Joint Beamforming and Power Control for Throughput Maximization in IRS-assisted MISO WPCNs
Yuan Zheng, Suzhi Bi, Ying-Jun Angela Zhang, Xiaohui Lin, Hui Wang
11 Joint Beamforming and Power Control forThroughput Maximization in IRS-assisted MISOWPCNs
Yuan Zheng, Suzhi Bi, Ying-Jun Angela Zhang, Xiaohui Lin, and Hui Wang
Abstract —Intelligent reflecting surface (IRS) is an emergingtechnology to enhance the energy- and spectrum-efficiency ofwireless powered communication networks (WPCNs). In thispaper, we investigate an IRS-assisted multiuser multiple-inputsingle-output (MISO) WPCN, where the single-antenna wirelessdevices (WDs) harvest wireless energy in the downlink (DL)and transmit their information simultaneously in the uplink(UL) to a common hybrid access point (HAP) equipped withmultiple antennas. Our goal is to maximize the weighted sumrate (WSR) of all the energy-harvesting users. To make fulluse of the beamforming gain provided by both the HAP andthe IRS, we jointly optimize the active beamforming of theHAP and the reflecting coefficients (passive beamforming) of theIRS in both DL and UL transmissions, as well as the transmitpower of the WDs to mitigate the inter-user interference atthe HAP. To tackle the challenging optimization problem, wefirst consider fixing the passive beamforming, and converting theremaining joint active beamforming and user transmit powercontrol problem into an equivalent weighted minimum meansquare error (WMMSE) problem, where we solve it using anefficient block-coordinate descent (BCD) method. Then, we fix theactive beamforming and user transmit power, and optimize thepassive beamforming coefficients of the IRS in both the DL andUL using a semidefinite relaxation (SDR) method. Accordingly,we apply a block-structured optimization (BSO) method toupdate the two sets of variables alternately. Numerical resultsshow that the proposed joint optimization achieves significantperformance gain over other representative benchmark methodsand effectively improves the throughput performance in mul-tiuser MISO WPCNs.
Index Terms —Wireless powered communication networks, in-telligent reflecting surface, multiuser MISO, resource allocation.
I. I
NTRODUCTION
With the advent of Internet-of-Things (IoT) era, tens ofbillions of wireless devices (WDs) are envisioned to be inter-connected, which inevitably induce the explosion of mobile datatraffic and the ever-growing demands for higher data rates. Thedemand for dramatic network capacity increase and ubiquitousconnectivity in IoT networks boosts the research on promisingwireless technologies, such as millimetre wave (mmWave),ultra-dense network (UDN) and massive multipleinput multiple-output (mMIMO) technologies [1]. However, their advanta-geous communication performance often comes at a cost of
Y. Zheng, S. Bi, X. Lin, and H. Wang are with the College of Electronicsand Information Engineering, Shenzhen University, Shenzhen, China, 518060(email: [email protected]; [email protected]; [email protected]). S. Bi isalso with the Peng Cheng Laboratory, Shenzhen, China, 518066.Y. J. Zhang is with the Department of Information Engineering, The ChineseUniversity of Hong Kong, Hong Kong (email: [email protected]). high network energy consumption and/or hardware expense.To address this problem, wireless powered communicationnetworks (WPCNs) have been proposed [2]–[4] to use dedicatedwireless energy transferring nodes to power the operationof communication devices. Compared with its conventionalbattery-powered counterpart, the WPCN has its advantages inlowering the operating cost and improving the robustness ofcommunication service especially in low power applications,such as sensor and IoT networks. However, the major technicalchallenge in WPCNs lies in the low power transfer efficiencyover long distance, resulting very limited harvested energyby the distributed WDs. Although several energy-efficienttechniques, including user cooperation [5], ambient backscattercommunication [6], multi-antenna technique [7], have beenproposed to address this problem, the low energy transferefficiency induced by the wireless channel attenuation is stilla fundamental performance bottleneck of WPCN systems.Recently, intelligent reflecting surface (IRS) technology hasreceived widespread attentions of its application in wirelesscommunications [8]. In particular, an IRS comprises a massivenumber of reconfigurable reflecting elements and a smartcontroller. Each element reflects impinging electromagneticwaves with a controllable amplitude variation and phase shiftusing the IRS controller. By properly adjusting the reflectingelements of IRS, the reflected signals are coherently combinedwith those from the other paths at the receiver to maximizethe signal strength. Compared to the use of conventionalamplify-and-forward (AF) or decode-and-forward (DF) relay,IRS merely changes the end-to-end channel through passivereflection without amplifying or re-encoding the received sig-nals. The recent advance in meta-surface technology [9] makesit feasible to reconfigure the reflecting coefficients in real time,thus greatly enhancing the applicability of IRS under wirelessfading channel. The integration of IRS technique in wirelesscommunication network leads to many new technologicalinnovations and networking paradigms. In terms of the circuitimplementations, practical IRS circuits include conventionalreflect-arrays [10], liquid crystal surfaces [11], and software-defined meta-materials [12], among others. For new networkingschemes, the utilization of IRS was extended to variouscommunication scenarios, such as backscatter communicationsystem [13], cognitive radio network [14], and the UAV-basedcommunication scenario [15].The essential advantage of deploying IRS lies in its ability toalter the wireless propagation environment to enhance the end-to-end channel strength in a passive and energy-efficient manner. a r X i v : . [ c s . N I] D ec HAP
IRS G WD K g g g K a a a K WD WD ... ... . .. Energy transfer Information transmission
Fig. 1. The network structure of our proposed IRS-aided MISO WPCN.
