# Uplink Performance of RIS-aided Cell-Free Massive MIMO System with
Electromagnetic Interference

Enyu Shi, , Jiayi Zhang, ,
Derrick Wing Kwan Ng, , and Bo Ai E. Shi and J. Zhang are with the School of
Electronics and Information Engineering, Beijing Jiaotong University, Beijing
100044, P. R. China. (e-mail: jiayizhang@bjtu.edu.cn).D. W. K. Ng is with the
School of Electrical Engineering and Telecommunications, University of New
South Wales, NSW 2052, Australia. (e-mail: w.k.ng@unsw.edu.au).B. Ai is with
the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong
University, Beijing 100044, China. (e-mail: boai@bjtu.edu.cn).

###### Abstract

Cell-free (CF) massive multiple-input multiple-output (MIMO) and
reconfigurable intelligent surface (RIS) are two promising technologies for
realizing future beyond-fifth generation (B5G) networks. In this paper, we
consider a practical spatially correlated RIS-aided CF massive MIMO system
with multi-antenna access points (APs) over spatially correlated fading
channels. Different from previous work, the electromagnetic interference (EMI)
at RIS is considered to further characterize the system performance of the
actual environment. Then, we derive the closed-form expression for the system
spectral efficiency (SE) with the maximum ratio (MR) combining at the APs and
the large-scale fading decoding (LSFD) at the central processing unit (CPU).
Moreover, to counteract the near-far effect and EMI, we propose practical
fractional power control (FPC) and max-min power control algorithms to further
improve the system performance. We unveil the impact of EMI, channel
correlations, and different signal processing methods on the uplink SE of user
equipments (UEs). The accuracy of our derived analytical results is verified
by extensive Monte-Carlo simulations. Our results show that the EMI can
substantially degrade the SE, especially for those UEs with unsatisfactory
channel conditions. Besides, increasing the number of RIS elements is always
beneficial in terms of the SE, but with diminishing returns when the number of
RIS elements is sufficiently large. Furthermore, the existence of spatial
correlations among RIS elements can deteriorate the system performance when
RIS is impaired by EMI.

###### Index Terms:

Reconfigurable intelligent surface, cell-free massive MIMO, electromagnetic
interference, spatial correlation, spectral efficiency.

##  I Introduction

The goal of the fifth-generation (5G) wireless networks is to increase the
network capacity by a factor of 1,000 compared to previous generations of
networks, providing ubiquitous wireless connectivity to at least 100 billion
devices worldwide [1]. Recently, with the large-scale commercial deployment of
5G around the world, the global industry has begun to conduct preliminary
research on the sixth-generation (6G) of mobile communication technology [2].
In the last few decades, several new technologies have been introduced
including millimeter-wave communications [3], massive multiple-input multiple-
output (MIMO)[4], and network densification [5]. Among them, massive MIMO is
the most appealing that has attracted much attention as it can guarantee
excellent quality of services to numerous users in the networks. Furthermore,
in various scenarios, the net throughput of massive MIMO systems can approach
the Shannon capacity via simple linear processing techniques such as zero-
forcing (ZF) or maximum ratio (MR) processing [6]. Besides, massive MIMO
architectures enjoy the advantage of low-backhaul requirements as the base
station (BS) antennas are installed in a form of a compact array. However,
traditional cellular networks suffer from severe inter-cell interference. In
particular, users at cell borders suffer from both high inter-cell
interference and path loss leading to degraded system performance [7]. As
such, novel advanced signal processing methods are needed to overcome the
inter-cell interference inherent in traditional cellular network deployments.

Cell-free (CF) massive MIMO has recently been introduced as a promising future
technology for realizing beyond-5G wireless communication systems [8]. Indeed,
CF massive MIMO is a network architecture consisting of a large number of
geographically distributed access points (APs), which are connected to the
central processing unit (CPU) and coherently serve all the user equipments
(UEs) by performing spatial multiplexing on the same time-frequency resources
[9]. Therefore, CF massive MIMO is a major leap of massive MIMO technology to
mitigate inter-cell interference, which is the fundamental performance
limitation of dense cellular networks [9, 10, 11]. Recently, a large number of
important aspects and fundamentals of CF massive MIMO have been studied to
unlock the potential of this architecture. For example, [12, 13] studied the
system performance that relies on a distributed implementation with the MR
processing. Also, the authors in [14] introduced the partially and fully
centralized signal processing in CF massive MIMO systems with minimum mean
square error (MMSE) combining schemes that can achieve higher spectral
efficiency (SE). Besides, the two-layer decoding method for CF massive MIMO is
considered an effective decoding scheme with the MR/MMSE combining methods
adopted in the first layer decoder and the large-scale fading decoding (LSFD)
method applied to the second layer decoder [15, 16]. Despite the various
advantage and potential, Cell-Free massive MIMO still cannot guarantee an
appropriate quality of service (QoS) under harsh propagation conditions, such
as in the presence of poor scattering environments or severe signal
attenuation caused by the presence of large obstacles [17]. Meanwhile, with
the skyrocketing demand for wireless data throughput, other advanced
technologies are also required to satisfy the ever-stringent QoS requirement.

Reconfigurable intelligent surface (RIS) is an emerging technology that is
capable of shaping the radio waves at the electromagnetic level without
applying sophisticated digital signal processing methods and active power
amplifiers [18]. Specifically, with massive reflective and refractive elements
at RIS, effective passive beamforming can be performed by altering the phase
of the reflected impinging signals to realize reconfigurable wireless
channels/radio propagation environments [19, 20, 21, 22]. Besides, RIS is
fabricated with low-power and low-cost which can be flexibly deployed to
assist users with unsatisfactory channel conditions to improve communication
quality [23, 24]. Due to its unique fabrication, RIS also poses many
properties that are different from the traditional uniform linear array (ULA).
For example, [25] proposed a sinc function-based model to capture the spatial
correlations that are determined by the spacing between two neighboring RIS
elements. Besides, in [26], the authors explored the scenario of having
electromagnetic interference (EMI) impinging on RIS and pointed out its
significant impact on the system performance. Therefore, in RIS-assisted
systems, it is particularly important to consider the existence of EMI and the
inherent characteristics of RIS, such as the spatial correlation caused by
subwavelength element sizes.

Recently, some works have focused on how to jointly exploit the advantages of
CF massive MIMO and RIS to further improve communication performance [27, 28,
29]. For instance, in [28], a joint precoding framework for RIS-aided CF
massive MIMO systems was proposed to improve the UE communication quality.
Besides, in [27] the authors studied an RIS-aided CF wireless energy transfer
(WET) framework to improve the system capacity and energy efficiency.
Furthermore, in [17] the author introduced an aggregated channel estimation
approach and investigate the RIS-aided CF massive MIMO systems with MR
combining over spatially-correlated channels. However, [17] considered the
Rayleigh fading model of UE-RIS and RIS-AP channels that is not generally
valid. In practice, the RIS is generally deployed such that a direct line-of-
sight path can be established to obtain better performance gains [30]. In
other words, the Rayleigh fading model adopted in [17, 31] fails to capture
this important characteristic. Also, [17] adopted single-layer signal
processing, which certainly limits the CF massive MIMO system performance. On
the other hand, a common practice in the design of RIS is to only consider the
signals generated by the desired system and thereby ignore the EMI or “noises”
that is inevitably presented in the communication environment [32]. Also, in
[33], the authors studied the RIS-assisted ultra-reliable low-latency
communication (URLLC) in the presence of EMI and the results showed that its
existence can reduce the system performance significantly. Besides, the
authors considered the physical layer security of RIS-assisted communications
and showed the potential of EMI-type interference in degrading the physical
layer security in [34]. In particular, there is a large amount of EMI in CF
massive MIMO systems, which would seriously affect the system performance [4].
In other words, existing theoretical analyzes from the previous research are
not applicable to RIS-aided CF massive MIMO systems with EMI. As such, the
performance limits of RIS-aided CF massive MIMO systems should be carefully
analyzed and designed.

Motivated by the aforementioned observation, we investigate the performance of
the RIS-aided CF massive MIMO system over spatially correlated channels with
spatial EMI impinging RIS. More specifically, we propose the MMSE-based
channel estimation to estimate the aggregated channels between the APs and the
UEs. Local processing with MR combining method at the APs and centralized LSFD
at CPU are considered in the uplink data transmission. In particular, the
uplink performance of the CF massive MIMO system and RIS-aided CF massive MIMO
system without EMI are analyzed for comparison. Then, we design two power
control optimization algorithms to compensate for the effects of EMI. The
specific contributions of the work are listed as follows:

  * •

We first derive the closed-form expression for the uplink SE of the RIS-aided
CF massive MIMO system over spatially correlated channels taking into account
spatial EMI. Our results show that the EMI at RIS has a significant impact on
the system performance, especially for the UEs with poor channel conditions.
Thus, effective interference cancellation schemes, e.g., LSFD cooperation, are
critical for improving the system performance. In addition, the RIS-aided CF
massive MIMO system achieves higher uplink SE performance than the counterpart
without RIS, especially for the edge UEs, the improvement is significant.

  * •

We analyze the effect of different system parameters and RIS characteristics
on the performance, such as the number of AP antennas, RIS elements, spatial
correlation, and RIS location. We find that increasing the number of AP
antennas magnifies the negative impact of EMI and increasing the number of RIS
elements are always beneficial, but with diminishing returns when the number
of RIS elements is sufficiently large. Besides, the existence of spatial
correlation of RIS elements degrades the system performance, which is even
more pronounced when EMI is considered.

  * •

We design the fractional power control method and the max-min power control
algorithm to alleviate the negative impacts of EMI to further improve the
system performance for practical implementation. The results show that the
proposed power control schemes can further improve the SE of the UEs with poor
performance. However, when the EMI power increases, the efficacy of the two
power controls decreases. As such, it is particularly important to deploy the
RIS in a low EMI environment for RIS-aided CF massive MIMO systems.

The rest of this paper is organized as follows. In Section II, we describe the
RIS-aided CF massive MIMO system model incorporating the combined effects of
spatial correlation, EMI, and channel estimation error. Next, Section III
presents the achievable uplink SE of the considered system in the existence of
EMI and designs two power control algorithms to compensate the performance
degradation caused by the EMI. Then, numerical results and performance
analysis are provided in Section IV. Finally, Section V concludes this paper.

Notation: The superscripts $\mathbf{x}^{H}$ and $x^{\mathrm{*}}$ are used to
represent conjugate transpose and conjugate, respectively. The matrices and
column vectors are denoted by boldface uppercase letters $\mathbf{X}$ and
boldface lowercase letters $\mathbf{x}$, respectively. The
${\rm{mod}}\left({\cdot,\cdot}\right)$, $\left\|\cdot\right\|$, and
$\left\lfloor\cdot\right\rfloor$ denote the modulus operation, the Euclidean
norm and the truncated argument, respectively. ${\rm{tr}}\left(\cdot\right)$,
$\mathbb{E}\left\\{\cdot\right\\}$, and ${\rm{Cov}}\left\\{\cdot\right\\}$ are
the trace, expectation and covariance operators. $\otimes$ denotes the
Kronecker products. We use
${\text{diag}}\left({{a_{1}},\cdots,{a_{n}}}\right)$ to denote a block-
diagonal matrix with the square matrices ${{a_{1}},\cdots,{a_{n}}}$ on the
diagonal. Finally, the circularly symmetric complex Gaussian random variable
$x$ with variance $\sigma^{2}$ is denoted by
$x\sim\mathcal{C}\mathcal{N}\left({0,{\sigma^{2}}}\right)$.

##  II System Model

As shown in Figure 1, we consider a RIS-aided CF mMIMO system consisting of
$M$ APs, $K$ UEs, one RIS, and a CPU. We consider that each AP is equipped
with $L$ antennas and each UE is equipped with a single antenna. All the APs
are connected to the CPU via fronthaul links and serve all the UEs via the
same time and frequency resources.111By adopting the central processing, the
CPU can utilize the global information from all the involved APs and optimize
the signal decoding coefficients with the global information to achieve a
better system performance at the expense of higher complexity and more
fronthaul signaling overhead [14]. The RIS has $N$ reflective elements that
can introduce some phase shifts to the incident signals [35]. Besides, we
consider that there exists some EMI at the RIS as EMI inevitably presents in
any environment [32]. Without loss of generality, the standard time division
duplex (TDD) protocol is adopted in our system, where the length of each
coherence time block is denoted as $\tau_{c}$. We consider that $\tau_{p}$
symbols are exploited for the channel estimation phase in the uplink and
$\tau_{u}=\tau_{c}-\tau_{p}$ symbols are utilized for data transmission. The
channel model, the channel estimation method, and the uplink data transmission
phase are then given in the following section.

Figure 1: An RIS-aided CF mMIMO system with EMI.

###  II-A Channel Model

We assume a quasi-static block fading model such that in each coherence time
block the channels are frequency flat and static. In Figure 1, the channels
are divided into two types: the direct link from the UE to the AP and the
cascaded link through RIS. Let ${{\bf{g}}_{mk}}\in{\mathbb{C}}{{}^{L}}$ denote
the direct link channel between UE $k$ and AP $m$. Variable
${{\bf{H}}_{m}}\in{\mathbb{C}}{{}^{N\times L}}$ denotes the channel matrix
from the RIS to AP $m$ and ${{\bf{z}}_{k}}\in{\mathbb{C}}{{}^{N}}$ denotes the
channel from UE $k$ to the RIS, respectively. We consider a realistic model to
capture the spatial correlation among the RIS scattering elements [25], which
is due to the sub-wavelength element sizes, sub-wavelength inter-element-
distances, and geometric layout. Meanwhile, the spatial correlation at the AP
multiple antennas is also considered in this paper. In [17], the authors
assume all the channels are Rayleigh fading, while in practice, RIS is usually
deployed such that there is a direct path experiencing Rician fading between
AP-RIS and UE-RIS for communication enhancement. On the other hand, the
channels between the UEs and the APs lack a direct path due to the multi-
scatterer distribution in an urban environment. Hence, we focus on a general
case that AP-UE channels are Rayleigh fading whereas RIS-UE and AP-RIS
channels are Rician fading. Then, the channels ${{\bf{g}}_{mk}}$,
${{\bf{z}}_{k}}$, and ${{\bf{H}}_{m}}$ can be modeled as

$\displaystyle{{\bf{g}}_{mk}}$ $\displaystyle\sim{\cal C}{\cal
N}\left({{\mathbf{0}},{{\mathbf{R}}_{mk}}}\right),\forall m,k,$ (1)
$\displaystyle{{\bf{H}}_{m}}$
$\displaystyle={{{\bf{\bar{H}}}}_{m}}+{{{\bf{\tilde{H}}}}_{m}},\;{{\bf{z}}_{k}}={{\bf{\Theta}}_{k}}{{{\bf{\bar{z}}}}_{k}}+{{{\bf{\tilde{z}}}}_{k}},\forall
m,k,$ (2)

where ${\mathbf{R}}_{mk}\in{\mathbb{C}}{{}^{L\times L}}$ is the spatial
correlation matrix of the antennas at AP $m$ and
${\beta_{mk}}={{{\rm{tr}}\left({{{\bf{R}}_{mk}}}\right)}\mathord{\left/{\vphantom{{{\rm{tr}}\left({{{\bf{R}}_{mk}}}\right)}L}}\right.\kern-1.2pt}L}$
is the large-scale fading coefficient between AP $m$ and UE $k$.
${{{\bf{\bar{H}}}}_{m}}\in{{\mathbb{C}}^{N\times L}}$ and
${{{\bf{\bar{z}}}}_{k}}\in{{\mathbb{C}}^{N}}$ represent the deterministic LoS
component of the corresponding AP-RIS and RIS-UE channels, respectively.
${{{\bf{\tilde{H}}}}_{m}}\sim{\cal C}{\cal
N}\left({{\bf{0}},{{{\bf{\tilde{R}}}}_{m}}}\right)$ and
${{{\bf{\tilde{z}}}}_{k}}\sim{\cal C}{\cal
N}\left({{\bf{0}},{{{\bf{\tilde{R}}}}_{k}}}\right)$ are the NLoS components,
where the spatial covariance matrices
${{\bf{\tilde{R}}}_{m}}\in{{\mathbb{C}}^{NL\times NL}}$ and
${{\bf{\tilde{R}}}_{k}}\in{{\mathbb{C}}^{N\times N}}$ are given by [16]

