Optimum Relay Position for Differential Amplify-and-Forward Cooperative Communications

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1 Optimum Relay Position for Differential Amplify-and-Forard Cooperative Communications Kazunori Hayashi #1, Kengo Shirai #, Thanongsak Himsoon 1, W Pam Siriongpairat, Ahmed K Sadek 3,KJRayLiu 4, and Hideaki Sakai #3 # Graduate School of Informatics, Kyoto University Yoshida Honmachi Sakyo-ku, Kyoto, , JAPAN 1 kazunori@ikyoto-uacjp shirai@sysikyoto-uacjp 3 hsakai@ikyoto-uacjp Department of Electrical and Computer Engineering, University of Maryland College Park, MD 74, USA 1 himsoon@glueumdedu ipaee@glueumdedu 3 aksadek@gmailcom 4 kjrliu@umdedu Abstract This paper derives an optimum relay position of amplify-and-forard AF cooperative communications ith the differential transmission As a performance benchmark, e utilize the probability of outage, hich is defined as a probability that the instantaneous signal-tonoise poer ratio SNR becomes loer than a certain threshold The optimum relay position is found based on the theoretical expression of the outage probability, Computer simulations are also conducted to verify the validity of the theoretical analysis Source Relay r s,r r r,d r s,d Destination :Transmission in Phase 1 :Transmission in Phase Keyords: cooperative communications, amplify-andforard protocol, differential modulation Fig 1 System Model 1 Introduction Cooperative communication concept has been draing much attention due to the potential impact on the various communications systems So far, considerable number of studies have been made on the cooperation protocols, such as amplify-and-forard AF and decode-andforard DF[1]-[3] Among them, the differential transmission ith the AF protocol[3] is of great interest from a vie point of sensor netorks[4] This is because the AF protocol does not require any decoding processing to the relay node, and the differential transmission means that the destination node is free from the burden of the estimation of channel state information CSI These features are extremely suited for the sensor netorks, here all the nodes should be simple and lo poer consumption In the previous orks, the relay node is usually assumed to exist or be predetermined, hoever, ho to assign the relay node ill be crucial problem in order to realize the cooperative communications systems Recently, the relay-assignment method has been proposed for the DF cooperative communications based on the analysis of the optimum relay position[5], but no discussion on the optimum relay position for the AF cooperative communications has been made In this paper, e derive the optimum relay position of the AF cooperative communications ith the differential transmission As a performance benchmark, e utilize the probability of outage, hich is defined as a probability that the instantaneous signal-to-noise poer ratio SNR becomes loer than a certain threshold Based on the theoretical expression of the outage probability, e can find the optimum relay position Computer simulations are also conducted to verify the validity of the theoretical analysis System Model Fig 1 shos a system model considered in this paper The to nodes source and destination nodes communicate ith the AF protocol using the differential M- ary PSK modulation scheme The information symbols transmitted by the source are ritten as v m expj φm, 1 here φ m πm/m for m, 1,,M 1 Then, the differentially encoded transmitted signals are given

2 by xτ v m xτ 1, here τ is the time index In the AF protocol, the signals are transmitted in to phases, and all the users transmit signals through orthogonal channels by using a multiple access scheme, such as TDMA, FDMA or CDMA schemes[1], [] In Phase 1, the source node transmit the symbol xτ ith poer P Assuming flat Rayleigh fading channels, the received signals at the destination node can be ritten as y s,d τ PK s,d h s,dτxτ+ s,d τ, 3 here K is a constant that depends on the antenna design, r s,d is the distance beteen the source and the destination nodes, h s,d τ is the channel fading gain beteen the nodes, hich is modeled as a zero mean circularly symmetric complex Gaussian random variable ith unit variance, η is the path loss exponent, and s,d τ is the additive hite Gaussian noise AWGN ith variance N In the same ay, the received signal at the relay node can be described as y s,r τ PKrs, h s,r τxτ+ s,r τ 4 In Phase, the relay node amplifies y s,r and transmits it to the destination ith transmission poer P Here, note that, in order to verify the effect of the relay position alone, e assume the same transmission poer as the source node The received signal at the destination node is given by y r,d τ PKr η r,d h r,dτy s,r τ+ r,d τ, 5 PK P PKrs, 6 + N The destination node combines the received signals in Phase 1 y s,d andphasey r,d Based on the multichannel differentially coherent detection MDCD[6], the combined signal is given by y a 1 y s,d τ 1 y s,d τ+a y r,d τ 1 y r,d τ, 7 here a 1 1 N, 8 PKrs, a + N N PKrs, + PK r,d + N 9 Finally, the information signal is obtained by ˆm arg max m,1,,m 1 R{v m y} 1 3 Optimum Relay Position Here, e derive optimum relay position based on the probability of outage, hich is defined as P O γ th P γ γ th, 11 here γ is the SNR of the combiner output y and γ th is a threshold of outage Hoever, the SNR formulation of the combiner output for the eights of 9 becomes very complex and is difficult to analyze For the simplicity, e utilize ideal maximum ratio combining eights[7] â 1 1 N, 1 PKrs, + N â N PKrs, + PK r,d h r,d + N, 13 for the derivation of the probability of outage Note that the time index of τ in the fading gain is omitted for the notational simplicity in the hereafter By using 13, the instantaneous SNR of the combiner output can be given by γ γ 1 + γ,here γ 1 PK h s,d s,d, 14 N P K rs, γ r η r,d h s,r h r,d N PKrs, + PK r,d h r,d + N 15 If e have the freedom to put the relay anyhere in the to-dimensional plane, hich include the source and the destination, then the optimum relay position ill be on the line joining the source and the destination This is because if the relay is at any position in the plane, then its distance to both the source and the destination are alays larger than their corresponding projections on the straight line joining the source and the destination Therefore, e can rerite and r s,r m r s,d, 16 r r,d 1 m r s,d, 17 here m [ : 1] Also, since e have assumed the ideal MRC and the difference of the relay position affects not γ 1 but γ, the relay position hich minimizes P Oγ γ th P γ γ th 18 also minimizes the outage probability P O γ th Therefore, the optimum relay position can be found via solving the folloing optimization problem m opt argmin m P Oγ γ th 19 Since h s,r and h r,d are independent exponential random variables, the probability density function PDF

