Hierarchical Passive Beamforming for Reconfigurable Intelligent Surface Aided Communications
Chang Cai, Xiaojun Yuan, Wenjing Yan, Zhouyang Huang, Ying-Chang Liang, Wei Zhang
11 Hierarchical Passive Beamforming for ReconfigurableIntelligent Surface Aided Communications
Chang Cai, Xiaojun Yuan,
Senior Member, IEEE,
Wenjing Yan,
Student Member, IEEE,
Zhouyang Huang,Ying-Chang Liang,
Fellow, IEEE, and Wei Zhang,
Fellow, IEEE
Abstract —This letter considers a multiple-input single-output(MISO) downlink communication assisted by dense deploymentof reconfigurable intelligent surfaces (RISs), where traditionalpassive beamforming (PB) design may be too complicated toimplement due to a massive number of reflecting elements.To overcome this difficulty, we propose a hierarchical passivebeamforming (HPB) design by fully exploiting the reflectionproperty of a RIS, in which the dimension of the variable spaceinvolved in optimization is significantly reduced by adopting thegeneralized Snell’s law. We develop efficient HPB algorithms tomaximize the achievable rate of the target user. Simulation resultsdemonstrate the appealing performance and low complexity ofthe proposed HPB design.
Index Terms —Reconfigurable intelligent surface, hierarchicalpassive beamforming.
I. I
NTRODUCTION
Reconfigurable intelligent surface (RIS) [1], a.k.a. intelli-gent reflecting surface (IRS) [2] and large intelligent meta-surface (LIM) [3], has been regarded as a promising newtechnology to revolutionize wireless networks. Typically, RISis a man-made planar array composed of a large number ofsub-wavelength metallic or dielectric scattering elements, eachof which is able to induce a certain amplitude and/or phasechange to the electromagnetic waves [4]. By appropriately anddynamically adjusting the reflecting coefficient of each RIS el-ement, impinging signals can be collaboratively combined andsteered to desired directions to achieve a passive beamforming(PB) gain.Recently, researches [1], [2], [5]–[7] focusing on PB designare springing up. The optimal/sub-optimal reflecting coeffi-cients are derived based on various optimization objectives,e.g., energy efficiency maximization [1], transmit power mini-mization [2], and sum-rate maximization [5], [6]. However,due to the non-convexity of the PB design problems, thecomputation complexity rises polynomially as the numberof RIS elements increases, causing unacceptable processingdelay. It becomes more severe in RIS-aided high-mobilitycommunications since the tolerance of delay is much morestringent. Moreover, the algorithms may fail to find a solutionfor dense deployment of large RISs, where hundreds of thou-sands reflecting elements are required to be properly adjusted.
C. Cai, X. Yuan, W. Yan, Z. Huang, and Y.-C. Liang are with theCenter for Intelligent Networking and Communications, University of Elec-tronic Science and Technology of China, Chengdu 611731, China (e-mail:[email protected]; [email protected]; [email protected];[email protected]; [email protected]).W. Zhang is with the School of Electrical Engineering and Telecommunica-tions, University of New South Wales, Sydney NSW 2052, Australia (e-mail:[email protected]). BS G G N f f N H UserRIS RIS N RIS controller
Reflecting element Δλ Δλ Fig. 1. A RIS-aided cellular downlink communication system.
