I Introduction
The number of InternetofThings (IoT) devices such as temperature sensors, humidity sensors, and illuminating light sensors, have rapidly skyrocketed recently due to their tremendous demands required in various application scenarios. Such a massive number of wireless devices thus requires a scalable solution for providing ubiquitous communication connectivity and perpetual energy supply in the future. Although the lifetimes of devices can be extended by replacing and/or recharging the embedded batteries, it is unsafe and inconvenient especially in a toxic environment and rural areas. To achieve selfsustainable communication, a variety of wireless technologies has been proposed in the past such as wireless power transfer and simultaneous information and power transfer (SWIPT) [1, 2]. However, the efficiencies of wireless energy transfer (WET) and wireless information transmission (WIT) are severely affected by the distance based path loss and/or the multipath fading. Although several technologies have been proposed to overcome this issue, which include massive multipleinput (MIMO), ultradense network (UDN), etc., [3, 4], they also face other challenges in practical implementation such as high power circuit consumption and high hardware cost.
To boost the spectral efficiency of wireless systems, fullduplex (FD) transmission is a promising technology to potentially double the spectral efficiency if the selfinterference (SI) is perfectly cancelled [5]. Different from the halfduplex (HD) mode that the wireless node operates in a time division manner, i.e., the transmitter either transmits or receives signals at one time, it is able to transmit signal and receive signal over the same frequency simultaneously for the FD mode. However, due to the simultaneous transmission and reception at the same wireless node, its receiver antenna will receive the undesirable signals transmitted by its nearby transmitter antennas, thus interfering with the desired signal received at the same time. In fact, the performance of an FD system may be even worse than that of an HD system if the SI is not well suppressed. Fortunately, several SI cancellation (SIC) techniques have been proposed in the literature [6, 7, 8, 9], which generally based on the analogdomain SIC and the digitaldomain SIC. It was reported in [9] that the artofthestate of SI suppression can be up to by combing the analog domain (i.e., SIC before analogtodigital conversion (ADC) by using analog signal processing techniques) with digital domain (SIC after ADC by using digital signal processing techniques). Various works have investigated the applications of FD in the different setups, such as WET and SWIPT systems [10, 11, 12, 13, 14]. In particular, the authors in [11] studied FD wirelesspowered communication network (WPCN) and aimed at maximizing the weighted sum rate over all users by jointly optimizing the time allocation and transmit power allocation at a hybrid access point (HAP). The results showed that the FDWPCN outperforms the HDWPCN if SI can be suppressed below a certain level. Subsequently, [12] derived closedform solutions in the FDWPCN with perfect SIC case.
Recently, intelligent reflecting surfaces (IRSs) have been proposed as a promising costeffective solution to improve both spectral and energyefficiency of wireless communication systems[15, 16, 17, 18, 19, 20]. IRS is composed of 2D planar arrays of subwavelength metallic or dielectric scatterers, each of which is able to independently and smartly induce different reflection amplitudes, phases, and polarization responses on the incident signals and thereby improving quality of services of users by forming the finegrained directions of beam towards to the desirable users. In addition, IRS also has several other appealing advantages such as low profile, lightweight, and conformal geometry. More importantly, IRS can be composed of large numbers of dielectric scatterers with a very limited size. For example, it was shown in [21] that for a large IRS with reflecting elements, the electrical sizes are about square meter, which is attractive for the practical deployment. As such, the IRS can be installed on the ceilings to enhance personal WiFi network and also can be attached to the facades of buildings to assist the cellular network. Due to the above advantages, IRS has been exploited for different applications by optimizing its reflection coefficients, such as physical layer security [22, 23, 24], multicell cooperation [25, 26, 27], and unmanned aerial vehicle communication [28, 29]. While the above works focused on leveraging IRS for enhancing information transmission, IRS is also beneficial for WET and SWIPT [30, 31, 32, 33, 34]. By exploiting its large aperture with grainedfine tunable phase shifts, high passive beamforming gains can be achieved by the IRS to effectively compensate the endtoend signal attenuation. For example, [33] studied an IRSassisted SWIPT system, where a set of IRSs are deployed to assist the downlink (DL) WIT and WET from a multiantenna AP to multiple information and energy users, respectively. The results showed that the transmission range is significantly enlarged with the help of IRSs.
