Aerial Intelligent Reflecting Surface: Joint Placement and Passive Beamforming Design with 3D Beam Flattening
aa r X i v : . [ ee ss . SP ] J u l Aerial Intelligent Reflecting Surface:Joint Placement and Passive BeamformingDesign with 3D Beam Flattening
Haiquan Lu, Yong Zeng,
Member, IEEE,
Shi Jin,
Senior Member, IEEE, and Rui Zhang,
Fellow, IEEE
Abstract —Intelligent reflecting surface (IRS) is a promisingtechnology to reconfigure wireless channels, which brings a newdegree of freedom for the design of future wireless networks. Thispaper proposes a new three-dimensional (3D) wireless passiverelaying system enabled by aerial IRS (AIRS). Compared tothe conventional terrestrial IRS, AIRS enjoys more deploymentflexibility as well as wider-range signal reflection, thanks to itshigh altitude and thus more likelihood of establishing line-of-sight(LoS) links with ground source/destination nodes. Specifically,we aim to maximize the worst-case signal-to-noise ratio (SNR)over all locations in a target area by jointly optimizing thetransmit beamforming for the source node and the placement aswell as 3D passive beamforming for the AIRS. The formulatedproblem is non-convex and thus difficult to solve. To gain usefulinsights, we first consider the special case of maximizing theSNR at a given target location, for which the optimal solutionis obtained in closed-form. The result shows that the optimalhorizontal AIRS placement only depends on the ratio betweenthe source-destination distance and the AIRS altitude. Then forthe general case of AIRS-enabled area coverage, we propose anefficient solution by decoupling the AIRS passive beamformingdesign to maximize the worst-case array gain , from its placementoptimization by balancing the resulting angular span and thecascaded channel path loss. Our proposed solution is based ona novel
3D beam broadening and flattening technique, wherethe passive array of the AIRS is divided into sub-arrays ofappropriate size, and their phase shifts are designed to forma flattened beam pattern with adjustable beamwidth cateringto the size of the coverage area. Both the uniform linear array(ULA)-based and uniform planar array (UPA)-based AIRSs areconsidered in our design, which enable two-dimensional (2D) and3D passive beamforming, respectively. Numerical results showthat the proposed designs achieve significant performance gainsover the benchmark schemes.
Index Terms —Aerial intelligent reflecting surface, 3D passivebeamforming, beam broadening and flattening, joint placementand beamforming design.
I. I
NTRODUCTION
While the fifth-generation (5G) wireless communicationnetwork is being deployed worldwide, research on thenext/sixth-generation (6G) wireless network has embarked. As
Part of this work has been presented at the IEEE ICC 2020 Workshop,Dublin, Ireland, 7-11 June 2020 [1].H. Lu, Y. Zeng, and S. Jin are with the National Mobile CommunicationsResearch Laboratory, Southeast University, Nanjing 210096, China, Y. Zengis also with the Purple Mountain Laboratories, Nanjing 211111, China (e-mail: { haiquanlu, yong zeng, jinshi } @seu.edu.cn). ( Corresponding author:Yong Zeng. )R. Zhang is with the Department of Electrical and Computer En-gineering, National University of Singapore, Singapore 117583 (e-mail:[email protected]). a key driver for the future intelligent information empoweredsociety, 6G is expected to provide pervasive connectivity withdata rate 100-1000 times higher than that of 5G, i.e., up to 1Tera-byte per second (Tbps) [2], [3]. To this end, several keywireless transmission technologies, such as Terahertz com-munication and ultra-massive multiple-input multiple-output(UM-MIMO) have received significant research attention [4].Despite of the great potential for drastic performance improve-ment by such technologies, their required large antenna arraysat high carrier frequency render practical implementationissues such as hardware cost, power consumption and signalprocessing complexity more severe. Therefore, developinghigh-capacity yet cost-effective communication techniques isof paramount importance for 6G.During the past years, various cost-effective wireless com-munication techniques have been proposed at the transmitterand/or receiver side, such as analog beamforming [5], hybridanalog/digital beamforming [6], [7], lens MIMO communi-cations [8], and low-resolution analog-to-digital converters(ADCs) [9], [10]. More recently, wireless communicationaided by intelligent reflecting surface (IRS) has emerged asa new promising technique for achieving cost-effective wire-less communications via proactively manipulating the radioenvironment [11]–[24]. IRS is a man-made reconfigurablemetasurface composed of a large number of regularly arrangedsub-wavelength passive elements and a smart controller [18],[19]. Through modifying the amplitude and/or phase of theimpinging radio waves, IRS is able to dynamically controlthe radio propagation for various purposes, such as signalenhancement, interference suppression and transmission secu-rity [19]. Different from the conventional active relays, theradio signal reflected by IRS is free from self-interferenceor noise corruption in an inherently full-duplex manner. IRS-aided wireless communication has been studied from differentaspects, such as energy efficiency maximization [15], secrecyrate maximization [17], joint active and passive beamformingdesign [18], and rate region characterization for IRS-aidedinterference channel [24], etc.However, most existing research on IRS-aided communi-cation focuses on terrestrial IRS that is deployed on e.g.,facades of buildings or indoor walls/ceilings. Such an IRSarchitecture poses fundamental limitations for several reasons.First, from the deployment perspective, finding the appropriateplace for IRS installation is usually difficult in practice, since itinvolves various issues like site rent, impact of urban landscapeand the willingness of owners to install large IRS on their ! "! " (cid:3) !" (cid:3) (a) Terrestrial IRS ! "! " $%& (cid:3) (cid:3) (b) AIRSFig. 1. Half-space reflection by terrestrial IRS versus panoramic/full-anglereflection by AIRS. !" (a) Terrestrial IRS !" (b) AIRSFig. 2. AIRS can reduce the number of reflections than terrestrial IRS. properties. Second, from the performance perspective, IRSdeployed on the walls or facades of buildings can at mostserve terminals located in half of the space, i.e., both the sourceand destination nodes must lie on the same side of the IRS, asillustrated in Fig. 1(a). Third, as shown in Fig. 2(a), in complexenvironment like urban areas, the radio signals originated froma source node typically have to undergo several reflectionsbefore reaching the desired destination. This thus leads tosignificant signal attenuation since each reflection, even withIRS-enabled passive beamforming, would still cause signalscattering to undesired directions.To address the above issues, we propose in this paper anovel three-dimensional (3D) wireless network enabled byaerial IRS (AIRS), where IRS is mounted on aerial platformslike balloon, unmanned aerial vehicle (UAV), so as to enableintelligent reflection from the sky. Compared to the conven-tional terrestrial IRS, AIRS has several appealing advantages.First, with elevated position, AIRS can more easily establishline-of-sight (LoS) links with the ground nodes [25], whichleads to stronger channel as compared to the terrestrial IRS.At the same time, the placement or trajectory of aerial plat-forms can be more flexibly optimized to further improve thecommunication performance, thereby offering a new degree offreedom (DoF) for performance enhancement via 3D networkdesign. Second, AIRS is able to achieve panoramic/full-anglereflection, i.e., one AIRS can in principle help reflect signalsbetween any pair of nodes on the ground, as illustrated inFig. 1(b). This is in a sharp contrast to the conventionalterrestrial IRS that can only serve nodes in half of the space.Last but not least, in contrast to the terrestrial IRS, AIRSis usually able to achieve desired signal manipulation byone reflection only, even in complex urban environment (seeFig. 