$\displaystyle{{{\bf{\tilde{R}}}}_{m}}=\frac{1}{{LN{\beta_{m}}}}\left({{\bf{R}}_{m}^{T}\otimes{{\bf{R}}_{r}}}\right),$
(3) $\displaystyle{{{\bf{\tilde{R}}}}_{k}}={\beta_{k}}{A_{r}}{\bf{R}},$ (4)

respectively, where ${{\beta_{m}}}$ and ${{\beta_{k}}}$ denote the large-scale
fading coefficients from AP $m$ and UE $k$ to RIS, respectively. Note that
${{\bf{R}}_{m}}\in{{\mathbb{C}}^{L\times L}}$ and
${{{\bf{R}}_{r}}={\beta_{m}}{A_{r}}\bf{R}}$ denote the correlation matrices of
the AP side and RIS side, respectively. ${A_{r}}={d_{H}}{d_{V}}$ denotes the
area of each RIS element, where ${d_{H}}$ and ${d_{V}}$ are the horizontal
width and the vertical height of each RIS element, respectively.
${\mathbf{R}}\in{{\mathbb{C}}^{N\times N}}$ denotes the spatial correlation of
the RIS which has the $\left({m^{\prime},n^{\prime}}\right)$-th element as
${{\left[\mathbf{R}\right]_{m^{\prime}n^{\prime}}}={\rm{sinc}}\left({{{2\left\|{{{\mathbf{u}}_{m^{\prime}}}-{{\mathbf{u}}_{n^{\prime}}}}\right\|}\mathord{\left/{\vphantom{{2\left\|{{{\mathbf{u}}_{m^{\prime}}}-{{\mathbf{u}}_{n^{\prime}}}}\right\|}\lambda}}\right.\kern-1.2pt}\lambda}}\right)}$,
where ${{\rm{sinc}}\left(y\right)={{\sin\left({\pi
y}\right)}\mathord{\left/{\vphantom{{\sin\left({\pi y}\right)}{\left({\pi
y}\right)}}}\right.\kern-1.2pt}{\left({\pi y}\right)}}}$ denotes the sinc
function and ${\lambda}$ denotes the carrier wavelength [25]. Besides,
${{\mathbf{u}_{x}}\\!=\\!{\left[{0,\bmod\\!\left({x\\!-\\!1,{N_{H}}}\\!\right){d_{H}},\left\lfloor{{{\left({x\\!-\\!1}\right)}\mathord{\left/{\vphantom{{\left({x-1}\right)}{{N_{H}}}}}\right.\kern-1.2pt}{{N_{H}}}}}\right\rfloor{d_{V}}}\right]^{T}}}$,
$x\\!\in\\!\left\\{{m^{\prime},n^{\prime}}\right\\}$ is the position vector,
where ${N_{V}}$ and ${N_{H}}$ are the numbers of elements at RIS in each
column and row, respectively, such that ${N={N_{H}}\times{N_{V}}}$. Moreover,
${{\bf{\Theta}}_{k}}={\rm{diag}}\left({{e^{j{\theta_{1k}}}},\cdots,{e^{j{\theta_{Nk}}}}}\right)\in{{\mathbb{C}}^{N\times
N}}$, where ${\theta_{nk}}\in\left[{-\pi,\pi}\right]$ is the phase-shift of
the LoS component between UE $k$ and the $n$-th element of RIS caused by the
small changes in the location of UE $k$ [15]. We assume that all the elements
of ${{\bf{\Theta}}_{k}}$ are identical [15] such that the LoS component in (2)
can be written as ${{{\bf{\bar{z}}}}_{k}}{e^{j{\theta_{k}}}}$.

Let
${\mathbf{\Phi}={\rm{diag}}\left({{e^{j{\varphi_{1}}}},{e^{j{\varphi_{2}}}},\cdots,{e^{j{\varphi_{N}}}}}\right)}$
denote the phase shift matrix of the RIS, where
${\varphi_{n}}\in\left[{-\pi,\pi}\right],\forall
n\in\left\\{{1,\ldots,N}\right\\}$, denotes the phase shift introduced by the
$n$-th RIS element. Thus, the total uplink aggregated channel between AP $m$
and UE $k$ can be formulated as

$\displaystyle{{\bf{o}}_{mk}}={{\bf{g}}_{mk}}+{\bf{H}}_{m}^{H}{\bf{\Phi}}{{\bf{z}}_{k}},\forall
m,k,$ (5)

which consists of a cascaded link reflected by the RIS and a direct link
between AP $m$ and UE $k$. Through simple mathematical derivation, the end-to-
end channel between AP $m$ and UE $k$ via the RIS is obtained as

$\displaystyle{{\bf{o}}_{mk}}$
$\displaystyle={{\bf{g}}_{mk}}+{\bf{H}}_{m}^{H}{\bf{\Phi}}{{\bf{z}}_{k}}$
$\displaystyle={{\bf{g}}_{mk}}+{\left({{{{\bf{\bar{H}}}}_{m}}+{{{\bf{\tilde{H}}}}_{m}}}\right)^{H}}{\bf{\Phi}}\left({{{{\bf{\bar{z}}}}_{k}}{{e^{j{\theta_{k}}}}}+{{{\bf{\tilde{z}}}}_{k}}}\right)$
$\displaystyle=\underbrace{{\bf{\bar{H}}}_{m}^{H}{\bf{\Phi}}{{{\bf{\bar{z}}}}_{k}}}_{{{{\bf{\bar{o}}}}_{mk}}}{e^{j{\theta_{k}}}}+\underbrace{{{\bf{g}}_{mk}}+{\bf{\bar{H}}}_{m}^{H}{\bf{\Phi}}{{{\bf{\tilde{z}}}}_{k}}+{\bf{\tilde{H}}}_{m}^{H}{\bf{\Phi}}{{{\bf{\bar{z}}}}_{k}}{e^{j{\theta_{k}}}}+{\bf{\tilde{H}}}_{m}^{H}{\bf{\Phi}}{{{\bf{\tilde{z}}}}_{k}}}_{{{{\bf{\tilde{o}}}}_{mk}}}.$
(6)

We assume that UEs move slowly such that the LoS component
${{{{\bf{\bar{o}}}}_{mk}}}$ is slow time-varying and is known at the APs [15].
In the following, we first analyze the second-order statistic of the unknown
${{\bf{\tilde{o}}}_{mk}}$ that will be handy for the analysis in the sequel.

######  Theorem 1.

The covariance matrix of the second term ${{{{\bf{\tilde{o}}}}_{mk}}}$ in
(II-A) can be obtained as

$\displaystyle\mathbb{E}\left\\{{{{{{\bf{\tilde{o}}}}_{mk}}{{{\bf{\tilde{o}}}}^{H}_{mk}}}}\right\\}\\!=\\!\underbrace{{{\bf{R}}_{mk}}\\!+\\!{\bf{\bar{H}}}_{m}^{H}\mathbf{\Phi}{{{\bf{\tilde{R}}}}_{k}}{\mathbf{\Phi}^{H}}{{{\bf{\bar{H}}}}_{m}}\\!+\\!{\bf{Q}}_{mk}^{1}\\!+\\!{\bf{Q}}_{mk}^{2}}_{{\bf{R}}_{mk}^{o}}.$
(7)

###### Proof:

The proof is given in Appendix A. ∎

######  Remark 1.

Note that since the mean and covariance matrix of the aggregated channel
${{{{\bf{o}}}_{mk}}}$ can be obtained at AP $m$ in each coherence time block,
we can utilize them to estimate the end-to-end aggregated channel in the later
section.

###  II-B Electromagnetic Interference Model

The EMI is produced by a superposition of a continuum of incoming plane waves
that are generated by some external sources. In [26], the authors have shown
that EMI at RIS has a significant impact on communication performance which
cannot be ignored. In particular, the EMI at RIS is modeled as

$\displaystyle{{\bf{n}}}\sim{\cal C}{\cal
N}\left({{\bf{0}},{A_{r}}{\sigma_{r}^{2}}{\mathbf{R}}}\right),$ (8)

where ${A_{r}}$ denote the area of each RIS element and ${{\mathbf{R}}}$ is
the spatial correlation matrix of RIS, respectively. ${\sigma_{r}^{2}}$ is the
EMI power at RIS. To proceed further, we define [26]

$\displaystyle\rho=\frac{{{p_{\max}}\sum\limits_{m=1}^{M}{{\beta_{m}}}}}{{M\sigma_{r}^{2}}},$
(9)

where ${{p_{\max}}}$ denotes the maximum transmit power of the UEs which have
the same maximum power. $\rho$ corresponds to the ratio between the received
signal power and the EMI power at each antenna element of the RIS, which is to
facilitate the description of the EMI power.

######  Remark 2.

The correlation of RIS in [25] shows that the correlation does not exist only
when the RIS are positioned along a straight line at a spacing of an integer
multiple of $\frac{\lambda}{2}$, which can never happen with a practical two-
dimensional RIS. Therefore, it is necessary to consider the effect of EMI with
spatial correlation at the RIS elements. Besides, from (8), we find that both
${A_{r}}$ and ${\mathbf{R}}$ are affected by the size of each RIS element and
the increases of spatial correlation can increase the impact of EMI.

###  II-C Uplink Pilot Training and Channel Estimation

We adopt $\tau_{p}$ pilot sequences which are mutually orthogonal for channel
estimation in each coherence time block. All the UEs share the same $\tau_{p}$
orthogonal uplink pilot sequences [14]. In particular, the pilot sequence of
UE $k$ is denoted by ${\bm{\phi}_{k}}\in{{\mathbb{C}}^{{\tau_{p}}}}$ and
satisfies ${{\left\|{{\bm{\phi}_{k}}}\right\|^{2}}={\tau_{p}}}$. Let ${{\cal
P}_{k}}$ denote the index subset of the UEs which adopts the same pilot
sequence as UE $k$ including itself as $K>{\tau_{p}}$. Different from the
ideal assumption adopted in [17, 28, 20], there are ambient electromagnetic
waves in the space and would also be reflected by the RIS and eventually
received by the APs [26]. Considering the EMI, the received signal
${\bf{X}}_{k}^{p}\in{\mathbb{C}^{N\times{\tau_{p}}}}$ at the RIS is

$\displaystyle{{\mathbf{X}}_{k}^{p}}={{\mathbf{z}}_{k}}{\bm{\phi}}_{k}^{T}+{\mathbf{N}},$
(10)

where ${\mathbf{N}}\in{\mathbb{C}^{N\times{\tau_{p}}}}$ denotes the EMI
collected over the ${\tau_{p}}$ samples, produced by the impinging and
uncontrollable electromagnetic waves, with
${\mathbf{n}}={\mathbf{N}}\left({:,}\right)\sim\mathcal{C}\mathcal{N}\left({0,{A_{r}}{\sigma_{r}^{2}}{\mathbf{R}}}\right)$.
The signal transmitted by UE $k$ is reflected by the RIS and the received
signal ${\mathbf{Y}}_{mk}^{p}\in{\mathbb{C}^{L\times{\tau_{p}}}}$ at AP ${m}$
can be written as

$\displaystyle{\mathbf{Y}}_{mk}^{p}$
$\displaystyle=\sqrt{{{\hat{p}}_{k}}}{\mathbf{H}}_{m}^{H}{\mathbf{\Phi}}{{\mathbf{X}}_{k}^{p}}+\sqrt{{{\hat{p}}_{k}}}{{\mathbf{g}}_{mk}}{\bm{\phi}}_{k}^{T}+{{\mathbf{N}}_{m}}$
$\displaystyle=\sqrt{{{\hat{p}}_{k}}}\left({{\mathbf{H}}_{m}^{H}{\mathbf{\Phi}}{{\mathbf{z}}_{k}}+{{\mathbf{g}}_{mk}}}\right){\bm{\phi}}_{k}^{T}+{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
N}}+{{\mathbf{N}}_{m}}$
$\displaystyle=\sqrt{{{\hat{p}}_{k}}}{{\mathbf{o}}_{mk}}{\bm{\phi}}_{k}^{T}+{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
N}}+{{\mathbf{N}}_{m}},$ (11)

where ${{{\hat{p}}_{k}}}$ is the pilot transmit power of UE $k$,
${{\mathbf{N}}_{m}}\in{\mathbb{C}^{L\times{\tau_{p}}}}$ is the additive noise
with independent ${\cal C}{\cal N}\left({0,{\sigma^{2}}}\right)$ entries, and
${\sigma^{2}}$ is the noise power. Then, the signals from all the UEs received
at AP $m$ can be written as

$\displaystyle{\mathbf{Y}}_{m}^{p}=\sum\limits_{k=1}^{K}{\sqrt{{{\hat{p}}_{k}}}{{\mathbf{o}}_{mk}}{\bm{\phi}}_{k}^{T}+}{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
N}}+{{\mathbf{N}}_{m}}.$ (12)

Then, we multiply the received signal by ${{\bm{\phi}}_{k}^{*}}$ at AP $m$ to
estimate ${{{\bf{o}}_{mk}}}$ and the results can be obtained as [16]

$\displaystyle{\mathbf{y}}_{m}^{p}$
$\displaystyle={\mathbf{Y}}_{m}^{p}{\bm{\phi}}_{k}^{*}=\sum\limits_{k=1}^{K}{\sqrt{{{\hat{p}}_{k}}}{{\mathbf{o}}_{mk}}{\bm{\phi}}_{k}^{T}{\bm{\phi}}_{k}^{*}+}\left({{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
N}}+{{\mathbf{N}}_{m}}}\right){\bm{\phi}}_{k}^{*}\hfill$
$\displaystyle=\sum\limits_{k\in{\mathcal{P}_{k}}}{\sqrt{{{\hat{p}}_{k}}}{\tau_{p}}{{\mathbf{o}}_{mk}}}+\left({{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
N}}+{{\mathbf{N}}_{m}}}\right){\bm{\phi}}_{k}^{*}\hfill$
$\displaystyle=\sqrt{{{\hat{p}}_{k}}}{\tau_{p}}{{\mathbf{o}}_{mk}}+\sum\limits_{i\in{\mathcal{P}_{k}}\backslash\left\\{k\right\\}}{\sqrt{{{\hat{p}}_{i}}}{\tau_{p}}{{\mathbf{o}}_{mi}}}+{\mathbf{n}}_{mk}^{p},$
(13)

where ${\mathbf{n}}_{mk}^{p}=\left({{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
N}}+{{\mathbf{N}}_{m}}}\right){\bm{\phi}}_{k}^{*}$ denotes the total noise at
AP $m$. Note that the noise here is different from traditional one as in [27,
28, 29]. The noise generated by EMI reaches AP $m$ through channel
${\mathbf{H}}_{m}$, so its covariance matrix is affected by the channel state
information (CSI) from RIS to APs. Then, we derive the covariance matrix of
the effective noise for later analysis.

######  Theorem 2.

The covariance matrix of the noise ${\mathbf{n}}_{mk}^{p}$ at AP $m$ in (II-C)
can be obtained as

$\displaystyle\mathbb{E}\left\\{{{\mathbf{n}}_{mk}^{p}{{\left({{\mathbf{n}}_{mk}^{p}}\right)}^{H}}}\right\\}={\tau_{p}}{{\mathbf{R}}_{mm}}+{\tau_{p}}{\sigma^{2}}{{\mathbf{I}}_{L}}.$
(14)

###### Proof:

The proof is given in Appendix B. ∎

######  Remark 3.

Note that the covariance matrix of the EMI ${\mathbf{n}}_{mk}^{p}$ including
two parts: matrix ${{\mathbf{R}}_{mm}}$ and Gaussian noise
${{\mathbf{I}}_{L}}$. ${{\mathbf{R}}_{mm}}$ is affected by RIS spatial
correlation and RIS-AP channels. It is clear that the increase of spatial
correlation will lead to the increase of ${{\mathbf{R}}_{mm}}$ and the
degradation of performance.

