# Energy Efficiency Maximization in the Wireless-Powered Backscatter Communication Networks with DF Relaying

Authors: Chuangming Zheng; Wengang Zhou; Xinxin Lu
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

This paper focuses on the design of an optimal resource allocation scheme to maximize the energy efficiency (EE) in the wireless-powered backscatter communication networks (WPBCN) with decode and forward (DF) relaying. The two different devices are supported to operate in different modes, the harvest-then-transmit (HTT) mode and backscatter communication (BackCom) mode, respectively. In particular, we formulate an optimization problem to maximize system EE by jointly optimizing the transmit power of hybrid access point (H-AP) and the system time resource allocation. To deal with the nonconvex problem, we investigate the characteristics of the EE expression and a variable substitution approach. Then, the optimal power allocation scheme and iterative optimization algorithm were derived for achieving maximum EE. Extensive simulation results have demonstrated that the system EE can be improved about 10% because the proposed scheme provides more flexibility to utilize the resource efficiently by employing the proposed scheme.


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## Body

## 1. Introduction

With the development of the Internet of Things (IoT), numerous wireless devices (WDs) have been deployed widely in the world to provide ubiquitous connectivity, which improves human being’s life greatly in all aspects [1, 2]. However, the sustainable energy supply is a major challenge for the evolvable IoT networks. Fortunately, a new technology named as wireless power transfer (WPT) has been deemed as an attractive promising approach to power WDs conveniently and steadily [3–8]. Wireless-powered communication network (WPCN) is a novel communication solution for IoT that using WPT techniques solves the problem of sustainable energy supply for the tiny WDs. Consequently, WPCN has been widely studied in recent years, especially in [5–9]. A well-known protocol for WPCN is named as “harvest-then-transmit (HTT)” protocol proposed in [8]. Following the HTT protocol, hybrid access point (H-AP) first broadcasts the wireless energy to its served users for energy harvesting, and then, users transmit their information actively to H-AP by utilizing the harvested energy.Recently, ambient backscatter communication (BackCom) [10] has been emerging as a promising technology for low-energy communication systems that was designed to communicate with WDs nearby without resorting to any existing energy supply or storage device. Different from the devices using the HTT mode, the BackCom devices transmit information by modulating and reflecting the instantaneous incident signals passively [11, 12]. Hence, the active RF components are not required at all, which can significantly decrease the circuit energy consumption and meet the low-power requirement of IoT devices. Furthermore, the dedicated energy harvesting (EH) time is also not necessary, and then, the information transmission time can be extended.Essentially, there exist some different tradeoffs between EH time and data transmission time for HTT protocol and BackCom. They may complement each other for increasing the data rates and energy utilization while they are used in IoT. Hence, applying BackCom in WPCNs and wireless-powered backscatter communication networks (WPBCNs) is a very efficient way to exploit the advantages of the HTT mode and the BackCom mode, which have gained extensive attention in academic field, especially in [12–16]. The authors in [12] presented an optimal time resource allocation method to achieve the maximal throughput for WPBCNs. The literatures [13–16] played an emphasis on optimizing the EE performance of WPBCNs.It is to be noted that EE is a crucial performance metric in IoT to achieve a better tradeoff between data rates and the overall energy consumption. Due to the characteristics of WPT, the energy cost of WPCNs, especially the energy consumption of the dedicated energy source devices, has drawn a great deal of attention [14, 15]. In [17], the authors proposed an EE resource allocation scheme which employed a Dinkelbach-based iterative algorithm to obtain the optimal time allocation, reflection coefficient, and transmit power of the dedicated RF energy source in BackCom networks. The literature [18] utilized energy beamforming communication and ambient BackCom to overcome the energy problem of network, and a EE cooperative communication scheme was presented. The EE maximum problem was investigated in cooperative sensor networks with bidirectional wireless information and power transfer [19]. By studying the derivative properties of the objective function, the authors derived the optimal power allocation and time allocation. The authors in [20] studied the EE performances for an unmanned aerial vehicle- (UAV-) assisted backscatter communication network. They maximized the EE of the network by jointly optimizing the UAV trajectory, the backscatter device scheduling, and the carrier emitter transmit power. The literature [14] studied the EE of WPBCNs which consist of two types of WDs that operate in different modes, the HTT mode and BackCom mode, respectively. The authors in [14] study the network which includes two different types WDs. One device named as HD, operating in the HTT mode, transmits its information directly to the access point (AP), and the other device named as BD, operating in the BackCom mode, backscatters its information directly to AP. By jointly optimizing the energy beamforming vector, power allocation, and time allocation, the system maximum EE is achieved. Obviously, the researchers have done many works in improving the EE of WPCNs and a lot of valuable achievements have been obtained. However, the EE optimization of WPBCNs that consists of two types of WDs has not been sufficiently exploited by the existing works.The authors in [12] proposed a new network structure in which HTT mode is adopted in the last hop and BackCom mode is used in the other hops. They assumed that the devices could only consume the energy harvested within the current time block. Therefore, based on the above assumption, the nodes do not harvest the energy after they finish the information forwarding in one time block, which leads to the reduction of harvested energy and the inefficient energy utilization. Both in [19, 21], the authors think that the devices can store the harvested energy in their battery and do not consume all the energy harvested in the current time block. Especially in [19], the remote device is designed to harvest the energy from the relay during the phase of relay transmitting information to the base station. Inspired by the literatures [19, 21], we develop a new network model and the corresponding communication protocol which improve the EE and throughput of new model.In this paper, we investigate a new WPBCN with DF relaying (WPB-R-CN) which is more suitable to the realistic IoT application scenario. WPB-R-CN consists of one hybrid access point (H-AP), one relay, and one backscatter user (SU) which is shown in Figure1. Following the literature [12], SU operates in BackCom mode and relay transmits the information by operating in HTT mode. Different from [12], SU is required to harvest energy from RF signal transmitted by a relay device during the phase of relay transmitting information to H-AP. The energy harvested during this time by SU will be stored in its battery for use in the next time block. Obviously, the proposed new model can improve system EE further.Figure 1
System model.Obviously, WPB-R-CN is a new network model which is different with the existing models, such as the models of literatures [12, 14], which are more similar to our proposed model. In [12, 14], BackCom mode and HTT mode are used simultaneously in the models. However, in [14], the device named as HD, operating in the HTT mode, transmits its information directly to access point (AP) and the other device named as BD, operating in the BackCom mode, backscatters its information directly to AP. The authors in [14] only study one-hop link performane for two different types WDs. Different from the model of [14], the authors in [12] study multihop hybrid backscatter communication network. However, the other nodes do not harvest the energy transmitted by the node operating in HTT mode which leads to the reduction of harvested energy and the inefficient energy utilization. Therefore, the model we studied in this paper is a new one. No existing algorithms can achieve the maximum EE of WPB-R-CN.We aim at WPB-R-CN EE maximization by optimizing its resource allocation and H-AP transmit power. To obtain the optimal parameters, we formulate a nonconvex system EE maximization problem. To make the optimization problem tractable, we investigate the characteristics of the EE expression. Then, the optimal power allocation scheme and iterative optimization algorithm were derived for solving the nonconvex optimal problem and achieving maximum EE. Our contribution can provide a useful insight for optimizing system performance in both system throughput and EE in WPBCN.The main contributions of our work are summarized as follows:(i)
We propose a new WPB-R-CN network model which consists of one hybrid access point (H-AP), one relay, and one backscatter user (SU). To improve the system EE performance, SU required to harvest energy from RF signal transmitted by relay during the phase of relay transmitting information to H-AP(ii)
We develop a new protocol for the WPB-R-CN network to achieve the maximum EE. In the new protocol, SU harvests wireless energy from the relay during relay transmitting the data to H-AP and stores its harvested energy in a battery. And the stored energy can be scheduled across different transmission blocks(iii)
We formulate a system EE maximization problem and obtain the joint power allocation scheme for the proposed model. By investigating the characteristics of the EE expression, the optimal power allocation scheme and iterative optimization algorithm were derived for achieving maximum EE(iv)
We investigate the performances of the optimal power allocation scheme and the proposed iterative optimization algorithm. Comparing their performances with the other two schemes, our proposed scheme is proved to be efficient in improving system EE performanceThe remainder of the paper is organized as follows. System model and communication protocol are described in Section2. The problem of jointly optimizing H-AP transmit power and time resource allocation scheme is formulated to maximize the system EE in Section 3. Section 4 gives the optimal time allocation and power allocation schemes for maximizing the system EE. Simulation experiments are conducted to verify the effectiveness of the proposed strategies in Section 5. Finally, we conclude the paper in Section 6.


