Big{(}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\mathbf{J}_{0,\theta}^{T}\Big{)}\mathbf{K}_{0,\theta}$
$\displaystyle+\mbox{Tr}\Big{(}\widetilde{\mathbf{V}}_{j}\mathbf{J}_{\theta,0}^{T}\Big{)}\mbox{Tr}\Big{(}\mathbf{\Theta}\mathbf{K}_{\theta,0}\mathbf{\Theta}^{H}\mathbf{J}_{0,\theta}^{T}\Big{)}\mathbf{K}_{0,\theta}+\sigma_{0}^{2}\mathbf{I},$
$\displaystyle\mathbf{\Sigma}_{\overline{j}}$
$\displaystyle=\mbox{Tr}\Big{(}\widetilde{\mathbf{V}}_{j}\mathbf{J}_{j}^{T}\Big{)}\mathbf{K}_{j}+\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\mbox{Tr}\Big{(}\widetilde{\mathbf{V}}_{j}\mathbf{J}_{\theta,0}^{T}\Big{)}\mathbf{K}_{\theta,0}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}+\mbox{Tr}\Big{(}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\mathbf{J}_{j,\theta}^{T}\Big{)}\mathbf{K}_{j,\theta}+\mbox{Tr}\Big{(}\widetilde{\mathbf{V}}_{j}\mathbf{J}_{\theta,0}^{T}\Big{)}$
(10b)
$\displaystyle\times\mbox{Tr}\Big{(}\mathbf{\Theta}\mathbf{K}_{\theta,0}\mathbf{\Theta}^{H}\mathbf{J}_{j,\theta}^{T}\Big{)}\mathbf{K}_{j,\theta}+\hat{\mathbf{H}}_{j,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{j,k}^{H}+\hat{\mathbf{H}}_{j,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}+\mbox{Tr}\Big{(}\widetilde{\mathbf{U}}_{k}\mathbf{J}_{j,k}^{T}\Big{)}\mathbf{K}_{j,k}+\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\mathbf{H}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{j,k}$
$\displaystyle+\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\mathbf{H}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}+\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\mbox{Tr}\Big{(}\widetilde{\mathbf{U}}_{k}\mathbf{J}_{\theta,k}^{T}\Big{)}\mathbf{K}_{\theta,k}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}+\mbox{Tr}\Big{(}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\mathbf{J}_{j,\theta}^{T}\Big{)}\mathbf{K}_{j,\theta}$
$\displaystyle+\mbox{Tr}\Big{(}\widetilde{\mathbf{U}}_{k}\mathbf{J}_{\theta,k}^{T}\Big{)}\mbox{Tr}\Big{(}\mathbf{\Theta}\mathbf{K}_{\theta,k}\mathbf{\Theta}^{H}\mathbf{J}_{j,\theta}^{T}\Big{)}\mathbf{K}_{j,\theta}+\sigma_{j}^{2}\mathbf{I}.$

###### Proof.

We prove the result for UL user $k$, and a similar reasoning can be carried
out for the DL user $j$. The total received covariance matrix $\mathbf{R}_{k}$
in (8a), given the CSI estimates and the CSI errors, can be written as

$\displaystyle\mathbf{R}_{k}$
$\displaystyle=\hat{\mathbf{H}}_{k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{k}^{H}+\Delta\mathbf{H}_{k}\widetilde{\mathbf{U}}_{k}\Delta\mathbf{H}_{k}^{H}+\hat{\mathbf{H}}_{k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}$
(11)
$\displaystyle+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{k}^{H}+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}$
$\displaystyle+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}$
$\displaystyle+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\Delta\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\Delta\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}$
$\displaystyle+\Delta\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\Delta\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\Delta\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\Delta\hat{\mathbf{H}}_{0,\theta}^{H}$
$\displaystyle+\Delta\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\Delta\hat{\mathbf{H}}_{0,\theta}^{H}$
$\displaystyle+\hat{\mathbf{H}}_{0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{0}^{H}+\Delta\mathbf{H}_{0}\widetilde{\mathbf{V}}_{j}\Delta\mathbf{H}_{0}^{H}+\hat{\mathbf{H}}_{0}\widetilde{\mathbf{V}}_{j}\mathbf{H}_{\theta,0}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}$
$\displaystyle+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\mathbf{H}_{\theta,0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{0}^{H}+\mathbf{H}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\mathbf{H}_{0,\theta}^{H}$
$\displaystyle+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\Delta\mathbf{H}_{\theta,0}\widetilde{\mathbf{V}}_{j}\Delta\mathbf{H}_{\theta,0}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}$
$\displaystyle+\Delta\mathbf{H}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\Delta\mathbf{H}_{0,\theta}^{H}$
$\displaystyle+\Delta\mathbf{H}_{0,\theta}\mathbf{\Theta}\Delta\mathbf{H}_{\theta,0}\widetilde{\mathbf{V}}_{j}\Delta\mathbf{H}_{\theta,0}^{H}\mathbf{\Theta}^{H}\Delta\mathbf{H}_{0,\theta}^{H}+\mathbf{L},$

where $\mathbf{L}$ includes terms which are linear in the CSI errors. Let
$\mathbf{\Sigma}_{k}=\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}}[\mathbf{R}_{k}]$
denote the expected signal plus interference plus noise covariance matrix,
where the expectation is carried out with respect to the channel responses,
including CSI errors. Consider the result given in [36, Lemma 1] for imperfect
CSI under the Gaussian-Kronecker model stating that: For
$\mathbf{H}\sim\mathcal{CN}(\hat{\mathbf{H}},\mathbf{J}\otimes\mathbf{K})$,
there is
$\mathbb{E}[\mathbf{H}\mathbf{X}\mathbf{H}^{H}]=\hat{\mathbf{H}}\mathbf{X}\hat{\mathbf{H}}^{H}+\mbox{Tr}(\mathbf{X}\mathbf{J}^{T})\mathbf{K}$
and
$\mathbb{E}[\mathbf{H}^{H}\mathbf{X}\mathbf{H}]=\hat{\mathbf{H}}^{H}\mathbf{X}\hat{\mathbf{H}}+\mbox{Tr}(\mathbf{K}\mathbf{X})\mathbf{J}^{T}$.
For the MIMO IRS-FD system, we first subtract the useful signal part from
$\mathbf{\Sigma}_{k}$ which depends only on the estimated channel responses
and then apply the result on each term of the expected interference plus noise
covariance matrix $\mathbf{\Sigma}_{\overline{k}}$. Ignoring the terms which
are linear in the CSI errors, and by applying the result above, it can be
shown that the ergodic WSR of the UL user $k$ under imperfect CSI has the
interference plus noise covariance matrix structure
$\mathbf{\Sigma}_{\overline{k}}$ given in (10a). ∎

It will be shown in Section V that the aforementioned approximation of the
ergodic rate achieves a high level of accuracy. It is important to note that
(9) serves as a lower bound on the achievable WSR for an IRS-FD system
operating under imperfect CSI conditions. Therefore, maximizing (9) is
equivalent to maximizing the worst-case WSR. Conversely, an upper bound on the
achievable ergodic WSR for the IRS-FD system is given by considering the ideal
CSI scenario, wherein the CSI error variance for all channels tends to zero.

##  IV Robust Joint Active and Passive Beamforming Via EWMMSE

The problem of maximizing the ergodic WSR as formulated in (7) is inherently
non-convex due to the presence of interference. However, in the ideal scenario
of perfect CSI, an equivalent problem formulation known as weighted minimum
mean squared error (WMMSE) can be employed, leveraging the established
relationship between WSR and MSE [43]. In the case of imperfect CSI, the WMMSE
problem can be further transformed into the framework of EWMMSE, which
incorporates the expectation with respect to the CSI errors [44, 42]. This
approach involves considering the average MSE covariance matrices and we adopt
this methodology throughout the paper.

