I Introduction
Next generation of wireless communication systems aims to address the everincreasing 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., 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 pathloss, and poor scattering. Since the channel in mmWave is mostly 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. Beamforming is primarily used to address the high attenuation in the channel. In addition to beamforming, relaying can potentially be designed to generate constructive superposition and enhance the received signals at the receiving nodes. Reconfigurable intelligent surface (RIS)[8][11][3] is a new paradigm with a great potential for stretching the coverage and enhancing the capacity of nextgeneration communication systems. Indeed, it is possible to shape the wireless channel by using RIS, e.g., by covering blind spots or providing diversity reception at a receiving node. In particular, passive RIS provide not only an energyefficient solution but also a costeffective one both in terms of the initial deployment cost and the operational costs. RIS are promising to be deployed in a wide range of communications scenarios and usecases, such as high throughput MIMO communications[7][14], adhoc networks, e.g., UAV communications[10], physical layer security[13], etc. Apart from the works focusing on theoretical performance analysis of RISenabled systems [6][14][9], considerable amount of work has been dedicated to optimizing such an integration, mostly focusing on the phase optimization of RIS elements [1][5][4][2] to achieve various goals such as maximum received signal strength, maximum spectral efficiency, etc. For more information on RIS, we refer the interested readers to [12] and the references therein.
In this paper, we consider a communication scenario between a transmitter, e.g., the base station (BS), and terrestrial endusers through a passive RIS that reflects 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 reflector (passive relay) maintaining communication links to the BS and to the users. Given the geospatial variance among the locations of the endusers served by the same wireless system, the RIS may have to accommodate users that lie in distant angular intervals simultaneously, with satisfactory quality of service (QoS). In what we refer to as
multibeamforming, we particularly address the design of beams consisting of multiple disjoint lobes using RIS in order to cover different blind spots using sharp and effective beam patterns. In the following, we summaries the main contributions of this paper:
We design the parameters of the RIS to achieve multiple disjoint beams covering various ranges of 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.

We formulate the multibeamforming design as an optimization problem for which we derive the optimal solution.

Thanks to the derived analytical closed form solutions for the optimal multibeamforming design, the proposed solution bears very low computational complexity even for RIS with massive array size.

Through numerical evaluation we show that by using passive RIS, multibeamforming can simultaneously cover multiple ACIs. Moreover, multibeamforming provides tens of dB power boost w.r.t. singlebeam RIS design even when the single beam is designed optimally.
Notation Throughout this paper, , , and denote the set of complex, real, and integer numbers, respectively,
denotes the circularly symmetric complex normal distribution with mean
and variance , is the closed interval between and is the all ones matrix, is the identity matrix, is the indicator function, is the norm, is the infinitynorm, may denote cardinality if applied to a set andnorm if applied to a vector,
is the Hadamard product, is the Kronecker product, and denote conjugate transpose, and entry of respectively.Ii System model
Iia Channel Model
Consider a communication system consisting of a multiantenna BS with antenna elements as a transmitter and a multiantenna receiver with antenna elements. The MIMO system is aided by a multielement RIS consisting of elements arranged in grid in the form of UPA as shown in figure 1 where and are the number of elements in the horizontal and vertical directions, respectively. The received signal as a function of the transmitted signal can be written as,
(1) 
where is the noise vector, with each element of
drawn from a complex Gaussian distribution
, and are the channel matrices between each party and the RIS. We assume that the RIS consists of elements for which both the phase and the gain (in form of attenuation of the reflected signal) of each element, say , may be controlled and is a diagonal matrix where the element denotes the coefficient of the element of the RIS. Assuming 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,(2)  
(3) 
where is the array response vector of an RIS with elements in a UPA structure (RISUPA), and are the solid angles of departure (AoD) of the transmitted beams from transmitter and the RIS and and are the solid angle of arrival (AoA) of the received beams at the RIS and the receiver, respectively. The gain of the LoS paths from the transmitter to the RIS and from the RIS to the receiver are denoted by and , respectively. Note that the solid angle specifies a pair of elevation and azimuth angles i.e. , . Further, assuming no pairing between the RIS elements, will be a diagonal matrix specified as
(4) 
where and .
