Enabling Panoramic Full-Angle Reflection via Aerial Intelligent Reflecting Surface
aa r X i v : . [ c s . I T ] J a n Enabling Panoramic Full-Angle Reflection viaAerial Intelligent Reflecting Surface
Haiquan Lu ∗ , Yong Zeng ∗ , Shi Jin ∗ , and Rui Zhang † *National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China † Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583Email: haiq [email protected], yong [email protected], [email protected], [email protected]
Abstract —This paper proposes a new three dimensional (3D)networking architecture enabled by aerial intelligent reflectingsurface (AIRS) to achieve panoramic signal reflection fromthe sky. Compared to the conventional terrestrial IRS, AIRSnot only enjoys higher deployment flexibility, but also is ableto achieve 360 ◦ panoramic full-angle reflection and requiresfewer reflections in general due to its higher likelihood ofhaving line of sight (LoS) links with the ground nodes. Wefocus on the problem to maximize the worst-case signal-to-noiseratio (SNR) in a given coverage area by jointly optimizing thetransmit beamforming, AIRS placement and phase shifts. Theformulated problem is non-convex and the optimization variablesare coupled with each other in an intricate manner. To tacklethis problem, we first consider the special case of single-locationSNR maximization to gain useful insights, for which the optimalsolution is obtained in closed-form. Then for the general case ofarea coverage, an efficient suboptimal solution is proposed byexploiting the similarity between phase shifts optimization forIRS and analog beamforming for the conventional phase array.Numerical results show that the proposed design can achievesignificant performance gain than heuristic AIRS deploymentschemes. I. I
NTRODUCTION
Wireless communication aided by intelligent reflecting sur-face (IRS) has been proposed as a promising technology torealize energy-efficient and cost-effective transmissions in thefuture [1]–[12]. IRS is a man-made reconfigurable metasurfacecomposed of a large number of regularly arranged passive re-flecting elements and a smart controller [1]. Through modify-ing the amplitude and/or phase shift of the radio signal imping-ing upon its reflecting elements, IRS is able to achieve highlyaccurate radio wave manipulation in desired manners, whichthus offers a new paradigm of wireless communication systemdesign via controlling the radio propagation environment forvarious purposes, such as signal enhancement, interferencesuppression and transmission security [1]. Thanks to thepassive array architecture, IRS-aided wireless communicationis able to reap the benefits of large antenna arrays with lowpower consumption and hardware cost. Furthermore, differentfrom the conventional relays, the radio signal reflected by IRSis free from self-interference and noise in an inherently full-duplex transmission manner [1].Most existing research mainly focuses on terrestrial IRSdeployed on facades of buildings or indoor walls, which, how-ever, poses fundamental performance limitations for severalreasons. First, from the deployment perspective, finding the ! "! " (cid:3) !" (cid:3) (a) Terrestrial IRS ! "! " $%& (cid:3) (cid:3) (b) AIRSFig. 1. 180 ◦ half-space reflection by terrestrial IRS versus 360 ◦ panoramicfull-angle reflection by AIRS. !" (a) Terrestrial IRS !" (b) AIRSFig. 2. AIRS reduces the number of reflections than terrestrial IRS. appropriate place for IRS installation is usually a difficulttask in practice. The installation process may also involveother issues, e.g., site rent, impact of urban landscape and thewillingness of owners to install large IRS on their properties.Second, from the performance perspective, IRS deployed onthe walls or facades of buildings can at most serve terminalslocated in half of the space, i.e., both the source and destinationnodes must lie on the same side of the IRS, as illustrated inFig. 1(a). Third, as shown in Fig. 2(a), in complex environmentlike urban areas, the radio signal originated from a source nodehas to be reflected many times before reaching the desireddestination node, even with the presence of sufficient numberof IRSs. This thus leads to significant signal attenuation sinceeach reflection, even by IRS, would cause signal scattering toundesired directions.To address the above issues, we propose in this paper anovel three dimensional (3D) networking architecture enabledby aerial IRS (AIRS), for which IRS is mounted on aerialplatforms like balloons, unmanned aerial vehicles (UAVs), soas to enable intelligent reflection from the sky. Compared tothe conventional terrestrial IRS, AIRS has several appealingadvantages. First, with elevated position, AIRS is able to estab-lish line-of-sight (LoS) links with the ground nodes with high (cid:17)(cid:17) !" , - R ! " T ! !" " *+,-( . ! ! H ! " T s
