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# Anomalous and specular reflections of reconfigurable intelligent surfaces: configuration strategies and system performance

Mattia Merluzzi, Antonio Clemente

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Mattia Merluzzi, Antonio Clemente. Anomalous and specular reflections of reconfigurable intelligent surfaces: configuration strategies and system performance. IEEE Wireless Communications Letters, 2024, 13 (10), pp.2707-2711. 10.1109/LWC.2024.3441933. cea-04760707

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Submitted on 6 Jan 2025

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# Anomalous and Specular Reflections of
Reconfigurable Intelligent Surfaces:
Configuration strategies and system performance

Mattia Merluzzi and Antonio Clemente

Univ. Grenoble Alpes, CEA, Leti, F-38000 Grenoble, France

***Abstract***—Reconfigurable intelligent surfaces (RISs) enable wireless communication in harsh propagation conditions. In fact, when impinging on an RIS, wireless signals can be anomalously reflected, i.e., not obeying to the Snell's law, to reach intended destinations at arbitrary locations in space and time. However, current reconfiguration strategies do not consider the presence of classical specular reflections (SRs), which exist due to hardware realizations/impairments and/or the fact that RISs are placed over larger non-reconfigurable surfaces in practical scenarios (e.g., walls). This phenomenon can harm wireless systems performance due to the energy spilled over non-intended directions, thus increasing interference and other metrics if not taken into account when configuring RISs. In this paper, we focus on the SRs generating from a surface behind the RIS, and we propose an SR-aware end-to-end system model and RIS reconfiguration that take into account this undesired effect. Numerical results show the effect of SRs on the achievable rate regions of an interference channel under various RIS configurations, with a considerable difference between SR-aware and SR-agnostic strategies.

***Index Terms***—Reconfigurable Intelligent Surfaces, spurious reflections, achievable rate regions, 6G

## I. INTRODUCTION

Reconfigurable Intelligent Surfaces (RISs) are largely recognized as a key technology enabler of 6G, as they can push wireless communication systems' performance towards new limits in terms of reliability, coverage, secrecy, and energy efficiency [1]. In fact, RISs can be dynamically reconfigured to opportunistically shape wireless signal reflection and/or transmission that do not obey the Snell's law. This opportunity paved the way towards a rich literature on RIS (re)configuration strategies and optimization [2], scenarios and use cases [1], channel models [3], and hardware realization [4], [5]. In most of the literature, RISs are modeled as arrays of nearly passive elements, whose phases can be dynamically reconfigured. Based on the signal direction of arrival and the configured phases, an RIS tipically exhibits an anomalous reflection (AR), i.e., a reflection that does not obey the Snell's law. For wireless communication, this is beneficial in scenarios in which the Line-of-Sight (LOS) path between a transmitter and a receiver is highly attenuated, or even not present due to dense environments.

Recently, it has been observed that RISs can be far from guaranteeing a single reflection, due to hardware realization constraints and impairments [6], but also due to the fact that, in practical scenarios, they are mounted on (possibly larger) non-reconfigurable surfaces, such as walls of different materials, i.e., with different reflection coefficients [7]. In these

cases, the typical (mirror-like) specular reflections (SRs) of these additional surfaces can harm system performance due to wireless power spilled over non-intended directions, thus increasing interference and spoofing capabilities of malicious agents among others. New link budget formulas have been proposed to integrate the presence of such phenomena [8].

However, while [7] provides useful models for RISs mounted on walls in practical scenarios, it does not investigate on the actual impact of SRs on a wireless communication system. The goal of this paper is to bridge the gap between the studies on such electromagnetic properties, and their actual effect on wireless communication systems. In particular, we study the impact of SRs on the performance of a wireless system in the presence of interference channels. First, we propose a simple to handle yet informative system model that integrates the presence of a larger surface (e.g., a wall) behind the RIS. Then, based on this model, we study the actual response of this RIS-over-wall system in all directions, showing the impact on the AR direction (ARD) and the SR direction (SRD). Further, considering a wireless system with two base stations (BSs), two user equipment (UE) and an RIS-over-wall, we explore the performance in terms of achievable rate regions, showing that an SR-agnostic RIS configuration can dramatically shrink the latter, depending on several parameters that include the wall relative size with respect to the RIS, and its reflection coefficient. The achievable rate regions obtained through various optimization strategies show how it is possible to overcome the SR drawback, under certain conditions and by paying different prices. Then, the main goal of this work is to warn the reader and the researchers who wish to give a contribution to the literature on RISs optimization, on how the expected performance can be harmed if all these propagation properties are not taken into account when modeling the system.

