Amplify and Forward Relaying; Channel Model and Outage Behaviour

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1 Amplify and Forward Relaying; Channel Model and Outage Behaviour Mehdi Mortazawi Molu Intitute of Telecommunication Vienna Univerity of Technology Guhautr. 5/E389, 4 Vienna, Autria mmortaza@nt.tuwien.ac.at Norbert Goertz Intitute of Telecommunication Vienna Univerity of Technology Guhautr. 5/E389, 4 Vienna, Autria norbert.goertz@nt.tuwien.ac.at Abtract Characteritic of receiver output aociated with Source-to-Relay-to-Detination link for Amplify and forward relaying ytem i tudied and theoretical epreion for probability denity function and cumulative ditribution function are derived. Thereby, cloed-form olution for outage probability of the ytem i derived. The analyi i then etended to a cenario where the detination combine the ignal correponding to the ource and the relay tranmiion uing maimal ratio combining technique. The tatitic of the equivalent i tudied and ome new cloed form epreion at high regime are derived. The comparion of theoretical and numerical reult how that the high epreion for outage probability are adequately tight in low too. The performance of the AF relaying ytem i further invetigated at low regime; the reult how that at low regime, the relay tranmiion doe not play role on the the performance of the overall ytem, therefore, pending reource (e.g. power or bandwidth) for the relay tranmiion i only wating the reource wherea at high regime the relay tranmiion provide full diverity. I. INTRODUCTION Fading i one of the mot evere channel impairment affecting the performance of wirele communication ytem. A variety of method aim to combat fading by time, frequency or patial diverity; multiple antenna are mot effective to eploit patial diverity []. However, depite the large theoretical performance gain of multiple antenna ytem, practical limit in implementing tranceiver (uer terminal in particular) call for other method of eploiting patial diverity. cooperative communication i another method of eploiting patial diverity which i obtained by cooperation among ditributed node. Several relaying protocol have been propoed in the literature, e.g. Amplify-and-Forward (AF), Decode-and-Forward (DF), Soft-DF, and Compre-and-Forward (CF) (e.g. [3], [4]), where, depending on the parameter of the network, each of them can be the method of choice. There i currently a lot of interet in AF relaying becaue of it implicity in term of analyi and it low compleity compared to other relaying protocol; hence, AF i alo the focu of thi work. For an analytical performance evaluation of a relaying cheme, the tatitical model of the Source-to-Relay-to- Detination (S-R-D) link i mot important. Several paper conider the problem: in [5], an equivalent S-R-D channel model ha been propoed in term of modified Beel function of the econd kind, but it i aumed that the direct Source-to- Detination (S-D) channel i in a deep fade, o the effect of the S-D link can be ignored. Moreover, only the high regime S Fig.. R Sytem Model i conidered in [5] uing the moment-generating function. A imilar approach i dicued in [6] too. In [3], the outage behaviour of different relaying protocol, including AF, ha been tudied at high and low tranmiion rate, and in [7] outage capacity of different protocol, including AF, i tudied in the low- regime. To the bet of our knowledge the problem ha not been invetigated at moderate regime. In fact, although, everal paper evaluate the performance of the AF relaying ytem under high or low aumption but, yet, there i not a general channel model which cover all the characteritic of the relaying ytem at low, moderate and high. In thi paper we tudy the outage behaviour of an AF relaying ytem auming a general S-R-D channel model when a direct link i available between the ource and the detination. The reminder of the paper i organized a follow. Section II introduce ytem model and derive a general equivalent channel model for S-R-D link. Section III tudie the outage behaviour of the cooperative ytem. Section IV provide aymptotic outage behaviour of the ytem and compare it with the already eiting reult and finally the reult are ummarized at ection V. II. SYSTEM AND CHANNEL MODEL We conider a two-hop Amplify-and-Forward (AF) communication ytem a illutrated by Fig.. The ource (S) end data to the detination (D) by the help of an intermediate relay node (R). The detination might hear both the ource and the relay tranmiion and apply Maimal Ratio Combining (MRC) of the available information in the detination, or it can only hear the relay tranmiion (e.g. due to deep fading on the S-D channel [5]): both the cenario are evaluated. It i aumed that the relay operate in half-duple mode, i.e. the relay can not receive and tranmit imultaneouly. Moreover, the overall ytem i orthogonal in time, i.e. the tranmiion D //$3. IEEE 6

