Joint Design of Hybrid Beamforming and Phase Shifts in RIS-Aided mmWave Communication Systems
aa r X i v : . [ c s . I T ] J a n Joint Design of Hybrid Beamforming and PhaseShifts in RIS-Aided mmWave CommunicationSystems
Bei Guo, Renwang Li and Meixia TaoDepartment of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, ChinaEmail: { guobei132, renwanglee, mxtao } @sjtu.edu.cn Abstract —This paper considers a reconfigurable intelligentsurface (RIS)-aided millimeter wave (mmWave) downlink com-munication system where hybrid analog-digital beamformingis employed at the base station (BS). We formulate a powerminimization problem by jointly optimizing hybrid beamformingat the BS and the response matrix at the RIS, under signal-to-interference-plus-noise ratio (SINR) constraints. The problem ishighly challenging due to the non-convex SINR constraints aswell as the non-convex unit-modulus constraints for both thephase shifts at the RIS and the analog beamforming at theBS. A penalty-based algorithm in conjunction with the manifoldoptimization technique is proposed to handle the problem,followed by an individual optimization method with much lowercomplexity. Simulation results show that the proposed algorithmoutperforms the state-of-art algorithm. Results also show thatthe joint optimization of RIS response matrix and BS hybridbeamforming is much superior to individual optimization.
I. I
NTRODUCTION
Reconfigurable Intelligent Surfaces (RISs) have emergedas a new technique to enhance wireless communications bymanipulating the radio propagation environment. An RIS is anartificial meta-surface consisting of a large number of passivereflection elements that can be programmed to control thephase of the incident electromagnetic waves [1]. It is appealingfor communications as it can create passive beamforming(BF) towards desired receivers without radio frequency (RF)components. Compared to traditional active multi-input multi-output (MIMO) relaying, RISs are more cost-effective and donot cause any noticeable processing delay.RISs bring a new degree of freedom to the optimizationof BF design. The work [2] studies the joint optimizationof active and passive BF in an RIS-aided multi-user systemfor transmit power minimization under signal-to-interference-plus-noise ratio (SINR) constraints. In [3], the joint optimiza-tion of active and passive BF is investigated for weighted-sum-rate (WSR) maximization under transmit power constraints.The work [4] considers the sum-rate maximization problemwhen only a limited number of discrete phase shifts can berealized by the RIS. Note that in all these works on jointactive-passive BF design, the active BF part is fully digitalas in most of the MIMO BF literature, which requires eachantenna to be connected to one RF chain.
This work is supported by the NSF of China under grant 61941106. (cid:37)(cid:54) (cid:50)(cid:69)(cid:86)(cid:87)(cid:68)(cid:70)(cid:79)(cid:72)(cid:86) (cid:56)(cid:86)(cid:72)(cid:85)(cid:3)(cid:20) (cid:56)(cid:86)(cid:72)(cid:85)(cid:3)(cid:46)(cid:53)(cid:44)(cid:54)(cid:3)(cid:38)(cid:82)(cid:81)(cid:87)(cid:85)(cid:82)(cid:79)(cid:79)(cid:72)(cid:85) (cid:53)(cid:44)(cid:54) (cid:41)(cid:3)(cid:72)(cid:79)(cid:72)(cid:80)(cid:72)(cid:81)(cid:87)(cid:86)
Fig. 1:
RIS-aided downlink mmWave communication system.
