# Physical Layer Security of Two-Way Ambient Backscatter Communication Systems

Authors: Hao Wang; Junjie Jiang; Gaojian Huang; Wenbin Wang; Dan Deng; Basem M. Elhalawany; Xingwang Li
Journal: Wireless Communications and Mobile Computing
 (2022)

---

## Abstract

Achieving high network traffic demand in limited spectrum resources is a technical challenge for the beyond 5G and 6G communication systems. To this end, ambient backscatter communication (AmBC) is proposed for Internet-of-Things (IoT), because the backscatter device (BD) can realize communication without occupying extra spectrum resources. Moreover, the spectral efficiency can be further improved by using two-way (TW) communication. However, secure communication is a great challenge for accessing massive IoT devices due to the broadcasting nature of wireless propagation environments. In light of this fact, this article proposed a two-way ambient backscatter communication (TW-AmBC) network with an eavesdropper. Specifically, the physical layer security (PLS) is studied through deriving the analytical/asymptotic expressions of the outage probability (OP) and intercept probability (IP). Moreover, the outage probabilities (OPs) in high signal-to-noise ratio (SNR) regions are studied for the asymptotic behavior and the intercept probabilities (IPs) in high main-to-eavesdropping ratio (MER) regions as well. Through analysis and evaluation of simulation performance, the results show that (i) when considering the target node, the OP of the BD decreases with increasing SNR, that is, enhancing the reliability; (ii) an optimal value of BD’s reflection coefficient that maximize the reliability of backscatter link can be obtained; (iii) in high SNR regions, the OPs approach a constant; thus, the diversity orders are zero; (iv) when increasing the MER, the IP of target node decreases, suggesting the security enhances; (v) a trade-off exists between reliability and security which can be optimized by carefully designing the parameters.


---

## Body

## 1. Introduction

Due to faster network traffic demand and limited spectrum resources, the next-generation wireless communication techniques urgently need to improve the spectrum efficiency (SE) and reduce latency [1]. The ambient backscatter communication (AmBC) that could utilize the environmental wireless signals for both powering low-power devices and backscattering signals was used to address the problem of limited spectrum resources [2]. The main advantages of AmBC for Internet-of-Things (IoT) can be summarized as follows: (i) the SE can be improved since information is conveyed without consuming additional bandwidth [3]; (ii) the expensive and energy-consuming components such as oscillators, filters, and amplifiers are not required for the backscatter device (BD); thus, it enjoys low cost and low power [4]. As a new green paradigm, AmBC has aroused great research interest. However, due to the simplicity of the components, BD cannot separate interfering signals from received signals and is vulnerable to security attacks.In particular, the performance of combining AmBC with other technologies was studied in much of the existing literature. For instance, in [5], the authors first studied the outage probability (OP) of symbiotic system incorporating the nonorthogonal multiple access (NOMA) and backscatter techniques. The authors in [6] acquired the analytical expressions of the OP and studied the outage performance of AmBC system with considering in-phase and quadrature-phase imbalance (IQI). In [7], the backscatter technique was studied with intelligent reflecting surface (IRS) which is known as another low-power technique; the proposed scheme can achieve a balance between the average throughput and coverage probability according to practical condition. In [8], the multiantenna backscatter tag AmBC system was proposed, wherein the multiantenna technique was used to enhance the reliability. In [9], an adaptive reflection coefficient scheme was proposed to minimize the OP of backscatter link, which provided a guideline to design the optimal reflection coefficient for practical AmBC systems.Meanwhile, in AmBC system, the power of BD is limited which is in contrast to the traditional key mechanism requiring high computing power; thus, some researchers focused on the security problems. Without consuming high computing power, the physical layer security (PLS) can be achieved by using the inherent randomness of physical media and the difference between legal channels and eavesdropping channels. Thus, it has attracted much attention in academia and industry and was studied in different scenarios [10–12]. For instance, in [13], the secrecy performance of AmBC systems considering IQI was investigated by deriving the analytical expressions of the OP and intercept probability (IP). In [14], the secrecy outage probability (SOP) of the multitag backscatter systems over the Rayleigh fading channels was given, where the channel correlation between the forward and backscatter links may exist. In [15], the authors studied the PLS for the relay selection schemes of NOMA systems, indicating that better security performance can be achieved when increasing the number of relays.On the other hand, due to the network congestion caused by information explosion, it is imperative to study the techniques increasing the throughput which is defined as the amount of data successfully transferred per unit time of a communication channel. The two-way (TW) communication method enables the capability of improving throughput and SE since the relay in the TW communication system can receive information from both nodes in a single time slot. The TW communication system therefore has been extensively studied specifically on the PLS aspect. For example, in [16], the OP of a TW relay system can be minimized by an improved dynamic scheme, which both the power-splitting ratios and the power allocation ratio can be dynamically adjusted. In [17], the OP of TW full-duplex relay system considering self-interference was evaluated, and asymptotic OP was presented for more insight. Different from the above works, reconfigurable intelligent surface (RIS) technique with great channel capacity advantage was employed in wireless communication systems to improve the performance [18]. Similarly, in [19], the performance of the RIS-assisted TW communication systems was studied wherein the channels can either be reciprocal or nonreciprocal, and the closed-form expressions of OP were derived for single-element RIS. Moreover, in [20], an artificial noise-assisted opportunistic relay selection scheme was proposed to enhance the security of underlay cognitive TW relay network.