2(b)), thanks to its high likelihood of having LoS links with the ground nodes. This thus greatly reduces the signalpower loss due to multiple reflections with the terrestrial IRS.Despite of the many advantages mentioned above, AIRSfaces several new challenges, such as the endurance, stabilityand controllability of the aerial platform carrying the IRS, theadditional safety measures required to deploy or fly AIRS,the impact of drift or vibration of the aerial platform on theperformance and design of AIRS, as well as the challenges inchannel estimation for AIRS-aided communication, which alldeserve further investigation.In this paper, we consider a basic setup of AIRS-enabledwireless relaying system, where an AIRS is deployed to helpextend the signal coverage from a ground source node (e.g.,base station (BS)/access point (AP)) to a given target area,say, a hot spot in cellular network or a remote area withoutcellular coverage. Our objective is to maximize the worst-case/minimum signal-to-noise ratio (SNR) in the target areaby jointly optimizing the transmit beamforming for the sourcenode and the placement as well as 3D passive beamform-ing for the AIRS. The formulated optimization problem isdifficult to be directly solved due to the following reasons.First, different from most of the existing research where thepassive beamforming of the terrestrial IRS is designed basedon the channel state information (CSI) of users at knownlocations, the beamforming optimization for the AIRS needsto balance the received SNRs at all locations in the target area,which results in a more complicated design problem. Second,different from terrestrial IRS deployed at fixed location, theAIRS placement is a new problem to solve, which affects notonly the source-AIRS-destination cascaded channel path loss,but also the angle of departures (AoDs) and angle of arrivals(AoAs) of the source-AIRS link as well as the angular span from the AIRS to the target area. The main contributions ofthis paper are summarized as follows.First, for the general uniform planar array (UPA)-basedAIRS, we develop the 3D channel model for our consideredsystem, which shows the effects of the AIRS placement onthe achievable SNR at any given location in the target area.Based on this model, an optimization problem is formulatedto maximize the worst-case SNR over all locations in thetarget area by jointly optimizing the transmit beamformingat the source node and the AIRS placement as well as 3Dpassive beamforming . We show that the optimal transmitbeamforming for the source node corresponds to the well-known maximum ratio transmission (MRT) towards the AIRS,regardless of the reflecting links from the AIRS to the targetcoverage area. Therefore, the studied problem is reduced tothe joint optimization of the AIRS placement and 3D passivebeamforming for the min-SNR maximization in the target area.Next, to draw essential insights, we consider the specialcase of SNR maximization at a given target location, forwhich the optimal AIRS placement and phase shifts for passivebeamforming are derived in closed-form. The solutions showthat the phase shifts of all AIRS elements should be designedto ensure that all rays reflected by the AIRS are coherently This facilitates the multiple access design for users at random locations inthe target area, via e.g., time-division multiple-access (TDMA) or frequency-division multiple-access (FDMA). combined at the single destination, as expected. Furthermore,the optimal horizontal AIRS placement depends on ρ , whichis the ratio between the source-destination distance and theAIRS altitude. For ≤ ρ ≤ , the AIRS should alwaysbe placed exactly above the midpoint between the sourcenode and the destination location. On the other hand, when ρ > , there exist two optimal horizontal AIRS placementlocations that are symmetric over the midpoint, and approachthe source/destination location as ρ increases.Last, for the general case of min-SNR maximization ina target area, we propose an efficient two-step solution bydecoupling the AIRS passive beamforming design and itsplacement optimization, where the former aims to maximizethe worst-case array gain and the latter balances the result-ing angular span and the cascaded path loss. The key ofthe proposed solution lies in a novel beam broadening andflattening technique, where the passive array is partitioned intomultiple sub-arrays with their phase shifts optimized to formone single flattened beam with its beamwidth properly tunedto match the size of the target coverage area. For ease ofexposition, we first consider the special case of uniform lineararray (ULA)-based AIRS, for which passive beam flatteningonly needs to be applied in one spatial frequency dimension.Then the proposed design is extended to the general UPA-based AIRS, which enables 3D passive beam flattening overtwo spatial frequency dimensions. By leveraging the proposedbeam broadening and flattening technique, an approximatelyequal array gain is achieved for all locations in the targetcoverage area, which subsequently simplifies the optimizationof AIRS placement for balancing the resulting angular spanand the cascaded path loss. Extensive numerical results areprovided, which show substantial performance gains of theproposed design over the benchmark schemes.Notice that wireless communication aided by aerial plat-forms (e.g., balloons and UAVs) has received significantattention recently (see [25] and references therein). However,most of such existing works are based on the conventionalactive communication techniques, such as active aerial signaltransmission/receiving/relaying. In fact, the appealing advan-tages of the passive IRS, such as its compact size, light weight,low energy consumption, and conformal geometry, make it apromising alternative for aerial platforms than conventionalmobile relays based on active communication techniques.On the other hand, beam broadening/flattening technique hasreceived increasing attention recently, mainly in the contextof analog beamforming for cost-effective designs at the trans-mitter side [5], [26]–[28]. Different from the existing work,in this paper, we provide a systematic and rigorous derivationfor the 3D passive beam broadening/flattening for AIRS-aidedcommunications, so as to form a 3D passive beam pattern thatmatches the size of the target coverage area with nearly equalgains over all locations therein. The derived results provideinsight into the achievable beamforming gain in terms of thesub-array grouping and phase shifts design, which greatlysimplifies the subsequent AIRS placement optimization.The rest of this paper is organized as follows. Section IIintroduces the system model and the problem formulation forthe AIRS-enabled wireless relaying system. In Section III and !" xyz H !" xyz ! " T R R T R Fig. 3. AIRS-enabled wireless communication system.
IV, we propose efficient algorithms to solve the formulatedproblems in the single target location and area coverage cases,respectively. Section V presents numerical results to evaluatethe performance of the proposed designs. Finally, we concludethe paper in Section VI.
Notations:
Scalars are denoted by italic letters. Vectorsand matrices are denoted by bold-face lower- and upper-caseletters, respectively. C M × N denotes the space of M × N complex-valued matrices. For a vector x , k x k denotes itsEuclidean norm. diag( x ) denotes a diagonal matrix with itsdiagonal elements given by x . The symbol j denotes theimaginary unit of complex numbers, with j = − . For a realnumber x , ⌈ x ⌉ denotes the smallest integer that is greater thanor equal to x . The symbol ⊗ denotes the Kronecker productoperation.II. S YSTEM M ODEL A ND P ROBLEM F ORMULATION
As shown in Fig. 3, we consider an AIRS-enabled wirelessrelaying system, where IRS mounted on an aerial platform(e.g., balloon or UAV) is deployed to assist a terrestrial sourcenode (e.g., BS/AP) to extend its communication coverage to agiven terrestrial area of interest, A . We assume that the directlink from the source node to the target area is negligible due tosevere terrestrial blockage/shadowing. Without loss of gener-ality, we assume that the source node is located at the origin ofa 3D Cartesian coordinate system and the center of the targetarea is on the x -axis, which is denoted as w = [ x , T on the x - y plane. For ease of exposition, we assume that A is a rectangular area on the x - y plane. Therefore, anylocation in A can be specified as w = [ w x , w y ] T on the x - y plane, w x ∈ (cid:2) x − D x , x + D x (cid:3) , w y ∈ h − D y , D y i , with D x and D y denoting the length and width of the rectangulararea, respectively. The AIRS consists of a sub-wavelengthUPA with N = N x N y passive reflecting elements, where N x and N y denote the number of elements along the x - and y -axis, respectively. The adjacent elements are separated by d x < λ and d y < λ , respectively, where λ denotes the signalwavelength. The source node is assumed to be equipped witha conventional UPA placed on the y - z plane, and the numberof antennas is M = M y M z with M y and M z denoting thenumber of elements along the y - and z -axis, respectively. !" xyz !" xyz R R ! !" ! " ! " . //0 x y x xn n y y n dn d ! " ! !" ( )’ * x y n x q y q H R Fig. 4. An illustration of the wave propagation direction from the sourcenode to the AIRS with zenith AoA, φ R , and azimuth AoA, η R . We assume that the altitude of the AIRS is fixed at