This makes IRS a promising solution to tackle the fundamentalperformance constraints of WPCN. A large body of researchon IRS-assisted WPCN has recently emerged in the literature[16]–[22]. For instance, the authors in [16] considered a jointdesign of active beamforming at the base station (BS) andpassive beamforming at the IRS to minimize the total transmitpower of the BS under the received user signal-to-noise ratio(SNR) constraints. [17] considered a downlink (DL) multiusermultiple-input single-output (MISO) scenario and maximizedthe energy efficiency of the BS by alternatively optimizing thetransmit beamforming at the BS and the phase shifts at theIRS. Besides, the weighted sum-rate maximization problem forIRS-assisted system was investigated in various scenarios, e.g.,MISO system [18], multicell multiple-input multiple-output(MIMO) network [19], and simultaneous wireless informationand power transfer (SWIPT) system [20]. The authors in [21]proposed an IRS-assisted mmWAVE communication system inwhich the IRS is used to overcome the impact of blockage. [22]designed a deep reinforcement learning (DRL)-based algorithmto jointly optimize the active and passive beamforming in theIRS-aided system.Most of the existing works adopt the IRS to assist eitherwireless energy transfer (WET) in the DL or wireless informa-tion transmission (WIT) in the UL. However, the UL and DLtransmissions in WPCNs are highly correlated by the deviceenergy causality. In this sense, a joint design of IRS-assistedDL and UL transmissions is needed to achieve the maximumcommunication performance in WPCNs. Although this jointdesign was recently studied in [23], it only considered theoptimization of passive beamforming of the IRS in both the DLand UL. The major challenge resides in the joint design of theactive beamforming of the HAP and the passive beamformingof the IRS in both DL and UL transmissions. Besides, theuser transmit power is affected by both the DL energy transferand the UL inter-user interference when spatial multiplexingis used. However, to the best of our knowledge, this importantresearch topic has not been studied so far.In this paper, we study the joint beamforming and usertransmit power control problem in an IRS-assisted multiuser MISO WPCN. As shown in Fig. 1, we consider a multi-antenna half-duplex HAP performing active beamformingto broadcast wireless energy to all WDs in the DL andthen receive information transmissions from the WDs in theUL. Specifically, the IRS performs passive beamforming byreflecting the transmitted energy (information) signals in theDL (UL) transmission. During the UL transmission, the WDsperform transmit power control to mitigate the multi-userinterference at the HAP. Our objective is to maximize theweighted sum communication rate of all the WDs. The maincontributions of this paper are summarized as follows: • With the proposed IRS-assisted MISO WPCN, we firstanalyze the achievable data rates of all WDs. Then,we formulate an optimization problem to maximize theweighted sum rate (WSR) of all WDs by jointly optimizingthe energy transmission time, the transmit power ofthe WDs, the active beamforming of the HAP and thepassive beamforming the of IRS in both the UL and DLtransmissions. The problem is highly non-convex becauseof the strong coupling of the design variables. • To tackle this non-convex problem, we first fix the energytransmission time and consider the joint beamformingand user transmit power control problem. Given passivebeamforming of the IRS, we convert the remaining activebeamforming and transmit power control problem intoan equivalent weighted minimum mean square error(WMMSE) problem, which can be efficiently solved byapplying a block-coordinate descent (BCD) method. Inparticular, we show that optimal energy beamformingmatrix during the DL energy transfer of the HAP isrank-one and aligned to the maximum eigenmode of theweighted sum of DL channel matrices. • Given the active beamforming and user transmit power,we then propose a semidefinite relaxation (SDR) methodto optimize the passive beamforming of the IRS, includingthe array reflecting coefficients in both DL and ULtransmissions. Accordingly, we devise a block-structuredoptimization (BSO) technique to update the two sets ofvariables alternately. Finally, we apply a one-dimensionalsearch method to obtain the optimal energy transmissiontime.We conduct extensive simulations to evaluate the per-formance of the proposed IRS-assisted MISO WPCN. Bycomparing with the other representative benchmark methods,we show that the proposed method achieves a significantthroughput performance gain in multiuser MISO WPCNs.The rest of the paper is organized as follows: In SectionII, we present the system model of the proposed IRS-assistedcommunication in multiuser MISO WPCN. We formulate theWSR optimization problem in Section III and propose anefficient algorithm to solve it in Section IV. In Section V,we perform simulations to evaluate the performance of theproposed method. Finally, Section VI concludes this paper.
Notations:
In this paper, vectors and matrices are denoted byboldface lowercase and uppercase letters, respectively. C m × n denotes the space of m × n complex-valued matrices. Theoperators | · | , (cid:107) · (cid:107) , ( · ) T and ( · ) H denote the absolute value, Euclidean norm, transpose and conjugate transpose, respectively.The symbols tr ( X ) and rank ( X ) denote the trace and rank ofmatrix X , respectively. E [ · ] stands for the statistical expectation. CN ( µ, σ ) denotes the distribution of a circularly symmetriccomplex Gaussian (CSCG) random vector with mean µ andcovariance σ . X (cid:23) means that X is positive semi-definite. arg( · ) denotes the phase extraction operation and [ x ] (1: N ) denotes the vector that contains the first N elements of x .diag ( x ) is a diagonal matrix withe the entries of the vector x .II. S YSTEM M ODEL
As show in Fig. 1, we consider a multiuser MISO WPCN,which consists of one HAP and K WDs. We define the setof WDs as K (cid:44) { , · · · , K } . It is assumed that the HAP isequipped with M antennas and each WD has a single antenna.Specifically, the HAP broadcasts wireless energy to the WDsin the DL and receives wireless information transmission fromthe WDs in the UL. All devices are assumed to operate overthe same frequency band, where a time-division-duplexing(TDD) circuit is implemented at both the HAP and the WDsto separate the energy and information transmissions. TheHAP performs energy beamforming in the DL and receivebeamforming (e.g., MMSE) in the UL information transmission.The HAP has stable power supplies and each WD has anenergy harvesting circuit and a rechargeable battery to storethe harvested energy to power its operations. To enhance thepropagation performance, we employ an IRS composed of N passive elements to assist the transmissions of the WPCN. TheIRS can dynamically adjust the phase shift of each reflectingelement based on the propagation environment [24]. Due tothe substantial path loss, we only consider one-time signalreflection by the IRS and ignore the signals that are reflectedthereafter [16].We assume that all channels follow a quasi-static flat fadingmodel, where all the channel coefficients remain constantduring each block transmission time, denoted by T , but varyfrom block to block. The baseband equivalent channels ofHAP-to-IRS, IRS-to-WD i , and HAP-to-WD i links are denotedas G ∈ C M × N , g i ∈ C N × and a i ∈ C M × , ∀ i ∈ K ,respectively. It is assumed that the channels of differenttransceiver pairs are independent to each other. Besides,the entries inside all channel vectors are modeled as zero-mean independent and identically distributed (i.i.d.) complexGaussian random variables with variance depending on the pathloss of the respective wireless links. The corresponding channelgains are denoted as g = (cid:107) G (cid:107) , g i = (cid:107) g i (cid:107) and h i = (cid:107) a i (cid:107) .With the IRS-aided channel, each element at the IRS firstcombines all the received multi-path signals and then re-scatters the combined signal with a certain phase shift. Let θ =[ θ , θ , · · · , θ N ] and Θ = diag( βe jθ , · · · , βe jθ n , · · · , βe jθ N ) denote the phase-shift matrix of the IRS, where θ n ∈ [0 , π ] and β ∈ [0 , are the phase shift and amplitude reflectioncoefficient of each element, respectively. In this paper, weset β = 1 for simplicity of in the following analysis, i.e., Θ = diag ( v , · · · , v n , · · · , v N ) with | v n | = 1 , n = 1 , · · · , N ,and use transmit power control to mitigate the inter-userinterference in the UL information transmission. HAP à WDs
HAP à IRS à WDs WD ,WD , ,WD K à HAP WD ,WD , ,WD K à IRS à HAP ... t T-t
WET WIT ...
