Physical-Layer Security for Two-Hop Air-to-Underwater Communication Systems With Fixed-Gain Amplify-and-Forward Relaying
Yi Lou, Ruofan Sun, Julian Cheng, Songzuo Liu, Feng Zhou, Gang Qiao
11 Physical-Layer Security for Two-HopAir-to-Underwater Communication SystemsWith Fixed-Gain Amplify-and-ForwardRelaying
Yi Lou,
Member, IEEE,
Ruofan Sun,
Student Member, IEEE,
Julian Cheng,
Senior Member, IEEE,
Songzuo Liu,
Member, IEEE,
Feng Zhou,
Member, IEEE, and Gang Qiao,
Member, IEEE
Abstract
We analyze a secure two-hop mixed radio frequency (RF) and underwater wireless optical com-munication (UWOC) system using a fixed-gain amplify-and-forward (AF) relay. The UWOC channel ismodeled using a unified mixture exponential-generalized Gamma distribution to consider the combinedeffects of air bubbles and temperature gradients on transmission characteristics. Both legitimate andeavesdropping RF channels are modeled using flexible α − µ distributions. Specifically, we first deriveboth the probability density function (PDF) and cumulative distribution function (CDF) of the receivedsignal-to-noise ratio (SNR) of the mixed RF and UWOC system. Based on the PDF and CDF expressions,we derive the closed-form expressions for the tight lower bound of the secrecy outage probability (SOP)and the probability of non-zero secrecy capacity (PNZ), which are both expressed in terms bivariateFox’s H -function. To utilize these analytical expressions, we derive asymptotic expressions of SOP andPNZ using only elementary functions. Also, we use asymptotic expressions to determine the optimaltransmitting power to maximize energy efficiency. Further, we thoroughly investigate the effect of levels Yi Lou, Ruofan Sun, Songzuo Liu, Feng Zhou, and Gang Qiao are with the Acoustic Science and Technology Laboratory,Harbin Engineering University, Harbin 150001, China, and also with the Key Laboratory of Marine Information Acquisitionand Security (Harbin Engineering University), Ministry of Industry and Information Technology, Harbin 150001, China (e-mail: { louyi,sunruofan,liusongzuo,zhoufeng,qiaogang } @hrbeu.edu.cn).Julian Cheng is with the School of Engineering, The University of British Columbia, Kelowna, BC, Canada (e-mail:[email protected]). a r X i v : . [ c s . I T ] S e p of air bubbles and temperature gradients in the UWOC channel, and study nonlinear characteristicsof the transmission medium and the number of multipath clusters of the RF channel on the secrecyperformance. Finally, all analyses are validated using Monte Carlo simulation. Index Terms
Amplify-and-forward (AF), α - µ distribution, non-zero capacity (PNZ), performance analysis, un-derwater wireless optical communication (UWOC), secrecy outage probability (SOP). I. I
NTRODUCTION
The rise of the underwater Internet of Things requires the support of a high-performanceunderwater communication network with high data rates, low latency, and long communicationrange. Underwater wireless optical communication (UWOC) is one of the essential technologiesfor this communication network. Unlike radio frequency (RF) and acoustic technologies, UWOCtechnology can achieve ultra-high data rates of Gpbs over a moderate communication range whenselecting blue or green light with wavelengths located in the transmission window [1]. Further,a light-emitting diode or laser diode as a light source provides the versatility to switch betweenmaximizing the communication range or the coverage area within the constraints of the range-beamwidth tradeoff to meet the needs of a specific application scenario.Using relay technology to construct a communication system in a multi-hop fashion is one ofthe primary techniques to extend the communication range. Based on the modality of processingand forwarding signals, relays can be divided into two main categories: decode-and-forwardrelays (DF) and amplify-and-forward (AF) relays. In DF relaying systems, the relay down-converts the received signals to the baseband, decodes, re-encodes, and up-converts them tothe RF band, which are then forwarded to the destination node. In AF relaying systems, therelay amplifies the received signals directly in the passband based on the amplification factor,then forwards them directly in the RF band. Since AF scheme does not require time-consumingdecoding and spectral shifting, it can significantly reduce complexity while still providing goodperformance [2]. Depending on the different CSI information required by the AF relay, AFrelaying can be divided into the variable gain AF (VG) one and fixed-gain AF (FG) one. Ina VG scheme, the relay requires instantaneous channel state information (CSI) of the source-to-relay link, whereas in an FG scheme, only statistical CSI of the SR link is required [3].
Therefore, from an engineering standpoint, the FG scheme is more attractive because of its lowimplementation complexity.To maximize the utilization of the different transmission environments of each hop andthus improve the overall performance of the multi-hop relaying system, mixed communicationsystems using different communication technologies have been proposed. For example, the mixedcommunication system using both RF and free space optical (FSO) technologies is proposed totake advantage of the robustness of the RF links and the high bandwidth characteristics ofthe FSO links. Further, RF sub-systems offer low-cost and non-line-of-sight communicationcapabilities, while FSO sub-systems offer low transmission latency and ultra-high transmissionrates. Therefore, a mixed RF/FSO system is a cost-effective solution to the last-mile problemin wireless communication networks, where the high-bandwidth FSO sub-system of a mixedRF/FSO system is used to connect seamlessly the fiber backbone and RF sub-system accessnetworks [4]–[7]. In the past, achieving ultra-high-speed communication between underwaterand airborne nodes or land-based stations across the sea surface medium has been a challengedue to the low data rate of underwater acoustic communications. To solve this problem, usingan ocean buoy or a marine ship as a relay node, the mixed RF/UWOC system is proposed, inwhich the high-speed UWOC is used instead of underwater acoustic communication, to achievehigher overall communication rates [8]–[12].Accurate modeling of the UWOC channel, including absorption, scattering, and turbulence, isa prerequisite for proper performance analysis and algorithm development of the UWOC system[13], [14]. Absorption and scattering have been extensively studied [15]–[17], where absorptionlimits the transmission distance of underwater light, while scattering diffuses the receiving radiusof underwater light transmission and deflects the transmission path, thus reducing the receivedoptical power. Due to changes in the random refractive index variation, turbulence can causefluctuations in the received irradiance, i.e., scintillation, which can limit the performance andaffect the stability of the UWOC system [1]. In early research, UWOC turbulence was modeled byborrowing models of atmospheric turbulence, e.g., weak turbulence is modeled by the Lognormaldistribution [18]–[21], and moderate-to-strong turbulence is modeled by the Gamma-Gammadistribution [22]–[25].However, the statistical distributions used to model atmospheric turbulence cannot accu-rately characterize UWOC systems due to the fundamental differences between aqueous andatmospheric mediums. Recently, based on experimental data, the mixed exponential-lognormal distribution has been proposed to model moderate to strong UWOC turbulence in the presence ofair bubbles in both fresh water and salty water [26]. Later, the mixture exponential-generalizedGamma (EGG) distribution was proposed to model turbulence in the presence of air bubblesand temperature gradients in either fresh or salt water [27]. The