# Security Analysis for Underlay Cognitive Network with Energy-Scavenging Capable Relay over Nakagami-m Fading Channels

Authors: Khuong Ho-Van; Thiem Do-Dac
Journal: Wireless Communications and Mobile Computing
 (2019)

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## Abstract

This study suggests an energy-scavenging capable unlicensed relay not only to retain communications between an unlicensed sender-recipient pair in underlay cognitive networks but also to secure these communications against eavesdropping of malicious users. Message-securing capability of such a network configuration is assessed through secrecy outage probability (SOP). For this purpose, a precise closed-form formula of the SOP accounting for interference power restriction, Nakagami-m fading, and maximum transmit power restriction is first proposed. Then, the proposed formula is validated by computer simulations. Ultimately, various results are supplied to contrive that the relay position, the time percentage, and the power percentage of the energy-scavenging technique should be appropriately chosen for achieving the best security performance. Moreover, the SOP decreases with lower severity fading level and is constant in the range of high maximum interference power or high maximum transmit power.


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## Body

## 1. Introduction

Thanks to the advantage of efficient radio spectrum utilization, the spectrum-sharing paradigm (SSP) has lately acquired significant interest from both industry and academia [1]. According to the SSP, unlicensed users (UUs) are granted opportunities to send their message in the spectrum of licensed users (LUs) [2]. Nonetheless, UUs must frequently adjust their transmit power in order to conform to the interference power restriction inflicted by the successful signal reception of LUs and the maximum transmit power restriction inflicted by hardware specifications. These restrictions limit the power of unlicensed transmitters, rendering direct message transmission between an unlicensed sender-recipient pair unsuccessful. Additionally, this direct message transmission is presumably blocked due to severe shadowing, heavy path loss, and deep fading. Accordingly, to retain communications between the unlicensed sender-recipient pair, an unlicensed relay between this pair should be exploited to connect the sender with the recipient [3]. Nevertheless, as a volunteer, the relay is probably unenthusiastic to consume its individual power to forward the message of the unlicensed sender. Propitiously, currently advanced technologies make possible for wireless devices to be self-powered by scavenging energy in radio frequency (RF) signals [4]. Accordingly, the energy which the relay scavenges is used for relaying to lengthen the radio coverage of the sender, enhancing the communication possibility between the unlicensed sender-recipient pair. Conditioned on the short supply of the scavenged energy, the question is if the relay is capable of securing message transmission for the sender against the eavesdropping of illegal users in the information-theoretic viewpoint. This viewpoint affirms that the secured message transmission is when the trusted channel has a larger capacity than the wiretap channel [5]. Our paper aims to find a suitable answer to such a question.


### 1.1. Literature Review

Although various works analyzed the security performance of the direct message transmission (i.e., no relay) in underlay cognitive networks with energy scavenging (e.g., [6–14] and references therein), few publications [15–20] have paid attention to security evaluation for the underlay cognitive network with the energy-scavenging capable relay (UCNwEScR). To be more specific, the authors in [15] investigated the UCNwEScR in which the decode-and-forward unlicensed relay consumes its energy scavenged from the RF signals of the unlicensed sender and the licensed transmitter to forward the unlicensed sender’s message to the unlicensed recipient as the direct message transmission is unreliable. Also, the relay applies the power-dividing method for energy scavenging. However, Raghuwanshi et al. [15] merely presented the simulated SOP results. The problem in [15] was reconsidered in [16] with distinct aspects in which the relay performs the amplify-and-forward operation and scavenges the energy from merely the unlicensed sender’s signals relied on the time-dividing method. For the security capability enhancement, Maji et al. [17] further developed the work in [16] with permitting both the relay and the sender to interfere the eavesdropper. The study in [18] kept improving [17] by selecting an appropriate relay for higher security capability. For further security capability improvement, Hieu et al. [19] suggested to select a multihop relaying link offering the largest spectral efficiency. Nonetheless, Hieu et al. [19] relied on committed beacons for the relays to harvest the energy with the time-dividing method. Moreover, Hieu et al. [19] merely implemented the analysis on the connection outage probabilities (the connection outage probability is the possibility which the signal-to-noise ratio (SNR) does not exceed a preset value) at the eavesdropper and the recipient. In [20], an analysis framework was proposed to assess the eavesdropping-decoding trade-off in the UCNwEScR where the unlicensed relay only harvests the energy in the unlicensed sender’s RF signals. The eavesdropping-decoding trade-off is represented by the relation between the possibilities which the unlicensed recipient and the eavesdropper unsuccessfully restore the sender’s message. Relied on the information-theoretic viewpoint, the SOP analysis is more vital than the eavesdropping-decoding trade-off analysis. Accordingly, the former should be carried out to evaluate the security capability of the UCNwEScR before practical deployment. It is reminded that all studies in [15–20] merely considered the UCNwEScR over Rayleigh fading channels and under the interference power restriction (it should be emphasized that Ho-Van and Do-Dac [21] provided the SOP analysis for the UCNwEScR with the system model similar to [15–20]; nonetheless, Ho-Van and Do-Dac [21] neglected the interference power restriction but considered the average scavenged energy and the outage probability restriction on the licensed receiver; although the interference power restriction advances the average scavenged energy and the outage probability restriction in ensuring the short-term signal reception quality at the licensed receiver (i.e., the latter ensures the long-term one), the former requires instantaneous channel information which complicates the SOP analysis; accordingly, the literature review [15–20] merely concentrates on the publications related to the interference power restriction which is also considered in this paper) and the maximum transmit power restriction. Also, it is common knowledge that the Nakagami-m fading channels are suitable and general to fit realistic ones (e.g., the Nakagami-m fading is reduced to the Rayleigh fading when m equals 1) [22]. Accordingly, it is of utmost importance to perform the SOP analysis for the UCNwEScR over Nakagami-m fading channels (it is noted that Lei et al. [23] performed the SOP analysis for underlay cognitive networks over Nakagami-m fading channels but did not study energy scavenging). The current paper aims to reach this objective.


