Reconfigurable Intelligent Surfaces: Bridging the Gap Between

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Reconﬁgurable Intelligent Surfaces: Bridging the gap between scattering and reﬂection Juan Carlos Bucheli Garcia, Student Member, IEEE, Alain Sibille, Senior Member, IEEE, and Mohamed Kamoun, Member, IEEE.

Abstract—In this work we address the distance dependence of is concentrated on algorithmic and signal processing aspects. reconfigurable intelligent surfaces (RIS). As differentiating factor In fact, the comprehension of the involved electromagnetic to other works in the literature, we focus on the array near- specificities has not been fully addressed. field, what allows us to comprehend and expose the promising potential of RIS. The latter mostly implies an interplay between More specifically, reconfigurable intelligent surfaces (RIS) the physical size of the RIS and the size of the Fresnel zones at correspond to the arrangement of a massive amount of in- the RIS location, highlighting the major role of the phase. expensive antenna elements with the objective of capturing To be specific, the point-like (or zero-dimensional) conventional and scattering energy in a controllable manner. Such a control scattering characterization results in the well-known dependence method varies widely in the literature [1]; among which PIN- with the fourth power of the distance. On the contrary, the characterization of its near-field region exposes a reflective diode and varactor based are popular. behavior following a dependence with the second and third In this context, the authors investigate an impedance con- power of distance, respectively, for a two-dimensional (planar) trolled RIS, although the addressed fundamentals are of a and one-dimensional (linear) RIS. Furthermore, a smart RIS much wider applicability. As a matter of fact, by characterizing implementing an optimized phase control can result in a power RIS in terms of the elements’ observed impedance, it is exponent of four that, paradoxically, outperforms free-space propagation when operated in its near-field vicinity. All these possible to study multiple variants. Nonetheless, with energy features have a major impact on the practical applicability of efficiency considerations in mind, the current work will focus the RIS concept. on the passive and non-dissipative purely reactive alternative. As one contribution of this work, the article concludes by It is well known from the radar community that, while a presenting a complete signal characterization for a wireless link mirror is a large reflecting surface with its reflected energy in the presence of RIS on all such regions of operation. decreasing with the 2nd power of distance, a scatterer is usually Index Terms—Reconfigurable Intelligent Surfaces, scattering, considered a near-point object with the scattered energy falling reflection, near-field, far-field, Fresnel, system model. with the 4th power of the distance [5]. As a matter of fact, to the authors knowledge, there is no I.INTRODUCTION consolidated concern in the literature about the nature of the RIS as a scatterer or as a mirror. In fact, although most works The enhancement of wireless connectivity in the last seem to use both terms interchangeably, the conventional view decades has radically changed the way we humans perceive has been through their characterization as scatterers. and interact with our surroundings. Nonetheless, the inter- Particularly, the authors of [1] departed from a general- facing network infrastructure has been mostly confined at ization of the two-ray channel model to argue that the RIS rooftops and distant-away serving sites. does not necessarily obey a path-loss dependence with the Recently, the ancient idea of improving wireless networks fourth power of distance. On the other hand, the authors of [6] by means of relays has been renovated through the concept concluded there that such a strong power law is probably arXiv:1912.05344v1 [cs.IT] 10 Dec 2019 of low-cost smart mirrors. As a consequence, nowadays’ unavoidable for a practical RIS. Nevertheless, no reference scientific literature is full of appellatives such as intelligent to the crucial role of the array near-field region was found as reflecting surfaces (IRS), large intelligent surfaces (LIS), re- a way to explain such discrepancies. configurable intelligent surfaces (RIS), passive relaying arrays Consequently, in this work, we offer a view that unifies (PRA), among others [1], [2], [3], [4]. the previous seemingly opposite scattering/reflection dual per- The underneath idea behind these is to control the char- spectives as a mean to identify scenarios and show the strong acteristics of the radio channel. Even-though such an idea potential behind the RIS concept. In particular, we show is in a sense revolutionary, most of the scientific literature how physical area and distance aspects become of paramount J. Bucheli is pursuing his doctoral degree with Huawei Technologies at the importance for the operation of such devices. Mathematical and Algorithmic Sciences Lab. Boulogne-Billancourt, France. The rest of the paper is organized as follows: the next and Telecom Paris. Institut Polytechnique de Paris. Palaiseau, France. (e-mail: section begins by deepening on the importance of the near- [email protected], [email protected]). A. Sibille is with the Laboratoire Traitement et Communication de field region for RIS and, subsequently, section III studies the l’Information, LTCI, Telecom Paris. Institut Polytechnique de Paris. Palaiseau, implications of operating it over such a region. France. (e-mail: [email protected]). In particular, the latter is based on the derived mathematical M. Kamoun is with Huawei Technologies at the Mathematical and Algorithmic Sciences Lab. Boulogne-Billancourt, France. (e-mail: model and, as well, on the understanding of the interacting [email protected]). Fresnel zones. Additionally, the model and established notions 2