Based on (II-C) and (14), if ${{{{\bf{\bar{o}}}}_{mk}}}$, ${\bf{R}}_{mk}^{o}$,
${\bf{R}}_{mm}$, and ${{\theta_{k}}}$ are available at AP $m$, we utilize the
MMSE channel estimator in [15] to estimate the effective channel
${{\bf{o}}_{mk}}$ as

$\displaystyle{{{\mathbf{\hat{o}}}}_{mk}}$
$\displaystyle={\mathbf{\bar{H}}}_{m}^{H}{\mathbf{\Phi}}{{{\mathbf{\bar{z}}}}_{k}}{e^{j{\theta_{k}}}}+\frac{{\sqrt{{{\hat{p}}_{k}}}{\mathbf{R}}_{mk}^{o}}\left({{\mathbf{y}}_{mk}^{p}-{\mathbf{\bar{y}}}_{mk}^{p}}\right)}{{\sum\limits_{i\in{\mathcal{P}_{k}}}{{p_{i}}{\tau_{p}}{\mathbf{R}}_{mi}^{o}}+{{\mathbf{R}}_{mm}}+{\sigma^{2}}{{\mathbf{I}}_{L}}}}$
$\displaystyle={{{\mathbf{\bar{o}}}}_{mk}}{e^{j{\theta_{k}}}}+\sqrt{{{\hat{p}}_{k}}}{\mathbf{R}}_{mk}^{o}{\mathbf{\Psi}}_{mk}^{-1}\left({{\mathbf{y}}_{mk}^{p}-{\mathbf{\bar{y}}}_{mk}^{p}}\right),$
(15)

where ${\bf{\bar{y}}}_{mk}^{p}=\sum\nolimits_{i\in{{\cal
P}_{k}}}{\sqrt{{{\hat{p}}_{i}}}}{\tau_{p}}{{{\bf{\bar{o}}}}_{mi}}{e^{j{\theta_{i}}}}$
and ${{\bf{\Psi}}_{mk}}=\sum\nolimits_{i\in{{\cal
P}_{k}}}{{{\hat{p}}_{i}}{\tau_{p}}{\bf{R}}_{mi}^{o}}+{\mathbf{R}}_{mm}+{\sigma^{2}}{{\bf{I}}_{L}}$.
The estimated channel ${{{\bf{\hat{o}}}}_{mk}}$ and the estimation error
${{{\bf{\tilde{o}}}}_{mk}}={{\bf{o}}_{mk}}-{{{\bf{\hat{o}}}}_{mk}}$ are
independent random variable with

$\displaystyle\mathbb{E}\left\\{{{{{\bf{\hat{o}}}}_{mk}}\left|{{\varphi_{k}}}\right.}\right\\}={{{\bf{\bar{o}}}}_{mk}}{e^{j{\varphi_{k}}}},\>\>{\rm{Cov}}\left\\{{{{{\bf{\hat{o}}}}_{mk}}\left|{{\varphi_{k}}}\right.}\right\\}={{\hat{p}}_{k}}{\tau_{p}}{{\bf{\Omega}}_{mk}},$
$\displaystyle\mathbb{E}\left\\{{{{{\bf{\tilde{o}}}}_{mk}}}\right\\}={\bf{0}},\quad\quad\quad\quad\;\;\;{\rm{Cov}}\left\\{{{{{\bf{\tilde{o}}}}_{mk}}}\right\\}={{\bf{C}}_{mk}},$
(16)

where
${{\bf{\Omega}}_{mk}}={\bf{R}}_{mk}^{o}{\bf{\Psi}}_{mk}^{-1}{\bf{R}}_{mk}^{o}$
and
${{\bf{C}}_{mk}}={\bf{R}}_{mk}^{o}-{{\hat{p}}_{k}}{\tau_{p}}{\bf{R}}_{mk}^{o}{\bf{\Psi}}_{mk}^{-1}{\bf{R}}_{mk}^{o}$.
222The MMSE channel estimator can minimize the channel estimation error and
provides a satisfactory system performance in CF massive MIMO system in [15].
The MMSE channel estimation still performs well when EMI is considered.

##  III Uplink Data Transmission and Performance Analysis

The existing CF massive MIMO system is a three-tier structure composed of the
CPU, APs, and UEs. The RIS-aided CF mMIMO system introduces an extra RIS layer
which adds a cascading link. To the best of the authors’ knowledge, recent
works for uplink analysis of RIS-aided CF mMIMO system consider the classical
AP local processing structure, e.g., [17, 28, 36], and the EMI at RIS is
ignored. In this section, we first consider the impact of EMI on RIS and
introduce a process of detecting the uplink transmitted signals by utilizing
the channel estimation method introduced in the previous section. Then, we
utilize the AP local MR combining and the LSFD at the CPU and derive an
asymptotic closed-form expression of the uplink SE.333There are many
performance indicators that can be adopted to describe the performance of the
system, e.g., bit error rate (BER), outage probability (OP), and SE. The BER
and OP are mainly used to obtain insights into the reliability of the system.
In this paper, we aim to explore the system effectiveness and SE is a suitable
metric to capture this important measure.

###  III-A Uplink Data Transmission

In the uplink, we consider that the UEs transmit the uplink data to all the
APs simultaneously. Due to the existence of EMI, we first obtain the received
signal ${\mathbf{x}}_{k}^{u}\in{\mathbb{C}^{N}}$ from UE $k$ at RIS as

$\displaystyle{\mathbf{x}}_{k}^{u}={{\mathbf{z}}_{k}}{s_{k}}+{\mathbf{n}},$
(17)

where ${s_{k}}\sim{\cal C}{\cal N}\left({0,{p_{k}}}\right)$ denotes the uplink
signal transmitted by UE $k$ with power
${p_{k}}=\mathbb{E}\\{{{{\left|{{s_{k}}}\right|}^{2}}}\\}$.
${{\bf{n}}}\sim{\cal C}{\cal
N}\left({{\bf{0}},{A_{r}}{{\sigma^{2}_{r}}}{{\bf{R}}}}\right)$ denotes the
additive EMI noise. Then, the received signal
${{\mathbf{y}}_{mk}}\in{\mathbb{C}^{L}}$ at AP $m$ can be divided into the
path reflected from the RIS and the one directly transmitted from UE $k$,
which is expressed as

$\displaystyle{{\mathbf{y}}_{mk}}$
$\displaystyle={\mathbf{H}}_{m}^{H}{\mathbf{\Phi
x}}_{k}^{u}+{{\mathbf{g}}_{mk}}{s_{k}}+{{\mathbf{n}}_{m}}$
$\displaystyle=\left({{\mathbf{H}}_{m}^{H}{\mathbf{\Phi}}{{\mathbf{z}}_{k}}+{{\mathbf{g}}_{mk}}}\right){s_{k}}+{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
n}}+{{\mathbf{n}}_{m}},$ (18)

where
${{\mathbf{n}}_{m}}\sim\mathcal{C}\mathcal{N}\left({0,{\sigma^{2}}{{\mathbf{I}}_{L}}}\right)$
denotes the additive noise. Since there are $K$ UEs, the total received signal
${{\mathbf{y}}_{m}}\in{\mathbb{C}^{L}}$ at AP $m$ can be obtained as

$\displaystyle{{\mathbf{y}}_{m}}$
$\displaystyle=\sum\limits_{k=1}^{K}{\left({{\mathbf{H}}_{m}^{H}{\mathbf{\Phi}}{{\mathbf{x}}_{k}}+{{\mathbf{g}}_{mk}}}\right){s_{k}}}+{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
n}}+{{\mathbf{n}}_{m}}$
$\displaystyle=\sum\limits_{k=1}^{K}{{{\mathbf{o}}_{mk}}{s_{k}}}+{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
n}}+{{\mathbf{n}}_{m}}.$ (19)

We consider that each AP can process the uplink data locally and
intermediately with a combining vector [14, 37]. We adopt
${{\bf{v}}_{mk}}\in{\mathbb{C}^{L}}$ denoting the combining vector which is
designed by AP $m$ for UE $k$ and then AP $m$ can obtain the local estimate of
${s_{k}}$ as

$\displaystyle{{\tilde{s}}_{mk}}={\mathbf{v}}_{mk}^{H}{{\mathbf{y}}_{m}}=\sum\limits_{k=1}^{K}{{\mathbf{v}}_{mk}^{H}{{\mathbf{o}}_{mk}}}{s_{k}}+{\mathbf{v}}_{mk}^{H}{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
n}}+{\mathbf{v}}_{mk}^{H}{{\mathbf{n}}_{m}}.$ (20)

Any linear combining vector is available for (20) and the local channel state
information (CSI) in AP $m$ can be used to design ${{\bf{v}}_{mk}}$. Here, we
consider the MR combining with ${{\bf{v}}_{mk}}={\bf{\hat{o}}}_{mk}$ due to
its simplicity [37]. After the MR combining, all the APs convey the local
estimates ${{\tilde{s}}_{mk}}$ in (20) to the CPU. To further mitigate the
inter-user interference [14], ${{\tilde{s}}_{mk}}$ are linearly weighted with
the weight coefficients ${a_{mk}^{*}}\in\mathbb{C}$ at the CPU as

$\displaystyle{{\hat{s}}_{k}}$
$\displaystyle=\sum\limits_{m=1}^{M}{a_{mk}^{*}}{{\tilde{s}}_{mk}}=\sum\limits_{m=1}^{M}{a_{mk}^{*}}{\mathbf{v}}_{mk}^{H}{{\mathbf{o}}_{mk}}{s_{k}}+\sum\limits_{m=1}^{M}{a_{mk}^{*}}\left({\sum\limits_{i\neq
k}^{K}{{\mathbf{v}}_{mk}^{H}{{\mathbf{o}}_{ml}}}{s_{i}}}\right)$
$\displaystyle+\sum\limits_{m=1}^{M}{a_{mk}^{*}}{\mathbf{v}}_{mk}^{H}{{\mathbf{n}}_{m}}+\sum\limits_{m=1}^{M}{a_{mk}^{*}}{\mathbf{v}}_{mk}^{H}{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
n}}$
$\displaystyle={\mathbf{a}}_{k}^{H}{{\mathbf{u}}_{kk}}{s_{k}}+\sum\limits_{i\neq
k}^{K}{{\mathbf{a}}_{k}^{H}{{\mathbf{u}}_{ki}}{s_{i}}}+{{\mathbf{n}}_{k}}+{{\mathbf{n}}_{\mathrm{EMI},k}},$
(21)

where
${{\bf{u}}_{ki}}={[{{\bf{v}}_{1k}^{H}{{\bf{o}}_{1i}},\cdots,{\bf{v}}_{Mk}^{H}{{\bf{o}}_{Mi}}}]^{T}}\in\mathbb{C}{{}^{M}}$
and ${{\bf{a}}_{k}}={[{{a_{1k}},\cdots,{a_{Mk}}}]^{T}}\in\mathbb{C}{{}^{M}}$
denotes the LSFD coefficient vector. The main difference in (III-A) compared
to [14, 38] is that the noise term consists of two parts.
${{\bf{n}}_{k}}=\sum\nolimits_{m=1}^{M}{a_{mk}^{*}{\bf{v}}_{mk}^{H}{{\bf{n}}_{m}}}$
and
${{\mathbf{n}}_{{\text{EMI}},k}}=\sum\nolimits_{m=1}^{M}{a_{mk}^{*}}{\mathbf{v}}_{mk}^{H}{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
n}}$ denote the noise at all the APs and the EMI reflected by the RIS,
respectively.

###  III-B Performance Analysis

In this section, we analyze the uplink performance of the RIS-aided CF massive
MIMO system with the MR combining scheme and LSFD, then investigate the impact
of spatial correlation and EMI. Based on (III-A), the uplink achievable SE
lower bound of UE $k$ can be obtained by utilizing the use-and-then-forget
(UatF) bound [14] as follows

$\displaystyle{\rm{S}}{{\rm{E}}_{k}}=\frac{{{\tau_{u}}}}{{{\tau_{c}}}}{\log_{2}}\left({1+{\gamma_{k}}}\right),$
(22)

with the effective signal-to-interference-plus-noise ratio (SINR)
${\gamma_{k}}$ given by

$\displaystyle{\gamma_{k}}\\!=\\!\frac{{{p_{k}}{{\left|{{\bf{a}}_{k}^{H}\mathbb{E}\left\\{{{{\bf{u}}_{kk}}}\right\\}}\right|}^{2}}}}{{{\bf{a}}_{k}^{H}\\!\\!\left({\sum\limits_{i=1}^{K}{{p_{i}}{{\bf{T}}_{ki}}}\\!-\\!{p_{k}}{{\mathbb{E}\\!\left\\{{{{\bf{u}}_{kk}}}\right\\}}{\mathbb{E}\\!\left\\{{{{\bf{u}}^{H}_{kk}}}\right\\}}}\\!+\\!{\sigma^{2}}{{\bf{D}}_{k}}}\\!+\\!{{{\mathbf{U}}^{\mathrm{EMI}}_{k}}}\\!\right)\\!{{\bf{a}}_{k}}}},$
(23)

where
${{\bf{T}}_{ki}}=[\mathbb{E}{\\{{{\bf{v}}_{mk}^{H}{{\bf{o}}_{mi}}{\bf{o}}_{m^{\prime}i}^{H}{{{\bf{v}}}_{m^{\prime}k}}}\\}:\forall
m,m^{\prime}}]\in{\mathbb{C}^{M\times M}}$ denotes the inter-user interference
caused by the pilot contamination and CSI imperfection. The noise at all the
APs is obtained as
${{\bf{D}}_{k}}={\rm{diag}}({\mathbb{E}\\{{{{\|{{{{\bf{v}}}_{1k}}}\|}^{2}}}\\},\cdots,\mathbb{E}\\{{{{\|{{{{\bf{v}}}_{Mk}}}\|}^{2}}}\\}})\in\mathbb{C}{{}^{M\times
M}}$, and the effect of EMI at RIS is denoted as
${{\mathbf{U}}^{\mathrm{EMI}}_{k}}={\text{diag}}({\mathbb{E}\\{{{{\|{{\mathbf{v}}_{1k}^{H}{\mathbf{H}}_{1}^{H}{\mathbf{\Phi
n}}}\|}^{2}}}\\},\cdots\\!,\mathbb{E}\\{{{{\|{{\mathbf{v}}_{Mk}^{H}{\mathbf{H}}_{M}^{H}{\mathbf{\Phi
n}}}\|}^{2}}}\\}})\\!\in\\!{\mathbb{C}^{M\times M}}$.

Various previous work on RIS-aided CF mMIMO system considered the application
of local processing at the APs and a simple centralized decoding at the CPU
[17, 28, 31] for simplicity. Yet, it greatly limits system performance. Here,
we adopt LSFD at the CPU compared with the simple centralized decoding to
further improve the system performance and derive the closed-form of uplink
SE.

####  III-B1 Large Scale Fading Decoding (LSFD)

In this case, each AP preprocess its signal by computing local channel
estimation of the received data that are then conveyed to the CPU for final
decoding. Notice that ${{{\bf{a}}_{k}}}$ can be optimized to maximize the SE
by the CPU, but only channel statistics are available since the CPU does not
have knowledge of the channel estimates. In [14], this approach is known as
LSFD in cellular massive MIMO which can be applied here.

