1

On Effective Secrecy Throughput of UnderlaySpectrum Sharing α− µ/ Malaga Hybrid Model

under Interference-and-Transmit Power ConstraintsMD. IBRAHIM1, A. S. M. BADRUDDUZA2, MD. SHAKHAWAT HOSSEN3, M. K. KUNDU4, AND

IMRAN SHAFIQUE ANSARI5

1Department of Electrical & Electronic Engineering, Rajshahi University of Engineering & Technology (RUET), Rajshahi-6204, Bangladesh2,3Department of Electronics & Telecommunication Engineering, RUET

4Department of Electrical & Computer Engineering, RUET5James Watt School of Engineering, University of Glasgow, Glasgow G12 8QQ, United Kingdom

Abstract: The underlay cognitive radio-based hybrid radio frequency / free-space optical (RF / FSO) systems have beenemerged as a promising technology due to its ability to eliminate spectrum scarcity and spectrum under-utilization problems.Consequently, this work analyzes the physical layer security aspects of a cognitive RF / FSO hybrid network that includesa primary user, a secondary source, a secondary receiver, and an eavesdropper where the secret communication takes placebetween two legitimate secondary peers over the RF and FSO links simultaneously, and the eavesdropper can overhear the RFlink only. In particular, the maximum transmit power limitation at the secondary user as well as the permissible interferencepower restriction at the primary user are also taken into consideration. All the RF links are modeled with α-µ fading whereas theFSO link undergoes Malaga (M) turbulence with link blockage and pointing error impairments. At the receiver, the selectioncombining diversity technique is utilized to select the signal with the best electrical signal-to-ratio (SNR). Moreover, theclosed-form expressions for the secrecy outage probability, probability of strictly positive secrecy capacity, and effective secrecythroughput are derived to analyze the secrecy performance. Besides, the impacts of fading, primary-secondary interference,detection techniques, link blockage probability, atmospheric turbulence, and pointing error are examined. Finally, Monte-Carlosimulations are performed to corroborate the derived expressions.

Keywords: Cognitive underlay network, effective secrecy throughput, hybrid RF / FSO system, Malaga turbulence, pointingerror, secure outage probability.

I. INTRODUCTION

A. Background and Related Works

With the rapid increase in wireless devices, the shortageof spectrum owing to the data traffic reflects a significantthreat in contemporary technology [1], [2]. In order to meetthe requirements, several optimistic technologies have beendeveloped recently. Among these, both cognitive radio network(CRN) and free-space optical (FSO) systems have receivedconsiderable attention from the research communities [3], [4].Although FSO communication appears with specific merits,such as high bandwidth, inherent security, and cost-effectiveoperation, several issues like atmospheric turbulence andpointing error can affect the secure transmission greatly [5].Furthermore, the probability of transmission blockage due toextreme line-of-site (LOS) necessity in FSO link leads tothe system failure [6]. Hence, utilizing radio frequency (RF)technology as an alternative option along with the FSO systemassures successful transmission since the FSO link delivershigh data speed while the RF link is independent of theatmospheric turbulence and weather conditions [7].

Due to the increased demand for faster data rates in wirelesscommunication, hybrid networks, which consist of RF andFSO systems, are deployed utilizing a variety of techniques.

An adaptive coding method in order to enhance the systemperformance of RF / FSO hybrid network was consideredin [8], [9]. Nonetheless, the performance of these systems ishighly dependent on the existence of feedback information atthe receiver [9]. Another attractive feature is the switchingmechanism that uses RF link when FSO link is down [10].Since one link is underutilized at a time, there is a significantamount of bandwidth wasted [11]. To extend the communi-cation range, several existing works like [12]–[15] utilize re-laying protocols in mixed RF-FSO models. However, weatherconditions (i.e. fog) can influence the performance of dual-hopnetworks badly. Unlike [10], [11], the considered hybrid RF /FSO model dispatches identical information via two separatelinks simultaneously and combines the received signal usingthe diversity combining method. Again, the mixed RF-FSOsystem differs completely from the hybrid RF / FSO modelin terms of system configuration. Recently, a comprehensiveanalysis on hybrid RF / FSO model was performed in [16],[17] where the hybrid system exhibits better link availabilitycompared to the individual links.

In CRN, an unlicensed user gets the permission of ac-cessing the same spectrum of the licensed user throughdifferent spectrum sharing methods [18]. More particularly,

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in the underlay approach, the secondary user (SU ) makesuse of the primary user’s (PU ) shared spectrum as long asthe interference occurring at SU is less than the permittedthreshold value [4]. Considering CR technology in the RFlinks, the system performance of mixed RF-FSO systems areinvestigated in [4], [18]–[22]. In [20], the performance analysiswas accomplished by developing analytical expressions ofoutage probability (OP) where RF-FSO links were subjectedto Nakagami-m fading and double generalized Gamma (DGG)turbulence, respectively. However, this model was extended in[18], [21] where the impact of antenna diversity in RF linkswas observed. In [22], the OP analysis was performed foramplify-and-forward (AF) relaying-based dual-hop networkwhile multi-users and multi-destination were taken into con-sideration. The error probability (EP) expression was derivedin [4] where the influence of RF and FSO parameters on thesystem performance was inspected.

Security has become a widespread concern in the next-generation wireless communications (5G and beyond) dueto its high risk of information leakage [23], [24]. In thisdomain, physical layer security (PLS) has been recognized as acomplementary choice to the traditional cryptography strategy[25], [26]. According to the comprehensive literature analysis,current PLS research has mainly limited on hybrid RF / FSOmodels [25], [27], [28] and mixed RF-FSO networks [29]–[31]. In [25], the PLS of a hybrid FSO / RF network wasexamined and it was found that secrecy performance is highlydependent on air turbulence, pointing error, and RF fadingparameters. In [27], authors obtained the secrecy diversity gainfor RF backhaul system along with parallel FSO link whereasthe impact of power allocation on secrecy performance wasinvestigated in [28] for the parallel mmWave and FSO links.Including link blockage (LB) probability in the FSO link dueto the sudden presence of moving objects (e.g. clouds, birds,insects, etc.), [32] introduced the strategy of enhancing secrecybehavior for CR-based hybrid RF / FSO system.

