# Sum-Throughput Maximization in Backscatter Communication-Based Cognitive Networks

Authors: Qian Li
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

In this paper, we consider a backscatter communication (BackCom) based cognitive network that consists of one primary transmitter, one primary receiver, multiple secondary transmitters (STs), and one secondary receiver (SR). Each ST operates in the BackCom or energy harvesting model. Our goal is to jointly optimize the energy harvesting and backscatter time, the transmit power of the primary transmitter, and the power reflection coefficient of each ST to maximize the sum throughput of all the STs under a nonlinear energy harvesting model while satisfying multiple constraints, i.e., the energy causality of each ST, the Quality of Service of the primary transmitter, etc. The formulated problem is nonconvex due to the coupled variables and hard to solve. In order to address this problem, we decouple partial coupled variables by using the properties of the objective function and constructing auxiliary variables, and the remaining coupled variables are decoupled via successive convex approximation (SCA). On this basis, a SCA based iterative algorithm is developed to solve the formulated problem. Simulation results are provided to support our work.


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## Body

## 1. Introduction

Internet of Things (IoT) is expected to deploy massive smart sensor nodes in the future communications to seamlessly connect the physical environment and the cyberspace for providing intelligent services [1]. However, such an approach requires huge spectrum resources, and this motivates us to consider high spectrum-efficient communication paradigms for IoT. In this context, cognitive radio has been proposed, whose key idea is to allow sensor nodes sharing spectrum with primary uses without causing any harmful factors to the primary transmission [2]. In cognitive radio, the smart sensor node (also called secondary user) transmits its own information by active radios that need the power-consuming components and consume a lot of energy, greatly shortening their lifespan and leading to an energy-constrained problem for smart sensor nodes.In addition to the cognitive radio that improves the spectrum efficiency, backscatter communication (BackCom) is another key technology, and its main purpose is to overcome the energy-constrained problem [3–5]. BackCom allows a smart sensor node modulating its information on the incident signals and backscattering the modulated signals to its associated receiver by changing the power reflection coefficient so that the power-consuming components can be avoided, while harvesting energy from the incident signal for realizing energy self-sustainability [3–5]. Despite these superiorities, the communication performance of BackCom is limited as it uses the ambient signals as the incident signals, and the ambient signals introduce serious cochannel interference to the BackCom receiver [4]. Accordingly, researchers proposed to use the controllable signals as the incident signals, but such an approach requires an extra cost to deploy RF sources. Recall that cognitive radio is able to provide controllable signals for the BackCom transmitter (also referred to as the secondary transmitter (ST) in this paper). Recent works have integrated BackCom into cognitive radio, yielding a spectrum- and energy-efficiency paradigm, called BackCom based cognitive networks.In [6], the authors formulated a problem to maximize the throughput of a BackCom link in a BackCom based cognitive network with a single ST by jointly optimizing the transmit power of the primary user and the power reflection coefficient of the ST. In [7], the authors proposed to maximize the energy efficiency of the secondary link by jointly optimizing the transmit power of the primary transmitter (PT), the power reflection coefficient of the ST, and the time for energy harvesting and BackCom. Extending the single ST scenario [6, 7] into multiple STs [8], the authors maximized the sum rate of STs by jointly optimizing the PT’s transmit power and the power reflection coefficient of each ST. The authors of [9] considered a full-duplex-enabled BackCom based cognitive network and proposed a joint time, transmit power, and power reflection coefficient scheduling to maximize the throughput of the BackCom system. In addition to the above works, the harvest-then-transmit (HTT) protocol has also been integrated into BackCom based cognitive networks, and various resource allocation schemes have been studied [10–12], where the main focus is to balance the time for energy harvesting, HTT, and backscattering.After carefully examining the existing resource allocation schemes, we note that all of them were based on a linear energy harvesting model. As pointed out by existing works, the linear energy harvesting model does not match the behavior of a practical energy harvester. More specifically, the harvest power is a nonlinear function with respect to the input power. The previous works [13–15] (in which BackCom based cognitive network has not been considered) have proved that the mismatch of energy harvesting models will lead to performance degradation. Accordingly, it is necessary to design resource allocation for BackCom based cognitive networks with nonlinear energy harvesting model. Motivated by this, in this paper, we consider a BackCom based cognitive network with multiple STs and aim to maximize the total throughput of STs by jointly optimizing the energy harvesting time, BackCom time, power reflection coefficient, and PT’s transmit power under a nonlinear energy harvesting model. Meanwhile, the energy-causality constraint of each ST and the Quality of Service (QoS) of both the primary link and the secondary links are considered. The main contributions of this paper are summarized as below.We formulate an optimization problem to maximize the sum throughput of STs, and propose an efficient iterative algorithm to solve the problem. Since the formulated problem is nonconvex, the main challenge is how to transform the original problem into a convex one. Towards this end, we firstly determine the optimal transmit power of the PT by using the properties of the objective function, then introduce some auxiliary variables to decouple coupled variables, and lastly employ the successive convex approximation (SCA) to transform a nonconvex constraint into a linear one. We also provide computer simulation results to verify the proposed iterative algorithm and show the advantages of the proposed scheme over the baseline schemes.