Fig. 3. AIRS-assisted wireless communication system. probability [13], which leads to stronger channel as comparedto the terrestrial IRS. At the same time, the placement ortrajectory of aerial platforms can be more flexibly optimizedto further improve the communication performance, therebyoffering a new degree of freedom (DoF) for performanceenhancement via 3D network design. Second, AIRS is able toachieve 360 ◦ panoramic full-angle reflection, i.e., one AIRScan in principle manipulate signals between any pair of nodeslocated on the ground, as illustrated in Fig. 1(b). This is in asharp contrast to the conventional terrestrial IRS that usuallycan only serve nodes in half of the space. Last but not least, incontrast to the terrestrial IRS, AIRS is usually able to achievedesired signal manipulation by one reflection only, even incomplex urban environment (see Fig. 2(b)), thanks to its highlikelihood of having LoS links with the ground nodes. Thisthus greatly reduces the signal power loss due to multiplereflections in the case of terrestrial IRS.In this paper, we consider a basic setup of an AIRS-assistedcommunication system, where an AIRS is deployed to enhancethe signal coverage of a given target area, say, a hot spot in thecellular network or a remote area without cellular coverage.Our objective is to maximize the minimum achievable signal-to-noise ratio (SNR) for the target area by jointly optimizingthe transmit beamforming of the source node, the placementand phase shifts of the AIRS. The formulated problem is non-convex and difficult to be optimally solved in general. To gainuseful insights at first, we consider the special case of single-location SNR maximization problem, for which the optimalAIRS placement and phase shifts are derived in closed-form.In particular, the optimal location of the AIRS is shown to onlydepend on the ratio between the AIRS height and the source-destination distance. For the general case of area coverage,we propose an efficient design by exploiting the similaritybetween phase shifts optimization for IRS and analog beam-forming for the conventional phase array. Numerical results arepresented which show the significant performance gain of theproposed design as compared to heuristic AIRS deploymentschemes.II. S YSTEM M ODEL A ND P ROBLEM F ORMULATION
As illustrated in Fig. 3, we consider an AIRS-assistedwireless communication system, where an AIRS is deployed toassist the source node (say a ground base station, access pointor a user terminal) to enhance its communication performance within a given area A (assumed to be rectangular for the pur-pose of exposition). We assume that the direct communicationlink from the source node to the target area is negligible due tosevere blockage. The source node is equipped with M transmitantennas, where the adjacent antenna elements are separatedby d . The AIRS comprises of a uniform linear array (ULA)with N passive reflecting elements, separated by the distance d < λ , where λ is the carrier wavelength. Without loss of gen-erality, we assume that the source node is located at the originin a Cartesian coordinate system and the center of the coveragearea is on the x-axis, which is denoted by w = [ x , T .Therefore, any location in the rectangular area A can bespecified as w = [ x a , y a ] T , x a ∈ (cid:2) x − D x , x + D x (cid:3) , y a ∈ h − D y , D y i , with D x and D y denoting the length and width ofthe rectangular area, respectively. For convenience, we assume D x ≥ D y .The AIRS is assumed to be placed at an altitude H . Inaddition, consider the first reflection element of the AIRS asthe reference point, whose horizontal coordinate is denoted by q = [ x, y ] T . Therefore, the distance from the source node tothe AIRS, and that from the AIRS to any location in 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 and ground nodes are LoS with high probability.Thus, for simplicity, we assume that the channel power gainsfollow the free-space path loss model, and the channel powergain from the source node to the AIRS can be expressed as β G ( q ) = 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 a location w ∈ A can be expressed as β h ( q , w ) = β H + k q − w k . (2)Let φ T,s ( q ) and φ R ( q ) be the angle of departure (AoD)and angle of arrival (AoA) of the signal from the source nodeto the AIRS, respectively. Then the channel matrix from thesource node to the AIRS, denoted as G ( q ) ∈ C N × M , can beexpressed as G ( q ) = p β G ( q ) e − j πd G λ a R ( φ R ( q )) a HT,s ( φ T,s ( q )) , (3)where a R ( φ ) and a T,s ( φ ) represent the receive array responseof the AIRS and the transmit array response of the sourcenode, respectively, which can be expressed as a R ( φ R ( q )) = h , e − j π ¯ d ¯ φ R ( q ) , · · · , e − j π ( N −
