# Relay Selection-and-Jamming Scheme with Nonlinear Energy Harvesting

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

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## Abstract

When direct source-destination communications are in outage, relay selection is a preferable solution to improve reliability for this communications. However, such a relay selection makes the eavesdropper better overhear source data through both source-relay and relay-destination communication hops, losing data security. To improve both reliability and security, this paper proposes a relay selection-and-jamming (RaJ) scheme to select one intermediate node as a conventional relay and another node as a jammer. To enhance energy efficiency, all intermediate nodes harvest radio frequency energy in source signals for their operations with nonlinear energy harvesting (NL-EH). The security and reliability of the RaJ scheme are assessed through suggested rigorous/asymptotic expressions and are significantly better than two benchmark schemes without neither jamming nor both relay selection and jamming. Additionally, they can be optimized with reasonable selection of specifications. Moreover, the NL property of the energy harvesters dramatically affects the reliability but negligibly degrades the security for the RaJ scheme. Furthermore, the linear EH (L-EH) is more reliable but less secure than the NL-EH.


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## Body

## 1. Introduction

### 1.1. Background

Radio frequency energy harvesting (RF EH), which exploits available RF signals to power communication devices, can solve problems of energy efficiency, energy shortage, and green communications in modern wireless systems [1–4]. RF EH is efficiently implemented through power-splitting (PS) and time-switching (TS) protocols in which the former carries out EH and data decoding in the same duration while the latter performs them separately in different durations (i.e., the latter requires lower circuitry implementation than the former) [5]. For performance analysis, EH has been modelled as linear (L) [6] or NL [7]. The linear EH (L-EH) model represents the linear increase of the scavenged energy with the input RF power. Nonetheless, nonlinear (NL) behaviors of EH circuit elements (e.g., diodes and inductors) induce the NL-EH model more realistic and precise than the L-EH one. As such, this paper is interested in the NL-EH model for practical-and-exact performance analysis.When direct source-destination communications are in outage due to severe propagation conditions (strong path-loss, heavy shadowing, and deep fading), relay selection in which only one intermediate node among all available nodes between the source-destination pair is selected to satisfy a preset criterion is regarded as a technique that is efficient in improving communication reliability and reducing complexity as well as economical in bandwidth and power [8]. Nonetheless, such a relay selection offers the eavesdropper more chances to overhear source message through both source-relay and relay-destination communication hops instead of merely one hop in direct source-destination communications, threatening data security. To conceal legitimate data, the jamming technique where jamming signals (or artificial noises) are purposively generated to impair solely the eavesdropper has been popularly exploited [9].This paper assumes two intermediate nodes, which are self-powered by scavenging RF energy in source signals with the practical NL-EH, are willing to ameliorate both reliability and security for data transmission between the source-destination pair. The question is which node plays a role as a traditional relay to enhance communication reliability and as a jammer to protect secret data over both hops. Our solution, relay selection-and-jamming (RaJ) scheme, solves this question.


### 1.2. Previous Works

Our proposed RaJ scheme differs [7, 10–22] which investigated problems of security and/or reliability for the NL-EH (References which studied the L-EH with/without relaying (e.g., [5, 6, 9]) or the NL-EH without relaying (e.g., [23–35]) must not be surveyed because this paper considers simultaneously the NL-EH, relaying, and jamming.) with relaying. More specifically, [7, 10] analyzed the reliability of the pure relaying (namely, the nonrelay selection nonjamming (nRnJ)) scheme in terms of outage probability (OP). In the nRnJ scheme in [7, 10], the relay performs the amplify-and-forward (AF) operation on source signals with energy harvested by the TS protocol. Reconsidering the nRnJ scheme and the AF relay in [7, 10], the authors in [11, 12] analyzed and simulated the reliability in terms of bit error rate and throughput, respectively. Additionally, both [11, 12] utilized the PS protocol for scavenging energy in source signals. Secrecy performance quantified by secrecy outage probability was simulated in [13], and the OP was analyzed in [14–20] for the same nRnJ scheme as [7]. Notwithstanding, all the works in [13–20] investigated the decode-and-forward (DF) relay, which is also a research object of this paper. As compared to the AF relay, the DF relay is advantageous in preventing noise enhancement, probably improving the overall system performance. In [21], the throughput was simulated for the nonrelay selection-and-jamming (nRaJ) scheme where one relay capable of EH is appointed as a conventional relay and another user is dedicated as an energy supplier as well as a jammer. Instead of dedicating an intermediate node as a jammer in [21], the authors in [22] availed the destination as a jammer. Further, [22] considered the AF relay and the energy harvesting based on the PS protocol. However, the performance analysis on the ergodic secrecy capacity and the total harvested energy was not included in [22].


### 1.3. Contributions

Beside proposing the RaJ scheme with the NL-EH, this paper suggests the rigourous/asymptotic formulas of the OP and the intercept probability (IP) to quickly assess both reliability and security. These expressions are then simplified to obtain the OP/IP of the nRnJ and RnJ (relay selection nonjamming) schemes for performance comparison and highlighting the efficacy of simultaneous relay selection and jamming. Moreover, the OP/IP of the RaJ scheme with the L-EH is derived. Monte-Carlo simulations validate these analyses and shed insights into the reliability/security of the considered schemes and the feature of the NL-EH in comparison to the L-EH.


### 1.4. Structure

The remainder of this paper is structured as follows. Part 2 describes the investigated system. Next, part 3 provides detailed derivations of reliability and security of the proposed RaJ scheme. Then, two (nRnJ and RnJ) benchmark schemes are discussed in part 4. Subsequently, some useful remarks are withdrawn in part 5, especially the remark on the L-EH. Finally, part 6 presents simulated/theoretical results, and part 7 closes the paper.


