Security Performance Analysis of Physical Layer over Fisher-Snedecor F Fading Channels
aa r X i v : . [ c s . I T ] M a y IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 00, NO. 00, MAY 2018 1
Security Performance Analysis of Physical Layerover Fisher-Snedecor F Fading Channels
Hussien Al-Hmood,
Member, IEEE, and H. S. Al-Raweshidy,
Senior Member, IEEE
Abstract —In this letter, the performance analysis of physicallayer security over Fisher-Snedecor F fading channels is inves-tigated. In particular, the average secrecy capacity (ASC), thesecure outage probability (SOP), the lower bound of the SOP(SOP L ), and the strictly positive secure capacity (SPSC) arederived in exact closed-from expressions. The Fisher-Snedecor F fading channel is a composite of multipath/shadowed fadingthat are represented by the Nakagami- m distribution. Moreover,it provides close results to the practical measurements than thegeneralised K ( K G ) fading channels. To validate our analysis, thenumerical results are affirmed by the Monte Carlo simulations. Index Terms —Fisher-Snedecor F fading, average secrecy ca-pacity, secure outage probability, strictly positive secure capacity. I. I
NTRODUCTION T HE physical layer security of the classic Wyner’s wiretapmodel has been widely analysed over multipath fadingchannels in the recent works [1]. For example, in [2] and ref-erences therein, both the main and the wiretap channels whichare the Alice/Bob and Alice/Eve channels are represented byusing various models of fading scenarios such as Rayleigh,Nakagami- m , and Rician.In a wireless communication, in addition to multipathfading, the channels may subject to the shadowing effect.Therefore, several efforts have been dedicated to study thephysical layer security under composite multipath/shadowingfading scenario [2]. For instance, in [3], the average securitycapacity (ASC), the secure outage probability (SOP), andthe strictly positive secure capacity (SPSC) over generalised- K ( K G ) fading model which is composite of Nakagami-m/Gamma distributions are derived in terms of the extendedgeneralized bivariate Meijer G-function (EGBMGF). This isbecause the statistical properties, namely, the probability den-sity function (PDF), cumulative distribution function (CDF),and the moment generating function (MGF), are derived interms of the modified Bessel functions. Therefore, to obtainsimple mathematical expressions of the performance metricsover generalised- K fading channel, a mixture gamma distri-bution is used as an approximate framework in [4]. However,the fading parameters are assumed to be integer values. Manuscript received May 20, 2018; xxxxx xxxxx xxxxx xxxxx xxxxxxxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxxxxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxxxxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx.Hussien Al-Hmood is with the Department of Electrical and Electron-ics Engineering, University of Thi-Qar, Thi-Qar, Iraq, e-mails: hussien.al-hmood@ { brunel.ac.uk, eng.utq.edu.iq } , [email protected]. S. Al-Raweshidy is with the Department of Electronic and Computer En-gineering, College of Engineering, Design and Physical Sciences, Brunel Uni-versity London, UB8 3PH, U.K., e-mail: [email protected]. More recent, the Fisher-Snedecor F fading channel has beenproposed as a composite of Nakagami- m /Nakagami- m [5]. Incontrast to the generalised- K fading channel, the statistics ofthe Fisher-Snedecor F fading channel are derived in simpleclosed-form expressions. Furthermore, the Fisher-Snedecor F fading channel includes Nakagami- m , Rayleigh, and one-sidedGaussian as special cases. Therefore, it can be employed forboth line-of-sight (LoS) and non-LoS (NLoS) communicationsscenarios with better fitting to the empirical measurementsthan the generalised- K ( K G ) fading model. However, it hasbeen utilised by one work in the open technical literature [6].Motivated by there is no work has been devoted to analysethe physical layer security over Fisher-Snedecor F fadingchannel, this paper investigates the aforementioned analysis.In particular, the ASC, the SOP, the lower bound of SOP(SOP L ), and the SPSC are derived when both the main andthe wiretap channels subject to the Fisher-Snedecor F fadingchannel. To this effect and the best of the authors’ knowledge,novel analytic results of the performance metrics are obtainedin exact closed-form mathematically tractable expressions.II. F ISHER -S NEDECOR F F ADING M ODEL
