Over-the-Air Computation via Intelligent Reflecting Surfaces
OOver-the-Air Computation via Intelligent ReflectingSurfaces
Tao Jiang and Yuanming Shi
School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, ChinaEmail: { jiangtao1, shiym } @shanghaitech.edu.cn Abstract —Over-the-air computation (AirComp) becomes apromising approach for fast wireless data aggregation via exploit-ing the superposition property in a multiple access channel. Tofurther overcome the unfavorable signal propagation conditionsfor AirComp, in this paper, we propose an intelligent reflectingsurface (IRS) aided AirComp system to build controllable wire-less environments, thereby boosting the received signal powersignificantly. This is achieved by smartly tuning the phase shiftsfor the incoming electromagnetic waves at IRS, resulting inreconfigurable signal propagations. Unfortunately, it turns outthat the joint design problem for AirComp transceivers andIRS phase shifts becomes a highly intractable nonconvex bi-quadratic programming problem, for which a novel alternatingdifference-of-convex (DC) programming algorithm is developed.This is achieved by providing a novel DC function representationfor the rank-one constraint in the low-rank matrix optimizationproblem via matrix lifting. Simulation results demonstrate thealgorithmic advantages and admirable performance of the pro-posed approaches compared with the state-of-art solutions.
I. I
NTRODUCTION
In the near future, it is anticipated that massive numberof Internet of things (IoT) devices and machines will beconnected to wireless networks to automate the operations ofour daily life, thereby providing intelligent services. To thisend, one critical challenge is the need of ultra-fast wirelessdata aggregation, which pervades a wide range of applicationsin massive machine type communication [1] and on-devicefederated machine learning [2]. In particular, we need tocollect and process data distributed among a huge numberof devices rapidly by wireless communication techniques.However, with enormous number of devices, conventionalinterference-avoiding channel access schemes become infea-sible since they normally result in low spectrum utilizationefficiency and excessive network latency [3]. To overcomethis challenge, a promising concurrent transmissions solutionnamed over-the-air computation (AirComp) was proposed viaexploiting the superposition property in a multiple accesschannel [4]–[6].There are extensive research works on investigating theAirComp systems from the point of view of information theory[4], signal processing [5], [7], and transceiver beamformingdesign [3], [6], [8]. In particular, a uniform-forcing transceiverdesign was developed in [6] via the successive convex ap-proximation method to compensate the non-uniform fading ofdifferent sensors. A novel transmitter design leveraging zero-forcing beamforming has recently been proposed in [8] tocompensate the non-uniform fading among different multiple antennas at IoT devices. In [3], a multiple-input-multipleoutput (MIMO) AirComp scheme was further investigated toenable high-mobility multi-modal sensing. It showed that moreantennas at access point (AP) is able to reduce the performancedegradation in terms of mean-squared-error (MSE). However,all these approaches are unable to control of the wirelessenvironments, where in some scenarios the harsh propagationenvironments may result in significant deterioration of thesystem performance [9]. For instance, high frequency (e.g.,millimeter wave or terahertz) signals, which are expected toplay a key role in future communication systems, however,may be blocked even by small objects [9].To overcome unfavorable signal propagation conditions forAirComp, in this paper, we propose to boost the performanceof AirComp by developing large intelligent reflecting surfaces (IRS), which is envisioned to achieve high spectrum andenergy efficiency by controlling the communication environ-ments [10], [11]. An IRS normally does not require anydedicated energy source and can be integrated easily in thesurrounding walls of the transmitters [12], [13]. Specifically,an IRS is generally composed of many small passive elements,each of which is able to reflect a phase-shifted version of theincident signal [12]–[15]. By intelligently tuning the phaseshifts, we are able to constructively combine reflected signalswith the non-reflected ones to boost the received signal powerdrastically, thereby improving the achievable performance ofAirComp.Although there is a growing body of recent works