Emilio Calvanese Strinati

George C Alexandropoulos

Vincenzo Sciancalepore

Marco Di Renzo

Henk Wymeersch

Dinh-Thuy Phan-Huy

Maurizio Crozzoli

Raffaele D'errico

Elisabeth De Carvalho

Petar Popovski

Paolo Di Lorenzo

Luca Bastianelli

Mathieu Belouar

Julien Etienne Mascolo Xi

Gabriele Gradoni

Sendy Phang

Geoffroy Lerosey

Benoît Denis

Wireless Environment as a Service Enabled by Reconfigurable Intelligent Surfaces: The RISE-6G Perspective

The design of 6th Generation (6G) wireless networks points towards flexible connect-and-compute technologies capable to support innovative services and use cases. Targeting the 2030 horizon, 6G networks are poised to pave the way for sustainable human-centered smart societies and vertical industries, such that wireless networks will be transformed into a distributed smart connectivity infrastructure, where new terminal types are embedded in the daily environment. In this context, the RISE-6G project aims at investigating innovative solutions that capitalize on the latest advances in the emerging technology of Reconfigurable Intelligent Surfaces (RISs), which offers dynamic and goal-oriented radio wave propagation control, enabling the concept of the wireless environment as a service. The project will focus on: i) the realistic modeling of RIS-assisted signal propagation, ii) the investigation of the fundamental limits of RIS-empowered wireless communications and sensing, and iii) the design of efficient algorithms for orchestrating networking RISs, in order to implement intelligent, sustainable, and dynamically programmable wireless environments enabling diverse services that go well beyond the 5G capabilities. RISE-6G will offer two unprecedented proof-of-concepts for realizing controlled wireless environments in near-future use cases.

I. INTRODUCTION

The 5th Generation (5G) of wireless networks is at an early deployment stage, providing a single platform for a variety of services and vertical applications [START_REF] Parkvall | 5G NR release 16: Start of the 5G evolution[END_REF]. However, novel concepts as well as new services and related use cases, incorporating new enabling technologies to satisfy future needs [START_REF] Strinati | 6G networks: Beyond Shannon towards semantic and goal-oriented communications[END_REF], are already being identified for addressing the current predictions for the performance requirements of the next 6th Generation (6G) of connect-and-compute networks around 2030. Among those predictions belong the up-to-10 Gbps/m 3 capacity, 100µs latency, 1 Tb/J energy efficiency, and 1 cm localization accuracy in 3 Dimensions (3D) [START_REF] Strinati | 6G: The next frontier: From holographic messaging to artificial intelligence using subterahertz and visible light communication[END_REF], which will need to be offered individually or in various combinations.

As a common view, wireless networks are designed considering the signal propagation environment as a black box that cannot be artificially controlled. This would exacerbate the simultaneous fulfilment of fundamentally conflicting targets, such as boosting quality of experience for different types of services at different places of intended users (e.g., ultrahigh data rate in hotspots, extreme reliability in factories, and massive connectivity in ultra dense areas), limiting the signal to non-intended users (e.g., users' ElectroMagnetic Field Exposure (EMFE) and eavesdroppers), and limiting the network and device energy consumption. Recently, there has been a surge of interest in Reconfigurable Intelligent Surfaces (RISs) [START_REF] Huang | Reconfigurable intelligent surfaces for energy efficiency in wireless communication[END_REF] as hardware-efficient and highly scalable means to realize desired dynamic transformations on the signal propagation environment in wireless communications [START_REF] Liaskos | A new wireless communication paradigm through software-controlled metasurfaces[END_REF], [START_REF] Renzo | Smart radio environments empowered by reconfigurable AI meta-surfaces: an idea whose time has come[END_REF]. RISs are man-made surfaces with thousands of reconfigurable elements, which can support various functionalities (e.g., control multipath geometry for localization, limit EMF exposure, mitigate obstructions, and extend radio coverage in dead zones). The RIS technology is envisioned to coat objects in the wireless environment [START_REF] Huang | Holographic MIMO surfaces for 6G wireless networks: Opportunities, challenges, and trends[END_REF] (e.g., building facades and room walls), and can operate either as a reconfigurable beyond-optics reflector [START_REF] Alexandropoulos | Reconfigurable intelligent surfaces and metamaterials: The potential of wave propagation control for 6G wireless communications[END_REF], or as a transceiver when equipped with active transmit [START_REF] Shlezinger | Dynamic metasurface antennas for 6G extreme massive MIMO communications[END_REF] and receive [START_REF] Alexandropoulos | A hardware architecture for reconfigurable intelligent surfaces with minimal active elements for explicit channel estimation[END_REF] radio-frequency elements.

The modeling of signal propagation in such novel RISempowere conditions appears very challenging [START_REF] Di Renzo | Smart radio environments empowered by reconfigurable intelligent surfaces: How it works, state of research, and road ahead[END_REF], [START_REF] Wu | Intelligent reflecting surface aided wireless communications: A tutorial[END_REF] involving a number of algorithmic solutions. In [START_REF] Gradoni | End-to-end mutual-coupling-aware communication model for reconfigurable intelligent surfaces: An electromagnetic-compliant approach based on mutual impedances[END_REF], a freespace model based on an impedance matrix formalism for RISs with discrete elements has been presented, while the corresponding optimization of the RIS settings for communication purposes is largely disussed in [START_REF] Huang | Reconfigurable intelligent surfaces for energy efficiency in wireless communication[END_REF], [START_REF] Wu | Intelligent reflecting surface aided wireless communications: A tutorial[END_REF]. Additional research directions focus on the challenging task of channel estimation in RIS-empowered communication systems, where multiple users and multiple RISs with large numbers of elements and non-linear hardware characteristics are considered [START_REF] Mursia | RISMA: Reconfigurable intelligent surfaces enabling beamforming for IoT massive access[END_REF]. The relevant literature includes pilot-assisted cascade channel estimation approaches [START_REF] Zheng | Intelligent reflecting surface-enhanced OFDM: Channel estimation and reflection optimization[END_REF] as well as deep learning frameworks [START_REF] Huang | Indoor signal focusing with deep learning designed reconfigurable intelligent surface[END_REF] that overcome the need for explicit channel estimation.

RISs will also have applications beyond communication, as in localization and sensing. 5G systems can make use of time, power, and angle measurements to accurately estimate the position and orientation of user devices [START_REF] Del Peral-Rosado | Survey of cellular mobile radio localization methods: From 1G to 5G[END_REF]. Localization algorithms can also turn received secondary paths in multipath channels into an advantage by using their estimated locationdependent radio parameters as a source of information. This improves localization continuity in highly obstructed scenarios, whenever a map/layout of the environment or a-priori parametric semi-deterministic models are available. If not, the map should be estimated using Simultaneous Localization And Mapping (SLAM) methods. However, even in such multipathaided cases, the exploited ElectroMagnetic (EM) interactions of the propagation environment remain uncontrolled. Despite its inherent advantages, the use of RIS for localization has received very limited treatment up to date. An overview of main research challenges of RISs from the perspectives of both localization and mapping was provided in [START_REF] Wymeersch | Radio localization and mapping with reconfigurable intelligent surfaces: Challenges, opportunities, and research directions[END_REF].

Beyond the above-mentioned state-of-the-art on the RIS technology, this paper highlights the RISE-6G project's perspective that will build a novel wireless connectivity paradigm empowered by RISs with a two-fold advantage: i) enable highly concentrated in time and space service delivery to intended end users and ii) remove energy from indoor/outdoor regions where non-intended users are present. This paradigm will pioneer the concept of the wireless environment as a service, which will promote dynamical trade-off of high capacity connectivity, Energy Efficiency (EE), EMFE, localization and sensing accuracy, and secrecy guarantees over eavesdroppers, while accommodating specific regulation on spectrum use, data protection, and EMF emission.

II. RIS-EMPOWERED SYSTEM ARCHITECTURE

Traditionally, a wireless network infrastructure is based on elements that operate at the PHY/MAC/link layers and above (e.g., base stations, access points) or even Layer 1 only (e.g., repeaters). Differently from this, an RIS which controls the propagation environment can be considered to operate at Layer 0. As such, RIS brings new directions in the evolution of the wireless architecture, whose investigation is one of the overarching objectives of the RISE-6G project. In this section, use cases that can benefit from deploying RISs are introduced with their associated Key Performance Indicators (KPIs) and the related network architectures and deployment strategies.

A. Scenarios and Use Cases

Considering both sub-6GHz and millimeter-Wave (mmWave) frequency bands, several scenarios and use cases will be identified and analyzed according to the following steps: i) specification of a reference scenario; ii) technological and economical analysis of use cases that can benefit from RIS operation; iii) analysis of the impact on spectrum and EMFE; iv) detailed specification of a use case that favorably illustrates the use of RISs in a given scenario.

Among the scenarios that are likely to benefit from the use of RISs, the following will be prioritized: • Enhanced connectivity and reliability. If the Quality of Experience (QoE)/Quality of Service (QoS) of a wireless network is below expectation, an RIS is applied to improve the radio coverage both in outdoor and indoor.

An RIS needs to be implemented in compliance with regulations and EMFE limits; • Localization and sensing. The RIS technology is expected to enable advanced sensing and localisation techniques for environment mapping, motion detection, opportunistic channel sounding and passive radar capabilities applied to various environments: industrial, high user-density and indoor; and • Sustainability and security. RIS-empowered networks are expected to enable the reduction of the energy spent to radiate the wireless signals, so as to improve the EE, reduce the EMFE, and increase the security due to highly directive and location-dependent communications. Two exemplary scenarios that have been selected for RISempowered networks are a train station and a vertical Industry 4.0 setup, respectively.

B. Metrics and KPIs for RIS-Empowered Wireless Systems

Relevant performance metrics and target KPIs to be used by the technical work packages will be identified and defined, and then re-assessed depending on the findings, in order to assess the performance of RIS-empowered wireless systems for the three types of scenarios defined in Sec.II-A: from the traditional coverage/capacity metrics, such as data rate and delay, to a set of KPIs related to localization and sensing, such as spatial accuracy, localization availability, and mapping latency. Furthermore, existing metrics on EE (delivered data versus spent energy) will be supplemented by two other ones: i) Spectral Secrecy Efficiency (SSE), initially defined as the spectral efficiency attained at the intended user normalized by the cost in terms of data rate successfully captured by the eavesdropper; and ii) EMFE Efficiency (EMFEE), defined as the data rate attained by intended users normalized by the cost in terms of exposure of the intended and non-intended users.