B. Motivation and Contributions

Despite the fact that a CR-based hybrid RF / FSO systemconfiguration is more robust and practical than a single RF orFSO link [27], there are very few studies that have lookedinto the secrecy analysis of such systems and majority ofthe previously mentioned research focused on performanceanalysis of hybrid RF / FSO systems only. As a result,the investigation of PLS over the CR-based hybrid model iscurrently an open concept. To the best of authors’ knowledge,there are no works that conduct secrecy analysis of a hybridCR RF / FSO model considering double power constraints(i.e. the transmit power constraints at the SU and maximumallowable interference power at the PU ). Being motivated bythis research gap, in this work, the security performance foran underlay CR based hybrid RF / FSO fading channel isdemonstrated taking into account the single power constraint(i.e. maximum permissible interference at the SU ) and thedouble power constrains in the presence of an eavesdropperthat can intercept the secure communication via a RF link.The generalized α-µ fading channel is considered for all the

RF links since it can characterize small scale fading perfectly[33]. The FSO link, on the other hand, is assumed to followMalaga fading distribution, since it is also a generalized modeland capable of accurately describing the pointing error andturbulence scenarios [34]. The contributions of this researchare highlighted as follows:

1) Firstly, the expressions of cumulative density function(CDF) for selection combining (SC) based hybrid cog-nitive spectrum-sharing RF / FSO network assumingtwo different scenarios (e.g. single and double powerconstraints) are derived in closed-form. It is noteworthythat most of the works in the literature explored secrecybehaviour of the CR RF / FSO schemes consideringmaximum interference power constrain at PU only,which is just a single part of the proposed model. In thisperspective, considering both single and double powerconstraints in such type of hybrid model is certainly anovel and more practical approach. Moreover, this ex-pressions of CDF also incorporates some other existingworks [16], [17], [25], [27], [32] as special cases.

2) Secondly, the secrecy characteristics are derived for theproposed system model utilizing the underlay approachover the RF links and developing the analytical expres-sions for secrecy outage probability (SOP), probabilityof strictly positive secrecy capacity (SPSC), and effec-tive secrecy throughout (EST) for the two scenarios.Since the considered α − µ and Malaga models aregeneralized, and combination of the two in such typeof hybrid model has been considered for the first timein the literature, according to the authors’ knowledge,the derived expressions are also novel and generalized.

3) The derived expressions are further utilized to realizesome numerical outcomes with some specific figuresdemonstrating impacts of each system parameters i.e.fading, primary-secondary interference, LB probability,pointing error, detection techniques, and various atmo-spheric turbulence conditions, etc. It is also notablethat both detection techniques, i.e. heterodyne detection(HD) and intensity modulation/direct detection (IM/DD)techniques, have not been considered yet in such typeof hybrid system model except in this work. Finally,the accuracy of the deduced analytical expressions arecorroborated via Monte-Carlo (MC) simulations.

C. Organization

The outline of this paper is arranged as follows: In section-II, III, and IV, the proposed framework along with the channelmodels is described. The novel analytical expressions of theperformance metrics (i.e. SOP, probability of SPSC, and ESTanalysis for the two scenarios) are demonstrated in Section-V. Section-VI incorporates some exclusive numerical resultsfollowed by insightful discussions. Finally, the concludingremarks of this work are presented in Section-VII.

II. SYSTEM MODEL

A hybrid cognitive underlay network (CUN) is illustratedin Fig. 1, which incorporates two parallel links: cognitive

3

: Pointing Error

: Turbulence: Link Blockage

FSO Link

Secondary User Transmitter ()

Secondary User Receiver ()

Primary User ()

Eavesdropper ()

Fig. 1. Proposed system model for hybrid RF / FSO based cognitive underlay network.

RF and FSO sub-systems, respectively. It is essential tomention that FSO allows the benefits of using a license-freespectrum with high data rates [12]. But in the FSO sub-system, the signal transmission can be substantially disruptedby atmospheric turbulence conditions and pointing errors.Hence, the RF sub-system is utilized to transmit the sameinformation concurrently with the FSO sub-system therebymaintaining a continuous data transmission. To prevent anydata loss, the receiver R exploits selection combining diversityand always picks the best signal, i.e. the signal with highest(signal-to-noise ratio) SNR, among the two sub-links. Sincethe secondary user (S) utilizes the given licensed spectrum ofthe primary user (P ), in the proposed underlay technique, thesystem requires that there must be no adverse interference atP .

1) Cognitive RF Sub-link: Based on the underlay approach,S transmits its concealed information to R under the followingconstraints:• The peak tolerable interference power impinged by S onP cannot exceed a predefined value PQ. In such a case,S is not considered as a power limited terminal and ithas the full freedom to utilize its power depending ona single power constraint, i.e. interference severity at P .This particular case is denoted as Scenario-I in the restof this work.

• The CUN in Scenario-I sometimes may not be practical,specifically when the channel coefficient of S − P link

rapidly fluctuates. This may lead to feedback burdensince the instantaneous feedback gain is difficult to track.To mitigate these types of feedback burden and feed-back errors, mean value power allocation strategy canbe adopted [35]. Hence, besides only interference/singlepower constraint case, in double power constraint case, Sis assumed to be a power limited terminal and allowed toexploit maximum transmit power PT . This particular caseis denoted as Scenario-II in the remaining manuscript.

For Scenario-II, utilizing mean value power allocation, thetransmit power of S is denoted as

Pt,II = min

(PQ|gp|2

, PT

), (1)

where gp denotes the channel gain of S−P interference link.Therefore, the SNR at R is expressed as

γr,II = min

(ΨQ

|gp|2,ΨT

)|gr|2, (2)

where ΨQ =PQNr

, ΨT = PTNr

, Nr represents the noise powerimposed on R, and gr is the channel gain of S−R RF sub-link.However, when S is not a power-limited terminal (Scenario-I),the transmit power at S is represented as

Pt,I =PQ|gp|2

. (3)

4

In this case, the SNR of S −R RF sub-link is expressed as

γr,I =ΨQ|gr|2

|gp|2. (4)

2) FSO Sub-link: The FSO sub-system comprises of S witha transmit aperture, FSO sub-link, and R with a receive photo-detector. The transmitter dispatches information in opticalform that is further converted into electrical one utilizing thephoto-detector at R. Thereby, instantaneous SNR for S − RFSO sub-link is written as

γo =PfNo‖go‖2, (5)

where go signifies the channel gain of S−R FSO sub-link, Nodenotes the optical noise power at R, and Pf is the transmitoptical power at S.

Finally, after selection combining at R, the received SNRis written as [25, Eq. (11)]

γf,j = max(γr,j , γo), (6)

where j ∈ I, II.3) Eavesdropper Link: In the proposed system, an eaves-

dropper (E) targets the RF sub-link to intercept the confiden-tial data stream. The instantaneous SNR at E is expressedas

γe,j =Pt,jNe‖ge‖2, (7)

where ge denotes the channel gain of S − E link and Ne isthe noise power at E.

III. CHANNELS REALIZATION

In this section, the channel models of the S − R (RFand FSO sub-links), S − E, and S − P links are realizedindividually for the further utilization in the mathematicalmodeling of the proposed CUN system.