Therefore, there is an urgent need to simplify the traditionalPB design to meet the real-time signal processing requirement.In this letter, we tackle this challenge from the perspective ofreducing the dimension of PB design variable space. Inspiredby the generalized Snell’s law [8] stating that the reflectionto any arbitrary directions can be achieved by setting thephases of each elements on a RIS to be arithmetic sequences,we propose a simplified PB design which dramatically re-duces the number of optimization variables by adopting thestructured phase shifts. We then clarify that optimizing thephase difference between two adjacent elements (microscopiclevel design) and optimizing the common phase offset ofa RIS (macroscopic level design) correspond to adjustingthe reflection angle and aligning the phases of the RIS-reflected beams, respectively, which implies a hierarchicalpassive beamforming (HPB) design. We develop efficient HPBalgorithms to maximize the achievable user rate. Numericalresults further verify that the proposed HPB design reaps mostof the performance gain of the traditional PB design and atthe same time significantly reduces the computation time.II. R
EFLECTION -A NGLE -B ASED C ASCADED C HANNEL M ODEL
As shown in Fig. 1, we consider a cellular downlinktransmission in a rich scattering environment, where one BSequipped with a half-wavelength uniform linear array (ULA)of M isotropically radiating elements serves a single-antennauser. The direct link between the BS and the user is blocked byobstacles. In order to provide reliable communication services, N RISs are deployed distributedly to reflect the signals from a r X i v : . [ c s . I T ] J a n the BS to the user. We assume that each RIS is composed of L reflecting elements with isotropic electromagnetic properties,which is arranged in an L × L uniform rectangular array(URA). With out loss of generality, we assume L is an evennumber, and thus the index of the central element is givenby (1 + L − , L − ) . The vertical and horizontal spacingbetween two adjacent elements is ∆ λ , where ∆ is the scalingfactor with respect to (w.r.t.) the signal carrier wavelength λ . A. Cascaded Channel Model
The geometric channel model [9] is adopted for the in-dividual BS-RIS and RIS-user links, which is based on theangle-of-departures (AoDs), angle-of-arrivals (AoAs), and thecomplex path gains of each path. In this letter, we assume thatthe channel state information (CSI), including all the anglesand other channel coefficients, are perfectly known at the BS.The CSI acquisition methods are studied in [10] by insertingactive elements that are connected to the baseband processingunit into the RISs. The multi-path channel G n ∈ C L × M fromthe BS to the n -th RIS can be expressed as G n = D n (cid:88) d =1 α n,d a ( ϑ RIS n,d , ψ
RIS n,d ) b H ( φ BS n,d ) , (1)where D n denotes the number of resolvable paths; α n,d denotes the complex path gain of the d -th path; ϑ RIS n,d and ψ RIS n,d denote the corresponding elevation and azimuth AoAs,respectively; and φ BS n,d denotes the corresponding AoD. Thesteering vector a ( ϑ RIS n,d , ψ
RIS n,d ) ∈ C L × is defined as a ( ϑ RIS n,d , ψ
RIS n,d ) = a x ( ϑ RIS n,d , ψ
RIS n,d ) ⊗ a y ( ϑ RIS n,d , ψ
RIS n,d ) , (2)where ⊗ denotes the Kronecker product; a x ( ϑ RIS n,d , ψ
RIS n,d ) ∈ C L × and a y ( ϑ RIS n,d , ψ
RIS n,d ) ∈ C L × are given as follows: (cid:2) a x ( ϑ RIS n,d , ψ
RIS n,d ) (cid:3) (cid:96) = 1 √ L e π ∆( L −
1) sin( ϑ RIS n,d ) cos( ψ RIS n,d ) × e − π ∆( (cid:96) −
1) sin( ϑ RIS n,d ) cos( ψ RIS n,d ) , (cid:96) ∈ [ L ] ; (3) (cid:2) a y ( ϑ RIS n,d , ψ
RIS n,d ) (cid:3) (cid:96) = 1 √ L e π ∆( L −
1) sin( ϑ RIS n,d ) sin( ψ RIS n,d ) × e − π ∆( (cid:96) −
1) sin( ϑ RIS n,d ) sin( ψ RIS n,d ) , (cid:96) ∈ [ L ] , (4)where denotes the imaginary unit. The steering vector b ( φ BS n,d ) ∈ C M × is given by (cid:2) b ( φ BS n,d ) (cid:3) m = 1 √ M e π ( M − sin( φ BS n,d ) × e − π ( m −