### 1.2. Contributions

The current paper differs from the studies [15–20] in the following matters:(i)
The relay is active merely as long as it exactly restores the unlicensed sender’s message. This (this paper and the previous studies [15, 18, 19] exploited the decode-and-forward relay while the studies in [16, 17, 20] employed the amplify-and-forward one) prevents the error propagation (the error propagation is due to the fact that the relay simply forwards the decoded information of the sender whether or not the decoded information is correct; this error propagation used to happen in noncognitive radio networks without energy scavenging) in [15, 18, 19] and the noise enhancement in [16, 17, 20].(ii)
This paper investigates the Nakagami-m fading, which generalizes the Rayleigh fading in [15–20].(iii)
The time percentage for energy scavenging is a variable, which is convenient for optimization of the SOP. It is reminded that Raghuwanshi et al. [15] fixed the time percentage of 0.5.(iv)
The SOP is analyzed in a precise closed form while Raghuwanshi et al., Benedict et al., and Maji et al. [15–18] merely presented the simulated SOP results and Hieu et al. [19] and Ho-Van and Do-Dac [20] merely analyzed the possibilities that the unlicensed recipient and the eavesdropper unsuccessfully restore the unlicensed sender’s message.Our contributions are listed as follows:(i)
Exploiting the unlicensed relay capable of energy scavenging to secure communications between the unlicensed sender-recipient pair as the direct message transmission is unreliable. The relay must assess the reliability of the received signal before agreeing to relay the sender’s message.(ii)
Proposing precise closed-form formulas for vital security capability measures such as the SOP, the intercept possibility (IP) (the IP is the possibility which the secrecy capacity is negative [24]), the possibility of positive secrecy capacity (PPSC) (the PPSC is the possibility which the secrecy capacity is positive [25]), under the maximum transmit power restriction, the Nakagami-m fading, and the interference power restriction to quickly assess the security capability of the UCNwEScR without the need of time-consuming computer simulations.(iii)
Optimizing key specifications based on the proposed formulas.(iv)
Providing numerous results to expose insights of the security capability, for example, the SOP saturation when one of the two power restrictions is ignored; considerable security capability improvement such as locating the relay and selecting the time percentage for energy scavenging and the power percentage for message decoding is appropriate.


### 1.3. Organization

This paper is organized as follows. The description of the system model under consideration is presented in Section2. Then, the precise closed-form formulas of the SOP, the IP, and the PPSC are derived in Section 3. Ultimately, results and conclusions are sequentially presented in Sections 4 and 5.


