Available online at ScienceDirect. Procedia Computer Science 40 (2014 ) 84 91

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1 Available online at ScienceDirect rocedia Computer Science ) Outage robability of Cognitive Relay Networks over Generalized Fading Channels with nterference Constraints etros S. Bithas a,b,, George. Efthymoglou c, Dimitris S. Kalivas b a nstitute for Astronomy, Astrophysics, Space Applications and Remote Sensing, National Observatory of Athens, GR-1536, enteli, Greece b Department of Electronics, Technological Educational nstitute of iraeus, Thivon 50 &. Ralli, GR-144, Egaleo, Greece c Department of Digital Systems, University of iraeus, Karaoli and Dimitriou 80, iraeus, Gr-18534, Greece Abstract n this paper, we evaluate the outage probability O) of a cognitive relay network operating over generalized fading channels, modeled by the generalized-k and generalized-gamma distributions. n particular, secondary users, based on the underlay approach, cooperate employing decode-and-forward protocol, satisfying in any case an interference constraint on the primary destination users. The derived results include exact expressions as well as approximated ones for high values of the maximum allowed transmitted power. Numerical evaluated results show that the O of cognitive relay networks is highly related with the fading/shadowing channel conditions as well as interfering constraints, with the latter resulting in higher O compared to the conventional relaying systems. c 014 The Authors. ublished by Elsevier B.V. This is an open access article under the CC BY-NC-ND license eer-review under reonsibility of organizing committee of Fourth nternational Conference on Selected Topics in Mobile & Wireless eer-review Networking under reonsibility MoWNet 014). of organizing committee of Fourth nternational Conference on Selected Topics in Mobile & Wireless Networking MoWNet 014) Keywords: Cognitive radio systems, decode-and-forward DF) relaying, generalized-gamma, generalized-k, outage probability, shadowing. 1. ntroduction Cognitive radio has been considered as an ideal candidate for improving the ectrum utilization and thus increasing the overall system throughput. However, in these networks, interfering effects play a dominant role and thus various approaches have been proposed to protect licensed primary) users data transmission, namely ectrum underlay, overlay and interweave 1. As far as the underlay approach is concerned, which is also the subject of this work, the basic idea is that as long as the interference generated by the secondary users does not exceed a prede ned threshold at the primary receivers, the secondary transmitters are allowed to transmit. Therefore, based on this approach and in order to extend the coverage of secondary transmission and further enhance the reliability in cognitive radio networks, cooperative relaying techniques have been adopted 3. The research area of underlay cognitive relay networks has recently gained an increased interest as it is proved by the numerous contributions that exist, for example 3,4,5,6,7,8,9. More eci cally, in 4, the outage probability O) of a cognitive relay network, employing secondary transmissions based on the underlay approach, i.e., constrained Correonding author. Tel.: ; fax: address: pbithas@ieee.org The Authors. ublished by Elsevier B.V. This is an open access article under the CC BY-NC-ND license eer-review under reonsibility of organizing committee of Fourth nternational Conference on Selected Topics in Mobile & Wireless Networking MoWNet 014) doi: /j.procs

2 etros S. Bithas et al. / rocedia Computer Science ) rimary D ) Relaying links nterfering links h rp h Relay R S ) h sr h rd Source S S ) Destination D S ) Fig. 1. The mode of operation of the underlay cognitive network under consideration. generated interference, and supporting a suitable selection criterion was investigated. t was shown that the O of the cognitive relay network is always higher than that of conventional one due to the interference constraint. n 5, this approach was extended to Nakagami-m fading channels, where the exact O was derived, while the impact of various key system parameters, such as interference temperature and fading severity, was also investigated. More recently, in 9, the approach of limited maximum transmit power at the secondary source was included in a cooperative diversity scheme for cyclic pre xed single-carrier systems in a ectrum sharing environment with the decode-andforward DF) relaying protocol. As a general observation it should be mentioned that the previously published papers, consider only small scale effects deite the fact that the large scale effects shadowing) become dominant in many land-mobile communication scenarios 10. Motivated by the preceding, in this paper we extend the analysis presented in these works to generalized fading environments modeled in our case using the generalized-k K G ) and generalized-gamma G G ) distributions 10,11.Both these fading distributions are very important since they model various fading as well as shadowing conditions, e.g., from worse than Rayleigh to no fading. n particular, we derive novel exact expressions for the O of an underlay cognitive network employing dual-hop DF relaying for both composite fading environments. n addition, considering higher values of the maximum available transmit power, convenient closed-form asymptotic expressions have been extracted. The rest of the paper is organized as follows. Section contains the general description of the system and channel model under consideration. n Section, the O analysis is performed for both generalized fading models under consideration. Section V presents some numerical results and Section V includes the concluding remarks.. System and Channel Model We consider a cognitive relay network, where primary users coexist with secondary ones, as shown in Fig. 1. At the rst phase, the secondary source S S ) transmits a signal only to the secondary relay R S ), since the direct path between S S and secondary destination D S ) is considered to be blocked. R S decodes the received data and at the second phase it forwards it to D S. n the adopted mode of operation, all secondary transmissions are allowed as long as they do not impose harmful interference at primary destination D ). n other words, they could utilize the primary users ectrum only in cases where the generated interference on the D remains below an interference threshold, which represents the maximum tolerable interference level at which the primary user can still maintain reliable communication 4. n addition, as in 4, it is assumed that R S and D S are not subject to any interfering effects coming from the primary source S ). Therefore, based on the above assumptions the following transmission power constraints exist for the source and relay, reectively, 5 ) s min h, r min h rp, ) 1) where is the maximum transmission power constraint, which without loosing the generality, has been considered equal to both links. n addition h is the channel gain from S S to D and h rp is the channel gain from R S to D.