In the project's vision, an RIS-empowered network will be able to dynamically generate "QoE boosted areas" where customers can get guaranteed QoE, "localization boosted areas" where they can enjoy high precision localization, "SSE boosted areas" where they can enjoy secured communications, "EMFEE boosted areas" where environment-conscious customers can avoid EMFE, "EE boosted areas" where they can improve their energy EE footprint. In such a way, the "boosted performance" concept, which do not exist today in this context, will be defined and implemented in areas identified by service providers on the basis of marketing insights. Potential tradeoffs will be proposed when multiple targets and conflicting KPIs are found. A high-speed train station will be one of the primary use cases where this concept will be applied.

C. Network Architectures and Deployment Strategies

Suitable network architectures will be specified by leveraging on the technology components developed within the project, as described in the subsequent sections. This will be done for each scenario defined in Sec.II-A, according to metrics and KPIs defined in Sec.II-B. Depending on scenarios and application needs, flexible RIS devices will be organized in a network of space-time reconfigurable mirrors, orchestrated together by advanced algorithms running on ad-hoc controllers so as to dynamically change the propagation settings of the installed RISs, based on real-time/predicted network dynamics.

A key new element of the network architecture is the communication channel through which an RIS is controlled and reconfigured. There are two generic ways for RIS control: i) in-band control, where the properties of the RIS are dynamically configured by using the same wireless signals whose propagation is affected by the RIS; ii) out-of-band control (green connectors on Fig. 1), which takes place over a wired or wireless communication interface not affected by the RIS operation. For each scenario we will determine the appropriate type/combination of RIS control channel to be used.

The most suitable RIS-empowered network architectures and deployment strategies will be selected to be at least partly implemented in a trial, as will be described in Section IV.

III. RESEARCH ON TECHNOLOGY COMPONENTS

A. Modeling, Design, and Characterization

Advanced EM-compliant and environment-aware RIS models will be developed within the RISE-6G project in order to predict the operation of the RIS in the scenarios of interest. The continuous models provided by metasurface theory will be extended to capture the effect of the tunable discrete structure of the RIS. Inherently, the fundamental research questions concerning RIS modeling are centered around the ability to include: i) the mutual coupling between unit cells for arbitrary cell topology and tuning circuity within; ii) the near-field interaction between transmitter/receiver and RIS; and iii) the integration of RISs within the overall multi-path propagation channel.

RISE-6G will consider impedance models that capture the scattered field behaviour of the discrete surface structure as well as the interaction among different close surfaces in multi-RIS scenarios. Transmit and receive antenna arrays will be also considered in the impedance formalism so that an end-toend channel model can be constructed between port voltages and currents. These impedance models will rely, on one hand, on the antenna theory with canonical elements, on the other hand, they will exploit full-wave simulation of the unit cells (hybrid model). These models will be exploited for design and evaluation of RISs in ideal free-space conditions.

Rich multi-path fading can be captured by a purely statistical model, the Random Coupling Model (RCM) [START_REF] Ma | Efficient statistical model for predicting electromagnetic wave distribution in coupled enclosures[END_REF], obtained for an antenna impedance operating in highly reflective, large, and complex environments. Deterministic tools will be used to predict the RIS-aided channel propagation. An energy flow method that computes densities of rays on numerical meshes, the Dynamical Energy Analysis (DEA), will be extended to include the RIS effect on multi-path propagation. The method represents ray densities in a joint space of position and direction of travel, the so called phase-space, through the Wigner function technique [START_REF] Gradoni | Wigner-function-based propagation of stochastic field emissions from planar electromagnetic sources[END_REF]. An averaging procedure will embed the phase-space density of the RIS scattered field into DEA. A model of the RIS scattered field obtained from the Green function approach will be used to integrate their behaviour within the DEA approach in order to have a powerful and unique tool for including the RIS within coverage planning tools. The model validation will be done through a rich activity of EM characterization of both RIS and the propagation environment itself [START_REF] Mudonhi | RIS-enabled mmwave channel sounding based on electronically reconfigurable transmitarrays[END_REF].

Different RISs (transmittive and reflective) will be designed in RISE-6G, spanning from sub-6 GHz to D-band. Different technologies (PIN-diodes, varactors, and MEMS) will be considered for unit-cell tunabilty, as well as realize space and polarization configurability. Depending on the selected frequency bands, different architectures will be proposed with full or discrete phase control. Large RIS structures with more than 400 unit elements will be fabricated.

B. RIS for Enhanced Connectivity and Reliability

A look at the current state-of-the-art in the RIS research paints a research landscape in which there is a wide range of assumptions. Those assumptions include different models for the surface, the propagation channel, and communication settings. This variety of assumptions makes it difficult to draw meaningful conclusions in terms of performance limits or system design choices. As advocated in RISE-6G, this observation calls for a rigorous framework, encompassing: a) an accurate communication model; b) an account of the control signaling necessary for RIS operations; and c) a performance evaluation framework assessing the essential features of RISempowered communications. The majority of the current research adopts simple communication models that fit farfield communications and incorporate simplified models of the surface [START_REF] Wu | Intelligent reflecting surface aided wireless communications: A tutorial[END_REF]. However, the way a metasurface affects wireless propagation properties depends on many characteristics, including the properties of the surface and its unit elements, as well as their interactions [START_REF] Gradoni | End-to-end mutual-coupling-aware communication model for reconfigurable intelligent surfaces: An electromagnetic-compliant approach based on mutual impedances[END_REF], their distance to the transmitter and receiver, and the characteristics of the channel. RISE-6G will develop communication models that are accurate and mathematically tractable, and that account for the circuit design of the surface and its EM properties. Such models are essential to design and optimize RIS-empowered systems, but also to assess their performance limits, their advantages, and limitations. Based on those accurate communication models, RISE-6G will investigate the connectivity and reliability properties of RIS-empowered systems, providing fundamentals in communication and assessing achievable performance. The project will also focus on the design and optimization of control signaling protocols, channel estimation methods, and RIS settings, as well as resource allocation and scheduling techniques to enable an effective and reliable support of RISs for seamless distributed smart end-to-end support of communication. Furthermore, RISE-6G will determine the best strategies according to communication metrics for an easy and flexible deployment of RISs considering boosted connectivity areas.

Mobile Edge Computing (MEC) will also be deeply investigated within the RISE-6G project. In particular, RISE-6G will design a dense network of RISs to enhance the performance of MEC systems, with the aim of delivering pervasive, lowlatency, and energy-efficient edge services to mobile users. The key research question is: How to exploit and design RISs in MEC to strike the best possible trade-off between energy, latency, and accuracy of edge computing tasks? The project will achieve this goal by jointly optimizing radio, computation, and RIS resources to enable energy-efficient edge computing, while keeping the overall QoS (i.e., delay, accuracy) within a prescribed value dictated by the specific application, and adapting on the fly to dynamic scenarios (e.g., mobility, blocking events). Machine learning will play a key role in RISE-6G designs, enabling to learn how to optimize edge resources over time, decide which users should be served through the available RISs, and provide automation and selfhealing capabilities for optimal operational decision.

C. RIS for Enhanced Localization and Sensing

Localization and sensing are important enablers for the RISE-6G vision: not only do they have inherent value in a wide range of location-based services and applications, but they can also support enhanced connectivity, highly localized services' provision, as well as opportunistic environmental and context awareness. As shown in Fig. 2, it is our vision that in RISE-6G we see localization and communication as mutually reinforcing, rather than competing for the same resources. Several fundamental research questions still need to be addressed in this field of RIS-aided localization, mapping, and sensing, in addition to the issues already faced by RISenabled communications (e.g., RIS EM and channel modeling, channel estimation). First of all is the issue of localizationoriented RIS control: where should RIS be placed, how should they be controlled and optimized, at which time scales, in order to provide optimal localization accuracy and/or coverage? Secondly, given a chosen RIS-based system architecture, along with suitable RIS control methods, how do we estimate the users' locations and possibly their orientations, in a resilient (e.g., against radio obstructions), highly flexible (e.g., on-demand) and scalable way (without necessitating redeployment or re-planning), with minimum footprint in terms of latency, overhead, and overall consumption? To address these questions, RISE-6G aims at developing localizationoriented architectures and control methods, operating jointly at RIS and system levels, benefiting from highly directional RIS operations, overall multipath channel reconfigurability, and a combination of model-based and data driven processing tools. Similarly to data communications, security, and low EMF, we envision the use of localization-boosted areas, where high performance could be guaranteed. We will also devise novel algorithms for positioning, tracking, as well active and passive detection, which relax the synchronization and overhead requirements inherent to conventional localization. This should lead to higher accuracy, service continuity, and flexibility than non-RIS solutions.

D. RIS for Enhanced Sustainability and Security

Enhancements in terms of sustainability and security will be attained by exploiting a novel and disruptive approach in the design and deployment of RIS-empowered networks, taking into account non-intended users such as EMF exposed users and eavesdroppers. This can be accomplished by leveraging sensing and EM-field 3D maps acquisitions enabled by RISs. We propose to feed the envisioned network with more knowledge on the non-intended users. Ideally, one would make a RIS smart enough to detect human bodies and configure itself to actively avoid EMF exposure (or eavesdropping) on them.

More precisely, we propose to jointly optimize RIS and beamforming in terms of EMFEE and SSE in addition to the more traditional EE metric. We will compare different levels of available knowledge for non-intended users. On one hand, EMF exposed users can fall in the category of overt non-intended users, i.e. users that want to be identified as non-intended users so that the RIS-empowered network avoids them. Such overt non-intended users may cooperate to provide knowledge. On the other hand, eavesdroppers fall in the category of covert non-intended users, i.e. users that do not want to be discovered; no cooperation is expected from such users. Thi terminology is illustrated in Fig. 3. The level of signal at non-intended users will be taken into account as a constraint for the design of the active transmitting nodes' beamforming and the RIS tunable configurations. We will also devise coordinating beamforming algorithms and compare centralized and distributed solutions. We will consider solutions based on deep learning using cutting-edge neural network structures (e.g., generative adversarial networks). We will also propose new schemes to optimize the trade-off between the efficiency metrics, in the particular cases where EMFEE (or SSE) is improved at the expense of EE.