Array far-field As a matter of fact, any array can be approached close enough to be in the far-field of each elementary unit but not Element in the far-field of the full array. The authors of this work refer near-field to that particular region of space as the array near-field. Linear The importance of the array near-field characterization is Antenna array that a RIS requires large antenna arrangements in the close proximity of users, unlike traditional transmitter-receiver links. Therefore, the term array near-field is extensively used here as a way to differentiate it, given its importance in the case of large enough RIS. In the array far-field, the directional characteristics of RIS Array naturally decouple from the separation distance. On the other near-field hand, in the array near-field, the previous dependencies are Figure 1: Pictorial representation of the field regions of a more intricate and call for a different understanding of the linear antenna arrangement. problem. For a simpler insight and self-consistency, appendix A and B are devoted to the fundamental derivations of the far-field and are verified through simulation results obtained from a com- array near-field based on elementary Maxwell’s equations. To mercial electromagnetic solver. be specific, Appendix A reviews the conventional solution Further, section IV explains how RIS offers the possibil- for far-field radiation problems, in order to familiarize with ity of outperforming free-space propagation over completely the adopted notation. Subsequently, Appendix B formally obstructed direct transmitter-receiver links. introduces the array near-field characterization via the here- Finally, the authors conclude by presenting in section V called generalized array manifold. a complete signal-level characterization for a transmitter- receiver link in the presence of RIS, which captures all the III.RIS-MEDIATED PROPAGATION described phenomena. In light of the difficulties of the conventional far-field view to capture the real scenarios of interest, the current section II.THE RIS FAR-FIELD DECOMPOSITION takes a look at how RIS can be characterized using the near- Let us begin by recalling that the far-field approximation field metrics described in Appendix B. imposes a minimum transmitter-receiver separation distance, More specifically, the section begins by reviewing existing notably, so that conventional antenna and propagation models work on modeling of a specific RIS-like architecture in its far- are valid. In fact, the far-field distance increases with the field. Subsequently, relying on the generalized array manifold, square of the largest dimension of the antenna. such model is extended to RIS over the array near-field region. To be clear, in the standard cell-centric network architecture, Altogether, the derived model is subsequently studied on 3 the far-field approximation has greatly sufficed as means of its canonical configuration in order to demonstrate its valid- characterization. Nevertheless, for the case of RIS, one large ity based on results obtained from a commercially-available issue at stake can be stated as a paradox and, also, related electromagnetic solver. to the fact that it has been mostly conceived as an entirely In what follows, we also expose how the Fresnel zone passive1 architecture. decomposition at the RIS interface can provide a powerful Specifically, a RIS must be large as a mean of capturing and intuitive understanding of its promising potential. enough energy; but as it grows, conventional far-field decomposition mandates that the transmitter and receiver must move A. RIS operating in the array far-field region away. Consequently, the larger the RIS gets, the farther the The authors of this paper derived in [4] the model for a transmitter and receiver must be and, therefore, the stronger dipole-based Passive Relaying Array (PRA). Particularly, a the path-loss of the transmitter-RIS-receiver link. PRA can be regarded as a bulky RIS in which the elements Fortunately, this paradox can be circumvented by operating 2 are not necessarily disposed conformal to a surface. Likewise, RIS over its near-field region. In order to understand the the control of PRA is assumed to be mediated through relevance of its near-field, consider the RIS essentially as the dynamically tunable reactive loads. arrangement of multiple antenna elements. Let us consider the derived model for the θ-polarized More specifically, Fig. 1 shows explicitly the array far-field scattered field in the array far-field region of the array: and the array near-field as disjoint regions. The element near- − kr field is also presented in dark blue for reference. Nonetheless, η e −1 E (rˆ , rˆ ) =  a˜ H (rˆ )Z +  X a˜ (rˆ ) E , the element near-field is not considered a region of interest in θ inc obs kπr obs L inc inc,θ this work as it is generally too close to the RIS. (1) where Z ∈ CN×N is the impedance matrix [5] of such an N N×N 1– in the sense that it does not inherently inject energy to the environment. element array; XL = diag(x) ∈ R is a diagonal matrix 2It must be stressed that we are referring exclusively to the near-field of the array and not to the near-field of the array elementary unit itself. 3– the one in which the RIS elements are short-circuited. 3 containing the values of the controllable reactive loads; r is the PRA-observer distance; a˜ (rˆ) is the there-called modified 4 steering vector ; and rînc and rôbs are two (outward) unitary vectors pointing towards the source and observer, respectively. In spite of PRA’s structural assumptions, the model derived in [4] is valid for architectures with arbitrarily disposed elements. Moreover, in the specific case of such a PRA, the architecture was based on dipoles as they are minimum scattering antennas (MSA) [9], allowing to entirely characterize its scattering behavior by the aid of the antenna circuit representation. Nonetheless, in the case of arbitrary array elementary units, the missing piece is utterly the contribution of the structural mode [10], which can be computed once and for all as it is constant with respect to the controllable loads (that are commonly assumed to be the available degrees of freedom). Note that, in the derivation of (1), the impact of the transmitting antenna pattern was assumed negligible as we were assuming it to be in the array far-field. In what follows, and for simplicity of presentation, we will Figure 2: Schematic view of Tx, Rx and RIS. resort to the assumption that the transmitter antenna is an isotropic source of fields. For simplicity, let us disregard the phenomenon of mutual coupling (i.e. Z = Z III × ) and, additionally, short circuit B. The field scattered by RIS at the array near-field region A N N all elements (x =  Ω). Under those circumstances, (3) can To begin with, it must be stressed that the array far-field ap- be simply expressed as: 5 proximation of both the reception and transmission processes 2 2 is implicitly captured in (1) by the modified steering vector k η H iso