EGG distribution not only canmodel turbulence of various intensities, but also has an analytically tractable mathematical form.Therefore, useful system performance metrics, such as ergodic capacity, outage probability, andBER, can be easily obtained.Due to the broadcast nature of RF signals, secrecy performance has always been one of themost important considerations for the mixed RF/FSO communication systems [5], [28]–[32]. In[29], the expressions of the lower bound of the secrecy outage probability (SOP) and averagesecrecy capacity (ASC) for mixed RF/FSO systems using VG or FG relaying schemes, were bothderived in closed-form, where the RF and FSO links are modeled by the Nakagami- m and GGdistributions, respectively. The authors in [30] used Rayleigh and GG distributions to model RFand FSO links, respectively. Considering the impact of imperfect channel state information (CSI),both the exact and asymptotic expressions of the lower bound for SOP of a mixed RF/FSO systemusing VG or FG relay are derived. The same authors then extended the analysis to multiple-input and multiple-output configuration and analyzed the impact of different transmit antennaselection schemes on the secrecy performance of the mixed RF/FSO system using a DF relay,where RF and FSO links are modeled by the Nakagami- m and M -distributions, respectively.Assuming the CSI of the FSO and RF links are imprecise and outdated, the authors derived thebound and asymptotic expressions of the effective secrecy throughput of the system. In [31],using more generalized η - µ and M -distributions to model RF and FSO links, respectively, andassuming that the eavesdropper is only at the relay location, the authors derived the analyticalresults for the SOP and the average secrecy rate of the mixed RF/FSO system using the FG orVG relaying scheme. To quantify the impact of the energy harvesting operation on the systemsecrecy performance, the authors in [5] derived exact closed-form and asymptotic expressions forthe SOP of the downlink simultaneous wireless information and power transfer system using DFrelaying scheme, under the assumption that RF and FSO links are modeled using the Nakagami- m and GG distributions, respectively.However, research on the secrecy performance of mixed RF/UWOC systems is still in itsinfancy despite the growing number of underwater communication applications. The authors in[10] investigated the secrecy performance of a two-hop mixed RF/UWOC system using a VG or FG multiple-antennas relay and maximal ratio combining scheme, where RF and UWOC linksare modeled by Nakagami- m and the mixed exponential-Gamma (EG) distributions, respectively.Assuming that only the source-to-relay link receives eavesdropping from unauthorized users, theauthors in [10] derived the exact closed expressions of the ASC and SOP of the mixed RF/UWOCsystems. Later, based on the same channel model as in [10], the same authors extended theanalysis to the mixed RF/UWOC system using a multi-antennas DF relay with the selectioncombining scheme [9]. Both the exact closed-form and asymptotic expressions of the SOP werederived.However, while the EG distribution is suitable for modeling turbulence of various intensitiesin both fresh water and salty water, this distribution fails to model the effects of air bubblesand temperature gradients on UWOC turbulence [27]. Further, the Nakagami- m distribution isonly applicable to certain specific scenarios and cannot accurately characterize the effects of theproperties of the transmission medium and multipath clusters on channel fading. It is shown thatthe impact of the medium on the signal propagation is mainly determined by the nonlinearitycharacteristics of the medium [33]. The α − µ distribution is a more general, flexible, andmathematically tractable model of channel fading whose parameters α and µ are correlated withthe nonlinearity of the propagation medium and the number of clusters of multipath transmission,respectively. Further, by setting α and µ to specific values, the α − µ distribution can be reduced toseveral classical channel fading models, including Nakagami- m , Gamma, one-sided Gaussian,Rayleigh, and Weibull distributions. Recently, the secrecy performance of a two-hop mixedRF/UWOC system using DF relaying has been analyzed in [34]; however, only the lower boundand asymptotic expressions of the SOP are derived. Furthermore, the overall end-to-end latencyof the mixed RF/UWOC communication system is increased by the decoding and forwardingand spectral shifting operations required by DF relaying.However, to the best of the authors’ knowledge, this is the first comprehensive secrecy per-formance analysis of the mixed RF/UWOC communications system using a low-complexity FGrelaying scheme. Unlike previous UWOC channel models that do not adequately characterize theunderwater optical propagation and RF channel models that use various simplifying assumptions,we model the RF channels and the UWOC channel using the more general and accurate α − µ and EGG distributions, respectively, to analyze the effects of a variety of real channel physicsphenomena, such as different temperature gradients and levels of air bubbles of UWOC channelsand different grades of medium nonlinearity, and the number of multipath clusters of the RF channels on the secrecy performance of the mixed RF/UWOC communication systems. Wepropose a novel analytical framework to derive the closed-form expressions of the SOP and thenon-zero secrecy capacity (PNZ) metrics by the bivariate Foxs H -function. Moreover, our secrecyperformance study provides a generalized framework for several fading models for both RF andUWOC channels, such as Rayleigh, Weibull for RF channels and EG and Generalized Gammafor UWOC channels. We first derive the probability density function (PDF) and cumulativedistribution function (CDF) of the end-to-end SNR for the mixed RF/UWOC communicationsystem in exact closed-form in terms of bivariate H -function. Depending on these expressions,we derive the exact closed-form expressions of the lower bound of the SOP and the PNZ.Furthermore, we also derive asymptotic expressions for both SOP and PNZ containing onlysimple functions at high SNRs. Also, based on the asymptotic expressions for SOP and PNZ,we provide a straightforward approach to determine the optimal source transmission power tomaximize energy efficiency for given performance goals of both SOP and PNZ. Finally, we useMonte Carlo simulation to validate all the derived analytical expressions and theoretical analyses.The rest of this paper is organized as follows. In Section II, the channel and system modelsare presented. In Section III, the end-to-end statistics are studied. Both exact and asymptoticexpressions for the SOP and PNZ are derived in Section IV. The numerical results and discussionsare discussed in Section V, which is followed by the conclusion in Section VI.II. S YSTEM AND C HANNEL M ODELS
A mixed RF/UWOC system is considered in Fig. 1 where the source node (S) in the air trans-mits its private data to the legitimate destination node (D) located underwater via a trusted relaynode (R), which can be a buoy or a surface ship. The RF channel from S to R and underwateroptical channel from the R to the D node is assumed to follow α − µ and EGG distributions,respectively. During transmission, one unauthorized receiver (E) attempts to eavesdrop on RFsignals received by the R. In this paper, we consider a VG AF relay where the relay amplifiesthe received signal by a fixed factor and then forwards the amplified message to the destinationnode. RS RF Links
UWOC Link D Eavesdropper RS RF Links
UWOC Link D Eavesdropper
Fig. 1. A mixed RF/UWOC two-hop communication system using an FG relaying scheme with one legitimate receiver in thepresence of eavesdropping.