### 1.1. Motivation and Contributions

In the existing works, in [21], an optimization algorithm was proposed to maximize throughput of relaying system where an unmanned aerial vehicle (UAV) enabled TW relay assist communication. In [22], the secrecy of TW relay NOMA systems was evaluated in terms of the ergodic secrecy sum rate, which degraded when the distance between the eavesdropper and either user becomes closer. In [23], the outage performance of a TW model wherein BD is embedded in the relay for backscattering was investigated. In [24], a BD cooperative relay communication network was proposed and the system reliability performance was studied. It is noted that all the reported works established models with relays and for the case of without relay was neglected. In this article, we first consider a two-way ambient backscatter communication (TW-AmBC) system without relay and study the reliability and security of the proposed system, where sources communicate with each other and also via BD with an eavesdropper. The main contributions of the article can be listed as follows:
(i)
We propose a novel TW-AmBC model without relay and study the PLS of target node, wherein BD and nodes transmit signals in the presence of an eavesdropper(ii)
We derive the analytical expressions of the outage probabilities (OPs) for the target node and BD and the analytical expressions of the intercept probabilities (IPs) for eavesdropper. It is found that the reflection coefficient and transmitted power have the opposite effect on OPs and IPs. The trade-off between security and reliability can be flexibly adjusted(iii)
We evaluate the performance of OPs and IPs in high signal-to-noise ratio (SNR) and main-to-eavesdropping ratio (MER) regions. Furthermore, we discuss the diversity orders of the target node and BD in high SNR regions. The diversity orders are zero owing to the OPs approach a constant, proving that the joint error-floor exists(iv)
We analyze the comprehensive influence factor on performance of both the reflection coefficient and transmitted power in the TW-AmBC network. When transmitted power is high, the system performance enjoys little fluctuation, and when the reflection coefficient increases, the security of the target node can be increased


### 1.2. Organization and Notations

The remainder of the article is composed as follows. In Section2, the TW-AmBC system without relay is first constructed, followed by the elaboration of deriving expressions of OPs and IPs in Section 3. In Section 4, the correctness of the theoretical analysis is validated via numerical results. Finally, conclusions are given in Section 5.E∙ is deemed as the expectation operation. Pr∙ denotes the probability and Kv∙ denotes the v-th order modified Bessel function of the second kind. CNμ,σ2 denotes the complex Gaussian random variable with mean μ and variance σ2. Besides, Ei∙ is the exponential integral function and Wu,v∙ is the Whittaker function. n denotes the natural number, and n! denotes the factorial operation. The cumulative distribution function (CDF) is expressed as F∙, and the probability dense function (PDF) is expressed as f∙.


## 1.1. Motivation and Contributions

In the existing works, in [21], an optimization algorithm was proposed to maximize throughput of relaying system where an unmanned aerial vehicle (UAV) enabled TW relay assist communication. In [22], the secrecy of TW relay NOMA systems was evaluated in terms of the ergodic secrecy sum rate, which degraded when the distance between the eavesdropper and either user becomes closer. In [23], the outage performance of a TW model wherein BD is embedded in the relay for backscattering was investigated. In [24], a BD cooperative relay communication network was proposed and the system reliability performance was studied. It is noted that all the reported works established models with relays and for the case of without relay was neglected. In this article, we first consider a two-way ambient backscatter communication (TW-AmBC) system without relay and study the reliability and security of the proposed system, where sources communicate with each other and also via BD with an eavesdropper. The main contributions of the article can be listed as follows:
(i)
We propose a novel TW-AmBC model without relay and study the PLS of target node, wherein BD and nodes transmit signals in the presence of an eavesdropper(ii)
We derive the analytical expressions of the outage probabilities (OPs) for the target node and BD and the analytical expressions of the intercept probabilities (IPs) for eavesdropper. It is found that the reflection coefficient and transmitted power have the opposite effect on OPs and IPs. The trade-off between security and reliability can be flexibly adjusted(iii)
We evaluate the performance of OPs and IPs in high signal-to-noise ratio (SNR) and main-to-eavesdropping ratio (MER) regions. Furthermore, we discuss the diversity orders of the target node and BD in high SNR regions. The diversity orders are zero owing to the OPs approach a constant, proving that the joint error-floor exists(iv)
We analyze the comprehensive influence factor on performance of both the reflection coefficient and transmitted power in the TW-AmBC network. When transmitted power is high, the system performance enjoys little fluctuation, and when the reflection coefficient increases, the security of the target node can be increased


