End-to-End Mutual Coupling Aware Communication Model for Reconfigurable Intelligent Surfaces: An Electromagnetic-Compliant Approach Based on Mutual Impedances
11 End-to-End Mutual-Coupling-Aware Communication Model for Reconfigurable IntelligentSurfaces: An Electromagnetic-Compliant Approach Based on Mutual Impedances
Gabriele Gradoni,
Member, IEEE and Marco Di Renzo,
Fellow, IEEE
Abstract —A reconfigurable intelligent surface (RIS) is anemerging technology for application to wireless networks. Weintroduce a physics and electromagnetic (EM) compliant commu-nication model for analyzing and optimizing RIS-aided wirelesssystems. The proposed model has four main characteristics: (i)it is end-to-end , i.e., it is formulated in terms of an equivalentchannel that yields a one-to-one mapping between the voltagesfed into the ports of a transmitter and the voltages measured atthe ports of a receiver; (ii) it is
EM-compliant , i.e., it accountsfor the generation and propagation of the EM fields; (iii) it is mutual-coupling-aware , i.e., it accounts for the mutual couplingamong the sub-wavelength unit cells of the RIS; and (iv) it is unit-cell-aware , i.e., it accounts for the intertwinement betweenthe amplitude and phase response of the unit cells of the RIS.
Index Terms —Wireless, reconfigurable intelligent surfaces.
I. I
NTRODUCTION
A reconfigurable intelligent surface (RIS) is an emergingand promising software-defined technology for enhancing theperformance of wireless networks at a low cost, power, andcomplexity [1], [2]. An RIS consists of a large number ofinexpensive and nearly-passive scattering elements that can beconfigured to customize the propagation of the radio waves [3].For analyzing and optimizing RIS-aided wireless systems,sufficiently realistic and accurate yet tractable communicationmodels that account for the physics and electromagnetic (EM)properties of the scattering elements of the RIS are needed.This is an open research issue, and only a few EM-awaremodels for RIS-aided wireless systems are available to date[4]-[7]. In [4], an experimentally-validated path-loss modelfor a non-homogenizable RIS is proposed. In [5], a path-lossmodel for a homogenizable RIS is introduced by using thevector theory of scattering. In [6], an approach for modelingthe interplay between the amplitude and phase response of alossy unit cell is proposed. In [7], the mutual coupling amongthe active radiating elements of a large surface is investigated.These research works are noteworthy but tackle specificissues in isolation, and do not provide an operational and end-to-end communication model for analyzing and optimizingRIS-aided wireless systems, in which the impact of the EMfields and of the currents impressed and induced on theradiating elements can be explicitly identified. In this letter,we tackle these fundamental issues by introducing a newphysics- and EM-compliant communication model for RIS-aided wireless systems. The proposed approach (i) has itsfoundation on the laws of electromagnetism for the generation,propagation, and scattering of the EM fields; (ii) accounts forthe intertwinement between the amplitude and phase responseand for the mutual coupling between the radiated EM fieldsand the current induced on closely spaced scatterers; and (iii)yields a one-to-one mapping between the voltages measured at
Manuscript received Sep. 6, 2020. G. Gradoni is with BT, the University ofNottingham and Cambridge University, UK. M. Di Renzo is with CNRS andParis-Saclay University, France (e-mail: [email protected]).
Fig. 1: System-model.the output ports of a multi-antenna receiver and the voltagesimpressed at the input ports of a multi-antenna transmitter.Therefore, the proposed communication model is end-to-end,EM-compliant, mutual-coupling-aware, and unit-cell-aware.More specifically, we introduce a circuit-based communi-cation model for RIS-aided wireless systems that is based onthe mutual impedances between all existing radiating elements(transmit/receive antennas, passive scatterers) [8]. We provethat the impact of (i) the incident EM fields (impressed by agenerator or scattered/induced); (ii) the input voltages at thetransmitter and the load impedances at the receiver; and (iii)the reconfigurable load impedances that control the passivescatterers of the RIS is jointly taken into account, and isexplicitly unfolded and individually observable in the proposedend-to-end communication model. To the best of our knowl-edge, this letter introduces the first complete EM-compliantcommunication model for RIS-aided wireless systems.
