Energy Harvesting Based Secure Two-Way Communication Using an Untrusted Relay

In this paper, we propose a two-way secure communication network where two sources exchange confidential messages via a wireless powered untrusted amplify-and-forward (AF) relay and a friendly jammer (FJ). By adopting the time switching (TS) architecture at the relay, the data transmission is accomplished in three phases. In the first phase, both the relay and FJ are charged by non-information radio frequency signals from the sources. In the second phase, the two sources transmit information signals and concurrently, the FJ transmits artificial noise to confuse the curious relay. Finally, the third phase is dedicated to forward the scaled version of the received signal from the relay to the sources. For the proposed network, we derive a new closed-form lower-bound expression for the ergodic secrecy sum rate (ESSR) in the high signal-to-noise ratio (SNR) regime. As a benchmark, we also examine the scenario of without friendly jamming (WoFJ) to highlight the performance advantages of the proposed with friendly jamming (WFJ) scenario. We further specify the high SNR slope and the high SNR power offset to determine the effect of friendly jamming on the ESSR. In the next step, novel compact closed-form expressions are derived for the intercept probability (IP) as a well-known security metric. Finally, numerical examples are provided to demonstrate the impacts of different system parameters such as energy harvesting time ratio, transmit SNR and the power allocation factor on the secrecy performance of both WFJ and WoFJ scenarios. The results illustrate that the proposed WFJ network outperforms the traditional one-way communication and the two-way WoFJ policy.


I. INTRODUCTION
R ELAYING is a promising approach to improve energy efficiency, extend coverage and increase the throughput of wireless communication networks. Recently, the benefits of relaying have been viewed from the viewpoint of wireless physical-layer security (PLS) [1] which has been recognized as an emerging design paradigm to provide security in next generation wireless networks [2]. The PLS solutions exploit the dynamics of fading channels to provide secure transmission. A key area of interest is the untrusted relaying scenario where the source-to-destination transmission is assisted by a relay which may also be a potential eavesdropper [2], [3]. This scenario occurs in large-scale wireless systems such as heterogeneous networks, device-to-device (D2D) communications and Internet-of-things (IoT) applications, where confidential messages are often retransmitted by intermediate nodes.
Secure transmission utilizing an untrusted relay was first studied in [4], where an achievable secrecy rate was derived. In [5], it was found that introducing a friendly jammer (FJ) could result in a positive secrecy rate for a one-way untrusted relay link with no direct source-destination transmission. Indeed, many recent papers on untrusted relay communications have focused on the one-way relaying scenario [6]- [10]. Recently, several works have considered the more interesting scenario of two-way untrusted relaying [11]- [15] where physical-layer network coding can provide security enhancement since the relay receives a superimposed signal from the two sources instead of each individual signal.
Wireless energy harvesting as a key promising technology to realize the next generation (5G) wireless networks have attracted significant interests in recent years [16]- [18]. Simultaneous wireless information and power transfer (SWIPT) techniques can enhance the lifetime of energy-constrained wireless networks especially in cooperative networks where the helper nodes can be fed by the main nodes [17], [18].