1) sin( φ BS n,d ) , m ∈ [ M ] . (5)Similarly, the multi-path channel f H n ∈ C × L from the n -thRIS to the user is modeled as f H n = K n (cid:88) k =1 β n,k u H ( θ RIS n,k , φ
RIS n,k ) , (6)where K n denotes the number of resolvable paths; β n,k denotes the complex path gain of the k -th path; θ RIS n,k and φ RIS n,k denote the corresponding elevation and azimuth AoDs;
Fig. 2. An illustration of the reflection property of a RIS. the steering vector u ( θ RIS n,k , φ
RIS n,k ) ∈ C L × is defined in thesame manner as a ( ϑ RIS n,d , ψ
RIS n,d ) in (2).By ignoring the direct link from the BS to the user, thereceived signal at the user can be expressed as y = N (cid:88) n =1 f H n Θ n G n w s + n = h H w s + n, (7)where Θ n = diag (cid:16) e θ n, , . . . , e θ n,L (cid:17) denotes the diagonalphase shifts matrix for the n -th RIS, with reflecting amplitudebeing ; w denotes the transmit beamforming vector at theBS; s denotes the transmit data symbol; n denotes additivewhite Gaussian noise (AWGN) with zero mean and variance σ ; and h H = (cid:80) Nn =1 f H n Θ n G n denotes the cascaded BS-RIS-user channel. B. Reflection-Angle-Based Cascaded Channel Representation
To reconfigure the wireless propagation environment, pas-sive beamforming (PB) design problems have been widelystudied in [1], [2], [5]–[7] and the references therein. However,the aforementioned work considers that the phase shift ofeach RIS element is independently adjusted, where a total of
N L variables should be jointly optimized. The complexity ofsolving such non-convex PB problems is sensitive to the sizeof the variable space and is in general very high. For example,the involved complexity is O (cid:0) ( N L ) (cid:1) via the semi-definiterelaxation (SDR) method [2], which is prohibitively high espe-cially for large surfaces consisting of thousands of reflectingelements. To meet the requirement of dense deployment oflarge RISs, we turn to explore the reflection property of eachRIS, so as to significantly reduce the dimension of the PBvariable space and thus simplify the optimization design.According to the generalized Snell’s law [8], to real-ize the electromagnetic wave deflection from the incidentelevation/azimuth angles (cid:0) ϑ in , ψ in (cid:1) to the reflection eleva-tion/azimuth angles ( ϑ rn , ψ rn ) in Fig. 2, the phase shifts func-tion of each RIS is affine along both the x - and y -axes . More The generalized Snell’s law is derived from infinite-size and continuoussurface to guarantee specular reflection. However, it is reported in [11] thatthe beamwidth of the reflected wave from a λ × λ RIS is about ◦ ,which indicates that a finite-size RIS with discrete elements is far from aspecular reflector. While in this case, following the generalized Snell’s lawstill guarantees that the main lobe of the beam is aligned with the target user. specifically, the phase shift of the ( i, j ) -th element of the n -thRIS satisfies [12] θ n, ( i − L + j = 2 π ∆ (cid:18) i − − L − (cid:19) q xn + 2 π ∆ (cid:18) j − − L − (cid:19) q yn + ϕ n , (8)where the design parameters include: 1) q xn and q yn , denotingrespectively the phase gradients of the x - and y -axes, whichcontrol the reflection angle by altering the phase differencebetween two adjacent elements; and 2) ϕ n , denoting thecommon phase offset of the n -th RIS, which aligns the phaseof the reflected beam. In order to steer the impinging signalfrom ( ϑ in , ψ in ) to ( ϑ rn , ψ rn ) , the phase gradient q n = [ q xn , q yn ] should be set as q xn = sin ϑ rn cos ψ rn + sin ϑ in cos ψ in ; (9) q yn = sin ϑ rn sin ψ rn + sin ϑ in sin ψ in . (10)Since the absolute values of both sin( · ) and cos( · ) are nolarger than 1, q xn , q yn ∈ [ − , always holds for any incidentand reflection angles. Moreover, due to the π -periodicity ofthe phase shift, q xn and q xn + ( q yn and q yn + ) yield the sameRIS response function. Therefore, we have q xn , q yn ∈ [ − ¯ q, ¯ q ] ,where ¯ q = min (cid:8) , (cid:9) .It is worth mentioning that the structured phase shifts in (8)fully realize the reflection capability of a