## 1.1. Background

Radio frequency energy harvesting (RF EH), which exploits available RF signals to power communication devices, can solve problems of energy efficiency, energy shortage, and green communications in modern wireless systems [1–4]. RF EH is efficiently implemented through power-splitting (PS) and time-switching (TS) protocols in which the former carries out EH and data decoding in the same duration while the latter performs them separately in different durations (i.e., the latter requires lower circuitry implementation than the former) [5]. For performance analysis, EH has been modelled as linear (L) [6] or NL [7]. The linear EH (L-EH) model represents the linear increase of the scavenged energy with the input RF power. Nonetheless, nonlinear (NL) behaviors of EH circuit elements (e.g., diodes and inductors) induce the NL-EH model more realistic and precise than the L-EH one. As such, this paper is interested in the NL-EH model for practical-and-exact performance analysis.When direct source-destination communications are in outage due to severe propagation conditions (strong path-loss, heavy shadowing, and deep fading), relay selection in which only one intermediate node among all available nodes between the source-destination pair is selected to satisfy a preset criterion is regarded as a technique that is efficient in improving communication reliability and reducing complexity as well as economical in bandwidth and power [8]. Nonetheless, such a relay selection offers the eavesdropper more chances to overhear source message through both source-relay and relay-destination communication hops instead of merely one hop in direct source-destination communications, threatening data security. To conceal legitimate data, the jamming technique where jamming signals (or artificial noises) are purposively generated to impair solely the eavesdropper has been popularly exploited [9].This paper assumes two intermediate nodes, which are self-powered by scavenging RF energy in source signals with the practical NL-EH, are willing to ameliorate both reliability and security for data transmission between the source-destination pair. The question is which node plays a role as a traditional relay to enhance communication reliability and as a jammer to protect secret data over both hops. Our solution, relay selection-and-jamming (RaJ) scheme, solves this question.


## 1.2. Previous Works

Our proposed RaJ scheme differs [7, 10–22] which investigated problems of security and/or reliability for the NL-EH (References which studied the L-EH with/without relaying (e.g., [5, 6, 9]) or the NL-EH without relaying (e.g., [23–35]) must not be surveyed because this paper considers simultaneously the NL-EH, relaying, and jamming.) with relaying. More specifically, [7, 10] analyzed the reliability of the pure relaying (namely, the nonrelay selection nonjamming (nRnJ)) scheme in terms of outage probability (OP). In the nRnJ scheme in [7, 10], the relay performs the amplify-and-forward (AF) operation on source signals with energy harvested by the TS protocol. Reconsidering the nRnJ scheme and the AF relay in [7, 10], the authors in [11, 12] analyzed and simulated the reliability in terms of bit error rate and throughput, respectively. Additionally, both [11, 12] utilized the PS protocol for scavenging energy in source signals. Secrecy performance quantified by secrecy outage probability was simulated in [13], and the OP was analyzed in [14–20] for the same nRnJ scheme as [7]. Notwithstanding, all the works in [13–20] investigated the decode-and-forward (DF) relay, which is also a research object of this paper. As compared to the AF relay, the DF relay is advantageous in preventing noise enhancement, probably improving the overall system performance. In [21], the throughput was simulated for the nonrelay selection-and-jamming (nRaJ) scheme where one relay capable of EH is appointed as a conventional relay and another user is dedicated as an energy supplier as well as a jammer. Instead of dedicating an intermediate node as a jammer in [21], the authors in [22] availed the destination as a jammer. Further, [22] considered the AF relay and the energy harvesting based on the PS protocol. However, the performance analysis on the ergodic secrecy capacity and the total harvested energy was not included in [22].


## 1.3. Contributions

Beside proposing the RaJ scheme with the NL-EH, this paper suggests the rigourous/asymptotic formulas of the OP and the intercept probability (IP) to quickly assess both reliability and security. These expressions are then simplified to obtain the OP/IP of the nRnJ and RnJ (relay selection nonjamming) schemes for performance comparison and highlighting the efficacy of simultaneous relay selection and jamming. Moreover, the OP/IP of the RaJ scheme with the L-EH is derived. Monte-Carlo simulations validate these analyses and shed insights into the reliability/security of the considered schemes and the feature of the NL-EH in comparison to the L-EH.


## 1.4. Structure

The remainder of this paper is structured as follows. Part 2 describes the investigated system. Next, part 3 provides detailed derivations of reliability and security of the proposed RaJ scheme. Then, two (nRnJ and RnJ) benchmark schemes are discussed in part 4. Subsequently, some useful remarks are withdrawn in part 5, especially the remark on the L-EH. Finally, part 6 presents simulated/theoretical results, and part 7 closes the paper.