######  Theorem 3.

The closed-form expression for the uplink SE of UE $k$ is given by (22), where
the SINR with LSFD is expressed as (24) at the top of the next page. Then, the
desired signal can be expressed by the following formula

$\displaystyle{{\bf{A}}_{k}}={\rm{diag}}\left({{a_{1k}},\cdots,{a_{Mk}}}\right)\in{\mathbb{C}^{M\times
M}},$ (25)
$\displaystyle{{\bf{Z}}_{k}}={\rm{diag}}\left({{z_{1k}},\cdots,{z_{Mk}}}\right)\in{\mathbb{C}^{M\times
M}},$ (26)
$\displaystyle{z_{mk}}={\rm{tr}}\left({{p_{k}}{\tau_{p}}{{\bf{\Omega}}_{mk}}+{{{\bf{\bar{o}}}}_{mk}}{{\left({{{{\bf{\bar{o}}}}_{mk}}}\right)}^{H}}}\right).$
(27)

Also, the definition of non-coherent interference ${{{\bf{\xi}}_{ki}}}$ is
shown as

$\displaystyle{{\bf{\Xi}}_{ki}}$
$\displaystyle={\rm{diag}}\left({{\xi_{1,ki}},\cdots,{\xi_{M,ki}}}\right)\in{\mathbb{C}^{M\times
M}},$ (28) $\displaystyle{\xi_{m,ki}}$
$\displaystyle={\hat{p}_{k}}{\tau_{p}}{\rm{tr}}\left({{\bf{R}}_{mi}^{o}{{\bf{\Omega}}_{mk}}}\right)+{\bf{\bar{o}}}_{mk}^{H}{\bf{R}}_{mi}^{o}{{{\bf{\bar{o}}}}_{mk}}$
$\displaystyle+{\hat{p}_{k}}{\tau_{p}}{\bf{\bar{o}}}_{mi}^{H}{{\bf{\Omega}}_{mk}}{{{\bf{\bar{o}}}}_{mi}}+{\left|{{\bf{\bar{o}}}_{mk}^{H}{{{\bf{\bar{o}}}}_{mi}}}\right|^{2}}.$
(29)

The coherent interference ${{{\bf{\Gamma}}_{ki}}}$ is shown as

$\displaystyle{{\bf{\Gamma}}_{ki}}={\hat{p}_{k}}{\hat{p}_{i}}\tau_{p}^{2}{\left|{{\rm{tr}}\left({{{\bf{A}}_{k}}{{\bf{\Delta}}_{ki}}}\right)}\right|^{2}},$
(30)
$\displaystyle{{\bf{\Delta}}_{ki}}={\rm{diag}}\left({{\varpi_{1,ki}},\cdots,{\varpi_{M,ki}}}\right)\in{{\mathbb{C}}^{M\times
M}},$ (31)
$\displaystyle{\varpi_{m,ki}}{\rm{=tr}}\left({{\bf{R}}_{mi}^{o}{\bf{\Psi}}_{mk}^{-1}{\bf{R}}_{mk}^{o}}\right).$
(32)

Finally, the noise at all the APs and the EMI at RIS is shown as

$\displaystyle{{\bf{J}}_{k}}={\rm{diag}}\left({{{\left\|{{{{\bf{\bar{o}}}}_{1k}}}\right\|}^{2}},\cdots,{{\left\|{{{{\bf{\bar{o}}}}_{Mk}}}\right\|}^{2}}}\right),$
(33)
$\displaystyle{{\mathbf{W}}_{k}}{{=\rm{diag}}}\left({{w_{1k}},\cdots,{w_{Mk}}}\right),$
(34)
$\displaystyle{w_{mk}}\\!=\\!{{{\mathbf{\bar{o}}}}^{H}}_{mk}{{\mathbf{R}}_{mm}}{{{\mathbf{\bar{o}}}}_{mk}}\\!+\\!{{\hat{p}}_{k}}\tau_{p}^{2}{\rm{tr}}\\!\left(\\!{{{\left({{\mathbf{\Psi}}_{mk}^{-1}}\\!\right)}^{H}}\\!{{\left({{\mathbf{R}}_{mk}^{o}}\right)}^{H}}{{\mathbf{R}}_{mm}}{\mathbf{R}}_{mk}^{o}}\\!\right).$
(35)

$\displaystyle\gamma_{k}^{{\mathrm{EMI}}^{(1)}}=\frac{{{p_{k}}{{\left|{{\text{tr}}\left({{\mathbf{A}}_{k}^{H}{{\mathbf{Z}}_{k}}}\right)}\right|}^{2}}}}{{\sum\limits_{i=1}^{K}{{p_{i}}{\text{tr}}\left({{\mathbf{A}}_{k}^{H}{{\mathbf{\Xi}}_{ki}}{{\mathbf{A}}_{k}}}\right)}+\sum\limits_{i\in{\mathcal{P}_{k}}\backslash\left\\{k\right\\}}{{p_{i}}{{\mathbf{\Gamma}}_{ki}}}+{\text{tr}}\left({{\mathbf{A}}_{k}^{H}{{\mathbf{W}}_{k}}{{\mathbf{A}}_{k}}}\right)+{\text{tr}}\left({{\mathbf{A}}_{k}^{H}\left({{\sigma^{2}}{{\mathbf{Z}}_{k}}-{p_{k}}{\mathbf{J}}_{k}^{2}}\right){{\mathbf{A}}_{k}}}\right)}}.$
(24)

###### Proof:

The proof is given in Appendix C. ∎

Furthermore, to maximize the effective SINR in (24), the CPU can utilize LSFD
to optimize ${{{\bf{a}}_{k}}}$ as

$\displaystyle{{\mathbf{a}}_{k}}$
$\displaystyle=\left({\sum\limits_{i=1}^{K}{{p_{i}}{{\mathbf{T}}_{ki}}}-{p_{k}}\mathbb{E}\left\\{{{{\mathbf{u}}_{kk}}}\right\\}\mathbb{E}\left\\{{{\mathbf{u}}_{kk}^{H}}\right\\}+{\sigma^{2}}{{\mathbf{D}}_{k}}}\right.$
$\displaystyle{\left.{+{\mathbf{U}}_{k}^{{\text{EMI}}}}\right)^{-1}}\mathbb{E}\left\\{{{{\mathbf{u}}_{kk}}}\right\\}.$
(36)

Then, the SINR in (23) can be derived as

$\displaystyle{\gamma_{k}}^{{\text{EM}}{{\text{I}}^{(1)}}}={p_{k}}\mathbb{E}\left\\{{{\mathbf{u}}_{kk}^{H}}\right\\}\times\left({\sum\limits_{i=1}^{K}{{p_{i}}{{\mathbf{T}}_{ki}}}}\right.$
$\displaystyle{\left.{-{p_{k}}\mathbb{E}\left\\{{{{\mathbf{u}}_{kk}}}\right\\}\mathbb{E}\left\\{{{\mathbf{u}}_{kk}^{H}}\right\\}+{\sigma^{2}}{{\mathbf{D}}_{k}}+{\mathbf{U}}_{k}^{{\text{EMI}}}}\right)^{-1}}\mathbb{E}\left\\{{{{\mathbf{u}}_{kk}}}\right\\}.$
(37)

###### Proof:

The proof of (III-B1) and (III-B1) can be easily derived following from the
standard results of matrix derivation in [15] and is therefore omitted. ∎

####  III-B2 Simple Centralized Decoding

In this case, the received signals at each AP were processed with MR
combining, then sent to the CPU where the signal from different APs was
processed with simple centralized decoding [14, 16]. Actually, it is just a
special case of LSFD and the coefficient is written as
${{\mathbf{a}}_{k}}={\left[{1,\cdots,1}\right]^{T}}\in{\mathbb{C}^{M\times
1}}$.

######  Corollary 1.

For adopting a simple centralized decoding at the CPU, the closed-form
expression for the uplink SE of UE $k$ is given by (22), where the SINR is
expressed as (38) at the top of this page. For comparison, we also provide a
closed-form expression (39) without EMI at the top of this page.

###### Proof:

The proof is given in Appendix C. ∎

######  Remark 4.

Note that the EMI and the spatial correlation via the RIS affect the system
performance via ${{\bf{R}}_{mm}}$ expressed in a closed-form in (24). By
reducing the spatial correlation of RIS elements or the power of EMI, the
reduction of the norm of ${{\bf{R}}_{mm}}$ leads to the reduction of the
denominator in (24), and the system performance is improved. Besides, reducing
the spatial correlation of RIS elements can increase ${{\bf{R}}_{mk}^{o}}$,
which leads to the increase of the numerator in (24), also, the system
performance is improved.

######  Remark 5.

In CF mMIMO systems, the spatial correlation of the AP antennas is beneficial
to the SE [37], while in our system, the aggregated channel and the EMI
consists the interaction between ${{\bf{R}}_{mk}}$ and ${{\bf{R}}}$. As such
the conclusion in [37] may no longer hold, especially when the RIS elements
are strongly correlated.

$\displaystyle\gamma_{k}^{{\mathrm{EMI}}^{(2)}}=\frac{{{p_{k}}{{\left|{{\text{tr}}\left({{{\mathbf{Z}}_{k}}}\right)}\right|}^{2}}}}{{\sum\limits_{i=1}^{K}{{p_{i}}{\text{tr}}\left({{{\mathbf{\Xi}}_{ki}}}\right)}+\sum\limits_{i\in{\mathcal{P}_{k}}\backslash\left\\{k\right\\}}{{p_{i}}{\hat{p}_{k}}{\hat{p}_{i}}\tau_{p}^{2}{{\left|{{\text{tr}}\left({{{\mathbf{\Delta}}_{ki}}}\right)}\right|}^{2}}}+{\text{tr}}\left({{{\mathbf{W}}_{k}}}\right)+{\text{tr}}\left({{\sigma^{2}}{{\mathbf{Z}}_{k}}-{p_{k}}{\mathbf{J}}_{k}^{2}}\right)}}.$
(38)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\gamma_{k}}=\frac{{{p_{k}}{{\left|{{\rm{tr}}\left({{\bf{A}}_{k}^{H}{{\bf{Z}}_{k}}}\right)}\right|}^{2}}}}{{\sum\limits_{i=1}^{K}{{p_{i}}{\rm{tr}}\left({{\bf{A}}_{k}^{H}{{\bf{\Xi}}_{ki}}{{\bf{A}}_{k}}}\right)}+\sum\limits_{i\in{{\cal
P}_{k}}\backslash\left\\{k\right\\}}{{p_{i}}{{\bf{\Gamma}}_{ki}}}+{\rm{tr}}\left({{\bf{A}}_{k}^{H}\left({{\sigma^{2}}{{\bf{Z}}_{k}}-{p_{k}}{\bf{J}}_{k}^{2}}\right){{\bf{A}}_{k}}}\right)}}.$
(39)

###  III-C Power Control Schemes

The EMI affects the performance of UEs differently in the RIS-aided CF network
because of the different locations of the UEs. Therefore, the impacts of EMI
can be reduced by the design of power allocation, exploiting the different
propagation conditions of the UEs.

####  III-C1 Fractional Power Control

In general, the system performance is limited by the near-far effects.
Transmitting signals at full power not only wastes energy resources, but also
causes severe inter-user interference. To address this issue, we extend the
FPC scheme proposed in [39] to the considered RIS-aided CF massive MIMO
system. Specifically, the FPC scheme for the RIS-aided CF massive MIMO system
based on the equivalent large-scale fading coefficients of the considered UE,
which captures the average signal strength from the UE to all the APs.
Inspired by this, we design the power control factor of UE $k$ as

$\displaystyle{\eta_{k}}={\left({\frac{{\mathop{\min}\limits_{k}\left({\sum\nolimits_{m=1}^{M}{{\text{tr}}\left({{\mathbf{R}}_{mk}^{o}}\right)}}\right)}}{{\sum\nolimits_{m=1}^{M}{{\text{tr}}\left({{\mathbf{R}}_{mk}^{o}}\right)}}}}\right)^{\alpha}},\forall
k,$ (40)

where ${{\mathbf{R}}_{mk}^{o}}$ is given in (7) and ${0<\alpha<1}$ denotes the
fractional power control parameter. Therefore, the transmission power
${p_{k}}$ of UE $k$ can be expressed as ${p_{k}}={\eta_{k}}{p_{\max}}$. For
the uplink transmission, the fairness of UEs is important and (40) aims to
improve the performance of UEs with poor connection strength.

Algorithm 1 Bisection Algorithm for Solving (III-C2)

1:Initialization: Choose the initial values of ${t_{\min}}$ and ${t_{\max}}$,
where ${t_{\min}}$ and ${t_{\max}}$ define a range of relevant values of the
objective function in (III-C2). Choose a tolerance $\varepsilon>0$.

2:Set $t:=\frac{{{t_{\min}}+{t_{\max}}}}{2}$. Solve the following convex
feasibility program:
$\displaystyle\left\\{{\begin{array}[]{*{20}{c}}{\frac{1}{t}{p_{k}}{{\left|{{\text{tr}}\left({{\mathbf{A}}_{k}^{H}{{\mathbf{Z}}_{k}}}\right)}\right|}^{2}}\geqslant{\zeta_{k}},\;k=1,\ldots
K,}\\\ {\;0\leqslant{p_{k}}\leqslant{p_{\max}},\;k=1,\ldots
K,}\end{array}}\right.$ (45) where
${\zeta_{k}}=\sum\limits_{i=1}^{K}{{p_{i}}{\text{tr}}\left({{\mathbf{A}}_{k}^{H}{{\mathbf{\Xi}}_{ki}}{{\mathbf{A}}_{k}}}\right)}+\sum\limits_{i\in{\mathcal{P}_{k}}\backslash\left\\{k\right\\}}{{p_{i}}{{\mathbf{\Gamma}}_{ki}}}+{\text{tr}}\left({{\mathbf{A}}_{k}^{H}{{\mathbf{W}}_{k}}{{\mathbf{A}}_{k}}}\right)+{\text{tr}}\left({{\mathbf{A}}_{k}^{H}\left({{\sigma^{2}}{{\mathbf{Z}}_{k}}-{p_{k}}{\mathbf{J}}_{k}^{2}}\right){{\mathbf{A}}_{k}}}\right)$.

3:If problem (45) is feasible, then set ${t_{\min}}:=t$, else set
${t_{\max}}:=t$.

4:Stop if ${t_{\max}}-{t_{\min}}<\varepsilon$. Otherwise, go to step 2.

####  III-C2 Max-min SE Power Control

To further improve the system performance of poor users, we adopt the max-min
SE power control and the optimization problem can be fromulated as

$\displaystyle\mathop{{\text{maxmize}}}\limits_{{p_{k}}}\;\mathop{{\text{min}}}\limits_{k=1,\cdots,K}\;\gamma_{k}^{{\mathrm{EMI}}^{\left(1\right)}}$
$\displaystyle{\text{subject}}\;{\text{to}}\quad
0\leqslant{p_{k}}\leqslant{p_{\max}},\;k=1,\ldots K,$ (41)

where $\gamma_{k}^{{\mathrm{EMI}}^{\left(1\right)}}$ is given by (24). Problem
(III-C2) can be equivalently reformulated as

$\displaystyle\mathop{{\text{maxmize}}}\limits_{{p_{k}},t}\quad t$
$\displaystyle{\text{subject}}\;{\text{to}}\quad
t\leqslant\gamma_{k}^{{\mathrm{EMI}}^{\left(1\right)}},\;k=1,\ldots K,$
$\displaystyle\quad\quad\quad\quad\;\;\;0\leqslant{p_{k}}\leqslant{p_{\max}},\;k=1,\ldots
K,$ (42)

with $t$ is an auxiliary optimization variable. Consequently, problem (III-C2)
can be efficiently solved by using bisection and solving a sequence of linear
feasibility problems as shown in Algorithm 1. Note that in Algorithm 1, if we
focus on a reasonable range of the initial upper and lower bounds (i.e.,
$t_{\min}=0$ and ${t_{\max}}\geqslant\max\left[{{\gamma_{k}}}\right]$), we can
effectively find a feasible solution within a few iterations.

######  Remark 6.

The computational complexity of Algorithm 1 is
$\mathcal{O}\left({2K^{2}+2K}\right)$ and it is clear that complexity
increases rapidly as the number of users increases. In contrast, the FPC
enjoys a linear complexity but at the expense of some performance loss. We can
find from Figure 2 that the running times of the two methods increase with the
increases of $K$, while that of the max-min method scales with $K$ much faster
than that of FPC. As such, in practice, when the number of UEs is small,
exploiting the max-min method is recommended. However, when the number of UEs
is large or the communication reliability requirement is not stringent, FPC
serves as a better choice.

Figure 2: Average running time against different number of UEs with different
power control methods (${M=10}$, ${L=1}$, ${N=16}$, ${\tau_{p}=3}$,
${{d_{\rm{V}}}={d_{\rm{H}}}=\frac{1}{2}\lambda}$).

