Beamforming Design for Multiuser Transmission Through Reconfigurable Intelligent Surface
Zhaohui Yang, Wei Xu, Chongwen Huang, Jianfeng Shi, Mohammad Shikh-Bahaei
aa r X i v : . [ ee ss . SP ] S e p Beamforming Design for MultiuserTransmission Through ReconfigurableIntelligent Surface
Zhaohui Yang, Wei Xu,
Senior Member, IEEE , Chongwen Huang, Jianfeng Shi,and Mohammad Shikh-Bahaei,
Senior Member, IEEE
Abstract
This paper investigates the problem of resource allocation for multiuser communication networkswith a reconfigurable intelligent surface (RIS)-assisted wireless transmitter. In this network, the sumtransmit power of the network is minimized by controlling the phase beamforming of the RIS andtransmit power of the BS. This problem is posed as a joint optimization problem of transmit power andRIS control, whose goal is to minimize the sum transmit power under signal-to-interference-plus-noiseratio (SINR) constraints of the users. To solve this problem, a dual method is proposed, where the dualproblem is obtained as a semidefinite programming problem. After solving the dual problem, the phasebeamforming of the RIS is obtained in the closed form, while the optimal transmit power is obtainedby using the standard interference function. Simulation results show that the proposed scheme canreduce up to 94% and 27% sum transmit power compared to the maximum ratio transmission (MRT)beamforming and zero-forcing (ZF) beamforming techniques, respectively.
Index Terms
Resource allocation, power minimization, reconfigurable intelligent surface, phase shift optimization,semidefinite programming, beamforming design.
Z. Yang and M. Shikh-Bahaei are with the Centre for Telecommunications Research, Department of Engineering, King’sCollege London, WC2R 2LS, UK. (Emails: [email protected], [email protected])W. Xu is with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China.(Email: [email protected].)C. Huang is with the Singapore University of Technology and Design, 487372 Singapore. (Email: chong-wen [email protected])J. Shi is with School of Electronic and Information Engineering, Nanjing University of Information Science and Technology,Nanjing 210096, China. (Email: [email protected])
I. I
NTRODUCTION
Driven by the rapid development of advanced multimedia applications, it is urgent for the next-generation wireless network to support high spectral efficiency and massive connectivity [1]. Dueto the demand of high data rate and serving a massive number of users, energy consumptionhas become a challenging problem in the design of the future wireless network [2]–[6].Recently, reconfigurable intelligent surface (RIS)-assisted wireless communication has beenproposed as a potential solution for enhancing the energy efficiency of wireless networks [7]–[16].RIS is a new paradigm that can flexibly manipulate electromagnetic (EM) waves. Researches ofRIS-assisted wireless communications mainly follow into two aspects: RIS as a passive reflectorand RIS as an active transceiver.On one hand, RIS can serve as a passive reflector. An RIS is a meta-surface equipped withlow-cost and passive elements that can be programmed to turn wireless channels into a partiallydeterministic space. In RIS-assisted wireless communication systems, a base station (BS) sendscontrol signals to an RIS controller so as to optimize the properties of incident waves and improvethe communication quality of users [17]–[19]. The RIS acting as a reflector does not perform anydecoding or digitalization operation. Hence, if properly deployed, the RIS promises much lowerenergy consumption than traditional amplify-and-forward (AF) relays [20]–[27]. A number of ex-isting works such as in [9], [28]–[36] have studied to optimize the deployment of RISs in wirelessnetworks. In [28], the downlink sum-rate of an RIS-assisted wireless communication system wascharacterized. Asymptotic analysis of uplink data rate in an RIS-based large antenna-array systemwas presented in [29]. Considering energy harvesting, an RIS was invoked for enhancing thesum-rate performance of a system with simultaneous wireless information and power transfer[9]. Instead of considering the availability of instantaneous channel state information (CSI),the authors in [30] proposed a two-time-scale transmission protocol to maximize the averageachievable sum-rate for an RIS-aided multiuser system under a general correlated Rician channelmodel. Taking the secrecy into consideration, the work in [31] investigated the problem of secrecyrate maximization of an RIS-assisted multi-antenna system. Further considering imperfect CSI,the physical layer security was enhanced by an RIS in a wireless channel [32]. Beyond theabove studies, the use of RISs for enhanced wireless energy efficiency has been studied in [37].In [37], authors proposed a new approach to maximize the energy efficiency of a multiusermultiple-input single-output (MISO) system by jointly controlling the transmit power of the
BS and the phase shifts of the RIS. The RIS-assisted simultaneous wireless information andpower transfer (SWIPT) system was studied in [33], where the sum transmit power at theBS was minimized via jointly optimizing its transmit precoders and the reflect phase shiftsat all RISs, subject to the quality-of-service (QoS) constraints at all users. The authors in [34]studied the resource allocation design for secure communication in RIS-assisted multiuser MISOcommunication systems by using artificial noise (AN). Considering both security and SWIPT,the energy efficiency maximization problem was studied in [35] for the secure RIS-aided SWIPT.For spectrum sensing, an RIS-assisted cognitive radio system was investigated in [36], where anRIS is deployed to assist in the spectrum sharing between a primary user link and a secondaryuser link.On the other hand, the RIS-assisted wireless transmitter [38]–[43] can directly perform mod-ulation on the EM carrier signals, without the need for conventional radio-frequency (RF)chains, which can be used for holographic multiuser multiple-input multiple-output (MIMO)technologies. In [39], authors investigated the RIS-based quadrature phase shift keying (QPSK)transmission over wireless channels. The RIS-based 8-phase shift keying (8PSK) was furtherstudied in [40]. The feasibility of using RIS for MIMO with higher-order modulations wasstudied in [43], which presented an analytical modelling of the RIS-based system. However,the above works [38]–[43] only considered the RIS-assisted