The next generation of wireless communication systems aims to address the ever-increasing demand for high throughput, low latency, better quality of service, and ubiquitous coverage. The abundance of bandwidth available at the mmWave frequency range, i.e., [20,100] GHz, is considered as a key enabler towards the realization of the promises of next generation wireless communication systems. However, communication in mmWave suffers from high path-loss and poor scattering and diffraction. Therefore, mmWave signals are vulnerable to blockages, especially in urban areas. In fact, since the channel in mmWave is mostly line-of-sight (LoS), i.e., a strong LoS path and very few and much weaker secondary components, the mmWave coverage map includes blind spots as a result of shadowing and blockage. Beam-forming employing massive MIMO is primarily used to address the high attenuation in the mmWave channel. In addition to beamforming, relaying can potentially be designed to generate constructive superposition and enhance the received signals at the receiving nodes. Equipping MIMO communication systems by Recon igurable Intelligent Surfaces (RISs) may extend the capacity of mmWave by covering the blind spots and providing diversity reception at the receiving nodes. RIS [1][2] [3] is denoted as a potential enabling technology for realizing 6G-and-beyond [4] [5] due to its great potential for manipulating the impinging electromagnetic waves and arti icially shaping the wireless propagation environment in a cost-effective and energy-ef icient manner. The wireless propagation environment in current communication systems is considered to be uncontrollable and stochastic, and therefore, the existing wireless system design technologies and principles (e.g., mmWave communication systems, massive MIMO, etc.) are traditionally developed on a reactive basis to adapt to this stochastic behavior. An RIS typically consists of a large number of low-cost and passive re lecting elements arranged in planar arti icial metasurfaces. Each RIS cell is capable of manipulating the phase and possibly the amplitude of incident electromagnetic waves in response to real-time external signals provided by a smart controller. The programmability of the RIS enables them to lexibly modulate RF signals without the need to use any mixers, analog phase-shifters, analog-to-digital/ digital-to-analog converter, etc. [6][7]. This not only drives the RIS hardware cost and energy consumption down, but also allows for the design of RF chain-free wireless transceivers [8] [9]. Therefore, either as active transceivers or passive re lectors, RIS may be a promising solution to revolutionize the design of the physical layer in next-generation wireless communication systems.
RISs have attracted a lot of attention both from industry and academia, over the past several years. The efforts towards realizing the prospects of achieving RIS-aided wireless communications initially started with theoretical developments based on mathematical models. More recently, real-world trials with prototyping and ield experiments have been pursued more seriously than before in the literature. Several funding agencies have heavily invested in theoretical development, designing, prototyping, and testing the intelligent metamaterial surfaces. Since 2012, the National Science Foundation (NSF) in the US, and the European Commission, Horizon 2020, have spent millions of dollars in projects tackling different aspects of the re lective metasurfaces and integrating the RIS with next generation networks [10].
With respect to experimental trials, extensive efforts have been made. The Japanese network operator NTT DoCoMo in collaboration with its partners has repeated a few successful trials over the past couple of years, demonstrating how their designed transparent dynamic metasurface can improve the power of the radio signal at 28 GHz [11]. In [12], the authors have proposed a prototype for RIS-enabled wireless communications with an RIS consisting of 1100 elements operating at 5.8 GHz. They show their passive re lect-array design provides significant gain improvement in point-to-point wireless communications through indoor and outdoor ield trials. Another prototype is proposed in [13] for modulating the incident waves based on single-carrier Quadrature Phase Shift Keying (QPSK) to design RIS-based transceivers achieving a 2.048 Mbps data rate for video-streaming.
The prototype in [8] achieves an RIS-based RF-chain free transceiver.
RISs are promising to be deployed in a wide range of communications scenarios, such as high throughput MIMO communications [9][14], ad-hoc networks, e.g., UAV communications [15][16], physical layer security [17], etc. In radar, deployment of RISs with judicious design of phase shifts has shown improvement in estimation of the radar cross-section [18] and moving target [19]. Apart from the work focusing on theoretical performance analysis of RIS-enabled systems [20][21], considerable amount of work has been dedicated to optimizing such an integration, mostly focusing on the phase optimization of RIS elements [22][23] [24] to achieve various goals such as maximum received signal strength, maximum spectral eficiency, etc. For more information on the challenges and opportunities associated with RIS, we refer the interested readers to [25][26] and the references therein.