By far, there are only a handful of works paying attention to studying IRSaided WPCN [35, 36, 37, 38, 39, 40]. The authors in [35] and [36] studied IRSaided WPCN, where an orthogonal protocol for DL WET and uplink (UL) WIT was proposed with the goal of maximizing the common and weighted sum throughput, respectively. The authors in [37] studied an IRSempowered WPCN, where the IRS is allowed to harvest energy from an HAP, and then considered two schemes, namely, timeswitching and power splitting, to support DL WET from the HAP to the distributed users and UL WIT from the users to the HAP. In [40], a new optimization framework on dynamic passive beamforming was firstly proposed to compromise the performance and complexity for implementing IRSaided WPCNs. However, all the above works focus on the HD system, which suffers from a low spectral and energy efficiencies.
In this paper, we study an IRSaided FD WPCN for further improving the system throughput. As shown in Fig. 1, the HAP operates in an FD mode with two antennas, which are use for DL WET and UL WIT, respectively. In addition, the IRS is deployed nearby the distributed devices to enhance DL WET from the HAP to the devices and UL WIT from the devices to the HAP simultaneously. Note that although there were several works [41, 42, 43] studied IRSaided FD systems for some applications, such as the cognitive radio system, the pointtopoint system, and the multiuser multipleinput singleoutput system, the above works assumed that the SI is either a constant or is perfectly canceled by ignoring the practical quantization error introduced by the strong SI under the limited dynamic range of the ADC converter. To the best of our knowledge, it is the first work to study the IRSaided FDWPCN with finite SI. It is worth pointing out that different from the WPCN without IRS where the channels of all devices remain static within a channel coherence block, we are able to proactively generate optimized artificial timevarying channels by properly designing the IRS reflection coefficients over different time slots within each channel coherence block, thus enhancing the multiuser diversity and improving the system throughput. The main contributions of this paper are summarized as follows.

First, we study an IRSaided FDWPCN and propose three types of IRS beamforming configurations based on how the IRS is allowed to adjust its phase shifts across time. We first consider the fully dynamic IRS beamforming, where the phaseshift vectors vary with each time slot for DL WET and UL WIT. To further reduce signaling overhead and implementation complexity, we then study two special cases, namely, partially dynamic IRS beamforming and static IRS beamforming. For the former case, there are two different phaseshift vectors during the whole period with one for DL WET and the other for UL WIT. For the latter case, DL WET and UL WIT are assumed to adopt the same phaseshift vector. For the above three cases, we formulate the corresponding system throughput maximization problems by jointly optimizing the time allocation, HAP transmit power, and IRS phase shifts. It is worth noting that such a thorough study in terms of different dynamic IRS beamforming configurations have not been studied yet in the literature.

Second, we consider fully dynamic IRS beamforming optimization with perfect SIC. Since the formulated problem is nonconvex due to the highly coupled optimization variables in the objective function and nonconvex unitmodulus constraints of phase shifts, there are no standard methods for solving such nonconvex optimization problem optimally. To solve this difficulty, we propose an efficient alternating optimization (AO) algorithm by first decomposing the entire variables into two blocks, namely, time allocation and phase shifts, and then optimize each block alternately, until convergence is achieved.

Third, we consider fully dynamic IRS optimization with imperfect SIC, where the corresponding problem is more challenging than the former one due to the additional HAP transmission power involved in the objective function. To address this issue, we propose a novel penaltybased algorithm, which includes a twolayer iteration, i.e., an inner layer iteration and an outer layer iteration. The inner layer solves the penalized optimization problem, while the outer layer updates the penalty coefficient over iterations to guarantee convergence.

Fourth, we respectively study partially dynamic IRS beamforming and static IRS beamforming, respectively. Since the formulated problems are different from that of the fully dynamic IRS beamforming, we extend the AO and penaltybased algorithms to solve them. In particular, a differenceofconvex (DC) optimization method is proposed to address unitmodulus phaseshift constraints, which guarantees to converge to locally optimal solutions.

Finally, simulation results demonstrate the benefits of the IRS used for enhancing the performance of the FDWPCN, especially when the fully dynamic IRS beamforming is adopted. We also show that the IRSaided FDWPCN is able to achieve significantly performance gain compared to the IRSaided halfduplex (HD)WPCN when the SI is well suppressed. Furthermore, it is found that the system with static IRS beamforming achieves almost the same performance as the case with partially dynamic IRS beamforming when the HAP has a large transmit power budget.
The rest of this paper is organized as follows. Section II introduces the system model and problem formulations for FDWPCN with three types of IRS beamforming configurations. In Section III, we propose an AO based algorithm to solve the problem with fully dynamic IRS beamforming under perfect SIC. In Section IV, we propose a penaltybased algorithm to solve the problem with fully dynamic IRS beamforming with imperfect SIC. In Section V, we extend the algorithms to solve the problems with partially dynamic IRS beamforming and static IRS beamforming, respectively. Numerical results are provided in Section VI and the paper is concluded in Section VII.