## 2. System Description

Figure1 sketches the considered RaJ scheme where the source S fails to convey secret messages directly to the destination D owing to bad propagation conditions (e.g., severe fading and strong shadowing). Therefore, two intermediate nodes, R0 and R1, are exploited with different roles as a traditional relay R to heal S⟶D communications and as a jammer J to protect secret data against the eavesdropper E. To improve energy efficiency, R0 and R1 scavenge RF energy in source signals through the TS protocol and utilize scavenged energy for relaying and jamming operations. Accordingly, secret data reaches D in three stages with an entire duration of T.Figure 1
System model.In stage I with (Stage I is just for energy harvesting. Consequently,S transmits an arbitrary signal, which carries RF energy, not necessarily the secret information or a deterministic signal.) a duration of αT, the nonlinear energy harvester of Ri, i=0,1, generates the power [7]:(1)Pri=AiPsgsri,Psgsri≤ζth,Aiζth,Psgsri>ζth,where Ai=2ϑiα/1−α; ϑi is energy converting efficiency; α is a time fraction; Ps is the transmission power of S; ζth is the saturation threshold; gsri=hsri2 is the S⟶Ri channel gain. Flat block Rayleigh fading channels are considered, and hence, hsri is modelled (hxy~CN0,λxy notates a zero-mean λxy-variance complex Gaussian random variable. Therefore, gxy=hxy2 obeys exponential distribution with mean λxy, shortly denoted as gxy~Eλxy, resulting in its cumulative distribution function (CDF) and the probability density function (PDF) as Fgxyw=1−e−w/λxy and fgxyw=e−w/λxy/λxy, respectively, where w≥0.) as hsri~CN0,λsri and is unchanged during T but changes independently in the next T. To guarantee S’s messages to be restored correctly at the intermediate nodes with the highest possibility, ultimately limiting error propagation as much as possible, Ri and Rj with j=1−i are selected as the conventional relay and the jammer, respectively, if gsri>gsrj (i.e., the S−Ri channel is better than the S⟶Rj channel).In stage II with a duration of1−αT/2, S transmits secret data while Rj jams E for securing S’s data. Therefore, signals received at E and Ri have an unique form as(2)ymII=hsmPsxs+hrjmPrjxrj+nm,where m∈ri,e; xs and xrj are the unit-power transmit symbols of S and Rj, correspondingly; gsm=hsm2~Eλsm and grjm=hrjm2~Eλrjm are the S⟶m and Rj⟶m channel gains, respectively; nm~CN0,σ2 is the additive noise at the receiver m.The jamming signalxrj created by Rj is deliberate to impair solely the wire-tapping of E without degrading signal reception of desired users (Ri and D). Such a special characteristic of xrj can come from pseudo-random signal generators whose seeds are securely shared only among D, Rj, and Ri [9, 21, 36–44]. Consequently, Ri and D can regenerate exactly xrj and entirely eliminate it, ultimately creating the signal received at Ri in stage II as(3)y~riII=hsriPsxs+nri,from which the signal-to-noise ratio (SNR) at Ri is established as(4)Ωsri=Psgsriσ2.Generatingxrj is solely known at D, Ri, and Rj for hiding xs but E is blind with it. As such, the signal-to-interference plus noise ratio (SINR) which E obtains for restoring xs in stage II is calculated from (2) to be(5)Ωse=PsgsePrjgrje+σ2.In stage III with a duration of1−αT/2, Ri decodes and forwards S’s data while Rj interrupts E for securing Ri’s transmission. Accordingly, signals received at D and E have an unique form as(6)ywIII=hriwPrixri+hrjwPrjxrj+nw,where w∈d,e; xri is the unit-power transmit symbol of Ri; grjw=hrjw2~Eλrjw and griw=hriw2~Eλriw are the Rj⟶w and Ri⟶w channel gains, correspondingly; nw~CN0,σ2 is the additive noise at the receiver w.Thanks to the property of the jamming signalxrj and processing it similarly as stage II, the SNR at D and the SINR at E in stage III are correspondingly given by(7)Ωrid=Prigridσ2,(8)Ωrie=PrigriePrjgrje+σ2.The decode-and-forward operation ofRi results in aggregated SINR at D for restoring xs as(9)Ωd=minΩsri,Ωrid.SinceE receives signals in both stages, it can perform selection-combining them for higher intercept possibility [45], yielding the aggregated SINR as(10)Ωe=maxΩse,Ωrie.The channel capacity available atw for restoring xs is represented to be(11)ℂw=1−α2log21+Ωw.


## 3. The Proposed RaJ Scheme

Communication reliability and data security can be measured through the OP atD and the IP at E, respectively. These probability expressions of the proposed RaJ scheme are first derived in this part to quickly evaluate both reliability and security without exhaustive simulations. Then, by simplifying them, the OP and the IP of two benchmark RnJ and nRnJ schemes are inferred in the next part to facilitate performance comparison and show the efficacy of simultaneous relay selection and jamming.