## 1.2. Organization and Notations

The remainder of the article is composed as follows. In Section2, the TW-AmBC system without relay is first constructed, followed by the elaboration of deriving expressions of OPs and IPs in Section 3. In Section 4, the correctness of the theoretical analysis is validated via numerical results. Finally, conclusions are given in Section 5.E∙ is deemed as the expectation operation. Pr∙ denotes the probability and Kv∙ denotes the v-th order modified Bessel function of the second kind. CNμ,σ2 denotes the complex Gaussian random variable with mean μ and variance σ2. Besides, Ei∙ is the exponential integral function and Wu,v∙ is the Whittaker function. n denotes the natural number, and n! denotes the factorial operation. The cumulative distribution function (CDF) is expressed as F∙, and the probability dense function (PDF) is expressed as f∙.


## 2. System Model

As illustrated in Figure1, we consider a TW-AmBC system which is composed of two source nodes (A and B), a BD, and an eavesdropper (E), wherein hn, h0, and he, respectively, denote the channel responses from A to BD, B, and E. gn and fe represent the channel responses from BD to B and E, respectively. In different time slots, B and A can directively transmit signals to each other, also via a BD. The eavesdropper can intercept signals from nodes and BD. We assume that (i) all nodes are equipped with a single antenna; (ii) all channel parameters are subjected to the independent Rayleigh fading. We have h0∼CN0,λh0, hn∼CN0,λhn, he∼CN0,λhe, gn∼CN0,λgn, ge∼CN0,λge, and fe∼CN0,λfe.Figure 1
Two-way ambient backscatter communication system model.


### 2.1. Received Signals atB

The received signals atB include signals from A and the backscattered signals from BD. The BD adds its own message ct to signals from A or B and then backscatters, where Ect2=1. Thus, the received signals yB at B can be expressed as
(1)yB=h0PsxAt+βhngnPsxAtct+n1t,where β is the complex reflection coefficient used to normalize ct and n1∼CN0,σ2 is the complex Gaussian noise with zero mean. xAt is the transmitted signal of A. Ps denotes the transmitted power. Furthermore, in order to extract the real-information from the scrambled signals, successive interference cancellation (SIC) technique is adopted at the receiver end [25]. Therefore, the received signal-to-interference-plus-noise ratio (SINR) extracting xAt at B can be written as
(2)γB=γh02γβ2hn2gn2+1,where γ=Ps/σ2 is the transmit SNR. By eliminating the interference from A, the SNR at B can be written as
(3)γBD=γβ2hn2gn2.


### 2.2. Wiretapped Signals

In the 1st time slot, the received signals atE can be expressed as
(4)yet=gePsxBt+βgnfePsxBtct+n2t,where n2t is the complex Gaussian noise at E and follows n2∼CN0,σ2. xBt is the transmitted signal of B. When only direct link signals are decoded at E, the SINR can be given by
(5)γe,B=γge2γβ2gn2fe2+1.It is noted thatE can eliminate direct link signals and extract the signals from BD with aid of SIC technique. In this case, the SNR can be expressed as
(6)γe,BD=γβ2gn2fe2.


## 2.1. Received Signals atB

The received signals atB include signals from A and the backscattered signals from BD. The BD adds its own message ct to signals from A or B and then backscatters, where Ect2=1. Thus, the received signals yB at B can be expressed as
(1)yB=h0PsxAt+βhngnPsxAtct+n1t,where β is the complex reflection coefficient used to normalize ct and n1∼CN0,σ2 is the complex Gaussian noise with zero mean. xAt is the transmitted signal of A. Ps denotes the transmitted power. Furthermore, in order to extract the real-information from the scrambled signals, successive interference cancellation (SIC) technique is adopted at the receiver end [25]. Therefore, the received signal-to-interference-plus-noise ratio (SINR) extracting xAt at B can be written as
(2)γB=γh02γβ2hn2gn2+1,where γ=Ps/σ2 is the transmit SNR. By eliminating the interference from A, the SNR at B can be written as
(3)γBD=γβ2hn2gn2.