wireless transmitter for single-usercases. In this paper, we investigate the beamforming design for multiuser transmission with theRIS-assisted wireless transmitter. The main contributions of this paper include: • We consider a downlink wireless communication system with one RIS-assisted wirelesstransmitter and multiple users. To minimize the sum transmit power of the BS, we jointlyoptimize phase shifts of the RIS and the multiuser power allocation at the BS. We formulatean optimization problem with the objective of minimizing the sum transmit power underindividual user constraints in terms of signal-to-interference-plus-noise ratio (SINR) andunit-modulus constraint of the RIS phase shifts. • To minimize the sum transmit power of the BS, a dual method is proposed. By using thedual method, the dual problem of the sum transmit power minimization problem is firstobtained. Then, phase shifts of the RIS can be obtained in the closed form. For the transmitpower of the BS, an iterative power control scheme based on the standard interferencefunction is proposed to obtain the optimal power control. • We consider both maximum ratio transmission (MRT) beamforming and zero-forcing (ZF) beamforming techniques when solving the sum transmit power minimization problem. Thepower scaling law performance of the multiuser communication is evaluated for the RIS-assisted wireless transmitter. Simulation results show that the proposed method saves up to94% and 27% sum transmit power compared to the conventional MRT and ZF schemes,respectively.The rest of this paper is organized as follows. System model and problem formulation aredescribed in Section II. Section III provides the algorithm design. Simulation results are presentedin Section IV. Conclusions are drawn in Section V.Notations: In this paper, the imaginary unit of a complex number is denoted by j = √− .Matrices and vectors are denoted by boldface capital and lower-case letters, respectively. Matrixdiag ( x , · · · , x N ) denotes a diagonal matrix whose diagonal components are x , · · · , x N . Thereal part of a complex number x is denoted by R ( x ) . X (cid:23) indicates that X is a positivesemidefinite matrix. x ∗ , x T , and x H respectively denote the conjugate, transpose, and conjugatetranspose of vector x . [ x ] n and [ X ] kn denote the n -th and ( k, n ) -th elements of the respectivevector x and matrix X . | x | stands for the module of a complex number x , while k x k denotesthe ℓ -norm of vector x . The identity matrix is denoted by I , while an all-one vector is denotedby . The distribution of a circularly symmetric complex Gaussian variable with mean x andcovariance σ is denoted by CN ( x, σ ) . The expectation operation is denoted by E . X † denotesthe Moore-Penrose pseudoinverse of matrix X . The optimal value of an optimization variable X is denoted by X ⋆ .II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
Consider an RIS-assisted multiuser wireless communication system that consists of one BSwith a single antenna, an RIS, and a set K of K users. The RIS consists of M rows and N columns of RIS units, while each user is equipped with one antenna. The RIS unit in the k -throw and n -th column is denoted by U kn . A. RIS-Assisted Modulation
RIS is indeed a programmable metasurface composed of sub-wavelength units, which canmanipulate EM waves. All RIS units are regularly arranged in a two-dimensional structure, asshown in Fig. 1. The length and width of each RIS unit are denoted by a and b , respectively. The control signal of each RIS unit can change the electrical parameter of the tunable component,such as the phase [43].Let ¯ E and ˆ E denote the incident and reflecting EM waves on RIS unit U kn , respectively.Following the definition of reflection phase of an EM, say φ kn as the reflection phase of RISunit U kn , it follows: ˆ E = e jφ kn ¯ E. (1)In particular, if the incident EM wave ¯ E is a single-tone EM wave (i.e., a carrier signal) withfrequency f c and amplitude A c , equation (1) can be further expressed by: ˆ E = A c e j (2 πf c t + φ kn ) . (2)From (2), it can be observed that the adjustable phase φ kn can achieve a phase modulation onthe carrier signal, which is referred to as the RIS-assisted modulation. B. RIS-Assisted Multiuser Communication
An RIS-assisted multiuser wireless communication system is illustrated in Fig. 1 [39], [43].The reflection phase of each RIS unit is controlled by the digital baseband through the digital-to-analog converters (DACs), i.e., each RIS unit is controlled by one dedicated DAC. Before thecarrier signal is reflected to each row of the RIS units, one narrow-band power amplifier (PA) isused to control the power of the carrier signal and the energy flux density on each RIS unit inthe k -th row is denoted as D k . The number of PAs equal to the number of served users. Notethat in the considered system, the information is actually embedded in the phase control signalsof each row of the RIS. As shown in Fig. 1, RIS is not connected with the digital basebanddirectly and RIS is used to reflect the baseband signal.Compared to the traditional hybrid mmWave system, the differences of the current RIS-transmitter setup include:1) Transmit antenna: There are multiple transmit antennas in the traditional hybrid mmWavesystem, while only one transmit antenna is needed for the RIS-assisted wireless transmittersystem in this paper. In system model, the authors in [44] considered RIS as a passive relay, while RIS is used as a passive transmitter in thispaper. Due to this difference, the mathematical signal model in this paper is different from [44].
RIS DA C (cid:258) (cid:258) Carrier signal Digital baseband (cid:258)(cid:258)(cid:258) (cid:258) PA PA PA User 1User 2
User K BS DA C DA C RIS (cid:258) (cid:258)(cid:258)(cid:258)
RIS unit U kn a Single-tone EM wave Modulated EM wave b M rows and N columns Control signals for phase values of all RIS units (cid:258) g KK g g Fig. 1. An RIS-assisted multiuser wireless communication system.
2) Modulation: For the traditional hybrid mmWave system, the signals are modulated inmultiple antennas before the baseband precoder. The RIS-assisted wireless transmitter candirectly perform modulation on the EM carrier signals, i.e., continuous phase modulationscheme is adopted.3) RF chain: For the traditional hybrid mmWave system, multiple RF chains are needed. Incontrast, the proposed RIS-assisted wireless transmitter is considered as an RF chain-freetransmitter.