## 2. System Model and Working Flow

As shown in Figure1, we consider a BackCom based cognitive network, which consists of one PT, one PR, K energy-constrained STs, and one SR. In this network, the PT transmits its own information to the PR, while the RF signals transmitted by the PT can also be exploited by the K STs for energy harvesting (EH) and information transmission. Assume that all the devices are equipped with a single antenna and work in the half-duplex mode. Suppose that each ST is equipped with both the EH circuit and the BackCom circuit so that it can harvest energy from the received RF signals and backscatter the received signals for information transmission. In order to prolong the operation time of each energy-constrained ST, we assume that each ST only uses its harvested energy instead of the energy early stored in its battery to cover the energy consumption during information backscattering. All channels including the PT-PR link and the ST-SR links are assumed to follow quasi-static fading. In the beginning of each transmission block, the channel state information (CSI) of all links can be perfectly achieved by the PT via the existing advanced channel estimation methods, so that the PT can determine the optimal resource allocation scheme based on the obtained CSI and feed the optimal scheme back to all the STs.Figure 1
BackCom based cognitive network and its frame structure.LetT denote the duration of the whole transmission block. For the PT-PR link, the PT may transmit its signals in the whole transmission block. For the ST-SR links, the whole transmission can be divided into two phases, which are the EH phase and the BackCom phase, respectively. In the EH phase, the PT broadcasts its signals to the PR while all the STs will perform EH. In the BackCom phase, the PT keeps broadcasting while all the STs take turns to perform BackCom in order to avoid the cochannel interference among STs.


### 2.1. EH Phase

Lette denote the duration of the EH phase. Denote Pt as the transmit power of the PT. Then, the received signal at the k-th k∈K=1,2,…,K ST is given by(1)ySTk=Pthkxp+NST,where hk denotes the channel gain of the PT–k-th ST link, xp with Exp2=1 is the information transmitted by the PT to the PR, and NST is the thermal noise at the k-th ST. Since the backscatter communication circuit consists only of passive components and takes few signal processing operations, the thermal noise is usually very small and can be ignored, i.e., NST≈0 [7].For EH, we consider a more practical nonlinear EH model since the linear EH model cannot characterize the nonlinearity of the practical EH circuit [16]. The reasons of considering the above nonlinear EH model instead of the one considered in [17, 18] are as follows. Firstly, the nonlinear EH model proposed in [16] is accurate enough for characterizing the nonlinearity of practical EH circuits. Secondly, the use of the nonlinear EH model proposed in [16] can simplify the difficulty and reduce the complexity of solving the formulated optimization problem. Accordingly, we compute the harvested energy of the k-th as(2)Eek=teakPthk+dkPthk+vk−dkvk,where ak,dk, and vk are the parameters of the nonlinear EH model.For the PT-PR link, the received signal at the PR is expressed as(3)yPRe=Ptfpxp+NPR,where fp denotes the channel gain of the PT-PR link and NPR is the additive white Gaussian noise (AWGN) at the PR with mean zero and variance σ2. Accordingly, the achievable throughput at the PR in this phase can be computed as(4)Rep=teBlog21+Ptfpσ2,where B is the system bandwidth.


### 2.2. BackCom Phase

The whole BackCom phase can be divided intoK subphases. In each subphase, each ST performs BackCom to transmitted information. Let tk denote the duration of the k-th subphase. Denote αk with 0≤αk≤1 as the power reflection coefficient of the k-th ST based on which the received RF signal at the k-th ST can be split into two parts: one part is used for BackCom and the other part is flowed into the EH circuit. Then, in the subphase tk, the received signal at the SR is given by(5)ySRk=αkPthkgkxpxs,k+Ptfsxp+NSR,where gk denotes the channel gain between the k-th ST and the SR, fs is the channel gain of the PT-SR link, xs,k with Exs,k2=1 is the transmitted information of the k-th ST, and NSR is the AWGN at the SR with mean zero and variance σ2.From (5), it can be observed that the cochannel interference from the PT-SR link always exists, which degrades the throughput achieved by the k-th ST via BackCom. In order to decode the transmitted information of the k-th ST correctly, the SR performs the successive interference cancellation (SIC) technology. Specifically, the SR first decodes the PT’s transmitted information xp by treating αkPthkgkxpxs,k as the cochannel interference and then uses the SIC technology to cancel the interference from the PT as well as decoding the transmitted information of the k-th ST xs,k. Therefore, the signal to interference plus noise ratio (SINR) at the SR for decoding xp is given by(6)γs,kp=PtfsαkPthkgk+σ2.After using SIC technology, the signal to noise ratio (SNR) at the SR for decodingxs,k is expressed as(7)γs,k=αkPthkgkηPtfs+σ2,where η with 0≤η≤1 is the interference cancellation factor.Accordingly, the achievable throughput of thek-th ST-SR link can be computed as(8)Rs,k=tkBlog21+ξγs,k,where ξ expresses the performance gap reflecting the real modulation [19–21]. In this subphase, the harvested energy of the k-th ST is determined by(9)Ekb=tkak1−αkPthk+dk1−αkPthk+vk−dkvk.At the end of the BackCom phase, the total harvested energy of thek-th ST is given by(10)Etotk=Ekb+te+∑i=1Kti−tkakPthk+dkPthk+vk−dkvk.For the PT-PR link, it also suffers from the cochannel interference from thek-th ST, and the received signal at the PR is given by(11)yPRk=Ptfpxp+αkPthkfkxpxs,k+NPR,where fk denotes the channel gain from the k-th ST to the PR. Then, the SINR at the PR for decoding xp is computed as(12)γkp=PtfpαkPthkfk+σ2.Correspondingly, the achievable throughput of the PT-PR link in this subphase is given byRkp=tkBlog21+γkp.