The key idea behind SWIPT is that a wireless node could capture radio-frequency (RF) signal sent by a source node and transform it into direct current to charge its battery, then making use of it for signal processing or information transmission. Accordingly, two main relaying protocols, i.e., time switching (TS) and power splitting (PS) policies were proposed [17]. In recent years, great efforts have been dedicated to the study of SWIPT for non-security based [19]- [21] and security based systems [22]- [24]. To be specific, the authors in [22] proposed to employ a wireless powered FJ to provide secure communication between a source node and destination node in the presence of an external eavesdropper. In untrusted relaying networks, for the first time, [23] studied secure one-way communication via a wireless energy harvesting untrusted relay, which utilizes destinationassisted jamming signal to provide secure transmission under either TS or PS policies at the relay. Then, [24] considered the maximization of secrecy rate of the wireless-powered relay networks by jointly designing power splitting and relay beamforming techniques. The authors in [25] and [26] investigated the SWIPT technology in multiple-input singleoutput downlink system in which a multi-antenna source transmits a confidential message to a destination equipped with one single antenna information receiver and multiple antenna untrusted wireless energy receivers (ERs). They maximized the harvested energy in regards of keeping the information signal secure from prospective eavesdropping by the ERs.
In contrast to the aforementioned works, we deal with the PLS of a two-way untrusted AF relaying system, where two sources exchange confidential messages in the presence of a FJ. A self-reliant cooperative wireless network is proposed in which the relay and FJ, as helper nodes, are powered with wireless energy of RF signals. The TS receiver architecture is employed at both the relay and the FJ. The role of the relay is to harvest the energy in order to forward the received information signal to the sources, while the mission of the FJ is to harvest the energy to confuse the untrusted relay and then to provide secure communication. For this proposed system, we derive tight lower-bound expressions for the ergodic secrecy sum rate (ESSR) in the high signal-to-noise ratio (SNR) regime. To highlight the secrecy performance advantages of the with friendly jamming (WFJ) scenario, we also investigate the case of without friendly jamming (WoFJ). We further characterize the high SNR slope and the high SNR power offset of the ESSR of the two scenarios, which explicitly capture the impact of network parameters on the ESSR [27]. In the following, we proceed to derive new exact closed-form expressions for the intercept probability (IP) of WoFJ and WFJ scenarios. Numerical results show that the proposed two-way secure communication WFJ provides numerous paramount advantages rather than its counterpart, one-way communication scheme proposed in [23] and the case of WoFJ. Specifically, the proposed two-way scenario achieves dramatically higher ESSR at the high SNR compared to the other transmission schemes. We further discuss on important design insights into the security impact of key system parameters including energy harvesting time, power allocation factor, transmit SNR, relay/FJ location and the path-loss exponent.