The millimeter wave (mmWave) communication over 30-300 GHz spectrum is a key technology in 5G networks toprovide high data-rate transmission. A fundamental issue ofmmWave communications is its sensitivity to signal blockagesdue to the high frequency band. Thus, an important use case ofRISs is to overcome the blockage effect in mmWave systems.Compared with the sub-6 GHz systems, mmWave systemssuffer much higher hardware cost and power consumptionon the RF circuits. Hybrid analog and digital (A/D) BF ismore favorable than fully digital BF since it allows multipleantennas to share one RF chain [5]. It is therefore desirableto consider hybrid BF for the active BF design in RIS-aidedmmWave communications. Recently, the work [6] focuses onWSR maximization in a nonorthogonal multiple access systemby jointly designing the RIS phase shifts and hybrid BF.The work [7] proposes an individual design algorithm for thehybrid beamformer, and the RIS response matrix to achievelow error rate in a wideband system.This paper considers the joint optimization of the hybrid BFat the BS and the phase shifts at the RIS in an RIS-aided multi-user mmWave system. We formulate an optimization problemfor minimizing the total transmit power at the BS subjectto individual SINR constraints for each user. This problemis highly non-convex and very challenging due to two mainobstacles. One is that all variables are tightly coupled in theconstraints. To tackle this issue, we reformulate the problemusing the penalty function method. More specifically, we intro-duce auxiliary variables to decouple these variables and thenadd the associated equality constraints to the objective functionas penalty terms. Another obstacle is that both the phase shiftsat the RIS and the analog beamformers at the BS have unit- modulus constraints. Unlike the conventional semidefinite re-laxation (SDR) method [2], we adopt a manifold optimizationtechnique to handle these unit-modulus constraints. Overall,we propose a two-layer penalty-based algorithm in conjunctionwith the Riemannian manifold optimization to find a stationarysolution to the original problem. Simulation results showthat the proposed penalty-based algorithm outperforms thetraditional SDR-based optimization algorithm. Results alsoshow that the proposed hybrid beamforming at the BS canperform closely to the fully digital beamforming.II. S
YSTEM M ODEL A ND P ROBLEM F ORMULATION
A. System Model
As shown in Fig. 1, we consider an RIS-aided downlinkmmWave system where one BS, equipped with M antennas,communicates with K single-antenna users via the help of oneRIS with F unit cells. The BS employs the sub-connectedhybrid A/D beamforming structure with N RF chains, eachconnected to D = M/N antennas. Let s j denote the infor-mation signal intended to user j , for j ∈ K , { , . . . , K } .It is assumed to be independent to each other and satisfies E [ | s j | ] = 1 . Each of these signals is first weighted by a digitalbeamforming vector, denoted as w j ∈ C N × . These weightedsignal vectors are summed together and each entry is sent to anRF chain, then multiplied by an analog beamforming vector,denoted as v n ∈ C D × , for n ∈ N , { , , · · · , N } . Eachentry of v n , denoted as v n,d , ∀ d ∈ D , { , . . . , D } is a phaseshifter, i.e., | v n,d | = 1 . The overall analog beamforming matrixcan be represented as V = diag { v , · · · , v N } ∈ C M × N . Atthe RIS, let F , { , , · · · , F } denote the set of total RIS unitcells, and define a diagonal matrix Θ = diag ( b , b , . . . , b F ) asthe response-coefficient matrix, where b f = e jθ f , θ f ∈ [0 , π ) being the phase shift of the f th unit cell. The total transmitpower of the BS is given by P total = K X k =1 k Vw k k = D K X k =1 k w k k . (1)We assume the BS-user link is blocked, and thus the directpath can be ignored. The channel state information (CSI) ofall links is assumed to be perfectly known at the BS and allthe channels experience quasi-static flat-fading.The received signal of user k can be represented as y k = h Hk ΘGV K X j =1 w j s j + n k , ∀ k ∈ K , (2)where G ∈ C F × M is the channel matrix from the BS to theRIS, h Hk ∈ C × F is the channel vector from the RIS to userk, and n k ∼ CN (0 , σ k ) is the additive white Gaussian noiseat the receiver of user k .The received SINR of user k can be expressed asSINR k = | h Hk ΘGVw k | P j = k | h Hk ΘGVw j | + σ k , ∀ k ∈ K . (3) B. mmWave Channel Model