### 3.1. Intercept Probability

The IP is addressed as the probability whichE restores xs successfully. According to information theory, given a preset transmission rate ℝ, the IP is the probability that ℝ is smaller than ℂe, namely,(12)IRaJ=ℙℂe≥ℝ=ℙΩe≥Ω¯,where Ω¯=22ℝ/1−α−1 and ℙ· is the probability operator.Inserting (10) into (12), one obtains(13)IRaJ=ℙmaxΩse,Ωrie≥Ω¯=1−ℙmaxΩse,Ωrie<Ω¯=a1−∑i=01ℙΩse<Ω¯,Ωrie<Ω¯,gsri>gsrj⏟I¯,where =a follows the total probability law and the event gsri>gsrj means Ri as a conventional relay while Rj as a jammer.LetX=gsri, Y=gsrj, and Z=Prjgrje+σ2. Then, plugging (5) and (8) into I¯ in (13) results in(14)I¯=ΞX,YℙPsgse<Ω¯Z,Prigrie<Ω¯Z⏟I~X>Y,where Ξ· is the expectation operator.I~ in (14) is expressed in closed-form as(15)I~=ΞZFgseΩ¯ZPsFgrieΩ¯ZPri=∫0∞1−e−Ω¯Prjx+σ2/Psλse1−e−Ω¯Prjx+σ2/Priλriee−x/λrjeλrjedx=1−e−Ω¯σ2/PsλseΩ¯Prjλrje/Psλse+1−e−Ω¯σ2/PriλrieΩ¯Prjλrje/Priλrie+1+e−Ω¯σ2/Psλse−Ω¯σ2/PriλrieΩ¯Prjλrje/Psλse+Ω¯Prjλrje/Priλrie+1.Based on (1), four combinations of Pri,Prj are considered when deriving (14) as follows.Combination 1:Pri,Prj=AiPsgsri,AjPsgsrjThis combination holds whenX≤ζth/Ps and Y≤ζth/Ps. Incorporating these conditions with X>Y in (14) results in existence region of X,Y as Y≤X and X≤ζth/Ps. By averaging I~ in (15) over this region, one obtains I¯ for this combination as(16)I¯1=1λsri∫0ζth/PsI1e−x/λsridx,where(17)I1=1λsrj∫0XI~e−y/λsrjdy.InvokingI~ in (15) with Pri,Prj=AiPsX,AjPsY and after some simplifications, the integral in (17) is solved as(18)I1=1−e−X/λsrj−λsee−Ω¯σ2/PsλseΩ¯AjλrjeλsrjΘX,1λsrj,λseΩ¯Ajλrje+AiλrieXλseeΩ¯σ2/Psλse−Ω¯σ2AiPsλrieXΘX,1λsrj,AiλrieXΩ¯Ajλrje−e−Ω¯σ2/AiPsλrieX1+λse/AiλrieXΘX,1λsrj,1Ω¯Ajλrje/λse+Ω¯Ajλrje/AiλrieX,where Θa,b,c=∫0ae−bx/x+cdx=ebcEi−ab−bc−Ei−bc with Ei· is the exponential integral [46].Combination 2:Pri,Prj=AiPsgsri,AjζthThis combination holds whenX≤ζth/Ps and Y>ζth/Ps. Incorporating these conditions with X>Y in (14) results in empty region of X,Y. Therefore, one obtains I¯ for this combination as(19)I¯2=0.Combination 3:Pri,Prj=Aiζth,AjPsgsrjThis combination holds whenX>ζth/Ps and Y≤ζth/Ps. Incorporating these conditions with X>Y in (14) results in existence region of X,Y as X>ζth/Ps and Y≤ζth/Ps. By averaging I~ in (15) over this region, one obtains I¯ for this combination as(20)I¯3=∫ζth/Ps∞fXxdx⏟I¯31∫0ζth/PsI~fYydy⏟I¯32,where I¯31=e−ζth/Psλsri and I¯32 is expressed in closed-form as follows after invoking I~ in (15), substituting Pri,Prj with Aiζth,AjPsY and executing simplifications:(21)I¯32=1−e−ζth/Psλsrj−λsee−Ω¯σ2/PsλseΩ¯AjλsrjλrjeΘζthPs,1λsrj,λseΩ¯Ajλrje+AiλrieζthλsePseΩ¯σ2/Psλse−Ω¯σ2/AiλrieζthΘζthPs,1λsrj,AiλrieζthΩ¯AjλrjePs−e−Ω¯σ2/Aiλrieζth1+λsePs/AiλrieζthΘζthPs,1λsrj,1Ω¯Ajλrje/λse+Ω¯AjλrjePs/Aiλrieζth.Combination 4:Pri,Prj=Aiζth,AjζthThis combination holds whenX>ζth/Ps and Y>ζth/Ps. Incorporating these conditions with X>Y in (14) results in existence region of X,Y as X>Y and Y>ζth/Ps. By averaging I~ in (15) over this region, one obtains I¯ for this combination as(22)I¯4=∫ζth/Ps∞∫y∞I~4fXxdxfYydy=e−λsri+λsrjζth/λsriλsrjPs1+λsrj/λsriI~4,where I~4 is I~ in (15) with Pri,Prj=Aiζth,Ajζth, which is a constant.Now it is ready to simplify (13) using the total probability law as(23)IRaJ=1−∑i=01∑k=14I¯k=1−∑i=01I¯1+I¯31I¯32+I¯4.The asymptotic IP,IasymRaJ, is obtained when Ps approaches infinity. In the asymptotic region, only the combination 4 happens and hence,(24)IasymRaJ=1−limPs⟶∞I¯4=1,which indicates a complete insecurity.


### 3.2. Outage Probability

The OP is addressed as the probability whichE restores xs unsuccessfully. Consequently, the OP is the probability that ℝ is greater than ℂd, namely,(25)ORaJ=ℙℂd≤ℝ=ℙΩd≤Ω¯.GivenΩd in (9), one rewrites (25) as(26)ORaJ=ℙminΩsri,Ωrid<Ω¯=1−ℙminΩsri,Ωrid>Ω¯=1−∑i=01ℙΩsri>Ω¯,Ωrid>Ω¯,X>Y⏟O¯.Plugging (4) and (7) into O¯ in (26) results in(27)O¯=ΞX,Yℙgrid>Ω¯σ2PriX>Y,X>Ω¯σ2Ps=ΞX,Ye−Ω¯σ2/PriλridX>Y,X>Ω¯σ2Ps.Based on (1), two cases of Pri are considered when deriving (27) as follows.Case 1.
Pri=AiPsX
This case holds whenX≤ζth/Ps. Incorporating this condition with X>Y and X>Ω¯σ2/Ps in (27) results in existence region of X,Y. More specifically, if Ω¯σ2>ζth, then the existence region is empty and hence, O¯ in (27) becomes O¯1=0. Otherwise, the existence region is Y<X and Ω¯σ2/Ps<X≤ζth/Ps. By averaging e−Ω¯σ2/AiPsλridx in (27) over this region, one obtains O¯ for this case as(28)O¯1=∫Ω¯σ2/Psζth/Pse−Ω¯σ2/AiPsλridx∫0XfYydyfXxdx=1λsriΨΩ¯σ2Ps,ζthPs,1λsri,Ω¯σ2AiPsλrid−ΨΩ¯σ2Ps,ζthPs,1λsri+1λsrj,Ω¯σ2AiPsλrid,where Ψu,v,k,l is given in (A.1) in Appendix A.Case 2.
Pri=Aiζth
This case holds whenX>ζth/Ps. Incorporating this condition with X>Y and X>Ω¯σ2/Ps in (27) results in existence region of X,Y as Y<X and X>B where B=maxΩ¯σ2,ζth/Ps. By averaging e−Ω¯σ2/Aiλridζth in (27) over this region, one obtains O¯ for this case as(29)O¯2=e−Ω¯σ2/Aiλridζth∫B∞∫0XfYydyfXxdx=e−Ω¯σ2/Aiλridζth−B/λsri1−λsrje−B/λsrjλsri+λsrj.
Now it is ready to simplify (26) using the total probability law as(30)ORaJ=1−∑i=01O¯1+O¯2.
The asymptotic OP,OasymRaJ, is obtained when Ps approaches infinity. In the asymptotic region, only the case 2 happens and hence,(31)OasymRaJ=1−∑i=01limPs⟶∞O¯2=1−∑i=01λsrie−Ω¯σd2/Aiλridζthλsri+λsrj,which indicates joint impact of three involved channels (λsri, λsrj, and λrid) on communication reliability.