In this paper, we present an RIS-aided architecture to facilitate two-way communications. Most of the prior art in RIS-aided two-way communications, focuses on singlebeam communications [27][28]. However, in this paper we propose the idea of RIS-aided multi-beamforming.
The idea of RIS-aided multi-beamforming, i. e., beamforming with multiple disjoint lobes, was irst introduced in [29], where the authors aimed at covering the mmWave blind spots by designing sharp beams covering various ranges of solid angle. The codebook design problem for such beam-forming was addressed in [30]. In [31], the authors employ dual beam-forming for short range target monitoring.
In this paper, we consider a communication scenario between a transmitter, e.g., the Base Station (BS), and terrestrial end users through a passive RIS that re lects the received signal from the transmitter towards the users. Hence, the users that are otherwise in blind spots of network coverage become capable of communicating with the base station through the RIS that is serving as a passive re lector (passive relay) maintaining communication links between the BS and the users. Given the geospatial variance among the locations of the end users served by the same wireless system, the RIS may have to simultaneously accommodate users that lie in different angular intervals that are widely separated with a satisfactory Quality of Service (QoS). In what we refer to as multi-beamforming, we particularly address the design of beams consisting of multiple disjoint lobes in order to cover different blind spots using sharp, high gain, and effective beam patterns. In the following, we summarize the main contributions of this paper:
• [RIS-UPA to UPA Transformation] We present simple yet important properties of RIS with UPA structure (RIS-UPA) when used as a beam-former in Section 3. As a consequence, we present a transformation between the beam-former design problem in UPA and RIS-UPA which allows us to directly borrow the design for UPA beam-forming and by the means of a transformation use it for RIS-UPA beam-forming.
We present a new beamforming design technique termed as multibeamforming aiming to design beams with multiple disjoint lobes. The multi-beamforming design inherently depends on the solid angle (say Ω 1 in Fig. 1) at which the incident wave activates the RIS elements. The proposed beam-forming design easily adapts to changes in Ω 1 and we provide a visualization as to how the beam would change in response to change in Ω 1 . • [Custom footprint] The proposed method has the lexibility to design a beam with a custom footprint.
The beam footprint may be de ined as the crosssection of the beam lobes with the sphere (say at beam gains within the half-power (3dB) point and maximum gain).
We design the parameters of the RIS to achieve multiple disjoint beams covering various ranges of a solid angle. The designed beams are fairly sharp, have almost uniform gain in the desired Angular Coverage Interval (ACI), and have negligible power transmitted outside the ACI.
• [Complexity] Thanks to the derived analytical closed form solutions for the multi-beamforming design, the proposed solution bears very low computational complexity even for an RIS with massive array sizes.
• [Multilink] We also provide RIS beam-forming design in multilink scenarios where different pairs of transmitters and receivers are communicating simultaneously with the help of the same RIS. Moreover, we establish a connection between multilink beam-forming and multi-beamforming.
• [Evaluation] Through numerical evaluation we show that by using a passive RIS, multi-beamforming can simultaneously cover multiple ACIs. Moreover, multi-beamforming provides tens of dB power boost w.r.t. the single-beam RIS design. set and 1-norm if applied to a vector, ⊙ is the Hadamard product, ⊗ is the Kronecker product, the operator diag{.} when applied to a square matrix takes the vector of its diagonal elements, and when applied to a vector of elements, forms a diagonal matrix of that vector, A H , and A a,b denote conjugate transpose, and (a, b) th entry of A respectively. We have summarized the list of variables and parameters frequently used in the paper in Table 1.
The remainder of the paper is organized as follows. Section 2 develops the system model by describing the channel model and the RIS con iguration. In Section 3 we present a few fundamental properties of the gain provided by the RIS-aided system that are central to our design in the subsequent sections. In Section 4 we describe the multi-pair two-way communications problem and illustrate how it can be posed as a MIMO beam-forming problem. Then we present the corresponding problem formulation. In Section 5 we propose our approach for solving the RIS-enabled multi-pair two-way communications problem resulting in a closed-form expression for the con iguration of the RIS. Section 6 presents the evaluation results for our numerical experiments verifying the effectiveness of our solution, and inally, we conclude in Section 7.