3 86 etros S. Bithas et al. / rocedia Computer Science ) The channel state information of h, h rp could be available at S S and R S by employing pilot signals from a primary receiver 1 or based on channel information estimator without feedback) 4,13. n this case the capacity of the secondary user based on a unit bandwidth is given by 14 C = 1 { min h sr ) h rd )} log 1 + s, log 1 + r where h sr is the channel gain from S S to R S, h rd is the channel from R S to D S and N 0 is the noise power. n this study, two wireless communication scenarios will be investigated as described below..1. Fading scenario A n this scenario, all links are subject to composite fading, modeled in our case using the K G distribution. Therefore, the power probability density function DF) of the channel gains h X, with X {sr, rd,, rp} is given by 15 γ β X 1)/ f X γ) = Ξβ X+1)/ X Γ m X ) Γ k X ) N 0 N 0 ) K αx [ ΞX γ) 1/], γ 0 3) with α X = k X m X,β X = k X + m X 1, where k X, m X are the distribution s shaping parameters, Ξ X = k X m X )/γ X, with γ X being the distribution s scaling parameter, Γ ) is the Gamma function 16 eq /1) and K v ) is the second kind modi ed Bessel function of vth order 16 eq /1). Since K G is a two parameters distribution, 3) can describe various fading and shadowing models by using different value combinations for k X and/or m X 17, e.g., as k X,it approximates Nakagami-m distribution or for m X = 1, it coincides with the K-distribution and approximately models Rayleigh-lognormal fading conditions 18. For integer values of the m X, the correonding cumulative distribution function CDF) of 3) is 19 F X γ) = 1 Ξ Xγ) kx Γ k X ) m X 1 q=0 Ξ X γ) q q! K kx q ΞX γ ). 4).. Fading scenario B For the second case, all links are subject to generalized fading, modeled in our case using the G G distribution. Therefore, the DF of the channel gain h X is given by 0 b X γ m Xb X ) / 1 bx / f X γ) = Γm X ) ) τ X γ exp mx b X / γ X τ X γ 5) X where b X > 0andm X 1/ are the distribution s shaping parameters related to the fading severity and τ X = Γm X )/Γm X +/b X ). For different values of m X and b X,3)simpli es to several important distributions used in fading channel modeling, e.g., Rayleigh, Nakagami-m and Weibull 0. ts correonding CDF can be expressed as F X γ) = 1 Γ [ m X, γ/ )) τ X γ ] bx / X 6) Γm X ) where Γ, ) is the upper incomplete Gamma function 16 eq /). 3. Outage robability Analysis n this section the O of the cognitive relay network under consideration for both fading channel models will be analyzed. The O is de ned as the probability that the instantaneous output signal to noise ratio SNR) falls below a

