This is a repository copy of Secure D2D Communication in Large-Scale Cognitive Cellular Networks with Wireless Power Transfer.

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1 This is a reository coy of Secure D2D Communication in Large-Scale Cognitive Cellular Networks with Wireless Power Transfer. White Rose Research Online URL for this aer: htt://erints.whiterose.ac.uk/683/ Version: Acceted Version Proceedings Paer: Liu, Y, Wang, L, Zaidi, SAR et al. 2 more authors) 25) Secure D2D Communication in Large-Scale Cognitive Cellular Networks with Wireless Power Transfer. In: ICC IEEE International Conference on Communications. ICC IEEE International Conference on Communications, 8-2 Jun 25, ExCel London, UK. IEEE, ISBN htts://doi.org/.9/icc c) 25 IEEE. Personal use of this material is ermitted. Permission from IEEE must be obtained for all other users, including rerinting/ reublishing this material for advertising or romotional uroses, creating new collective works for resale or redistribution to servers or lists, or reuse of any coyrighted comonents of this work in other works. Reuse Unless indicated otherwise, fulltext items are rotected by coyright with all rights reserved. The coyright excetion in section 29 of the Coyright, Designs and Patents Act 988 allows the making of a single coy solely for the urose of non-commercial research or rivate study within the limits of fair dealing. The ublisher or other rights-holder may allow further reroduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the ublisher as the coyright holder, users can verify any secific terms of use on the ublisher s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, lease notify us by ing erints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. erints@whiterose.ac.uk htts://erints.whiterose.ac.uk/

2 Secure D2D Communication in Large-Scale Cognitive Cellular Networks with Wireless Power Transfer Yuanwei Liu, Lifeng Wang, Syed Ali Raza Zaidi, Maged Elkashlan, and Trung Q. Duong Queen Mary University of London, London, UK University of Leeds, Leeds, UK Queen s University Belfast, Belfast, UK Abstract In this aer, we investigate secure device-to-device D2D) communication in energy harvesting large-scale cognitive cellular networks. The energy constrained D2D transmitter harvests energy from multi-antenna equied ower beacons PBs), and communicates with the corresonding receiver using the sectrum of the cellular base stations BSs). We introduce a ower transfer model and an information signal model to enable wireless energy harvesting and secure information transmission. In the ower transfer model, we roose a new ower transfer olicy, namely, best ower beacon BPB) ower transfer. To characterize the ower transfer reliability of the roosed olicy, we derive new closed-form exressions for the exact ower outage robability and the asymtotic ower outage robability with large antenna arrays at PBs. In the information signal model, we resent a new comarative framework with two receiver selection schemes: ) best receiver selection BRS), and 2) nearest receiver selection NRS). To assess the secrecy erformance, we derive new exressions for the secrecy throughut considering the two receiver selection schemes using the BPB ower transfer olicies. We show that secrecy erformance imroves with increasing densities of PBs and D2D receivers because of a larger multiuser diversity gain. A ivotal conclusion is reached that BRS achieves better secrecy erformance than NRS but demands more instantaneous feedback and overhead. I. INTRODUCTION Wireless ower transfer WPT) has recently received significant attention for its attractive energy harvesting caabilities and rolonging the life-time of the wireless network []. The motivation behind it is the most devices surrounded by the ambient radio-frequency RF) signals which can carry energy and information together during transmission. Two ractical receiver designs namely time switching TS) receiver and ower slitting PS) receiver were roosed in a multileinut multile-outut MIMO) system in [2], which laid a solid foundation on the research of WPT. In [3], a new concet based on ower beacons PBs) that deloy dedicated ower stations to charge the nearby mobile devices with WPT was roosed. In [3], based on stochastic geometry, the ulink erformance in cellular networks was investigated under an outage constraint. Along with imroving the energy efficiency through energy harvesting [4], another key design objective is to maximize the sectral efficiency. Cognitive radio CR) [5] and deviceto-device D2D) technology [6], have rekindled the interest of researchers to achieve a more sectrally efficient cellular networks. In [7], a wireless ower transfer rotocol for a two-ho decode-and-forward relay system is roosed in a cognitive radio network. In [8], D2D communication in energy harvesting