###  IV-A Digital Combining

Assume that the FD BS for UL user $k$ and the DL user $j$ apply the combiners
$\mathbf{F}_{k}$ and $\mathbf{F}_{j}$ to estimate their data streams as

$\hat{\mathbf{s}}_{k}=\mathbf{F}_{k}\mathbf{y}_{k},\quad\hat{\mathbf{s}}_{j}=\mathbf{F}_{j}\mathbf{y}_{j}.$
(12)

Given (12), let $\mathbf{E}_{\tilde{k}}$ and $\mathbf{E}_{\tilde{j}}$ denote
the MSE error matrices for instantaneous CSI for $k$-th UL user and $j$-th DL
user, respectively, which can be written as

$\mathbf{E}_{\tilde{k}}=\mathbb{E}\Big{[}(\mathbf{F}_{k}\mathbf{y}_{k}-\hat{\mathbf{s}}_{k})(\mathbf{F}_{k}\mathbf{y}_{k}-\hat{\mathbf{s}}_{k})^{H}\Big{]},$
(13a)
$\mathbf{E}_{\tilde{j}}=\mathbb{E}\Big{[}(\mathbf{F}_{j}\mathbf{y}_{j}-\hat{\mathbf{s}}_{j})(\mathbf{F}_{j}\mathbf{y}_{j}-\hat{\mathbf{s}}_{j})^{H}\Big{]}.$
(13b)

Let $\mathbf{Q}_{k},\mathbf{T}_{j},\mathbf{T}_{0}$, and $\mathbf{Q}_{j,k}$
denote the matrices defined as

$\mathbf{Q}_{k}=\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}}\Big{(}\overline{\mathbf{H}}_{k}\widetilde{\mathbf{U}}_{k}\overline{\mathbf{H}}_{k}^{H}\Big{)},\;\mathbf{T}_{j}=\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}}\Big{(}\overline{\mathbf{H}}_{j}\widetilde{\mathbf{V}}_{j}\overline{\mathbf{H}}_{j}^{H}\Big{)},$
(14a)
$\mathbf{T}_{0}=\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}}\Big{(}\overline{\mathbf{H}}_{0}\widetilde{\mathbf{V}}_{j}\overline{\mathbf{H}}_{0}^{H}\Big{)},\;\mathbf{Q}_{j,k}=\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}}\Big{(}\overline{\mathbf{H}}_{j,k}\widetilde{\mathbf{U}}_{k}\overline{\mathbf{H}}_{j,k}^{H}\Big{)},$
(14b)

which are given in Appendix A. The optimization of the combiners can be based
on the expected MSE (EMSE) covariance matrices, given below

$\displaystyle\mathbf{E}_{k}$
$\displaystyle=\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}}\Big{[}\mathbf{E}_{\tilde{k}}\Big{]}$
(15a)
$\displaystyle=\mathbf{F}_{k}\mathbf{Q}_{k}\mathbf{F}_{k}^{H}-\mathbf{F}_{k}(\hat{\mathbf{H}}_{k}+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k})\mathbf{U}_{k}+\mathbf{F}_{k}\mathbf{T}_{0}\mathbf{F}_{k}^{H}$
$\displaystyle\quad+\sigma_{0}^{2}\mathbf{F}_{k}\mathbf{F}_{k}^{H}-\mathbf{U}_{k}^{H}\hat{\mathbf{H}}_{k}^{H}\mathbf{F}_{k}^{H}-\mathbf{U}_{k}^{H}\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}\mathbf{F}_{k}^{H}-\mathbf{I},$
$\displaystyle\mathbf{E}_{j}$
$\displaystyle=\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}}\Big{[}\mathbf{E}_{\tilde{j}}\Big{]}$
(15b)
$\displaystyle=\mathbf{F}_{j}\mathbf{T}_{j}\mathbf{F}_{j}^{H}-\mathbf{F}_{j}(\hat{\mathbf{H}}_{j}+\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0})\mathbf{V}_{j}+\mathbf{F}_{j}\mathbf{Q}_{j,k}\mathbf{F}_{j}^{H}$
$\displaystyle\quad+\sigma_{j}^{2}\mathbf{F}_{j}\mathbf{F}_{j}^{H}-\mathbf{V}_{j}^{H}\hat{\mathbf{H}}_{j}^{H}\mathbf{F}_{j}^{H}-\mathbf{V}_{j}^{H}\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}\mathbf{F}_{j}^{H}-\mathbf{I}.$

The optimization problem of the combiners can be stated as the minimization of
the error covariance matrices (15) as

$\underset{\begin{subarray}{c}\mathbf{F}_{l},\mathbf{F}_{r}\end{subarray}}{\min}\quad\mbox{Tr}\Big{(}\mathbf{E}_{k}\Big{)}+\mbox{Tr}\Big{(}\mathbf{E}_{l}\Big{)}.$
(16)

By solving (16), we get the following optimal combiners

$\mathbf{F}_{k}=\mathbf{U}_{k}^{H}\Big{(}\hat{\mathbf{H}}_{k}^{H}+\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}\Big{)}\Big{(}\mathbf{Q}_{k}+\mathbf{T}_{0}+\sigma_{0}^{2}\mathbf{I}\Big{)}^{-1},$
(17a)
$\mathbf{F}_{j}=\mathbf{V}_{j}^{H}\big{(}\hat{\mathbf{H}}_{j}^{H}+\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}\big{)}\big{(}\mathbf{T}_{j}+\mathbf{Q}_{j,k}+\sigma_{j}^{2}\mathbf{I}\big{)}^{-1}.$
(17b)

###  IV-B Active Digital Beamforming Under Imperfect CSI

Given the optimal combiners, the EWMMSE problem with respect to the digital
beamformers under the average total sum-power constraint and given the IRS
phase response $\mathbf{\Theta}$ fixed, can be formally stated as

$\underset{\begin{subarray}{c}\mathbf{V}_{j},\mathbf{U}_{k}\end{subarray}}{\min}\quad\mbox{Tr}\Big{(}\mathbf{W}_{k}\mathbf{E}_{k}\Big{)}+\mbox{Tr}\Big{(}\mathbf{W}_{j}\mathbf{E}_{j}\Big{)},$
(18a) $\text{s.t.}\hskip
5.69054pt\mbox{Tr}\Big{(}\mathbf{U}_{k}\mathbf{U}_{k}^{H}\Big{)}\preceq\alpha_{k},$
(18b)
$\quad\;\;\mbox{Tr}\Big{(}\mathbf{V}_{j}\mathbf{V}_{j}^{H}\Big{)}\leq\alpha_{0},$
(18c)

where $\mathbf{W}_{i}$ is a constant weight matrix associated with node $i$.
Problem (18) and (7) are equivalent if their gradients are the same, which can
be ensured by selecting the weight matrices as follows:

$\mathbf{W}_{k}=\frac{w_{k}}{\mbox{ln}\;2}\Big{(}\mathbf{E}_{k}\Big{)}^{-1},\quad\mathbf{W}_{j}=\frac{w_{j}}{\mbox{ln}\;2}\Big{(}\mathbf{E}_{j}\Big{)}^{-1}.$
(19)

The equivalence between the two problems can be demonstrated by following a
similar proof as provided in [45, Appendix A], but adapted for the case of
expected MSE under imperfect CSI.

To optimize the digital beamformers $\mathbf{V}_{j}$ and $\mathbf{U}_{k}$, we
calculate the partial derivative of the Lagrangian function of (18) with
respect to their conjugate. This calculation yields the following optimal
beamformers.