1) ¯ d ¯ φ R ( q ) i T , (4) a T,s ( φ T,s ( q )) = h , e − j π ¯ d ¯ φ T,s ( q ) , · · · , e − j π ( M −
1) ¯ d ¯ φ T,s ( q ) i T , (5) with ¯ φ R ( q ) ∆ = sin ( φ R ( q )) , ¯ φ T,s ( q ) ∆ = sin ( φ T,s ( q )) , ¯ d = dλ and ¯ d = d λ . Note that the AIRS placement q not only affectsthe path loss β G ( q ) , but also the AoD/AoA of the source-AIRS link. Similarly, denote φ T ( q , w ) as the AoD for thecommunication link from the AIRS to a location w ∈ A . Thenthe corresponding channel, denoted as h H ( q , w ) ∈ C × N ,an be expressed as h H ( q , w ) = p β h ( q , w ) e − j πd h λ a HT ( φ T ( q , w )) , (6)where a T ( φ ) is the transmit (reflect) array response at theAIRS, which is given by a T ( φ T ( q , w )) = h , e − j π ¯ d ¯ φ T ( q , w ) , · · · , e − j π ( N −
1) ¯ d ¯ φ T ( q , w ) i T , (7) with ¯ φ T ( q , w ) ∆ = sin ( φ T ( q , w )) .Then the received signal at each location w ∈ A is y ( q , Θ , w , v ) = h H ( q , w ) ΘG ( q ) v √ P s + n, (8)where Θ = diag (cid:0) e jθ , · · · , e jθ N (cid:1) is a diagonal phase-shiftmatrix with θ n ∈ [0 , π ) denoting the phase shift of the n threflection element; P and s are the transmit power and signalat the source node, respectively; v is the transmit beamformingvector at the source node with k v k = 1 ; n ∈ CN (cid:0) , σ (cid:1) is theadditive white Gaussian noise (AWGN). The received SNR atthe location w ∈ A can be written as γ ( q , Θ , w , v ) = P (cid:12)(cid:12) h H ( q , w ) ΘG ( q ) v (cid:12)(cid:12) σ . (9)By denoting θ = [ θ , · · · , θ N ] , our objective is to maximizethe minimum SNR within the rectangular area A (since inpractice the destination nodes can be randomly located in it),by jointly optimizing the AIRS placement q , the phase shifts θ and the transmit beamforming vector v . This optimizationproblem can be 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 . Problem (P1) is difficult to solve optimally in general dueto the following reasons. First, the objective function is theminimum SNR over a 2D area, which is difficult to express interms of the optimization variables. Second, the optimizationproblem is highly non-convex and the optimization variables q , θ and v are intricately coupled with each other in theobjective function. In the following, we first rigorously showthat the optimal transmit beamforming vector v is simplythe maximum ratio transmission (MRT) towards the AIRS,regardless of the reflected link from the AIRS to the ground.Furthermore, for the optimization of the AIRS placement q and phase shifts θ , we first consider the special casewith one single-location SNR maximization to gain usefulinsights, for which the optimal solution can be obtained inclosed-form. Then for the general case for area coverageenhancement, an efficient algorithm is proposed by exploitingthe similarity between phase shifts optimization for IRS andanalog beamforming for the conventional phase array.III. P ROPOSED S OLUTIONS
First, by exploiting the structure of the concatenated channel ˜h H , h H ( q , w ) ΘG ( q ) , the optimal transmit beamformingvector at the source node can be obtained in the followingproposition. Proposition 1:
The optimal transmit beamforming vector v to (P1) is v ∗ = a T,s ( φ T,s ( q )) k a T,s ( φ T,s ( q )) k . Proof:
For any given AIRS placement q , destination nodelocation w and phase shifts θ , it can be shown that the optimalbeamforming vector to maximize γ ( q , Θ , w , v ) in (9), 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 )) Θ H h ( q , w ) (cid:12)(cid:12)(cid:12) a T,s ( φ T,s ( q )) a HT,s ( φ R ( q )) . (10) It then follows that ˜h˜h H is a rank-one matrix, whose eigen-vector is simply a T,s ( φ T,s ( q )) k a T,s ( φ T,s ( q )) k . More importantly, this eigen-vector is independent of the destination node location w . Thus,it is optimal regardless of w to set transmit beamforming as a T,s ( φ T,s ( q )) k a T,s ( φ T,s ( q )) k . The proof of Proposition 1 is thus completed.By substituting v ∗ ( q , Θ , w ) to (9), the resulting SNR atthe destination node location w ∈ A can be expressed as γ ( q , Θ , w ) = ¯ P (cid:12)(cid:12)(cid:12)(cid:12) h H ( q , w ) ΘG ( q ) a T,s ( φ T,s ( q )) k a T,s ( φ T,s ( q )) k (cid:12)(cid:12)(cid:12)(cid:12) = ¯ P β M (cid:12)(cid:12)(cid:12)(cid:12) N P n =1 e j ( θ n +2 π ( n −