A. RF Channel Model
The RF SR link is modeled by α - µ flat fading models, where the PDF of the received SNR,denoted by γ , can be expressed as f γ ( γ )= α µ ) µ µ (¯ γ ) αµ γ αµ − exp (cid:32) − µ (cid:18) γ ¯ γ (cid:19) α (cid:33) (1)where γ ≥ , µ ≥ , α ≥ , and Γ( · ) denotes the gamma function. The fading model parameters α and µ are associated with the non-linearity and multi-path propagation of the channel. Further,the PDF of the received SNR at the eavesdropping node E, denoted by f γ e ( γ e ) , also follows α - µ with parameters α e and µ e .Based on the definition of the H -function, the CDF of γ , which is defined as F γ ( γ ) = (cid:82) γ f γ ( γ ) dγ , can be expressed as F γ ( γ ) ( a ) = κ (cid:90) γ H , , γ Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) − α + µ, α (cid:1) dγ = − iκ π (cid:90) s L Λ − s Γ (cid:18) sα + µ − α (cid:19)(cid:90) γ γ − s dγds = iκ π (cid:90) s L γ − s Λ − s s − (cid:18) sα + µ − α (cid:19) ds = κ Λ H , , γ Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 , (cid:0) µ, α (cid:1) , (0 , (2)where we use [35, Eq. (1.60)] and [35, Eq. (1.125)] to express f γ ( γ ) in the right side of equity(a) into the form of H -function, where H · , ·· , · [ ·|· ] is the H -Function [35, Eq. (1.2)], κ = β Γ( µ )¯ γ , Λ = β ¯ γ , and β = Γ ( α + µ ) Γ( µ ) . Note that, the present form of F γ ( γ ) in (2) is more suitable forderiving secrecy performance of a two-hop mixed RF/UWOC than the form proposed in [36,Eq. (2)] for the point-to-point system over single-input multiple-output α − µ channels. B. UWOC channel model
To characterize the combined effects of different levels of air bubbles and temperature gradientson the light intensity received at underwater node D, we model the UWOC channel from R to Dusing the EGG distribution [27], where the PDF of the received SNR, defined as γ = ( ηI ) r /N ,has been derived in closed-form in terms of Meijer- G functions [37, Eq. (3)]. Based on [35, Eq.(1.112)], we can re-write the PDF of γ using H -functions as f γ ( γ )= c (1 − ω ) γr Γ( a ) H , , b − c (cid:18) γ µ r (cid:19) cr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( a, + ωγ r H , , λ (cid:18) γ µ r (cid:19) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 , (3)where the parameters ω , a, b and c can be estimated using the maximum likelihood criterion withexpectation maximization algorithm. The parameter ω is the mixed weight of the distribution; λ is the parameter related to the exponential distribution; parameters a , b , and c are related to theexponential distribution; r is a parameter dependent on the detection scheme, specifically, r = 1 for heterodyne detection and r = 2 for intensity modulation and direct detection [38, Eq. (31)]. The EGG distribution can provide the best fit with the measured data form laboratory watertank experiments in the presence of temperature gradients and air bubbles [37]. Therefore,by using the EGG distribution to model the UWOC link, we can gain more insight into therelationship between characteristics of the UWOC link and the secrecy performance of themixed RF/UWOC communication system.Using the definition of CCDF, i.e., ¯ F γ ( γ ) = (cid:82) γ f γ ( γ ) dγ , and an approach similar to thatused to derive (2), we can derive the CCDF of γ as ¯ F γ ( γ )= − i (1 − ω )2 π Γ( a ) (cid:90) s L Γ (cid:16) a + rsc (cid:17) b rs µ sr (cid:90) ∞ γ γ − s − dγds − iω π (cid:90) s L Γ( rs + 1) λ rs µ sr (cid:90) ∞ γ γ − s − dγds = − i (1 − ω )2 π Γ( a ) (cid:90) s L γ − s s Γ (cid:16) a + rsc (cid:17) ( b r µ r ) s ds − iω π (cid:90) s L γ − s s Γ( rs + 1) ( λ r µ r ) s ds = (1 − ω )Γ( a ) H , , b − r γµ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 , , , ( a, rc ) + rωH , , γλ − r µ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (0 , r ) . (4)It is worth to mention that the newly derived expression in (4) is useful to derive the closed-formCDF expression of the end-to-end SNR of the mixed RF/UWOC communication system.III. E ND -T O -E ND SNRIn this section, we derive the exact closed-form expressions for PDF and CDF of the end-to-end SNR of mixed RF/UWOC communication system. We then use these expressions to deriveclosed-form and asymptotic expressions for the system secrecy metrics in the following section.The end-to-end instantaneous SNR of the mixed RF/UWOC system using the FG relayingscheme is given as γ eq = γ γ γ + C (5) where C denotes the FG amplifying constant and is inversely proportional to the square ofthe relay transmitting power, and this constant is defined as C = 1 / ( G N ) , where the FGamplifying factor G is defined as G = (cid:115) P P | h | + N . (6)Using the definition of H -function, we can express G in terms of the H -functions G = κH , , Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (0 , , , ( − α + µ, α ) . (7) Theorem 1.
The CDF of the end-to-end SNR of the mixed RF/UWOC communcation systemusing the FG relaying scheme F γ eq ( γ eq ) , dened in (5) , can be obtained in exact closed-form asshown in (8) in terms of bivariate H -functions, where H · , · : · , · ; · , ·· , · : · , · ; · , · [ ·|· ] is the bivariate H -Functiondefined as [35, Eq. (2.55)]Proof. See Appendix A.Note that the current implementation of bivariate H -function for numerical computation ismature and efficient, including GPU-accelerated versions, and has been implemented using themost popular software, including MATLAB (cid:114) [39], Mathematica (cid:114) [40], and Python [41]. Also,the exact-closed expression for the CDF in (8) is a key analytical tool to derive the SOP metricof the mixed RF/UWOC system. Theorem 2.
