We compare the signal-to-noise ratio performances of an amplify-forward relay link and a direct link in a wireless sensor network. For a slow Rayleigh fading channel, an exact expression for the probability that the SNR of an amplify-forward relay link exceeds the SNR of a direct link is obtained. For a Rician fading channel, an upper bound for the corresponding probability is obtained. Numerical results indicate that, among the two fading channels considered, Rayleigh fading is more detrimental to the SNR performance of the amplify-forward relaying scheme

Viswanathan, R.

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Southern Illinois University CarbondaleOpenSIUCConference Proceedings Department of Electrical and ComputerEngineering1-2007Signal-to-Noise Ratio Comparison of Amplify-Forward and Direct Link in Wireless SensorNetworksR. ViswanathanSouthern Illinois University Carbondale, viswa@engr.siu.eduFollow this and additional works at: http://opensiuc.lib.siu.edu/ece_confsPublished in Viswanathan, R. (2007). Signal-to-noise ratio comparison of amplify-forward anddirect link in wireless sensor networks. 2nd International Conference on Communication SystemsSoftware and Middleware, 2007 (COMSWARE 2007), 1-4. doi: 10.1109/COMSWA.2007.382467©2007 IEEE. Personal use of this material is permitted. However, permission to reprint/republishthis material for advertising or promotional purposes or for creating new collective works for resaleor redistribution to servers or lists, or to reuse any copyrighted component of this work in otherworks must be obtained from the IEEE. This material is presented to ensure timely dissemination ofscholarly and technical work. Copyright and all rights therein are retained by authors or by othercopyright holders. All persons copying this information are expected to adhere to the terms andconstraints invoked by each author's copyright. In most cases, these works may not be repostedwithout the explicit permission of the copyright holder.This Article is brought to you for free and open access by the Department of Electrical and Computer Engineering at OpenSIUC. It has been acceptedfor inclusion in Conference Proceedings by an authorized administrator of OpenSIUC. For more information, please contact opensiuc@lib.siu.edu.Recommended CitationViswanathan, R., "Signal-to-Noise Ratio Comparison of Amplify-Forward and Direct Link in Wireless Sensor Networks" (2007).Conference Proceedings. Paper 86.http://opensiuc.lib.siu.edu/ece_confs/86Signal-to-Noise Ratio Comparison of Amplify-Forward and Direct Link in Wireless SensorNetworksR. ViswanathanDepartment of Electrical & Computer EngineeringSouthern Illinois University at CarbondaleCarbondale, IL 62901-6603Email: viswagengr.siueduAbstract-We compare the signal-to-noise ratio performances of quality is bad, the relay does not transmit but conserves it'san amplify-forward relay link and a direct link in a wireless transmit power and lets the destination to decode the directsensor network. For a slow Rayleigh fading channel, an exact reception from the source. In AF, the relay simply amplifiesexpression for the probability that the SNR of an amplify- its received signal from the source and then transmits it to theforward relay link exceeds the SNR of a direct link is obtained. destination. Since background noise (and any interference) isFor a Rician fading channel, an upper bound for the....corresponding probability is obtained. Numerical results picked upalong withe tesignalditioby therel t ificatoindicate that, among the two fading channels considered, process amplifes the noisein addition to the receivedsignal.Rayleigh fading is more detrimental to the SNR performance of Despite this noise amplification, the overall gain in thethe amplify-forward relaying scheme. performance (i.e., channel capacity or outage performance) ofthe AF relay scheme is documented in the literature [1], [4].Keywords- cooperative relaying; amplify-forward; Rayleigh In this paper, we present an analysis that shows when afading; Rician fading. direct transmission from S to D may be preferable to the signalfrom the AF relay