We adopt the widely used Saleh-Valenzuela channel model[8] for mmWave communications. Specifically, the channelmatrix between BS and RIS can be written as G = s MFN cl N ray N cl X i =1 N ray X l =1 α i l a R ( φ Rri l , δ Rri l ) a B ( φ Bi l , δ Bi l ) H . (4) Here, N cl denotes the number of scattering clusters, N ray denotes the number of rays in each cluster, α i l denotes thechannel coefficient of the l th ray in the i th propagationcluster. Moreover, a R ( φ Rri l , δ Rri l ) and a B ( φ Bi l , δ Bi l ) repre-sent the receive array response vectors of the RIS and thetransmit array response vectors of the BS respectively, where φ Rri l ( φ Bi l ) and δ Rri l ( δ Bi l ) represent azimuth and elevationangles of arriving at the RIS (or departing from the BS).The channel vector between the RIS and the k -th user canbe represented as h Hk = s FN cl N ray N cl X i =1 N ray X l =1 β i l a R ( φ Rti l , δ Rti l ) H . (5)Here, N cl , N ray , β i l , φ Rti l and δ Rti l are defined in the sameway as above.We consider the uniform planar array (UPA) structure atboth BS and RIS. The array response vector can be denotedas a z ( φ, δ ) = 1 √ A A h , . . . , e j πλ d ( o sin φ sin δ + p cos δ ) . . . , e j πλ d (( A −
1) sin φ sin δ )+( A −
1) cos δ ) i T , (6)where z ∈ { R, B } , λ is the signal wavelength, d is theantenna or unit cell spacing which is assumed to be halfwavelength distance, ≤ o < A and ≤ p < A , A and A represent the number of rows and columns of the UPA inthe 2D plane, respectively. C. Problem Formulation
We aim to minimize the transmit power by jointly optimiz-ing the digital beamforming matrix W = [ w , · · · , w K ] ∈ C N × K and the analog beamforming matrix V at the BS, aswell as the overall response-coefficient matrix Θ at the RIS,subject to a minimum SINR constraint for each user. Thus,the optimization problem can be formulated as P : min { V , W , Θ } D K X k =1 k w k k (7a)s.t. SINR k ≥ γ k , ∀ k ∈ K , (7b) | v n,d | = 1 , ∀ n ∈ N , ∀ d ∈ D , (7c) | b f | = 1 , ∀ f ∈ F , (7d)where γ k > is the minimum SINR requirement of user k .The problem is non-convex due to the non-convex SINRconstraints (7b) and the unit-modulus constraints (7c), (7d). A commonly used approach to solve this type of optimiza-tion problems approximately is to apply the block coordi-nate descent (BCD) techniques in conjunction with the SDRmethod. More specifically, the digital beamforming matrix W ,the analog beamforming matrix V , and the RIS response-coefficient matrix Θ are updated in an alternating manner ineach iteration. The sub-problem of finding W can be solved bysecond-order cone program (SOCP) method, and both the sub-problems of finding V and finding Θ can be solved by SDR.However, the solution obtained by SDR is not guaranteed tobe rank-one and additional randomization approach is needed.In addition, when the number of users is close to the numberof RF chains at the BS, the randomization procedure may failto find a feasible solution.III. P ENALTY - BASED J OINT O PTIMIZATION A LGORITHM
In this section, we propose a two-layer penalty-based al-gorithm for the considered problem P . The BCD method isadopted in the inner layer to solve a penalized problem andthe penalty factor is updated in the outer layer until converge.Specifically, we firstly introduce auxiliary variables { t k,j } torepresent h Hk ΘGVw j such that variables W , V and Θ canbe decoupled. Then, the non-convex constraints (7b) can beequivalently written as | t k,k | P Kj = k | t k,j | + σ k ≥ γ k , ∀ k ∈ K , (8a) t k,j = h Hk ΘGVw j , ∀ k, j ∈ K . (8b)Then, the equality constraints (8b) is relaxed and added tothe objective function as a penalty term. Thereby, the originalproblem P can be converted to P : min V , W , Θ , { t k,j } D K P k =1 k w k k + ρ K P j =1 K P k =1 (cid:12)(cid:12) h Hk ΘGVw j − t k,j (cid:12)(cid:12) s.t. (8a) , (7c) , (7d) , (9)where ρ > is the penalty factor. The choice of ρ is crucialto balance the original objective function and the equalityconstraints. It is seen that the objective function in P isdominated by the penalty term when ρ is large enough andconsequently, the equality constraints (8b) can be well met bythe solution. Thus, we can start with a small value of ρ toget a good start point, and then by gradually increasing ρ , ahigh precision solution can be obtained. Similar approach isadopted in [9]. A. Inner Layer: BCD Algorithm for Solving Problem P For any given ρ , the problem P is non-convex but all theoptimization variables { V , W , Θ , { t k,j }} are decoupled in theconstraints. We therefore adopt BCD method to optimize eachof them alternately while keeping the others fixed.