Fig. 2. The transmit protocol of the proposed IRS-assisted MISO WPCN.
As shown in the Fig. 2. we consider a harvest-then-transmitprotocol that operates in two phases. In the first phase ofduration t , the HAP transfers wireless energy in the DL for allthe WDs to harvest. Meanwhile, the IRS scatters the incidentsignal from the HAP to the WDs, such that the WDs receivesignals from both the direct-path and reflect-path channels. Theremaining time of the block is assigned for the UL informationtransmission, during which WDs transmit their independentinformation to the HAP. Likewise, the IRS simultaneouslyscatters the signals transmitted by all WDs to the HAP.We assume that the channel state information (CSI) of allchannels are perfectly known at the HAP. We jointly optimizethe active beamforming of the HAP and passive beamformingof the IRS in both the UL and DL, the user transmit power,and the transmission time allocation between the UL and DLtransmissions, to maximize the WSR of all the users. In thefollowing, we formulate the WSR maximization problem andpropose an efficient method to solve it.III. P
ROBLEM F ORMULATION
In this section, we derive the throughput of each WD andformulate the WSR maximization problem.
A. Phase I: Energy Transfer
In the first WET stage of duration t , we denote x ( t ) ∈ C M × as the pseudo-random baseband energy signal transmitted bythe HAP [2]. The transmit power is constrained by E (cid:2) | x ( t ) | (cid:3) = tr (cid:0) E (cid:2) x ( t ) x ( t ) H (cid:3)(cid:1) (cid:44) tr ( W ) ≤ P . (1)where W (cid:23) is the energy beamforming matrix, and P denotes the maximum transmit power.Then, the received signal by the i -th WD is expressed as[16] y (1) i ( t ) = ( GΘ g i + a i ) T x ( t ) + n i ( t ) , ∀ i ∈ K , (2)where Θ = diag ( v , , · · · , v ,N ) denotes the energy reflectioncoefficient matrix at the IRS with v ,n = e jθ ,n , n = 1 , · · · , N ,which satisfies | v ,n | = 1 . n i ( t ) denotes additive white Gaus-sian noise (AWGN) at the receiver with n i ( t ) ∼ CN (0 , N ) .By neglecting the noise power, the amount of energyharvested by the i -th WD is E (1) i = η tr (cid:16) ( GΘ g i + a i )( GΘ g i + a i ) H W (cid:17) t, ∀ i ∈ K , (3) The CSI can be precisely estimated by the channel estimation methods forIRS system proposed in [25] and [26], which is out of the scope of this paper. where < η < denotes the fixed energy harvesting efficiencyfor all the WDs. Accordingly, the residual energy of the i -thWD is E i = min { E + E (1) i , E max } , ∀ i ∈ K , (4)where E is the known residual energy at the beginning of thecurrent time slot, and E max is the battery capacity. B. Phase II: Information Transmission
In the subsequent WIT phase of duration T − t , all WDstransmit their independent information simultaneously to theHAP using the harvested energy in phase I. Meanwhile,the IRS reflects signal of the WDs to the HAP. Let s i ( t ) denote the information signal transmitted by the i -th WD with E [ | s i ( t ) | ] = 1 , and P i denote the transmit power of WD i ,which is restricted by ( T − t ) P i + E (2) i ≤ E i , ∀ i ∈ K , (5)where E (2) i ≥ denotes the fixed energy consumption of WD i within a transmission block, such as the data processing unitand passive circuitry power consumption.Then, the received signals at the HAP from the i -th WD inthe UL is y (2) i ( t ) = ( GΘ g i + a i ) (cid:112) P i s i ( t )+ (cid:88) j ∈K\ i ( GΘ g j + a j ) (cid:112) P j s j ( t ) + n ( t ) , ∀ i ∈ K , (6)where Θ = diag ( v , , · · · , v ,N ) denotes the reflection-coefficient matrix at the IRS with | v ,n | = 1 , n = 1 , · · · , N . n ( t ) ∈ C M × denotes the AWGN vector at the HAP with n ( t ) ∼ CN ( , N I ) .It is assumed that the signals of different users are indepen-dent. In this paper, we consider linear receive beamforming atthe HAP by treating the interference as noise. The estimatedsignal is expressed as ˆ s i = f Hi y (2) i ( t ) , ∀ i ∈ K , (7)where f i ∈ C M × denotes the receiver beamforming vector.Then, the interference-plus-noise ratio (SINR) at the HAPfor decoding the signal of the i -th WD is γ i = (cid:107) f Hi ( GΘ g i + a i ) (cid:107) P i (cid:80) j ∈K\ i (cid:107) f Hi ( GΘ g j + a j ) (cid:107) P j + (cid:107) f Hi (cid:107) N , ∀ i ∈ K . (8)Thus, the achievable rate for information transmission of WD i in the UL is given by R i = T − tT log (1 + γ i ) , ∀ i ∈ K . (9)where it is assumed without loss of generality that T = 1 ,such that T is not present in the data rate expressions in Although a single energy harvesting circuit exhibits non-linear energyharvesting property due to the saturation effect of circuit, it is shown in [27]and [28] that the non-linear effect is effectively rectified by using multipleenergy harvesting circuits concatenated in parallel, resulting in a sufficientlylarge linear conversion region. the remainder of this paper. Note that a trade-off exists inthe WET duration t to maximize R i . Specifically, a larger t leads to higher allowable transmit power but shorter remaininginformation transmission time. C. Problem Formulation