scheme. We are not combining the signalsfrom the direct path and the relay path, but we are simplytrying to determine, given the power constraints, when aIn cooperative relaying, the availability of relay channels relaying may or may not be useful.between a source (sensor) and a destination is exploited byappropriately combining signals arriving via various relaychannels. A summary of cooperative diversity along with a II. ACHIEVABLE SNR IN S-TO-D AND AF SCHEMES INchronological order of developments is provided by Laneman FADING CHANNELSet al. in [1]. In a two-part paper, Sendonaris and Aazhang Let PT denote the transmit power from the source andhave provided the concept and the implementation aspects of define the channel gains (coefficients) in different links ascooperative diversity for wireless networks of mobile users follows: a,, a2 ad for S to R, R to D, and S to D,[2]-[3].Laneman et al. propose cooperative diversity for relaying respectively. If the power amplification gain in the AFin wireless sensor networks [1]. Their analysis primarily scheme is g and if the noise power picked up by the relayconsidered a single relay terminal (R) helping the receiver (destination receiver) is denoted as N1 (N2), thencommunications between a source (S) and a destination (D), the effective received powers at the destination produced byeven though they point out possible extensions to situations AF and direct reception are given byinvolving multiple relays. They had formulated two schemes, PRAF = a2 {g(Si + Ni)} (1)namely, amplify and forward (AF) and decode and forward2(DF) under fixed relaying procedures and two other schemes, PR,d = d PT, (2)namely, selection relaying and incremental relaying under where S1 is the received power at the relay and is given byadaptive relaying procedures. In selection relaying, depending S_ T (3)on the quality of the link between the source and the relay, a 51(IP()decision as to whether the relay would retransmit the message The effective SNR (i.e., the ratio of the signal power from theit had received from the source would be made. When the signal term in (1) and the noise power, which is the sum of the1-4244-0614-5/07/$20.00 ©C2007 IEEE.Authorized licensed use limited to: Southern Illinois University Carbondale. Downloaded on May 30, 2009 at 16:16 from IEEE Xplore.  Restrictions apply.amplified noise power of the relay node, as received at the ()destination, and the noise power picked up by the destination I fP(g at x - ad (I + gx) > |a,2 = X)f,x (x)dxnode) of the AF relay path is given by 0 (8)SNRAF = gUt2NPT (4) where f,, (x) is the probability density function of a2gar2N +N22Similarly, for the direct path After some manipulations of the integral (8), we get (seead PT Appendix)SNRd = (5) = 1 (1BeB Ei(B)), (9)N2 9Then, from (4) and (5), for fixed channel coefficients, AF is 1 + dbetter than the direct link (in terms of SNR), if K 2Ni where(g al a(2-ad) > g (2a. With the reasonable i / ikN2 B = 1 (10)assumption of Ni = N2, AF has better SNR than the direct g 22 (I + Ad / 2i)link when the following inequality is satisfied: and E1 (B) is the exponential integral defined by [5]ga 2 (al -ad) > ad (6) E (B)= jed (11With a reasonable assumption of a > ad (which B Zcorresponds to the signal power at R being larger than the As g -> oo, B -> 0, and B E (B) -> 0. Hence, thesignal power at D from a direct reception), it is clear that (6) 1can be satisfied by adequately increasing the amplifier gain g probability I is upper bounded by I <I + (Ad/Al)at R. That is, if there were no power limitation at R, increasingthe power at R will eventually make the relay link better than That is, the probability that the SNR of AF exceeds the SNRthe direct link. This happens irrespective of the fact that the of the direct link cannot approach 1 even if the amplificationnoise picked up by the relay is also amplified by the relay (if at the relay is infinite (unless Ad /2A is arbitrarily close to 0).then at has to be> ad N1 for this to be t) This result has some similarity to the well known result in anN2 adaptive power control scheme, which attempts to compensateIf