Sum Rate Maximization in IRS-assisted Wireless Power Communication Networks

—Wireless powered communication networks (WPC-Ns) are a promising technology supporting resource-intensive devices in the Internet of Things (IoT). However, their transmission efﬁciency is very limited over long distances. The newly emerged intelligent reﬂecting surface (IRS) can effectively mitigate the propagation-induced impairment by controlling the phase shifts of passive reﬂection elements. In this paper, we integrate IRS into WPCNs to assist both the energy and information transmission. We aim to maximize the uplink (UL) sum rate of all IoT devices (IoTDs) by jointly optimizing the time allocation variable, energy beam matrix at the power transmitting base station (PTBS), receive beamforming matrix at the information receiving base station (IRBS), and the phase shifts of the IRS both in the UL and downlink (DL) subject to time allocation constraint, together with transmit power constraint for the PTBS and unit modulus constraints. This problem is very difﬁcult to solve directly due to the highly coupled variables, which results in the optimization problem taking neither linear nor convex form. Hence, we decouple this problem into three subproblems by using the block coordinate descent (BCD) method. The UL receive beamforing matrix and phase shift are alternatively optimized in the UL optimization subproblem with ﬁxed time allocation and the DL variables. The DL optimization subproblem is solved by the proposed successive convex approximation (SCA) algorithm. Simulation results demonstrate that the performance of integrating IRS and WPCNs outperforms traditional WPCNs. Besides, the results show that IRS is an effective method to preserve the tradeoff of energy efﬁciency and transmission efﬁciency in the IoT.

W ITH the explosive development of the big data era, the number of Internet of Things (loT) devices is rapidly growing.According to the report of Cisco, the number of connected IoT devices (IoTDs) will attain up to 29.3 billion by the year of 2023, in which more than half are mobile devices [1].This brings more challenges for wireless communication systems to enhance mobile broadband, together with providing ultra-reliable and low-latency communications.More and more wireless devices not only need higher communication connectivity but also sustained energy supply in the future.The advent of the 5th generation (5G) of wireless communication technology has met these requirements to some extent, but there is no single enabling technology that can support the demands of all 5G applications in the process of standardization [2].Hence, the technology of wireless powered communication networks (WPCNs) has attracted much attention [3]- [8].In WPCNs, IoTDs first harvest energy from a power transmitting base station (PTBS) and then use the harvested energy to transmit a signal to the information receiving base station (IRBS).However, WPCNs still suffer high attenuation of microwave energy and information signals over distance [9], [10], even though there are some measurements were used to improve the performance of WPCNs.In [11], a non-linear energy harvesting model was used, and resource allocation was investigated to improve the performance of WPCNs.In [12] and [13], the authors improved energy efficiency and sum-throughput by using orthogonal frequency division multiple access (OFDMA) and non-orthogonal multiple access (NOMA) techniques.However, these works mainly focus on resource allocation in WPCNs, and also suffer high attenuation in energy and information transfer.Therefore, more efficient technologies to guarantee that WPCNs can be adapted to the loT environment must be proposed.
More-recently, intelligent reflecting surface (IRSs) have been proposed as one of the beyond-5G techniques to improve both the spectrum and energy efficiency of wireless communication systems [14]- [18].An IRS is one kind of manmade metasurface that is composed of many passive reflecting elements.These elements can independently adjust the phase shift of incident signals under the instructions of an IRS controller, thereby providing secondary transmission channels to enhance the received signals [19], [20].Recent research about IRS-aided wireless communication has attracted extensive attention, such as weighted sum rate maximization [21]- [23], assisted unmanned aerial vehicle (UAV) communications [24], energy efficiency maximization [25], phase shift design of IRS [26]- [31].Specifically, in [26], Wu et al. minimized the total transmit power of an access point (AP) by jointly optimizing the AP's transmit beamforming and IRS's reflecting beamforming, which guaranteed the constraints of the users' individual signal-to-interference-plus-noise ratio (SINR).An alternative optimization design for the phase shift of an IRS and the precoder at the transmitter for minimizing the symbol error rate was studied in [27].Zhou et al. maximized the sum rate of all the multicasting groups in [28] by jointly optimizing the precoding matrix at the base station (BS) and the reflection coefficients at the IRS satisfying both the power and unit-modulus constraints.Existing works for IRSaided wireless communication mainly considered frequencyflat channels and perfect channel state information (CSI) was known at the transmitter.Yang et al. [29] studied an IRS enhanced OFDM system with frequency-selective channels by exploiting a novel IRS elements grouping method and proposed a practical transmission protocol with channel estimation.For an IRS-enhanced OFDM system, Zheng et al. [30] executed channel estimation by proposing a practical transmission protocol and discussed reflection optimization under unit-modulus constraints.Furthermore, IRSs were also used in improving physical layer security by maximizing the secrecy sum rate of IRS-aided secure wireless systems [32]- [36].
However, little work has studied IRS-assisted WPCNs.Wu et al. [37] investigated weighted sum power maximization for IRS-aided simultaneous wireless information and power transfer (SWIPT) by jointly optimizing the AP's transmit precoders and the IRS's reflect phase shifts, which is very different from WPCN systems.Some sensors are only equipped with limited batteries in practical IoT networks, it is very difficult for them to operate continuously.Hence, in this paper, we propose an IRS-aided WPCN system as shown in Fig. 1, in which IoTDs are only equipped as an energy storage device (ESD) and harvest energy from the PTBS before transmitting information to the IRBS.With the aid of the IRS, both the energy harvesting and information transmitting processes can be enhanced by carefully adjusting the phase shifts of the reflecting elements of the IRS.We study the sum rate in the uplink (UL) maximization problem by jointly optimizing the time allocation variable, energy beam matrix and receive beamforming variables, and phase shifts of the IRS both in the downlink (DL) and the UL.This problem is challenging to solve due to the highly coupled optimization variables.The main contributions of this paper can be summarized as follows: 1) It is the first work to introduce IRS into WPCNs to assist the communications of IoTDs in the proposed system.
In particular, we maximize the UL sum rate by jointly optimizing the time allocation variable, the energy beam matrix and phase shifts in the DL, receive beamforming matrix and phase shifts in the UL as well as guaranteeing the limitation of the transmit power of the PTBS and the unit modulus constraints of phase shifts both in the DL and the UL.