In this paper, we focus on maximizing the WSR of the K WDs by jointly optimizing the transmit time t , user transmitpower P = [ P , · · · , P K ] , the active beamforming of the HAP(including the energy beamforming matrix W and the receiverbeamforming matrix F = [ f , · · · , f K ] ), and the passivebeamforming of the IRS (i.e., the phase shift matrices Θ , Θ ). Mathematically, it is formulated as (P1) : max t, P , W , F , Θ , Θ (cid:88) i ∈K ω i R i s. t. (1) , (3) , (4) and (5) ,t, P i ≥ , ∀ i ∈ K , | v i,n | = 1 , i = 1 , , n = 1 , · · · , N. (10)where ω i ≥ is the weighting factor controlling the schedulingpriority of WD i .Notice that the objective function in (9) is not a concavefunction in the optimizing variables. Besides, due to themodulus constraints and the multiplicative terms of (3), (4) and(5), (P1) is highly non-convex in its current form. In the nextsection, we first transform (P1) into an equivalent problem andpropose an efficient optimization algorithm to solve it.IV. P ROPOSED S OLUTION TO (P1)We first fix t = ¯ t in (P1), partition the remaining optimizationvariables into two blocks, and alternatively optimize the twoblocks of variables in an iterative manner [29]. Specifically,the optimization variables are partitioned as { W , F , P } and { Θ , Θ } . Then, we propose efficient algorithms to solvethe joint active beamforming and transmit power control sub-problem (to optimize { W , F , P } ) and passive beamformingproblem (to optimize { Θ , Θ } ) separately in the following.By assuming that the information messages of differentusers are independent, i.e., E [ s i s j ] = 0 if i (cid:54) = j , we write themean-square error (MSE) as e i = E s i (cid:8) | ˆ s i − s i | (cid:9) = | f Hi ( GΘ g i + a i ) (cid:112) P i − | + (cid:88) j ∈K\ i | f Hi ( GΘ g j + a j ) | P j + N (cid:107) f Hi (cid:107) , ∀ i ∈ K . (11)Following the celebrated rate-MMSE equivalence establishedin [30], we show in the following proposition that the originalmaximization problem can be transformed to a more tractableoptimization problem. Proposition 1 : Given t = ¯ t , the WSR maximization problemis equivalent to the following WMMSE problem, (P2) : min W , F , P , q , Θ , Θ (cid:88) i ∈K ω i (1 − ¯ t ) (cid:16) q i e i − log( q i ) − (cid:17) s. t. (1) , (3) , (4) and (5) ,P i , q i ≥ , ∀ i ∈ K , | v i,n | = 1 , i = 1 , , n = 1 , · · · , N, (12)where q = [ q , · · · , q K ] and q i is a positive weight variablefor i = 1 · · · , K . Proof : Please refer to Appendix 1.With the above transformation, we design an efficient alter-nating optimization algorithm to solve the active beamformingand transmit power control sub-problem described as follows.
A. Optimizing the active beamforming matrices and transmitpower { W , F , P } We first optimize { W , F , P } under fixed Θ and Θ .Define b i = GΘ g i + a i and ˜ b i = GΘ g i + a i , ∀ i ∈ K .Then, (3) and (8) are respectively expressed as, E (1) i = η tr ( b i b Hi W )¯ t, ∀ i ∈ K , (13) γ i = (cid:107) f Hi ˜ b i (cid:107) P i (cid:80) j ∈K\ i (cid:107) f Hi ˜ b j (cid:107) P j + (cid:107) f Hi (cid:107) N , ∀ i ∈ K . (14)Accordingly, we reformulate problem (P2) into the followingequivalent problem (P3) : min F , W , P , q (cid:88) i ∈K ω i (1 − ¯ t ) (cid:16) q i e i − log( q i ) − (cid:17) s. t. (1) , (4) , (5) and (13) ,P i , q i ≥ , ∀ i ∈ K , (15)where e i is re-written as e i = | f Hi ˜ b i (cid:112) P i − | + (cid:88) j ∈K\ i | f Hi ˜ b j | P j + N (cid:107) f Hi (cid:107) . (16)Note that the objective function of problem (P3) is convexover each of the optimization variable f i , P i and q i forall i ∈ K . Following [30], we employ a block-coordinatedescent (BCD) approach to tackle this problem. Specifically,we optimize one of the block variables in { f i , P i , q i } with theother two fixed. To obtain some insights on the optimal solutionstructure, we apply the Lagrange duality method to solve (P3).The partial Lagrangian of problem (15) is formulated as L ( F , W , P , q , µ ) =(1 − ¯ t ) (cid:88) i ∈K ω i (cid:16) q i e i − log( q i ) − (cid:17) + (cid:88) i ∈K µ i (cid:16) (1 − ¯ t ) P i + E (2) i (cid:17) − µ P − tr ( AW ) , (17)where A = (cid:80) i ∈K µ i η ¯ t b i b Hi − µ I , µ and µ i , ∀ i ∈ K are thenonnegative dual variables corresponding to the constraints (1) and (5), respectively. For convenience, we denote µ =[ µ , µ , · · · , µ K ] . Then, the dual function of (P3) is d ( µ ) = min F , W , P , q L ( F , W , P , q, µ ) s. t. F , W (cid:23) , P , q ≥ . (18)and the dual problem is (P3a) : max µ d ( µ ) s. t. µ ≥ . (19)Therefore, we first investigate the optimal solution of thedual function in (18) given a set of dual variables. Secondly,we determine the optimal dual variables µ ∗ and µ ∗ i , ∀ i ∈ K tomaximize the dual function. Proposition 2 : The optimal energy bemaforming matrix W ∗ for problem (P3) is expressed as W ∗ = P u u H . (20)where u is the unit-norm eigenvector of a matrix B = (cid:80) i ∈K µ ∗ i η ¯ t b i b Hi corresponding to the maximum eigenvalue λ . Moreover, the optimal dual variables must satisfy µ ∗ i > for i = 1 , · · · , K and µ ∗ = λ . Proof : Please refer to Appendix 2.