Consider a communication system consisting of a multiantenna BS with M t antenna elements as a transmitter and a multi-antenna receiver with M r antenna elements.
The MIMO system is aided by a multi-element RIS consisting of M elements arranged in an M h × M v grid in the form of a UPA as shown in Fig. 1, where M h and M v are the number of elements in the horizontal and vertical directions, respectively. The received signal y ∈ C Mr as a function of the transmitted signal x ∈ C Mt can be written as,
where z is the noise vector, where each element of z is drawn from a complex Gaussian distribution CN 0, σ 2 n , H t ∈ C M ×Mt and H r ∈ C Mr×M are the channel matrices between each party and the RIS. We assume that the RIS consists of elements for which both the phase θ m and the gain β m (in form of the attenuation of the relected signal) of each element, say m, may be controlled and Θ ∈ C M ×M is a diagonal matrix where the element (m, m) denotes the coef icient β m e jθm of the m th element of the RIS. Assuming a LoS channel model both between the transmitter and the RIS and between the RIS and the receiver and using the directivity vectors at the transmitter, the RIS, and the receiver, the effective channel matrices can be written as, The decision variables representing the coef icients of RIS elements manipulating the impinging electromagnetic waves λ
The decision variables representing the overall impact of the RIS when excited from an incident solid AoA of
The normalized version of λ, or equivalently, the normalized beam-forming vector corresponding to a Uniform Planar Array (UPA) of antennas g
Equal-gain vector that has to be optimally chosen when deciding the optimal con iguration c.
The gain of an RIS con igured by Θ at AoD Ω when excited from an incident angle Ω 1 .
The beam-forming gain of a UPA of antennas con igured by
The design parameter embedded in g determining the quality of the beams formed by the multi-beamforming approach. Parameters Description H t
The effective channel matrix from the transmitter to the RIS H r
The effective channel matrix from the RIS to the receiver a M (Ω)
The array response vector of an array of M antennas or an RIS consisting of M elements at the solid angle Ω ρ t
The gain of the Line-of-Sight (LoS) path from the transmitter to the RIS ρ r
The gain of the Line-of-Sight (LoS) path from the RIS to the receiver Ω t
The Angle of Departure (AoD) from the transmitter towards the RIS Ω r
The Angle of Arrival (AoA) from the RIS at the receiver Ω 1
The Angle of Arrival (AoA) from the transmitter at the RIS Ω 2
The Angle of Departure (AoD) from the RIS towards the receiver τ (Ω)
The operator τ maps the solid angle
The directivity vector of an array of antennas at direction ψ B
The total angular range under study in the Ω domain B ψ
The total angular range under study in the ψ domain D n
The n-th receive zone in the n-th communication pair
The minimal index set of the particles covering the n-th receive zone
The area covered by the (p, q)-th particle in the ψ-domain e p,q
The vector with a 1 in the (p, q)-th entry out of
ones and zero everywhere else where a M (Ω) is the array response vector of an RIS with elements in a UPA structure (RIS-UPA), Ω t and Ω 2 are the solid Angles of Departure (AoD) of the transmitted beams from the transmitter and the RIS and Ω 1 and Ω r are the solid Angles of Arrival (AoA) of the received beams at the RIS and the receiver, respectively. The gains of the LoS paths from the transmitter to the RIS and from the RIS to the receiver are denoted by ρ t and ρ r , respectively. Note that an arbitrary solid angle Ω speci ies a pair of elevation and azimuth angles (ϕ, θ). Therefore, in the above de initions we set Ω a = [ϕ a , θ a ], a ∈ {1, 2, t, r}. Further, assuming no pairing between the RIS elements, Θ will be a diagonal matrix speci ied as
between the transmitter and the RIS [32]. RIS-enabled communications in the near ield requires a thoroughly different design. For an example of such a speci ication, please refer to [33]. The array response vector of an RIS-UPA can be found in a similar way to that of a UPA. At a solid angle Ω = [φ, θ], we have
where we de ine r Ω = [cos ϕ cos θ, cos ϕ sin θ, sin ϕ], and r m = (m h d x , 0, m v d z ) to respectively denote the direction corresponding to the solid angle Ω, and the location of the m-th RIS element corresponding to the antenna placed at the position
Further, we de ine a transformation of variables as follows. For a solid angle
Introducing the new variables into equation (6), it is straightforward to write,
where d M denotes the directivity vector of the RIS, and the directivity vectors d Ma , a ∈ {v, h} are de ined as follows.