To construct the multiple data streams for multiple users, the N units in the k -th row of theRIS are allocated to user k , i.e., the phases of RIS units U k , U k , · · · , U kN are used to modulatethe transmitted signal for user k . Let the transmitted signal s k for user k be: s k = e jϕ k . (3)Through changing the value of ϕ k , signal s k can be modulated by phase shift-keying (PSK). In[43], it was shown that quadrature amplitude modulation (QAM) can also by achieved by equation(3) of proper designs. As a result, the transmitted signal by the RIS units U k , U k , · · · , U kN canbe presented by: [ e jφ k , e jφ k , · · · , e jφ kN ] T = [ e jϕ k , e jϕ k , · · · , e jϕ kN ] T e jϕ k = [ e jϕ k , e jϕ k , · · · , e jϕ kN ] T s k , (4)where ϕ kn = φ kn − ϕ k . For notational simplicity, we introduce: θ kn , e jϕ kn , (5)and equation (4) can be rewritten as [ e jϕ k , e jϕ k , · · · , e jϕ kN ] T s k = θ k s k , (6)where θ k = [ θ k , · · · , θ kN ] T is the phase beamformer of user k which can be adjusted by theRIS.Assume that the carrier signal with frequency f c is a uniform plane wave. In consequence,the transmitted signal at the BS is x = [ √ p s θ ; √ p s θ ; · · · ; √ p K s K θ K ] , (7)where p k = ( abD k ) is the transmit power of the BS for user k . Equation (7) is a mathematicalsignal model.Since the source is close to the RIS, as shown in Fig. 1, the channel from the source to theRIS can be precisely measured, which can be regarded as a constant as in [43]. The receivedsignal at user k can be given by: y k = g Hk x + n k = K X i =1 √ p i g Hki θ i s i + n k , (8) In this paper, the RIS unit allocation for each user is assumed to be fixed for the convenience of deployment, i.e., K = M . where g k is the channel gain from all RIS elements to user k , g ki is the channel gain from N RIS elements in the i -th row to user k , and n k ∼ CN (0 , σ ) is the additive white Gaussiannoise. Assume that the signals are only reflected by the RIS once. Since both the phase andtransmit power can be changed, we can construct that the input term √ p i s i follows the Gaussiandistribution, and the channel capacity can be achieved. Based on this consideration on (8), theSINR at user k is γ k = p k (cid:12)(cid:12) g Hkk θ k (cid:12)(cid:12) P Ki =1 ,i = k p i | g Hki θ i | + σ . (9)Note that the required CSI for the proposed RIS-assisted wireless transmitter only includeschannel gains from the BS (i.e., RIS) to users. In contrast, for the conventional RIS-assistedwireless transmission where the RIS acts as a passive reflector, the required CSI includes channelgains from the BS to the RIS, the RIS to users, and the BS to users. As a result, comparedto the conventional RIS-assisted wireless transmission, one key novelty of the proposed RIS-assisted wireless transmitter is that the required amount of CSI is smaller. There are alreadymany significant methods that are proposed in existing works for obtaining CSI in RIS-basedcommunication systems. For example, the works [45] and [46] presented compressive sensingand deep learning approaches for recovering the involved channels and designing the RIS phasematrix. Based on the parallel factor framework, the authors in [47] proposed an alternating leastsquare method, which continuously estimates the all channels without too high complexity. C. Problem Formulation
Given the considered system model, our objective is to jointly optimize the phase beamforming θ k and transmit power p k so as to minimize the sum transmit power under individual minimumSINR requirements. Mathematically, the problem for the RIS-assisted multiuser transmission canbe formulated as: min θ , p K X k =1 p k (10)s.t. p k (cid:12)(cid:12) g Hkk θ k (cid:12)(cid:12) P Ki =1 ,i = k p i | g Hki θ i | + σ ≥ Γ k , ∀ k ∈ K , (10a) | θ kn | = 1 , ∀ k ∈ K , n ∈ N , (10b)where θ = [ θ , · · · , θ N , · · · , θ KN ] T , p = [ p , · · · , p K ] T , N = { , · · · , N } , and Γ k is theminimum SINR requirement of user k . The minimum SINR constraints for all users are given in (10a), and (10b) presents the unit-modulus constraints. Different from the conventional RFbeamforming design, the phase beamforming problem (10) introduces the unique unit-modulusconstraints (10b). Due to the nonconvex constraints in (10b), the problem (10) is nonconvex.III. A LGORITHM D ESIGN
To solve the nonconvex problem in (10), the dual method is first applied, where the dualproblem of (10) is always a convex problem, which can be effectively solved. For comparisons,two conventional techniques, MRT beamforming and ZF beamforming, are also provided tosolve the problem (10).