## 2.1. EH Phase

Lette denote the duration of the EH phase. Denote Pt as the transmit power of the PT. Then, the received signal at the k-th k∈K=1,2,…,K ST is given by(1)ySTk=Pthkxp+NST,where hk denotes the channel gain of the PT–k-th ST link, xp with Exp2=1 is the information transmitted by the PT to the PR, and NST is the thermal noise at the k-th ST. Since the backscatter communication circuit consists only of passive components and takes few signal processing operations, the thermal noise is usually very small and can be ignored, i.e., NST≈0 [7].For EH, we consider a more practical nonlinear EH model since the linear EH model cannot characterize the nonlinearity of the practical EH circuit [16]. The reasons of considering the above nonlinear EH model instead of the one considered in [17, 18] are as follows. Firstly, the nonlinear EH model proposed in [16] is accurate enough for characterizing the nonlinearity of practical EH circuits. Secondly, the use of the nonlinear EH model proposed in [16] can simplify the difficulty and reduce the complexity of solving the formulated optimization problem. Accordingly, we compute the harvested energy of the k-th as(2)Eek=teakPthk+dkPthk+vk−dkvk,where ak,dk, and vk are the parameters of the nonlinear EH model.For the PT-PR link, the received signal at the PR is expressed as(3)yPRe=Ptfpxp+NPR,where fp denotes the channel gain of the PT-PR link and NPR is the additive white Gaussian noise (AWGN) at the PR with mean zero and variance σ2. Accordingly, the achievable throughput at the PR in this phase can be computed as(4)Rep=teBlog21+Ptfpσ2,where B is the system bandwidth.


## 2.2. BackCom Phase

The whole BackCom phase can be divided intoK subphases. In each subphase, each ST performs BackCom to transmitted information. Let tk denote the duration of the k-th subphase. Denote αk with 0≤αk≤1 as the power reflection coefficient of the k-th ST based on which the received RF signal at the k-th ST can be split into two parts: one part is used for BackCom and the other part is flowed into the EH circuit. Then, in the subphase tk, the received signal at the SR is given by(5)ySRk=αkPthkgkxpxs,k+Ptfsxp+NSR,where gk denotes the channel gain between the k-th ST and the SR, fs is the channel gain of the PT-SR link, xs,k with Exs,k2=1 is the transmitted information of the k-th ST, and NSR is the AWGN at the SR with mean zero and variance σ2.From (5), it can be observed that the cochannel interference from the PT-SR link always exists, which degrades the throughput achieved by the k-th ST via BackCom. In order to decode the transmitted information of the k-th ST correctly, the SR performs the successive interference cancellation (SIC) technology. Specifically, the SR first decodes the PT’s transmitted information xp by treating αkPthkgkxpxs,k as the cochannel interference and then uses the SIC technology to cancel the interference from the PT as well as decoding the transmitted information of the k-th ST xs,k. Therefore, the signal to interference plus noise ratio (SINR) at the SR for decoding xp is given by(6)γs,kp=PtfsαkPthkgk+σ2.After using SIC technology, the signal to noise ratio (SNR) at the SR for decodingxs,k is expressed as(7)γs,k=αkPthkgkηPtfs+σ2,where η with 0≤η≤1 is the interference cancellation factor.Accordingly, the achievable throughput of thek-th ST-SR link can be computed as(8)Rs,k=tkBlog21+ξγs,k,where ξ expresses the performance gap reflecting the real modulation [19–21]. In this subphase, the harvested energy of the k-th ST is determined by(9)Ekb=tkak1−αkPthk+dk1−αkPthk+vk−dkvk.At the end of the BackCom phase, the total harvested energy of thek-th ST is given by(10)Etotk=Ekb+te+∑i=1Kti−tkakPthk+dkPthk+vk−dkvk.For the PT-PR link, it also suffers from the cochannel interference from thek-th ST, and the received signal at the PR is given by(11)yPRk=Ptfpxp+αkPthkfkxpxs,k+NPR,where fk denotes the channel gain from the k-th ST to the PR. Then, the SINR at the PR for decoding xp is computed as(12)γkp=PtfpαkPthkfk+σ2.Correspondingly, the achievable throughput of the PT-PR link in this subphase is given byRkp=tkBlog21+γkp.


## 3. Throughput Maximization for BackCom-Based Cognitive Networks

In this section, we design an optimal resource allocation scheme to maximize the total throughput of all the STs for the BackCom based cognitive network. In particular, we formulate a throughput maximization problem by jointly optimizing the transmit power of the PT, BackCom time, and power reflection coefficients of STs, as well as the EH time, subject to QoS, energy causality, latency, transmit power, and power reflection coefficient constraints, and then use the existing convex tools to solve it.


### 3.1. Problem Formulation

Before formulating the throughput maximization problem, we should determine the optimization objective and constraints.


#### 3.1.1. Optimization Objective

The optimization objective is to maximize the total throughput of the STs which can be computed as(13)Rtots=∑k=1KRs,k=∑k=1KtkBlog21+ξαkPthkgkηPtfs+σ2.


#### 3.1.2. QoS Constraints

There are two QoS constraints to constrain the throughput of the PT and each ST, respectively. For the QoS constraint of each ST, we should guarantee that the achievable throughput of each ST is not less than its minimum required throughput. LetCmin,k denote the minimum required throughput of the k-th ST. Then, the QoS constraint of the k-th ST can be expressed as(14)Rs,k≥Cmin,k,∀k,⟺tkBlog21+ξαkPthkgkηPtfs+σ2≥Cmin,k,∀k.For the QoS constraint of the PT, the total achievable throughput of the PT should not be less than the PT’s minimum required throughput, denoted byCmin. Therefore, the QoS constraint of the PT can be expressed as(15)Rep+∑k=1KRkp=teBlog21+Ptfpσ2+∑k=1KtkBlog21+PtfpαkPthkfk+σ2≥Cmin.


