# Empirical Validation of the Impedance-Based RIS Channel Model in an Indoor
Scattering Environment

Placido Mursia1, Taghrid Mazloum2, Frédéric Munoz2, Vincenzo Sciancalepore1,
Gabriele Gradoni3,
Raffaele D’Errico2, Marco Di Renzo4, Xavier Costa-Pérez15, Antonio Clemente2,
Geoffroy Lerosey6  1NEC Laboratories Europe GmbH, Heidelberg, Germany, Email:
<EMAIL_ADDRESS>2CEA-LETI, Univ. Grenoble Alpes, Grenoble, France
3University of Surrey, Guildford, United Kingdom  4Université Paris-Saclay,
CNRS, Centrale Supélec, Gif-sur-Yvette, France  5i2CAT Foundation and ICREA,
Barcelona, Spain  6Greenerwave, Paris, France

###### Abstract

Ensuring the precision of channel modeling plays a pivotal role in the
development of wireless communication systems, and this requirement remains a
persistent challenge within the realm of networks supported by Reconfigurable
Intelligent Surfaces (RIS). Achieving a comprehensive and reliable
understanding of channel behavior in RIS-aided networks is an ongoing and
complex issue that demands further exploration. In this paper, we empirically
validate a recently-proposed impedance-based RIS channel model that accounts
for the mutual coupling at the antenna array and precisely models the presence
of scattering objects within the environment as a discrete array of loaded
dipoles. To this end, we exploit real-life channel measurements collected in
an office environment to demonstrate the validity of such a model and its
applicability in a practical scenario. Finally, we provide numerical results
demonstrating that designing the RIS configuration based upon such model leads
to superior performance as compared to reference schemes.

###### Index Terms:

RIS, mutual coupling, antennas, electromagnetics, propagation, measurements.

##  I Introduction

Reconfigurable intelligent surfaces (RISs) are widely considered as one of the
ongoing revolutions in the network design due to their property to
programmatically alter the propagation properties of the radio environment
while keeping the overall manufacturing cost affordable. As such, RISs are a
candidate technology for the next-generation wireless networks [1]. In this
regard, one of the main research topics has been on exploring and further
developing accurate channel modelling, which is essential to fully unlock the
potentials of this technology.

Recently, the authors in [2] have proposed a mutual coupling and unit cell-
aware electromagnetically-consistent channel model based on mutual impedances.
Therein, the unit cells of the RIS are modelled as arbitrarily-spaced wire
dipoles, whose loads can be tuned to alter the propagation conditions. Such a
model has been exploited in [3] to derive the optimal RIS configuration that
maximizes the received power in a single-user and single-antenna setup.
Moreover, the authors in [4] have generalized this optimization framework for
the case of multi-input multiple-output (MIMO) systems. Therein, the multipath
components originated by scattering objects in the environment are modelled as
an additive statistical component. On the other hand, the authors of [5]
exploit the discrete dipole approximation (DDA) [6] to model the scattering
objects present in the environment as loaded wire dipoles, thus effectively
capturing its interaction with the transmitted signals. Under this setting,
the authors of [7] developed a provably convergent and nearly-optimal RIS
optimization algorithm based on Gram-Schmidt’s orthogonalization method.

In this paper, we go one-step beyond and provide an empirical validation of
the model in [5] by showcasing how its parameters can be tuned to effectively
recreate a real-life environment. To this purpose, we exploit a set of channel
measurements [8] performed in an office environment wherein a reflective RIS
operating at $28$ GHz [9] aids in the communication between a transmitter [10]
and a single-antenna user equipment (UE), in the presence of several
scattering objects. Additionally, we demonstrate how the considered RIS
channel model effectively captures such physical propagation scenario with few
parameters. Finally, numerical results demonstrate that performing the RIS
optimization using such a tailored channel model can bring significant gains
in terms of received power at the UE.

Notation. We use bold font lower and upper case for vectors and matrices,
respectively. $(\cdot)^{\mathrm{H}}$ denotes the hermitian transpose operator,
while $j=\sqrt{-1}$ is the imaginary number.