## 1.1. Literature Review

Although various works analyzed the security performance of the direct message transmission (i.e., no relay) in underlay cognitive networks with energy scavenging (e.g., [6–14] and references therein), few publications [15–20] have paid attention to security evaluation for the underlay cognitive network with the energy-scavenging capable relay (UCNwEScR). To be more specific, the authors in [15] investigated the UCNwEScR in which the decode-and-forward unlicensed relay consumes its energy scavenged from the RF signals of the unlicensed sender and the licensed transmitter to forward the unlicensed sender’s message to the unlicensed recipient as the direct message transmission is unreliable. Also, the relay applies the power-dividing method for energy scavenging. However, Raghuwanshi et al. [15] merely presented the simulated SOP results. The problem in [15] was reconsidered in [16] with distinct aspects in which the relay performs the amplify-and-forward operation and scavenges the energy from merely the unlicensed sender’s signals relied on the time-dividing method. For the security capability enhancement, Maji et al. [17] further developed the work in [16] with permitting both the relay and the sender to interfere the eavesdropper. The study in [18] kept improving [17] by selecting an appropriate relay for higher security capability. For further security capability improvement, Hieu et al. [19] suggested to select a multihop relaying link offering the largest spectral efficiency. Nonetheless, Hieu et al. [19] relied on committed beacons for the relays to harvest the energy with the time-dividing method. Moreover, Hieu et al. [19] merely implemented the analysis on the connection outage probabilities (the connection outage probability is the possibility which the signal-to-noise ratio (SNR) does not exceed a preset value) at the eavesdropper and the recipient. In [20], an analysis framework was proposed to assess the eavesdropping-decoding trade-off in the UCNwEScR where the unlicensed relay only harvests the energy in the unlicensed sender’s RF signals. The eavesdropping-decoding trade-off is represented by the relation between the possibilities which the unlicensed recipient and the eavesdropper unsuccessfully restore the sender’s message. Relied on the information-theoretic viewpoint, the SOP analysis is more vital than the eavesdropping-decoding trade-off analysis. Accordingly, the former should be carried out to evaluate the security capability of the UCNwEScR before practical deployment. It is reminded that all studies in [15–20] merely considered the UCNwEScR over Rayleigh fading channels and under the interference power restriction (it should be emphasized that Ho-Van and Do-Dac [21] provided the SOP analysis for the UCNwEScR with the system model similar to [15–20]; nonetheless, Ho-Van and Do-Dac [21] neglected the interference power restriction but considered the average scavenged energy and the outage probability restriction on the licensed receiver; although the interference power restriction advances the average scavenged energy and the outage probability restriction in ensuring the short-term signal reception quality at the licensed receiver (i.e., the latter ensures the long-term one), the former requires instantaneous channel information which complicates the SOP analysis; accordingly, the literature review [15–20] merely concentrates on the publications related to the interference power restriction which is also considered in this paper) and the maximum transmit power restriction. Also, it is common knowledge that the Nakagami-m fading channels are suitable and general to fit realistic ones (e.g., the Nakagami-m fading is reduced to the Rayleigh fading when m equals 1) [22]. Accordingly, it is of utmost importance to perform the SOP analysis for the UCNwEScR over Nakagami-m fading channels (it is noted that Lei et al. [23] performed the SOP analysis for underlay cognitive networks over Nakagami-m fading channels but did not study energy scavenging). The current paper aims to reach this objective.


## 1.2. Contributions

The current paper differs from the studies [15–20] in the following matters:(i)
The relay is active merely as long as it exactly restores the unlicensed sender’s message. This (this paper and the previous studies [15, 18, 19] exploited the decode-and-forward relay while the studies in [16, 17, 20] employed the amplify-and-forward one) prevents the error propagation (the error propagation is due to the fact that the relay simply forwards the decoded information of the sender whether or not the decoded information is correct; this error propagation used to happen in noncognitive radio networks without energy scavenging) in [15, 18, 19] and the noise enhancement in [16, 17, 20].(ii)
This paper investigates the Nakagami-m fading, which generalizes the Rayleigh fading in [15–20].(iii)
The time percentage for energy scavenging is a variable, which is convenient for optimization of the SOP. It is reminded that Raghuwanshi et al. [15] fixed the time percentage of 0.5.(iv)
The SOP is analyzed in a precise closed form while Raghuwanshi et al., Benedict et al., and Maji et al. [15–18] merely presented the simulated SOP results and Hieu et al. [19] and Ho-Van and Do-Dac [20] merely analyzed the possibilities that the unlicensed recipient and the eavesdropper unsuccessfully restore the unlicensed sender’s message.Our contributions are listed as follows:(i)
Exploiting the unlicensed relay capable of energy scavenging to secure communications between the unlicensed sender-recipient pair as the direct message transmission is unreliable. The relay must assess the reliability of the received signal before agreeing to relay the sender’s message.(ii)
Proposing precise closed-form formulas for vital security capability measures such as the SOP, the intercept possibility (IP) (the IP is the possibility which the secrecy capacity is negative [24]), the possibility of positive secrecy capacity (PPSC) (the PPSC is the possibility which the secrecy capacity is positive [25]), under the maximum transmit power restriction, the Nakagami-m fading, and the interference power restriction to quickly assess the security capability of the UCNwEScR without the need of time-consuming computer simulations.(iii)
Optimizing key specifications based on the proposed formulas.(iv)
Providing numerous results to expose insights of the security capability, for example, the SOP saturation when one of the two power restrictions is ignored; considerable security capability improvement such as locating the relay and selecting the time percentage for energy scavenging and the power percentage for message decoding is appropriate.


## 1.3. Organization

This paper is organized as follows. The description of the system model under consideration is presented in Section2. Then, the precise closed-form formulas of the SOP, the IP, and the PPSC are derived in Section 3. Ultimately, results and conclusions are sequentially presented in Sections 4 and 5.