$\displaystyle\mathbf{X}_{k}=$
$\displaystyle\Big{(}\hat{\mathbf{H}}_{k}^{H}+\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}\Big{)}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\hat{\mathbf{H}}_{k}+\mbox{Tr}\Big{(}\mathbf{K}_{k}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\Big{)}\mathbf{J}_{k}^{T}+\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}$
(20a)
$\displaystyle+\mathbf{H}_{k}^{H}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}+\mbox{Tr}\Big{(}\mathbf{K}_{\theta,k}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\Big{)}\mathbf{J}_{\theta,k}^{T}+\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\mbox{Tr}(\mathbf{K}_{0,\theta}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k})\mathbf{J}_{0,\theta}^{T}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}$
$\displaystyle+\hat{\mathbf{H}}_{j,k}^{H}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}+\Big{(}\hat{\mathbf{H}}_{j,k}^{H}+\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}\Big{)}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\hat{\mathbf{H}}_{j,k}+\mbox{Tr}\Big{(}\mathbf{K}_{j,k}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\Big{)}\mathbf{J}_{j,k}^{T}$
$\displaystyle+\mbox{Tr}\Big{(}\mathbf{K}_{0,\theta}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k})\mbox{Tr}(\mathbf{K}_{\theta,k}\mathbf{\Theta}^{H}\mathbf{J}_{0,\theta}^{T}\mathbf{\Theta}\Big{)}\mathbf{J}_{\theta,k}^{T}+\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}$
$\displaystyle+\mbox{Tr}\Big{(}\mathbf{K}_{\theta,k}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\Big{)}\mathbf{J}_{\theta,k}^{T}+\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\mbox{Tr}\Big{(}\mathbf{K}_{j,\theta}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\Big{)}\mathbf{J}_{j,\theta}^{T}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}$
$\displaystyle+\mbox{Tr}\Big{(}\mathbf{K}_{j,\theta}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\Big{)}\mbox{Tr}\Big{(}\mathbf{K}_{\theta,k}\mathbf{\Theta}^{H}\mathbf{J}_{j,\theta}^{T}\mathbf{\Theta}\Big{)}\mathbf{J}_{\theta,k}^{T},$
$\displaystyle\mathbf{X}_{j}=$
$\displaystyle\Big{(}\hat{\mathbf{H}}_{j}^{H}+\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}\Big{)}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\hat{\mathbf{H}}_{j}+\mbox{Tr}\Big{(}\mathbf{K}_{j}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\Big{)}\mathbf{J}_{j}^{T}+\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}$
(20b)
$\displaystyle+\hat{\mathbf{H}}_{j}^{H}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}+\mbox{Tr}\Big{(}\mathbf{K}_{\theta,0}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\Big{)}\mathbf{J}_{\theta,0}^{T}+\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\mbox{Tr}\Big{(}\mathbf{K}_{j,\theta}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\Big{)}\mathbf{J}_{j,\theta}^{T}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}$
$\displaystyle+\mathbf{H}_{0}^{H}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}+\Big{(}\hat{\mathbf{H}}_{0}^{H}+\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}\Big{)}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\hat{\mathbf{H}}_{0}+\mbox{Tr}\Big{(}\mathbf{K}_{0}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\Big{)}\mathbf{J}_{0}^{T}$
$\displaystyle+\mbox{Tr}\Big{(}\mathbf{K}_{j,\theta}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\Big{)}\mbox{Tr}\Big{(}\mathbf{K}_{\theta,0}\mathbf{\Theta}^{H}\mathbf{J}_{j,\theta}^{T}\mathbf{\Theta}\Big{)}\mathbf{J}_{\theta,0}^{T}+\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}+\mbox{Tr}\Big{(}\mathbf{K}_{\theta,0}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}$
$\displaystyle\;\;\;\;\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\Big{)}\mathbf{J}_{\theta,0}^{T}+\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\mbox{Tr}\Big{(}\mathbf{K}_{0,\theta}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\Big{)}\mathbf{J}_{0,\theta}^{T}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}+\mbox{Tr}\Big{(}\mathbf{K}_{0,\theta}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\Big{)}\mbox{Tr}\Big{(}\mathbf{K}_{\theta,0}\mathbf{\Theta}^{H}\mathbf{J}_{0,\theta}^{T}\mathbf{\Theta}\Big{)}\mathbf{J}_{\theta,0}^{T}.$

$\mathbf{U}_{k}=\Big{(}\mathbf{X}_{k}+\lambda_{k}\mathbf{I}\Big{)}^{-1}\Big{(}\hat{\mathbf{H}}_{k}^{H}+\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}\Big{)}\mathbf{F}_{k}^{H}\mathbf{W}_{k},$
(21a)
$\mathbf{V}_{j}=\Big{(}\mathbf{X}_{j}+\lambda_{0}\mathbf{I}\Big{)}^{-1}\Big{(}\hat{\mathbf{H}}_{j}^{H}+\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}\Big{)}\mathbf{F}_{j}^{H}\mathbf{W}_{j},$
(21b)

where $\mathbf{X}_{k}$ and $\mathbf{X}_{j}$ are defined in (20a) and (20b),
respectively, and the scalars $\lambda_{k}$ and $\lambda_{0}$ denote the
Lagrange multipliers for the uplink user $k$ and the FD BS. The multipliers
can be searched while performing power allocation for the users given the
average total sum-power constraints. Namely, consider the singular value
decomposition (SVD) of the matrices as
$\mathbf{X}_{k}=\mathbf{A}_{k}\mathbf{\Lambda}_{k}\mathbf{B}_{k}$ and
$\mathbf{X}_{j}=\mathbf{A}_{j}\mathbf{\Lambda}_{j}\mathbf{B}_{j}$, where
$\mathbf{A}_{i}$ and $\mathbf{B}_{i}$ denote the left and right unitary
matrices obtained with SVD and $\mathbf{\Lambda}_{i}$ denote the singular
values. The average power constraints (7b) and (7c), after some
simplifications, can be written as

$\mbox{Tr}\Big{(}\mathbf{V}_{j}\mathbf{V}_{j}^{H}\Big{)}=\frac{\sum_{i=i}^{M_{0}}\mathbf{S}_{j}(i,i)}{(\lambda_{0}+\mathbf{\Lambda}_{j}(i,i))^{2}},$
(22a)
$\mbox{Tr}\Big{(}\mathbf{U}_{k}\mathbf{U}_{k}^{H}\Big{)}=\frac{\sum_{i=i}^{M_{k}}\mathbf{S}_{k}(i,i)}{(\lambda_{j}+\mathbf{\Lambda}_{k}(i,i))^{2}}.$
(22b)

where the matrices $\mathbf{S}_{k}$ and $\mathbf{S}_{j}$ are defined as

$\displaystyle\mathbf{S}_{j}=\mathbf{B}_{j}\Big{(}\hat{\mathbf{H}}_{j}+\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}\Big{)}^{H}$
$\displaystyle\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{W}_{j}\mathbf{F}_{j}$
(23a)
$\displaystyle\Big{(}\hat{\mathbf{H}}_{j}+\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}\Big{)}\mathbf{A}_{j},$
$\displaystyle\mathbf{S}_{k}=\mathbf{B}_{k}\Big{(}\hat{\mathbf{H}}_{k}+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}\Big{)}^{H}$
$\displaystyle\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{W}_{k}\mathbf{F}_{k}$
(23b)
$\displaystyle\Big{(}\hat{\mathbf{H}}_{k}+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}\Big{)}\mathbf{A}_{k}.$

The optimal Lagrange multipliers that satisfy the average total power
constraints can be determined using a linear search technique, such as the
Bisection method, which we employ in this study. If the calculated values of
the multipliers are found to be negative, we set them to zero. It is important
to note that the average power constraint is fulfilled based on the estimated
CSI.

###  IV-C Passive Beamforming Under Imperfect CSI

In this section, we delve into the optimization of the phase response of the
IRS with the objective of simultaneously enhancing the UL and DL channels
while mitigating the impact of SI and interference in the presence of
imperfect CSI.