## 2. System Model and Communication Protocol

### 2.1. System Model

We consider a WPB-R-CN illustrated in Figure1, which consists of one H-AP, one relay, and one SU. Each device is equipped with one antenna. H-AP is assumed to have sustainable power supply acting as a wireless power provider and information receiver. Relay and SU operate in the HTT mode and the BackCom mode, respectively. They do not have embedded energy supplies. Both relay and SU can harvest the energy from RF signal using their EH circuit and store the energy into a rechargeable battery. The stored energy can be consumed while relay and SU transmit their information. Moreover, by using the existing energy in SU battery, the information transmission can be initialized before energy harvesting and SU can consume all the harvested energy during the frame [19]. Each channel among nodes is assumed to be undergoing independent identically distributed (i.i.d) quasistatic block fading [22]. And the channels are reciprocal in two directions [19, 22]. Assume that no direct link exists between nodes H-AP and SU due to severe path loss and/or shadowing. Therefore, relay node has the obligation to relay the data of SU to H-AP.Leth0,h1,h2 denote the channel response of the relay–SU link, the H-AP–relay link, and the H-AP–SU link, respectively. Similarly, d0,d1,d2 are the distance of relay and SU, H-AP and relay, and H-AP and SU. Let n1,n2,n3 denote the additive white Gaussian noise at the receiver of SU, relay, and H-AP, respectively, ni~CN0,σ2,i∈1,2,3. Without loss of generality, relay is closer to H-AP than SU in the system which means d1<d2, resulting in h12>h22. The duration of each frame Tf is one second, and the system bandwidth is B Hz. Pcr is the circuit power consumed while the H-AP and relay work as the receiver. Pct is denoted as the circuit power consumed while SU and relay work as the transmitter.