g is unbounded, could we still Nxpect2a similar resultin.. for deep channel fades: in Rayleigh fading, the capacity ofexpect similar result insuch an adaptive scheme is zero. Since AF relaying involvesfading channel? Before we answer this question, we first additional hardware and energy consumption, one canobserve that the probability, P(ga2alt> (i+gag2)ad) I reasonable argue that the AF should be preferred only if thewhich equals g a > is a monotonic ireprobability I exceeds some value, say 3/4? We will examinewceulg>at increasing specific numerical solution of I, as a function of g, in the next-'+ghX2) section.function of g, since all the channel gains are positive. Hence,an upper bound to this probability is achieved at g = o0 . The{ A ~~~B.Rician Fadingcorresponding upper bound is given by P'al > ad) Probability that the SNR of AF is better than the SNR ofthe direct link is given by the probability of the event specifiedA. Rayleigh Fading Channels by (6). Numerical calculation of this probability, when all theConsider a slow Rayleigh fading channel where the signal amplitudes of channel coefficients (amplitude coefficient ispower r v rsquare root of the power coefficient) fade according to Ricianpower received durings e teransmission of a symole(ora distributions, is possible but complicated. Instead, we willpacket) can be considered to be a sample of an exponential coptthupebun,wihsatiedfrniierandom variable (corresponding to a Rayleigh amplitude). cHence, let ai , i = 1,2, d be independently distributed as amplification gain at a relay. Let Zt = /at, Zd = ad,exponential with mean values, Ai i i = 1,2, d, respectively. where ZI, Zd are distributed as Rice with parametersFor a given value of g, the probability that the SNR of AF is (Al , ac2), (Ad, ), respectively [6]. Here Ai representsbetter than that of the direct link is given by (upon rearranging the amplitude of the line of sight (los) component of RiceI = P(g at ac2-ad (i ± g a2)> o). (7) distribution and vi- represents the average power of theUsng the ndependence of varous lnks and by condtonng a in component oon the variable, ar2,(7) can be evaluated as PuRice =P(t1tn>and) is given byAuthorized licensed use limited to: Southern Illinois University Carbondale. Downloaded on May 30, 2009 at 16:16 from IEEE Xplore.  Restrictions apply.72~ ~ ~ a) Fig. 1. With sufficient strength in the direct component ofPu Rice =- 2(1-Q( b, ,a+- Rician amplitude in the S-to-R link, it is possible to achieve1 + y the upper bound of one to the probability. Certainly, the(12) presence of direct component is beneficial to the relayinga Q(;,} hiprocedure. Fig. 3 corresponds to the case where the ratio of+ 2 the average SNR of the fading component of S-to-R and that2 of S-to-D is one. When the ratio of the direct componentwhere a Ad amplitudes of S-to-R and S-to-D is one, irrespective of the2 ' - 2 ' relative strengths of the direct and the fading components, they+1 Jdupper bound to probability, which is achieved with infinite2A1 U1giathrlybeoeO5b = a r rd = y = . When the los components gain at the relay, b c m s 0.5.Ad Cdvanish, Rice distributions become Rayleigh distributions and IV. CONCLUSIONAid o sc 1 In this paper, for an amplify-forward relaying scheme, wethe ratio f ection II (i) iS equivalent to -. Hence, theiIy2 S evaluated and studied the variation of the probability that theSNR of amplify-forward link exceeds the SNR of the directupper bound is a function of the three parameters, link. While an exact expression for the probability was, rd and y . A numerical study of the behaviors of this derived for the Rayleigh fading case, only an upper bound,upper bound and the exact probability for the Rayleigh case is corresponding to an infinite power gain at the relay, waspresented in the following section. calculated for the Rician fading case. Of the two fadingchannels, it is observed that the Rayleigh fading is moredetrimental to the relaying scheme. Specific sensor networkUsing MATLABW we numerically evaluated the exact configuration and specific modulation / coding scheme mayprobability that the SNR of AF is greater than