CR networks was roosed using stochastic geometry. Furthermore, it is currently noted that CR networks are also confronted with security issues since the broadcast nature of the wireless medium is suscetible to otential security threats such as eavesdroing and imersonation. Physical PHY) layer security, which is initialed by Wyner [9] and recently aroused wide-sread interest, has been considered in CR networks []. In [], the authors revealed the imact of the rimary network on the secondary network in the resence of a multi-antenna wireta channel and resented closed-form exressions for the exact and the asymtotic secrecy outage robability in cognitive secure communications. In this aer, we consider secure communication underlay cognitive cellular networks with an energy constrained D2D transmitter. A statistical model based on stochastic geometry is used to describe and evaluate the roosed D2D communication in energy harvesting large-scale cognitive cellular networks. Differing from [3] which neglects the small-scale fading and requires energy storage units at the mobile terminals, we deloy a battery-free design [2, 3] for the energy constrained D2D transmitter. Considering the imact of smallscale fading, we roose a new WPT olicy, namely, best ower beacon BPB) ower transfer, where the transmitter selects the PB with the strongest channel to harvest the energy. We also resent a new comarative framework with the best receiver selection BRS) and the nearest receiver selection NRS) schemes. For the roosed BPB, we derive a new closed-form exression for the ower outage robability. We also derive new analytical exressions for the secrecy throughut with BRS and NRS. Our analytical and numerical results show that BRS achieves higher secrecy throughut than NRS at the cost of more instantaneous feedback and overhead. A. Network Descrition II. NETWORK MODEL We consider secure cognitive D2D communication in cellular networks, where the energy constrained D2D transmitter Alice) communicates with D2D receivers Bobs) under

3 Y meters) Alice X meters) 8 Fig.. An examle of a art of a network snashot considering that the satial distributions of PBs ink diamonds), Bobs emty circles), BSs blue five-ointed stars), and Eves red stars) follow homogeneous oisson oint rocesses PPP). malicious attemt of D2D eavesdroers Eves). The eavesdroers are assive and interret the signal without trying to modify it. It is assumed that Alice is energy constrained, i.e., the transmission can only be scheduled by utilizing ower harvested from PBs. The satial toology of all PBs, cellular base stations BSs), Bobs, and Eves, are modeled using homogeneous oisson oint rocess PPP) Φ, Φ l, Φ b, and Φ e with density λ, λ l, λ b, and λ e, resectively. As shown in Fig., we consider that Alice is located at the origin in a two-dimensional lane. For Alice, Bob, and Eve, each node is equied with a single antenna. Each PB is furnished with M antennas and maximal ratio transmission MRT) is emloyed at PBs to erform WPT to the energy constrained Alice. All channels are assumed to be quasi-static fading channels where the channel coefficients are constant for each transmission block but vary indeendently between different blocks. In this network, we assume that the time of each frame is T, which includes two time slots: ) ower transfer time slot, in which Alice harvests the ower from PBs during the β)t time, with β being the fraction of the information rocessing time; and 2) information rocessing time slot, in which Alice transmits the information signal to the corresonding Bob using the harvested energy during the βt time. B. Power Transfer Model We consider a simle yet efficient ower transfer model. It is assumed that PBs oerate on a frequency band which is isolated from the communication band where BSs and D2D transceivers schedule their transmission. Secifically, the ower transmitted by PBs does not interfere with the cellular and D2D communication. We also consider that Alice is a battery-free user, which means that there is no battery storage energy for future use and all the harvested energy during the ower transfer time slot is used to transmit the information signal [2, 3]. We roose a new best ower beacon BPB) ower transfer olicy in the ower transfer model, where Alice selects the strongest PB to harvest energy. The harvested energy of Alice from the PB can be obtained as follows E H = ηp S max h 2 Lr ) β)t, ) where η is the ower conversion efficiency of the receiver, P S is the transmit ower of PBs. Here, h is C M vector, whose entries are indeendent comlex Gaussian distributed with zero mean and unit variance emloyed to cature the effect of small-scale fading between PBs and Alice. Lr ) = Kr α is the ower-law ath-loss exonent. The ath-loss function deends on the distance r, a frequency deendent constant K, and an environment/terrain deendent ath-loss exonent α 2. All the channel gains are assumed to be indeendent and identically distributed i.i.d.). Based on ), the maximum transmit ower at Alice is given by P H = max h 2 Lr ) C. Information Signal Model ηps β). 