### 2.2. Proposed New Communication Protocol

As shown in Figure2, one time-frame duration Tf is divided into three time phases that are denoted by τ0,τ1, and τ2, respectively. During the first time phase τ0, H-AP transfers its wireless energy to relay and SU by transmitting the modulated signal st=PHxt, where PH denotes the transmit power of H-AP and xt is a known signal with Ext2=1. Correspondingly, both relay and SU work as the energy harvester and harvest the energy from the ambient RF signal using their EH circuit and store the energy into a rechargeable battery. The harvested energy ER0 by relay and the harvested energy EU0 by SU are expressed as
(1)ER0=ηPHh12τ0,(2)EU0=ηPHh22τ0,where η∈0,1 denotes the energy harvesting efficiency. The harvested energy of relay is split into two parts. One part energy is used to maintain the circuit consumption of the relay during the time τ1, and the other part is used to transmit information during the time τ2. The harvested energy of SU is used to maintain its circuit consumption while SU backscatters the information during the time τ1.Figure 2
Framework of the communication protocol.During the second time phaseτ1, H-AP continues to transmit the modulated signal st. SU enters the active state to backscatter its information to relay by utilizing incident signals from H-AP, and relay also enters the active state to decode the information of SU. The received signal y1t by SU is given by
(3)y1t=PHh2xt+n1t.Then, SU modulates its own signalc1t on the received signal y1t and c1t satisfies Ec1t2=1. Therefore, the backscattered signal y2t by SU is written as
(4)y2t=PHh2xtc1t+n1tc1t.The signal received by relay is denoted asy3t which is expressed as
(5)y3t=h0y2t+h1st+n2t=PHh2h0xtc1t+h0n1tc1t+PHh1xt+n2t.Obviously, The first term ofy3t is the received desired signal by relay. The second term of y3t is the noise caused by backscattering process. The third term of y3t is the interference from the H-AP, the power of which is typically larger than that of the desired signal. Following the literatures [23, 24], successive interference cancellation (SIC) technique is used to remove it from the y3t since relay has known the information st well. Thus, the signal-noise-ratio (SNR) at relay during the second time phase τ1 is given by
(6)γ1=PHh02h22h02+1σ2.Whileh02<<1, then equation (6) can be written as
(7)γ1=PHh02h22σ2.During the third time phaseτ2, H-AP stops transmitting the modulated signal st and acts as a information receiver. Relay operates in HTT mode and transmits the information decoded during the time τ1. SU enters sleep state and acts as energy harvester. It harvests the RF energy transmitted by relay. Let PR denote the transmit power of relay which is written as
(8)PR=ER0−Pcrτ1−Pctτ2τ2.Relay transmits the informationc2t to H-AP, and c2t satisfies Ec2t2=1. The received signal of H-AP y4t and SNR γ2 at H-AP is, respectively, given by
(9)y4t=PRh1c2t+n2t,γ2=PRh12σ2.The harvested energyEU1 by SU is expressed as
(10)EU1=ηPRh02τ2.Thus, the total energy harvestedEU by SU is written as
(11)EU=EU0+EU1=ηPHh22τ0+ηPRh02τ2.Consequently, the SU–relay link and relay–H-AP link throughput are, respectively, calculated as(12)R1=τ1Blog21+γ1=τ1Blog21+PHh02h22σ2,(13)R2=τ2Blog21+γ2=τ2Blog21+PRh12σ2.Following the literature [19], the system throughput R is given by
(14)R=minR1,R2.Obviously, since both SU and relay are powered by harvested energy, only H-AP device consumes the energy in the system. Therefore, the total energy consumption of the whole system is also the energy consumption of H-AP which consists of two parts: the energy consumed in H-AP transmitting RF signal phase and the energy consumed in H-AP decoding information phase. Then, the total energy consumption of the whole system is written as(15)Ec=PHζτ0+τ1+Pcrτ2,where ζ∈0,1 is the power amplifier efficiency.


## 2.1. System Model

We consider a WPB-R-CN illustrated in Figure1, which consists of one H-AP, one relay, and one SU. Each device is equipped with one antenna. H-AP is assumed to have sustainable power supply acting as a wireless power provider and information receiver. Relay and SU operate in the HTT mode and the BackCom mode, respectively. They do not have embedded energy supplies. Both relay and SU can harvest the energy from RF signal using their EH circuit and store the energy into a rechargeable battery. The stored energy can be consumed while relay and SU transmit their information. Moreover, by using the existing energy in SU battery, the information transmission can be initialized before energy harvesting and SU can consume all the harvested energy during the frame [19]. Each channel among nodes is assumed to be undergoing independent identically distributed (i.i.d) quasistatic block fading [22]. And the channels are reciprocal in two directions [19, 22]. Assume that no direct link exists between nodes H-AP and SU due to severe path loss and/or shadowing. Therefore, relay node has the obligation to relay the data of SU to H-AP.Leth0,h1,h2 denote the channel response of the relay–SU link, the H-AP–relay link, and the H-AP–SU link, respectively. Similarly, d0,d1,d2 are the distance of relay and SU, H-AP and relay, and H-AP and SU. Let n1,n2,n3 denote the additive white Gaussian noise at the receiver of SU, relay, and H-AP, respectively, ni~CN0,σ2,i∈1,2,3. Without loss of generality, relay is closer to H-AP than SU in the system which means d1<d2, resulting in h12>h22. The duration of each frame Tf is one second, and the system bandwidth is B Hz. Pcr is the circuit power consumed while the H-AP and relay work as the receiver. Pct is denoted as the circuit power consumed while SU and relay work as the transmitter.