#### 3.1.3. Energy-Causality Constraint

The energy-causality constraint states that the energy consumption of each ST should not be larger than its harvested energy during the whole transmission block. Note that a rechargeable battery is equipped in each ST and that each ST may first use the energy stored in its battery to support BackCom and then use the harvested energy to power the battery. The energy-causality constraint ensures that the energy early stored in the battery is not reduced. Here, we consider fixed power consumption for BackCom by following [19–21]. Let Pb,k denote the power consumption for the k-th ST when performing BackCom. Therefore, the energy-causality constraint for the k-th ST is given by(16)Pb,ktk≤Etotk=tkfk1−αkPthk+te+∑i=1Kti−tkfkPthk,∀k,where fkx=akx+dk/x+vk−dk/vk. Please note that the power consumption for the EH circuit has been included in the EH model and thus has not been considered in (16).


#### 3.1.4. Transmit Power Constraint

LetPmax denote the maximum allowed transmit power of the PT. Then, the PT’s transmit power constraint can be expressed as(17)0≤Pt≤Pmax.Based on (13), (14), (15), (16), and (17), the throughput maximization problem can be formulated as(18)P1:maxPt,te,tkk=1K,αkk=1KRtots,s.t.C1:14,15,C2:16,C3:17,C4:γs,kp≥γth,∀k,C5:te+∑k=1Ktk≤T,te,tk≥0,∀k,C6:0≤αk≤,∀k,where C1 denotes the QoS constraints for each ST and the PT, C2 is the energy-causality constraint for each ST, C3 constrains the maximum transmit power of the PT, C4 ensures that each ST can decode xp successfully and γth is the threshold required for decoding xp, C5 is the latency constraint, and C6 is the constraint for the power reflection coefficient of each ST.It can be observed thatP1 is a highly nonconvex optimization problem and is difficult to solve due to the following reasons. Firstly, there exist several coupled relationships between multiple optimization variables, i.e., Pt, tk, αk, etc., leading to the nonconvex objective function and constraints, i.e., C1, C2, etc. Secondly, the cochannel interference causes the difference of convex (DC) structures in the objective function and C1, bringing new challenges to solving P1. Thirdly, the consideration of the nonlinear EH model is another difficulty for solving P1 since the nonlinear EH model makes C1 more complex. Therefore, it is hard to solve P1.


### 3.2. Solution

In order to remove DC structures existing in the objective function andC1 and simplify P1, the following lemma is introduced to determine the optimal transmit power of the PT.Lemma 1.
The maximum throughput of all the STs for the considered network is achieved when the PT transmits its signals with its maximum transmit power, i.e.,Pt∗=Pmax.Proof.
Please see Appendix A.
By substitutingPt=Pmax into P1, P1 can be reformulated as(19)P2:maxte,tkk=1K,αkk=1K∑k=1KtkBlog21+ζαkPmaxhkgkηPmaxfs+σ2,s.t.C1′:tkBlog21+ζαkPmaxhkgkηPmaxfs+σ2≥Cmin,k,∀k,∑k=1KtkBlog21+PmaxfpαkPmaxhkfk+σ2+teBlog21+Pmaxfpσ2≥Cmin,C2′:Pb,ktk≤te+∑i=1Kti−tkfkPmaxhk+tkfk1−αkPmaxhk,∀k,C5,C7:0≤αk≤minPmaxfs−γthσ2Pmaxhkgkγth,1,∀k,where C7 is the combination of C4 and C6.
P2 is still nonconvex since the coupled relationships between different variables, e.g., αk and tk, still exist in the objective function and several constraints. To address this issue, we introduce the following auxiliary variables: zk=αktk,∀k, to replace the variables αk,∀k, and rewrite P2 as(20)P2:maxte,tkk=1K,zkk=1K∑k=1KtkBlog21+ζzkPmaxhkgktkηPmaxfs+σ2,s.t.C1″:tkBlog21+ζzkPmaxhkgktkηPmaxfs+σ2≥Cmin,k,∀k,∑k=1KtkBlog21+PmaxfpzkPmaxhkfk+σ2+teBlog21+Pmaxfpσ2≥Cmin,C2″:Pb,ktk≤tkfktk−zkPmaxhktkte+∑i=1Kti−tkfkPmaxhk,∀k,C5,C7′:0≤zk≤tk×minPmaxfs−γthσ2Pmaxhkgkγth,1,where αk=zk/tk,∀k.Proposition 1.
InP3, the objective function and all the constraints except C1′′ are convex.Proof.
Please see Appendix B.
In order to handle the nonconvex constraintC1′′ and solve P3, the SCA method is used, where the first-order Taylor expression is used to approximate the function tkBlog21+tkPmaxfp/zkPmaxhkfk+σ2tk in C1′′ and turn this nonconvex function into a linear function. Specifically, let Fkαk=tkBlog21+Pmaxfp/αkPmaxhkfk+σ2. By taking the first-order derivative of Fkαk with respect to αk, we have(21)∂Fkαk∂αk=−Pmax2fphkfkBtkαkPmaxhkfk+σ2+PmaxfpαkPmaxhkfk+σ2In2.
Using the first-order Taylor expression,Fkαk can be approximated as(22)Fkαk≈∂Fkαk0∂αk0αk−αk0+Fkαk0=−Pmax2fphkfkBtkαk−αk0αk0Pmaxhkfk+σ2+Pmaxfpαk0Pmaxhkfk+σ2In2+Fkαk0,=−Pmax2fphkfkBzk−αk0tkαk0Pmaxhkfk+σ2+Pmaxfpαk0Pmaxhkfk+σ2In2+Fkαk0,where αk0 is the given value for αk and can be updated iteration by iteration.
By substituting (22) into C1′′, the QoS constraint for the PT’s transmission can be rewritten as(23)∑k=1K−Pmax2fphkfkBzk−αk0tkαk0Pmaxhkfk+σ2+Pmaxfpαk0Pmaxhkfk+σ2In2+Fkαk0+teBlog21+Pmaxfpσ2≥Cmin.
Accordingly,P3 can be transformed into the following subproblem, given by(24)P4:maxte,tkk=1K,zkk=1K∑k=1KtkBlog21+ζzkPmaxhkgktkηPmaxfs+σ2,s.t.C1‴:tkBlog21+ζzkPmaxhkgktkηPmaxfs+σ2≥Cmin,k,∀k,∑k=1KtkBlog21+−Pmax2fphkfkBzk−αk0tkαk0Pmaxhkfk+σ2+Pmaxfpαk0Pmaxhkfk+σ2In2+Fkαk0+teBlog21+Pmaxfpσ2≥Cmin,C2‴,C5,C7′.Proposition 2.
P4 is proved to be convex, which can be efficiently solved by using the existing convex optimization tools.Proof.
After using the first-order Taylor expression, the nonconvex QoS constraint for the PT’s transmission inC1′′′ can be turned into a linear constraint. Combining with Proposition 1, P4 can be proved to be convex and can be efficiently solved by using the existing convex optimization tools.