1) Optimize W : When V , Θ and { t k,j } are fixed, problem P becomes a non-constraint convex optimization problem. Thus, the optimal W can be obtained by the first-orderoptimality condition, i.e., w k = ρ A − K X j =1 ˜ h Hj t j,k , ∀ k ∈ K , (10)where ˜ h j = h Hj ΘGV and A = 2 D I N + ρ K P j =1 ˜ h Hj ˜ h j .
2) Optimize Θ : Let b = [ b , b , . . . , b F ] H . When othervariables are fixed, problem P is reduced to (with constantterms ignored) min b f ( b ) = K X j =1 K X k =1 (cid:12)(cid:12) b H c k,j − t k,j (cid:12)(cid:12) (11a)s.t. | b f | = 1 , ∀ f ∈ F , (11b)where c k,j = diag ( h Hk ) GVw j ∈ C F × . Although the objec-tive function is convex for b , the problem (11) is still non-convex due to the unit-modulus constraints (11b). To handlethis problem, one way is to alternately optimize the F unitsone by one as in [3], [9]. Although closed-form expressionis available for each unit, this method is still inefficient sincethe unit number F is usually very large. Another way is toadopt the SDR technique as in [2]. But its complexity ishigh and additional randomization procedure is needed. Notethat the unit-modulus constraints (11b) form a complex circlemanifold M = { b ∈ C F : | b | = · · · = | b F |} [10]. Therefore,different from the above approaches, we adopt the manifoldoptimization technique to solve this problem efficiently andoptimally. In specific, we adopt the Riemannian conjugate gra-dient (RCG) algorithm. The RCG algorithm is widely appliedin hybrid beamforming design [11] and recently applied inRIS-aided systems as well [12], [13]. Each iteration of theRCG algorithm involves three key steps, namely, to computeRiemannian gradient, to find search direction and retraction.The Riemannian gradient grad b f ( b ) of the function f ( b ) is defined as the orthogonal projection of the Euclideangradient ∇ f ( b ) onto the tangent space T b M of the manifold M , which can be expressed as T b M = (cid:8) z ∈ C M : ℜ { z ⊙ b ∗ } = M (cid:9) , (12)where ⊙ denotes the Hadamard product. The Euclidean gra-dient of f ( b ) over b is given by ∇ f ( b ) = 2 K X j =1 K X k =1 c k,j ( c Hk,j b − t Hk,j ) . (13)Then, the Riemannian gradient is given by grad b f ( b ) = ∇ f ( b ) − Re {∇ f ( b ) ⊙ b } ⊙ b . (14)With the Riemannian gradient, we can update the searchdirection d by conjugate gradient method, i.e., d = − grad f b + λ T ( d ) , (15)where λ is the update parameter, d is the previous searchdirection and T ( d ) = d − Re (cid:8) d ⊙ b ∗ (cid:9) ⊙ b . Since the updated point may leave the previous manifoldspace, a retraction operation
Retr b is needed to project thepoint to the manifold itself: Retr b : b f ← ( b + λ d ) f (cid:12)(cid:12)(cid:12) ( b + λ d ) f (cid:12)(cid:12)(cid:12) , (16) where λ is the Armijo backtracking line search step size.