A. PDF and CDF of SNRs for the RF Links

The S − R RF sub-link, S − P , and S − E links areassumed to follow α − µ distribution, where α denotes thenon-linearity parameter of propagation environment (α > 0)and µ represents the number of multi-path clusters (µ > 0).Since α − µ model exhibits generalized characteristics [36],several multi-path models [37], e.g. exponential, Rayleigh,Weibull, Gaussian, Gamma, and Nakagami-m, etc., can beobtained by setting different values of α and µ. The PDF ofγi, i ∈ r, p, e, where r, p, and e correspond to the S − R(RF), S − P , and S − E links, respectively, is given as [38,Eq. (2)]

fγi(γ) =αiδ

µii

2Γ(µi)e−δiγ

αiγΘi . (8)

Here δi = Φ−αii , Θi = αiµi − 1, αi = αi2 , Φi is the average

SNR of the RF links, and Γ(·) denotes the Gamma operator.The CDF of γi is defined as

Fγi(γ) =

∫ γ

0

fγi(γ)dγ. (9)

Plugging (8) into (9), and after performing integration uponutilizing [39, Eq. (3.381.8)], the CDF leads to

Fγi(γ) =γ(µi, δiγ

αi)

Γ(µi), (10)

where γ(, ., ) denotes the lower incomplete Gamma function.Finally, with the help of [39, Eq. (8.352.6)], the CDF of γican be expressed in an alternative form as

Fγi(γ) = 1− e−δiγαi

µi−1∑mi=0

(δiγαi)mi

mi!. (11)

B. PDF and CDF of SNR for FSO SUB-Link

The S − R FSO sub-link is considered to follow Malaga(M) turbulence. Since Malaga turbulence is a generalizedFSO channel, it incorporates some other distributions such asGamma-Gamma, lognormal, Rice-Nakagami, and Gamma asspecial cases [31, Table II]. Hence, the PDF of γo is expressedas [40, Eq. (9)]

fγo(γ) =ε2χo2sγ

βo∑mo=0

ϑnG3,01,3

[$

(γ

µs

) 1s∣∣∣∣ ε2 + 1ε2, αo,mo

],

(12)

where

χo =2α

αo2o

g1+αo2 Γ(αo)

(gβo

gβo + Ωo

)βo+αo2

,

$ =ε2αoβo(g + Ωo)

(ε2 + 1)(gβo + Ωo),

ϑn = υn

(αoβo

gβo + Ωo

)−αo+mo2

,

υn =

(βo − 1

mo − 1

)(gβo + Ωo)

1−mo2

(mo − 1)!

(Ωog

)mo−1(αoβo

)mo2

,

αo and βo denote the fading parameters, g = 2bo(1 − ρ)represents the average power of scattering components re-ceived through off-axis eddies, 2bo denotes the total averagepower of scattering components, ε (0 ≤ ρ ≤ 1) signifies theamount of scattering power coupled to the LOS component,the average power realized from coherent contributions is

symbolized by Ωo = Ωo + 2boρ+ 2

√2boρΩo cos(φA − φB),

Ωo is termed as the average power of LOS components, φAand φB represent the deterministic phases of LOS, µ1 = Φo,µ2 = αoε

2(ε2+1)−2(ε2+2)(g+Ωo)Φo(αo+1)[2g(g+2Ωo)+Ω2

o(1+1/βo)] , s denotes the detectiontechnique (i.e. s = 1 refers to HD technique and s = 2 refersto IM/DD technique), Φo denotes the average SNR of FSOsub-link that is related to electrical SNR µs, and G[·] representsthe Meijer’s G function [41]. Plugging (12) into (9), the CDFof γo is written as [40, Eq. 11]

Fγo(γ) = K

βo∑mo=0

ςnG3s,1s+1,3s+1

[V γ

µs

∣∣∣∣ 1, q1

q2, 0

], (13)

where K = ε2χo2s(2π)s−1 , ςn = ϑns

αo+mo−1, V = $s

s2s ,

q1 = ε2+1s , · · · , ε

2+ss including s number of terms, and

5

q2 = ε2

s , · · · ,ε2+s−1

s , αos , · · · ,αo+s−1

s , mos , · · · ,mo+s−1

s including 3s number of terms.

The FSO link can be temporarily blocked due to its extremeLOS requirements. Hence, including link blocking probability(Po) in FSO and assuming SNR of the S − R FSO sub-linkas γo∗ , the PDF of γo∗ is expressed finally as [6, Eq. (10)]

fγo∗ (γ) = Poδ(γ) + (1− Po)fγo(γ), (14)

where δ(·) denotes a Dirac delta function [42,Eq. (14.03.02.0001.01)]. Now, the CDF of γo∗ is expressedas

Fγo∗ (γ) =

∫ γ

0

fγo∗ (γ)dγ ∼= Po + (1− Po)Fγo(γ). (15)

Substituting (13) into (15), we obtain

Fγo∗ (γ) = Po + (1− Po)Kβo∑

mo=0

ςnG3s,1s+1,3s+1

[V γ

µs

∣∣∣∣ 1, q1

q2, 0

].

(16)

IV. COGNITIVE UNDERLAY NETWORK

This section demonstrates the CDFs of SNRs for the hybridRF / FSO CUN considering both scenarios (Scenarios- I & II).

A. Scenario I

Considering the interference of S − P link and denotingSNR of the S − R RF sub-link as γr,I , the CDF of γr,I iswritten as [4, Eq. (4)]

Fγr,I (γ) =

∫ ∞0

Fγr

(γ

ΨQx

)fγp(x)dx. (17)

Now substituting (11) and (8) into (17), the CDF is expressedfurther as

Fγr,I (γ) = 1−µr−1∑mr=0

αpδmrr δ

µpp

2Γ(µp)mr!

(γ

ΨQ

)αrmr×∫ ∞

0

e−(δpxαp+ΞI1γ

αrr xαr )xΘp+αrmrdx, (18)

where ΞI1 = δrΨ−αrQ . Now, following some algebraic ma-

nipulations, assuming αp = αr, and performing integrationutilizing [39, Eq. (3.326.2)], the CDF of S −R RF sub-linkis finally obtained as

Fγr,I (γ) = 1−µr−1∑mr=0

Γ(ΞI2)αpδmrr δ

µpp

αp2Γ(µp)mr!

(γ

ΨQ

)αrmr× (ΞI1γ

αr + Φ−αpp )−ΞI2 , (19)

where ΞI2 =αrmr+µpαp

αp. According to (6), the CDF of end-

to-end SNR at R is written as

Fγf,I (γ) = Fγr,I (γ)× Fγo∗ (γ). (20)

By replacing (19) and (16) into (20), the CDF of γf,I is derivedin (21).

