# Simultaneous Jamming-and-Transmitting Scheme for Spectrum-Sharing Relaying Networks with Nonlinear Energy Scavenging

Authors: Triet Pham-Minh; Khuong Ho-Van; Khanh Nghi-Vinh
Journal: Wireless Communications and Mobile Computing
 (2021)

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## Abstract

This paper recommends a simultaneous jamming-and-transmitting scheme for spectrum-sharing relaying networks with nonlinear energy scavenging. More specifically, spectrum-sharing relaying networks include a secondary source which expects to communicate with a secondary destination but impossible due to communication blockage between them, a secondary relay which helps the source overcome this blockage, a primary receiver, and a wiretapper which steals secret messages from both source and relay. To motivate assistance, the relay scavenges radio frequency energy from the source with practical nonlinear energy scavenger (NL-ES) instead of linear energy scavenger (L-ES) as in previous publications. Additionally, both source and relay perform simultaneous jamming-and-transmitting to secure their communication. The intercept and outage probabilities of the recommended scheme are evaluated through exact closed-form formulas, which are corroborated by Monte-Carlo simulations. Illustrative results show that the proposed scheme offers the reliability-security trade-off yet suffers error floor at large maximum transmit/interference power. Moreover, its performance can be optimally set with appropriate system parameters. Notably, the proposed scheme can guarantee absolute security with proper parameter setting. Furthermore, the NL-ES is practical but performs significantly worse than the L-ES.


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## Body

## 1. Introduction

### 1.1. Background

Energy in radio frequency (RF) signals can be scavenged to power wireless users for improving energy efficiency, solving energy shortage, and offering green communication in advanced communication networks [1–4]. Two typical paradigms, expressly power-splitting (PS) and time-switching (TS), efficiently implement RF energy scavenging (ES) where the former performs ES and data restoring at the same time while the latter carries out these operations independently in distinct times [5]. As a result, the former demands lower circuitry implementation than the latter. Energy scavenger can be classified as linear (L) [6] or nonlinear (NL) [7]. Practically, circuit components of the energy scavenger include diodes and inductors. These NL components explicitly mean the NL-ES paradigm is more practical than the L-ES one. Therefore, this paper focuses on a class of NL energy scavengers, namely, piecewise linearity in [7], to assess performance of RF ES systems.Spectrum sharing (the literature names the spectrum-sharing operation mode differently as the underlay one; moreover, cognitive radios can operate in other two modes (interweave and overlay) which are beyond our scope) is the typical operation mode of cognitive radios aiming at improving significantly spectrum utilization efficiency [8]. This mode restricts transmit power of cognitive radios by the tolerable interference power at the primary users and their maximum transmit power [9, 10]. These power constraints limit their transmit power, shortening their coverage range and causing discontinuity in direct transmission between cognitive radios. Also, unfavorable propagation conditions (e.g., severe fading and heavy channel loss) are other causes to discontinuity in direct transmission. Consequently, a secondary relay, which can amplify-and-forward (AF) or decode-and-forward (DF) secondary source signals to a secondary destination, should be utilized to ensure continuity in direct transmission. Nevertheless, the relay, which participates in relaying source signals as a volunteer, may be staggering to spend its personal energy for this operation. To raise the motivation for the relay, the energy scavenged from the source by recent advanced energy scavenging technologies should power its operation [11–13]. As such, the relay not only extends the coverage range of the secondary source but also keeps direct transmission continuous without sacrificing its personal energy. Nonetheless, the energy harvested by the relay is typically limited, and thus, the question is if the relay can assist direct transmission reliably as well as secure its transmission to the secondary destination against the overhearing of wiretappers. Another concern is that wiretappers also overhear communication through the source-relay hop and hence how to secure this hop. The current paper is aimed at solving these concerns. Before presenting the related works, it should be recalled that the jamming method, which produces intentionally jamming signals to interrupt solely the wiretappers, is popular and efficient in securing information transmission [14].


### 1.2. Related Works

Direct communication with the NL-ES in both cognitive and noncognitive radio networks has been extensively considered in [15–28]. In addition, a variety of publications investigated relaying networks with the NL-ES but for noncognitive users (e.g., [14, 29–36]) or non-spectrum-sharing users (e.g., overlay users [7, 37–40]). Nonetheless, few works studied performance analysis for spectrum-sharing relaying networks with nonlinear energy scavenging (SSRNwNLES). Specifically, Ref. [41] considered the basic system model for SSRNwNLES where a secondary DF relay aids direct secondary source-destination transmission. Secondary communication interferes signal reception of a primary receiver. Nevertheless, Ref. [41] assumed an eavesdropper wiretaps legitimate information over solely the second hop (from the relay to the destination). Moreover, Ref. [41] only simulated the secrecy outage probability under the maximum transmit power constraint (MTPC) and the interference power restriction (IPR) without proposing any technique to protect legitimate information over both (source-relay and relay-destination) hops. In [42], the authors reconsidered the system model in [41] for maximizing the throughput but without the eavesdropper. The authors in [43] considered SSRNwNLES with several secondary relays and primary users. Although [43] presented the bit error rate analysis, the spectrum-sharing mode imposes implicit constraints on the operations of the cognitive radios without explicitly investigating the IPR and the MTPC, simplifying significantly the analysis. Moreover, Ref. [43] did not study the security aspect of the considered system model. In [44], the authors studied SSRNwNLES with several secondary relays and destinations. However, only relays are constrained by both maximum transmit and interference powers while the source is imposed by the interference power restriction. Furthermore, Refs. [42, 44] were not interested in performance analysis and security issue.


### 1.3. Contributions

The above literature survey indicates that concurrent considerations of performance analysis and security issue in SSRNwNLES have been still an open topic so far. This paper is the first attempt to research that topic with the following contributions:(i)
Propose a simultaneous jamming-and-transmitting scheme to secure message transmission over both source-relay and relay-destination hops in the system model in [41] by asking both the relay and the source to broadcast simultaneously legitimate and jamming signals. Apparently, the proposed scheme solved the security issue left by previous works [41–44](ii)
Recommend the relay to scavenge energy in source signals with the nonlinear energy scavenger to increase its motivation in relaying source messages(iii)
Propose precise formulas of intercept probability (IP) and outage probability (OP) for evaluating rapidly both security and reliability of the recommended scheme without invoking exhaustive Monte-Carlo simulations. The derivation of the OP and the IP accounts for both the MTPC and the IPR. These analyses were obviously ignored in [41–44](iv)
Optimize crucial system parameters using the recommended IP/OP expressions(v)
Prove the absolute security achievable with proper system parameter setting(vi)
Illustrate multifarious results of the reliability and security performances to make helpful conclusions; for instance, the unchanged IP/OP as either the MTPC or the IPR is forsaken, considerable security/reliability improvement as setting properly system parameters, the reliability-security trade-off, and the inferiority of the NL-ES to the L-ES


### 1.4. Organization

The next section describes the recommended simultaneous jamming-and-transmitting scheme for SSRNwNLES. Then, Section3 details the derivation of the accurate closed-form IP/OP expressions. Next, Section 4 illustrates analytical/simulated results on the recommended scheme. Eventually, Section 5 ends the paper with insightful comments.