the SNR of ultimately determine if amplify-forward relaying in certaindirect link for the Rayleigh fading case and also the upper paths in such a network.bound (Eq. 12) for the Rician fading case. Fig.1 shows theprobability as a function of g 22, for Rayleigh fading, when APPENDIXthe ratio of the average SNR of S-to-R and that of the direct Here we show the steps that lead to the derivation oflink from S-to-D is 10 dB. As to be expected, this probability (9). Since al , a2i ad are all independent, theincreases monotonically as the relay amplification gain g conditional probability function in the integrand of (8)(more precisely, g i22) increases and reaches the maximum becomes the unconditional probability,value of 1/(1+1/10) at g =oo. In order to justify the PI = P(gaIx - ad((+±gx)> 0).additional hardware and energy expenses associated with If we let YI, Yd to denote g x atl, ( + g x) ad,relaying, a practical criterion may be to employ relaying only respectively, then YI, Yd are independent and distributedif this probability exceeds certain value, say 0.75. For l of as exponentials with meansAd Mi = gx Md = m(1±gx)2d, respectively. Hence,10 dB, this probability is 0.75 or more if g 22 is at least 0.58 1dB. When l is only 5 dB, in order that this probability is PI = P(YI> Yd) +( gx ( d AfterAd I......................................+ gx ) 2i)0.75 or higher, g 22 has to exceed 20.1 dB. We can observesubstituting this expression for P1 in (8) and executinga nonlinear dependence of g on ,with lower values of the simple algebraic manipulations, we get an equivalentAd ........................expressionfor (8):latter dictating much larger values of the power gain at the 122Brelay, in order to achieve a specific relay link quality. x+i2BI fI x 2 -X11,2 dxFor Rician fading, the upper bound on probability, 0 (12Ad 2equation (12), is plotted against rd for different values of r Ai+ j22in Fig. 2 (y = 10 dB ) and in Fig. 3 (y = 0 dB ). When where B is defined in (10). Using the definition (11) inrd=0 dB and M/ approaches zero (i.e., yi in dB the evaluation of the above integral leads to (9).approaches -o~), the fading channel becomes Rayleigh andthe upper bound in Fig. 2 matches with the asymptotic value inAuthorized licensed use limited to: Southern Illinois University Carbondale. Downloaded on May 30, 2009 at 16:16 from IEEE Xplore.  Restrictions apply.REFERENCES Aazhang, "User cooperation diversity, Part II:Implementation aspects and performance analysis,"[1] J.N. Laneman, D.N.C. Tse, and G.W. Womell, IEEE Trans. Commun., vol. 51, pp. 1939-1948, Nov."Cooperative diversity in wireless networks: 2003.efficient protocols and outage behavior," IEEE [4] M. Yu and J. Li, "Is amplify-and-Trans. Inform. Theory, vol. 50, pp. 3062-3080, Dec. forward practically better than decode-and-forward or2004. vice-versa?" in Proc. ICASSP, pp. 111-365- 368, 2005.[2] A. Sendonaris, E. Erkip, and B. [5] M. Abromowitz and I.A. Stegun,Aazhang, "User cooperation diversity, Part 1: System Editors, Handbook of Mathematical Functions,description," IEEE Trans. Commun., vol. 51, pp. Dover Pub., NY, 1972.1927-1938, Nov. 2003. [6] M. Schwartz, W. R. Bennett and S.[3] A. Sendonaris, E. Erkip, and B. Stein, Communication Systems and Techniques,McGraw-Hill, NY, 1966.0.9 '0.9|68 6-2061 612M_v ---0.7 0.70.6 0/6.4 6.46.3 6.30.2 6l26.1 6.1-36 26 16 6 1 2-4 -2 0 2 4 6 8 16gIambda2, dB rd, dBFig. 1 Probability that SNR of AF is greater than the SNR of direct link, Fig. 3 Upper bound on probability, Eq. (12) Vs rd dB, Rician fading,Rayleigh Fading 0 dB6.7-6.4-6.3--4 -2 6 2 4 6 8 160 1 2 14 1 6rd, dBFig. 2 Upper bound on probability, Eq. (12) Vs rd dB, Rician fading,y/ 10 dBAuthorized licensed use limited to: Southern Illinois University Carbondale. Downloaded on May 30, 2009 at 16:16 from IEEE Xplore.  Restrictions apply.

Signal-to-Noise Ratio Comparison of Amplify-Forward and Direct Link in Wireless Sensor Networks

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Signal-to-Noise Ratio Comparison of Amplify-Forward and Direct Link in Wireless Sensor Networks

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