A. System Model
We consider a two-way communication scenario illustrated in Fig. 1, where two source nodes, (S 1 ) and (S 2 ) communicate with each other via an untrusted amplify-and-forward (AF) relay (R) which acts as both a requisite low complex helper node as well as a potential passive eavesdropper. Two secure transmission protocols are taken into account; i) WFJ in which one friendly jammer (F J ) is employed to enhance the security of the network by degrading the relay channel through sending its jamming signal, and ii) WoFJ. Note that both R and F J are assumed to be energy-starved nodes, yet equipped with rechargeable batteries with infinite capacity, that are wirelessly powered by the sources to enjoy their utility. It is assumed that most of the nodes' energy are consumed for data transmission, and energy consumption for signal processing is ignored for simplicity [19].
In the WFJ scenario, the data exchange between two sources is implemented in three phases. In the first phase, shown with solid lines, S 1 and S 2 transmit non-information signals to F J and R, to charge them via the received RF signals. During the second phase, the source nodes send their information signals to the relay. Simultaneously, F J deteriorates the channel capacity of R by transmitting the jamming signal powered by the sources in the first stage, as demonstrated with dashed lines. Finally, in the third phase, R broadcasts the scaled version of the received signals to S 1 and S 2 , and then each source extracts their corresponding information signal after self-interference cancellation. Note that we assume the sources have perfect knowledge of the jamming signals transmitted by the FJ for they have paid for the jamming service 1 [11]. The WoFJ scenario follows the same three phases as the WFJ scheme assuming the absence of F J .

