Secrecy Outage Analysis over Correlated Composite Nakagami-m/Gamma Fading Channels

Transcription

1 Secrecy Outage Anaysis over Correated Composite Nakagami-m/Gamma Fading Channes George C. Aexandropouos, Senior Member, IEEE and Kostas P. Peppas, Senior Member, IEEE arxiv:7.9v [cs.it] 3 Oct 7 Abstract The secrecy outage performance of wireess communication systems operating over spatiay correated composite fading channes is anayzed in this paper. We adopt a mutipicative composite channe mode for both the egitimate communication ink and the ink between the eavesdropper and the egitimate transmitter, consisting of Nakagami-m distributed sma-scae fading and shadowing arge-scae fading) modeed by the Gamma distribution. We consider the reaistic case where sma-scae fading between the inks is independent, but shadowing is arbitrariy correated, and present nove anaytica expressions for the probabiity that the secrecy capacity fas beow a target secrecy rate. The presented numericay evauated resuts, verified by equivaent computer simuations, offer usefu insights on the impact of shadowing correation and composite fading parameters on the system s secrecy outage performance. Index Terms Fading correation, Gamma distribution, Nakagami-m fading, physica ayer security, secrecy capacity. I. INTRODUCTION Physica Layer Security PLS) has been recenty considered as a companion technoogy to conventiona cryptography offering the potentia to significanty enhance the quaity of secure communication in fifth generation 5G) wireess networks []. In the pioneering work of Wyner in information theoretic security [], it was shown that secure communication is feasibe when the channe quaity of egitimate parties is better than that of the eavesdropper. However, in practice, there are certain cases where the atter channes may experience correated conditions, which wi intuitivey render the performance of PLS schemes imited. Spatia fading correation highy depends on antenna depoyments, proximity of the egitimate receiver and eavesdropper, as we as scatters around them [3]. Assuming that the egitimate transmitter knows the channe gains towards the egitimate receiver and eavesdropper in [4], the oss of the secrecy capacity due to spatia correation was quantified. Infinite series expressions for both the average secrecy capacity and outage probabiity were obtained in [5] for correated Rayeigh fading channes. By considering that the egitimate communication ink and the ink between the eavesdropper and the egitimate transmitter are arbitrariy correated and both modeed by the og-norma distribution, [6] studied the Probabiity of the Non Zero Secrecy Capacity PNZSC). The PLS of Mutipe-Input Mutipe-Output G. C. Aexandropouos is with the Mathematica and Agorithmic Sciences Lab, Paris Research Center, Huawei Technoogies France SASU, 9 Bouogne-Biancourt, France. The views expressed here are his own and do not represent Huawei s ones. e-mai: george.aexandropouos@huawei.com). K. P. Peppas is with the Department of Teecommunication Science and Technoogy, University of Peoponnese, Tripoi, Greece e-mai: peppas@uop.gr). MIMO) wiretap channes with orthogona space-time bock codes was investigated in [7]. In that work, the fading channes between the egitimate ink and the ink between the eavesdropper and the egitimate transmitter were assumed to be independent and modeed as Ricean and Rayeigh distributed, respectivey. However, within each communication ink the mutipe fading channes, resuting from the utiization of mutipe antennas, were assumed to be arbitrariy correated. Recenty in [8], the average secrecy capacity and Secrecy Outage Probabiity SOP) were studied for the cases where egitimate and eavesdropper inks experience independent ognorma fading, correated og-norma fading, or independent composite fading conditions. In the context of underying cognitive radio networks, the SOP performance was aso atey investigated in [9] considering correated Rayeigh fading. Motivated by the atest advances in the secrecy capacity anaysis [] and aiming at studying PLS performance under more reaistic fading conditions, we adopt in this paper a correated composite fading channe mode for the egitimate and eavesdropping inks. Our mode comprises of independent sma-scae fading and arbitrariy correated shadowing. For the sma-scae fading we consider the versatie Nakagamim fading mode [], whie shadowing arge-scae fading) is modeed by the Gamma distribution. We first present a nove anaytica expression for the numerica SOP evauation. Then, for the