Let $\mathbf{S},\mathbf{T}$, and $\mathbf{Z}$ denote the matrices independent
of the IRS phase response, given in Appendix B. The expected WMMSE
optimization problem for the IRS phase response $\mathbf{\Theta}$ under
imperfect CSI, given the matrices $\mathbf{S},\mathbf{T}$, and $\mathbf{Z}$,
can be formally stated as

$\underset{\begin{subarray}{c}\mathbf{\Theta}\end{subarray}}{\min}\quad\mbox{Tr}\Big{(}\mathbf{\Theta}^{H}\mathbf{Z}\mathbf{\Theta}\mathbf{T}\Big{)}+\mbox{Tr}\Big{(}\mathbf{\Theta}^{H}\mathbf{S}^{H}\Big{)}+\mbox{Tr}\Big{(}\mathbf{\Theta}\mathbf{S}\Big{)}+c,$
(24a) $\text{s.t.}\quad\quad\quad\Big{|}\bm{\theta}(i)\Big{|}=1,\quad\forall
i,$ (24b)

where the scalar $c$ denotes the constant terms, independent of
$\mathbf{\Theta}$. In (24a), the problem is stated with respect to the matrix
$\mathbf{\Theta}$. However, we wish to maximize only the diagonal response of
such matrix to maximize the ergodic WSR or minimize the expected MSE, because
the off-diagonal elements result to be zero. Therefore, we first consider
restating the problem (24a) with respect to $\bm{\theta}$ instead of
$\mathbf{\Theta}$. To this end, based on the result [46, identity 1.10.6], we
write the first term of (24a) as

$\mbox{Tr}\Big{(}\mathbf{\Theta}^{H}\mathbf{Z}\mathbf{\Theta}\mathbf{T}\Big{)}=\bm{\theta}^{H}\mathbf{\Sigma}\bm{\theta},\quad\mbox{where}\;\mathbf{\Sigma}=\mathbf{Z}\odot\mathbf{T}^{T}.$
(25)

Let $\mathbf{s}$ denote the vector containing only the diagonal elements of
the matrix $\mathbf{S}$. Then, the second and the third term of the problem
(24a) can be restated as a function of the vector $\bm{\theta}$ as

$\mbox{Tr}\Big{(}\mathbf{\Theta}^{H}\mathbf{S}^{H}\Big{)}={\mathbf{s}}^{H}\bm{\theta}^{*},\quad\mbox{Tr}\Big{(}\mathbf{\Theta}\mathbf{S}\Big{)}={\mathbf{s}}^{T}\bm{\theta}.$
(26)

By using the results stated above, the overall optimization problem (24a) can
be restated with respect to the vector $\bm{\theta}$ as

$\underset{\begin{subarray}{c}\bm{\theta}\end{subarray}}{\min}\quad\bm{\theta}^{H}\mathbf{\Sigma}\bm{\theta}+{\mathbf{s}_{r}}^{H}\bm{\theta}^{*}+{\mathbf{s}}^{T}\bm{\theta}_{l},$
(27a) $\text{s.t.}\quad\Big{|}\bm{\theta}(i)\Big{|}=1,\quad\forall i.$ (27b)

Problem (27) is still very challenging as it is a non-convex problem due to
the unit-modulus constraint. To solve it we adopt the majorization-
maximization method [47]. Such method aims to solve a more difficult problem
by constructing a series of more tractable problems stated with the upper
bound. Let $\mathcal{R}(\bm{\theta}^{(n)})$ denote the function evaluating
(27) at the $n$-th iteration for computed $\bm{\theta}$. Let
$\mathcal{R}_{u}(\bm{\theta}|\bm{\theta}^{(n)})$ denote an upper bound
constructed at $n$-th iteration for $\mathcal{R}$. According to [47], for the
problem of the form (27), an upper bound can be constructed as

$\mathcal{R}_{u}(\bm{\theta}|\bm{\theta}^{(n)})=2\mbox{Re}\\{{\mathbf{s}}^{H}\mathbf{q}^{(n)}\\}+c_{u},$
(28)

where $c_{u}$ denote constant terms in the upper bound independent of
$\bm{\theta}$ and $\mathbf{q}^{(n)}$ is given by

$\mathbf{q}^{(n)}=\Big{(}\lambda^{max}\mathbf{I}-\mathbf{\Sigma}\Big{)}\bm{\theta}^{(n)}-{\mathbf{s}}^{*},$
(29)

with $\lambda^{max}$ denoting the maximum eigenvalues of $\mathbf{\Sigma}$.
Based on the result above, the hard non-convex optimization problem (27a)
simplifies to a series of the following problem

$\underset{\begin{subarray}{c}\bm{\theta}\end{subarray}}{\min}\quad
2\mbox{Re}\\{{\mathbf{s}}^{H}\mathbf{q}^{(n)}\\},$ (30a)
$\text{s.t.}\quad\Big{|}\bm{\theta}(i)\Big{|}=1,\quad\forall i,$ (30b)

which need to be solved iteratively until convergence for each update of
$\bm{\theta}$. By solving (30a), we get the optimal solution of $\bm{\theta}$
at $(n+1)$-th iteration, given the $\bm{\theta}$ at the $n$-th iteration as

$\bm{\theta}^{(n+1)}=e^{i\angle\mathbf{q}^{(n)}}.$ (31)

When the remaining variables are held fixed during the alternating
optimization process, a single update of $\bm{\theta}$ involves solving
equation (31) until convergence. This solution provides the value of
$\bm{\theta}$ that maximizes the rate. The steps for optimizing the phase
response of the IRS using the majorization-maximization technique are
described in detail in Algorithm 1. The overall procedure consists of
optimizing the digital beamformers, updating the weight matrices, digital
combiners, and the IRS phase response through alternating optimization. The
formal description of this procedure can be found in Algorithm 2.

Algorithm 1 Robust Optimization of IRS Phase Response

Given: $\bm{\theta}^{0}$.
Initialize: iteration index $n=1$, accuracy $\epsilon$.
Evaluate: $\mathcal{R}(\bm{\theta}^{0})$.
Repeat until convergence

Calculate $\mathbf{q}_{i}^{(n)}$ with (29).

Update $\bm{\theta}_{i}^{(n+1)}$ with (31).

if
$|\mathcal{R}^{(n+1)}-\mathcal{R}(\bm{\theta}_{i}^{(n)})|/\mathcal{R}(\bm{\theta}_{i}^{(n+1)})\leq\epsilon$

Stop and return $\bm{\phi}_{i}^{(n+1)}$.

else n=n+1 and repeat.

Algorithm 2 Robust Beamforming for MIMO IRS-FD system

Given: The CSI estimates, errors statistics and the rate weights.
Initialize the iteration index $n$, accuracy $\epsilon$, beamformers and
combiners.
Repeat until convergence

for $i$, where $i=k$ or $i=j$

Update $\mathbf{F}_{i}$ with (17).

Update $\mathbf{W}_{i}$ with (19).

if i=k

Update $\mathbf{U}_{k}$ with (21a).

else

Update $\mathbf{V}_{j}$ with (21b).

Update $\mathbf{\Theta}$ with Algorithm 2.

if convergence condition is satisfied

Stop and return the optimized variables.

else repeat.

###  IV-D Convergence

To prove the convergence, we consider an equivalent optimization problem for
ergodic WSR maximization. For this, we consider the MSE weights and expected
receive filters as new variables to be optimized, together with the digital
beamformer for the DL and UL users and the IRS phase response based on the
average MSE covariance matrices. The overall optimization problem as a
function of the new variables can be formally stated as [45]

$\displaystyle\underset{\begin{subarray}{c}\mathbf{F}_{k},\mathbf{F}_{j},\mathbf{U}_{k}\\\
\mathbf{V}_{j},\mathbf{\Theta},\mathbf{W}_{k},\mathbf{W}_{j}\end{subarray}}{\min}$
$\displaystyle\mbox{Tr}(\mathbf{W}_{k}\mathbf{E}_{k})-w_{k}\mbox{logdet}(\frac{\mbox{ln
2}}{w_{k}}\mathbf{W}_{k})+u_{k}\frac{w_{k}}{\mbox{ln}2}$ (32)
$\displaystyle+\mbox{Tr}(\mathbf{W}_{j}\mathbf{E}_{j})-w_{j}\mbox{logdet}(\frac{\mbox{ln
2}}{w_{j}}\mathbf{W}_{j})+v_{j}\frac{w_{j}}{\mbox{ln}2}.$

Consider the digital beamformers and $\mathbf{\Theta}$ to be fixed, the
combiners can be chosen as the EMMSE combiners as in (17). By doing so, the
new cost function has the same structure as above but with the updated average
error covariance matrices, obtained by substituting the EMMSE combiners. By
optimizing the new cost function with respect to the weight matrices, we can
conclude that they can be optimized as

$\mathbf{W}_{k}=\frac{w_{k}}{\mbox{ln}2}\Big{(}\mathbf{E}_{k}\Big{)}^{-1},\quad\mathbf{W}_{j}=\frac{w_{j}}{\mbox{ln}2}\Big{(}\mathbf{E}_{j}\Big{)}^{-1}.$
(33)

Figure 2: Convergence behaviour of the proposed design.