Eθ (rrx, rtx) = aθ (rrx) aθ (rtx) Fθ Itx. (4) a˜ (rˆ) and, as well, by the distance dependence. ZA Nonetheless, as exposed in Appendix B, an expression valid If, additionally, we assume that the RIS is provided with ˆ iso for the array near-field can be obtained by identifying the θ-polarized isotropic elementary units, i.e. F⊥(rˆ) = θFθ , the steering vector as a special case of the generalized array expression (4) can be simplified further to: manifold, the latter of which is given by: 2 2 N k η 3 Õ ( ) iso − − r − rn Eθ rrx, rtx = Fθ Itx G rrx rn G rtx rn , (5) ( ) ( − ) ≤ ZA ap r n := G r rn F0,p n N, (2) n=1 |r − rn | ∀ G(r) with G(r) being the free-space Green function of Appendix A where is, once more, the Green function of Appendix A. Even-though the expression in (5) corresponds to the array and F (rˆ) the radiation vector of the array elementary unit 0,p near-field electric field intensity, it characterizes the linear along the p direction of polarization. By doing so, it can be shown that the scattered electric field combination of (locally) far-field sources. Thus, the radiation intensity at the array near-field reads: density (power per unit area) resulting from the RIS can be related simply to its scattered field through P = (2η)−1|E|2. 2 2 H −1 iso Eθ (rrx, rtx) = k η aθ (rrx) Z +  XL aθ (rtx) Fθ Itx, (3) In particular, introducing rt,n:=|rtx − rn | and rr,n:=|rrx − rn | as the distances from the transmitter and receiver to every where r and r are the complete coordinates6 of the trans- tx rx RIS element, respectively, the radiation density of the scattered mitter and receiver relative to RIS’ coordinate reference (e.g. field reads: the RIS center as in Fig. 2) and, as we are dealing with zˆ- N 2 oriented dipoles, only the θ polarization is considered. 4 3 − krr, n − krt, n k η iso 6 2 Õ e e P(rrx, rtx) = F |Itx| . (6) Moreover, note that the dependence on the input current to 2|Z |2 θ 4πr 4πr A n=1 r,n t,n the transmitter antenna Itx exposes the role of its respective iso Observe that, even-though we have assumed hypothetical radiation vector (i.e. Fθ ). −1 Observe also that Z +  XL is a transpose symmetric isotropic elements, (6) allows to analyze the radiation density matrix, which has often (e.g. [1], [2], [3]) been characterized as regards the transmitter and receiver locations relative to the as a diagonal matrix containing complex exponential factors element’s disposition, i.e. rn n ≤ N. ∀ that account for digitally-tunable phase shifts. In the scenario of Fig. 2 is shown a symmetrical setup in which the transmitter and receiver are both at a distance r from 4The modified denomination was added given that, unlike the conventional the center of the RIS, which is composed of a 2D periodic steering vector, the one in [4] includes the impact of the element pattern. structure along its planar surface with a total of N elements. 5– those which jointly compose the scattering process. 6 Moreover, in light of the loading condition under evaluation – as opposed to (1) where |rtx | > rFF and |rrx | > rFF and, therefore, there was only a dependence on the directions of incidence and observation. (x =  Ω) in (6), the transmitter and receiver are positioned 4 symmetrically (45◦ from the vector normal to RIS’ surface) as Rx' required by the Snell-Decartes law of reflection. The latter, in order to expose a case in which waves are naturally interfering constructively7 towards the receiver side. r For simplicity, an odd number of 2Kh+1 horizontal elements C 45° Infinite PEC plate and 2Kv + 1 vertical elements are disposed on the surface of RIS; i.e. for a total of N = (2Kh + 1)(2Kv + 1) elements. 45° 45° In what follows, two configurations will be evaluated: r r 1) A linear RIS with Kh = 10 and Kv = 0 for a total number of 21 λ/2-spaced elements. Tx Rx 2) A planar RIS with Kh = Kv = 10 for a total number of λ/ 441 2-spaced elements. Figure 4: Schematic view of the Tx, Rx for mirror reflection. As observed in Fig. 3, the radiation density clearly exposes different behaviors for the array near-field (r < rFF) and its far-field region (r > rFF). C. The Fresnel zone perspective More specifically, the array far-field region unsurprisingly Previously, it was shown that the operation of a planar RIS exposes a path-loss related to r−4 in both the linear and in its near-field uncovers a behavior that resembles free-space planar configurations. The operation over such a sector can propagation, in particular, under a non-line of sight transmitter- be characterized through metrics used in the radar community receiver link. In spite of this, the derived mathematical model (such as the radar cross section) as done in [4]. Nonetheless, does not immediately offer an insightful understanding from the latter is clearly not the most interesting region of operation the perspective of the propagating wave. for the RIS. Thus, the aim of this section is to develop an intuition on On the other hand, the array near-field exposes a seemingly how this is possible and, more specifically, on the interaction oscillatory behavior around r−2 and r−3 for the planar and of the size of RIS and its near-field region in relation to the linear configurations, respectively. The latter is explained by Fresnel zones at the RIS interface. the fact that, through its finite number of antenna elements, As a preamble, such an intuition will be confirmed in the RIS is sampling the field at discrete points in space. section III-D through a simple yet insightful simulation on a In particular, the oscillations illustrate the constructive and commercial electromagnetic solver8, based on the well known destructive interference caused by the complex exponential method of moments (MoM). In addition, the latter will not terms in (6) as induced by the Green function. Nonetheless, as only corroborate the predictions of section III-B, but it will it is shown in section IV, such oscillations can be compensated open the possibility for a path-loss better than the free-space for through smart dephasing. one, as revealed in section IV. Let us begin by considering Fig. 4, where an unobstructed 7 In fact, they don’t interfere perfectly (as it will be clear later) but such a transmitter-receiver link is presented. Specifically, note the setup serves to illustrate the point the authors want to make. presence of an infinite perfect electrically conducting (PEC) plane parallel to the line joining the transmitter and receiver Scattered radiation density vs. distance to RIS sides. Note that, by virtue of the well-known principle of images, -4 ∝ r the PEC plane can be removed to study separately the line- of-sight and the reflection. 100 In fact, the contribution of the reflection is obtained by mirroring the receiver side and studying the equivalent envi- Planar RIS ronment. Thus, the equivalent setup of Fig. 5 is used in what follows to uncover the spatial distribution of the fields around -2 ∝ r such an interfacing plane. Linear RIS More specifically, Fig. 5 shows the transmitter as well as 10-5