The PDF of the received instantaneous SNR, γ , usingFisher-Snedecor F distribution is expressed as [5, (5)] f γ i ( γ i ) = Ξ m i i B ( m i , m s i ) γ m i − i (1 + Ξ i γ i ) − ( m i + m si ) (1)where i ∈ { D, E } , Ξ i = m i m si ¯ γ i , m i , m s i , ¯ γ i and B ( ., . ) arethe multipath index, the shape parameter, the average SNRand the beta function defined in [7, (8.380.1)], respectively.The CDF of γ using Fisher-Snedecor F distribution is givenas [5, (4)] F γ i ( γ i ) = Ξ m i i m i B ( m i , m s i ) γ m i i × F ( m i + m s i , m i ; 1 + m i ; − Ξ i γ i ) (2)where F ( ., . ; . ; . ) is the hypergeometric function defined in[7, (9.14.1)].III. A VERAGE S ECRECY C APACITY
The ASC can be calculated by ¯ C s = I + I − I [4, (6)]where I , I , and I are given as I = Z ∞ ln (1 + γ D ) f D ( γ D ) F E ( γ D ) dγ D . (3) I = Z ∞ ln (1 + γ E ) f E ( γ E ) F D ( γ E ) dγ E . (4) EEE WIRELESS COMMUNICATIONS LETTERS, VOL. 00, NO. 00, MAY 2018 2 I = 1Γ( m D )Γ( m s D )Γ( m E )Γ( m s E ) (cid:18) Ξ E Ξ D (cid:19) m E G , , , , , , − m D − m E m s D − m E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − m E + m s E , − m E , − m E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D , Ξ E Ξ D ! (6) I = 1Γ( m E )Γ( m s E )Γ( m D )Γ( m s D ) (cid:18) Ξ D Ξ E (cid:19) m D G , , , , , , − m E − m D m s E − m D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − m D + m s D , − m D , − m D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E , Ξ D Ξ E ! (7)SOP = Ξ m D D Γ(1 − m E )Γ( m D )Γ( m s D )Γ( m E )Γ( m s E )(1 − θ ) (cid:18) θ Ξ E (cid:19) m D (cid:18) θ (1 − θ )Ξ E (cid:19) − m E (cid:18) − θθ Ξ E (cid:19) m D − m sE × G , , , , , , − m D m s E + m E − m D − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − m D + m s D , − m D , − m D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ (1 − θ )Ξ E , Ξ D (1 − θ ) (cid:18) θ (1 − θ )Ξ E (cid:19)! (13) I = Z ∞ ln (1 + γ E ) f E ( γ E ) dγ E . (5)Accordingly, I and I over Fisher-Snedecor F fadingscenarios are given in (6) and (7) at the top of this page.In addition, I is expressed as I = 1Γ( m E )Γ( m s E ) G , , − m E , , m s E , , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ! (8)where G a,bc,d ( . ) and G a ,b : ... : a n : b n c ,d : ... : c n : d n ( . ) are Meijer G-function andEGBMGF, respectively. Proof:
Substituting (1) and (2) in (3), we have I = Ξ m D D Ξ m E E m E B ( m D , m s D ) B ( m E , m s E ) × Z ∞ ln (1 + γ D ) γ m D + m E − D (1 + Ξ D γ D ) − ( m D + m sD ) × F ( m E + m s E , m E ; m E + 1; − Ξ E γ D ) dγ D (9)Invoking the identities [8, (11)], [8, (10)], and [8, (17)] withsome mathematical manipulations, (9) can be rewritten as I = Ξ m D D Ξ m E E Γ( m D )Γ( m s D )Γ( m E )Γ( m s E ) × Z ∞ γ m D + m E − D G , , , , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ D ! × G , , − m D − m s D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ξ D γ D ! × G , , − m E − m s E , − m E , − m E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ξ E γ D ! dγ D (10)Using [9, (9)] to compute the integral in (10) and doingsome mathematical simplifications, (6) is yielded which com-pletes the proof of I .Following the same steps that are employed to derive I , I can be deduced in closed-from expression as given in (7).To obtain I , we substitute (1) in (5) and recall the identity [8, (11)]. Thus, this yields I = Ξ m E E B ( m E , m s E ) × Z ∞ γ m E − E (1 + Ξ E γ E ) − ( m E + m sE ) G , , , , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ E ! dγ E (11)Employing [10, (2.24.2.4)], (8) is yielded which completesthe proof of I .IV. S ECURE O UTAGE P ROBABILITY
