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Int. J. Communications, Network and System Sciences, 2010, 3, 430-433 doi:10.4236/ijcns.2010.35056 Published Online May 2010 (http://www.SciRP.org/journal/ijcns/) Copyright © 2010 SciRes. IJCNS Outage Performance of Opportunistic Amplify-and-Forward Relaying over Asymmetric Fading Environments Sudhan Majhi1,2,3, Youssef Nasser1,2,3, Jean François Hélard1,2,3 1European University of Brettany, Rennes, France 2Institut National des Sciences Appliquees, Rennes, France 3Institute of Electronics and Telecommunications of Rennes, Rennes, France E-mail: {sudhan.majhi, youssef.nasser, jean-francois.helard}@insa-rennes.fr Received February 16, 2010; revised March 21, 2010; accepted April 22, 2010 Abstract This letter analyzes the outage probability of opportunistic amplify-and-forward relaying over asymmetric and independent but non-identically distributed (i.n.d) fading environments. The work investigates the sce- narios where cooperative nodes are located at different geographical locations. As a result, the different sig- nals are affected by different i.n.d fading channels, one may undergo Rician fading distribution and others may undergo Rayleigh fading distribution. In this letter, a lower bound of the outage probability for various asymmetric fading environments is derived at high SNR by applying the initial value theorem. The analytical model is validated through Monte-Carlo simulation results. Keywords: Outage Probability, Opportunistic Relaying, Amplify-and-Forward Relaying, Rayleigh and Rician Fading Channels, Asymmetric Fading Channels, Independent and Non-Identically Distributed 1. Introduction Cooperative relaying is a promising technology for fu- ture wireless communications. It can benefit most of the leverages of multiple input multiple output (MIMO) wit- hout using the conventional MIMO schemes [1]. Among the cooperative techniques, the opportunistic relaying, in which only one relay (R) node forwards the source’s (S) data to the destination (D) has shown its efficiency com- pared to other techniques [2]. The outage performance of opportunistic amplify-and- forward (AF) relaying over a symmetric fading environ- ment is widely investigated in [1,3,4]. However, in prac- tice, cooperative nodes are usually located in different geographical location environments and at different dis- tances with respect to S and D. Therefore, one link could be either in line-of-sight (LOS) situation or in non-LOS (NLOS) situation. For example, the fixed relay nodes used for forwarding source’s data to a specific region (e.g. tunnel, behind the building) often use directional antenna, so the R-D link is usually in a LOS situation. However, we cannot assume such a situation in all transmission environments especially when D is in a deep shadowing region with respect to S. The outage performance analysis of opportunistic relaying for mixed and i.n.d fading environments is, therefore, of practical importance. The asymmetric fading channel is introduced in [5]. However, the authors of this work assume additive white Gaussian noise (AWGN) channel of the R-D link. In [6], an approximation of the outage performance over asym- metric fading channel, i.e., Rayleigh and Rician, is given. However, to the best knowledge of the authors, no clo- sed-form expression is provided. In this letter, the ana- lytical model of the outage probability of opportunistic AF relaying over asymmetric and i.n.d fading environm- ents is given. Then, the lower bound of the outage prob- ability for high SNR values is deduced and verified thro- ugh Monte-Carlo simulations. 2. System Model and SNR Evaluation In this framework, we consider a general 2-hop AF re- laying network consisting of S, m relays, Ri, 12i,,...,m and D. We assume that D performs maximum ratio com- bining at the receiving side. The equivalent instantaneous end-to-end signal-to-noise ratio (SNR) for opportunistic S. MAJHI ET AL. Copyright © 2010 SciRes. IJCNS 431 AF relaying is given as [3]: 22 2 22 1 ii ii ii ii ssr srd srr d ssd di sd ssr srd srr d Ph Ph NN Ph SNR max NPh Ph NN (1) where 2 ab hrepresents the channel gain of the a-b link, a P is the power transmitted by the node a. As mentioned in [3], we assume that AWGN variance is 0 1 ab N/ , a,b where 0 is proportional to the system SNR. For simplicity reasons, we use different notations of the random variables of the different fading distributions. For the Rayleigh fading, let 2 aba ab Ph be the instant- taneous signal power and for the Rician fading, the in- stantaneous signal power is denoted as ab . The prob- ability density function (PDF) of ab and ab are ex- pressed respectively as: 1ab ab x/ ab fx e (2) 1 0 41 1abab ab ab K/K ab ab ab ab ab KK K feI (3) where, {} ab ab E {} ab ab E and ab K is the Rician factor. {.