The remainder of this paper is organized as follows: in Section II, we present the system model, including the end-to-end (E2E) channel model that comprises RIS and wall reflections. Section III illustrates the RIS-over-wall responses under various conditions and parameters, using SR-agnostic RIS codebooks. Section IV presents the interference-prone system scenario under investigation, and the performance in terms of achievable rate regions under different RIS configurations. Finally, Section V draws the conclusions and presents the potential impact of this work with the challenges ahead.

***Notation:*** bold upper and lower case letters denote matrices and vectors, respectively; $A_{kn}$ denotes the $(n, k)$-th element of matrix **A**, while Tr($\mathbf{A}$) its trace; whereas, $| \cdot |$ de notes the absolute value of a complex number; finally, $j = -1$.

<footnote>This work was supported by the H2020 project RISE-6G no. 101017011, by the SNS JU project TERRAMETA no. 101097101, and by the ANR under the France 2030 program, grant "NF-YACARI: ANR-22-PEFT-0005".</footnote>
## II. SYSTEM MODEL

In this section, we present the channel, transceivers, and RIS models. Let us first focus on the general channel model between two network nodes, being them a BS, UE, or RISs. We present the general case of Multiple-Input-Multiple-Output (MIMO) channels. Considering a transmitting network node k equipped with $N_k$ antenna elements, and a receiving network node p equipped with $N_p$ antenna elements, we denote by $\mathbf{H}^{p,k} \in \mathbb{C}^{N_p \times N_k}$ the matrix containing the wireless channel coefficients between k and p. We assume a channel model with a Line-of-Sight (LOS) component $\mathbf{H}^{p,k}(\text{LOS})$ and a random non-LOS (NLOS) component $\mathbf{H}^{p,k}(\text{NLOS})$. For the sake of simplicity, we assume network nodes to lay on the same elevation plane, and we focus on azimuth angles between them. The system model can be straightforwardly generalized to integrate elevation angles, but this goes beyond the scope of this work and does not change its messages. Denoting by $d_{mn}^{p,k}$ the distance between the n-th element of transmitter k and the m-th element of receiver p, and by $\phi_{mn}^{p,k(k,p)}$ the azimuth angle between the broadside direction of array k(p) and array p(k) and the line passing through the two elements, the LOS component can be written as follows [9]:

$$H_{mn}^{p,k(\text{LOS})} = \frac{q}{\pi^2} \frac{\cos(\phi_{mn}^{p,k}) \cos(\phi_{mn}^{k,p}) e^{-jkd_{mn}^{p,k}}}{\sqrt{\kappa}} \quad (1)$$

where $\kappa = \frac{2\pi}{\lambda}$ the wave number, with $\lambda$ denoting the wavelength. The $\theta$ terms represent the non-isotropic response of the array elements. Instead, the NLOS components are modeled as independent and identically distributed according to $\sim CN(0, 1)$ [10]. Finally, denoting by $K$ the Rice factor, the wireless channel between transmitting network node k and receiving network node p can be written as

$$\mathbf{H}^{p,k} = \frac{\mathbf{s}^{-1} \frac{\theta_{p,k}}{K+1}}{\sqrt{-K} \mathbf{H}^{p,k}(\text{LOS}) + \mathbf{H}^{p,k}(\text{NLOS})} , \quad (2)$$

where $\theta_{p,k}$ denotes the average path loss. This channel definition and modeling applies to all propagation paths involved in the wireless communication between a transmitter Tx and a receiver Rx. In the following, we will mostly exploit LOS channels, although keeping the model as generic as possible. Finally, for a wideband system, the proposed model needs to be extended, e.g., as in [11], in order to take into account the selective responses over different sub-carriers. In the following, we present the RIS model and the E2E channel model, which involves all channels between a transmitter and a receiver (i.e., Tx $\to$ RIS and RIS $\to$ Rx).