By plugging the new weight matrices into the cost function above, it can be
shown that the ergodic WSR maximization problem is equivalent to the original
WSR cost function under the imperfect CSI, function of the average error
covariance matrices. By considering the statistical distribution of the CSI
errors and the CSI estimates, the proposed robust beamforming design optimizes
the digital beamformers and the IRS phase response which leads to a monotonic
increase in the ergodic WSR sum rate metric, which assures convergence to a
local optimum [44, 42].

The convergence behaviour of the robust joint active and passive beamforming
design is illustrated in Fig. 2. The figure clearly demonstrates that the
proposed method leads to a consistent and monotonic improvement in the WSR
with each iteration, ensuring convergence. Additionally, Fig. 2 depicts the
convergence pattern when there is a slight or significant mismatch between the
true and the estimated CSI available. It is observed that larger uncertainties
lead to less WSR and as the mismatch increases, a higher number of iterations
is needed for the proposed design to converge.

###  IV-E Complexity Analysis

In this Section, we present the computational complexity analysis for the
proposed statistically robust beamforming design. Each update consists of
updating the digital beamformers for the DL and UL users, searching for the
Lagrange multipliers and finally updating the IRS phase response based on
solving sub-problems.

The complexity of updating the digital beamformers for the DL and UL users can
be expressed as $\mathcal{O}(N_{j}^{3})$ and $\mathcal{O}(N_{0}^{3})$,
respectively. The complexity of the Lagrange multipliers search is considered
negligible since it is linear. In order to update the phase response of the
IRS, the majorization-minimization approach requires the initial calculation
of the maximum eigenvalue for the matrix $\mathbf{\Sigma}$, which has a
complexity of $(RC)^{3}$. During each update of the phase response, the main
computational burden lies in computing $\mathbf{q}$ at each iteration, which
has a complexity of $(RC)^{2}$. Let $N_{max}$ denote the total number of
iterations required for each IRS update when updating the digital beamformers.
The overall complexity can be expressed as
$\mathcal{O}(N_{j}^{3}+N_{0}^{3}+(RC)^{3}+N_{max}(RC)^{2})$.

##  V Numerical Results

In this section, we present extensive simulation results to evaluate the
performance of the proposed robust beamforming design. The FD BS and the IRS
are positioned at $(0\text{m},0\text{m},0\text{m})$ and
$(20\text{m},10\text{m},0\text{m})$, respectively, in three-dimensional
coordinates. The UL and DL users are randomly distributed within circular
regions of radius $r=8$ m centered at $(20\text{m},0\text{m},30\text{m})$ and
$(30\text{m},0\text{m},20\text{m})$, respectively. Both the FD BS and the
users are equipped with uniform linear arrays (ULAs) positioned at a half-
wavelength apart. The IRS consists of $10\times 10=100$ elements and assists
in MIMO FD communication unless stated otherwise. The MIMO FD BS has
$M_{0}=15$ transmit and $N_{0}=8$ receive antennas. The UL user $k$ and the DL
user $j$ are assumed to have $M_{k}=5$ transmit and $N_{j}=5$ receive
antennas, respectively, and each user is served with $d_{k}=d_{j}=2$ data
streams. For simulations, the digital beamformers are initialized as the
dominant eigenvectors of the effective channel covariance matrix of the
intended user and the IRS phase response is initialized to be random.

In general, the wireless channel models can be obtained using stochastic or
deterministic methods, or by using their combination. Deterministic modelling
usually employs ray tracing-based methods, as in [48, 49]. These deterministic
channel models can achieve high accuracy, yet require substantial data for the
characterization of real-world surroundings. Therefore, we adopt stochastic
modelling for channel characterization, which is also widely adopted in the
literature. More specifically, to model the large-scale fading, we adopt the
following path-loss model [47]

$PL=PL_{0}-10\alpha\log_{10}\left(\frac{d}{d_{0}}\right),$ (34)

where $PL_{0}=-30\leavevmode\nobreak\ $dB is the pathloss at the reference
distance $d_{0}=1\leavevmode\nobreak\ $m, and $\alpha$ is the path-loss
exponent set to be $2$. The SI channel is modelled according to the Rician
fading channel model [8]

$\mathbf{H}_{0}=\sqrt{\frac{\kappa_{0}}{\kappa_{0}+\kappa_{0}}}\mathbf{H}_{0}^{LoS}+\sqrt{\frac{1}{\kappa_{0}+1}}\mathbf{H}_{0}^{NLoS}$
(35)

where $\kappa_{0}=1\leavevmode\nobreak\ $ denotes the Rician factor,
$\mathbf{H}_{0}^{LoS}$ is the deterministic line-of-sight (LoS) component and
$\mathbf{H}_{0}^{Ref}$ denote the non-LoS (NLoS) component which is Rayleigh
fading. The direct links between the users and the FD BS are also modelled
with a Rician fading channel model with a Rician factor of $1$. The channels
involving the IRS are modelled according to Rayleigh fading [50].

We define the signal-to-noise (SNR) of our system as

$SNR=\frac{\alpha_{0}}{\sigma_{j}^{2}}=\frac{\alpha_{k}}{\sigma_{0}^{2}},$
(36)

where $\alpha_{0}$ and $\alpha_{k}$ are the average total transmit power at
the FD BS and the multi-antenna UL user $k$, respectively. We assume that the
CSI errors to be i.i.d zero-mean circularly symmetric complex Gaussian
distribution with the same variance $\sigma_{csi}^{2}$, and therefore, the
error covariance matrices set chosen as

$\begin{cases}\mathbf{J}_{k}=\mathbf{I},\;\mathbf{K}_{k}=\sigma_{csi}^{2}\mathbf{I},\quad\quad\quad\quad\mathbf{J}_{\theta,0}=\mathbf{I},\;\mathbf{K}_{\theta,0}=\sigma_{csi}^{2}\mathbf{I},\\\
\;\mathbf{J}_{0}=\mathbf{I},\;\mathbf{K}_{0}=\sigma_{csi}^{2}\mathbf{I},\quad\quad\quad\quad\mathbf{J}_{j,k}=\mathbf{I},\mathbf{K}_{j,k}=\sigma_{csi}^{2}\mathbf{I},\\\
\;\mathbf{J}_{j}=\mathbf{I},\mathbf{K}_{j}=\sigma_{csi}^{2}\mathbf{I},\quad\quad\quad\quad\;\mathbf{J}_{0,\theta}=\mathbf{I},\mathbf{K}_{0,\theta}=\sigma_{csi}^{2}\mathbf{I}\\\
\;\mathbf{J}_{j,\theta}=\mathbf{I},\mathbf{K}_{j,\theta}=\sigma_{csi}^{2}\mathbf{I}\quad\quad\quad\mathbf{J}_{\theta,k}=\mathbf{I},\mathbf{K}_{\theta,k}=\sigma_{csi}^{2}\mathbf{I}\\\
\end{cases}$ (37)

Figure 3: Average WSR as a function of $\rho$ with SNR$=30\leavevmode\nobreak\
$dB.