## 1.1. Background

Energy in radio frequency (RF) signals can be scavenged to power wireless users for improving energy efficiency, solving energy shortage, and offering green communication in advanced communication networks [1–4]. Two typical paradigms, expressly power-splitting (PS) and time-switching (TS), efficiently implement RF energy scavenging (ES) where the former performs ES and data restoring at the same time while the latter carries out these operations independently in distinct times [5]. As a result, the former demands lower circuitry implementation than the latter. Energy scavenger can be classified as linear (L) [6] or nonlinear (NL) [7]. Practically, circuit components of the energy scavenger include diodes and inductors. These NL components explicitly mean the NL-ES paradigm is more practical than the L-ES one. Therefore, this paper focuses on a class of NL energy scavengers, namely, piecewise linearity in [7], to assess performance of RF ES systems.Spectrum sharing (the literature names the spectrum-sharing operation mode differently as the underlay one; moreover, cognitive radios can operate in other two modes (interweave and overlay) which are beyond our scope) is the typical operation mode of cognitive radios aiming at improving significantly spectrum utilization efficiency [8]. This mode restricts transmit power of cognitive radios by the tolerable interference power at the primary users and their maximum transmit power [9, 10]. These power constraints limit their transmit power, shortening their coverage range and causing discontinuity in direct transmission between cognitive radios. Also, unfavorable propagation conditions (e.g., severe fading and heavy channel loss) are other causes to discontinuity in direct transmission. Consequently, a secondary relay, which can amplify-and-forward (AF) or decode-and-forward (DF) secondary source signals to a secondary destination, should be utilized to ensure continuity in direct transmission. Nevertheless, the relay, which participates in relaying source signals as a volunteer, may be staggering to spend its personal energy for this operation. To raise the motivation for the relay, the energy scavenged from the source by recent advanced energy scavenging technologies should power its operation [11–13]. As such, the relay not only extends the coverage range of the secondary source but also keeps direct transmission continuous without sacrificing its personal energy. Nonetheless, the energy harvested by the relay is typically limited, and thus, the question is if the relay can assist direct transmission reliably as well as secure its transmission to the secondary destination against the overhearing of wiretappers. Another concern is that wiretappers also overhear communication through the source-relay hop and hence how to secure this hop. The current paper is aimed at solving these concerns. Before presenting the related works, it should be recalled that the jamming method, which produces intentionally jamming signals to interrupt solely the wiretappers, is popular and efficient in securing information transmission [14].


## 1.2. Related Works

Direct communication with the NL-ES in both cognitive and noncognitive radio networks has been extensively considered in [15–28]. In addition, a variety of publications investigated relaying networks with the NL-ES but for noncognitive users (e.g., [14, 29–36]) or non-spectrum-sharing users (e.g., overlay users [7, 37–40]). Nonetheless, few works studied performance analysis for spectrum-sharing relaying networks with nonlinear energy scavenging (SSRNwNLES). Specifically, Ref. [41] considered the basic system model for SSRNwNLES where a secondary DF relay aids direct secondary source-destination transmission. Secondary communication interferes signal reception of a primary receiver. Nevertheless, Ref. [41] assumed an eavesdropper wiretaps legitimate information over solely the second hop (from the relay to the destination). Moreover, Ref. [41] only simulated the secrecy outage probability under the maximum transmit power constraint (MTPC) and the interference power restriction (IPR) without proposing any technique to protect legitimate information over both (source-relay and relay-destination) hops. In [42], the authors reconsidered the system model in [41] for maximizing the throughput but without the eavesdropper. The authors in [43] considered SSRNwNLES with several secondary relays and primary users. Although [43] presented the bit error rate analysis, the spectrum-sharing mode imposes implicit constraints on the operations of the cognitive radios without explicitly investigating the IPR and the MTPC, simplifying significantly the analysis. Moreover, Ref. [43] did not study the security aspect of the considered system model. In [44], the authors studied SSRNwNLES with several secondary relays and destinations. However, only relays are constrained by both maximum transmit and interference powers while the source is imposed by the interference power restriction. Furthermore, Refs. [42, 44] were not interested in performance analysis and security issue.


## 1.3. Contributions

The above literature survey indicates that concurrent considerations of performance analysis and security issue in SSRNwNLES have been still an open topic so far. This paper is the first attempt to research that topic with the following contributions:(i)
Propose a simultaneous jamming-and-transmitting scheme to secure message transmission over both source-relay and relay-destination hops in the system model in [41] by asking both the relay and the source to broadcast simultaneously legitimate and jamming signals. Apparently, the proposed scheme solved the security issue left by previous works [41–44](ii)
Recommend the relay to scavenge energy in source signals with the nonlinear energy scavenger to increase its motivation in relaying source messages(iii)
Propose precise formulas of intercept probability (IP) and outage probability (OP) for evaluating rapidly both security and reliability of the recommended scheme without invoking exhaustive Monte-Carlo simulations. The derivation of the OP and the IP accounts for both the MTPC and the IPR. These analyses were obviously ignored in [41–44](iv)
Optimize crucial system parameters using the recommended IP/OP expressions(v)
Prove the absolute security achievable with proper system parameter setting(vi)
Illustrate multifarious results of the reliability and security performances to make helpful conclusions; for instance, the unchanged IP/OP as either the MTPC or the IPR is forsaken, considerable security/reliability improvement as setting properly system parameters, the reliability-security trade-off, and the inferiority of the NL-ES to the L-ES


## 1.4. Organization

The next section describes the recommended simultaneous jamming-and-transmitting scheme for SSRNwNLES. Then, Section3 details the derivation of the accurate closed-form IP/OP expressions. Next, Section 4 illustrates analytical/simulated results on the recommended scheme. Eventually, Section 5 ends the paper with insightful comments.