Remark 1 : Note that the optimal energy matrix in (20) is rank-one such that transmitting a single energy stream is the optimalstrategy for the DL energy transfer. Besides, u is aligned withthe maximum eigenmode of matrix B . Accordingly, the optimalenergy signal x ( t ) is determined as x ( t ) = √ P u x ( t ) , where x ( t ) denotes an arbitrary random scalar with unit variance.Furthermore, by checking the first-order optimality condi-tions for maximizing dual function with respect to q i , f i and P i , respectively, we have e i − q i = 0 , (21) (cid:88) j ∈K f Hi (cid:107) ˜ b j (cid:107) P j − b Hi (cid:112) P i + 2 N f Hi = , (22) (cid:88) j ∈K ω j q j | f Hj ˜ b i | − ω i q i f Hi ˜ b i √ P i + µ i = 0 , (23)for all i ∈ K . Then, we update each block variable in a closed-form manner given by q ∗ i = 1 e i , ∀ i ∈ K , (24) f ∗ i = ˜ b i √ P i (cid:80) j ∈K (cid:107) ˜ b j (cid:107) P j + N , ∀ i ∈ K , (25) P ∗ i = (cid:32) ω i q i f Hi ˜ b i (cid:80) j ∈K ω j q j | f Hj ˜ b i | + µ ∗ i (cid:33) , ∀ i ∈ K . (26)After solving the dual function, we obtain the optimaldual variables µ ∗ i by sub-gradient based algorithms, e.g., theellipsoid method. The subgradient of d ( λ ) is denoted as ς = [ ς , · · · , ς K ] , where ς i = (1 − ¯ t ) P ∗ i + E (2) i − η tr ( b i b Hi W ∗ )¯ t, ∀ i ∈ K . (27)The detailed description of the BCD method for problem(P3) is summarized in Algorithm 1. Algorithm 1:
Proposed BCD method for problem (P3)
Input: P , G , t , Θ , Θ , N , { g i , h i , ∀ i ∈ K} ; Output: q ∗ , P ∗ , W ∗ , F ∗ ; Initialize : j ← , µ (0) > , feasible q (0) , F (0) , P (0) ; repeat Calculate W ( j +1) using (20) with given µ ( j ) ; Calculate e i in (16) with given f ( j ) i and P ( j ) i ; Calculate q ( j +1) i using (24) with given f ( j ) i and P ( j ) i ; Calculate f ( j +1) i using (25) with given q ( j +1) i and P ( j ) i ; Calculate P ( j +1) i using (26) with given q ( j +1) i and f ( j +1) i ; Calculate the sub-gradient of µ ( j ) using (27); Update µ ( j +1) by using the ellipsoid method; j ← j + 1 ; until The optimal objective value of primal problem (P3) converges; Return { q ∗ , P ∗ , W ∗ , F ∗ } as a solution to (P3). B. Optimizing the passive beamforming matrices { Θ , Θ } Now, we optimize the phase shift matrices { Θ , Θ } givenfixed { W , F , P } . Let v n = [ v n, , · · · , v n,N ] T , n = 1 , .Define ζ i = G diag ( g Ti ) ∈ C M × N , ∀ i ∈ K . Then, we have GΘ n g i + a i = G diag ( g Ti ) v n + a i = ζ i v n + a i . (28)To tackle the non-convex modulus constraint in (P2), we firstdefine ¯ v n = (cid:20) v n (cid:21) ∈ C ( N +1) × , n = 1 , , ¯ ζ i = (cid:2) ζ i , a i (cid:3) ∈ C M × ( N +1) and V n = ¯ v n ¯ v Hn ∈ C ( N +1) × ( N +1) . Thus, wehave (cid:107) f Hi ( GΘ g i + a i ) (cid:107) = (cid:107) f Hi ( ζ i v + a i ) (cid:107) = (cid:107) f Hi ¯ ζ i ¯ v (cid:107) = tr ( V ¯ ζ Hi f i f Hi ¯ ζ i ) . (29)Accordingly, we rewrite (3) as E (1) i = η ¯ t tr ( V ¯ ζ Hi W ¯ ζ i ) , ∀ i ∈ K . (30)Consider the following transformation | f Hi ( GΘ g i + a i ) (cid:112) P i − | = | f Hi ( ζ i v + a i ) (cid:112) P i − | = | f Hi ζ i v (cid:112) P i + f Hi a i (cid:112) P i − | = (cid:107) (cid:98) ζ i ¯ v (cid:107) = tr ( V ψ i ) , (31)where (cid:98) ζ i = (cid:2) f Hi ζ i √ P i , f Hi a i √ P i − (cid:3) ∈ C × ( N +1) and ψ i = (cid:98) ζ iH (cid:98) ζ i ∈ C ( N +1) × ( N +1) , ∀ i ∈ K . Then, the MSE in(11) is re-expressed as e i ( V ) = | f Hi ( GΘ g i + a i ) (cid:112) P i − | + (cid:88) j ∈K\ i | f Hi ( GΘ g j + a j ) | P j + N (cid:107) f Hi (cid:107) = (cid:107) (cid:98) ζ i ¯ v (cid:107) + (cid:88) j ∈K\ i (cid:107) f Hi ¯ ζ j ¯ v (cid:107) P j + N (cid:107) f Hi (cid:107) = tr ( V ψ i ) + (cid:88) j ∈K\ i tr ( V ¯ ζ Hj f i f Hi ¯ ζ j ) P j + N tr ( f i f Hi ) . (32)Note that [ V i ] n,n = 1 , i = 1 , , n = 1 , · · · , N +1 hold fromthe modulus constraint of v i,n ( [ X ] m,n denotes the element in the m -th row and n -th column of matrix X ). Besides, V i must satisfy rank ( V i ) = 1 . Thus, we rewrite problem (P2) as (P4) : min V , V K (cid:88) i =1 ω i (1 − ¯ t ) (cid:16) q i e i ( V ) − log( q i ) − (cid:17) s. t. (4) , (5) and (30) , [ V i ] n,n = 1 , i = 1 , , n = 1 , · · · , N + 1 , rank ( V i ) = 1 , V i (cid:23) . (33)Dropping the non-convex rank-one constraint and removingthe terms irrelevant to V , V , we reduce prblem (P4) to (P4a) : min V , V (cid:23) K (cid:88) i =1 ω i q i e i ( V ) s. t. (4) , (5) and (30)[ V i ] n,n = 1 , i = 1 , , n = 1 , · · · , N + 1 . (34)Note that problem (P4a) is a standard semidefinite program-ming (SDP) and it can be efficiently solved by the optimizationtools such as CVX [31]. Let’s denote the optimal solution toproblem (P4a) as { V ∗ , V ∗ } . Generally, the relaxed problem(P4a) may not yield a rank-one solution. To recover v i from V ∗ i for i = 1 , , we obtain the eigenvalue decomposition of V ∗ i as V ∗ i = U i Σ i U iH , where U i ∈ C ( N +1) × ( N +1) and Σ i ∈ C ( N +1) × ( N +1) denote a unitary matrix and diagonalmatrix, respectively. Then, we apply the standard Gaussianrandomization method [32] to obtain a suboptimal solution ¯ v i , i.e., ¯ v i = U i Σ / i r i , i = 1 , , where r i ∈ C ( N +1) × is arandom vector generated from r i ∼ CN ( , I N +1 ) . With manycandidate solutions r i ’s, we select the best one ¯ v i among all r i which minimizes the objective of (P4a). Finally, we obtain v ∗ i = e j arg([¯ v i ] (1: N ) / ¯ v i,N +1 ) , the optimal Θ ∗ and Θ ∗ can beobtained from v ∗ and v ∗ , respectively. Algorithm 2:
Proposed BSO-based alternating iterativealgorithm to problem (P1) with given t = ¯ t Input: P , N , G , t , N , { g i , h i , ∀ i ∈ K} ; Output: P ∗ , W ∗ , F ∗ , Θ ∗ , Θ ∗ ; Initialize : k ← , t ← ¯ t , Θ (0)1 and Θ (0)2 ; repeat Calculate W ( k +1) , F ( k +1) and P ( k +1) from Algorithm 1; Update V ( k +1) i by solving SDP in (34) and recover v ( k +1) i ( Θ ( k +1) i ) from V ( k +1) i ; k ← k + 1 ; until The optimal objective value of (P1) converges; Return { P ∗ , W ∗ , F ∗ , Θ ∗ , Θ ∗ } as a solution to (P1). Based on the solutions to the two sub-problems (P3) and(P4), we devise an efficient iterative algorithm summarized inAlgorithm 2. Specifically, given t = ¯ t , the algorithm starts withcertain feasible values of Θ (0)1 and Θ (0)2 . Next, given a fixedsolution { Θ ( k )1 , Θ ( k )2 } in the k -th iteration, we first obtain theoptimal W ( k +1) , F ( k +1) and P ( k +1) from Algorithm 1. Then,we update the phase shift matrices Θ ( k +1)1 and Θ ( k +1)2 usingthe SDR technique to solve problem (P4a) in the ( k + 1) -thiteration. The process repeats until convergence. At last, weobtain the optimal energy transmission time t ∗ via a simple one-dimensional search method over t ∈ (0 , , e.g., golden-section search [33] or the data-driven-based search [34], whichis omitted here for brevity. C. Convergence and Complexity Analysis
The proposed BSO algorithm alternatingly solves twosub-problems (P3) and (P4) that optimize { W , F , P } and { Θ , Θ } , respectively. Following the Theorem 3 in [30],the BCD method used in Algorithm 1 converges and theobjective value of (P3) after optimization is non-increasingcompared to that achieved by the initial input parameter.Besides, by our design, the randomization method used tosolve (P4a) also guarantees that the objective is non-increasingafter optimization. Due to the equivalence in Proposition 1 ,now that the objective of (P1) is non-decreasing in both thealternating steps and the optimal value of (P1) is boundedabove, we conclude that the proposed Algorithm 2 convergesasymptotically. In a practical setup, we will show the numberof alternating iterations consumed by Algorithm 2 untilconvergence in simulation section.We then analyze the complexity of Algorithm 2. Here, weconsider
N > M ≥ K in a practical IRS-assisted multiuserMISO WPCN. The complexity of problem (P3) is dominatedby the calculation of W ∗ , which requires calculating theeigenvalue decomposition of an M × M matrix B withcomplexity of O ( M ) [35]. The SDP problem (P4a) can besolved with a worst-case complexity of O (( N +1) . ) [36]. Aswe will show later in Fig. 9, the number of alternating iterationsused by Algorithm 2 until convergence is of constant order,i.e., O (1) , regardless of the value of M and N . Therefore, theoverall complexity of Algorithm 2 is O ( M + ( N + 1) . ) .V. S IMULATION R ESULTS
In this section, we provide numerical results to evaluate theperformance of the proposed IRS-assisted MISO WPCN. In allsimulations, we consider a two-dimensional (2D) coordinatesystem as shown in Fig. 3, where the HAP and IRS are locatedat (0 , and (4 , , the WDs are uniformly and randomlyplaced in a circle centered at ( d c , with radius equal to 2m [23]. To account for the small-scale fading, we assumethat all channels follow Rayleigh fading and the distance-dependent path loss is modeled as L = C ( dd ) − α , where C is the constant path loss at the reference distance d , d denotesthe link distance, and α denotes the path loss exponent. Toaccount for the heterogeneous channel conditions and avoidsevere signal blockage, we set different path loss exponents ofthe HAP-IRS, IRS-WD i and HAP-WD i channels as . , . and . , respectively. For simplicity, we assume equal weights ω i = 1 in all simulations.Unless otherwise stated, the parameters used in the simu-lations are listed in Table I, which corresponds to a typicaloutdoor wireless powered sensor network similar to the setupsin [18] and [23]. The number of random vector for Gaussianrandomization is set as 100 and the stopping criteria for theproposed algorithm is set as − . All the simulation resultsare obtained by averaging over 1000 independent channelrealizations. HAP WD IRS ... WD WD K-1 WD K (0,0) y x(4,3) Fig. 3. The placement model of simulation setup.TABLE IS
YSTEM P ARAMETERS
Parameter Description Value P Maximum transmission power of HAP dBm η Energy harvesting efficiency . C Fixed path loss at reference distance dB N Noise power at receiver antenna − dBm K Number of WDs 4 M Number of HAP antennas 6 N Number of reflecting elements 30 d c Distance between the HAP and WDs m E (2) i Circuit energy consumption of WD i − J [37] ω i Weight factor of WD i In addition, we select three representative benchmark meth-ods for performance comparison:1)
Passive beamforming optimization (PBO) : We set uni-form energy beamforming (i.e., W = I M ) and MMSEreceive beamforming in line 3 of Algorithm 2. Then, theuser transmit power P i and passive beamforming of theIRS { Θ , Θ } are optimized alternatively in an iterativemanner similarly to our proposed method. This methodcorresponds to the case that optimizes only the passivebeamforming of the considered WPCN.2) Active beamforming optimization (ABO) : In this case,the phase shifts of all reflecting elements at the IRS forboth WET and WIT are fixed and uniformly generatedas θ i,n ∈ [0 , π ] . The other variables are optimized usingour proposed method. This method corresponds to thecase where only the active beamforming of the HAP isoptimized.3) Without IRS : All WDs first harvest energy from theHAP and then transmit independently to the HAP. Thiscorresponds to the method in [7].For fair comparison, we optimize the resource allocations inall the benchmark schemes. The details are omitted due to thepage limit.Fig. 4 shows the impact of the maximum transmit powerof the HAP (i.e., P ) to the WSR performance. As excepted,the WSR of all schemes increases with P because the WDsare able to harvest more energy when the transmit power ofHAP is higher. The joint optimization method achieves evidentperformance advantage over the other methods. In particular,
10 15 20 25 30 35 P (dBm) W e i gh t ed s u m r a t e ( bp s / H z ) Proposed schemePBO schemeABO schemewithout IRS
Fig. 4. The WSR performance versus the maximum transmit power of theHAP.