Since the variance of the CSI errors strictly depend on the average transmit
power (used also to estimate the channels) and the noise variance, we assume
that $\sigma_{csi}^{2}$ decays as $\mathcal{O}(SNR^{-\alpha})$, for some
constant $\alpha$ [44], satisfying the CSI error decay rate inversely
proportional to the SNR. As the SNR represents the transmit SNR, note that
$SNR\rightarrow\infty$ is equivalent to
$\alpha_{k}=\alpha_{0}\rightarrow\infty$, which enhance the CSI quality and
reduces the CSI error variance as $\sigma_{csi}^{2}\rightarrow 0$. The CSI
errors variance is set as $\sigma_{csi}^{2}=\rho/SNR^{\alpha}$, where $\rho$
denotes a scale factor and $\alpha\in[0,1]$ determine the quality of the CSI.
Maintaining the quality of the CSI with $\alpha=1$ could be exhausting in
terms of resources required to acquire the CSI and therefore we set
$\alpha=0.6$ [44].

We label our proposed design as a _FD-IRS-RB_. For comparison, we define the
following benchmark schemes:

  1. 1.

_FD-IRS-Non-RB_ : FD system assisted with IRS and non robust beamforming,
i.e., the available CSI is treated as perfect;

  2. 2.

_FD-No-IRS-RB_ : FD system with no IRS and robust beamforming;

  3. 3.

_FD-No-IRS-Non-RB_ : FD system with no IRS and non robust beamforming;

  4. 4.

_HD-IRS-RB_ : HD system assisted with IRS and robust beamforming;

  5. 5.

_HD-IRS-Non-RB_ : HD system assisted with IRS and non robust beamforming;

  6. 6.

_HD-No-IRS-RB_ : HD system with no IRS and robust beamforming;

  7. 7.

_HD-No-IRS-Non-RB_ : HD system with no IRS and non robust beamforming.

We also compare our approximation proposed for the ergodic WSR in Theorem 1
given the transmit covariance matrices of the proposed robust beamforming
design which we label as _Analytical FD-IRS-RB_ and compare it with the
ergodic rate achieved with the EWMMSE.

Figure 4: Average WSR as a function of SNR with $\rho=0.4$.

Figure 5: Average WSR as a function of SNR with $\rho=0.9$.

Figure 6: CDF of the various schemes with $\rho=0.4$.

Figure 7: CDF of the various schemes with $\rho=0.9$.

Fig. 3 shows the performance of the proposed robust joint beamforming design
as a function of the scale factor $\rho$ dictating the CSI error variance, by
means of Monte Carlo simulations at SNR$=30\leavevmode\nobreak\ $dB. It is
shown that the proposed robust design achieves significant performance gain
compared to the naive FD-IRS-Non-RB scheme, which does not account for the CSI
errors. Moreover, we can also see that the achieved ergodic WSR accurately
matches the result stated in Theorem 1. It is to be noted that as the CSI
error variance gets extremely large, the proposed statistically robust
beamforming design preserves significant robustness against the uncertainties
in the CSI quality. Namely, for $\rho\geq 0.4$, the FD-IRS-Non-RB scheme
achieves less gain than the IRS-aided HD system deploying robust beamforming.
Such a result showcases the importance of adopting a robust beamforming
approach for the IRS-FD systems, as, in practice, their performance could
degrade significantly due to imperfect CSI.

Fig. 5 and Fig. 5 present the achieved ergodic WSR of our proposed design as a
function of the transmit SNR, compared to benchmark schemes with $\rho=0.4$
and $\rho=0.9$, respectively. The results clearly demonstrate that our design
achieves significant gains in the presence of CSI errors when compared to the
other schemes. Theoretically, FD systems offer a twofold improvement in the
WSR compared to HD systems. However, the corresponding curves for the FD-IRS-
Non-RB scheme reveal that with high CSI error variance, the IRS-FD system may
exhibit no gain in comparison to the IRS-aided HD system at any SNR level if a
non-robust beamforming approach is employed in the presence of large CSI
uncertainties. This can be attributed to the crucial role played by residual
SI in determining the IRS-FD system’s gain. With a large CSI error variance,
these interference components cannot be adequately handled. On the other hand,
adopting robust beamforming for IRS-FD systems can be promising in terms of
robustness. We can also observe that as the CSI error variance decreases, the
performance of the non-robust beamforming scheme gradually converges towards
that of the robust beamforming schemes, approaching the desired twofold
improvement in the WSR due to quasi-ideal simultaneous transmission and
reception.

Fig. 7 and Fig. 7 show the cumulative distribution function (CDF) of the
proposed statistically robust beamforming design, in comparison with the
benchmark schemes at $SNR=30\leavevmode\nobreak\ $dB. It is clearly visible
that the proposed method achieves significant average WSR compared to the
other schemes and the reported gains are stable with different values of the
scale factor $\rho$, and therefore with the different amount of CSI error
variance.

Based on the presented numerical results, it can be concluded that CSI errors
play a critical role in determining the performance of IRS-FD systems.
Neglecting these errors can significantly impact the system’s performance,
leading to suboptimal beamforming decisions. Consequently, the potential of
the IRS-FD can not be fully exploited, limiting its ability to enhance channel
quality for UL and DL users and mitigate interference. In the UL direction,
uncertainties in the SI channel result in residual SI that cannot be
adequately suppressed even with joint active and passive beamforming,
resulting in a low signal-to-total-interference plus noise ratio. In the DL
direction, FD systems experience interference from nearby UL users, which can
be as detrimental as well. Therefore, effective mitigation of interference
through robust beamforming decisions is crucial for achieving satisfactory UL
and DL performance. The results demonstrate that adopting a statistically
robust beamforming approach is essential for effectively managing SI and
cross-interference in practical IRS-FD systems with imperfect CSI. By doing
so, the full potential of IRS-FD systems can be harnessed, leading to highly
efficient communication systems in terms of spectral and energy efficiency.

##  VI Conclusions

In this paper, the authors introduced a novel statistically robust joint
beamforming design for maximizing the ergodic WSR of an IRS-FD system in the
presence of imperfect CSI. The CSI errors were modelled using the Gaussian-
Kronecker model, and an approximation for the ergodic WSR was derived based on
the statistical distribution of the errors given the channel estimates. To
address the ergodic WSR maximization problem, an iterative approach based on
the EWMMSE method was employed, which involved solving two layers of sub-
problems through alternating optimization. Simulation results demonstrated
that imperfect CSI could significantly impact the performance of IRS-FD
systems, and adopting a robust beamforming strategy led to substantial
improvements compared to the naive approach. Furthermore, the derived
approximation for the ergodic WSR aligned well with the achieved results using
the expected WMMSE method.

##  Appendix A Auxilary Matrices for (14)

The matrices $\mathbf{Q}_{k},\mathbf{T}_{j},\mathbf{T}_{0},\mathbf{Q}_{j,k}$
are defined as follows

$\displaystyle\mathbf{Q}_{k}$
$\displaystyle=\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}}\Big{(}\overline{\mathbf{H}}_{k}\widetilde{\mathbf{U}}_{k}\overline{\mathbf{H}}_{k}^{H}\Big{)}$
(38)
$\displaystyle=\hat{\mathbf{H}}_{k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{k}^{H}+\hat{\mathbf{H}}_{k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}+\mbox{Tr}\Big{(}\widetilde{\mathbf{U}}_{k}\mathbf{J}_{k}^{T}\Big{)}\mathbf{K}_{k}$
$\displaystyle\;\;+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{k}^{H}$
$\displaystyle\;\;+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\mbox{Tr}\Big{(}\widetilde{\mathbf{U}}_{k}\mathbf{J}_{\theta,k}^{T}\Big{)}\mathbf{K}_{\theta,k}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}$
$\displaystyle\;\;+\mbox{Tr}\Big{(}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\mathbf{J}_{0,\theta}\Big{)}\mathbf{K}_{0,\theta}$
$\displaystyle\;\;+\mbox{Tr}\Big{(}\widetilde{\mathbf{U}}_{k}\mathbf{J}_{\theta,k}^{T}\Big{)}\mbox{Tr}\Big{(}\mathbf{\Theta}\mathbf{K}_{\theta,k}\mathbf{\Theta}^{H}\mathbf{J}_{0,\theta}^{T}\Big{)}\mathbf{K}_{0,\theta},$
$\displaystyle\mathbf{T}_{j}$
$\displaystyle=\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}}\Big{(}\overline{\mathbf{H}}_{j}\widetilde{\mathbf{V}}_{j}\overline{\mathbf{H}}_{j}^{H}\Big{)}$
(39)
$\displaystyle=\hat{\mathbf{H}}_{j}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{j}^{H}+\hat{\mathbf{H}}_{j}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}+\mbox{Tr}\Big{(}\widetilde{\mathbf{V}}_{j}\mathbf{J}_{j}^{T}\Big{)}\mathbf{K}_{j}$
$\displaystyle\;\;+\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{j}^{H}+\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}$
$\displaystyle\;\;\;+\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\mbox{Tr}\Big{(}\widetilde{\mathbf{V}}_{j}\mathbf{J}_{\theta,0}^{T}\Big{)}\mathbf{K}_{\theta,0}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}$
$\displaystyle\;\;\;+\mbox{Tr}\Big{(}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\mathbf{J}_{j,\theta}^{T}\Big{)}\mathbf{K}_{j,\theta}$
$\displaystyle\;\;\;+\mbox{Tr}\Big{(}\widetilde{\mathbf{V}}_{j}\mathbf{J}_{\theta,0}^{T}\Big{)}\mbox{Tr}\Big{(}\mathbf{\Theta}\mathbf{K}_{\theta,0}\mathbf{\Theta}^{H}\mathbf{J}_{j,\theta}^{T}\Big{)}\mathbf{K}_{j,\theta},$