H > . Without loss of generality, as shown in Fig. 4, we takethe bottom-left element of the AIRS as the reference pointto represent the AIRS’s horizontal location, whose coordinateis denoted by q = [ q x , q y ] T on the x - y plane. Therefore, thedistance from the source node to the AIRS, and that fromthe AIRS to any location of the target area w ∈ A can beexpressed as d G = q H + k q k and d h = q H + k q − w k ,respectively.In practice, the communication links between the aerialplatform (with sufficiently large H ) and ground nodes aredominated by LoS with high probability [25]. Thus, forsimplicity, we assume that the aerial-ground channel followsthe free-space path loss model, and the channel power gainfrom the source node to the AIRS can be expressed as β G ( q ) = β d − G = β H + k q k , (1)where β represents the channel power at the reference dis-tance d = 1 m. Similarly, the channel power gain from theAIRS to any location w ∈ A can be expressed as β h ( q , w ) = β d − h = β H + k q − w k . (2)Note that in practice, the AIRS size is much smaller thanthe link distances between the AIRS and source/destinationnodes. Therefore, the signal from the source node to the AIRSand that from the AIRS to the destination node can be wellapproximated as uniform plane waves. As illustrated in Fig. 4,denote by φ R ( q ) the zenith AoA of the signal from the sourcenode to the AIRS, i.e., the angle between the wave propagationdirection and the positive z -axis, and η R ( q ) the azimuth AoA,i.e., the angle between the horizontal projection of the wavepropagation direction and the positive x -axis. The receive arrayresponse of the AIRS, denoted as a R ( φ R ( q ) , η R ( q )) , is thusdependent on the AIRS (horizontal) placement q , which isderived as follows. As shown in Fig. 4, with the AIRS locatedat ¯q = [ q x , q y , H ] T in 3D, the wave propagation direction ofthe signal from the source node to the AIRS is k = ¯q k ¯q k , andthe coordinate of the ( n x , n y ) th AIRS element is c n x ,n y = ¯q +[( n x − d x , ( n y − d y , T , ≤ n x ≤ N x , ≤ n y ≤ N y .Then the phase delay of the ( n x , n y ) th element relative to the reference element at ¯q is ψ n x ,n y = 2 πλ k T (cid:0) c n x ,n y − ¯q (cid:1) = 2 πλ ¯q T k ¯q k ( n x − d x ( n y − d y = 2 πλ (cid:18) q x k ¯q k ( n x − d x + q y k ¯q k ( n y − d y (cid:19) . (3)Furthermore, it is observed from Fig. 4 that the horizon-tal coordinates of the AIRS can be expressed as q x = k ¯q k sin ( φ R ) cos ( η R ) and q y = k ¯q k sin ( φ R ) sin ( η R ) . Wethus have ψ n x ,n y = 2 πλ (( n x − d x sin ( φ R ) cos ( η R ) +( n y − d y sin ( φ R ) sin ( η R )) . (4)Thus, the corresponding complex coefficients for the ( n x , n y ) th element is e − jψ nx,ny , ≤ n x ≤ N x , ≤ n y ≤ N y . By concatenating such complex coefficients of all the N = N x N y elements, the receive array response vector ofthe AIRS can be expressed as a R ( φ R ( q ) , η R ( q )) = h , · · · , e − j π ( N x −
1) ¯ d x ¯Φ R ( q ) i T ⊗ h , · · · , e − j π ( N y −
1) ¯ d y ¯Ω R ( q ) i T , (5)where ¯ d x ∆ = d x λ , ¯ d y ∆ = d y λ , ¯Φ R ( q ) ∆ =sin ( φ R ( q )) cos ( η R ( q )) = q x k ¯q k can be interpretedas the spatial frequency along the x -dimensioncorresponding to AoAs φ R ( q ) and η R ( q ) , and ¯Ω R ( q ) ∆ = sin ( φ R ( q )) sin ( η R ( q )) = q y k ¯q k as the spatialfrequency along the y -dimension. Similarly, the transmitarray response with respect to AoDs from the source nodeto the AIRS can be obtained, which is compactly denoted as a T,s ( q ) for convenience, with k a T,s ( q ) k = M . Thus, thechannel matrix from the source node to the AIRS, denoted as G ( q ) ∈ C N × M , can be expressed as G ( q ) = p β G ( q ) e − j πd G λ a R ( φ R ( q ) , η R ( q )) a HT,s ( q ) , (6)Note that the AIRS placement q affects not only the path loss β G ( q ) , but also the AoDs and AoAs of the source-AIRS link.Similarly, denote by φ T ( q , w ) and η T ( q , w ) the zenith andazimuth AoDs for the communication link from the AIRS toany location w ∈ A . The reflect array response at the AIRScan be similarly obtained as a T ( φ T ( q , w ) , η T ( q , w )) = h , · · · , e − j π ( N x −
1) ¯ d x ¯Φ T ( q , w ) i T ⊗ h , · · · , e − j π ( N y −
1) ¯ d y ¯Ω T ( q , w ) i T , (7)where ¯Φ T ( q , w ) ∆ = sin ( φ T ( q , w )) cos ( η T ( q , w )) = w x − q x k ¯w − ¯q k with ¯w = [ w x , w y , T can be interpreted asthe spatial frequency along the x -dimension correspond-ing to AoDs φ T ( q , w ) and η T ( q , w ) , and ¯Ω T ( q , w ) ∆ =sin ( φ T ( q , w )) sin ( η T ( q , w )) = w y − q y k ¯w − ¯q k as the spatial fre-quency along the y -dimension. Then the channel from the AIRS to a location w ∈ A , denoted as h H ( q , w ) ∈ C × N ,can be expressed as h H ( q , w ) = p β h ( q , w ) e − j πd h λ a HT ( φ T ( q , w ) , η T ( q , w )) . (8)As a result, the received signal at each location w ∈ A is y ( q , Θ , w , v ) = h H ( q , w ) ΘG ( q ) v √ P s + n, (9)where Θ = diag (cid:0) e jθ , · · · , e jθ N (cid:1) is a diagonal phase-shiftmatrix with θ n = θ n x ,n y = θ ( n x − N y + n y ∈ [0 , π ) denotingthe phase shift of the n th reflecting element that is located atthe n x th column and n y th row on the IRS; P and s are thetransmit power and information-bearing signal at the sourcenode, respectively; v ∈ C M × is the transmit beamformingvector at the source node with k v k = 1 ; n is the additivewhite Gaussian noise (AWGN) with zero mean and power σ .The received SNR at the location w ∈ A is thus expressed as γ ( q , Θ , w , v ) = ¯ P (cid:12)(cid:12) h H ( q , w ) ΘG ( q ) v (cid:12)(cid:12) , (10)where ¯ P = Pσ . By denoting θ = [ θ , · · · , θ N ] , our objectiveis to maximize the worst-case/minimum SNR within the targetarea A , by jointly optimizing the transmit beamforming vector v of the source node, as well as the AIRS placement q andits 3D passive beamforming with phase shifts θ . The problemis formulated as (P1) max q , θ , v min w ∈A γ ( q , Θ , w , v )s . t . ≤ θ n < π, n = 1 , · · · , N, k v k = 1 . Note that in practice, a separate reliable wireless controllink could be used for the control of the AIRS. Further-more, it can be seen from (P1) that the AIRS placementand beamforming design mainly depends on the location andsize of the target area A , rather than the instantaneous CSIof the users, as in most prior work on IRS passive beam-forming design. This eases the control and synchronizationrequirements between the AIRS and ground source node. Byexploiting the special structure of the concatenated channel ˜h H , h H ( q , w ) ΘG ( q ) , we first show that the optimaltransmit beamforming vector v corresponds to the simpleMRT towards the AIRS, regardless of the reflecting link fromthe AIRS to the target area. Proposition 1:
The optimal transmit beamforming vector v to (P1) is v ∗ = a T,s ( q ) √ M . Proof:
For any given AIRS placement q , target location w and AIRS phase shifts θ , it is known that the optimal transmitbeamforming vector to maximize γ ( q , Θ , w , v ) in (10), de-noted as v ∗ ( q , Θ , w ) , is the eigenvector corresponding to thelargest eigenvalue of the channel matrix ˜h˜h H . Furthermore, ˜h˜h H can be simplified as ˜h˜h H = G H ( q ) Θ H h ( q , w ) h H ( q , w ) ΘG ( q )= β G ( q ) (cid:12)(cid:12)(cid:12) a HR ( φ R ( q ) , η R ( q )) Θ H h ( q , w ) (cid:12)(cid:12)(cid:12) a T,s ( q ) a HT,s ( q ) . (11) It readily follows that ˜h˜h H is a rank-one matrix, whose eigen-vector is v ∗ = a T,s ( q ) √ M . More importantly, this eigenvector isindependent of the target location w . Thus, regardless of w , it is optimal to set the transmit beamforming as v ∗ = a T,s ( q ) √ M in (P1). The proof is thus completed.By substituting the optimal v ∗ into (10) and after somemanipulations, the corresponding SNR at the target location w ∈ A can be written as (12), shown at the top of the nextpage. As a result, problem (P1) reduces to (P2) max q , θ min w ∈A γ ( q , Θ , w )s . t . ≤ θ n < π, n = 1 , · · · , N. Problem (P2) is difficult to be directly solved due to the fol-lowing reasons. First, the objective function is the worst-caseSNR over a two-dimensional (2D) area, which is difficult tobe explicitly expressed in terms of the optimization variables.Second, the optimization problem is highly non-convex andthe optimization variables q and θ are intricately coupled witheach other, as shown in (12). To tackle this problem, we firstconsider the special case of (P2) for a given location w ∈ A ,for which the optimal solutions for the AIRS phase shifts andplacement are derived in closed-form. Then for the generalcase of (P2), we first consider the simplified ULA-based AIRS,i.e., N y = 1 and N = N x , where the passive beamformingonly involves the beam steering for the spatial frequency ¯Φ along the x -dimension. However, even for this simplified case,problem (P2) is still non-convex and difficult to solve. We thuspropose an efficient suboptimal solution by decoupling the 2Dpassive beamforming design of the AIRS phase-shift vector θ and its placement optimization q , with the former aiming tomaximize the worst-case array gain and the latter to balancethe angular span and the cascaded path loss. In particular, sub-array based beam flattening technique is applied to form oneflattened beam with its beamwidth catering to the size of thetarget area, thus achieving an approximately equal array gainfor all locations in it. Finally, the proposed design for ULA-based AIRS is extended to the general case of UPA-basedAIRS, which enables 3D passive beam flattening to cover thetarget area efficiently.III. O PTIMIZATION FOR
SNR M
AXIMIZATION AT S INGLE T ARGET L OCATION
In this section, we consider the special case of (P2) where A degenerates to one single point, denoted as w . In thiscase, the inner minimization of the objective function in (P2)is irrelevant, and problem (P2) reduces to (P3) max q , θ γ ( q , Θ )s . t . ≤ θ n < π, n = 1 , · · · , N. It is not difficult to see that at the optimal solution to (P3),the different rays reflected by the AIRS should be coherentlyadded at the designated location w . Therefore, based on (12),for any given AIRS placement q , the optimal phase shifts forpassive beamforming are given by θ ∗ n x ,n y ( q ) = ¯ θ − π ( n x −
1) ¯ d x (cid:2) ¯Φ T ( q , w ) − ¯Φ R ( q ) (cid:3) − π ( n y −