IiB RIS Model
Suppose an RIS consisting of antenna elements forming a UPA structure is placed at the  plane, where and axis corresponds to horizon. Let , and denote the distance between the antennas elements in and axis, respectively. The directivity of a RISUPA can be found in similar way to that of a UPA. At a solid angle , we have,
(5) 
where respectively, , and denote the direction corresponding to the solid angle and the location of the th RIS element corresponding to the antenna placed at the position . Further, we define a transformation of variables as follows. For a solid angle , define as follows,
(6) 
Introducing the new variables into equation (5), it is straightforward to write,
(7) 
where we define for the directivity vectors as follows, and denote by the directivity vector corresponding to the RIS.
(8) 
Let be the angular range under cover defined as
(9) 
We note that there is a onetoone correspondence between the solid angle and its representation after change of variable as . Accordingly, let be the angular range under cover in the domain given by
(10) 
In this paper, we set , , and , hence , and . Note that, the dependence between variables and can be resolved using the approximation in [15]. Let us uniformly divide into subregions, where and are the number of division in horizontal and vertical directions, respectively. A subregion is denoted by
where , and defining,
(11) 
with , and . In the next section, we define the multibeamforming design problem as the core of our proposed RIS structure.
Iii Problem Formulation
Prior to formulating the multibeamforming design problem, we proceed with a few preliminary definitions. Let us define the multibeam as collection of compound beams where and , with being the set of all pairs that all beams cover . The union of is in fact approximating the shape of the solid angle for the desired compound beam corresponding to . By using larger number of division, i.e., finer beams, one can make the approximation better. We have
(12) 
Further define . We aim to design a beamforming vector such that the multibeam is covered when the RIS is excited by an incident wave received at solid angle . Using (1)(3), the contribution of the RIS in the channel matrix for a receiver at the solid angle is given by
(13) 
where is defined as follows. For antenna element located at position in the UPA grid, we have
(14) 
where is the representation of in the domain, and hence the vector is given by
(15) 
We note that depends on the AoA of the incident beams at the RIS, i.e., , as well as the RIS parameters. The reference gain of RIS in direction in terms of is given by
(16) 
On the other hand, the gain of UPA antenna with the feed coefficients is given by
(17) 
that has a clear similarity. This means that to design the RISUPA for the STMR problem with receive zone we can use the multibeamforming design framework to cover the ACI’s included in for the UPA antenna. In particular, a RISUPA with parameters and a UPAantenna with beamforming parameters have the same beamforming gain pattern if UPA structures are the same and . Hence, a RISUPA which is excited from the solid angle has the same beamforming gain as its UPA antenna counterpart if . For any normalized beamforming vector , it is straightforward to show that,
(18) 
We wish to design beamformers that provide high, sharp, and constant gain within the desired ACI’s and zero gain everywhere else. We have then for the ideal gain corresponding to such beamformer that,
(19) 
where denotes the area of the th beam in the domain. Therefore, we can derive . It holds that,
(20) 
Using the beamformer we wish to mimic the deal gain in equation (20). Therefore, we formulate the following optimization problem,
(21) 
By partitioning the range of into the predefined intervals, and then uniformly sampling with the rate per interval along both axis, we can rewrite the optimization problem as follows,
(22) 
where,
(23) 
with , for . Note that it holds for all pairs that, . We can rewrite equation (22) as,
(24) 
where,
(25) 
and,
(26) 
Unfortunately, the optimization problem in (24) does not admit an optimal closedform solution as is, due to the absolute values of the complex numbers existing in the formulation. However, note that,
(27) 
with being the standard basis vector for the th axis among pairs. Now, note that for any equal gain where . An equalgain vector is a vector where all elements have equal absolute values (in this case, equal to ). Therefore, we can write:
(28) 
Also, it is straightforward to write,
(29) 
where, , and for , and we have,
(30) 
where,
(31)  
(32) 
Comparing the expressions (24), (28), and (29), one can show that the optimal choice of in (21) is the solution to the following optimization problem for proper choices of .