$\displaystyle\mathbf{T}_{0}$
$\displaystyle=\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}}\Big{(}\overline{\mathbf{H}}_{0}\widetilde{\mathbf{V}}_{j}\overline{\mathbf{H}}_{0}^{H}\Big{)}$
(40)
$\displaystyle=\hat{\mathbf{H}}_{0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{0}^{H}+\hat{\mathbf{H}}_{0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}+\mbox{Tr}\Big{(}\widetilde{\mathbf{V}}_{j}\mathbf{J}_{0}^{T}\Big{)}\mathbf{K}_{0}^{T}$
$\displaystyle\;\;+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{0}^{H}+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}$
$\displaystyle\;\;+\hat{\mathbf{H}}_{0,\theta}\mathbf{\Theta}\mbox{Tr}\Big{(}\widetilde{\mathbf{V}}_{j}\mathbf{J}_{\theta,0}^{T}\Big{)}\mathbf{K}_{\theta,0}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{0,\theta}^{H}$
$\displaystyle\;\;+\mbox{Tr}\Big{(}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{\theta,0}^{H}\mathbf{\Theta}^{H}\mathbf{J}_{0,\theta}^{T}\Big{)}\mathbf{K}_{0,\theta}$
$\displaystyle\;\;+\mbox{Tr}(\widetilde{\mathbf{V}}_{j}\mathbf{J}_{\theta,0}^{T})\mbox{Tr}\Big{(}\mathbf{\Theta}\mathbf{K}_{\theta,0}\mathbf{\Theta}^{H}\mathbf{J}_{0,\theta}^{T}\Big{)}\mathbf{K}_{0,\theta},$

$\displaystyle\mathbf{Q}_{j,k}$
$\displaystyle=\mathbb{E}_{\mathbf{H}|\hat{\mathbf{H}}}\Big{(}\overline{\mathbf{H}}_{j,l}\widetilde{\mathbf{U}}_{l}\overline{\mathbf{H}}_{j,k}^{H}\Big{)}$
(41)
$\displaystyle=\hat{\mathbf{H}}_{j,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{j,k}^{H}+\hat{\mathbf{H}}_{j,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}+\mbox{Tr}\Big{(}\widetilde{\mathbf{U}}_{k}\mathbf{J}_{j,k}^{T}\Big{)}\mathbf{K}_{j,k}$
$\displaystyle\;\;+\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{j,k}^{H}+\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}$
$\displaystyle\;\;++\hat{\mathbf{H}}_{j,\theta}\mathbf{\Theta}\mbox{Tr}\Big{(}\widetilde{\mathbf{U}}_{k}\mathbf{J}_{\theta,k}^{T}\Big{)}\mathbf{K}_{\theta,k}\mathbf{\Theta}^{H}\hat{\mathbf{H}}_{j,\theta}^{H}$
$\displaystyle\;\;+\mbox{Tr}\Big{(}\mathbf{\Theta}\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}\mathbf{\Theta}^{H}\mathbf{J}_{j,\theta}^{T}\Big{)}\mathbf{K}_{j,\theta}$
$\displaystyle\;\;+\mbox{Tr}\Big{(}\widetilde{\mathbf{U}}_{k}\mathbf{J}_{\theta,k}^{T}\Big{)}\mbox{Tr}\Big{(}\mathbf{\Theta}\mathbf{K}_{\theta,k}\mathbf{\Theta}^{H}\mathbf{J}_{j,\theta}^{T}\Big{)}\mathbf{K}_{j,\theta}.$

##  Appendix B Matrices to optimize $\mathbf{\Theta}$

The matrices $\mathbf{S},\mathbf{T}$, and $\mathbf{Z}$ to optimize the IRS
phase response $\mathbf{\Theta}$ are as follows:

$\displaystyle\mathbf{S}=$
$\displaystyle\hat{\mathbf{H}}_{0,\theta}^{H}\mathbf{F}_{k}^{H}\Big{(}\mathbf{W}_{k}\mathbf{F}_{k}\hat{\mathbf{H}}_{k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}+\mathbf{W}_{k}\mathbf{F}_{k}\hat{\mathbf{H}}_{0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{\theta,0}^{H}$
(42)
$\displaystyle+\mathbf{W}_{k}\mathbf{U}_{k}^{H}\hat{\mathbf{H}}_{\theta,k}^{H}\Big{)}+\hat{\mathbf{H}}_{j,\theta}^{H}\mathbf{F}_{j}^{H}\Big{(}\mathbf{W}_{j}\mathbf{F}_{j}\mathbf{H}_{j}\widetilde{\mathbf{V}}_{j}\mathbf{H}_{\theta,0}^{H}$
$\displaystyle+\mathbf{W}_{j}\mathbf{F}_{j}\mathbf{H}_{j,k}\widetilde{\mathbf{U}}_{k}\mathbf{H}_{\theta,k}^{H}-\mathbf{W}_{j}\mathbf{V}_{j}^{H}\hat{\mathbf{H}}_{\theta,0}^{H}\Big{)},$

$\displaystyle\mathbf{Z}=$
$\displaystyle\hat{\mathbf{H}}_{0,\theta}^{H}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\mathbf{H}_{0,\theta}+\hat{\mathbf{H}}_{0,\theta}^{H}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\mathbf{H}_{0,\theta}$
(43)
$\displaystyle+\mbox{Tr}\Big{(}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\mathbf{J}_{0,\theta}^{T}\Big{)}\mathbf{K}_{0,\theta}+\mbox{Tr}\Big{(}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\mathbf{J}_{0,\theta}^{T}\Big{)}\mathbf{K}_{0,\theta}$
$\displaystyle+\mbox{Tr}\Big{(}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\mathbf{J}_{0,\theta}^{T}\Big{)}\mathbf{K}_{0,\theta}+\hat{\mathbf{H}}_{0,\theta}^{H}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\hat{\mathbf{H}}_{0,\theta}$
$\displaystyle+\hat{\mathbf{H}}_{0,\theta}^{H}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\hat{\mathbf{H}}_{0,\theta}+\mbox{Tr}\Big{(}\mathbf{F}_{k}^{H}\mathbf{W}_{k}\mathbf{F}_{k}\mathbf{J}_{0,\theta}^{T}\Big{)}\mathbf{K}_{0,\theta}$
$\displaystyle+\hat{\mathbf{H}}_{j,\theta}^{H}\mathbf{F}_{j}\mathbf{W}_{j}\mathbf{F}_{j}\hat{\mathbf{H}}_{j,\theta}+\hat{\mathbf{H}}_{j,\theta}^{H}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\hat{\mathbf{H}}_{j,\theta}$
$\displaystyle+\mbox{Tr}\Big{(}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\mathbf{J}_{j,\theta}^{T}\Big{)}\mathbf{K}_{j,\theta}+\mbox{Tr}\Big{(}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\mathbf{J}_{j,\theta}^{T}\Big{)}\mathbf{K}_{j,\theta}$
$\displaystyle+\hat{\mathbf{H}}_{j,\theta}^{H}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\hat{\mathbf{H}}_{j,\theta}+\hat{\mathbf{H}}_{j,\theta}^{H}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\hat{\mathbf{H}}_{j,\theta}$
$\displaystyle+\mbox{Tr}\Big{(}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\mathbf{J}_{j,\theta}^{T}\Big{)}\mathbf{K}_{j,\theta}+\mbox{Tr}\Big{(}\mathbf{F}_{j}^{H}\mathbf{W}_{j}\mathbf{F}_{j}\mathbf{J}_{j,\theta}^{T}\Big{)}\mathbf{K}_{j,\theta},$