## 2. Simultaneous Jamming-and-Transmitting Scheme

Figure1 sketches the system model under consideration where the secondary destination D expects to get signals from the secondary source S. The eavesdropper E tries to extract the source message. Because of bad propagation conditions such as dire fading and severe channel loss, direct S⟶D transmission may be discontinuous. Therefore, the DF relay R between S and D should be utilized to relay S’s signal to D for keeping the S⟶D transmission continuous. To reward for R’s assistance, R only relays the source message with the energy scavenged in the source signals. Accordingly, secret messages reach D in three phases with a total duration of T as shown in Figure 2. Also, due to the spectrum-sharing operation mode, cognitive radios (S and R) interfere (in the spectrum-sharing operation mode, primary users interfere cognitive radios; nonetheless, such an interference is forsaken under scenarios that it follows the Gaussian distribution or the distance between the primary user and the cognitive radio is adequately large; forsaking such an interference is admitted extensively in the literature of the spectrum-sharing networks (please refer to [45–47])) the primary receiver P.Figure 1
System model.Figure 2
Phase times.Figure1 notates channel gain hbc, b∈S,R, c∈E,R,D,P. This paper considers all independent flat block Rayleigh fading channels, and hence, hbc is represented by a zero-mean θbc-variance complex Gaussian random variable, namely, hbc~CN0,θbc, and is unchanged during T but changes independently in the next T. Additionally, state information of all channels is assumed to be perfectly (channel state information (CSI) may be imperfectly estimated [48–50]; we reserve the case of imperfect CSI in our future works; instead, we focus on proposing a solution to secure SSRNwNLES in both transmission phases and analyze its intercept/outage performance in this paper) acquired, for example, through pilot estimation [51]. Furthermore, the path loss can be considered in θbc as θbc=dbc−δ where dbc and δ notate the b⟶c distance and the path-loss exponent, correspondingly. The following notates the channel power gain as gbc=hbc2 whose pdf (probability density function) and cdf (cumulative distribution function) are addressed, respectively, as
(1)fgbcx=1θbce−x/θbcΦx,Fgbcx=1−e−x/θbcΦx,whereΦx is the unit step function.We notatePs as the secondary source’s transmit power. Then, the spectrum-sharing operation mode regulates Ps such that S interferes P with the amount of the interference power Psgsp below the tolerable interference threshold It of P, i.e., Psgsp≤It. Also, Ps is restricted by the maximum transmit power P˘s according to hardware specification, i.e., Ps≤P˘s. Consequently, combining these power constraints regulates Ps as
(2)Ps=minItgsp,P˘s.In (2), the equality is set for maximizing the cognitive radios’ transmission range [52].In Phase 1 with a duration ofαT, the nonlinear energy harvester at R generates the following power [7].
(3)P˘r=LPsgsr,Psgsr≤ρ,Lρ,Psgsr>ρ,whereL=2ψα/1−α with ψ∈0,1 being the energy converting efficiency and α∈0,1 being a time fraction; ρ is the power saturation threshold.Different from the conventional direct communication scheme whereS sends only its data to D in Phase 2, our scheme allows S to use a fraction of its power 1−βPs to send its message xs and the remaining power βPs to broadcast jamming signal xa in order to protect its data against the eavesdropper. Here, β∈0,1 stands for the power distribution coefficient for the legitimate message and the jamming signal at S. As such, the signal transmitted by S can be expressed as us=1−βPsxs+βPsxa and the received signals at R and E are uniquely expressed to be
(4)ym=hsm1−βPsxs+βPsxa+εm,where the additive noise at the receiverm∈r,e is denoted as εm~CN0,εm.The a priori characteristic of the jamming signal is assumably known atR; for example, xa can be a pseudorandom signal whose seed is shared with R [14]. As such, the jamming signal merely interrupts the eavesdropper and can be definitely abolished from the received signal of R. After annihilating the jamming signal, R further processes the following signal for decoding xs:
(5)y~r=hsr1−βPsxs+εr.R obtains the signal-to-noise ratio (SNR) from (5) for recovering xs to be
(6)Γsr=gsrPs1−βεr.Generatingxa is merely shared between S and R but E is blind with it. Accordingly, the signal-to-noise plus interference ratio (SNIR) which E can obtain for recovering xs in Phase 2 is deduced from (4) to be
(7)Γse=gsePs1−βgsePsβ+εe.In Phase 3 with a duration of1−αT/2, R broadcasts the restored signal x^s as well as the jamming signal x^a to secure x^s against the eavesdropper. Consequently, the signals received at E and D have the same form as
(8)yw=hrw1−λPrx^s+λPrx^a+εw,where the receive antenna ofw∈d,e causes the additive noise εw~CN0,εw; λ∈0,1 stands for the power distribution factor for the restored signal and the jamming signal at R; Pr is the transmit power of R. For the spectrum-sharing operation mode, Pr is constrained as
(9)Pr=minP˘r,Itgrp.Thanks to the property of the jamming signalx^a and processing it similarly to Phase 2, the SNR at D and the SNIR at E in Phase 3 are correspondingly given by
(10)Γrd=grdPr1−λεd,(11)Γre=grePr1−λgrePrλ+εe.It is seen from the denominators of (7) and (11) that E suffers the amount of jamming power in both last phases. This amount mitigates the probability of recovering precisely xs at E, eventually improving the message security.The DF operation ofR results in the aggregated SNIR at D for recovering xs as
(12)Γd=minΓsr,Γrd.In order forE to improve its intercept capability, it needs to combine its received signals in Phase 2 and Phase 3. We assume E prefers affordable complexity. Then, it can employ the signal selection combining method (other combining methods which E can employ are maximum ratio combining and equal gain combining [53]; nonetheless, although these methods offer better performance, they require higher implementation complexity), which produces the entire SNIR to be
(13)Γe=maxΓse,Γre.The channel capacity whichw∈d,e acquires to recover xs is represented to be
(14)Rw=1−α2log21+Γw.


## 3. Security and Reliability Analyses

Communication reliability and message security can be measured through the OP atD and the IP at E, respectively. These probability expressions of the proposed simultaneous jamming-and-transmitting scheme for SSRNwNLES are derived to quickly evaluate both reliability and security without exhaustive simulations.