### 3.3. Comment

BothIRaJ and ORaJ in (23) and (30) are expressed in novel-and-exact forms, facilitating in evaluating swiftly both security and reliability of the proposed RaJ scheme with the NL-EH without exhaustive simulations. In addition, they are leveraged to derive performance measures for benchmark schemes as well as linear energy harvesters.


## 3.1. Intercept Probability

The IP is addressed as the probability whichE restores xs successfully. According to information theory, given a preset transmission rate ℝ, the IP is the probability that ℝ is smaller than ℂe, namely,(12)IRaJ=ℙℂe≥ℝ=ℙΩe≥Ω¯,where Ω¯=22ℝ/1−α−1 and ℙ· is the probability operator.Inserting (10) into (12), one obtains(13)IRaJ=ℙmaxΩse,Ωrie≥Ω¯=1−ℙmaxΩse,Ωrie<Ω¯=a1−∑i=01ℙΩse<Ω¯,Ωrie<Ω¯,gsri>gsrj⏟I¯,where =a follows the total probability law and the event gsri>gsrj means Ri as a conventional relay while Rj as a jammer.LetX=gsri, Y=gsrj, and Z=Prjgrje+σ2. Then, plugging (5) and (8) into I¯ in (13) results in(14)I¯=ΞX,YℙPsgse<Ω¯Z,Prigrie<Ω¯Z⏟I~X>Y,where Ξ· is the expectation operator.I~ in (14) is expressed in closed-form as(15)I~=ΞZFgseΩ¯ZPsFgrieΩ¯ZPri=∫0∞1−e−Ω¯Prjx+σ2/Psλse1−e−Ω¯Prjx+σ2/Priλriee−x/λrjeλrjedx=1−e−Ω¯σ2/PsλseΩ¯Prjλrje/Psλse+1−e−Ω¯σ2/PriλrieΩ¯Prjλrje/Priλrie+1+e−Ω¯σ2/Psλse−Ω¯σ2/PriλrieΩ¯Prjλrje/Psλse+Ω¯Prjλrje/Priλrie+1.Based on (1), four combinations of Pri,Prj are considered when deriving (14) as follows.Combination 1:Pri,Prj=AiPsgsri,AjPsgsrjThis combination holds whenX≤ζth/Ps and Y≤ζth/Ps. Incorporating these conditions with X>Y in (14) results in existence region of X,Y as Y≤X and X≤ζth/Ps. By averaging I~ in (15) over this region, one obtains I¯ for this combination as(16)I¯1=1λsri∫0ζth/PsI1e−x/λsridx,where(17)I1=1λsrj∫0XI~e−y/λsrjdy.InvokingI~ in (15) with Pri,Prj=AiPsX,AjPsY and after some simplifications, the integral in (17) is solved as(18)I1=1−e−X/λsrj−λsee−Ω¯σ2/PsλseΩ¯AjλrjeλsrjΘX,1λsrj,λseΩ¯Ajλrje+AiλrieXλseeΩ¯σ2/Psλse−Ω¯σ2AiPsλrieXΘX,1λsrj,AiλrieXΩ¯Ajλrje−e−Ω¯σ2/AiPsλrieX1+λse/AiλrieXΘX,1λsrj,1Ω¯Ajλrje/λse+Ω¯Ajλrje/AiλrieX,where Θa,b,c=∫0ae−bx/x+cdx=ebcEi−ab−bc−Ei−bc with Ei· is the exponential integral [46].Combination 2:Pri,Prj=AiPsgsri,AjζthThis combination holds whenX≤ζth/Ps and Y>ζth/Ps. Incorporating these conditions with X>Y in (14) results in empty region of X,Y. Therefore, one obtains I¯ for this combination as(19)I¯2=0.Combination 3:Pri,Prj=Aiζth,AjPsgsrjThis combination holds whenX>ζth/Ps and Y≤ζth/Ps. Incorporating these conditions with X>Y in (14) results in existence region of X,Y as X>ζth/Ps and Y≤ζth/Ps. By averaging I~ in (15) over this region, one obtains I¯ for this combination as(20)I¯3=∫ζth/Ps∞fXxdx⏟I¯31∫0ζth/PsI~fYydy⏟I¯32,where I¯31=e−ζth/Psλsri and I¯32 is expressed in closed-form as follows after invoking I~ in (15), substituting Pri,Prj with Aiζth,AjPsY and executing simplifications:(21)I¯32=1−e−ζth/Psλsrj−λsee−Ω¯σ2/PsλseΩ¯AjλsrjλrjeΘζthPs,1λsrj,λseΩ¯Ajλrje+AiλrieζthλsePseΩ¯σ2/Psλse−Ω¯σ2/AiλrieζthΘζthPs,1λsrj,AiλrieζthΩ¯AjλrjePs−e−Ω¯σ2/Aiλrieζth1+λsePs/AiλrieζthΘζthPs,1λsrj,1Ω¯Ajλrje/λse+Ω¯AjλrjePs/Aiλrieζth.Combination 4:Pri,Prj=Aiζth,AjζthThis combination holds whenX>ζth/Ps and Y>ζth/Ps. Incorporating these conditions with X>Y in (14) results in existence region of X,Y as X>Y and Y>ζth/Ps. By averaging I~ in (15) over this region, one obtains I¯ for this combination as(22)I¯4=∫ζth/Ps∞∫y∞I~4fXxdxfYydy=e−λsri+λsrjζth/λsriλsrjPs1+λsrj/λsriI~4,where I~4 is I~ in (15) with Pri,Prj=Aiζth,Ajζth, which is a constant.Now it is ready to simplify (13) using the total probability law as(23)IRaJ=1−∑i=01∑k=14I¯k=1−∑i=01I¯1+I¯31I¯32+I¯4.The asymptotic IP,IasymRaJ, is obtained when Ps approaches infinity. In the asymptotic region, only the combination 4 happens and hence,(24)IasymRaJ=1−limPs⟶∞I¯4=1,which indicates a complete insecurity.