The PDF of the end-to-end SNR, which is defined in (5) , of the mixed RF/UWOCcommunication system using the FG relaying scheme, denoted by f γ eq ( γ eq ) , can be obtained inexact closed-form as shown in (9) .Proof. See Appendix B.It is worth noting that the PDF expression in (9) is the most critical step required to evaluatethe PNZ performance metric, as will be shown in the next section.IV. PERFORMANCE METRICSThis section presents analytical expressions for the critical secrecy performance metrics of amixed RF/UWOC communication system, including both SOP and PNZ, in the presence of airbubbles and temperature gradients in the UWOC channel and medium nonlinearity in the RFchannel. F γ eq ( γ eq )=1 − γκ (1 − ω )Γ( a ) H , , , , , , b − r Cµ r γ Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 , ,
1) : ; (cid:0) α − µ, α (cid:1) : (0 , , ( a, rc ) ; (1 , − γκrωH , , , , , , Cλ − r µ r γ Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 , ,
1) : ; (cid:0) α − µ, α (cid:1) : (1 , , (0 , r ) ; (1 , . (8) f γ eq ( γ eq )= κ (1 − ω )Γ( a ) H , , , , , , γ Λ b − r Cµ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 , ,
1) : (cid:0) α − µ, α (cid:1) ;: (2 ,
1) ; (0 , , ( a, rc ) + κrωH , , , , , , γ Λ Cλ − r µ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 , ,
1) : (cid:0) α − µ, α (cid:1) ;: (2 ,
1) ; (1 , , (0 , r ) . (9)SOP L =1 − Θ κ (1 − ω ) κ e Γ( a )Λ e H , , , , , , Λ e ΘΛ b − r Cµ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 , ,
1) : (cid:0) α − µ, α (cid:1) ;: (cid:16) α e µ e α e , α e (cid:17) , (1 ,
1) ; (0 , , ( a, rc ) − Θ κrωκ e Λ e H , , , , , , Λ e ΘΛ Cλ − r µ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 , ,
1) : (cid:0) α − µ, α (cid:1) ;: (cid:16) α e µ e α e , α e (cid:17) , (1 ,
1) ; (1 , , (0 , r ) . (10)SOP a = 1 − κ (1 − ω ) κ e ΛΓ( a )Λ e H , , b r Λ e µ r C ΘΛ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 , , (1 − a, rc ) , (1 − µ, α ) (cid:16) µ e , α e (cid:17) , (0 , − κrωκ e ΛΛ e H , , λ r Λ e µ r C ΘΛ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 , r ) , (1 − µ, α ) (cid:16) µ e , α e (cid:17) . (11)SOP ae =1 − (1 − ω ) α e Γ( a )Γ( µ )Γ ( µ e ) Γ ( α e µ e + 1) Γ (cid:16) µ + α e µ e α (cid:17) (cid:18) Λ e ΘΛ (cid:19) α e µ e H , , b r µ r C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 , , (1 − a, rc )( α e µ e , − rωα e Γ( µ )Γ ( µ e ) Γ ( α e µ e + 1) Γ (cid:16) µ + α e µ e α (cid:17) (cid:18) Λ e ΘΛ (cid:19) α e µ e H , , λ r µ r C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (0 , , (1 , r )( α e µ e , . (12) A. SOP
SOP is defined as the probability that the secrecy capacity C s falls below a target rate ofcondential information R s and it can be expressed as SOP ( R s )=Pr (cid:26) log (cid:18) γ eq γ e (cid:19) < R s (cid:27) =Pr { γ eq ≤ Θ γ e + Θ − } = (cid:90) ∞ F eq (Θ γ e + Θ − f e ( γ e ) dγ e (13)where Θ = e R s .
1) Lower bound:
Referring to [42], [43], a tight lower bound for the SOP can be given asderived as
SOP L = (cid:90) ∞ F γ eq (Θ γ ) f γ e ( γ ) dγ. (14) Theorem 3.
The lower bound for the SOP of the mixed RF/UWOC communication system usingthe FG relaying scheme dened in (14) can be obtained in exact closed-form as shown in (10) .Proof.
See Appendix C.
2) Asymptotic results:
To gain more insight into the SOP performance and the dependencybetween the link quality of both RF and UWOC channels, we now derive asymptotic expressionsfor SOP. We consider two scenarios, namely γ → ∞ and γ e → ∞ . Corollary 3.1.
For scenarios γ → ∞ and γ e → ∞ , the asymptotic expressions of SOP of amixed RF/UWOC communication system using FG relaying scheme can be given as (11) and (12) in terms of H -functions, respectively.Proof. See Appendix D.Note that in contrast to the closed expression of the lower bound of the SOP in (10) interms of bivariate H -functions, which requires numerical evaluation of double line integrals,the asymptotic expressions in (11) and (12) only require numerical calculation of single lineintegrals, thus reducing the complexity of the calculations. Furthermore, as shown in SectionV, for a target SOP performance, the asymptotic expressions in (11) and (12) can be used todetermine rapidly the optimal transmitting power to maximize energy efficiency.In the special case of a two-hop mixed RF/UWOC communication system over RayleighRF links and a thermally uniform UWOC channel, we can further simplify the asymptotic expressions in (11) and (12) by setting c = 1 , α = α e = 1 , µ = µ e = 1 . For example, eq. (11)can be simplified into SOP a = κκ e λ ΛΛ e µ r − aλ (1 − ω )Λ e µ r exp (cid:18) C ΘΛ b Λ e µ r (cid:19) × E a +1 (cid:18) C ΘΛ b Λ e µ r (cid:19) − C ΘΛ ω exp (cid:18) C ΘΛ λ Λ e µ r (cid:19) × E i (cid:18) − C ΘΛ λ Λ e µ r (cid:19) + ΛΛ e − κωκ e ΛΛ e (15)where E i ( x ) and E n ( x ) both denote the exponential integral [44, Eq. (8.211.1)] B. PNZ
PNZ is another critical metric for to evaluate the secrecy performance of a communicationsystem, which is defined as
Pr ( C s > , where C s is the secrecy capacity. PNZ is generallyrelated to channel conditions of all the channels in the mixed RF/UWOC systems. In this section,we derive the exact closed-form and asymptotic expressions for PNZ and analyze the relationshipbetween channel parameters and PNZ performance.
1) Exact results:
According to [43], PNZ can be reformed as P nz = Pr ( γ eq > γ e ) = (cid:90) ∞ f eq ( γ eq ) F e ( γ eq ) dγ eq . (16) Theorem 4.
The exact PNZ of the mixed RF/UWOC communication system using the FG relayingscheme dened in (16) can be obtained in exact closed-form as shown in (17) .Proof.
See Appendix E.
2) Asymptotic results:
To gain more insight into the PNZ performance and the dependencybetween the link quality of both RF and UWOC channels, we now derive asymptotic expressionsfor PNZ. We consider two scenarios, namely γ → ∞ and γ e → ∞ . Corollary 4.1.