## 2. System Model

Relevant users in the considered system model of the underlay cognitive radio with the energy-scavenging capable unlicensed relay R are denoted in Figure1. The unlicensed sender S broadcasts legitimate message to the unlicensed recipient D. This transmission is wire-tapped by the eavesdropper E and interferes the signal reception of the licensed receiver P (it is almost assumed in publications related to the SSP (e.g., [26, 27] and references therein) that licensed transmitters may inflict the ignorable amount of interference to unlicensed recipients; this can be justified due to long distance between licensed transmitters and unlicensed recipients or such Gaussian-distributed interferences). Due to unexpected causes (e.g., severe shadowing, deep fading, and heavy path loss), D and E may receive the signals of S not strong enough for successful decoding. Consequently, S should ask R, which is between S and D, for help in relaying its message to D. As such, two transmission stages are required for message from S to reach D as shown in Figure 2(a).Figure 1
System model.Figure 2
Stage times and signal processing at the relay. (a) Stage times. (b) Signal processing at R.
(a)
(b)To facilitate performance optimization, durations of two stages are different:μψ for Stage 1 and 1−μψ for Stage 2, where 0<μ<1 is the time percentage and ψ is the time to finish message transmission from S to D. Stage 1 is for S to broadcast its message while Stage 2 is for R to forward its received message to D. To avoid noise enhancement of the amplify-and-forward operation, R chooses to operate in the decode-and-forward manner. Moreover, R has self-powered capability by harvesting energy in RF signals. For system resource saving, R employs the power-dividing method [28, 29] as shown in (the power-dividing model is linear in Figure 2, which was assumed in most recent publications [6–21, 26, 29]) Figure 2(b) for energy scavenging. This method divides the signal received at R into two fractions with different powers which are represented by the power percentage 0<ρ<1: one fraction for restoring the sender’s message (it is common in existing works (e.g., [6–12, 15] and references therein) that the message decoder consumes the negligible amount of power for its operation; therefore, its energy consumption can be ignored) and the other for scavenging the energy.Fading channels under consideration are independent and Nakagami-m distributed. Channel coefficients are denoted in Figure 1 where ghk with h∈S,R and k∈R,E,D,P stands for the coefficient of the h-k channel. For the Nakagami-m distribution, ghk is represented by a parameter pair mhk,βhk where βhk=mhk/κhk with κhk=Eghk2ghk2 being the fading power and EX⋅ being the average operator with respect to the random variable X. Under investigation of the path loss, κhk can be represented as κhk=dhk−ϕ where dhk is the h-k distance and ϕ is the path-loss exponent. Accordingly, the channel gain ghk2 has the cumulative distribution function (CDF) and the probability density function (PDF), correspondingly, as(1)Fghk2x=γmhk,βhkxΓmhk,(2)fghk2x=βhkmhkΓmhkxmhk−1e−βhkx,where x≥0, Γ⋅ is the gamma function and γ⋅,⋅ is the lower incomplete gamma function ([30], equation (8.350.1)).In Stage 1, S performs message transmission while R processes the received signal from S. Therefore, R receives the following signal:(3)qr=gsrHsts+εr,where Hs is S’s transmit power, ts is the unity-power transmit symbol of S, and εr∼CN0,Nr is the additive noise produced by the receive antenna at R.For the SSP [31], S’s transmit power must comply with the interference power restriction and the maximum transmit power restriction. The former guarantees that S does not interfere the licensed receiver with the interference power exceeding a tolerable threshold, namely, the maximum interference power T, i.e., Hsgsp2≤T. In the meantime, the latter limits S’s transmit power by the maximum transmit power K inflicted by hardware specifications, i.e., Hs≤K. Accordingly, these two restrictions establish S’s transmit power as(4)Hs=minTgsp2,K.Processing the received signalqr at R follows the diagram in Figure 2(b): qr is divided into two fractions. The first fraction of ρqr is for scavenging the energy and the second fraction of 1−ρqr is for restoring S’s message. The energy which R scavenges from the first fraction is given by(5)Ir=φEts,εrρqr2μψ=φρHsgsr2+Nrμψ,where φ, 0<φ<1, is an energy conversion efficiency of the energy scavenger at R.The time of Stage 2 is1−μψ; hence, the power which R consumes in Stage 2 is given by(6)Hr=Ir1−μψ=μφρ1−μHsgsr2+Nr.The input of the message decoder in Figure2(b) is represented as(7)q^r=1−ρqr+ε^r,where ε^r∼CN0,N^r is the additive noise inflicted by the passband-to-baseband signal conversion.Embedding (3) in (7), one can rewrite (7) as(8)q^r=1−ρHsgsrts+1−ρεr+ε^r.Based on (8), the performance of the message decoder relies on the following SNR:(9)θr=1−ρHsgsr21−ρNr+N^r=Hsgsr2N˜r,where(10)N˜r=Nr+N^r1−ρ.The channel capacity which R can obtain isΛr=μlog21+θr bits/s/Hz where the pre-logarithm factor μ appears since the time of Stage 1 is μψ. Relied on the communication theory, R correctly restores S’s message as its channel capacity exceeds the preset spectral efficiency Λ1, i.e., Λr≥Λ1. Equivalently, ts is successfully recovered at R if θr≥θ1 where θ1=2Λ1/μ−1.In Stage 2, the relay whose transmit power is denoted asHr sends the restored symbol tr as long as it correctly recovers S’s message (i.e., θr≥θ1 and tr=ts). Otherwise, it is in the standby mode. As such, D and E receive the following signals, respectively:(11)qd=grdH^rtr+εd,θr≥θ1,εd,θr<θ1,(12)qe=greH^rtr+εe,θr≥θ1,εe,θr<θ1,where εe∼CN0,Ne and εd∼CN0,Nd are the additive noises produced by the receive antenna at E and D, respectively, and H^r is R’s transmit power, which is given by(13)H^r=minTgrp2,Hr,because R operates in the SSP.Based on (11) and (12), one infers the SNRs at D and E as(14)θd=grd2H^rNd,θr≥θ1,0,θr<θ1,(15)θe=gre2H^rNe,θr≥θ1,0,θr<θ1.Then, E and D can achieve the following channel capacities, respectively [32]:(16)Λe=1−μlog21+θe,(17)Λd=1−μlog21+θd.In (16) and (17), the time of Stage 2 is 1−μψ which introduces the pre-logarithm coefficient of 1−μ.According to [5], the secrecy capacity of the UCNwEScR is the capacity difference between the R-D trusted channel and the R-E wiretap channel:(18)Λ=Λd−Λe+=1−μlog21+θd1+θe+,θr≥θ1,0,θr<θ1,where x+ represents maxx,0.