important specia case of non zero secrecy capacity, a nove infinite series representation for PNZSC is deduced. Finay, in order to obtain further insights on the key factors affecting PLS performance, a simpe cosed form expression for PNZSC that becomes asymptoticay tight for high vaues of the Signa-Noise-Ratio SNR) is presented. A derived anaytica resuts are substantiated with equivaent ones obtained by means of computer simuations. Notations: E{ } denotes expectation, Γ ) is the Gamma function [, eq. 8.3/)], x) i Γx + i)/γx) is the Pochhammer s symbo [, p. xiii], u ) is the unit step function [, p. xiv], K a ) is the modified Besse function of the second kind and order a [, eq. 8.47/)], U,, ) is the Kummer hypergeometric function [, eq. 9./)], and G m,n p,q[ ] is the Meijer s G-function [, eq. 9.3)]. II. SYSTEM AND CHANNEL MODELS We consider a egitimate wireess communication ink where a egitimate transmitter sends a message to the egitimate receiver B, whie the eavesdropper E attempts to decode this message from its received signa through the wireess ink between itsef and the egitimate transmitter. The channe inks

2 are assumed to be arbitrariy correated due to either cose proximity ofb ande or simiarity of the scatters around them. In addition, we assume that both channes experience ergodic bock fading, where channe coefficients remain constants during a bock period and vary independenty from one bock to the next one. We aso consider, simiar to [4] [9], [], [3] [7], that the channe coefficients from the egitimate transmitter to B and to E are ideay estimated in B and E, respectivey. In cases of active eavesdropping, E is capabe of estimating its corresponding channe as B does, whereas in other cases, it needs to eavesdrop characteristics of the channe estimation process e.g., the egitimate transmitter s piots signas). Assuming narrowband communication inks, the baseband received compex-vaued signas at B and E, respectivey, can be mathematicay expressed as y B = ph B s+n B, y E = ph E s+n E, a) b) where p denotes the fixed average power of the egitimate transmitter and s is its unit power compex-vaued information message chosen from a discrete moduation set. In ), h B and h E represent the compex channe gains from the egitimate transmitter to B and to E, respectivey. Aso, n B and n E denote the zero mean Additive White Gaussian Noises AWGNs) at B and E, respectivey, with variances σ B and σ E. Both wireess channes are assumed to be subject to composite propagation conditions incorporating mutipath fading and shadowing. The former is modeed by the versatie Nakagami-m distribution, whie the atter by the Gamma distribution. In mathematica representation, we mode the ampitudes of the channe gains as g B h B = b w and g E h E = b w, where b and b are Gamma random variabes RVs) with shaping parameters k and k and scaing parameters θ and θ, respectivey. In addition, w and w are assumed to be Nakagami-m RVs with shaping parameters m and m and average powers Ω E{w } and Ω E{w }, respectivey. Due to either cose proximity of B and E and/or simiarity of the scatters around them, we consider the reaistic case whereg B andg E are correated RVs resuting from correated shadowing, but sma-scae fading is assumed to be independent betweenb ande. As such,w and w are assumed to be independent Nakagami-m RVs, whereas b and b are modeed as correated Gamma RVs. Capitaizing on the system mode of a), the instantaneous received SNR at B is given by γ pgb /σ B with average vaue derived as γ pe{gb }/σ B, where E{g B } = k θ Ω. Simiary from b), the instantaneous received SNR at E and its average vaue are given by γ pge /σ E and γ pk θ Ω /σb, respectivey. The joint Probabiity Density Function PDF) of γ and γ for the considered arbitariy correated composite Nakagami-m/Gamma fading channe mode can be obtained by empoying [8, eq. 3)] for the specia case of independent Nakagami-m RVs and after using a standard transformation of RVs, yieding x,x ) = 4 ρ)k Γm )Γm ) = i,j= k ) i k k ) j ρ i+j i!j!i+k ) j A ξ xξ Γi+k ) K ψ ) A x, ) where ξ m + k + i)/, ξ m + k + i + j)/, ψ m k i, ψ m k i j, and A m k ρ)γ. In the atter PDF expression, ρ [, ) represents the correation coefficient between the RVs b and b [8, Sec. II]. III. SECRECY PERFORMANCE ANALYSIS In this section, we present nove anaytica expressions for the SOP and PNZSC performance of the considered PLS communication system operating over arbitrariy correated composite Nakagami-m/Gamma fading channes. A. Secrecy Outage Probabiity SOP) The SOP performance of