3 of γ is obtained as see Appendix m ηγ s,dm η p γ λ 1 m η γ s,d m η γ s,d m η γ s,d 1 m η γ s,d m η λ d, here γ s,d PKr η s,d 1 N is the average SNR of the received signal of the direct path Using, P Oγ γ th is obtained as P Oγ γ th γth p γ λdλ m ηγ s,dm η 1 m η γ s,d m η γ s,d m η γ s,d 1 m η γ s,d m η d m ηγ s,dm η 1 m η γ s,d m η γ s,d m η γ s,d 1 m η γ s,d m η γ th d 3 While e are able to have the closed form of the first term of 3, it is difficult to have the form of the second term So as to have the closed form of the second term, e consider the approximation of the term By observing the argument of the exponential, e can see that γ th / is dominant for small, hile γ s,d m η + 1/γ s,d 1 m η γ s,d m η dominates the argument for large Therefore, the second term of 3 can be approximated or upper bounded as m ηγ s,dm η 1 m η γ s,d m η γ s,d m η γ s,d 1 m η γ s,d m η γ th m ηγ s,dm η 1 m η γ s,d m η exp γ th m ηγ s,dm η d d + R 1 m η γ s,d m η γ s,d m η γ s,d 1 m η γ s,d m η d, 4 here R is a certain threshold, hich satisfies R γ s,d m η 5 By using 4, P Oγ γ th can be loer bounded by POγ l γ th as P Oγ γ th POγ l γ th m ηγ s,dm η 1 m η γ s,d m η γ s,d m η γ s,d 1 m η γ s,d m η d m ηγ s,dm η + 1 m η γ s,d m η exp γ th d { Rγ s,d m η } 1 exp γ s,d 1 m η γ s,d m η R γ thγ s,d m η { Rγs,d m η exp γ th /R γs,d m η 1 m η γ th γ s,d m η R exp γ th γth γ s,d m η } R γ s,d m η E 1 Rγ s,d m η, 6 here E 1 is the exponential integral defined as[9] exp zt E n z 1 t n dt 7 Finally, e determine the threshold R in order to achieve good approximation The approximation of 4 is an upper bound of the second term of 3, because the omitted portions, in both the first and second terms of 4, are alays in the range of [:1] for the interval of the integration Therefore, the tightest approximation can be achieved by utilizing such R that maximizes POγ l γ th By differentiating POγ l γ th ith respect to R and solving POγ l γ th, 8 R the optimum R is obtained as R opt 1+ γ s,d m η m η γ s,d γ th 1 m η 4 Numerical Results Numerical calculation and computer simulations are conducted ith DQPSK modulation scheme Fig shos POγ l γ th of 6 ith R of 9 versus the relay position m The average received SNR of the direct path γ s,d, the path loss exponent η, and the threshold of γ th ere set to be 1[dB], 3, and 3[dB], respectively From the figure, e can see that the optimum relay position is around m 8in this case This means that the optimum relay position of the AF protocol is closer to the destination than the source, hile it has been shon in [5] that the optimum relay position for the DF protocol is the midpoint of the source and the destination From a viepoint of relay-assignment algorithm, it can be said that the nodes closer to the destination node are more suited for the relay than