## 2.2. Proposed New Communication Protocol

As shown in Figure2, one time-frame duration Tf is divided into three time phases that are denoted by τ0,τ1, and τ2, respectively. During the first time phase τ0, H-AP transfers its wireless energy to relay and SU by transmitting the modulated signal st=PHxt, where PH denotes the transmit power of H-AP and xt is a known signal with Ext2=1. Correspondingly, both relay and SU work as the energy harvester and harvest the energy from the ambient RF signal using their EH circuit and store the energy into a rechargeable battery. The harvested energy ER0 by relay and the harvested energy EU0 by SU are expressed as
(1)ER0=ηPHh12τ0,(2)EU0=ηPHh22τ0,where η∈0,1 denotes the energy harvesting efficiency. The harvested energy of relay is split into two parts. One part energy is used to maintain the circuit consumption of the relay during the time τ1, and the other part is used to transmit information during the time τ2. The harvested energy of SU is used to maintain its circuit consumption while SU backscatters the information during the time τ1.Figure 2
Framework of the communication protocol.During the second time phaseτ1, H-AP continues to transmit the modulated signal st. SU enters the active state to backscatter its information to relay by utilizing incident signals from H-AP, and relay also enters the active state to decode the information of SU. The received signal y1t by SU is given by
(3)y1t=PHh2xt+n1t.Then, SU modulates its own signalc1t on the received signal y1t and c1t satisfies Ec1t2=1. Therefore, the backscattered signal y2t by SU is written as
(4)y2t=PHh2xtc1t+n1tc1t.The signal received by relay is denoted asy3t which is expressed as
(5)y3t=h0y2t+h1st+n2t=PHh2h0xtc1t+h0n1tc1t+PHh1xt+n2t.Obviously, The first term ofy3t is the received desired signal by relay. The second term of y3t is the noise caused by backscattering process. The third term of y3t is the interference from the H-AP, the power of which is typically larger than that of the desired signal. Following the literatures [23, 24], successive interference cancellation (SIC) technique is used to remove it from the y3t since relay has known the information st well. Thus, the signal-noise-ratio (SNR) at relay during the second time phase τ1 is given by
(6)γ1=PHh02h22h02+1σ2.Whileh02<<1, then equation (6) can be written as
(7)γ1=PHh02h22σ2.During the third time phaseτ2, H-AP stops transmitting the modulated signal st and acts as a information receiver. Relay operates in HTT mode and transmits the information decoded during the time τ1. SU enters sleep state and acts as energy harvester. It harvests the RF energy transmitted by relay. Let PR denote the transmit power of relay which is written as
(8)PR=ER0−Pcrτ1−Pctτ2τ2.Relay transmits the informationc2t to H-AP, and c2t satisfies Ec2t2=1. The received signal of H-AP y4t and SNR γ2 at H-AP is, respectively, given by
(9)y4t=PRh1c2t+n2t,γ2=PRh12σ2.The harvested energyEU1 by SU is expressed as
(10)EU1=ηPRh02τ2.Thus, the total energy harvestedEU by SU is written as
(11)EU=EU0+EU1=ηPHh22τ0+ηPRh02τ2.Consequently, the SU–relay link and relay–H-AP link throughput are, respectively, calculated as(12)R1=τ1Blog21+γ1=τ1Blog21+PHh02h22σ2,(13)R2=τ2Blog21+γ2=τ2Blog21+PRh12σ2.Following the literature [19], the system throughput R is given by
(14)R=minR1,R2.Obviously, since both SU and relay are powered by harvested energy, only H-AP device consumes the energy in the system. Therefore, the total energy consumption of the whole system is also the energy consumption of H-AP which consists of two parts: the energy consumed in H-AP transmitting RF signal phase and the energy consumed in H-AP decoding information phase. Then, the total energy consumption of the whole system is written as(15)Ec=PHζτ0+τ1+Pcrτ2,where ζ∈0,1 is the power amplifier efficiency.


## 3. Problem Formulation and Analysis

In this section, we formulate an optimization problem to maximize the system EE by jointly optimizing time resource allocation and H-AP transmit power. The system EEψPH,τ0,τ1,τ2 is defined as the ratio of the achievable system throughput to the total energy consumption [14, 16], which is given by
(16)ψPH,τ0,τ1,τ2=REc=minR1,R2PH/ζτ0+τ1+Pcrτ2.Then, the optimization problem is formulated as(17)P1:maxPH,τ0,τ1,τ2minR1,R2PH/ζτ0+τ1+Pcrτ2s.t.C1:τ0+τ1+τ2=TfC2:τ0,τ1,τ2≥0C3:0≤PH≤PmaxC4:EU≥Pctτ1C5:ER0≥Pcrτ1In problem P1, C1 indicates that the summation of three time variables equals to the duration of one frameTf. C2 limits that each time variables must be nonnegative. C3 constrains the transmit power range of H-AP. C4 and C5 guarantee that the total energy consumed does not exceed the total energy harvested for SU and relay, respectively. Notice that the WPB-R-CN is a new network which is distinguished from the conventional relaying and the main difference can be found in Section 2. These differences make the formulated EE problem noticeably different from that of the conventional relaying network.Obviously, the above-formulated EE optimization problem is appealing in practice. For one thing, the time resource allocation can be exploited to satisfy the maximum system throughput requirement. For another, the EE can be further improved by optimizing the transmit power of H-AP. However, problem P1 cannot be solved directly for the following two main challenges. First, both the nominator and denominator of problem P1 include the variablesτ0,τ1,τ2,PH . Second, the optimization variables τ0,τ1,τ2,PH are coupled in both objective function and the constrains C4 and C5. Consequently, P1 is a nonconvex problem which cannot be solved directly. In general, there are no standard methods to solve the nonconvex optimization problems efficiently. Note that when PH remains unchanged, the bigger nominator of objective function leads to the bigger EE. Then, the original problem P1 can be written as
(18)P2:maxPH,τ0,τ1,τ2maxminR1,R2PH/ζτ0+τ1+Pcrτ2s.t.C1−C5Obviously, the problem P2 is also nonconvex optimization problem which is too difficult to obtain a globally optimal solution.


## 4. Energy Efficiency Maximization Resource Allocation

To solve problem P2 for obtaining its optimal solutionτ0∗,τ1∗,τ2∗,PH∗, we decompose the problem P2 into two subproblems to make it more tractable according to reference [19].


