Joint Dynamic Passive Beamforming and Resource Allocation for IRS-Aided Full-Duplex WPCN

08/15/2021 ∙ by Meng Hua, et al. ∙ 0

This paper studies intelligent reflecting surface (IRS)-aided full-duplex (FD) wireless-powered communication network (WPCN), where a hybrid access point (HAP) broadcasts energy signals to multiple devices for their energy harvesting in the downlink (DL) and meanwhile receives information signals in the uplink (UL) with the help of IRS. Particularly, we propose three types of IRS beamforming configurations to strike a balance between the system performance and signaling overhead as well as implementation complexity. We first propose the fully dynamic IRS beamforming, where the IRS phase-shift vectors vary with each time slot for both DL wireless energy transfer (WET) and UL wireless information transmission (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, two different phase-shift vectors can be exploited for the DL WET and the UL WIT, respectively, whereas for the latter case, the same phase-shift vector needs to be applied for both DL and UL transmissions. We aim to maximize the system throughput by jointly optimizing the time allocation, HAP transmit power, and IRS phase shifts for the above three cases. Two efficient algorithms based on alternating optimization and penalty-based algorithms are respectively proposed for both perfect self-interference cancellation (SIC) case and imperfect SIC case by applying successive convex approximation and difference-of-convex optimization techniques. Simulation results demonstrate the benefits of IRS for enhancing the performance of FD-WPCN, and also show that the IRS-aided FD-WPCN is able to achieve significantly performance gain compared to its counterpart with half-duplex when the self-interference (SI) is properly suppressed.

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I Introduction

The number of Internet-of-Things (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 self-sustainable 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 multi-path fading. Although several technologies have been proposed to overcome this issue, which include massive multiple-input (MIMO), ultra-dense 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, full-duplex (FD) transmission is a promising technology to potentially double the spectral efficiency if the self-interference (SI) is perfectly cancelled [5]. Different from the half-duplex (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 analog-domain SIC and the digital-domain SIC. It was reported in [9] that the art-of-the-state of SI suppression can be up to by combing the analog domain (i.e., SIC before analog-to-digital 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 wireless-powered 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 FD-WPCN outperforms the HD-WPCN if SI can be suppressed below a certain level. Subsequently, [12] derived closed-form solutions in the FD-WPCN with perfect SIC case.

Recently, intelligent reflecting surfaces (IRSs) have been proposed as a promising cost-effective solution to improve both spectral- and energy-efficiency of wireless communication systems[15, 16, 17, 18, 19, 20]. IRS is composed of 2-D planar arrays of sub-wavelength 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 fine-grained 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], multi-cell 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 grained-fine tunable phase shifts, high passive beamforming gains can be achieved by the IRS to effectively compensate the end-to-end signal attenuation. For example, [33] studied an IRS-assisted SWIPT system, where a set of IRSs are deployed to assist the downlink (DL) WIT and WET from a multi-antenna 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 IRS-aided WPCN [35, 36, 37, 38, 39, 40]. The authors in [35] and [36] studied IRS-aided 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 IRS-empowered WPCN, where the IRS is allowed to harvest energy from an HAP, and then considered two schemes, namely, time-switching 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 IRS-aided 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 IRS-aided 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 IRS-aided FD systems for some applications, such as the cognitive radio system, the point-to-point system, and the multi-user multiple-input single-output 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 IRS-aided FD-WPCN 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 time-varying 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 IRS-aided FD-WPCN 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 phase-shift 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 phase-shift 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 phase-shift 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 non-convex due to the highly coupled optimization variables in the objective function and non-convex unit-modulus constraints of phase shifts, there are no standard methods for solving such non-convex 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 penalty-based algorithm, which includes a two-layer 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 penalty-based algorithms to solve them. In particular, a difference-of-convex (DC) optimization method is proposed to address unit-modulus phase-shift 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 FD-WPCN, especially when the fully dynamic IRS beamforming is adopted. We also show that the IRS-aided FD-WPCN is able to achieve significantly performance gain compared to the IRS-aided half-duplex (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 FD-WPCN 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 penalty-based 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 upper-case and lower-case letter denote matrix and vector, respectively. stands for the set of complex matrices. For a complex-valued 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 semi-definite. A circularly symmetric complex Gaussian random variable

with mean

and variance

is denoted by . is the big-O computational complexity notation.