## 2.2. Wiretapped Signals

In the 1st time slot, the received signals atE can be expressed as
(4)yet=gePsxBt+βgnfePsxBtct+n2t,where n2t is the complex Gaussian noise at E and follows n2∼CN0,σ2. xBt is the transmitted signal of B. When only direct link signals are decoded at E, the SINR can be given by
(5)γe,B=γge2γβ2gn2fe2+1.It is noted thatE can eliminate direct link signals and extract the signals from BD with aid of SIC technique. In this case, the SNR can be expressed as
(6)γe,BD=γβ2gn2fe2.


## 3. Performance Analysis

In this section, we derive the analytical expressions of OPs and IPs to investigate the security and reliability of the TW-AmBC system. Furthermore, the asymptotic behaviors of OPs and IPs are studied by asymptotic expressions, and the diversity orders are derived in high SNR regions.


### 3.1. Outage Performance Analysis

#### 3.1.1. Outage Probability Expressions forB

Considering the direct link only, the outage event would occur when signalxAt is unsuccessfully decoded at B. Thus, the OP at B can be given by
(7)PoutB=1−PrγB>γthB,where γthB is the SNR threshold at B.Theorem 1.
For the direct link, we can get the OP analytical expression, which can be expressed as(8)PoutB=1+Q1eQ1−γthB/γλh0Ei−Q1,where Q1=λh0/γthBβ2λhnλgn, Eix represents the exponential integral function and Eix=∫−∞xet/tdt, where t denotes the variable. Eix can be expanded to a series form which is given by
(9)Eix=ν+lnx+∑n=1∞xnn·n!,where ν is the Euler constant, ν≈0577. n denotes the natural number, and n! denotes the factorial operation for n.Proof.
See AppendixA.Corollary 1.
Under high signal-to-noise ratio (SNR) case, the OP asymptotic expression of direct link can be given as(10)Pout,∞B=1+Q1eQ11−γthBγλh0Ei−Q1.Proof.
According to asymptotic principle, Equation (10) can be derived from Equation (8) by using approximate equation e−x≈1−x when γ⟶∞.


#### 3.1.2. Outage Probability Expressions for BD

For the backscatter link, the outage event occurs when direct link signals are successfully decoded but failing with backscattered signals. Therefore, the OP of backscattered signals can be expressed as(11)PoutBD=1−PrγB>γthB,γB>γthBD,where γthBD is the SNR threshold for backscattered signals.Theorem 2.
For the backscatter link, we can obtain the OP analytical expression, which can be expressed as(12)PoutBD=1+Q1eQ1−γthB/γλh0Ei−Q1+πΔ2N∑i=1Ne−Δ1K02Δ2δi+11−δi2,where Δ1=γthBγthBDδi+1+2γthB/2γλh0, Δ2=γthBD/γβ2λhnλgn, δi=cos2i−1/2Nπ, N is a trade-off parameter between accuracy and complexity, and K0∙ is the 0th order modified Bessel function of the second kind.Proof.
See AppendixB.Corollary 2.
Under high SNRs case, the OP asymptotic expression for backscatter link in the proposed system can be given as(13)Pout,∞BD=1+Q1eQ11−γthBγλh0Ei−Q1−πΔ2N∑i=1N1−Δ1lnΔ2δi+121−δi2.Proof.
According to the asymptotic principle, through using two approximate equations, i.e.,K0x≈−lnx/2 and ex=1+x, Equation (13) can be obtained.
For further insights on the backscatter link and direct link, we also study the diversity orderdψ, ψ∈B,BD. The diversity order is given [26]:
(14)dψ=−limγ⟶∞logPout,∞ψlogγ.Corollary 3.
The diversity orders of the two links can be calculated as(15)dB=dBD=0.Remark 1.
From Corollaries1–3 and Theorems 1 and 2, we can know that (1) when increasing Ps, the OPs decrease, enhancing the reliability of the system; (2) when increasing β, the OP of direct link increases; (3) when γ is large, the latter of Equation (12) is almost zero; thus, when γ⟶∞, Pout,∞BD approximates Pout,∞B; (4) when γ⟶∞, asymptotic OPs exist the joint error-floor, causing that the diversity orders are zero.


### 3.2. Security Performance Analysis

When the SINR or SNR atE surpasses the secrecy SNR threshold, direct link signals or backscattered signals would be intercepted. In further detail below, the expressions of IPs are derived to investigate the system security.