ontransmit beamforming and IRS phase shifts design [12], [13],the transceiver design for AirComp raises unique challengesdue to the coupled design of the optimal phase shifts of alarge IRS. In this paper, we propose to jointly optimize thetransceiver and the phase shifts to minimize the MSE for Air-Comp. However, it turns out that the joint design problem forAirComp transceivers and IRS phase shifts becomes a highlyintractable nonconvex bi-quadratic programming problem . Inorder to address the coupled issue, we propose to optimize thephase shifts and the decoding vector at the AP alternatively.It turns out that the decoding vector design problem forAirComp [3] and the phase shifts matrix design problemfor IRS [13] are both nonconvex quadratically constrainedquadratic programming (QCQP) problems.A popular way to convexify the nonconvex QCQP problemis to reformulate it as a rank-one constrained matrix optimiza-tion problem via matrix lifting, followed by the semidefinite a r X i v : . [ c s . I T ] A p r elaxation (SDR) technique to drop the nonconvex rank-oneconstraint [16]. However, it was observed that the performanceof SDR approach degenerates in the scenarios with largenumber of antennas due to its low probability of returningrank-one solutions [2], [6], [17]. To address the limitationsof the popular SDR technique, in this paper, we develop ageneral framework to solve the rank-one constrained matrixoptimization problem via difference-of-convex (DC) program-ming. This is achieved by providing a novel DC functionrepresentation for the rank-one constraint, followed by amajorization-minimization algorithm to solve the resulting DCproblem. Furthermore, simulation results demonstrate that theproposed approach outperforms the SDR method significantly,and large IRS is able to dramatically enhance the AirCompperformance. Notations : (cid:107) · (cid:107) , ( · ) T , ( · ) H and Tr( · ) denote Euclidian norm,transpose, conjugate transpose and trace operators, respec-tively. Q ∼ CN ( µ, σ I ) stands for each element in Q followingi.i.d. normal distribution with mean µ and variance σ .II. S YSTEM M ODEL AND P ROBLEM FORMULATION
A. System Model
We consider a multi-user MISO communication systemconsisting of K single-antenna users and an AP with N antennas. In the scenario of over-the-air computation, the APaims to compute a target function of the aggregated data fromall users [3], [6] , as shown in Fig. 1. Specifically, let x k ∈ C denote data generated at user k and ψ k ( · ) : C → C denote thepre-processing function of user k , the target function computedat AP can be written in the form as f = φ (cid:32) K (cid:88) k =1 ψ k ( x k ) (cid:33) , (1)where φ ( · ) is the post-processing function of AP. Denote s k := ψ ( x k ) as the transmitted symbols at user k . The transmittedsymbols are assumed to be normalized to have unit variance,i.e., E ( s k s H k ) = 1 , and E ( s k s H j ) = 0 , ∀ k (cid:54) = j . To computethe target function f , AP needs to obtain the target-functionvariable defined as s := K (cid:88) k =1 s k . (2)In this paper, we aim to recover this target-function variable byexploiting the superposition property of a wireless multiple-access channel.To enhance the performance for over-the-air computationby controlling the signal propagation environment, we shallpropose to deploy an intelligent reflecting surface (IRS) ona surrounding wall, thereby dynamically adjusting the phaseshift of each reflecting elements according to the channelstate information (CSI). In particular, the IRS controller canswitch between two operational modes, i.e., the receivingmode for sensing the environment (e.g., CSI estimation) andthe reflecting mode for scattering the incident signals from theusers [13], [14]. The IRS has M elements, each of which re-scatters the received incident signals with a phase shift and a …… … … Data aggregation