4 5 Conclusion We have derived the theoretical outage probability in order to optimize the relay position of the differential transmission ith the AF protocol in the todimensional space Unlike the DF protocol, the optimum relay position of the AF protocol is not the midpoint of the source and the destination nodes but the closer point to the destination Also, computer simulation results are conducted to support the theoretical analysis From all the results, it can be concluded that our analytical results can be applied for the optimization of the relay position of the MDCD ith the AF protocol The investigation of the relay assignment method for the MDCD ith the AF protocol in multi-node scenario ill be our future study Bit Error Rate Fig Optimization of Relay Position MDCD Ideal MRC Relay Position m Fig 3 Bit Error Rate Performance References [1] J N Laneman, D N C Tse, and G W Wornell, Cooperative diversity in ireless netorks: efficient protocols and outage behavior, IEEE Trans Inf Theory, vol 5, no 1, pp36-38, Dec 4 [] J N Laneman and G W Wornell, Distributed space-time coded protocols for exploiting cooperative diversity in ireless netorks, IEEE Trans Inf Theory, vol 49, no 1, pp415-55, Oct 3 [3] T Himsoon, W Su, and K J Ray Liu, Differential transmission for amplify-and-forard cooperative communications, IEEE Signal Processing Letters, vol1, no 9, pp597-6, Sept 5 [4] I Akyildiz, W Su, Y Sankarasubramaniam, and E Cayirci, A survey on sensor netorks, IEEE Commun Mag, vol4, no 8, pp1-114, Aug [5] AKSadek,ZHan,andKJRayLiu, Relay-assignment protocols for coverage extension in cooperative communications over ireless netorks, submitted to IEEE Trans Wireless Commun [6] JGProakis,Digital communications, third Edition, McGra- Hill, 1995 [7] D G Brennan, Linear diversity combining techniques, Proc of IEEE, vol 91, no, pp , Feb 3 [8] M K Simon and M-S Alouini, A unified approach to the probability of error for noncoherent and differentially coherent modulations over generalized fading channels, IEEE Trans Commun, vol 46, no 1, pp , Dec 1998 [9] M Abramoitz and I A Stegun, Handbook of mathematical functions, Dover Publications, 1965 the nodes closer to the source node This asymmetry property comes from the fact that not only the desired signal but also the additive noise is also amplified at the relay node in the AF protocol Fig 3 shos the computer simulation results of the bit error rate BER performance of the MDCD ith the AF protocol versus the relay position m for γ s,d [db] and η 3The BER performance of the ideal MRC combining ith the AF protocol is also plotted in the same figure We can see that the analytical result is ell accorded ith the simulation results and, although e have utilized the ideal MRC in the analysis for the tractability, our analytical results can be also applied for the MDCD ith the AF protocol Appendix Here, e derive the PDF of γ Firstly, e define to random variables α and β as P K rs, α r η r,d h r,d N PKrs, + PK r,d h r,d + N, 3 β h s,r, 31 so that γ α β Since h s,r is an exponential random variable, the PDF of β is given by p β β exp β 3 Defining ζ h r,d, hose PDF is given by p ζ ζ exp ζ, 33

5 α can be reritten as α P K s,r r η r,d ζ N PK s,r + PK r,d ζ + N Note that the range of α is α PKr η s,r N that of ζ is ζ Here, e have and f ζ def fζ 34 because P K rs,r η r,d PKr η s,r + N N PKrs, + PK r,d ζ + N, 35 Hence, the PDF of γ is obtained as p γ γ PKr η s,r N PKr η s,r N p γγ,dt 4 N rs,pkr η s,r + N r,d PKr η s,r N N PK s,r + N PKPKrs, r,d N r,d γ 43 ζ N PKrs, + N α PKPKrs, r,d N, 36 r,d α therefore, the PDF of α can be obtained as p α α p ζζ f ζ N PKrs, + PKr η r,d ζ + N P K rs, r,d PKr η s,r + N exp ζ 37 N rs, PKrs, + N r,d PKr η s,r N α N PKrs, + N α PKPKrs, r,d N r,d α 38 Since α and β are independent because h r,d and h s,r are assumed to be independent, the joint PDF of α and β is simply given by p αβ α, β p α αp β β 39 In order to obtain the PDF of γ, e further consider a transformation of { γ αβ, 4 α Then the system has a single solution of α, β γ / and the Jacobian of Jα, β Therefore, the joint PDF of γ and is given by p γγ, 1 p αβ, γ N rs,pkr η s,r + N r,d PKr η s,r N N PK s,r + N PKPKrs, r,d N r,d γ 41