Notation : Vectors and matrices are denoted in bold font; ˆz is the unit-norm vector z ; j = √− is the imaginary unit; ε and µ are the permittivity and permeability in vacuum; c = 1 / √ ε µ is the speed of light; η = (cid:112) µ /ε is thecharacteristic independence in vacuum; f is the frequency; ω = 2 πf is the angular frequency; λ is the wavelength; k =2 π/λ is the wavenumber; δ ( · ) is the Dirac delta function; ∂ z denotes the second-order partial derivative with respect to z ;and · denotes the scalar product between vectors.II. S YSTEM M ODEL
We consider the wireless system in Fig. 1, in which atransmitter with N t antennas and a receiver with N r antennascommunicate through an RIS that is made of N ris = M × N passive scatterers. The locations of the t th transmit antenna,the r th receive antenna, and the s th passive scatter are r ξ = x ξ ˆx + y ξ ˆy + z ξ ˆz for ξ = { t, r, s } . For illustrativepurposes, the transmit antennas, the receive antennas, andthe passive scatterers are cylindrical thin wires of perfectlyconducting material whose length is l and whose radius a (cid:28) l is finite but negligible with respect to l (thin wire regime). Theinter-distance between adjacent radiating elements is denotedby d . Since the transmitter, the receiver, and the RIS may a r X i v : . [ c s . I T ] S e p have different implementation requirements, we consider threedistinct triplets ( l ξ , a ξ , d ξ ) for each of them. All thin wires areparallel to the ˆz -axis. The proposed model is, however, generalenough for application to different types of antennas, passivescatterers, spatial distributions, etc., provided that the radiatingelements are minimum scattering antennas [8]. A. Transmitter Modeling
Each transmit antenna is fed by an independent voltagegenerator V Gt , t = 1 , , . . . , N t , according to the delta-gap model [9, Eq. 8.28] that imitates an antenna fed bya transmission line. As shown in Fig. 1, the generator isapplied between the lower and upper halves of the transmitantenna across a short gap of negligible size. The voltagegenerator has an internal impedance Z Gt , and, therefore, thecurrent flowing through the port of the transmit antenna is I Tt = ( V Gt − V Tt ) /Z Gt , where V Tt is the voltage at the inputof the port. In our system model, V Gt and Z Gt are given, and I Tt and V Tt are variables to be determined as elaborated next. B. Receiver Modeling
Each receive antenna is connected to an independent loadimpedance Z Lr , r = 1 , , . . . , N r . Therefore, the relationbetween the voltage drop across the load, V Lr , and the currentflowing through the load, I Lr , is I Lr = − V Lr /Z Lr . Sincethe load impedance is assumed to be infinitesimally small,this model is usually referred to as the impedance delta-gapmodel. In our system model, Z Lr is given, and I Lr and V Lr are variables to be determined as elaborated next. C. RIS Modeling
The RIS is modeled as a collection of N ris passive scat-terers. Each passive scatterer is a cylindrical thin wire ofperfectly conducting material, which is connected, betweenthe lower and upper halves of the wire, to a tunable load Z Smn , m = 1 , , . . . , M and n = 1 , , . . . , N . Similar tothe receiver, the relation between the voltage drop across theload, V Smn , and the current flowing through the load, I Smn ,is I Smn = − V Smn /Z Smn . Also, Z Smn is given, and I Smn and V Smn are variables to be determined as elaborated next.By appropriately optimizing the tunable loads Z Smn atthe ports of the RIS, one can adaptively configure the wavetransformations applied to the incident EM fields. Assuming,e.g., that each tunable load is a positive intrinsic negative(PIN) diode, the corresponding impedance is Z Smn = R Smn + jωL Smn and Z Smn = Z (cid:107) Smn + jωL Smn with (cid:46) Z (cid:107) Smn =1/ R Smn + jωC Smn for forward and negative bias implemen-tations, respectively, where R Smn , L Smn , and C Smn are theresistance, inductance, and capacitance of the PIN diode. Byvarying the inductance and/or the capacitance, one can recon-figure the response of each passive scatterer. The resistanceusually accounts for the internal losses of the PIN diode.