3) Optimize V : Define x , (cid:2) v T , v T , . . . , v TN (cid:3) T ∈ C M × , and Z j , diag { w j, I D , . . . , w j,N I D } ∈ C M × M , where | x m | = 1 , ∀ m ∈ M , { , , · · · , M } and w j,n denotes the n -th entry of w j . Then, we have Vw j = Z j x ∈ C M × . Whenother variables are fixed, problem P is given by min x f ( x ) = K X j =1 K X k =1 | d k,j x − t k,j | (17a)s.t. | x ( m ) | = 1 , ∀ m ∈ M , (17b)where d k,j = b H diag ( h Hk ) GZ j ∈ C × M . Similar to SectionIII-A2, it can be effectively solved by the RCG algorithm andthe details are skipped.
4) Optimize { t k,j } : With other variables fixed, problem P can be reduced to min { t k,j } K X j =1 K X k =1 (cid:12)(cid:12) h Hk ΘGVw j − t k,j (cid:12)(cid:12) (18a)s.t. | t k,k | P Kj = k | t k,j | + σ k ≥ γ k , ∀ k ∈ K . (18b)The objective function is convex over { t k,j } . Althoughconstraints (18b) are still non-convex, they can be translatedto the form of second-order cone, which can be effectivelyand optimally solved by SOCP method [14]. B. Outer Layer: Update Penalty factor
The penalty factor ρ is initialized to be a small number tofind a good start point, then gradually increased to tighten thepenalty. Specifically, ρ := ρc , < c < , (19)where c is a scaling parameter. A larger c may lead to a moreprecise solution with higher running time. C. Algorithm
The overall penalty-based algorithm is summarized in
Al-gorithm
1. Define the stopping indicator ξ as following ξ , max (cid:8) | h Hk ΘGVw j − t k,j | , ∀ k, j ∈ K (cid:9) . (20)When ξ is below a pre-defined threshold ǫ > , the equalityconstraints (8b) are considered to be satisfied and the pro-posed algorithm is terminated. Since we start with a smallpenalty and gradually increase its value, the objective valueof problem P is finally determined by the penalty part andthe equality constraints are guaranteed to be satisfied. Notethat, for any given ρ , problem P is solved through theBCD method and each subproblem can obtain an optimalsolution. Thus, Algorithm
Algorithm 1
Penalty-based Algorithm with Manifold Opti-mization Initialize V , Θ , ρ and { t k,j } , ∀ k, j ∈ K . repeat repeat Update W by (10); Update Θ by solving problem (11); Update V by solving problem (17); Update { t k,j } by solving problem (18); until The decrease of the objective value of problem P is below threshold ǫ > . Update ρ by (19). until The stopping indicator ξ is below threshold ǫ > .a stationary point. The total complexity of Algorithm O ( I O I J ( KN + I Q K F + I V K M + K )) , where I O , I J , I Q , and I V denote the iteration times of the outer loop, theinner loop, the inner RCG algorithm to update Θ , and theinner RCG algorithm to update V , respectively.IV. I NDIVIDUAL O PTIMIZATION
To reduce the complexity of solving problem P , wedevelop an individual optimization approach in this section,where the RIS response matrix Θ , the analog beamformer V , and the digital beamformer W are obtained sequentiallywithout alternating optimization. A. RIS design
The equivalent channel between the BS and the k th user viathe RIS can be represented as h Hk ΘG . To ensure the receivesignal quality of each user, we aim to find the optimal RISresponse matrix for maximizing the equivalent channel gainof the user who has the worst channel state, i.e., max Θ min k ∈K k h Hk ΘG k (21a)s.t. | b f | = 1 , ∀ f ∈ F . (21b)This problem can be effectively solved by SDR. B. Analog BF design