2) We apply the block coordinate descent (BCD) technique
to solve the optimization problem with highly coupled variables.Then, we optimize the UL related variables, wherein the receive beamforming matrix can be obtained by exploiting the equivalent relationship between the sum rate maximization and its weighted minimum mean-square error (WMMSE).Besides, due to the nonconvexity of the unit modulus constraint, we propose an efficient iterative algorithm based on the majorizationminimization (MM) algorithm to obtain the optimal UL phase shifts of the IRS. 3) Given a fixed time allocation variable, received beamforming matrix and UL phase shifts for the IRS in the information transmitting stage, the original problem also can be transformed into its WMMSE.After introducing the slack variables, we transform this problem into a convex problem for the energy beam matrix, for which it is straightforward to obtain the optimal solutions.Furthermore, the successive convex approximation (SCA) method is adopted to obtain the optimal DL phase shifts for the IRS.4) Simulation results demonstrate the convergence of the proposed algorithm, and the UL sum rate of the IoTDs in the WPCNs can be significantly improved by the aid of the IRS when compared to traditional single WPCNs.The rest of this paper is organized as follows.In Section II, the IRS-assisted WPCNs communication model is presented and the UL sum rate maximization problem is formulated.In Section III, the UL sum rate maximization problem is decoupled into three subproblems, which are separately solved.Then the BCD algorithm is used to obtain the optimal solutions for the original problem.In Section IV, simulation results demonstrate the effectiveness of introducing an IRS into WPCNs.Finally, we conclude the paper in Section V.
Notations: Re{a} represents the real part of the complex value a. Boldface upper case H and lower case h separately denote matrix H and vector h.C N ×1 denotes the set of N ×1 complex vectors.x denotes the 2-norm of vector x.A B denotes the Hadamard product of matrix A and matrix B.  I.