}E holds for expectation value. The upper bound of the instantaneous SNR of (1) for the opportunistic AF relaying is defined as: 22 2 0ii ubs sds srs rd i SNRP hmaxmin P h,Ph (4) This instantaneous SNR value will be used in the fol- lowing section to evaluate the outage probability. 3. Analysis of Outage Probability In this section, we provide the lower bound of the outage probability of opportunistic AF relaying for different channels given in Figure 1. 3.1. Asymmetric Channel I Theorem 1: If S-D link is Rayleigh fading channel and S-R and R-D links are Rician fading channels, then the lower bound of the outage probability over asymmetric channel I is: 1 1 11 1 1 ii srrd ii ii msrr d Im out KK i sd srrd KK pmee (5) Figure 1. Different asymmetric fading channels of a coop- erative network. Proof: By using2 s dssd Ph , 2 ii srs sr Ph and 2 ii rds rd Ph in [4], the outage probability over the asymmetric channel I can be written as: [] II out ub pPr (6) where ub SNR is derived toI ubsdmax , 2 0 21 R/ , 12 max max m , ,..., and i min ii s rrd ξ,ξ. The cumulative distribution function (CDF) of the random variable i over i.n.d is given as: 1 1 11 []1 [] 21 12 21 2 iii i i i i i i srr d sr sr sr rd rd rd FPrPr K QK, K QK, (7) where 1 Q, is the 1st order Marcum Q-function and the PDF of i is obtained by differentiating above as: 1 21 2 21 2 i ii rd i i i isr i i sr sr sr rd rd rd K fK, f K QK, f (8) The CDF of the random variable max over i.n.d fad- ing channel can be expressed as: 1 max i m i FF (9) and the corresponding PDF of max is obtained by dif- ferentiating the above as: S. MAJHI ET AL. Copyright © 2010 SciRes. IJCNS 432 1 1 maxi j m m j i ji ffF (10) Since 00 i F , the 1thm order derivative of (10) at high SNR i.e., at 0 as 0 , can be written as: 1 0 1 1 !0 max i mm m i f|mf (11) The outage probability given in (6) is a CDF of I ub , which can be evaluated by using the initial value theorem (IVT) of the Laplace Transformation (LT). The LT of the PDF of the random variable I ub can be expressed by using Equation 15 in [3] and, then (11), as: 1 0 11 1 1 1 () 0 !00 ubsd max sd i m I mm m m i Lfff | s mff s (12) Since 0 sd f and 0 i f are constant with respect to the variables, the PDF of I ub is obtained by apply- ing the inverse LT (ILT) on (12) as: 1 ()0 0 ubsd i m Im i fff (13) We complete the proof by integrating (13) and substi- tuting the vale of 0 sd f and 0 i f . 3.2. Asymmetric Channel II Theorem 2: If S-D and S-R links are Rician fading channels and R-D link is Rayleigh fading channel, then the lower bound of the outage probability over asymmet- ric channel II is: 1 1 1 11 1 i sd sr i i i msr sd II m out KK ird sd sr K K pme e (14) Proof: For the asymmetric channel II, we use 2 s dssd Ph , 2 ii srs sr P h and 2 ii rds rd Ph . The outage probability can be expressed as: [] II II out ub pPr (15) where II ub sdmax g , 12 max max m g g ,g,...,g and min ii isrrd g, . The CDF of the random variable i g over i.n.d fading channel can be written as: 1 21 12 1 i ii rd i i sr gsr sr K FQK, F (16) where rd i F is the CDF of the random variable i rd . The corresponding PDF of i g is expressed as: 1 21 2 1 i ii rd i i rd sr ii sr gsr sr K fQK, f Ff (17) Similarly, by using the IVT and the ILT, the PDF of I I ub can be derived as: 1 00 ubsd i m II m g i fff (18) By integrating (18), we complete the proof. 3.3. Asymmetric Channel III Theorem 3: If S-D link is Rician fading channel and S-R and R-D link are Rayleigh fading channels, the corre- sponding lower bound of outage probability is: 1 1 111 1sd ii m sd III m out K isrrd sd K pme (19) Proof: For the asymmetric channel III, we use 2 s dssd Ph , 2 ii srs sr Ph and 2 ii rds rd Ph . The outage probability can be written as: [] III III out ub pPr (20) where III ubsd max , 12max m max,,..., and ii isrrd min , . The corresponding PDF of the random variable i is given by: 11 isrrdrdsr iiii fFf Ff (21) Again by using the IVT and ILT, the PDF of I II ub is obtained as: 1 00 ubsd i m III m i fff (22) By integrating (22), we complete the proof. Similarly, the outage probability of other possible asymmetric channels can be derived by using the above procedure. The upper bound of the outage probability of the opportunistic AF relaying can be derived simply by using the above method and Equation 8 in [7]. 