For scenarios γ → ∞ and γ e → ∞ , the asymptotic expressions of PNZ of amixed RF/UWOC communication system using the FG relaying scheme are given as (18) and (19) in terms of H -functions, respectively.Proof. Observing that the expressions for the lower bound of the SOP in (10) and exact PNZin (17) have a similar structure; therefore, eqs. (18) and (19) can be easily obtained using thesame techniques as those used for deriving (11) and (12), and the proof is complete. Note that, similar to the asymptotic expressions of the SOP in (11) and (12), for a targetPNZ performance, the asymptotic expressions of PNZ in (18) and (19) are also suitable forfast numerical calculations and useful to determine the optimal transmitting power to maximizeenergy efficiency. P nz = κ (1 − ω ) κ e Γ( a )Λ e H , , , , , , b − r Cµ r Λ e Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 , ,
1) : ; (cid:0) α − µ, α (cid:1) : (0 , , ( a, rc ) ; (cid:16) α e µ e α e , α e (cid:17) , (1 , + κrωκ e Λ e H , , , , , , Cλ − r µ r Λ e Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 , ,
1) : ; (cid:0) α − µ, α (cid:1) : (1 , , (0 , r ) ; (cid:16) α e µ e α e , α e (cid:17) , (1 , . (17) P nza = κrωκ e ΛΛ e H , , λ r Λ e µ r C Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 , r ) , (1 − µ, α ) (cid:16) µ e , α e (cid:17) + κ (1 − ω ) κ e ΛΓ( a )Λ e H , , b r Λ e µ r C Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 , , (1 − a, rc ) , (1 − µ, α ) (cid:16) µ e , α e (cid:17) , (0 , . (18) P nzae = κrωα e κ e Λ − α e µ e − Λ α e µ e − e Γ (cid:0) µ + α e µ e α (cid:1) Γ ( α e µ e + 1) H , , λ r µ r C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (0 , , (1 , r )( α e µ e , + κ (1 − ω ) α e κ e Λ − α e µ e − Λ α e µ e − e Γ (cid:0) µ + α e µ e α (cid:1) Γ( a )Γ ( α e µ e + 1) H , , b r µ r C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 , , (1 − a, rc )( α e µ e , . (19)V. N UMERICAL R ESULTS A ND D ISCUSSION
In this section, we provide some numerical results to verify the analytic and asymptoticexpressions of SOP and PNZ derived in Section IV, and thoroughly investigate the combinedeffect of the channel quality of both RF and UWOC channels on the secrecy performance of thetwo-hop mixed RF/UWOC communication system. All practical environmental physical factorsthat can affect channel quality, including levels of air bubbles, temperature gradients, and salinityof the UWOC channel, as well as the medium nonlinearity and multipath cluster characteristicsof the RF channel, are taken into account. For brevity, we use [ · , · ] to denote the value set of [ air bubbles level , temperature gradient ] in this section.In Fig. 2 – Fig. 6, we investigate the combined effect of the channel quality of both RF andUWOC channels on the SOP metric of the two-hop mixed RF/UWOC communication system. S O P Fig. 2. SOP versus ¯ γ with various fading parameters when α = α e = 1 . , µ = µ e = 1 . , R s = 0 . , and ¯ γ e = ¯ γ = 10 dB S O P Fig. 3. SOP versus ¯ γ with various fading parameters when α = α e = 1 . , µ = µ e = 1 . , R s = 0 . , and ¯ γ = 0 dB Figure 2 shows the lower bound and the asymptotic SOP with average SNR of the SR link γ for a mixed two-hop RF/UWOC system under different quality scenarios of UWOC channel.Both RF SR and SE links follow the α - µ distribution and have the same parameters, where α = α e = 1 . , µ = µ e = 1 . . The average SNR of the SE and RD links are both set as ¯ γ = ¯ γ e = 10 dB. As shown in Fig. 2, the exact theoretical results are almost identical to thesimulation results, and both closely agree with the derived lower bound. Asymptotic results are S O P Fig. 4. SOP versus ¯ γ with various fading parameters R s = 0 . , ¯ γ = ¯ γ e = 10 dB, and UWOC channel parameter is [2.4,0.05] S O P Fig. 5. SOP versus ¯ γ e with various fading parameters when α = α e = 1 . , µ = µ e = 1 . , R s = 0 . , and ¯ γ = ¯ γ = 10 dB tight when the average SNR is greater than 30 dB. Further, when the average SNR increases from0 to 30 dB, SOP rapidly decreases. Also, SOP tends to saturate when the average SNR is between30 and 40 dB. Given the cost of the relay battery replacement and engineering difficulties, thecommunication system should guarantee the SOP while cutting down on energy consumption.In practice, one should therefore select the optimal transmission power corresponding to thesaturation starting point.Figure 3 depicts the SOP variation versus the SR average SNR γ for the mixed two-hop S O P Fig. 6. SOP versus ¯ γ e with various fading parameters when α = α e = 1 . , µ = µ e = 1 . , R s = 0 . , and ¯ γ = 20 dB RF/UWOC system under three different eavesdropper interference levels, i.e., ¯ γ e = 3 , , − dB.Parameters in Fig. 3 are set as follows: α = α e = 1 . , µ = µ e = 1 . , UWOC channel parameteris [2.4,0.05], and ¯ γ = 0 dB. It can be observed that the lower bounds closely match the exactresults in the whole SNR region. The asymptotic result curve gradually coincides with the exactresult curve when ¯ γ takes higher values starting from 20 dB. We can also observe that theSOP is monotonically increasing with ¯ γ , assuming that the SNR of the SE link is a fixedvalue. Holding the other parameters constant, the larger the ¯ γ e , the lower the system SOP. Inshort, as the quality of the eavesdropping channel improves, the SOP performance of the systemdeteriorates.Figure 4 indicates the effect of the variation in average SNR of the SR link on the SOPmetric of a two-hop mixed RF/UWOC, with three different RF channel qualities. Evidently,SOP monotonically decreases with the increase of ¯ γ , and SOP tends to saturate when ¯ γ ≥ α − µ value increases, the two-hop mixed RF/UWOCsystem secrecy performance worsens, and vice versa. This is because of the phenomena ofsevere nonlinearity and sparse clustering when the signals are propagating in a high α − µ value RF channel, and poor RF channel quality makes it easier for eavesdroppers to interceptsignals. As shown in Fig. 5, as the ¯ γ e progressively increases, the SOP value increases, theinformation