B. Scenario II

Assuming double power constraints in the RF sub-link, theCDF of Fγr,II is written as

Fγr,II (γr) = Pr

min

(ΨQ

|gp|2,ΨT

)|gr|2 ≤ γr

= Pr

|gr|2 ≤

γrΨT

,ΨQ

|gp|2≥ ΨT

︸ ︷︷ ︸

Λ1

+ Pr

|gr|2

|gp|2≤ γr

ΨQ,

ΨQ

|gp|2≤ ΨT

︸ ︷︷ ︸

Λ2

. (22)

Since gr and gp are completely independent to one other,hence, Λ1 is expressed in an alternative form as

Λ1 = Pr

|gp|2 ≤

ΨQ

ΨT

Pr

|gr|2 ≤

γrΨT

= Fγp

(ΨQ

ΨT

)Fγr

(γ

ΨT

). (23)

Substituting (11) into (23), Λ1 can be finally written as

Λ1 = 1−µp−1∑mp=0

ΞII1 −µr−1∑mr=0

ΞII2 γαrmre−δrΨ−αr

T γαr

+

µp−1∑mp=0

µr−1∑mr=0

ΞII3 γαrmre−(δpΨ

αpQ Ψ

−αpT +δrΨ−αr

T γαr ), (24)

where ΞII1 =δmpp

mp!

(ΨQΨT

)αpmpe−δpΨ

αpQ Ψ

−αpT , ΞII2 =

δmrrmr! Ψ−αrmrT , and ΞII3 =

δmpp δmrrmr!mp!

(ΨQΨT

)αpmpΨ−αrmrT . Now,

according to the concept of probability theory [43], Λ2 iswritten as

Λ2 =

∫ ∞ΨQΨT

fγp(y)

∫ γrΨQ

y

0

fγr (x)dxdy

=

∫ ∞ΨQΨT

fγp(y)Fγr

(γy

ΨQ

)dy. (25)

Replacing (8) and (11) into (25), Λ2 is obtained as

Λ2 = ΞII5 −µr−1∑mr=0

Ω−1∑m3=0

m3∑m4=0

∞∑m5=0

(m3

m4

)(Ω +m5 − 1

m5

)× ΞII6 γ

αr(mr+m4+m5)e−(δpΨαrQ Ψ−αrT +δrΨ−2αr

T Ψ−αrQ γαr ),

(26)

where ΞII5 =αpδ

µpp

2Γ(µp)αpδΞII4p

Γ

[ΞII4 , δp

(ΨQΨT

)αp],

ΞII4 =Θp+1αp

, Ω =Θp+αrmr+1

αr, and ΞII6 =

αpδmr+m4+m5r (Ω−1)!(−1)m5

2Γ(µp)mr!αrm3! δ−(Ω+m5−m3+m4−µp)p

Ψ−αr(mr−m3+m5)Q (Ψ−αrT )m3+m4 . For proof, please see the

Appendix. Finally, utilizing (24) and (26) into (22), andperforming some simple mathematical operations, Fγr,II (γ)is written as shown in (27). Now, similar to (20), the CDF ofγf,II is expressed as

Fγf,II (γ) = Fγr,II (γ)× Fγo∗ (γ). (28)

6

Fγf,I (γ) = Po + (1− Po)βo∑

mo=0

KςnG3s,1s+1,3s+1

[V γ

µs

∣∣∣∣ 1, q1

q2, 0

]− Po

µr−1∑mr=0

δmrr αpδµpp Γ(ΞI2)

2Γ(µp)αpmr!

(γ

ΨQ

)αrmr(ΞI1γ

αp + Φ−αpp )−ΞI2

− (1− Po)µr−1∑mr=0

βo∑mo=0

KςnΓ(ΞI2)αpδ

µpp δmrr

αp2Γ(µp)mr!

(γ

ΨQ

)αrmr(ΞI1γ

αp + Φ−αpp )−ΞI2G3s,1s+1,3s+1

[V γ

µs

∣∣∣∣ 1, q1

q2, 0

]. (21)

Fγr,II (γ) = 1 + ΞII5 −µp−1∑mp=0

ΞII1 −µr−1∑mr=0

ΞII2 γαrmre−δrΨ−αr

T γαr +

µp−1∑mp=0

µr−1∑mr=0

ΞII3 γαrmre−(δpΨ

αpQ Ψ

−αpT +δrΨ−αr

T γαr )

−µr−1∑mr=0

Ω−1∑m3=0

m3∑m4=0

∞∑m5=0

(m3

m4

)(Ω +m5 − 1

m5

)ΞII6 γ

αr(mr+m4+m5)e−(δpΨαrQ Ψ−αrT +δrΨ−2αr

T ΨαrQ γαr ). (27)

Fγf,II (γ) = Po

X −

µr−1∑mr=0

ΞII2 γαrmre−δrΨ−αr

T γαr +

µp−1∑mp=0

µr−1∑mr=0

ΞII3 γαrmre−(δpΨ

αpQ Ψ

−αpT +δrΨ−αr

T γαr ) −µr−1∑mr=0

Ω−1∑m3=0

×m3∑m4=0

∞∑m5=0

(m3

m4

)(Ω +m5 − 1

m5

)ΞII6 γ

αr(mr+m4+m5)e−(δpΨαrQ Ψ−αrT +δrΨ−2αr

T ΨαrQ γαr )

+(1− Po)K

βo∑mo=0

ςn

×G3s,1s+1,3s+1

[V γ

µs

∣∣∣∣ 1, q1

q2, 0

]X −

µr−1∑mr=0

ΞII2 γαrmre−δrΨ−αr

T γαr +

µp−1∑mp=0

µr−1∑mr=0

ΞII3 γαrmre−(δpΨ

αpQ Ψ

−αpT +δrΨ−αr

T γαr )

−µr−1∑mr=0

Ω−1∑m3=0

m3∑m4=0

∞∑m5=0

(m3

m4

)(Ω +m5 − 1

m5

)ΞII6 γ

αr(mr+m4+m5) e−(δpΨαrQ Ψ−αrT +δrΨ−2αr

T ΨαrQ γαr )

. (29)

Now, substituting (27) and (16) into (28), the CDF of γf,II isobtained as shown in (29), where X = 1+ΞII5 −

∑µp−1mp=0 ΞII1 .

V. PERFORMANCE ANALYSIS

In this section, we demonstrate the novel analytical expres-sions for SOP, probability of SPSC, and EST utilizing (8),(21), and (29).

A. Secure Outage Probability Analysis

In order to investigate the secrecy behaviour of the proposedhybrid system, SOP is one of the fundamental performancemetrics. SOP particularly depends on whether the target se-crecy rate (Υe) is greater than secrecy capacity (Cs), whereCs = [log2(1+γf,j)−log2(1+γe,j)]

+ and [z]+ = maxz, 0.So, mathematically, SOP can be described as [44, Eq. (14)]

SOP = Pr Cs(γf,j , γe,j) ≤ Υe

=

∫ ∞0

Fγf,j (σγ + σ − 1)fγe,j (γ)dγ, (30)

where σ = 2Υe and Υe > 0. Since Meijer’s G function existsin (21) and (29), derivation of SOP utilizing (30) is almostimpossible. Hence, the lower bound of SOP can be derived as

SOP ≥ SOPL =

∫ ∞0

Fγf,j (σγ)fγe,j (γ)dγ. (31)

1) Scenario I: Substituting (21) and (8) into (31), SOP iswritten as

SOP IL =αeδ

µee Po

2Γ(µe)=1 + (1− Po)Kςn

×βo∑

mo=0

αeδ

µee

2Γ(µe)=2 − ΞI5=4

−Po

µr−1∑mr=0

ΞI5=3, (32)

where ΞI5 =(

ΞI2+m2−1m2

) αpδµpp2Γ(µp)

αeδµee

2Γ(µe)δmrrmr!