A. System Model
In this paper, we consider an-IRS assisted WPCNs consisting of one PTBS, one IRBS, k ∈ {1, ..., K} IoTDs and one IRS, which is deployed to assist in the communications of K IoTDs with single antenna over a given frequency band.The number of antennas at the PTBS and the IRBS, and reflecting units at the IRS are denoted by N b and N r , respectively.The system model is shown in Fig. 1, in which the PTBS exploits wireless power transfer, and the other IRBS is responsible for the information processing of the IoTDs.We assume that all IoTDs are only equipped with an ESD and need to replenish energy from the energy transmit signals that are sent by the PTBS.The baseband DL channels from the PTBS to the kth The DL channel from the PTBS to the kth IoTD G The DL channel from the PTBS to the IRS g r,k The DL channel from the IRS to the kth IoTD h k The UL channel from the kth IoTD to the IRBS H The UL channel from the IRS to the IRBS h r,k The UL channel from the kth IoTD to the IRS V The energy beam matrix of the PTBS W The receive beamforming matrix of the IRBS α Time allocation variable Φ d The DL diagonal reflection-coefficients matrix Φu The UL diagonal reflection-coefficients matrix θ d The DL phase shifts vector of the IRS θ u The UL phase shifts vector of the IRS IoTD, PTBS to IRS and IRS to the kth IoTD are denoted by g k ∈ C 1×N b , G ∈ C Nr×N b , g r,k ∈ C 1×Nr , respectively.The UL channels from the kth IoTD to the IRBS, IRS to the IRBS and the kth IoTD to the IRS are denoted by respectively.In particular, we adopt the time split protocol proposed in [38], [39].In the first αT (0 < α < 1) amount of time, the PTBS broadcasts energy signals in the DL to transfer energy to all IoTDs.In the second (1 − α)T amount of time, all IoTDs transmit their independent signals to the IRBS in the UL, in which the IRS can boost both the information and energy signals by designing the phase shifts of the reflective elements in the DL and the UL.Without loss of generality, the time block T is normalized to unit in the rest of this paper.Let θ l ∈ [0, 2π] be the phase shift of the ith reflection element of the IRS which is separately denoted as θ d l and θ u l in the DL and the UL phases.Define the diagonal reflection-coefficients matrix as Nr } for the UL phase, respectively.
Specifically, in the DL, the PTBS broadcasts energy to all IoTDs through N b energy beams.The energy signal can be expressed as where v l ∈ C N b ×1 denotes the lth energy beam and s l denotes the energy carrying signal that are assumed as independent and identically distributed (i.i.d.) random variables (RVs) with zero mean and unit variance.Then the transmit power of the PTBS can be expressed as where n is the additive white Gaussian noise (AWGN) and n ∼ CN (0, σ 2 ).
The kth IoTD harvests energy in the DL which can be expressed as where g k = g k + g r,k Φ d G denotes the channel between the PTBS and the kth IoTD, and 0 < ζ k < 1 is the energy harvest efficiency of the kth IoTD.The average transmit power1 of the kth IoTD in the UL phase of information transmission is where In the UL phase, each IoTD transmits information by using the harvested energy.Then the transmit signal of the kth IoTD is given as where s k denotes the information signal transmitted by the kth IoTD to the IRBS, and E[s k 2 ] = 1 and E[s k s l ] = 0, for l = k.The received signal at the IRBS in the UL phase can be expressed as where n ∼ CN (0, Then, the transmit rate (in bit/s/Hz) of the kth IoTD can be expressed as where h k h k + HΦ u h r,k denotes the channel between the kth IoTD and the IRBS in the DL, denotes the receive beamforming matrix.