2) β We consider the cognitive underlay scheme [4], and assume that the instantaneous CSI of the links between Alice and cellular BSs are available at Alice. Consequently, the transmit ower P A at Alice is strictly constrained by the maximum transmit ower P t at Alice and the eak interference ower I at cellular BSs according to P A = min max l Φ l I,P t h l 2 Lr l ), 3) where h l 2 Lr l ) is the overall channel gain from Alice to the BS l. Here, h l is the small-scale fading coefficient with h l CN,) and Lr l ) = Kr α l is the ower-law athloss exonent. The ath-loss function deends on the distance r l. All the channel gains are assumed to be i.i.d.. For D2D communication, we consider two receiver selection schemes. ) Best Receiver Selection BRS) scheme: Under BRS, Alice selects one Bob with the strongest channel as the desired receiver. The instantaneous signal-to-noise ratio SNR) at the selected Bob is exressed as γ B = P A max h b 2 Lr b ) N b Φ b = ζmax h b 2 Lr b ), 4) γ whereζ = min max h l 2 Lr l ), γ,n is the noise ower, l Φ l γ = I /N, γ = P t /N, h b 2 Lr b ) is the channel ower gain between Alice and Bobs, h b is the small-scale fading coefficient with h b CN,), r b is the distance between Alice and Bobs.

4 2) Nearest Receiver Selection NRS) scheme: Under NRS, Alice selects the nearest Bob as the desired receiver. The advantage of this scheme is that it reduces the system comlexity since no instantaneous CSI and feedback from Bobs are required. Then the instantaneous SNR at the selected Bob can be exressed as γ B = P A N h b 2 max Lr b ) = ζ h b 2 max Lr b ), 5) where h b is the small-scale fading coefficient of Alice to the nearest Bob with h b CN,). For the eavesdroers, the instantaneous SNR at the most detrimental eavesdroer that has the strongest SNR between itself and Alice is exressed as γ E = P A max N e Φ e = ζmax h e 2 Lr e ) e Φ e h e 2 Lr e ), 6) where h e CN,), r e is the distance between Alice and Eves. III. POWER OUTAGE PROBABILITY We assume there exists a threshold transmit owerp t, below which the transmission cannot be scheduled, the transmission cannot be scheduled and Alice is considered to be in a ower limited regime. In order to characterize the ower limited regime of Alice, we introduce ower outage robability, i.e., robability that the harvested ower is not sufficient to carry out the transmission at a certain desired quality-of-service QoS) level. The objective of this section is to quantify the ower outage robability using BPB olicy. In ractical scenario, we exect a constant ower for the information transmission. Therefore, we also denote the ower threshold P t as the transmit ower of Alice when erforming information transmission to Bobs. A. Exact Analysis for Power Transfer In this subsection, we rovide exact analysis for the roosed BPB ower transfer olicy. In this olicy, only the PB with the strongest channel transfers ower to Alice. Theorem : The ower outage robability of BPB olicy can be exressed in closed-form as λπ µ H out = e M m= Γm+) m! ), 7) βp where µ = t ηp S K β), = 2/α, and Γ.) is Gamma function. Proof: Based on 2), the ower outage robability of BPB olicy can be exressed as PrP H P t = Pr max h 2 α r µ = E Φ Pr h 2 r α µ = E Φ F h 2 r α µ), 8) where F h 2 is the CDF of h 2 and is exressed as M ) F h 2 x) = x m e x. 9) m! m= Alying the generating functional given by [5], we rewrite 8) as ] H out = ex [ λ F h 2 r α µ) )dr. ) R 2 Then changing to olar coordinates and substituting 9) into ), the ower outage robability of BPB is given by [ M µ m r mα+ e r αµ ] dr H out = ex 2πλ. m! m= ) Then alying [6, Eq )] and calculating the integral in ), we obtain the closed form exression in 7). B. large antenna array analysis for Power Transfer In this subsection, we resent large antenna array analysis for ower transfer. We first examine the distribution of h 2 when M. Since h 2 is i.i.d. exonential random variables RVs), using law of large numbers, we have h 2 a.s. M, 2) where a.s. denotes the almost sure convergence. Theorem 2: The ower outage robability of large antenna array analysis for the BPB ower transfer olicy is given by H large out βp t MηP S K β). = e λπ θ, 3) where θ = Proof: The ower outage robability of BPB for large antenna arrays analysis can be exressed as ), 4) H large out = PrP H P t = F r α θ where F r is the cumulative distribution function CDF) of r and can be exressed as F r x) = x f r )dr = e λ πx 2, 5) where r reresenting the distance from the nearest PB to Alice and its robability density function PDF) is given by f r ) = 2λ πr e λπr2. Substituting 5) into 4), we obtain 3).