### 3.1. Intercept Probability

The intercept event occurs when the channel capacity ofE surpasses the target spectral efficiency R~e. Consequently, the IP is expressed to be
(15)Ψ=PrRe≥R~e=PrΓe≥Γ~e,whereΓ~e=22R~e/1−α−1.Inserting (13) into (15) yields
(16)Ψ=PrmaxΓse,Γre≥Γ~e=1−PrmaxΓse,Γre<Γ~e=1−EPsPrΓse<Γ~ePs⏟Ψ1PrΓre<Γ~ePs⏟Ψ2,whereE· is the expectation operator.Plugging (7) into Ψ1 in (16) results in
(17)Ψ1=PrgsePs1−βgsePsβ+εe<Γ~ePs=PrgsePs1−β−Γ~eβ<Γ~eεePs=Ψ¯1,Γ~e<1−ββ,1,Γ~e≥1−ββ,where
(18)Ψ¯1=Prgse<Γ~eεePs1−β−Γ~eβPs=1−e−N/Ps,withN=Γ~eεe/1−β−Γ~eβθse.SinceΓ~e=22R~e/1−α−1, (17) shows that the eavesdropping of E in Phase 2 can be completely eliminated by splitting appropriately times for energy harvesting and signal transmission (i.e., selecting α), allocating properly S’s power to the jamming signal (i.e., selecting β), and setting reasonably the target spectral efficiency R~e such that the inequality Γ~e≥1−β/β holds.Inserting (11) into Ψ2 in (16) yields
(19)Ψ2=PrgrePr1−λgrePrλ+εe<Γ~ePs=PrgrePr1−λ−λΓ~e<Γ~eεePs=Ψ¯2,Γ~e<1−λλ,1,Γ~e≥1−λλ,where
(20)Ψ¯2=Prgre<Γ~eεePr1−λ−λΓ~ePs=EPrPrgre<Γ~eεePr1−λ−λΓ~ePr,Ps=EPr1−e−Γ~eεe/Pr1−λ−λΓ~eθrePs.BecauseΓ~e=22R~e/1−α−1, (19) indicates that the eavesdropping of E in Phase 3 can be completely eliminated by splitting appropriately times for energy harvesting and signal transmission (i.e., selecting α), allocating properly R’s power to the jamming signal (i.e., selecting λ), and setting reasonably the target spectral efficiency R~e such that the inequality Γ~e≥1−λ/λ holds.GivenPr in (9), (20) is further simplified as
(21)Ψ¯2=Egsr,grp1−e−Γ~eεe/minP˘r,It/grp1−λ−λΓ~eθrePs=Egsr1−∫0It/P˘re−Γ~eεe/P˘r1−λ−λΓ~eθree−x/θrpθrpdx−∫It/P˘r∞e−Γ~eεe/It1−λ−λΓ~eθre/xe−x/θrpθrpdxPs=Egsr1−e−Q/P˘r+He−Q/P˘rH⏟APs,whereQ=Γ~eεe/1−λ−λΓ~eθre and H=θrpQ/It+θrpQ.Based on (3), two cases of P˘r are considered when deriving (21) as follows.Case 1.
(P˘r=Lρ).
This case holds whengsr>ρ/Ps. By averaging A in (21) over this region, one obtains Ψ¯2 for this case as
(22)Ψ¯21=∫ρ/Ps∞1−e−Q/Aρ+He−Q/AHρe−x/θsrθsrdx=Me−ρ/θsrPs,
whereM=1−e−Q/Aρ+He−Q/AHρ.Case 2.
(P˘r=LPsgsr).
This case holds whengsr≤ρ/Ps. By averaging A in (21) over gsr≤ρ/Ps, one obtains Ψ¯2 for this case as
(23)Ψ¯22=∫0ρ/Ps1−e−Q/LPsx+He−Q/LHPsxe−x/θsrθsrdx=1−e−ρ/Psθsr+1θsr∫0ρ/Pse−x/θsrHe−Q/LHPsx−e−Q/LPsxdx.Using the series expansion fore−x/θsr, one rewrites (23) as
(24)Ψ¯22=1−e−ρ/Psθsr+1θsr∫0ρ/Ps∑k=0∞1k!−xθsrkHe−Q/LHPsx−e−Q/LPsxdx.Performing the variable changet=1/x, one reduces (24) to
(25)Ψ¯22=1−e−ρ/Psθsr+1θsr∑k=0∞−θsr−kk!∫Psρ∞t−k−2He−Qt/LHPs−e−Qt/LPsdt=1−e−ρ/Psθsr+∑k=0∞JPsk+1,where J=Q/θsrLk+1/k!k+1!∑m=1k+1−Lρ/Qmm−1!e−Q/Lρ+−1m+1Hm−ke−Q/LHρ−Ei−Q/Lρ+H−kEi−Q/LHρ with Ei· being the exponential integral.The total probability law simplifies (21) as
(26)Ψ¯2=Ψ¯21+Ψ¯22=1+M−1e−ρ/Psθsr+∑k=0∞JPsk+1.Inserting (18) into (17) and (26) into (19), one obtains Ψ1 and Ψ2 as
(27)Ψ1=1−e−N/Ps,Γ~e<1−ββ,1,Γ~e≥1−ββ,(28)Ψ2=1+M−1e−ρ/θsrPs+∑k=0∞JPsk+1,Γ~e<1−λλ,1,Γ~e≥1−λλ.Plugging (27) and (28) into (16), one obtains
(29)Ψ=Ψ˘1,Γ~e≥max1−ββ,1−λλ,Ψ˘2,1−ββ≤Γ~e<1−λλ,Ψ˘3,1−λλ≤Γ~e<1−ββ,Ψ˘4,Γ~e<min1−ββ,1−λλ.SinceΓ~e=22R~e/1−α−1, it is indicated from (29) that the security of the proposed simultaneous jamming-and-transmitting scheme experiences four different levels dependent on adjusting the system parameters (R~e, α, β, λ).The following sequentially computesΨ˘1, Ψ˘2, Ψ˘3, and Ψ˘4 to numerically evaluate (29). First, we start with Ψ˘1. It is straightforward to infer that
(30)Ψ˘1=0.BecauseΓ~e=22R~e/1−α−1, (30) indicates that the eavesdropping of E can be completely eliminated in the proposed simultaneous jamming-and-transmitting scheme where both legitimate messages (of S and R) are protected in Phase 2 and Phase 3 by splitting appropriately times for energy harvesting and signal transmission (i.e., selecting α), distributing reasonably S’s power to the jamming signal (i.e., selecting β), allocating properly R’s power to the jamming signal (i.e., selecting λ), and setting the target spectral efficiency R~e such that the inequality, Γ~e≥max1−β/β,1−λ/λ, holds.Second,Ψ˘2 is written explicitly as
(31)Ψ˘2=1−EPs1+M−1e−ρ/θsrPs+∑k=0∞JPsk+1.GivenPs in (2), one solves (31) in closed form as
(32)Ψ˘2=−∫0It/P˘sM−1e−ρ/θsrP˘s+∑k=0∞JP˘sk+1e−x/θspθspdx−∫It/P˘s∞M−1e−ρx/θsrIt+∑k=0∞xItk+1Je−x/θspθspdx=−M−1e−ρ/θsrP˘s+∑k=0∞JP˘sk+11−e−It/θspP˘s−M−1θspρθsrIt+1θsp−1e−ρ/θsrIt+1/θspIt/P˘s−e−It/P˘sθsp∑k=0∞∑n=0k+1k+1!θspk+1Jn!Itk+1−nθspP˘sn.Third,Ψ˘3 has an explicit form as
(33)Ψ˘3=1−EPs1−e−N/Ps.WithPs in (2), the expectation in (33) is evaluated as
(34)Ψ˘3=∫0It/P˘se−N/P˘se−x/θspθspdx+∫It/P˘s∞e−N/It/xe−x/θspθspdx=e−N/P˘s1−e−It/θspP˘s+θspNIt+1−1e−N/It+1/θspIt/P˘s.Finally,Ψ˘4 is given by
(35)Ψ˘4=1−EPs1−e−N/Ps1+M−1e−ρ/θsrPs+∑k=0∞JPsk+1Ps=Ψ˘2+Ψ˘3+EPsM−1e−ρ/θsr+N1/Ps+∑k=0∞Je−N/PsPsk+1Ps.WithPs in (2), the expectation in (35) is evaluated as
(36)Ψ˘4=Ψ˘2+Ψ˘3+∫0It/P˘sM−1e−ρ/θsr+N1/P˘s+∑k=0∞Je−N/P˘sP˘sk+1e−x/θspθspdx+∫It/P˘s∞M−1e−ρ/θsr+NxIt+∑k=0∞JxItk+1e−Nx/Ite−x/θspθspdx=Ψ˘2+Ψ˘3+M−1e−ρ/θsr+N1/P˘s+e−N/P˘s∑k=0∞JP˘sk+11−e−It/θspP˘s+M−1θspItN+ρθsr+1−1e−N+ρ/θsr+It/θsp1/P˘s+e−N+It/θsp/P˘s1+θspN/It∑k=0∞∑i=0k+1N+Itθspi−k−1k+1!Ji!P˘si.


### 3.2. Outage Probability

The outage event happens as the channel capacity ofD subceeds the target spectral efficiency R~d. Accordingly, the OP is addressed to be
(37)Y=PrRd≤R~d=PrΓd≤Γ~d,whereΓ~d=22R~d/1−α−1.EmployingΓd in (12) reduces (37) to
(38)Y=PrminΓsr,Γrd≤Γ~d=1−PrminΓsr,Γrd>Γ~d=1−EPsEgsrY~gsr>Γ~dεrPs1−βPs⏟Y¯,whereY~=Prgrd>Γ~dεd/Pr1−λ.GivenPr in (9), Y~ is rewritten as
(39)Y~=EgrpPrgrd>Γ~dεd1−λminP˘r,It/grp=Egrpe−Γ~dεd/1−λθrdminP˘r,It/grp=∫0It/P˘re−Γ~dεd/1−λθrdP˘re−x/θrdθrpdx+∫It/P˘r∞e−Γ~dεdx/1−λθrdIte−x/θrpθrpdx=e−W/P˘r−Ge−W/P˘rG,whereW=Γ~dεd/1−λθrd and G=θrpW/θrpW+It.Based on (3), two cases of P˘r are considered when deriving Y¯ in (38) as follows.Case 1.
(P˘r=LPsgsr).
This case holds whengsr≤ρ/Ps. Incorporating this condition with gsr>εrΓ~d/Ps1−β in (38) results in existence region of gsr to be D/Ps<gsr≤ρ/Ps where D=εrΓ~d/1−β. By averaging Y~ in (39) over this region, one obtains Y¯ for this case as Y¯1=∫D/Psρ/Pse−W/LPsx−Ge−W/LGPsxe−x/θsr/θsrdx. Applying the series expansion for e−x/θsr and then performing the variable change t=1/x, one solves the integral in Y¯1 as
(40)Y¯1=1θsr∫D/Psρ/Ps∑k=0∞1k!−xθsrke−W/LPsx−Ge−W/LGPsxdx=∑k=0∞−θsr−k−1k!∫Ps/DPs/ρe−Wt/LPs−Ge−Wt/LGPs1tk+2dt=∑k=0∞−θsr−k−1k!∫Ps/D∞e−Wt/LPs−Ge−Wt/LGPs1tk+2dt−∫Ps/ρ∞e−Wt/LPs−Ge−Wt/LGPs1tk+2dt=∑k=0∞VPsk+1,
whereV=1/k!k+1!W/θsrLk+1Ei−W/Lρ−Ei−W/LD+G−kEi−W/LDG−Ei−W/LGρ+∑u=1k+1u−1!−L/WuDue−W/LD−ρue−W/Lρ−Gu−kDue−W/LDG−ρue−W/LGρ.Case 2.
(P˘r=Lρ).
This case holds whengsr>ρ/Ps. Incorporating this condition with gsr>εrΓ~d/Ps1−β in (38) results in existence region of gsr>B/Ps where B=maxεrΓ~d/1−β,ρ. By averaging Y~ in (39) over this region, one obtains Y¯ for this case as
(41)Y¯2=∫B/Ps∞e−W/Lρ−Ge−W/LGρe−x/θsrθsrdx=Ue−B/θsrPs,
whereU=e−W/Lρ−Ge−W/LGρ.The total probability law simplifiesY¯ in (38) as
(42)Y¯=Y¯1+Y¯2=Ue−B/θsrPs+∑k=0∞VPsk+1.Plugging (42) into Y¯ in (38) with a note that Ps is given in (2) results in
(43)Y=1−EPsUe−B/θsrPs+∑k=0∞VPsk+1=1−∫0It/P˘sUe−B/θsrP˘s+∑k=0∞VP˘sk+1e−x/θspθspdx−∫It/P˘s∞Ue−Bx/θsrIt+∑k=0∞VxItk+1e−x/θspθspdx=1−Ue−B/θsrP˘s+∑k=0∞VP˘sk+11−e−It/θspP˘s−UθspBθsrIt+1−1e−B/θsr+It/θsp1/P˘s−∑k=0∞∑i=0k+1θspItk+1−ie−It/θspP˘sk+1!Vi!P˘si.