##  IV Numerical Results and Discussion

In this section, we provide some numerical results to verify the accuracy of
the derived analysis and evaluate the performance of the RIS-aided CF mMIMO
system over EMI and the spatial correlation of RIS elements. We assume that
the APs and UEs are uniformly distributed in the $0.1\times
0.1\,{\rm{k}}{{\rm{m}}^{2}}$ area with a wrap-around scheme [14],
respectively. Moreover, the RIS is located at the regional center. The height
of each AP, UE, and RIS is $15$ m, $1.65$ m, and $30$ m, respectively. We set
the carrier frequency as 1.9 GHz. Besides, each coherence block consists of
${\tau_{c}}=200$ corresponding to a coherence bandwidth of 200 kHz and
${\tau_{p}}=3$ are reserved for pilot transmission. For the path loss, we take
AP-RIS as an example which consists of an LoS path and we utilize the COST 321
Walfish-Ikegami model [16] to compute the path loss as

$\displaystyle{\beta_{m}}\left[{{\rm{dB}}}\right]=-30.18-26{\log_{10}}\left({\frac{{{d_{m}}}}{{1\>{\rm{m}}}}}\right)+{F_{m}},$
(44)

where $d_{m}$ denotes the distance between AP $m$ and RIS. The Rician
$\kappa$-factor is denoted as ${\kappa_{m}}={10^{1.3-0.003{d_{m}}}}$. The
shadow fading $F_{m}$ and other parameters is similar to [9] with
${F_{m}}=\sqrt{{\delta_{f}}}{a_{m}}+\sqrt{1-{\delta_{f}}}{b_{{\text{RIS}}}}$,
where ${a_{m}}\sim{\cal N}\left({0,\delta_{{\text{sf}}}^{2}}\right)$ and
${b_{\rm{RIS}}}\sim{\cal N}\left({0,\delta_{{\text{sf}}}^{2}}\right)$ are
independent random variables and ${{\delta_{f}}}$ is the shadow fading
parameter. The variances of $a_{m}$ and $b_{\rm{RIS}}$ are
$\mathbb{E}\left\\{{{a_{m}}{a_{m^{\prime}}}}\right\\}={2^{-\frac{{{d_{mm^{\prime}}}}}{{{d_{{\text{dc}}}}}}}}$,
$\mathbb{E}\left\\{{{b_{{\text{RIS}}}}{b_{{\text{RIS'}}}}}\right\\}={2^{-\frac{{{d_{{\text{RISRIS}}^{\prime}}}}}{{{d_{{\text{dc}}}}}}}}$,
respectively, where ${{d_{mm^{\prime}}}}$ are the geographical distances
between AP $m$-AP $m^{\prime}$ and ${{d_{{\text{RISRIS}}^{\prime}}}}=0$,
respectively, $d_{\rm{dc}}$ is the decorrelation distance depending on the
environment. We set ${{\delta_{f}}}=0.5$, $d_{\rm{dc}}=100$ m and
${\delta_{{\text{sf}}}}=8$ in this paper. The large-scale coefficients of
${{\bf{H}}_{m}}$ are given by

$\displaystyle\beta_{m}^{{\rm{LoS}}}=\frac{{{\kappa_{m}}}}{{{\kappa_{m}}+1}}{\beta_{m}},{\kern
1.0pt}\;\;\beta_{m}^{{\rm{NLoS}}}=\frac{1}{{{\kappa_{m}}+1}}{\beta_{m}}.$ (45)

Each AP is equipped with a uniform linear array (ULA) with omnidirectional
antennas so the $n$-th element of the deterministic LoS component
${{{\bf{\bar{H}}}}_{m}}\in{\mathbb{C}^{N\times L}}$ can be written as
${\left[{{{{\bf{\bar{H}}}}_{m}}}\right]_{n,l}}=\sqrt{\beta_{m}^{{\rm{LoS}}}}{e^{j2\pi{d_{H}}\left({n-1}\right)\sin\left({{\theta_{m}}}\right)}}$,
where ${\theta_{m}}$ is the angle of arrival from AP $m$ to the RIS and
${d_{H}}$ denotes the antenna spacing parameter (in fractions of the
wavelength).

Figure 3: CDF of the uplink average SE against different systems over the MR
and LSFD combining methods with EMI (${M=10}$, ${K=5}$, ${L=1}$, ${N=16}$,
${\tau_{p}=3}$, ${{d_{\rm{V}}}={d_{\rm{H}}}=\frac{1}{2}\lambda}$,
$\rho=20\>\rm{dB}$). Figure 4: Uplink average SE against the number of AP $M$
over different numbers of AP antennas with LSFD ($K=5$, $N=16$, $\tau_{p}=3$,
${{d_{\rm{V}}}={d_{\rm{H}}}=\frac{1}{2}\lambda}$, $\rho=20\>\rm{dB}$).

The Gaussian local scattering model in [40] is utilized to generate the
spatial correlation matrix ${{\bf{R}}_{mk}}$. The ${{\bf{R}}_{mk}}$
$\left({l,n}\right)$-th element can be written as

$\displaystyle{\left[{{{\bf{R}}_{mk}}}\right]_{ln}}=\frac{{\beta_{mk}^{{\rm{NLoS}}}}}{{\sqrt{2\pi}{\sigma_{\varphi}}}}\int_{-\infty}^{+\infty}{{e^{j2\pi{d_{H}}\left({l-n}\right)\sin\left({{\theta_{mk}}+\delta}\right)}}}{e^{-\frac{{{\delta^{2}}}}{{2\sigma_{\varphi}^{2}}}}}d\delta,$
(46)

where $\delta\sim{\cal N}\left({0,\sigma_{\varphi}^{2}}\right)$ is the
distributed deviation from ${\theta_{mk}}$ with angular standard deviation
(ASD) ${{\sigma_{\varphi}}}$. Every UE transmits with a power of $23$ dBm and
the noise power ${\sigma^{2}}=94$ dBm. The fractional power control parameter
$\alpha=0.6$. As for the phase shift design of RIS, we take a fixed value that
the $N$ elements phase shift is set equal to
${\pi\mathord{\left/{\vphantom{\pi 4}}\right.\kern-1.2pt}4}$ [17].444We use
UatF bound to derive closed solutions. Since we calculate the expected value
as shown in (23), the phase shift effect of RIS is not significant, especially
when the spatial correlation is weak. Note that only Figures 9 and 10 consider
the power control schemes.

Figure 3 compares the CDF of the uplink average SE for RIS-aided CF and CF
massive MIMO systems under the random locations of UEs and APs with MR/LSFD
combining methods with EMI. We illustrate the CDF of the uplink SE by using
Monte-Carlo simulations and the proposed analytical framework. It is clear
that the RIS-aided CF system has a significant gain over the conventional CF
system. Meanwhile, the LSFD achieves a 1.86 times improvement at 95%-likely
points compared with MR combining. Besides, from the 95%-likely points, we
know that the EMI harms the system performance, especially for the UEs with
poor performance while its impact on those UEs with high SE is minimal.

Figure 5: Uplink average SE for the CF and RIS-aided CF mMIMO systems against
the number of AP $M$ with different numbers of RIS elements with LSFD ($K=5$,
$L=1$, $\tau_{p}=3$, ${{d_{\rm{V}}}={d_{\rm{H}}}=\frac{1}{2}\lambda}$,
$\rho=20\>\rm{dB}$). Figure 6: Uplink average SE for the CF and RIS-aided CF
mMIMO systems against the number of UE $K$ with different numbers of RIS
elements with LSFD ($M=10$, $L=1$, $\tau_{p}=3$,
${{d_{\rm{V}}}={d_{\rm{H}}}=\frac{1}{2}\lambda}$, $\rho=20\>\rm{dB}$).

Figure 4 shows the average SE per UE as a function of the number of AP $M$
with the different numbers of per AP’s antennas $L$ with LSFD. It is clear
that increasing the numbers of AP $M$ and AP antennas $L$ can improve the
system performance as the increased number of spatial degrees of freedom
facilitates more efficient beamforming. Moreover, it is interesting to find
that when $M=20$, the average SE of $L=2$ achieves a 12.5% gain compared to
that of $L=1$. Yet, for $L=8$, only 10% of the gain is achieved compared to
that of $L=4$, which is much smaller. This reveals that the potential gains
due to the AP antennas are diminishing when the number of AP antennas is
sufficiently large.

Figure 7: Uplink average SE against the number of AP antennas $L$ with
different EMI powers with LSFD ($M=10$, $K=5$, $N=16$, $\tau_{p}=3$,
${{d_{\rm{V}}}={d_{\rm{H}}}=\frac{1}{2}\lambda}$). Figure 8: Uplink average SE
against the number of RIS elements $N$ with MR combining and LSFD ($M=10$,
$L=1$, $K=5$, $\tau_{p}=3$, ${{d_{\rm{V}}}={d_{\rm{H}}}=\frac{1}{2}\lambda}$,
$\rho=20\>\rm{dB}$).

Figure 5 and Figure 6 show the average SE per UE as a function of the number
of AP $M$ and UE $K$ with different RIS elements $N$ with LSFD. It is clear
that increasing the number of RIS elements $N$ can improve the system
performance as it increases the amount of reflected energy in the reflected
link. Figure 5 shows that as the number of UEs increases, the system
performance decreases, which is caused by the increased interference among the
UEs. Note that the performance degradation due to the increased number of UEs
can be relieved by increasing the number of RIS elements. In fact, passive
beamforming can achieve better interference management. This reveals the
importance of deploying RIS in hotspot areas.

Figure 7 shows the average SE as a function of the number of AP antennas $L$
with different EMI powers with LSFD. It is clear that the performance gap is
enhanced with the increase of $L$ since an increasing number of AP antennas
will cause the increased EMI power from RIS. In particular, when the number of
AP antennas $L$ is sufficiently large, for example, compared with $L=14$,
$L=16$ only offers a marginal performance gain. It reveals that continuously
increasing the number of AP antennas in a CF system to achieve further system
performance improvements is not cost-effective unless more efficient combining
methods are employed at the AP rather than the MR combining.

Figure 9: CDF of the uplink average SE for the RIS-aided CF massive system
over the MR combining and the LSFD with EMI (${M=10}$, ${K=5}$, ${L=1}$,
${N=16}$, ${\tau_{p}=3}$, ${{d_{\rm{V}}}={d_{\rm{H}}}=\frac{1}{2}\lambda}$,
$\rho=20\>\rm{dB}$). Figure 10: CDF of the uplink average SE for the RIS-aided
CF massive system against different EMI power with the LSFD (${M=10}$,
${K=5}$, ${L=1}$, ${N=16}$, ${\tau_{p}=3}$,
${{d_{\rm{V}}}={d_{\rm{H}}}=\frac{1}{2}\lambda}$).

Figure 8 shows the average SE as a function of the number of RIS elements $N$
with MR combining and LSFD. It is clear that the performance gap becomes
smaller and is less sensitive to the increases of $N$. It reveals that
increasing the element number of RIS in RIS-aided CF massive MIMO systems is
beneficial to naturalizing the impairment caused by EMI. Moreover, compared
with the MR combining, when $N$ is larger than 16, the system performance
barely increases and the LSFD behaves similarly after $N$ is larger than 64.
This reveals that with a large number of RIS elements, the LSFD can still
exploit the spatial DoF offered by the RIS in CF massive MIMO system.

Figure 11: Average SE against different sizes of RIS element with and without
EMI over the LSFD (${M=10}$, ${L=1}$, ${K=5}$, ${N=64}$, ${\tau_{p}=3}$,
$\rho=20\>\rm{dB}$).  Figure 12: Average SE against different RIS positions
with and without EMI over the LSFD (${M=10}$, ${L=1}$, ${K=5}$, ${N=16}$,
${\tau_{p}=3}$, ${{d_{\rm{V}}}={d_{\rm{H}}}=\frac{1}{2}\lambda}$).

Figure 9 compares the CDF of the uplink average SE for RIS-aided CF massive
MIMO system with MR combining and the LSFD under different power control
schemes. It is clear that compared with the proposed fractional power control
and the max-min SE power control, we can significantly improve the SE for the
UEs with poor performance. Moreover, jointly with the max-min power control,
the LSFD and MR combining achieve 25.1% and 20% gains compared with the full
power method at 90%-likely points, respectively. Hence, the max-min SE power
control with the addition of excellent signal processing methods can further
achieve user fairness in the RIS-aided CF massive MIMO system.

Figure 10 compares the CDF of the uplink average SE for the RIS-aided CF
massive MIMO system against different EMI powers with LSFD. It is clear that
the same as in Figure 3, the EMI has a severer negative impact on the UEs with
poor performance. In particular, when the EMI is serious, i.e.,
$\rho=10\>\rm{dB}$, the performance degradation of poor users is significant
even though the max-min SE power control is applied. This result suggests that
when the EMI is strong, it is not a sensible option for RIS to continue
serving all UEs.

Figure 11 shows the effect of the spatial correlation brought by the size of
the RIS elements on the average SE of the RIS-aided CF massive MIMO system. It
is clear that for the cases with or without EMI, the system performance is the
best when the element size is of a half wavelength. Moreover, when EMI is
considered, the spatial correlation of RIS has a severer impact on system
performance. This can be explained by (8) that the power distribution of EMI
is affected by the spatial correlation of RIS. Therefore, the element size of
RIS needs to be carefully designed in practice to reduce the impact of spatial
correlation and EMI on the system performance.

Figure 12 shows the impact of different RIS deployment locations on the system
performance with and without EMI over LSFD. It is clear that deploying RIS at
the center of the service area can obtain satisfactory system performance
regardless of the presence of EMI. In the absence of EMI, the system
performance in the region is almost centrosymmetric. However, when there is
EMI at RIS, the degradation of the system performance at the edge is
magnified, which reveals the importance of deploying RIS at the center of the
service area.

##  V Conclusions

In this paper, we investigated the uplink SE of RIS-aided CF massive MIMO
systems over spatially correlated Rician channels with EMI. Furthermore, we
analyzed the uplink SE with the MR combining at the APs and the LSFD at the
CPU and obtained the analytical closed-form expression with EMI for
characterizing the performance. It is important that the EMI degrades the
performance of the considered systems, especially for the UEs with
unsatisfactory channel conditions, while the LSFD cooperation is less
sensitive to the EMI compared with MR combining. As such, it is particularly
important to deploy the RIS in a low EMI environment for RIS-aided CF massive
MIMO systems. In addition, increasing the number of APs and RIS elements
significantly improve the average SE, with diminishing returns when the number
of AP antennas and RIS elements is sufficiently large. Besides, increasing the
number of AP antennas also increases the negative impact of EMI. Also,
increasing the number of RIS elements is always beneficial as the impairment
caused by EMI can be relieved by the extra spatial DoF brought by the
increasing number of RIS elements. Meanwhile, considering the existence of
EMI, the spatial correlations among RIS elements impose a severer negative
impact on the system performance. Also, we find that designing the element
spacing of RIS as half wavelength can suppress the spatial correlation and EMI
to the greatest extent. Finally, for practical application in RIS-aided CF
scenarios, we proposed the practical fractional power control in the
considered systems to improve the SE performance of the worst AP and max-min
SE power control scheme for preliminary EMI improvement. In future work, we
will consider the EMI-aware beamforming design to enable the implementation of
RIS-aided CF mMIMO networks.