Finally, RISE-6G will use the beyond state-of-the-art physics-based modeling and simulation tools for RISs, described in Sec. III-A, to assess RIS-empowered networks in terms of EE, EMFEE, and SSE, taking into account the nonintended users in the assessment in various propagation environments. Measurements with a commercial 5G base station connected to the live network of Telecom Italia [START_REF] Micheli | Power boosting and compensation during OTA testing of a real 4G LTE base station in reverberation chamber[END_REF] will also pave the way for in-situ tests at the field trial facilities.

IV. RIS-EMPOWERED TRIALS FOR 6G USE CASES

Realistic use cases will substantiate the attainable performance of RIS-empowered environments, while pursuing different objectives, such as overall performance optimization and localization accuracy maximization. This will be performed and hereafter described by means of two different field-trials that cover public and private scenarios: i) a public train station; and ii) a private smartly connected factory.

A. Use Case 1: Train Station

An RIS-empowered solution will be properly and carefully set up in the train station located in Rennes, France, as depicted in Fig. 4. To facilitate the deployment of such proof-of-concept, generic distribution control networks will be developed to centrally control all RIS-based equipment considered in the scenario. We will identify key validation points as the following: i) interfacing with RISs in terms of maximal density of independent channels over each RIS interface; and ii) cost, power consumption, and complexity scaling functions with desired system performance.

Three relevant use cases have been identified that would greatly benefit from an RIS-empowered solution: i) coverage optimization for indoor shops; ii) dedicated resting areas for instant video download and cloud gaming ensuring the highest bandwidth and network efficiency; and iii) EMF limitations in private areas or for workers protection purposes. For High Definition (HD) video connections, an RIS-empowered network will work at 26 GHz by means of station infrastructure. At 70 GHz, short-range high data rate video streaming will be tested with a compact RIS-based reflector, and a 77 GHz radio module to support at least 1.488 Gbps uncompressed HD video streaming with commercial video encoder 4K transmission will be demonstrated.

B. Use Case 2: Connected Factory

The RISE-6G project will deploy RISs within a smartly connected factory scenario. In particular, it will focus on the industrial operations in an automotive plant of Fiat-Chrysler Automobiles (FCA). Inside the plant, a variety of processes (logistics, for example, for sub-assembly, preparation, and transport of components or group of components (called as kits) from logistics areas to lineside) require precise positioning of their moving assets: components, containers, racks, kit handlers, Autonomous Guided Vehicles (AGV), unmanned aerial vehicles, and robots. In the case of the logistics process of kitting, the composition of an assembly kit includes: i) containers of different components brought to a common area located near the final assembly line, stored and moving through the area; and ii) components related to a single final product (e.g. vehicle, engine) picked and housed together, forming a kit.

The state-of-the-art defines operations performed in a mixed way, partially automated (via AGVs) and partially manual (via human operators) so that the complete kit is brought to the assembly line in an automated way (via AGVs). The RISE-6G advanced solutions will deploy RISs to enable a complete mapping of the area (e.g., components, containers) in order to coordinate the AGVs' motion through continuous communication, facilitating the mission management. In particular, the component and container precise positioning and location in the rack, are not fixed on beforehand (that is, when the mission is given to the AGV) and might lead to a mission failure and a in-situ action of a logistics operator. In order to avoid potential failures it is, thus, crucial to achieve extreme accuracy in the localization process of the components, containers, or AGVs that must be carried out by means of advanced indoor localization techniques that cannot rely on cameras and distributed sensors. A novel framework to automatically control signal propagation properties by means of RISs will be delivered. Such RISs will complement the existing wireless network and enable a continuous communication channel between access points and the containers/AGVs via sub-6GHz and Ka-band (26 GHz) links. Additionally, cm-level localization demonstrations will be performed at 70 GHz and 130 GHz carrier frequencies. Finally, the proposed solution will target at minimizing the number of installed RISs required to seamlessly cover entire working areas, as well as at identifying the best localization algorithms offering resolution/refresh rate trade-offs.

V. CONCLUSIONS

This paper thoroughly overviewed the RISE-6G vision which capitalizes on the emerging RIS technology. The main technical challenges to properly tackle the unprecedented 6G network design requirements via the project's perspective were described: i) definition of a novel network architecture and operation strategies incorporating multiple RISs; ii) characterization of the fundamental limits of RIS-empowered communications, enabled by our proposed realistic and experimentally validated radio wave propagation models; iii) design of efficient solutions to enable online trading between highcapacity connectivity, EE, EMFE, and localization accuracy based on dynamically programmable wireless propagation environments, while accommodating specific legislation and regulation requirements on spectrum use, data protection, and EMF emission; and iv) prototype-benchmark proposed innovation via two complementary field trials with verticals.

VI. ACKNOWLEDGEMENT

This work has been supported by H2020 RISE-6G project.

Fig. 1 .

 1 Fig. 1. The RISE-6G system architecture showing the inband and outband Control Planes (C/P) among RISs and the edge Data Centers (DC).

Fig. 2 .

 2 Fig. 2. Localization and communication in RISE-6G will reinforce each other rather than compete for the same resources.

Fig. 3 .

 3 Fig. 3. Illustrating the concepts of EE-boosted, EMFEE-boosted, and SSEboosted areas, as well as the intended users and the overt and covert nonintended users.

Fig. 4 .

 4 Fig. 4. Envisioned RIS-empowered enhanced connectivity field trial at the Rennes, France train station.

Fig. 5 .

 5 Fig. 5. Envisioned RIS-empowered localization of AGVs equipped with a camera in kitting operations in a FCA automotive plant.


R Fara

D.-T Phan-Huy

P Ratajczak

A Ourir

M Di Renzo

J De Rosny

Reconfigurable Intelligent Surface-Assisted Ambient Backscatter Communications -Experimental Assessment

Keywords: Ambient backscattering, 6G, Internet of things, Reconfigurable intelligent surface, Metasurface

Sixth generation (6G) mobile networks may include new passive technologies, such as ambient backscatter communication or the use of reconfigurable intelligent surfaces, to avoid the emission of waves and the corresponding energy consumption. On the one hand, a reconfigurable intelligent surface improves the network performance by adding electronically controlled reflected paths in the radio propagation channel. On the other hand, in an ambient backscatter system, a device, named tag, communicates towards a reader by backscattering the waves of an ambient source (such as a TV tower). However, the tag's backscattered signal is weak and strongly interfered by the direct signal from the ambient source. In this paper, we propose a new reconfigurable intelligent surface assisted ambient backscatter system. The proposed surface allows to control the reflection of an incident wave coming from the exact source location towards the tag and reader locations (creating hot spots at their locations), thanks to passive reflected beams from a predefined codebook. A common phase-shift can also be applied to the beam. Thanks to these features, we demonstrate experimentally that the performance of ambient backscatter communications can be significantly improved.

I. INTRODUCTION

The Internet of Things (IoT) is continuously expanding and connecting more people and objects, through wireless links, resulting in a considerable growth of the power consumption due to the emission of radio frequency (RF) waves. Recently, technologies such as ambient backscatter (AmB) communications have been proposed to reduce the power consumption and the emission of RF waves of IoT devices [START_REF] Gati | Key technologies to accelerate the ict green evolution: An operators point of view[END_REF]. In an AmB communication system, tags send messages to readers by backscattering RF waves generated by an ambient RF source such as a TV tower, a Wi-Fi access point or a base station from the cellular network. As tags avoid generating additional RF waves, their energy consumption is very low, and solar or RF waves energy harvesting is sufficient to power them [START_REF] Liu | Ambient backscatter: Wireless communication out of thin air[END_REF]. Such tags are thus promising devices for IoT applications [START_REF] Zhang | A green paradigm for Internet of Things: Ambient backscatter communications[END_REF].

More precisely, in its simplest implementation [START_REF] Gati | Key technologies to accelerate the ict green evolution: An operators point of view[END_REF], the tag is a dipole antenna (designed to resonate at the carrier frequency of the RF ambient source) whose two branches are either connected or disconnected thanks to a switch. The tag is illuminated by an ambient RF source. Hence, if the branches are connected, the tag backscatters the ambient waves and the tag is said to be in a backscattering state. If the branches are disconnected, the tag is almost transparent to the ambient waves and the tag is said to be in a transparent state. One state corresponds to bit '0' and the other to bit '1'. A tag sends a binary message simply by switching itself between the two states. A reader nearby receives the ambient waves, and only needs to detect variations in the received power to distinguish the two states and demodulate the binary message (using a noncoherent energy detector) from the tag [START_REF] Gati | Key technologies to accelerate the ict green evolution: An operators point of view[END_REF].

Unfortunately, the reliability in terms of bit error rate (BER) of such a simple system is limited by the weakness of the backscattered signal and the simplicity of the non-coherent energy detector. As a first limitation, the tag may be in a deep fade of the ambient signal. In that case, it cannot backscatter any signal and it is undetectable by the reader. To overcome this first limitation, in [START_REF] Fara | Ambient backscatters-friendly 5G networks: creating hot spots for tags and good spots for readers[END_REF] a massive multiple input multiple output (M-MIMO) antenna is used to beamform the ambient signal and create a hot spot on the tag, to boost the backscattered waves. As a second limitation, the ambient signal interferes with the backscattered signal of the tag and can degrade the BER. Indeed, the authors of [START_REF] Rachedi | Demo Abstract : Real-Time Ambient Backscatter Demonstration[END_REF] show that, even in line-of-sight (LOS), the interferences and the desired signal can combine in such a way that the received signals at the reader side in the two states are close in power and only differ by their phases. In this case, the performance of a non-coherent detector collapses. To overcome this second limitation, the authors of [START_REF] Fara | Ambient backscatters-friendly 5G networks: creating hot spots for tags and good spots for readers[END_REF] create another additional hot spot, on the reader, this time, with a controlled phase shift, to make sure that both the signals of the source and the tag combine coherently when they arrive at the reader. Such scheme maximises the received contrast (between the two states of the tag) in terms of power.