D. Modeling Assumptions and Methodology
Our objective is to introduce a physics- and EM-compliantcommunication model for RIS-aided wireless systems thatis accurate enough, but, at the same time, is tractable andinsightful enough for application to wireless communications.Due to the complexity of the generation, propagation, andscattering of the EM waves, some assumptions are necessary. In this section, we clarify the assumptions of our systemformulation, by focusing on the transmission, interaction, anddetection of the EM waves in free space. The generalizationto random media is postponed to a future research work.An antenna, whether transmitting or receiving, is driven byan external electric field that acts as a source. Based on Sec.II-A, a transmit antenna is driven by a voltage generator that isapplied to its port. Thus, the source electric field is determinedby the voltage generator. A receive antenna, e.g., for thereceiver and the RIS in Secs. II-B and II-C, respectively, isdriven by the external electric field (generated by any physicalsource, e.g., a transmit antenna driven by a voltage generator)that impinges upon it. In both cases, the source electric fieldis referred to as the incident field and is denoted by E (inc) .This incident field induces a current on the antenna, either intransmit or receive mode, and the induced current generates,in turn, its own electric field that is radiated away from theantenna itself. This electric field is referred to as the radiated field and is denoted by E (rad) . For any antenna, thus, theelectric field at any observation point, r , is the sum of theincident and radiated fields, i.e., E ( r ) = E (inc) ( r )+ E (rad) ( r ) .Given E (inc) , the key component to develop an EM-compliant communication model lies in determining the dis-tribution of the current induced on (the surface of) eachradiating element. This current distribution is determined bythe boundary conditions of the electric field on the surfaceof each antenna. When the radiating elements are made ofperfectly conducting material, the boundary conditions imposethat the tangential component of the electric field vanishes onthe surface of the antenna. In Fig. 1, the thin wire antennas areparallel to the ˆz -axis. Thus, the tangential component of theelectric field is E z ( r ) = E ( r ) · ˆz = E (inc) ( r ) · ˆz + E (rad) ( r ) · ˆz = E (inc) z ( r )+ E (rad) z ( r ) , and the boundary conditions at any pointon the surface, r S , of any (transmit or receive) radiating ele-ment impose E z ( r S ) = 0 , i.e., E (rad) z ( r S ) = − E (inc) z ( r S ) .Consider a generic radiating element (antenna) located at r ξ . The tangential component of the incident field on thesurface of the antenna, E (inc) z ( r S ) , is, thus, sufficient forcomputing the distribution of the current on the antenna itself.In particular, let I ( z (cid:48) ) , for z ξ − l /2 ≤ z (cid:48) ≤ z ξ + l /2 , denote the(unknown) surface current. Under the thin wire regime, I ( z (cid:48) ) is distributed only along the ˆz -axis, i.e., I ( z (cid:48) ) ≈ I z ( z (cid:48) ) ˆz , andit is solution of Pocklington’s equation [9, Eq. (8.22)]: (cid:90) z ξ + l/ z ξ − l/ I z ( z (cid:48) ) (cid:0) ∂ z + k (cid:1) G ( z, z (cid:48) ) dz (cid:48) = c P E (inc) z ( r S ) (1)where G ( z, z (cid:48) ) = exp ( − jk R ( z, z (cid:48) ))/ R ( z, z (cid:48) ) is the Greenfunction, R ( z, z (cid:48) ) = (cid:113) ( z − z (cid:48) ) + a , c P = − j πωε , andthe constraint that I z ( z (cid:48) ) vanishes at the two ends of theantenna, i.e., I z ( − l/
2) = I z ( l/
