Performance Analysis of the Dual-Hop Asymmetric Fading Channel

Transcription

1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 8, NO 6, JUNE Performance Analysis of e Dual-Hop Asymmetric Fading Channel Himal A Suraweera, Member, IEEE, George K Karagiannidis, Senior Member, IEEE, and Peter J Smi, Senior Member, IEEE Abstract In real wireless communication environments, it is highly likely at different channels associated wi a relay network could experience different fading phenomena In is paper, we investigate e end-to-end performance of a dualhop fixed gain relaying system when e source-relay and e relay-destination channels experience Rayleigh/Rician and Rician/Rayleigh fading scenarios respectively Analytical expressions for e cumulative distribution function of e end-to-end signal-to-noise ratio are derived and used to evaluate e outage probability and e average bit error probability of M-QAM modulations Numerical and simulation results are presented to illustrate e impact of e Rician factor on e end-toend performance Furermore, ese results confirm at e system exhibits an improved performance in a Rician/Rayleigh source-relay link/relay-destination link environment compared to a Rayleigh/Rician environment Index Terms Wireless relays, amplify-and-forward, Rayleigh fading, Rician Fading, outage probability, average bit error probability I INTRODUCTION RECENTLY, wireless relaying techniques have become a major topic in e wireless research community due to eir possible application in cellular, ad-hoc networks and military communications [] Amplify-and-Forward AF relaying is a popular meod for implementing relay systems Compared to Decode-and-Forward DF which employs full decoding and re-encoding at e relay, AF requires less processing power at e relay [] An AF relay amplifies e received signal from e source and en retransmits towards e destination [3] Therefore, at e relay no decoding is performed AF relays can be broadly categorized as: channel state information CSI assisted relays, and fixedgainor blind relays [4] Fixed gain relays use power amplifiers wi a constant gain to amplify e signals and exhibit a performance close to at of e systems wi CSI-assisted relays [3] Several works have analysed e performance of AF relay networks operating under different fading conditions using Manuscript received March 7, 8; revised July 5, 8, October 8, 8, and December 7, 8; accepted January 7, 9 The associate editor coordinating e review of is letter and approving it for publication was S Ghassemzadeh This work was supported by a Discovery Grant DP from e Australian Research Council H A Suraweera is wi e Center for Telecommunications and Microelectronics, Victoria University, PO Box 448, Melbourne, Victoria 8, Australia himalsuraweera@vueduau G K Karagiannidis is wi e Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, GR-544, Greece geokarag@augr P J Smi is wi e Department of Electrical and Computer Engineering, The University of Canterbury, Private Bag 48, Christchurch, New Zealand psmi@eleccanterburyacnz Digital Object Identifier 9/TWC /9$5 c 9 IEEE e outage probability of e end-to-end signal-to-noise ratio SNR and e average bit error probability ABEP See [3] [7] and references erein In [3], Hasna and Alouini have studied e ABEP of dual-hop systems wi AF relaying over Rayleigh fading channels In [5] and [6], Karagiannidis et al have studied e performance bounds of AF multihop transmissions over non-identically distributed Nakagami-m fading channels More recently, in [4], Mheidat and Uysal have investigated e impact of receive diversity on e performance of a relay-assisted network in which e relay is operating under e fixed gain constraint Zhao et al [7] presented an asymptotic SER study of a selection AF network where e best relay which