where ψ v = ξ, and ψ h = ζ. Finally, let B be the angular range for Ω under our interest de ined as follows, 3), the contribution of the RIS to the channel matrix when excited from incident angle Ω 1 for a receiver at a given solid angle solid angle Ω is given by
where we de ine
Consider an RIS consisting of M v × M h antenna elements forming a UPA structure that is placed at the x-z plane, where M . M v M h and the z-axis corresponds to the horizon. Let d z , and d x denote the distance between the antenna elements in the z and x axis, respectively. Throughout this paper, we assume that the transmitter and the receiver are within the far ield of the RIS. As opposed to the near-ield regime, the diversity gain distribution in the far ield does not depend on the distance Accordingly, let B ψ be the angular range under interest in the (ξ, ζ) domain given by
In this paper, we set
2 ), and ζ ∈ [-π, π). To formalize the variable transformation introduced in (7), we de ine the transformation operator τ :
For a RIS-UPA excited by an emission from solid angle Ω 1 , we can write for the reference gain at any direction Ω,
where we de ine m = [m v , m h ] T . In the following we present three fundamental facts together with their interpretation, that are useful for our subsequent developments in the next sections. The proofs for all these facts is straightforward and can be achieved by basic calculus. First of all, from the identity,
we note that the gain of a UPA with beam-forming matrix Θ at any solid angle Ω is equal to that of a RIS-UPA excited from an AOA Ω 1 with parameter matrix diag(d H M (ψ 1 )Θ)). Second, for any solid angle Ω 2 , with τ (Ω 2 ) = ψ 2 , it holds that,
where τ (Ω ′ ) = τ (Ω)τ (Ω 2 ). The last identity implies that for two RIS-UPAs with parameter matrices Θ and diag{d H M (ψ 2 )Θ} that are excited from the same AoA Ω 1 , the gain patterns are just a rotation of each other such that the gain at direction Ω for the irst one is equal to the gain at the direction of Ω ′ for the second one, where Ω ′ is as denoted above. The same holds for a UPA by setting Ω 1 = 0., i.e., the gain pattern of two UPAs with parameters Θ and diag{d H M (ψ 2 )Θ} are related by a rotation of each other such that the gain at direction Ω for the irst one is equal to the gain at the direction of Ω ′ for the second one. Finally, it is straightforward to show,
where τ (Ω ′′ ) = τ (Ω) + τ (Ω 2 )τ (Ω 1 ). By virtue of the last identity, we note that for two RIS-UPAs with the same parameters Θ that are excited from two different Please note that if a user is located at the solid angle Ω with respect to the RIS, then the AoA of the incident wave at the RIS when the user is transmitting is de ined as Ω = [ϕ, θ]. However, due to the change in the direction of the beam, the re lected beam from an RIS towards the user is directed at angle Ω r = [-ϕ, θ +π]. Hence, from (7) and by the de inition of the operator τ , we have τ (Ω r ) = -τ (Ω).
From (12), for an RIS which is excited from an incident angle with AoA of
where
. which means that the RIS has the same gain at the AoD of Ω r 1 when it is excited from the incident angle with AoA of Ω r 2 . This is trivially equivalent to the channel reciprocity property which indicates the possibility of two-way communications between each pair.
In this section, we show how RIS can be used to simultaneously facilitate two-way communications between multiple pairs. We will irst present the description of the problem and explain our approach. We then proceed with the formulation of the problem and propose our solution.