B. Channel Model
In the proposed scenario, we assume that all the nodes are equipped with a single antenna and operate in half-duplex mode. The direct link between S 1 and S 2 is assumed to be broken, which is common in the scenarios where two sources are located far away from each other or within heavily shadowed areas to the extent that using the relay service is mandatory [19]. The channels are assumed to be reciprocal and follow a quasi-static block-fading Rayleigh model [23]. Furthermore, a key assumption is that the sources have perfect knowledge of the jamming signals transmitted by F J as well as the channel state information (CSI) of the links S 1 -R, S 2 -R, and F J -R [11]. Let us denote h ij as the channel coefficient between nodes i and j, with channel reciprocity where h ij = h ji . The channel power gain |h ij | 2 follows an exponential distribution with mean µ ij as where f |hij| 2 (x) is the probability density function (pdf) of r.v |h ij | 2 . Fig. 2 describes the proposed wireless energy harvesting two-way relaying transmission protocol. Using the TS policy, the relay switches from energy harvesting to information encoding, and completes a round of data exchange in three phases over a period of T . To be specific, in the first phase with the duration of T 1 = αT (0 < α < 1), both R and F J harvest the energy of the RF signals transmitted by S 1 and S 2 . In the second time slot which lasts T 2 = (1 − α) T 2 , S 1 and S 2 send their information signals to R, and simultaneously F J transmits its jamming signal, powered by the received RF signals during the first phase of communication. Finally, in the third phase, R broadcasts the scaled version of the received signal.

D. Energy Harvesting at the Relay and Friendly Jammer
In the first phase, two source nodes send non-information signals, powering both R and F J to obtain the required power for activity. The received power at R and F J are respectively, given by P R = P S1 |h S1R and P J = P S1 |h S1J | 2 + P S2 |h S2J | 2 .
Note that P R and P J should be more than the minimum threshold power (Θ) to activate the harvesting circuitry, unless the helper nodes will remain inactive. In the TS protocol, the harvested energy E HR and E HJ in αT duration at R and F J are respectively, given by and E HJ = ηαT (P S1 |h S1J where η represents the energy conversion efficiency factor. The relay uses the harvested energy obtained in the first phase (4) to retransmit source signals in the third phase with power P T R which can be written as During the second phase, F J uses the harvested energy in (5) to transmit its jamming signal with the power of P T J which can be expressed as 2ηα(P S1 |h S1J | 2 + P S2 |h S2J | 2 ) 1 − α .

E. Signal Representation at the Relay
Let us denote x Si , i ∈ {1, 2}, and x J as the information signals and the jamming signal with the powers of P Si and P T J , respectively. Then, the received signal at R can be expressed as where n R is considered as the additive white Gaussian noise (AWGN) at the relay and for simplicity the processing noise at the relay is ignored [23]. Note that F J sends its jamming signal x J with total power harvested in the first phase, which is higher that the minimum threshold power for circuitry activation.
Based on the received signal y R in (8) and considering the multi-user decoding (MUD) performs at the relay, the SNR at R can be obtained as where N 0 denotes the noise power at R, and β = 1−α 2ηα . Finally, R broadcasts the amplified version of the received signals in the second phase which is given by where G is the scaling factor of R as

F. Signal Representation at the Sources
Next, we focus on the received signal at S 2 , from which similar expressions can be derived for the received signal at S 1 . By using (8) and (10), the received signal at S 2 can be expressed as where n S2 is the AWGN at S 2 with the power N 0 . Since we assume that S 1 and S 2 have the perfect knowledge of jamming signal as well as the CSI of the links S 1 ↔R, S 2 ↔R, and R↔F J , this means that S 1 and S 2 are able to cancel the self-interference and the jamming signals in (12). Accordingly, the received signal at S 2 can be simplified as Substituting (11) into (13), and then substituting (7) the received instantaneous end-to-end SNR at S 2 after some algebraic manipulations can be obtained as where ǫ = Following the same procedure of S 2 , the received instantaneous end-to-end SNR at S 1 is also given by To make the further analysis tractable, we proceed to examine high SNR relaying regime by replacing ǫ = 0 in (14) and (15).