contributes most to received SNR is chosen for retransmission In addition to e widely used Rayleigh and Nakagami-m fading assumptions, Rician fading is often used in e technical literature to model wireless propagation comprising wi a line-of-sight LoS component and a scattered component [8] A recently released WINNER II project deliverable [9] also documents Rician propagation characteristics in micro/macro cellular multi-hop transmissions Despite e importance of e Rician model, only a few works have analyzed e performance of relays under LoS fading conditions [], [] For example, e performance of Decode-and-Forward DF relay networks under Rician fading has been considered in [] The symbol error probability of a multiple-input multiple-output semi-blind relay system has been studied in [] Recently, some papers have also studied e performance of relay networks under asymmetric fading scenarios [], [3] Asymmetric fading conditions, where links associated in e relay network are subject to different fading distributions, may arise due to e close proximity of e terminals or shadowing effects In [], Adinoyi and Yanikomeroglu have studied e error performance for a DF relay network by considering different signal strengs/severities such as non-los Rayleigh and LoS Rician fading for e network hops The information eoretic work of [3] has also been considered an asymmetric scenario where e source s transmission experiences Rayleigh fading, while e relay transmission experiences a fixed amplitude gain additive white Gaussian noise AWGN channel In is work, e outage probability and e ABEP of a dual-hop relay system in an asymmetric fading environment is investigated We assume at e source-relay and relaydestination links experience Rayleigh or Rician fading First, using e cumulative distribution functions cdfs of e endto-end signal-to-noise ratio SNR, e outage probability is derived Next, using e cdf, e ABEP of M-ary square QAM modulation is derived Simulation results are also presented to verify e eoretical analysis Auorized licensed use limited to: University of Canterbury Downloaded on November 9, 9 at :37 from IEEE Xplore Restrictions apply

2 784 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 8, NO 6, JUNE 9 The rest of e paper is organized as follows: In Section II we outline e system and channel model In Section III, we derive outage probability expressions for e asymmetric channel The ABEP of M-QAM modulated signals is computed in Section IV Finally, we conclude wi a brief summary of results in Section V II SYSTEM AND CHANNEL MODEL Assume a dual-hop fixed gain AF relay system cf [3, Fig ] operating in an asymmetric fading environment The source, S communicates wi e destination, D using a relay, R Each transmission period is divided into two signalling intervals: In e first signalling interval, e received signal at R is multiplied by a gain factor G and in e second signalling interval, it is retransmitted to D Assuming at S transmits a signal wi an average power normalized to unity, e instantaneous end-to-end SNR at e destination, γ eq,is [3] γ eq = α /N α /N α /N +/G N where α,α are e fading amplitudes of e wireless channels in e S R and R D links respectively, N and N are e power of e AWGN component at e input of e relay and e destination, and G is e relay gain If C =/G N, γ i = α i /N i for i =, and en simplifies to γ eq = γ γ C + γ In is work, we consider two cases for e fading distributions of e S R and R D links, namely: The S R link is subject to Rayleigh fading and e R D link is subject to Rician fading In e following, is Rayleigh/Rician fading condition will be identified as scenario a The S R link is subject to Rician fading and e R D link is subject to Rayleigh fading In e following, is Rician/Rayleigh fading condition will be