the PLS system described in Section II is given by the foowing probabiity [3, eq. 7)] P o r) Pr[γ > r +γ ) ] = hx,r) x,x )dx dx, 3) where r denotes the target secrecy rate in bps/hz and hx,r) + x ) r. Based on the atter integra expression, we estabish in the foowing proposition a method for the efficient numerica SOP evauation. Proposition. The SOP of the considered PLS system can be tighty approximated numericay using the expression given by 4) top of next page), wherew k andt k fork =,,...,5 are the weights and abscissas given in [9, Tabs. II and III]. Proof. Substituting the joint PDF of γ and γ given by ) into 3), the foowing two-fod integra is deduced I= x u ) hx,r) = x ξ K ψ A x )dx. 5) The inner integra, i.e., the one with respect to x, can be computed in cosed form by expressing the Besse and unit step functions in terms of Meijer s G-functions, i.e., as K ν x) =.5 πg,, [ x ν/,ν/] [, eq /)] and ux ) = G,, [x ] [, eq. 8.4./)], respectivey. Then, by empoying the integra expression [, eq..4./)], 5) can be simpified to the foowing singe integra I = A ξ x ξ G,3 3, [ A hx,r) ], ψ /+ξ,ψ /+ξ K ψ A x )dx. It is noted that a necessary conditions for the existence of [, eq..4./)] are satisfied throughout this paper s anaysis. The integra in 6) cannot be in genera soved in cosed form when r > hods. However, by empoying the identity K ν x) = πe x x) ν U.5+ν,+ν,x) [, eq. 6)

3 3 P o r) = 4 π ρ)k Γm )Γm ) i,j= k ) i k k ) j i k+ ρ i+j i!j!i+k ) j Γi+k )Γi+k ) 5 k= w k t 4m k [ G 3, r k m t 4 k γ,3 + r )k m,i+j+k k m γ ρ)γ,m ]U m i k +/,m i k +,t k ) 4) P o ) = ρ)k Γm )Γm ) i,j= k ) i k k ) j ρ i+j [ m k γ ] i!j!i+k ) j Γi+k )Γi+k ) G,3, m, i j k 3,3 i+k m k γ,m, 7) 9.38/3)] as we as the change of variabes A x = y, the resuting integra can be efficienty evauated numericay by using the modified Gauss-Chebyshev quadrature technique described in [9]. Foowing this technique, SOP can be numericay evauated as in 4), thus, competing the proof. B. Probabiity of Non Zero Secrecy Capacity PNZSC) PNZSC defined using 3) as P o ) often serves as a fundamenta benchmark on the secrecy performance of PLS systems [3]. Athough it can be numericay approximated for the considered PLS system from 4) after setting r =, we next present a nove anaytica PNZSC infinite series representation. Proposition. An infinite series expression for PNZSC for the considered PLS system is given by 7) top of this page). Proof. Starting from 3), the PNZSC P o ) is obtained as P o r) = x x,x )dx dx. 8) Substituting the joint PDF expression ) into 8), the foowing two-fod integra appears in the PNZSC expression J = x ξ K ψ A x )dx. 9) x = The atter inner integra with respect to x can be soved using [, eqs /), 8.4./), and.4./)] yieding J = A ξ x ξ G 3, [ ],3 A x, ψ /+ξ,ψ /+ξ K ψ A x )dx. Finay, 7) is deduced after using [, eq..4./)]. C. Asymptotic Anaysis for PNZSC ) To gain further insights on the impact of the composite fading parameters as we as of shadowing correation on the considered PLS system s performance, we next present a cosed form asymptotic expression for PNZSC that is vaid for high vaues of the average received SNRs. Proposition 3. For high vaues of γ, PNZSC can be obtained from the foowing expression with α min{k,m }: P o ) γ = Γ k ρ)k m )Γk +α ) α = Γm )Γk ) α m k γ Γm +α ). ) m k γ Proof. When γ, hods θ. For this asymptotic case, the joint Moment Generating Function MGF) of b and b obtained using [, eq. 7)] can be approximated as M b,b s,s ) γ = = θk ρ) k s + θ ρ) ) k. ) By taking the inverse Lapace transform of the atter MGF, the joint PDF of b and b can be asymptoticay approximated as f b,b y,y ) γ = y k ρ) k Γk )Γk )θ k θk y k e y = θ ρ). 3) The joint PDF of RVs g B and g E can be derived as foows f gb,g E x,x ) = f gb b x y )f ge b x y ) f b,b y,y )dy dy, 4) where f gb b ) denotes the PDF of g B conditioned on b and f ge b ) denotes the PDF of g E conditioned on b. Based on the channe mode in Section II, the atter PDFs are the margina Nakagami-m PDFs with average powers E{g B b } = b Ω and E{g E b } = b Ω, respectivey. Using the transformations of RVs γ = g B γ /k θ Ω and γ = g E γ /k θ Ω in the joint PDF definition 4) yieds after some agebraic manipuations the foowing asymptoticay approximate bivariate PDF expression x,x ) γ = 4 ρ) k [ x ξ A ξ Γk )Γm ) K ψ ) ] A x, 5) = where ξ m +k )/ and ψ m k. The proof competes by using the identityk α x) x = /x) α Γ α )/, a simiar ine of arguments as in the proof of Proposition, and [, eq..4./)] for evauating the finay resuting singe integra with respect to x.