III. ERGODIC SECRECY SUM RATE ANALYSIS
In this section, we first derive closed-form expressions for the power outage probability at the relay and the FJ. Then, analytical as well as closed-form lower-bound expressions are evaluated for the ESSR of both WFJ and WoFJ cases.
The received power at R must be greater than the minimum required power Θ to activate the energy harvesting circuitry [12]. If the received power P R in (2) is less than the threshold power Θ, the harvesting circuitry at R maintains inactive, leading to the power outage. As such, the probability of power outage at R is defined as which the expression for P por is obtained in the following proposition.
Proposition 1. The power outage probability at the relay is calculated as is the lower incomplete Gamma function [28].
Proof. See Appendix A.

Remark 1:
The expression for the power outage probability at F J (P poj ) can be obtained similar to P por by replacing µ S1R and µ S2R with µ S1J and µ S2J into (17), respectively.
In principle, the ergodic secrecy rate (ESR) characterizes the rate below which any average secure transmission is accessible [1]. Since we assume the MUD is performed at the untrusted relay to decode both the signals x S1 and x S2 , the integrated secrecy rate of the communication network, the ESSR, is considered as [11]. Therefore, the instantaneous secrecy sum rate R sec is evaluated by where and K ∈ {S 1 , R, S 2 }. By combining (18) and (19), R sec can be rewritten as where [x] + = max(x, 0) and the pre-log factor 1−α 2 is due to the efficient time of information exchange between the two sources. Moreover, γ R , γ S1 and γ S2 are given by (9), (14) and (15), respectively. In the following, we proceed to derive the ESSR performance of WFJ and WoFJ scenarios.

A. Without Friendly Jamming
By plugging P T J = 0 into (26) and regarding the power outage probability at the relay, the analytical expression of the ESSR yields (21) and the corresponding lower-bound can be formulated as where for i ∈ {1, 2}, where Φ ≈ 0.577215 is the Euler's constant [29], m Ri = dt is the exponential integral [28]. Furthermore, the term I 3 is given by where m x = and m y = Proof. See Appendix B.

B. With Friendly Jamming
Based on (20) and considering the power outage probability at the relay and FJ, the exact expression of the ESSR can be written asR which can be further expressed as and we define X = |h S1R | 2 , Y = |h S2R | 2 , Z = |h S1J | 2 , W = |h S2J | 2 , and U =|h RJ | 2 in the r.v.s of γ S1 , γ S2 , and γ R . Although the multiple integral expression in (26) can be evaluated numerically, a closed-form expression is not straightforward to obtain. As such, we proceed by deriving a new compact lower-bound expression for the ESSR in the following proposition.
Proposition 2. The lower-bound expression for the ESSR of WFJ scenario (R W F J LB ) can be expressed as with and where A is given by Proof. See Appendix C.
As shown in the numerical results, the lower-bound expression in (27) is tight in the high SNR regime.

IV. THE ASYMPTOTIC ESSR ANALYSIS
In this section, we proceed to analyze the asymptotic ESSR when the transmit SNR by each node goes to infinity by deriving the high SNR slope in bits/s/Hz (S ∞ ) and the high SNR power offset in 3 dB units (L ∞ ), which are defined respectively as is the general asymptotic form of the ESSR performance [27].
Lemma 1. Let C be a strictly positive constant, and let X and Y be two exponential random variables with mean m x and m y , respectively. Therefore, we have the following results.
E{ln(X)} = ln(m x ) − Φ, Proof. This lemma can be simply proved using [ Applying Lemma 1 to (34), (35), and (36) and then substituting them into (22), the closed-form expression for the asymptotic ESSR of the WoFJ network can be obtained as By plugging (37) into (32), we obtain the high SNR slope and the power offset respectively, as and 2) With Friendly Jamming: Similar to the WoFJ case, the asymptotic ESSR for WFJ scenario becomes as where using (15), (14) and (9), the terms J 1 , J 2 and J 3 are derived as follows: where (a) follows from Jensen's inequality, and (b) follows from using Lemma 1 and expression (31). Similar to J 1 , we obtain J 2 as Ultimately, the term J 3 is derived as where (a) follows from setting ǫ = 0; this means that the untrusted relay is considered as an ideal eavesdropper with the capability of noise cancellation which is the worst case assumption at the untrusted relay. Furthermore, (b) follows from Lemma 1. Consequently, plugging (41)-(43) into (40) and then plugging to (32), the high SNR power offset and the high SNR slope for the WFJ case are respectively, obtained as and Remark 2: By comparing (38) and (45), we can obtain . This result expresses that the WFJ scenario can achieve more high SNR slope compared to the WoFJ scenario when a FJ with low threshold to activate the energy harvesting circuitry is exploited.