1) ¯ d y (cid:2) ¯Ω T ( q , w ) − ¯Ω R ( q ) (cid:3) , (13) γ ( q , Θ , w ) = ¯ P β G ( q ) M (cid:12)(cid:12) h H ( q , w ) Θa R ( φ R ( q ) , η R ( q )) (cid:12)(cid:12) = ¯ P β M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N x P n x =1 N y P n y =1 e j π ( n x −
1) ¯ d x [ ¯Φ T ( q , w ) − ¯Φ R ( q ) ] e j π ( n y −
1) ¯ d y [ ¯Ω T ( q , w ) − ¯Ω R ( q ) ] e jθ nx,ny (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) H + k q − w k (cid:17) (cid:16) H + k q k (cid:17) . (12)where ≤ n x ≤ N x , ≤ n y ≤ N y , and ¯ θ is an arbitrary phaseshift that is common to all reflecting elements. As a result, thereceived SNR at the target location w is simplified as γ ( q ) = ¯ P β M N (cid:16) H + k q − w k (cid:17) (cid:16) H + k q k (cid:17) . (14)To maximize the received SNR given in (14) at the singletarget location w , problem (P3) reduces to (P4) min q (cid:16) H + k q − w k (cid:17) (cid:16) H + k q k (cid:17) . Proposition 2:
For the single-location SNR maximizationproblem (P4), the optimal AIRS 2D placement solution isgiven by q ∗ = ξ ∗ ( ρ ) w , (15)where ρ ∆ = k w k H , and ξ ∗ ( ρ ) = , if 0 ≤ ρ ≤ ± r − ρ , otherwise . (16) Proof:
The result can be obtained by checking the first-order derivative, which is omitted for brevity.Proposition 2 shows that the optimal horizontal placementof the AIRS only depends on ρ , i.e., the ratio between thesource-destination distance k w k and AIRS altitude H . For ≤ ρ ≤ , the AIRS should always be placed exactlyabove the midpoint between the source node and the targetlocation. On the other hand, for ρ > , there exist two optimalhorizontal AIRS placement locations that are symmetric overthe midpoint, as shown in Fig. 5. Note that the above result isdifferent from the conventional active relay placement [29],whose optimal solution generally depends on the transmitpower and the relay processing noise power.With the optimal AIRS placement location in (15), theoptimal SNR at the single target location w is given by (17),shown at the top of the next page. Remark 1:
It is observed from (17) that since the opti-mal SNR decreases with the increase of the AIRS altitude H , a relatively small H should be selected to enhance theperformance. However, such a result is based on the LoSlink assumptions. In order to obtain LoS channels with bothsource/target location, we need to choose sufficiently large H . Therefore, this is also an important trade-off for AIRSdeployment in practice for altitude selection. Distance-versus-altitude ratio T he op t i m a l A I R S dep l o y m en t c oe ff i c i en t * () Fig. 5. The optimal AIRS deployment coefficient ξ ∗ ( ρ ) against distance-versus-altitude ratio ρ = k w k H . IV. O
PTIMIZATION FOR M IN -SNR M AXIMIZATION IN A REA C OVERAGE
In this section, we study the general case of (P2) forAIRS-enabled area coverage. To gain useful insights, we firstconsider the special case of ULA-based AIRS, i.e., N y = 1 and N = N x , for which passive beamforming is only appliedto the x -dimension for steering over the spatial frequency ¯Φ .Then we extend our proposed solution to the general case ofUPA-based AIRS with 3D passive beamforming over both ¯Φ and ¯Ω dimensions. A. The Special Case of ULA-Based AIRS
For ULA-based AIRS, by substituting N y = 1 and N = N x into (12), the SNR reduces to γ ( q , Θ , w ) = ¯ P β M (cid:12)(cid:12)(cid:12)(cid:12) N P n =1 e j ( θ n +2 π ( n −
1) ¯ d x [ ¯Φ T ( q , w ) − ¯Φ R ( q ) ]) (cid:12)(cid:12)(cid:12)(cid:12) (cid:16) H + k q − w k (cid:17) (cid:16) H + k q k (cid:17) , (18)where θ n = θ n x , ≤ n ≤ N .Note that even for the ULA case with the SNR given by(18), solving problem (P2) by standard optimization tech-niques is difficult in general. By exploiting the fact that thephase shifts optimization for IRS resembles the extensivelystudied phase array design or analog beamforming, we proposean efficient two-step solution to (P2) by decoupling the phaseshifts optimization and AIRS placement optimization. First, it γ ∗ ( ρ ) = ¯ P β M N (cid:16) H + k w k (cid:17) , if 0 ≤ ρ ≤ P β M N (cid:18) H + (cid:16) + q − ρ (cid:17) k w k (cid:19) (cid:18) H + (cid:16) − q − ρ (cid:17) k w k (cid:19) , otherwise . (17)is noted that by discarding constant terms, problem (P2) with γ ( q , Θ , w ) given in (18) can be equivalently written as (P5) max q , θ min w ∈A f ( q , θ , w ) f ( q , w )s . t . ≤ θ n < π, n = 1 , · · · , N, where f ( q , θ , w ) ∆ = (cid:12)(cid:12)(cid:12)(cid:12) N P n =1 e j ( θ n +2 π ( n −
1) ¯ d x [ ¯Φ T ( q , w ) − ¯Φ R ( q ) ]) (cid:12)(cid:12)(cid:12)(cid:12) accounts for the array gain due to the passive beamforming bythe AIRS, and f ( q , w ) ∆ = (cid:16) H + k q − w k (cid:17) (cid:16) H + k q k (cid:17) accounts for the concatenated path loss from the source nodeto a location w ∈ A via the AIRS located at q .With the proposed solution, for any given AIRS place-ment q , the phase shifts in θ are designed to maximize theminimum/worst-case array gain among all locations in A bysolving the following problem (P5 .
1) max θ min w ∈A f ( q , θ , w )s . t . ≤ θ n < π, n = 1 , · · · , N. Note that (P5.1) is an approximation of the original problem(P5) with given q , since f ( q , w ) is ignored in the innerminimization of the objective function in (P5.1). Such anapproximation is reasonable since the array gain f ( q , θ , w ) is usually more sensitive than the concatenated path loss f ( q , w ) to the location variation of w in the target area A ,especially when A is small or H ≫ D x and D y . After solving(P5.1), in the second step, the obtained solution to (P5.1),denoted as θ ∗ ( q ) , is substituted into the objective functionof (P5), based on which the AIRS placement is optimized bysolving the following problem (P5 .
2) max q min w ∈A f ( q , θ ∗ ( q ) , w ) f ( q , w ) . As such, the obtained max-min SNR within the target area A after solving the two-step problems will be a lower bound ofthe optimal value of (P5). In the following, we present thedetails for solving (P5.1) and (P5.2), respectively.
1) Beam Broadening and Flattening for Passive Beamform-ing:
For any given AIRS placement q , let ∆ min ( q ) and ∆ max ( q ) denote the minimum and maximum deviation of thespatial frequency ¯Φ T ( q , w ) corresponding to the AoDs alongthe x -axis from ¯Φ R ( q ) in the target area A , respectively, i.e., ∆ min ( q ) ∆ = min w ∈ A ¯Φ T ( q , w ) − ¯Φ R ( q ) , (19) ∆ max ( q ) ∆ = max w ∈A ¯Φ T ( q , w ) − ¯Φ R ( q ) . (20) Then (P5.1) can be equivalently written as max θ min ∆ min ( q ) ≤ ∆ ≤ ∆ max ( q ) g ( θ , ∆) ∆ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X n =1 e j ( θ n +2 π ( n −
1) ¯ d x ∆ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s . t . ≤ θ n < π, ≤ n ≤ N. (21) While directly solving the above optimization problem ischallenging due to its non-convexity, we propose an efficientsolution based on the beam flattening technique. Specifically,for the above max-min problem, θ should be designed suchthat the array gain g ( θ , ∆) defined in (21) is approximatelyequal for all ∆ within the interval [∆ min ( q ) , ∆ max ( q )] ,which implies a flattened beam pattern of the AIRS in thisinterval. Towards this end, the N -element array is partitionedinto L sub-arrays with N s = N/L elements in each sub-array. For notational convenience, we assume that
N/L is aninteger. By using the array manifold concept, the design ofphase shifts vector θ can be transformed into that of AIRS’sarray manifold. Specifically, the manifold of the l th sub-array,denoted as a l (cid:0) ¯Φ l (cid:1) , aims to direct its sub-beam towards thespatial frequency direction ¯Φ l , which can be expressed as a l (cid:0) ¯Φ l (cid:1) = e jα l h , e − j π ¯ d x ¯Φ l , · · · , e − j π ( N s −
1) ¯ d x ¯Φ l i T , l = 1 , · · · , L, (22) where ¯Φ l ∆ = sin ( φ l ) cos ( η l ) with φ l and η l denoting the zenithand azimuth angles of beam direction for the l th sub-array,respectively; e jα l is a common phase term for the l th sub-array. Thus, the phase shifts θ n in (21) can be expressed as θ n = θ ( l − N s + i = α l − π ( i −
1) ¯ d x ¯Φ l , l = 1 , · · · , L, i =1 , · · · , N s . Then the array gain g ( θ , ∆) in (21) can be furtherexpressed as a function of L , N s and (cid:8) α l , ¯Φ l (cid:9) Ll =1 , i.e., g (cid:0) L, N s , (cid:8) α l , ¯Φ l (cid:9) , ∆ (cid:1) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L X l =1 N s X i =1 e jθ ( l − Ns + i e j π (( l − N s + i −
1) ¯ d x ∆ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L X l =1 e j ( α l +2 π ( l − N s ¯ d x ∆ ) N s X i =1 e j π ( i −
1) ¯ d x ( ∆ − ¯Φ l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L X l =1 e j ( α l + π ((2 l − N s −
1) ¯ d x ∆ − π ( N s −
1) ¯ d x ¯Φ l ) sin (cid:0) πN s ¯ d x (cid:0) ∆ − ¯Φ l (cid:1)(cid:1) sin (cid:0) π ¯ d x (cid:0) ∆ − ¯Φ l (cid:1)(cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L X l =1 e jh ( α l , ∆ , ¯Φ l ) sin (cid:0) πN s ¯ d x (cid:0) ∆ − ¯Φ l (cid:1)(cid:1) sin (cid:0) π ¯ d x (cid:0) ∆ − ¯Φ l (cid:1)(cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (23) where h (cid:0) α l , ∆ , ¯Φ l (cid:1) ∆ = α l + π ((2 l − N s −
1) ¯ d x ∆ − π ( N s −