### 3.3. Asymptotic Analysis

The asymptotic intercept and outage probabilities are analyzed under consideration of the following extreme cases: (1) Case 1: high maximum transmit power (P˘s⟶∞) and (2) Case 2: high maximum interference power (It⟶∞). Case 2 corresponds to noncognitive networks (i.e., no interference power constraint) or no interference caused by secondary users on primary users (e.g., secondary users are distant from primary users or secondary-primary channels are blocked).Case 1.
(P˘s⟶∞).
The intercept probability in this scenario becomes(44)ΨP˘s⟶∞=0,Γ~e≥max1−ββ,1−λλ,Ψ˘2P~s⟶∞,1−ββ≤Γ~e<1−λλ,Ψ˘3P~s⟶∞,1−λλ≤Γ~e<1−ββ,Ψ˘4P~s⟶∞,Γ~e<min1−ββ,1−λλ,
whereΨ˘2P˘s⟶∞, Ψ˘3P˘s⟶∞, and Ψ˘4P˘s⟶∞ are obtained from Ψ˘2 in (32), Ψ˘3 in (34), and Ψ˘4 in (36), respectively, as Ψ˘2P˘s⟶∞=limP˘s⟶∞Ψ˘2=1+θspρ/θsrIt−11−M−∑k=0∞θsp/Itk+1k+1!J, Ψ˘3P˘s⟶∞=limP˘s⟶∞Ψ˘3=θspN/It+1−1, and Ψ˘4P˘s⟶∞=limP˘s⟶∞Ψ˘4=Ψ˘2P˘s⟶∞+M−1θsp/Itρ/θsr+N+1−1+Ψ˘3P˘s⟶∞+It/θsp∑k=0∞k+1!N+It/θsp−k−2J.Likewise, the outage probability in this scenario is obtained from (43) to be
(45)YP˘s⟶∞=limP˘s⟶∞Y=1−θspBθsrIt+1−1U−∑k=0∞k+1!θspItk+1V.It is observed from (44) and (45) that when the maximum transmit power is large, both IP and OP converge, creating the error floors (or zero diversity orders). This result will be validated in Figure 3.Case 2.
(It⟶∞).
The intercept probability in this case becomes(46)ΨIt⟶∞=0,Γ~e≥max1−ββ,1−λλ,Ψ˘2It⟶∞,1−ββ≤Γ~e<1−λλ,Ψ˘3It⟶∞,1−λλ≤Γ~e<1−ββ,Ψ˘4It⟶∞,Γ~e<min1−ββ,1−λλ,
whereΨ˘2It⟶∞, Ψ˘3It⟶∞, and Ψ˘4It⟶∞ are obtained from Ψ˘2 in (32), Ψ˘3 in (34), and Ψ˘4 in (36), respectively, as Ψ˘2It⟶∞=limIt⟶∞Ψ˘2=1−M1e−ρ/θsrP˘s−∑k=0∞J1/P˘sk+1, Ψ˘3It⟶∞=limIt⟶∞Ψ˘3=e−N/P˘s, and Ψ˘4It⟶∞=limIt⟶∞Ψ˘4=Ψ˘2It⟶∞+Ψ˘3It⟶∞+M1−1e−N+ρ/θsr1/P˘s+e−N/P˘s∑k=0∞J1/P˘sk+1 where J1=limIt⟶∞J=1/k!k+1!Q/Lθsrk+1e−Q/Lρ∑u=1k+1u−1!−Lρ/Qu−Ei−Q/Lρ and M1=limIt⟶∞M=1−e−Q/Lρ.Figure 3
IP and OP versusΘ=P˘s/ε. “Ana.” and “Sim.” represent the simulation result and the analytical result in Section 3, respectively, while “Asym.” represents the asymptotic result.Likewise, the outage probability in this scenario is obtained from (38) to be
(47)YIt⟶∞=limIt⟶∞Y=1−U1e−B/θsrP˘s−∑k=0∞V1P˘sk+1,whereU1=limIt⟶∞U=e−W/Lρ and V1=limIt⟶∞V=1/k!k+1!W/θsrLk+1Ei−W/Lρ−Ei−W/LD+∑u=1k+1u−1!−L/Wue−W/LDDu−e−W/Lρρu.It is seen from (46) and (47) that when the maximum interference power is large, both IP and OP also converge, creating the error floors (or zero diversity orders). This result will be validated in Figure 4.Figure 4
IP and OP versusIt/ε.