Notations: Boldface uppercase and lowercase letter denote matrix and vector, respectively. stands for the set of complex matrices. For a complexvalued vector , represents the Euclidean norm of , denotes the phase of , and denotes a diagonal matrix whose main diagonal elements are extracted from vector . For a vector , and stand for its conjugate and conjugate transpose respectively. For a square matrix , , and respectively stand for its trace, Euclidean norm and rank, while indicates that matrix
is positive semidefinite. A circularly symmetric complex Gaussian random variable
with meanand variance
is denoted by . is the bigO computational complexity notation.Ii System Model and Problem Formulation
Iia System Model
Consider an IRSaided fullduplex WPCN consisting of an HAP, singleantenna devices, and an IRS, as shown in Fig. 1. We assume that the HAP operates in the FD mode to enhance the spectral efficiency and is equipped with two antennas, i.e., a transmitter antenna and a receiver antenna. The transmitter antenna broadcasts energy to the distributed devices in the DL and meanwhile the receiver antenna receives the information from the distributed devices in the UL simultaneously over the same frequency band. In addition, the distributed devices are assumed to operate in a timedivision HD mode due to their limited signal processing capability, where the devices first harvest energy in the DL and then transmit information in the UL.
We consider a quasistatic flatfading channel in which the channel state information remains constant in a channel coherence frame, but may change in the subsequent frames. As shown in Fig. 2, each channel coherence frame consists of multiple blocks and each transmission period of one block of interest denoted by is divided into time slots each with duration of , where . The th slot is a dedicated time slot used to broadcast energy to all distributed devices in the DL, while the th () time slot is used for both DL WET and UL WIT. Since there are two antennas at the HAP, the DL and UL channels are different in general. In the DL transmission, denote by , , and the complex equivalent baseband channel between the HAP and the IRS, between the IRS and the th device, and between the HAP and the th device, , where , respectively. In the UL transmission, denote by , , and the complex equivalent baseband channel between the HAP and the IRS, between the IRS and the th device, and between the HAP and the th device, , respectively. In addition, the channel reciprocity is assumed and thus we have for IRSdevice links.
In this paper, we consider three types of IRS beamforming configurations, namely, fully dynamic IRS, partially dynamic IRS, and static IRS. In the following, we first provide the details for modeling the fully dynamic IRS case, and the other two cases are discussed in Section IIB.
IiA1 Dl Wet
During DL WET, the received signal by device in time slot is given by
(1) 
where is a pseudorandom signal which is a prior known at the HAP satisfying , represents the HAP’s transmit power at time slot , represents a diagonal reflection coefficient matrix in the th time slot, where the reflection amplitude is fixed as and denotes the th reflecting element of the IRS phase shifts in the th time slot, and stands for the additive white Gaussian noise at device in the th time slot. For ease of exposition, we assume that the th time slot is occupied by device for UL WIT, .^{1}^{1}1Note that in practice, the device scheduling order for the UL WIT may be affected by many other factors. Thus, in Section VI, we study two scheduling strategies based on the power gain of direct links, and evaluate their impacts on the system performance. It is worth noting that as in [11], since we consider a periodic transmission protocol shown in Fig. 2, each device can harvest energy during all time slots except its own transmission time slot . Particularly, the energy harvested by device after its own WIT in the previous transmission block (e.g., th) will be used for its WIT in the next transmission block (e.g., th). Under this protocol, the amount of harvested energy by device during can be equivalently expressed as^{2}^{2}2 The energy harvested from the noise and the received UL WIT signals from other devices are assumed to be negligible, since both the noise power and device transmit power are much smaller than the transmit power of the HAP in practice [44],[45].
(2) 
where denotes the energy conversion coefficient at device .
On the other hand, when device is scheduled to transmit in the th time slot, device will exhaust the total harvested energy for UL WIT. The average transmit power of device for UL WIT is thus given by
(3) 
IiA2 Ul Wit
During UL WIT, device transmits its own data to the HAP, the received signal by the HAP in time slot is given by
(4) 
where denotes device ’s transmit signal, which is assumed to be of zero mean and unit power, i.e., and , represents the effective loopback channel at the HAP that satisfies , and denotes the received additive white Gaussian noise. It can be observed that the interference term in (4) consists of two parts. The first denotes the SI resulting from the DL transmission by the HAP and the second part
denotes the interference introduced by the reflection of the DL transmit signal by the IRS. We assume that the IRS is deployed close to the devices and is far from the HAP by considering the following two facts. First, when the IRS is deployed close to the devices, the double propagation loss of the cascaded HAPIRSdevice link will be substantially reduced, which is beneficial for improving the signaltonoise ratio (SNR). Second, when the IRS is deployed far from the HAP, the power of interference introduced by the reflection link, i.e., HAPIRSHAP link, will be significantly small compared to that of SI resulting from the DL transmission. As such, the second part of interference is neglected in the sequel of this paper.