A. Dual Method
To rewrite sum power minimization problem (10) in a simplifier manner, we introduce w k = √ p k θ k . Replacing θ k with w k , the problem (10) is equivalent to: min w , p K X k =1 p k (11)s.t. (cid:12)(cid:12) g Hkk w k (cid:12)(cid:12) P Ki =1 ,i = k | g Hki w i | + σ ≥ Γ k , ∀ k ∈ K , (11a) [ w k w Hk ] nn = p k , ∀ k ∈ K , n ∈ N , (11b)where w = [ w , · · · , w N , · · · , w KN ] T .Denote Γ = [Γ , · · · , Γ K ] T and we define the following time-sharing condition. Definition 1:
Let ( w ( a ) , p ( a ) ) and ( w ( b ) , p ( b ) ) be the optimal solutions to the optimizationproblem (11) with Γ = Γ ( a ) and Γ = Γ ( b ) . An optimization problem of the form (11) is said tosatisfy the time-sharing condition if for any Γ ( a ) , Γ ( b ) and any κ ∈ [0 , , there always exists afeasible solution ( w ( c ) , p ( c ) ) such that (cid:12)(cid:12)(cid:12) g Hkk w ( c ) k (cid:12)(cid:12)(cid:12) P Ki =1 ,i = k (cid:12)(cid:12)(cid:12) g Hki w ( c ) i (cid:12)(cid:12)(cid:12) + σ ≥ κ Γ ( a ) k + (1 − κ )Γ ( b ) k , ∀ k ∈ K , w ( c ) k ( w ( c ) k ) H ] nn = p ( c ) k , ∀ k ∈ K , n ∈ N , and K X k =1 p ( c ) k ≥ κ K X k =1 p ( a ) k + (1 − κ ) K X k =1 p ( b ) k . The time-sharing condition has the following intuitive interpretation. Consider the maximumvalue of the optimization problem (11) as a function of the constraint Γ . Clearly, a smaller Γ k implies a more relaxed constraint. So, roughly speaking, the maximum value is an increasingfunction of Γ . The time-sharing condition implies that the maximum value of the optimizationproblem is a concave function of Γ . To show the gap between problem (11) and its dual problem,we provide the following lemma. Lemma 1:
The duality gap for multiuser RIS-assisted optimization (11) always tends to zeroas the number of users K or the number of RIS unit elements N goes to infinity, regardless ofwhether the original problem is convex. Proof:
This result holds by directly applying Theorems 1 and 2 in [48]. According to [48,Theorem 2], the time-sharing property holds for all optimization problems with infinite channelgains, i.e., K → ∞ or N → ∞ . Based on [48, Theorem 1], the optimization problem has a zeroduality gap If it satisfies the time-sharing property. (cid:3) According to Lemma 1, the near optimal solution of problem (11) can be obtained by solvingits dual problem if the number of users is high or the number of RIS units is large. In practice,for example, since RIS unit is low-cost, a large number of RIS units can be deployed at the BS.Consequently, the sum power minimization problem (11) can be effectively solved via its dualproblem.
Theorem 1:
The dual problem of problem (11) is: max q , α K X k =1 α k σ (12)s.t. N X n =1 q kn ≤ , ∀ k ∈ K , (12a) Q k + K X i =1 ,i = k α i g ik g Hik (cid:23) α k Γ k g kk g Hkk , ∀ k ∈ K , (12b) α k ≥ , ∀ k ∈ K , (12c)where q = [ q , · · · , q N , · · · , q KN ] T , α = [ α , · · · , α K ] T , q kn and α k are the Lagrange multiplierscorresponding to power constraints (11b) and SINR constraints (11a), respectively, and Q k isdefined in (14). Proof:
The Lagrangian function for the optimization problem (11) is given by: L ( w , p , q , α ) = K X k =1 p k + K X k =1 N X n =1 q kn ([ w k w Hk ] n,n − p k )+ − K X k =1 α k k (cid:12)(cid:12) g Hkk w k (cid:12)(cid:12) − K X i =1 ,i = k (cid:12)(cid:12) g Hki w i (cid:12)(cid:12) − σ ! . (13)Denoting Q k = diag ( q k , · · · , q kN ) , (14)we can rewrite Lagrangian function (13) by: L ( w , p , q , α ) = K X k =1 α k σ − K X k =1 p k N X n =1 q kn − ! + K X k =1 w Hk Q k + K X i =1 ,i = k α i g ik g Hik − α k Γ k g kk g Hkk ! w k . (15)The dual objective can be given by [49]: D ( q , α ) = min w , p L ( w , p , q , α ) . (16)Since p k must be positive and there are no constraints on the beamforming w k , we have D ( q , α ) = −∞ if P Nn =1 q kn ≥ or Q k + P Ki =1 ,i = k α i g ik g Hik − α k Γ k g kk g Hkk is not positive semidef-inite.Due to the fact that q and α should be selected that the dual objective is finite. As a result,constraints that P Nn =1 q kn ≤ and Q k + P Ki =1 ,i = k α i g ik g Hik − α k Γ k g kk g Hkk is positive semidefiniteshould be satisfied. Formally, the Lagrangian dual problem can be stated as (12). (cid:3)
Since the objective is linear and the constraints are either linear or linear matrix inequalities,dual problem (12) is a semidefinite programming (SDP) problem, which can be solved by usingthe standard CVX toolbox [50], [51]. Having obtained the dual variables by solving dual problem(12), it remains to obtain the optimal beamforming w and transmit power p . To find the optimal w , we calculate the gradient of the Lagrangian function for problem (11) with respect to w andset it to zero: ∂ L ( w , p , q , α ) ∂ w k =2 Q k + K X i =1 ,i = k α i g ik g Hik − α k Γ k g kk g Hkk ! w k = . (17)Based on (17), we have: Q k + K X i =1 α i g ik g Hik ! w k = 1 + Γ k Γ k α k g kk g Hkk w k . (18) Solving equation (18) yields: w k = Q k + K X i =1 α i g ik g Hik ! † k Γ k α k g kk g Hkk w k . (19)Since g Hkk w k is a scalar, the optimal w has the following