Radiation density the mirrored image of the receiver in a perfectly unobstructed environment. Additionally, the first three Fresnel zones are pre- r-3 ∝ sented as a way to understand the most contributing regions. The lth Fresnel zone is a region of space whose boundaries r = 100λ FF are ellipsoids defined as the paths with (l−1) π and l π propagation phase-shifts with respect to the shortest central path [11] 100 101 102 103 Distance to RIS (r/λ) for its inner and outer boundaries, respectively. Additionally, the transmitter and receiver locations are the focal points of Figure 3: Radiation density (power per unit area) of the field scattered by RIS versus distance for the setup of Fig. 2. 8– i.e. WIPL-D, https://wipl-d.com/ 5

Phase distribution on the first four Fresnel zones π

π/2

-π/2 Figure 5: View of the equivalent Tx-Rx’ scenario for the contribution of the reﬂector.

such ellipsoids, whose boundaries at the midpoint C can be -π approximated by: Figure 7: Phase distribution on the first four Fresnel zones. r lλ White and orange represent constructive and destructive R (r) r , r lλ, (7) l u 2 interference, respectively. th where Rl(r) corresponds to the radius of the l Fresnel zone for the symmetrical (in r) arrangement of Fig. 4. On the contrary, for smaller values of r, high order Fresnel It is well known that the contribution of the first Fresnel zones are almost equally strong due to 2) but, at the same zone is the most important one. In order to understand why time, they interfere destructively with their successive one as that is the case, note the following: a result of 3). 1) As shown in Fig. 6, the Fresnel zones are defined as In particular, note that even numbered Fresnel zones always ellipsoids with constant propagation phase relative to the interfere destructively; as opposed to odd numbered ones that shortest path between the transmitter and receiver. As a interfere constructively as shown in Fig. 7. The reader might consequence, the closer the transmitter and receiver sides, realize at this point how, by a proper dephasing, RIS could in the smaller the Fresnel zones and vice-versa. principle outperform free-space propagation. 2) The Fresnel zone boundaries get closer for increasing l: To continue, it might be useful to look at an infinitely large r λ √ mirror as spatially integrating the fields over such an infinite R (r) − R (r) = r O l l+1 l 2 aperture. As explained by the equivalence principle over the scenario of Fig. 4, such a spatial integration converges to the −2 3) There is a phase difference of 2π between any pair of r path-loss dependence we are so familiarized to. paths distanced two Fresnel zones from each other. In fact, if the mirror were finite and centered in the shortest Therefore, in the extreme of large r, high order Fresnel transmitter-receiver path, the spatial integration would be zones (i.e. higher than one) are significantly weak relative to truncated. At the same time, due to the Fresnel zone resizing, the first zone and, thus, they do not contribute significantly to such a truncation would expose oscillations if the transmitter the received power; see 1). and receiver were symmetrically moved. The latter gives an intuitive understanding to the array near- field behavior of Fig. 3. Lth Fresnel zone Finally, as a mean of linking RIS’ field decomposition and the Fresnel zone perspective, observe from (7) (as well as