4 etros S. Bithas et al. / rocedia Computer Science ) predetermined threshold γ th and is given by 5 h sr h rd out = r min s, r N } {{ 0 N }} {{ 0} Z s Z r <γ th = F Zs γ th ) + F Zr γ th ) F Zs γ th )F Zr γ th ). 7) n 7), the de nition of two new random variables RV)s Z s, Z r is provided. t has been proved that the CDF of Z s is given by 15 γth N ) 0 ) F Zs γ th ) =F sr F + γth N 0 x ) f x)f sr dx. / } {{ } t is noted that F Zr γ th ) can be evaluated by substituting in 8) sr, with rd, rp, reectively. Next, integral will be evaluated for both scenarios under consideration Fading scenario A Substituting 3) and 4) in of 8), integrals of the following form appear 1 = = / / x k+m 1 K α [ Ξ x ) 1/ ] dx k+m+ksr +q x 1 K α [ Ξ x ) 1/ ] K ksr q Ξsr γ th N 0 x 8) ) 1/ dx. 9) ntegral 1 can be evaluated in closed form by making a change of variables of the form y = x 1/, changing the integration limits and employing 16 eq /16) as well as 1 eq ), to yield 1 = Γk )Γm ) + π csc [ π )] k m Ξ Ξ k+m Γm ) ) m Ξ k+m Ξ F Q m ;1+ m, 1 k + m ; Ξ ) k Γk ) F Q k ;1+ k, 1 + k m ; Ξ ) where F Q ) is the regularized generalized hypergeometric function 1 eq ). Obtaining a closedform expression for seems to be impossible. An alternative approach is to employ the in nite series representation for Bessel K v z) function, i.e., K v z) = π cscπv) z/) i v i=0 Γi v+1)i! ) z/) i+v i=0 Γi+v+1)i! 1 eq ). Substituting this in nite series expression in, integrals of the same form as 1 appear, which can be solved by following a similar approach as the one for deriving 10). Therefore, based on these solutions the nal expression for is given by = 1 + π csc[πk m )] Γm )Γk ) Ξ F Q m ;1+ m, 1 k + m ; Ξ ) Γm ) ) k Γk ) F Q k ;1+ k, 1 + k m ; Ξ ) Ξ )] ) m ] π csc[πk m m )] sr 1 Γm )Γk )Γk sr ) q=0 10) [ F k, m ) F m, k )] q! 11)

5 88 etros S. Bithas et al. / rocedia Computer Science ) where Ξ i+y [ /i! Γi + ksr + y)γi + y + q) F x, y) = Γi x + y + 1) i=0 Ξ sr N 0 γ th /) i+y F Q ) ksr ) q Ξsr N 0 γ th Γq ksr ) i + k sr + y Ξsr N 0 γ th ) i+y Γk sr q) i + y + q F Q with F Q ) being the generalized hypergeometric function 16 eq. 9.14). i + k sr + y;1+ i + k sr + y, 1 + k sr q; Ξ srn 0 γ th i + q + y;1 k sr + q, 1 + i + y + q; Ξ srn 0 γ th ) ) i+y Asymptotic analysis n order to better understand the impact of the interference temperature constraint on the O, similar to 5,we will present an asymptotic analysis. n particular, we consider the case of higher values of. Following such an approach and using 1 eq ), the hypergeometric function simpli es to F Q a 1 ; b 1, b ; z) 1, for z 0). Therefore, based on this simpli cation using the de nitions of the Gauss hypergeometric function as well as the generalized hypergeometric function, 16 eqs ) and 9.14), reectively, and after some mathematical manipulations, 11) simpli es to the following closed-form expression ) 1) where 1 1 m Ξ π csc [ π k m )] Γk sr )Γm )Γk ) ) m Γk m ) Γm )Γk ) 1 k m sr 1 q=0 Ξ ) k Γm k ) Γm )Γk ) 1 [ ) )] Q k, m Q m, k q! ) Ξ y ) Γk sr + y)γy + q) Ξ Qx, y) = F 1 k sr + y, y + q; y x + 1; Ξ sr N 0 γ th Γy x + 1) Ξ sr N 0 γ th ) ksr ) Ξsr N 0 γ th Ξ y ) q ) Γk sr + y)γq k sr ) Γk sr + y + 1)Γy x + 1) Ξsr N 0 γ th Ξ y Γy + q)γk sr q) Γy + q + 1)Γy x + 1). 13) 14) 3.. Fading scenario B Substituting 5) and 6) in of 8), and using the in nite series representation for the upper incomplete gamma function 16 eq /), i.e., Γa, z) =Γa) z a i=0, integrals of the following form appear 3 = / z) i a+i)i! ) b / x t exp x τ γ dx 15) where t = m b 1ort = m b +ib sr +m sr b sr 1. These integrals can be solved by making a change of variables of the form y = x b/ and employing 16 eq /). Based on this solution and after some mathematical manipulations yields the following exact expression for of 8) = 1 ) b / / Γm ) Γ m, τ γ [1 Γm sr)] ) 1 τ γ N 0 γ mb th ) τ γ N 0 γ ib th 1) i + Γm ) τ sr γ sr τ sr γ sr m sr + i)i! Γ m + ib sr + m ) 16) srb sr / b, b b τ γ. i=0