Problem 1.
Given equalgain vectors , for find vector such that
(33) 
However, we now need to find the optimal choices of that minimize the objective in (24). Using (28), and (29), we have the following optimization problem.
Problem 2.
Find equalgain vectors , such that
(34) 
where denotes the elementwise absolute value of a vector.
Iv Proposed Multibeam Design
Note that the solution to problem 1 is the limit of the sequence of solutions to a leastsquare optimization problem as goes to infinity. For each we find that,
(35)  
(36) 
where , noting that it holds that,
(37) 
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 convexification of the objective problem (34) in the form of
(38) 
and using from (36) leads to an effective solution for the original problem. Indeed, it can be verified by solving the optimization problem (38) numerically that a closetooptimal solution admits the following form.
(39) 
for some , where , . In the following, we use the the analytical form (39) for for the rest of our derivations. This solution would not be the optimal solution for the original problem (34). However, it provides a near optimal solution with added benefits of allowing to (i) find the limit of the solution as goes to infinity, and (ii) express the beamforming vectors in closed form, as it will be revealed in the following discussion. An analytical closed form solution for can be found as follows. It holds that,
(40) 
where . We can then write for the component of the beamformer ,
(41) 
to get,
(43) 
with , and , for . Now that the closedform expression for , and therefore, is known, the RIS parameters at the antenna placed at location can be easily computed. More precisely, we get,
(44)  
(45) 
In the case that gain control (attenuation) at the RIS elements is not feasible, will be replaced by the derivation for the absolute value of the RIS parameters. Next, we verify the effectiveness of our multibeamforming design approach by means of numerical experiments.
V Performance Evaluation
In this section, we evaluate the performance of our multibeam design framework. We aim to design a dualbeam which comprises of two lobes with centers at and in the domain, and with the beamwidth equal to . We divide both the , and the range uniformly into , and regions resulting in equallyshaped units in domain. We cover each desired beam with the smallest number of the designed units to provide uniform gain at the desired angular regions. Figures 7(a)(c) depict the beam pattern of the dual beam obtained through our design where all angles are measured in radians. Figure 7(a), shows the heatmap corresponding to the gain of the reflected beam from RIS for the designed dualbeam. The gains are computed in dB. It can be seen that the designed beamformer generates two disjoint beams with an almost uniform gain over the desired ACIs. It is also observed that the beams sharply drop outside the desired ACIs and effectively suppress the gain everywhere outside ACI. In order to quantify the suppression we depict the crosssection of the gain pattern at a fixed elevation angle for two values of located inside the two lobes of the designed dual beam in Figure 7(b). Similarly, Figure 7(c) shows the crosssection of the beam pattern at a fixed azimuth angle for two values of . Both Figures 7(b) and (c) confirm the sharpness of both lobes of the designed dualbeam and can be used to find the beamwidth of each lobes at an arbitrary fraction from its maximum values, e.g,, the 3dB beamwidth or 10dB beamwidth. Indeed, there is negligible difference between 3dB and 10dB beamwidth which clarifies the sharpness of the beams. From Figures 7(b) and(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 angle may suffer from slight deformation as seen by Figure 7(a). This phenomenon worsens as the corresponding lobes of the beams get too close to the plane of the RIS.
Finally, in order to compare the performance of our multibeam design to a singlebeam design, we consider a beam with single lobe which is capable of covering the same two regions as in the dualbeam design. Figure 7(d), shows the heatmap corresponding to the gain of the reflected beam from RIS for the corresponding single beam that is optimized based on our design. As it was the case for multibeam, this figure also shows that for a single beam our design generates an almost uniform and fairly sharp beam. However, comparing Figures 7(a) and(d), we observe that in the desired ACI, the multibeamforming procedure, enhances the gain by about dB over the beams with optimized single lobe.