##  II System model

We consider the scenario depicted in Fig. 1, wherein a transmitter (TX)
equipped with $M=400$ antenna elements is located in the origin of the
reference system [10], an RIS equipped with $N=1600$ elements is placed (i.e.,
its center-point) at coordinates $\begin{bmatrix}-2&-3&0\end{bmatrix}$ m [9],
a single-antenna UE is placed at coordinates
$\begin{bmatrix}1&-3.6&0\end{bmatrix}$ m, and a number of scattering objects
are present in the environment.Moreover, we employ the impedance-based RIS
channel model in [5], such that the scattering objects in the environment are
modelled as discrete arrays of half-wavelength loaded dipoles. Hence, the
$1\times M$ channel vector is given by

$\displaystyle\mathbf{h}^{\mathrm{H}}=Z_{\scriptscriptstyle\mathrm{RL}}\Big{[}\mathbf{z}_{\scriptscriptstyle\mathrm{ROT}}-\mathbf{z}_{\scriptscriptstyle\mathrm{ROS}}\Big{(}\mathbf{Z}_{\scriptscriptstyle\mathrm{SS}}+\mathbf{Z}_{\scriptscriptstyle\mathrm{SOS}}+\mathbf{Z}_{\scriptscriptstyle\mathrm{RIS}}\Big{)}^{-1}\mathbf{Z}_{\scriptscriptstyle\mathrm{SOT}}\Big{]}\mathbf{Z}_{\scriptscriptstyle\mathrm{TG}},$
(1)

where $Z_{\scriptscriptstyle\mathrm{RL}}\in\mbox{$\mathbb{C}$}$
($\mathbf{Z}_{\scriptscriptstyle\mathrm{TG}}\in\mbox{$\mathbb{C}$}^{M\times
M}$) accounts for the load and self-impedance at the UE (transmitter);
$\mathbf{z}_{\scriptscriptstyle\mathrm{ROT}}\in\mbox{$\mathbb{C}$}^{1\times
M}$ includes the mutual impedances between the transmitter and the UE, both
directly and indirectly through the scattering objects;
$\mathbf{z}_{\scriptscriptstyle\mathrm{ROS}}\in\mbox{$\mathbb{C}$}^{1\times
N}$
($\mathbf{Z}_{\scriptscriptstyle\mathrm{SOT}}\in\mbox{$\mathbb{C}$}^{N\times
M}$) models the mutual impedance between the RIS and the UE (transmitter),
including the effect due to the scattering objects;
$\mathbf{Z}_{\scriptscriptstyle\mathrm{SS}}\in\mbox{$\mathbb{C}$}^{N\times N}$
and
$\mathbf{Z}_{\scriptscriptstyle\mathrm{SOS}}\in\mbox{$\mathbb{C}$}^{N\times
N}$ represent the self-impedance among the RIS elements and the self and
mutual impedance among the scattering objects and the RIS, respectively;
lastly,
$\mathbf{Z}_{\scriptscriptstyle\mathrm{RIS}}\in\mbox{$\mathbb{C}$}^{N\times
N}$ stands for the diagonal matrix of tunable RIS impedances modelled as
$\mathbf{Z}_{\scriptscriptstyle\mathrm{RIS}}=\mathrm{diag}(R_{0}+jx_{\ell}),\quad\ell=1,\ldots,N$,
where $R_{0}\geq 0$ is the constant resistance of each RIS load and
$x_{\ell}\in\mbox{$\mathbb{R}$}$ is the tunable reactance at the $\ell$-th
element. The precoding vector at the transmitter is denoted by
$\mathbf{w}\in\mbox{$\mathbb{C}$}^{M\times 1}$. As depicted in Fig. 1, such
vector is defined in order to produce a beam pointing at $-35^{\circ}$ in
azimuth and $0^{\circ}$ in elevation. Specifically, its beampattern is given
in [10]. We remark that the considered measurements collected in [8] involve
arrays of patch antennas with a uniform gain of $6$ dBi, while the adopted RIS
channel model in [5] assumes dipole antennas. Hence, to compensate for the
reduced antenna gain, we design the precoder vector such that
$\|\mathbf{w}\|^{2}\leq M\,G$, with $G=6$ dBi. Moreover, in order to reproduce
the setting in [8], we fix the RIS configuration in
$\mathbf{Z}_{\scriptscriptstyle\mathrm{RIS}}$ by feeding the channel vector in
(1) in the absence of both the scattered paths and the direct link, to the
_SARIS_ algorithm in [5] (see Section III-B2 for details). The resulting beam
at the RIS exhibits a main lobe pointing at $-10^{\circ}$ in azimuth and
$0^{\circ}$ in elevation. As previously done for the precoder vector, we then
compensate for the patch antenna gain at the RIS by multiplying the channel
coefficient corresponding to the path reflected upon the RIS by $\sqrt{G}$.
Lastly, the receive signal at the UE is thus given by

$\displaystyle y=\mathbf{h}^{\mathrm{H}}\mathbf{w}x+n\in\mbox{$\mathbb{C}$},$
(2)

where $x\in\mbox{$\mathbb{C}$}$ is the (known) transmit signal, and $n$ is the
additive white Gaussian noise coefficient.