#### 3.2.1. Intercept Probability Expressions forB

The IP expression of direct link signals can be written as(16)PintB=Prγe,B>γthe,B,where γthe,B is the secrecy SNR threshold for B.Theorem 3.
For the direct link, we can get the IP analytical expression, which can be expressed as(17)PintB=−e−γthe,B/γλgeQ2eQ2Ei−Q2,where Q2=λge/γthe,Bβ2λgnλfe.Proof.
Substituting Equation (5) into Equation (16), the IP of direct link signals can be expressed as
(18)PintB=e−γthe,B/γλge∫0∞e−γthe,Bβ2y/λgefgn2fe2ydy,where fgn2fe2y=2/λgnλfeK02y/λgnλfe. The integral can be resolved by Appendix A, and Equation (17) can be derived.Here, we use the MER metric to study the asymptotic behavior of IPs, which depends on the ratio of the main link to eavesdropping link [27]. The main link and eavesdropping link are B to BD and BD to E, respectively. The MER can be expressed as λme=λgn/λfe. In the next corollary, the asymptotic analysis for IPs in high MER regions is derived.Corollary 4.
Under high main-to-eavesdropping ratio (MER) case, the IP asymptotic expression can be given by(19)Pint,∞B=−e−γthe,B/γλgeQ2∗1+Q2∗ln−Q2∗,where Q2∗=λge/γthe,Bβ2λmeλfe2.Proof.
According to the asymptotic principle, Equation (19) can be obtained by substituting ex≈1+x and Eix≈lnx+C into Equation (17), where C is a constant.


#### 3.2.2. Intercept Probability Expressions for BD

The IP expression of backscattered signals can be written as(20)PintBD=Prγe,BD>γthe,BD,where γthe,BD is the secrecy SNR threshold of BD.Theorem 4.
For the backscatter link, we can get the IP analytical expression, which can be expressed as(21)PintBD=2Q3K12Q3,where Q3=γthe,BD/γβ2λgnλfe and K1∙ is the 1st order modified Bessel function of the second kind.Proof.
Substituting Equation (6) into Equation (20), PintBD can be expressed as
(22)PintBD=1λfe∫0∞e−γthe,BD/γβ2λgny−y/λfedy,where the integral can be calculated by Equation (3.324.1) [28].Corollary 5.
Under high MER case, the IP asymptotic expression for backscatter link can be written as(23)Pint,∞BD=1+2Q3∗lnQ3∗,where Q3∗=γthe,BD/γβ2λmeλfe2.Proof.
By using approximate equationK1x≈1/x+x/2lnx/2, we can get Equation (23).Remark 2.
From Theorems3 and 4 and Corollaries 4 and 5, we can know that (1) with increasing reflection coefficient β, the security of B gets enhanced owing to PintB decreases; (2) the security of B and BD decrease when Ps increases; (3) when λme increases, the security of B is enhanced; (4) when λme⟶∞, Pint,∞BD is close to 1, suggesting λme takes small value to ensure BD security.


## 3.1. Outage Performance Analysis

### 3.1.1. Outage Probability Expressions forB

Considering the direct link only, the outage event would occur when signalxAt is unsuccessfully decoded at B. Thus, the OP at B can be given by
(7)PoutB=1−PrγB>γthB,where γthB is the SNR threshold at B.Theorem 1.
For the direct link, we can get the OP analytical expression, which can be expressed as(8)PoutB=1+Q1eQ1−γthB/γλh0Ei−Q1,where Q1=λh0/γthBβ2λhnλgn, Eix represents the exponential integral function and Eix=∫−∞xet/tdt, where t denotes the variable. Eix can be expanded to a series form which is given by
(9)Eix=ν+lnx+∑n=1∞xnn·n!,where ν is the Euler constant, ν≈0577. n denotes the natural number, and n! denotes the factorial operation for n.Proof.
See AppendixA.Corollary 1.
Under high signal-to-noise ratio (SNR) case, the OP asymptotic expression of direct link can be given as(10)Pout,∞B=1+Q1eQ11−γthBγλh0Ei−Q1.Proof.
According to asymptotic principle, Equation (10) can be derived from Equation (8) by using approximate equation e−x≈1−x when γ⟶∞.


### 3.1.2. Outage Probability Expressions for BD

For the backscatter link, the outage event occurs when direct link signals are successfully decoded but failing with backscattered signals. Therefore, the OP of backscattered signals can be expressed as(11)PoutBD=1−PrγB>γthB,γB>γthBD,where γthBD is the SNR threshold for backscattered signals.Theorem 2.
For the backscatter link, we can obtain the OP analytical expression, which can be expressed as(12)PoutBD=1+Q1eQ1−γthB/γλh0Ei−Q1+πΔ2N∑i=1Ne−Δ1K02Δ2δi+11−δi2,where Δ1=γthBγthBDδi+1+2γthB/2γλh0, Δ2=γthBD/γβ2λhnλgn, δi=cos2i−1/2Nπ, N is a trade-off parameter between accuracy and complexity, and K0∙ is the 0th order modified Bessel function of the second kind.Proof.
See AppendixB.Corollary 2.
Under high SNRs case, the OP asymptotic expression for backscatter link in the proposed system can be given as(13)Pout,∞BD=1+Q1eQ11−γthBγλh0Ei−Q1−πΔ2N∑i=1N1−Δ1lnΔ2δi+121−δi2.Proof.
According to the asymptotic principle, through using two approximate equations, i.e.,K0x≈−lnx/2 and ex=1+x, Equation (13) can be obtained.
For further insights on the backscatter link and direct link, we also study the diversity orderdψ, ψ∈B,BD. The diversity order is given [26]:
(14)dψ=−limγ⟶∞logPout,∞ψlogγ.Corollary 3.
The diversity orders of the two links can be calculated as(15)dB=dBD=0.Remark 1.
From Corollaries1–3 and Theorems 1 and 2, we can know that (1) when increasing Ps, the OPs decrease, enhancing the reliability of the system; (2) when increasing β, the OP of direct link increases; (3) when γ is large, the latter of Equation (12) is almost zero; thus, when γ⟶∞, Pout,∞BD approximates Pout,∞B; (4) when γ⟶∞, asymptotic OPs exist the joint error-floor, causing that the diversity orders are zero.