Γ(ΞI2)αp

(σ

ΨQ

)αrmr(−1)m2

(ΞI1σ

αr)m2

(Φ−αpp

)−ΞI2−m2

and the four integralterms, =1, =2, =3, and =4 are derived as follows.

Derivation of =1: Utilizing the formula of [39, Eq.(3.326.2)], =1 is expressed as

=1 =

∫ ∞0

e−δeγαeγΘedγ =

Γ(

Θe+1αe

)αe

δ−Θe+1

αee . (33)

Derivation of =2: =2 is written as

=2 =

∫ ∞0

e−δeγαeγΘeG3s,1

s+1,3s+1

[V σγ

µs

∣∣∣∣ 1, q1

q2, 0

]dγ. (34)

7

Now, with the help of [41, Eqs. (2.24.1.1) and (8.4.3.1)], =2

is converted as

=2 =

∫ ∞0

γΘeG1,00,1

[δeγ

αe

∣∣∣∣ −0]G3s,1s+1,3s+1

[V σγ

µs

∣∣∣∣ 1, q1

q2, 0

]dγ

=αe

s(2Θe+1)(V σµs

)−(Θe+1)

(2π)s(αe−1)

×G1+αe, ˜3sαeαe(3s+1),1+αe(s+1)

δeαe2sαe(V σµs

)αe∣∣∣∣∣∣∣ ζ1,∆(αe,−Θe)

0, ζ2, ζ3

,(35)

where ζ1 = ∆(αe,−Θe − q2), ζ2 = ∆(αe,−Θe − 1), ζ3 =∆(αe,−Θe− q1), and ∆(c, r) = r

c ,r+1c , ..., r+c−1

c as definedin [45, Eq. (22)].

Derivation of =3: =3 is expressed as

=3 =

∫ ∞0

(ΞI1σαrγαp + Φ−αpp )−ΞI2 γΘe+αrmre−δeγ

αedγ.

(36)

Utilizing binomial expansion and [39, Eq. (3.326.2)], =3 isexpressed in alternative form as

=3 =

∞∑m2=0

∫ ∞0

γΞI3e−δeγαedγ =

∞∑m2=0

Γ(ΞI4)

αeδ−ΞI4e , (37)

where ΞI3 = Θe + αrmr + αpm2 and ΞI4 =ΞI3+1αe

.Derivation of =4: Following the similar approach of =2

and =3, =4 is illustrated as

=4 =

∞∑m2=0

∫ ∞0

e−δeγαeγΞI3G3s,1

s+1,3s+1

[V σγ

µs

∣∣∣∣ 1, q1

q2, 0

]dγ

=

∞∑m2=0

αes(2 ΞI3+1)

(V σµs

)−(ΞI3+1)

(2π)s(αe−1)

×G1+αe, ˜3sαeαe(3s+1),1+αe(s+1)

δeαe2sαe(V σµs

)αe∣∣∣∣∣ ζ4,∆(αe,−ΞI3)

0, ζ5, ζ6

.(38)

where ζ4 = ∆(αe,−ΞI3 − q2), ζ5 = ∆(αe,−ΞI3 − 1), andζ6 = ∆(αe,−ΞI3 − q1).

2) Scenario II: Substituting (8) and (29) into (31), SOP fordouble power constraints scenario is derived as (39), where theeight integral terms, R1, R2, R3, R4, R5, R6, R7, and R8

are derived as follows.Derivation of R1: Utilizing [39, Eq. (3.326.2)], R1 is

expressed as

R1 =

∫ ∞0

e−δeγαeγΘedγ =

Γ(

Θe+1αe

)αe

δ−Θe+1

αee . (40)

Derivation of R2 and R3: R2 and R3 are expressed as

R2 = R3 =

∫ ∞0

γαrmr+Θee−δrσαrΨ−αr

T γαr e−δeγαedγ.

(41)

Following some mathematical manipulations defined in [46]and utilizing identities [41, Eqs. (8.4.3.1) and (2.24.1.1)], R2

and R3 are further derived as

R2 = R3 =

∫ ∞0

γαrmr+Θe

αr e−δrΨ−αrT σαrγe−δeγ

αeαr d

(γ

1αr

)=

1

αr

∫ ∞0

γΞII7 −1e−δrΨ−αrT σαrγe−δeγ

αeαr dγ

=1

αr

∫ ∞0

γΞII7 −1G1,00,1

[δrΨ

−αrT σαrγ

∣∣∣∣ −0]

×G1,00,1

[δeγ

αeαr

∣∣∣∣ −0]dγ =

αeΞII7 − 1

2 (δrΨ−αrT σαr )−ΞII7

αr12 (2π)

12 (αe+αr−2)

×Gαr,αeαe,αr

[δαre αr

−αr

(δrΨ−αrT σαr )αe αe

−αe

∣∣∣∣ ∆(αe, 1− ΞII7 )0

],

(42)

where ΞII7 = αrmr+Θe+αrαr

.

Derivation of R4: By following the similar approach as ofR2 and R3, R4 is expressed as

R4 =

∫ ∞0

γαr(mr+m4+m5)+Θee−δrΨ−2αrT ΨαrQ σαrγαr e−δeγ

αedγ

=

∫ ∞0

γαr(mr+m4+m5)+Θe

αr e−δrΨ−2αrT ΨαrQ σαrγe−δeγ

αeαr d

(γ

1αr

)=

1

αr

∫ ∞0

γΞII8 −1e−δrΨ−2αrT ΨαrQ σαrγe−δeγ

αeαr dγ

=1

αr

∫ ∞0

γΞII8 −1G1,00,1

[δrΨ

−2αrT Ψαr

Q σαrγ

∣∣∣∣ −0]

×G1,00,1

[δeγ

αeαr

∣∣∣∣ −0]dγ =

αeΞII8 − 1

2 (δrΨ−2αrT Ψαr

Q σαr )−ΞII8

αr12 (2π)

12 (αe+αr−2)

×Gαr,αeαe,αr

[δαre αr

−αr

(δrΨ− ˜2αrT Ψαr

Q σαr )αe αe−αe

∣∣∣∣ ∆(αe, 1− ΞII8 )0

],

(43)

where ΞII8 = αr(mr+m4+m5)+Θe+αrαr

.

