2G

    Second Generation
3G

    3${}^{\text{rd}}$ Generation
3GPP

    3${}^{\text{rd}}$ Generation Partnership Project
4G

    4${}^{\text{th}}$ Generation
5G

    5${}^{\text{th}}$ Generation
AA

    Antenna Array
AC

    Admission Control
AD

    Attack-Decay
ADSL

    Asymmetric Digital Subscriber Line
AHW

    Alternate Hop-and-Wait
AMC

    Adaptive Modulation and Coding
AP

    access point
AoA

    angle of arrival
AoI

    age of information
APA

    Adaptive Power Allocation
AR

    autoregressive
ARMA

    Autoregressive Moving Average
ATES

    Adaptive Throughput-based Efficiency-Satisfaction Trade-Off
AWGN

    additive white Gaussian noise
BB

    Branch and Bound
BD

    Block Diagonalization
BER

    bit error rate
BF

    Best Fit
BLER

    BLock Error Rate
BPC

    Binary power control
BPSK

    binary phase-shift keying
BPA

    Best pilot-to-data power ratio (PDPR) Algorithm
BRA

    Balanced Random Allocation
BS

    base station
CAP

    Combinatorial Allocation Problem
CAPEX

    Capital Expenditure
CAZAC

    Constant Amplitude Zero Autocorrelation
CB

    codebook
CBF

    Coordinated Beamforming
CBR

    Constant Bit Rate
CBS

    Class Based Scheduling
CC

    Congestion Control
CDF

    Cumulative Distribution Function
CDMA

    Code-Division Multiple Access
CL

    Closed Loop
CLPC

    Closed Loop Power Control
CNR

    Channel-to-Noise Ratio
CPA

    Cellular Protection Algorithm
CPICH

    Common Pilot Channel
CoMP

    Coordinated Multi-Point
CQI

    Channel Quality Indicator
CRLB

    Cramér-Rao Lower Bound
CRM

    Constrained Rate Maximization
CRN

    Cognitive Radio Network
CS

    Coordinated Scheduling
CSI

    channel state information
CSIR

    channel state information at the receiver
CSIT

    channel state information at the transmitter
CUE

    cellular user equipment
D2D

    device-to-device
DCA

    Dynamic Channel Allocation
DE

    Differential Evolution
DFT

    Discrete Fourier Transform
DISCOVER

    Deep Intrinsically Motivated Exploration
DIST

    Distance
DL

    downlink
DMA

    Double Moving Average
DMRS

    Demodulation Reference Signal
D2DM

    D2D Mode
DMS

    D2D Mode Selection
DPC

    Dirty Paper Coding
DRA

    Dynamic Resource Assignment
DRL

    deep reinforcement learning
DDPG

    Deep Deterministic Policy Gradient
RL

    reinforcement learning
SAC

    Soft Actor-Critic
DSA

    Dynamic Spectrum Access
DSM

    Delay-based Satisfaction Maximization
ECC

    Electronic Communications Committee
ECRB

    expectation of the conditional Cramér-Rao bound
EFLC

    Error Feedback Based Load Control
EI

    Efficiency Indicator
eNB

    Evolved Node B
EPA

    Equal Power Allocation
EPC

    Evolved Packet Core
EPS

    Evolved Packet System
E-UTRAN

    Evolved Universal Terrestrial Radio Access Network
ES

    Exhaustive Search
FDD

    frequency division duplexing
FDM

    Frequency Division Multiplexing
FER

    Frame Erasure Rate
FF

    Fast Fading
FIM

    Fisher Information Matrix
FSB

    Fixed Switched Beamforming
FST

    Fixed SNR Target
FTP

    File Transfer Protocol
GA

    Genetic Algorithm
GBR

    Guaranteed Bit Rate
GLR

    Gain to Leakage Ratio
GOS

    Generated Orthogonal Sequence
GPL

    GNU General Public License
GRP

    Grouping
HARQ

    Hybrid Automatic Repeat Request
HMS

    Harmonic Mode Selection
HOL

    Head Of Line
HSDPA

    High-Speed Downlink Packet Access
HSPA

    High Speed Packet Access
HTTP

    HyperText Transfer Protocol
ICMP

    Internet Control Message Protocol
ICI

    Intercell Interference
ID

    Identification
IETF

    Internet Engineering Task Force
ILP

    Integer Linear Program
JRAPAP

    Joint RB Assignment and Power Allocation Problem
UID

    Unique Identification
i.i.d.

    independent and identically distributed
IIR

    Infinite Impulse Response
ILP

    Integer Linear Problem
IMT

    International Mobile Telecommunications
INV

    Inverted Norm-based Grouping
IoT

    Internet of Things
IP

    Internet Protocol
IPv6

    Internet Protocol Version 6
IRS

    intelligent reflecting surface
ISD

    Inter-Site Distance
ISI

    Inter Symbol Interference
ITU

    International Telecommunication Union
JOAS

    Joint Opportunistic Assignment and Scheduling
JOS

    Joint Opportunistic Scheduling
JP

    Joint Processing
JS

    Jump-Stay
KF

    Kalman filter
KKT

    Karush-Kuhn-Tucker
L3

    Layer-3
LAC

    Link Admission Control
LA

    Link Adaptation
LC

    Load Control
LMMSE

    linear minimum mean squared error
LOS

    line of sight
LP

    Linear Programming
LS

    least squares
LTE

    Long Term Evolution
LTE-A

    LTE-Advanced
LTE-Advanced

    Long Term Evolution Advanced
M2M

    Machine-to-Machine
MAC

    Medium Access Control
MANET

    Mobile Ad hoc Network
MC

    Modular Clock
MCS

    Modulation and Coding Scheme
MDB

    Measured Delay Based
MDI

    Minimum D2D Interference
MF

    Matched Filter
MG

    Maximum Gain
MH

    Multi-Hop
MIMO

    multiple-input multiple-output
MINLP

    Mixed Integer Nonlinear Programming
MIP

    Mixed Integer Programming
MISO

    Multiple-input single-output
ML

    maximum likelihood
MLWDF

    Modified Largest Weighted Delay First
MME

    Mobility Management Entity
MML

    misspecified maximum likelihood
MMSE

    minimum mean squared error
MOS

    Mean Opinion Score
MPF

    Multicarrier Proportional Fair
MRA

    Maximum Rate Allocation
MR

    Maximum Rate
MRC

    maximum ratio combining
MRT

    Maximum Ratio Transmission
MRUS

    Maximum Rate with User Satisfaction
MS

    mobile station
MSE

    mean squared error
MSI

    Multi-Stream Interference
MTC

    Machine-Type Communication
MTSI

    Multimedia Telephony Services over IMS
MTSM

    Modified Throughput-based Satisfaction Maximization
MU-MIMO

    multiuser multiple-input multiple-output
MU-MISO

    multiuser multiple-input single-output
MU

    multi-user
NAS

    Non-Access Stratum
NB

    Node B
NE

    Nash equilibrium
NCL

    Neighbor Cell List
NLP

    Nonlinear Programming
NLOS

    Non-Line of Sight
NMSE

    normalized mean squared error
NOMA

    non-orthogonal multiple access
NORM

    Normalized Projection-based Grouping
NP

    Non-Polynomial Time
NR

    New Radio
NRT

    Non-Real Time
NSPS

    National Security and Public Safety Services
O2I

    Outdoor to Indoor
OFDMA

    orthogonal frequency division multiple access
OFDM

    orthogonal frequency division multiplexing
OFPC

    Open Loop with Fractional Path Loss Compensation
O2I

    Outdoor-to-Indoor
OL

    Open Loop
OLPC

    Open-Loop Power Control
OL-PC

    Open-Loop Power Control
OPEX

    Operational Expenditure
ORB

    Orthogonal Random Beamforming
JO-PF

    Joint Opportunistic Proportional Fair
OSI

    Open Systems Interconnection
PAIR

    D2D Pair Gain-based Grouping
PAPR

    Peak-to-Average Power Ratio
P2P

    Peer-to-Peer
PC

    Power Control
PCI

    Physical Cell ID
PDF

    Probability Density Function
PDPR

    pilot-to-data power ratio
PER

    Packet Error Rate
PF

    Proportional Fair
P-GW

    Packet Data Network Gateway
PL

    Pathloss
PPR

    pilot power ratio
PRB

    physical resource block
PROJ

    Projection-based Grouping
ProSe

    Proximity Services
PS

    Packet Scheduling
PSAM

    pilot symbol assisted modulation
PSO

    Particle Swarm Optimization
PZF

    Projected Zero-Forcing
QAM

    Quadrature Amplitude Modulation
QoS

    Quality of Service
QPSK

    Quadri-Phase Shift Keying
RAISES

    Reallocation-based Assignment for Improved Spectral Efficiency and Satisfaction
RAN

    Radio Access Network
RA

    Resource Allocation
RAT

    Radio Access Technology
RATE

    Rate-based
RB

    resource block
RBG

    Resource Block Group
REF

    Reference Grouping
RIS

    reconfigurable intelligent surface
RLC

    Radio Link Control
RM

    Rate Maximization
RNC

    Radio Network Controller
RND

    Random Grouping
RRA

    Radio Resource Allocation
RRM

    Radio Resource Management
RSCP

    Received Signal Code Power
RSRP

    Reference Signal Receive Power
RSRQ

    Reference Signal Receive Quality
RR

    Round Robin
RRC

    Radio Resource Control
RSSI

    Received Signal Strength Indicator
RT

    Real Time
RU

    Resource Unit
RUNE

    RUdimentary Network Emulator
RV

    Random Variable
SAC

    Soft Actor-Critic
SCM

    Spatial Channel Model
SC-FDMA

    Single Carrier - Frequency Division Multiple Access
SD

    Soft Dropping
S-D

    Source-Destination
SDPC

    Soft Dropping Power Control
SDMA

    Space-Division Multiple Access
SE

    spectral efficiency
SER

    Symbol Error Rate
SES

    Simple Exponential Smoothing
S-GW

    Serving Gateway
SINR

    signal-to-interference-plus-noise ratio
SI

    Satisfaction Indicator
SIP

    Session Initiation Protocol
SISO

    single input single output
SIMO

    single input multiple output
SIR

    signal-to-interference ratio
SLNR

    Signal-to-Leakage-plus-Noise Ratio
SMA

    Simple Moving Average
SNR

    signal-to-noise ratio
SORA

    Satisfaction Oriented Resource Allocation
SORA-NRT

    Satisfaction-Oriented Resource Allocation for Non-Real Time Services
SORA-RT

    Satisfaction-Oriented Resource Allocation for Real Time Services
SPF

    Single-Carrier Proportional Fair
SRA

    Sequential Removal Algorithm
SRS

    Sounding Reference Signal
SU-MIMO

    single-user multiple input multiple output
SU

    Single-User
SVD

    Singular Value Decomposition
TCP

    Transmission Control Protocol
TDD

    time division duplexing
TDMA

    Time Division Multiple Access
TETRA

    Terrestrial Trunked Radio
TP

    Transmit Power
TPC

    Transmit Power Control
TTI

    Transmission Time Interval
TTR

    Time-To-Rendezvous
TSM

    Throughput-based Satisfaction Maximization
TU

    Typical Urban
UAV

    unmanned aerial vehicle
UE

    user equipment
UEPS

    Urgency and Efficiency-based Packet Scheduling
UL

    uplink
ULA

    uniform linear array
UMTS

    Universal Mobile Telecommunications System
URA

    uniform rectangular array
URI

    Uniform Resource Identifier
URM

    Unconstrained Rate Maximization
UT

    user terminal
VAR

    vector autoregressive
VR

    Virtual Resource
VoIP

    Voice over IP
WAN

    Wireless Access Network
WCDMA

    Wideband Code Division Multiple Access
WF

    Water-filling
WiMAX

    Worldwide Interoperability for Microwave Access
WINNER

    Wireless World Initiative New Radio
WLAN

    Wireless Local Area Network
WMPF

    Weighted Multicarrier Proportional Fair
WPF

    Weighted Proportional Fair
WSN

    Wireless Sensor Network
WSS

    wide-sense stationary
WWW

    World Wide Web
XIXO

    (Single or Multiple) Input (Single or Multiple) Output
ZF

    zero-forcing
ZMCSCG

    Zero Mean Circularly Symmetric Complex Gaussian

# Combating Inter-Operator Pilot Contamination in Reconfigurable Intelligent
Surfaces Assisted Multi-Operator Networks

Doğa Gürgünoğlu, _Student Member, IEEE_ , Emil Björnson,  _Fellow, IEEE_ ,
Gábor Fodor, _Senior Member, IEEE_ D. Gürgünoglu, E. Björnson and G. Fodor are
with the Faculty of Electrical Engineering and Computer Science, KTH Royal
Institute of Technology, Stockholm 100 44, Sweden (e-mails:
{dogag,emilbjo,gaborf}@kth.se). G. Fodor is also with Ericsson Research,
Stockholm 164 80, Sweden (e-mail: {gabor.fodor}@ericsson.com). This study is
supported by EU Horizon 2020 MSCA-ITN-METAWIRELESS, Grant Agreement 956256. E.
Björnson is funded by the FFL18-0277 grant from SSF. The preliminary version
[1] of this work was presented at IEEE BlackSeacom 2023, Istanbul, Turkiye.