from (19) in Appendix B) that Rl=1(rFF) = D; with rFF being b a the low limit of the array far-field and D being the visible 9 e dimension of the array. Tx Rx In other words, the RIS is being operated in its near-field region when at least the first Fresnel zone of the transmitter- d c receiver equivalent path (see Fig. 5 and Fig. 7) is perfectly contained within the RIS itself.

9By visible dimension we refer to the smallest diameter of a circle located th on the plane transversal to the direction of propagation and containing the Figure 6: Geometrical representation of the l Fresnel zone. array. 6

|S |2 vs. distance for direct and scattered paths Scattered radiation density vs. distance to RIS 2,1 -20 Planar RIS (short circuit) Reflected path (WIPL based) Planar RIS (smart dephasing) Direct path (WIPL based) -30 102 Planar RIS (Model based)

-40 100 -50

(dB) -60 2 | 10-2 2,1

|S -70 Radiation density -80 10-4

-90

-6 -100 r = 100λ 10 r = 100λ FF FF 100 101 102 103 100 101 102 103 Distance to plane-RIS (D/λ) Distance to RIS (r/λ) | |2 Figure 8: S2,1 quantifying the power received through Figure 9: Radiation density (power per unit area) of the field reflection from a finite metal plate vs. transmitter-receiver scattered by the RIS with smart dephasing versus distance. distance.