Derivation of R5: With some mathematical manipulationsand following identities [41, Eqs. (8.4.3.1) and (2.24.1.1)], R5

is derived as

R5 =

∫ ∞0

γΘee−δeγαeG3s,1s+1,3s+1

[V σγ

µs

∣∣∣∣ 1, q1

q2, 0

]dγ

=

∫ ∞0

γΘeG1,00,1

[δeγ

αe

∣∣∣∣ −0]G3s,1s+1,3s+1

[V σγ

µs

∣∣∣∣ 1, q1

q2, 0

]dγ

=αe

s(2Θe+1)(V σµs

)−(Θe+1)

(2π)s(αe−1)

×G1+αe, ˜3sαeαe(3s+1),1+αe(s+1)

δeαe2sαe(V σµs

)αe∣∣∣∣∣ ζ1,∆(αe,−Θe)

0, ζ2, ζ3

.(44)

8

SOP IIL =αeδ

µee

2Γ(µe)

[Po

XR1 −

µr−1∑mr=0

ΞII2 σαrmrR2 +

µp−1∑mp=0

µr−1∑mr=0

ΞII3 σαrmre−δpΨ

αpQ Ψ

−αpT R3 −

µr−1∑mr=0

Ω−1∑m3=0

m3∑m4=0

∞∑m5=0

×(

Ω +m5 − 1

m5

)(m3

m4

)ΞII6 σ

αr(mr+m4+m5)e−δpΨαrQ Ψ−αrT R4

+(1− Po)K

βo∑mo=0

ςn

XR5 −

µr−1∑mr=0

ΞII2 R6 σαrmr +

µp−1∑mp=0

×µr−1∑mr=0

ΞII3 R7σαrmre−δpΨ

αpQ Ψ

−αpT −

µr−1∑mr=0

Ω−1∑m3=0

m3∑m4=0

∞∑m5=0

(m3

m4

)(Ω +m5 − 1

m5

)ΞII6 R8 σ

αr(mr+m4+m5)e−δpΨαrQ Ψ−αrT

].

(39)

Derivation of R6 and R7: R6 and R7 are derived byfollowing [41, Eq. (8.4.3.1)] as

R6 = R7 =

∫ ∞0

γαrmr+Θee−δrΨ−αrT σαrγαr e−δeγ

αe

×G3s,1s+1,3s+1

[V σγ

µs

∣∣∣∣ 1, q1

q2, 0

]dγ

=

∫ ∞0

γαrmr+ΘeG1,00,1

[δrΨ

−αrT σαrγαr

∣∣∣∣ −0]

×G1,00,1

[δeγ

αe

∣∣∣∣ −0]G3s,1s+1,3s+1

[V σγ

µs

∣∣∣∣ 1, q1

q2, 0

]dγ.

(45)

Now, for tractable analysis, transforming Meijer’s G functionsinto Fox’s H functions via utilizing [47, Eqs. (6.2.3) and(6.2.8)] and performing integration by means of [48, Eq. (2.3)]and [36, Eq. (3)], R6 and R7 are obtained as

R6 = R7 =

∫ ∞0

γαrmr+ΘeH3s,1s+1,3s+1

[V σγ

µs

∣∣∣∣ (1, 1), (q1, 1)(q2, 1), (0, 1)

]

×H1,00,1

[ΞII10γ

αr

∣∣∣∣ −(0, 1)

]H1,0

0,1

[δeγ

αe

∣∣∣∣ −(0, 1)

]dγ =

δ−ΞII9αe

e

αe

×H1,0:1,0:3s,11,0:0,1:s+1,3s+1

[J1

J2

∣∣∣∣ −(0, 1)

∣∣∣∣ (1, 1), (q1, 1)(q2, 1), (0, 1)

∣∣∣∣J3, J4

],

(46)

where ΞII9 = αrmr + Θe + 1, ΞII10 = δrΨ−αrT σαr , J1 =(

1− ΞII9αe

; αrαe ,1αe

), J2 = (1;−), J3 = ΞII10δ

− αrαee , J4 =

V σµ−1s δ− 1αe

e , and Hm,np,q [·] is the Fox’s H function introduced

in [49, Eq. (1.2)], and Hx1,y1:x2,y2:x3,y3

c1,d1:c2,d2:c3,d3[·] is the extended gen-

eralized bivariate Fox’s H function (EGBFHF) as explainedin [49, Eq. (2.57)].

Derivation of R8: R8 is derived by following the similaridentities from R6 and R7 as

R8 =

∫ ∞0

γαr(mr+m4+m5)+Θee−δrΨ−2αrT ΨαrQ σαrγαr e−δeγ

αe

×G3s,1s+1,3s+1

[V σγ

µs

∣∣∣∣ 1, q1

q2, 0

]dγ =

∫ ∞0

γαr(mr+m4+m5)+Θe

×G1,00,1

[ΞII12γ

αr

∣∣∣∣ −0]G1,0

0,1

[δeγ

αe

∣∣∣∣ −0]

×G3s,1s+1,3s+1

[V σγ

µs

∣∣∣∣ 1, q1

q2, 0

]dγ = αe

−1δ−ΞII11αe

e

×H1,0:1,0:3s,11,0:0,1:s+1,3s+1

[J5

J2

∣∣∣∣ −(0, 1)

∣∣∣∣ (1, 1), (q1, 1)(q2, 1), (0, 1)

∣∣∣∣J6, J4

],

(47)

where ΞII11 = αr(mr + m4 + m5) + Θe + 1, ΞII12 =

δrΨ−2αrT Ψαr

Q σαr , J5 = (1 − ΞII11

αe; αrαe ,

1αe

), and J6 =

ΞII12 δ− αrαee .

B. Strictly Positive Secrecy Capacity Analysis

In order to ensure secrecy in a wiretap prone model, theprobability of SPSC acts as one of the crucial performancemetric that ensures uninterrupted data stream conveyance onlywhen secrecy behaviour of the system remains up to scratch(γf,j > γe,j). Mathematically, the analytical expression ofprobability of SPSC can be easily deducted from the SOPthat is given as [31, Eq. (25)]

SPSC = Pr Cs > 0= 1− Pr Cs ≤ 0 .

Hence, SPSCI = 1− SOPIL|Υe=0, (Scenario-I) (48)

and SPSCII = 1− SOPIIL |Υe=0. (Scenario-II) (49)

C. Effective Secrecy Throughput Analysis

EST is another secrecy performance metric that ensuresachievable secure average throughput measurements. Whenthe reliability issue is considered equally important alongwith the security issue then this parameter is considered.Mathematically, EST is formulated as the product of target

9

secrecy rate and the probability of successful transmission thatis given as [50, Eq. (5)]

EST = Υe(1− SOP ).