5 IV. SECRECY THROUGHPUT In this section, a comarative framework is resented with two receiver selection schemes, namely, best receiver selection scheme and nearest receiver selection scheme. We use secrecy throughut as a metric to characterize the secrecy erformance. A. New Statistics Theorem 3: The PDF of ζ = P A N is given by ωl x )e ) ω l x γ γ f ζ x) =, < x < γ, 6) γ Diracx γ ),x γ e ωl γ where ω l = K πλ l Γ), Dirac ) is the Dirac delta function. Proof: See Aendix A. Theorem 4: For BRS scheme, the CDF of γ B conditioned on ζ is given by F γb ζ z) = e ω B ζ z, 7) where ω B = K πλ b Γ). For NRS scheme, the CDF of γ B conditioned on ζ is given by F γb ζ z) = 2λ b π r b e λ bπr 2 b z Kζ rα b dr b. 8) Proof: See Aendix B. Similar to 7), we can obtain the CDF of γ E conditioned on ζ as where ω E = K 2 πλ e Γ). F γe ζ z) = e ω E ζ z, 9) B. Best Receiver Selection BRS) scheme In this scheme, the instantaneous secrecy rate is defined as C BRS s = [log 2 +γ B ) log 2 +γ E )] +, 2) where [x] + = maxx,. The secrecy throughut is the average of the instantaneous secrecy rate Cs BRS over γ B and γ E. As such, the secrecy throughut using BPB ower transfer olicy is given by C BRS s = H out ) β F γe ζ x 2 ) ln2 +x 2 F γb ζ x 2 ) ) f ζ x )dx 2 dx. 2) where F γb and F γe can be obtained in 7) and 9), searately, H out is the ower outage robability in the ower transfer model. Substituting 6), 7), and 9) into 2), after some maniulation, the secrecy throughut is derived as 22) on the to ) of next age, where Q 2 = ω E x 2 + ω l γ and Q 3 = ωb x 2 + ω E x 2 + ω l γ. C. Nearest Receiver Selection NRS) Scheme In this scheme, the instantaneous secrecy rate is defined as C NRS s = [log 2 +γ B ) log 2 +γ E )] +. 23) As such, the secrecy throughut is given by C NRS s = H out ) β F γe ζ x 2 ) ln2 +x 2 F γb ζ x 2 ) ) f ζ x )dx 2 dx. 24) Substituting 6), 8), and 9) into 24), we can obtain the secrecy throughut of NRS scheme. V. NUMERICAL RESULTS In this section, reresentative numerical results are resented to illustrate erformance evaluations including ower outage robability secrecy throughut for BPB ower transfer olicy in the ower transfer model and two receiver selection schemes in the information signal model. In the considered network, we set the transmit ower of PBs as P S = 43 dbm. The carrier frequency for ower transfer and information transmission is set as 8 MHz and 9 MHz resectively. Furthermore, the bandwidth of the information transmission signal is assumed to be MHz and the information receiver noise is assumed to be white Gaussian noise with average ower -55dBm. In addition, we assume that the energy conversion efficiency of WPT is η =.8. In each figure, we see recise agreement between the Monte Carlo simulation oints marked as and the analytical curves, which validates our derivation. Fig. 2 lots the ower outage robability versus density of PBs with different ower threshold P t. The black solid curve, reresenting the BPB olicy, is obtained from 7). We observe that as density of PBs increases, the ower outage robability dramatically decreases. This is because the multiuser diversity gain is imroved with increasing number of PBs when charging with WPT. We also see that as the ower threshold increases, the outage occurs more frequently. Fig. 3 lots the ower outage robability versus M of PBs using the exact analysis and the large antenna array analysis. The dashed curve, reresenting the large antenna array analysis of BPB is obtained from 3). We see that the ower outage robability decreases with increasing M. This is because larger antenna array gain is achieved with increasing M. As M increases, the large antenna array analysis and the exact analysis have recise agreement. This is due to the fact that when M grows large, the effect of small-scale fading is averaged out. Fig. 4 lots the secrecy throughut versus density of the receivers. The solid and dashed curves, reresenting the BRS and NRS schemes, are obtained from 22) and 24), searately. Several observations are drawn as follows: ) the secrecy throughut increases with increasing density of Bobs, this is because multiuser diversity gain is imroved with increasing number of Bobs; 2) the secrecy throughut also increases with number of antennas at PBs M since lower ower outage