Figure 1: System model

##  III Empirical validation procedure

In this section, we firstly describe the considered measurements collected in
the real-life environment described in Fig. 1. Subsequently, focusing on the
channel model in (1), we describe the procedure to fit the associated relevant
parameters in order to match the underlying physical propagation environment.

###  III-A Channel measurements

In the indoor environment depicted in Fig. 1, the propagation channel was
measured using a vector network analyzer (VNA) operating within the frequency
range of $25$-$35$ GHz. The setup included the use of an RIS as an extender, a
TX, and a wideband monopole antenna serving as UE. All of these terminals were
placed at a height of about $1.6$ m above the ground, corresponding to the
azimuth plane. Further details on the measurement setup and the analysis of
the channel characteristics are found in [8]. The TX was configured to steer
the beam towards the RIS, which, in turn, was configured to redirect the
received beam towards the intended UE. The UE, placed on a 2D positioner, was
moved on a $3\times 3$ spatial grid of positions, with a step equivalent to
half a wavelength, thus emulating a $3\times 3$ antenna array. Across a
bandwidth of $2$ GHz, this spatial grid allows the extraction of the multipath
components (MPCs) in both the time delay (or alternatively the distance) and
the angular domains, where the latter corresponds specifically to the azimuth
angle-of-arrival (AoA) at the UE-side. This is achieved by applying a wideband
high resolution algorithm such as the Space-Alternating Generalized
Expectation-Maximization algorithm (SAGE) [11].

The TX and the RIS operate within the Ka band at the resonance frequency of
$28$ GHz. The linearly-polarized TX [10] consists of $20\times 20$ unit cells
(UCs) and is illuminated with a $10$-dBi horn antenna. The employed RIS [9] is
an assembly of four rectangular lattices of $20\times 20$ UCs each, resulting
on a total of $40\times 40$ UCs. The one-bit UCs of the TX and the RIS are
arranged at regular intervals of half a wavelength and the configuration is
realized through electronic control of integrated PIN diodes. This enables ON
and OFF states with a relative phase-shift of approximately $180^{\circ}$ at
the chosen operating frequency. Further details on the $1$-bit UC architecture
and performance are provided in [10] for the TX and in [9] for the RIS.

The heatmap of the MPCs as perceived by the UE, extracted using the SAGE
algorithm, is illustrated in Fig. 2, where each path is graphically
represented by a point on the angle-distance axes, and its color is related to
its amplitude (in dB). The identification of the MPCs on the heatmap is
simplified with geometrical calculations based on the dimension of the floor
plan. Accordingly, the MPC reflected by the RIS appears at $7.2$ m with an AoA
of $76^{\circ}$, and the direct TX to UE path is located at $4$ m with an AoA
of $14^{\circ}$. Their powers are respectively $-72.4$ dB and $-68.7$ dB.
Also, the strongest scattered path due to the environmental objects is
represented by the one corresponding to a distance of $5.69$ m, AoA of
$-50^{\circ}$ and a power of $-79.8$ dB. All other MPCs are due to scatterers
that are far-away from the UE or to measurement noise. In this paper for model
validation we limit our analysis to the main considered MPCs summarized in
Table I.

TABLE I: Main MPCs in terms of distance travelled, AoA at the UE and measured
power.

MPC | Distance | AoA | Power
---|---|---|---
Direct path (TX/UE LoS) |  $4$ m | $14^{\circ}$ |  $-68.7$ dB
Reflected path (TX/RIS/UE) |  $7.2$ m | $76^{\circ}$ |  $-72.4$ dB
Secondary Cluster |  $5.69$ m | $-50^{\circ}$ |  $-79.8$ dB

###  III-B Channel model fitting

In order to tune the parameters in the model described in Section II and
replicate the setup in [8], we distinguish three cases, namely i) the direct
path from the transmitter to the receiver i.e. the TX/UE Line of Sight (LoS)
path, ii) the reflected path from the transmitter to the UE via the RIS, i.e.
the TX/RIS/UE path, and iii) the scattered path from the transmitter to the
receiver via a cluster of reflectors in the vicinity of the UE.