V. INTERCEPT PROBABILITY ANALYSIS
When the capacity of main link falls below the wiretap link's capacity, the eavesdropper can intercept the source signal and an intercept event occurs. Thus, the probability that the eavesdropper can successfully intercepts the source signal is called intercept probability (IP) and denoted as P int . The IP is a key metric in evaluating the grade of the security of networks [30]. According to the definition, for the data rate R t , P int of the system is given by [30] In what follows, we derive the IP of both WoFJ and WFJ secure transmission schemes.

A. Without Friendly Jamming
When the all nodes are active, the IP for the WoFJ scenario, can be represented by where ν = 2 2R t 1−α − 1. Substituting (9) into (47) yields where F S (.) is the cdf of S = P S1 |h S1R | 2 +P S2 |h S2R | 2 . This cdf has been presented in Lemma 2 in Appendix A. Note that X = P S1 |h S1R | 2 and Y = P S2 |h S2R | 2 are two exponential r.v.s. We can conclude from (48) that by increasing the data rate R t or energy harvesting time ratio α, the probability of interception is reduced. Furthermore, as the transmit SNR or the average channel gain grows, the IP is increased dramatically, i.e., security of communication is harmed.

B. With Friendly Jamming
Before deriving the IP of WFJ scenario, we present the following useful proposition.
where h holds non-negative values, and K ν (x) is the νth order modified Bessel function of the second kind [29,Eq. (8.432)].
Proof. See Appendix D.
Now, based on Lemma 3 we proceed the evaluation of the probability P W F J int as follows where δ = 2ηα 1−α ,ν = 2 2R t 1−α − 1 and Q = P S1 |h S1R | 2 + P S2 |h S2R | 2 with the distribution characteristics as given in Lemma 2 (See appendix A) with m x = P S1 µ S1R and m y = P S2 µ S2R . Moreover, H = (P S1 |h S1J | 2 + P S2 |h S2J | 2 )|h J R | 2 has the pdf given by Proposition 3, with m z = P S1 µ S1J , m w = P S2 µ S2J , and m u = µ J R . Finally, after calculating the expression in (50), the exact closed-form expression for the IP of WFJ scenario can be represented by In this section, we provide some numerical results to verify the accuracy of the provided expressions. Furthermore, we reveal the impact of different system parameters on the ESSR and the IP. In the simulations, unless otherwise stated, we set the following practical system parameters [18].  Fig. 3 plots the ESSR versus transmit SNR of one-way, two-way WFJ, and two-way WoFJ scenarios. From the figure, we observe that: 1) The exact numerical expressions in (26) and (21) are well-approximated in the high SNR regime by the closed-form lower-bound expressions in (27) and (22), respectively, 2) The ESSR is significantly enhanced as the transmit SNR increases for both the one-way scenario [23] and the proposed two-way communication scenarios, 3) In the high SNR regime, the proposed two-way WFJ outperforms the oneway scenario. For example, in SNR = 50 dB, the ESSR of twoway WFJ scenario provides approximately 1 bit/s/Hz more than the one-way transmission scenario and, 4) Evidently, the high SNR-slope of the curve belongs to the proposed two-way WFJ is twice as much as the slope of the WoFJ scenario as we pointed out this result via the mathematical analysis in Remark 2. Fig. 4 plots the probability of interception versus transmit SNR of the network. The noteworthy result is that increasing transmit SNR has two contradictory impact on the IP. The more power the sources use to transmit their signals, the more the probability of interception increases, yet in the middle range of transmit SNR; since the untrusted relay harvests more power, increasing the node's ability to decode the received data. On the other hand, as the transmit power increases, the jamming signal power also grows proportionally. Accordingly, as seen from Fig. 4, the IP experiences a saturation zone in which increasing transmit SNR above a threshold has no effect on the IP, so the optimal transmit power by each nodes can be considered for the WFJ scenario in order to use power resources efficiently. Furthermore, we can conclude from Fig.  4 that employing a FJ significantly enhances performance of the system in terms of decreasing the IP. This effect is becoming more noticeable when the wireless energy harvesting time ratio (α) grows, i.e. from α = 0.2 to α = 0.75. This means by increasing the time allocated to harvest energy by the helper nodes, the IP is decreased. Finally, another key point is that the expressions we derived for both the WoFJ Ergodic Secrecy Sum Rate (bits/s/Hz) Simulation of WoFJ using (21) Lower-bound of WoFJ using (22) Simulation of WFJ using (26) Lower-bound of WFJ using (27) Simulation of One-way Lower-bound of One-way  This finding reveals the importance of TS ratio which should be taken into account in the system design. This observation says that the secrecy performance of the network is highly dependent on both the jamming strategies (WFJ or WoFJ) and the TS ratio. If the TS ratio is too low, the harvested energy at the relay (and the FJ) may be too low and then, power outage may occur or the received SNR at the sources may be too low. On the other hand, if the TS ratio is too high, insufficient time is dedicated for the relay to broadcast the information signal and hence, the received instantaneous SNR at the receivers may be too low. As a consequence, the reliable communication is influenced. As such, there is a trade-off between a secure transmission and a reliable communication. We consider this issue in our future works. Fig. 6 depicts the impact of distance between the network nodes on the ESSR performance. We assume that all the nodes, except the two sources, are located in equal distances from each other denoted by d. One interesting result from Fig. 6 is that the proposed two-way WFJ significantly outperforms the one-way and two-way WoFJ communication when the network's nodes are close together, i.e. d = 2m. It is also observed that in this topology, the energy harvesting time should be much less than the broadcasting time to achieve the optimal ESSR. Moreover, by extending the network scale to d = 5m, the best ESSR is achievable if more time is dedicated to RF energy harvesting.