6 C s BRS = H out ) β ln2 ω l γ +x 2 ) e γ + Q 2 Q 3 Q 3 Q 3 e γ Q 2 Q 2 l γ )+ e ω γ ωe γ x 2 +x 2 ωb γ x e 2 ) dx 2. 22) Analytical Simulation Pt =,5, dbm) 2 Density of PBs Secrecy throughut bits/s/hz) BRS NRS Simulation M =2,8,32 2 Density of Bobs Fig. 2. Power outage robability versus density of PBs with M = 32, P S = 43 dbm, P t = dbm, and β =.5. Fig. 4. Secrecy throughut versus density of Bobs λ b with different M, where P S = 43 dbm, P t = dbm, β =.5, λ =,λ e = 3, and λ l = 3. Power outage robability Exact analytical Large analytical Large simulation M Fig. 3. Power outage robability versus M for large antenna array analysis with P S = 43 dbm, P t = dbm, and β =.5. robability is achieved with larger antenna array gain, which results in imroving secrecy throughut; and 3) BRS achieves better secrecy erformance than NRS but demands more instantaneous feedbacks and overheads. Fig. 5 shows the secrecy throughut versus P t and β for BRS and NRS schemes using BPB ower transfer olicy. We see that β and P t have joint effects on the secrecy throughut. By jointly considering β and P t, we observe that there exits an otimal value for each of these two receiver selection schemes. This behavior is exlained as follows: ) as β increases, the time for ower transfer decreases and the transmitter receives less ower, but the time for information transmission increases; and 2) on the one hand, the ower outage robability increases with increasing ower threshold. On the other hand, the transmit ower of Alice also increases since the ower threshold is the transmit ower of Alice, which results in a lower ower outage robability. As such, there exits a tradeoff between the ower outage robability and the transmit ower. In this case, it is of significance to select a suitable P t and β to transmit information to maximize the secrecy throughut. These results rovide us guidelines when roceeding the system arameters in the networks. VI. CONCLUSIONS In this aer, secure transmission in large-scale cognitive cellular networks with an energy constrained device-to-device transmitter was considered. We roosed a novel wireless ower transfer olicy in the ower transfer model, namely, best ower beacon ower transfer. We also considered best receiver selection and nearest receiver selection schemes in the information signal model. We used stochastic geometry

7 Average secrecy throughut bits/s/hz) Pt dbm) 5.5 β a) BRS scheme.5..5 Average secrecy throughut bits/s/hz) Pt dbm) 5.5 β b) NRS scheme Fig. 5. Secrecy throughut of BRS and NRS versus β and ower threshold P t, with M = 32, P S = 43 dbm, λ =,λ b = 2,λ e = 3, and λ l = 3. aroach to rovide a comlete framework to model, analyze, and evaluate the erformance of the roosed network. New analytical exressions in terms of ower outage robability and secrecy throughut are derived to determine the system security erformance. Numerical results were resented to verify our analysis and rovide useful insights into ractical design. We concluded that by carefully setting the network design arameters, along with wireless ower transfer, an accetable secure transmission can be achieved in device-todevice networks without affecting the base stations. APPENDIX A: PROOF OF