Fig. 1. Over-the-air computation with intelligent reflecting surface. magnitude loss. Specifically, let Θ = diag( βe jθ , · · · , βe jθ M ) represent the diagonal phase shifts matrix of the IRS with θ m ∈ [0 , π ] and β ∈ [0 , as the amplitude reflectioncoefficient on the incident signals. In this paper, we assume β = 1 without loss of generality. Furthermore, it is well knownthat the power of signals reflected by twice or more times canbe ignored due to significant propagation loss [13]. Therefore,the equivalent uplink channel between users and AP consistsof three components, i.e., AP-user link, IRS-user link, andIRS-AP link as show in Fig. 1. Additionally, we assume all theinvolved channels are constant during a block of transmission.Let h dk ∈ C N , h rk ∈ C M , and G ∈ C N × M be the equivalentchannels from user k to the AP, from user k to the IRS, andfrom IRS to the AP, respectively. The received signal at APis thus given by y = K (cid:88) k =1 ( G Θ h rk + h dk ) w k s k + n , (3)where w k ∈ C is the transmitter scalar and n ∈ C N ∼CN (0 , σ I ) is the additive white Gaussian noise. Transmis-sion power at each user can not exceed a given positive value P , namely, | w k | ≤ P , ∀ k. (4)Given a decoding vector m ∈ C N at the AP, the estimatedtarget function variable is given by ˆ s = 1 √ η m H y = 1 √ η m H K (cid:88) k =1 ( G Θ h rk + h dk ) w k s k + 1 √ η m H n , (5)where η is a normalizing factor. B. Problem Formulation
In this paper, we aim to minimize the distortion afterdecoding, which is measured by the MSE defined as follows
MSE := E ( | ˆ s − s | )= K (cid:88) k =1 (cid:12)(cid:12)(cid:12)(cid:12) √ η m H h ek w k − (cid:12)(cid:12)(cid:12)(cid:12) + σ (cid:107) m (cid:107) η , (6)here h ek = G Θ h rk + h dk denotes the combined AP-userchannel vector. To minimize the MSE, we need to seek theoptimal transceivers w k ’s, m , and the phase shifts matrix Θ .Given the decoding vector m and the phase shifts matrix Θ , the optimal transmitter scalars can be designed as [2], [6] w k = √ η ( m H h ek ) H (cid:107) m H h ek (cid:107) , ∀ k, (7)where η is calculated by η = P min k (cid:107) m H h ek (cid:107) , (8)to satisfy the power constraint (4) on each transmitter scalar.Given the optimal transmitter scalars w k ’s and the normal-izing factor η , the MSE can be further rewritten as MSE = σ (cid:107) m (cid:107) P min k (cid:107) m H ( G Θ h rk + h dk ) (cid:107) . (9)We thus propose to jointly optimize the phase shifts matrix Θ and the decoding vector m to minimize the MSE as follows: minimize m , Θ (cid:18) max k (cid:107) m (cid:107) (cid:107) m H ( G Θ h rk + h dk ) (cid:107) (cid:19) subject to 0 ≤ θ n ≤ π, ∀ n = 1 , · · · , N. (10)We further equivalently reformulate problem (10) as the fol-lowing bi-quadratic programming problem: P : minimize m , Θ (cid:107) m (cid:107) subject to (cid:107) m H ( G Θ h rk + h dk ) (cid:107) ≥ , ∀ k, ≤ θ n ≤ π, ∀ n = 1 , · · · , N. (11)However, problem P turns out to be highly intractable due tothe nonconvex quadratic constraints with respect to m and Θ .In Section III, we shall leverage the alternating minimizationapproach to solve this problem. A novel alternating DCprogramming algorithm is further developed in Section IV.III. A LTERNATING M INIMIZATION
In this section, we propose to solve problem P by thealternating minimization approach. Specifically, the decodingvector m at AP and the phase shifts matrix Θ at the IRSare optimized in an alternative manner until the algorithmconverges. A. Alternating Minimization
For given phase shifts matrix Θ , problem P becomes thefollowing nonconvex QCQP problem minimize m (cid:107) m (cid:107) subject to (cid:107) m H h ek (cid:107) ≥ , ∀ k. (12)On the other hand, for a given decoding vector m , problem P is reduced to a feasibility detection problem. Specifically,let m H h dk = c k and v m = e jθ m , m = 1 , · · · , M , we have m H G Θ h rk = a H k v , where v = [ e jθ , · · · , e jθ M ] T and a H k = m H G diag( h rk ) . Therefore, problem P can be written as find v subject to | a H k v + c k | ≥ , ∀ k, | v n | = 1 , ∀ v = 1 , · · · , N. (13)Although problem (13) is nonconvex and inhomogeneous, itcan be reformulated as a homogeneous nonconvex QCQPproblem [13]. Specifically, by introducing an auxiliary variable t , we can equivalently rewrite problem (13) as find v subject to ˜ v H R k ˜ v + c k ≥ , ∀ k, | v n | = 1 , ∀ v = 1 , · · · , N, (14)where R k = (cid:20) a k a H k , a k c k c H k a H k , (cid:21) , ˜ v = (cid:20) v t (cid:21) . (15)Obviously, if ˜ v ∗ = [ v , t ] T is a feasible solution to problem(14), then we can obtain a feasible solution to problem (13)as v ∗ = v /t . The phase shifts matrix Θ ∗ can be recoveredfrom v ∗ trivially. Note that problem (12) is always feasible,while the feasibility of problem (14) may not be guaranteed.We thus terminate the alternating algorithm either problem(14) becomes infeasible during the iterative procedure or thedifference between the MSE of consecutive iterations is lessthan a predefined threshold.To summarize, we propose to solve the nonconvex bi-quadratic problem P by seeking the optimal solution toproblem (12) and problem (14) in an alternative manner.Although both problem (12) and problem (14) are still nonconvex, we shall reveal the algorithmic advantages in thefollowing sections. B. Matrix Lifting