B. Problem Formulation
We aim to maximize the sum rate of all IoTDs in the UL phase by jointly optimizing the time allocation variable α, energy beam matrix V, receive beamforming matrix W and phase shifts in the DL and the UL, which can be formulated as where Nr } denote the phase shifts vectors of the IRS in the DL and UL, respectively.P d max denotes the maximum transmit power of the PTBS.Notice that there exists the coupling effect among the beamforming matrices W, V and the phase shifts θ d , θ u , and as such this optimization problem is difficult to solve.In the following, we propose an effective algorithm for solving Problem P 0 .

III. JOINT OPTIMIZING SUM RATE OF ALL IOTDS
In this section, in order to solve Problem P 0 , the BCD technique is used to decouple the optimization variables.Specifically, it can be decoupled into three more tractable subproblems.Then, the formulated problem can be solved by alternately optimizing these subproblems.

A. Optimizing the Time Allocation Variable α
Given W, V, θ u ,θ d , Problem P 0 for optimizing time allocation variable α can be transformed as where Therefore, Problem P 1 is a convex optimization problem, which can be solved by the interior point method.

B. Optimizing the Receive Beamforming Matrix W and the UL Phase Shift θ u
In this section, we aim to optimize the receive beamforming matrices W and θ u with fixed α, V, θ d .The original problem can be reformulated as By exploiting the relationship between the data rate maximization problem and its WMMSE problem, and introducing a set of auxiliary variables , Problem P 2 is equivalent to a more tractable problem, which is written as where e u k (W, θ u ) denotes the mean-square error (MSE) of the kth IoTD, which are respectively given by Problem P 2−E1 is more tractable than Problem P 2 .This is because with given IRS phase shift vector θ u and Υ u , the objective function of Problem P 2−E1 is a convex function for the receive beamforming matrix variables.While fixing the other one, we can exploit the BCD algorithm to solve P 2−E1 and the specific process is as follows.
1) Optimizing the Receive Beamforming Matrix: Once the phase shift θ u and the auxiliary variable Υ u are fixed, we can obtain the receive beamforming matrix by taking the firstorder derivative of the objective function in Problem P 2−E1 with respect to w k and setting it equal to zero.So the receive beamforming vector is expressed as where 2) Optimizing the UL Auxiliary Variable: With fixed uplink phase shift θ u and receive beamforming matrix W, we can obtain the solution by setting the first-order derivative of the objective function of Problem P 2−E1 with respect to Υ u k to zero.
3) Optimizing the UL Uplink Phase Shift of the IRS: Under the condition of fixing the auxiliary variable Υ u and the receive beamforming matrix W, we now focus on the IRS uplink phase shift θ u .After substituting (12) into the objective function of Problem P 2−E1 , Problem P 2−E1 can be reformulated as where and every part of it can be respectively expanded We collect the diagonal elements of T .The first term of the objective function of Problem P 2−E3 can be formulated as where denotes the Hadamard product.The second and third terms of the objective function of Problem P 2−E3 can also be rewritten as respectively.Moreover, define Ψ A B. Then Problem P 2−E3 can be simplified as Problem P 2−E4 is non-convex due to the unit modulus constraint.In the following, the MM algorithm is exploited to solve this non-convex problem.In the first step of the MM algorithm, we need to find a surrogate function g(φ u |φ t u ) as the upperbound of f (φ u ), where t denotes the iteration index.Then, in the second step, we update φ u according to ).We will construct g(φ u |φ t u ) by exploiting the proposition in [41], which can be constructed as ) It's evident that the surrogate function g(φ u |φ t u ) is more tractable than the original objective function f (φ u ).Then Problem P 2−E4 at the tth iteration can be rewritten as From the expression of g(φ u |φ t u ), we can see that (φ t u ) H (λ max I Nr −Ψ)φ t u is a constant for a given φ t u in the tth iteration and φ H u λ max I Nr φ u = N r λ max because φ H u φ u = N r .So Problem P 2−E5 is equivalent to the following problem We can obtain the optimal solution of the above problem Then, the optimal solution of Problem P 2−E3 can be obtained as According to the above discussions, the specific procedure of solving Problem P 2 by using the MM algorithm is concluded in Algorithm 1.
The complexity of Algorithm 1 mainly lies in Step 2. In this step, the complexity of calculating W n+1 in ( 13) is O(max{KN 3 b , KN b N 2 r }) and the complexity of calculating Υ u in ( 14) is O(K).By using the MM algorithm to solve Problem P 2−E3 , the complexity of calculating the eigenvalue λ max of Ψ is O(N 3 r ).In each iteration of the MM algorithm, the main complexity lies in the calculation of φ t+1 u in (27), which is O(N 2 r ).Then, the complexity of the MM algorithm is O(N 3 r + t mm max N 2 r ), where t mm max is the total iteration number of the MM algorithm.Then, the complexity of Algorithm 1 can be obtained as Algorithm 1 Joint optimization of W and θ u > ξ && n < t max Calculate W n+1 by using (13), Calculate (Υ u ) n+1 by using ( 14), Solving Problem P 2−E3 to obtain (θ u ) (n+1) by exploiting the MM algorithm, , n = n + 1. end while 3: Output optimal W and θ u .