## 3.1. Intercept Probability

The intercept event occurs when the channel capacity ofE surpasses the target spectral efficiency R~e. Consequently, the IP is expressed to be
(15)Ψ=PrRe≥R~e=PrΓe≥Γ~e,whereΓ~e=22R~e/1−α−1.Inserting (13) into (15) yields
(16)Ψ=PrmaxΓse,Γre≥Γ~e=1−PrmaxΓse,Γre<Γ~e=1−EPsPrΓse<Γ~ePs⏟Ψ1PrΓre<Γ~ePs⏟Ψ2,whereE· is the expectation operator.Plugging (7) into Ψ1 in (16) results in
(17)Ψ1=PrgsePs1−βgsePsβ+εe<Γ~ePs=PrgsePs1−β−Γ~eβ<Γ~eεePs=Ψ¯1,Γ~e<1−ββ,1,Γ~e≥1−ββ,where
(18)Ψ¯1=Prgse<Γ~eεePs1−β−Γ~eβPs=1−e−N/Ps,withN=Γ~eεe/1−β−Γ~eβθse.SinceΓ~e=22R~e/1−α−1, (17) shows that the eavesdropping of E in Phase 2 can be completely eliminated by splitting appropriately times for energy harvesting and signal transmission (i.e., selecting α), allocating properly S’s power to the jamming signal (i.e., selecting β), and setting reasonably the target spectral efficiency R~e such that the inequality Γ~e≥1−β/β holds.Inserting (11) into Ψ2 in (16) yields
(19)Ψ2=PrgrePr1−λgrePrλ+εe<Γ~ePs=PrgrePr1−λ−λΓ~e<Γ~eεePs=Ψ¯2,Γ~e<1−λλ,1,Γ~e≥1−λλ,where
(20)Ψ¯2=Prgre<Γ~eεePr1−λ−λΓ~ePs=EPrPrgre<Γ~eεePr1−λ−λΓ~ePr,Ps=EPr1−e−Γ~eεe/Pr1−λ−λΓ~eθrePs.BecauseΓ~e=22R~e/1−α−1, (19) indicates that the eavesdropping of E in Phase 3 can be completely eliminated by splitting appropriately times for energy harvesting and signal transmission (i.e., selecting α), allocating properly R’s power to the jamming signal (i.e., selecting λ), and setting reasonably the target spectral efficiency R~e such that the inequality Γ~e≥1−λ/λ holds.GivenPr in (9), (20) is further simplified as
(21)Ψ¯2=Egsr,grp1−e−Γ~eεe/minP˘r,It/grp1−λ−λΓ~eθrePs=Egsr1−∫0It/P˘re−Γ~eεe/P˘r1−λ−λΓ~eθree−x/θrpθrpdx−∫It/P˘r∞e−Γ~eεe/It1−λ−λΓ~eθre/xe−x/θrpθrpdxPs=Egsr1−e−Q/P˘r+He−Q/P˘rH⏟APs,whereQ=Γ~eεe/1−λ−λΓ~eθre and H=θrpQ/It+θrpQ.Based on (3), two cases of P˘r are considered when deriving (21) as follows.Case 1.
(P˘r=Lρ).
This case holds whengsr>ρ/Ps. By averaging A in (21) over this region, one obtains Ψ¯2 for this case as
(22)Ψ¯21=∫ρ/Ps∞1−e−Q/Aρ+He−Q/AHρe−x/θsrθsrdx=Me−ρ/θsrPs,
whereM=1−e−Q/Aρ+He−Q/AHρ.Case 2.
(P˘r=LPsgsr).
This case holds whengsr≤ρ/Ps. By averaging A in (21) over gsr≤ρ/Ps, one obtains Ψ¯2 for this case as
(23)Ψ¯22=∫0ρ/Ps1−e−Q/LPsx+He−Q/LHPsxe−x/θsrθsrdx=1−e−ρ/Psθsr+1θsr∫0ρ/Pse−x/θsrHe−Q/LHPsx−e−Q/LPsxdx.Using the series expansion fore−x/θsr, one rewrites (23) as
(24)Ψ¯22=1−e−ρ/Psθsr+1θsr∫0ρ/Ps∑k=0∞1k!−xθsrkHe−Q/LHPsx−e−Q/LPsxdx.Performing the variable changet=1/x, one reduces (24) to
(25)Ψ¯22=1−e−ρ/Psθsr+1θsr∑k=0∞−θsr−kk!∫Psρ∞t−k−2He−Qt/LHPs−e−Qt/LPsdt=1−e−ρ/Psθsr+∑k=0∞JPsk+1,where J=Q/θsrLk+1/k!k+1!∑m=1k+1−Lρ/Qmm−1!e−Q/Lρ+−1m+1Hm−ke−Q/LHρ−Ei−Q/Lρ+H−kEi−Q/LHρ with Ei· being the exponential integral.The total probability law simplifies (21) as
(26)Ψ¯2=Ψ¯21+Ψ¯22=1+M−1e−ρ/Psθsr+∑k=0∞JPsk+1.Inserting (18) into (17) and (26) into (19), one obtains Ψ1 and Ψ2 as
(27)Ψ1=1−e−N/Ps,Γ~e<1−ββ,1,Γ~e≥1−ββ,(28)Ψ2=1+M−1e−ρ/θsrPs+∑k=0∞JPsk+1,Γ~e<1−λλ,1,Γ~e≥1−λλ.Plugging (27) and (28) into (16), one obtains
(29)Ψ=Ψ˘1,Γ~e≥max1−ββ,1−λλ,Ψ˘2,1−ββ≤Γ~e<1−λλ,Ψ˘3,1−λλ≤Γ~e<1−ββ,Ψ˘4,Γ~e<min1−ββ,1−λλ.SinceΓ~e=22R~e/1−α−1, it is indicated from (29) that the security of the proposed simultaneous jamming-and-transmitting scheme experiences four different levels dependent on adjusting the system parameters (R~e, α, β, λ).The following sequentially computesΨ˘1, Ψ˘2, Ψ˘3, and Ψ˘4 to numerically evaluate (29). First, we start with Ψ˘1. It is straightforward to infer that
(30)Ψ˘1=0.BecauseΓ~e=22R~e/1−α−1, (30) indicates that the eavesdropping of E can be completely eliminated in the proposed simultaneous jamming-and-transmitting scheme where both legitimate messages (of S and R) are protected in Phase 2 and Phase 3 by splitting appropriately times for energy harvesting and signal transmission (i.e., selecting α), distributing reasonably S’s power to the jamming signal (i.e., selecting β), allocating properly R’s power to the jamming signal (i.e., selecting λ), and setting the target spectral efficiency R~e such that the inequality, Γ~e≥max1−β/β,1−λ/λ, holds.Second,Ψ˘2 is written explicitly as
(31)Ψ˘2=1−EPs1+M−1e−ρ/θsrPs+∑k=0∞JPsk+1.GivenPs in (2), one solves (31) in closed form as
(32)Ψ˘2=−∫0It/P˘sM−1e−ρ/θsrP˘s+∑k=0∞JP˘sk+1e−x/θspθspdx−∫It/P˘s∞M−1e−ρx/θsrIt+∑k=0∞xItk+1Je−x/θspθspdx=−M−1e−ρ/θsrP˘s+∑k=0∞JP˘sk+11−e−It/θspP˘s−M−1θspρθsrIt+1θsp−1e−ρ/θsrIt+1/θspIt/P˘s−e−It/P˘sθsp∑k=0∞∑n=0k+1k+1!θspk+1Jn!Itk+1−nθspP˘sn.Third,Ψ˘3 has an explicit form as
(33)Ψ˘3=1−EPs1−e−N/Ps.WithPs in (2), the expectation in (33) is evaluated as
(34)Ψ˘3=∫0It/P˘se−N/P˘se−x/θspθspdx+∫It/P˘s∞e−N/It/xe−x/θspθspdx=e−N/P˘s1−e−It/θspP˘s+θspNIt+1−1e−N/It+1/θspIt/P˘s.Finally,Ψ˘4 is given by
(35)Ψ˘4=1−EPs1−e−N/Ps1+M−1e−ρ/θsrPs+∑k=0∞JPsk+1Ps=Ψ˘2+Ψ˘3+EPsM−1e−ρ/θsr+N1/Ps+∑k=0∞Je−N/PsPsk+1Ps.WithPs in (2), the expectation in (35) is evaluated as
(36)Ψ˘4=Ψ˘2+Ψ˘3+∫0It/P˘sM−1e−ρ/θsr+N1/P˘s+∑k=0∞Je−N/P˘sP˘sk+1e−x/θspθspdx+∫It/P˘s∞M−1e−ρ/θsr+NxIt+∑k=0∞JxItk+1e−Nx/Ite−x/θspθspdx=Ψ˘2+Ψ˘3+M−1e−ρ/θsr+N1/P˘s+e−N/P˘s∑k=0∞JP˘sk+11−e−It/θspP˘s+M−1θspItN+ρθsr+1−1e−N+ρ/θsr+It/θsp1/P˘s+e−N+It/θsp/P˘s1+θspN/It∑k=0∞∑i=0k+1N+Itθspi−k−1k+1!Ji!P˘si.