RIS. Furthermore,for URA-shaped RISs, the variable space reduces from N L to N , where only q xn , q yn and ϕ n need to be customized forthe n -th RIS. Motivated by all above, we adopt the phase shiftsstructure specified in (8) for PB design. Plugging (8) into thecascaded channel h H , we have h H = N (cid:88) n =1 (cid:32) K n (cid:88) k =1 β n,k u H n,k (cid:33) Θ n (cid:32) D n (cid:88) d =1 α n,d a n,d b H n,d (cid:33) = N (cid:88) n =1 K n (cid:88) k =1 D n (cid:88) d =1 α n,d β n,k p n,k,d b H n,d , (11)where p n,k,d = (cid:16) u xn,k ⊗ u yn,k (cid:17) H Θ n (cid:16) a xn,d ⊗ a yn,d (cid:17) = L (cid:88) i =1 L (cid:88) j =1 (cid:2) u xn,k (cid:3) ∗ i (cid:2) u yn,k (cid:3) ∗ j θ n, ( i − L + j (cid:2) a xn,d (cid:3) i (cid:2) a yn,d (cid:3) j = e ϕ n L (cid:32) e π ∆( L − s xn,k,d L (cid:88) i =1 e − π ∆( i − s xn,k,d (cid:33) × e π ∆( L − s yn,k,d L (cid:88) j =1 e − π ∆( j − s yn,k,d , (12)with s xn,k,d = sin θ RIS n,k cos φ RIS n,k + sin ϑ RIS n,d cos ψ RIS n,d − q xn ; (13) s yn,k,d = sin θ RIS n,k sin φ RIS n,k + sin ϑ RIS n,d sin ψ RIS n,d − q yn . (14) The steering vectors u ( θ RIS n,k , φ
RIS n,k ) , a ( ϑ RIS n,d , ψ
RIS n,d ) , and b ( φ BS n,d ) areabbreviated to u n,k , a n,d , and b n,d in the sequel of this paper. By applying the sum of the geometric progression [13], p n,k,d in (12) can be further expressed as p n,k,d = e ϕ n sinc (cid:16) ∆ Ls xn,k,d (cid:17) sinc (cid:16) ∆ s xn,k,d (cid:17) sinc (cid:16) ∆ Ls yn,k,d (cid:17) sinc (cid:16) ∆ s yn,k,d (cid:17) , (15)As expected, (15) shows the beam pattern of a finite-size RISreflector, where the reflection angle should be fine-tuned byadjusting q n . We rewrite (11) in a more compact form as h H = v H H , (16)where v = (cid:2) e ϕ , . . . , e ϕ N (cid:3) H , H = (cid:2) r H1 B , . . . , r H N B N (cid:3) , B n = (cid:2) b n, , . . . , b n,D n (cid:3) H , r n = (cid:2) r n, , . . . , r n,D n (cid:3) H , and r n,d = α n,d (cid:80) K n k =1 β n,k | p n,k,d | .It is observed that the phase shift expression in (8) isdivided into two separate parts in (16), i.e., Q = [ q , . . . , q N ] inside H and v outside H . The wireless channel thus can bereconfigured from two aspects. Firstly, from the element level,the reflection angle of each RIS can be controlled by adjustingthe phase difference, i.e., Q , between two adjacent elements,which can be regarded as a microscopic PB design. Secondly,from the RIS level, the common phase offset of each RIS,i.e., ϕ n , is adjusted to align the phase of each reflected beamarrived at the user, which can be regarded as a macroscopic PBdesign. This leads to a reflection-angle-based HPB problem,as formally stated in the following section.III. P ROBLEM S TATEMENT
We aim to maximize the achievable rate of the user subjectto the maximum transmit power constraint at the BS. In thesingle-user setup in this letter, the objective reduces to themaximization of the received signal power. Thus, the problemcan be formulated as max w , Q , v (cid:12)(cid:12) v H Hw (cid:12)(cid:12) (17a) s . t . (cid:107) w (cid:107) ≤ p, (17b) | v n | = 1 , ∀ n, (17c) q xn , q yn ∈ [ − ¯ q, ¯ q ] , ∀ n, (17d)where Q and v are the HPB variables. Maximum-ratio trans-mission (MRT) is applied to obtain the optimal transmit beam-forming solution, i.e., w (cid:63) = √ p H H v (cid:107) v H H (cid:107) . By substituting w (cid:63) into the above problem, it can be simplified to the followingequivalent problem: max Q , v f ( Q , v ) (cid:44) (cid:13)(cid:13) v H H (cid:13)(cid:13) s . t . (17c) , (17d) . (18)IV. H IERARCHICAL P ASSIVE B EAMFORMING D ESIGN
In this section, we first present an alternating optimization(AO) based algorithm, i.e., HPB-AO algorithm, to optimize theHPB variables in (18). Then, to avoid recursively optimizing Q and v , a heuristic algorithm is proposed to split the HPBdesign into two stages. Algorithm 1:
SA Algorithm
Input:
Initial temperature T max ; terminate temperature T min ; annealing parameter γ . Initialize: Q ← Q ; Q (cid:63) ← Q ; T ← T max . while T > T min do Obtain a random neighbour Q new ← Q + ∆ Q ; if dE (cid:44) f ( Q ) − f ( Q new ) < then Q ← Q new ; if f ( Q (cid:63) ) − f ( Q new ) < then Q (cid:63) ← Q new ; endelse if exp (cid:0) − dE T (cid:1) > random(0 , then Q ← Q new ; end Decrease the current temperature T ← γT ; endOutput: Q (cid:63) . A. HPB-AO Algorithm
For given v , the problem in (18) reduces to max Q f ( Q ) s . t . (17d) . (19)It can be observed from (11) to (16) that f ( Q ) is highlynon-convex w.r.t. Q due to the involvement of sinc functions.Thus, to solve problem (19), the gradient descent algorithmsare likely to be stuck in local optimum. Inspired by thermo-dynamic principles, we adopt the simulated annealing (SA)algorithm [14], which models the physical process of heating amaterial and then slowly lowering the temperature to decreasedefects, so as to find a near-optimal solution.The pseudo-code of the SA algorithm is shown in Algorithm1. Specifically, the algorithm begins with temperature T max . Ateach iteration, a new point Q new is randomly generated, wherethe distance from the current point Q is given by ∆ Q . Thealgorithm accepts all new points that raise the objective, butalso, with a certain probability, points that lower the objective.By accepting the points that lower the objective, the algorithmavoids being trapped in local maximum, and is able to ex-plore globally for a more appealing solution. The annealingschedule systematically decreases the temperature from T to γT at each iteration, and the probability of accepting a worsesolution also decreases as the algorithm proceeds. Finally, thealgorithm terminates when the temperature decreases to T min and chooses the best solution Q (cid:63) that explored during theannealing procedure as the algorithm output.For given Q , the problem in (18) reduces to max v f ( v ) s . t . (17c) . (20)This is a non-convex quadratically constrained quadratic pro-gram (QCQP) problem and can be iteratively solved by thesuccessive convex approximation (SCA) algorithm. For given v t − at the t -th iteration, we obtain from convexity that f ( v ) ≥ (cid:8) ( v t − ) H HH H v (cid:9) − ( v t − ) H HH H v t − , (21)which gives an approximation of f ( v ) and the equality holdsat point v = v t − . By maximizing this approximation subject Algorithm 2:
SCA Algorithm
Input:
Maximum iteration number I sca ; relativedecrease criterion (cid:15) sca . Initialize: v ; t ← . while t < I sca do Update v according to (22); if | f ( v t ) − f ( v t − ) f ( v t − ) | < (cid:15) sca thenbreak ; end t ← t + 1 ; endOutput: v t .to the constraint (17c), the closed-form solution at the t -thiteration is readily given by v t = e arg( HH H v t − ) . (22)Based on the above discussions, we provide the implementa-tion details of the SCA algorithm in Algorithm 2.In HPB-AO, Q and v are alternately optimized until con-vergence. The iteration involved may be still time-consuming.As such, we next propose a two-stage algorithm, termed HPB-Strongest-Path-Pairing (HPB-SPP), to strike a more appealingbalance between complexity and performance. B. HPB-SPP Algorithm
The two-stage algorithm is originated from the physicalsignificance of HPB variables. In the first stage, we propose todesign Q that adjusts the reflection angle of the beam reflectedby RIS, aiming at steering right towards the target user throughthe strongest paths of the BS-RIS and the RIS-user links. Inthe second stage, the design of v is considered to align thephases of beams that reflected by each RIS. The algorithmdetails are specified as follows.
1) Stage I:
For the n -th RIS, we assume that the d -thpath and the k -th path are the strongest paths of G n and f H n ,respectively. Then, the gradient q n is set by letting s xn,k,d and s yn,k,d in (13) and (14) to be , which activates the strongestpaths. We emphasize that this is inconsistent with the originaloptimization objective in problem (18). However, simulationresults demonstrate a marginal performance gap comparedwith the result of the HPB-AO algorithm.