### 4.1. Optimal Resource Allocation Scheme

First, we formulate one subproblem to achieve the maximum system throughput by optimizing the resource allocation whilePH is considered as remaining unchanged. The optimal resource allocation is denoted as τ0∗,τ1∗,τ2∗. Let R∗ denote the system throughput which corresponds to the optimal resource allocation τ0∗,τ1∗,τ2∗. It means that R∗>R+,∀τ0+,τ1+,τ2+≠τ0∗,τ1∗,τ2∗ when PH remains unchanged, where R+ denotes the system throughput which corresponds to the resource allocation τ0+,τ1+,τ2+.Therefore, the first subproblem denoted as P2a is formulated as(19)P2a:maxτ0,τ1,τ2minR1,R2s.t.C1−C5Accordingly, P2a is still a nonconvex problem because there are coupling relationships among different optimization variables. In order to solve it, we present the following lemmas.Lemma 1.
R1=R2 is a necessary but insufficient condition for problem P2a obtaining the optimal solution.Proof.
It is assumed thatR1>R2. Then, R=R2 is the maximum system throughput. In order to cut down the SU–relay link throughput R1, we reduce SU backscattering time τ1 to τ1′ for achieving R1′=R2. Let τ1=τ1′+Δ. Thus, we divide Δ into three parts Δ0′,Δ1′,Δ2′ in the ratio of τ0 : τ1′ : τ2 and let τ0′=τ0+Δ0′, τ1′′=τ1′+Δ1′, and τ2′=τ2+Δ2′. Obviously, the achieved system throughput R′ is greater than R which is result from the new resource allocation scheme τ0′,τ1′′,τ2′, which contradicts with the assumption. Then, Lemma 1 is proved.We apply Lemma1 to the problem P2a. Then, the new optimization problem P3 is formulated as
(20)P3:maxτ0,τ1,τ2R1=R2s.t.C1−C5Similar to problem P2a, problem P3 is also a nonconvex problem. To tackle this problem, we first divide the problem P3 into two subproblems and solve two subproblems sequently. Finally, We use the optimal solutions of the two subproblems to obtain the optimal solution of problem P3 by iterative optimization and proportional compression algorithm.At first, we relax the variableτ2 and make τ2=0. Correspondingly, the constrain C1 becomes the new constrain τ0+τ1=Tf. Let τ0=αTf and τ1=1−αTf, where 0<α<1 is the factor of SU transmission time. The first subproblem P3a is formulated as
(21)P3a:maxαR1s.t.C3−C5,C6:0<α<1According to the references [4, 17, 22], Pcr≤Pct is always satisfied in WPCN. Obviously, ER0>EU is always satisfied according to the system model. Thus, the constrain C5 can be satisfied while the constrain C4 is satisfied. Based on the above conclusion, we formulate problem P4 as follows:
(22)P4:maxαR1s.t.C3,C4,C6Obviously, P4 is a more tractable problem. By use of the optimization method in [12, 17], P4 can obtain the maximum throughput while the constrain C4 satisfies EU=Pctτ1. Substituting τ0=αTf, τ1=1−αTf and equation (2) into EU=Pctτ1, the following equation can be obtained shown as follows:
(23)ηPHh22αTf=Pct1−αTf.The optimal solutionα∗ can be easily calculated from equation (16) which is given by
(24)α∗=PctηPHh22+Pct.Substituting the optimal solutionα∗ into the objective function, the maximum throughput R^1 of problem P3a is achieved which is written as
(25)R^1=ηPHh22ηPHh22+PctTfBlog21+PHh02h22σ2.Secondly, we relax the variableτ1 to satisfy τ1=0 and relax the variable Tf to satisfy Tf=TR, where 0<TR<Tf denotes the transmission cycle time of relay. Additionally, to simplify the optimization process, we make τ0=βTR and τ2=1−βTR, where 0<β<1 is the factor of relay transmission cycle time. Thus, another subproblem P3b of problem P3 is formulated as
(26)P3b:maxβR2s.t.C3,C7:0<β<1Substitutingτ0=βTR, τ2=1−βTR and equation (7) into equation (12), R2 can be written as
(27)R2=1−βTRBlog21+ηPHh14β/1−β1−β−Pcth12σ2.Letc=ηPHh14/σ2 and b=1−Pcth12/σ2. Then, the objective function R2 can be rewritten as
(28)R2=1−βTRBlog2cβ1−β+b.Lemma 2.
The optimization problem P3b is convex, and the optimal parameterβ∗=x∗−b/x∗−b+c and x∗ is the solution of equation cxlnx−x+b−c=0.Proof.
See the appendix.
Substituting the optimal solutionβ∗ into the objective function of P3b, the maximum throughput R^2 of problem P3b is achieved which is given by
(29)R^2=1−β∗TRBlog2cβ∗1−β∗+b.
In order to solve problem P3, we makeR^2=R^1 to obtain TR which is expressed as
(30)TR=R^11−β∗Blog2cβ∗/1−β∗1−β∗+b.
According to our model, we can easily get0<TR<Tf for the cause of h12>h22. To find the optimal solution τ0∗,τ1∗,τ2∗, we define τ01=α∗Tf, τ11=1−α∗Tf, τ02=β∗TR, and τ22=1−β∗TR. Obviously, we select τ00=maxτ01,τ02 which can satisfy R1=R2. Then, we devise a iterative optimization algorithm to obtain the optimal solution τ0∗,τ1∗,τ2∗ of problem P3 which is given in Algorithm 1 with an error ε.Algorithm 1: Iterative optimization algorithm for solving problem P3.
1: Initialize the parameterε2: Divide the problem P3 into two subproblems by relaxing variableτ2 and τ1, respectively.3: Solve problem P3a to obtainα∗ and solve problem P3b to obtain β∗.4: CalculateTR by use of (29).5: Letτ01=α∗Tf, τ11=1−α∗Tf, τ02=β∗TR and τ22=1−β∗TR6:ifτ01>τ02then7:  Letτ0=τ01, τ1=τ11 and τ2=τ228:  CalculatePR and R19:  Updateτ2=R1/Blog21+PRh12/σ210:whileτ22−τ2>εdo11:τ22=τ212:   Calculateτ^01=τ01−PRh02/PHh12τ2213:   Letτ0 = maxτ^01,τ0214:   UpdatePR15:   Calculateτ2=R1/Blog21+PRh12/σ216:end while17:else18:  Letτ0=τ02, τ1=τ11 and τ2=τ2219:end if20: Thenτ0∗=τ0Tf/τ0+τ1+τ2, τ1∗=τ1Tf/τ0+τ1+τ2, τ2∗=τ2Tf/τ0+τ1+τ2.