IV. FREE-SPACE PROPAGATION D. Model and intuition corroboration OUTPERFORMEDTHROUGH RIS In order to confirm that the last assertions are indeed correct, and that the model of section III-B predicts the right Recall that, from section III-C, the contribution of the phenomena, we have evaluated a very simple scenario using even numbered Fresnel zones is always destructive. As a 11 WIPL-D. consequence, if such zones are contained within the RIS , More specifically, a two port setup with two vertically- a dense enough architecture might in principle be able to polarized half-wave dipoles (acting as the transmitter and compensate for their destructive nature. receiver) in the presence of a finite metal plate was simulated. As a matter of fact, in connection with the notion estab- The size of the plate was fixed to 10λ × 10λ and the dipoles lished in section III-C, the model in section III-B allows to were positioned symmetrically (45◦ from the vector normal to transparently predict the maximum obtainable power for an the plate’s surface) a distance r; exactly like for the RIS setup architecture-specific RIS. of section III-B. In particular, a smart-enough RIS would, at its best, com- 2 In particular, the power transmission coefficient |S2,1| was pensate for the path-related phase shift; giving for the received computed as we are technically operating over the near-field radiation density: of the plate (i.e. making far-field metrics such as the radar 2 4 3 N cross section invalid). k η iso 6 2 Õ 1 1 Pmax(rrx, rtx) = F |Itx| , (8) Additionally, as the aforementioned setup computes the 2|Z |2 θ 4πr 4πr A n=1 r,n t,n net (direct+reflected) fields, separate simulations (with and without the plate) were done in order to subtract the the direct using the notation of (6) to represent all involved quantities. path and obtain the reflected contribution. Such a radiation density for the architecture of section III-B As observed in Fig. 8, the reflected path exhibits the is shown and compared to the short-circuit, i.e. x =  Ω, in expected oscillatory behavior at the near-field region of the the following. plate. Note as well that, in spite of the discrete nature of the As observed in Fig. 9, the path-loss for the mentioned smart- RIS, the model is able to capture with fair accuracy the details dephasing technique does deviate from the conventional r−2 of its continuous-equivalent10. Nonetheless, it can be shown in the array near-field region. In fact, it exposes a behavior that the WIPL numerical results and the model converge when that outperforms free-space propagation for a completely- the element density of the model is increased within such a obstructed (NLOS) transmitter-receiver link. confined region of space. Note, as well, that the lack of oscillations in the dashed curve is a direct consequence of the removal of the (even- numbered) Fresnel zone destructive nature; as expected from section III-C. It must be stressed that an element spacing of λ/2 was enough in Fig. 9 to maintain a constant log-log increase of 10Naturally, RIS’ curve was vertically shifted to make it coincide with WIPL’s results in the array far-field as, in particular, the multiplicative coefficients in (6) cannot be determined for hypothetical isotropic antennas. 11Namely, if it is being operated in the array near-field region. 7 receive power with decreasing distance12 for a significant part higher order scattering might strongly attenuate contributions of the array near-field region. other than the line-of-sight ones. More importantly, the behavior observed in Fig. 9 implies that the dependence with the fourth power of the distance is VI.CONCLUSION a consequence of the constant phase along the RIS. Typically, In this work, we have presented a view that unifies the the latter manifests at the array far-field region where, as opposite behavior of RIS as a scatterer and as a mirror. In explained in section III-C, the RIS is fully contained within particular, relying on fundamental electromagnetics, we do so the first Fresnel zone. Nonetheless, as explored in this section, as a mean to identify scenarios and show the strong potential such a behavior can be artificially enforced in the array near- behind the RIS concept. field. We have shown that, depending on its size and distance, the In particular, while a r−4 dependence at large distance RIS can be observed as a zero, one or two dimensional object may be seen as poor, it turns out to be advantageous when whose radiated power exposes a dependence with the fourth, approaching the mirror below r as a way to avoid the FF third or second power of the distance, respectively. transition to the (inferior) r−2 regime. Additionally, we have employed the Fresnel zone decomposition to build an intuitive understanding of the interplay of V. COMPLETELINK-LEVELSYSTEMCHARACTERIZATION the involved quantities. More specifically, we have uncovered To finish up, the authors would to like briefly summarize the role of the phase in determining the ultimate path-loss the implications of section III to the familiar link-level char- exponent and to show how, through smart dephasing, free- acterization of communication systems. space propagation can be outperformed. For simplicity, in what follows, we will assume single- Moreover, as one of the main contributions of the current element, single-polarization and isotropic transmitter and re- work, we have presented a model for the signal-level char- ceiver sides. On the other hand, the RIS elementary unit can acterization of a transmitter-receiver link in the presence of be arbitrarily defined as its impact is accounted for in the RIS. Particularly, such a model concisely captures all described generalized array manifold of (2). phenomena over all the regions of operation. It must also be added that the multi-antenna transmit- To finalize, even though the amount of elements studied in ter/receiver extension can be straightforwardly envisaged by this work might seem tremendous, it must be kept in mind that virtue of the superposition principle. Nonetheless, its math- the described planar architecture would barely occupy 1 m2 ematical representation can easily become cumbersome as a at a central frequency of 3 GHz. More importantly, it would result of the multi-location dependencies13. present a path-loss more favorable than free-space (under a Therefore, relying on the RIS model derivation of sec- completely obstructed direct transmitter-receiver link) for a tion III-B, the complete link-level system model can be shown distance up to rFF = 10 m, on such a frequency of operation. equivalent to: ˜ ˜ H −1 APPENDIX A y = htr G(rtr) + hRIS ap (rrx) Z +  XL ap(rtx) s + n, (9) THE FAR-FIELD RADIATED BY AN ANTENNA where s, y and n are the conventional input, output and additive Recall that the electric field intensity radiated by a source white Gaussian noise at the receiver side, respectively; G(r) current density is given in terms of the so-called a magnetic is the free-space Green function of Appendix A; rtr is the potential vector A [7]: shortest-path transmitter-receiver distance; rtx and rrx are the 1 h i transmitter and receiver locations relative to RIS’ coordinate E = ∇∇ · A + k2A , (10) N reference, respectively; and ap(r) ∈ C is the generalized  ωµ array manifold of (2) (see Appendix B for its derivation) where and µ are the electric permittivity and magnetic polarized along p for an N element RIS. permeability of the propagation medium, and ω = 2π f . Additionally, in (9), h˜tr and h˜RIS are spatially-flat chan- Particularly, the magnetic potential vector (used as a conve- nel coefficients that represent a scenario in which all links nient step to obtain E) is obtained as the convolution of the (transmitter-receiver, transmitter-RIS and RIS-receiver) are source current density J(r) with the free-space Green function dominated by their line-of-sight components. These channel of the Helmholtz equation [7]: coefficients also absorb all physical quantities that are not of e− k |r | concern for link-level characterization; allowing to introduce ∇2G(r) + k2G(r) = −δ(r), G(r) := , (11) the dimensionless signal denomination14. 4π|r| Note that, if the transmitter-receiver link is either obstructed where G(r) is the Green function, k = 2π/λ is the wave-number or suffers from strong multi-path propagation, its impact shall and δ(r) is the three-dimensional delta function. ˜ be embedded onto htr. On the other hand, we do not expect In mathematical form, the magnetic potential vector reads: ˜ hRIS to be greatly impacted by multi-path propagation as ¹ A(r) I µ J(r0) G(r − r0) d3r0, 12 = (12) – to be specific, of 40 dB per decade. V 0 13– in particular for the case in which the array elementary units of the transmitter and receiver sides are not isotropic anymore. where I is the input current, J(r) is the current distribution of 14– where a signal is simply defined as an observable change in a quantity. the element normalized to such an input current, r is the field 8