### A. RIS modeling and end-to-end channel

In this work, we consider transceivers communicating through an RIS without a direct path, i.e., we assume an obstruction with virtually infinite attenuation between them. This is a typical assumption in literature around RISs, since a LOS component, if present, is generally much stronger than the RIS-assisted path, due to a multiplicative path loss law affecting the latter [10]. Then, for a wireless communication involving an RIS r, the end-to-end channel $\mathbf{H}_{\text{E2E}}^{\text{Rx,Tx}}$ between the transmitter and the receiver entails the channel

<figure><img src="image_3.png" /><figcaption>(a) Generic transmitter and receiver</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(b) RIS placed over a wall</figcaption></figure>

<caption>Fig. 1. Transceivers and RIS-over-wall modeling parameters</caption>

$\mathbf{H}_{\text{Rx,Tx}}^{r}$ between the transmitter and the RIS, and the channel $\mathbf{H}_{\text{Rx,r}} \in \mathbb{C}^{N_{\text{Rx}} \times N_r}$ between the RIS and the receiver, with $N_r$ the number of RIS elements. Neglecting the mutual coupling between RIS elements (the latter could be included in the analysis [3]), the RIS response is modeled through a diagonal reflectivity matrix $\mathbf{\Theta} \in \mathbb{C}^{N_r \times N_r}$. In particular, the l-th element of the diagonal can be modeled as $\mathbf{\Theta}_{ll} = \sqrt{\alpha_l} e^{j\theta_l}$, where $\alpha_l \in [0, 1]$ is a reflection coefficient that depends on the hardware, and $\mathbf{\Theta} \in [0, 2\pi]$ is a reconfigurable phase shift, which is generally quantized in practical realizations [12].

Besides bouncing on the RIS with controlled reflection (AR), the transmitted signal generally impinges on the area that surrounds it, incurring in more classical reflection phenomena that obey the Snell's law. We model this additional reflection (SR) through a fictitious surface surrounding the RIS, and whose phases cannot be controlled (i.e., phase shift are equal to 0). In particular, we consider the area as a regular grid of $N_w$ scatterers, and we denote by $\mathbf{\Theta}^w \in \mathbb{C}^{N_w \times N_w}$ the reflectivity matrix, with diagonal elements $\mathbf{\Theta}^w = \frac{\alpha_w}{\sqrt{\alpha_w}}$, where $\alpha_w \in [0, 1]$ denotes the reflection coefficient. The latter depends on the material of the surface surrounding the RIS. We use the same channel model integrating the same element responses as in (1) to approximate the diffuse scattering from the wall [13]. Then, the E2E channel $\mathbf{H}_{\text{E2E}}^{\text{Rx,Tx}}$ reads as follows

$$\mathbf{H}_{\text{E2E}}^{\text{Rx,Tx}} = \frac{\mathbf{H}_{\text{Rx,r}} \mathbf{\Theta} \mathbf{H}_{\text{Rx,Tx}}^{r}}{\text{AR term (typical literature)}} + \frac{\mathbf{H}_{\text{Rx,w}} \mathbf{\Theta}^{w}}{\text{SR term}} \frac{\mathbf{H}_{\text{w,Tx}}^{w}}{\mathbf{X}} \quad (3)$$