2) = 0 needs to be imposed.The exact solution of (1) usually requires numerical meth-ods, e.g., the method of moments [9]. For the consideredsystem model, however, an approximate closed-form solutionis available under the thin wire regime for minimum scatteringradiating elements, since the current density is not influencedby the proximity of other elements (even if they are termi-nated in an open circuit). In particular, I z ( z (cid:48) ) in (1) can be approximated with the sinusoidal function: I z ( z (cid:48) ) ≈ I ( z ξ ) sin ( k ( l/ − | z (cid:48) − z ξ | )) / sin ( k l/ (2)where z ξ − l /2 ≤ z (cid:48) ≤ z ξ + l /2 and I ( z ξ ) is the current thatflows through the port of the antenna. Based on Secs. II-A-II-C, I ( z ξ ) is equal to I Tt , I Lr , and I Smn for the transmitter,receiver, and RIS, respectively. In particular, I ( z ξ ) is unknownand is a variable to be estimated. This is discussed next.It is worth mentioning that I ( z ξ ) in (2) can be applied toisolated antennas and to closely spaced antennas as shownin Fig. 1. Numerical results for thin wire antennas havesubstantiated that (2) is sufficiently accurate provided that theinter-distance between the antennas is d > λ/ or d > λ/ [8]. This is usually sufficient for application to RIS-aidedwireless systems, since the inter-distance between the unit cellsis usually in the range λ/ ≤ d ≤ λ/ [3].III. I MPEDANCE -B ASED M UTUAL C OUPLING M ODELING
Based on (2), we introduce three enabling results for Sec.IV: (i) we compute the electric field that is radiated by anisolated antenna; (ii) we introduce the concept of self/mutualimpedances; and (iii) we calculate the voltage observed at theports of a pair of coupled (transmit and receive) antennas.We consider two arbitrary antennas χ = { p, q } , which arecharacterized by the pair ( l χ , a χ ) and whose locations are r χ = x χ ˆx + y χ ˆy + z χ ˆz . χ operates in transmit or receive mode,and, in either cases, the current on its surface is I z,χ ( z (cid:48) ) ≈I ( z χ ) sin ( k ( l χ / − | z (cid:48) − z χ | )) / sin ( k l χ / in (2). A. Isolated Antenna: Radiated Electric Field
Based on Sec. II-D, we need to compute only the tangentialcomponent of the electric field that is radiated by χ . This fieldevaluated at the observation point r is denoted by E (rad) z,χ ( r ) . Lemma 1:
Consider the transmit antenna p . Under the thinwire regime, E (rad) z,p ( r ) is equal to ( c E = − jη / (4 πk ) ): E (rad) z,p ( r ) = c E (cid:90) z p + l p / z p − l p / F p ( r , z (cid:48) ) G p ( r , z (cid:48) ) I z,χ ( z (cid:48) ) dz (cid:48) (3)where G p ( r , z (cid:48) ) = exp ( − jk R p ( r , z (cid:48) ))/ R p ( r , z (cid:48) ) with R p ( r , z (cid:48) ) = (cid:113) a p + ( z − z (cid:48) ) if ( x, y ) = ( x p , y p ) and R ( r , z (cid:48) )= (cid:113) ( x − x p ) + ( y − y p ) + ( z − z (cid:48) ) if ( x, y ) (cid:54) =( x p , y p ) , and F p ( r , z (cid:48) ) = ( z − z (cid:48) ) R p ( r , z (cid:48) ) (cid:18) R p ( r , z (cid:48) ) + 3 jk R p ( r , z (cid:48) ) − k (cid:19) − ( jk + 1/ R p ( r , z (cid:48) ))/ R p ( r , z (cid:48) ) + k (4) Proof:
From Maxwell’s equations, the radiated electricfield is E (rad) z,p ( r ) = − c E (cid:0) ∂ z + k (cid:1) V z,p ( r ) [9, Eq. (3.15)],where V z,p ( r )= (cid:82) z p + l p / z p − l p / G p ( r , z (cid:48) ) I z,p ( z (cid:48) ) dz (cid:48) is the normal-ized magnetic potential, and, under the thin wire regime, G p ( r , z (cid:48) ) ≈ (2 π ) − (cid:82) π exp ( − jk R p ( r , z (cid:48) ))/ R p ( r , z (cid:48) ) dz (cid:48) . B. Self and Mutual Impedances
Consider the radiating elements p and q , either in transmitor receive mode, and let I z,p ( z (cid:48) ) and I z,q ( z (cid:48)(cid:48) ) be the currentson the surface of p and q , respectively. When p and q areclose to each other, whether in transmit or receive mode, someenergy created locally within a specific antenna may reach andinfluence the boundary field of the other antenna. For example, part of the energy radiated from p impinges upon q and excitessome currents, which, in turn, generate a scattered field thatimpinges upon p and excites other currents. This interchangeof energy is known as mutual coupling. This effect, which isespecially relevant in RISs whose passive scatterers are closelyspaced, can be quantified through the mutual independence Z qp that characterizes the mutual coupling induced by p on q . Definition 1:
Let E (rad) qp ( z (cid:48)(cid:48) ) = E (rad) z,p (cid:0) r S q (cid:1) be the electricfield in (3) that is radiated by p and is observed on the surface S q of q , where r S q = x q ˆx + y q ˆy + z (cid:48)(cid:48) ˆz with z q − l q /2 ≤ z (cid:48)(cid:48) ≤ z q + l q /2 under the thin wire regime. Z qp is defined as: Z qp = − I ( z q ) I ( z p ) (cid:90) z q + l q / z q − l q / E (rad) qp ( z (cid:48)(cid:48) ) I z,q ( z (cid:48)(cid:48) ) dz (cid:48)(cid:48) (5)The self impedance Z pp is obtained from (5) by setting q = p . By virtue of reciprocity, Z pq = Z qp in vacuum.An important property of Z qp is that it depends only on thegeometry of the radiating elements, as formalized below. Lemma 2: Z qp can be explicitly formulated as follows: Z qp = (cid:90) z q + l q / z q − l q / (cid:90) z p − l p / z p − l p / g qp ( z (cid:48) , z (cid:48)(cid:48) ) ˜ I z,p ( z (cid:48) ) ˜ I z,q ( z (cid:48)(cid:48) ) dz (cid:48) dz (cid:48)(cid:48) (6)where ˜ I z,χ ( z (cid:48) ) = sin ( k ( l χ / − | z (cid:48) − z χ | )) / sin ( k l χ / and g qp ( z (cid:48) , z (cid:48)(cid:48) ) = jη (4 πk ) − F p (cid:0) r S q , z (cid:48) (cid:1) G p (cid:0) r S q , z (cid:48) (cid:1) . Proof:
It follows by inserting (3) in (5).From (6), we evince that Z qp is independent of the (un-known) currents I ( z χ ) that flow through the ports of χ = { p, q } . This confirms that Z qp depends only on the geometry ofthe antennas and is independent of the voltage generators andthe (tunable) loads. Thus, it can be precomputed once and thenused for different RIS-aided wireless systems. The usefulnessof Z qp will become apparent in the next sub-sections. C. Pair of Antennas: Two Transmit Antennas
Consider the antenna elements p and q , and assume thatthey both operate in transmit mode. Thus, they are fed bytwo generators V Gp and V Gq as described in Sec. II-A. Theobjective of this section is to compute an explicit analyticalrelation between the voltages V Tp and V Tq and the currents I Tp = I ( z p ) and I Tq = I ( z q ) as a function of the mutualcoupling between p and q . This is possible by using the selfand mutual impedances Z qp , as stated in the following lemma. Lemma 3:
Consider two transmit antennas p and q . Thevoltages and currents at their ports are interwoven through thefollowing linear system (known as constitutive equations): (cid:26) V Tp = Z pp I Tp + Z pq I Tq V Tq = Z qp I Tp + Z qq I Tq (7)where Z qp is the mutual impedance in (6). Proof:
From Maxwell’s equations, the radiated electricfield on the surface S q of q , i.e., evaluated at r S q = x q ˆx + y q ˆy + z (cid:48)(cid:48) ˆz with z q − l q /2 ≤ z (cid:48)(cid:48) ≤ z q + l q /2 under the thinwire regime, is equal to E (rad , mc) z,q (cid:0) r S q (cid:1) = E (rad) z,p (cid:0) r S q (cid:1) + E (rad) z,q (cid:0) r S q (cid:1) in the presence of p , where “mc” stands formutual coupling and the two addends can be computed from(3). By definition of delta-gap model, the voltage generatoris modeled as an incident electric field that is non-zero justoutside the surface of the gap, while the gap itself is filled with perfectly conducting material. Therefore, the incidentelectric field on S q is E (inc) z,q (cid:0) r S q (cid:1) = V Tq δ ( z (cid:48)(cid:48) − z q ) , andthe boundary conditions for perfectly conducting wires ap-ply to the entire surface S q of q . For perfectly conductingwires, in particular, the boundary conditions on S q impose E (inc) z,q (cid:0) r S q (cid:1) + E (rad , mc) z,q (cid:0) r S q (cid:1) = 0 , which yields E (rad) z,p (cid:0) r S q (cid:1) + E (rad) z,q (cid:0) r S q (cid:1) = − V Tq δ ( z (cid:48)(cid:48) − z q ) . Consider the integral: J q = − I z,q ( z q ) (cid:90) z q + l q / z q − l q / E (rad , mc) z,q (cid:0) r S q (cid:1) I z,q ( z (cid:48)(cid:48) ) dz (cid:48)(cid:48) (8)From (5), by definition, we obtain J q = Z qp I Tp + Z qq I Tq . From the boundary conditions, we obtain J q = − I z,q ( z q ) (cid:82) z q + l q / z q − l q / ( − V Tq δ ( z (cid:48)(cid:48) − z q )) I z,q ( z (cid:48)(cid:48) ) dz (cid:48)(cid:48) = V Tq .By comparing these two results, the second equation in (7)is proved. The first equation follows mutatis mutandis. D. Pair of Antennas: One Transmit and One Receive Antenna
Consider the antenna elements p and q that operate intransmit and receive mode, respectively. Therefore, p is drivenby a voltage generator according to Sec. II-A and the port of q is connected to a load impedance according to Secs. II-Cand II-B. The relation between the voltages V Tp and V Lq or V Smn and the currents I Tp = I ( z p ) and I Lq = I ( z q ) or I Smn = I ( z q ) at the ports of p and q is given as follows. Lemma 4:
Consider one transmit antenna p and one receiveantenna q . The voltages and currents at their input ports areinterwoven through the following system of linear equations: (cid:26) V Tp = Z pp I Tp + Z pq I q V q = Z qp I Tp + Z qq I q (9)where V q = V Lq and I q = I Lq for the receiver, V q = V Smn and I q = I Smn for the RIS, and Z qp is given in (6). Proof:
It is similar to the proof of Lemma 3 with themain difference that q operates in receive mode, i.e., q isnot driven by a voltage generator but is connected to a loadimpedance. The voltage drop across the load impedance canbe viewed as generated by an incident electric field that isnon-zero just outside the surface of the gap occupied by theload, while the gap itself is filled with perfectly conductingmaterial. By definition, the equivalent incident electric field is E (inc) z,q (cid:0) r S q (cid:1) = V q δ ( z (cid:48)(cid:48) − z q ) with V q given in Secs. II-B andII-C. The rest of the proof is the same as for Lemma 3.IV. E ND - TO -E ND C OMMUNICATION M ODEL
Based on the enabling results proved in Lemmas 3 and 4,we introduce an end-to-end communication model for RIS-aided wireless systems that accounts for an arbitrary numberof coupled antenna elements at the transmitter and the receiver,and passive scatterers at the RIS. Some special system con-figurations are analyzed in order to gain engineering insights.
A. End-to-End Equivalent Channel Model
Let V G be the N t × vector whose t th entry is V Tt and V L be the N r × vector whose r th entry is V Lr . With end-to-end communication model, we refer to an analyticalframework that formulates V L as a function of V G . Moreprecisely, we wish to find an N r × N t matrix, H E2E , suchthat V L = H E2E V G . The matrix, H E2E is referred to asend-to-end equivalent communication channel, which accounts for the transmit and receive antenna elements, the passivescatterers of the RIS, the load impedances at the receiver, and,more importantly, the tunable load impedances of the RIS. Thematrix H E2E is given in the following theorem.
Theorem 1:
Let Z XY be the N x × N y matrix whose (x , y) thentry is the mutual impedance in (6) between the x th and y thradiating elements of X and Y , where X , Y = { T , S , R } and N x , N y = { N t , N ris , N r } , and T , S , R identify the transmitter,the RIS, and the receiver, respectively. H E2E is as follows: H E2E = (cid:0) I L + P RSR Z − − P RST P − P TSR Z − (cid:1) − + P RST P − (10)where Z G is the N t × N t diagonal matrix whose ( t, t ) thentry is Z Gt , Z RIS is the N ris × N ris diagonal matrix whose ( mn, mn ) th entry is Z Smn , Z L is the N r × N r diagonal matrixwhose ( r, r ) th entry is Z Lr , P GTST = Z G + P TST , and: P XSY = Z XY − Z XS ( Z RIS + Z SS ) − Z SY (11) Proof:
Let V T and I T be the N t × vectors whose t th entries are V Tt and I Tt , so that V T = V G − Z G I T ; I L be the N r × vector whose r th entry is I Lr so that V L = − Z L I L ; and V S and I S be the N ris × vectors whose mn th entries are V Smn and I Smn , so that V S = − Z RIS I S . Theproof is obtained in two steps. (i) The proofs of Lemmas 3and 4 are applied to the system with N t transmit antennas, N r receive antennas, and N ris passive scatterers. This yields V = Z I , where V = [ V T , V S , V L ] , I = [ I T , I S , I L ] , and Z = [ Z TT , Z TS , Z TR ; Z ST , Z SS , Z SR ; Z RT , Z RS , Z RR ] areblock matrices. (ii) The system of equations V = Z I is solvedusing V T = V G − Z G I T , V L = − Z L I L , and V S = − Z RIS I S .The matrix H E2E is an EM-compliant end-to-end multiple-input multiple-output communication channel model that canbe utilized to quantify the advantages and limitations of RISs.By computing, e.g., the rank, the eigenvalues, and the eigen-vectors of H E2E , the multiplexing vs. diversity tradeoff ofRIS-aided systems can be unveiled. In particular, the matrix oftunable load impedances Z RIS can be appropriately optimizedfor realizing the desired multiplexing vs. diversity tradeoff.