C. Optimizing the Energy Beam Matrix V and the DL Phase Shift θ d
In this section, we focus on optimizing the energy beam matrix V and phase shift θ d in the DL energy transmission with given α, W, θ u .So the original problem (8) can be formulated as As in Problem P 2 , Problem P 3 is equivalent to the following problem where denotes auxiliary variables and e d k (V, θ d ) represents the MSE of the kth IoTD in the DL, which is given by where ).By substituting (4) into (31), we can obtain where Optimizing the DL Auxiliary Variable: When we fix the phase shift θ d and receive beamforming matrix V, the DL auxiliary variables Υ d can be obtained by setting the firstorder derivative of the objective function of Problem P 3−E1 with respect to Υ d k to zero.Then we have 2) Optimizing the Energy Beam Matrix : When we fix the phase shift θ d and the auxiliary variable Υ d , Problem P 3−E1 can be transformed into where The second constraint of Problem P 3−E3 holds with equality at the optimum solution.
Proof: Assuming (S , V ) as the optimal solution of Problem P 3−E3 , and that V * denotes the optimal solution of Problem P 3−E2 , then we have S k ≤ g k V .If S k < g k V , the objective function value of Problem P 3−E3 at (S , V ) is larger than the objective function value of Problem P 3−E2 at V * .Hence V is optimal for P 3−E2 if and only if V = V * is optimal for Problem P 3−E3 , where However, Problem P 3−E3 is still non-convex because the second constraint is still non-convex.We assume g k V = 0 and V t denotes the energy beam matrix in the tth iteration.This constraint can be approximated by (36) Then, Problem P 3−E2 can be transformed into Problem P 3−E3 is convex for V and S. Hence it can be solved by using the interior point method.
3) Optimizing the DL Phase Shift : Given energy beam matrix V and auxiliary variable Υ d , Problem P 3−E2 can be rewritten as By substituting It is clear that e d k ( V, θ d ) is non-convex for θ d .But from its expression, we recognize that it has the form of a convex function minus a convex function for φ d .Then, we can adopt the SCA method to approximate it, which is given by which is the first-order Taylor series expansion of e d k ( V, θ d ) and (φ d ) t represents the value of the variable at the tth iteration.
Hence, Problem P 3−E5 can be approximated by the following problem where By removing the other constants, Problem P 3−E6 can be transformed as where Then, the optimal solution of Problem P 3−E6 is given by The optimal solution of Problem P 3−E5 can be given by We conclude the SCA algorithm for solving Problem P 3−E5 in Algorithm 2.
The complexity of Algorithm 2 is mainly dominated by the calculation of q t in step 2, the complexity of which is O(K 2 N 2 r ).Then, the total complexity of Algorithm 2 is O(t max 2 K 2 N 2 r ), where t max 2 denotes the total number of iterations that guarantees the convergence of Algorithm 2.
The specific procedure of solving Problem P 3 is concluded in Algorithm 3.