###### Abstract

In this paper, we study a new kind of pilot contamination appearing in multi-
operator reconfigurable intelligent surfaces (RIS) assisted networks, where
multiple operators provide services to their respective served users. The
operators use dedicated frequency bands, but each RIS inadvertently reflects
the transmitted uplink signals of the user equipment devices in multiple
bands. Consequently, the concurrent reflection of pilot signals during the
channel estimation phase introduces a new inter-operator pilot contamination
effect. We investigate the implications of this effect in systems with either
deterministic or correlated Rayleigh fading channels, specifically focusing on
its impact on channel estimation quality, signal equalization, and channel
capacity. The numerical results demonstrate the substantial degradation in
system performance caused by this phenomenon and highlight the pressing need
to address inter-operator pilot contamination in multi-operator RIS
deployments. To combat the negative effect of this new type of pilot
contamination, we propose to use orthogonal RIS configurations during uplink
pilot transmission, which can mitigate or eliminate the negative effect of
inter-operator pilot contamination at the expense of some inter-operator
information exchange and orchestration.

###### Index Terms:

Reconfigurable intelligent surface, channel estimation, pilot contamination.

##  I Introduction

Pilot contamination is a key problem that frequently arises in wireless
communication systems [2]. When multiple users use the same pilot sequences
simultaneously in the same band, due to the limited channel coherence time,
the base station (BS) cannot distinguish their channels. This typically
results in poor channel estimates and extra beamformed interference from or
towards the user equipments that reuse the same pilot sequence. Therefore,
pilot contamination adversely affects the coherent reception of data, and
methods to mitigate pilot contamination—including adaptive pilot reuse, power
control, user grouping, multi-cell coordination, and coded random access
techniques—have been widely studied in the communication literature [2, 3, 4,
5, 6].

In recent years, reconfigurable intelligent surfaces have arisen as a new
technology component for 6G [7]. An RIS is a surface consisting of multiple
reflecting elements that have sub-wavelength spacing and controllable
reflection properties [8]. This feature provides partial control of the
propagation environment that can lead to better services for users, especially
when their serving BS is not in their line of sight (LOS). By adjusting the
impedances of the individual elements via a RIS controller, the elements are
capable of adding desired phase shifts to the reflected signals, thereby
forming reflected beams in desired directions that can significantly boost the
signal-to-interference-plus-noise ratio (SINR) and reduce the symbol
estimation error at the receiver [7, 9].

On the other hand, the addition of RISs to existing systems introduces new
design and operational challenges [10]. For example, the length of the pilot
signal required by a single UE is proportional to the number of RIS elements
(e.g., tens or hundreds), because the RIS must change its configuration to
explore all channel dimensions [11, 12]. In addition, the path loss of the
reflected path through a passive RIS is proportional to the multiplication of
the path losses to and from the RIS [13], so a larger surface is needed to
achieve a decent signal-to-noise ratio (SNR) improvement. Active RISs, on the
other hand, use amplifiers to overcome the large path loss but have the
traditional issues of relays: increased power consumption, higher cost, and
additional noise [14, 15]. While the aforementioned problems caused by the RIS
have been recognized [16], pilot contamination caused by the presence of
multiple RISs has not been studied in the literature.

Wireless communication systems use standardized protocols, interfaces, and
well-defined pilot sequences and codebooks to ensure inter-operability [17].
While employing RISs in cellular networks have not been studied by the
relevant standards organizations yet, it may be expected that the
configuration sequences that facilitate the deployment of RISs while
maintaining interoperability will be specified. Consequently, when multiple
cellular networks are deployed by different network operators in overlapping
geographical areas, the RISs may adopt identical or overlapping pilot
sequences and cause pilot contamination. The number of orthogonal pilot
sequences is limited by the length of the pilot sequence, and increasing the
pilot sequence length not only creates more channel estimation overhead but
also is infeasible due to the limited coherence budget of the channel. As a
consequence, the need for repeating pilot sequences comes up very often.

In this paper, we argue that when multiple RISs are deployed for the purpose
of shaping the propagation characteristics of the environment, the propagation
characteristics might change in unintended ways. For example, an RIS belonging
to another operator might modify the propagation of a UE’s own pilot signal,
leading to pilot contamination even in the absence of any interfering signals
or intra-band pilot reuse. The underlying reason is that an RIS
element—although designed for a particular frequency—does not act as a
bandpass filter, but reflects all frequencies with varying amplitude and
phase. Indeed, as pointed out in [18], due to the lack of baseband signal
processing, the RIS reflects the impinging broadband signal with frequency-
flat reflection coefficients. Therefore, in realistic system models of, for
example, passive RIS assisted 5G New Radio systems, we need to take into
account that the RIS inadvertently reflects the transmitted uplink signals of
the user equipment devices in multiple bands as in [19], [20].

Specifically, in this paper, we identify this new pilot contamination
phenomenon as a major practical challenge when multiple RIS assisted operator
networks are deployed over a geographical area, including the important
practical scenario of inter-operator site sharing [21, 22]. In such an
environment, a UE that transmits pilots to its serving BS via multiple RIS,
which may change their configurations simultaneously, is exposed to new pilot-
related ambiguities that have not been studied before. Since this phenomenon
exacerbates the pilot contamination problem, it is clear that pilot
contamination due to the presence of multiple RISs must be dealt with.

To the best of the authors’ knowledge, the problem of inter-operator pilot
contamination in RIS-aided wireless communication systems has not been
addressed before, except in the preliminary version of this manuscript [1],
which assumed deterministic rather than stochastically fading channels. In
this paper, we derive the capacity lower bound of the system under pilot
contamination and imperfect channel state information (CSI) assuming Rayleigh
fading. Our major contributions can be summarized as follows:

  * •

For the case when inter-operator pilot contamination is neglected, we provide
a misspecified maximum likelihood (ML) estimator under the assumption that all
the channels in the system setup are deterministic. We also derive the
resulting channel estimation mean squared error (MSE) for different choices of
the RIS configurations.

  * •

Based on the obtained results for the channel estimation error under inter-
operator pilot contamination, we provide the data signal estimation MSE for a
misspecified minimum mean squared error (MMSE) estimator.

  * •

In addition to deterministic channels, we also consider the case where the
channels are Rayleigh fading with spatial correlation. For generic channel
spatial covariances, we derive the misspecified MMSE estimator and the
resulting MSE.

  * •

Based on the channel estimation error model, obtained for spatially correlated
Rayleigh fading channels, we derive a capacity lower bound given the channel
estimates. Our numerical results show that the choice of RIS configurations
during channel estimation makes a significant impact on the capacity lower
bound.

The rest of the manuscript is organized as follows: in Section II, we provide
the received signal model, in Section III, we provide the misspecified ML
estimator where the inter-operator pilot contamination is neglected and the
channels are assumed to be deterministic. Section IV builds on top of section
III by providing the data estimation MSE as a result of inter-operator pilot
contamination. In Section V, we derive the impact of inter-operator pilot
contamination in closed-form for the case of spatially correlated Rayleigh-
fading channels by considering the channel estimation MSE as our performance
metric. Since the data estimation MSE has a dependence on individual channel
realizations, an alternative performance metric for data transmission is
needed to capture the behavior of fading channels. To this end, we derive the
capacity lower bound under channel side information in Section VI. We provide
the numerical results in Section VII, and conclude the manuscript in Section
VIII.

##  II System Model

In this paper, we study the pilot contamination caused by the presence of
multiple RISs by considering the uplink of a system consisting of two wide-
band RISs, two single-antenna UEs, and two co-located single-antenna BSs. The
UEs are subscribed to different operators and use non-overlapping frequency
bands. Each RIS is dedicated to and controlled by a single operator, but both
UE signals are reflected from both RISs. In this scenario, although there is
no interference between the two UEs, both RISs affect both frequency bands.

In Fig. 1, we graphically describe the system that we consider. The components
associated with the two different operators are depicted in two different
colors: blue BS, RIS, UE and the channels belong to operator 1, while the red
ones belong to operator 2. The operators can potentially use site-sharing (as
in the figure) to reduce deployment costs but transmit over two disjoint
narrow frequency bands to their respective serving BSs. Each RIS has $N$
reflecting elements, and is dedicated to and controlled by a single operator
but affects both bands. To focus on the fundamentals of pilot contamination,
we consider an environment where the direct UE-BS paths are blocked, while the
UE-RIS and RIS-BS paths are operational. Since the BSs and RISs have fixed
deployment locations, we assume the RIS-BS channels
$\mathbf{h}_{k}\in\mathbb{C}^{N}$ are known, while the UE-RIS channels
$\mathbf{g}_{k}\in\mathbb{C}^{N}$ are unknown and to be estimated, for
$k=1,2$.

Figure 1: The considered setup with two UEs, two RISs, and two co-located
single-antenna BSs. The blue channels correspond to frequency band 1, and the
red channels correspond to frequency band 2, subscribed by UEs $1$ and $2$,
respectively. The desired channels are denoted by solid lines, while the
undesired channels (whose existences might be unknown to the BSs) are denoted
by dashed lines. Each channel vector is $N$-dimensional because each RIS has
$N$ elements.

The signal transmitted by UE $k$ reaches its serving BS through the channels
$\mathbf{h}_{k}$ and $\mathbf{g}_{k}$, for $k=1,2$. Importantly, each UE’s
transmitted signal is also reflected by the non-serving operator’s RIS and
parts of the reflected signal will reach the serving BS. This effect
contaminates the pilot signal reflected by the serving RIS and we will study
the implications. The phenomenon is illustrated in Fig. 1, where the resulting
UE-RIS and RIS-BS channels are denoted by $\mathbf{p}_{k}\in\mathbb{C}^{N}$
and $\mathbf{q}_{k}\in\mathbb{C}^{N}$, respectively, for $k=1,2$. Defining the
pilot signal of UE $k$ as $s_{k}\in\mathbb{C}$, the received signals on bands
$1$ and $2$ at the BSs can be expressed as

$\displaystyle
y_{p1}=\sqrt{P_{p}}\mathbf{h}_{1}^{T}\boldsymbol{\Phi}_{1}\mathbf{g}_{1}s_{1}+\sqrt{P_{p}}\mathbf{q}_{1}^{T}\boldsymbol{\Phi}_{2}\mathbf{p}_{1}s_{1}+w_{p1},$
(1a) $\displaystyle
y_{p2}=\sqrt{P_{p}}\mathbf{h}_{2}^{T}\boldsymbol{\Phi}_{2}\mathbf{g}_{2}s_{2}+\sqrt{P_{p}}\mathbf{q}_{2}^{T}\boldsymbol{\Phi}_{1}\mathbf{p}_{2}s_{2}+w_{p2},$
(1b)

where $y_{pk}\in\mathbb{C}$ denotes the received signal,
$w_{pk}\sim\mathcal{CN}(0,1)$ denotes the receiver noise for band $k$, and
$\boldsymbol{\Phi}_{k}=\text{diag}(e^{-j\phi_{k1}},\dots,e^{-j\phi_{kN}})$
denotes the $k$th RIS’s reflection matrix, and $P_{p}$ denotes the pilot
signal’s transmission power. We assume $s_{1}=s_{2}=1$ without loss of
generality. When analyzing channel estimation, it is more convenient to
rewrite (1) as

$\displaystyle
y_{p1}=\sqrt{P_{p}}\boldsymbol{\phi}_{1}^{T}\mathbf{D}_{\mathbf{h}_{1}}\mathbf{g}_{1}+\sqrt{P_{p}}\boldsymbol{\phi}_{2}^{T}\mathbf{D}_{\mathbf{q}_{1}}\mathbf{p}_{1}+w_{p1},$
(2a) $\displaystyle
y_{p2}=\sqrt{P_{p}}\boldsymbol{\phi}_{2}^{T}\mathbf{D}_{\mathbf{h}_{2}}\mathbf{g}_{2}+\sqrt{P_{p}}\boldsymbol{\phi}_{1}^{T}\mathbf{D}_{\mathbf{q}_{2}}\mathbf{p}_{2}+w_{p2},$
(2b)

where $\mathbf{D}_{\mathbf{h}_{k}}$ and $\mathbf{D}_{\mathbf{q}_{k}}$
represent the $N\times N$ diagonal matrices containing the elements of
$\mathbf{h}_{k}$ and $\mathbf{q}_{k}$, and
$\boldsymbol{\phi}_{k}\in\mathbb{C}^{N}$ denotes the column vectors containing
the diagonal entries of $\boldsymbol{\Phi}_{k}$ for $k=1,2$.

As there are $N$ parameters in $\mathbf{g}_{1}$ and $\mathbf{g}_{2}$, at least
$N$ linearly independent observations are needed to estimate them uniquely. To
this end, we perform $L\geq N$ pilot transmissions over time, and we
vertically stack the received signals to obtain

$\displaystyle\mathbf{y}_{p1}=\sqrt{P_{p}}\mathbf{B}_{1}\mathbf{D}_{\mathbf{h}_{1}}\mathbf{g}_{1}+\sqrt{P_{p}}\mathbf{B}_{2}\mathbf{D}_{\mathbf{q}_{1}}\mathbf{p}_{1}+\mathbf{w}_{p1},$
(3a)
$\displaystyle\mathbf{y}_{p2}=\sqrt{P_{p}}\mathbf{B}_{2}\mathbf{D}_{\mathbf{h}_{2}}\mathbf{g}_{2}+\sqrt{P_{p}}\mathbf{B}_{1}\mathbf{D}_{\mathbf{q}_{2}}\mathbf{p}_{2}+\mathbf{w}_{p2},$
(3b)

where $\mathbf{y}_{pk}=[y_{pk}[1],\dots,y_{pk}[L]]^{T}\in\mathbb{C}^{L}$
denotes the sequence of received signals from the $k$th UE over $L$ time
instances, and the matrices $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$ represent
the sequence of RIS configurations over $L$ time instances; that is,
$\mathbf{B}_{k}\triangleq\begin{bmatrix}\boldsymbol{\phi}_{k}[1]&\dots&\boldsymbol{\phi}_{k}[L]\end{bmatrix}^{T}\in\mathbb{C}^{L\times
N}$ for $k=1,2$.