© International Telecommunication Union, 2023
We study the RIS design problem for RIS-enabled twoway communications where multiple communications pairs are involved. Each pair realizes a MIMO system consisting of a multi-antenna transmitter (e.g. multi-antenna BS), and a possibly multi-antenna receiver (e.g. multiantenna mobile user). We assume there is no LoS channel between the pairs and the mmWave channel is assisted by an RIS with elements that are arranged according to a UPA structure. Formally, we consider N communications pairs speci ied by (Ω n , D n ), n = 1, . . . , N . The n-th transmitter is emitting at the RIS with an AoA of Ω n , and the n-th receiver may reside within the receive zone D n , i.e., D n represents the range of AoDs from the RIS that may cover the n-th receiver. For the n-th receive zone, we denote each continuous range of such AoDs by an Angular Coverage Interval (ACI). We note that each receive zone may comprise of one or more ACIs. Fig. 2 depicts an example of such a setup for N = 2 pairs, where receive zone D 1 = D 1,1 ∪ D 1,2 consists of two ACIs. As far as the n-th communications pair is concerned, the ideal RIS must be con igured in such a way that when excited from solid angle Ω n it covers the receive zone D n uniformly, with high and sharp gains, while leaving minimal gain leakage to other intervals. In RIS-aided MTMR two-way communications, we aim to design the RIS such that all pairs are satis ied simultaneously with a high QoS. The simplest instance of the above problem is when N = 1, i.e. when there is only one pair. We denote this instance by STMR. Further, we note that (i) the boundaries of the ACIs are determined based on the potential locations of the receivers, and, (ii) we wish to consider the ACIs of minimal size to avoid sacri icing the gain. Therefore, the footprint of the receive zones on spherical coordinates may become of arbitrary shape. This may introduce additional complexity into the RIS design problem. To resolve this issue, let us uniformly divide B ψ into Q = Q v Q h subregions, where Q v and Q h determine the division resolution in the vertical and horizontal directions, respectively. We denote each such subregion by a particle B ψ p,q that is speci ied as,
where
with δ v = 2ξ B Qv , and δ h = 2ζ B Q h . Further, de ine for all (p, q) pairs the notation δ p,q = δ v δ h . We wish to cover each receive zone D n in the RIS-enabled MTMR problem with the smallest set of particles, i.e., D n ∼ (p,q)∈An B ψ p,q , with A n being the smallest set of index pairs (p, q) that beams B ψ p,q collectively cover D n . The union of B ψ p,q is in fact approximating the shape of the desired receive zone D n , where the resolution of the approximation is set by the pair (Q v , Q h ). By using larger values of Q v , and Q h , one can opt for iner particles and boost the quality of the approximation at the expense of solving a larger optimization problem. Explicitly, we have
Further de ine A = N n=1 A n . Next, we will show how every instance of MTMR can be cast as an STMR problem. Then we pose the two-way communications problem in the STMR case, as a composite beam-forming problem under the UPA antenna structure. We then propose a lowcomplexity closed-form for the optimal solution to the last problem.
Consider N communications pairs (Ω n , D n ), n = 1, . . . , N , where Ω n is the AoA of the n-th transmitter to the RIS-UPA plane, and D n denotes the n-th receive zone.
For an arbitrary RIS-UPA con iguration Θ, and for any communications pair, by equation ( 15), we get
where τ ( Ω) = τ (Ω) + τ (Ω 1 )τ (Ω n ). This would mean under the ψ-domain that if Θ is optimized to cover the angular interval D n , when excited from an AoA of Ω n , same con iguration when excited from an AoA of Ω Θ * be the optimal con iguration of the RIS-UPA derived for solving the STMR problem. Then a straightforward use of equation (15) in the reverse order for each n, will verify that Θ = Θ * is the optimal RIS-UPA con iguration for the MTMR case. An example of the above transformation is illustrated for N = 2 in Fig. 3.