Currently, another promising technology, named reconfigurable intelligent surface (RIS), is being investigated for future 6 th generation (6G) mobile networks [START_REF] Renzo | Smart Radio Environments Empowered by Reconfigurable Intelligent Surfaces: How It Works, State of Research, and The Road Ahead[END_REF]. An RIS is an electronically controlled metasurface composed of a large array of passive reflecting elements, which can reflect RF waves in a controlled manner, to allow operators improve the efficiency and the coverage of their network, without requiring any network densification [START_REF] Renzo | Smart Radio Environments Empowered by Reconfigurable Intelligent Surfaces: How It Works, State of Research, and The Road Ahead[END_REF][START_REF] Renzo | Smart radio environments empowered by reconfigurable AI meta-surfaces: An idea whose time has come[END_REF]. More precisely, an RIS is an array of backscatters, where each element controls electronically and individually the phase-shift (and/or an amplitude and/or a polarization rotation) with which it backscatters an incident wave [START_REF] Basar | Index Modulation Techniques for Next-Generation Wireless Networks[END_REF][START_REF] Zhao | Metasurface-assisted massive backscatter wireless communication with commodity Wi-Fi signals[END_REF][START_REF] Park | Intelligent Reflecting Surface-aided Phase-Shift Backscatter Communication[END_REF][START_REF] Karasik | Single-RF Multi-User Communication Through Reconfigurable Intelligent Surfaces:An Information-Theoretic Analysis[END_REF]. By applying a particular set of phase-shifts, the RIS controls the direction of the reflection, i.e. it creates a reflected beam. Phase-shifts can hence be viewed as reflection beamforming weights, similar to active beamforming weights. Such an RIS improves the coverage by creating a reflected beam towards a coverage hole.

In [START_REF] Nemati | Short-Range Ambient Backscatter Communication Using Reconfigurable Intelligent Surfaces[END_REF], an RIS has been proposed to increase the quality of the signal at the reader location in an AmB communication system. More precisely, it has been shown by means of simulations in rich-scattering environments that an RIS can modify the multipath channel and enhances the system performance. Similarly, in [START_REF] Liang | Symbiotic Radio: Cognitive BackscatteringCommunications for Future Wireless Networks[END_REF] it has been proposed to use an RIS to assist a cognitive backscattering communication system. However, the solutions proposed in [START_REF] Nemati | Short-Range Ambient Backscatter Communication Using Reconfigurable Intelligent Surfaces[END_REF] and [START_REF] Liang | Symbiotic Radio: Cognitive BackscatteringCommunications for Future Wireless Networks[END_REF] have not been tested and verified experimentally.

In this paper, we propose a new practical RIS-assisted AmB system, to be deployed on an information desk exposing tags for the purpose of reading. The idea of using RISs in AmB systems has been experimentally studied from the information-theroric standpoint in [START_REF] Karasik | Single-RF Multi-User Communication Through Reconfigurable Intelligent Surfaces:An Information-Theoretic Analysis[END_REF], where an RIS communicates a message to a reader, in our system the RIS does not transmit any data but helps existing tags, carrying information, to transmit them to the reader. In our proposed system, the source is the room Wi-Fi access point, it is fixed and in LOS of the desk, and its location is registered and known at the RIS. On the contrary, the exact locations of the tag and reader on the desk are unknown. However, the desk area is split into pre-defined locations. The RIS is close to the desk and has two features. First, the RIS can pick a beam among a pre-defined beam of a codebook. Each beam is designed to reflect waves coming from the exact source location towards one of the pre-defined locations on the desk. Second, the RIS can apply a common phase shift to all its elements. The RIS tests several pairs of beam and common phase-shift. Each time, the reader reports a performance quality feedback (a quantized contrast for instance). The RIS and the reader together search for the beam and common phase-shift pair that maximizes the performance. We expect, with such approach, to improve the BER, with respect to the scenario without RIS, with a similar approach as in [START_REF] Fara | Ambient backscatters-friendly 5G networks: creating hot spots for tags and good spots for readers[END_REF]: i.e., either by creating a "hot spot" on the tag (and boosting the backscattered wave), or by creating "coherent spots" both on the tag and the reader, in such a way that all waves reflected by the RIS combine coherently with all other waves, when arriving at the reader (to boost the received contrast). Finally, through experiments, we validate our approach and show a significant performance improvement.

The paper is organized as follows: Section II introduces the model of our proposed new system, Section III describes our experimental setup for validation, Section IV illustrates the obtained experimental results and Section V concludes this paper.

Notations: bold letters denotes matrices and vectors. We use |ℎ| and arg ℎ to denote the absolute and argument value of the complex number ℎ, respectively. erfc • is the complementary error function.

II. SYSTEM MODEL

A. Channel modeling

We consider the communication between one tag and one reader assisted by an RIS in a LOS environment. Our system model, illustrated in Fig. 1, does not take into account the mutual coupling between the RIS elements (cells) [START_REF] Abrardo | MIMO Interference Channels Assisted by Reconfigurable Intelligent Surfaces: Mutual Coupling Aware Sum-Rate Optimization Based on a Mutual Impedance Channel Model[END_REF]. First, the source transmits a signal originally destined to a legacy device. The signal is then reflected by the RIS with cells and also backscattered by the tag to the reader.

The reader (R) receives the signal from the source (S) through the path composed of four different paths between S and R. Among them, two paths are controlled by the RIS and two are modulated by the tag (T). The direct path is denoted by "S-R". The multi-hop path that is modulated by the tag is denoted by "S-T-R". The multi-hop path that is reflected by the RIS is denoted by "S-R, RIS". Finally, the path that is reflected by the RIS and is modulated by the tag is denoted by "S-T-R, RIS". Other paths, with more hops, are neglected. Multi-hop paths are composed of single-hop paths: "S-T", "T-R". Table I, defines, for each path, the path-specific notation of the channel coefficient ℎ, the amplitude and the distance . With such notation and definitions, is given by:

= ℎ + ℎ + ℎ , + ℎ , , (1)
where is a scalar that can take two different values depending on the state of the tag ( = 1 if the tag is in the backscattering state and = 0 if the tag is in the transparent state).

∈ ℝ Amplitude ∈ ℝ + Distance ∈ ℝ + S-R ℎ S-T ℎ T-R ℎ S-T-R ℎ S-R,RIS ℎ , , NA NA S-T-R,RIS ℎ , ,
NA NA Whatever the path without RIS reflection is, the coefficient ℎ of a propagation channel can be approximated by:

ℎ = ! "# , (2) with
= $%& '
, where ( corresponds to the wavelength of the signal. The channel coefficient of a path between a point A and a point B, assisted by the reflection of the RIS, depends on the channel coefficients of the sub-paths between A and each cell of the RIS, and the sub-paths between each cell of the RIS and B. Hence we define the channel vectors ) ∈ ℂ *+, , ) ∈ ℂ ,+* and ) ∈ ℂ *+, with sub-paths defined in Table II. For each sub-path, the sub-path-specific notation for the channel coefficient ℎ, its amplitude and its phase , is provided. Again, whatever the sub-path is, we have: ℎ = ! "# .

) -∈ ℂ Phase of the propagation channel coefficient ∈ ℝ Amplitude ∈ ℝ + Distance ∈ ℝ + S-cell m of the RIS ) - - - - cell m of the RIS-R ) - - - - cell m of the RIS-T ) - - - -
With the notations defined in Table I and Table II, we obtain:

ℎ = ℎ ℎ , ℎ , = ) ./012 3) 012/0 , ℎ , = ) ./012 3) 012/4 ℎ 4/0 , (3-a) (3-b) (3-c)
where 3 is the matrix of the reflection beamforming phase-shift coefficients applied to the cells of the RIS. In this paper, we consider an RIS that can only phase shifts the signal (without amplitude modulation). Therefore, 3 is a diagonal matrix, of size , of unitary complex coefficients.

B. Energy detector : performance metrics

On the reader side, we consider an energy detector to detect the tag. The performance of such detector depends on the amplitude of the signal contrast received by the reader Δ between the two states of the tag, i.e. Δ = 67 89, 7 / 7 89: 76.

The signal level received by the reader for each state of the tag ( = 1 or = 0) can be determined with the equation ( 1) and the amplitude of the contrast can be expressed, with the RIS Δ and without the RIS Δ ;<= , as follows:

Δ ;<= = 7|ℎ + ℎ | / |ℎ |7, Δ = 67ℎ + ℎ + ℎ , + ℎ , 7
/ 7ℎ + ℎ , 76. Considering an energy detector used by the reader to decode the tag message, the performance is given, as in [START_REF] Rachedi | Demo Abstract : Real-Time Ambient Backscatter Demonstration[END_REF] by:

?@0 = 0.5 erfc B Δ C D, (5)
where C is the square root of the noise average power. We propose to improve Δ with the RIS by using a codebook-based reflection beamforming approach. We first split the considered area (information desk) on which the tag and the reader are placed into E predefined locations. For each location F, we compute the matrix of predefined reflection beamforming coefficients G H that must be applied to the RIS, to create a reflected beam targeting the considered location F, as illustrated in Fig. 2. G H is a diagonal matrix of unitary complex coefficients of size . The codebook I is defined as the set of reflection beamforming matrices I = JG , … G H , … , G L M, each matrix corresponds to a reflection beamformer that targets a predefined location. We build 3 as follows:

C. Codebook-based reflection beamforming

3 = G! "N , (6)
where G ∈ O. With this notation, G -is a cell-specific beam coefficient to be applied to the cell P of the RIS, whereas Q ∈ Δ is a beam phase-shift applied to all cells, that is chosen among a set of R pre-defined beam phase-shifts Δ = S0, … ,

$% T , … , $% T , T U.
In order to compensate the phase shifts induced by the propagation channel (see equation ( 2) ) and therefore ensure a coherent focusing towards the target exact location F (as illustrated in Fig. 3), the predefined reflected beamforming coefficient G - H of the m th cell of the RIS is computed from:

G - H = ! "$% & V WXYZ[ \& V YZ[X] ' , (7)
Note that to compute the predefined coefficient G - H the location of the source and the RIS, and the propagation channels, must be known. The chosen spherical wave assumption is suitable for the considered scenario, where the source and the target are close to the RIS. Indeed, as illustrated in Fig. 3, in this case, the difference of beam coefficients between two successive cells is not constant as it takes into account the exact distances between all elements. Note that this simple computation method of reflecting beamforming coefficients does not take into account the mutual coupling between the cells of the RIS. We thus expect this method to be accurate enough to control for the direction of the main lobe of the reflected beam and less accurate for the control of the side lobes of the reflected beams. The RIS tests G and Q pairs and the reader reports a quality feedback (a measured Δ for instance). Together, the RIS and the reader find, through an exhaustive search, the G and Q pairs that maximise the performance. We expect at least two types of such pairs to exist such as Δ >Δ ;<= .