1) ¯ d ( ¯ φ T ( q , w ) − ¯ φ R ( q ) )) (cid:12)(cid:12)(cid:12)(cid:12) (cid:0) H + k q − w k (cid:1) (cid:0) H + k q k (cid:1) , (11) where ¯ P = Pσ . 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. A. Single-Location SNR Maximization
In this subsection, we consider the special case of (P2) withone single destination node of known location in A . Denoteby ˆw the destination node location and D = k ˆw k the source-destination distance. In this case, the inner minimization ofthe objective function in (P2) is not needed, and problem (P2)reduces to (P3) max q , θ γ ( q , Θ , ˆw )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 elements should becoherently added at the receiver, that is, θ ∗ n ( q ) = ¯ θ − π ( n −
1) ¯ d (cid:0) ¯ φ T ( q , ˆw ) − ¯ φ R ( q ) (cid:1) , n = 1 , · · · , N, (12)where ¯ θ is an arbitrary phase shift that is common to allreflecting elements. As a result, the received SNR at the targetlocation is simplified to γ ( q , ˆw ) = ¯ P β M N (cid:16) H + k q − ˆw k (cid:17) (cid:16) H + k q k (cid:17) . (13)After some manipulations, problem (P2) can be reformulatedas (P4) min q (cid:16) H + k q − ˆw k (cid:17) (cid:16) H + k q k (cid:17) . Proposition 2:
For the single-location SNR maximizationproblem (P4), the optimal horizontal placement of the AIRSis q ∗ = ξ ∗ ( ρ ) ˆw , (14)ith ξ ∗ ( ρ ) = , if ρ ≥ − r − ρ or 12 + r − ρ , otherwise . (15) where ξ is called the ratio coefficient and ρ = HD . Proof:
Please refer to Appendix A.Proposition 2 shows that the optimal horizontal placementof AIRS only depends on ρ = HD , that is, the ratio of AIRSheight H and source-destination distance D . For ρ ≥ , theAIRS should always be placed above the middle point betweenthe source and destination nodes. On the other hand, for ρ < ,there exist two optimal horizontal placement locations forthe AIRS that are symmetrical about the midpoint, as shownin Fig. 4. Note that the above result is different from theconventional relay placement [14], whose optimal solution alsodepends on the transmit SNR due to the relay receiver noise. Height-versus-distance ratio T h e op ti m a l A I R S d e p l oy m e n t c o e ff i c i e n t * () Fig. 4. The optimal AIRS deployment coefficient ξ ∗ ( ρ ) against height-versus-distance ratio ρ . B. Area Coverage Enhancement
Next, we study the general case of (P2) for area coverageenhancement. However, solving problem (P2) by the standardoptimization techniques is difficult in general. On one hand,the AIRS placement q not only affects the link distancesto/from the AIRS, but also its AoA and AoD as shown in(4), (7) and (11). On the other hand, the design of the phaseshifts vector θ needs to balance the SNRs at different locations w in the target area. In this paper, by exploiting the factthat the phase shifts optimization for IRS resembles that forthe extensively studied phase array or analog beamforming,we propose an efficient two-step suboptimal solution to (P2)by decoupling the phase shifts optimization from the AIRSplacement design. To this end, it is noted that problem (P2)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 ( ¯ φ T ( q , w ) − ¯ φ R ( q ) )) (cid:12)(cid:12)(cid:12)(cid:12) is the array gain due to the passive beamforming by the AIRS,and f ( q , w ) ∆ = (cid:16) H + k q − w k (cid:17) (cid:16) H + k q k (cid:17) accountsfor the concatenated path loss.For the proposed design, for any given AIRS placement q ,we design the phase shifts θ in the first step to maximize the worst-case array gain by solving 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. Such an approximationis reasonable since in general, the array gain f ( q , θ , w ) ismore sensitive than the concatenated path loss f ( q , w ) tothe location variation of w in the target area A , especiallywhen the source-destination distance D ≫ D x and D y . Thenin the second step, the obtained solution to (P5.1), denoted as θ ∗ ( q ) , is substituted into the objective function of (P5). Notethat even after obtaining θ ∗ ( q ) , the worst-case SNR in thistarget area A is still unknown, thus the AIRS placement needsto be optimized to maximize the worst-case SNR in A , whichcan be expressed as (P5 .