In the remainder of this paper, we will analyze channel estimation and the
resulting communication performance for deterministic and fading channels,
respectively.

##  III Maximum Likelihood Estimation of Deterministic Channels

In this section, we will consider channel estimation for deterministic
channels. The same assumptions and results will then be considered in Section
IV for data transmission.

We assume that $\mathbf{g}_{k}$ is a deterministic and unknown channel without
any known structure. That is, $\mathbf{g}_{k}$ is an $N\times 1$ vector of
complex deterministic parameters to be estimated. In addition, we assume that
$\mathbf{h}_{k}$ is perfectly known. On the other hand, the BSs do not know
the existence of $\mathbf{q}_{k}$ and $\mathbf{p}_{k}$. Consequently, we can
denote the received signal models assumed by the BSs as

$\displaystyle\hat{\mathbf{y}}_{p1}=\sqrt{P_{p}}\mathbf{B}_{1}\mathbf{D}_{\mathbf{h}_{1}}\mathbf{g}_{1}+\mathbf{w}_{p1},$
(4a)
$\displaystyle\hat{\mathbf{y}}_{p2}=\sqrt{P_{p}}\mathbf{B}_{2}\mathbf{D}_{\mathbf{h}_{2}}\mathbf{g}_{2}+\mathbf{w}_{p2}.$
(4b)

Since $\mathbf{g}_{k}$ does not have a known structure and hence consists of
$N$ scalars, BS $k$ requires at least $N$ independent observations to estimate
it. To this end, both $\mathbf{B}_{1},\mathbf{B}_{2}\in\mathbb{C}^{L\times N}$
must have full column rank. Furthermore, we require that the RIS
configurations used at different time instances are mutually orthogonal and
contain entries on the unit circle that can be realized using a reflecting
element. These assumptions result in
$\mathbf{B}_{k}^{H}\mathbf{B}_{k}=L\mathbf{I}_{N}$. In classical non-Bayesian
parameter estimation, the ML estimator is widely used, which maximizes the
likelihood function of the received observation over the unknown parameter.
Since the BSs have misspecified received pilot signal models, they will
instead maximize the likelihood functions obtained from the misspecified
model, leading to misspecified maximum likelihood (MML) estimators. For (4),
the MML estimator can be expressed as

$\displaystyle\hat{\mathbf{g}}_{k}$
$\displaystyle=\arg\max_{\mathbf{g}_{k}}f(\mathbf{y}_{pk};\mathbf{g}_{k})$
$\displaystyle=\arg\max_{\mathbf{g}_{k}}\frac{1}{(\pi\sigma_{w}^{2})^{L}}\exp{\left(-\frac{\|\mathbf{y}_{pk}-\sqrt{P_{p}}\mathbf{B}_{k}\mathbf{D}_{\mathbf{h}_{k}}\mathbf{g}_{k}\|^{2}}{\sigma_{w}^{2}}\right)}$
$\displaystyle=\arg\min_{\mathbf{g}_{k}}\left\|\mathbf{y}_{pk}-\sqrt{P_{p}}\mathbf{B}_{k}\mathbf{D}_{\mathbf{h}_{k}}\mathbf{g}_{k}\right\|^{2}$
$\displaystyle=\frac{1}{\sqrt{P_{p}}}\mathbf{D}_{\mathbf{h}_{k}}^{-1}(\mathbf{B}_{k}^{H}\mathbf{B}_{k})^{-1}\mathbf{B}_{k}^{H}\mathbf{y}_{pk}$
$\displaystyle=\frac{1}{L\sqrt{P_{p}}}\mathbf{D}_{\mathbf{h}_{k}}^{-1}\mathbf{B}_{k}^{H}\mathbf{y}_{pk}.$
(5)

In the following subsections, we describe the behavior of this estimator for
two different choices of the $\mathbf{B}_{k}$ matrices.

###  III-A Case 1: The RISs Adopt the Same Configuration Sequence

We discussed in the introduction that in the absence of inter-operator
cooperation, it is highly likely that the RISs will use the same standardized
sequence of configurations during the channel estimation phase, which
corresponds to $\mathbf{B}_{1}=\mathbf{B}_{2}=\mathbf{B}$.111The analysis in
this paper can be easily extended to the case when
$\mathbf{B}_{1}=\mathbf{U}\mathbf{B}_{2}$ for some unitary matrix
$\mathbf{U}$, so that the configuration sequences have identical spans. It is
the overlap of the spans that can cause issues. In this case, (5) becomes

$\displaystyle\hat{\mathbf{g}}_{k}=\mathbf{g}_{k}+\mathbf{D}_{\mathbf{h}_{k}}^{-1}\mathbf{D}_{\mathbf{q}_{k}}\mathbf{p}_{k}+\frac{1}{L\sqrt{P}_{p}}\mathbf{D}_{\mathbf{h}_{k}}^{-1}\mathbf{B}^{H}\mathbf{w}_{pk}.$
(6)

Since we consider the channels as deterministic parameters, we obtain the
probability distribution

$\hat{\mathbf{g}}_{k}\sim\mathcal{CN}\\!\left(\mathbf{g}_{k}+\mathbf{D}_{\mathbf{h}_{k}}^{-1}\mathbf{D}_{\mathbf{q}_{k}}\mathbf{p}_{k},\frac{\sigma_{w}^{2}}{LP_{p}}(\mathbf{D}_{\mathbf{h}_{k}}^{H}\mathbf{D}_{\mathbf{h}_{k}})^{-1}\\!\right).$
(7)

We notice that $\hat{\mathbf{g}}_{k}$ is biased; that is,
$\mathbf{b}_{k}\triangleq\mathbb{E}[\hat{\mathbf{g}}_{k}-\mathbf{g}_{k}]=\mathbf{D}_{\mathbf{h}_{k}}^{-1}\mathbf{D}_{\mathbf{q}_{k}}\mathbf{p}_{k}\neq\boldsymbol{0}$.
The estimator bias does not vanish when increasing $P_{p}$ or $L$, or
decreasing $\sigma_{w}^{2}$. Hence, this estimator is not asymptotically
unbiased. This is a new instance of an extensively studied phenomenon in the
massive multiple-input multiple-output (MIMO) literature, namely pilot
contamination [2, 3]. Interestingly, the RISs cause pilot contamination even
between two non-overlapping frequency bands, which has not been recognized in
the previous literature.

###  III-B Case 2: The RISs Adopt Different Configuration Sequences

In this section, we consider the generic case of
$\mathbf{B}_{1}\neq\mathbf{B}_{2}$. To motivate the proposed method for
configuring $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$, we first consider the case
where the BSs are aware of the true signal model in (3), and therefore can
estimate both $\mathbf{g}_{k}$ and
$\mathbf{r}_{k}\triangleq\mathbf{D}_{\mathbf{q}_{k}}\mathbf{p_{k}}$. The
resulting system model can be expressed as

$\displaystyle\mathbf{y}_{p1}=\sqrt{P}_{p}\begin{bmatrix}\mathbf{B}_{1}\mathbf{D}_{\mathbf{h}_{1}}&\mathbf{B}_{2}\end{bmatrix}\begin{bmatrix}\mathbf{g}_{1}\\\
\mathbf{r}_{1}\end{bmatrix}+\mathbf{w}_{p1},$ (8a)
$\displaystyle\mathbf{y}_{p2}=\sqrt{P}_{p}\begin{bmatrix}\mathbf{B}_{2}\mathbf{D}_{\mathbf{h}_{2}}&\mathbf{B}_{1}\end{bmatrix}\begin{bmatrix}\mathbf{g}_{2}\\\
\mathbf{r}_{2}\end{bmatrix}+\mathbf{w}_{p2}.$ (8b)

In (8), a known linear transformation is applied to the parameter vector of
interest in the presence of additive noise. Consequently, the ML estimates of
UE 1’s channels become

$\begin{bmatrix}\hat{\mathbf{g}}_{1}\\\
\hat{\mathbf{r}}_{1}\end{bmatrix}=\frac{1}{\sqrt{P_{p}}}\begin{bmatrix}L\mathbf{D}_{\mathbf{h}_{1}}^{H}\mathbf{D}_{\mathbf{h}_{1}}&\mathbf{D}_{\mathbf{h}_{1}}^{H}\mathbf{B}_{1}^{H}\mathbf{B}_{2}\\\
\mathbf{B}_{2}^{H}\mathbf{B}_{1}\mathbf{D}_{\mathbf{h}_{1}}&L\mathbf{I}_{N}\end{bmatrix}^{-1}\begin{bmatrix}\mathbf{D}_{\mathbf{h}_{1}}^{H}\mathbf{B}_{1}^{H}\\\
\mathbf{B}_{2}^{H}\end{bmatrix}\mathbf{y}_{p1}.$ (9)

Note that in this case, the total dimension of the unknown parameter vector is
$2N$, hence, at least $2N$ independent observations are required for the
matrix inverse to exist.222The pseudo-inverse could be used when there are
fewer observations, but it will not provide a useful estimate. The structure
in (9) applies to UE 2 with alternated indices, and it gives the ML estimator,
which is both unbiased and efficient, since (8) is a linear observation model
with additive Gaussian noise [23, Th. 7.3]. Hence, (9) is unbiased
irrespective of other parameters such as $\sigma_{w}^{2}$, $L$, and $P_{p}$,
and it achieves the Cramér-Rao Lower Bound (CRLB), which provides a lower
bound on the MSE of any unbiased estimator [24]. It has to be noted that when
$\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$, (9) becomes

$\displaystyle\begin{bmatrix}\hat{\mathbf{g}}_{1}\\\
\hat{\mathbf{r}}_{1}\end{bmatrix}$
$\displaystyle=\frac{1}{L\sqrt{P_{p}}}\begin{bmatrix}\mathbf{D}_{\mathbf{h}_{1}}^{H}\mathbf{D}_{\mathbf{h}_{1}}&\boldsymbol{0}\\\
\boldsymbol{0}&\mathbf{I}_{N}\end{bmatrix}^{-1}\begin{bmatrix}\mathbf{D}_{\mathbf{h}_{1}}^{H}\mathbf{B}_{1}^{H}\\\
\mathbf{B}_{2}^{H}\end{bmatrix}\mathbf{y}_{p1}$
$\displaystyle=\frac{1}{L\sqrt{P_{p}}}\begin{bmatrix}\mathbf{D}_{\mathbf{h}_{1}}^{-1}\mathbf{B}_{1}^{H}\\\
\mathbf{B}_{2}^{H}\end{bmatrix}\mathbf{y}_{p1}.$ (10)

Note that the expression for $\hat{\mathbf{g}}_{1}$ in (III-B) is the same as
in (5). This shows that when
$\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$, the MML in (5) coincides
with the ML estimator; that is, the misspecified model is sufficient when the
configuration sequences are designed to alleviate pilot interference because
the missing terms anyway vanish in the receiver processing. The probability
distribution of $\hat{\mathbf{g}}_{k}$ in this case can be expressed as

$\hat{\mathbf{g}}_{k}\sim\mathcal{CN}\left(\mathbf{g}_{k},\frac{\sigma_{w}^{2}}{LP_{p}}(\mathbf{D}_{\mathbf{h}_{k}}^{H}\mathbf{D}_{\mathbf{h}_{k}})^{-1}\right),$
(11)

which shows that choosing the RIS configuration sequences such that
$\mathbf{B}_{1}$ and $\mathbf{B}_{2}$ remove the bias from the MML estimator.
However, the major setback of this approach is that the minimum number of
observations required for this channel estimation procedure is $2N$ instead of
$N$, due to the fact that the $2N$-many $L$-dimensional columns must all be
mutually orthogonal, for which $L\geq 2N$ is required. Considering that the
estimator bias in (7) does not vanish with increasing $L$, this is a necessary
sacrifice. Hence, it has to be noted that the number of pilot transmissions
increases linearly with the number of RISs deployed in proximity.