RIS-UPA refers to an RIS with UPA structure while UPAantenna refers to a regular multi-element antenna array where the elements are arranged in the same UPA structure. In this section, we clarify the relation between the We start by explicitly writing λ in equation ( 5) in terms of its elements as follows
where λ mv,m h corresponds to the element located at position (m v , m h ) in the UPA grid. Using the expressions ( 6)-( 9), for an RIS-UPA that is excited from an incident angle Ω 1 , we have
We note that λ depends on the AoA of the incident beams at the RIS, i.e., Ω 1 , as well as the RIS parameters Θ. Using equations ( 5) and ( 8), we can restate the reference gain of the RIS in direction ψ = [ξ, ζ] by explicitly de ining G (ξ, ζ, λ) as follows
On the other hand, the gain of UPA-antenna and the feed coef icients c is given by
that has a clear similarity. This means that to design the RIS-UPA for the STMR problem with receive zone D we can use the multibeamforming design framework to cover the ACIs included in D for the UPA-antenna [29]. In particular, an RIS-UPA with parameters λ and a UPA-antenna with beam-forming parameters c have the same beam-forming gain pattern if UPA structures are the same and λ = c. Hence, an RIS-UPA which is excited from the solid angle Ω 1 has the same beam-forming gain as its UPA-antenna counterpart if Θ = diag{c T ⊙ a M (Ω 1 )}. In the following, we address the design of beam-forming coef icients of c for an antenna with UPA structure which can then be used to design a RIS-UPA. For any normalized beam-forming vector c, it is straightforward to show that
that can be interpreted as the power conservation law [34]. Please note that the dependence between variables ξ and ζ can be resolved using the approximation in [35].
The power conservation law (25) allows us to de ine an optimization problem for the UPA multi-beamforming design in terms of a normalized gain pattern where the total gain is divided by (2π) 2 .
In the next section, we de ine the multi-beamforming design problem as the core of designing our proposed RIS structure.
We wish to design beam-formers that provide high, sharp, and constant gain within the desired ACIs and zero gain everywhere else. We have then for the ideal gain corresponding to such beam-former c that,
where δ p,q = δ v δ h denotes the area of the (p, q)-th beam in the (ξ, ζ) domain. Therefore, we can derive t = (2π) 2 |A|δp,q . It holds that,
© International Telecommunication Union, 2023
Using the beam-former c we wish to mimic the deal gain in equation (27). Therefore, we formulate the following optimization problem,
By partitioning the range of (ξ, ζ) into prede ined intervals and then uniformly sampling with the rate (L v , L h ) per interval along both axis, we can rewrite the optimization problem as follows,
where,
We can rewrite equation (29) as,
where,
and,
Unfortunately, the optimization problem in (31) does not admit an optimal closed-form solution as is, due to the absolute values of the complex numbers existing in the formulation. However, note that,
with e p,q ∈ Z Q being the standard basis vector for the (p, q)-th axis among (Q v , Q h ) pairs. Now, note that 1 L,1 = g ⊙ g * for any equal gain g ∈ C L where L = L h L v . An equal-gain vector g ∈ C L is a vector where all elements have equal absolute values (in this case, equal to 1). Therefore, we can write:
Also, it is straightforward to write,
where,
, and for a ∈ {v, h}, and b ∈ [Q a ] we have,
where,
Comparing the expressions (31), (35), and (36), one can show that the optimal choice of c D in (28) is the solution to the following optimization problem for proper choices of g p,q .
Problem 1. Given equal-gain vectors g p,q ∈ C L , for (p, q) ∈ A ind vector c D ∈ C M such that
However, we now need to ind the optimal choices of g p,q that minimize the objective in (31). Using (35) and (36), we have the following optimization problem.
where abs(.) denotes the element-wise absolute value of a vector.
In the next section we continue with the solution of problems 1 and 2.
Note that the solution to Problem 1 is the limit of the sequence of solutions to a least-square optimization problem as L goes to in inity. For each L we ind that,
where
, noting that it holds that,
Even though Problem 1 admits a nice analytical closed form solution, doing so for the Problem 2 is not a trivial task, especially due to the fact that the objective function is not convex. However, the convexi ication of the objective problem in the form of
= arg min <gp,q> (p,q)∈A κD H D -I LQ (p,q)∈A e p,q ⊗ g p,q 2 (45) leads to an effective solution for the original problem. Indeed, it can be veri ied by solving the optimization problem (45) numerically that the solution admits the form (46) in conjecture.
Conjecture 3. The minimizer of ( 45) is in the form of
for some η v , η h where α a = e j( ηa La ) , a ∈ {v, h}.