2) max q min w ∈A f ( q , θ ∗ ( q ) , w ) f ( q , w ) .
1) Phase Shifts Optimization:
In order to solve (P5.1), wefirst give the following result.
Proposition 3:
For any AIRS sub-array with ¯ N ≤ N elements and placement q , assuming that its phase shifts aredesigned to maximize the receive SNR at a location ¯w in A ,then its array gain at any other location w in A is g (cid:0) ∆ ¯ φ (cid:1) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin (cid:0) π ¯ N ¯ d ∆ ¯ φ (cid:1) sin (cid:0) π ¯ d ∆ ¯ φ (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (16)where ∆ ¯ φ = ¯ φ T ( q , w ) − ¯ φ T ( q , ¯w ) is the deviation of thesin-AoD (also called spatial frequency) from ¯ w . Proof:
According to (12), to make all reflected signalscoherently combined at the location ¯w , the phase shifts of the ¯ N elements of the AIRS are given by θ n ( q ) = ¯ θ − π ( n −
1) ¯ d (cid:0) ¯ φ T ( q , ¯w ) − ¯ φ R ( q ) (cid:1) , n = 1 , · · · , ¯ N. (17)By substituting θ n ( q ) into f ( q , θ , w ) with ¯ N ≤ N , we have g ( q , w ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¯ N X n =1 e j ( π ( n −
1) ¯ d ( ¯ φ T ( q , w ) − ¯ φ T ( q , ¯w ) )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin (cid:0) π ¯ N ¯ d (cid:0) ¯ φ T ( q , w ) − ¯ φ T ( q , ¯w ) (cid:1)(cid:1) sin (cid:0) π ¯ d (cid:0) ¯ φ T ( q , w ) − ¯ φ T ( q , ¯w ) (cid:1)(cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (18)By letting ∆ ¯ φ = ¯ φ T ( q , w ) − ¯ φ T ( q , ¯w ) , the proof of Propo-sition 3 is thus completed.Fig. 5 shows the array gain in (16) versus the deviation ∆ ¯ φ for different ¯ N . It is observed that at the target location ¯w , the received power is magnified by ¯ N times, which is inaccordance with the single-location SNR maximization in (13).As ∆ ¯ φ increases, the power reduces in general. By setting π ¯ N ¯ d ∆ ¯ φ = kπ, k = 1 , · · · , ¯ N − , we have the array gainnulls, and the beamwidth can be obtained by letting k = 1 ,i.e., ∆ ¯ φ BW = N ¯ d . This reflects the well-known fact thatthe beamwidth of phase array is inversely proportional to thearray aperture ¯ N ¯ d . Furthermore, the half-power beamwidth isknown as 3-dB beamwidth, i.e., the deviation ∆ ¯ φ at which thearray gain g (cid:0) ∆ ¯ φ (cid:1) drops to half of its peak value. According to ig. 5. AIRS array gain versus spatial frequency deviation ∆ ¯ φ . [15], for a large ¯ N , the 3-dB beamwidth can be approximatedas ∆ ¯ φ ≈ . N ¯ d .For the given AIRS with a total of N elements andplacement q , assume that the phase shifts θ of all the N elements are designed to maximize the SNR at ¯w = w ,i.e., the center of the rectangular area A . Then the maximumspatial frequency deviation ∆ ¯ φ in A can be derived as ∆ ¯ φ max ( q ) = max w ∈A (cid:0)(cid:12)(cid:12) ¯ φ T ( q , w ) − ¯ φ T ( q , w ) (cid:12)(cid:12)(cid:1) . (19)Intuitively, to achieve SNR enhancement for the entire area A , the 3-dB beamwidth of the AIRS should be sufficientlylarge so that all locations in A lie within the main lobe of theAIRS, i.e., ∆ ¯ φ ≥ ∆ ¯ φ max ( q ) . Particularly, it is observedfrom Fig. 5 that the 3-dB beamwidth can be increased byreducing ¯ N of the sub-array. This, however, decreases the peakgain of the sub-array. Therefore, there exists a design trade-offfor the partition of N reflecting elements into sub-arrays. Tothis end, we need to consider two cases depending on whether ∆ ¯ φ ≥ ∆ ¯ φ max ( q ) holds, to determine whether the sub-arrayarchitecture should be used. Case 1:
When ∆ ¯ φ ≥ ∆ ¯ φ max ( q ) , the 3-dB beamwidthcan cover the entire area. Therefore, the AIRS with the fullarray architecture should be used for maximal area coverage.In this case, the optimal phase shifts θ ∗ ( q ) for (P5.1) areobtained by setting ¯w = w in (17). Case 2:
When ∆ ¯ φ < ∆ ¯ φ max ( q ) , the resulting 3-dBbeamwidth using the full array architecture cannot cover theentire area. To tackle this problem, a sub-array architecture ofthe AIRS is proposed in this case. Specifically, the maximumspatial frequency deviation ∆ ¯ φ max ( q ) and the full array with N elements are both equally partitioned into L parts resultingin L sub-arrays, each to serve one sub-area correspondingto one of the L spatial frequency partitions, as illustratedin Fig. 6. The equal maximum spatial frequency deviationfor each partition, denoted as ∆ ¯ φ max ,l ( q ) , l = 1 , · · · , L , isreduced by L times, whereas the 3-dB beamwidth of eachsub-array with N/L (assumed to be an integer) elements isincreased by L times. Notice that when reducing the numberof elements, the peak sub-array gain is also reduced, asshown in Fig. 5. We set L as the minimum integer to ensure ∆ ¯ φ max ( q ) L ≤ ∆ ¯ φ L . Since D x ≥ D y , by adjusting thephase shifts of each sub-array to achieve coherent signalsuperposition at the corresponding horizontal location in A , !" ! $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ !!!!"!!!! % &’( ! (a) Spatial frequency !" % & ’ ( ) * (b) Horizontal locationFig. 6. An illustration of sub-array partition with respect to spatial frequencydeviation and horizontal location, respectively. the 3-dB beamwidth of each sub-array can cover the sub-areawith spatial frequency deviation ∆ ¯ φ max ,l ( q ) . The phase shifts θ ∗ ( q ) for problem (P5.1) are then obtained according to theselocations.
2) AIRS Placement Optimization:
With the above obtainedphase shifts θ ∗ ( q ) at a given AIRS placement q , the worst-case SNR in A can be obtained, which occurs at the boundarypoint (cid:0) x + D x , (cid:1) for both the cases of full array and sub-array architecture, which has the smallest array gain but thelargest concatenated path loss. Based on the obtained worst-case SNR for any given q , the AIRS placement q is thenoptimized. It is observed that the maximum spatial frequencydeviation ∆ ¯ φ max ( q ) depends on q , and different array archi-tectures should be used according to the relationship between ∆ ¯ φ max ( q ) and ∆ ¯ φ , as shown in the above. However,since it is difficult to obtain the closed-form expression of theobjective function of (P5.2) for any given q , (P5.2) cannot beanalytically solved. Fortunately, since the optimal placement ofAIRS should lie in the x-axis, i.e., q = [ x, T , the horizontalplacement x can be found via the one-dimensional search.Thus, (P5.2) is solved and a suboptimal solution is obtainedfor (P5). IV. N UMERICAL R ESULTS
In this section, numerical results are provided to evaluate theperformance of our proposed design. The altitude of AIRS isset as H = 100 m. The length and width of the rectangulararea are D x = 1000 m and D y = 600 m, respectively, and thecenter is located at (1000 , , m. Unless otherwise stated,the noise and transmit power are set as σ = − dBm and P = 20 dBm, respectively, and the reference channel poweris β = − dB. The number of transmit antennas at sourcenode is M = 16 . Furthermore, the separation of antennas atthe source node and that of reflecting elements at the AIRSare d = λ/ and d = λ/ , respectively. Number of passive reflecting elements -15-10-5051015 S N R a t t h e s i ng l e l o ca ti on ( d B ) Optimal placementBenchmark placement with q =[500,0] T m Fig. 7. SNR versus number of AIRSreflecting elements.