intercepted by the eavesdropper increases, and the system’s secrecy performancegradually decreases. Moreover, the asymptotic result is more accurate at ¯ γ e greater than 15 dB. In Fig. 6, we set the same channel parameters as in Fig. 3, except for setting the UWOC averageSNR, i.e., ¯ γ = 20 dB. Fig. 6 shows that SOP increases with ¯ γ e when the other parameters remainunchanged. The same interpretation of Fig. 5 can also be applied to Fig. 6. Additionally, therate at which the asymptotic results approach exact results varies for different SR average SNR.For ¯ γ = 20 dB, the asymptotic results begin to match the exact result starting at ¯ γ e = 5 dB.Besides, the close match of the lower bound and the exact results demonstrates the robustnessand accuracy of (10).In Fig. 7 – Fig. 10, We investigate the combined effect of the channel quality of both RF andUWOC channels on the PNZ metric of the two-hop mixed RF/UWOC communication system. Fig. 7. P nz versus ¯ γ with various fading parameters when α = α e = 2 . , µ = µ e = 1 . and ¯ γ e = ¯ γ = 0 dB Figure 7 shows the effect of the SR link average SNR ¯ γ on the PNZ of the mixed RF/UWOCfor different UWOC channel parameters. PNZ increases incrementally as ¯ γ increases, whichindicates an increase in secrecy performance. It can be observed that PNZ decreases as the degreeof turbulence increases, i.e., the higher the level of air bubbles and the larger the temperaturegradient, the worse the secrecy performance in the system. Additionally, we depict the effects ofsalinity on UWOC performance in Fig. 7. The salinity affects the system secrecy performanceto a much lesser extent than the level of air bubble and temperature gradient. This is becausethe generation and break-up of the air bubbles in the UWOC channels causes dramatic andrandom fluctuations of the underwater optical signals, which can significantly deteriorate thesecrecy performance of the system. Fig. 7 shows that eavesdroppers may benefit from a low Fig. 8. P nz versus ¯ γ with various fading parameters when α = α e = 1 . , µ = µ e = 0 . and ¯ γ = 0 dB Fig. 9. P nz versus ¯ γ e with various fading parameters when α = α e = 2 . , µ = µ e = 1 . and ¯ γ = ¯ γ = 20 dB UWOC channel quality. On the contrary, in a high quality UWOC channel, the likelihood of aneavesdropper successfully eavesdropping is greatly reduced. Therefore, in practical applications,increasing the channel quality can increase the system transmission capacity and thus improvethe system secrecy performance. Fig. 7 also shows that asymptotic results can quickly approachthe exact result for poorer channels. For example, for a UWOC channel with channel parametersof [16.5,0], the asymptotic result can achieve a match with the exact value at ¯ γ ≥ dB. Whenthe channel parameter set is [2.4, 0.05], the asymptotic result can only be accurate at ¯ γ > Fig. 10. P nz versus ¯ γ e with various fading parameters when α = α e = 1 . , µ = µ e = 0 . and ¯ γ = 20 dB dB. The remaining parameters are set as follows, ¯ γ e = ¯ γ = 0 dB, α = α e = 2 . , µ = µ e = 1 . .In Fig. 8, the RF channel parameters are α = α e = 1 . , µ = µ e = 0 . , and the UWOCchannel parameters are [2.4,0.05]. We can explain the curves in Fig. 8 using a principle similarto Fig. 7. Furthermore, in the case where ¯ γ remains unchanged, the smaller the ¯ γ e , the worse isthe quality of the eavesdropping channel; therefore, leading to a PNZ performance degradation.In addition to Fig. 8, we analyzed the effect of the average SNR ¯ γ on the PNZ, as shown byFig. 9 and Fig.10. The difference is that in Fig. 9 α = α e = 2 . , µ = µ e = 1 . and ¯ γ = ¯ γ = 20 dB. whereas the RF channel parameters in Fig. 10 are α = α e = 1 . and µ = µ e = 0 . . It canbe inferred from Fig. 9 and Fig. 10 that the asymptotic result only matches the exact value when ¯ γ e is large, and the PNZ gradually decreases until it reaches zero.VI. C ONCLUSION
We investigated the secrecy performance of a two-hop mixed RF/UWOC communicationsystem using fixed-gain AF relaying. To allow the results to be more generic and applicableto more realistic physical scenarios, we model RF channels using the α - µ distribution, whichconsiders both the nonlinear of the transmission medium and multipath cluster characteristics, andmodel UWOC channels using the laboratory EGG distribution, which can account for differentlevels of air bubbles, temperature gradients, and salinity. Closed-form expressions for the PDFand the CDF of the two-hop end-to-end SNR were both derived in terms of the bivariate H - function. Based on these results, we obtained a tight closed-form expression of the lower boundof the SOP and the exact closed-form expression of the PNZ. Furthermore, we also derivedasymptotic expressions in simple functions for both SOP and PNZ to allow rapid numericalevaluation. Moreover, based on the asymptotic results, we presented an approach to determinethe optimal transmitting power to maximize the energy efficiency, for given target performanceof both SOP and PNZ. We fully investigated the effects of various existing phenomena of bothRF and UWOC channels on the secrecy performance of the mixed RF/UWOC communicationsystem. Also, our generalized theoretical framework is also applicable to various classical RF andunderwater optical channel models including Rayleigh and Nakagami for RF channels and EGand Generalized Gamma for UWOC channels. Our results can be used in practical mixed securityRF/UWOC communication systems design. The interesting topics for future work include: (i)to investigate the secrecy performance of a mixed RF/UWOC communication system using aenergy-harvesting enabled relay with the aim of improving the system life time; (ii) to investigatethe secrecy performance of a mixed RF/UWOC communication system using multiple relays withappropriate relaying selection algorithms.A PPENDIX AP ROOF OF