D. Overall Algorithm to Solve Problem P 0
According to the above investigations, we propose the specific procedures of the BCD algorithm in Algorithm 4 to solve Problem P 0 .We can determine that the objective function value of Problem P 0 is increasing, which is guaranteed in Step 3 and Step 4.Moreover, the IoTDs exist maximum transmit power constraints in the UL phase, and the objective function value of Problem P 0 has an upper bound.Then, Algorithm 4 is guaranteed to converge.
Based on the complexity of Algorithm 1-3, the complexity of the proposed algorithm mainly depends on Steps 3 and 4. Hence, the complexity of Algorithm 4 is on the order of O(max{KN In this section, numerical simulations are provided to inthe performance of the proposed algorithms for improving the UL sum rate of the IoTDs in the IRS-assisted WPCNs.The simulated system model for the IRS-assisted WPCNs is shown in Fig. 2, where the PTBS and IRBS are located at (0,0) m and (800,0) m, respectively.The IRS is located at (400, 20) m.The IoTDs are uniformly and randomly distributed in a circle, in which the center is located at (400,0) Algorithm 4 Joint Optimization of α, W, V, θ d and θ u 1: Initialization, initialize maximum number of iterations t max

5: ξ
where PL 0 denotes the path lose at the reference distance of d 0 , d and α are the communication distance and the path loss exponent, respectively.The parameters that are used in the simulation are listed in the Table II.
A. Convergence of the Proposed Algorithms Fig. 3 shows the convergence of the proposed algorithm under different number of reflecting elements in the IRS.It can be seen that our proposed algorithm converges after several iterations, which demonstrates the effectiveness of our proposed algorithm.Besides, it is apparent that the sum rate of the IoTDs can be improved by increasing the number of reflecting elements in the IRS.

B. Performance Comparsion
In this subsection, we compare the IRS assisted WPCNs with the following schemes.
• RandPhase: In this scheme, the phase shifts of the IRS are uniformly and independently generated in [0, 2π].We drop the phase shifts optimization both in Steps 3 and 4, and only optimize the time allocation variable and beamforming matrices both in the DL and the UL.
• No-IRS: We consider the optimization problem in traditional WPCNs, which means the IRS related channels are equal to zero.Then, in this scheme, the BCD algorithm is used to obtain the optimal time allocation and beamforming matrices both in the DL and the UL by removing the phase shifts optimization in Steps 3 and 4. Fig. 4 shows the uplink sum rate versus the number of IRS's reflecting elements M in different schemes.We know that the sum rate increases with the number of IRS's reflecting elements by using our proposed algorithm, which significantly outperforms the other schemes.Moreover, the performance gain becomes more obvious by increasing M .Specifically, the sum rate gain of the IRS assisted scheme over the No-IRS scheme is approximately 0.57 × 10 4 bits/s, and over the RandPhase scheme nearly 0.37 × 10 4 bits/s when M = 10.And this gain becomes almost 13 × 10 4 bits/s and 6 × 10 4 bits/s when M = 100.We find that the reflecting signal power can be boosted by increasing M , which can vastly improve the sum rate.Furthermore, it can bring higher reflecting beamforming gain by carefully designing the phase shifts of the IRS, and increasing the number of IRS's reflecting elements leads to much higher reflection based beamforming gain.By combining these two gains, the IRS can effectively improve the UL sum rate of the IoTDs.Sum rate (bits/s) No-IRS RandPhase IRS-Aided Fig. 5. Achievable sum rate versus the maximum transmit power.