THEOREM 3 We comute the CDF of ζ as follows: F ζ x) = Prζ x = Pr min γ, γ max h l 2 Lr l ) x l Φ l γ = Pr max h l 2 Lr l ) max l Φ l x, γ γ +Pr max h l 2 Lr l ) γ, γ x l Φ l γ, γ x = Pr max h l 2 Lr l ) γ, γ > x. A.) l Φ l x G l Following the similar rocedure getting 7), we obtain G l as πλl G l = e K Γ)x γ. A.2) Substituting A.2) into A.), we obtain, γ x F ζ x) = πλl e K Γ)x γ, γ > x πλl = U γ x)e K Γ)x γ, A.3),x > where Ux) is the unit ste function as Ux) =,x. By taking the derivative of F ζ x) in A.3), we obtain the PDF of ζ in 6) APPENDIX B: PROOF OF THEOREM 4 The CDF of γ B conditioned on ζ is given by F γb ζ z) = Prγ B z = Pr max h b 2 Lr b ) ζ z. B.) Following the similar rocedure getting A.2), we obtain 7). The CDF of γ B conditioned on ζ is given by F γb ζ z) = Prγ B z = Pr h b 2 rα b z Kζ ) α = e r b z Kζ f r b )dr b = 2λ b π r b e λ bπr 2 b z Kζ rα b dr b, B.2) wherer b reresents the distance form the nearest Bob to Alice with the PDF given by f r b ) = 2λ b πr b e λ bπr 2 b. Thus, we can obtain 8). REFERENCES [] T. Le, K. Mayaram, and T. Fiez, Efficient far-field radio frequency energy harvesting for assively owered sensor networks, IEEE J. Solid-State Circuits, vol. 43, no. 5, , 28. [2] R. Zhang and C. K. Ho, MIMO broadcasting for simultaneous wireless information and ower transfer, IEEE Trans. Commun., vol. 2, no. 5, , 23. [3] K. Huang and V. Lau, Enabling wireless ower transfer in cellular networks: architecture, modeling and deloyment, IEEE Trans. Wireless Commun., vol. 3, no. 2, , 24. [4] D. Liu, Y. Chen, K. K. Chai, and T. Zhang, Otimal user association for delay-ower tradeoffs in hetnets with hybrid energy sources, In Proc. IEEE 25th Annual International Symosium on Personal, Indoor, and Mobile Radio Communications PIMRC), Washington, DC, USA 24. [5] A. Goldsmith, S. A. Jafar, I. Maric, and S. Srinivasa, Breaking sectrum gridlock with cognitive radios: An information theoretic ersective, Proceedings of the IEEE, vol. 97, no. 5, , 29. [6] K. Doler, M. Rinne, C. Wijting, C. B. Ribeiro, and K. Hugl, Deviceto-device communication as an underlay to LTE-advanced networks, IEEE Commun. Mag., vol. 47, no. 2, , 29. [7] S. A. Mousavifar, Y. Liu, C. Leung, M. Elkashlan, and T. Q. Duong, Wireless energy harvesting and sectrum sharing in cognitive radio, in in Proc. IEEE Vehicular Technology Conference VTC Fall), 8th, 24,. 5. [8] A. H. Sakr and E. Hossain, Cognitive and energy harvesting-based D2D communication in cellular networks: Stochastic geometry modeling and analysis, htt://arxiv.org/abs/45.23, 24. [9] A. D. Wyner, The wire-ta channel, Bell Syst. Tech. J., vol. 54, no. 8, , 975. [] Y. Pei, Y.-C. Liang, L. Zhang, K. C. Teh, and K. H. Li, Secure communication over MISO cognitive radio channels, IEEE Trans. Wireless Commun., vol. 9, no. 4, , 2. [] M. Elkashlan, L. Wang, T. Q. Duong, G. K. Karagiannidis, and A. Nallanathan, On the security of cognitive radio networks, Acceted by IEEE Trans. Veh. Technol., 24. [2] I. Krikidis, S. Sasaki, S. Timotheou, and Z. Ding, A low comlexity antenna switching for joint wireless information and energy transfer in mimo relay channels, IEEE Trans. Commun., 24. [3] Y. Liu, L. Wang, M. Elkashlan, T. Q. Duong, and A. Nallanathan, Two-way relaying networks with wireless ower transfer: Policies design and throughut analysis, In Proc. IEEE Global Commun. Conf. GLOBECOM), Austin Texas, USA 24. [4] S. A. R. Zaidi, D. C. McLernon, and M. Ghogho, Breaking the area sectral efficiency wall in cognitive underlay networks, IEEE J. Sel. Areas Commun., vol. 32, no., 24. [5] W. K. D. Stoyan and J. Mecke, Stochastic Geometry and its Alications, 2nd ed. John Wiley and Sons, 996. [6] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. New York, NY, USA: Academic Press, 2.