$\displaystyle\mathbf{T}=$
$\displaystyle\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}+\mbox{Tr}\Big{(}\widetilde{\mathbf{U}}_{k}\mathbf{J}_{\theta,k}^{T}\Big{)}\mathbf{K}_{\theta,k}+\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}$
(44)
$\displaystyle+\mbox{Tr}\Big{(}\widetilde{\mathbf{U}}_{k}\mathbf{J}_{\theta,k}^{T}\Big{)}\mathbf{K}_{\theta,k}+\hat{\mathbf{H}}_{\theta,0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{\theta,0}^{H}+\mbox{Tr}\Big{(}\widetilde{\mathbf{V}}_{j}\mathbf{J}_{\theta,0}^{T}\Big{)}\mathbf{K}_{\theta,0}$
$\displaystyle+\hat{\mathbf{H}}_{\theta,0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{\theta,0}^{H}+\mbox{Tr}\Big{(}\widetilde{\mathbf{V}}_{j}\mathbf{J}_{\theta,0}^{T}\Big{)}\mathbf{K}_{\theta,0}+\hat{\mathbf{H}}_{\theta,0}\widetilde{\mathbf{V}}_{j}\hat{\mathbf{H}}_{\theta,0}^{H}$
$\displaystyle+\mbox{Tr}\Big{(}\widetilde{\mathbf{V}}_{j}\mathbf{J}_{\theta,0}^{T}\Big{)}\mathbf{K}_{\theta,0}+\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}+\mbox{Tr}\Big{(}\widetilde{\mathbf{U}}_{k}\mathbf{J}_{\theta,k}^{T}\Big{)}\mathbf{K}_{\theta,k}$
$\displaystyle+\hat{\mathbf{H}}_{\theta,k}\widetilde{\mathbf{U}}_{k}\hat{\mathbf{H}}_{\theta,k}^{H}+\mbox{Tr}\Big{(}\widetilde{\mathbf{U}}_{k}\mathbf{J}_{\theta,k}^{T}\Big{)}\mathbf{K}_{\theta,k}.$

## Acknowledgement

This work is supported by the Luxembourg National Fund (FNR)-RISOTTI–the
Reconfigurable Intelligent Surfaces for Smart Cities under Project
FNR/C20/IS/14773976/RISOTTI.

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|  Chandan Kumar Sheemar (Member, IEEE) received his PhD from EURECOM, Sophia
Antipolis, France, in 2022. He is currently a research associate at the
Interdisciplinary Centre for Security, Reliability and Trust (SnT), University
of Luxembourg, Luxembourg., within the SIGCOM research group. His main
interests include full duplex, joint communications and sensing, massive MIMO,
reflecting intelligent surfaces, optimization theory, and non-terrestrial
networks.
---|---
|  Sourabh Solanki (Member, IEEE) received the M. Tech. degree in
Communication and Signal Processing, and a PhD degree in Electrical
Engineering, both from the Indian Institute of Technology (IIT) Indore, India,
in 2015 and 2019, respectively. From 2019 to 2020, he was a Research Professor
at the Korea University, Seoul, South Korea, where he received the Brain Korea
21 Postdoctoral Fellowship from the National Research Foundation, Government
of Korea. Since 2021, he has been working as a Research Associate with the
Interdisciplinary Centre for Security, Reliability and Trust (SnT), University
of Luxembourg, Luxembourg. His main research interests include cognitive
radio, terahertz communications, mmWave networks, energy harvesting, and non-
terrestrial networks. He has been serving as a technical program committee
member of various conferences and has also been involved in the peer review
process of major IEEE journals and conferences. He was a co-recipient of the
Best Paper Award at the International Conference on ICT Convergence (ICTC),
Jeju Island, South Korea, in October 2020. He also received an exemplary
reviewer award for IEEE Transactions on Communications in 2020.
---|---
|  Jorge Querol (Member, IEEE) received the B.Sc. (+5) degree in
telecommunication engineering, the M.Sc. degree in electronics engineering,
the M.Sc. degree in photonics, and the PhD degree (cum laude) in signal
processing and communications from Universitat Politècnica de Catalunya-
BarcelonaTech (UPC), Barcelona, Spain, in 2011, 2012, 2013, and 2018,
respectively. Since 2018, he has been with the SIGCOM Research Group,
Interdisciplinary Centre for Security, Reliability, and Trust (SnT),
University of Luxembourg, Luxembourg, and the Head of the Satellite
Communications Laboratory. He is involved in several ESA and Luxembourgish
national research projects dealing with signal processing and satellite
communications. His research interests include SDR, real-time signal
processing, satellite communications, 5G non-terrestrial networks, satellite
navigation, and remote sensing. He received the Best Academic Record Award of
the year in electronics engineering from UPC, in 2012, the First Prize of the
European Satellite Navigation Competition (ESNC) Barcelona Challenge from the
European GNSS Agency (GSA), in 2015, the Best Innovative Project of the Market
Assessment Program (MAP) of EADA Business School, in 2016, the Award Isabel P.
Trabal from Fundacióó Caixa d’Enginyers for its quality research during the
Ph.D., in 2017, and the Best Ph.D. Thesis Award in remote sensing from the
IEEE Geoscience and Remote Sensing (GRSS) Spanish Chapter, Spain, in 2019.
---|---
|  Sumit Kumar (S’14-M’19) received the B.Tech and M.S. in Electronics $\&$
Communication Engineering from Gurukula Kangri University, Haridwar, India
(2008) and the International Institute of Information Technology, Hyderabad,
India (2014), respectively, and the Ph.D. from EURECOM (France) in 2019.
Currently, he is working as a Research Associate at the Interdisciplinary
Centre for Security, Reliability, and Trust (SnT), University of Luxembourg.
His research interests are in wireless communication, interference management,
Integration of 5G with Non-Terrestrial-Networks and Software Defined Radio
prototyping.
---|---
|  Symeon Chatzinotas (Fellow, IEEE) is currently Full Professor / Chief
Scientist I and Head of the research group SIGCOM in the Interdisciplinary
Centre for Security, Reliability and Trust, University of Luxembourg. In
parallel, he is an Adjunct Professor in the Department of Electronic Systems,
Norwegian University of Science and Technology and a Collaborating Scholar of
the Institute of Informatics & Telecommunications, National Center for
Scientific Research “Demokritos”. In the past, he has lectured as Visiting
Professor at the University of Parma, Italy and contributed in numerous R&D
projects for the Institute of Telematics and Informatics, Center of Research
and Technology Hellas and Mobile Communications Research Group, Center of
Communication Systems Research, University of Surrey. He has received the
M.Eng. in Telecommunications from Aristotle University of Thessaloniki, Greece
and the M.Sc. and Ph.D. in Electronic Engineering from University of Surrey,
UK in 2003, 2006 and 2009 respectively. He has authored more than 700
technical papers in refereed international journals, conferences and
scientific books and has received numerous awards and recognitions, including
the IEEE Fellowship and an IEEE Distinguished Contributions Award. He is
currently in the editorial board of the IEEE Transactions on Communications,
IEEE Open Journal of Vehicular Technology and the International Journal of
Satellite Communications and Networking.
---|---