The SOP can be evaluated by [2, (14)]SOP = Z ∞ F D ( θγ E + θ − f E ( γ E ) dγ E (12)where θ = exp( R s ) ≥ with R s ≥ is the target secrecythreshold.The SOP can be expressed in exact closed-form as given in(13) at the top of the this page. Proof:
Inserting (1) and (2) in (12), the result isSOP = Ξ m D D Ξ m E E m D B ( m D , m s D ) B ( m E , m s E ) × Z ∞ γ m E − E ( θγ E + θ − m D (1 + Ξ E γ E ) − ( m E + m sE ) × F ( m D + m s D , m D ; m D + 1; − Ξ D ( θγ E + θ − dγ E (14)Assuming x = θγ E + θ − and dx = θdγ E and performingsome mathematical simplifications, (14) becomes as followsSOP = Ξ m D D m D B ( m D , m s D ) B ( m E , m s E )(1 − θ ) × (cid:18) θ (1 − θ )Ξ E (cid:19) − m E (cid:18) − θθ Ξ E (cid:19) − m sE × Z ∞ x m D (cid:16) − θ x (cid:17) m E − × (cid:16) E θ + (1 − θ )Ξ E x (cid:17) − ( m E + m sE ) × F ( m D + m s D , m D ; m D + 1; − Ξ D x ) dx (15) EEE WIRELESS COMMUNICATIONS LETTERS, VOL. 00, NO. 00, MAY 2018 3
SOP L = 1Γ( m E )Γ( m s E )Γ( m D )Γ( m s D ) (cid:18) θ Ξ D Ξ E (cid:19) m D G , , − m E − m D , − m D − m s D , − m D m s E − m D , , − m D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ Ξ D Ξ E ! (18)Utilising the identities [8, (10)] and [8, (17)], (15) isexpressed asSOP = Ξ m D D Γ(1 − m E )Γ( m D )Γ( m s D )Γ( m E )Γ( m s E )(1 − θ ) × (cid:18) θ (1 − θ )Ξ E (cid:19) − m E (cid:18) − θθ Ξ E (cid:19) − m sE × Z ∞ x m D G , , m E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − θ x ! × G , , − m E − m s E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ξ E θ + (1 − θ )Ξ E x ! × G , , − m D − m s D , − m D , − m D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ξ D x ! dx (16)Making use of [9, (9)], the derived result in (13) is yieldedV. L OWER B OUND OF THE S ECURE O UTAGE P ROBABILITY
The SOP L can be computed by [2, (17)]SOP L = Z ∞ F D ( θγ E ) f E ( γ E ) dγ E (17)The SOP L over Fisher-Snedecor F fading scenarios can bederived as given in (18) at the top of this page. Proof:
Plugging (1) and (2) in (17) and doing somemathematical manipulations, we haveSOP L = θ m D Ξ m D D B ( m E , m s E )Γ( m D )Γ( m s D )Ξ m sE E × Z ∞ γ m E + m D − E (cid:16) E + γ E (cid:17) − ( m E + m sE ) × F ( m D + m s D , m D ; m D + 1; − θ Ξ D x ) dγ E (19)With the help of [8, (17)], (19) can be rewritten asSOP L = θ m D Ξ m D D B ( m E , m s E )Γ( m D )Γ( m s D )Ξ m sE E × Z ∞ γ m E + m D − E (cid:16) E + γ E (cid:17) − ( m E + m sE ) × G , , − m D − m s D , − m D , − m D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ξ D θγ E ! dγ E (20)Utilising [10, (2.23.2.4)] to compute the integral in (20), theresult in (18) is deduced and this completes the proof.VI. S TRICTLY P OSITIVE S ECURE C APACITY
The SPSC is expressed as [2, (20)]SPSC = 1 − SOP for θ = 1 (21)Consequently, the SPSC over Fisher-Snedecor F fadingchannels can be obtained by using (13) and θ = 1 and insertingthe result in (21). −1 λ = ¯ γ D / ¯ γ E A S C ( m E ,m s E ) = (3 , . , (1 , . , (0 . , . , (0 . ,
50) AnalysisSimulation
Fig. 1. ASC over Fisher-Snedecor F fading channels versus λ for differentvalues of ( m E , m s E ), ¯ γ E = 5 dB, m D = 2.5, and m s D = 5 . VII. A
NALYTICAL AND S IMULATION R ESULTS
In this section, to validate our derived expressions ofthe physical layer security over Fisher-Snedecor F fadingchannels, the Monte Carlo simulations that are obtained viagenerating realizations are compared with the analyticalresults. In all figures, the simulations and the numerical resultsof the performance metrics that are plotted versus λ = ¯ γ D / ¯ γ E for m D = 2 . and m s D = 5 (moderate shadowing) arerepresented by the solid lines and the stars, respectively.Moreover, two different scenarios of the shadowing impactat the eavesdropper which are light and heavy shadowing arestudied by using m s E = 0 . and m s E = 50 , respectively.In all results, a MATHEMATICA code that is provided in[9] has been used to calculate the EGBMGF. This is becauseit is not available as a built in function in MATLAB andMATHEMATICA software packages.Figs. 1-5 show the ASC, the SOP, the SOP L , and the SPSCover Fisher-Snedecor