## 3.1.1. Outage Probability Expressions forB

Considering the direct link only, the outage event would occur when signalxAt is unsuccessfully decoded at B. Thus, the OP at B can be given by
(7)PoutB=1−PrγB>γthB,where γthB is the SNR threshold at B.Theorem 1.
For the direct link, we can get the OP analytical expression, which can be expressed as(8)PoutB=1+Q1eQ1−γthB/γλh0Ei−Q1,where Q1=λh0/γthBβ2λhnλgn, Eix represents the exponential integral function and Eix=∫−∞xet/tdt, where t denotes the variable. Eix can be expanded to a series form which is given by
(9)Eix=ν+lnx+∑n=1∞xnn·n!,where ν is the Euler constant, ν≈0577. n denotes the natural number, and n! denotes the factorial operation for n.Proof.
See AppendixA.Corollary 1.
Under high signal-to-noise ratio (SNR) case, the OP asymptotic expression of direct link can be given as(10)Pout,∞B=1+Q1eQ11−γthBγλh0Ei−Q1.Proof.
According to asymptotic principle, Equation (10) can be derived from Equation (8) by using approximate equation e−x≈1−x when γ⟶∞.


## 3.1.2. Outage Probability Expressions for BD

For the backscatter link, the outage event occurs when direct link signals are successfully decoded but failing with backscattered signals. Therefore, the OP of backscattered signals can be expressed as(11)PoutBD=1−PrγB>γthB,γB>γthBD,where γthBD is the SNR threshold for backscattered signals.Theorem 2.
For the backscatter link, we can obtain the OP analytical expression, which can be expressed as(12)PoutBD=1+Q1eQ1−γthB/γλh0Ei−Q1+πΔ2N∑i=1Ne−Δ1K02Δ2δi+11−δi2,where Δ1=γthBγthBDδi+1+2γthB/2γλh0, Δ2=γthBD/γβ2λhnλgn, δi=cos2i−1/2Nπ, N is a trade-off parameter between accuracy and complexity, and K0∙ is the 0th order modified Bessel function of the second kind.Proof.
See AppendixB.Corollary 2.
Under high SNRs case, the OP asymptotic expression for backscatter link in the proposed system can be given as(13)Pout,∞BD=1+Q1eQ11−γthBγλh0Ei−Q1−πΔ2N∑i=1N1−Δ1lnΔ2δi+121−δi2.Proof.
According to the asymptotic principle, through using two approximate equations, i.e.,K0x≈−lnx/2 and ex=1+x, Equation (13) can be obtained.
For further insights on the backscatter link and direct link, we also study the diversity orderdψ, ψ∈B,BD. The diversity order is given [26]:
(14)dψ=−limγ⟶∞logPout,∞ψlogγ.Corollary 3.
The diversity orders of the two links can be calculated as(15)dB=dBD=0.Remark 1.
From Corollaries1–3 and Theorems 1 and 2, we can know that (1) when increasing Ps, the OPs decrease, enhancing the reliability of the system; (2) when increasing β, the OP of direct link increases; (3) when γ is large, the latter of Equation (12) is almost zero; thus, when γ⟶∞, Pout,∞BD approximates Pout,∞B; (4) when γ⟶∞, asymptotic OPs exist the joint error-floor, causing that the diversity orders are zero.


## 3.2. Security Performance Analysis

When the SINR or SNR atE surpasses the secrecy SNR threshold, direct link signals or backscattered signals would be intercepted. In further detail below, the expressions of IPs are derived to investigate the system security.