##  Appendix A Proof of Theorem 1

This appendix calculates the covariance matrix in (5). To start with, we
express

$\displaystyle\mathbb{E}\left\\{{{{{{\bf{\tilde{o}}}}_{mk}}{{{\bf{\tilde{o}}}}^{H}_{mk}}}}\right\\}={{\bf{R}}_{mk}}+{\bf{\bar{H}}}_{m}^{H}{\bf{\Phi}}{{{\bf{\tilde{R}}}}_{k}}{{\bf{\Phi}}^{H}}{{{\bf{\bar{H}}}}_{m}}$
$\displaystyle\\!+\\!\underbrace{\mathbb{E}\\!\left\\{\\!{{\bf{\tilde{H}}}_{m}^{H}{\bf{\Phi}}{{{\bf{\bar{z}}}}_{k}}{\bf{\bar{z}}}_{k}^{H}{{\bf{\Phi}}^{H}}{{{\bf{\tilde{H}}}}_{m}}}\\!\right\\}}_{{\bf{Q}}^{1}_{mk}}\\!+\\!\underbrace{\mathbb{E}\\!\left\\{\\!{{\bf{\tilde{H}}}_{m}^{H}{\bf{\Phi}}{{{\bf{\tilde{z}}}}_{k}}{\bf{\tilde{z}}}_{k}^{H}{{\bf{\Phi}}^{H}}{{{\bf{\tilde{H}}}}_{m}}}\\!\right\\}}_{{\bf{Q}}^{2}_{mk}}\\!.$
(47)

To calculate ${{\bf{Q}}^{1}_{mk}}$, we let
${{\bf{B}}_{k}}={\bf{\Phi}}{{{\bf{\bar{z}}}}_{k}}{\bf{\bar{z}}}_{k}^{H}{{\bf{\Phi}}^{H}}$,
and ${{\bf{Q}}^{1}_{mk}}$ is derived as

$\displaystyle{{\bf{Q}}^{1}_{mk}}$
$\displaystyle=\mathbb{E}\left\\{{{\bf{\tilde{H}}}_{m}^{H}{{\bf{B}}_{k}}{{{\bf{\tilde{H}}}}_{m}}}\right\\}=\mathbb{E}\left\\{{\begin{array}[]{*{20}{c}}{{\bf{\tilde{H}}}_{1m}^{H}{{\bf{B}}_{k}}{{{\bf{\tilde{H}}}}_{1m}}}&\cdots&{{\bf{\tilde{H}}}_{1m}^{H}{{\bf{B}}_{k}}{{{\bf{\tilde{H}}}}_{Lm}}}\\\
\vdots&\ddots&\vdots\\\
{{\bf{\tilde{H}}}_{Lm}^{H}{{\bf{B}}_{k}}{{{\bf{\tilde{H}}}}_{1m}}}&\cdots&{{\bf{\tilde{H}}}_{Lm}^{H}{{\bf{B}}_{k}}{{{\bf{\tilde{H}}}}_{Lm}}}\end{array}}\right\\}.$
(51)

For each element in ${{\bf{Q}}^{1}_{mk}}$, we can obtain
$\mathbb{E}\\!\left\\{\\!{{\bf{\tilde{H}}}_{lm}^{H}{{\bf{B}}_{k}}{{{\bf{\tilde{H}}}}_{l^{\prime}m}}}\\!\right\\}\\!{\buildrel(a)\over{=}}{\rm{tr}}\\!\left(\\!\\!{{{\bf{B}}_{k}}{{\left[{{{{\bf{\tilde{R}}}}_{m}}}\right]}_{\left({lN-N+1\sim
lN,l^{\prime}N-N+1\sim l^{\prime}N}\right)}}}\\!\right)$, where (a) follows by
applying the trace of product property,
${\rm{tr}}\left({{\bf{XY}}}\right)={\rm{tr}}\left({{\bf{YX}}}\right)$, for
some given size-matched matrices ${\bf{X}}$ and ${\bf{Y}}$. We calculate
${{\bf{Q}}^{2}_{mk}}$ by applying the same method as above, then $Q_{mk}^{2}$
can be obtained as (55) at the top of this page, where the $ll^{\prime}$-th
element is obtained as
${\left[{{\bf{Q}}_{mk}^{2}}\right]_{ll^{\prime}}}={\rm{tr}}\left({{\bf{\Phi}}{{{\bf{\tilde{R}}}}_{k}}{{\bf{\Phi}}^{H}}{{\left[{{{{\bf{\tilde{R}}}}_{m}}}\right]}_{\left({lN-N+1\sim
lN,l^{\prime}N-N+1\sim l^{\prime}N}\right)}}}\right)$.

$\displaystyle
Q_{mk}^{2}\\!=\\!\\!\\!\left[{\begin{array}[]{*{20}{c}}{{\text{tr}}\left({{\mathbf{\Phi}}{{{\mathbf{\tilde{R}}}}_{k}}{{\mathbf{\Phi}}^{H}}{{\left[{{{\mathbf{\tilde{R}}}_{m}}}\right]}_{\left({1\sim
n,1\sim
n}\right)}}}\right)}&\cdots&{{\text{tr}}\left({{\mathbf{\Phi}}{{{\mathbf{\tilde{R}}}}_{k}}{{\mathbf{\Phi}}^{H}}{{\left[{{{\mathbf{\tilde{R}}}_{m}}}\right]}_{\left({1\sim
n,LN-N+1\sim LN}\right)}}}\right)}\\\ \vdots&\ddots&\vdots\\\
\\!\\!\\!{{\text{tr}}\\!\left(\\!\\!{{\mathbf{\Phi}}{{{\mathbf{\tilde{R}}}}_{k}}{{\mathbf{\Phi}}^{H}}\\!{{\left[{{{\mathbf{\tilde{R}}}_{m}}}\right]}_{\left({LN-N+1\sim
LN,1\sim
n}\right)}}}\right)}&\cdots&{{\text{tr}}\\!\left(\\!\\!{{\mathbf{\Phi}}{{{\mathbf{\tilde{R}}}}_{k}}{{\mathbf{\Phi}}^{H}}\\!{{\left[{{{\mathbf{\tilde{R}}}_{m}}}\right]}_{\left({LN\\!-\\!N\\!+\\!1\sim
LN,LN\\!-\\!N\\!+\\!1\sim LN}\right)}}}\right)}\end{array}}\\!\\!\right],$
(55)

##  Appendix B Proof of Theorem 2

The covariance matrix in (14) is calculated in this appendix. First, we
express

$\displaystyle{\mathbf{n}}_{mk}^{p}=\left({{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
N}}+{{\mathbf{N}}_{m}}}\right){\bm{\phi}}_{k}^{*}={\mathbf{H}}_{m}^{H}{\mathbf{\Phi
N}}{\bm{\phi}}_{k}^{*}+{{\mathbf{N}}_{m}}{\bm{\phi}}_{k}^{*}.$ (50)

Since the noise at different pilot symbol instants is independent and
uncorrelated, the covariance matrix of ${\mathbf{n}}_{mk}^{p}$ can be written
as

$\displaystyle\mathbb{E}\\!\left\\{{{\mathbf{n}}_{mk}^{p}{{\left({{\mathbf{n}}_{mk}^{p}}\right)}^{H}}}\\!\right\\}\\!=\\!{\tau_{p}}\mathbb{E}\\!\left\\{{{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
n}}{{\mathbf{n}}^{H}}{{\mathbf{\Phi}}^{H}}{{\mathbf{H}}_{m}}}\\!\right\\}\\!+\\!{\tau_{p}}\mathbb{E}\\!\left\\{{{{\mathbf{n}}_{m}}{\mathbf{n}}_{m}^{H}}\right\\}\\!,$
(51)

where ${{\bf{n}}}\sim{\cal C}{\cal
N}\left({{\bf{0}},{A_{r}}{{\sigma^{2}_{r}}}{{\bf{R}}}}\right)$ denotes the EMI
and
${{\mathbf{n}}_{m}}\sim\mathcal{C}\mathcal{N}\left({0,{\sigma^{2}}{{\mathbf{I}}_{L}}}\right)$
denotes the noise. The first part can be written as

$\displaystyle\mathbb{E}\left\\{{{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
n}}{{\mathbf{n}}^{H}}{{\mathbf{\Phi}}^{H}}{{\mathbf{H}}_{m}}}\\!\right\\}$
$\displaystyle=\mathbb{E}\left\\{{{{\left({{{{\mathbf{\bar{H}}}}_{m}}+{{{\mathbf{\tilde{H}}}}_{m}}}\right)}^{H}}{\mathbf{\Phi
n}}{{\mathbf{n}}^{H}}{{\mathbf{\Phi}}^{H}}\left({{{{\mathbf{\bar{H}}}}_{m}}+{{{\mathbf{\tilde{H}}}}_{m}}}\\!\right)}\right\\}$
$\displaystyle=\mathbb{E}\left\\{{{\mathbf{\bar{H}}}_{m}^{H}{\mathbf{\Phi
n}}{{\mathbf{n}}^{H}}{{\mathbf{\Phi}}^{H}}{{{\mathbf{\bar{H}}}}_{m}}}\right\\}+\mathbb{E}\left\\{{{\mathbf{\tilde{H}}}_{m}^{H}{\mathbf{\Phi
n}}{{\mathbf{n}}^{H}}{{\mathbf{\Phi}}^{H}}{{{\mathbf{\tilde{H}}}}_{m}}}\right\\}$
$\displaystyle=\sigma_{r}^{2}{A_{r}}{\mathbf{\bar{H}}}_{m}^{H}{\mathbf{\Phi
R}}{{\mathbf{\Phi}}^{H}}{{{\mathbf{\bar{H}}}}_{m}}+\underbrace{\mathbb{E}\left\\{{{\mathbf{\tilde{H}}}_{m}^{H}{\mathbf{\Phi
n}}{{\mathbf{n}}^{H}}{{\mathbf{\Phi}}^{H}}{{{\mathbf{\tilde{H}}}}_{m}}}\right\\}}_{{\mathbf{Q}_{m}}}.$
(52)

Using the same method as (55), we can obtain

$\displaystyle{\left[{{\mathbf{Q}_{m}}}\right]_{ll^{\prime}}}={\left[{\mathbb{E}\left\\{{{\mathbf{\tilde{H}}}_{m}^{H}{\mathbf{\Phi
n}}{{\mathbf{n}}^{H}}{{\mathbf{\Phi}}^{H}}{{{\mathbf{\tilde{H}}}}_{m}}}\right\\}}\right]_{ll^{\prime}}}$
$\displaystyle=\sigma_{r}^{2}{A_{r}}{\text{tr}}\left({{\mathbf{\Phi
R}}{{\mathbf{\Phi}}^{H}}{{\left[{{{{\mathbf{\tilde{R}}}}_{m}}}\right]}_{\left({lN-N+1\sim
lN,l^{\prime}N-N+1\sim l^{\prime}N}\right)}}}\right).$ (53)

Then, we can obtain
$\mathbb{E}\left\\{{{\mathbf{n}}_{mk}^{p}{{\left({{\mathbf{n}}_{mk}^{p}}\right)}^{H}}}\right\\}={\tau_{p}}{{\mathbf{R}}_{mm}}+{\tau_{p}}{\sigma^{2}}{{\mathbf{I}}_{L}}$,
where
${{\mathbf{R}}_{mm}}=\sigma_{r}^{2}{A_{r}}{\mathbf{\bar{H}}}_{m}^{H}{\mathbf{\Phi
R}}{{\mathbf{\Phi}}^{H}}{{{\mathbf{\bar{H}}}}_{m}}+{\mathbf{Q}_{m}}$.

##  Appendix C Proof of Theorem 3

The expectations in (23) are calculated here. We begin with the numerator term
as

$\displaystyle\mathbb{E}\left\\{{{{\bf{u}}_{kk}}}\right\\}=\mathbb{E}\left\\{{{{\left[{{\bf{\hat{o}}}_{1k}^{H}{{{\bf{\hat{o}}}}_{1k}},\cdots,{\bf{\hat{o}}}_{Mk}^{H}{{{\bf{\hat{o}}}}_{Mk}}}\right]}^{T}}}\right\\}$
$\displaystyle\\!=\\!{\left[{{\rm{tr}}\left({{\hat{p}_{k}}{\tau_{p}}{{\bf{\Omega}}_{1k}}}\right)\\!+\\!{{\left\|{{{{\bf{\bar{o}}}}_{1k}}}\right\|}^{2}},\cdots\\!,{\rm{tr}}\left({{\hat{p}_{k}}{\tau_{p}}{{\bf{\Omega}}_{Mk}}}\right)\\!+\\!{{\left\|{{{{\bf{\bar{o}}}}_{Mk}}}\right\|}^{2}}}\right]^{T}}\\!\\!.$
(54)

Similarly, we compute the noise term as

$\displaystyle{\bf{a}}_{k}^{H}{{\bf{D}}_{k}}{{\bf{a}}_{k}}$
$\displaystyle={\bf{a}}_{k}^{H}{\rm{diag}}\left({\mathbb{E}\left\\{{{{\left\|{{{{\bf{\hat{o}}}}_{1k}}}\right\|}^{2}}}\right\\},\cdots,\mathbb{E}\left\\{{{{\left\|{{{{\bf{\hat{o}}}}_{Mk}}}\right\|}^{2}}}\right\\}}\right){{\bf{a}}_{k}}$
$\displaystyle={\bf{A}}_{k}^{H}{\rm{diag}}\left({{z_{1k}},\cdots,{z_{Mk}}}\right){{\bf{A}}_{k}},$
(55)

where
${z_{mk}}={\text{tr}}\left({{{\hat{p}}_{k}}{\tau_{p}}{{\mathbf{\Omega}}_{mk}}+{{\left\|{{{{\mathbf{\bar{o}}}}_{mk}}}\right\|}^{2}}}\right)$.

The interference term expectation in the denominator of (23) is

$\displaystyle\mathbb{E}\left\\{{{{\left|{\sum\limits_{m=1}^{M}{a_{mk}{\bf{\hat{o}}}_{mk}^{H}{{\bf{o}}_{mi}}}}\right|}^{2}}}\right\\}=\sum\limits_{m=1}^{M}{\sum\limits_{n=1}^{M}{{a_{mk}}a_{nk}^{*}}}\mathbb{E}\left\\{{{{\left({{\bf{\hat{o}}}_{mk}^{H}{{\bf{o}}_{mi}}}\right)}^{H}}\left({{\bf{\hat{o}}}_{nk}^{H}{{\bf{o}}_{ni}}}\right)}\right\\},$
(56)

where
$\mathbb{E}\\{{{{({{\bf{\hat{o}}}_{mk}^{H}{{\bf{o}}_{mi}}})}^{H}}({{\bf{\hat{o}}}_{nk}^{H}{{\bf{o}}_{ni}}})}\\}$
is computed for all the possible APs and UEs combinations. We utilize the
independence of channel estimation at different APs. When $m\neq
n,i\notin{{\cal P}_{k}}$, we obtain
$\mathbb{E}\left\\{{{{\left({{\bf{\hat{o}}}_{mk}^{H}{{\bf{o}}_{mi}}}\right)}^{H}}\left({{\bf{\hat{o}}}_{nk}^{H}{{\bf{o}}_{ni}}}\right)}\right\\}=0$.
For $m\neq n,i\in{{\cal P}_{k}}\backslash\left\\{k\right\\}$, we derive
$\mathbb{E}\left\\{{{{\left({{\bf{\hat{o}}}_{mk}^{H}{{\bf{o}}_{mi}}}\right)}^{H}}\left({{\bf{\hat{o}}}_{nk}^{H}{{\bf{o}}_{ni}}}\right)}\right\\}=\mathbb{E}\left\\{{{\bf{\hat{o}}}_{mi}^{H}{{{\bf{\hat{o}}}}_{mk}}}\right\\}\left\\{{{\bf{\hat{o}}}_{nk}^{H}{{{\bf{\hat{o}}}}_{ni}}}\right\\}$,
where

$\displaystyle\mathbb{E}\left\\{{{\bf{\hat{o}}}_{mi}^{H}{{{\bf{\hat{o}}}}_{mk}}}\right\\}=\sqrt{{{\hat{p}}_{k}}{{\hat{p}}_{i}}}{\tau_{p}}{\rm{tr}}\left({{\bf{R}}_{mk}^{o}{\bf{\Psi}}_{mk}^{-1}{\bf{R}}_{mi}^{o}}\right),$
(57)

since
$\mathbb{E}\left\\{{{{{\bf{\bar{o}}}}^{H}_{mk}}{{{\bf{\bar{o}}}}_{mi}}{e^{-j{\varphi_{mk}}}}{e^{j{\varphi_{mi}}}}}\right\\}=0$
and
$\mathbb{E}\left\\{{\sqrt{{\hat{p}_{k}}{\tau_{p}}}\left({\bf{R}}_{mk}^{o}{\bf{\Psi}}_{mk}^{-1}\left({{\bf{y}}_{mk}^{p}-{\bf{\bar{y}}}_{mk}^{p}}\right)\right)^{H}{{{\bf{\bar{o}}}}_{mi}}{e^{j{\varphi_{mi}}}}}\right\\}=0$.
We repeat the same calculation for AP $n$ and obtain