###  III-C MSE During Channel Estimation

The estimation error can be quantified through the MSE, which is the trace of
the error covariance matrix. We will derive the MSE in this section. In this
derivation, we do not assume a particular choice of $\mathbf{B}_{1}$ and
$\mathbf{B}_{2}$, but we utilize the basic assumption
$\mathbf{B}_{k}\mathbf{B}_{k}^{H}=L\mathbf{I}_{N}$. Consequently, we use
$\mathbf{b}_{k}$ to denote the potential estimator bias. We can then compute
the error covariance matrix as

$\displaystyle\boldsymbol{\Sigma}_{e,k}$
$\displaystyle=\mathbb{E}\left[(\hat{\mathbf{g}}_{k}-\mathbf{g}_{k})(\hat{\mathbf{g}}_{k}-\mathbf{g}_{k})^{H}\right]$
$\displaystyle=\mathbf{b}_{k}\mathbf{b}_{k}^{H}+\frac{1}{LP_{p}}\mathbb{E}\left[\mathbf{D}_{\mathbf{h}_{k}}^{-1}\mathbf{w}_{pk}\mathbf{w}_{pk}^{H}\mathbf{D}_{\mathbf{h}_{k}}^{-H}\right]$
$\displaystyle=\mathbf{b}_{k}\mathbf{b}_{k}^{H}+\frac{\sigma_{w}^{2}}{LP_{p}}\left(\mathbf{D}_{\mathbf{h}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{H}\right)^{-1}.$
(12)

Consequently, the trace of the error covariance matrix becomes

$\mathrm{tr}(\boldsymbol{\Sigma}_{e,k})=\|\mathbf{b}_{k}\|^{2}+\frac{\sigma_{w}^{2}}{LP_{p}}\sum_{n=1}^{N}\frac{1}{|h_{kn}|^{2}}.$
(13)

Note that for high $P_{p}$, $L$, and/or low $\sigma_{w}^{2}$, the second term
in (13) vanishes, and the trace of the error covariance converges to
$\|\mathbf{b}_{k}\|^{2}$ which depends on the configurations of
$\mathbf{B}_{1},\mathbf{B}_{2}$. For the two previously considered cases, we
have

$\|\mathbf{b}_{k}\|^{2}=\begin{cases}\sum_{n=1}^{N}\frac{|r_{kn}|^{2}}{|h_{kn}|^{2}}&\mathbf{B}_{1}=\mathbf{B}_{2},\\\
0&\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}.\end{cases}$ (14)

This result shows that configuring the RISs such that
$\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$ removes the asymptotic floor
on the average MSE, which comes from the energy of the estimator bias. On the
other hand, when the intended RIS-BS links $\mathbf{h}_{1},\mathbf{h}_{2}$ are
strong relative to the unintended and unknown overall link $\mathbf{r}_{k}$,
the estimator bias will be weaker and the performance loss associated with
choosing $\mathbf{B}_{1}=\mathbf{B}_{2}$ will be lower. Nevertheless, pilot
contamination results in a fundamental error floor, even if the RISs are
utilized in different bands. In the next section, we consider the estimation
of data based on the channel estimation performed in this section and analyze
the consequence of pilot contamination in this phase.

##  IV Data Signal Estimation with Deterministic Channels

The channel estimation is followed by data transmission over the same
deterministic channel as in Section III. The receiver can use the channel
estimate derived in the last section when determining the transmitted signal.
Practical channels are never fully deterministic but might have a long
coherence time. Moreover, the impact of estimation errors is only relevant
when the data packet has a modest size so we cannot afford to spend much
resources on pilots. For this reason, we cannot consider the channel capacity
as performance metric but will instead consider the MSE.

Defining the data signal transmitted by the $k$th UE as
$x_{k}\sim\mathcal{CN}(0,1)$, we can express the received data as

$\displaystyle
y_{1}=\sqrt{P_{d}}(\mathbf{h}_{1}^{T}\hat{\boldsymbol{\Phi}}_{1}\mathbf{g}_{1}+\mathbf{q}_{1}^{T}\hat{\boldsymbol{\Phi}}_{2}\mathbf{p}_{1})x_{1}+w_{1},$
(15a) $\displaystyle
y_{2}=\sqrt{P_{d}}(\mathbf{h}_{2}^{T}\hat{\boldsymbol{\Phi}}_{2}\mathbf{g}_{2}+\mathbf{q}_{2}^{T}\hat{\boldsymbol{\Phi}}_{1}\mathbf{p}_{2})x_{2}+w_{2},$
(15b)

where $w_{k}\sim\mathcal{CN}(0,\sigma_{w}^{2})$ denotes the receiver noise,
$P_{d}$ denotes the data transmission power, and the RIS configuration
matrices $\hat{\boldsymbol{\Phi}}_{k}$ are selected based on the estimated
channels to maximize the average gain of the desired cascaded channel as shown
in [12, Sec. II]:

$\displaystyle\hat{\phi}_{kn}=\arg(h_{kn})+\arg(\hat{g}_{kn}),$
$\displaystyle\hat{\boldsymbol{\Phi}}_{k}=\mathrm{diag}\left(e^{-j\hat{\phi}_{k1}},\dots,e^{-j\hat{\phi}_{kN}}\right).$
(16)

However, since the BSs are unaware of the unintended reflections and base
their data reception on the previously obtained channel estimates, they assume
the following misspecified received data signal models:

$\displaystyle\hat{y}_{1}=\sqrt{P_{d}}\mathbf{h}_{1}^{T}\hat{\boldsymbol{\Phi}}_{1}\hat{\mathbf{g}}_{1}x_{1}+w_{1},$
(17a)
$\displaystyle\hat{y}_{2}=\sqrt{P_{d}}\mathbf{h}_{2}^{T}\hat{\boldsymbol{\Phi}}_{2}\hat{\mathbf{g}}_{2}x_{2}+w_{2}.$
(17b)

Introducing the notation
$m_{k}\triangleq\sqrt{P_{d}}(\mathbf{h}_{k}^{T}\hat{\boldsymbol{\Phi}}_{k}\mathbf{g}_{k}+\mathbf{q}_{k}^{T}\hat{\boldsymbol{\Phi}}_{j}\mathbf{p}_{k})$
for $j,k\in\\{1,2\\},j\neq k$, and
$\hat{m}_{k}\triangleq\sqrt{P_{d}}\mathbf{h}_{k}^{T}\hat{\boldsymbol{\Phi}}_{k}\hat{\mathbf{g}}_{k}$,
(15) and (17) can be expressed as

$\displaystyle y_{k}=m_{k}x_{k}+w_{k},\quad k=1,2,$ (18a)
$\displaystyle\hat{y}_{k}=\hat{m}_{k}x_{k}+w_{k},\quad k=1,2.$ (18b)

Based on the misspecified observation model in (18b), the BSs estimate $x_{k}$
by using the misspecified MMSE estimator

$\hat{x}_{k}=\frac{\hat{m}_{k}^{*}}{|\hat{m}_{k}|^{2}+\sigma_{w}^{2}}y_{k},\quad
k=1,2.$ (19)

In this section, we consider the MSE between $x_{k}$ and $\hat{x}_{k}$ as the
performance metric for the data transmission. We derive the data estimation
MSE for UE $k$ as

$\displaystyle\mathbb{E}\left[|x_{k}-\hat{x}_{k}|^{2}\right]=1+\mathbb{E}\left[|\hat{x}_{k}|^{2}\right]-2\mathrm{Re}(\mathbb{E}[x_{k}\hat{x}_{k}^{*}])$
$\displaystyle=1+\mathbb{E}\left[\frac{|\hat{m}_{k}|^{2}(|m_{k}|^{2}+\sigma_{w}^{2})}{(|\hat{m}_{k}|^{2}+\sigma_{w}^{2})^{2}}\right]$
$\displaystyle-2\mathrm{Re}\left(\mathbb{E}\left[\frac{\hat{m}_{k}m_{k}^{*}}{|\hat{m}_{k}|^{2}+\sigma_{w}^{2}}\right]\right)$
$\displaystyle=1+\mathbb{E}\left[\frac{|\hat{m}_{k}|^{2}(|m_{k}|^{2}+\sigma_{w}^{2})-2\mathrm{Re}(\hat{m}_{1}m_{1}^{*})(|\hat{m}_{k}|^{2}+\sigma_{w}^{2})}{(|\hat{m}_{k}|^{2}+\sigma_{w}^{2})^{2}}\right]$
$\displaystyle=1+\mathbb{E}\left[\frac{|\hat{m}_{k}|^{2}(|m_{k}|^{2}+\sigma_{w}^{2})-2\mathrm{Re}(\hat{m}_{1}m_{1}^{*})(|\hat{m}_{k}|^{2}+\sigma_{w}^{2})}{(|\hat{m}_{k}|^{2}+\sigma_{w}^{2})^{2}}\right]$
$\displaystyle+\mathbb{E}\left[\frac{\sigma_{w}^{2}(|m_{k}|^{2}+\sigma_{w}^{2})-\sigma_{w}^{2}(|m_{k}|^{2}+\sigma_{w}^{2})}{(|\hat{m}_{k}|^{2}+\sigma_{w}^{2})^{2}}\right]$
$\displaystyle=1+\mathbb{E}\left[\frac{(|\hat{m}_{k}|^{2}+\sigma_{w}^{2})(|m_{k}|^{2}+\sigma_{w}^{2}-2\mathrm{Re}(\hat{m}_{k}m_{k}^{*}))}{(\hat{m}_{k}^{2}+\sigma_{w}^{2})^{2}}\right]$
$\displaystyle-\mathbb{E}\left[\frac{\sigma_{w}^{2}(|m_{k}|^{2}+\sigma_{w}^{2})}{(\hat{m}_{k}^{2}+\sigma_{w}^{2})^{2}}\right]$
$\displaystyle=\mathbb{E}\left[\frac{|m_{k}-\hat{m}_{k}|^{2}+2\sigma_{w}^{2}}{|\hat{m}_{k}|^{2}+\sigma_{w}^{2}}\right]-\mathbb{E}\left[\frac{\sigma_{w}^{2}(|m_{k}|^{2}+\sigma_{w}^{2})}{(|\hat{m}_{k}|^{2}+\sigma_{w}^{2})^{2}}\right].$
(20)

Defining $\epsilon_{k}\triangleq m_{k}-\hat{m}_{k}$, (20) can be rewritten as

$\mathbb{E}[|x_{k}-\hat{x}_{k}|^{2}]=\mathbb{E}\left[\frac{|\epsilon_{k}|^{2}+2\sigma_{w}^{2}}{|m_{k}-\epsilon_{k}|^{2}+\sigma_{w}^{2}}-\frac{\sigma_{w}^{2}(|m_{k}|^{2}+\sigma_{w}^{2})}{(|m_{k}-\epsilon_{k}|^{2}+\sigma_{w}^{2})^{2}}\right].$
(21)

To examine the impact of pilot contamination on the data estimation
performance more clearly, we now consider channel estimation at high SNRs, so
that the estimation error only comes from the estimator bias, i.e., pilot
contamination. This happens when $L$ or $P_{p}$ is high and/or
$\sigma_{w}^{2}$ is low, which results in the estimator covariances in (7) and
(11) becoming zero. For notational convenience, we consider the case where
$P_{p}$ is arbitrarily large so that
$\lim_{P_{p}\to\infty}\hat{\mathbf{g}}_{k}=\mathbf{g}_{k}+\mathbf{b}_{k}$,
where

$\mathbf{b}_{k}=\begin{cases}\mathbf{D}_{\mathbf{h}_{k}}^{-1}\mathbf{D}_{\mathbf{q}_{k}}\mathbf{p}_{k}&\mathbf{B}_{1}=\mathbf{B}_{2},\\\
\boldsymbol{0}&\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}.\end{cases}$
(22)

###  IV-A Data MSE with Channel Estimation at High SNR

In (21), $\epsilon_{k}$ and $m_{k}$ are functions of $\hat{\mathbf{g}}_{1}$
and $\hat{\mathbf{g}}_{2}$, therefore as $\hat{\mathbf{g}}_{1}$ and
$\hat{\mathbf{g}}_{2}$ converge to their means, $\epsilon_{k}$ and $m_{k}$
become

$\displaystyle\overline{m}_{k}=\sqrt{P_{d}}(\mathbf{h}_{k}^{T}\bar{\boldsymbol{\Phi}}_{k}\mathbf{g}_{k}+\mathbf{q}_{k}^{T}\bar{\boldsymbol{\Phi}}_{j}\mathbf{p}_{k}),$
(23a)
$\displaystyle\overline{\epsilon}_{k}=\sqrt{P_{d}}(\mathbf{q}_{k}^{T}\bar{\boldsymbol{\Phi}}_{j}\mathbf{p}_{k}-\mathbf{h}_{k}^{T}\bar{\boldsymbol{\Phi}}_{k}\mathbf{b}_{k}),$
(23b)

for $j,k\in\\{1,2\\}$ and $j\neq k$. $\bar{\boldsymbol{\Phi}}_{k}$ denotes the
RIS configuration computed according to (16) with
$\hat{\mathbf{g}}_{k}=\mathbf{g}_{k}+\mathbf{b}_{k}$. At high SNR, the MSE in
(21) can be rewritten as

$\text{MSE}=\frac{|\overline{\epsilon}_{k}|^{2}+2\sigma_{w}^{2}}{|\overline{m}_{k}-\overline{\epsilon}_{k}|^{2}+\sigma_{w}^{2}}-\frac{\sigma_{w}^{2}(|\overline{m}_{k}|^{2}+\sigma_{w}^{2})}{(|\overline{m}_{k}-\overline{\epsilon}_{k}|^{2}+\sigma_{w}^{2})^{2}}.$
(24)

This is a practically achievable limit since RIS-aided systems require long
pilot sequences over a narrow bandwidth, thus, the effective SNR (proportional
to $P_{p}L$) during pilot transmission can be much larger than in the data
transmission phase.