1) ¯ d x ¯Φ l . As such, problem (21) can be transformed ... ... intersection point intersection point Fig. 6. An illustration of beam flattening via sub-array grouping and steering. to max L,N s , { α l , ¯Φ l } min ∆ min ( q ) ≤ ∆ ≤ ∆ max ( q ) g (cid:0) L, N s , (cid:8) α l , ¯Φ l (cid:9) , ∆ (cid:1) s . t . LN s = N, − ≤ ¯Φ l ≤ , l = 1 , · · · , L, ≤ α l < π, l = 1 , · · · , L. (24)To solve (24), we first study the property of the followingfunction that appears in (23): s (∆) = sin (cid:0) πN s ¯ d x ∆ (cid:1) sin (cid:0) π ¯ d x ∆ (cid:1) . (25)It is observed that s (∆) has a peak at ∆ = 0 with s (0) = N s ,and nulls at ∆ = ± kN s ¯ d x , k = 1 , · · · , N s − . The null-to-null beamwidth is thus given by ¯∆ BW = N s ¯ d x , as illustratedin Fig. 6. This corroborates the fact that the beamwidth ofphase array is inversely proportional to the array aperture N s ¯ d x [30]–[32]. Furthermore, we take the spatial frequencyinterval h − N s ¯ d x , N s ¯ d x i as the beam coverage interval [33],so that the coverage beamwidth (CVBW) with sub-array size N s is ¯∆ CVBW ( N s ) = 1 N s ¯ d x . (26)Next, it is observed that with the array manifold designgiven in (22), the resulting array gain in (23) is given bya superposition of L copies of s (∆) , each shifted by aspatial frequency ¯Φ l with a phase coefficient h (cid:0) α l , ∆ , ¯Φ l (cid:1) .To obtain a flattened beam, the sub-array beam directions ¯Φ l , l = 1 , · · · , L , should be carefully designed. Inspiredby the subcarrier spacing for Orthogonal Frequency DivisionMultiplexing (OFDM), as illustrated in Fig. 6, ¯Φ l is designedso that the adjacent spatial frequency shift is separated bythe spatial frequency resolution of the subarray, namely N s ¯ d x .Thus, we have ¯Φ l = ¯Φ + l − N s ¯ d x , l = 1 , · · · , L, (27)for some starting spatial frequency ¯Φ , and the resultingbeam pattern will be centered at 0 when ¯Φ = − L − N s ¯ d x .Furthermore, it is seen that at ∆ = ¯Φ l , only the l th sub-array would contribute to the array gain since it correspondsto the nulls of all other sub-arrays. By substituting (27)into (23), the resulting array gain for ∆ = ¯Φ l is givenby g (cid:0) L, N s , (cid:8) α l , ¯Φ l (cid:9) , ∆ = ¯Φ l (cid:1) = (cid:12)(cid:12)(cid:12)(cid:12) sin ( πN s ¯ d x ( ∆ − ¯Φ l )) sin ( π ¯ d x ( ∆ − ¯Φ l )) (cid:12)(cid:12)(cid:12)(cid:12) = N s , l = 1 , · · · , L . -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8051015202530354045 A rr a y ga i n ( d B ) Sub-array beam 1Sub-array beam 2Sub-array beam 3Sub-array beam 4Resulting beam
Fig. 7. Array gain of the broadened and flattened beam versus spatialfrequency.
For ∆ = ¯Φ l , the array gain is in general given bythe superposition of the contributions from all the L sub-arrays. However, since sin ( πN s ¯ d x ∆ ) sin ( π ¯ d x ∆ ) is relatively small for | ∆ | ≥ N s ¯ d x , a closer look at (23) and (27) reveals that for anygiven ∆ value, those two adjacent sub-arrays would have themost significant contributions to the array gain. Furthermore,in order to mitigate the array fluctuation, the phase shift α l of each sub-array is designed so that the resulting array gainat the intersection points of the beam pattern of two adjacentsub-arrays is maximized. Specifically, for the intersection point ∆ p = ¯Φ + p − N s ¯ d x + N s ¯ d x , p = 1 , · · · , L − , the array gaincan be expressed as (28), shown at the top of the next page,where in ( a ) we have only retained the two most significantadjacent terms for l = p, p +1 , since when (cid:12)(cid:12) ∆ p − ¯Φ l (cid:12)(cid:12) > N s ¯ d x ,the effect of the l th sub-array on the intersection point ∆ p is small, as shown in Fig. 6. To maximize (28), we shouldhave h (cid:0) α p , ∆ p , ¯Φ p (cid:1) = h (cid:0) α p +1 , ∆ p , ¯Φ p +1 (cid:1) + 2 kπ, k ∈ Z . Bysubstituting ¯Φ p and ¯Φ p +1 into this equation, it follows that α p +1 − α p = − πN s ¯ d x (cid:18) ¯Φ + 2 p − N s ¯ d x (cid:19) + π ( N s − N s + 2 kπ = − πN s ¯ d x ¯Φ − πN s + 2 kπ. (29)Let k = 0 , the common phase term for each sub-array can begiven by α l = − (cid:18) πN s ¯ d x ¯Φ + πN s (cid:19) l, l = 1 , · · · , L. (30)With the obtained sub-array common phase terms, itis observed from (28) that for sufficiently large N s ,the resulting array gain at the intersection points is g (cid:0) L, N s , (cid:8) α l , ¯Φ l (cid:9) , ∆ = ∆ p (cid:1) = ( π Ns ) ≈ π N s , whichis also proportional to N s , similar to that for ∆ = ¯Φ l . Inaddition, with such a design, the array gain at other spatialfrequencies can be obtained numerically. Fig. 7 shows oneexample of the resulting broadened and flattened beam fora ULA with N = 512 elements partitioned into L = 4 g (cid:18) L, N s , (cid:8) α l , ¯Φ l (cid:9) , ∆ p = ¯Φ + 2 p − N s ¯ d x (cid:19) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L X l =1 e jh ( α l , ∆ p , ¯Φ l ) sin (cid:16) πN s ¯ d x (cid:16) p − l )+12 N s ¯ d x (cid:17)(cid:17) sin (cid:16) π ¯ d x (cid:16) p − l )+12 N s ¯ d x (cid:17)(cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ) ≈ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e jh ( α p , ∆ p , ¯Φ p ) sin (cid:16) πN s ¯ d x N s ¯ d x (cid:17) sin (cid:16) π ¯ d x N s ¯ d x (cid:17) + e jh ( α p +1 , ∆ p , ¯Φ p +1 ) sin (cid:16) − πN s ¯ d x N s ¯ d x (cid:17) sin (cid:16) − π ¯ d x N s ¯ d x (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) e jh ( α p , ∆ p , ¯Φ p ) + e jh ( α p +1 , ∆ p , ¯Φ p +1 ) (cid:12)(cid:12)(cid:12) sin (cid:16) π N s (cid:17) . (28)sub-arrays by setting ¯Φ = − L − N s ¯ d x , where the dotted lineindicates the array gain of (cid:0) N s (cid:1) dB. It is observed thatthe array gain within the spatial frequency interval (cid:2) ¯Φ , ¯Φ (cid:3) slightly fluctuates around (cid:0) N s (cid:1) , and the maximumfluctuation is only about 1.5 dB. Thus, with the proposedbeam flattening design, the array gain within the main lobeis approximated by g (cid:0) L, N s , (cid:8) ¯Φ l , α l (cid:9) , ∆ (cid:1) ≈ N s = N L , ¯Φ ≤ ∆ ≤ ¯Φ L . (31)Besides, Fig. 7 also shows that the coverage beamwidth of theflattened beam is approximately h ¯Φ − N s ¯ d x , ¯Φ L + N s ¯ d x i ,within which the worst-case array gain occurs at the boundaryspatial frequencies ¯Φ − N s ¯ d x and ¯Φ L + N s ¯ d x . A closerlook at Fig. 7 shows that only one sub-array has the mostcontribution to the array gain of the two spatial frequencies.Thus, for sufficiently large N s , by setting ∆ = N s ¯ d x in (25),the worst-case array gain is g worst2 (cid:0) L, N s , (cid:8) ¯Φ l , α l (cid:9) , ∆ (cid:1) ≈ π N s = 4 π N L , ¯Φ − N s ¯ d x ≤ ∆ ≤ ¯Φ L + 12 N s ¯ d x , (32)which is inversely proportional to L with given total numberof elements N . Furthermore, for the proposed beam flatteningtechnique with sub-array manifold using L sub-arrays eachwith N s elements, the coverage beamwidth in terms of L isgiven by ¯∆ broad ( L ) = L ¯∆ CVBW ( N s ) ≈ L N s ¯ d x = L N ¯ d x , (33)where N ¯ d x corresponds to the coverage beamwidth of the N -element full array without sub-array partition or beambroadening/flatenning. Therefore, with the proposed beambroadening and flattening technique, it is observed from (32)and (33) that the coverage beamwidth is broadened by a factorof L , but at the cost of the same proportional reduction of theworst-case array gain. Therefore, there exists a design trade-off for choosing the optimal number of sub-arrays L , so asto maximize the worst-case array gain while ensuring that thebeamwidth is sufficiently large to cover the entire target area.Based on the above discussions, an efficient solution isproposed to solve problem (24). Specifically, for the givenAIRS placement q , based on (19) and (20), we define thespan of the spatial frequency deviation ∆ span ( q ) associatedwith A as ∆ span ( q ) ∆ = ∆ max ( q ) − ∆ min ( q ) . (34) Intuitively, to maximize the worst-case array gain within theinterval [∆ min ( q ) , ∆ max ( q )] , L should be large enough sothat all locations in A lie within the coverage beamwidth of theAIRS. Based on (33), we should have ¯∆ broad ( L ) ≥ ∆ span ( q ) .On the other hand, in order to maximize the approximateworst-case array gain π N L , L should be as small as possible.Thus, we set L ∗ ( q ) = (cid:24)q ∆ span ( q ) N ¯ d x (cid:25) . (35)The number of elements in each sub-array is thus given by N ∗ s ( q ) = N/L ∗ ( q ) . It is observed that when ∆ span ( q ) ≤ N ¯ d x , we have L = 1 and N s = N . In other words, when thetarget coverage area A and/or the AIRS size N is relativelysmall, no beam flatenning/broadening is needed, and the fullarray architecture is sufficient to cover the entire area. On theother hand, when the coverage area A and/or the AIRS size N is large, we have L > in general. Furthermore, it isnot difficult to see that the starting spatial frequency shouldbe set as ¯Φ ∗ ( q ) = ∆ min ( q ) + N ∗ s ( q ) ¯ d x . Substituting ¯Φ ∗ ( q ) into (27) and (30), the sub-array beam directions and commonphase terms are given by ¯Φ ∗ l ( q ) = ∆ min ( q ) + 2 l − N ∗ s ( q ) ¯ d x , l = 1 , · · · , L ∗ ( q ) , (36) α ∗ l ( q ) = − (cid:18) πN ∗ s ( q ) ¯ d x ∆ min ( q ) + π + πN ∗ s ( q ) (cid:19) l,l = 1 , · · · , L ∗ ( q ) . (37)Based on the sub-array manifold, the phase shifts θ ∗ ( q ) for(P5.1) can be obtained as θ ∗ ( l − N ∗ s ( q )+ i ( q ) = − (cid:18) πN ∗ s ( q ) ¯ d x ∆ min ( q ) + π + πN ∗ s ( q ) (cid:19) l − π ( i −
1) ¯ d x (cid:18) ∆ min ( q ) + 2 l − N ∗ s ( q ) ¯ d x (cid:19) ,l = 1 , · · · , L ∗ ( q ) , i = 1 , · · · , N ∗ s ( q ) . (38) The proposed beam broadening and flattening technique forpassive beamforming design for ULA-based AIRS is summa-rized in the following Proposition.