To address the nonconvexity issue of problem (12) and prob-lem (14), a natural way is to reformulate them as semidefiniteprogramming (SDP) problems by the matrix lifting technique[16]. Specifically, by defining M = mm H , which lifts thevector m into a positive semidefinite (PSD) matrix with rank( M ) = 1 , problem (12) can be equivalently reformulatedas the following low-rank matrix optimization problem P : minimize M Tr( M )subject to Tr( M H k ) ≥ , ∀ k, M (cid:23) , rank( M ) = 1 , (16)where H k = h ek ( h ek ) H .Similarly, we also adopt the matrix lifting technique toreformulate the nonconvex quadratic constraints in problem(14). Specifically, let V = ˜ v ˜ v H and note that ˜ v H R k ˜ v =r( R k ˜ v ˜ v H ) . Problem (14) can be equivalently written as thefollowing low-rank matrix optimization problem P : find V subject to Tr( R k V ) + c k ≥ , ∀ k, V n,n = 1 , ∀ n = 1 , · · · , N, V (cid:23) , rank( V ) = 1 . (17) C. Problem Analysis
To further address the nonconvexity in problem P andproblem P , one popular way is to simply drop the nonconvexrank-one constraints via the SDR technique [16]. The resultingSDP problems can be solved efficiently by existing convexoptimization solvers such as CVX [18]. If the optimal solutionto the SDP problem is rank-one, the optimal solution to theoriginal problem can be recovered by rank one decomposition.On the other hand, if the optimal solution of the SDP problemfails to be rank-one, additional steps such as Gaussian random-ization [16] need to be applied to extract a suboptimal solutionfor the original problem. However, it was observed that for thehigh-dimensional optimization problems (e.g., the number ofantennas N increases), the probability of returning a rank-onesolution for the SDR approach becomes low, which yieldssignificant performance deterioration [6], [17]. To overcomethe limitations of the SDR methods, we instead propose anovel DC framework in the following section to solve problem P and problem P .IV. A LTERNATING
DC A
LGORITHM
In this section, we will introduce a novel DC represen-tation for the rank function, following by leveraging themajorization-minimization technique to iteratively solve theresulting DC problems.
A. DC Framework for Rank-One Constraint Problems
For ease of presentation, we first consider the DC algorithmfor general low-rank matrix optimization problems with arank-one constraint as follows, minimize X ∈C Tr( A X )subject to Tr( A k X ) ≥ d k , ∀ k, X (cid:23) , rank( X ) = 1 , (18)where the constraint set C is convex. A key observation onthe rank-one constraint is that it can be equivalently writtenas a DC function constraint, which is formally stated in thefollowing proposition [2]. Proposition 1.
For positive semidefinite (PSD) matrix X ∈ C N × N and Tr( X ) ≥ , we have rank( X ) = 1 ⇐⇒ Tr( X ) − (cid:107) X (cid:107) = 0 , (19) where trace norm Tr( X ) = (cid:80) Ni =1 σ i ( X ) and spectral norm (cid:107) X (cid:107) = σ ( X ) with σ i ( X ) denoting the i -th largest singularvalue of matrix X . In order to enhance a low rank solution for problem (18),instead of removing the nonconvex rank-one constraint via the SDR method, we propose to add the DC function in (19) intothe objective function as a penalty component, yielding minimize X ∈C Tr( A X ) + ρ · (Tr( X ) − (cid:107) X (cid:107) )subject to Tr( A k X ) ≥ d k , ∀ k, X (cid:23) , (20)where ρ > is the penalty parameter. Note that we are able toobtain an exact rank-one solution X ∗ when the nonnegativecomponent (Tr( X ∗ ) − (cid:107) X ∗ (cid:107) ) in the objective function isenforced to be zero. B. DC Algorithm
Although problem (20) is still nonconvex, it can be solved inan iterative manner by leveraging majorization-minimizationtechniques, yielding a DC algorithm [19]. The main idea is totransform problem (20) into a series of simple subproblemsby linearizing the concave term − ρ (cid:107) X (cid:107) in the objectivefunction. Specifically, we need to solve the subproblem atiteration t which is given by minimize X ∈C Tr( A X ) + ρ · (cid:104) X , I − ∂ (cid:107) X t − (cid:107) (cid:105) subject to Tr( A k X ) ≥ d k , ∀ k, X (cid:23) , (21)where X t − is the optimal solution of the subproblem atiteration t − . It is clear that the subproblem (21) is convexand can be solved efficiently by existing solvers such as CVX[18]. In addition, the subgradient ∂ (cid:107) X (cid:107) can be computedefficiently by the following proposition [2]. Proposition 2.