### 3.3. Design of a SCA Based Iterative Algorithm

In this subsection, we propose a SCA based iterative algorithm to solveP3 efficiently. The detailed process of the proposed algorithm is shown in Algorithm 1.Algorithm1: SCA based iterative algorithm.
(1) Set the maximum toleranceε and the maximum number of iterations Imax;(2) Set the iteration indexi=1 and the initial given values αk0,∀k;(3) Based on Lemma1, the optimal transmit power of the PT Pt∗ is set as Pmax;(4) repeat(5) Solve the optimization problemP4 with given αk0,∀k, to obtain the optimal solutions, denoted by te∗,tk∗k=1K,zk∗k=1K;(6) Computeαk∗ as zk∗/tk∗,∀k;(7) Compute the value ofRtots based on (13);(8) ifαk∗−αk0≤ε then(9)  SetFlag=1;(10) else(11)  SetFlag=0 and i=i+1;(12)  Updateαk0 as αk0=αk∗,∀k;(13) end if(14)
untilFlag=1 or i=Imax.(15)
OutputPt∗, te∗,tk∗k=1K,αk∗k=1K and Rtots.As shown in Algorithm1, in each iteration, we should use the existing convex tools, i.e., CVX, to optimally solve the subproblem P4 with given αk0,∀k. Then the optimal solutions to P4 are obtained, denoted by te∗,tk∗k=1K,αk∗k=1K, where αk∗ is computed as zk∗/tk∗,∀k. If the stop condition, namely, |αk∗−αk0|≤ε with the maximum tolerance ε, is satisfied, then the solution to P3 is te∗,tk∗k=1K,αk∗k=1K. Otherwise, we should update the value of αk0 as αk∗ and repeat the above steps until the stop condition is satisfied.


## 3.1. Problem Formulation

Before formulating the throughput maximization problem, we should determine the optimization objective and constraints.


### 3.1.1. Optimization Objective

The optimization objective is to maximize the total throughput of the STs which can be computed as(13)Rtots=∑k=1KRs,k=∑k=1KtkBlog21+ξαkPthkgkηPtfs+σ2.


### 3.1.2. QoS Constraints

There are two QoS constraints to constrain the throughput of the PT and each ST, respectively. For the QoS constraint of each ST, we should guarantee that the achievable throughput of each ST is not less than its minimum required throughput. LetCmin,k denote the minimum required throughput of the k-th ST. Then, the QoS constraint of the k-th ST can be expressed as(14)Rs,k≥Cmin,k,∀k,⟺tkBlog21+ξαkPthkgkηPtfs+σ2≥Cmin,k,∀k.For the QoS constraint of the PT, the total achievable throughput of the PT should not be less than the PT’s minimum required throughput, denoted byCmin. Therefore, the QoS constraint of the PT can be expressed as(15)Rep+∑k=1KRkp=teBlog21+Ptfpσ2+∑k=1KtkBlog21+PtfpαkPthkfk+σ2≥Cmin.


### 3.1.3. Energy-Causality Constraint

The energy-causality constraint states that the energy consumption of each ST should not be larger than its harvested energy during the whole transmission block. Note that a rechargeable battery is equipped in each ST and that each ST may first use the energy stored in its battery to support BackCom and then use the harvested energy to power the battery. The energy-causality constraint ensures that the energy early stored in the battery is not reduced. Here, we consider fixed power consumption for BackCom by following [19–21]. Let Pb,k denote the power consumption for the k-th ST when performing BackCom. Therefore, the energy-causality constraint for the k-th ST is given by(16)Pb,ktk≤Etotk=tkfk1−αkPthk+te+∑i=1Kti−tkfkPthk,∀k,where fkx=akx+dk/x+vk−dk/vk. Please note that the power consumption for the EH circuit has been included in the EH model and thus has not been considered in (16).