Proposition 3:
To design a passive beamforming of an N -element ULA-based AIRS located at q with the beamwidthmatching with the size of a target area A , the following beambroadening and flattening technique is proposed:1. Calculate the span of the spatial frequency deviation ∆ span ( q ) associated with A based on (19), (20), and (34). Number of AIRS reflecting elements, N T he w o r s t - c a s e a rr a y ga i n ( d B ) Fig. 8. The worst-case array gain versus the number of AIRS reflectingelements for area coverage.
2. Determine the number of sub-arrays L ∗ ( q ) based on(35).3. Set the sub-array beam directions (cid:8) ¯Φ ∗ l ( q ) (cid:9) and commonphase terms { α ∗ l ( q ) } based on (36) and (37).4. Obtain the phase-shift solution θ ∗ ( q ) based on (38).It is worth pointing out that the worst-case array gain withinthe beam coverage h ¯Φ ∗ ( q ) − N ∗ s ( q ) ¯ d x , ¯Φ ∗ L ( q ) + N ∗ s ( q ) ¯ d x i is given by π ( N ∗ s ( q )) . Thus, with the proposed design, theobjective value of (P5.1) is approximated as f ( q , θ ∗ ( q ) , w ) ≈ π N ( L ∗ ( q )) = 4 π N lp ∆ span ( q ) N ¯ d x m , ∀ w ∈ A . (39) It is observed that when the number of reflecting elements N and/or the spatial frequency span ∆ span ( q ) is small suchthat ∆ span ( q ) N ¯ d x < , the beamwidth of the full array candirectly match the size of the target coverage area, and (39)reduces to f ( q , θ ∗ ( q ) , w ) ≈ π N , ∀ w ∈ A . (40)This implies that the worst-case array gain of all the locationsin the target area quadratically increases with N . This resultis in accordance with the single-location SNR maximizationproblem with the full array (see (14)).On the other hand, when N and/or ∆ span ( q ) is sufficientlylarge such that ∆ span ( q ) N ¯ d x ≫ , the ceiling operator in(39) is negligible, and we have f ( q , θ ∗ ( q ) , w ) ≈ π N ∆ span ( q ) N ¯ d x = 4 π N ∆ span ( q ) ¯ d x , ∀ w ∈ A . (41) The result in (41) shows that for the area coverage setup withrelatively wide target area size and/or large AIRS elementsnumber N , the worst-case array gain over all the locationsin the target area increases linearly with N , which is incontrast to (14) or (40). This is expected since for areacoverage with relatively wide area and/or large N , the N AIRS elements need to be partitioned into sub-arrays to formbroadened and flattened beams, at the cost of reduced arraygain along the main beam. For a fixed spatial frequency span ∆ span ( q ) = 0 . , Fig. 8 plots the worst-case array gain (39)versus N , together with its approximations (40) and (41). Itis observed that (40) and (41) well approximate (39) at thesmall and large N regimes, respectively, with the worst-caseSNR of the target area first increasing quadratically with N ,and then only linearly when N becomes large.
2) AIRS Placement Optimization:
With the proposed pas-sive beamforming design in Proposition 3 and the corre-sponding worst-case array gain in (39), the AIRS placementoptimization problem (P5.2) is greatly simplified. Specifically,by substituting (39) into (P5.2), we have max q min w ∈A π N lp ∆ span ( q ) N ¯ d x m (cid:0) H + k q − w k (cid:1) (cid:0) H + k q k (cid:1) . (42) Since only the term k q − w k in (42) depends on w , by letting d max ( q ) ∆ = max w ∈ A k q − w k , problem (42) can be equivalentlytransformed to min q (cid:24)q ∆ span ( q ) N ¯ d x (cid:25) (cid:0) H + d ( q ) (cid:1) (cid:16) H + k q k (cid:17) . (43)Problem (43) shows that the optimal AIRS placement shouldachieve a balance between minimizing the spatial frequency(or angular) span associated with the target coverage area A , as well as minimizing the cascaded path loss from thesource node to the worst-case destination node. Note that ifthe target area A is sufficiently small so that the first termin (43) is always 1 for all q , then problem (43) degrades to(P4). Therefore, the AIRS placement optimization in (43) canbe regarded as a generalization of that for the single-locationSNR maximization. Meanwhile, it is observed from (43) thatfor a given AIRS placement q , the worst-case SNR in A occurs at the location that is farthest from the AIRS placement,i.e., arg max w ∈A k q − w k , which should be one of the boundarypoints in A .Furthermore, since the target area A is symmetric about the x -axis, it can be shown that the optimal q to (43) should liein the x -axis, i.e., we should have q = [ q x , T . Thus, (43)reduces to the univariate optimization problem, i.e., min q x (cid:24)q ∆ span ( q x ) N ¯ d x (cid:25) (cid:0) H + d ( q x ) (cid:1) (cid:16) H + | q x | (cid:17) . (44)It can be observed that starting from q x = 0 , as q x increasesso that the AIRS moves closer to the target area, the firstterm and the third term in (44) increase, while the secondterm decreases. As a result, there in general exists a non-trivial optimal AIRS placement q ∗ x to balance the above threeterms, which can be efficiently found via a one-dimensionalsearch. As such, the AIRS placement problem (P5.2) andhence (P5) are solved. The main procedures for solving (P5.2)are summarized in Algorithm 1. γ ( q , Θ , w ) = ¯ P β M (cid:12)(cid:12)(cid:12)(cid:12) N x P n x =1 e j ( θ nx +2 π ( n x −
1) ¯ d x [ ¯Φ T ( q , w ) − ¯Φ R ( q ) ]) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N y P n y =1 e j ( θ ny +2 π ( n y −
1) ¯ d y [ ¯Ω T ( q , w ) − ¯Ω R ( q ) ]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) H + k q − w k (cid:17) (cid:16) H + k q k (cid:17) . (45) Algorithm 1
AIRS Placement Optimization for (P5 . Initialization:
Set the search regime of AIRS placement q x ∈ [ q min , q max ] . For all candidate placement q x , obtain the maximumdistance d max ( q x ) , and the spatial frequency span basedon (34). Calculate the cost value in (44). Choose q x that gives the minimum cost value in (44) as theoptimal placement, and return the corresponding passivebeamforming based on Proposition 3. B. The General Case of UPA-Based AIRS
Based on the results obtained for ULA-based AIRS, in thissubsection, we extend our proposed solution to the generalcase of UPA-based AIRS by solving problem (P2) with thegeneral SNR expression (12). Different from the ULA-basedAIRS, the UPA-based AIRS is able to achieve passive beamsteering over both spatial frequency dimensions ¯Φ and ¯Ω ,as evident from (5), (7) and (12). Therefore, the difficultyin solving problem (P2) for UPA-based AIRS lies in that aphase shift θ n x ,n y by the ( n x , n y ) th passive element would ingeneral impact the beam steering for both ¯Φ and ¯Ω dimensions(see (12)). Fortunately, motivated by the result obtained in theprevious subsection, we are able to treat the UPA-based AIRSas two decoupled ULA-based AIRS along the x - and y -axis,respectively.The key to achieve such a decoupling is to restrict thephase shift design θ n x ,n y as a sum of two phase shifts, i.e., θ n x ,n y = θ n x + θ n y , ≤ n x ≤ N x , ≤ n y ≤ N y .As such, the resulting SNR in (12) can be re-expressedas (45), shown at the top of this page. Note that such asimplification is sub-optimal in general, since it narrows downthe optimization variable space of problem (P2), where thenumber of independent phase-shift variables to be optimizedis reduced from N x N y to N x + N y . However, it serves as auseful technique for efficient 3D passive beamforming designin practice. A closer look at the numerator of (45) reveals thatthe resulting SNR can be treated as that determined by thearray gains achieved by two independent ULAs, along x - and y -axis, respectively. Therefore, by applying the similar phase-shift design in Proposition 3 along the x - and y -axis for beamsteering over spatial frequencies ¯Φ and ¯Ω , respectively, a 3Dbroadened and flattened beam pattern can be achieved with itsbeamwidth matching the size of the target coverage area.Specifically, for any given AIRS placement q , denote by ∆ span ,x ( q ) and ∆ span ,y ( q ) the span of the spatial frequencydeviation associated with the target area A along the x - and y -axis, respectively, where ∆ span ,x ( q ) can be obtained based on(19), (20) and (34), and ∆ span ,y ( q ) can be similarly obtainedas ∆ span ,y ( q ) = ∆ max ,y ( q ) − ∆ min ,y ( q )= max w ∈A ¯Ω T ( q , w ) − min w ∈A ¯Ω T ( q , w ) , (46) where ∆ max ,y ( q ) and ∆ min ,y ( q ) are the maximum andminimum deviation of ¯Ω T ( q , w ) from ¯Ω R ( q ) , respectively.Similar to (35), to ensure that the coverage beamwidth of thetwo decoupled beams match the size of the target coveragearea, the required number of sub-arrays along the x - and y -axis are given by L ∗ a ( q ) = (cid:24)q ∆ span ,a ( q ) N a ¯ d a (cid:25) , a ∈ { x, y } . (47)With the obtained number of sub-arrays, the sub-array beamdirection, denoted as ¯ S ∗ l,a ( q ) , can be determined based on(36), i.e., ¯ S ∗ l,a ( q ) = ∆ min ,a ( q ) + 2 l − N ∗ s,a ( q ) ¯ d a , l = 1 , · · · , L ∗ a ( q ) , (48)where N ∗ s,a ( q ) = N a /L ∗ a ( q ) denotes the number of elementsin each sub-array along the a -axis, a ∈ { x, y } . Similarly,let α ∗ l.a ( q ) , a ∈ { x, y } denote the sub-array common phaseterms, which can be obtained as α ∗ l.a ( q ) = − (cid:18) πN ∗ s,a ( q ) ¯ d a ∆ min ,a ( q ) + π + πN ∗ s,a ( q ) (cid:19) l,l = 1 , · · · , L ∗ a ( q ) . (49)Furthermore, based on (38), the phase shifts θ ∗ n a ( q ) along the a -axis can be obtained as (50), shown at the top of the nextpage.Therefore, the phase shifts for UPA-based AIRS can beobtained accordingly as θ ∗ n x ,n y ( q ) = θ ∗ n x ( q ) + θ ∗ n y ( q ) , ≤ n x ≤ N x , ≤ n y ≤ N y . Based on the above discussions, forany given AIRS placement q , the 3D passive beamformingbased on the proposed beam broadening and flattening tech-nique is summarized as Algorithm 2. Algorithm 2
3D Passive Beam Design for UPA-based AIRS Input:
The AIRS placement q , and the target coveragearea A . Calculate the span of the spatial frequency deviations ∆ span ,x ( q ) and ∆ span ,y ( q ) based on (34) and (46),respectively. Determine the required number of sub-arrays L x and L y based on (47). Set the sub-array beam directions n ¯ S ∗ l,x ( q ) o and n ¯ S ∗ l,y ( q ) o based on (48), and common phase terms n α ∗ l,x ( q ) o , n α ∗ l,y ( q ) o based on (49). Obtain the phase shifts θ ∗ n x ( q ) and θ ∗ n y ( q ) based on (50). Output:
The phase shifts θ ∗ n x ,n y ( q ) = θ ∗ n x ( q ) + θ ∗ n y ( q ) .Similar to (39), the array gain for the entire area A canbe further characterized by (51), shown at the top of the nextpage. θ ∗ ( l − N ∗ s,a ( q )+ i ( q ) = − (cid:18) πN ∗ s,a ( q ) ¯ d a ∆ min ,a ( q ) + π + πN ∗ s,a ( q ) (cid:19) l − π ( i −
1) ¯ d a ∆ min ,a ( q ) + 2 l − N ∗ s,a ( q ) ¯ d a ! ,l = 1 , · · · , L ∗ a ( q ) , i = 1 , · · · , N ∗ s,a ( q ) . (50) f ( q , θ ∗ ( q ) , w ) ≈ π N x (cid:24)q ∆ span ,x ( q ) N x ¯ d x (cid:25) N y (cid:24)q ∆ span ,y ( q ) N y ¯ d y (cid:25) , ∀ w ∈ A . (51) min q (cid:24)q ∆ span ,x ( q ) N x ¯ d x (cid:25) (cid:24)q ∆ span ,y ( q ) N y ¯ d y (cid:25) (cid:0) H + d ( q ) (cid:1) (cid:16) H + k q k (cid:17) . (52)Based on (51), the AIRS placement optimization problemcan be similarly solved as that in Section IV-A2. Specifi-cally, similar to (43), the problem can be first equivalentlytransformed to (52), shown at the top of this page. It isobserved from (52) that the placement optimization for UPA-based AIRS with 3D passive beamforming needs to strike abalance between minimizing the angular spans along both x -and y -axis, as well as minimizing the cascaded path loss.Furthermore, as the target area A is symmetric over the x -axis, so is the optimized AIRS placement, i.e., q ∗ y = − N y d y .In addition, similar to Algorithm 1, the optimized placement q ∗ x can also be found via a one-dimensional search. Thus, anefficient solution is obtained for (P2) via the joint placementand 3D beamforming design for the general UPA-based AIRS.V. N UMERICAL R ESULTS
In this section, numerical results are provided to evaluate theperformance of our proposed design. The altitude of the AIRSis set as H = 100 m. Unless otherwise stated, the noise andtransmit power are set as σ = − dBm and P = 20 dBm,respectively, and the reference channel power gain is β = − dB, corresponding to a carrier frequency of 2.4 GHz. Thenumber of transmit antennas at the source node is M = 64 .The separation of adjacent elements along the x - and y -axisare d x = d y = λ/ .We first consider the single-location SNR maximizationproblem studied in Section III. Fig. 9 shows the achievableSNR for the target destination location at [1000 , T m versusthe number of AIRS reflecting elements, N . For the consideredsetup, it follows from Proposition 2 that ρ = 10 , and theoptimal AIRS deployment coefficients are ξ ∗ ( ρ ) = 0 . and . , i.e., the AIRS should be placed either close to thesource node or to the target location. As a comparison, we alsoconsider a benchmark AIRS placement scheme with the AIRSsimply placed above the midpoint between the source and thetarget location, i.e., q = [500 , T m. For both the proposedoptimal placement and benchmark placement, the optimalpassive beamforming with phase shifts given in (13) is appliedat the AIRS to achieve coherent reflected signal combinationat the target location. It is observed that for both placementschemes, the optimal SNR increases quadratically with thenumber of AIRS elements, as expected. In addition, the
200 300 400 500 600 700 800
Number of AIRS reflecting elements, N S NR a t t he s i ng l e t a r ge t l o c a t i on ( d B ) Optimal placementBenchmark placement with q =[500,0] T m Fig. 9. SNR versus the number of AIRS reflecting elements for single-locationSNR maximization. optimal placement significantly outperforms the benchmarkplacement. For example, to achieve a target SNR of 15 dB, thenumber of AIRS elements required for the benchmark place-ment is about 580, while this number is significantly reducedto about 225 for the proposed optimal placement. This exampledemonstrates the importance of exploiting the flexible AIRSplacement for communication performance optimization.Next, we consider the AIRS-enabled area coverage with thesimplified ULA-based AIRS by assuming that A is also a one-dimensional (1D) line segment along the x -axis with interval [ x l , x u ] . Three different setups are considered, with [ x l , x u ] =[250 , m, [500 , m, and [155 , m, respectively.The number of reflecting elements is set as N = N x = 256 .To illustrate the impact of AIRS placement q on each of thethree factors in (43), Fig. 10 plots the required number of sub-arrays L ∗ ( q x ) in (35) and the worst-case concatenated pathloss versus the AIRS placement q x along the x -axis, togetherwith the corresponding worst-case SNR plotted in Fig. 11. Let x c = x l + x u be the center of the target line segment for AIRScoverage. It can be shown that the optimal AIRS placementshould satisfy q ∗ x ≤ x c . Therefore, the maximum values overthe x -axis in the plots of Fig. 10 and Fig. 11 are set to x c . Itis firstly observed from Fig. 10 that as the AIRS moves from -150-100 0 100 200 300 400 500 AIRS placement along the x-axis, q x (m) T he r equ i r ed nu m be r o f s ub - a rr a ys T he w o r s t - c a s e c on c a t ena t ed pa t h l o ss ( d B ) (a) x l = 250 m, x u = 750 m -150 0 200 400 600 800 1000 AIRS placement along the x-axis, q x (m) T he r equ i r ed nu m be r o f s ub - a rr a ys T he w o r s t - c a s e c on c a t ena t ed pa t h l o ss ( d B ) (b) x l = 500 m, x u = 1500 m -150 -100 -50 0 50 100 150 200 240 AIRS placement along the x-axis, q x (m) T he r equ i r ed nu m be r o f s ub - a rr a ys T he w o r s t - c a s e c on c a t ena t ed pa t h l o ss ( d B ) (c) x l = 155 m, x u = 325 mFig. 10. The required number of sub-arrays and the worst-case concatenated path loss versus the AIRS placement along the x -axis for 1D coverage with theULA-based AIRS. -150-100 0 100 200 300 400 500 AIRS placement along the x-axis, q x (m) -10-5051015 T he w o r s t - c a s e S NR ( d B ) (a) x l = 250 m, x u = 750 m -150 0 200 400 600 800 1000 AIRS placement along the x-axis, q x (m) -25-20-15-10-50510 T he w o r s t - c a s e S NR ( d B ) (b) x l = 500 m, x u = 1500 m -150 -100 -50 0 50 100 150 200 240 AIRS placement along the x-axis, q x (m) T he w o r s t - c a s e S NR ( d B ) (c) x l = 155 m, x u = 325 mFig. 11. The worst-case SNR versus the AIRS placement along the x -axis for 1D coverage with the ULA-based AIRS. the left of the source node towards x c , the required numberof