A. Transmit SNR
In Fig. 7, we demonstrate that the WFJ scheme again outperforms the WoFJ scenario in terms of IP for equal environments and network data rates. It is observed that as the energy harvesting time ratio grows, the IP decreases. Specifically, reducing the IP for high data rate, i.e. R t =2.5 bits/s/Hz, is noticeable in the middle range of α values, while for R t =1 bit/s/Hz, the IP decreases dramatically in the high α values. This illustrates the fact that for different network data rates, the required time for energy harvesting should be chosen intelligently in order to achieve the low IP.

C. Power Allocation Factor (λ)
We provide Fig. 8 to observe the impact of relay position and power allocation on the achievable ESSR of two-way WFJ, two-way WoFJ, and one-way scenarios. Let define the power allocation factor λ (0 < λ < 1) such that P S1 = λP and P S2 = (1 − λ)P . We can observe from Fig. 8  of the transmission schemes, when the untrusted relay is close to one node, little amount of power budget is to be allocated to that node to maximize the secrecy rate. It should be pointed out that for the one-way communication the higher secrecy performance is obtained when the source to the untrusted relay is located close to the destination, while for the two-way WFJ scenario, the more closer the source to the untrusted relay becomes, the less power it should consume to provide the higher ESSR.

D. Path Loss Exponent (κ)
We plot Fig. 9 to illustrate the impact of path-loss exponent on the secrecy rate with different R-F J distances. When the environmental path-loss increases, both the one-way scenario and our proposed two-way relaying protocols suffer from decreasing ESSR. However, as can be seen, the FJ location  Two-way, WFJ One-way Two-way, WoFJ   Fig. 8. ESSR versus the power allocation factor for one-way, twoway WoFJ, and two-way WFJ scenarios with respect to the location of the untrusted relay. We set dRJ =d/2 and P = 10 dBW.   plays a significant role in improving the ESSR performance.
To be specific, for the d JR = 2d = 10m, the two-way scenario provides the worse ESSR in contrast to the one-way scenario, either in urban area (small κ) or in sub-urban (large κ). The priority of two-way transmission is disclosed when d JR = d/2 = 2.5m. This result expresses that by intelligently choosing the optimal FJ from a group of FJs, the proposed two-way communication scenario, incontrovertibly, presents higher secrecy rate compared to the conventional one-way transmission scheme.

VII. CONCLUSION
In this paper, we proposed a wireless powered two-way cooperative network in which two sources communicate via an untrusted relay. To enhance the secrecy rate, we proposed to employ a friendly jammer (FJ). By adapted the time switching (TS) protocol at the untrusted relay, we investigated the ergodic secrecy sum rate (ESSR) and the intercept probability (IP) for both the cases of with friendly jamming (WFJ) as well as without friendly jamming (WoFJ) schemes. Tight lowerbound expressions were derived for the ESSR of the scenarios, and the asymptotic ESSR analysis were also investigated. We further derived exact closed-form expressions for the IP of two transmission scenario. Numerical examples revealed the priority of the proposed two-way WFJ. They illustrated that exploiting a FJ significantly enhances both the ESSR and the IP performances. Furthermore, several engineering insights were presented regarding the impact of different system parameters such as TS ratio, signal-to-noise ratio (SNR), power allocation factor, path loss exponent, and nodes' location on the ESSR and the IP secrecy performances.

APPENDIX A
Power outage probability at R is represented by To evaluate P por , we present the following useful lemma. where Υ(s, x) = x 0 t (s−1) e −t dt is the lower incomplete Gamma function [28].
Proof. We commence from evaluating the pdf of S as Evaluating the integral in (54) yields the expression as in (53). Note that the r.v. S is non-negative and (a) follows from the fact that two r.v.s X and Y are independent.

APPENDIX B
In the following, we proceed to evaluate the parts of I 1 , I 2 and I 3 , respectively. We commence from I 1 as follows where R = . Furthermore, (a) follows from this fact that ln 1+exp(x) is a convex function for x > 0 and then applying Jensen's inequality. The results in [31] express that this lower bound is sufficiently tight. Using [26, (4 Note that the averages of R and S are equal to m R = where (a) follows the fact that γ R is an exponential r.v introduced in Lemma 2 with the pdf given by (53)  The lower-bound expression of the ESSRR sec in the presence of the FJ can be obtained as where inequality (a) follows from the fact that E{max(X, Y )} ≥ max(E{X}, E{Y }) [28]. Moreover, for any positive r.v.s X and Y , the following approximation can be used [32] E ln 1 + Based on the approximation (60) and using the expression (14), the part T 1 is approximated as T 1 ≈ ln 1 + E{P S1 XY } Y N 0 + N 0 β + N 0 (P S1 Z + P S2 W )E{H} .
Finally, to evaluate E {H} where H = 1 PS 1 X+PS 2 Y as a new r.v. Let us denote R = P S1 X, S = P S2 Y as two r.v.s with exponential distribution and the means equal to µ R = P S1 µ S1R and µ S = P S2 µ S2R , respectively. Using (54), APPENDIX D For any two r.v.s X and Y , let define Z = XY . Therefore, the cdf of Z is readily obtained as Now, let X = Z + W be a r.v as stated in Lemma 2 and Y = U be an exponential r.v with mean µ RJ . After tedious manipulations and using Eq. (8.432.6) in [29], F Z (z) can be obtained. Thereafter, by considering the fact that f Z (z) = dFZ (z) dz , we reach the final expression as presented in (49). It is worth pointing out that (a) is concluded based on the lemma below.
By taking average over X, the integral expression in (67) is obtained.