6 etros S. Bithas et al. / rocedia Computer Science ) Fig.. K G fading: Outage probability as a function of /N 0 for various values of the interference constraints Asymptotic analysis For higher values of γ sr and, the upper incomplete Gamma function of the CDF expression in 6), simpli es as follows Γa, z) =Γa) za a,forz 0) 1 eq ). Based on this expression and following a similar approach for deriving 16), the following closed-form expression has been obtained 1 Γm sr) Γm ) Γm ) τ γ 1 τ γ N 0 γ th + Γm )m sr τ sr γ sr ) msr bsr ) mb 1 m Γ m + m srb sr b ) τ γ ) ) m + msrbsr b b 1 m + m srb sr b. 17) 4. Numerical Results n this section, using the previously derived expressions for the instantaneous output SNR, the O of the underlay dual-hop DF relay scheme will be studied. n Fig., considering K G fading/shadowing environment and assuming k = 0.9, m = 0.6, γ = 0dB, k sr =.9, m sr =, γ sr = 0dB, γ th = 10dB, the O is plotted as a function of the ratio /N 0 for various values of /N 0.nthesame gure, for comparison purposes, the asymptotic O, obtained using 13) as well as the correonding performance of the system under consideration without interference temperature constraint, are also plotted. n this gure, it is shown that the performance improves as /N 0 increases, with however a decreased rate for higher values of /N 0. n addition, as /N 0 increases, i.e., the interference temperature constraint becomes loose, the performance also improves, whilst the performance of the system without constraints is always better, as expected. t is interesting to note that the curves, based on the asymptotic expression given in 13), are quite close to the exact ones, eecially for higher values of the /N 0. n Fig. 3, for the same composite fading environment and in order to better understand the impact of the fading/shadowing parameters to the system performance, the O is plotted as a function of /N 0 for various values of the secondary transmission channel parameters as well as under different interference temperature constraints. n particular, we have assumed three communication scenarios, with severe, i.e., k sr = k rd = 1.5; m sr = m rd = 1), moderate, i.e., k sr = k rd =.5; m sr = m rd = ) and light, i.e., k sr = k rd = 3.5; m sr = m rd = 3) fading/shadowing conditions. For all scenarios we have also assumed k = k rp = 0.6; m = m rp = 0.5). As previously, it is depicted that the performance improves as /N 0 and/or /N 0 increase. An

7 90 etros S. Bithas et al. / rocedia Computer Science ) Fig. 3. K G fading: Outage probability as a function of /N 0 for different fading/shadowing environments. interesting observation is that the performance gap among the three channel conditions under consideration increases for higher values of /N 0. Finally, in Fig. 4, considering G G fading channels the O is plotted as a function of the /N 0 for various values of the average input SNR for both secondary wireless links. n this gure we have assumed, m = m rp = 3, b = b rp = 1.5, γ = γ rp = 10dB, m sr = m rd =, b sr = b rd = 1, = 15dB, γ th = 10dB. n addition, in the same gure, the correonding curves, based on the asymptotic expression for the O derived in 17), are also included. n this gure, it is depicted that the O improves as /N 0 and/or γ X, with X {sr, rd}, increase. t is interesting to note that even for moderate values of γ X, the O plots based on the approximated expression are quite close to the ones obtained using the exact, which however becomes almost equal for higher values of the /N 0 and/or γ X. For comparison purposes, computer simulation performance results are also included in all gures, verifying the validity of the proposed theoretical approach. 5. Conclusions n this paper, the O of a cognitive relaying network, which is based on the underlay approach, operating over generalized fading channels, modeled by the K G and G G distributions, is investigated. Exact as well as approximated expressions of the O have been derived, for both communication scenarios. Various numerical evaluated results with parameters of interest the fading/shadowing severity, the interference constraint and the maximum allowed transmission power have been presented. n all cases, it was shown that the performance decreases as the interference constraint becomes tighter or fading/shadowing conditions worsen. Acknowledgements This research has been co- nanced by the European Union European Social Fund - ESF) and Greek national funds through the Operational rogram Education and Lifelong Learning of the National Strategic Reference Framework NSRF) - Research Funding rogram: Thales. nvesting in knowledge society through the European Social Fund. References 1. Goldsmith, A., Jafar, S.A., Maric,., Srinivasa, S.. Breaking ectrum gridlock with cognitive radios: An information theoretic perective. roceedings of the EEE 009;975):

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