## 3. Security Analysis

This section aims at deriving a precise closed-form formula for the SOP of the UCNwEScR. The SOP is the possibility which the secrecy capacityΛ is smaller than a preset security level Λ¯0. Accordingly, the smaller SOP means more secure message transmission. As such, the communication theory claims the SOP as a vital measure for assessing the security capability of wireless communications. The SOP analysis is helpful not only for assessing message securing capability without exhaustive simulations but also to obtain other useful security capability measures such as the IP and the PPSC.By representingPrℒ as the possibility of the event ℒ, one expresses the SOP as(19)ΦΛ¯0=PrΛ<Λ¯0.As shown in (18), Λ exists for θr≥θ1; hence, (19) can be computed in two different cases as(20)ΦΛ¯0=Pr1−μlog21+θd1+θe+<Λ¯0,θr≥θ1+Pr0<Λ¯0,θr<θ1.Furthermore, the required security level is positive (i.e.,Λ¯0>0); hence, (20) can be simplified as(21)ΦΛ¯0=Prlog21+θd1+θe+<Λ0,θr≥θ1︸Q+Prθr<θ1,where Λ0=Λ¯0/1−μ.In order to analytically evaluateΦΛ¯0, the term Q must be first computed as(22)Q=Prlog21+θd1+θe<Λ0,θr≥θ1∣log21+θd1+θe≥0Prlog21+θd1+θe≥0+Pr0<Λ0,θr≥θ1log21+θd1+θe<0Prlog21+θd1+θe<0=Prθd<2Λ01+θe−1,θr≥θ1θd≥θePrθd≥θe+Prθr≥θ1θd<θePrθd<θe=Prθd<2Λ01+θe−1,θr≥θ1.Observing (14) and (15), one recognizes that θe and θd have a common term H^r as θr≥θ1. Accordingly, θe correlates θd. To decorrelate them, Q in (22) should be computed through the conditional probability as(23)Q=εH^rPrθd<2Λ01+θe−1H^r︸Iθr≥θ1.To deriveI, one further rewrites it in an explicit form as(24)I=∫0∞∫02Λ01+x−1fθe,θdx,yH^rdydx.Conditioned onH^r, it is apparent that θe and θd are statistically independent. Consequently, fθe,θdx,yH^r can be represented as fθexH^rfθdyH^r. Using this fact, (24) can be further simplified as(25)I=∫0∞∫02Λ01+x−1fθdyH^rdyfθexH^rdx=∫0∞Fθd2Λ01+x−1H^rfθexH^rdx.In the sequel,FθdxH^r and fθexH^r are computed to completely derive (25). First, based on the definition of the CDF, FθdxH^r is derived as(26)FθdxH^r=Prθd<xH^r.Then, substituting (14) into (26), one achieves(27)FθdxH^r=PrH^rgrd2Nd<xH^r=Prgrd2<NdxH^rH^r=γmrd,βrdNdx/H^rΓmrd.By transforming the variable,fθexH^r can be deduced as(28)fθexH^r=NeH^rfgre2NeH^rx=βreNeH^rmrexmre−1Γmree−βreNe/H^rx.By changing variables reasonably in (27) and (28) and then substituting the result into (25), one achieves(29)I=∫0∞γmrd,βrdNd2Λ01+x−1/H^rΓmrdβreNeH^rmrexmre−1Γmree−βreNex/H^rdx.Using(30)γn,x=1−e−x∑i=0n−1xii!