expression: w ⋆k = p p ⋆k θ ⋆k , (20)where θ ⋆k = √ N (cid:16) Q k + P Ki =1 α i g ik g Hik (cid:17) † g kk (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) Q k + P Ki =1 α i g ik g Hik (cid:17) † g kk (cid:13)(cid:13)(cid:13)(cid:13) . (21)Note that the ℓ -norm of phase vector θ ⋆k is k θ ⋆k k = √ N since the module of each element inphase vector is unit. To obtain the value of power p ⋆k , we find that the minimum SINR constraints(11a) must hold with equality for all users at the optimum solution. Substituting (20) into SINRconstraints (11a) and setting them with equality, we can obtain: (cid:12)(cid:12) g Hkk θ ⋆k (cid:12)(cid:12) Γ k p ⋆k = K X i =1 ,i = k (cid:12)(cid:12) g Hki θ ⋆i (cid:12)(cid:12) p ⋆i + σ . (22)By using the concept of standard interference function, equation (22) can be written in thefollowing form: p ⋆ = f ( p ⋆ ) , (23)where f = [ f , · · · , f K ] T and f k ( p ⋆ ) = Γ k | g Hkk θ ⋆k | K X i =1 ,i = k (cid:12)(cid:12) g Hki θ ⋆i (cid:12)(cid:12) p ⋆i + σ ! . (24)By checking the positivity, monotonicity, and scalability properties, we can prove that function f ( p ⋆ ) is always a standard interference function [52], which allows us to use the iterative powercontrol scheme to solve equation (23). The iterative power control scheme is given by: p ( t ) = f ( p ( t − ) , (25)where the superscript ( t ) means the value of the variable in the t -th iteration. According to [52,Theorem 2], the iterative power control scheme (25) always converges to the unique fixed point p ⋆ if (23) is feasible.The dual method for solving problem (12) is summarized in Algorithm 1. Since the dualitygap is zero for the large number of RIS units, the solution ( w ⋆k , p ⋆k ) obtained by the dual methodin Algorithm 1 is the optimal solution of the original problem in (11). Algorithm 1
Dual Method for Problem (11) Solve the dual problem (12) by using SDP. Calculate the optimal phase beamforming vector θ ⋆k according to (21). Initialize p (0) = , iteration number t = 1 , and set the accuracy ǫ . repeat for k = 1 : K do Update p ( t ) k = Γ k | g Hkk θ ⋆k | (cid:16)P Ki =1 ,i = k (cid:12)(cid:12) g Hki θ ⋆i (cid:12)(cid:12) p ( t − i + σ (cid:17) . end for Set t = t + 1 and p ( t ) = [ p ( t )1 , · · · , p ( t ) K ] T . until k p ( t ) − f ( p ( t ) ) k < ǫ . Output w ⋆k = q p ( t ) k θ ⋆k , p ⋆k = q p ( t ) k , ∀ k ∈ K . B. SDR Approach
In this section, we apply the semidefinite relaxation (SDR) technique to solve problem (11).We introduce matrix W k = w k w Hk , which needs to satisfy W k (cid:23) and the rank of W k is one.As a result, problem (11) is equivalent to min W , p K X k =1 p k (26)s.t. g Hkk W k g kk ≥ Γ k K X i =1 ,i = k g Hki W i g ki + σ ! , ∀ k ∈ K , (26a) [ W k ] nn = p k , ∀ k ∈ K , n ∈ N , (26b) rank ( W k ) = 1 , ∀ k ∈ K , (26c) W k (cid:23) , ∀ k ∈ K , (26d)where W = [ W , W , · · · , W K ] . However. (26) is still a nonconvex problem due to thenonconvex rank-one constraint (26c). We apply SDR to relax this rank-one constraint and problem (26) can be represented as: min W , p K X k =1 p k (27)s.t. g Hkk W k g kk ≥ Γ k K X i =1 ,i = k g Hki W i g ki + σ ! , ∀ k ∈ K , (27a) [ W k ] nn = p k , ∀ k ∈ K , n ∈ N , (27b) W k (cid:23) , ∀ k ∈ K . (27c)Problem (27) is a standard convex problem, which can be solved by existing toolboxes such asCVX. Since the rank-one constraint is relaxed, the solution of problem (27) is not necessarily tothe optimal solution of the original problem (27), which implies that the optimal objective valueof problem (27) only serves an upper bound of problem (26). Thus, we can use the method in[53] to construct a rank-one solution from the obtained higher-rank solution to problem (26).According to [54], the SDR approach followed by a sufficiently large number of randomiza-tions guarantees at least a π -approximation of the optimal objective value of problem (26). Sinceconstructing a feasible rank-one solution requires a lot of randomizations, the complexity of theSDR approach is higher than the proposed dual method. Besides, the proposed dual method canbe guaranteed to be the optimal for large number of users or RIS unit elements. C. MRT and ZF Beamforming
To solve the sum power minimization problem (10) in the conventional ways, we provide theMRT and ZF beamforming methods. Note that the optimal power control can be obtained byusing the iterative power control scheme (25), this section only provides the method of optimizingthe phase beamforming θ k .
1) MRT beamforming:
In MRT beamforming, the beamforming is designed such that thereceived signal at each user is maximized. Mathematically, the MRT beamforming problem isformulated as: max θ k | g Hkk θ k | (28)s.t. | θ kn | = 1 , ∀ n ∈ N . (28a)To solve problem (28), the MRT beamforming can be expressed as: θ ⋆kn = [ g kk ] n | [ g kk ] n | , ∀ n ∈ N . (29) From (29), we can see that the optimal phase vector θ ⋆k should be tuned such that the signalthat passes through all RIS units is aligned to be a signal vector with the equal phase at eachelement, i.e., θ ⋆kn [ g kk ] n is a real number for all n .