Fig. 1: An IRS-aided FD-WPCN.

Fig. 2: Transmission protocol for FD energy harvesting and information transmission.

Ii System Model and Problem Formulation

Ii-a System Model

Consider an IRS-aided full-duplex WPCN consisting of an HAP, single-antenna 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 time-division 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 quasi-static flat-fading 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 IRS-device 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 II-B.

Ii-A1 Dl Wet

During DL WET, the received signal by device in time slot is given by

(1)

where is a pseudo-random 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, .111Note 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 as222 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)

Ii-A2 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 HAP-IRS-device link will be substantially reduced, which is beneficial for improving the signal-to-noise ratio (SNR). Second, when the IRS is deployed far from the HAP, the power of interference introduced by the reflection link, i.e., HAP-IRS-HAP 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.

Ii-A3 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 pilot-based 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.

Ii-B 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 IRS-aided FD-WPCN by jointly optimizing the time allocation, HAP transmit power, and IRS phase shifts.333Note 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.

Ii-B1 Fully Dynamic IRS Beamforming

In this case, the phase-shift 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.

Ii-B2 Partially Dynamic IRS Beamforming

In this case, two different phase-shift 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 .

Ii-B3 Static IRS Beamforming

In this case, we adopt the same phase-shift vector for both UL and DL transmission, i.e., . Denote by the phase-shift 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 non-convex since the optimization variables are highly coupled in the objective functions, there are no standard methods for solving such non-convex 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 phase-shift vector in (10a). Therefore, different algorithms are required. In the following two sections, we propose to two framework algorithms, namely AO and penalty-based 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 non-convex optimization problem since the optimization variables are coupled in (11a) and (8e) involves unit-modulus 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 non-convex 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) phase-shift vector , and then optimize each block in an iterative way, until convergence is achieved.

Iii-a Time Allocation Optimization

For any given phase-shift 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.

Based on Lemma 1 and the fact that all constraints in (13) are linear, problem (13) is thus a convex optimization problem, which can be solved by the standard convex optimization techniques, such as interior-point method [47].

Iii-B IRS Phase Shift Optimization

For any given time allocation , the IRS phase shift optimization is given by

(14a)
(14b)

Problem (14) involves non-convex unit-modulus constraint (8e). To address this non-convexity, we relax the unit-modulus 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 lower-bounded by its first-order 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 phase-shift vector over each UL time slot while others being fixed. The problem is given by

(19a)
(19b)

Similar to (17), taking the first-order 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 interior-point method [47].

Iii-C Overall Algorithm and Computational Complexity Analysis

1:  Initialize IRS phase-shift vector , and threshold .
2:  repeat
3:   Update time allocation by solving (13).
4:   Update DL WET phase shift vector by solving (18).
5:   for
6:    Update UL WIT phase shift vector by solving (21).
7:   end
8:  until the fractional increase of the objective function of (11) is below a threshold .
9:  Reconstruct phase shifts based on (22).
10:  Update time allocation by solving (13) based on the newly obtained phase shifts.
11:  Output: time allocation , and phase-shift vector .
Algorithm 1 AO for solving (11).

Finally, we need to reconstruct the obtained phase shifts as unit-modulus 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 interior-point method, whose complexity is [48], where denotes the number of variables. In steps and , the complexity for solving (18) and (21) by the interior-point 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 non-decreasing 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 non-decreasing 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 penalty-based algorithm, which includes a two-layer 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 non-convex optimization problem due to the non-convex objective function as well as non-convex unit-modulus 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 unit-modulus 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.

Iv-1 Optimizing for given

Ignoring irrelevant terms w.r.t. , this subproblem can be expressed as

(25a)
(25b)

Problem (25) has a similar form to problem (13) discussed in Section III-A, which thus can be solved similarly.

Iv-2 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 interior-point method [47].

Iv-3 Optimizing for given

This subproblem can be expressed as

(27)

It is observed that (IV-3) has no constraints and the auxiliary optimization variables