### 3.1.4. Transmit Power Constraint

LetPmax denote the maximum allowed transmit power of the PT. Then, the PT’s transmit power constraint can be expressed as(17)0≤Pt≤Pmax.Based on (13), (14), (15), (16), and (17), the throughput maximization problem can be formulated as(18)P1:maxPt,te,tkk=1K,αkk=1KRtots,s.t.C1:14,15,C2:16,C3:17,C4:γs,kp≥γth,∀k,C5:te+∑k=1Ktk≤T,te,tk≥0,∀k,C6:0≤αk≤,∀k,where C1 denotes the QoS constraints for each ST and the PT, C2 is the energy-causality constraint for each ST, C3 constrains the maximum transmit power of the PT, C4 ensures that each ST can decode xp successfully and γth is the threshold required for decoding xp, C5 is the latency constraint, and C6 is the constraint for the power reflection coefficient of each ST.It can be observed thatP1 is a highly nonconvex optimization problem and is difficult to solve due to the following reasons. Firstly, there exist several coupled relationships between multiple optimization variables, i.e., Pt, tk, αk, etc., leading to the nonconvex objective function and constraints, i.e., C1, C2, etc. Secondly, the cochannel interference causes the difference of convex (DC) structures in the objective function and C1, bringing new challenges to solving P1. Thirdly, the consideration of the nonlinear EH model is another difficulty for solving P1 since the nonlinear EH model makes C1 more complex. Therefore, it is hard to solve P1.


## 3.1.1. Optimization Objective

The optimization objective is to maximize the total throughput of the STs which can be computed as(13)Rtots=∑k=1KRs,k=∑k=1KtkBlog21+ξαkPthkgkηPtfs+σ2.


## 3.1.2. QoS Constraints

There are two QoS constraints to constrain the throughput of the PT and each ST, respectively. For the QoS constraint of each ST, we should guarantee that the achievable throughput of each ST is not less than its minimum required throughput. LetCmin,k denote the minimum required throughput of the k-th ST. Then, the QoS constraint of the k-th ST can be expressed as(14)Rs,k≥Cmin,k,∀k,⟺tkBlog21+ξαkPthkgkηPtfs+σ2≥Cmin,k,∀k.For the QoS constraint of the PT, the total achievable throughput of the PT should not be less than the PT’s minimum required throughput, denoted byCmin. Therefore, the QoS constraint of the PT can be expressed as(15)Rep+∑k=1KRkp=teBlog21+Ptfpσ2+∑k=1KtkBlog21+PtfpαkPthkfk+σ2≥Cmin.


## 3.1.3. Energy-Causality Constraint

The energy-causality constraint states that the energy consumption of each ST should not be larger than its harvested energy during the whole transmission block. Note that a rechargeable battery is equipped in each ST and that each ST may first use the energy stored in its battery to support BackCom and then use the harvested energy to power the battery. The energy-causality constraint ensures that the energy early stored in the battery is not reduced. Here, we consider fixed power consumption for BackCom by following [19–21]. Let Pb,k denote the power consumption for the k-th ST when performing BackCom. Therefore, the energy-causality constraint for the k-th ST is given by(16)Pb,ktk≤Etotk=tkfk1−αkPthk+te+∑i=1Kti−tkfkPthk,∀k,where fkx=akx+dk/x+vk−dk/vk. Please note that the power consumption for the EH circuit has been included in the EH model and thus has not been considered in (16).


## 3.1.4. Transmit Power Constraint

LetPmax denote the maximum allowed transmit power of the PT. Then, the PT’s transmit power constraint can be expressed as(17)0≤Pt≤Pmax.Based on (13), (14), (15), (16), and (17), the throughput maximization problem can be formulated as(18)P1:maxPt,te,tkk=1K,αkk=1KRtots,s.t.C1:14,15,C2:16,C3:17,C4:γs,kp≥γth,∀k,C5:te+∑k=1Ktk≤T,te,tk≥0,∀k,C6:0≤αk≤,∀k,where C1 denotes the QoS constraints for each ST and the PT, C2 is the energy-causality constraint for each ST, C3 constrains the maximum transmit power of the PT, C4 ensures that each ST can decode xp successfully and γth is the threshold required for decoding xp, C5 is the latency constraint, and C6 is the constraint for the power reflection coefficient of each ST.It can be observed thatP1 is a highly nonconvex optimization problem and is difficult to solve due to the following reasons. Firstly, there exist several coupled relationships between multiple optimization variables, i.e., Pt, tk, αk, etc., leading to the nonconvex objective function and constraints, i.e., C1, C2, etc. Secondly, the cochannel interference causes the difference of convex (DC) structures in the objective function and C1, bringing new challenges to solving P1. Thirdly, the consideration of the nonlinear EH model is another difficulty for solving P1 since the nonlinear EH model makes C1 more complex. Therefore, it is hard to solve P1.