The channel model in (3) can be generalized and enriched by adding other reflections from other walls that are located, e.g., in the direction of secondary lobes. In this work, we only consider one wall behind the RIS, being the most impactful one in terms of spilled energy, due to directive communications. Now, denoting by $\mathbf{w}_{\text{Tx}}(\text{Rx} \in \mathbb{C}^{N_{\text{Tx}}(\text{Rx})} \times 1}$ the transmitter precoding (receiver combining) vector, the expected channel power gain between Tx and Rx reads as

$$\Gamma^{\text{Rx,Tx}} = E \frac{\mathbf{w}^{\text{Rx}} \mathbf{H}_{\text{Rx,Tx}}^{\text{Rx}} \mathbf{w}^{\text{Tx}}}{\mathbf{w}^{\text{Rx}} \mathbf{H}_{\text{Rx,Tx}}^{\text{Rx}} \mathbf{w}^{\text{Rx}}}, \quad (4)$$
<figure><img src="image_5.png" /><figcaption>Fig. 2. Normalized resulting RIS patterns for different wall relative extensions and reflection coefficients</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 3. Channel gain in the ARD normalized to the SRD as a function of wall relative extension, for different wall reflection coefficients</figcaption></figure>

where $E \text{ Tr } \mathbf{w}_{\text{TX}}(\text{Rx})\mathbf{w}_{\text{TX}}^{H}(\text{Rx}) \begin{Bmatrix} 1 \\ \vdots \\ 1 \end{Bmatrix} = 1$. The model in (3) considers the simplest case of a purely reflecting RIS, while novel architectures are being studied to perform simultaneous transmission/reception/reflection, a.k.a. STAR-RIS [14]. These results are complementary to the STAR-RIS model, as the latter considers the same elements used, simultaneously, for transmission and reflections. In our model, different elements belong to either the RIS or to an uncontrollable surface (e.g., the wall), thus simultaneously reflecting in different directions.

### III. ANOMALOUS AND SPECULAR REFLECTIONS

In this section, we show the effect of the introduced system model in terms of beam patterns obtained from a signal reflection from an RIS surrounded by an uncontrollable surface (e.g., a wall). We consider a carrier frequency $f_c = 28$ GHz, with a source illuminating the RIS (and the wall) from an angle of $-45^\circ$ with respect to their broadside direction. The source is a Uniform Planar Array (UPA) with $10 \times 10$ elements and inter-element distance $A/2$, transmitting towards the RIS. The RIS is a UPA with $20 \times 20$ elements and the same inter-element distance, configured through a predefined codebook, maximizing the gain in its broadside direction. We consider a pure LOS channel with a single path, i.e., the one passing through the RIS (and the wall). As already mentioned, the wall is modeled as set of scatterers surrounding the RIS, organized into a regular grid, and behaving as the elements of the RIS in terms of angular response, although being non-reconfigurable. In Fig. 2, we show the beam pattern generating from the RIS

in all direction $\in [-90^\circ, 90^\circ]$, normalized to the maximum gain obtained in the broadside direction in the absence of the wall. The patterns result from the described RIS-over-wall structure, and they are obtained for different wall reflection coefficients $\alpha_w$ (across the different figures 2a, 2b, and 2c) and wall extensions (encoded by the parameter $\gamma_w$ within the same plot- with different colors). In particular, for the latter, denoting by $A = \frac{\lambda N_r}{2}$ the area of the RIS, we define the area of the wall as $A^w = \frac{\gamma^w \lambda N_r}{2} = \frac{\lambda N_r}{2}$ with $\gamma^w \ge 1$ a parameter that determines the wall relative size with respect to the RIS. With $\gamma_w = 1$, the system model boils down to the case without the wall, i.e., the typically exploited model. Also, Fig. 3 shows the ratio between the channel gain in the ARD and the SRD, as a function of $\gamma^w$, with different wall reflection coefficients.

As visible from the figures, the SR term can play a significant role in the resulting pattern, depending on the relative size of the wall and the reflection coefficient $\alpha_w$. Looking at Fig. 2a, we can notice how, for $\gamma_w = 1.2$, the specular reflection is already non-negligible. Furthermore, for higher wall relative extensions (e.g., $\gamma_w = 2$), the gain in the SRD overcomes the one in the ARD. This effect is mitigated for lower reflection coefficients (cf. Figs. 2b, 2c, 3).