10 20 30 40 50 60 N W e i gh t ed s u m r a t e ( bp s / H z ) Proposed schemePBO schemeABO schemewithout IRS
Fig. 5. The WSR performance versus the number of IRS reflecting elements. the performance gap between the proposed scheme with thebenchmark methods increases with P , which demonstratesits efficient usage of the harvested energy. It is also worthmentioning that even the IRS-assisted method with fixed phaseshifts achieves better performance than that without the IRSthanks to the array energy gain provided by the IRS.In Fig. 5, we study the impact of number of reflectingelements N on the WSR performance when the value of N varies from 10 to 60. We observe an evident increase ofthe WSR for the three IRS-assisted methods. In particular,compared to the ABO scheme, the slope of increase is largerfor the proposed scheme and the PBO methods, becausethey can achieve extra beamforming gain besides the arraygain of the IRS. By jointly optimizing the active and passivebeamforming, our proposed scheme significantly outperforms Distance between the HAP and WDs, d c (m) -2 -1 W e i gh t ed s u m r a t e ( bp s / H z ) Proposed schemePBO schemeABO schemewithout IRS
Fig. 6. The WSR performance versus the distance between the HAP andWDs.
Number of antenna, M W e i gh t ed s u m r a t e ( bp s / H z ) Proposed schemePBO schemeABO schemewithout IRS
Fig. 7. The WSR performance versus the number of HAP antennas. the PBO and ABO schemes. On average, the proposedjoint optimization method achieves . , . and . higher throughput than the three benchmark methods,respectively.Fig. 6 investigates the impact of the WDs deploymentlocation to the WSR performance by varying d c . We alsosee that the proposed scheme achieves evident performanceadvantages over the three benchmark methods. As expected, theperformance gain of all the methods decreases as d c increases,because as the WDs move further away from both the HAPand IRS, and suffering from more severe signal attenuationin both energy harvesting and information transmission. Theperformance gain is especially evident when d c is large, e.g. d c > m, where the without-IRS scheme achieves very lowrate (less than − ) while the proposed scheme still maintains Number of WDs, K W e i gh t ed s u m r a t e ( bp s / H z ) Proposed schemePBO schemeABO schemewithout IRS
Fig. 8. The WSR performance versus the number of WDs. relatively high rate (around ten times larger). This is becausethe WDs are unable to efficiently harvest sufficient energy forinformation transmission without the assistance of the IRS.Then, we compare in Fig. 7 the WSR performance of allthe schemes when the number of the HAP antennas (i.e., M ) changes. It is observed that the WSR performance ofall the methods increases with M because of the higherspatial diversity gain. We also notice that our proposed schemeand the PBO scheme produce much better performance thanthe other two schemes due to the higher beamforming gain.Meanwhile, the performance of ABO scheme even performsclose to the without-IRS scheme, which implies the importanceof optimizing the passive beamforming to the throughputperformance.In Fig. 8, we evaluate the WSR performance versus thenumber of WDs (i.e., K ) for all the methods. Here, we vary K from 2 to 10. It can be observed that the WSR performanceincreases with the number of WDs for all methods due to thebenefit of multiuser diversity. Meanwhile, the performance gapbetween the three IRS-assisted methods and the without-IRSscheme gradually increases with K . This is because the uplinkinformation transmission becomes interference-limited whenthe number of WDs is large. As a result, the optimal solutionwill allocate more time for transmitting information, which inconsequence decreases the WET phase duration. In this case,the IRS becomes a critical factor that effectively increases theharvested energy of the WDs within the limited energy transfertime.We then show in Fig. 9 the convergence rate of Algorithm 2,for which the convergence is proved in Section IV.C. In partic-ular, we plot the average number of iterations required until thealgorithm converges under 100 independent simulations. Here,we investigate the convergence rate when either the numberof HAP antennas (i.e., M ) or IRS reflecting elements (i.e., N )varies. With fixed N = 30 in Fig. 9(a), we see that the numberof iterations used till convergence does not vary significantly M i t e r a t i on s (a) Average number of iterations till convergence (N=30)
20 30 40 50 60 70 N i t e r a t i on s (b) Average number of iterations till convergence (M=6) Fig. 9. The average iteration number of convergence of Algorithm 2, (a) as afunction of M under fixed N = 30 ; and (b) as a function of N under fixed M = 6 . as M increases. Similarly in Fig. 9(b), with a fixed M = 6 ,we do not observe significant increase of iterations when N increases from 20 to 70. Besides, all the simulations performedin Fig. 9 require at most 20 iterations to converge. Therefore,we can safely estimate that the number of iterations used tillconvergence is of constant order, i.e., O (1) . This indicatesthat the proposed method enjoys fast convergence even in anetwork with a large number of active antennas at the HAP orpassive reflecting elements at the IRS.To sum up, our simulation results show that the proposedjoint beamforming and power control optimization achievessuperior throughput performance in MISO WPCNs undervarious setups. Meanwhile, we observe that, between thetwo better performing benchmark method, the PBO methodoutperforms the ABO scheme in all simulations. This indicatesthe importance of a refined passive beamforming design toachieve the high beamforming gain provided by the massivereflecting elements. Nonetheless, the significant performancegap between the PBO scheme and our proposed schemeconfirms the benefit of joint active and passive beamformingoptimization in enhancing the throughput performance of IRS-assisted WPCNs.VI. C ONCLUSIONS AND F UTURE W ORK