For given PSD matrix X ∈ C N × N , the sub-gradient ∂ (cid:107) X (cid:107) can be computed as v v H , where v ∈ C N is the leading eigenvector of matrix X . The presented DC algorithm converges to critical pointsof problem (20) from arbitrary initial points [19]. We thussummarize the presented DC algorithm in Algorithm 1.
Algorithm 1:
DC algorithm for solving problem (20).
Input :
Initial point X , threshold (cid:15) dc . for t = 1 , , · · · do Compute a subgradient: ∂ (cid:107) X t − (cid:107) .Solve the convex subproblem (21), and obtain X t . if The decrease of the objective function in (20) is below (cid:15) dc thenbreakendend C. Proposed Alternating DC Approach
In this subsection, we apply the proposed DC framework toproblem P and problem P . Specifically, to find a rank-onesolution to problem P , we propose to solve the followingDC programming problem minimize M Tr( M ) + ρ (Tr( M ) − (cid:107) M (cid:107) )subject to Tr( M H k ) ≥ , ∀ k, M (cid:23) , (22)here ρ > is the penalty parameter. When the penaltycomponent is enforced to be zero, problem (22) shall induce arank-one solution M (cid:63) , we can thus recover the solution m toproblem (12) through Cholesky decomposition M (cid:63) = mm H .To detect feasibility for problem P , we propose to mini-mize the difference between trace norm and spectral norm asfollows, minimize V Tr( V ) − (cid:107) V (cid:107) subject to Tr( R k V ) + c k ≥ , ∀ k, V n,n = 1 , ∀ n = 1 , · · · , N, V (cid:23) . (23)When the objective value of problem (23) becomes zero, weshall find an exact rank-one optimal solution V (cid:63) . By Choleskydecomposition V (cid:63) = ˜ v ˜ v H , we can obtain a feasible solution ˜ v to problem (14). If the objective value fails to be zero, weclaim that problem P (i.e., problem (14)) is infeasible.In summary, the proposed alternating DC algorithm forsolving problem P can be presented in Algorithm 2. Algorithm 2:
Proposed Alternating DC Algorithm forProblem P . Input :
Initial point Θ , threshold (cid:15) > . for t = 1 , , · · · do For given Θ t , solve problem P by Algorithm 1 to obtainthe solution M t .For given M t , solve problem P by Algorithm 1 toobtain the solution Θ t +1 . if The decrease of the MSE is below (cid:15) or problem P becomes infeasible. thenbreakendend V. S
IMULATIONS
In this section, we conduct numerical experiments to evalu-ate the performance of the proposed alternating DC algorithmfor solving problem P and the effectiveness of IRS for theover-the-air computation systems. Fig. 2. Layout of AP, IRS and users.