B. Design Insights H E2E accounts for the mutual coupling among the N r N ris N t available radiating elements. To substantiate theconsistency of H E2E and to gain engineering insights forsystem design, we analyze the following relevant case study.
Corollary 1:
Consider (i) N t = N r = 1 and (ii) large trans-mission distances between the transmitter and the receiver, thetransmitter and the RIS, and the RIS and the receiver. Then, H E2E in (10) is a scalar and simplifies as follows: H E2E = Y (cid:16) Z RT − Z RS ( Z RIS + Z SS ) − Z ST (cid:17) (12)where Y = Z L (Z L + Z RR ) − (Z G + Z TT ) − , and Z G , Z L , Z TT , Z RR , Z RT are scalar versions of the associated matrices. Proof:
In the large distance regime, based on (3), theabsolute value of Z qp in (6) for p (cid:54) = q decays linearly withthe radial distance between the antenna elements. On the otherhand, Z pp is independent of the radial distance. Then, from(11), we have P TST ≈ Z TT and P RSR ≈ Z RR . Also, Z RR Z − − ( Z G + Z TT ) − Z − P RST P TSR ≈ Z RR Z − .The proof follows from (10), by setting N t = N r = 1 .From Corollary 1, the following remarks can be made. Far-Field Path-Loss . H E2E in (12) is the sum of the line-of-sight (LOS) link H LOS = Z RT and the virtual LOS link H VLOS = Z RS ( Z RIS + Z SS ) − Z ST . Since the received poweris proportional to P L ∝ |H E2E | , we retrieve the expectedscaling laws as a function of the transmitter-receiver distance( r RT ), i.e., H LOS ∝ r − , and the transmitter-RIS ( r ST ) andRIS-receiver ( r RS ) distances, i.e., H VLOS ∝ r − r − [5]. Reconfigurability of the RIS . H E2E in (12) explicitly dependson the tunable load impedances Z RIS , which can be appropri-ately optimized for obtaining the desired performance. It isworth noting that the intertwinement between the amplitudeand phase response of the individual passive scatterers of theRIS is inherently accounted for in H E2E through Z RIS [6],and it depends on the circuital model of the tuning circuit.
Conventional vs. Mutual Coupling Modeling . H E2E in (12)explicitly accounts for the mutual coupling among the passivescatterers of the RIS through the mutual impedances Z SS . Inparticular, H VLOS = (cid:80) N ris u =1 (cid:80) N ris v =1 Φ ( v, u ) Z ST ( u ) Z RS ( v ) ,where Z ST ( u ) and Z RS ( v ) are the u th and v th entriesof Z ST and Z RS , respectively, and Φ ( v, u ) is the ( v, u ) thentry of Φ = ( Z RIS + Z SS ) − . In the absence of mutualcoupling, Z SS is a diagonal matrix and H VLOS simplifies to H (no coupling)VLOS = (cid:80) N ris u =1 Φ ( u, u ) Z ST ( u ) Z RS ( u ) . The obtained H E2E subsumes current frameworks that are mutual-coupling-unaware, but it allows for the optimization of the tunable loadsof the RIS by accounting for their mutual coupling.