Γn,in [30], equation (8.352.1), (29) can be expanded as(31)I=∫0∞1−e−βrdNd2Λ01+x−1/H^r∑i=0mrd−1βrdNd2Λ01+x−1/H^rii!βreNeH^rmrexmre−1Γmree−βreNex/H^rdx=1−βreNeH^rmree−βrdNd2Λ0−1/H^rΓmre∑i=0mrd−11i!βrdNdH^r2−Λ0i1i!βrdNdH^r2−Λ0i∫0∞x+1−2−Λ0ixmre−1e−βreNe+βrdNd2Λ0x/H^rdx.Applying the binominal expansionu+vi=∑l=0iilulvi−l with il=i!/l!i−l! being the binomial coefficient, (31) can be expanded as(32)I=1−βreNeH^rmree−βrdNd2Λ0−1/H^rΓmre∑i=0mrd−11i!βrdNdH^r2−Λ0i∫0∞∑l=0iilxl1−2−Λ0i−lxmre−1e−βreNe+βrdNd2Λ0x/H^rdx=1−βreNeH^rmree−βrdNd2Λ0−1/H^rΓmre∑i=0mrd−1∑l=0iil1−2−Λ0i−li!βrdNdH^r2−Λ0i∫0∞xmre+l−1e−βreNe+βrdNd2Λ0x/H^rdx.By exerting [30], equation (3.351.3), one computes the last integral in (32) as(33)I=1−βreNeH^rmree−βrdNd2Λ0−1/H^rΓmre∑i=0mrd−1∑l=0iilmre+l−1!i!1−2−Λ0l−iβrdNdH^r2−Λ0iH^rβreNe+βrdNd2Λ0mre+l=1−βreNemreΓmre∑i=0mrd−1∑l=0iilβrdNd2Λ0imre+l−1!e−βrdNd2Λ0−1/H^ri!βreNe+βrdNd2Λ0mre+l1−2−Λ0l−iH^ri−l︸K.Substituting (33) into (23), one obtains(34)Q=εH^r1−Kθr≥θ1.GivenQ in (34), one can rewrite (21) as(35)ΦΛ¯0=EH^r1−Kθr≥θ1+Prθr<θ1=EH^r1θr≥θ1−εH^rKθr≥θ1+Prθr<θ1=∫θ1∞fθrzdz+Prθr<θ1−εH^rKθr≥θ1=Prθr≥θ1+Prθr<θ1−εH^rKθr≥θ1=1−εH^rKθr≥θ1.PluggingK in (33) into (35), one obtains(36)ΦΛ¯0=1−NeβremreΓmre∑i=0mrd−1∑l=0iil1−2−Λ0i−l×βrdNd2Λ0imre+l−1!i!βreNe+βrdNd2Λ0mre+lEH^re−βrdNd2Λ0−1/H^rH^ri−lθr≥θ1︸X.Using (13), the term X in (36) is simplified as(37)X=EHr,grp2minTgrp2,Hrl−ie−βrdNd2Λ0−1/minT/grp2,Hrθr≥θ1=ℰHr∫T/Hr∞Txl−ie−βrdNd2Λ0−1/T/xfgrp2xdx+∫0T/HrHrl−ie−βrdNd2Λ0−1/Hrfgrp2xdxθr≥θ1.Inserting the PDF ofgrp2, which has a general form as (2), into (37), one simplifies (37) as(38)X=EHrTl−iβrpmrpΓmrp∫T/Hr∞xmrp+i−l−1e−βrp+βrdNd2Λ0−1/Txdx+Hrl−ie−βrdNd2Λ0−1/Hrγmrp,βrpT/HrΓmrpθr≥θ1.The last integral in (38) is computed with the help of [30], equation (3.351.2); hence, one simplifies (38) as(39)X=ℰHrTl−iβrpmrpΓmrp∑k=0mrp+i−l−1THrkβrdNd2Λ0−1T+βrpl+k−mrp−imrp+i−l−1!k!eβrdNd2Λ0−1+βrpT/Hr+Hrl−iγmrp,βrpT/HreβrdNd2Λ0−1/HrΓmrpθr≥θ1.Using (30) to decompose the γ⋅,⋅ function in (39), one obtains(40)X=EHrTl−iβrpmrpΓmrp∑k=0mrp+i−l−1mrp+i−l−1!Tke−βrdNd2Λ0−1+βrpT/Hrk!βrp+βrdNd2Λ0−1/Tmrp+i−l−kHrk+Hrl−ie−βrdNd2Λ0−1/Hr1−e−βrpT/Hr∑j=0mrp−1βrpT/Hrjj!θr≥θ1.After some manipulations, one rewrites (40) in a compact form as(41)X=Tl−iβrpmrpΓmrp∑k=0mrp+i−l−1mrp+i−l−1!Tkk!βrp+βrdNd2Λ0−1/Tmrp+i−l−kℳA,k+ℳB,i−l−∑j=0mrp−1βrpTjj!