4 4 Secrecy Outage Probabiity SOP) k =, m =, =. k =, m = 4, =. k = 8, m = 4, =. k =, m =, =.6 k =, m = 4, =.6 k = 8, m = 4, =.6 k =, m = 4, =.9 Simuations Probabiity of Non Zero Secrecy Capacity PNZSC) =. =.6 =.9 Asymptotics Simuations k = k = k = k = 8 Target Secrecy Rate in bps/hz Figure. SOP vs r in bps/hz for various vaues of the correation coefficient ρ, the sma-scae shaping parameter m, and the shadowing parameter k. Average Received SNR in db Figure. PNZSC vs γ in db for γ = db and m = 4, different vaues of the correation coefficient ρ, and the shadowing parameters k and k. IV. NUMERICAL RESULTS AND DISCUSSION In this section, we numericay evauate the anaytica expressions 4), 7), and ) for the secrecy outage performance of the considered PLS communication system that operates over arbitrariy correated composite Nakagami-m/Gamma fading channes. In the two figures that foow we aso incude equivaent resuts obtained by means of computer simuations in order to verify the correctness of the presented mathematica formuas. For the numerica evauation of the doube infinite series appearing in 4) and 7), we have truncated both series in each expression to the same finite number of terms N eading to a perfect match with equivaent computer simuations up to the third significant digit. In genera, N increases with increasing vaues of any of the parameters ρ, m, m, k, and k, and decreases as the average SNR increases. To further decrease the computationa compexity of 4) and 7), the incuded Kummer hypergeometric function and the Meijer G- function have been first precomputed and then stored, and finay used in the evauation of the respective truncated series. Figure iustrates SOP versus r in bps/hz for the common average SNR vaues γ = γ = 4dB, various vaues of the correation coefficient ρ, of the common shadowing shaping parameter k k = k, and of the common sma-scae shaping parameter m m = m. PNZSC as a function of γ in db is depicted in Fig. for γ = db, m = 4, as we as different vaues of ρ and the shadowing parameters k and k. For the SOP resuts in Fig. we have used from N = for k =, m =, and ρ =.) to N = 45 for k =, m = 4, and ρ =.9) terms to truncate both infinite series incuded in 4). The corresponding range of terms in Fig. for the PNZSC curves obtained using 7) is from N = to N = 3. As shown in both figures and as expected, SOP degrades with increasing r and PNZSC improves with increasing γ. In addition, increasing ρ and/or the shadowing parameters degrades SOP for the potted range of r in Fig., and improves PNZSC as γ increases as depicted in Fig.. This trend for the SOP and PNZSC performance agrees with that in [4] [8], [3] either correated sma-scae fading or correated shadowing was considered. The numericay evauated performance resuts of the anaytica expressions 4), 7), and ) incuded in Figs. and revea that arge and severey correated shadowing in the egitimate receiver and eavesdropper might have a detrimenta effect in the secrecy outage performance, even if sma-scae fading between these nodes is independent. Future extensions of our framework incude the consideration of MIMO techniques at some or a communications ends and the anaysis of the impact of imperfect channe estimation. REFERENCES [] N. Yang, L. Wang, G. Geraci, M. Ekashan, J. Yuan, and M. Di Renzo, Safeguarding 5G wireess communication networks using physica ayer security, IEEE Commun. Mag., vo. 53, no. 4, pp. 7, Apr. 5. [] A. D. Wyner, The wire-tap channe, Be Syst. Tech. Journ., vo. 54, pp , Jan [3] D.-S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, Fading correation and its effect on the capacity of mutieement antenna systems, IEEE Trans. Commun., vo. 48, no. 3, pp. 5 53, Mar.. [4] H. Jeon, N. Kim, J. Choi, H. Lee, and J. Ha, Bounds on secrecy capacity over correated ergodic fading channes at high SNR, IEEE Trans. Inf. Theory, vo. 57, no. 4, pp , Apr.. [5] X. Sun, J. Wang, W. Xu, and C. Zhao, Performance of secure communications over correated fading channes, IEEE Signa Process. Lett., vo. 9, no. 8, pp , Aug.. [6] X. Liu, Outage probabiity of secrecy capacity over correated ognorma fading channes, IEEE Commun. Lett., vo. 7, no., pp. 89 9, Feb. 3. [7] N. S. Ferdinand, D. B. da Costa, and M. Latva-aho, Physica ayer security in MIMO OSTBC ine-of-sight wiretap channes with arbitrary transmit/receive antenna correation, IEEE Wireess Commun. Lett., vo., no. 5, pp , Oct. 3. [8] G. Pan, C. Tang, X. Zhang, T. Li, Y. Weng, and Y. Chen, Physicaayer security over non-sma-scae fading channes, IEEE Trans. Veh. Techno., vo. 65, no. 3, pp , Mar. 6. [9] J. Zhang, H. Zhao, and G. Pan, Secrecy outage anaysis for underay cognitive radio networks over correated channes, SCIENCE CHINA Inf. Sci., vo. 6, pp. 9, Feb. 7. [] K. Cumanan, G. C. Aexandropouos, Z. Ding, and G. K. Karagiannidis, Secure communications with cooperative jamming: Optima power aocation and secrecy outage anaysis, IEEE Trans. Veh. Techno., vo. 66, no. 8, pp , Aug. 7.

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