###  IV-B Data MSE with Transmission at High SNR

In the previous section, we obtained the expression for the data MSE when the
channels are estimated with a high pilot SNR, while the data transmission is
done at an arbitrary SNR. To study the case when also the data transmission is
conducted at a high SNR, we let $\sigma_{w}^{2}\to 0$, which results in the
limit

$\lim_{\sigma_{w}^{2}\to
0}\text{MSE}=\frac{|\overline{\epsilon}_{k}|^{2}}{|\overline{m}_{k}-\overline{\epsilon}_{k}|^{2}}.$
(25)

Note that the resulting expression denotes the ratio between the estimated
overall single input single output (SISO) channel $\hat{m}_{k}$’s power and
the mismatch parameter $\epsilon_{k}$’s power. Recall that $\mathbf{b}_{k}$
depends on which sequence of RIS configurations is utilized. For
$\mathbf{B}_{1}=\mathbf{B}_{2}$, we can obtain $\epsilon_{k}$ as

$\displaystyle\epsilon_{k}$
$\displaystyle=\sqrt{P_{d}}\mathbf{q}_{k}^{T}\hat{\boldsymbol{\Phi}}_{j}\mathbf{p}_{k}-\sqrt{P_{d}}\mathbf{h}_{k}^{T}\hat{\boldsymbol{\Phi}}_{k}\mathbf{D}_{\mathbf{h}_{1}}^{-1}\mathbf{D}_{\mathbf{q}_{1}}\mathbf{p}_{1}$
$\displaystyle=\sqrt{P_{d}}\mathbf{q}_{k}^{T}\hat{\boldsymbol{\Phi}}_{j}\mathbf{p}_{k}-\sqrt{P_{d}}\hat{\boldsymbol{\phi}}_{k}\mathbf{D}_{\mathbf{h}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{-1}\mathbf{D}_{\mathbf{q}_{1}}\mathbf{p}_{1}$
$\displaystyle=\sqrt{P_{d}}\mathbf{q}_{k}^{T}\hat{\boldsymbol{\Phi}}_{j}\mathbf{p}_{k}-\sqrt{P_{d}}\hat{\boldsymbol{\phi}}_{k}\mathbf{D}_{\mathbf{q}_{1}}\mathbf{p}_{1}$
$\displaystyle=\sqrt{P_{d}}\mathbf{q}_{k}^{T}\hat{\boldsymbol{\Phi}}_{j}\mathbf{p}_{k}-\sqrt{P_{d}}\mathbf{q}_{k}^{T}\hat{\boldsymbol{\Phi}}_{k}\mathbf{p}_{k}$
$\displaystyle=\sqrt{P_{d}}\mathbf{q}_{k}^{T}(\hat{\boldsymbol{\Phi}}_{j}-\hat{\boldsymbol{\Phi}}_{k})\mathbf{p}_{k}.$
(26)

On the other hand, $\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$ removes
$\mathbf{b}_{k}$ for $k\in\\{1,2\\}$. Consequently, we have

$\epsilon_{k}=\begin{cases}\sqrt{P_{d}}\mathbf{q}_{k}^{T}(\hat{\boldsymbol{\Phi}}_{j}-\hat{\boldsymbol{\Phi}}_{k})\mathbf{p}_{k}&\mathbf{B}_{1}=\mathbf{B}_{2},\\\
\sqrt{P_{d}}\mathbf{q}_{k}^{T}\hat{\boldsymbol{\Phi}}_{j}\mathbf{p}_{k}&\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}.\end{cases}$
(27)

We notice that $\epsilon_{k}$ corresponds to only the unintended reflection
path when $\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$. On the other
hand, $\mathbf{B}_{1}=\mathbf{B}_{2}$ yields an expression depending on the
difference between the two RISs’ configurations during data transmission.
Since each RIS is configured based on the (estimated) channels of their
respective users, it is highly unlikely that the configurations will be close.
Moreover, it has to be noted that the RIS configuration of the non-serving RIS
is different between the two cases since the channel estimates are also
different.

##  V Channel Estimation based on Correlated Rayleigh Fading Priors

We now switch focus to consider fading channels that can be modeled using the
Bayesian framework. In this section, we consider channel estimation and assume
that all the UE-RIS channels exhibit spatially correlated Rayleigh fading:
$\mathbf{g}_{k}\sim\mathcal{CN}(\boldsymbol{0},\boldsymbol{\Sigma}_{\mathbf{g}_{k}})$
and
$\mathbf{p}_{k}\sim\mathcal{CN}(\boldsymbol{0},\boldsymbol{\Sigma}_{\mathbf{p}_{k}})$.
The covariance matrices
$\boldsymbol{\Sigma}_{\mathbf{g}_{k}},\boldsymbol{\Sigma}_{\mathbf{p}_{k}}\in\mathbb{C}^{N\times
N}$ are generic positive semi-definite matrices. In addition, the BSs know
$\mathbf{h}_{k}$ perfectly while they consider $\mathbf{q}_{k}$ as
deterministic and unknown channels for $k=1,2$. The pilot transmission model
assumed by the BSs for $k=1,2$ can be expressed as

$\displaystyle\hat{\mathbf{y}}_{p1}=\sqrt{P_{p}}\mathbf{B}_{1}\mathbf{D}_{\mathbf{h}_{1}}\mathbf{g}_{1}+\mathbf{w}_{p1}\in\mathbb{C}^{L},$
(28a)
$\displaystyle\hat{\mathbf{y}}_{p2}=\sqrt{P_{p}}\mathbf{B}_{2}\mathbf{D}_{\mathbf{h}_{2}}\mathbf{g}_{2}+\mathbf{w}_{p2}\in\mathbb{C}^{L}.$
(28b)

Based on (28), the BSs can estimate $\mathbf{g}_{k}$ via a misspecified MMSE
estimator, which can be expressed as

$\displaystyle\hat{\mathbf{g}}_{1}$
$\displaystyle=\frac{1}{\sqrt{P_{p}}}\boldsymbol{\Sigma}_{\mathbf{g}_{1}}\mathbf{D}_{\mathbf{h}_{1}}^{H}\mathbf{B}_{1}^{H}$
$\displaystyle\times\left(\mathbf{B}_{1}\mathbf{D}_{\mathbf{h}_{1}}\boldsymbol{\Sigma}_{\mathbf{g}_{1}}\mathbf{D}_{\mathbf{h}_{1}}^{H}\mathbf{B}_{1}^{H}+\frac{\sigma_{w}^{2}}{P_{p}}\mathbf{I}_{L}\right)^{-1}\mathbf{y}_{p1},$
(29a) $\displaystyle\hat{\mathbf{g}}_{2}$
$\displaystyle=\frac{1}{\sqrt{P_{p}}}\boldsymbol{\Sigma}_{\mathbf{g}_{2}}\mathbf{D}_{\mathbf{h}_{2}}^{H}\mathbf{B}_{2}^{H}$
$\displaystyle\times\left(\mathbf{B}_{2}\mathbf{D}_{\mathbf{h}_{2}}\boldsymbol{\Sigma}_{\mathbf{g}_{2}}\mathbf{D}_{\mathbf{h}_{2}}^{H}\mathbf{B}_{2}^{H}+\frac{\sigma_{w}^{2}}{P_{p}}\mathbf{I}_{L}\right)^{-1}\mathbf{y}_{p2}.$
(29b)

The diagonal entries of the error covariance matrix represent the MSEs of the
corresponding channel entry. We first define
$\mathbf{r}_{k}\triangleq\mathbf{D}_{\mathbf{q}_{k}}\mathbf{p}_{k}$, which
results in
$\mathbf{r}_{k}\sim\mathcal{CN}(\boldsymbol{0},\boldsymbol{\Sigma}_{\mathbf{r}_{k}})$
where
$\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\triangleq\mathbf{D}_{\mathbf{q}_{k}}\boldsymbol{\Sigma}_{\mathbf{p}_{k}}\mathbf{D}_{\mathbf{q}_{k}}^{H}$.
To simplify the representation of the channel estimation error covariance
matrix, we introduce the following notation:

$\displaystyle\mathbf{C}_{\mathbf{g}\hat{\mathbf{y}}}\triangleq\mathbb{E}[\mathbf{g}_{k}\hat{\mathbf{y}}_{pk}^{H}]=\mathbb{E}[\mathbf{g}_{k}\mathbf{y}_{pk}^{H}]=\sqrt{P_{p}}\boldsymbol{\Sigma}_{\mathbf{g}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{H}\mathbf{B}_{k}^{H},$
(30a)
$\displaystyle\mathbf{C}_{\hat{\mathbf{y}}\hat{\mathbf{y}}}\triangleq\mathbb{E}[\hat{\mathbf{y}}_{pk}\hat{\mathbf{y}}_{pk}^{H}]=P_{p}\mathbf{B}_{k}\mathbf{D}_{\mathbf{h}_{k}}\boldsymbol{\Sigma}_{\mathbf{g}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{H}\mathbf{B}_{k}^{H}+\sigma_{w}^{2}\mathbf{I}_{L},$
(30b)
$\displaystyle\mathbf{C}_{\mathbf{y}\mathbf{y}}\triangleq\mathbb{E}[\mathbf{y}_{pk}\mathbf{y}_{pk}^{H}]=\mathbf{C}_{\hat{\mathbf{y}}\hat{\mathbf{y}}}+P_{p}\mathbf{B}_{j}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{B}_{j}^{H}.$
(30c)

With this notation, the error covariance matrix can be represented as

$\displaystyle\mathbb{E}[(\mathbf{g}_{k}-\hat{\mathbf{g}}_{k})(\mathbf{g}_{k}-\hat{\mathbf{g}}_{k})^{H}]=$
$\displaystyle=\boldsymbol{\Sigma}_{\mathbf{g}_{k}}+\mathbf{C}_{\mathbf{g}\hat{\mathbf{y}}}\mathbf{C}_{\hat{\mathbf{y}}\hat{\mathbf{y}}}^{-1}\mathbf{C}_{\mathbf{y}\mathbf{y}}\mathbf{C}_{\hat{\mathbf{y}}\hat{\mathbf{y}}}^{-1}\mathbf{C}_{\mathbf{g}\hat{\mathbf{y}}}^{H}-2\mathbf{C}_{\mathbf{g}\hat{\mathbf{y}}}\mathbf{C}_{\hat{\mathbf{y}}\hat{\mathbf{y}}}^{-1}\mathbf{C}_{\mathbf{g}\hat{\mathbf{y}}}^{H}$
$\displaystyle=\boldsymbol{\Sigma}_{\mathbf{g}_{k}}+\mathbf{C}_{\mathbf{g}\hat{\mathbf{y}}}\mathbf{C}_{\hat{\mathbf{y}}\hat{\mathbf{y}}}^{-1}(\mathbf{C}_{\hat{\mathbf{y}}\hat{\mathbf{y}}}+P_{p}\mathbf{B}_{j}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{B}_{j}^{H})\mathbf{C}_{\hat{\mathbf{y}}\hat{\mathbf{y}}}^{-1}\mathbf{C}_{\mathbf{g}\hat{\mathbf{y}}}^{H}$
$\displaystyle-2\mathbf{C}_{\mathbf{g}\hat{\mathbf{y}}}\mathbf{C}_{\hat{\mathbf{y}}\hat{\mathbf{y}}}^{-1}\mathbf{C}_{\mathbf{g}\hat{\mathbf{y}}}^{H}$
$\displaystyle=\underbrace{\boldsymbol{\Sigma}_{\mathbf{g}_{k}}-\mathbf{C}_{\mathbf{g}\hat{\mathbf{y}}}\mathbf{C}_{\hat{\mathbf{y}}\hat{\mathbf{y}}}^{-1}\mathbf{C}_{\mathbf{g}\hat{\mathbf{y}}}^{H}}_{\text{error
covariance for
}\mathbf{y}=\hat{\mathbf{y}}}+\underbrace{P_{p}\mathbf{C}_{\mathbf{g}\hat{\mathbf{y}}}\mathbf{C}_{\hat{\mathbf{y}}\hat{\mathbf{y}}}^{-1}\mathbf{B}_{j}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{B}_{j}^{H}\mathbf{C}_{\hat{\mathbf{y}}\hat{\mathbf{y}}}^{-1}\mathbf{C}_{\mathbf{g}\hat{\mathbf{y}}}^{H}}_{\text{term
coming from pilot contamination}}.$ (31)

Note that the additive term in (31) coming from pilot contamination depends on
$\boldsymbol{\Sigma}_{\mathbf{r}_{k}}$. If we consider the case where the
channel $\mathbf{r}_{k}$ does not exist, that is,
$\boldsymbol{\Sigma}_{\mathbf{r}_{k}}=\boldsymbol{0}$, then we would obtain
the error covariance in the form of a typical MMSE estimation error covariance
matrix. It is also important to address the dependence of the term coming from
pilot contamination on $P_{p}$: while the terms
$\mathbf{C}_{\mathbf{g}\hat{\mathbf{y}}}$ scale with $\sqrt{P_{p}}$ each, the
terms $\mathbf{C}_{\hat{\mathbf{y}}\hat{\mathbf{y}}}^{-1}$ scale with
$1/P_{p}$ each, and along with the leading $P_{p}$ multiplier, we can see that
$P_{p}$-dependent terms cancel each other out, hence leaving a non-vanishing
pilot contamination term.