(observation) point and r0 is the source (integration) point (i.e. Therefore, as a ﬁrst step, consider the source current density over V 0 that is a volume containing all sources). of a multiple antenna architecture such as:

At this point, we ought to highlight the linearity of the N integro-differential operator in (12) with the input current that Õ J(r) = In J0(r − rn), (17) propagates until E as per (10). n=1 In particular, it is because of such a linearity that we are where N is the number of elements, r is the location of the allowed to resort to tools such as the array factor as a mean of n nth element with respect to a common reference, I is the input describing the behavior of arrays in terms of their elementary n current at the nth element and J (r) is the (identical) current unit. 0 distribution of the array elementary unit normalized to such an input current. A. Far-field approximation Based on Appendix A, the magnetic potential vector for 15 The shifted argument of the Green function, i.e. G(r − r0) such an architecture can be written as : in (12), can be approximated for the region known as far-field ¹ N Õ 0 0 3 0 through: A(r) = µ In J0(r − rn) G(r − r ) d r , (18) V 0 0 n=1 e− k (|r |−rˆ ·r ) 2l2 G(r − r0) , for |r| l and |r| , (13) where V 0 should include all the sources represented by (17). u 4π|r| λ Recall that the most commonly used antenna metrics (di- where l is the largest dimension of the smallest integration rectivity, gain, antenna aperture, etc.) give an approximately volume V 0 in (12) containing all sources and rˆ is a unitary correct characterization for the far-field region of the antenna vector pointing at the far-field observation point. or antenna array under consideration. Therefore, in (13), the dependence with the shifted obser- Nonetheless, note that the lower limit of the array and vation point in the numerator was replaced by a first order element far-field regions are given in terms of the largest approximation whereas, in the denominator, it was replaced dimension of the array D and its elementary unit D0 by: by an approximation of order zero. 2 2D2 Moreover, solving (12) with (13) and plugging it in (10), the 2D elem 0 rFF = , r = , (19) radiated electric field intensity E can be shown equal to [7]: λ FF λ c/f e− kr where λ = corresponds to the wavelength of operation and E(r) = − Ikη F⊥(rˆ), F⊥(rˆ) = F (rˆ) × rˆ, (14) c to the speed of light. 4πr As a consequence, the far-field approximation of the Green where F (rˆ) is a far-field measure known as the radiation function of (13) cannot be used in (18) to compute A over vector and is explicitly defined as: elem the array near-field region of interest (i.e. rFF < |r| < rFF). ¹ On the other hand, in the following, we will resort to a 0  krˆ ·r0 3 0 F (rˆ) := J(r ) e d r , (15) different strategy as a mean of approximating it at the region V 0 of interest. 0 with V fully containing the source current distribution. The far-field condition revisited: if the integration and To conclude, it must be noted that F⊥(rˆ) = F (rˆ)×rˆ in (14) summation are swapped in (18), such an expression can be is also known as the effective length (vector) of the antenna. rewritten as: In particular, the effective length vector is of relevance when N ¹ studying far-field radiation incident to the antenna. Õ 00 00 3 00 A(r) = µ In J0(r ) G (r − rn) − r d r , (20) More specifically, the electro-motive force E at the antenna 00 n=1 V terminals can be written simply as [5]: 00 0 where the substitution r = r − rn was used and accounted E = F⊥(rˆ)· Einc, (16) for in the volume of integration. Note that, if J0(r) is concentrated in a closed domain over where Einc is the electric field intensity characterizing the 3 r ∈ R and |rn−rm| > dmax n , m with dmax being the largest incident radiation at the location of the receiving antenna under dimension of such a closed∀ domain, (20) can be expressed as: consideration. N ¹ Õ 00 00 3 00 A(r) = µ In J0(r ) G (r − rn) − r d r , (21) APPENDIX B 00 n=1 Vn THE ARRAY NEAR-FIELD RADIATION where V 00 = ÐN V 00 with V 00 tightly enclosing the domain Let us continue by considering the problem of determining n=1 n n over which J (r − r ) is concentrated and, more importantly, the electric field intensity E at the array near-field region 0 n such regions are disjoint, i.e. V 00 Ñ V 00 = i j. resulting from a multi-antenna arrangement as source of fields. i j ∅ , The importance of the previous result lies on∀ that, while the In particular, as it will be clear in what follows, the far-field Green function cannot be used in (20), it can be used characterization at the array near-field region captures the behavior over the array far-field as a particular case. Thus, 15– observe that we don’t resort directly to the radiation vector as this one allowing to model RIS on both such regions of interest. would inherently solve the problem in the array far-field region. 9