Orthogonal match pursuit (OMP) method is widelyadopted to design the analog beamformer [8]. If theBS adopts the fully digital BF structure, the opti-mal digital BF under the zero-forcing (ZF) schemeis given by F opt = ˜ H † diag ( p γ σ , . . . , p γ K σ K ) ,where ˜ H = (cid:2) ( h H ΘG ) T , . . . , ( h HK ΘG ) T (cid:3) T and † de-notes the pseudo-inverse. Define the overlapping coef-ficient as µ , and denote the codebook as A =[ a B ( ψ , φ ) , . . . , a B ( ψ , φ µN z ) , . . . , a B ( ψ µN y , φ µN z )] , where N y and N z denote the horizontal and vertical length, ψ i = π ( i − µN y , i = 1 , , . . . , µN y and φ j = π ( j − µN z , j =1 , , . . . , µN z , respectively. Then, we can use a selection ma- trix T ∈ R µ N y N z × N to select proper columns. Specifically,the analog BF problem can be formulated as T ∗ = arg min T k F opt − A t TF BB k F (22a)s.t. (cid:13)(cid:13) diag (cid:0) TT H (cid:1)(cid:13)(cid:13) = N, (22b) A t = I t ⊙ A , t ∈ N , (22c)where I t is a M × zero-vector with the entry from ( t − D +1 to tD being one; ⊙ denotes the Hadamard product. Since thestructure of analog BF is sub-connected, we use I t to modifythe codebook. Then, the OMP method can be applied to obtainthe optimal T ∗ . The analog BF can be recovered, i.e., V = A t T ∗ . C. Digital BF design
After obtaining the RIS phase shifts and the analog beam-former, we need to obtain the optimal digital BF vector bysolving following problem, min W D K P k =1 k w k k s.t. (7b) . (23)Note that it is the conventional power minimization problemin the multi-input single-output (MISO) downlink system,which can be effectively solved by SOCP method.V. S IMULATION R ESULTS
We consider a × UPA structure at the BS with atotal of M = 36 antennas and N = 6 RF chains locatedat (0 m, 0 m). The RIS is located at ( d RIS m, 10 m)and equipped with F × F unit cells where F = 6 and F can vary. Users are uniformly and randomly distributedin a circle centered at (100 m, 0 m) with radius 5 m.As for the channel, we set N cl = N cl = 2 clusters, N ray = N ray = 5 rays; the complex gain α il and β il followthe complex Gaussian distribution CN (0 , − . P L ( d ) ) , where P L ( d ) = ϕ a + 10 ϕ b log ( d ) + ξ ( dB ) with ξ ∼ N (cid:0) , σ (cid:1) , ϕ a = 72 . , ϕ b = 2 . , σ = 8 . dB [15]. The auxiliary vari-ables { t k,j } are initialized following CN (0 , . The penaltyfactor is initialized by ρ = 10 − . Other system parameters areset as follows unless specified otherwise later: K = 3 , F =6 , d RIS = 50 , c = 0 . , ǫ = 10 − , ǫ = 10 − , γ k = 10 dB, σ k = − dBm, ∀ k ∈ K . All simulation curves are averagedover independent channel realizations. A. Convergence Performance
We show the stopping indicator (20) of the penalty-basedalgorithm in Fig. 2 and the average convergence of the penalty-based algorithm in Fig. 3 . These curves are plotted withthe average plus and minus the standard deviation. It isobserved that the stopping indicator can always meet thepredefined accuracy − after about 110 outer layer iterationsin Fig. 2. Thus, solutions obtained by the Algorithm
Iteration Number of Outer Layer -8 -7 -6 -5 -4 -3 -2 -1 S t opp i ng I nd i c a t o r , Fig. 2:
Stopping indicator of Penalty-based Algorithm
Total Iteration Number -50510152025 T r an s m i t P o w e r / d B m
195 200 2052121.5
Fig. 3:
Convergence of Penalty-based Algorithm
B. Performance Comparison with other schemes
To demonstrate the efficiency of the proposed algorithmsand to reveal some design insights, we compare the perfor-mance of the following algorithms: • Penalty-Manifold joint design with hybrid BF structure(Penalty-Manifold HB): This is the proposed
Algorithm • Penalty-Manifold joint design with fully digital BF struc-ture (Penalty-Manifold FD): This is the proposed