2) ZF bemaforming:
The idea of ZF beamforming is to invert the channel matrix at thetransmitter in order to create orthogonal channels between the BS and users. The ZF beamformingproblem can be formulated as: max θ k | g Hkk θ k | (30)s.t. g Hik θ k = 0 , ∀ i = k, (30a) | θ kn | = 1 , ∀ n ∈ N . (30b)Due to the nonconvex unit-modulus constraints (30b), the conventional ZF method cannot bedirectly applied. With new nonconvex constraints (30b), it is of importance to investigate thefeasibility of (30). For the feasibility of problem (30), we have the following lemma. Lemma 2:
Problem (30) is feasible only if the following constraints are satisfied: n ∈N | [ g ki ] n | ≤ X n ∈N | [ g ki ] n | , ∀ i = k. (31) Proof:
According to constraints (30a), we have: [ g ki ] l θ ∗ kl = − X n ∈N ,n = l [ g ki ] n θ ∗ kn , ∀ i = k. (32)Taking the absolute value at both sides yields: | [ g ki ] l θ ∗ kl | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ∈N ,n = l [ g ki ] n θ ∗ kn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (33)Further combining constraints (30b), we have: | [ g ki ] l | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ∈N ,n = l [ g ki ] n θ ∗ kn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X n ∈N ,n = l | [ g ki ] n θ ∗ kn | = X n ∈N ,n = l | [ g ki ] n | , (34)where the inequality follows from the triangle inequality. Adding | [ g ki ] l | on both sides, (34)becomes: | [ g ki ] l | ≤ X n ∈N | [ g ki ] n | . (35)Since inequality (35) should be satisfied for any l ∈ N , conditions (31) are obtained by usingthe max l ∈N operation on both sides of (35). (cid:3) XY XY (a) (b)[ g ik ] (cid:169) k [ g ik ] (cid:169) k [ g ik ] (cid:169) k [ g ik ] (cid:169) k [ g ik ] (cid:169) k [ g ik ] (cid:169) k *** ** * Fig. 2. An example of N = 3 and max n ∈N | [ g ki ] n | = | [ g ki ] | . (a) | [ g ki ] | ≤ | [ g ki ] | + | [ g ki ] | and (b) | [ g ki ] | > | [ g ki ] | + | [ g ki ] | . To elaborate lemma 2 further, an example of N = 3 and max n ∈N | [ g ki ] n | = | [ g ki ] | is shownin Fig. 2. In this figure, we consider the special case that there are N = 3 reflecting elementsand the channel gain with the maximum amplitude is max n ∈N | [ g ki ] n | = | [ g ki ] | . If the condition(31) is satisfied, we have n ∈N | [ g ki ] n | = 2 | [ g ki ] | ≤ X n =1 | [ g ki ] n | , i.e., | [ g ki ] | ≤ | [ g ki ] | + | [ g ki ] | , as shown in Fig. 2(a). In this case, we can always construct a triangle with the length of threeedges are respectively | [ g ki ] | , | [ g ki ] | , and | [ g ki ] | , which means that there always exists θ k such that P Nn =1 θ ∗ kn [ g ki ] n = 0 . However, if the condition (31) is not satisfied, i.e., | [ g ki ] | > | [ g ki ] | + | [ g ki ] | , as shown in Fig. 2(b). In this case, we cannot construct a triangle with thelength of three edges that are respectively | [ g ki ] | , | [ g ki ] | , and | [ g ki ] | . As a result, there is nosolution for equation P Nn =1 θ ∗ kn [ g ki ] n = 0 . After checking the feasibility, we find that ZF beamforming problem (30) is still difficult tosolve (i.e., it is nonconvex) because of nonconvex constraints (30b). In the following, an iterativealgorithm is proposed to effectively solve ZF beamforming problem (30) with low complexity.Without loss of generality, the term g Hkk θ k in the objective function (30) can be expressed asa real number through an arbitrary rotation to phase beamforming θ k . As a result, the objectivefunction (30) can be equivalent to max θ k R ( g Hkk θ k ) . (36)According to constraint (30a), θ k must lie in the orthogonal complement of the subspacespan { g ik , ∀ i = k } and the orthogonal projector matrix on this orthogonal complement is [55] Z k = I − G k ( G Hk G k ) † G Hk , (37)where G k = [ g k , · · · , g ( k − k , g ( k +1) k · · · , g Kk ] . (38)For any vector θ k satisfying constraint (30a), θ k can be expressed by: θ k = Z k v k , (39)where v k is an N × complex vector to be optimized.Based on (36) and (39), ZF beamforming optimization problem (30) is equivalent to: max θ k , v k R ( g Hkk θ k ) (40)s.t. θ k = Z k v k , (40a) | θ kn | = 1 , ∀ n ∈ N . (40b)Since both variables θ k and v k are coupled in the constraint (40a), we introduce the barriermethod to transform problem (40) as follows max θ k , v k R ( g Hkk Z k v k ) − λ k θ k − Z k v k k (41)s.t. | θ kn | = 1 , ∀ n ∈ N , (41a)where λ > is a large penalty factor [49]. To solve problem (41), an iterative algorithm isproposed via alternatingly optimizing θ k with fixed v k , and updating v k with optimized θ k inthe previous step, which admits efficient closed-form solutions in each step. Algorithm 2
Iterative Optimization for Problem (41) Initialize v (0) k . Set iteration number t = 1 . repeat Given v ( t − k , obtain the optimal solution of problem (42), which is denoted by θ ( t ) k . Given θ ( t ) k , obtain the optimal solution of problem (44), which is denoted by v ( t ) k . Set t = t + 1 . until the objective value (41) converges.In the first step, given v k , problem (41) can be formulated as: max θ k − k θ k − Z k v k k = 2 θ k Z k v k − N − v Hk Z Hk Z k v k (42)s.t. | θ kn | = 1 , ∀ n ∈ N . (42a)To maximize (42), it is easy to get θ ⋆kn = [ Z k v k ] ∗ n | [ Z k v k ] n | , ∀ n ∈ N . (43)In the second step, we update the value of v k with the optimized θ k in (43). Then, problem(41) becomes max v k R ( g Hkk Z k v k ) − λ k θ k − Z k v k k . (44)Problem (44) is convex and thus the optimal solution can be obtained by setting the gradient tozero. We calculate the gradient of (44) with respect to v k and set it to zero, i.e., R ( Z Hk g kk + 2 λ Z Hk ( θ k − Z k v k )) = , (45)which gives v k = ( Z Hk Z k ) † (cid:18) λ Z Hk g kk + Z Hk θ k (cid:19) . (46)Note that the iterative algorithm for solving problem (41) is summarized in Algorithm 2. Dueto the fact that the optimal solution of problem (42) and (44) can be obtained, the objectivevalue (41) is always increasing at each iteration. Since the objective value of problem (41) isincreasing at each iteration and the objective value of problem (41) always has a finite upperbound, Algorithm 2 always converges. D. Power Scaling Law with Infinite RIS Units