### 3.2.1. Intercept Probability Expressions forB

The IP expression of direct link signals can be written as(16)PintB=Prγe,B>γthe,B,where γthe,B is the secrecy SNR threshold for B.Theorem 3.
For the direct link, we can get the IP analytical expression, which can be expressed as(17)PintB=−e−γthe,B/γλgeQ2eQ2Ei−Q2,where Q2=λge/γthe,Bβ2λgnλfe.Proof.
Substituting Equation (5) into Equation (16), the IP of direct link signals can be expressed as
(18)PintB=e−γthe,B/γλge∫0∞e−γthe,Bβ2y/λgefgn2fe2ydy,where fgn2fe2y=2/λgnλfeK02y/λgnλfe. The integral can be resolved by Appendix A, and Equation (17) can be derived.Here, we use the MER metric to study the asymptotic behavior of IPs, which depends on the ratio of the main link to eavesdropping link [27]. The main link and eavesdropping link are B to BD and BD to E, respectively. The MER can be expressed as λme=λgn/λfe. In the next corollary, the asymptotic analysis for IPs in high MER regions is derived.Corollary 4.
Under high main-to-eavesdropping ratio (MER) case, the IP asymptotic expression can be given by(19)Pint,∞B=−e−γthe,B/γλgeQ2∗1+Q2∗ln−Q2∗,where Q2∗=λge/γthe,Bβ2λmeλfe2.Proof.
According to the asymptotic principle, Equation (19) can be obtained by substituting ex≈1+x and Eix≈lnx+C into Equation (17), where C is a constant.


### 3.2.2. Intercept Probability Expressions for BD

The IP expression of backscattered signals can be written as(20)PintBD=Prγe,BD>γthe,BD,where γthe,BD is the secrecy SNR threshold of BD.Theorem 4.
For the backscatter link, we can get the IP analytical expression, which can be expressed as(21)PintBD=2Q3K12Q3,where Q3=γthe,BD/γβ2λgnλfe and K1∙ is the 1st order modified Bessel function of the second kind.Proof.
Substituting Equation (6) into Equation (20), PintBD can be expressed as
(22)PintBD=1λfe∫0∞e−γthe,BD/γβ2λgny−y/λfedy,where the integral can be calculated by Equation (3.324.1) [28].Corollary 5.
Under high MER case, the IP asymptotic expression for backscatter link can be written as(23)Pint,∞BD=1+2Q3∗lnQ3∗,where Q3∗=γthe,BD/γβ2λmeλfe2.Proof.
By using approximate equationK1x≈1/x+x/2lnx/2, we can get Equation (23).Remark 2.
From Theorems3 and 4 and Corollaries 4 and 5, we can know that (1) with increasing reflection coefficient β, the security of B gets enhanced owing to PintB decreases; (2) the security of B and BD decrease when Ps increases; (3) when λme increases, the security of B is enhanced; (4) when λme⟶∞, Pint,∞BD is close to 1, suggesting λme takes small value to ensure BD security.


## 3.2.1. Intercept Probability Expressions forB

The IP expression of direct link signals can be written as(16)PintB=Prγe,B>γthe,B,where γthe,B is the secrecy SNR threshold for B.Theorem 3.
For the direct link, we can get the IP analytical expression, which can be expressed as(17)PintB=−e−γthe,B/γλgeQ2eQ2Ei−Q2,where Q2=λge/γthe,Bβ2λgnλfe.Proof.
Substituting Equation (5) into Equation (16), the IP of direct link signals can be expressed as
(18)PintB=e−γthe,B/γλge∫0∞e−γthe,Bβ2y/λgefgn2fe2ydy,where fgn2fe2y=2/λgnλfeK02y/λgnλfe. The integral can be resolved by Appendix A, and Equation (17) can be derived.Here, we use the MER metric to study the asymptotic behavior of IPs, which depends on the ratio of the main link to eavesdropping link [27]. The main link and eavesdropping link are B to BD and BD to E, respectively. The MER can be expressed as λme=λgn/λfe. In the next corollary, the asymptotic analysis for IPs in high MER regions is derived.Corollary 4.
Under high main-to-eavesdropping ratio (MER) case, the IP asymptotic expression can be given by(19)Pint,∞B=−e−γthe,B/γλgeQ2∗1+Q2∗ln−Q2∗,where Q2∗=λge/γthe,Bβ2λmeλfe2.Proof.
According to the asymptotic principle, Equation (19) can be obtained by substituting ex≈1+x and Eix≈lnx+C into Equation (17), where C is a constant.