ℳA,i+j−l,where(42)A=βrdNd2Λ0−1+βrpT,(43)B=βrdNd2Λ0−1,(44)ℳa,n=EHre−a/HrHrnθr≥θ1.Obviously,X in (41) is represented in closed form if ℳa,n in (44) is also solved in closed form. Next, the derivation of ℳa,n in (44) is presented.The scavenged power of R,Hr, in (6) can be rewritten as(45)Hr=Uθr+V,where(46)U=μφρN˜r1−μ,V=NrN˜r.Using (45) to rewrite ℳa,n in (44) gives(47)ℳa,n=Eθre−a/Uθr+VUθr+Vnθr≥θ1=1Un∫θ1∞e−a/U/x+Vx+Vnfθrxdx.In order to complete the computation of the last integral in (47), the PDF of θr must be derived first. Towards this end, the CDF of θr, Fθrx, is derived as follows and then the derivative of Fθrx is taken to result in fθrx.Theorem 1.
F
θ
r
x has a precise closed form as(48)Fθrx=1−βspmspΓmsp∑w=0msr−1Cww!xwe−CxL+∑z=0msp+w−1Dx+Mmsp+w−z,where(49)C=βsrN˜rK,(50)D=msp+w−1!Tmspz!Kz−wβsrN˜rmsp+w−ze−βspT/K,(51)M=βspTβsrN˜r,(52)L=msp−1!βspmsp1−e−βspT/K∑z=0msp−1βspT/Kzz!.Proof.
See AppendixA.
The PDF ofθr is produced from the derivative of Fθrx as(53)fθrx=dFθrxdx=−βspmspΓmsp∑w=0msr−1Cww!e−CxLw−Cxxw−1+∑z=0msp+w−1xw−1x+Mmsp+w−zwD+z−msp−wDxx+M−CDx.Theorem 2.
ℳ
a
,
n has a precise closed form as(54)ℳa,n=−βspmspUnΓmsp∑w=0msr−1Cww!LwTw−1,n,C,a/U−LCTw,n,C,a/U+∑z=0msp+w−1wDYw−1,msp+w−z,n,C,a/U+z−msp−wDYw,msp+w+1−z,n,C,a/U−CDYw,msp+w−z,n,C,a/U,where(55)Ta,b,c,g=∑u=0∞∑z=0a−1a+u−zguecVu!Vz−aazJθ1+V,c,b+u−z−1,(56)Ju,p,n=−1n+1pnn!Ei−pu+e−puun∑k=0n−1−1kpkuk∏j=0kn−j,(57)Ya,b,v,c,g=∑u=0∞∑z=0∞−1a+u−zguecVu!Vz−aaz×∑q=1bGb+1−qecM−VJθ1+M,c,q−1+∑t=1v+u−zQv+u−z+1−tJθ1+V,c,t−1,(58)Gq=V−Mz−v−u−q+1q−1!∏w=0q−2z−v−u−w,(59)Qt=M−V1−b−tt−1!∏w=0t−2−b−w,and Ei⋅ denotes the exponential integral [30].Proof.
See AppendixB.The precise closed-form representation ofℳa,n in (54) makes X in (41) to be represented in the precise closed form. Accordingly, ΦΛ¯0 in (36) is also derived in the precise closed form, which is favourable in assessing the security performance of the UCNwEScR under the interference power restriction, the Nakagami-m fading, and the maximum transmit power restriction without long computer simulations. Relied on our knowledge, this formula of ΦΛ¯0 has not been reported yet. Additionally, some vital security performance measures (e.g., the PPSC or the IP) can be straightforwardly derived from this analysis. Specifically, the PPSC is the possibility which the secrecy capacity is greater than zero, i.e.,(60)Ψ=PrΛ>0=1−PrΛ<0=1−Φ0.Furthermore, the IP is the possibility which the secrecy capacity is negative [24], i.e.,(61)ϒ=PrΛ<0=Φ0.