In the first type of pairs, a "hot spot" beamformer G improves Δ by boosting ℎ , and reducing ℎ , ^0: thus, it creates a hot spot on the tag as in [START_REF] Fara | Ambient backscatters-friendly 5G networks: creating hot spots for tags and good spots for readers[END_REF]. In this case, equation (4b) becomes:

Δ ~67ℎ + ℎ + ℎ , 7 / |ℎ |6. (8)
In this case, Q must be chosen such that:

arg ℎ + ℎ = arg ℎ , `2bc, (9)
which is equivalent to:

Q = arg ℎ + ℎ / arg ) G) ℎ `2bc. (10)
In the second type of pair, a "coherent spot" beamformer would attain the coherence condition as in [START_REF] Fara | Ambient backscatters-friendly 5G networks: creating hot spots for tags and good spots for readers[END_REF]. More precisely, the coherence condition illustrated in Fig. 4 is obtained when the sum of the signals controlled by the RIS arrive at the reader location in phase with the sum of all the other signals. Such condition ensures that the two sums combine coherently and boost Δ

. This coherence condition is expressed as follows:

arg ℎ + ℎ = arg ℎ , + ℎ , `2bc. ( 11
)
We expect to find in the codebook a G that boosts ℎ , + ℎ , , for instance, by targeting at the same time the tag and the reader locations. For example, in some cases, this term can be boosted by two lobes of the same beam, pointing at the tag and the reader. Then, the following Q satisfies condition [START_REF] Park | Intelligent Reflecting Surface-aided Phase-Shift Backscatter Communication[END_REF]:

Q = arg ℎ + ℎ /arg ) G) + ) G) ℎ . (12)

III. EXPERIMENTAL SETUP

We conduct experimental measurements to validate the proposed RIS-assisted AmB communication system presented in Section II. As illustrated in Fig. 5, the measurements are set up on a desk, with all the devices (source, tag, reader and RIS) in the LOS of each other.

In order to evaluate the benefit of the RIS, two different configurations are tested and compared: with and without RIS.

We evaluate the performance of the AmB communication system in terms of BER for various G and Q pairs.

A. Experimental setup

In the experimental setup, the source, the tag and the reader are all equipped with dipole antennas. The source generates an ambient signal with a 500kHz bandwidth at 5.2GHz. The reader receives the combined signals of the source, the RIS and the tag. A non-coherent energy detector is implemented in the reader to measure, in real time, the contrast Δ defined in Section II. Both the source and the reader are controlled by an universal software radio peripheral (USRP) B210 on two separated channels. The software-defined radio "GNU Radio" is used to process the signals and acquire the data.

The tag switches between two states by disconnecting or connecting the two branches of its dipole as in [START_REF] Gati | Key technologies to accelerate the ict green evolution: An operators point of view[END_REF]. It is electronically controlled thanks to a pin diode connected between the two branches of the antenna.

The coordinates of the location of each element of the system are denoted by d e , f e , g e , where the element E can be the source (S), the tag (T), the reader (R) or the RIS. All the elements are located in the same horizontal desk, with (g = g = g = g ) as vertical coordinates.

B. RIS hardware prototype

The RIS hardware prototype is a two-dimensional array composed of 14 rows and 14 columns, and is constituted, in total, by 196 cells. A single cell is illustrated in the Fig. 6. All cells have identical dimensions, i.e. 14×14mm and are designed for a frequency range between 5.15 to 5.75 GHz [START_REF] Baracco | Reconfigurable Antennas Demonstrators Using Varactors Technology[END_REF][START_REF] Ratajczak | C-band active reflectarray based on high impedance surface[END_REF]. The size of the cell is approximatively a quarter of the wavelength. The phase shift of each cell is controlled by applying a voltage between 0V and 5V, to a varactor implemented in each cell. The correlation between the phase shift and the voltage has been studied for this RIS and has been well characterized in [START_REF] Ratajczak | C-band active reflectarray based on high impedance surface[END_REF]. In particular, the 14 columns of the RIS are controlled with a microcontroller connected to a digital-to-analog converter in order to provide the 0-5V control voltage to each column of cells.

In the experimental setup, the devices lie in the same horizontal plane, as shown in Fig. 5. Hence, we propose to exploit the symmetry of this configuration and limit the control of the RIS beamforming to the horizontal plane only. More precisely, the same beam coefficient is applied to each cell in the same column, and the 14 columns are controlled independently.

C. Codebook experimental implementation

We first split the desk into E pre-defined locations illustrated in Fig. 5. For each location F, we compute the corresponding codebook according to Section II and to the locations of the elements detailed in Table III. Note that as the inter-cell distance is a quarter of the wavelength, the mutual coupling between the RIS elements is not negligible and will impact the radiation pattern. However, the simple codebook computation method of Section II, i.e. without mutual coupling taken into account, is expected to be accurate enough for the control of the main direction and less accurate for the control of side lobes. The E target locations are chosen along the y axis. As the elements are located on that same y axis, we vary the f coordinate of the targets between f = 0.1m and f = 0.4m with a step of 2mm. This results in E = 151 different locations and a codebook of 151 different reflected beams.

Regarding the beam phase-shift Q, we select R = 18 phase shift values between 0 and 2b.

IV. RESULTS

This section reports the simulation results obtained by using the simple model described in Section II and the experimental results obtained from the setup described in Section III.

The following parameters are set to the same values for the simulation and the experiments: the locations of the source, tag, reader and RIS, the carrier frequency, the pre-defined codebook and the predefined set of phase shifts (given in Section III-C).

For both the simulations and the experiments, all reflection beams of the codebook and all phase shifts are tested. For each pair of G and Q, the contrast is assessed either by simulation or by direct experimental measurement at the reader side. In each case, the corresponding simulated BER and measured BER is derived. In addition, we conduct measurements in the absence of the RIS to provide a reference for comparison.

In Fig. 7-a) and b), we observe the simulated BER and measured BER, respectively, as a function of the index of the reflected beam applied G and the applied phase shift Q. As expected, we observe, in Fig. 7, by simulations and experiments, that some pairs of reflected beam and beam phase-shift improve the performance: i.e., these correspond to the yellow pairs that minimize the BER. As expected, hot spots on the tag can be created with some reflected beams between 1 and 31 and with an optimised beam phase-shift fulfilling the condition in [START_REF] Basar | Index Modulation Techniques for Next-Generation Wireless Networks[END_REF]. One example of such reflected beam (the reflected beam 1) is illustrated in Fig. 8-a), where the simulated spatial map of the reflected signal strength is plotted in dB color scale. We observe, in this example, that the reflected beam 1 creates a hot spot on the tag thanks to its main lobe, whereas the reader receives almost no signal from the RIS. We also observe in Fig. 7.b) that, for the reflected beam 1, the condition ( 8) is approximatively obtained for Q = 140°.

As expected, coherent spots can be obtained with several beams between 60 and 151 and with an optimized beam phaseshift verifying the condition in [START_REF] Basar | Index Modulation Techniques for Next-Generation Wireless Networks[END_REF]. Two examples of such beams (the reflected beams 68 and 145, respectively) are illustrated in Fig. 8-b) and c), with simulated maps of the reflected signal strength in dB color scale. As illustrated in Fig. 8-b) and c), respectively, the reflected beams 68 and 145, respectively, create coherent spots on both the tag and the reader, thanks to their main and side lobes. For both the reflected beams 68 and 145, we observe in Fig. 7.b) that the coherent condition in [START_REF] Park | Intelligent Reflecting Surface-aided Phase-Shift Backscatter Communication[END_REF] is approximatively obtained for Q = 0°.

We note that, in the simulated reflected beams, we did not take into account the mutual coupling. Hence, accurate information on the direction of the main lobe is obtained whereas the side lobes may be overestimated. Finally, in Fig. 9, we provide a performance comparison of the AmB communication system with and without RIS. In the absence of the RIS, the system attains a reference BER value of 5.37×10 -2 that has been assessed through experimental measurements. In the presence of the RIS, the system attains a BER that is plotted in Fig. 9 as a function of the beam and phase shift pairs, in color scale. The chosen color scale highlights BER values exceeding the reference BER value (in blue) and BER values better than the reference BER value (in yellow). In yellow, we hence observe the pairs of G and Q that improve the BER compared to the configuration without RIS, whereas, in blue we have the RIS configurations that degrade the system performance.

These experimental results demonstrate that, assisted by an RIS that uses a codebook-based reflection beamforming scheme whose beams are chosen among a set of pre-defined beams for pre-defined locations and whose beam phase shifts are selected among a set of predefined values, the BER performance of an AmB communication system can be improved.

V. CONCLUSION

In this paper, for the first time, we have demonstrated experimentally that reconfigurable intelligent surfaces can improve the performance of ambient backscatter systems. The proposed reconfigurable intelligent surface is able to realize pairs of a reflected beam (belonging to a codebook) and a beam phase-shift that improve the tag-to-reader bit-error-rate. Since the reconfigurable intelligent surface controls the reflected signal, a hot spot on the tag or coherent spots on the tag and the reader can be created to assist the communication between the tag and the reader. Future studies will evaluate more general codebooks that could be applied in different setups and also machine learning algorithms to optimize the phase reflection coefficients.

Fig. 1 .

 1 Fig. 1. System model with the links between the source (S), the RIS, the tag (T) and the reader (R).

  these expressions, only the terms ℎ , and ℎ , are controllable by the RIS.

Fig. 2 .

 2 Fig. 2. Codebook principle.

Fig. 3 .

 3 Fig. 3. Illustration of the model for the cells coherently targeting the location n, knowing the exact location of the source S.

Fig. 4 .

 4 Fig. 4. Illustration of the coherent spots with the signals controlled by the RIS in phase with the direct signals (not passing through the RIS) at the reader.

Fig. 5 .

 5 Fig. 5. Experimental setup.

Fig. 6 .

 6 Fig. 6. Structure of the elementary cell of the RIS [15,16].

Fig. 7 .