$\displaystyle\mathbb{E}\left\\{{{{\left({{\bf{\hat{o}}}_{mk}^{H}{{\bf{o}}_{mi}}}\right)}^{H}}\left({{\bf{\hat{o}}}_{nk}^{H}{{\bf{o}}_{ni}}}\right)}\right\\}={{\hat{p}}_{k}}{{\hat{p}}_{i}}\tau_{p}^{2}{\rm{tr}}\left({{\bf{R}}_{mk}^{o}{\bf{\Psi}}_{mk}^{-1}{\bf{R}}_{mi}^{o}}\right){\rm{tr}}\left({{\bf{R}}_{ni}^{o}{\bf{\Psi}}_{nk}^{-1}{\bf{R}}_{nk}^{o}}\right).$
(58)

For another case $m\neq n,i=k$, we obtain

$\displaystyle\mathbb{E}\left\\{{{{\left({{\bf{\hat{o}}}_{mk}^{H}{{\bf{o}}_{mi}}}\right)}^{H}}\left({{\bf{\hat{o}}}_{nk}^{H}{{\bf{o}}_{ni}}}\right)}\right\\}=\hat{p}_{k}^{2}\tau_{p}^{2}{\rm{tr}}\left({{{\bf{\Omega}}_{mk}}}\right){\rm{tr}}\left({{{\bf{\Omega}}_{nk}}}\right)+{\rm{tr}}\left({{{{\bf{\bar{o}}}}_{mk}}{\bf{\bar{o}}}_{mk}^{H}}\right){\rm{tr}}\left({{{{\bf{\bar{o}}}}_{nk}}{\bf{\bar{o}}}_{nk}^{H}}\right)$
$\displaystyle+{{\hat{p}}_{k}}{\tau_{p}}{\rm{tr}}\left({{{\bf{\Omega}}_{nk}}}\right){\rm{tr}}\left({{{{\bf{\bar{o}}}}_{mk}}{\bf{\bar{o}}}_{mk}^{H}}\right)+{{\hat{p}}_{k}}{\tau_{p}}{\rm{tr}}\left({{{\bf{\Omega}}_{mk}}}\right){\rm{tr}}\left({{{{\bf{\bar{o}}}}_{nk}}{\bf{\bar{o}}}_{nk}^{H}}\right).$
(59)

Similarly for $m=n,i=k$, we adopt the same method as in [15] and calculate the
following equations

$\displaystyle\mathbb{E}\left\\{{{\mathbf{\tilde{o}}}_{mk}^{H}{{{\mathbf{\hat{o}}}}_{mk}}{\mathbf{\hat{o}}}_{mk}^{H}{{{\mathbf{\tilde{o}}}}_{mk}}}\right\\}={\text{tr}}\\!\left({\left({{{\hat{p}}_{k}}{\tau_{p}}{{\mathbf{\Omega}}_{mk}}+{{{\mathbf{\bar{o}}}}_{mk}}{\mathbf{\bar{o}}}_{mk}^{H}}\right){{\mathbf{C}}_{mk}}}\right),$
(60)
$\displaystyle\mathbb{E}\left\\{{{\mathbf{\hat{o}}}_{mk}^{H}{{{\mathbf{\hat{o}}}}_{mk}}{\mathbf{\hat{o}}}_{mk}^{H}{{{\mathbf{\hat{o}}}}_{mk}}}\right\\}={\left({{{\hat{p}}_{k}}{\tau_{p}}}\right)^{2}}{\text{tr}}\left({{{\mathbf{\Omega}}_{mk}}}\right){\text{tr}}\left({{{\mathbf{\Omega}}_{mk}}}\right)$
$\displaystyle+{{\hat{p}}_{k}}{\tau_{p}}{\text{tr}}\left({\left({{\mathbf{R}}_{mk}^{o}-{{\mathbf{C}}_{mk}}}\right){{\mathbf{\Omega}}_{mk}}}\right)+{{\hat{p}}_{k}}{\tau_{p}}{\mathbf{\bar{o}}}_{mk}^{H}{{\mathbf{\Omega}}_{mk}}{{{\mathbf{\bar{o}}}}_{mk}}$
$\displaystyle+{\mathbf{\bar{o}}}_{mk}^{H}\left({{\mathbf{R}}_{mk}^{o}-{{\mathbf{C}}_{mk}}}\right){{{\mathbf{\bar{o}}}}_{mk}}+{\mathbf{\bar{o}}}_{mk}^{H}{{{\mathbf{\bar{o}}}}_{mk}}{\mathbf{\bar{o}}}_{mk}^{H}{{{\mathbf{\bar{o}}}}_{mk}}+2{{\hat{p}}_{k}}{\tau_{p}}{\mathbf{\bar{o}}}_{mk}^{H}{{{\mathbf{\bar{o}}}}_{mk}}{\text{tr}}\left({{{\mathbf{\Omega}}_{mk}}}\right).$
(61)

Combining the above two formulas yields the result for $m=n,i=k$ as

$\displaystyle\mathbb{E}\left\\{{{{\left|{{\mathbf{\hat{o}}}_{mk}^{H}{{\mathbf{o}}_{mk}}}\right|}^{2}}}\right\\}$
$\displaystyle=\hat{p}_{k}^{2}\tau_{p}^{2}{\left|{{\text{tr}}\left({{{\mathbf{\Omega}}_{mk}}}\right)}\right|^{2}}+{{\hat{p}}_{k}}{\tau_{p}}{\text{tr}}\left({{{\mathbf{\Omega}}_{mk}}{\mathbf{R}}_{mk}^{o}}\right)$
$\displaystyle+{\mathbf{\bar{o}}}_{mk}^{H}{\mathbf{R}}_{mk}^{o}{{{\mathbf{\bar{o}}}}_{mk}}+{{\hat{p}}_{k}}{\tau_{p}}{\mathbf{\bar{o}}}_{mk}^{H}{{\mathbf{\Omega}}_{mk}}{{{\mathbf{\bar{o}}}}_{mk}}$
$\displaystyle+2{{\hat{p}}_{k}}{\tau_{p}}{\text{tr}}\left({{{\mathbf{\Omega}}_{mk}}}\right){\mathbf{\bar{o}}}_{mk}^{H}{{{\mathbf{\bar{o}}}}_{mk}}+{\text{tr}}{\left({{{{\mathbf{\bar{o}}}}_{mk}}{\mathbf{\bar{o}}}_{mk}^{H}}\right)^{2}}.$
(62)

Then, for the case $m=n,i\notin{{\cal P}_{k}}$, we obtain

$\displaystyle\mathbb{E}\left\\{{{{\left|{{\bf{\hat{o}}}_{mk}^{H}{{\bf{o}}_{mi}}}\right|}^{2}}}\right\\}$
$\displaystyle={\hat{p}_{k}}{\tau_{p}}{\rm{tr}}\left({{\bf{R}}_{mi}^{o}{{\bf{\Omega}}_{mk}}}\right)+{\bf{\bar{o}}}_{mk}^{H}{\bf{R}}_{mi}^{o}{{{\bf{\bar{o}}}}_{mk}}+{\hat{p}_{k}}{\tau_{p}}{\bf{\bar{o}}}_{mi}^{H}{{\bf{\Omega}}_{mk}}{{{\bf{\bar{o}}}}_{mi}}+{\left|{{\bf{\bar{o}}}_{mk}^{H}{{{\bf{\bar{o}}}}_{mi}}}\right|^{2}}.$
(63)

For $m=n,i\in{{\cal P}_{k}}\backslash\left\\{k\right\\}$, we obtain

$\displaystyle\mathbb{E}\left\\{{{{\left|{{\bf{\hat{o}}}_{mk}^{H}{{\bf{o}}_{mi}}}\right|}^{2}}}\right\\}$
$\displaystyle={{\hat{p}}_{k}}{\tau_{p}}{\rm{tr}}\left({{{\bf{\Omega}}_{mk}}{\bf{R}}_{mi}^{o}}\right){\rm{+tr}}\left({{{{\bf{\bar{o}}}}_{mk}}{\bf{\bar{o}}}_{mk}^{H}{\bf{R}}_{mi}^{o}}\right)$
$\displaystyle+\\!{\left|{{\bf{\bar{o}}}_{mk}^{H}{{{\bf{\bar{o}}}}_{mi}}}\right|^{2}}\\!+\\!{{\hat{p}}_{k}}{{\hat{p}}_{i}}\tau_{p}^{2}{\left|{{\rm{tr}}\left({{\bf{R}}_{mk}^{o}{\bf{\Psi}}_{mk}^{-1}{\bf{R}}_{mk}^{o}}\right)}\right|^{2}}\\!.$
(64)

Finally, arranging all these equations yields (C) at the top of this page.

$\displaystyle\mathbb{E}\left\\{{{{\left|{\sum\limits_{m=1}^{M}{a_{mk}^{*}{\mathbf{v}}_{mk}^{H}{{\mathbf{o}}_{mi}}}}\right|}^{2}}}\right\\}$
$\displaystyle=\sum\limits_{m=1}^{M}{{{\left|{{a_{mk}}}\right|}^{2}}\left[{{p_{k}}{\tau_{p}}{\text{tr}}\left({{\mathbf{R}}_{mi}^{o}{{\mathbf{\Omega}}_{mk}}}\right)+{\mathbf{\bar{o}}}_{mk}^{H}{\mathbf{R}}_{mi}^{o}{{{\mathbf{\bar{o}}}}_{mk}}+{p_{k}}{\tau_{p}}{\mathbf{\bar{o}}}_{mi}^{H}{{\mathbf{\Omega}}_{mk}}{{{\mathbf{\bar{o}}}}_{mi}}+{{\left|{{\mathbf{\bar{o}}}_{mk}^{H}{{{\mathbf{\bar{o}}}}_{mi}}}\right|}^{2}}}\right]}$
$\displaystyle+\left\\{{\begin{array}[]{*{20}{c}}{\sum\limits_{m=1}^{M}{{{\left|{{a_{mk}}}\right|}^{2}}{{\hat{p}}_{k}}{{\hat{p}}_{i}}\tau_{p}^{2}{{\left|{{\text{tr}}\left({{\mathbf{R}}_{mi}^{o}{\mathbf{\Psi}}_{mk}^{-1}{\mathbf{R}}_{mk}^{o}}\right)}\right|}^{2}},}}&{i\in{\mathcal{P}_{k}}\backslash\left\\{k\right\\}},\\\
{\sum\limits_{m=1}^{M}{{{\left|{{a_{mk}}}\right|}^{2}}\left[{\hat{p}_{k}^{2}\tau_{p}^{2}{{\left|{{\text{tr}}\left({{{\mathbf{\Omega}}_{mk}}}\right)}\right|}^{2}}+2\sqrt{{{\hat{p}}_{k}}{{\hat{p}}_{i}}}{\tau_{p}}{\text{tr}}\left({{{\mathbf{\Omega}}_{mk}}}\right){\mathbf{\bar{o}}}_{mk}^{H}{{{\mathbf{\bar{o}}}}_{mk}}}\right],}}&{i=k},\\\
{0,}&{i\notin{\mathcal{P}_{k}}},\end{array}}\right.$ (66)
$\displaystyle+\left\\{{\begin{array}[]{*{20}{c}}{\sum\limits_{m=1}^{M}{\sum\limits_{n=1}^{M}{{a_{mk}}a_{nk}^{*}\left[{{{\hat{p}}_{k}}{{\hat{p}}_{i}}\tau_{p}^{2}{\text{tr}}\left({{\mathbf{R}}_{mk}^{o}{\mathbf{\Psi}}_{mk}^{-1}{\mathbf{R}}_{mi}^{o}}\right){\text{tr}}\left({{\mathbf{R}}_{nk}^{o}{\mathbf{\Psi}}_{nk}^{-1}{\mathbf{R}}_{ni}^{o}}\right)+{{\hat{p}}_{k}}{\tau_{p}}{\text{tr}}\left({{{\mathbf{\Omega}}_{nk}}}\right){\text{tr}}\left({{{{\mathbf{\bar{o}}}}_{mk}}{\mathbf{\bar{o}}}_{mk}^{H}}\right)}\right.}}}&{}\hfil\\\
{\left.{+{{\hat{p}}_{k}}{\tau_{p}}{\text{tr}}\left({{{\mathbf{\Omega}}_{mk}}}\right){\text{tr}}\left({{{{\mathbf{\bar{o}}}}_{nk}}{\mathbf{\bar{o}}}_{nk}^{H}}\right)+{\text{tr}}\left({{{{\mathbf{\bar{o}}}}_{mk}}{\mathbf{\bar{o}}}_{mk}^{H}}\right){\text{tr}}\left({{{{\mathbf{\bar{o}}}}_{nk}}{\mathbf{\bar{o}}}_{nk}^{H}}\right)}\right],}&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{i\in{\mathcal{P}_{k}}},\\\
{0,}&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{i\notin{\mathcal{P}_{k}}}.\end{array}}\right.$
(70)

Then, for the EMI term
${{\mathbf{U}}_{k}}={\text{diag}}({\mathbb{E}\\{{{{\|{{\mathbf{\hat{o}}}_{1k}^{H}{\mathbf{H}}_{1}^{H}{\mathbf{\Phi
n}}}\|}^{2}}}\\},\cdots,\mathbb{E}\\{{{{\|{{\mathbf{\hat{o}}}_{Mk}^{H}{\mathbf{H}}_{M}^{H}{\mathbf{\Phi
n}}}\|}^{2}}}\\}})$, we calculate as follow

$\displaystyle\mathbb{E}\left\\{{{{\left\|{{\mathbf{\hat{o}}}_{mk}^{H}{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
n}}}\right\|}^{2}}}\right\\}=\mathbb{E}\left\\{{{\mathbf{\hat{o}}}_{mk}^{H}{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
n}}{{\mathbf{n}}^{H}}{{\mathbf{\Phi}}^{H}}{{\mathbf{H}}_{m}}{{{\mathbf{\hat{o}}}}_{mk}}}\right\\}$
$\displaystyle=\sigma_{r}^{2}{A_{r}}\mathbb{E}\left\\{{{{\left({{{{\mathbf{\bar{o}}}}_{mk}}{e^{j{\varphi_{k}}}}+\sqrt{{{\hat{p}}_{k}}}{\mathbf{R}}_{mk}^{o}{\mathbf{\Psi}}_{mk}^{-1}\left({{\mathbf{y}}_{mk}^{p}-{\mathbf{\bar{y}}}_{mk}^{p}}\right)}\right)}^{H}}{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
R}}{{\mathbf{\Phi}}^{H}}{{\mathbf{H}}_{m}}}\right.$
$\displaystyle\times\left.{\left({{{{\mathbf{\bar{o}}}}_{mk}}{e^{j{\varphi_{k}}}}+\sqrt{{{\hat{p}}_{k}}}{\mathbf{R}}_{mk}^{o}{\mathbf{\Psi}}_{mk}^{-1}\left({{\mathbf{y}}_{mk}^{p}-{\mathbf{\bar{y}}}_{mk}^{p}}\right)}\right)}\right\\}=\sigma_{r}^{2}{A_{r}}\mathbb{E}\left\\{{{{{\mathbf{\bar{o}}}}^{H}}_{mk}{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
R}}{{\mathbf{\Phi}}^{H}}{{\mathbf{H}}_{m}}{{{\mathbf{\bar{o}}}}_{mk}}}\right\\}$
$\displaystyle+\sigma_{r}^{2}{A_{r}}\mathbb{E}\left\\{{\sqrt{{{\hat{p}}_{k}}}{{\left({{\mathbf{y}}_{mk}^{p}-{\mathbf{\bar{y}}}_{mk}^{p}}\right)}^{H}}{{\left({{\mathbf{\Psi}}_{mk}^{-1}}\right)}^{H}}{{\left({{\mathbf{R}}_{mk}^{o}}\right)}^{H}}{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
R}}{{\mathbf{\Phi}}^{H}}{{\mathbf{H}}_{m}}\sqrt{{{\hat{p}}_{k}}}{\mathbf{R}}_{mk}^{o}{\mathbf{\Psi}}_{mk}^{-1}\left({{\mathbf{y}}_{mk}^{p}-{\mathbf{\bar{y}}}_{mk}^{p}}\right)}\right\\}.$
(66)