Algo-rithm D = 1 . • Penalty-Manifold joint design with random Θ (Random Θ ): The phase shifts at the RIS are randomly selected tobe feasible values. Then the hybrid beamforming matrices { W , V } at the BS are obtained by using the penalty-manifold joint algorithm as in Algorithm
1, where theupdate of Θ is skipped. This is to find out the significanceof optimizing the phase shifts at the RIS. • Penalty-Manifold joint design with SDR Θ (SDR Θ ):The phase shifts at the RIS are designed for maximizingthe effective channel of the worse-cast user by using theSDR approach based on (21). Then the hybrid beamform-ing matrices { W , V } at the BS are obtained by using thepenalty-manifold joint algorithm as in Algorithm
1, wherethe udpate of Θ is skipped. This is again to find out thesignificance of optimizing the phase shifts at the RIS. • BCD-SDR joint design (BCD-SDR): The conventionalBCD method in conjunction with the SDR method, asmentioned in the end of Section II-C. • Individual design: the proposed individual design whereRIS phase shifts, analog BF, and digital BF are optimizedsequentially in Section IV.
SINR Targets/dB T r an s m i t P o w e r / d B m Random Individual designSDR BCD-SDRPenalty-based HBPenalty-based FD
Fig. 4:
Transmit power versus SINR targets
12 24 36 48 60
RIS element number, F T r an s m i t po w e r / d B m RandomIndividual designBCD-SDRPenalty-Manifold HBPenalty-Manifold FD
Fig. 5:
Transmit power versus the element number of RIS
Fig. 4 illustrates the transmit power versus SINR targets.We first observe that the Penalty-Manifold joint design out-performs the start-of-the-art BCD-SDR joint design. Second,the Penalty-Manifold joint design with random Θ performs theworst among all the considered schemes. By simply changingthe random Θ to the SDR Θ (while keeping the joint designof { W , V } unchanged), the transmit power consumption canbe reduced by 5 dB. If Θ is involved in the Penalty-Manifoldjoint design, another 10dB power reduction can be obtained.These observations indicate that the design of RIS phase shiftsplays the crucial role for performance optimization. Third, theindividual design is about 2dB worse than the joint design withSDR Θ . This suggests that, when the RIS response matrixis designed individually for maximizing the effective channelgain of the worse-case user, further optimizing the hybrid BFat the BS can only bring marginal improvement. Last but notleast, the power consumed by Penalty-Manifold HB is about2.5dB higher than the power consumed by Penalty-ManifoldFD. Note that the hybrid BF has much lower hardware costsince it only employs N = 6 RF chains, while the fully digitalBF has M = 36 RF chains.The influence of the RIS element number, F , is consideredin Fig. 5. When F increases from 12 to 60, the transmit powerdrops about 15dB. Thus, we conclude that the RIS can greatlyreduce the transmit power by installing a large number ofelements.Fig. 6 illustrates the influence of the RIS location. It isseen that as the RIS horizontal distance d RIS increases, thetransmit power increases firstly, and reaches the peak at 50 m,then decreases. This can be explained that the received powerthrough the reflection of the RIS in the far field is proportional
10 20 30 40 50 60 70 80 90
Horizontal distance, d
RIS T r an s m i t P o w e r / d B m RandomIndividual designBCD-SDRPenalty-Manifold HBPenalty-Manifold FD
Fig. 6:
Transmit power versus horizontal distance of RIS to d d , where d and d denote the distance between the BS-RIS and RIS-user, respectively. It is found that the RIS canbe located near the BS or users to save energy.VI. C ONCLUSION