## 3.2. Outage Probability

The outage event happens as the channel capacity ofD subceeds the target spectral efficiency R~d. Accordingly, the OP is addressed to be
(37)Y=PrRd≤R~d=PrΓd≤Γ~d,whereΓ~d=22R~d/1−α−1.EmployingΓd in (12) reduces (37) to
(38)Y=PrminΓsr,Γrd≤Γ~d=1−PrminΓsr,Γrd>Γ~d=1−EPsEgsrY~gsr>Γ~dεrPs1−βPs⏟Y¯,whereY~=Prgrd>Γ~dεd/Pr1−λ.GivenPr in (9), Y~ is rewritten as
(39)Y~=EgrpPrgrd>Γ~dεd1−λminP˘r,It/grp=Egrpe−Γ~dεd/1−λθrdminP˘r,It/grp=∫0It/P˘re−Γ~dεd/1−λθrdP˘re−x/θrdθrpdx+∫It/P˘r∞e−Γ~dεdx/1−λθrdIte−x/θrpθrpdx=e−W/P˘r−Ge−W/P˘rG,whereW=Γ~dεd/1−λθrd and G=θrpW/θrpW+It.Based on (3), two cases of P˘r are considered when deriving Y¯ in (38) as follows.Case 1.
(P˘r=LPsgsr).
This case holds whengsr≤ρ/Ps. Incorporating this condition with gsr>εrΓ~d/Ps1−β in (38) results in existence region of gsr to be D/Ps<gsr≤ρ/Ps where D=εrΓ~d/1−β. By averaging Y~ in (39) over this region, one obtains Y¯ for this case as Y¯1=∫D/Psρ/Pse−W/LPsx−Ge−W/LGPsxe−x/θsr/θsrdx. Applying the series expansion for e−x/θsr and then performing the variable change t=1/x, one solves the integral in Y¯1 as
(40)Y¯1=1θsr∫D/Psρ/Ps∑k=0∞1k!−xθsrke−W/LPsx−Ge−W/LGPsxdx=∑k=0∞−θsr−k−1k!∫Ps/DPs/ρe−Wt/LPs−Ge−Wt/LGPs1tk+2dt=∑k=0∞−θsr−k−1k!∫Ps/D∞e−Wt/LPs−Ge−Wt/LGPs1tk+2dt−∫Ps/ρ∞e−Wt/LPs−Ge−Wt/LGPs1tk+2dt=∑k=0∞VPsk+1,
whereV=1/k!k+1!W/θsrLk+1Ei−W/Lρ−Ei−W/LD+G−kEi−W/LDG−Ei−W/LGρ+∑u=1k+1u−1!−L/WuDue−W/LD−ρue−W/Lρ−Gu−kDue−W/LDG−ρue−W/LGρ.Case 2.
(P˘r=Lρ).
This case holds whengsr>ρ/Ps. Incorporating this condition with gsr>εrΓ~d/Ps1−β in (38) results in existence region of gsr>B/Ps where B=maxεrΓ~d/1−β,ρ. By averaging Y~ in (39) over this region, one obtains Y¯ for this case as
(41)Y¯2=∫B/Ps∞e−W/Lρ−Ge−W/LGρe−x/θsrθsrdx=Ue−B/θsrPs,
whereU=e−W/Lρ−Ge−W/LGρ.The total probability law simplifiesY¯ in (38) as
(42)Y¯=Y¯1+Y¯2=Ue−B/θsrPs+∑k=0∞VPsk+1.Plugging (42) into Y¯ in (38) with a note that Ps is given in (2) results in
(43)Y=1−EPsUe−B/θsrPs+∑k=0∞VPsk+1=1−∫0It/P˘sUe−B/θsrP˘s+∑k=0∞VP˘sk+1e−x/θspθspdx−∫It/P˘s∞Ue−Bx/θsrIt+∑k=0∞VxItk+1e−x/θspθspdx=1−Ue−B/θsrP˘s+∑k=0∞VP˘sk+11−e−It/θspP˘s−UθspBθsrIt+1−1e−B/θsr+It/θsp1/P˘s−∑k=0∞∑i=0k+1θspItk+1−ie−It/θspP˘sk+1!Vi!P˘si.


## 3.3. Asymptotic Analysis

The asymptotic intercept and outage probabilities are analyzed under consideration of the following extreme cases: (1) Case 1: high maximum transmit power (P˘s⟶∞) and (2) Case 2: high maximum interference power (It⟶∞). Case 2 corresponds to noncognitive networks (i.e., no interference power constraint) or no interference caused by secondary users on primary users (e.g., secondary users are distant from primary users or secondary-primary channels are blocked).Case 1.
(P˘s⟶∞).
The intercept probability in this scenario becomes(44)ΨP˘s⟶∞=0,Γ~e≥max1−ββ,1−λλ,Ψ˘2P~s⟶∞,1−ββ≤Γ~e<1−λλ,Ψ˘3P~s⟶∞,1−λλ≤Γ~e<1−ββ,Ψ˘4P~s⟶∞,Γ~e<min1−ββ,1−λλ,
whereΨ˘2P˘s⟶∞, Ψ˘3P˘s⟶∞, and Ψ˘4P˘s⟶∞ are obtained from Ψ˘2 in (32), Ψ˘3 in (34), and Ψ˘4 in (36), respectively, as Ψ˘2P˘s⟶∞=limP˘s⟶∞Ψ˘2=1+θspρ/θsrIt−11−M−∑k=0∞θsp/Itk+1k+1!J, Ψ˘3P˘s⟶∞=limP˘s⟶∞Ψ˘3=θspN/It+1−1, and Ψ˘4P˘s⟶∞=limP˘s⟶∞Ψ˘4=Ψ˘2P˘s⟶∞+M−1θsp/Itρ/θsr+N+1−1+Ψ˘3P˘s⟶∞+It/θsp∑k=0∞k+1!N+It/θsp−k−2J.Likewise, the outage probability in this scenario is obtained from (43) to be
(45)YP˘s⟶∞=limP˘s⟶∞Y=1−θspBθsrIt+1−1U−∑k=0∞k+1!θspItk+1V.It is observed from (44) and (45) that when the maximum transmit power is large, both IP and OP converge, creating the error floors (or zero diversity orders). This result will be validated in Figure 3.Case 2.
(It⟶∞).
The intercept probability in this case becomes(46)ΨIt⟶∞=0,Γ~e≥max1−ββ,1−λλ,Ψ˘2It⟶∞,1−ββ≤Γ~e<1−λλ,Ψ˘3It⟶∞,1−λλ≤Γ~e<1−ββ,Ψ˘4It⟶∞,Γ~e<min1−ββ,1−λλ,
whereΨ˘2It⟶∞, Ψ˘3It⟶∞, and Ψ˘4It⟶∞ are obtained from Ψ˘2 in (32), Ψ˘3 in (34), and Ψ˘4 in (36), respectively, as Ψ˘2It⟶∞=limIt⟶∞Ψ˘2=1−M1e−ρ/θsrP˘s−∑k=0∞J1/P˘sk+1, Ψ˘3It⟶∞=limIt⟶∞Ψ˘3=e−N/P˘s, and Ψ˘4It⟶∞=limIt⟶∞Ψ˘4=Ψ˘2It⟶∞+Ψ˘3It⟶∞+M1−1e−N+ρ/θsr1/P˘s+e−N/P˘s∑k=0∞J1/P˘sk+1 where J1=limIt⟶∞J=1/k!k+1!Q/Lθsrk+1e−Q/Lρ∑u=1k+1u−1!−Lρ/Qu−Ei−Q/Lρ and M1=limIt⟶∞M=1−e−Q/Lρ.Figure 3
IP and OP versusΘ=P˘s/ε. “Ana.” and “Sim.” represent the simulation result and the analytical result in Section 3, respectively, while “Asym.” represents the asymptotic result.Likewise, the outage probability in this scenario is obtained from (38) to be
(47)YIt⟶∞=limIt⟶∞Y=1−U1e−B/θsrP˘s−∑k=0∞V1P˘sk+1,whereU1=limIt⟶∞U=e−W/Lρ and V1=limIt⟶∞V=1/k!k+1!W/θsrLk+1Ei−W/Lρ−Ei−W/LD+∑u=1k+1u−1!−L/Wue−W/LDDu−e−W/Lρρu.It is seen from (46) and (47) that when the maximum interference power is large, both IP and OP also converge, creating the error floors (or zero diversity orders). This result will be validated in Figure 4.Figure 4
IP and OP versusIt/ε.