We characterize the scaling law of the average received power at the user with respect to thenumber of RIS units, i.e., N → ∞ . For simplicity, we consider the single-user case with K = 1 .The received power at the user is P = P | g T θ | , where P is the transmitted power at the BS, g is the channel gain between the RIS and the user, and θ is the phase beamfoming of theBS. We compare two phase beamforming solutions: (i) θ = and (ii) the MRT beamforming [ θ ] n = [ g ] n | [ g ] n | . Theorem 2:
Assume that g ∼ CN ( , ρ I ) . If N → ∞ , we have: P = N ρP , if θ = , ( π − π +16)4 N ρP , if [ θ ] n = [ g ] n | [ g ] n | , ∀ n ∈ N . (47) Proof:
When θ = , we have g T θ ∼ CN (0 , N ρ ) . Thus, we have P = E ( P | g T θ | ) = N ρP .When [ θ ] n = [ g ] n | [ g ] n | , we have g T θ = N X n =1 [ θ ] ∗ n [ g ] n = N X n =1 | [ g ] n | . (48)Since g ∼ CN ( , ρ I ) , | [ g ] n | follows the Rayleigh distribution with mean √ πρ and variance (4 − π ) ρ . According to the central limit theorem, P Nn =1 | [ g ] n | ∼ CN ( N √ πρ , N (4 − π ) ρ ) . As a result,we have P = E ( P | g T θ | ) = ( π − π + 16)4 N ρP . (49)The proof completes. (cid:3) According to Theorem 2, it is obtained that the received power only linearly increase withthe number of RIS units if there is no optimization on the phase beamforming. However, underthe proposed phase beamforming, the received power quadratically increase with the number ofRIS units, which can greatly increase the received signal strength especially for large number ofRIS units. For the conventional massive MIMO, the order of transmit beamforming gain is N [3], [44]. The gain of RIS as a transmitter lies in the fact that RIS is used to modulate the signalvia reflection and increasing the number of RIS elements, which does not require any additionaltransmit power of the transmitter. E. Complexity Analysis
The complexity of Algorithm 1 lies in solving the SDP problem (12). Since the total numberof variables is S = ( K + 1) N and the total number of constraints is S = 3 K , the complexityof solving problem (12) is O ( S S ) = O ( K N ) according to [56]. Iteration number S u m t r an s m i t po w e r ( W ) =1=2=3 Fig. 3. Convergence behaviour of Algorithm 1 under different SINR requirements.
The complexity of Algorithm 2 lies in solving problem (44) at each step. According to (46), thecomplexity of solving problem (44) is O ( N ) . The total complexity of Algorithm 2 is O ( IN ) ,where I denotes the number of iterations for Algorithm 2.IV. N UMERICAL R ESULTS
In this section, we evaluate the performance of the proposed algorithm. There are K usersuniformly distributed in a square area of size m × m with the BS located at its center.The large-scale pathloss model is − . d − α ( d is in m), where α is the pathloss factor. Thenoise power is -114 dBm. For the channel gain g ki , we set [ g ki ] n ∼ CN (0 , , ∀ k, i ∈ K , n ∈ N [57], [58]. Unless specified otherwise, we choose a pathloss factor α = 3 , a total of K = 8 users, a number of N = 20 RIS units allocated for each user, a penalty factor λ = 10 , and anequal SINR requirement Γ = · · · Γ K = Γ = 2 . Additionally, the effectiveness of the proposeddual method (labeled as “DM”) is verified by comparing with the MRT and ZF methods.Fig. 3 illustrates the convergence of Algorithm 1 under different SINR requirements. It canbe seen that the proposed algorithm converges fast, and six iterations are sufficient to converge,which shows the effectiveness of the proposed algorithm in terms of convergence performance.The sum transmit power versus the minimum SINR requirement for continuous and discretephase shift schemes is shown in Fig. 4. In this figure, b denotes the number of bits used toindicate the number of phase shift levels L where L = 2 b . For simplicity, we assume that suchdiscrete phase-shift values are obtained by uniformly quantizing the interval [0 , π ) . Thus, the Mininum SINR value S u m t r an s m i t po w e r ( W ) Continuousb=1b=2b=4
Fig. 4. Sum transmit power versus the minimum SINR requirement for continuous and discrete phase shift schemes. set of discrete phase-shift values at each element is given by F = (cid:26) , πL , πL , · · · , L − πL (cid:27) . (50)Denote θ ∗ as the obtained result of considering continuous phase shifts. We use the roundingmethod to obtain discrete phase shifts solution ˆ θ , where ˆ θ kn = arg min θ kn ∈F | θ kn − θ ∗ kn | , ∀ k ∈ K , n ∈ N . (51)With the discrete phase shifts solution ˆ θ , the power control can be obtained by using the iterativepower control scheme (25). According to Fig. 4, it is observed that the performance loss dueto the rounding is small for large b and small minimum SINR value Γ , which indicates that theproposed approach is also suitable to discrete phase shifter with large number of phase shiftlevels.The sum transmit power versus the minimum SINR requirement for DM, SDR, and nearoptimal solution is given in Fig. 5. In this figure, the near optimal solution is obtained bythe algorithm with two steps. In the first step, the nonconvex unit module constraint is addedin the objective function by using the penalty method in [33] and then the successive convexapproximation method is used to solve the modified optimization problem in the second step.The near optimal solution is calculated by using the successive convex approximation methodwith multiple initial solutions and the solution with the best objective function is regarded asthe near optimal solution. From Fig. 5, it is shown that the DM always achieves the betterperformance than that of SDR. The reason is that the SDR scheme requires the randomizations Mininum SINR value S u m t r an s m i t po w e r ( W ) DMSDRNear Optimal
Fig. 5. Sum transmit power versus the minimum SINR requirement for DM, SDR, and near optimal solution.