4. Numerical Examples In this section, analytical and Monte-Carlo simulation results are presented. Since the channels are i.n.d, we set different means for different S-Ri/Ri-D links. In the S. MAJHI ET AL. Copyright © 2010 SciRes. IJCNS 433 Rician fading channel, the Rician factor ab K is uni- formly distributed in [2,3] and the mean ab of the NLOS components are uniformly distributed in [0,1]. The LOS components are derived for a given ab K and ab . It is clear from (5), (14) and (19) that the outage probability over Rician fading channel is obtained by substituting 11 sd K s dsd sd /Ke/ in (5) and the outage probability over Rayleigh fading channel is ob- tained by substituting 0 sd K in (19). Figure 2 shows the lower bound of the outage probability over the sym- metric and asymmetric fading environments. Due to the presence of LOS signal, the outage performance over Rician fading channel outperforms all other scenarios. Inversely, due to the absence of direct signal, the Ray- leigh fading channel has poorer outage performance than the other scenarios. The opportunistic relaying provides better outage per- formance than without cooperation. It implies that the outage performance of opportunistic relaying depends mainly on cooperative links (S-R and R-D links). For this reason, asymmetric channel I provides better outage performance than the asymmetric channel II and asym- metric channel III due to the presence of LOS signal inboth S-R and R-D links. We also note that asymmetric channel II provides bett- er outage performance than asymmetric channel III. Si- nce S-D and R-D links undergo the same fading in both scenarios, the LOS component existing in S-R link of Rician, Analytical Rayleigh, Analytical Asymmetric I, Analytical Equation (5) Asymmetric II, Analytical Equation (14) Asymmetric III, Analytical Equation (19) Simulation Outage probability 10 0 10 -1 10 -2 10 -3 10 -4 10 -5 10 -6 10 -7 SNR [dB] 0 2 4 6 8 10 12 14 16 18 20 Figure 2. The outage probability over asymmetric channel I, asymmetric channel II and asymmetric channel III. Due to the high SNR approximation for the analysis, analytical results converge with Monte-Carlo simulation results at medium and high SNR regime. scenario II highly improves the outage performance. It is clear from the above discussion that S-R is a dominating link, therefore, it is better to localize the opportunistic relay node in LOS environment with respect to S in order to improve the overall outage performance. Finally, the Monte-Carlo simulation results provided in Figure 2 shows that the analytical outage probabilities are a tight bound at medium and high SNR regime. 5. Conclusions In this letter, the outage performance of opportunistic AF relaying over asymmetric and i.n.d fading environments has been investigated. A lower bound of the outage probability has been derived and validated through Monte-Carlo simulation results. We show that the outage performance is better when the relay is in LOS situation with respect to the source rather than to the destination. 6. Acknowledgements The authors would like to thank the European IST-FP7 WHERE project and the European Network of Excel- lence NEWCOM++ for support of this work. 7. References [1] J. N. Laneman, D. N. C. Tse and G. W. Wornell, “Coop- erative Diversity in Wireless Networks: Efficient Proto- cols and Outage Behavior,” IEEE Transactions of Infor- mation Theory, Vol. 50, No. 12, 2004, pp. 3062-3080. [2] Y. Zhao, R. Adve and T. J. Lim, “Outage Probability at Arbitrary SNR with Cooperative Diversity,” IEEE Com- munications Letters, Vol. 9, No. 8, 2005, pp. 700-703. [3] Y. Zhao, R. Adve and T. J. Lim, “Symbol Error Rate of Selection Amplify-and-For- ward Relay Systems,” IEEE Communications Letters, Vol. 10, No. 11, 2006, pp. 757- 759. [4] A. Bletsas, H. Shin and M. Z. Win, “Cooperative Com- munication with Outage Optimal Opportunistic Relay- ing,” IEEE Transactions on Wireless Communications, Vol. 6, No. 9, 2007, pp. 3450-3459. [5] M. Katz and S. Shamai, “Relaying Protocols for two Co- located Users,” IEEE Transactions on Information The- ory, Vol. 52, No. 6, 2009, pp. 2329-2344. [6] H. Suraweera, G. Karagiannidis and P. Smith, “Perform- ance Analysis of the Dual-Hop Asymmetric Fading Ch- annel,” IEEE Transactions on Wireless Communications Letters, Vol. 8, No. 6, 2009, pp. 2783-2788. [7] K.-S. Hwang, Y.-C. Ko and M.-S. Alouini, “Performance Analysis of Incremental Opportunistic Relaying over Identically and Non-Identically Distributed Cooperative Paths,” IEEE Transactions on Wireless Communications, Vol. 8, No. 4, April 2009, pp. 1953-1961. |