From this first results we can deduce that, with respect to the currently exploited models that consider a single AR, additional EMF exposure, interference, and other undesired effects may arise in practical RIS deployment scenarios in the SRD. In the next section, we present a simple yet informative system scenario in which such effects have a considerable impact on the achievable rate regions of interference channels.

### IV. INVESTIGATED WIRELESS SYSTEM: INTERFERENCE CHANNELS AND ACHIEVABLE RATE REGIONS

<figure><img src="image_7.png" /><figcaption>Fig. 4. Scenario: interference channels in a RIS-over-wall aided system</figcaption></figure>
is a reduction of the maximum spectral efficiency of UE₁ with respect to the max gain in ARD solution (orange solid line), as visible from the right hand points of the curves. This is due to the choice of the objective function, which promotes law interference, but it could also decrease the channel gain in the ARD. The above observations also apply to larger wall relative sizes (e.g., γ<sub>w</sub> = 1.4), while larger walls (see γ<sub>w</sub> = 2 - purple lines) make it difficult to improve performance. Finally, a dramatic performance degradation is obtained with the SR-agnostic (GA-based opt.) (dotted lines), due to its inherent in-capability of predicting mutual interference, especially the one coming from the wall. More reconfiguration strategies can be envisioned to explore different achievable rate regions. From the proposed ones, we can draw the following conclusions:

* the presence of an SR can have a strong impact on the performance of a wireless communication system with interference, namely on the achievable rate regions;

* an SR-aware configuration strategy can extend the achievable rate regions at the potential cost of a reduced maximum spectral efficiency of a sub-set of users. In [20], the authors show how it is possible to affect the SRs from walls, also validating the model with EM simulations;

* an SR-agnostic optimization has detrimental effects on system performance in terms of achievable rate regions.

* irrespective of the strategies, a system dynamic optimization should certainly take into account the presence of SRs (and the resulting achievable rate regions) to guarantee wireless system performance and constraints (e.g., interference, EMF exposure, secrecy, etc.)

Also, by further increasing the wall size, impact on performance diminishes, since only the illuminated part has to be considered for the cascade channel model in (3), especially in the considered scenario of directive communications. Finally, another key role in performance is represented by the distance between the transmitter and the RIS, and between the RIS and the receiver. Although not presented in this work, the former mostly affects UE₁ performance, while the latter affects both UE's performance due to interference reduction when increasing the distance. Eventually, the case with the UE in the near field region of the RIS should also be investigated.

## V. CONCLUSIONS AND FUTURE DIRECTIONS

In this paper, we built on recent observations on how RISs could exhibit spurious reflections beyond the optimized ones, which may generate additional undesired interference. We exploited the proposed model in the context of a typical wireless system with interference channels, to study the effect of various RIS configurations on the achievable rate regions of the system. Simulation results show that agnosticism of SRs has dramatic effects on the achievable rate regions. Future works include the dynamic optimization of RISs to best exploit these properties in different contexts including interference management, communication secrecy and EMF-awareness, but also the improvement of performance through other techniques including, e.g., reflection modulation. SRs could also be exploited as a benefit, for example for serving multiple users, with better performance through active RISs.

<table><caption>TABLE I
SIMULATION PARAMETERS</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>B5(BS) position/Ce position/Eg(RIS) position (x,y,z)</td><td>[0.5,1.5][0.3,3.0][1.5,2.5][0.05,1.5][0.1,1.5][1.5,1.5] m</td></tr><tr><td>B5(RIS) arrays, carrier freq., noise power, PL exp.</td><td>UPA (10 × 10/20 × 20), 28 GHz, -97 dBm, 3</td></tr><tr><td>BSs transmit power</td><td>1000 linearly spaced values between 0.1 mW and 1 W</td></tr></tbody></table>

Machine Learning techniques also represent a promising way of handling the configuration problem with a data-driven approach. Finally, EM-based simulations are fundamental to validate the proposed model and corresponding performance.

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