This paper proposed a two layer penalty-based algorithm tosolve the RIS-aided hybrid beamforming optimization prob-lem in mmWave systems. In the inner layer, we alternatelyoptimize the digital beamforming and analog beamforming atthe BS and the response coefficient at the RIS. The outerlayer updates the penalty factor to obtain a high precisionsolution. A low-complexity individual optimization method isalso proposed. Extensive simulation results demonstrate thatthe proposed algorithm has a good performance and the RIScan significantly improve the energy efficiency.R
EFERENCES[1] T. Cui, D. Smith, and R. Liu,
Metamaterials: Theory, Design, andApplications . Springer, 2010.[2] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wirelessnetwork via joint active and passive beamforming,”
IEEE Trans. WirelessCommun. , vol. 18, no. 11, pp. 5394–5409, 2019.[3] H. Guo, Y.-C. Liang, J. Chen, and E. G. Larsson, “Weighted sum-rate optimization for intelligent reflecting surface enhanced wirelessnetworks,” 2019. [Online]. Available: https://arxiv.org/abs/1905.07920[4] B. Di, H. Zhang, L. Song, Y. Li, Z. Han, and H. V. Poor, “Hybridbeamforming for reconfigurable intelligent surface based multi-usercommunications: Achievable rates with limited discrete phase shifts,”2019. [Online]. Available: https://arxiv.org/abs/1910.14328[5] A. F. Molisch, V. V. Ratnam, S. Han, Z. Li, S. L. H. Nguyen, L. Li,and K. Haneda, “Hybrid beamforming for massive MIMO: A survey,”
IEEE Commun. Mag. , vol. 55, no. 9, pp. 134–141, 2017.[6] Y. Xiu, J. Zhao, W. Sun, M. D. Renzo, G. Gui, Z. Zhang, and N. Wei,“Reconfigurable intelligent surfaces aided mmwave noma: Joint powerallocation,phase shifts, and hybrid beamforming optimization,” 2020.[Online]. Available: https://arxiv.org/abs/2007.05873[7] K. Ying, Z. Gao, S. Lyu, Y. Wu, H. Wang, and M. Alouini, “GMD-based hybrid beamforming for large reconfigurable intelligent surfaceassisted millimeter-wave massive MIMO,”
IEEE Access , vol. 8, pp.19 530–19 539, 2020.[8] O. E. Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. W. Heath,“Spatially sparse precoding in millimeter wave MIMO systems,”
IEEETrans. Wireless Commun. , vol. 13, no. 3, pp. 1499–1513, 2014.[9] Q. Wu and R. Zhang, “Joint active and passive beamformingoptimization for intelligent reflecting surface assisted SWIPT under QoSconstraints,” 2019. [Online]. Available: https://arxiv.org/abs/1910.06220[10] P.-A. Absil, R. Mahony, and R. Sepulchre,
Optimization algorithms onmatrix manifolds . Princeton University Press, 2009.[11] X. Yu, J.-C. Shen, J. Zhang, and K. B. Letaief, “Alternating minimizationalgorithms for hybrid precoding in millimeter wave MIMO systems,”
IEEE J. Sel. Topics Signal Process. , vol. 10, no. 3, pp. 485–500, 2016.[12] X. Yu, D. Xu, and R. Schober, “MISO wireless communicationsystems via intelligent reflecting surfaces,” 2019. [Online]. Available:https://arxiv.org/abs/1904.12199 [13] H. Guo, Y.-C. Liang, J. Chen, and E. G. Larsson, “Weighted sum-rate maximization for reconfigurable intelligent surface aided wirelessnetworks,”
IEEE Trans. Wireless Commun. , vol. 19, no. 5, pp. 3064–3076, 2020.[14] A. Wiesel, Y. C. Eldar, and S. Shamai, “Linear precoding via conicoptimization for fixed MIMO receivers,”
IEEE Trans. Signal Process. ,vol. 54, no. 1, pp. 161–176, 2006.[15] M. R. Akdeniz, Y. Liu, M. K. Samimi, S. Sun, S. Rangan, T. S.Rappaport, and E. Erkip, “Millimeter wave channel modeling andcellular capacity evaluation,”