####  III-B1 Direct path

Regarding the direct path from the transmitter to the UE, we modify the
channel vector as

$\displaystyle\mathbf{h}_{d}^{\mathrm{H}}=Z_{\scriptscriptstyle\mathrm{RL}}\mathbf{z}_{\scriptscriptstyle\mathrm{RT}}\mathbf{Z}_{\scriptscriptstyle\mathrm{TG}},$
(3)

where $\mathbf{z}_{\scriptscriptstyle\mathrm{RT}}$ is the mutual impedance
between the transmitter and the UE. As stated above, the transmitter
configuration is set as described in [10].

####  III-B2 Reflected path

In this case, we consider only the path that reaches the UE from the
transmitter via the RIS. Hence, we modify the channel vector in (1) as

$\displaystyle\mathbf{h}_{\scriptscriptstyle\mathrm{RIS}}^{\mathrm{H}}=-Z_{\scriptscriptstyle\mathrm{RL}}\Big{[}\mathbf{z}_{\scriptscriptstyle\mathrm{RS}}\Big{(}\mathbf{Z}_{\scriptscriptstyle\mathrm{SS}}+\mathbf{Z}_{\scriptscriptstyle\mathrm{RIS}}\Big{)}^{-1}\mathbf{Z}_{\scriptscriptstyle\mathrm{ST}}\Big{]}\mathbf{Z}_{\scriptscriptstyle\mathrm{TG}},$
(4)

with $\mathbf{z}_{\scriptscriptstyle\mathrm{RS}}$ and
$\mathbf{Z}_{\scriptscriptstyle\mathrm{ST}}$ the mutual impedance between the
receiver and the RIS and between the latter and the transmitter in the absence
of scattering objects, respectively. Here, all parameters can be computed
given the (fixed) location of the transmitter, the RIS and the receiver [2].
The TX configurations is set as described in Section II, while the RIS
configuration is obtained by feeding (4) to the SARIS algorithm [5].

####  III-B3 Secondary cluster

Lastly, we focus on the path that reaches the UE from the TX via a cluster of
scatterers, which is located in the vicinity of the UE. In this case, the
channel vector is given by

$\displaystyle\mathbf{h}_{ref}^{\mathrm{H}}=-Z_{\scriptscriptstyle\mathrm{RL}}\mathbf{z}_{\scriptscriptstyle\mathrm{RO}}(\mathbf{Z}_{\scriptscriptstyle\mathrm{OO}}+\mathbf{Z}_{\scriptscriptstyle\mathrm{US}})^{-1}\mathbf{z}_{\scriptscriptstyle\mathrm{RO}}\mathbf{Z}_{\scriptscriptstyle\mathrm{TG}}.$
(5)

As described in [5], $\mathbf{z}_{\scriptscriptstyle\mathrm{RO}}$ represents
the mutual impedance between the UE and the cluster of scatterers,
$\mathbf{Z}_{\scriptscriptstyle\mathrm{OO}}$ the self and mutual impedances
among the scatterers, $\mathbf{Z}_{\scriptscriptstyle\mathrm{US}}$ the
diagonal matrix modelling the load attached to each scattering dipole, and
$\mathbf{z}_{\scriptscriptstyle\mathrm{RO}}$ stands for the mutual impedance
between the cluster of scatterers and the TX. Note that the dimension of such
quantities depends on $N_{c}$, i.e., the number of scattering dipoles, which
needs to be fitted to accturately represent the cluster. In particular, based
upon the floor plan of the channel measurements described in Section III-A,
such cluster consists of a cylindrical pillar with a width of approximately
$1$ m and a height of approximately $3$ m. However, given the narrow shape in
3D of the beam pattern at the transmitter, it is sufficient to model only a
small section of $16\times 8$ cm. Specifically, by applying the DDA [6], we
model such object as a cylindrical array of radius $0.5$ m, consisting of
$N_{c}=N_{x}\times N_{y}$ loaded dipoles of length $h=\lambda/4$ whose inter-
element distance is given by $d$, with $N_{x}$ the number of dipoles on the
horizontal plane, $N_{y}$ the number of dipoles on the vertical plane, and
$\lambda$ the signal wavelength. The load on each $t$-th dipole is given by
$Z_{\scriptscriptstyle\mathrm{US}t}=y_{0}+jx_{t}$, where
$y_{0}\in\mbox{$\mathbb{R}$}$ is constant over the array, and it depends on
the specific absorption and reflection properties of the material, while
$x_{t}\in\mbox{$\mathbb{R}$}$ models the scattering properties of the object.
To this end, the values of $N_{x}$, $N_{y}$, $d$,
$\\{x_{t}\\}_{t=1}^{N_{x}\times N_{y}}$, as well as the location of the object
need to be fitted to reproduce the results in Table I.