###  V-A High-SNR Channel Estimation

Now, we investigate the behavior of the error covariance matrix in (31) when
$\sigma_{w}^{2}$ is low or $P_{p}$ is high. Note that the first term in (31)
is already the error covariance matrix for an MMSE estimator without
misspecification that estimates $\mathbf{g}_{k}$ based on the observation
$\hat{\mathbf{y}}_{pk}$. Therefore, we know that this term vanishes at high
SNR. Consequently, the asymptotic error covariance matrix is governed by the
high-SNR behavior of the term coming from pilot contamination, that is

$\displaystyle\lim_{\sigma_{w}^{2}\to
0}\mathbb{E}[(\mathbf{g}_{k}-\hat{\mathbf{g}}_{k})(\mathbf{g}_{k}-\hat{\mathbf{g}}_{k})^{H}]$
$\displaystyle=\lim_{\sigma_{w}^{2}\to
0}P_{p}\mathbf{C}_{\mathbf{g}\hat{\mathbf{y}}}\mathbf{C}_{\hat{\mathbf{y}}\hat{\mathbf{y}}}^{-1}\mathbf{B}_{j}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{B}_{j}^{H}\mathbf{C}_{\hat{\mathbf{y}}\hat{\mathbf{y}}}^{-1}\mathbf{C}_{\mathbf{g}\hat{\mathbf{y}}}^{H}$
$\displaystyle=P_{p}^{2}\boldsymbol{\Sigma}_{\mathbf{g}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{H}\mathbf{B}_{k}^{H}\left(P_{p}\mathbf{B}_{k}\mathbf{D}_{\mathbf{h}_{k}}\boldsymbol{\Sigma}_{\mathbf{g}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{H}\mathbf{B}_{k}^{H}+\sigma_{w}^{2}\mathbf{I}_{L}\right)^{-1}$
$\displaystyle\times\mathbf{B}_{j}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{B}_{j}^{H}\left(P_{p}\mathbf{B}_{k}\mathbf{D}_{\mathbf{h}_{k}}\boldsymbol{\Sigma}_{\mathbf{g}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{H}\mathbf{B}_{k}^{H}+\sigma_{w}^{2}\mathbf{I}_{L}\right)^{-1}\mathbf{B}_{k}\mathbf{D}_{\mathbf{h}_{k}}\boldsymbol{\Sigma}_{\mathbf{g}_{k}}$
$\displaystyle=\lim_{\sigma_{w}^{2}\to
0}\boldsymbol{\Sigma}_{\mathbf{g}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{H}\mathbf{B}_{k}^{H}\left(\mathbf{B}_{k}\mathbf{D}_{\mathbf{h}_{k}}\boldsymbol{\Sigma}_{\mathbf{g}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{H}\mathbf{B}_{k}^{H}+\frac{\sigma_{w}^{2}}{P_{p}}\mathbf{I}_{L}\right)^{-1}$
$\displaystyle\times\mathbf{B}_{j}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{B}_{j}^{H}\left(\mathbf{B}_{k}\mathbf{D}_{\mathbf{h}_{k}}\boldsymbol{\Sigma}_{\mathbf{g}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{H}\mathbf{B}_{k}^{H}+\frac{\sigma_{w}^{2}}{P_{p}}\mathbf{I}_{L}\right)^{-1}\mathbf{B}_{k}\mathbf{D}_{\mathbf{h}_{k}}\boldsymbol{\Sigma}_{\mathbf{g}_{k}}.$
(32)

Note that since $\mathbf{B}_{k}^{H}\mathbf{B}_{k}=L\mathbf{I}_{N}$, the
pseudoinverse corresponds to
$\mathbf{B}_{k}^{\dagger}=\frac{1}{L}\mathbf{B}_{k}^{H}$. Consequently, we
have

$\displaystyle\lim_{\sigma_{w}^{2}\to
0}\mathbb{E}[(\mathbf{g}_{k}-\hat{\mathbf{g}}_{k})(\mathbf{g}_{k}-\hat{\mathbf{g}}_{k})^{H}]$
$\displaystyle=\frac{1}{L^{4}}\boldsymbol{\Sigma}_{\mathbf{g}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{H}\mathbf{B}_{k}^{H}\mathbf{B}_{k}\mathbf{D}_{\mathbf{h}_{k}}^{-H}\boldsymbol{\Sigma}_{\mathbf{g}_{k}}^{-1}\mathbf{D}_{\mathbf{h}_{k}}^{-1}\mathbf{B}_{k}^{H}$
$\displaystyle\quad\times\mathbf{B}_{j}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{B}_{j}^{H}\mathbf{B}_{k}\mathbf{D}_{\mathbf{h}_{k}}^{-H}\boldsymbol{\Sigma}_{\mathbf{g}_{k}}^{-1}\mathbf{D}_{\mathbf{h}_{k}}^{-1}\mathbf{B}_{k}^{H}\mathbf{B}_{k}\mathbf{D}_{\mathbf{h}_{k}}\boldsymbol{\Sigma}_{\mathbf{g}_{k}}$
$\displaystyle=\frac{1}{L^{2}}\mathbf{D}_{\mathbf{h}_{k}}^{-1}\mathbf{B}_{k}^{H}\mathbf{B}_{j}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{B}_{j}^{H}\mathbf{B}_{k}\mathbf{D}_{\mathbf{h}_{k}}^{-H}.$
(33)

From (33), one can observe that the asymptotic behavior of the pilot
contamination depends on the choice of $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$.
To analyze this, we consider two cases.

####  V-A1 Case 1: $\mathbf{B}_{1}=\mathbf{B}_{2}$

In this case, we have that $\mathbf{B}_{k}^{H}\mathbf{B}_{j}=L\mathbf{I}_{N}$
for $j,k=1,2$ and $j\neq k$. Consequently, (33) becomes

$\displaystyle\frac{1}{L^{2}}\mathbf{D}_{\mathbf{h}_{k}}^{-1}\mathbf{B}_{k}^{H}\mathbf{B}_{j}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{B}_{j}^{H}\mathbf{B}_{k}\mathbf{D}_{\mathbf{h}_{k}}^{-H}=\mathbf{D}_{\mathbf{h}_{k}}^{-1}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{-H}.$
(34)

Note that this result does not depend on $L$, which implies that this channel
estimation error caused by pilot contamination cannot be eliminated by
increasing the number of pilots when $\mathbf{B}_{1}=\mathbf{B}_{2}$.

####  V-A2 The RISs are configured such that
$\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$

In this case, $\mathbf{B}_{k}^{H}\mathbf{B}_{j}=\boldsymbol{0}$ for $k\neq j$
and $k,j\in\\{1,2\\}$. This implies that (33) is zero; thus, BS $k$ can
estimate $\mathbf{g}_{k}$ even without being aware of $\mathbf{r}_{k}$. Also
note that (33) implies that any choice of $\mathbf{B}_{1}$ and
$\mathbf{B}_{2}$ that does not satisfy
$\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$ will result in pilot
contamination while estimating correlated Rayleigh fading channels. We can
summarize the high-SNR behavior of the channel estimation error covariance
matrix as

$\displaystyle\lim_{\sigma_{w}^{2}\to
0}\mathbb{E}[(\mathbf{g}_{k}-\hat{\mathbf{g}}_{k})$
$\displaystyle(\mathbf{g}_{k}-\hat{\mathbf{g}}_{k})^{H}]$
$\displaystyle=\begin{cases}\mathbf{D}_{\mathbf{h}_{k}}^{-1}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{-H}&\mathbf{B}_{1}=\mathbf{B}_{2},\\\
\boldsymbol{0}&\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}.\\\
\end{cases}$ (35)

This result shows that in order to estimate $\mathbf{g}_{k}$ reliably, it is
necessary to configure the RISs such that
$\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$.

##  VI Capacity Lower Bound for Reliable Communication under Imperfect CSI

In this section, we compute a lower bound on the ergodic capacity based on the
imperfect CSI obtained in the previous section via channel estimation. In
particular, we consider the impact of pilot contamination and the effect of
the signal model misspecification on the channel capacity. We derive the
channel capacity lower bound for the two cases considered in Section V.

###  VI-A Capacity Lower Bound of a SISO Channel with Channel Side Information

Consider a generic SISO system with the following received signal model:

$y=hx+w$ (36)

with $w\sim\mathcal{CN}(0,\sigma_{w}^{2})$ and $x\sim\mathcal{CN}(0,1)$.
Suppose that the receiver has partial information on $h$, denoted by $\Omega$.
Then the capacity lower bound is [25, Eq. 2.46]

$C\geq\mathbb{E}_{\Omega}\left[\log_{2}\left(1+\frac{|\mathbb{E}[h|\Omega]|^{2}}{\text{Var}(h|\Omega)+\text{Var}(w|\Omega)}\right)\right].$
(37)

This bound is valid under certain conditions, which can be listed as follows
[25, Section 2.3.5]:

  * •

The noise $w$ has zero-mean conditioned on $\Omega$, that is,
$\mathbb{E}[w|\Omega]=0$.

  * •

The transmitted signal $x$ and the noise $w$ are uncorrelated conditioned on
$\Omega$, that is,
$\mathbb{E}[xw^{*}|\Omega]=\mathbb{E}[x|\Omega]\mathbb{E}[w^{*}|\Omega]$.

  * •

The received signal $hx$ and the noise $w$ are uncorrelated conditioned on
$\Omega$, that is,
$\mathbb{E}[hxw^{*}|\Omega]=\mathbb{E}[hx|\Omega]\mathbb{E}[w^{*}|\Omega]$.

In our setup, the data signal model for the two users can be expressed as

$\displaystyle
y_{1}=\sqrt{P_{d}}(\mathbf{h}_{1}^{T}\boldsymbol{\Phi}_{1}\mathbf{g}_{1}+\mathbf{q}_{1}^{T}\boldsymbol{\Phi}_{2}\mathbf{p}_{1})x_{1}+w_{1},$
(38a) $\displaystyle
y_{2}=\sqrt{P_{d}}(\mathbf{h}_{2}^{T}\boldsymbol{\Phi}_{2}\mathbf{g}_{2}+\mathbf{q}_{2}^{T}\boldsymbol{\Phi}_{1}\mathbf{p}_{2})x_{2}+w_{2},$
(38b)

where $x_{k}\sim\mathcal{CN}(0,1)$ denotes the transmitted data for $k=1,2$
and $P_{d}$ denotes the data transmission power. During the data transmission
phase, both RISs are configured to phase-align the cascaded channel, that is:

$\phi_{kn}=\arg(h_{kn})+\arg(\hat{g}_{kn}).$ (39)

For both BSs, we consider the side information $\Omega$ as the knowledge of
$\boldsymbol{\Phi}_{1}$, $\boldsymbol{\Phi}_{2}$, $\mathbf{h}_{1}$,
$\mathbf{h}_{2}$, $\hat{\mathbf{g}}_{1}$, and $\hat{\mathbf{g}}_{2}$.
Consequently, the outer expectation in (37) refers to the expectation with
respect to the marginal distributions of $\hat{\mathbf{g}}_{1}$ and
$\hat{\mathbf{g}}_{2}$.

######  Lemma 1.

The system setup described by the signal model in (38) that uses the RIS
configurations described in (39) satisfies the three regularity conditions
required by the capacity bound in (37).

###### Proof.

First, let us identify the $h$ and $w$ that we had defined in our system setup
in (38):

$\displaystyle
h=\sqrt{P_{d}}(\mathbf{h}_{k}^{T}\boldsymbol{\Phi}_{k}\mathbf{g}_{k}+\mathbf{q}_{k}^{T}\boldsymbol{\Phi}_{j}\mathbf{p}_{k})$
(40a) $\displaystyle w=w_{k}$ (40b)

We can now prove that the conditional mean of the noise conditioned on the
channel side information, which we can describe as
$\Omega=\hat{\mathbf{g}}_{k}$, is zero. Note that $\hat{\mathbf{g}}_{k}$ and
$w_{k}$ are independent: for $\mathbf{B}_{1}=\mathbf{B}_{2}$, the less trivial
case, we have that
$\hat{\mathbf{g}_{k}}=\mathbf{g}_{k}+\mathbf{D}_{\mathbf{h}_{k}}^{-1}\mathbf{r}_{k}$,
that is, no dependence on the noise. At lower SNRs, the vanishing components
contain the realizations of the noise received during channel estimation, and
considering the fact that the receiver noise is white over time, that does not
affect the independence between $\hat{\mathbf{g}}_{k}$ and $w_{k}$ either.
Therefore, $\mathbb{E}[w_{k}|\hat{\mathbf{g}}_{k}]=\mathbb{E}[w_{k}]=0$.

It is also straightforward to prove that the transmitted signal is
uncorrelated with the noise conditioned on $\Omega$ due to the fact that
$x_{k}$ and $w_{k}$ are independent of $\Omega$ individually. So the
expression $\mathbb{E}[x_{k}w_{k}^{*}|\Omega]$ does not have anything that
depends on $\Omega$, i.e.,
$\mathbb{E}[x_{k}w_{k}^{*}|\Omega]=\mathbb{E}[x_{k}w_{k}^{*}]=\mathbb{E}[x_{k}]\mathbb{E}[w_{k}^{*}]=\mathbb{E}[x_{k}|\Omega]\mathbb{E}[w_{k}^{*}|\Omega]$
can be obtained, proving that $x$ and $w$ are uncorrelated with each other
conditioned on $\Omega$.