IiA3 Sic
In practice, the FD operation is often precluded by the significant quantization error introduced by the strong SI under the limited dynamic range of ADC at the receiver. Following [11], [46], this quantization error after ADC can be modeled as an independent white Gaussian noise denoted by , where and are given by
(5) 
where holds due to the fact that the power of SI is generally much larger than that of the received signal from UL devices and the received noise. Expression (5) shows that the power of quantization error is proportional to that of transmit power at the HAP. In addition, we assume that the perfect loopback channel
is available at the receiver of the HAP via using pilotbased channel estimation methods
[46]. Therefore, subtracting SI from (4), the received signal at the HAP can be recast as(6) 
As a result, the achievable throughput of device in bits/Hz in time slot can be expressed as
(7) 
where represents the gap from channel capacity owing to the practical modulation and coding scheme.
IiB Problem Formulation
Denote by sets , , , and , where and . Let and . We thus have and . The objective of this paper is to maximize the sum throughput of the IRSaided FDWPCN by jointly optimizing the time allocation, HAP transmit power, and IRS phase shifts.^{3}^{3}3Note that the considered problem formulation can be readily extended to take into account the fairness among devices by adding the different weighting factors on each device in the objective function, which does not affect our proposed algorithms. As such, we focus on the achievable sum throughput maximization problem instead. We consider three types of IRS beamforming configurations, which are specified as follows.
IiB1 Fully Dynamic IRS Beamforming
In this case, the phaseshift vectors change over each time slot during period of . Mathematically, the problem is formulated as follows
(8a)  
(8b)  
(8c)  
(8d)  
(8e) 
where in (8d) denotes the HAP transmit power budget.
IiB2 Partially Dynamic IRS Beamforming
In this case, two different phaseshift vectors can be allowed to be used for DL WET and UL WIT, which are denoted by and , respectively. We thus have . Define . Accordingly, the problem can be formulated as follows
(9a)  
(9b)  
(9c) 
where denotes the th entry of .
IiB3 Static IRS Beamforming
In this case, we adopt the same phaseshift vector for both UL and DL transmission, i.e., . Denote by the phaseshift vector for static IRS. Accordingly, the problem can be formulated as follows
(10a)  
(10b)  
(10c) 
where denotes the th entry of .
The above three problems (8), (9) and (10) are all nonconvex since the optimization variables are highly coupled in the objective functions, there are no standard methods for solving such nonconvex optimization problems optimally in general. Although it looks like (8) and (10) have the similar objective function, the hidden structures are fundamentally different. Specifically, in (8a), and are not coupled with respect to (w.r.t.) . While and have the same phaseshift vector in (10a). Therefore, different algorithms are required. In the following two sections, we propose to two framework algorithms, namely AO and penaltybased algorithms, based on the successive convex approximation (SCA) and DC optimization techniques to solve problem (8). The extension to solve problems (9) and (10) will be studied in Section V.
Iii Proposed Solution For Fully Dynamic IRS Beamforming with Perfect SIC
In this section, we study the fully dynamic IRS beamforming with perfect SIC, i.e., , which also provides the performance upper bound for the case with imperfect SIC. Problem (8) can be simplified to
(11a)  
(11b) 
Problem (11) is a nonconvex optimization problem since the optimization variables are coupled in (11a) and (8e) involves unitmodulus constraint, which makes it difficult to solve. We can readily prove that at the optimal solution to (11), we must have since (11a) is monotonically increasing w.r.t. . As such, problem (11) can be simplified to
(12a)  
(12b) 
Although problem (12) is still a nonconvex optimization problem, it has less constraints and variables compared to (11). Additionally, it is observed that each optimization variable in (12) is involved in at most one constraint, which motivates us to apply AO method to solve it. Specifically, we divide all the variables into two blocks, i.e., 1) time allocation , and 2) phaseshift vector , and then optimize each block in an iterative way, until convergence is achieved.
Iiia Time Allocation Optimization
For any given phaseshift vector , the time allocation optimization problem is given by
(13a)  
(13b) 
Lemma 1: The objective function of (13) is a jointly concave function of .
Proof: It can be proved via checking its Hessian matrix, please refer to Appendix B of reference [12] for details.