## 3.2. Solution

In order to remove DC structures existing in the objective function andC1 and simplify P1, the following lemma is introduced to determine the optimal transmit power of the PT.Lemma 1.
The maximum throughput of all the STs for the considered network is achieved when the PT transmits its signals with its maximum transmit power, i.e.,Pt∗=Pmax.Proof.
Please see Appendix A.
By substitutingPt=Pmax into P1, P1 can be reformulated as(19)P2:maxte,tkk=1K,αkk=1K∑k=1KtkBlog21+ζαkPmaxhkgkηPmaxfs+σ2,s.t.C1′:tkBlog21+ζαkPmaxhkgkηPmaxfs+σ2≥Cmin,k,∀k,∑k=1KtkBlog21+PmaxfpαkPmaxhkfk+σ2+teBlog21+Pmaxfpσ2≥Cmin,C2′:Pb,ktk≤te+∑i=1Kti−tkfkPmaxhk+tkfk1−αkPmaxhk,∀k,C5,C7:0≤αk≤minPmaxfs−γthσ2Pmaxhkgkγth,1,∀k,where C7 is the combination of C4 and C6.
P2 is still nonconvex since the coupled relationships between different variables, e.g., αk and tk, still exist in the objective function and several constraints. To address this issue, we introduce the following auxiliary variables: zk=αktk,∀k, to replace the variables αk,∀k, and rewrite P2 as(20)P2:maxte,tkk=1K,zkk=1K∑k=1KtkBlog21+ζzkPmaxhkgktkηPmaxfs+σ2,s.t.C1″:tkBlog21+ζzkPmaxhkgktkηPmaxfs+σ2≥Cmin,k,∀k,∑k=1KtkBlog21+PmaxfpzkPmaxhkfk+σ2+teBlog21+Pmaxfpσ2≥Cmin,C2″:Pb,ktk≤tkfktk−zkPmaxhktkte+∑i=1Kti−tkfkPmaxhk,∀k,C5,C7′:0≤zk≤tk×minPmaxfs−γthσ2Pmaxhkgkγth,1,where αk=zk/tk,∀k.Proposition 1.
InP3, the objective function and all the constraints except C1′′ are convex.Proof.
Please see Appendix B.
In order to handle the nonconvex constraintC1′′ and solve P3, the SCA method is used, where the first-order Taylor expression is used to approximate the function tkBlog21+tkPmaxfp/zkPmaxhkfk+σ2tk in C1′′ and turn this nonconvex function into a linear function. Specifically, let Fkαk=tkBlog21+Pmaxfp/αkPmaxhkfk+σ2. By taking the first-order derivative of Fkαk with respect to αk, we have(21)∂Fkαk∂αk=−Pmax2fphkfkBtkαkPmaxhkfk+σ2+PmaxfpαkPmaxhkfk+σ2In2.
Using the first-order Taylor expression,Fkαk can be approximated as(22)Fkαk≈∂Fkαk0∂αk0αk−αk0+Fkαk0=−Pmax2fphkfkBtkαk−αk0αk0Pmaxhkfk+σ2+Pmaxfpαk0Pmaxhkfk+σ2In2+Fkαk0,=−Pmax2fphkfkBzk−αk0tkαk0Pmaxhkfk+σ2+Pmaxfpαk0Pmaxhkfk+σ2In2+Fkαk0,where αk0 is the given value for αk and can be updated iteration by iteration.
By substituting (22) into C1′′, the QoS constraint for the PT’s transmission can be rewritten as(23)∑k=1K−Pmax2fphkfkBzk−αk0tkαk0Pmaxhkfk+σ2+Pmaxfpαk0Pmaxhkfk+σ2In2+Fkαk0+teBlog21+Pmaxfpσ2≥Cmin.
Accordingly,P3 can be transformed into the following subproblem, given by(24)P4:maxte,tkk=1K,zkk=1K∑k=1KtkBlog21+ζzkPmaxhkgktkηPmaxfs+σ2,s.t.C1‴:tkBlog21+ζzkPmaxhkgktkηPmaxfs+σ2≥Cmin,k,∀k,∑k=1KtkBlog21+−Pmax2fphkfkBzk−αk0tkαk0Pmaxhkfk+σ2+Pmaxfpαk0Pmaxhkfk+σ2In2+Fkαk0+teBlog21+Pmaxfpσ2≥Cmin,C2‴,C5,C7′.Proposition 2.
P4 is proved to be convex, which can be efficiently solved by using the existing convex optimization tools.Proof.
After using the first-order Taylor expression, the nonconvex QoS constraint for the PT’s transmission inC1′′′ can be turned into a linear constraint. Combining with Proposition 1, P4 can be proved to be convex and can be efficiently solved by using the existing convex optimization tools.


## 3.3. Design of a SCA Based Iterative Algorithm

In this subsection, we propose a SCA based iterative algorithm to solveP3 efficiently. The detailed process of the proposed algorithm is shown in Algorithm 1.Algorithm1: SCA based iterative algorithm.
(1) Set the maximum toleranceε and the maximum number of iterations Imax;(2) Set the iteration indexi=1 and the initial given values αk0,∀k;(3) Based on Lemma1, the optimal transmit power of the PT Pt∗ is set as Pmax;(4) repeat(5) Solve the optimization problemP4 with given αk0,∀k, to obtain the optimal solutions, denoted by te∗,tk∗k=1K,zk∗k=1K;(6) Computeαk∗ as zk∗/tk∗,∀k;(7) Compute the value ofRtots based on (13);(8) ifαk∗−αk0≤ε then(9)  SetFlag=1;(10) else(11)  SetFlag=0 and i=i+1;(12)  Updateαk0 as αk0=αk∗,∀k;(13) end if(14)
untilFlag=1 or i=Imax.(15)
OutputPt∗, te∗,tk∗k=1K,αk∗k=1K and Rtots.As shown in Algorithm1, in each iteration, we should use the existing convex tools, i.e., CVX, to optimally solve the subproblem P4 with given αk0,∀k. Then the optimal solutions to P4 are obtained, denoted by te∗,tk∗k=1K,αk∗k=1K, where αk∗ is computed as zk∗/tk∗,∀k. If the stop condition, namely, |αk∗−αk0|≤ε with the maximum tolerance ε, is satisfied, then the solution to P3 is te∗,tk∗k=1K,αk∗k=1K. Otherwise, we should update the value of αk0 as αk∗ and repeat the above steps until the stop condition is satisfied.