## 4. Demonstrative Results

The current section demonstrates simulated/theoretical results to assess both security and reliability of the proposed jamming-and-transmitting scheme for spectrum-sharing relaying networks with nonlinear energy scavenging via essential parameters. Monte-Carlo simulations (the Monte-Carlo simulation is widely accepted, e.g., [54]; therefore, an explanation of its operation should not be included in this paper) produce simulated results while the derived expressions in Section 3 are calculated to achieve theoretical ones. For illustration intention, users are arbitrarily positioned in the 2-dimensional plane as E at 0.6,−0.5, P at 0.9,0.8, D at 1.0,0.0, R at 0.5,0.0, and S at 0.0,0.0. Unless otherwise stated, the following parameters are considered: the target spectral efficiency R~e=R~d=0.3 bps/Hz, the path-loss exponent δ=3, the energy converting efficiency ψ=0.9, and equal noise variances εd=εr=εe=ε.Figure3 plots the IP and the OP versus Θ=P˘s/ε for α=0.35, ρ/ε=10 dB, β=0.48, and λ=0.63. The results demonstrate the coincidence between the simulation and the analysis, validating the accurateness of the analytical results. Moreover, the reliability of the system is enhanced and the security is decreased with increasing P˘s, which comes from the reality that increasing P˘s facilitates R in recovering successfully the source data and scavenging more energy in the source signals, hence mitigating the outage probability in Phase 3. Likewise, E is also beneficial in both Phase 2 and Phase 3 when P˘s increases, so the intercept probability also grows with P˘s. However, both IP and OP converge at high P˘s. Such a convergence is owing to the power distribution of the spectrum-sharing operation mode for S and R in (2) and (9) where transmit powers are uncorrelated with P˘s for large P˘s (i.e., as P˘s is adequately large, the MTPC is not necessary), causing the constant IP/OP. Such a constant IP/OP is named as the asymptotic IP/OP, which was analyzed in Subsection 3.3. Furthermore, the OP goes down and the IP goes up with increasing It. This observation is comprehended similarly to the case of increasing P˘s owing to the power distribution of the spectrum-sharing operation mode for S and R. Due to the opposite trends of the IP and the OP, the security-reliability trade-off is observed. By the exhaustive search, any security-reliability trade-off level can be found straightforwardly; for example, Figure 3 reveals that the best security-reliability trade-off wherein the OP and the IP are equal occurs at Θ=3.84 dB for It/ε=10 dB.Figure4 plots the IP/OP versus It/ε for the same specifications as Figure 3. The results reveal the coincidence between the simulation and the analysis, asserting the correctness of the analytical results. Moreover, the reliability/security is improved/degraded with increasing It and P˘s, which shows the security-reliability trade-off. By the exhaustive search, any security-reliability trade-off level can be found easily; for instance, Figure 4 unveils that the best security-reliability trade-off wherein the IP and the OP are equal happens at It/ε=1.46 dB for Θ=10 dB. Nonetheless, the reliability and the security converge at large It, which validates the asymptotic analysis in Subsection 3.3. The results in Figure 4 are comprehended similarly to those in Figure 3.Figure5 exposes the IP/OP versus the power distribution coefficient β at S, which presents the percentage of power that S uses to transmit the jamming signal, for P˘s/ε=20 dB, ρ/ε=15 dB, α=0.35, and λ=0.48. The results unveil the coincidence between the analysis and the simulation, affirming the correctness of the analytical results. It is noted that β is the superposition coefficient which is inversely proportional to S’s signal but proportional to the jamming signal in Phase 2. Consequently, the increase in β creates less energy for S to transmit information signal and less chance for R to decode successfully the message, which leads to the growth of the OP and the slight decline of the IP. However, the IP remains constant for Γ~e≥1−β/β (or β≥1/1+Γ~e) as seen from (27). This is because E suffers a complete outage in Phase 2, and hence, the IP only depends on Phase 3. Moreover, the reliability trades off with the security owing to the opposite trends of the IP and the OP, which was also observed similarly to Figure 3. Additionally, the security/reliability is degraded/enhanced with the increase in It, which is similar to results in previous figures.Figure 5
IP and OP versusβ.Figure6 demonstrates the IP/OP versus the power distribution coefficient λ at R, which presents the percentage of power that R uses to transmit the jamming signal, for P˘s/ε=20 dB, ρ/ε=15 dB, α=0.35, and β=0.63. The results reveal the coincidence between the analysis and the simulation, asserting the correctness of the analytical results. Moreover, it is seen that the reliability is deteriorated while the security is better with increasing λ. This can be comprehended by the power allocation of the spectrum-sharing operation mode at R for the legitimate signal and the jamming signal, similar to Figure 5. Notably, the IP reduces to zero and the absolute security is achievable when λ is greater than a certain value; for example, λ>0.5 as in the case of Figure 6. This can be interpreted as follows. Since β=0.63 and α=0.35, the inequality β≥1/1+Γ~e holds where Γ~e=22R~e/1−α−1. Therefore, when λ≥1/1+Γ~e, E suffers a complete outage in both phases (2 and 3) as seen in (29), causing Ψ=0. In addition, the security trades off with the reliability owing to the opposite trends of the IP and the OP, which was also observed similarly to Figure 3. Additionally, the security/reliability is deteriorated/enhanced with respect to the increase in It, as anticipated.Figure 6
OP and IP versusλ.Figure7 plots the IP/OP versus the time fraction α for P˘s/ε=20 dB, ρ/ε=15 dB, β=0.48, and λ=0.63. The results unveil the coincidence between the analysis and the simulation, affirming the correctness of the analytical results. Further, α can be selected optimally for the best reliability; for instance, αopt=0.3 is seen in Figure 7. αopt is available owing to the following reality. The increase in α extends the duration of Phase 1 (energy harvesting stage), facilitating R in harvesting more energy and recovering successfully the source message with a higher possibility. Notwithstanding, the increase in α also deteriorates the channel capacities in Phase 2 and Phase 3 (information transmission stage) because of the factor 1−α/2 before the logarithm in (14). Accordingly, α can be optimally adopted to balance the durations of energy scavenging and data transmission stages for the best reliability. Interestingly, the security is enhanced with increasing α. Some reasons lead to this observation as follows. Firstly, increasing α reduces the channel capacities at E, which can be explained as above. Secondly, although increasing α helps R collect more energy, E suffers the increase of the jamming power from R, eventually reducing the IP. Moreover, the IP reduces to zero for large α. This is because large α makes the inequality Γ~e≥max1−λ/λ,1−β/β holds where Γ~e=22R~e/1−α−1, and hence, the IP reduces to zero according to (29), which is similar to the results in Figure 6. In addition, the security/reliability is degraded/enhanced with the increase in It, as anticipated.Figure 7
IP and OP versusα.Figure8 exposes the IP/OP versus ρ/ε for P˘s/ε=20 dB, β=0.63, and λ=0.48. The results reveal the coincidence between the analysis and the simulation, affirming the correctness of the analytical results. Further, the reliability-security trade-off of the proposed jamming-and-transmitting scheme is observed in this figure. Nonetheless, the reliability gain increases faster than the security loss with increasing the power saturation threshold of the NL energy scavenger ρ, exposing the advantage of both relaying and jamming in our scheme in ensuring high reliability with affordable security threat. Moreover, the starting points at which the IP and the OP start to be saturated are coincident at ρ/ε of approximately 15 dB. That the IP and the OP are saturated at large ρ is because large ρ reduces the NL energy harvester to the linear one. In addition, the security/reliability is degraded/enhanced with the increase in It, as anticipated.Figure 8
IP and OP versusρ/ε.


## 5. Conclusion

This paper recommended the simultaneous jamming-and-transmitting scheme for spectrum sharing relaying networks with the nonlinear energy scavenger. Its security and reliability were analyzed and assessed with the intercept and outage probabilities under the IPR and the MTPC. The recommended analysis is validated by multifarious Monte-Carlo simulations. Illustrative results expose that the relay capable of scavenging RF energy enhances drastically the reliability even when the direct secondary source-destination link is unreliable because of bad propagation conditions. Moreover, jamming in both signal transmission phases achieve better security even with increasing transmission power. Additionally, the reliability of the proposed scheme can be optimized with selecting appropriately the time fraction. Notably, the networks can achieve the absolute security with adopting a suitable set ofα, β, λ, and R~e. Further, the proposed scheme offers the security-reliability compromise but suffers saturated performance at large maximum transmit/interference power. Furthermore, the reliability/security of the NL-ES is significantly worse/better than that of the L-ES.



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*Source: 2368201-2021-10-19.xml*