identified as scenario b If a link experiences Rayleigh fading, γ i, wi i =or, is an exponentially distributed random variable RV That is, its probability density function pdf is given by p γi γ = γ i e γ/ γi 3 where γ i =Ω i /N i and Ω i is e average fading power of at link If a link experiences Rician fading, e pdf of α i is given by p αi α = K +e K α e K+α Ω i I α Ω i KK + Ω i where K is e Rician K-factor defined as e ratio of e powers of e LoS component to e scattered components and I is e zero order modified Bessel function of e first 4 kind γ i is distributed according to a noncentral-χ distribution given by K +e K p γi γ = e K+γ KK +γ γ i I γ i γ i 5 Observe at when K =e Rician distribution becomes e Rayleigh distribution As K, e distribution approximates at of an AWGN no fading channel Values of e K-factor in indoor/outdoor land mobile applications normally range from db [8] III OUTAGE PROBABILITY Outage probability is an important performance measure at is commonly used to characterize a wireless communication system It is defined as e probability at e instantaneous end-to-end SNR, γ eq, falls below a reshold γ Therefore maematically, e outage probability is given by [3] [ ] γ γ P out = F γeq γ =Pr <γ 6 C + γ where F γeq γ is e cdf of e end-to-end SNR Next, we calculate e outage probability applicable to scenarios a and b A Scenario a In e case of scenario a we can express P out as [ P out = Pr γ < γ ] C + γ γ p γ γ dγ 7 γ The cdf of γ is e γ/ γ, and using 5, 7 can be written as K [ +e K P out = e γ/ γ+c/γ] 8 γ e K+γ γ I K +e K = γ e K+γ γ I KK +γ γ dγ e γ/ γ+c/γ KK +γ γ dγ The integral required to compute e outage probability in 8 is I = e Cγ / γ K+γ / γ KK +γ I dγ γ 9 We are unaware of a closed-form analytical solution to is integral Nevereless, using e infinite-series representation of I x [4, Eq 8447-] I x = l= x l l l! Auorized licensed use limited to: University of Canterbury Downloaded on November 9, 9 at :37 from IEEE Xplore Restrictions apply

3 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 8, NO 6, JUNE we can rewrite I as K l K + l I = l! γ l l= γ l e Cγ/ γγ K+γ/ γ dγ The integral in can be evaluated using [4, Eq 347-9] We write I as I = K l K + l! γ l= K l+ l CK +γ γ Cγ l+ for γ > and K ν is e ν-order modified Bessel function of e second kind Finally, P out, for scenario a is given by P out = e K γ/ γ K l+ l= CK +γ γ K l CK +γ l! γ l+ 3 Concerning e convergence of e infinite series in 3, e truncation error if T terms are used is R = e K γ/ γ K l+ l=t + K l l! CK +γ γ CK +γ γ l+ 4 For ν> and fixed x, K ν x can be asymptotically approximated as [5] ν! x ν K ν x 5 Substituting 5 in 4 and after simplifications we get T R e e K γ/ γ K K l 6 l! l= Finally using [6, Eq 47] R e γ/ γ ΓT +,K 7 T! where Γα, x = x tα e t dt is e complementary incomplete gamma function [6, p 79] B Scenario b In e case of scenario b we can express P out as [ P out = Pr γ < Cγ ] γ γ γ p γ γ dγ 8 and after some manipulations 8 can be reexpressed as P out = γ e Cγ/ γ γ γ p γ γ dγ 9 Using e Rician pdf of 5, e integral in 9 is given by K +e K I = e Cγ/ γ γ γ e K+ γ γ γ KK +γ I dγ Let u = γ γ,en I = K + e K K+γ γ e Cγ/ γu e K+u KK +γ + u I du Invoking again e infinite series representation of I x we can expand as K + I = e K K+γ K l K + l l! γ l l= e Cγ/ γu e K+u u + γ l du Using e binomial expansion and e integral result of [4, Eq 347-9] I can be solved Therefore, P out for scenario b is expressed as P out = e K K+γ K + l r+ C γ l= r+ l γ l r+ 3 r r= CK +γ K r+ γ K l l! For numerical verification, assume at bo e S R and R D links experience Rayleigh fading, ie, e scenario considered in [3] Substituting K =in 3 and 3 we get [3, Eq 9] In all cases of practical significance, e infinite series representations involved in 3 can be truncated wiout