2) Stage II:
For given well designed Q , we consider thesame problem in (20) to align the phases.V. N UMERICAL R ESULTS
In this section, we use simulation results to evaluate theproposed HPB design. The BS is equipped with M = 8 antennas, emitting signals with carrier wavelength λ = 0 . m. The parameters related to RIS configuration are ∆ = and L = 900 . Without loss of generality, D n and K n areassumed to be the same and denoted by P . The AoAs/AoDsare Laplacian distributed with an angular spread σ AS = 10 ◦ .The complex path gain α n,d of the BS-RIS channel is circu-larly symmetric complex Gaussian (CSCG) distributed, i.e., α n,d ∼ CN (0 , σ n,d ) , where σ n,d s are randomly generated Fig. 3. Pseudo-color image of received signal power, with P = 9 , N = 1 . form an exponential distribution and normalized such that (cid:80) D n d =1 σ n,d = 1 . The complex path gain β n,k of the RIS-userchannel is similarly modeled and here we omit the details.Additionally, the path-loss coefficient of the cascaded channelis given by [13, Propersition 1] P L n = G BS G RIS,n G User ∆ L λ π d ,n d ,n , (23)where G BS = 5 dBi, G RIS,n = 5 dBi, and G User = 0 dBi are the antenna gains at the BS, the n -th RIS, and theuser, respectively; d ,n = 50 m and d ,n = 50 m denotethe distance from the BS to the n -th RIS, and from the n -thRIS to the user, respectively. The remaining parameters areset as follows: p = 0 . W, σ = − dBm, T max = 10 , T min = 10 − , γ = 0 . , I sca = 10 , (cid:15) sca = 10 − . Theexperiments are carried out on a Windows x64 machine with . GHz CPU and
GB RAM. All the simulation curves areobtained by averaging independent channel realizations.The following PB algorithm and other comparison schemesare considered. • PB-SCA: the SCA method is adopted to solve the tra-ditional PB problem, where the closed-form expressionscan be derived in each iteration. • HPB-ES: for the single-RIS setup, i.e., N = 1 , a × gird is established to exhaustively search (ES) thebest solutions of q x and q y . • Random Phase Shifts: for each channel realization,the result is averaged with randomly generatedreflection coefficients.Fig. 3 considers the single-RIS setup and plots the pseudo-color image of the received signal power w.r.t. the phasegradient components q x and q y , with P = 9 . It is observedthat the solution of HPB-AO almost overlaps with the globaloptimum of HPB obtained by exhaustive search. Moreover, asshown in the red circle label, if we select the strongest paths ofthe BS-RIS channel and the RIS-user channel as the incidentand reflection paths directly, the solution is also reasonablegood and achieves near-optimum.In Fig. 4, the achievable rate is evaluated over the numberof channel paths when N = 1 , and over the number of RISswhen P = 9 , respectively. We see that the performance ofthe three HPB algorithms, namely HPB-ES, HPB-AO, and Fig. 4. Achievable rate versus (a) P , with N = 1 ; (b) N , with P = 9 . -3 Fig. 5. Computation time versus N , with P = 9 . HPB-SPP, are far beyond the random phase shifts settings.Also, the performance gap between the three HPB algorithmsand PB-SCA is acceptable small. When N = 1 , HPB-AOachieves the same favourable performance as HPB-ES, whilethe latter incurs unaffordable complexity when N is large.Compared with HPB-SPP, as N increases, the achievable rateimprovement of HPB-AO decreases since the performance ofthe SA procedure in HPB-AO is sensitive to the dimension ofvariable space.We remark that although the PB design performs slightlybetter than the HPB designs, their computation complexitiesare quite different. It is seen from Fig. 5 that the computationtime of PB-SCA rises very fast as N increases, and requiresabout seconds to obtain a PB solution when N = 20 .However, the computation times of both HPB-AO and HPB-SPP are low, and it is easy to meet the real-time signalprocessing requirement especially for the latter algorithm (lessthan . second even when N = 20 ).VI. C ONCLUSION
This letter proposed a new concept, i.e., HPB, for RIS re-flecting coefficients design by assuming arithmetic-sequence-structured phase shifts. Experimental results evidence that theproposed solution achieves a close-to-ideal performance withsignificantly reduced computation complexity in PB design.R
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