IiiB IRS Phase Shift Optimization
For any given time allocation , the IRS phase shift optimization is given by
(14a)  
(14b) 
Problem (14) involves nonconvex unitmodulus constraint (8e). To address this nonconvexity, we relax the unitmodulus constraint (8e) as
(15) 
which is convex.
In addition, it is also observed that does not appear in and simultaneously, which motivates us to apply AO algorithm to solve it. As such, we can partition the entire phase shifts into blocks, i.e., , and then alternately optimize each block until convergence is achieved. However, optimizing phase shift for DL WET and , for UL WIT are different. As such, we solve them separately with two cases in the following.
Case 1: Define and , . The phase shift optimization problem for DL WET, i.e., for , is given by
(16a)  
(16b) 
It is observed that is a convex quadratic function of . Recall that any convex function is globally lowerbounded by its firstorder Taylor expansion at any feasible point [47]. As a result, the SCA technique is applied. Specifically, for any local point in the th iteration, we have
(17) 
which is linear and convex w.r.t. .
For , is a lower bound of . Substituting into (16a), we have the following convex optimization problem
(18a)  
(18b) 
which can be solved by the interior point method [47].
Case 2: The phase shift optimization problem for UL WIT, i.e., for . We alternately update phaseshift vector over each UL time slot while others being fixed. The problem is given by
(19a)  
(19b) 
Similar to (17), taking the firstorder Taylor expansion of at any feasible point , we have its lower bound given by
(20) 
It can be readily checked that is linear and convex w.r.t. . Based on (17) and (20), problem (19) can be approximated by
(21a)  
(21b) 
which is convex and can be solved by interiorpoint method [47].
IiiC Overall Algorithm and Computational Complexity Analysis
Finally, we need to reconstruct the obtained phase shifts as unitmodulus solutions. The reconstruction scheme is given by
(22) 
Based on the solutions to the above subproblems, an AO algorithm is proposed, which is summarized in Algorithm 1.
The mainly computational complexity lies from steps , , and . Specifically, in step , (13) can be solved by the interiorpoint method, whose complexity is [48], where denotes the number of variables. In steps and , the complexity for solving (18) and (21) by the interiorpoint method is the same with , where denotes the number of variables. Therefore, the total complexity of Algorithm 1 is , where stands for the number of iterations required to reach convergence. Since at steps , , and , each subproblem is optimally solved, the objective function is nondecreasing over iterations. In addition, the maximum objective value of (11) is upper bounded by a finite value due to the limited HAP transmit power. As such, by applying the proposed Algorithm 1, the objective value is guaranteed to be nondecreasing over the iterations and terminated finally.
Iv Proposed Solution for Fully Dynamic IRS Beamforming with Imperfect SIC
For the fully dynamic IRS beamforming with imperfect SIC, i.e., , the formulated problem is more challenging than that with perfect SIC since the HAP transmission power is involved in the objective function of (8). To solve this problem, we extend the proposed AO algorithm in Section III to a novel penaltybased algorithm, which includes a twolayer iteration, i.e., an inner layer iteration and an outer layer iteration. The inner layer solves the penalized optimization problem by exploiting the AO algorithm, while the outer layer updates the penalty coefficient over iterations to guarantee convergence. Specifically, by introducing a new auxiliary variable , problem (8) is rewritten as
(23a)  
(23b)  
(23c) 
We then use (23b) as a penalty term that is added to the objective function of (23), yielding the following optimization problem
(24a)  
(24b) 
where is a penalty coefficient that penalizes the violation equality constraint (23b). By gradually decreasing the value of in the outer layer, as , it follows that . In this case, equality in (23b) is guaranteed in the optimal solution to problem (24). For any given , (23) is still a nonconvex optimization problem due to the nonconvex objective function as well as nonconvex unitmodulus constraint (8e). However, it is observed that each optimization variable is involved in at most one constraint, which motivates us to apply AO algorithm to solve it in the inner layer. Specifically, we first relax unitmodulus constraint (8e) as (15). We then divide all the optimization variables into four blocks, i.e., 1) time allocation , 2) HAP transmission power , 3) auxiliary variable , and 4) IRS phase shift , and then alternately optimize each block, until convergence is achieved.
Iv1 Optimizing for given
Iv2 Optimizing for given
This subproblem is written as
(26a)  
(26b) 
It is observed that both the objective function and constraints are convex, which thus can be efficiently solved by interiorpoint method [47].
Iv3 Optimizing for given
This subproblem can be expressed as
(27) 
It is observed that (IV3) has no constraints and the auxiliary optimization variables
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