How to Utilize the Proposed Communication Model? H (no coupling)VLOS resembles communication-theoretic models thatare formulated in terms of received signal-to-noise ratio (SNR)and that are typically utilized in wireless communications [10,Eq. (1)]. In particular, Φ ( u, u ) can be viewed as the reflectioncoefficient of the u th passive scatterer (unit-cell) of the RIS. Φ ( v, u ) in H VLOS has a similar meaning, but it accounts forthe mutual coupling between the unit cells u and v . The maindifferences with SNR-based models are: (i) H VLOS is based onmutual impedances; (ii) H VLOS yields a one-to-one mappingamong the voltages and the currents that can be measured atthe ports of the transmitter and receiver; and (iii) H VLOS stemsdirectly from Maxwell’s equations. The analytical similaritywith widespread EM-unaware communication models allowsus to utilize H VLOS for system optimization by using algo-rithms similar to those utilized for the SNR, with the advantagethat the proposed model is end-to-end, EM-compliant, unit-cell-aware, and mutual-coupling-aware. In principle, thus, theoptimization and performance analysis frameworks proposedto date for conventional SNR-based models can be generalizedand enhanced by using the proposed impedance-based end-to-end model, in which Φ , V G , and Z L play the roles of thereflection matrix of the RIS, the beamforming vector of thetransmitter, and decoding vector of the receiver, respectively.V. N UMERICAL R ESULTS
In Fig. 2, we illustrate |H VLOS | = (cid:12)(cid:12)(cid:12) Z RS ( Z RIS + Z SS ) − Z ST (cid:12)(cid:12)(cid:12) in (12) as a function ofthe inter-distance and the number N ris of nearly-passive /16 /8 /4 /2 Inter-distance [m] -20 -19 -18 -17 -16 -15 -14 -13 N ris = 16N ris = 64N ris = 256 Fig. 2: Impact of mutual coupling in RIS-aided transmission.scattering elements of the RIS. The setup is the following: f = 28 GHz, N t = N r = 1 , r t = (5 , − , , r r = (5 , , ,the RIS is centered at (0 , , with M = N = √ N ris ,the transmit and receive antennas and the passive scatterersof the RIS are identical with a = λ/ , l = λ/ , and Z Smn = R Smn + jωL Smn with R Smn = 1 Ω , L Smn = 1 nH.In the considered setup, we observe that the mutual couplingamong the nearly-passive scatterers of the RIS has a noticeableimpact on the received power |H VLOS | . The results in Fig. 2substantiate the need of mutual-coupling-aware RISs, in whichthe tunable loads Z Smn are optimized to maximize |H VLOS | .VI. C ONCLUSION
We have introduced an end-to-end, EM-compliant, mutual-coupling-aware, and unit-cell-aware communication model forRIS-aided wireless systems. The proposed model is inherentlycompatible with conventional communication-theoretic frame-works and can be leveraged for the physics-compliant model-ing, analysis, and optimization of RIS-aided communications.R
EFERENCES[1] M. Di Renzo et al., “Smart radio environments empowered by reconfig-urable AI meta-surfaces: An idea whose time has come”,
EURASIP J.Wireless Commun. Net. , vol. 129, 20 pages, May 2019.[2] M. Di Renzo et al., “Reconfigurable intelligent surfaces vs. relaying:Differences, similarities, and performance comparison”,
IEEE Open J.Commun. Society , vol. 1, pp. 798-807, Jun. 2020.[3] M. Di Renzo et al., “Smart radio environments empowered by recon-figurable intelligent surfaces: How it works, state of research, and theroad ahead”,
IEEE J. Sel. Areas Commun. , 2020 (arXiv:2004.09352).[4] W. Tang et al., “Wireless communications with reconfigurable intelligentsurface: Path loss modeling and experimental measurement”,
IEEETrans. Wireless Commun. , 2020 (arXiv:1911.05326).[5] F. H. Danufane et al., “On the path-loss of reconfigurable intelligentsurfaces: An approach based on Green’s theorem applied to vectorfields”,
IEEE Trans. Wireless Commun. , submitted (arXiv:2007.13158).[6] S. Abeywickrama et al., “Intelligent reflecting surface: Practical phaseshift model and beamforming optimization”,
IEEE Trans. Commun. ,submitted (arXiv:2002.10112).[7] R. J. Williams et al., “A communication model for large intelligentsurfaces”,
IEEE Int. Conf. Commun. , Jun. 2020.[8] S. Phang et al., “Near-field MIMO communication links”,
IEEE Trans.Circuits Syst. I Regul. Pap. , vol. 65, no. 9, pp. 3027-3036, Sep. 2018.[9] C. A. Balanis,
Antenna Theory: Analysis and Design , John Wiley, 2016.[10] X. Qian et al., “Beamforming through reconfigurable intelligent surfacesin single-user MIMO systems: SNR distribution and scaling laws in thepresence of channel fading and phase noise”,