AIRS placement along x-axis(m) -35-30-25-20-15-10-505 T h e w o r s t - ca s e S N R ( d B ) Fig. 8. The worst-case SNR versus AIRSplacement along the x-axis.
20 22 24 26 28 30
Transmit power at the source node(dBm) -30-20-10010 T h e w o r s t - ca s e S N R ( d B ) Optimized schemeBenchmark scheme with q=[1000,0] T m Fig. 9. The worst-case SNR versus transmitpower at the source node.
Fig. 7 shows the achievable SNR for the special caseof single-location SNR maximization versus the number ofpassive reflecting elements. The single target location forSNR enhancement is set as [1000 , T m. We also considerthe benchmark placement with the AIRS placed above themidpoint between the source and the target location, i.e., [500 , T m. For both the optimal placement and benchmarkplacement, the optimal phase shifts are applied at the AIRS toachieve coherent signal superposition at the target location. Itis observed that the achievable SNR increases with the numberof passive reflecting elements for both placements, as ex-pected. In addition, the performance of the optimal placementsignificantly outperforms that of the benchmark placement,which shows the great benefit of placement optimization ofthe AIRS. For example, to achieve a target SNR of 5 dB, thenumber of elements required for the benchmark placement isabout 360, while this number is significantly reduced to 140for the optimal placement.For the more general area coverage or min-SNR maxi-mization problem, Fig. 8 plots the worst-case SNR versusAIRS placement along the x-axis. The number of reflectingelements is set as N = 200 . It is observed that different fromthat for single-location SNR maximization case, as the AIRSmoves from the source node to the target area, the performancedegrades in general, although there is some small fluctuation.This can be explained by the following two reasons. First, themaximum spatial frequency deviation (or angle separation) isrelatively small when the AIRS is far away from the targetarea, and in this case, it is more likely that the 3-dB beamwidthof the AIRS with the full array architecture is sufficient tocover the entire target area. Second, when the AIRS is nearthe source node, the concatenated path loss is small (seeProposition 2 and Fig. 4). In contrast, when the AIRS is closeto the target area, the maximum spatial frequency deviationincreases significantly, thus the sub-array architecture needsto be applied to achieve area coverage by sacrificing the peakgain of each sub-array (although the concatenated path loss issimilar to the case when the AIRS is near the source node).As a result, it is optimal to place the AIRS above the sourcenode.Last, Fig. 9 shows the worst-case SNR versus transmitpower at the source node. The number of the AIRS elements is also set as N = 200 . For comparison, we consider in this casethe benchmark scheme with the AIRS placed above the centerof the rectangular area, i.e., q = [1000 , T m. It is observedthat the optimal AIRS placement above the source node (seeFig. 8) significantly outperforms the benchmark placement.The above results show the importance of our proposed jointAIRS deployment and active/passive beamforming design.V. C ONCLUSION
This paper proposed a new 3D networking architecture withthe AIRS to achieve efficient signal coverage from the sky.The worst-case SNR in a given target area was maximized byjointly optimizing the transmit beamforming, AIRS placementand phase shifts. We first investigated the special case ofsingle-location SNR maximization and derived the optimalAIRS placement in closed-form, which depended on the ratioof AIRS height and source-destination distance only. Then forthe general case of area coverage, we proposed an efficientsuboptimal solution based on the sub-array design. Numericalresults demonstrated that the proposed design can significantlyimprove the performance over heuristic AIRS deploymentschemes. A