### 4.2. Optimal Transmit Power Allocation Scheme

The optimal power allocation scheme will be aimed at maximizing the energy efficiency in this subsection while the optimal resource allocation solution is obtained. Then, another subproblem P5 of problem P2 is formulated as(31)P5:maxPHψPH=τ1∗Blog21+PHh02h22PHh02h22/σ2PH/ζτ0∗+τ1∗+Pcrτ2∗s.t.C3Letλ=σ2/h02h22 and φ=1/ζτ0∗+τ1∗. By taking the derivative of ψPH with respect to PH, we get
(32)∂ψPH∂PH=τ1∗BλPHφ+Pcrτ2∗τ1∗BλPHφ+Pcrτ2∗/ln21+PHλln21+PHλ−τ1∗Blog21+PHλφPHφ+Pcrτ2∗2.In equation (30), the denominator PHφ+Pcrτ2∗2 is greater than zero. Therefore, the sign of ∂ψPH/∂PH depends on its numerator. We define the numerator as a new function gPH which is given by
(33)gPH=τ1∗BλPHφ+Pcrτ2∗ln21+PHλ−τ1∗Blog21+PHλφ.Similarly, the derivative ofgPH with respect to PH can be written as
(34)∂gPH∂PH=−τ1∗Bλ2PHφ+Pcrτ2∗ln2ln21+PHλ2.It is evident that∂gPH/∂PH is less than zero, which means that gPH is a monotone decreasing function of PH while 0<PH<Pmax. In addition, g0 is greater than zero. Then, when gPmax≥0, the optimal solution PH∗=Pmax. The reason is that ∂ψPH/∂PH is always greater than zero while 0<PH<Pmax. It means that ψPH is a monotone increasing function while 0<PH<Pmax. Therefore, the maximum energy efficient can be obtained while PH∗=Pmax. Otherwise, there exists PH+ satisfying ∂ψPH/∂PH=0. And the energy efficient ψPH is a monotone increasing function while 0<PH<PH+ and ψPH is a monotone decreasing function while PH+<PH<Pmax. Therefore, we can solve the equation gPH∗=0 to obtain the optimal solution PH∗. Thus, the optimal solution PH∗ is given by
(35)PH∗=PmaxgPmax≥0PH+gPmax<0,where PH+ is the unique solution satisfying gPH+=0.To solve the optimal solutionPH∗, we devise a power allocation algorithm which is described in Algorithm 2 with the given error δ.Algorithm 2: Power allocation algorithm.
1: LetP1=0, P2=Pmax and δ>02: Calculatev=gPmax based on Eq.(32).3:ifv<0then4:whileP2−P1>δdo5:  LetPH+=P1+P2/26:  Calculateu=gPH+7:ifu≥0then8:   LetP1=P1+P2/29:else10:   LetP2=P1+P2/211:end if12:end while13:PH∗=PH+14:else15:PH∗=Pmax.16:end if


## 4.1. Optimal Resource Allocation Scheme

First, we formulate one subproblem to achieve the maximum system throughput by optimizing the resource allocation whilePH is considered as remaining unchanged. The optimal resource allocation is denoted as τ0∗,τ1∗,τ2∗. Let R∗ denote the system throughput which corresponds to the optimal resource allocation τ0∗,τ1∗,τ2∗. It means that R∗>R+,∀τ0+,τ1+,τ2+≠τ0∗,τ1∗,τ2∗ when PH remains unchanged, where R+ denotes the system throughput which corresponds to the resource allocation τ0+,τ1+,τ2+.Therefore, the first subproblem denoted as P2a is formulated as(19)P2a:maxτ0,τ1,τ2minR1,R2s.t.C1−C5Accordingly, P2a is still a nonconvex problem because there are coupling relationships among different optimization variables. In order to solve it, we present the following lemmas.Lemma 1.
R1=R2 is a necessary but insufficient condition for problem P2a obtaining the optimal solution.Proof.
It is assumed thatR1>R2. Then, R=R2 is the maximum system throughput. In order to cut down the SU–relay link throughput R1, we reduce SU backscattering time τ1 to τ1′ for achieving R1′=R2. Let τ1=τ1′+Δ. Thus, we divide Δ into three parts Δ0′,Δ1′,Δ2′ in the ratio of τ0 : τ1′ : τ2 and let τ0′=τ0+Δ0′, τ1′′=τ1′+Δ1′, and τ2′=τ2+Δ2′. Obviously, the achieved system throughput R′ is greater than R which is result from the new resource allocation scheme τ0′,τ1′′,τ2′, which contradicts with the assumption. Then, Lemma 1 is proved.We apply Lemma1 to the problem P2a. Then, the new optimization problem P3 is formulated as
(20)P3:maxτ0,τ1,τ2R1=R2s.t.C1−C5Similar to problem P2a, problem P3 is also a nonconvex problem. To tackle this problem, we first divide the problem P3 into two subproblems and solve two subproblems sequently. Finally, We use the optimal solutions of the two subproblems to obtain the optimal solution of problem P3 by iterative optimization and proportional compression algorithm.At first, we relax the variableτ2 and make τ2=0. Correspondingly, the constrain C1 becomes the new constrain τ0+τ1=Tf. Let τ0=αTf and τ1=1−αTf, where 0<α<1 is the factor of SU transmission time. The first subproblem P3a is formulated as
(21)P3a:maxαR1s.t.C3−C5,C6:0<α<1According to the references [4, 17, 22], Pcr≤Pct is always satisfied in WPCN. Obviously, ER0>EU is always satisfied according to the system model. Thus, the constrain C5 can be satisfied while the constrain C4 is satisfied. Based on the above conclusion, we formulate problem P4 as follows:
(22)P4:maxαR1s.t.C3,C4,C6Obviously, P4 is a more tractable problem. By use of the optimization method in [12, 17], P4 can obtain the maximum throughput while the constrain C4 satisfies EU=Pctτ1. Substituting τ0=αTf, τ1=1−αTf and equation (2) into EU=Pctτ1, the following equation can be obtained shown as follows:
(23)ηPHh22αTf=Pct1−αTf.The optimal solutionα∗ can be easily calculated from equation (16) which is given by
(24)α∗=PctηPHh22+Pct.Substituting the optimal solutionα∗ into the objective function, the maximum throughput R^1 of problem P3a is achieved which is written as
(25)R^1=ηPHh22ηPHh22+PctTfBlog21+PHh02h22σ2.Secondly, we relax the variableτ1 to satisfy τ1=0 and relax the variable Tf to satisfy Tf=TR, where 0<TR<Tf denotes the transmission cycle time of relay. Additionally, to simplify the optimization process, we make τ0=βTR and τ2=1−βTR, where 0<β<1 is the factor of relay transmission cycle time. Thus, another subproblem P3b of problem P3 is formulated as
(26)P3b:maxβR2s.t.C3,C7:0<β<1Substitutingτ0=βTR, τ2=1−βTR and equation (7) into equation (12), R2 can be written as
(27)R2=1−βTRBlog21+ηPHh14β/1−β1−β−Pcth12σ2.Letc=ηPHh14/σ2 and b=1−Pcth12/σ2. Then, the objective function R2 can be rewritten as
(28)R2=1−βTRBlog2cβ1−β+b.Lemma 2.
The optimization problem P3b is convex, and the optimal parameterβ∗=x∗−b/x∗−b+c and x∗ is the solution of equation cxlnx−x+b−c=0.Proof.
See the appendix.
Substituting the optimal solutionβ∗ into the objective function of P3b, the maximum throughput R^2 of problem P3b is achieved which is given by
(29)R^2=1−β∗TRBlog2cβ∗1−β∗+b.
In order to solve problem P3, we makeR^2=R^1 to obtain TR which is expressed as
(30)TR=R^11−β∗Blog2cβ∗/1−β∗1−β∗+b.
According to our model, we can easily get0<TR<Tf for the cause of h12>h22. To find the optimal solution τ0∗,τ1∗,τ2∗, we define τ01=α∗Tf, τ11=1−α∗Tf, τ02=β∗TR, and τ22=1−β∗TR. Obviously, we select τ00=maxτ01,τ02 which can satisfy R1=R2. Then, we devise a iterative optimization algorithm to obtain the optimal solution τ0∗,τ1∗,τ2∗ of problem P3 which is given in Algorithm 1 with an error ε.Algorithm 1: Iterative optimization algorithm for solving problem P3.
1: Initialize the parameterε2: Divide the problem P3 into two subproblems by relaxing variableτ2 and τ1, respectively.3: Solve problem P3a to obtainα∗ and solve problem P3b to obtain β∗.4: CalculateTR by use of (29).5: Letτ01=α∗Tf, τ11=1−α∗Tf, τ02=β∗TR and τ22=1−β∗TR6:ifτ01>τ02then7:  Letτ0=τ01, τ1=τ11 and τ2=τ228:  CalculatePR and R19:  Updateτ2=R1/Blog21+PRh12/σ210:whileτ22−τ2>εdo11:τ22=τ212:   Calculateτ^01=τ01−PRh02/PHh12τ2213:   Letτ0 = maxτ^01,τ0214:   UpdatePR15:   Calculateτ2=R1/Blog21+PRh12/σ216:end while17:else18:  Letτ0=τ02, τ1=τ11 and τ2=τ2219:end if20: Thenτ0∗=τ0Tf/τ0+τ1+τ2, τ1∗=τ1Tf/τ0+τ1+τ2, τ2∗=τ2Tf/τ0+τ1+τ2.