## 4. Numerical Results

In this section, both the effectiveness and the superiority of the proposed algorithm are verified via computer simulations. The key simulation parameters, unless otherwise specified, are provided in Table1. Following [16], the parameters of the considered nonlinear EH model at the k-th ST are set as ak=2.463, dk=1.635, and vk=0.826,∀k. For the settings of all channels, we consider a standard channel fading model. Specifically, the channel gain of the PT-PR link is modeled by fp=fp′Dp−β, where fp′ denotes the small-scale fading of the PT-PR link, Dp is the distance from the PT to the PR, and β denotes the path loss exponent. The channel gain of the PT–k-th ST link is given by gk=gk′D1,k−β,∀k, where gk′ and D1,k are the small-scale fading and the distance from the PT to the k-th ST, respectively. The channel gain of the k-th ST–SR link is modeled by hk=hk′D2,k−β,∀k, where hk′ and D2,k are the small-scale fading and the distance from the k-th ST to the SR, respectively. The channel gain of the k-th ST–PR link is fk=fk′D3,k−β,∀k with the small-scale fading fk′ and the distance D3,k. The channel gain of the PT-SR link is modeled by fs=fs′Ds−β,∀k, with the small-scale fading fs′ and the distance Ds. In the simulations, we set β=2.7, Dp=30 m, Ds=30 m, D1,1=12 m, D1,2=10 m, D1,3=15 m, D1,4=13 m, D2,1=20 m, D2,2=15 m, D2,3=20 m, D2,4=15 m, D3,1=25 m, D3,2=30 m, D3,3=30 m, and D3,4=30 m.Table 1
Key simulation settings.
ParameterNotationValueThe entire time blockT1 secondThe communication bandwidthB100 kHzThe constant circuit power consumption for BackCom at thek-th STPb,k10μ WThe maximum transmit power at the PTPmax1 WThe performance gap reflecting the real modulation for BackComξ−15 dBThe noise powerσ2−60 dBmThe number of STsK4The minimum required throughput of thek-th STCmin,k10 bitsThe minimum required throughput of the PTCmin100 bitsThe threshold required for decodingxpγth20Figure2 demonstrates the convergence of Algorithm 1, where different settings of Cs,min are considered and Cmin1=Cmin2=Cmin3=Cmin4=Cs,min. We set Cs,min as 10 bits, 20 bits, and 30 bits. It can be observed that the proposed algorithm in Algorithm 1 can always converge to a certain value after only a few iterations, i.e., 3 iterations. This indicates that the proposed algorithm is convergent and computationally efficient. Besides, it can also be seen that a larger Cs,min brings a lower total throughput of all STs. This is because a larger Cs,min means a higher QoS requirement for the ST’s transmission, and more resources will be allocated to the STs with worse channels, leading to a reduction in the total throughput of all STs.Figure 2
The convergence of Algorithm1 under different settings of Cs,min, where Cmin1=Cmin2=Cmin3=Cmin4=Cs,min.Figure3 plots the total throughput of all STs versus the minimum required throughput for each ST Cs,min, and Cs,min is varied from 5 bits to 30 bits. In order to illustrate the advantages of the proposed scheme, we compare the performance achieved by the proposed scheme with the fixed scheme, where the power reflection coefficient of each ST is fixed as 0.5, 0.8, and 0.9, respectively. As shown in this figure, the total throughput of all STs decreases with the increase of Cs,min, since a larger Cs,min brings a higher QoS requirement for the ST’s transmission, and more resources will be allocated to the STs with worse channels, leading to a reduction in the total throughput of all STs. By comparisons, we can see that the total throughput of all STs under the proposed scheme is higher than that under the fixed scheme, as the proposed scheme provides more flexibilities to utilize resources efficiently for maximizing the total throughput of all STs. This also demonstrates the superiority of the proposed scheme.Figure 3
Total throughput of all STs versus the minimum required throughput for each ST.Figure4 shows total throughput of all STs versus the minimum SINR threshold required for decoding xp, γth, where γth ranges from 20 to 100. The power reflection coefficient of each SN under the fixed scheme is set as 0.5, 0.8, and 0.9. It can be observed from this figure that the total throughput of all STs decreases when γth increases. This is because a higher γth brings a higher requirement for decoding xp, leading to a reduction in the total throughput of all STs. By comparisons, we can observe that the proposed scheme outperforms the other schemes in terms of the total throughput of all STs, which illustrates the advantages of the proposed scheme.Figure 4
Total throughput of all STs versus the minimum SINR threshold required for decodingxp.Figure5 shows the total throughput of all STs versus the maximum allowed transmit power at the PT Pmax under different schemes. Here, Pmax varies from 1 W to 3 W. It can be observed that the total throughput of all STs under all the schemes increase with the increase of Pmax. Based on Lemma 1, the optimal transmit power of the PT is determined by Pmax, and a higher transmit power of the PT allows STs to harvest more energy for supporting the energy consumption of BackCom and to backscatter signals with a higher power, resulting in an improvement for the total throughput of all STs. Besides, we also observe that the total throughput of all STs under the proposed scheme is the highest among these schemes, which also demonstrates the superiority of the proposed scheme in terms of total throughput of all STs.Figure 5
Total throughput of all STs versus the maximum allowed transmit power at the PT.


## 5. Conclusions

In this paper, we have studied the throughput maximization for the BackCom based cognitive network while considering a nonlinear EH model. Specifically, we have formulated an optimization problem to maximize the total throughput of all STs by jointly optimizing the EH time, the transmit power of the PT, the BackCom time, and the power reflection coefficient of each ST under the QoS, energy causality, latency, transmit power, and power reflection coefficient constraints. In order to solve the nonconvex problem, we first determined the optimal transmit power of the PT by using the properties of the objective function and then proposed a SCA based iterative algorithm to obtain the proposed scheme. The simulation results verified the quick convergence of the proposed algorithm and showed that the proposed scheme outperforms the other schemes in terms of total throughput of all STs.



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*Source: 7768588-2022-02-04.xml*