## 3.2. Outage Probability

The OP is addressed as the probability whichE restores xs unsuccessfully. Consequently, the OP is the probability that ℝ is greater than ℂd, namely,(25)ORaJ=ℙℂd≤ℝ=ℙΩd≤Ω¯.GivenΩd in (9), one rewrites (25) as(26)ORaJ=ℙminΩsri,Ωrid<Ω¯=1−ℙminΩsri,Ωrid>Ω¯=1−∑i=01ℙΩsri>Ω¯,Ωrid>Ω¯,X>Y⏟O¯.Plugging (4) and (7) into O¯ in (26) results in(27)O¯=ΞX,Yℙgrid>Ω¯σ2PriX>Y,X>Ω¯σ2Ps=ΞX,Ye−Ω¯σ2/PriλridX>Y,X>Ω¯σ2Ps.Based on (1), two cases of Pri are considered when deriving (27) as follows.Case 1.
Pri=AiPsX
This case holds whenX≤ζth/Ps. Incorporating this condition with X>Y and X>Ω¯σ2/Ps in (27) results in existence region of X,Y. More specifically, if Ω¯σ2>ζth, then the existence region is empty and hence, O¯ in (27) becomes O¯1=0. Otherwise, the existence region is Y<X and Ω¯σ2/Ps<X≤ζth/Ps. By averaging e−Ω¯σ2/AiPsλridx in (27) over this region, one obtains O¯ for this case as(28)O¯1=∫Ω¯σ2/Psζth/Pse−Ω¯σ2/AiPsλridx∫0XfYydyfXxdx=1λsriΨΩ¯σ2Ps,ζthPs,1λsri,Ω¯σ2AiPsλrid−ΨΩ¯σ2Ps,ζthPs,1λsri+1λsrj,Ω¯σ2AiPsλrid,where Ψu,v,k,l is given in (A.1) in Appendix A.Case 2.
Pri=Aiζth
This case holds whenX>ζth/Ps. Incorporating this condition with X>Y and X>Ω¯σ2/Ps in (27) results in existence region of X,Y as Y<X and X>B where B=maxΩ¯σ2,ζth/Ps. By averaging e−Ω¯σ2/Aiλridζth in (27) over this region, one obtains O¯ for this case as(29)O¯2=e−Ω¯σ2/Aiλridζth∫B∞∫0XfYydyfXxdx=e−Ω¯σ2/Aiλridζth−B/λsri1−λsrje−B/λsrjλsri+λsrj.
Now it is ready to simplify (26) using the total probability law as(30)ORaJ=1−∑i=01O¯1+O¯2.
The asymptotic OP,OasymRaJ, is obtained when Ps approaches infinity. In the asymptotic region, only the case 2 happens and hence,(31)OasymRaJ=1−∑i=01limPs⟶∞O¯2=1−∑i=01λsrie−Ω¯σd2/Aiλridζthλsri+λsrj,which indicates joint impact of three involved channels (λsri, λsrj, and λrid) on communication reliability.


## 3.3. Comment

BothIRaJ and ORaJ in (23) and (30) are expressed in novel-and-exact forms, facilitating in evaluating swiftly both security and reliability of the proposed RaJ scheme with the NL-EH without exhaustive simulations. In addition, they are leveraged to derive performance measures for benchmark schemes as well as linear energy harvesters.


## 4. Benchmark Schemes (RnJ and nRnJ)

### 4.1. The RnJ Scheme

To evaluate the efficacy of the jamming operation in our scheme, we compare it with the only relay selection scheme (e.g., [6, 47]) which lets Rj be idle in our scheme, namely, the RnJ scheme. The OP of the RnJ scheme is the same as that of ours (i.e., ORnJ=ORaJ) but the IP of the former is different from that of the latter. Following the derivation of ORaJ in (30) with the note that Prj=0, one obtains the IP of the RnJ scheme as(32)IRnJ=1−1−e−Ω¯σ2/Psλse∑i=01λsriλsri+λsrj−1λsriΦζthPs,1λsri,Ω¯σ2AiPsλrie+1λsriΦζthPs,1λsri+1λsrj,Ω¯σ2AiPsλrie−e−ζth/Psλsri−Ω¯σ2/Aiλrieζth1−λsrje−ζth/Psλsrjλsri+λsrj,where Φv,k,l is given in (B.1) in Appendix B.The asymptotic OP of the RnJ scheme is(33)IasymRnJ=limPs⟶∞IRnJ=1,which indicates a complete insecurity.