 7 Fig. 7. Simulation (a) and experimental (b) results of the BER achievable depending on the beam of the codebook and the phase shift applied to the beam.

Fig. 8 .

 8 Fig. 8. Map illustrations of the beamformed reflected signal of the RIS for the reflection phase coefficient of beam 1 (a) and beam 68 (b) beam 145 (c) from the codebook.

Fig. 9 .

 9 Fig. 9. Experimental results of the BER improvement/degradation due to the RIS depending on the beam of the codebook and the phase shift applied to the beam.

TABLE I

 I

	.	NOTATIONS OF THE PROPAGATION CHANNEL COEFFICIENTS
			FOR ALL PATHS
			Phase of the
		Channel	propagation
	Path	coefficient ℎ ∈ ℂ	channel coefficient

TABLE II .

 II

NOTATIONS FOR THE RIS CHANNEL SUB-PATHS Sub-Path Channel coefficient

ACKNOWLEDGMENT

This work has been partially supported by EU H2020 RISE-6G project.


Student Member, IEEE Roy Karasik
email: royk@campus.technion.ac.il

Fellow, IEEE Osvaldo Simeone
email: osvaldo.simeone@kcl.ac.uk

Fellow, IEEE Marco Di Renzo
email: marco.di-renzo@universite-paris-saclay.fr

Shlomo Shamai

Single-RF Multi-User Communication Through Reconfigurable Intelligent Surfaces: An Information-Theoretic Analysis

Reconfigurable intelligent surfaces (RISs) are typically used in multi-user systems to mitigate interference among active transmitters. In contrast, this paper studies a setting with a conventional active encoder as well as a passive encoder that modulates the reflection pattern of the RIS. The RIS hence serves the dual purpose of improving the rate of the active encoder and of enabling communication from the second encoder. The capacity region is characterized, and informationtheoretic insights regarding the trade-offs between the rates of the two encoders are derived by focusing on the high-and low-power regimes.

I. INTRODUCTION

A reconfigurable intelligent surface (RIS) is a nearly-passive device that can shape the wireless propagation channel by applying phase shifts to the incident signals [1]-[7]. In multiuser (MU) systems, RISs can help mitigate inter-user interference and obtain beamforming gain for standard active transmitters [8]-[15].

kept fixed for the duration of a coherence interval and optimized to maximize some function of the achievable rates [START_REF] Guo | Weighted sum-rate maximization for intelligent reflecting surface enhanced wireless networks[END_REF]- [START_REF] Zhou | Fairness-oriented multiple RISs-aided MmWave transmission: Stochastic optimization approaches[END_REF]. In this paper, we study a different use of RISs, whereby a single active transmitter coexists with a passive user, having no direct radio frequency (RF) chain, that conveys its own message by modulating the reflection pattern of the RIS (see Fig. 1). With reference to Fig. 1, the RIS is accordingly used for the dual purpose of enhancing the rate of the active encoder (Encoder 1) and of enabling communication for the passive encoder (Encoder 2). Unlike prior work [START_REF] Yang | A novel RIS-assisted modulation scheme[END_REF] that focused on a specific transmission strategy, this paper concentrates on the information-theoretic analysis of the rate trade-offs between the two encoders, providing fundamental insights.

Related Work: A comprehensive survey of the state-of-the-art on RIS-aided MU systems is available in [START_REF] Di Renzo | Smart radio environments empowered by reconfigurable intelligent surfaces: How it works, state of research, and road ahead[END_REF]. As notable representative examples of works involving active transmitters, the maximization of the weighted sum-rate in RIS-aided MU systems was studied in [START_REF] Guo | Weighted sum-rate maximization for intelligent reflecting surface enhanced wireless networks[END_REF]- [START_REF] Mu | Joint deployment and multiple access design for intelligent reflecting surface assisted networks[END_REF], whereas references [START_REF] Huang | Reconfigurable intelligent surfaces for energy efficiency in wireless communication[END_REF], [START_REF] Han | Intelligent reflecting surface aided network: Power control for physical-layer broadcasting[END_REF] focused on optimizing the energy efficiency, and papers [START_REF] Zhang | Physical layer security enhancement with reconfigurable intelligent surface-aided networks[END_REF], [START_REF] Zhou | Fairness-oriented multiple RISs-aided MmWave transmission: Stochastic optimization approaches[END_REF] on physical-layer security and outage-probability enhancements. A MU system with an active transmitter and a passive encoder, akin to Fig. 1, was proposed in [START_REF] Yang | A novel RIS-assisted modulation scheme[END_REF] by assuming binary modulation, a single receiver antenna, and a specific successive interference cancellation decoding strategy.

From an information-theoretic perspective, the single-RF MU communication system depicted in Fig. 1 can be viewed as a multiple access channel (MAC) with both multiplicative and additive elements. The capacity of the Gaussian multiplicative MAC was derived in [START_REF] Pillai | On the capacity of multiplicative multiple access channels with awgn[END_REF] for two active encoders. The capacity region of a backscatter multiplicative MAC, which can be viewed as a special case of the RIS-aided MU communication system in Fig. 1 with one reflecting element, was studied in [START_REF] Liu | Backscatter multiplicative multiple-access systems: Fundamental limits and practical design[END_REF]. Under the assumptions of a single receiver antenna and Gaussian codebooks, this work shows that conventional time-sharing schemes are suboptimal in the high-power and weak-backscatter regimes. The capacity of an RIS-aided single-user channel was derived in [START_REF] Karasik | Adaptive coding and channel shaping through reconfigurable intelligent surfaces: An information-theoretic analysis[END_REF].

Main Contributions:

In this paper, we study the RIS-aided MU system illustrated in Fig. 1, in which Encoder 1 is active, whereas Encoder 2 can only alter the reflection pattern of the RIS in a passive manner. We derive the capacity region under the practical assumptions of a multi-antenna decoder, a finite-input constellation, and a set of discrete phase shifts at the RIS. Then, we specialize the results for the high-and low-power regimes, showing that: (i) for sufficiently high transmission power, both encoders can simultaneously communicate at maximum rate; and (ii) in the low-power regime, Encoder

( 1 bits) 2 ( 2 bits) ( ˆ 1 , ˆ 2 )
Control Link (Rate = 1/ ) Fig. 1. In the system under study, Encoder 1 is active and it encodes its message 1 into a codeword of symbols sent on the wireless link; whereas Encoder 2 is passive and it encodes the message 2 into a control action, which is sent on the control link to the RIS at a rate of one action every channel symbols.

while still enabling non-zero-rate communication for Encoder 1. Finally, numerical examples demonstrate the dual role of the RIS as means to enhance the transmitted signal on the one hand and as the enabler of MU communication on the other hand.

Notation: Random variables, vectors, and matrices are denoted by lowercase, boldface lowercase, and boldface uppercase Roman-font letters, respectively. Realizations of random variables, vectors, and matrices are denoted by lowercase, boldface lowercase, and boldface uppercase italic-font letters, respectively. For example, is a realization of random variable x, is a realization of random vector x, and is a realization of random matrix X. For any positive integer , we define the set [ ] {1, 2, . . . , }. The cardinality of a set A is denoted as |A|. The ℓ 2 -norm and the conjugate transpose of a vector are denoted as and * , respectively. diag( ) represents a diagonal matrix with diagonal given by the vector . The vectorization of matrix , i.e., the operator that stacks the columns of on top of one another, is denoted by vec( ). The Kronecker product ⊗ of the identity matrix of size and matrix is denoted as ⊗ .

II. SYSTEM MODEL

We consider the system depicted in Fig. 1

( ) = (s 1 ( ), . . . , s ( )) ⊺ ∈ S ×1
, and is assumed to satisfy the power constraint

1 [s * ( )s( )] ≤ 1. (1)
The phase shifts applied by the RIS in the th block are denoted by the vector

θ θ θ( ) ( θ 1 ( ) , . . . , θ ( ) ) ⊺ , (2)
with θ ( ) ∈ A being the phase shift applied by the th RIS element, ∈ [ ].

We assume quasi-static flat-fading wireless channels, which remain fixed throughout a coding slot. Specifically, the channel from Encoder 1 to the decoder is denoted by vector h ∈ ℂ ×1 ; the channel from Encoder 1 to the RIS is denoted by the vector h ∈ ℂ ×1 ; and the channel from the RIS to the receiving antennas is denoted by the matrix H ∈ ℂ × . Furthermore, we assume that h , h , and H are drawn from a continuous distribution.

Finally, we denote the signal received by the antennas for the th transmitted symbol

in block ∈ [ / ] by y ( ) ∈ ℂ ×1 , ∈ [ ]. The overall received signal matrix Y( ) =
(y 1 ( ), . . . , y ( )) ∈ ℂ × in the th sub-block can hence be written as

Y( ) = √ H diag θ θ θ( ) h s ⊺ ( ) + h s ⊺ ( ) + Z( ) = √ H θ θ θ( ) + h s ⊺ ( ) + Z( ), (3)
where > 0 denotes the transmission power of Encoder 1; the matrix H H diag(h ) ∈ ℂ × , combines the channels h and H ; and the matrix Z( ) ∈ ℂ × , whose elements are independent and identically distributed (i.i.d.) as CN(0, 1), denotes the additive white Gaussian noise at the receiving antennas. In order to characterize the distribution of the output signal Y( ) in ( 3), we vectorize it as

y( ) vec(Y( )) = √ H θ θ θ( ) + h ⊗ s( ) + z( ), (4)
where we have defined the vector z( ) vec(Z( )) ∈ ℂ ×1 .

We assume that both the encoders and the decoder have perfect channel state information (CSI), in the sense that the channel matrix H and channel vector h are known.

Having received signal y( ) in ( 4) for ∈ [ / ], the decoder produces the estimates

ˆ ℓ = ˆ ℓ (y(

III. CAPACITY REGION

In this section, we first derive a general characterization of the capacity region C( ,

)
for the channel in (4). Then, we leverage this result to provide theoretical insights into the trade-offs between the rate of the two encoders in Fig. 1 by focusing on the low-and highpower regimes.