sacrificing numerical accuracy Suppose at we truncate 3 after T terms Therefore, e remainder, R after applying 5 becomes asymptotically R e K K+γ l=t + l! l KK +γ 4 r K +γ l r! γ r= and simplifying furer R e K K l l! Γ l +, K +γ 5 l=t + We are unable to obtain a closed-form solution for e summation in 5 However note at, using e L Hopital rule it can be shown at lim x Kx /x! = Therefore, R asymptotically converges to a finite number Figs and show e outage probability for scenarios a and b respectively In all cases, we have employed e fixed gain G assumed in [4] under e so-called average power scaling The eoretical results were plotted using e first terms of e infinite series l =, where ey match exactly Auorized licensed use limited to: University of Canterbury Downloaded on November 9, 9 at :37 from IEEE Xplore Restrictions apply

4 786 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 8, NO 6, JUNE 9 outage probability 3 4 Fig outage probability 3 4 Fig Simulated K = db, γ Simulated K = 6 db, γ = db Simulated K = 6 db, γ Rayleigh/Rayleigh, γ = db Rayleigh/Rayleigh, γ Average SNR per hop db Outage probability in Rayleigh/Rician fading Simulated K = db, γ Simulated K = 6 db, γ = db Simulated K = 6 db, γ Rayleigh/Rayleigh, γ = db Rayleigh/Rayleigh, γ Average SNR per hop db Outage probability in Rician/Rayleigh fading wi e simulated results The plots in Figs and show similar trends, ie, wi an increasing K-factor, outage events decrease However, in all cases, e outage probability of Rician/Rayleigh fading is lower an Rayleigh/Rician fading Better channel conditions in e first hop Rician fading leads to an outage performance improvement because e end-to-end SNR probabilistically reaches higher values more often an in e first hop Rayleigh fading scenario IV AVERAGE BIT ERROR PROBABILITY In is section we evaluate e ABEP of e dual-hop relay network assuming at e transmitted data are modulated using a M-ary square QAM alphabet which transmits data by changing e amplitude of two carrier signals, and is widely used in wireless communication systems The ABEP is a useful measure of evaluating e performance of wireless communication applications Traditionally e ABEP is computed by determining e pdf of γ eq and en averaging e conditional BEP in AWGN, P b e γ, over is pdf Maematically, P b e in e relay channel is given by P b e = P b e γp γeq γdγ 6 Using e BEP in an AWGN channel [7, Eq 4], e k- bit ABEP of Gray bit-mapped M-ary square QAM is given by P b e k = M k M i= i k k + E M { i k M [ Q i + 3γ M 7 ]} where E[ ] denotes e statistical expectation operator, Qx being e Gaussian Q-function defined as Qx = / π / x e t dt and x is e largest integer smaller or equal to x Finally, e ABEP of M-ary square QAM is calculated from, P b e = log M log M k= P b e k 8 In order to compute e ABEP for scenarios a and b we identify at e integral at needs to be computed is of e form: J = Q βγp γeq γdγ 9 where β is a constant Note, at J can be computed using e meod presented in [7] After integration by parts and e change of variable t = βγ, 9 can be rewritten as J = t F γeq e t dt 3 π β Consider scenario a: Using 3, J, can be written as J = e K K l π l! l= t l+ e + γ β β CK + γ β t K l+ l+ CK + γ β t dt 3 Using [4, Eq 663-3], 3 can be evaluated as J = K l e K l! +/ γ l= β Γ l l+ CK + Ψ l + 3 CK +,l+; + β γ + β γ where Ψa, b; z is e Tricomi function also known as e Kummer U function [8], defined as Ψa, c; z = Γa e zt t a + t c a dt [9, p 54] and Γx = t z e t dt is e gamma function We have employed e well known relationship between e Whittaker function and Ψa, b; z [9, p 55] to arrive at 3 Note, at Ψ, ; z can be evaluated using popular symbolic software such as MATLAB, MAPLE and MATHEMATICA Finally, after substituting 3 wi β =3i + /M in 7 and using 8 we arrive at e ABEP Auorized licensed use limited to: University of Canterbury Downloaded on November 9, 9 at :37 from IEEE Xplore Restrictions apply