### 4.2. The nRnJ Scheme

To evaluate the efficacy of simultaneous relay selection and jamming in our scheme, we compare it with the nRnJ scheme (e.g., [7]) which lets Ri always relay the source data and Rj be idle in our scheme (i.e., pure relaying). Then, the IP of the nRnJ scheme is(34)InRnJ=1−ℙΩse<Ω¯,Ωrie<Ω¯=1−ℙPsgse<Ω¯σ2,Prigrie<Ω¯σ2.By considering two cases ofPri as for deriving ORaJ in (30), one obtains the IP of the nRnJ scheme as(35)InRnJ=1−1−e−Ω¯σ2/Psλse1−e−ζth/Psλsri−Ω¯σ2/Aiλrieζth−1λsriΦζthPs,1λsri,Ω¯σ2AiPsλrie,from which the asymptotic IP of the nRnJ scheme is given by(36)IasymnRnJ=limPs⟶∞InRnJ=1,which indicates a complete insecurity.The OP of the nRnJ scheme is expressed to be(37)OnRnJ=1−ℙΩsri>Ω¯,Ωrid>Ω¯=1−ℙPsgsri>Ω¯σ2,Prigrid>Ω¯σ2.By considering two cases ofPri as for deriving ORaJ in (30), one obtains the OP of the nRnJ scheme as(38)OnRnJ=1−e−Ω¯σ2/Aiζthλrid−B/λsri−1λsriΨΩ¯σ2Ps,ζthPs,1λsri,Ω¯σ2AiPsλrid,Ω¯σ2<ζth,1−e−Ω¯σ2/Aiζthλrid−B/λsri,Ω¯σ2>ζth,from which the asymptotic OP of the nRnJ scheme is given by(39)OasymnRnJ=limPs⟶∞OnRnJ=1−e−Ω¯σ2/Aiζthλrid,which indicates the dependence of communication reliability on only the Ri⟶D channel.


## 4.1. The RnJ Scheme

To evaluate the efficacy of the jamming operation in our scheme, we compare it with the only relay selection scheme (e.g., [6, 47]) which lets Rj be idle in our scheme, namely, the RnJ scheme. The OP of the RnJ scheme is the same as that of ours (i.e., ORnJ=ORaJ) but the IP of the former is different from that of the latter. Following the derivation of ORaJ in (30) with the note that Prj=0, one obtains the IP of the RnJ scheme as(32)IRnJ=1−1−e−Ω¯σ2/Psλse∑i=01λsriλsri+λsrj−1λsriΦζthPs,1λsri,Ω¯σ2AiPsλrie+1λsriΦζthPs,1λsri+1λsrj,Ω¯σ2AiPsλrie−e−ζth/Psλsri−Ω¯σ2/Aiλrieζth1−λsrje−ζth/Psλsrjλsri+λsrj,where Φv,k,l is given in (B.1) in Appendix B.The asymptotic OP of the RnJ scheme is(33)IasymRnJ=limPs⟶∞IRnJ=1,which indicates a complete insecurity.


## 4.2. The nRnJ Scheme

To evaluate the efficacy of simultaneous relay selection and jamming in our scheme, we compare it with the nRnJ scheme (e.g., [7]) which lets Ri always relay the source data and Rj be idle in our scheme (i.e., pure relaying). Then, the IP of the nRnJ scheme is(34)InRnJ=1−ℙΩse<Ω¯,Ωrie<Ω¯=1−ℙPsgse<Ω¯σ2,Prigrie<Ω¯σ2.By considering two cases ofPri as for deriving ORaJ in (30), one obtains the IP of the nRnJ scheme as(35)InRnJ=1−1−e−Ω¯σ2/Psλse1−e−ζth/Psλsri−Ω¯σ2/Aiλrieζth−1λsriΦζthPs,1λsri,Ω¯σ2AiPsλrie,from which the asymptotic IP of the nRnJ scheme is given by(36)IasymnRnJ=limPs⟶∞InRnJ=1,which indicates a complete insecurity.The OP of the nRnJ scheme is expressed to be(37)OnRnJ=1−ℙΩsri>Ω¯,Ωrid>Ω¯=1−ℙPsgsri>Ω¯σ2,Prigrid>Ω¯σ2.By considering two cases ofPri as for deriving ORaJ in (30), one obtains the OP of the nRnJ scheme as(38)OnRnJ=1−e−Ω¯σ2/Aiζthλrid−B/λsri−1λsriΨΩ¯σ2Ps,ζthPs,1λsri,Ω¯σ2AiPsλrid,Ω¯σ2<ζth,1−e−Ω¯σ2/Aiζthλrid−B/λsri,Ω¯σ2>ζth,from which the asymptotic OP of the nRnJ scheme is given by(39)OasymnRnJ=limPs⟶∞OnRnJ=1−e−Ω¯σ2/Aiζthλrid,which indicates the dependence of communication reliability on only the Ri⟶D channel.