## 4.2. Optimal Transmit Power Allocation Scheme

The optimal power allocation scheme will be aimed at maximizing the energy efficiency in this subsection while the optimal resource allocation solution is obtained. Then, another subproblem P5 of problem P2 is formulated as(31)P5:maxPHψPH=τ1∗Blog21+PHh02h22PHh02h22/σ2PH/ζτ0∗+τ1∗+Pcrτ2∗s.t.C3Letλ=σ2/h02h22 and φ=1/ζτ0∗+τ1∗. By taking the derivative of ψPH with respect to PH, we get
(32)∂ψPH∂PH=τ1∗BλPHφ+Pcrτ2∗τ1∗BλPHφ+Pcrτ2∗/ln21+PHλln21+PHλ−τ1∗Blog21+PHλφPHφ+Pcrτ2∗2.In equation (30), the denominator PHφ+Pcrτ2∗2 is greater than zero. Therefore, the sign of ∂ψPH/∂PH depends on its numerator. We define the numerator as a new function gPH which is given by
(33)gPH=τ1∗BλPHφ+Pcrτ2∗ln21+PHλ−τ1∗Blog21+PHλφ.Similarly, the derivative ofgPH with respect to PH can be written as
(34)∂gPH∂PH=−τ1∗Bλ2PHφ+Pcrτ2∗ln2ln21+PHλ2.It is evident that∂gPH/∂PH is less than zero, which means that gPH is a monotone decreasing function of PH while 0<PH<Pmax. In addition, g0 is greater than zero. Then, when gPmax≥0, the optimal solution PH∗=Pmax. The reason is that ∂ψPH/∂PH is always greater than zero while 0<PH<Pmax. It means that ψPH is a monotone increasing function while 0<PH<Pmax. Therefore, the maximum energy efficient can be obtained while PH∗=Pmax. Otherwise, there exists PH+ satisfying ∂ψPH/∂PH=0. And the energy efficient ψPH is a monotone increasing function while 0<PH<PH+ and ψPH is a monotone decreasing function while PH+<PH<Pmax. Therefore, we can solve the equation gPH∗=0 to obtain the optimal solution PH∗. Thus, the optimal solution PH∗ is given by
(35)PH∗=PmaxgPmax≥0PH+gPmax<0,where PH+ is the unique solution satisfying gPH+=0.To solve the optimal solutionPH∗, we devise a power allocation algorithm which is described in Algorithm 2 with the given error δ.Algorithm 2: Power allocation algorithm.
1: LetP1=0, P2=Pmax and δ>02: Calculatev=gPmax based on Eq.(32).3:ifv<0then4:whileP2−P1>δdo5:  LetPH+=P1+P2/26:  Calculateu=gPH+7:ifu≥0then8:   LetP1=P1+P2/29:else10:   LetP2=P1+P2/211:end if12:end while13:PH∗=PH+14:else15:PH∗=Pmax.16:end if