5 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 8, NO 6, JUNE Tricomi function Ψa, b, z is a monotonically decreasing function respect to a and b This means at when a and b increase Tricomi decreases Therefore, e truncation error, R 3 in 3 after T 3 terms can be upper bounded as R 3 < Ψ T 3 + 5,T CK ; + β γ l=t 3+ e K 33 K l Γ l + 3 l+ CK + l! +/ β + β γ Using e identity Γ l + 3 = πl+! [6, p 773] and l+ l! expressing e summation of 33 in closed-form we finally obtain K T3+ Γ T T3+ CK + R 3 < T 3 +! +/ β + β γ 34 e K Ψ T 3 + 5,T CK ; + β γ F,T ; T CKK + 3 +,T 3 +; + β γ where p F q a,,a p ; b,,b q ; z is e generalized hypergeometric function [6, p 788] Consider scenario b Substituting F γeq γ of 3 in 3 we obtain J = e K π l= l r K l l! C r= r+ K + γ e K++ γ β β t K r+ l r β r l CK + γ β t t l r+ dt 35 Using [4, Eq 663-3], J can be simplified as shown at e top of e next page Note at by exploiting e monotonic behavior of e Tricomi function as in 33 and using e L Hopital rule, we can show at 36 converge This proof is omitted here due to limited space Figs 3 and 4 show e ABEP performance of e dualhop relay network over Rayleigh/Rician and Rician/Rayleigh fading All cases correspond to 4-QAM and 6-QAM Theoretical results were obtained by truncating e infinite series of 3 and 36 to terms, ie, l = The number of terms required in 3 and 36 to achieve a given figure accuracy depend on e SNR and e K-factor For example, to achieve six significant figure accuracy at K =db, only eight terms are necessary, while at K =db, 5 terms are required However compared to time consuming computer simulations, in all cases, e computational load required for calculating e eoretical ABEP is marginal In Fig 3 we have also plotted e performance curves for Rayleigh/Rayleigh fading for comparison purposes Under Rayleigh/Rayleigh fading, J simplifies to J = ϕ K ϕ K ϕ e ϕ +/ γ β, 37 ABEP 3 4 K = db, 4 QAM K = db, 6 QAM K = 6 db, 4 QAM K = 6 db, 6 QAM Rayleigh/Rayleigh, 4 QAM Rayleigh/Rayleigh, 6 QAM Average SNR per hop db Fig 3 ABEP of 4 and 6-QAM in Rayleigh/Rician fading Dashed lines and markers denote eoretical and simulated ABEPs respectively ABEP 3 4 K = db, 4 QAM K = db, 6 QAM K = 6 db, 4 QAM K = 6 db, 6 QAM Average SNR per hop db Fig 4 ABEP of 4 and 6-QAM in Rician/Rayleigh fading Dashed lines and markers denote eoretical and simulated ABEPs respectively where ϕ = C/ + β γ Clearly, Rayleigh/Rayleigh fading deteriorates e ABEP of e relay system compared to bo asymmetric fading conditions In all cases, eoretical ABEPs and simulations match very well For a small K-factor db, Figs 3 and 4 illustrate at ABEP of scenarios a and b are almost identical However, for a typical K-factor 6 db and at high SNR, Rician/Rayleigh fading is able to provide a slightly better error performance compared to Rayleigh/Rician fading This is due to e fact in AF systems wi fixed-gain, e endto-end performance is dominated by e first hop channel gain This is evident from Therefore, better channel conditions in e first hop Rician fading leads to improved performance compared to e first hop Rayleigh fading scenario V CONCLUSIONS In is paper, we have presented two infinite-series representations for e outage probability of a dual-hop communication system equipped wi a single fixed-gain amplifyand-forward relay in Rayleigh/Rician and Rician/Rayleigh Auorized licensed use limited to: University of Canterbury Downloaded on November 9, 9 at :37 from IEEE Xplore Restrictions apply