Hence, EST I = Υe(1− SOP IL), (Scenario-I) (50)

and EST II = Υe(1− SOP IIL ). (Scenario-II) (51)

VI. NUMERICAL RESULTS

In this section, the impacts of all the system parameters(e.g., αi, µi, Φi, ΨQ, ΨT , Po, αo, βo, ε, and s) on the secrecybehaviour of the proposed scenarios are illustrated graphicallyutilizing the expressions in (32), (39), (48), (49), (50), and(51). Note that the univariate Meijer’s G and Fox’s H func-tions can be easily computed utilizing standard Mathematicapackages whereas the EGBFHF is computed utilizing [36,Table I]. Moreover, all the mathematical expressions are alsovalidated via MC simulations averaging 106 random samplesof α−µ and Malaga random variables. It can clearly be seenthat MC simulation results are in a good agreement with theanalytical results that strongly corroborate the analysis of thispaper. All the numerical results are demonstrated assuming(αo, βo) = (2.296, 2) for strong, (αo, βo) = (4.2, 3) formoderate, and (αo, βo) = (8, 4) for weak turbulences, r = 1for HD and r = 2 for IM/DD techniques, and Υe = 0.05bits/sec/Hz.

The impacts of αi on the EST and SPSC are demonstratedin Figs. 2 and 3 for Scenario-I.

0 5 10 15 20 25 30 35

alP=E=2alP=E=3alP=E=4alP=E=5alP=E=6alP=E=6

10

10

10

Ψ

2 (Sim) 3 (Sim) 4 (Sim) 5 (Sim) 6 (Sim)Analysis

Fig. 2. The EST versus ΨQ for selected values of αp and αe with αo =2.296, βo = 2, Ωo = 1, g = 2, s = 2, ε = 1, Po = 0.5, αr = 2,µr = µp = µe = 2, Φr = 15dB, Φp = 10dB, Φe = µs − 5dB, andΥe = 0.05 bits/sec/Hz.

It is observed that EST gradually increases with ΨQ. Similarobservations are also seen in [51]. Again, since with theincrease in non-linearity parameter, the communication linkbecomes better [5] (i.e. the link will undergo less amount offading with higher αi), the EST becomes better for lower αpand αe. On the other hand, as expected, the SPSC is enhancedas Φo and αr increase from a lower to a higher value.

3 6 9 12 15 18 21 24 27 30

aR=1aR=3aR=33

10

Φo

10

10

HD

IM/DD

1 (Sim) 3 (Sim)Analysis

Fig. 3. The SPSC versus Φo for selected values of αr and s with αo =2.296, βo = 2, Ωo = 1, g = 2, ε = 1, Po = 0.5, αp = αe = 5,µr = µp = µe = 6, Φr = Φp = 15dB, Φe = 0dB, and ΨQ = 15dB.

-10 -5 0 5 10 15 20 25 30 35

Mue=1Mue=2Mue=3Mue=3

Φ

Φe 10

Φe 0

1 (Sim) 2 (Sim) 3 (Sim)Analysis

10

10

10

10

10

Fig. 4. The SOP versus Φp for selected values of µe and Φe with αo =2.296, βo = 2, Ωo = 1, g = 2, s = 1, ε = 1, αr = αp = αe = 2,µr = µp = 2, Po = 0.1, Φr = 15dB, µr = 12dB, ΨQ = −5dB, andΥe = 0.05 bits/sec/Hz.

The SOP is plotted against Φp,ΨQ, and Φr consideringScenario-I in Figs. 4, 5, and 6, respectively, with a view toobserve the impacts of the number of multipath clusters.

It can clearly be noted that the SOP performance improveswith ΨQ and Φr, and degrades with Φp as testified in [4].This is as expected because the larger ΨQ indicates the highertransmitting power at S. On the other hand, Φr improves theS − R RF sub-link whereas Φp strengths the S − P link. It

10

-5 0 5 10 15 20

muP=2muP=3muP=4muP=5muP=5

Ψ

2 (Sim) 3 (Sim) 4 (Sim) 5 (Sim)Analysis

10

10

10

Fig. 5. The SOP versus ΨQ for selected values of µp with αo = 2.296,βo = 2, Ωo = 1, g = 2, ε = 1, s = 1, αr = αp = αe = 2, µr = µe = 2,Po = 0.1, Φp = 10dB, Φr = 15dB Φo = 10dB, Φe = 10dB, andΥe = 0.05 bits/sec/Hz.

25 30 35 40 45

muR=1muR=2muR=3muR=4muR=4

1 (Sim) 2 (Sim) 3 (Sim) 4 (Sim)Analysis

Φr

10

10

10

Fig. 6. The SOP versus Φr for selected values of µr with αo = 2.296,βo = 2, Ωo = 1, g = 2, εo = 1, s = 1, αr = αp = αe = 2, µp = µe = 2,Po = 0.1, Φp = 15dB, Φo = 10dB, Φe = 15dB, ΨQ = −5dB, andΥe = 0.05 bits/sec/Hz.

is obvious that an increase in µi will increase the channeldiversity thereby mitigating the channel fading. Hence, theSOP decreases with µr, and increases with µp and µe. Similarresults were also demonstrated in [5] that corroborate theoutcomes in this work. In Fig. 7, impact of µp is demonstratedon SOP considering Scenario-II. Similar to Scenario-I, in thisparticular case, it is also observed that the SOP drasticallydegrades with µp.

In Fig. 8, the probability of link blockage is observed viaplotting SOP against average SNR of FSO link under variouslevels of pointing error. It is observed that probability of link

0.4

0.45

0.5

0.55

0.6

-5 0 5 10 15 20 25

muP=2muP=3mup=4muP=4

Ψ

2 (Sim)

3 (Sim)

4 (Sim)Analysis

Fig. 7. The SOP versus ΨT for selected values of µp with αo = 2.296,βo = 2, Ωo = 1, g = 2, ε = 6.7, Po = 0.2, s = 1, αr = αp = αe = 2,µr = µe = 2, Φp = −5dB, µs = 10dB, Φr = −5dB, ΨQ = 5dB,Φe = 5dB, and Υe = 0.05 bits/sec/Hz.

-5 0 5 10 15 20 25 30 35

Po=0.5 (Sim)Po=0.5 (Sim)Po=0.5

1 (Sim)

6.7 (Sim)Analysis

0.5

0.3

0.1

10

10

Fig. 8. The SOP versus µs for selected values of Po and ε with αo = 2.296,βo = 2, Ωo = 1, g = 2, s = 1, αr = αp = αe = 2, µr = µp = µe =6, Φr = Φp = 15dB, Φe = −5dB, ΨQ = −10dB, and Υe = 0.05bits/sec/Hz.

blockage impose significant impact on the SOP performance,that is, the SOP degrades with the increase in Po (increasedPo signifies a stronger link blockage). It is obvious becausesimilar to the observations in [6], the atmospheric turbulencedegrades with the increase in link blockage, which yields to adegraded secrecy performance.