The last point is also quite straightforward since $\Omega$ does not contain
any relations between the overall channel, the receiver noise, and the
transmitted signal. Therefore it is easy to claim that this regularity
condition also holds, hence the capacity bound provided in (37) is applicable
to our system setup. ∎

###  VI-B Capacity Lower Bound with High-SNR Channel Estimates Available at
the BSs

If we assume that the channel estimation is performed at a high SNR, we can
model the channel estimation error according to (35). Consequently, we can
express the channel estimates in terms of the true channels and the channel
estimation error as

$\displaystyle\hat{\mathbf{g}}_{k}=\mathbf{g}_{k}+\mathbf{e}_{k},$ (41)

where the channel estimation error is

$\displaystyle\mathbf{e}_{k}=\begin{cases}\boldsymbol{0}&\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0},\\\
\mathbf{D}_{\mathbf{h}_{k}}^{-1}\mathbf{r}_{k}&\mathbf{B}_{1}=\mathbf{B}_{2}.\end{cases}$
(42)

Hence, BS $k$ knows $\mathbf{g}_{k}$ perfectly if the RISs are configured such
that $\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$ during channel
estimation, and
$\mathbf{e}_{k}=\mathbf{D}_{\mathbf{h}_{k}}^{-1}\mathbf{r}_{k}$ when
$\mathbf{B}_{1}=\mathbf{B}_{2}$. Also note that even when we have
$\mathbf{B}_{1}=\mathbf{B}_{2}$, the channel and the channel estimation error
are independent. Consequently, we can rewrite the overall SISO channel as

$v_{k}\triangleq\sqrt{P_{d}}(\boldsymbol{\phi}_{k}^{T}\mathbf{D}_{\mathbf{h}_{k}}\mathbf{g}_{k}+\boldsymbol{\phi}_{j}^{T}\mathbf{r}_{k}).$
(43)

The mean of the overall SISO channel conditioned on the side information can
be expressed as

$\displaystyle\mathbb{E}[v_{k}|\Omega]$
$\displaystyle=\sqrt{P_{d}}(\boldsymbol{\phi}_{k}^{T}\mathbf{D}_{\mathbf{h}_{k}}\mathbb{E}[\mathbf{g}_{k}|\hat{\mathbf{g}}_{k}]+\boldsymbol{\phi}_{j}^{T}\mathbb{E}[\mathbf{r}_{k}|\hat{\mathbf{g}}_{k}])$
(44)

Here, we can utilize the channel estimate structure provided by (41) and (42).
Note that when we consider $\mathbb{E}[\mathbf{g}_{k}|\hat{\mathbf{g}}_{k}]$,
it can be thought as estimating $\mathbf{g}_{k}$ based on observing
$\hat{\mathbf{g}}_{k}$ since both $\mathbf{g}_{k}$ and $\mathbf{e}_{k}$ and
they are independent from each other. The same goes for computing
$\mathbb{E}[\mathbf{r}_{k}|\hat{\mathbf{g}}_{k}]$. Since the
$\mathbf{g}_{k}-\hat{\mathbf{g}}_{k}$ and
$\mathbf{r}_{k}-\hat{\mathbf{g}}_{k}$ are jointly Gaussian, the MMSE estimate,
also known as the conditional mean estimate coincides with the linear minimum
mean squared error (LMMSE) estimate, therefore, we can use the LMMSE
formulation here:

$\displaystyle\mathbb{E}[\mathbf{g}_{k}|\hat{\mathbf{g}}_{k}]=\mathbb{E}[\mathbf{g}_{k}\hat{\mathbf{g}}_{k}^{H}](\mathbb{E}[\hat{\mathbf{g}}_{k}\hat{\mathbf{g}}_{k}^{H}])^{-1}\hat{\mathbf{g}}_{k}$
(45a)
$\displaystyle\mathbb{E}[\mathbf{r}_{k}|\hat{\mathbf{g}}_{k}]=\mathbb{E}[\mathbf{r}_{k}\hat{\mathbf{g}}_{k}^{H}](\mathbb{E}[\hat{\mathbf{g}}_{k}\hat{\mathbf{g}}_{k}^{H}])^{-1}\hat{\mathbf{g}}_{k}$
(45b)

which can be expressed more explicitly as

$\displaystyle\mathbb{E}[\mathbf{g}_{k}|\hat{\mathbf{g}}_{k}]=\boldsymbol{\Sigma}_{\mathbf{g}_{k}}(\boldsymbol{\Sigma}_{\mathbf{g}_{k}}+\mathbf{D}_{\mathbf{h}_{k}}^{-1}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{-H})^{-1}\hat{\mathbf{g}}_{k},$
(46a) $\displaystyle\mathbb{E}[\mathbf{r}_{k}|\hat{\mathbf{g}}_{k}]=$
$\displaystyle\begin{cases}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{-H}(\boldsymbol{\Sigma}_{\mathbf{g}_{k}}+\mathbf{D}_{\mathbf{h}_{k}}^{-1}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{-H})^{-1}\hat{\mathbf{g}}_{k}&\mathbf{B}_{1}=\mathbf{B}_{2},\\\
\boldsymbol{0}&\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}.\end{cases}$
(46b)

On the other hand, the variance of $v_{k}$ conditioned on
$\hat{\mathbf{g}}_{k}$ can be expressed as

$\displaystyle\text{Var}(v_{k}|\hat{\mathbf{g}}_{k})=P_{d}\boldsymbol{\phi}_{k}^{T}\mathbf{D}_{\mathbf{h}_{k}}\text{Var}(\mathbf{g}_{k}|\hat{\mathbf{g}}_{k})\mathbf{D}_{\mathbf{h}_{k}}^{H}\boldsymbol{\phi}_{k}^{*}$
$\displaystyle+P_{d}\boldsymbol{\phi}_{j}^{T}\text{Var}(\mathbf{r}_{k}|\hat{\mathbf{g}}_{k})\boldsymbol{\phi}_{j}^{*}+2\text{Re}(\boldsymbol{\phi}_{k}^{T}\mathbf{D}_{\mathbf{h}_{k}}\mathbb{E}[\mathbf{g}_{k}\mathbf{r}_{k}^{H}|\hat{\mathbf{g}}_{k}]\boldsymbol{\phi}_{j}^{*})$
$\displaystyle-2\text{Re}(\boldsymbol{\phi}_{k}^{T}\mathbf{D}_{\mathbf{h}_{k}}\mathbb{E}[\mathbf{g}_{k}|\hat{\mathbf{g}}_{k}]\mathbb{E}[\mathbf{r}_{k}^{H}|\hat{\mathbf{g}}_{k}]\boldsymbol{\phi}_{j}^{*})$
(47)

where $\mathbb{E}[\mathbf{g}_{k}|\hat{\mathbf{g}}_{k}]$ and
$\mathbb{E}[\mathbf{r}_{k}|\hat{\mathbf{g}}_{k}]$ are provided by (46). In
addition, we can use LMMSE formulation results for
$\text{Var}(\mathbf{g}_{k}|\hat{\mathbf{g}}_{k})$ and
$\text{Var}(\mathbf{r}_{k}|\hat{\mathbf{g}}_{k})$ which correspond to the
error covariance matrices as a result of estimating $\mathbf{g}_{k}$ and
$\mathbf{r}_{k}$ with an LMMSE estimator based on the observation
$\hat{\mathbf{g}}_{k}$. Consequently, these two terms can be expressed as

$\displaystyle\text{Var}(\mathbf{g}_{k}|\hat{\mathbf{g}}_{k})=\mathbb{E}[\mathbf{g}_{k}\mathbf{g}_{k}^{H}]-\mathbb{E}[\mathbf{g}_{k}\hat{\mathbf{g}}_{k}^{H}](\mathbb{E}[\mathbf{g}_{k}\hat{\mathbf{g}}_{k}^{H}])^{-1}\mathbb{E}[\hat{\mathbf{g}}_{k}\mathbf{g}_{k}^{H}],$
(48a)
$\displaystyle\text{Var}(\mathbf{r}_{k}|\hat{\mathbf{g}}_{k})=\mathbb{E}[\mathbf{r}_{k}\mathbf{r}_{k}^{H}]-\mathbb{E}[\mathbf{r}_{k}\hat{\mathbf{g}}_{k}^{H}](\mathbb{E}[\mathbf{r}_{k}\hat{\mathbf{g}}_{k}^{H}])^{-1}\mathbb{E}[\hat{\mathbf{g}}_{k}\mathbf{r}_{k}^{H}].$
(48b)

Computing the expectations above and also the cross-term
$\mathbb{E}[\mathbf{g}_{k}\mathbf{r}_{k}^{H}|\hat{\mathbf{g}}_{k}]$, we can
obtain the implicit expressions in (47) as

$\displaystyle\text{Var}(\mathbf{g}_{k}|\hat{\mathbf{g}}_{k})=\boldsymbol{\Sigma}_{\mathbf{g}_{k}}-\boldsymbol{\Sigma}_{\mathbf{g}_{k}}(\boldsymbol{\Sigma}_{\mathbf{g}_{k}}+\mathbf{D}_{\mathbf{h}_{k}}^{-1}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{-H})^{-1}\boldsymbol{\Sigma}_{\mathbf{g}_{k}}$
(49a) $\displaystyle\text{Var}(\mathbf{r}_{k}|\hat{\mathbf{g}}_{k})$
$\displaystyle=\boldsymbol{\Sigma}_{\mathbf{r}_{k}}-\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{-H}(\boldsymbol{\Sigma}_{\mathbf{g}_{k}}+\mathbf{D}_{\mathbf{h}_{k}}^{-1}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{-H})^{-1}\mathbf{D}_{\mathbf{h}_{k}}^{-1}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}$
(49b)
$\displaystyle\mathbb{E}[\mathbf{g}_{k}\mathbf{r}_{k}^{H}|\hat{\mathbf{g}}_{k}]$
$\displaystyle=\hat{\mathbf{g}}_{k}\hat{\mathbf{g}}_{k}^{H}(\boldsymbol{\Sigma}_{\mathbf{g}_{k}}+\mathbf{D}_{\mathbf{h}_{k}}^{-1}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{-H})^{-1}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}\mathbf{D}_{\mathbf{h}_{k}}^{-H}-\mathbf{D}_{\mathbf{h}_{k}}^{-1}\boldsymbol{\Sigma}_{\mathbf{r}_{k}}$
(49c)

for $\mathbf{B}_{1}=\mathbf{B}_{2}$ and

$\displaystyle\text{Var}(\mathbf{g}_{k}|\hat{\mathbf{g}}_{k})=\boldsymbol{0}$
(50a)
$\displaystyle\text{Var}(\mathbf{r}_{k}|\hat{\mathbf{g}}_{k})=\boldsymbol{\Sigma}_{\mathbf{r}_{k}}$
(50b)
$\displaystyle\mathbb{E}[\mathbf{g}_{k}\mathbf{r}_{k}^{H}|\hat{\mathbf{g}}_{k}]=\boldsymbol{0}$
(50c)

for $\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$. Obtaining
$\text{Var}(w_{k}|\Omega)=\sigma_{w}^{2}$ is straightforward since $w_{k}$ is
independent from $\Omega$. Also note that $\hat{\mathbf{g}}_{k}$ depends on
the choice of $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$. As a result, the capacity
lower bound can be expressed as

$\displaystyle
C_{k}\geq\mathbb{E}_{\Omega}\left[\log_{2}\left(1+\frac{|\mathbb{E}[v_{k}|\Omega]|^{2}}{\text{Var}(v_{k}|\Omega)+\sigma_{w}^{2}}\right)\right]$
(51)

with $\mathbb{E}[v_{k}|\Omega]$ and $\text{Var}(v_{k}|\Omega)$ taking values
according to the choice of $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$.

##  VII Numerical Results

In this section, we provide numerical examples to demonstrate the implications
of the analytical results obtained in Sections III-VI. First, we provide the
numerical results for the channel and data estimation MSEs when deterministic
channels are considered. For correlated Rayleigh fading, we demonstrate the
impact of pilot contamination on channel estimation MSE and the resulting
capacity lower bound.

###  VII-A Estimation Performance With Deterministic Channels

For deterministic channel estimation, we consider the normalized mean squared
error (NMSE) as our performance metric. Moreover, we consider the results for
a single UE, since the results for different UEs only differ by the channel
realizations. For the deterministic channel $\mathbf{g}_{k}$, we obtain the
NMSE as

$\text{NMSE}=\frac{\text{MSE}}{\|\mathbf{g}_{k}\|^{2}}.$ (52)

Table I: Parameters used in Figures 2 and 3. Parameter | Value
---|---
$P_{p}$ or $P_{d}$ |  $-30,-25,\dots,40$ dBm333The results for $P_{p}=45,50,55,\text{ and }60$ dBm are also demonstrated in Fig. 2 to display the high SNR floor more clearly.
UE-RIS path loss |  $-80$ dB
RIS-BS path loss |  $-60$ dB
$\sigma_{w}^{2}$ |  $-90$ dBm
$N$ | $256$
$L$ | $513$

In Fig. 2, we plot (13) for different values of $P_{p}$, and we also provide
the high-SNR floor for the case where $\mathbf{B}_{1}=\mathbf{B}_{2}$. The set
of parameters used to generate Fig. 2 and 3 are provided in Table I. On the
other hand, the range of transmission power in Fig. 6 is between $-10$ and
$60$ dBm. In addition, for Fig. 4, 4 and 4, we consider a $8\times 8$ uniform
rectangular array (URA) geometry with $\lambda/2$ spacing in both vertical and
horizontal axes. Therefore, the parameter values $N=64$ and $L=128$ apply to
those figures. Note that at lower transmission powers, the covariance matrix
of the estimator acts dominantly, hence, both RIS configurations perform
nearly the same. However, after $P_{p}=20$ dBm, the power of the estimator
bias starts to dominate, and the average MSE for
$\mathbf{B}_{1}=\mathbf{B}_{2}$ converges to the floor denoted by the black
dashed line, which is given by (14). On the other hand, the average MSE for
$\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$ does not stop there but
keeps decreasing towards zero. As mentioned before, the MML estimators used by
the BSs coincide with the true ML estimators when the RISs are configured such
that $\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$.