Except for some special cases, we have not been able to analytically prove this conjecture in its entirety. In the following, we use the the analytical form (46) for g * p,q for the rest of our derivations. This solution would not be the optimal solution for the original problem (41). However, it provides a near-optimal solution with added bene its of allowing us to (i) ind the limit of the solution as L goes to in inity, and (ii) express the beam-forming vectors in closed form, as it will be revealed in the following discussion. An analytical closed form solution for c D can be found as follows. It holds that,
σg p,q,lv,l h d Mt (ξ p,lv , ζ q,l h )
where µ mv,m h p,q,lv,l h = (m v ξ p,lv + m h ζ q,l h ). We can then write for the (m v , m h ) th component of the beam-former c D ,
Using equation (30), we can rewrite (48) as,
with χ mv,m h p,q = (m v ξ p + m h ζ q ), and ξ a = δ a m a + η a , for a ∈ {v, h}. Now that the closed-form expression for c D , and therefore, λ is known, for an RIS that is excited from the solid angle Ω 1 , the RIS parameters at the antenna placed at location (m v , m h ) can be easily computed. More precisely, we get,
The solution given by (51) and (52) is optimized to ind a beam-forming pattern which is the closest to the desired normalized beam pattern. However, in practice the norm of c depends on the power P that can be inserted by ampli iers (active elements), say ∥c∥ ≤ P . Hence for an RIS-UPA with active elements the gains β mv,m h is scaled by √ P . Also, the solution given by (51) and (52) may be further tailored for the case that the RIS elements are passive.
© International Telecommunication Union, 2023 If the gain control for the passive RIS elements is still possible, the absolute value of the gain for each element may not exceed 1 (a value less than 1 corresponds to an attenuation). In order to maximize the power re lected by the RIS, we scale the gains β mv,m h so that their maximum is equal to one, i.e., β mv,m h = |c D,mv,m h |/∥c∥ ∞ . Finally, in the case that gain control (attenuation) at the RIS with passive elements is not feasible, we have β mv,m h = 1.
In the next section, we evaluate the effectiveness of our RIS beam-forming design approach by means of numerical experiments.
In this section, evaluate the performance of our multibeam design framework.
First, we consider a dual-beam design which comprises of two lobes with centers at directions (-8π/32, -5π/32) and (7π/32, π/32) for the pairs of the solid angle (ϕ, θ) with the beamwidth equal to π/16. We divide both the ψ h , and the ψ v range uniformly into Q h = 16, and Q v = 16 regions resulting in Q = 256 equally-shaped units in (ψ v , ψ h ) domain. We cover each desired beam with the smallest number of the designed units to provide uniform gain at the desired angular regions. Figures 4(a In order to compare the performance of our multi beam design to a single-beam design, we consider a beam with a single lobe which is capable of covering the same two regions as in the dual-beam design. Fig. 4(d) shows the heat map corresponding to the gain of the re lected beam from an RIS for the corresponding single beam that is optimized based on our design. As was the case for multibeam, this igure also shows that for a single beam our design generates an almost uniform and fairly sharp beam. However, comparing Fig. 4(a) and Fig. 4(d), we observe that in the desired ACI the multi-beamforming procedure enhances the gain by about 20 dB over the beams with an optimized single lobe. The reason for this difference is that according to the power conservation axiom, with a constant input power level, the narrower each beam is the higher will be the reference gain over the area it covers.
In fact, if we target two disjoint ACIs with a single beam, the beam must be wide enough to cover both areas. This will result in dissipating the power in angles that are not intended. However, the muti-beam design allows for generating two disjoint beams, each being narrow enough to only cover the intended ACI. Therefore, the power will not be wasted in undesired directions.