F fading channels for ¯ γ E = 5 dB anddifferent values of the fading parameters m E and m s E . Inthese figures, it can be observed that the performance becomesbetter, when m s E increases. This is because small and largevalues of m s E correspond to light and heavy shadowing, re-spectively. For instance, in Fig. 1, when λ = 6 and m E = 0 . (fixed), the ASC for m s E = 50 is approximately higherthan m s E = 0 . . In the same context, when m E increases, theASC decreases. This refers to less impact of the multipath onthe Eve which would lead to reduce the total ASC.In Figs. 2 and 4 that are plotted for R s = 1 bit/s/Hz, onecan see that the values of SOP are greater than or equal to theSOP L which confirms our derived expressions. Furthermore,another confirmation that proves the validation of our analysisis the perfect matching between the numerical results and theirMonte Carlo simulation counterparts in all provided figures. EEE WIRELESS COMMUNICATIONS LETTERS, VOL. 00, NO. 00, MAY 2018 4 −2 −1 λ = ¯ γ D / ¯ γ E S O P ( m E ,m s E ) = (3 , . , (1 , . , (0 . , . , (0 . , Fig. 2. SOP over Fisher-Snedecor F fading channels versus λ for differentvalues of ( m E , m s E ), ¯ γ E = 5 dB, m D = 2.5, m s D = 5 , and R s = 1. −2 −1 λ = ¯ γ D / ¯ γ E S O P ( m E ,m s E ) = (3 , . , (1 , . , (0 . , . , (0 . , Fig. 3. SOP over Fisher-Snedecor F fading channels versus λ for differentvalues of ( m E , m s E ), ¯ γ E = 5 dB, m D = 2.5, m s D = 5 , and R s = 2. VIII. C
ONCLUSIONS
In this letter, the secrecy performance of physical layer overFisher-Snedecor F fading channels is analysed. Specifically,the ASC, the SOP, the SOP L , and the SPSC are derived inexact mathematically tractable closed-form expressions. Theresults of this work provide a good insight about the securityof the physical layer over composite multipath/shadowingfading channels when the wireless channels subject to heavy,moderate, or light shadowing. Moreover, the analysis of thephysical layer security over different scenarios can be deducedfrom the derived expressions by setting m and m s for specificvalues such as the Nakagami- m fading condition is obtainedby inserting m s → ∞ and m = m where m is the Nakagami- m multipath index. R EFERENCES[1] A. D. Wyner, “ The wire-tap channel, ” Bell Syst. Tech. J. , vol. 54, no. 8,pp. 1355-1387, Oct. 1975.[2] H. Al-Hmood, and H. Al-Raweshidy, “ Secrecy analysis of physical layerover κ − µ shadowed fading scenarios, ” IEEE Access , Submitted April2018, https://arxiv.org/abs/1804.09208. −2 −1 λ = ¯ γ D / ¯ γ E S O P L ( m E ,m s E ) = (3 , . , (1 , . , (0 . , . , (0 . , Fig. 4. SOP L over Fisher-Snedecor F fading channels versus λ for differentvalues of ( m E , m s E ), ¯ γ E = 5 dB, m D = 2.5, m s D = 5 , and R s = 1. −10 −8 −6 −4 −2 0 2 4 6 8 1010 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 λ = ¯ γ D / ¯ γ E S P S C ( m E ,m s E ) = (3 , . , (1 , . , (0 . , . , (0 . , Fig. 5. SPSC over Fisher-Snedecor F fading channels versus λ for differentvalues of ( m E , m s E ), ¯ γ E = 5 dB, m D = 2.5, and m s D = 5 .[3] H. Lei, H. Zhang, I. S. Ansari, C. Gao., Y. Guo, G. Pan, and K. A.Qaraqe, “ Physical layer security over generalized- K fading channels, ” IET Commun. , vol. 10, no. 16, pp. 2233-2237, July 2016.[4] H. Lei, H. Zhang, I. S. Ansari, C. Gao., Y. Guo, G. Pan, and K. A. Qaraqe, “ Performance analysis of physical layer security over generalized- K fading channels using a mixture gamma distribution, ” IEEE Commun.Lett. , vol. 20, no. 2, pp. 408-411, Feb. 2016.[5] S. K. Yoo, S. L. Cotton, P. C. Sofotasios, M. Matthaiou, M. Valkama,and G. K. Karagiannidis, “ The Fisher-Snedecor F distribution: A simpleand accurate composite fading model ” IEEE Commun. Lett. , vol. 21, no.7, pp. 1661-1664, March 2017.[6] H. Al-Hmood, “ Performance of cognitive radio systems over κ − µ shadowed with integer µ and Fisher-Snedecor F fading channels, ” in Proc. IEEE IICETA , Najaf, Iraq, May 2018, To be appear.[7] I. S. Gradshteyn, and I. M. Ryzhik,
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