In this paper, we have studied an IRS-assisted multiuserMISO WPCN. Specifically, the WSR optimization problemwas formulated to jointly optimize the energy transmissiontime, the user transmit power, the active beamforming of theHAP and passive beamforming of IRS in both the UL andDL transmissions. To tackle this non-convex problem, wefixed the passive beamforming of the IRS and converted theoriginal problem to an equivalent WMMSE problem, whichwas efficiently solved by a BCD method. Likewise, givenuser transmit power and active beamforming of the HAP, weoptimized the passive beamforming of the IRS by the SDR technique. This leads to a BSO-based iterative algorithm toupdate the two sets of variables alternately. At last, we appliedan one-dimensional search method to obtain the optimal WETtime. By comparing with representative benchmark methods,we showed that the proposed joint optimization achievessignificant performance advantage and effectively enhancesthe throughput performance in multi-user MISO WPCNs underdifferent practical network setups.Finally, we conclude the paper with some interesting futureworking directions. First, it is interesting to consider a practicalnon-linear energy harvesting model, such that the active andpassive beamforming design in the DL energy transfer mustbe adapted to improve the energy harvesting efficiency ofall users. In addition, it is also promising to consider morerealistic imperfect CSI case, where the knowledge of thecascaded HAP-IRS-WD channels are under uncertainty dueto channel estimation error. To tackle the problem, we mayinvestigate the robust transmission design for IRS-assistedMISO communication systems under a stochastic CSI errormodel. Moreover, although some recent works have consideredIRS with only a finite number of phase shifts at each element. Inthis case, the beamforming problem becomes very challengingdue to the combinatorial phase shift variables and the strongcoupling with the other system design parameters. One possibleway is to introduce a learning-based discrete beamformingmethod for reducing the computational complexity. At last, itis also challenging to extend the considered network model toother practical setups, such as full-duplex transmission, cluster-based cooperation, hardware-constrained reflection at the IRS,and non-interference scenario, etc.A PPENDIX ROOF OF P ROPOSITION Proof:
Firstly, by employing the well-known rate-MMSEequivalence established in [30], we have log(1 + γ i ) = log([ e MMSE i ] − ) , (35)where e MMSE i denotes the MMSE of received signal s i fromWD i , and it is expressed as e MMSE i = min e i , (36)where e i is defined as (11). Then, by substituting it into (35),we have log(1 + γ i ) = log([min e i ] − ) = max log( e i − ) . (37)Consider the following equality log( x − ) = max y ≥ (log( y ) − ( xy ) + 1) , (38)where the optimal solution is achieved at y ∗ = x − . Thus, werewrite (37) as log(1 + γ i ) = max log( e i − )= max q i ≥ (log( q i ) − ( q i e i ) + 1)= min q i ≥ (cid:0) ( q i e i ) − log( q i ) − (cid:1) . (39)Accordingly, given t = ¯ t , problem (P1) can be transformedinto the equivalent problem as (P2). A PPENDIX ROOF OF P ROPOSITION Proof:
The Karush-Kuhn-Tucker (KKT) conditions of (P3)with respect to W ∗ are ¯ AW ∗ = , (40) µ ∗ i ≥ , µ ∗ ≥ , W ∗ (cid:23) , ∀ i ∈ K , (41) µ ∗ (cid:16) tr ( W ∗ ) − P (cid:17) = 0 , (42) µ ∗ i (cid:16) (1 − ¯ t ) P i + E (2) i − η ¯ t tr ( b i b Hi W ∗ ) (cid:17) = 0 , ∀ i ∈ K , (43)where ¯ A = (cid:80) i ∈K µ ∗ i η ¯ t b i b Hi − µ ∗ I .In practice, we always find a rank-one energy beamformingmatrix by using the derived optimal conditions in (40)-(43). Wefirst consider the case of µ ∗ i = 0 and µ ∗ > . In this case, wehave W ∗ = from (40) since ¯ A = − µ ∗ I , which contradictsthe complementary slackness condition (42). Also, for thecase where µ ∗ i > and µ ∗ = 0 , W ∗ = from (40) since ¯ A = (cid:80) i ∈K µ ∗ i η ¯ t b i b Hi , which contradicts the complementaryslackness condition (43). Hence, both µ ∗ and µ ∗ i are greaterthan zero, i.e., µ ∗ > and µ ∗ i > , ∀ i ∈ K .Next, we denote B = (cid:80) i ∈K µ ∗ i η ¯ t b i b Hi . Let the eigenvaluedecomposition of matrix ¯ A be ¯ A = U ( Λ − µ ∗ I ) U H , where U ∈ C M × M and Λ = diag ( λ , · · · , λ M ) ∈ C M × M with λ ≥ · · · ≥ λ M are the eigenvector matrix and eigenvaluematrix of B , respectively. Since µ ∗ i > , ∀ i ∈ K , B isalways a positive semidefinite, and resulting in the non-negativeeigenvalues λ j , for j = 1 , · · · , M . Thus, for ¯ A to have non-positive eigenvalues, i.e., λ j − µ ∗ ≤ , we obtain ≤ λ j ≤ µ ∗ .When µ ∗ i > , ∀ i ∈ K , we have rank ( B ) > , the maximumeigenvalue λ > .Note that if µ ∗ > λ , ¯ A becomes a full-rank and negative-definite matrix. Thus, we obtain W = from (40), whichcontradicts the complementary slackness condition (42) since µ ∗ > . Therefore, we obtain the optimal dual variable µ ∗ as µ ∗ = λ . We define ¯ Au = , where u is the unit-normeigenvector of B corresponding to the maximum eigenvalue λ . From (40) and (41), we obtain the optimal W ∗ = (cid:15) u u H for any (cid:15) ≥ . Next, we find (cid:15) from (42), i,e., P − tr ( W ∗ ) = 0 due to µ ∗ > , which leads to tr ( W ∗ ) = (cid:15) = P . (cid:4) R EFERENCES[1] L. Chettri and R. Bera, “A comprehensive survey on Internet of Things(IoT) toward 5G wireless systems,”
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