## 5. Simulation Results

In this section, the performance of WPB-R-CN is evaluated by the system-level simulation. The following parameters are used for simulation unless stated otherwise. The simulation duration is ten time frames, and the channel model between arbitrary nodes is modeled ashi2=gi2di−3 [22], where gi~CN0,1 denotes the channel coefficient between two nodes and is set as gi =1 for simplicity [14]; di is the distance between adjacent nodes. System bandwidth is set B =100 Hz. The distance between relay and SU is set d0 =20 m, the distance between H-AP and relay is set d1 =14 m, and the distance between H-AP and SU is set d2 =30 m. Pmax =30 dBm [14], σ2=−70 dBm, η=0.6 [12], Tf=1.0 s, Pct=−16.5 dBm [12], and Pcr=−20.5 dBm [12].Assume thatτ0∗,τ1∗,τ2∗,PH∗ is an optimal solution for achieving maximum EE of the system. According to Algorithm 1, if the energy EU0 harvested by SU during the time τ0∗ is greater than the energy consumed by SU maintaining the circuit normal operation during the time τ1∗, SU does not require to harvest the energy from the relay during the time τ2∗ . Therefore, it does not need to execute iteration. This scenario is not considered in this section. We mainly play an emphasis on the other scenario that includes EU0<Pctτ1∗ and the energy ER0 harvested by relay during the time τ0∗ being enough to transmit the required data to H-AP. Therefore, SU needs to harvest the energy from the relay during the time τ2∗ to store it into the rechargeable battery for use in the next frame.Figure3 depicts the EE of the proposed Algorithm 1 versus the number of iterations under different H-AP transmit powers PH. It can be seen that Algorithm 1 converges after only three iterations, which demonstrate the efficiency of Algorithm 1. Simultaneously, Figure 3 shows that system EE increases while H-AP transmit power PH increases from 19 dBm to 23 dBm. It shows that the optimal H-AP transmit power PH∗ is greater than 23 dBm.Figure 3
The convergence of Algorithm1.Figure4 shows the system EE performance versus H-AP maximum transmit power Pmax. Four different schemes have been simulated to verify the predominance of the proposed scheme. “Proposed scheme with iteration” denotes as “the proposed Algorithm 1” and the optimal H-AP transmit power PH∗ is obtained by use of Algorithm 2. “Proposed scheme without iteration” denotes as the “scheme 2” that the optimal τ0∗ equals to the maximal value of τ01 and τ02 in the proposed Algorithm 1, and the optimal H-AP transmit power PH∗ is obtained by the use of Algorithm 2. “Throughput maximization” denotes “scheme 3” that τ0∗,τ1∗,τ2∗ are solved by the use of the proposed Algorithm 1, and PH∗ is set to Pmax for achieving the maximal system throughput. To show the superiority of the framework design and proposed scheme, we have simulated the performance of the existing scheme proposed in reference [12], which is denoted as “scheme 4.” In “scheme 4,” we aim at the maximum EE by solving for the optimal H-AP transmitting power. It can be seen from Figure 4 that the system EE of four schemes is increasing with the increasing Pmax while Pmax≤PH∗. “The proposed Algorithm 1,” “scheme 3,” and “scheme 4” behave exactly the same in system EE when Pmax≤PH∗. However, the system EE of “scheme 3” decreases sharply and the other two schemes keep unchangeable while Pmax≥PH∗. We conclude that the throughput maximization scheme may achieve the lower EE than “the proposed Algorithm 1” due to the fact that the optimal resource allocation for throughput maximization is not energy efficient. It illustrates the importance of maximizing the EE. It also can be observed that “the proposed Algorithm 1” always achieves the better system EE than that of the other three schemes, because SU can harvest energy from the relay during the time τ2∗ and the τ0∗ is reduced by the iteration process.Figure 4
System EE versus H-AP maximum transmit powerPmax.In addition, we have also simulated the system throughput performance of four different schemes shown in Figure5, which shows the system throughput performance versus H-AP maximum transmit power Pmax. “The proposed Algorithm 1,” “scheme 2,” and “scheme 4” behave similarly in the system throughput. Their system throughput is increasing when Pmax≤PH∗ and kept unchanged when Pmax≥PH∗, whereas the system throughput of “scheme 3” always increases with the increasing Pmax. The reason is that “the proposed Algorithm 1,” “scheme 2,” and “scheme 4” are aimed at the maximum system EE. When Pmax≥PH∗, the algorithms always select PH=PH∗ as the optimal transmit power. However, “scheme 3” always sets PH=Pmax to achieve the maximum system throughput. Equations (11), (12), and (13) show that the system throughput is increasing function with the variant PH. Therefore, When Pmax≥PH∗, H-AP transmit power is kept unchanged in Algorithm 1” and “scheme 2” and increasing in “scheme 3,” which leads to the different throughput performances.Figure 5
System throughout versus H-AP maximum transmit powerPmax.Simultaneously, Figures4 and 5 tell us that our proposed scheme can always achieve the better performance both in EE performance and throughput performance. The reason is that the optimal parameters are obtained by searching algorithm while our scheme adopts more accurate iterative optimization method. Therefore, the proposed scheme can achieve the better performance compared with the scheme in reference [12]. Simulation results demonstrate that the proposed algorithm is efficient and achieves the more system EE due to adopting the proposed new communication protocol. Figures 4 and 5 illustrate the importance of considering the EE. Another observation is that our proposed scheme can achieve the highest EE among these schemes since the proposed scheme provides more flexibility to utilize the resource efficiently.


## 6. Conclusion

In this article, a new WPBCN with DF relaying has been studied which is more suitable to the realistic IoT application scenario. In order to achieve high system EE in this network, a new communication protocol was developed, in which SU harvests wireless energy from the relay during relay transmitting the data to H-AP and stores its harvested energy in a battery. And the stored energy can be scheduled across different transmission blocks. To maximize the system EE, we obtained the joint power allocation scheme for the proposed model. Then, by investigating the derivative of the EE expression, the optimal power allocation scheme and iterative optimization algorithm were derived for achieving maximum EE. Extensive simulation results have demonstrated that the system EE can be improved about 10% because the proposed scheme provides more flexibility to utilize the resource efficiently by employing the proposed scheme. Our contribution can provide a useful insight for optimizing system performance in both system throughput and EE in WPBCN.



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*Source: 2806423-2022-04-16.xml*