Most existing works on the multiplicative Gaussian MAC [START_REF] Pillai | On the capacity of multiplicative multiple access channels with awgn[END_REF], [START_REF] Liu | Backscatter multiplicative multiple-access systems: Fundamental limits and practical design[END_REF] and on RIS-aided systems (see, e.g., [START_REF] Guo | Weighted sum-rate maximization for intelligent reflecting surface enhanced wireless networks[END_REF]) consider Gaussian codebooks for the transmitted signal s( ). This implies that the resulting achievable rates are formulated in the standard form "log 2 (1+SNR)".

In contrast, as described in Section II, we focus our attention on the more practical model in which the transmitted symbols and the RIS elements' phase response take values from finite sets [START_REF] Wu | A survey on MIMO transmission with finite input signals: Technical challenges, advances, and future trends[END_REF]. Therefore, in a manner similar to [START_REF] Karasik | Adaptive coding and channel shaping through reconfigurable intelligent surfaces: An information-theoretic analysis[END_REF], the expressions for the achievable rates that we present in this section are more complex, and require the following definition.

Definition 1: The cumulant-generating function (CGF) of a random variable u conditioned on a random vector x is defined as

(u|x) log 2 ( [ u |x]) , for ∈ ℝ, (5)
and the value of the conditional CGF for = 1 is denoted as (u|x) 1 (u|x). We now characterize the capacity region in the form of a union of rate regions, with each region corresponding to rates achievable for a specific choice of encoding distributions s ( ) and θ θ θ ( ) of the transmitted symbols s( ) and RIS phase shifts θ θ θ( ) in (4), respectively. Proposition 1: For input distributions s ( ) and θ θ θ ( ), let R( s , θ θ θ , , ) be the set of rate pairs ( 1 ( , ), 2 ( , )) such that the inequalities

ℓ ( , ) ≤ -log 2 ( ) - 1 (u ℓ |s 1 , θ θ θ 1 , z), ℓ ∈ {1, 2}, (6a)
and

1 ( , ) + 2 ( , ) ≤ -log 2 ( ) - 1 (u 3 |s 1 , θ θ θ 1 , z) (6b)
hold, where random variable u 1 , u 2 , and u 3 are defined as

u 1 -z + √ θ θ θ 1 + ⊗ (s 1 -s 2 ) 2 , (7a)
u 2 -z + √ θ θ θ 1 -θ θ θ 2 ⊗ s 1 2 , ( 7b
)
u 3 -z + √ θ θ θ 1 + ⊗ s 1 - √ θ θ θ 2 + ⊗ s 2 2 , (7c)
respectively, with independent random vectors s 1 , s 2 ∼ s ( ), θ θ θ 1 , θ θ θ 2 ∼ θ θ θ ( ), and z ∼

CN(0,

). The capacity region C( , ) is the convex hull of the union of the regions R( s , θ θ θ , , ) over all input distributions s ( ) and θ θ θ ( ) with ∈ S ×1 , ∈ A ×1 , such that [s * s] ≤ .

Proof: See Appendix A.

Next, we specialize the results in Proposition 1 to characterize the capacity region in the high-and low-power regimes.

A. High-Power Regime

The following corollary shows that the capacity region C( , ) converges to a rectangle as the power of Encoder 1 increases.

C ( 1 , 2 ) : 1 ≤ log 2 ( ), 2 ≤ log 2 ( ) . (8)
The capacity region C( , ) converges to C as the power increases in the sense that C( , ) ⊆ C, and there exists a sequence of achievable rate pairs ( 1 ( , ), 2 ( , )) ∈ C( , ) such that, almost surely,

lim →∞ 1 ( , ) = log 2 ( ), (9a)
lim →∞ 2 ( , ) = log 2 ( ). (9b)
Proof: See Appendix B.

Corollary 1 implies that, for sufficiently high power , both encoders can simultaneously achieve their maximum rates. As a result, while not useful in increasing the high-power rate of Encoder 1, the presence of the RIS enables communication at the maximum rate for Encoder 2 without creating deleterious interference on Encoder 1's transmission.

B. Low-Power Regime

In this section, we characterize the capacity region C( , ) in the low-power regime.

To simplify the analysis, we focus on a system with one receiver antenna, = 1, and an RIS control ratio of = 1. For this special case, the channel (4) can be written as

y( ) = √ h ⊺ θ θ θ( ) + h s( ) + z( ), (10)
where h ∈ ℂ ×1 and h ∈ ℂ denote the reflected and direct channel paths, respectively, and z( ) ∼ CN(0, 1) denotes the additive white Gaussian noise. Furthermore, we assume that the phase shift applied by each element of the RIS is chosen from a finite set of uniformly spaced phases, i.e., A = {0, 2 / , . . . , 2 ( -1)/ }; and that S is a zero-mean input constellation, i.e.,

∈S = 0, (11)
which is known to achieve the minimum energy per bit in many single-user channels [START_REF] Verdu | Spectral efficiency in the wideband regime[END_REF]- [START_REF] Lapidoth | Fading channels: how perfect need "perfect side information" be?[END_REF].

In order to formulate the capacity region in the low-power regime, we define the normalized rate ℓ ( , ℎ ), ℓ ∈ {1, 2}, for unit of power as

ℓ ( , ℎ ) lim →0 ℓ ( , ℎ ) . (12)
The capacity region in the low-power regime C( , ℎ ) is accordingly defined as the closure of the set of achievable normalized rate pairs (see, e.g., [START_REF] Caire | Suboptimality of TDMA in the low-power regime[END_REF]).

Proposition 2: For input distributions s ( ) and θ θ θ ( ), let R( s , θ θ θ , , ℎ ) be the set of normalized rate pairs ( 1 ( , ℎ ), 2 ( , ℎ )) such that the inequalities

ℓ ( , ℎ ) ≤ [u ℓ ] ln(2) , ℓ ∈ {1, 2}, (13a)
and

1 ( , ℎ ) + 2 ( , ℎ ) ≤ [u 3 ] ln(2) (13b)
hold, where random variable u 1 , u 2 , and u 3 are defined as

u 1 ⊺ θ θ θ 1 + ℎ (s 1 -s 2 ) 2 , (14a)
u 2 ⊺ θ θ θ 1 -θ θ θ 2 s 1 2 , (14b)
u 3 ⊺ θ θ θ 1 + ℎ s 1 -⊺ θ θ θ 2 + ℎ s 2 2 , (14c)
respectively, with independent random variables s 1 , s 2 ∼ s ( ) and random vectors θ θ θ 1 , θ θ θ 2 ∼ θ θ θ ( ). The capacity region in the low-power regime C( , ℎ ) is the convex hull of the union of the regions R( s , θ θ θ , , ℎ ) over all input distributions s ( ) and θ θ θ ( ) with

∈ S, ∈ A ×1 , such that [|s| 2 ] ≤ 1. Proof: See Appendix C.
Unlike the high-power regime, the low-power capacity region ( 13) is not a rectangle, implying that it is not possible for both encoders to communicate at their respective maximum rates. The next corollary elaborates on this point.

Corollary 2: Let ˜ be the beamforming phase-shift vector maximizing Encoder 1's rate, i.e.,

˜ arg max

∈A ×1 ⊺ + ℎ 2 . ( 15
)
In the low-power regime, Encoder 1 can achieve its maximum normalized rate

1 ( , ℎ ) = 2 ln(2) ⊺ ˜ + ℎ 2 (16)
if and only if Encoder 2 does not communicate, i.e., 2 ( , ℎ

) = 0. In contrast, if | ⊺ ˜ + ℎ | 2 > 2 , Encoder 2 can achieve its maximum normalized rate 2 ( , ℎ ) = 2 ln(2) 2 , ( 17
)
while Encoder 1 communicates at a normalized rate of

1 ( , ℎ ) = 2 ln(2) ⊺ ˜ + ℎ 2 - 2 . ( 18
)
Proof: See Appendix D.

The asymmetry between Encoder 1 and Encoder 2 revealed by Corollary 2 stems from the fact the, in order for Encoder 1 to obtain its maximum rate in the low-power regime, Encoder 2 needs to steer its phases according to the beamforming solution [START_REF] Zhou | Fairness-oriented multiple RISs-aided MmWave transmission: Stochastic optimization approaches[END_REF]. This in turn makes it impossible to encode additional information for Encoder 2. In contrast, Encoder 2's maximum rate can be obtained as long as Encoder 1's signal is transmitted at the maximum power and can be decoded while treating the modulation of the RIS's phases by Encoder 2 as a nuisance.

We finally remark that Corollary 1 and Corollary 2 imply that time-sharing, which would yield a triangular rate region, is suboptimal in both high-and low-power regimes. This is in contrast to the multiplicative MAC studied in [START_REF] Pillai | On the capacity of multiplicative multiple access channels with awgn[END_REF] that assumes two standard active encoders with separate power constraints.

IV. EXAMPLES

In this section, we provide numerical examples for the capacity region derived in Section III. For the phase response set, we consider uniformly spaced phases in the set A = {0, 2 / , . . . , 2 ( -1)/ }, whereas, for the input constellation, we consider amplitude shift keying (ASK) and phase-shift keying (PSK) modulations. In addition, we assume a channel vector with elements having amplitude 1, and a channel matrix with elements having amplitude > 0, where denotes the path-loss ratio between the reflected and direct paths. The phases of and used in this section are summarized in Table I. Furthermore, the expectation over Gaussian random vectors, e.g., z in Proposition 1, is evaluated via a Monte Carlo empirical averages.

In Fig. 2, we plot the capacity region for an average power constraint of = -20 dB, = 2 receiver antennas, = 4 RIS elements, = 2 available phase shifts, a symbol-to-RIS control rate = 2, input constellation given by BPSK, i.e., S = {-1, 1}, and a path-loss ratio of = 0.5 or = 1. In addition, we plot for reference the maximum rate achievable by Encoder 1 for a channel with no RIS, i.e., for = 0. By comparing with the capacity of the channel with no RIS, Fig. 2 illustrates the two roles of the RIS: The RIS can be used to increase the rate of Encoder 1 by beamforming the transmitted signal, and it can enable communication from a passive secondary user. In this regard, Fig. 2 demonstrates that the insights obtained in Corollary 2 by studying the low-power regime carry over to more general conditions. In particular, the maximum rate for Encoder 1 is achieved if and only if Encoder 2 does not communicate, while Encoder 2's maximum rate can coexist with a non-zero rate for Encoder 1.