Then first term can be obtain by Appendix B as

$\displaystyle\mathbb{E}\\!\left\\{{{{{\mathbf{\bar{o}}}}^{H}}_{mk}{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
R}}{{\mathbf{\Phi}}^{H}}{{\mathbf{H}}_{m}}{{{\mathbf{\bar{o}}}}_{mk}}}\right\\}\\!\\!$
$\displaystyle=\\!\sigma_{r}^{2}{A_{r}}{{{\mathbf{\bar{o}}}}^{H}}_{mk}{\mathbf{\bar{H}}}_{m}^{H}{\mathbf{\Phi
R}}{{\mathbf{\Phi}}^{H}}{{{\mathbf{\bar{H}}}}_{m}}{{{\mathbf{\bar{o}}}}_{mk}}$
$\displaystyle\\!+\\!\sigma_{r}^{2}{A_{r}}{{{\mathbf{\bar{o}}}}^{H}}_{mk}\\!{\left[\\!{{\text{tr}}\left(\\!\\!{{\mathbf{\Phi
R}}{{\mathbf{\Phi}}^{H}}\\!{{\left[{{{{\mathbf{\tilde{R}}}}_{m}}}\\!\right]}_{\left(\\!{lN\\!-\\!N\\!+\\!1\sim
lN,l^{\prime}N\\!-\\!N\\!+\\!1\sim
l^{\prime}N}\\!\right)}}}\\!\right)}\\!\right]_{\\!ll^{\prime}}}\\!\\!{{{\mathbf{\bar{o}}}}_{mk}}.$
(67)

The second term is derived as

$\displaystyle\mathbb{E}\left\\{{\sqrt{{{\hat{p}}_{k}}}{{\left({{\mathbf{y}}_{mk}^{p}-{\mathbf{\bar{y}}}_{mk}^{p}}\right)}^{H}}{{\left({{\mathbf{\Psi}}_{mk}^{-1}}\right)}^{H}}{{\left({{\mathbf{R}}_{mk}^{o}}\right)}^{H}}{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
R}}{{\mathbf{\Phi}}^{H}}{{\mathbf{H}}_{m}}\sqrt{{{\hat{p}}_{k}}}{\mathbf{R}}_{mk}^{o}{\mathbf{\Psi}}_{mk}^{-1}\left({{\mathbf{y}}_{mk}^{p}-{\mathbf{\bar{y}}}_{mk}^{p}}\right)}\right\\}$
$\displaystyle={{\hat{p}}_{k}}\tau_{p}^{2}{{\mathbf{\Psi}}_{mk}}{\left({{\mathbf{\Psi}}_{mk}^{-1}}\right)^{H}}{\left({{\mathbf{R}}_{mk}^{o}}\right)^{H}}{\mathbf{\bar{H}}}_{m}^{H}{\mathbf{\Phi
R}}{{\mathbf{\Phi}}^{H}}{{{\mathbf{\bar{H}}}}_{m}}{\mathbf{R}}_{mk}^{o}{\mathbf{\Psi}}_{mk}^{-1}$
$\displaystyle+{{\hat{p}}_{k}}\mathbb{E}\left\\{{{{\left({{\mathbf{y}}_{mk}^{p}-{\mathbf{\bar{y}}}_{mk}^{p}}\right)}^{H}}{{\left({{\mathbf{\Psi}}_{mk}^{-1}}\right)}^{H}}{{\left({{\mathbf{R}}_{mk}^{o}}\right)}^{H}}{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
R}}{{\mathbf{\Phi}}^{H}}{{\mathbf{H}}_{m}}{\mathbf{R}}_{mk}^{o}{\mathbf{\Psi}}_{mk}^{-1}\left({{\mathbf{y}}_{mk}^{p}-{\mathbf{\bar{y}}}_{mk}^{p}}\right)}\right\\}$
$\displaystyle={{\hat{p}}_{k}}\tau_{p}^{2}{\text{tr}}\left({{{\left({{\mathbf{\Psi}}_{mk}^{-1}}\right)}^{H}}{{\left({{\mathbf{R}}_{mk}^{o}}\right)}^{H}}{\mathbf{\bar{H}}}_{m}^{H}{\mathbf{\Phi
R}}{{\mathbf{\Phi}}^{H}}{{{\mathbf{\bar{H}}}}_{m}}{\mathbf{R}}_{mk}^{o}}\right)$
$\displaystyle+{{\hat{p}}_{k}}\tau_{p}^{2}{\text{tr}}\left({{{\left({{\mathbf{\Psi}}_{mk}^{-1}}\right)}^{H}}{{\left({{\mathbf{R}}_{mk}^{o}}\right)}^{H}}{{\left[{{\text{tr}}\left({{\mathbf{\Phi
R}}{{\mathbf{\Phi}}^{H}}{{\left[{{{{\mathbf{\tilde{R}}}}_{m}}}\right]}_{\left({lN-N+1\sim
lN,l^{\prime}N-N+1\sim
l^{\prime}N}\right)}}}\right)}\right]}_{ll^{\prime}}}{\mathbf{R}}_{mk}^{o}}\right).$
(68)

By combining the first and second terms, we can derive the expectation of EMI
as

$\displaystyle\mathbb{E}\left\\{{{{\left\|{{\mathbf{\hat{o}}}_{mk}^{H}{\mathbf{H}}_{m}^{H}{\mathbf{\Phi
n}}}\right\|}^{2}}}\right\\}={{{\mathbf{\bar{o}}}}^{H}}_{mk}{{\mathbf{R}}_{mn}}{{{\mathbf{\bar{o}}}}_{mk}}+{{\hat{p}}_{k}}\tau_{p}^{2}{\text{tr}}\left({{{\left({{\mathbf{\Psi}}_{mk}^{-1}}\right)}^{H}}{{\left({{\mathbf{R}}_{mk}^{o}}\right)}^{H}}{{\mathbf{R}}_{mn}}{\mathbf{R}}_{mk}^{o}}\right).$
(69)

Finally, combining the above cases, we can derive the expectation of (23).

## References

  * [1] W. Saad, M. Bennis, and M. Chen, “A vision of 6G wireless systems: Applications, trends, technologies, and open research problems,” _IEEE network_ , vol. 34, no. 3, pp. 134–142, Mar. 2019.
  * [2] M. Giordani, M. Polese, M. Mezzavilla, S. Rangan, and M. Zorzi, “Toward 6G networks: Use cases and technologies,” _IEEE Commun. Mag._ , vol. 58, no. 3, pp. 55–61, Mar. 2020.
  * [3] T. S. Rappaport, Y. Xing, G. R. MacCartney, A. F. Molisch, E. Mellios, and J. Zhang, “Overview of millimeter wave communications for fifth-generation (5G) wireless networks¡ªwith a focus on propagation models,” _IEEE Trans. Antennas and Propag._ , vol. 65, no. 12, pp. 6213–6230, Dec. 2017.
  * [4] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” _IEEE Commun. Mag._ , vol. 52, no. 2, pp. 186–195, Feb. 2014.
  * [5] E. Björnson, L. Sanguinetti, and M. Kountouris, “Deploying dense networks for maximal energy efficiency: Small cells meet massive MIMO,” _IEEE J. Sel. Areas Commun._ , vol. 34, no. 4, pp. 832–847, Apr. 2016.
  * [6] T. Van Chien, T. N. Canh, E. Björnson, and E. G. Larsson, “Power control in cellular massive MIMO with varying user activity: A deep learning solution,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 9, pp. 5732–5748, Sep. 2020.
  * [7] E. Shi, J. Zhang, R. He, H. Jiao, Z. Wang, B. Ai, and D. W. K. Ng, “Spatially correlated reconfigurable intelligent surfaces-aided cell-free massive MIMO systems,” _IEEE Trans. Veh. Technol._ , vol. 71, no. 8, pp. 9073–9077, Aug. 2022.
  * [8] J. Zhang, E. Björnson, M. Matthaiou, D. W. K. Ng, H. Yang, and D. J. Love, “Prospective multiple antenna technologies for beyond 5G,” _IEEE J. Sel. Areas Commun._ , vol. 38, no. 8, pp. 1637–1660, Aug. 2020.
  * [9] H. Q. Ngo, A. Ashikhmin, H. Yang, E. G. Larsson, and T. L. Marzetta, “Cell-free massive MIMO versus small cells,” _IEEE Trans. Wireless Commun._ , vol. 16, no. 3, pp. 1834–1850, Mar. 2017.
  * [10] J. Zhang, J. Zhang, D. W. K. Ng, S. Jin, and B. Ai, “Improving sum-rate of cell-free massive MIMO with expanded compute-and-forward,” _IEEE Trans. Signal Process._ , vol. 70, pp. 202–215, Nov. 2021.
  * [11] S. Buzzi, C. D¡¯Andrea, A. Zappone, and C. D¡¯Elia, “User-centric 5G cellular networks: Resource allocation and comparison with the cell-free massive MIMO approach,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 2, pp. 1250–1264, Feb. 2019.
  * [12] H. Q. Ngo, L.-N. Tran, T. Q. Duong, M. Matthaiou, and E. G. Larsson, “On the total energy efficiency of cell-free massive MIMO,” _IEEE Trans. Green Commun. Netw._ , vol. 2, no. 1, pp. 25–39, Jan. 2017.
  * [13] L. Du, L. Li, H. Q. Ngo, T. C. Mai, and M. Matthaiou, “Cell-free massive MIMO: Joint maximum-ratio and zero-forcing precoder with power control,” _IEEE Trans. Commun._ , vol. 69, no. 6, pp. 3741–3756, Jun. 2021.
  * [14] E. Björnson and L. Sanguinetti, “Making cell-free massive MIMO competitive with MMSE processing and centralized implementation,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 1, pp. 77–90, Jan. 2019.
  * [15] Ö. Özdogan, E. Björnson, and J. Zhang, “Performance of cell-free massive MIMO with Rician fading and phase shifts,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 11, pp. 5299–5315, Nov. 2019.
  * [16] Z. Wang, J. Zhang, B. Ai, C. Yuen, and M. Debbah, “Uplink performance of cell-free massive MIMO with multi-antenna users over jointly-correlated Rayleigh fading channels,” _IEEE Trans. Wireless Commun._ , vol. 21, no. 9, pp. 7391–7406, Sep. 2022.
  * [17] T. Van Chien, H. Q. Ngo, S. Chatzinotas, M. Di Renzo, and B. Ottersten, “Reconfigurable intelligent surface-assisted cell-free massive MIMO systems over spatially-correlated channels,” _IEEE Trans. Wireless Commun._ , vol. 21, no. 7, pp. 5106–5128, Jul. 2021.
  * [18] Y. Zhang, J. Zhang, M. Di Renzo, H. Xiao, and B. Ai, “Reconfigurable intelligent surfaces with outdated channel state information: Centralized vs. distributed deployments,” _IEEE Trans. Commun._ , vol. 70, no. 4, pp. 2742–2756, Apr. 2022.
  * [19] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network,” _IEEE Commun. Mag._ , vol. 58, no. 1, pp. 106–112, Jan. 2019.
  * [20] M. Di Renzo, A. Zappone, M. Debbah, M.-S. Alouini, C. Yuen, J. de Rosny, and S. Tretyakov, “Smart radio environments empowered by reconfigurable intelligent surfaces: How it works, state of research, and the road ahead,” _IEEE J. Sel. Areas Commun._ , vol. 38, no. 11, pp. 2450–2525, Nov. 2020\.
  * [21] Q. Wu, S. Zhang, B. Zheng, C. You, and R. Zhang, “Intelligent reflecting surface aided wireless communications: A tutorial,” _IEEE Trans. Commun._ , 2021.
  * [22] B. Di, H. Zhang, L. Song, Y. Li, Z. Han, and H. V. Poor, “Hybrid beamforming for reconfigurable intelligent surface based multi-user communications: Achievable rates with limited discrete phase shifts,” _IEEE J. Sel. Areas Commun._ , vol. 38, no. 8, pp. 1809–1822, Aug. 2020.
  * [23] L. Yang, Y. Yang, M. O. Hasna, and M.-S. Alouini, “Coverage, probability of SNR gain, and DOR analysis of RIS-aided communication systems,” _IEEE Wireless Commun. Lett._ , vol. 9, no. 8, pp. 1268–1272, Aug. 2020.
  * [24] Y. Chen, Y. Wang, J. Zhang, and Z. Li, “Resource allocation for intelligent reflecting surface aided vehicular communications,” _IEEE Trans Veh. Technol._ , vol. 69, no. 10, pp. 12 321–12 326, Oct. 2020.
  * [25] E. Björnson and L. Sanguinetti, “Rayleigh fading modeling and channel hardening for reconfigurable intelligent surfaces,” _IEEE Wireless Commun. Lett._ , vol. 10, no. 4, pp. 830–834, Apr. 2020.
  * [26] A. de Jesus Torres, L. Sanguinetti, and E. Björnson, “Electromagnetic interference in RIS-aided communications,” _IEEE Wireless Commun. Lett._ , vol. 11, no. 4, pp. 668–672, 2021.
  * [27] E. Shi, J. Zhang, S. Chen, J. Zheng, Y. Zhang, D. W. Kwan Ng, and B. Ai, “Wireless energy transfer in RIS-aided cell-free massive MIMO systems: Opportunities and challenges,” _IEEE Commun. Mag._ , vol. 60, no. 3, pp. 26–32, Mar. 2022.
  * [28] Z. Zhang and L. Dai, “A joint precoding framework for wideband reconfigurable intelligent surface-aided cell-free network,” _IEEE Trans. Signal Process._ , vol. 69, pp. 4085–4101, Jun. 2021.
  * [29] X. Ma, D. Zhang, M. Xiao, C. Huang, and Z. Chen, “Cooperative beamforming for RIS-aided cell-free massive MIMO networks,” _IEEE Trans. Wireless Commun., early access_ , 2023.
  * [30] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 11, pp. 5394–5409, Nov. 2019.
  * [31] E. Shi, J. Zhang, Y. Du, Z. Wang, B. Ai, and D. W. K. Ng, “Spatially correlated RIS-aided CF massive MIMO systems with generalized MR combining,” _IEEE Trans. Veh. Technol._ , vol. 71, no. 10, pp. 11 245–11 250, Oct. 2022.
  * [32] D. Middleton, “Statistical-physical models of electromagnetic interference,” _IEEE Trans. Electromagn. Compat._ , no. 3, pp. 106–127, Mar. 1977.
  * [33] G. S. Chandra, R. K. Singh, S. Dhok, P. K. Sharma, and P. Kumar, “Downlink URLLC system over spatially correlated RIS channels and electromagnetic interference,” _IEEE Wireless Commun. Lett._ , vol. 11, no. 9, pp. 1950–1954, Sep. 2022.
  * [34] J. D. Vega-Sánchez, G. Kaddoum, and F. J. López-Martínez, “Physical layer security of RIS-assisted communications under electromagnetic interference,” _IEEE Commun. Lett._ , vol. 26, no. 12, pp. 2870–2874, Dec. 2022.
  * [35] X. Yu, V. Jamali, D. Xu, D. W. K. Ng, and R. Schober, “Smart and reconfigurable wireless communications: From IRS modeling to algorithm design,” _IEEE Wireless Commun._ , vol. 28, no. 6, pp. 118–125, Jun. 2021\.
  * [36] M. Bashar, K. Cumanan, A. G. Burr, P. Xiao, and M. Di Renzo, “On the performance of reconfigurable intelligent surface-aided cell-free massive MIMO uplink,” in _Proc. IEEE GLOBECOM_ , 2020, pp. 1–6.
  * [37] Z. Wang, J. Zhang, E. Björnson, and B. Ai, “Uplink performance of cell-free massive MIMO over spatially correlated Rician fading channels,” _IEEE Commun. Lett._ , vol. 25, no. 4, pp. 1348–1352, Apr. 2020\.
  * [38] J. Zheng, J. Zhang, E. Björnson, and B. Ai, “Impact of channel aging on cell-free massive MIMO over spatially correlated channels,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 10, pp. 6451–6466, Oct. 2021.
  * [39] R. Nikbakht and A. Lozano, “Uplink fractional power control for cell-free wireless networks,” in _Proc. IEEE ICC_ , 2019, pp. 1–5.
  * [40] X. Li, J. Zhang, Z. Wang, B. Ai, and D. W. K. Ng, “Cell-free massive MIMO with multi-antenna users over Weichselberger Rician channels,” _IEEE Trans. Veh. Technol._ , vol. 71, no. 11, pp. 12 368–12 373, Nov. 2022\.