Figure 2: Pilot transmission power versus the channel estimation NMSE for
deterministic channels. Since the prior distribution of the parameter vector
is not considered in the non-random parameter estimation framework, it is
highly likely to obtain NMSEs greater than $1$.

###  VII-B Data Estimation with Deterministic Channels

In Fig. 3, the data estimation MSE performance with the two RIS pilot
configurations is analyzed when the channel estimation SNR is high as in
Section IV-A. That is, (24) is plotted for $\mathbf{B}_{1}=\mathbf{B}_{2}$ and
$\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$. In addition, the case where
all of the channels are perfectly known is plotted to serve as the golden
standard, labeled as _Perfect CSI_. However, even when all the channels are
perfectly known, each RIS is assumed to be optimized independently according
to the subscribed UE’s CSI. Note that although the channel estimation SNR is
high, $\mathbf{B}_{1}=\mathbf{B}_{2}$ yields biased estimates of
$\mathbf{g}_{1}$ due to pilot contamination caused by self-interference. On
the other hand, $\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$ yields the
true $\mathbf{g}_{1}$ as the estimate, however, since BS $1$ is unaware of the
path through the second RIS, the data estimate is biased, hence, there is
still a high data transmission SNR floor. At around $P_{d}=5$ dBm,
$\mathbf{B}_{1}=\mathbf{B}_{2}$ starts to approach the high-SNR floor. On the
other hand, $\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$ does not suffer
from the lack of awareness of the second RIS path until around $P_{d}=20$ dBm.
Hence, Fig. 3 clearly shows the benefit of configuring the RIS pilot
configurations sequences orthogonally.

Figure 3: Data transmission power versus the data estimation NMSE for
deterministic channels with high channel estimation SNR.

###  VII-C Channel Estimation Based on Correlated Rayleigh Fading Priors

In Fig. 4, the channel estimation performance with the two RIS pilot
configurations are analyzed for correlated Rayleigh fading channels. Fig. 4 is
generated by computing the trace of the channel estimation error covariance
matrix provided in (31) and normalizing it by the factor of
$\text{tr}(\boldsymbol{\Sigma}_{\mathbf{g}_{k}})$ for different $P_{p}$ values
and different spatial channel correlation matrices. The spatial channel
correlation matrices are computed for isotropic scattering based on the
different RIS element geometries according to [26, Prop. 1].

Figure 4: Pilot transmission power versus channel estimation MSE for different
RIS geometries.

Note that configuring the RISs such that $\mathbf{B}_{1}=\mathbf{B}_{2}$
causes severe problems in channel estimation, that is, the MSE increases as
pilot transmission power increases for all geometries while
$\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$ completely eliminates pilot
contamination. In addition, one can note that as the spatial correlation
increases, channel estimation performance also increases since different
channel parameters contain more information from one another.

Furthermore, in Fig. 5, we demonstrate the two different components of the
channel estimation error in the presence of inter-operator pilot
contamination. While the red curve corresponds to the total NMSE and the blue
curve corresponds to the NMSE in the absence of pilot contamination as usual,
the green curve demonstrates the term coming from pilot contamination, as in
(31). On the other hand, the black dashed line represents the asymptote of the
channel estimation NMSE in the presence of pilot contamination, which is
provided by (35). Note that as the transmission power increases, the NMSE
coming from pilot contamination also increases up to a certain point, and
converges to the trace of (35) for $\mathbf{B}_{1}=\mathbf{B}_{2}$ due to the
fact that at very high transmission powers, the increase in pilot
contamination cancels out with the increasing ability to estimating the
channel.

Figure 5: Different components of the channel estimation NMSE

###  VII-D Capacity Lower Bound for Reliable Communication Under Imperfect CSI

In Fig. 6, the capacity lower bound derived in Section VI is plotted against
the data transmission power. This is performed by generating several channel
realizations and computing (37). Note that when
$\mathbf{B}_{1}=\mathbf{B}_{2}$, the capacity lower bound stops increasing
after $P_{d}=30$ dBm while this happens at around $P_{d}=40$ dBm for
$\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$ when the effect of the
misspecified channel during data transmission starts to appear. In any case,
it is clear that configuring the RISs such that
$\mathbf{B}_{1}^{H}\mathbf{B}_{2}=\boldsymbol{0}$ almost doubles the capacity
lower bound.

Figure 6: Data transmission power versus capacity lower bound for the two
users.

##  VIII Conclusions

In this paper, we have studied the impact of pilot contamination in a system
consisting of two wide-band RISs, two single-antenna UEs, and two co-located
single-antenna BSs. We have demonstrated that the presence of multiple RISs in
the same area causes pilot contamination, although the UEs are subscribed to
different operators and transmit over disjoint narrow frequency bands. To
combat this new type of pilot contamination, we have proposed the use of
orthogonal RIS configurations during pilot transmission. For two different
sets of assumptions, that is, deterministic and correlated Rayleigh-fading
channel models, we have derived the channel and data estimation MSEs and the
capacity lower bound in closed-form. In the numerical results, we have clearly
shown that the proposed approach eliminates pilot contamination completely,
and decreases data estimation MSE significantly for deterministic channels. On
the other hand, we have also shown that the capacity lower bound almost
doubles when the RISs are configured orthogonally during the pilot
transmission step. While one might argue that this doubling comes at the
expense of doubling the number of pilots, the estimates can be used for many
data transmissions if the channel is static enough, resulting in a higher
overall data rate. While this study covers the channel estimation performance
in multi-operator RIS-based pilot contamination scenarios for both
deterministic and stochastic channels, further analysis is needed for
parametric channel models, which opens a new set of possibilities.

## References

  * [1] D. Gürgünoğlu, E. Björnson, and G. Fodor, “Impact of pilot contamination between operators with interfering reconfigurable intelligent surfaces,” 2023.
  * [2] T. L. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” _IEEE Trans. Wireless Commun._ , vol. 9, no. 11, pp. 3590–3600, 2010.
  * [3] L. Sanguinetti, E. Björnson, and J. Hoydis, “Toward Massive MIMO 2.0: Understanding spatial correlation, interference suppression, and pilot contamination,” _IEEE Trans. Commun._ , vol. 68, no. 1, 2020.
  * [4] J. Jose, A. Ashikhmin, T. L. Marzetta, and S. Vishwanath, “Pilot contamination and precoding in multi-cell TDD systems,” _IEEE Trans. Wireless Commun._ , vol. 10, no. 8, pp. 2640–2651, 2011.
  * [5] V. Saxena, G. Fodor, and E. Karipidis, “Mitigating pilot contamination by pilot reuse and power control schemes for massive MIMO systems,” in _2015 IEEE 81st Vehicular Technology Conference (VTC Spring)_ , 2015, pp. 1–6.
  * [6] G. Fodor, N. Rajatheva, W. Zirwas, L. Thiele, M. Kurras, K. Guo, A. Tolli, J. H. Sorensen, and E. De Carvalho, “An overview of massive MIMO technology components in METIS,” _IEEE Communications Magazine_ , vol. 55, no. 6, pp. 155–161, 2017.
  * [7] C. Pan, H. Ren, K. Wang, J. F. Kolb, M. Elkashlan, M. Chen, M. Di Renzo, Y. Hao, J. Wang, A. L. Swindlehurst, X. You, and L. Hanzo, “Reconfigurable intelligent surfaces for 6G systems: Principles, applications, and research directions,” _IEEE Communications Magazine_ , vol. 59, no. 6, pp. 14–20, 2021.
  * [8] E. Björnson, Ö. Özdogan, and E. G. Larsson, “Reconfigurable intelligent surfaces: Three myths and two critical questions,” _IEEE Commun. Mag._ , no. 12, pp. 90–96, 2020.
  * [9] G. T. de Araújo, A. L. F. de Almeida, R. Boyer, and G. Fodor, “Semi-blind joint channel and symbol estimation for IRS-assisted MIMO systems,” _IEEE Transactions on Signal Processing_ , vol. 71, pp. 1184–1199, 2023.
  * [10] R. Liu, M. Li, H. Luo, Q. Liu, and A. L. Swindlehurst, “Integrated sensing and communication with reconfigurable intelligent surfaces: Opportunities, applications, and future directions,” _IEEE Wireless Communications_ , vol. 30, no. 1, pp. 50–57, 2023.
  * [11] L. Wei, C. Huang, G. C. Alexandropoulos, C. Yuen, Z. Zhang, and M. Debbah, “Channel estimation for RIS-empowered multi-user MISO wireless communications,” _IEEE Transactions on Communications_ , vol. 69, no. 6, pp. 4144–4157, 2021.
  * [12] E. Björnson and P. Ramezani, “Maximum likelihood channel estimation for RIS-aided communications with LOS channels,” in _Asilomar Conference on Signals, Systems and Computers_ , 2022.
  * [13] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah, and C. Yuen, “Reconfigurable intelligent surfaces for energy efficiency in wireless communication,” _IEEE Trans. Commun._ , vol. 18, no. 8, pp. 4157–4170, 2019.
  * [14] Z. Zhang, L. Dai, X. Chen, C. Liu, F. Yang, R. Schober, and H. V. Poor, “Active RIS vs. passive RIS: Which will prevail in 6g?” _IEEE Transactions on Communications_ , vol. 71, no. 3, pp. 1707–1725, 2023.
  * [15] M. Rihan, A. Zappone, S. Buzzi, G. Fodor, and M. Debbah, “Passive vs. active reconfigurable intelligent surfaces for integrated sensing and communication: Challenges and opportunities,” _IEEE Network_ , pp. 1–1, 2023.
  * [16] N. Garg, H. Ge, and T. Ratnarajah, “Generalized superimposed training scheme in IRS-assisted cell-free massive mimo systems,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 16, no. 5, pp. 1157–1171, 2022.
  * [17] 3GPP, “NR; Physical channels and modulation,” 3rd Generation Partnership Project (3GPP), Technical Specification (TS) 38.211, 09 2022, version 17.4.0.
  * [18] S. Zhang and R. Zhang, “Capacity characterization for intelligent reflecting surface aided MIMO communication,” _IEEE Journal on Selected Areas in Communications_ , vol. 38, no. 8, pp. 1823–1838, 2020.
  * [19] X. Pei, H. Yin, L. Tan, L. Cao, Z. Li, K. Wang, K. Zhang, and E. Björnson, “RIS-aided wireless communications: Prototyping, adaptive beamforming, and indoor/outdoor field trials,” pp. 8627–8640, 2021.
  * [20] V. Tapio, A. Shojaeifard, I. Hemadeh, A. Mourad, and M. Juntti, “Reconfigurable intelligent surface for 5g NR uplink coverage enhancement,” in _2021 IEEE 94th Vehicular Technology Conference (VTC2021-Fall)_ , 2021, pp. 1–5.
  * [21] M. Vincenzi, A. Antonopoulos, E. Kartsakli, J. Vardakas, L. Alonso, and C. Verikoukis, “Cooperation incentives for multi-operator C-RAN energy efficient sharing,” in _2017 IEEE International Conference on Communications (ICC)_ , 2017, pp. 1–6.
  * [22] H.-T. Chien, Y.-D. Lin, H.-W. Chang, and C.-L. Lai, “Multi-operator fairness in transparent RAN sharing by soft-partition with blocking and dropping mechanism,” _IEEE Transactions on Vehicular Technology_ , vol. 67, no. 12, pp. 11 597–11 605, 2018.
  * [23] S. M. Kay, _Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory_. Prentice Hall, 1993\.
  * [24] H. V. Poor, _An Introduction to Signal Detection and Estimation (2nd Ed.)_. Berlin, Heidelberg: Springer-Verlag, 1994.
  * [25] T. L. Marzetta, E. G. Larsson, H. Yang, and H. Q. Ngo, _Fundamentals of Massive MIMO_. Cambridge University Press, 2016.
  * [26] E. Björnson and L. Sanguinetti, “Rayleigh fading modeling and channel hardening for reconfigurable intelligent surfaces,” _IEEE Wireless Communications Letters_ , vol. 10, no. 4, pp. 830–834, 2021.

  *[PDPR]: pilot-to-data power ratio
  *[BS]: base station
  *[RIS]: reconfigurable intelligent surface
  *[LOS]: line of sight
  *[SINR]: signal-to-interference-plus-noise ratio
  *[RISs]: reconfigurable intelligent surface
  *[UE]: user equipment
  *[SNR]: signal-to-noise ratio
  *[CSI]: channel state information
  *[ML]: maximum likelihood
  *[MSE]: mean squared error
  *[MMSE]: minimum mean squared error
  *[UEs]: user equipment
  *[BSs]: base station
  *[MML]: misspecified maximum likelihood
  *[MIMO]: multiple-input multiple-output
  *[CRLB]: Cramér-Rao Lower Bound
  *[SNRs]: signal-to-noise ratio
  *[SISO]: single input single output
  *[MSEs]: mean squared error
  *[LMMSE]: linear minimum mean squared error
  *[NMSE]: normalized mean squared error
  *[URA]: uniform rectangular array
  *[NMSEs]: normalized mean squared error
  *[ CSI]: channel state information