We consider two links. The irst link is between the transmitter located as
) and a receiver that is in the ACI D 1 = [-0.36, 0] × [-0.56, 0] and the second link is between the transmitter located as Ω 2 = ( π 32 , π 8 ) and a receiver that is in the ACI D 1 = [-0.18, 0.18]×[0.26, 0.86]. Each of the receive zones D 1 and D 2 is comprised of 4 particles in an Q v × Q h = 8 × 8 grid. Fig. 5(a) depicts the angular location of the two transmitters and the ACI for the receivers. In order to design a beam-former that covers D 1 when transmitting from angular position Ω 1 and covers D 2 when transmitting from angular position Ω 2 , we irst ind a composite beam with a uni ied incident angle, say Ω 1 . This means that we ind the ACI D ′ 2 when the RIS array is excited by a beam at incident angle Ω 1 which is equivalent to the ACI D 2 where the same array is excited from the incident angle Ω 2 . The transformed ACI D ′ 2 is depicted in Fig 5(a). The heat map of the designed beams is depicted in igures 5(b)-(d) where the RIS is excited from angles Ω 1 , Ω 2 , and Ω 4 , respectively. Figures 5(b)-(c) illustrate that when the RIS is excited from an incident angle Ω 1 and Ω 2 , the corresponding ACI D 1 and D 2 are respectively covered by the designed beam as illustrated in the respective igures, however, in either case another angular region is also covered by the beam which is not necessary and could be considered as possible wastage of the power. We note that this happens due to the fact that the unwanted ACI when the RIS is excited from the incident angle Ω 1 is indeed generating the desired ACI when the RIS is excited from the incident angle Ω 2 . Finally, we consider Ω 4 to be an angular point in the ACI D 2 , e.g., we take Ω 4 to be the center of the ACI D 2 . Fig. 5(d) shows that if the RIS is excited from an incident angle Ω 4 , e.g., when a user in ACI D 2 is replying, then it will be received by the corresponding transmitter which is located at angular position Ω 2 . In igures 5(b)-(d) the windows show the positioning of the grids that lie on the particles covering the corresponding desired ACIs.
Here, we illustrate the possibility of designing a beam with an arbitrary pattern, or more precisely, arbitrary footprint. We aim to design a beam which is covering the receive zone D that is speci ied as follows. We consider two possible quantization levels of the beam footprint with resolution Q h = Q v = 12 and Q h = Q v = 24. Fig. 6(a) and Fig. 6(b) illustrate the heat map of the designed beam. The iner the resolution the better the approximation of the shape of the beam. Fig. 6(c) provides the quantitative comparison between these two resolutions; it shows the higher resolution increases the overall beam-forming gain, lowers the leakage and generates smoother gain.
An RIS can be incorporated into mmWave communications to ill the coverage gaps in the blind spots of the mmWave system. We proposed a novel approach for designing RISs, namely RIS-UPA, where the RIS elements are arranged according to a UPA structure. We proposed a con iguration for the elements of an RIS-UPA that enables the coverage of multiple disjoint angular intervals simultaneously. On this ground, we showed that an RIS-UPA assisted MIMO system can support multiple two-way communication pairs simultaneously. We established that the RIS-aided multiple-pair scenario can be transformed to a single-pair scenario and then by values of θ c ∈ {-5π/32, π/32)}. Both Fig. 4(b) and Fig. 4(c) con irm the sharpness of both lobes of the designed dualbeam and can be used to ind the beamwidth of each lobes at an arbitrary fraction from its maximum values, e.g., the 3dB beamwidth or 10dB beamwidth. Indeed, there is a negligible difference between 3dB and 10dB beamwidth which clari ies the sharpness of the beams. From Fig. 4(b) and Fig. 4(c), it is also observed that the gain within the ACI is almost uniform. Nonetheless, we should emphasize the fact that the shape of the lobes of the beam that are centered at different solid angles may suffer from slight deformation as seen by Fig. 4(a). This phenomenon worsens as the corresponding lobes of the beams get too close to the plane of the RIS.
appealing to the similarities of RIS-UPA and UPA beamforming we argued that we can borrow the principles of UPA multi-beamforming design to obtain closed-form low-complexity solutions for the RIS design problem. Both our theoretical results and numerical experiments demonstrate that our RIS con iguration can form beams of custom footprints and will result in sharp, high, and stable gains within the desired ACIs regardless of their spatial locations, while effectively suppressing all the undesired out-of-band components.
© International Telecommunication Union, 2023 ITU Journal on Future and Evolving Technologies, Volume 4, Issue 1, March 2023
© International Telecommunication Union, 2023 Torkzaban et al.: RIS-aided mmWave beam-forming for two-way communications of multiple pairs