## 5. Remarks

Remark 1.
Three (RaJ, RnJ, and nRnJ) schemes are completely insecure asPs⟶∞ (please refer to (24), (33), and (36)). This is reasonable since E receives strong signals from S as Ps⟶∞, making E decode successfully source data.Remark 2.
For the L-EH (i.e.,ζth in (1) is infinite), the IP and the OP of the RaJ scheme are, respectively, addressed as(40)ILinRaJ=limζth⟶∞IRaJ=1−1λsri∑i=01∫0∞I1e−x/λsridx,(41)OLinRaJ=limζth⟶∞ORaJ=1−1λsri∑i=01ΛΩ¯σ2Ps,1λsri,Ω¯σ2AiPsλrid−ΛΩ¯σ2Ps,1λsri+1λsrj,Ω¯σ2AiPsλrid,where Λ·,·,· is expressed in (A.4) in Appendix B.


## 6. Demonstrative Results

This part illustrates theoretical/simulated results to evaluate both reliability and security of the considered schemes via pivotal parameters. Monte-Carlo simulations produce simulated results while the derived expressions in parts 3–5 are calculated to achieve theoretical ones. Path-loss is accounted by modelling fading power ofx⟶y channel as λxy=dxy−3 where dxy is the x⟶y distance. For illustration purpose, users are located on a 2-dimension plane where S at 0.0,0.0, D at 1.0,0.0, E at 0.5,0.3, R0 at 0.4877,0.0503, and R1 at 0.4763,−0.0490; ϑi=0.9, i∈0,1; ℝ=0.3 bits/s/Hz. Since both RaJ and RnJ schemes achieve the same OP (i.e., ORnJ=ORaJ), the following results only expose ORaJ.Results in Figures2–4 show coincidences between theory and simulation and between asymptote and theory at large Ps, validating both rigorous and asymptotic analyses. Moreover, these figures demonstrate that without jamming, two (RnJ and nRnJ) benchmark schemes suffer a complete insecurity (IRnJ=InRnJ=1) since secret data is not protected in both stages (II and III) as in the proposed RaJ scheme which is drastically secured (i.e., IRaJ<1). Furthermore, with relay selection, both (RaJ and RnJ) schemes achieve significantly better reliability than the nRnJ scheme (i.e., ORaJ=ORnJ<OnRnJ). Therefore, the proposed RaJ scheme, which exploits simultaneous jamming and relay selection, considerably outperforms the (RnJ and nRnJ) benchmark schemes in terms of security as well as reliability.Figure 2
IP or OP versusPs/σ2. “The.” and “Sim.” represent the theoretical and the simulated results, correspondingly.Figure 3
IP or OP versusα.Figure 4
IP or OP versusζth/σ2.Figure2 unveils the OP/IP via Ps/σ2 for ζth/σ2=10 dB and α=0.4. It is observed that ORaJ decreases with increasing Ps, which makes senses because of increasing the harvested energy. Nevertheless, IRaJ is minimum (i.e., the security reaches the peak) at a certain value of Ps. This implies that increasing Ps does not always improve security because E also benefits from receiving strong signals from S and Ri, eventually wire-tapping more source data. Interestingly, the minimum IRaJ happens at the asymptotic OP OasymRaJ, and hence, Ps can be optimized to obtain the best reliability and security performances. Compared to the NL-EH, the L-EH apparently offers considerably better reliability (OLinRaJ<ORaJ). However, the security of the former (IRaJ) fluctuates around that of the latter (ILinRaJ) with increasing Ps.Figure3 plots the IP/OP versus α for Ps/σ2=ζth/σ2=10 dB. It is noted that α is proportional to the EH time but inversely proportional to the signal processing time (stages II and III). Accordingly, α should be optimally adopted to balance these times for optimum reliability. Figure 3 shows that the RaJ and nRnJ schemes reach the lowest OP at α=0.6 and α=0.4, respectively. Interestingly, the security of the proposed RaJ scheme is considerably enhanced (i.e., IRaJ significantly decreases) with increasing α. Some reasons lead to this observation as follows. Firstly, increasing α reduces the channel capacity at E because of the factor 1−α/2 before the logarithm in (11), causing the decrease of IRaJ. Secondly, although increasing α helps Ri and Rj collect more energy, E suffers the increase of the jamming power from Rj, eventually reducing IRaJ. Compared to the NL-EH, the L-EH is drastically more reliable (OLinRaJ<ORaJ) but less secure (ILinRaJ>IRaJ).Figure4 exposes the IP/OP versus ζth/σ2 for Ps/σ2=5 dB and α=0.4. The reliability-and-security trade-off of the proposed RaJ scheme is observed in this figure. Nonetheless, the reliability gain increases faster than the security loss with increasing the saturation threshold of the NL energy scavenger ζth, exposing the advantage of both relay selection and jamming in our scheme in ensuring high reliability with affordable security threat. Such a trade-off with increasing ζth is reasonable since the NL-EH operates in the linear mode with higher harvested energy more frequently in the range of large ζth. Indeed, the performances of the NL-EH, IRaJ and ORaJ, reach those of the L-EH, ILinRaJ and OLinRaJ, at large ζth.


## 7. Conclusions

The current paper recommended the relay selection-and-jamming scheme for radio frequency energy harvesting networks with the nonlinear energy harvester. Its security and reliability were also analyzed via the intercept and outage probabilities. Thanks to selecting the relay with the highest probability of decoding the source data and jamming the eavesdropper in both signal transmission stages, the proposed RaJ scheme achieved better reliability and security than the (RnJ and nRnJ) benchmark schemes without neither jamming nor both relay selection and jamming. Additionally, the best performance of the recommended scheme is achievable with choosing properly the source power and the time fraction, respectively. Moreover, the nonlinearity property of the energy harvesters significantly affects the reliability yet slightly degrades the security for the proposed scheme. Furthermore, the nonlinear energy harvester is less reliable yet more secure than the linear energy harvester.



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*Source: 1717585-2021-10-11.xml*