## 4. Illustrative Results

This section evaluates the SOP of the UCNwEScR through vital specifications. For demonstration purpose, the path-loss exponent is exemplified asϕ=4 and the coordinates of S, D, R, E, and P are randomly adopted as 0.0,0.0, 0.8,0.0, (d, 0.0), 0.9,0.5, and 0.5,0.7, respectively. Thereafter, “Sim.” and “Ana.” stand for the analytical result in (36) and the simulated result, respectively; the energy conversion efficiency φ=0.8 is adopted; without loss of generality, equal noise variances and equal severity levels are assumed, i.e., Nd=Ne=Nr=N^r=N0 and mhk=m for h∈S,R and k∈R,P,E,D.The SOP with respect to (w.r.t) the maximum transmit power-to-noise variance ratioK/N0 is illustrated in Figure 3 for Λ1=0.4 bits/s/Hz, μ=0.6, d=0.5, ρ=0.7, Λ¯0=0.2 bits/s/Hz, and T/N0=15 dB. This figure shows the coincidence of the simulation with the analysis, verifying the exactness of (36). Moreover, the security performance is enhanced with increasing K/N0. This is obvious because increasing K/N0 offers R more opportunities to harvest more energy in S’s transmit signal and to correctly restore S’s message, ultimately declining the outage possibility in Stage 2. Nonetheless, the security capability becomes constant at large K/N0. The unchanged security capability is due to the power allotment of R and S (observe (4) and (13)) where the powers of R and S do not depend on K/N0 for large K/N0 (i.e., the maximum transmit power restriction can be ignored in the case of large K/N0), rendering the SOP unchanged. Furthermore, the SOP decreases with lower fading severity level (i.e., increasing m) as expected.Figure 3
SOP w.r.tK/N0.For the similar specifications as Figure3 with the exception of K/N0=15 dB, Figure 4 illustrates the SOP w.r.t the maximum interference power-to-noise power ratio T/N0. The simulated/analytical results confirm the coincidence of the analysis with the simulation, again justifying the derivation of (36). Additionally, the security performance is improved with increasing T/N0 and is constant at large T/N0. These results are comprehended from the power allotment of R and S, identical to Figure 3. Furthermore, the SOP reduces with lower fading severity level.Figure 4
SOP w.r.tT/N0.Figure5 shows the SOP w.r.t the S-R distance d for Λ¯0=0.2 bits/s/Hz, T/N0=9dB, K/N0=13 dB, μ=0.6, ρ=0.7, and Λ1=0.4 bits/s/Hz. The results expose the coincidence of the analysis with the simulation, proving the exactness of (36). It is recalled that as R does not exactly restore S’s message (i.e., the S-R distance is long) or R does not reliably forward S’s recovered message to D (i.e., the R-D distance is long), the secrecy outage happens. Accordingly, an optimal relay position that well balances the possibility which R can correctly recover S’s message and the possibility that R can reliably forward S’s recovered message to D to minimize the SOP will exist. Figure 5 illustrates this reasoning where the security performance is optimized when R is dopt=0.57,0.55,0.54 away from S for m=1,2,3, respectively. Furthermore, lower fading severity level improves the security performance as expected.Figure 5
SOP w.r.td.Figure6 demonstrates the SOP w.r.t the time percentage μ for ρ=0.7, Λ¯0=0.2 bits/s/Hz, K/N0=13 dB, d=0.5, Λ1=0.4 bits/s/Hz, and T/N0=9dB. The results demonstrate the coincidence of the analysis with the simulation, confirming the accuracy of (36). Additionally, optimal μ (e.g., μopt=0.62,0.56,0.52 corresponding to m=1,2,3 as illustrated in Figure 6) for the best security performance exists. The presence of μopt is comprehended as follows. The increment of μ stretches out the time of Stage 1; therefore, R harvests more energy and restores S’s message more exactly. Nonetheless, the increment of μ also decreases the secrecy capacity in Stage 2; therefore, the security capability reduces. As such, μ should be optimized to balance the times of two stages for the best security capability. Moreover, the best security performance is enhanced with lower fading severity level.Figure 6
SOP w.r.tμ.Figure7 shows the SOP w.r.t the power percentage ρ for μ=0.6, d=0.5, Λ¯0=0.2 bits/s/Hz, T/N0=9dB, Λ1=0.4 bits/s/Hz, and K/N0=13dB. The results expose the coincidence of the analysis with the simulation, certifying the accuracy of (36). In addition, optimal ρ (e.g., ρopt=0.05,0.1,0.15 corresponding to m=1,2,3 as demonstrated in Figure 7) for the best security capability exists. The presence of ρopt is explained as follows. The increment of ρ creates more opportunities for R to harvest higher energy; therefore, R increases the signal reception quality in Stage 2, eventually improving the security capability. Nonetheless, the increment of ρ also declines the energy for the message decoder, thus mitigating the possibility which R exactly recovers S’s message and leading to more secrecy outage in Stage 2. Accordingly, ρ should be optimized to trade off the transmission reliability of R and S in two stages. Moreover, the best security performance is enhanced with lower fading severity level.Figure 7
SOP w.r.tρ.Figure8 illustrates the SOP w.r.t the preset spectral efficiency Λ1 for ρ=0.7, μ=0.6, K/N0=15dB, d=0.5, T/N0=12dB, and Λ¯0=0.2 bits/s/Hz. This figure exposes the coincidence of the analysis with the simulation, confirming the accuracy of (36). Additionally, the SOP rises with Λ1. This makes senses because the higher Λ1, the lower the possibility which R successfully restores S’s message, leading to the higher SOP. Furthermore, the SOP is reduced with lower fading severity level.Figure 8
SOP w.r.tΛ1.For the similar specifications as Figure8 with the exception of Λ1=0.4 bits/s/Hz, Figure 9 plots the SOP w.r.t the preset security level Λ¯0. The results confirm the coincidence of the simulation with the analysis, again validating the derivation of (36). Additionally, the security performance is degraded with increasing Λ¯0. This is reasonable because under fixed system parameters, the network only obtains a certain security level. Therefore, the higher Λ¯0, the more the outage events. Moreover, the security performance is improved with lower fading severity level.Figure 9
SOP w.r.tΛ¯0.


## 5. Conclusion

The precise closed-form formula of the SOP for the underlay cognitive network with the energy-scavenging capable relay under the maximum interference power restriction, the Nakagami-m fading, and the transmit power restriction was proposed in the current paper. The proposed formula was certified by computer simulations. Multiple results demonstrated that the deployment of the self-powered relay can drastically enhance the security capability even when the sender-recipient channel is inapplicable due to heavy shadowing, deep fading, severe path loss (i.e., the direct communication suffers the SOP of 1). Additionally, the best security performance can be achieved by appropriately locating the relay and selecting the parameters of energy-scavenging and message transmission processes (i.e., the time percentage and the power percentage). Moreover, the security capability is unchanged at high maximum interference power or high maximum transmit power. Furthermore, the security performance is improved with lower severity fading level.



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*Source: 5080952-2019-09-25.xml*