6 788 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 8, NO 6, JUNE 9 J = e K π + K +/ γ β K l l! + β/k + l+ Γ l= l C r γ r= l + 3 r+ Γ l r + Ψ l + 3,r+; CK + K ++ β γ 36 fading environments respectively Based on ese expressions, e average bit error probability ABEP of M-ary square QAM was also derived These results demonstrate at e system exhibits an improved performance in a Rician/Rayleigh source-relay link/relay-destination link environment compared to a Rayleigh/Rician environment Furermore, bo Rayleigh/Rician and Rician/Rayleigh asymmetric fading environments, compared to e scenario where bo links suffer only from Rayleigh fading Rayleigh/Rayleigh, produce improved ABEPs Simulation results for e outage probability and e ABEP are in excellent agreement wi e eoretical results obtained in is paper Since it has been found in practice at different links of a relay system could experience line-of-sight/non line-of-sight fading conditions, is analysis is useful to e system design engineer for performance evaluation purposes REFERENCES [] R Pabst, B H Walke, D C Schultz, et al, Relay-based deployment concepts for wireless and mobile broadband radio, IEEE Commun Mag, vol 4, pp 8-89, Sept 4 [] Y Fan and J Thompson, MIMO configurations for relay channels: eory and practice, IEEE Trans Wireless Commun, vol 6, pp , May 7 [3] M O Hasna and M-S Alouini, A performance study of dual-hop transmissions wi fixed gain relays, IEEE Trans Wireless Commun, vol 3, pp , Nov 4 [4] H Mheidat and M Uysal, Impact of receive diversity on e performance of amplify-and-forward relaying under APS and IPS power constraints, IEEE Commun Lett, vol, pp , June 6 [5] G K Karagiannidis, T A Tsiftsis, and R K Mallik, Bounds of multihop relayed communications in Nakagami-m fading, IEEE Trans Commun, vol 54, pp 8-, Jan 6 [6] G K Karagiannidis, Performance bounds of multihop wireless communications wi blind relays over generalized fading channels, IEEE Trans Wireless Commun, vol 5, pp , Mar 6 [7] Y Zhao and R Adve, Symbol error rate of selection amplify-andforward relay systems, IEEE Commun Lett, pp , Nov 6 [8] J D Parsons, The Mobile Radio Propagation Channel New York: Wiley, 99 [9] P Kyösti, J Meinilä, L Hentilä, et al, WINNER II Interim Channel Models IST WINNER II D V [Online] Available: [] J Adeane, M R D Rodrigues, and I J Wassell, Characterisation of e performance of cooperative networks in Rician fading channels, in Proc Intl Conf Telecommun ICT 5, CapeTown,Sou Africa, May 5 [] T Q Doung, H Shin, and E-K Hong, Effect of line-of-sight on dualhop nonregenerative relay wireless communications, in Proc IEEE Vehicular Technol Conf VTC 7 Fall, Baltimore, MD, Sept/Oct 7, pp [] A Adinoyi and H Yanikomeroglu, On e performance of cooperative wireless fixed relays in asymmetric channels, in Proc IEEE GLOBE- COM 6, San Franscisco, CA, Nov/Dec 6, pp -5 [3] M Katz and S Shamai Shitz, Relaying protocols for two colocated users, IEEE Trans Inform Theory, vol 5, pp , June 6 [4] I S Gradshteyn and I M Ryzhik, Table of Integrals, Series and Products, 6 ed San Diego: CA, Academic Press, [5] R Penfold, J-M Vanden-Broeck, and S Grandison, Monotonicity of some modified Bessel function products, Integral Transforms and Special Functions, vol 8, no, pp 39-44, Feb 7 [6] A P Prudnikov, Y A Brychkov, and O I Marichev, Integrals and Series, vol New York: Gordon and Breach Science Publishes, 986 [7] K Cho and D Yoon, On e general BER expression of one and two dimensional amplitude modulations, IEEE Trans Commun, vol 5, pp 74-8, July [8] I C Sikaneta, Numerically computing e KummerU function and a special case of e Gauss hypergeometric function, Defence R&D Canada Technical Report TR 6-94, Dec 6 [Online] Available: ehtml [9] M Abramowitz and I A Stegun, Handbook of Maematical Functions wi Formulas, Graphs, and Maematical Tables, ed New York: Dover, 97 Auorized licensed use limited to: University of Canterbury Downloaded on November 9, 9 at :37 from IEEE Xplore Restrictions apply