##  IV Numerical results and discussion

In order to validate the effectiveness of the channel model in (1) and [5], we
evaluate the (normalized) power received at the UE-side in dB, which is given
by

$\displaystyle P=10\log_{10}(|\mathbf{h}^{\mathrm{H}}\mathbf{w}|^{2}),$ (6)

in the various cases highlighted in Section III-B. The obtained results for
the reflected and direct paths are shown in Table II, demonstrating an error
of just $1.16\%$ and $0.35\%$ for the two cases, respectively. For the case of
the path including the cylindrical reflector, we generate $10^{3}$ different
combinations of the reactance $x_{t}$ for the load of each $t$-th dipole in
the cylindrical array by drawing values from the uniform distribution
$\mathcal{U}[-330,100]$. Furthermore, we vary the total number of dipoles
$N_{x}\times N_{y}$ with fixed ratio $N_{y}=2N_{x}$, the inter-element
distance $d$, and the real part of the dipole loads $y_{0}$. For each
combination, we select the maximum value obtained over the random reactance.

In Fig. 3, we show the receive power at the UE versus the total number of
dipoles for the case of $d=\lambda/4$, and for different values of $y_{0}$. We
thus conclude that approximately $2450$ dipoles, corresponding to a $35\times
70$ configuration, with $y_{0}=100~{}\Omega$ is a precise enough fitting of
the measured data, resulting in an error of $0.14\%$. Moreover, in Table III,
we provide the simulated received power for the case of a $35\times 70$
cylindrical array with $y_{0}=100~{}\Omega$ for different values of the inter-
element $d$, thus confirming that a dense array accurately represents the
physical propagation properties of a real-life object.

While so far we have successfully emulated the measurements in [8] by
configuring the RIS as described in Section II, in the following we
demonstrate how the algorithm in [5] can lead to superior performance by
designing an RIS configuration that explicitly interacts with the scattering
environment and creates constructive interference at the UE. In this regard,
we set all channel parameters according to the values that provide the best
fitting of the physical propagation environment, as described above, and
compare the RIS configuration used during the measurements described in
Section III-A versus the SARIS algorithm in [5]. As shown in Table IV, the
proposed RIS optimization algorithm achieves significant gains in the received
power thanks to an accurate modelling and exploitation of the physical
properties of the channel.

TABLE II: Simulated MPCs for the reflected and direct paths versus the
measured data.

MPC | Measured power | Simulated power
---|---|---
Direct path |  $-68.7$ dB |  $-68.5$ dB
Reflected path |  $-72.4$ dB |  $-73.3$ dB

Figure 2: Heatmap of the time delay (or alternatively the distance) and
azimuth AoA for the multipath in the considered scenario as perceived by the
UE. Figure 3: Received power versus the number of dipoles in the cylindrical
array with inter-element spacing of $d=\lambda/4$, and for different values of
$y_{0}$. TABLE III: Simulated MPC involving the close-by reflector versus
different inter-element spacings $d$ in a $35\times 70$ cylindrical array and
with $y_{0}=100~{}\Omega$.

Simulated power | Inter-element spacing $d$ | Simulated power | Inter-element spacing $d$
---|---|---|---
$-113.85$ dB | $5\lambda$ | $-89.45$ dB | $\lambda$
$-86.64$ dB | $0.5\lambda$ | $-79.93$ dB | $0.25\lambda$

TABLE IV: Received power in the considered scenario under different RIS
configurations.

RIS configuration | Simulated power
---|---
$-10^{\circ}$ azimuth, $0^{\circ}$ elevation |  $-63.27$ dB
SARIS [5] |  $-52.34$ dB

##  V Conclusion

In this paper, we shed experimentally validate a recently-introduced
electromagnetic-compliant channel model for RIS, which takes into account
mutual coupling effects and considers the presence of scattering objects. To
this end, we have validated its accuracy in representing the physical
characteristics of real-world scenarios. Interestingly, we have fine-tuned the
parameters of model using a dataset of channel measurements collected from an
office environment. By recreating the channel propagation with the DDA, we
have demonstrated that the proposed modeling can be leveraged to derive
enhanced RIS configurations, leading to substantial improvements in the
overall system performance.

## Acknowledgment

This work was supported in part by the EU H2020 RISE-6G project under grant
101017011.

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