PPENDIX AP ROOF OF P ROPOSITION ¯q as the projection of reference point on the straightline connecting the source node with the target location. Itcan be shown that k q − ˆw k = k ¯q − ˆw k + k q − ¯q k , and k q k = k ¯q k + k q − ¯q k . Obviously, by letting k q − ¯q k =0 , i.e., q = ξ ˆw , the minimum k q − ˆw k and k q k can beobtained, where ξ is the ratio coefficient. Thus, problem (P4)can be further reduced to: min ξ k ˆw k (cid:0) ξ + ρ (cid:1) (cid:16) ( ξ − + ρ (cid:17) , (20)where ρ = H k ˆw k . Defining f ( ξ ) = (cid:0) ξ + ρ (cid:1) (cid:16) ( ξ − + ρ (cid:17) ,the first-order derivative of f ( ξ ) can be expressed as f ′ ( ξ ) = 4 ξ − ξ + (cid:0) ρ (cid:1) ξ − ρ . (21)By substituting ξ = ζ + to (21), we have f ′ ( ζ ) = ζ + aζ + b, (22)where a = ρ − and b = 0 . According to the value of ∆ = (cid:0) b (cid:1) + (cid:0) a (cid:1) = (cid:16) ρ − (cid:17) , known as the discriminantf the cubic equation, the solutions to f ′ ( ζ ) = ζ + aζ + b = 0 can be obtained under the following three cases. Case 1:
When ∆ > , that is, ρ > , there is only one realsolution, which is given by ζ = vuut − b s(cid:18) b (cid:19) + (cid:16) a (cid:17) + vuut − b − s(cid:18) b (cid:19) + (cid:16) a (cid:17) = 0 . (23) then ξ = ζ + = . Furthermore, by checking the second-order derivative of f ( ξ ) , we have f ′′ ( ξ ) = 12 (cid:18) ξ − ξ + 16 + ρ (cid:19) > (cid:18) ξ − ξ + 16 + 112 (cid:19) = 12 (cid:18) ξ − (cid:19) ≥ . (24) Since the first-order derivative f ′ ( ξ ) a monotonically increas-ing function of ξ and f ′ (cid:0) (cid:1) = 0 , the monotonicity of f ( ξ ) in this case is plotted in Fig. 10. ! Fig. 10. The monotonicity of f ( ξ ) in Case 1. Therefore, f ( ξ ) first decreases and then increases withrespect to ξ , and the minimum value is obtained when ξ = . Case 2:
When ∆ = 0 , namely, ρ = , there are three equalreal solutions, i.e., ζ = ζ = ζ = 0 . Similarly, the minimumvalue can be obtained when ξ = . Case 3:
When ∆ < , there are three real solutions, whichare given by ζ = 2 r − a ϑ r − ρ ,ζ = 2 r − a (cid:18) ϑ ◦ (cid:19) = − r − ρ ,ζ = 2 r − a (cid:18) ϑ − ◦ (cid:19) = 0 , (25) where ϑ = arccos − b √− a a = 90 ◦ . Thus, ξ = + q − ρ , ξ = − q − ρ and ξ = . First, by letting the second-order derivative of f ′′ ( ξ ) be equal to 0, we obtain (cid:18) ξ − ξ + 16 + ρ (cid:19) = 12 (cid:18) ξ − (cid:19) −
112 + ρ ! = 0 , (26) Since ξ ′′ = + q − ρ and ξ ′′ = − q − ρ , f ′ ( ξ ) increases in the interval (cid:18) −∞ , − q − ρ (cid:21) and (cid:20) + q − ρ , + ∞ (cid:19) , and decreases in the interval (cid:18) − q − ρ , + q − ρ (cid:19) .Thus, f ( ξ ) decreases in the interval (cid:16) −∞ , − q − ρ i and (cid:16) , + q − ρ i , and increases in the interval ! ! " ! " ! ! " " " Fig. 11. The monotonicity of f ( ξ ) in Case 3. (cid:16) − q − ρ , i and (cid:16) + q − ρ , + ∞ (cid:17) . The monotonicityof f ( ξ ) in this case is plotted in Fig. 11. Furthermore, bysubstituting − q − ρ and + q − ρ to f ( ξ ) , wehave f (cid:16) − q − ρ (cid:17) = f (cid:16) + q − ρ (cid:17) . Thus, theminimum value of (20) is obtained when ξ = − q − ρ or ξ = + q − ρ . The proof of Proposition 2 is thuscompleted. R EFERENCES[1] Q. Wu, and R. Zhang, “Towards smart and reconfigurable environment:Intelligent reflecting surface aided wireless network,”
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