Figure 9 depicts the impact of ΨT on the EST performancefor Scenario-II. It can be noted that an increased ΨT is ben-eficial for EST performance, since for that case the increasedtransmit power yields a better received SNR at the destination.After 15dB, an error floor is observed, as expected. This

11

0

0.01

0.02

0.03

0.04

0.05

-10 -5 0 5 10 15 20

PSI_P=-3PSI_P=-10PSI_P=-12PSI_P=-15PSI_P=-18PSI_P=-18

Φ

Ψ 03 (Sim)Ψ 10 (Sim)Ψ 12 (Sim)Ψ 15 (Sim)Ψ 18 (Sim)Analysis

Fig. 9. The EST versus Φp for selected values of ΨT with αo = 2.296,βo = 2, Ωo = 1, g = 2, s = 2, ε = 6.7, Po = 0.1, αr = αp = αe = 2,µr = µp = µe = 2, Φo = −5dB, Φr = 10dB, ΨQ = −10dB, Φe =−5dB, and Υe = 0.05 bits/sec/Hz.

0 5 10 15 20 25 30 35 40 45

strmodweakstr

Φo

Strong turbulence (Sim)Moderate turbulence (Sim)Weak turbulence (Sim)Analysis

10

10

IM/DD

HD

Fig. 10. The SOP versus Φo for selected values of αo, βo, and s with Ωo =1, g = 2, ε = 6.7, Po = 0.1, αr = αp = αe = 2, µr = µp = µe = 6,Φr = Φp = 15dB, Φe = −5dB, ΨT = ΨQ = −10dB, and Υe = 0.05bits/sec/Hz.

occurs as the transmit power at S is limited by a thresholdvalue.

The FSO link performance is significantly affected by theatmospheric turbulence. Based on the values of αo and βo,three turbulence conditions (e.g. strong, moderate, and weak)are considered in this work. Figures 10 and 11 demonstratehow the secrecy performance is affected with various turbu-lence conditions under Scenarios- I & II.

Since stronger turbulence yields a weaker SNR at thedestination, the weaker the turbulence, the better the secrecyperformance that is also demonstrated in [31]. Moreover, a

-5 0 5 10 15 20 25 30 35

Strong (Sim)Moderate (Sim)Weak (Sim)Weak

Φo

10

10

IM/DD

HD

Strong turbulence (Sim)Moderate turbulence (Sim)Weak turbulence (Sim)Analysis

Fig. 11. The SOP versus Φo for selected values of αo, βo and s withΩo = 1, g = 2, ε = 1, Po = 0.1, αr = αp = αe = 2, µr = µp =µe = 6,Φr = Φp = 15dB, Φe = −5dB, ΨQ = −10dB, and Υe = 0.05bits/sec/Hz.

-5 0 5 10 15 20 25 30 35

Strong (Sim)Modeate (Sim)Weak (Sim)Analytical

1

Φo

6.7

Strong turbulence (Sim)Moderate turbulence (Sim)Weak turbulence (Sim)Analysis

10

10

Fig. 12. The SOP versus Φo for selected values of αo, βo and ε withΩo = 1, g = 2, s = 2, Po = 0.1, αr = αp = αe = 2, µr = µp =µe = 6,Φr = Φp = 15dB, Φe = −5dB, ΨQ = −10dB, and Υe = 0.05bits/sec/Hz.

comparison between two detection techniques is also pre-sented that clearly reveals that HD technique overcomes theatmospheric turbulence more appropriately than the IM/DDtechnique. The results in [50] also completely match with thisoutcome.

Besides the atmospheric turbulence, pointing error alsoplays a vital role on the FSO link performance that is illus-trated in Fig. 12 by depicting SOP as a function of the averageSNR of the FSO channel experiencing Scenario-I.

The result exhibits that higher pointing error (lower value ofε) is more detrimental for better SOP than the lower pointing

12

error (higher value of ε). This happens since lower pointingerror designates a better pointing accuracy that yields a betterdetection of the received signal. This statement can be easilyjustified with similar results presented in [50].

VII. CONCLUSIONS

In this work, the secrecy performance of a CUN basedhybrid RF / FSO network has been investigated wherein bothmaximum transmit power and interference power constraintswithin the CR spectrum are taken into consideration. Assum-ing all the RF and FSO links were subjected to α− µ fadingand Malaga (M) turbulence, respectively, the expressions ofSOP, probability of SPSC, and EST were derived in closed-form. The tightness of the simulation and analytical resultswere also demonstrated via MC simulations. Furthermore, theimpacts of the system parameters on the security performancewere analyzed capitalizing on the derived expressions. Numer-ical outcomes revealed that with decrease in fading severity ofprimary interference link (S−P link) and wiretap link (S−Elink), and increase in link blockage probability, atmosphericturbulence and pointing error, the secrecy performance deteri-orates significantly where the HD technique overcomes thosesecrecy threats more strongly relative to the IM/DD technique.

APPENDIX

A. Proof of Λ2

Λ2 is given as

Λ2 =

∫ ∞ΨQΨT

αpδµpp

2Γ(µp)yΘpe−δpy

αpdy︸ ︷︷ ︸

P1

−∫ ∞

ΨQΨT

µr−1∑mr=0

αpδµpp

2Γ(µp)

δmrrmr!

Ψ−αrmrQ γαrmrr yΘp+αrmr

e(δpyαp+δrΨ−αr

Q γαrr yαr )dy︸ ︷︷ ︸

P2

.

By using [39, Eq. (3.381.9)], P1 is expressed as

P1 =αpδ

µpp

2Γ(µp)

∫ ∞ΨQΨT

yΘpe−δpyαpdy = ΞII5 .

P2 is written as

P2 =

µr−1∑mr=0

αpδµpp δmrr

2Γ(µp)mr!Ψ−αrmrQ γαrmrr

∫ ∞ΨQΨT

yΘp+αrmr

× e−δpyαpe−δrΨ−αr

Q γαrr yαr dy. (52)

Due to tractable analysis, assuming αp = αr and using similarformula as for P1, P2 is expressed as

P2 =

µr−1∑mr=0

αpδµpp δmrr

2Γ(µp)mr!Ψ−αrmrQ γαrmrr

∫ ∞ΨQΨT

yΘp+αrmr

× e−yαr (δp+δrΨ−αr

Q γαrr )dy =

µr−1∑mr=0

αpδµpp δmrr

2Γ(µp)mr!

×Ψ−αrmrQ γαrmrr

Γ

[Ω,(δp + δrΨ

−αrT γαrr

) (ΨQΨT

)αr]αr(δp + δrΨ

−αrQ γαrr )Ω

.

Utilizing the integral formula of [39, Eq. (8.352.7)] andperforming some mathematical manipulations, P2 is demon-strated in an alternative form as

P2 =

µr−1∑mr=0

Ω−1∑m3=0

m3∑m4=0

∞∑m5=0

(m3

m4

)(Ω +m5 − 1

m5

)× ΞII6 γ

αr(mr+m4+m5)r e−(δpΨαrQ Ψ−αr

T +δrΨ−2αrT ΨαrQ γαrr ).

(53)

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