A. Simulation Settings
We consider a three-dimensional (3D) coordinate systemwith a uniform linear array of antennas at the AP and a uniform rectangular array of passive reflecting elements at the IRS,respectively. The AP and the IRS are respectively located at (0 , , meters and (50 , , meters, while the users areare uniformly located at region [ − , × [50 , metersas illustrated in Fig. 2. We consider the following pass lossmodel L ( d ) = T ( dd ) − α , (24)where T is the path loss at the reference distance d = 1 m , d is the link distance and α is the path loss exponent. In simula-tions, we set T = 30 dB, and the path loss exponent α for AP-user link, AP-IRS link and IRS-user link are respectively set tobe . , . , . . In addition, we assume Rayleigh fading for allthe considered channels. Specifically, the channel coefficientsare given by h dk = (cid:113) L ( d dk ) γ d and h dk = (cid:112) L ( d rk ) γ r , where γ d ∼ CN (0 , I ) , γ r ∼ CN (0 , I ) , Here, d dk and d rk are thedistance between user k and AP, the distance between user k and IRS, respectively. The channel matrix G = (cid:112) L ( d ) Γ ,where Γ ∼ CN (0 , I ) and d is the distance between AP andIRS. The average transmit signal-to-noise-ratio (SNR) P /σ is set to be dB. The other parameters are set as follows: ρ = 5 , (cid:15) = 10 − , (cid:15) dc = 10 − .We compare our proposed alternating DC algorithm withthe alternating SDR method for solving problem P , i.e., theSDR method is adopted to solve both problem P [16] andproblem P [13]. For the SDR method, we remove the rank-one constraint in problem P and problem P and solvethem alternatively via CVX [18], and we stop it when thedifference between the MSE of consecutive iterations is below (cid:15) or the SDR approach fails to return a feasible solutionto problem (14). Since the SDR method does not generallyreturn a rank-one solution, we apply Gaussian randomizationtechniques [16] when we fail to obtain a rank-one solution. Wealso illustrate the results given by random phase shift methodas the baseline. That is, to solve problem P , we choose afixed random phase shifts matrix Θ and minimize the MSEby solving problem P via proposed DC Algorithm 1. B. Simulation Results
We show the convergence behavior of the proposed alter-nating DC algorithm and alternating SDR method in Fig. 3(a)under the setting: K = 16 , M = 30 , N = 20 . It demonstratesthat the alternating SDR method stops at the third iterationsince it fails to find a feasible solution to problem (14)even with Gaussian randomization techniques. However, theproposed alternating DC algorithm is able to induce exactrank-one optimal solutions, thereby accurately detecting thefeasibility of problem (14). This yields good performance witha small MSE overall.We compare in Fig. 3(b) the MSE versus different numbersof the AP antennas N . The number of elements at IRS is fixedto M = 15 and the number of users is K = 8 . Each pointin Fig. 3(b) is averaged over channel realizations. As canbe seen from Fig. 3(b), the MSE decreases significantly as N increases, which indicates more antennas at AP will bringbetter performance for AirComp. Furthermore, the proposed Iterations -1 M SE Proposed alternating DCAlternating SDR (a) Convergence results.
Number of antennas at AP M SE Proposed alternating DCRandom phase shiftAlternating SDR (b) MSE vs. number of AP antennas.
10 20 30 40 50 60
Number of elements at IRS -2 -1 M SE Proposed alternating DCRandom phase shiftAlternating SDR (c) MSE vs. number of IRS elements.
Number of users -6 -4 -2 M SE With IRSWithout IRS (d) MSE vs. number of users.Fig. 3. Performance of different algorithms for solving problem P . alternating DC approach significantly outperforms alternatingSDR methods and the baseline.We further compare in Fig. 3(c) the MSE versus differentnumber of IRS elements M with fixed N = 10 , K = 8 . Eachpoint in Fig. 3(c) is averaged over channel realizations.From Fig. 3(c), it illustrates that as M increases, the MSEdecreases significantly, which indicates that IRS with largernumber of elements is able to achieve smaller MSE. Inaddition, the proposed alternating DC approach outperformsalternating SDR methods significantly.Finally, we compare the performance between AirCompwith IRS and the one without IRS. We fix the number ofthe AP antennas N = 8 . For the case without IRS, we setthe phase shifts matrix Θ = in problem P , and minimizethe MSE by solving problem (22) using the proposed DCAlgorithm 1. For the case with IRS, we fix the the numberof the IRS elements M = 15 . We illustrate the results ofthe MSE versus the number of users in Fig. 3(d), and eachpoint is averaged over channel realizations. It showsthat the MSE performance of the case without IRS is bad,which suggests that deploying IRS in AirComp system cansignificantly enhance its performance.VI. C ONCLUSION
In this paper, we proposed to leverage the large intelligentsurfaces to boost the performance for over-the-air computation,thereby achieving ultra-fast data aggregation. To find theoptimal transceiver and phase shifts design, we proposedan alternating minimization based approach, which resultsin solving the nonconvex QCQP problems alternatively. Toovercome the nonconvexity issue, we further reformulated theQCQP problems as a rank-one constrained matrix optimizationproblem via matrix lifting, followed by a novel DC frameworkto address the nonconvex rank-one constraints. Simulation re-sults demonstrated the admirable performance of the proposedapproaches compared with the state-of-the-art algorithms.R
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