In contrast, by Corollary 1, for sufficiently high power , both encoders can communicate with the decoder at their respective maximum rates. This is verified by Fig. 3, where we plot the capacity region for an average power constraint of = 40 dB, = 2 receiver antennas, = 4 RIS elements, = 2 available phase shifts, a symbol-to-RIS control rate = 1, input constellation given by 4-ASK, i.e., S = { , 3 , 5 , 7 } with = 1/ √ 21, and a path-loss ratio of = 1. Although Encoder 1 does not gain from the existence of the RIS in the highpower regime, the RIS enables MU communication with a single transmitter in a manner that resembles the single-RF multiple-input multiple-output (MIMO) system [START_REF] Karasik | Adaptive coding and channel shaping through reconfigurable intelligent surfaces: An information-theoretic analysis[END_REF], [START_REF] Li | Single-RF MIMO: From spatial modulation to metasurface-based modulation[END_REF].

V. CONCLUSION

In this work, we have studied the finite-input capacity region of an RIS-aided MU communication system, in which the RIS is not used solely for increasing the rate of an active encoder, but also for enabling communication for a secondary passive encoder. The fundamental trade-offs between the rates of the two encoders were characterized. It was shown that, for sufficiently high power, both users can communicate at their respective maximum rates. Furthermore, in the low-power regime, the maximum rate for the active encoder is achieved if and only if the passive encoder does not communicate, while the passive encoder's maximum rate can coexist with a non-zero rate for the active encoder. Finally, time-sharing was demonstrated to be suboptimal.

APPENDIX

A. Proof of Proposition 1

The model ( 4) can be viewed as a MAC with inputs (s, θ θ θ) and output y.

( , ) ≤ 1 (s; y|θ θ θ), (19a) 2
( , ) ≤ 1 (θ θ θ; y|s), (19b)
and 1 ( , ) + 2 ( , ) ≤ 1 (s, θ θ θ; y)

hold. Since inputs s and θ θ θ are selected from finite sets, the mutual information (s; y|θ θ θ) in ( 19) can be written as (see, e.g., [19, App. A])

(s; y|θ θ θ) = - log 2 ( ) - ∫ ℂ ×1 z ( ) 1 ∈S ×1 s ( 1 ) 1 ∈A ×1 θ θ θ ( 1 ) log 2 2 ∈S ×1 s ( 2 ) 1 d (20)
with z ∼ CN(0, ) and where we have defined the scalar

1 -+ √ 1 + ⊗ ( 1 -2 ) 2 . (21)
By applying the conditional CGF definition in ( 5) to [START_REF] Wu | A survey on MIMO transmission with finite input signals: Technical challenges, advances, and future trends[END_REF], we get

(s; y|θ θ θ) = - log 2 ( ) -(u 1 |s 1 , θ θ θ 1 , z). (22)
Similarly, we also have

(θ θ θ; y|s) = - log 2 ( ) -(u 2 |s 1 , θ θ θ 1 , z), (23a)
(s, θ θ θ; y) = - log 2 ( ) -(u 3 |s 1 , θ θ θ 1 , z). (23b)
Therefore, the region R( s , θ θ θ , , ) in ( 19) is identical to the region R( s , θ θ θ , , )

in [START_REF] Yuan | Reconfigurable-intelligent-surface empowered 6G wireless communications: Challenges and opportunities[END_REF].

B. Proof of Corollary 1

The inclusion C( , ) ⊆ C is trivial since, for all input distributions s ( ) and θ θ θ ( ) with ∈ S ×1 and ∈ A ×1 we have (s) ≤ log 2 ( ) and (θ θ θ) ≤ log 2 ( ). In addition, in the high-power regime, we have the limits

(s; y|θ θ θ) ----→ →∞ (s) ≤ log 2 ( ), (24a)
(θ θ θ; y|s) ----→ →∞ (θ θ θ) ≤ log 2 ( ), (24b)
where equality is achieved for a uniform distributions s ( ) and θ θ θ ( ). Next, note that the noiseless received signal y( ) -z( ) in ( 4) takes values from a discrete set. Furthermore, since channel matrix H and channel vector h are drawn from a continuous distribution, almost surely, for all ∈ [ / ], there exist unique inputs ˆ ( ) ∈ S ×1 and ˆ ( ) ∈ A ×1 such that (see, e.g., [START_REF] Motahari | Real interference alignment: Exploiting the potential of single antenna systems[END_REF])

y( ) -z( ) = √ ˆ ( ) + ⊗ ˆ ( ). (25)
Therefore, for all input distributions s ( ) and θ θ θ ( ), the transmitted signal s( ) and reflection pattern θ θ θ( ) can be correctly jointly decoded in the high-power regime, i.e., we have the limit

Let 1 ( , ), 2 ( , ) ∈ C( , ) be the rate pair achieved using uniform distributions s ( ) and θ θ θ ( ). It hence follows from the region in [START_REF] Karasik | Adaptive coding and channel shaping through reconfigurable intelligent surfaces: An information-theoretic analysis[END_REF] and limits [START_REF] Caire | Suboptimality of TDMA in the low-power regime[END_REF] and [START_REF] Gamal | Network information theory[END_REF] that, almost surely, we have the limits

lim →∞ 1 ( , ) = log 2 ( ), (27a)
lim →∞ 2 ( , ) = log 2 ( ). (27b)

C. Proof of Proposition 2

For input distributions s ( ) and θ θ θ ( ), let functions ˜ ℓ ( , , ℎ ), ℓ ∈ {1, 2, 3}, be defined as

˜ ℓ ( , , ℎ ) -log 2 ( ) -(u ℓ |s 1 , θ θ θ 1 , z), (28)
where (u ℓ |s 1 , θ θ θ 1 , z) are the conditional CGFs in Proposition 1 for the special case in which = = 1. By calculating the derivative of ˜ ℓ ( , , ℎ ) with respect to the power and taking the limit → 0, we get

lim →0 ˜ ℓ ( , , ℎ ) = [u ℓ ] ln(2) , (29)
where random variables u ℓ are defined in [START_REF] Zhang | Physical layer security enhancement with reconfigurable intelligent surface-aided networks[END_REF]. Therefore, it follows from Proposition 1 that the normalized rate pairs ( 1 ( , ℎ ), 2 ( , ℎ )) satisfy

and similarly we have

1 ( , ℎ ) + 2 ( , ℎ ) ≤ [u 3 ] ln(2) . ( 31
)

D. Proof of Corollary 2

Since s 1 , s 2 , and θ θ θ 1 in Proposition 2 are all independent, we have

[u 1 ] = ⊺ θ θ θ 1 + ℎ (s 1 -s 2 ) 2 = ⊺ θ θ θ 1 + ℎ 2 |s 1 -s 2 | 2 ≤ 2 ⊺ θ + ℎ 2

Corollary 1 :

 1 For any finite constellation S of = |S| points and any set A of = |A| phases, let C be the set of rate pairs ( 1 , 2 ) such that

1Fig. 2 . 2 , = 4 , = 2 , = 2 ,

 22422 Fig. 2. Capacity region for = -20 dB, = 2, = 4, = 2, = 2, and BPSK input constellation. The dashed line illustrates the capacity of Encoder 1 for a channel with no RIS.

Fig. 3 . 2 , = 4 , = 2 , = 1 ,

 32421 Fig. 3. Capacity region for = 40 dB, = 2, = 4, = 2, = 1, and 4-ASK input constellation. The dashed line illustrates the capacity of Encoder 1 for a channel with no RIS.

  (s, θ θ θ; y) ----→ →∞ (s) + (θ θ θ) ≤ log 2 ( ) + log 2 ( ).

  1 can achieve maximum rate if and only if Encoder 2 does not communicate, while Encoder 2 can achieve its maximum rate

			RIS
		Encoder 2			elements
		h	h	H	antennas
	1	Encoder 1	Wireless Link	Decoder

  |S| points. Furthermore, the RIS is controlled by Encoder 2 by selecting the phase shift applied by each of the elements of the RIS from a finite set A of = |A| distinct hardware-determined values as a function of the message 2 . Due to practical limitations on the RIS control rate, the phase shifts can only be modified once for each block

	the impinging wireless signal. Encoder 2 represents, for example, a sensor embedded in the
	RIS that applies metasurface-based modulation in order to convey its sensed data without
	emitting a new radio wave [5, Sec. 3.3].
	A coding slot consists of symbols, which are divided into / blocks of	symbols
	each, with / assumed to be integer. Specifically, the codeword transmitted by Encoder
	1 as a function of message 1 occupies the entire coding slot, and it includes symbols
	from a constellation S of =
		in which two encoders communicate with a
	decoder equipped with	antennas over a quasi-static fading channel in the presence of an
	RIS that comprises	nearly-passive reconfigurable elements. Encoder 1 is equipped with
	a single-RF transmitter and encodes its message 1 ∈ [2 1 ] of rate 1 [bits/symbol] into
	a codeword of symbols sent on the wireless link to the decoder. In contrast, Encoder 2
	encodes its message 2 ∈ [2 2 ] of rate 2 [bits/symbol] in a passive manner by modulating
	the reflection pattern of the RIS. The reflection pattern is controlled through a rate-limited
	control link, and is defined by the phase shifts that each of the	RIS elements applies to

of consecutive transmitted symbols. During the th block, the fraction of the codeword of Encoder 1 consisting of transmitted symbols is denoted by s

  1), . . . , y( / ), H , h ), for ℓ = 1, 2, using knowledge of the CSI. Given channel realizations and , a rate pair ( 1 ( , ), 2 ( , )) is said to be achievable if the probability of error satisfies the limit Pr( ˆ 1 ≠ 1 , ˆ 2 ≠ 2 ) → 0 when the codeword length

grows large, i.e., → ∞. The corresponding capacity region C( , ) is defined as the closure of the set of achievable rate pairs.

  Therefore, it follows from the capacity region of the MAC [26, Thm. 4.2] that C( , ) is the convex hull of the union of regions R(

s , θ θ θ , , ) over all input distributions s ( ) and θ θ θ ( ) such that

[s * s] ≤ , where R( s , θ θ θ , ,

) is the set of rate pairs ( 1 ( , ), 2 ( , )) such that inequalities 1

To this end, the configuration of the RIS is This work has been supported by the European Research Council (ERC) and by the Information and Communication Technologies (ICT) under the European Union's Horizon 2020 Research and Innovation Programme (Grant Agreement Nos. 694630, 725731, and 871464).