THEOREM 1Using (5), we write the CDF of the end-to-end SNR in the following form F γ eq ( γ eq )= (cid:90) ∞ Pr (cid:20) γ γ γ + C ≤ γ | γ (cid:21) f γ ( γ ) dγ =1 − (cid:90) ∞ γ ¯ F γ (cid:18) Cγx − γ (cid:19) f γ ( x ) dx. (A.1)Substituting (1) and (4) into (A.1) and replacing the integral variable x with z = x + γ , aftersome simplifications, we can express (A.1) as F γ eq ( γ eq ) = 1 + I + I (A.2)where I = − κ (1 − ω )Γ( a ) (cid:90) ∞ H , , ( z + γ )Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) − α + µ, α (cid:1) × H , , b − r Cγzµ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 , , , ( a, rc ) dz (A.3) and I = − κrω (cid:90) ∞ H , , Cγλ − r zµ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (0 , r ) × H , , ( z + γ )Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) − α + µ, α (cid:1) dz. (A.4)To solve (A.3), we convert all the H -functions in (A.3) into a line integral, and place theintegral with respect to x in the innermost part by rearranging the order of multiple integrals.Then, we have I = κ (1 − ω )4 π Γ( a ) (cid:90) t L Γ( t )Γ( t + 1) Γ (cid:18) a + rtc (cid:19) (cid:18) b r µ r Cγ (cid:19) t × (cid:90) s L Λ − s Γ (cid:18) sα + µ − α (cid:19)(cid:90) ∞ z t ( z + γ ) − s dzdsdt. (A.5)By utilizing [44, Eq. (3.197/1)] to solve the integration of z , after some simplifications andusing the definition of the bivariate H -function [35, Eq. (2.57)], we can finally express I in(A.3) in the following form I = − γκ (1 − ω )Γ( a ) × H , , , , , , b − r Cµ r γ Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 , , (cid:0) α − µ, α (cid:1) :(0 , , ( a, rc ); (1 , . (A.6)We can solve (A.4) in a similar way as we have solved (A.3). All H -functions are convertedto the form of the line integrals and by rearranging the multiple integrals, the integral regarding z is placed in the innermost part of the expression. Then, we have I = κrω π (cid:90) s L Γ( rs ) (cid:18) λ r µ r Cγ (cid:19) s (cid:90) t L Λ − t Γ (cid:18) tα + µ − α (cid:19) × (cid:90) ∞ z s ( z + γ ) − t dzdtds. (A.7)Again, we use [44, Eq. (3.197/1)] to solve the integration regarding z . Then use [35, Eq.(2.57)] and some simplification, we obtain the following expression I = − γκrω × H , , , , , , Cλ − r µ r γ Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 , , (cid:0) α − µ, α (cid:1) :(1 , , (0 , r ); (1 , . (A.8) Substituting (A.6) and (A.8) into (A.2), we obtian the exact closed-form expression for theCDF as shown by (8). A
PPENDIX BP ROOF OF
THEOREM 2The PDF of the end-to-end SNR can be obtained by using f ( γ eq ) = dF ( γ eq ) dγ eq . (B.1)Substituting (8) into (B.1), after some simplifications, we have f γ eq ( γ eq )= d J d γ eq + d J d γ eq (B.2)where J = γ eq κ (1 − ω )4 π Γ( a ) (cid:90) t L (cid:90) s L t ) (cid:18) γ eq Λ (cid:19) t Γ( − s )Γ (cid:16) a − rsc (cid:17) × Γ( s + t − (cid:18) tα + µ − α (cid:19) (cid:18) b − r Cµ r (cid:19) s dsdt (B.3)and J = γ eq κrω π (cid:90) t L (cid:90) s L t ) (cid:18) γ eq Λ (cid:19) t Γ(1 − s )Γ( − rs )Γ( s + t − × Γ (cid:18) tα + µ − α (cid:19) (cid:18) Cλ − r µ r (cid:19) s dsdt. (B.4)By enabling the differential operation in (B.2), after some rearrangements, we can representthe first and the second terms on the right of the equation (B.2) as d J d γ eq = κ (1 − ω )4 π Γ( a ) (cid:90) t L (cid:90) s L (1 − t )Γ( t ) C s Γ( − s )( γ eq Λ) − t b − rs µ − sr × Γ( s + t − (cid:18) tα + µ − α (cid:19) Γ (cid:16) a − rsc (cid:17) dsdt (B.5)and d J d γ eq = κrω π (cid:90) t L (cid:90) s L t ) (1 − t ) C s Γ(1 − s )( γ eq Λ) − t λ − rs × µ − sr Γ( − rs )Γ( s + t − (cid:18) tα + µ − α (cid:19) dsdt, (B.6)respectively.After substitute (B.5) and (B.6) to (B.2), and use the definition of bivariate H -function, wecan derive the exact closed-form expression of PDF as shown in (9). A PPENDIX CP ROOF OF
THEOREM 3Substituting (1) and (8) into (14), after some rearrangements, we haveSOP L = 1 + Q + Q (C.1)where Q = − Θ κ (1 − ω ) κ e Γ( a ) (cid:90) ∞ γH , , γ Λ e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) − α e + µ e , α e (cid:17) × H , , , , , , b − r Cµ r γ ΘΛ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 , , (cid:0) α − µ, α (cid:1) :(0 , , ( a, rc ); (1 , dγ (C.2)and Q = − rγ Θ κωκ e (cid:90) ∞ γH , , γ Λ e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) − α e + µ e , α e (cid:17) × H , , , , , , Cλ − r µ r γ ΘΛ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 , , (cid:0) α − µ, α (cid:1) :(1 , , (0 , r ); (1 , dγ. (C.3)To simplify (C.2) further, we first express the bivariate H -functions in (C.2) into the form ofa double line integral, and then place the curve integral regarding γ to the innermost level byrearranging (C.2), we have Q = Θ κ (1 − ω ) κ e π Γ( a ) (cid:90) t L (ΘΛ) − t Γ( t ) Γ (cid:18) tα + µ − α (cid:19) × (cid:90) s L Γ( − s )Γ (cid:16) a − rsc (cid:17) Γ( s + t − (cid:18) b − r Cµ r (cid:19) s × (cid:90) ∞ γ − t H , , γ Λ e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) − α e + µ e , α e (cid:17) dγdsdt. (C.4)Then, using [45, Eq. 2.25.2/1], we can transform (C.4) into Q = Θ κ (1 − ω ) κ e π Γ( a ) (cid:90) t L (ΘΛ) − t Γ( t ) Γ (cid:18) tα + µ − α (cid:19) × (cid:90) s L Γ( − s )Γ (cid:16) a − rsc (cid:17) Γ( s + t − × Γ (cid:18) − tα e + µ e − α e (cid:19) Λ t − e (cid:18) b − r Cµ r (cid:19) s dsdt. (C.5) Finally, converting the double curve integral into bivariate H -function using [35, Eq. (2.57)],after some simplifications, we obtain from (C.5) in an exact closed-form as Q = − Θ κ (1 − ω ) κ e Γ( a )Λ e × H , , , , , , Λ e ΘΛ b − r Cµ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 , , (cid:0) α − µ, α (cid:1) ;: (cid:16) α e µ e α e , α e (cid:17) , (1 , , , ( a, rc ) . (C.6)To process (C.3) further, we first convert the bivariate H -function in (C.3) into the form ofone double curve integral using [35, Eq. (2.55)]. After placing the line integral of γ into theinnermost layer, we can transform (C.3) into Q = Θ κrωκ e π (cid:90) t L (ΘΛ) − t Γ( t ) Γ (cid:18) tα + µ − α (cid:19) × (cid:90) s L Γ(1 − s )Γ( − rs )Γ( s + t − (cid:18) Cλ − r µ r (cid:19) s × (cid:90) ∞ γ − t H , , γ Λ e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) − α e + µ e , α e (cid:17) dγdsdt. (C.7)Subsequently, using [45, Eq. 2.25.2/1], we express the innermost curve integral in (C.7) inthe form of the product of Gamma functions. Then, we can write (C.7) as Q = Θ κrωκ e π (cid:90) t L (ΘΛ) − t Γ( t ) Γ (cid:18) tα + µ − α (cid:19) × (cid:90) s L Γ(1 − s )Γ( − rs )Γ( s + t − × Γ (cid:18) − tα e + µ e − α e (cid:19) Λ t − e (cid:18) Cλ − r µ r (cid:19) s dsdt. (C.8)Subsequently, based on the same steps as for the derivation of (C.6), eq. (C.8) can be expressedin exact closed-form as Q = − Θ κrωκ e Λ e × H , , , , , , Λ e ΘΛ Cλ − r µ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 , , (cid:0) α − µ, α (cid:1) ;: (cid:16) α e µ e α e , α e (cid:17) , (1 , , , (0 , r ) . (C.9)After substituting (C.6) and (C.9) into (C.1), we can finally obtain the closed-form expressionof SOP L in (10). A PPENDIX DP ROOF OF C OROLLARY H -function on the right-hand side of (10), which are denotedby O and O , respectively. We consider two cases: (a) γ → ∞ and (b) γ e → ∞ . A. Case γ → ∞ For the case γ → ∞ , we first focus on deriving asymptotic expression for O . Observe thatas γ tends to infinity, θ ΛΛ e tends to zero. Thus, we first express the bivariate H -function in theform of one double curve integral, and express the curve integral containing θ ΛΛ e in the form ofan H - function. Then, we have O = i Θ κ (1 − ω ) κ e π Γ( a )Λ e (cid:90) t L Γ( − t )Γ (cid:18) a − rtc (cid:19) (cid:18) b − r Cµ r (cid:19) t × H , , ΘΛΛ e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) − α e − µ e , α e (cid:17) , (0 , − t, , ( − α + µ, α ) dt. (D.1)It is easy to observe that the H -function in (D.1) contains two poles: (1 − t ) and (1 − αµ ) .According to [41], when the argument tends to zero, the asymptotic value of the H -function canbe expressed as the residue of the closest pole to the left of the integration path l . Therefore,by utilizing [46, Eq. (1.8.4)], we can express (D.1) as O = iκ (1 − ω ) κ e π ΛΓ( a )Λ e (cid:90) t L Γ( − t )Γ(1 − t ) Γ (cid:18) a − rtc (cid:19) Γ (cid:18) µ − tα (cid:19) × Γ (cid:18) tα e + µ e (cid:19) (cid:18) b − r C ΘΛΛ e µ r (cid:19) t dt. (D.2)Following some simplifications, and using the definition of the H -function, we can transform(D.2) into the following form O = − κ (1 − ω ) κ e ΛΓ( a )Λ e H , , b r Λ e µ r C ΘΛ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 , , (1 − a, rc ) , (1 − µ, α ) (cid:16) µ e , α e (cid:17) , (0 , . (D.3) Next, we derive the asymptotic expression for O . Observing that O and O have a similarstructure, we can readily transform O into the following form O = − i Θ κrωκ e π Λ e (cid:90) t L Γ(1 − t )Γ( − rt ) (cid:18) Cλ − r µ r (cid:19) t × H , , ΘΛΛ e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) − α e − µ e , α e (cid:17) , (0 , − t, , ( − α + µ, α ) dt. (D.4)Similarly, we again use the residue of the pole (1 − t ) to represent the asymptotic value ofthe H -function in (D.4) as the argument tends to zero. Then, we have O = iκrωκ e π ΛΛ e (cid:90) t L Γ( − rt )Γ (cid:18) µ − tα (cid:19) Γ (cid:18) tα e + µ e (cid:19) × (cid:18) C Θ λ − r ΛΛ e µ r (cid:19) t dt. (D.5)By using the definition of the H -function, we can transform (D.5) into the following form O = − κrωκ e ΛΛ e H , , λ r Λ e µ r C ΘΛ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 , r ) , (1 − µ, α ) (cid:16) µ e , α e (cid:17) . (D.6)Substituting (D.3) and (D.6) into (10), we obtain the asymptotic expression for SOP for thecase γ → ∞ as shown in (11). B. Case γ e → ∞ Now, we focus on the case γ e → ∞ . Obviously, as γ e tends to infinity, θ ΛΛ e tends to infinity.Thus, using [46, Eq. (1.5.9)] and a similar approach to that used in case γ → ∞ , we can easilyobtain closed-form expressions for O and O for case γ e → ∞ , as O = − (1 − ω ) α e Γ( a )Γ( µ )Γ ( µ e ) Γ ( α e µ e + 1) Γ (cid:16) µ + α e µ e α (cid:17) × (cid:18) Λ e ΘΛ (cid:19) α e µ e H , , b r µ r C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 , , (1 − a, rc )( α e µ e , and O = − rωα e Γ( µ )Γ ( µ e ) Γ ( α e µ e + 1) Γ (cid:16) µ + α e µ e α (cid:17) × (cid:18) Λ e ΘΛ (cid:19) α e µ e H , , λ r µ r C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (0 , , (1 , r )( α e µ e , respectively.Substituting (D.7) and (D.7) into (10), we obtain the asymptotic expression for SOP for thecase γ e → ∞ as shown in (12). A PPENDIX EP ROOF OF
THEOREM 4Substituting (2) and (9) into (16), after some simplifications, we can transform the PNZexpression in (16) to P nz = T + T (E.1)where T = (cid:90) ∞ κ (1 − ω ) κ e Γ( a )Λ e H , , γ Λ e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 , (cid:16) µ e , α e (cid:17) , (0 , × H , , , , , , γ Λ b − r Cµ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 , , (cid:0) α − µ, α (cid:1) ;: (2 ,
1) ;(0 , , ( a, rc ) dγ (E.2)and T = (cid:90) ∞ rκωκ e Λ e H , , γ Λ e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 , (cid:16) µ e , α e (cid:17) , (0 , × H , , , , , , γ Λ Cλ − r µ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 , , (cid:0) α − µ, α (cid:1) ;: (2 ,
1) ;(1 , , (0 , r ) dγ. (E.3)Representing the bivariate H -function into the form of one double line integral and movingthe line integral regarding γ to the innermost level, we can re-write (E.2) as T = − κ (1 − ω ) κ e π Γ( a )Λ e (cid:90) t L Γ( − t )Γ (cid:18) a − rtc (cid:19) (cid:18) b − r Cµ r (cid:19) t × (cid:90) s L Λ − s Γ( s −
1) Γ( s + t − (cid:18) sα + µ − α (cid:19) × (cid:90) ∞ γ − s H , , γ Λ e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 , (cid:16) µ e , α e (cid:17) , (0 , dγdsdt. (E.4) Afterwards, using the same technique as that used for deducing (C.6) and (C.9), we canexpress (E.4) as T = κ (1 − ω ) κ e Γ( a )Λ e × H , , , , , , b − r Cµ r Λ e Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 , , (cid:0) α − µ, α (cid:1) :(0 , , ( a, rc ); (cid:16) α e µ e α e , α e (cid:17) , (1 , . (E.5)Similarly, T in (E.3) can be transformed into T = − κrωκ e π Λ e (cid:90) t L Γ(1 − t )Γ( − rt ) (cid:18) Cλ − r µ r (cid:19) t × (cid:90) s L Λ − s Γ( s −
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