over the separate domains of integration in (21). The latter, as NOTETOTHEREADER | − | elem elem long as r rn > rFF n with rFF given by (19). Observe that the definition of the radiation vector in terms ∀ In particular, the expression (21) can be largely simplified of the normalized current density makes the radiation vector by identifying the radiation vector of (15) through: be slightly different to the one in [7]. As a matter of fact, their definition is equal to ours mul- AFF (r−r ) 0 n tiplied by the input current. Nonetheless, this makes the link z }| { N − k |r−rn | between transmission and reception modes straightforward; i.e. Õ e r − rn A(r) = I µ F , (22) through the same metric (not having to introduce a different n 4π|r − r | 0 |r − r | n=1 n n notation for the effective length vector as these become | {z } G(r−rn ) essentially equivalent).

FF where A0 (r) is identified as the far-field approximation of the REFERENCES magnetic potential vector of the array elementary unit, G(r) [1] E. Basar, M. Di Renzo, J. De Rosny, M. Debbah, M. Alouini and R. as the Green function in (11) and F0(rˆ) is the radiation vector Zhang, "Wireless Communications Through Reconfigurable Intelligent of the elementary array unit of Appendix A. Surfaces," in IEEE Access, vol. 7, pp. 116753-116773, 2019. ( ) [2] Y. Han, W. Tang, S. Jin, C. Wen and X. Ma, "Large Intelligent Surface- Note in (22) that F0 rˆ depends exclusively on the direction Assisted Wireless Communication Exploiting Statistical CSI," in IEEE th of the observation point relative to the location of the n Transactions on Vehicular Technology, vol. 68, no. 8, pp. 8238-8242, element16. Moreover, unlike the conventional array far-field, Aug. 2019. [3] E. Bjornson, O. Ozdogan, E. Larsson, “Intelligent Reflecting Surface the radiation vector cannot be factored out of the summation vs. Decode-and-Forward: How Large Surfaces Are Needed to Beat and, therefore, an array factor cannot be defined anymore. Relaying?,” June 2019. [Online]. Available: arXiv:1906.03949. By properties of the operators in (10), given that the [4] J. C. Bucheli Garcia, M. Kamoun and A. Sibille, "Reconfigurable passive FF relaying array for coverage enhancement," 2019 IEEE Wireless Commu- argument of A0 (r) in (22) is simply translated on every nications and Networking Conference (WCNC), Marrakesh, Morocco, summation term, the total radiated field in the array near-field 2019, pp. 1-6. region can be written as: [5] C. A. Balanis, Antenna Theory Analysis and Design. Chapters 2 and 8. Wiley, 2005. [6] O. Ozdogan, E. Bjornson, E. Larsson, “Intelligent Reflecting Surfaces: N Õ r − rn Physics, Propagation, and Pathloss Modeling 2019,” October 2019. [On- E(r) = − kη I G(r − r ) F ⊥ . (23) line]. Available: arXiv:1911.03359. n n 0, |r − r | n=1 n [7] S. J. Orfanidis, Electromagnetic Waves and Antennas. Rutgers University, 2016. Note also that, for the general case of dual polarized [8] R. C. Hansen, "Relationships between antennas as scatterers and as radiators," in Proceedings of the IEEE, vol. 77, no. 5, pp. 659-662, May transmitting antennas, such a total radiated field can be written 1989. in terms of its p polarization as: [9] B. A. Munk, Finite Antenna Arrays and FSS. Chapter 2. Wiley, 2003. [10] R. C. Hansen, "Relationships between antennas as scatterers and as N radiators," Proceedings of the IEEE, vol. 77, pp. 659-662, May 1989. Õ S. J. Orfanidis, "Electromagnetic waves and antennas" Ep(r) = − kη In an,p(r). (24) [11] T. Wayne, "Electronic Communication Systems - Fundamentals Through n=1 Advanced," Pearson, p. 1023. where an,p(r) is an order-2 tensor quantity called here the generalized array manifold; formally defined as: r − rn an,p(r) := G(r − rn) F0,p n ≤ N, (25) |r − rn | ∀ with G(r − rn) being the translation of the Green function of (11) and F0,p(r) denoting the radiation vector along the p direction of polarization17. Particularly, observe that if single p-polarized radiation is considered, the array manifold in (25) collapses into a vector ( ) ≤ that can be simply denoted as ap r n = an,p n N; for N array elements. ∀ Moreover, if (23) is to be evaluated in the array far-field region (i.e. |r| > rFF), the conventional array factor can be recovered by replacing the Green function with its far- field approximation of (13). Thus, showing that (25) indeed generalizes the array manifold with the array far-field region as a special case.

16– as this last one is a far-ﬁeld measure with respect to such an element. 17note that pˆ must always be orthogonal to rˆ.