## 3.2.2. Intercept Probability Expressions for BD

The IP expression of backscattered signals can be written as(20)PintBD=Prγe,BD>γthe,BD,where γthe,BD is the secrecy SNR threshold of BD.Theorem 4.
For the backscatter link, we can get the IP analytical expression, which can be expressed as(21)PintBD=2Q3K12Q3,where Q3=γthe,BD/γβ2λgnλfe and K1∙ is the 1st order modified Bessel function of the second kind.Proof.
Substituting Equation (6) into Equation (20), PintBD can be expressed as
(22)PintBD=1λfe∫0∞e−γthe,BD/γβ2λgny−y/λfedy,where the integral can be calculated by Equation (3.324.1) [28].Corollary 5.
Under high MER case, the IP asymptotic expression for backscatter link can be written as(23)Pint,∞BD=1+2Q3∗lnQ3∗,where Q3∗=γthe,BD/γβ2λmeλfe2.Proof.
By using approximate equationK1x≈1/x+x/2lnx/2, we can get Equation (23).Remark 2.
From Theorems3 and 4 and Corollaries 4 and 5, we can know that (1) with increasing reflection coefficient β, the security of B gets enhanced owing to PintB decreases; (2) the security of B and BD decrease when Ps increases; (3) when λme increases, the security of B is enhanced; (4) when λme⟶∞, Pint,∞BD is close to 1, suggesting λme takes small value to ensure BD security.


## 4. Numerical Results

In this section, the correctness of previous analysis is validated via numerical simulation results. In these simulation examples, we consider105 Monte Carlo trials. It is assumed that λh0=6, λhn=4, λgn=5, λfe=6, λge=4, β=0.2, and σ2=1.The simulation results of OPs versusPs for B and BD are plotted in Figure 2. It can be seen that in high Ps regions, the OPs are getting close to constant with increasing Ps, resulting in 0 diversity orders. From Remark 1, we can know that the value of the joint error-floor depends on the SNR threshold at B. Furthermore, it can be seen that the OPs increase when the SNR thresholds, i.e., γthB and γthBD, increase.Figure 2
OPs versusPs for different SNR threshold values.The IPs versusPs for B and BD are plotted in Figure 3. The IPs decrease as the secrecy SNR thresholds increase. In low Ps regions, it is shown that the IP decreases significantly, indicating that the TW-AmBC system has higher security performance. In high Ps regions, the IPs are close to 1; thus, the security is impaired. Meanwhile, from Figures 2 and 3, it can be observed that when Ps increases, the OPs decrease but the IPs increase. Thus, it can be inferred that a trade-off between reliability and security exists in this system.Figure 3
IPs versusPs for different secrecy SNR threshold values.Figure4 illustrates the IPs versus OPs for different β values. It can be observed that the system outage performance gets impaired when β changes from 0.1 to 0.6. This is because the backscattered signals make the interference stronger when B decodes signals. In addition, it can be observed that the secrecy performance of direct link is enhanced when β increases. It can be explained that the backscattered signals increase with increasing β; thus, it can project more interference when E decodes the signals from B. Meanwhile, when increasing β, the scopes of OP and IP are obviously reduced, which indicates that the trade-off between security and reliability degrades. In summary, high reliability and high security cannot be achieved simultaneously, which indicates that the reliability is compromised to get better security and vice versa.Figure 4
IPs versus OPs for differentβ values.The OPs and IPs versusβ for different Ps values are plotted in Figure 5. In low β regions, the IP of BD decreases significantly, which greatly improves the system security performance. Thus, when β is small, the eavesdropper can easily intercept the messages from B, but it is difficult to decode the messages from BD. The IP trend curves of BD and B are opposite, indicating that high security of one link can be achieved with compromised security performance of the other link. Thus, a trade-off of security exists between the two links. Meanwhile, the curve of BD’s OP firstly decreases and then increases, suggesting that we can get an optimal value β to minimize OP.Figure 5
OPs and IPs versusβ for different Ps values.Figure6 reveals the variations of IPs versus MER in conditions of different Ps values, with β=0.1. We consider 106 Monte Carlo trials to reduce the impact of randomness on experimental accuracy. It can be observed that when MER increases, the IP of B decreases and that of BD increases, and in high MER regions, the IP of BD is close to 1, suggesting that the security of backscatter link can only be achieved when MER is low. When Ps decreases, the IPs decrease, indicating that the security gets enhanced. Beyond that, there is the strict approximation relationship between the theoretical value and the asymptotic value.Figure 6
IPs versus MER for differentPs values.


## 5. Conclusions

In this article, a novel TW-AmBC network structure was proposed, which integrated the benefits of both TW communication and AmBC. Through analyzing the PLS performance of the proposed TW-AmBC, it is found that the parameters can be designed to achieve an optimal trade-off between reliability and security, such as reflection coefficientβ and the transmitted power Ps. Specifically, the secrecy performance of B can be enhanced when reflection coefficient β increases. When the transmitted power Ps is high, the asymptotic OPs approach a joint error-floor, indicating that the reliability of two links is similar. These findings would be useful in instructing to apply BD in limited spectrum resource application scenarios.



---

*Source: 5445676-2022-03-29.xml*