Mininum SINR value S u m t r an s m i t po w e r ( W ) DMMRTZF
Fig. 6. Sum transmit power versus the minimum SINR requirement for DM, MRT, and ZF. to construct a rank-one solution, which can lead to the performance degradation. It can be alsoseen that DM achieves similar performance with the near optimal solution, which verifies thetheoretical findings in Theorem 1.We compare the sum transmit power and energy efficiency performance of DM, MRT, andZF. Figs. 6 and 7 show the sum transmit power and energy efficiency versus the minimum SINRrequirement. In Fig. 7, the energy efficiency is calculated by P Kk =1 B log (1 + Γ k ) µP + P B + P Kk =1 P k + N KP R , (52)where µ = ν − with ν = 0 . being the power amplifier efficiency of the BS, P is the sumtransmit power of the BS, P B = 29 dBm is the circuit power consumption of the BS, P k = 5 dBm is the circuit power consumption of user k , P R = 5 dBm is the power consumption of Mininum SINR value E ne r g y e ff i c i en cy ( M b i t s / J ou l e ) DMMRTZF
Fig. 7. Energy efficiency versus the minimum SINR requirement. each reflecting element in the RIS, and B = 1 MHz is the bandwidth of the BS. From thesetwo figures, DM achieves the best performance. In particular, DM can reduce up to 94% and23% sum transmit power compared to MRT and ZF, respectively. Besides, DM can increase upto 61% and 27% energy efficiency compared to MRT and ZF, respectively. According to Fig.6, the sum transmit power increases slightly with the minimum SINR requirement for DM andZF, while it increases rapidly with the minimum SINR requirement for MRT. This is due to thefact that MRT only maximizes the received signal strength without considering the multiuserinterference, which indicates that MRT is not suitable for high SINR requirement. It is alsofound that the energy efficiency of DM and ZF increases with the minimum SINR requirementfrom Fig. 7. From both Figs. 6 and 7, it is observed that MRT is superior over ZF for low SINRrequirement, while ZF is better than MRT for high SINR requirement.Fig. 8 shows how the sum transmit power changes as the pathloss factor varies. We can seethat the sum transmit power of all schemes increases with the pathloss factor. This is becausethe large pathloss factor results in poor channel gains for the users. It is found that DM achievesthe best performance among all schemes. From Fig. 8, DM can achieve the best sum transmitpower performance especially for the case that the pathloss factor is high.Fig. 9 shows the sum transmit power versus the number of RIS units N . From this figure, wecan see that the sum transmit power of all schemes monotonically decreases with the numberof RIS units. This is because large number of RIS units can lead to high spectral efficiency,which can reduce the transmit power of the system. According to Fig. 9, it can be shown thatthe decrease speed of sum transmit power for MRT is faster than that for ZF and DM, which Pathloss factor S u m t r an s m i t po w e r ( W ) DMMRTZF
Fig. 8. Sum transmit power versus the pathloss factor.
10 15 20 25 30
Number of RIS units N S u m t r an s m i t po w e r ( W ) DMMRTZF
15 20 250.050.10.15
Fig. 9. Sum transmit power versus the number of RIS units N with K = 3 . indicates that MRT is suitable for the case with large number of RIS units.Figs. 10 and 11 depict the sum transmit power versus number of users with N = 30 and N = 3 K , respectively. From both figures, DM achieves the best performance. In Fig. 10, the sumtransmit power increases with the number of users. This is because the multiuser interferenceis serious for large number of users. It can also be shown that the increase speed of DM isslower than that for MRT or ZF, which shows that DM can save more power than MRT orZF. Clearly, the DM is always better than MRT and ZF especially when the number of users islarge. According to Fig. 11, the sum transmit power tends to decrease with the number of users,which shows the different trends compared to Fig. 10. The reason is that the number of RISunits is fixed in Fig. 10, while the number of RIS units also increases linearly with the number Number of users K S u m t r an s m i t po w e r ( W ) DMMRTZF
Fig. 10. Sum transmit power versus the number of users K with N = 30 . Number of users K S u m t r an s m i t po w e r ( W ) DMMRTZF
Fig. 11. Sum transmit power versus the number of users K with N = 3 K . of users in Fig. 11. When the ratio between the number of users and the number of RIS units isfixed, the sum transmit power first decreases with the number of users and then remains stablefor high number of users.In Fig. 12, we show the sum transmit power versus the number of users K for distributeddeployment of RIS units. In this figure, the centralized deployment of all RIS units is labeled“CRIS”, while the distributed deployment of RIS units is labeled “DRIS”. In DRIS, we considerthe case that there are K RISs and the location of RIS l is given by (cos(2 lπ/L ) , sin(2 lπ/L )) × m. For each RIS, it is equipped with N RIS units. It is found that DRIS achieves thebetter performance than CRIS, which indicates the benefit of distributed deployment of RISs.Multiple RISs are spatially distributed in DRIS, which can decrease the transmit power between Number of users K S u m t r an s m i t po w e r ( W ) CRIS-DM, =1CRIS-DM, =1.5DRIS-DM, =1DRIS-DM, =1.5
Fig. 12. Sum transmit power versus the number of users K for distributed deployment of RIS units with N = 30 . the transceivers. V. C ONCLUSIONS
In this paper, we have investigated the resource allocation problem for a wireless communi-cation network with an RIS-assisted wireless transmitter. The RIS phase shifts and BS transmitpower were jointly optimized to minimize the sum transmit power while satisfying minimumSINR requirements and unit-modulus constraints. To solve this problem, we have proposed thedual method, compared to the SDR, MRT, and ZF beamforming techniques. Moreover, we haveanalyzed the asymptotic performance of the RIS-assisted communication system with infinitelynumber of RIS units. Numerical results have shown that the dual method outperforms MRT andZF schemes in terms of sum transmit power and energy efficiency, especially for high SINRrequirements. Furthermore, the distributed deployment of RIS units was shown to be favorablefor decreasing the sum transmit power. The optimization of RIS elements allocation is left forour future work. R
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