Vi Conclusions
RIS can be incorporated into mmWave communications to fill the coverage gaps in the blindspots of the mmWave system. We proposed a novel approach for designing RIS employing a UPA antenna structure, that is capable of covering multiple disjoint angular intervals simultaneously. Both our theoretical results and numerical experiments demonstrate that our technique termed as multibeamforming will result in sharp, high, and stable gains within the desired ACI’s regardless of their spatial locations, while effectively, suppressing all the undesired outofband components.
References
 [1] (2020) Intelligent reflecting surface: practical phase shift model and beamforming optimization. IEEE Trans. on Comm. 68 (9), pp. 5849–5863. External Links: Document Cited by: §I.
 [2] (2020) Reconfigurable intelligent surface assisted two–way communications: performance analysis and optimization. IEEE Transactions on Communications 68 (10), pp. 6552–6567. External Links: Document Cited by: §I.
 [3] (2019) Wireless communications through reconfigurable intelligent surfaces. IEEE Access 7 (), pp. 116753–116773. External Links: Document Cited by: §I.
 [4] (2020) Practical hybrid beamforming with finiteresolution phase shifters for reconfigurable intelligent surface based multiuser communications. IEEE Transactions on Vehicular Technology 69 (4), pp. 4565–4570. External Links: Document Cited by: §I.
 [5] (2020) Weighted sumrate maximization for reconfigurable intelligent surface aided wireless networks. IEEE Transactions on Wireless Communications 19 (5), pp. 3064–3076. External Links: Document Cited by: §I.
 [6] (2019) Large intelligent surfaceassisted wireless communication exploiting statistical csi. IEEE Trans. on Vehicular Tech. 68 (8), pp. 8238–8242. External Links: Document Cited by: §I.

[7]
(2020)
Reconfigurable intelligent surface assisted multiuser miso systems exploiting deep reinforcement learning
. IEEE Journal on Selected Areas in Communications 38 (8), pp. 1839–1850. External Links: Document Cited by: §I.  [8] (2019) Reconfigurable intelligent surfaces for energy efficiency in wireless communication. IEEE Transactions on Wireless Communications 18 (8), pp. 4157–4170. External Links: Document Cited by: §I.
 [9] (2020) Performance analysis of large intelligent surfaces (liss): asymptotic data rate and channel hardening effects. IEEE Transactions on Wireless Communications 19 (3), pp. 2052–2065. External Links: Document Cited by: §I.
 [10] (2020) Reconfigurable intelligent surface assisted uav communication: joint trajectory design and passive beamforming. IEEE Wireless Communications Letters 9 (5), pp. 716–720. External Links: Document Cited by: §I.
 [11] (2018) A new wireless communication paradigm through softwarecontrolled metasurfaces. IEEE Communications Magazine 56 (9), pp. 162–169. External Links: Document Cited by: §I.
 [12] (2021) Reconfigurable intelligent surfaces: principles and opportunities. IEEE Communications Surveys Tutorials (), pp. 1–1. External Links: Document Cited by: §I.
 [13] (2020) Physical layer security in vehicular networks with reconfigurable intelligent surfaces. In 2020 IEEE 91st Vehicular Technology Conference (VTC2020Spring), Vol. , pp. 1–6. External Links: Document Cited by: §I.
 [14] (2020) Asymptotic maxmin sinr analysis of reconfigurable intelligent surface assisted miso systems. IEEE Transactions on Wireless Communications 19 (12), pp. 7748–7764. External Links: Document Cited by: §I.
 [15] (2017) Common codebook millimeter wave beam design: designing beams for both sounding and communication with uniform planar arrays. IEEE Transactions on Communications 65 (4), pp. 1859–1872. External Links: Document Cited by: §IIB.
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