Number of IRS elements M
Fig. 5 illustrates the uplink sum rate versus the maximum transmit power in different schemes.We find that the sum rate increases with the maximum transmit power in all the schemes.The sum rate gain in the IRS-assisted scheme outperforms the other two schemes.Specifically, the sum rate gain obtained by our proposed algorithm is above the No-IRS scheme when the maximum transmit power is increased, which verifies the effectiveness of introducing the IRS.For example, the sum rate of our proposed algorithm is approximately 16.9% and 24.2% higher than the RandPhase scheme and No-IRS scheme when the transmit power limit is 5dB.We find that the sum rate of the RandPhase scheme is slightly better than the No-IRS scheme.This is mainly because only a few of the reflecting signals are reflected towards the receivers.By using our proposed algorithm, after carefully designing the phase shifts of the IRS elements, most of the reflecting signals can be beamed to the receivers to improve the sum rate of the IoTDs.
Fig. 6 compares the sum rate in different schemes with the path loss exponents.Different from the free space PTBS-IRS and IoTD-IRS channel, we investigate the influence of the path loss exponents of the IRS-related channels.It is seen that the sum rate decreases with increasing path loss exponents α IRS .The reason is that when α IRS increases, the signal No-IRS RandPhase BCD-MM Fig. 6.Achievable sum rate versus the path loss exponent of the IRS-related channel.
attenuation in the IRS-related channel becomes serious, and the signal reflected by the IRS becomes weaker until it has the same performance as the No-IRS scheme.It also shows that the sum rate gain in the IRS-assisted scheme, by carefully optimizing the phase shifts of the IRS is much better than the RandPhase scheme, which demonstrates optimizing the phase shifts of the IRS is an effective way to alleviate attenuation and improve the performance of the IRS-assisted systems.From Fig. 6, we find that when α IRS is very small, there exists free space links in the IRS-related channels.Hence, the IRSassisted sum rate of the WPCNs can be effectively improved.Specifically, when the path loss exponent α IRS = 2, the sum rate gain of the IRS-assisted scheme is up to 4.4 × 10 5 bits/s, which is much larger than the No-IRS scheme.This illustrates that the IRS should be deployed in open space, such as at the roof of buildings and advertisement panels to make full use of its advantages.Fig. 7 shows the sum rate of all IoTDs achieved by different schemes versus the distance between the PTBS and the IoTD circle central position.We can assume this is equivalent to changing locations of IoTDs because all IoTDs are randomly distributed in this circle.It is easy to observe that our proposed algorithm outperforms the other two schemes when the circle center moves away from the PTBS.Besides, the IoTDs can harvest more energy in the IRS-assisted WPCNs model, which is mainly caused by the energy signal gain from the IRS even when IoTDs are far away from the PTBS.

V. CONCLUSIONS
In this paper, we improved the UL sum rate of WPCNs by using an IRS.Specifically, IRS-assisted WPCNs can enhance both the energy transfer and information transfer by tuning the phase shifts of the reflecting elements of the IRS.We investigated the UL sum rate maximization problem by jointly optimizing the time allocation variable, energy beam matrix and phase shift of the IRS in the DL, and receiving beamforming vector and phase shift of the IRS in the UL as well as guaranteeing the constraint of maximum transmit power of the PTBS and unit-modulus constraints of the IRS, which is a non-convex optimization problem.To tackle this challenging problem, we alternatively optimized them by using the BCD algorithm, in which we proposed a SCA algorithm to optimize the phase shift in the DL.Simulation results demonstrated that the proposed algorithms achieved better performance than other benchmarks.These results motivate us to keep exploring how to improve the performance of multi-UEs in WPCNs and how to satisfy the need for wireless battery charging which arises for devices to achieve sustainable access in energyconstrained wireless networks such as IoT networks.
Tr(A) denotes the trace operation of matrix A. (•) T , (•) * , (•) H represent the transpose, conjugate and Hermitian operators, respectively.E[•] denotes the expectation operation.diag{•} and arg{•} denote the diagonalization operation and extraction phase information extraction operation, respectively.The main notations used in this paper are illustrated in Table
The IoTDs receive the energy signals directly from the PTBS and the reflection energy signals from the IRS.Thus the kth IoTD's received energy signals from the PTBS-IoTD and the PTBS-IRS-IoTD channels are modeled as

Fig. 4 .
Fig. 4. Achievable sum rate versus the number of IRS's reflecting elements M .