sub-arrays L ∗ ( q x ) increases in a staircase manner. This isexpected since when q x increases or the AIRS moves closer tothe target coverage area, the angular span ∆ span ( q x ) increases,which thus requires broader beam (or equivalently more sub-array partitions) to cover the target area, though this comes atthe cost of reduced worst-case array gain, as can be inferredfrom (39). It is also observed from Fig. 10 that the worst-caseconcatenated path loss generally first decreases, then increasesand finally decreases with the increase of q x .With the effects of both the angular span and concatenatedpath loss taken into account, the worst-case SNR versus q x based on (39) is shown in Fig. 11 for the three cases consideredin Fig. 10. The optimal AIRS placement that leads to the bestworst-case SNR is also labelled in the figure. It is observedthat different from the single-location SNR maximization case,where the optimal AIRS placement is always located betweenthe source node and the target location, the optimal AIRSplacement q ∗ x for area coverage critically depends on the sizeof the target area, for which we might have q ∗ x < , q ∗ x ≈ ,or < q ∗ x ≤ x c , corresponding to the three cases in Fig. 11,respectively. In particular, in Fig. 11(a), the optimal AIRS liesto the left of the source node, since it can result in smallerangular span and hence higher worst-case array gain, though atthe cost of higher concatenated path loss. This implies that thearray gain is the dominating factor for the considered setup.Similar observations can be made for the two other cases in
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Transmit power at the source node, P (dBm) -25-20-15-10-5051015 T he w o r s t - c a s e S NR ( d B ) Optimized placementBenchmark placement with q =[1000,0] T m Fig. 12. The worst-case SNR versus transmit power for the ULA-based AIRScoverage.
Fig. 11. These three different setups show that it is a non-trivialtask to find the optimal AIRS placement for area coverage ingeneral, as it needs to achieve an optimal balance betweenminimizing the angular span and the concatenated path loss.Next, for the ULA-based AIRS, we consider the AIRS-enabled coverage for a rectangular area A , with the length andwidth given by D x = 1000 m and D y = 600 m, respectively,
400 500 600 700 800 900 1000
Number of AIRS reflecting elements, N -30-20-1001020 T he w o r s t - c a s e S NR ( d B ) Proposed 3D passive beamformingBenchmark: 1D passive beamforming
Fig. 13. The worst-case SNR versus the number of AIRS reflecting elementsfor the UPA-based AIRS coverage. and the center at w = [1000 , T m. The number of AIRSelements is set as N = 256 . For comparison, we consider thebenchmark scheme with the AIRS placed above the center ofthe rectangular target area, i.e., q = [1000 , T m. Fig. 12shows the worst-case SNR versus the transmit power P forboth the optimized and benchmark schemes. It is observedthat the optimized AIRS placement significantly outperformsthe benchmark placement, with an about 25 dB SNR gain.This demonstrates the importance of our proposed joint AIRSdeployment and beamforming design.Last, we consider the general case of UPA-based AIRS asstudied in Section IV-B. The number of elements along the y -axis is fixed as N y = 20 , and N is varied by varying thenumber of elements along the x -axis, N x . The size of therectangular target area is the same as the ULA-based AIRScase considered previously. Besides the proposed scheme inSection IV-B, we also consider a benchmark scheme with1D passive beamforming applied to UPA-based AIRS, bygrouping the N y reflecting elements in each column into onesub-surface and applying an identical phase shift [13]. In thiscase, the phase shift design of the AIRS with N = N x N y elements follows the pattern θ n x ,n y = θ n x , ∀ n y . Such adesign further restricts the optimization space in (P2), sincethe number of independent phase shift design variables is N x , instead of N x + N y in our proposed 3D beam design inSection IV-B. By following the similar design of Proposition 3,the received SNR with the above 1D beamforming applied toUPA-based AIRS can be expressed as γ ( q , Θ , w ) ≈ π ¯ P β MN x (cid:12)(cid:12)(cid:12)(cid:12) sin ( πN y ¯ d y [ ¯Ω T ( q , w ) − ¯Ω R ( q ) ]) sin ( π ¯ d y [ ¯Ω T ( q , w ) − ¯Ω R ( q ) ]) (cid:12)(cid:12)(cid:12)(cid:12) lp ∆ span ,x ( q ) N x ¯ d x m (cid:0) H + k q − w k (cid:1) (cid:0) H + k q k (cid:1) . (53) In contrast to the broadened and flattened beam in the proposed3D beamforming design, due to the inability of phase steeringover the dimension of ¯Ω , the SNR in (53) varies significantlywith the target location w and can be even zero when
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Transmit power at the source node, P (dBm) -20-15-10-5051015 T he w o r s t - c a s e S NR ( d B ) Optimized placementBenchmark placement with q =[1000,0] T m Fig. 14. The worst-case SNR versus transmit power for the UPA-based AIRScoverage. πN y ¯ d y (cid:2) ¯Ω T ( q , w ) − ¯Ω R ( q ) (cid:3) = kπ, k = 1 , · · · , N y − .Fig. 13 shows the worst-case SNR versus the numberof AIRS reflecting elements N for both the proposed andbenchmark schemes. It is observed that the worst-case SNRincreases with the number of AIRS reflecting elements for bothschemes, as expected. Furthermore, the proposed 3D passivebeamforming with beam broadening and flattening drasticallyoutperforms the benchmark 1D passive beamforming appliedto UPA-based AIRS, with an about 30 dB SNR gain. Thisis expected since the benchmark scheme is unable to achievebeam steering over ¯Ω dimension (see (53)), and its resultingSNR is thus limited by the insufficient array gain along the y -axis, especially for a large coverage area. In contrast, thanksto the 3D beam broadening and flattening over both ¯Φ and ¯Ω dimensions, approximately equal array gain can be achievedfor all locations in the target coverage area, thus leading tosignificant performance gain over the benchmark scheme.Fig. 14 shows the worst-case SNR versus transmit powerat the source node for the UPA-based AIRS area coverage.The number of AIRS elements along x - and y -axis areset as N x = N y = 20 . We also consider the benchmarkplacement scheme with the AIRS placed above the centerof the rectangular target area, i.e., q = [1000 , T m, andthe proposed 3D beamforming in Section IV-B is applied forboth the optimized and benchmark placement schemes. It isobserved that the optimized placement achieves significantperformance gains over the benchmark placement. Again, thisshows the importance of our proposed joint AIRS placementand beamforming design for AIRS-enabled wireless commu-nications. VI. C ONCLUSION
This paper proposed a new wireless relaying system enabledby passive AIRS for coverage extension over the sky. Theworst-case SNR in a target coverage area was maximized byjointly optimizing the transmit beamforming for the sourcenode as well as the placement and 3D passive beamforming for the AIRS. We first studied the special case of single-locationSNR maximization and derived the optimal solution in closed-form. It was shown that the optimal horizontal AIRS place-ment only depends on the ratio between the source-destinationdistance and the AIRS altitude. Then for the general case ofAIRS-enabled area coverage, we proposed an efficient solutionby decoupling the passive beamforming design for the AIRS tomaximize the worst-case array gain from the AIRS placementoptimization via balancing the resulting angular span andcascaded path loss. In particular, a novel beam broadeningand flattening technique was applied to form a flattened beampattern with broadened/adjustable beamwidth catering to thesize of the target coverage area. Numerical results demon-strated that the proposed design can significantly improvethe performance over the benchmark beamforming/placementschemes, and the importance of exploiting both the AIRSpassive beamforming and flexible deployment in designingAIRS-enabled wireless communications.R EFERENCES[1] H. Lu, Y. Zeng, S. Jin, and R. Zhang, “Enabling panoramic full-anglereflection via aerial intelligent reflecting surface,” in
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