2 X PDF() X.5.5 Fig.. The area under the curve correpond to different. X X X+X+ < for time i divided into two periodically repeated lot: the wirele channel i allocated for the ource tranmiion during the firt time lot and for the relay tranmiion in the econd time lot. Of coure, the orthogonality contraint induce the crucial need for full ynchronization among the node (which i aumed). The channel are ubject to Rayleigh fading and AWGN receiver noie. The ignal correponding to the ource tranmiion received at the detination (y d ) and the relay (y r ) are y d p P h d + n d y r p P h r + n r () where i tranmit ignal vector. The parameter P i the ource power contraint, and h d and h r repreent the channel coefficient correponding to the S-D and the S-R link, repectively. The channel coefficient, which capture the effect of path-lo and fading, are zero-mean, white comple Gauian procee with variance d and r. The coefficient are contant during every time lot (or tranmit block) and they vary independently from one block to another (blockfading model). Additive receiver noie i modelled by n d and n r, which are ample-vector from zero-mean, white comple Gauian procee, for implicity both with variance N. For Amplify and Forward (AF), the relay amplifie (without any further proceing) the ignal received from the ource uch that it fulfil the relay power contraint, P r, and retranmit the ignal toward the detination; the channel coefficient h r i aumed to be available to the relay. The ignal received at the detination correponding to the relay tranmiion i given by y rd P r E( y r ) h rdy r + n d () P r P P h r + h P r rh rd + P h r + h rdn r + n d The ignal y d and y r are vector of matched-filter output ample; their dimenion depend on the number of channel ue within a time lot. Fig. 3. The PDF of RV X, (6), for variou value of and. The R-D channel (Rayleigh fading with variance rd ) and the noie characteritic (N ) are imilar to thoe of the S-D and the S-R link. From inpection of () it i clear that the equivalent S-R- D link can not be modelled a a Rayleigh fading channel. However, due to the block-fading aumption, the equivalent noie at the detination correponding to the relay tranmiion (middle term in the econd line of ()) i Gauian per block and another Gauian receiver noie n d i added. Hence, a ubtitute additive Gauian noie model can be ued. The correponding equivalent receiver-output Signal-to-Noie Ratio () at the detination will be one of the major parameter governing the performance of the overall ytem, a thi output can directly be related to the capacity, diverity, throughput, error rate and other performance meaure of the overall ytem. Therefore, the tatitic of the equivalent S-R- D ( rd ) will be derived. Auming P P r P for implicity, the intantaneou rd uing () i rd P h r h rd h r + h rd + (3). N /P /P. With the where N i aumed and channel coefficient known at the receiver (both at the relay and the detination) coherent detection can be ued, and the quared magnitude h ij of the Rayleigh-ditributed channel coefficient that appear in (3) are eponentially ditributed. with parameter ij / ij, i {,r}, j {r,d}, i 6 j. In the ret of thi ection the cumulated denity function (CDF) and the probability denity function (PDF) of the h random variable (RV) r h rd h r + h rd + are derived, negelecting the factor P in (3). Theorem. CDF of the RV X XX X +X + Let X and X be two independent eponential RV with the PDF f Xi ( i ) i e i i, i, i {, }, and the parameter, >, and let > be a real contant. Then, the CDF of the RV X i given by XX X +X + F X () e ( + )p ( + ) (4) K p ( + ) 7

3 with K ( ) the modified Beel function of the econd kind and -th order. Proof: Fig. illutrate the RV X<for variou value of, where, accordingly, one of the RV X or X can take any value larger than ; e.g. let ay X [, ), then X [, ). Therefore we have (+ ) ( ) Without S D link X X F X () P ( X+X+ Z Z Z ( + ) ( ) e Z Z u <) e e d d e d ( + ) ( ) e ( + ) Z e ( + ) e d (u++ ) e u e (u+) du u (+ ) e u e u du e ( + )p ( + ) (5) K p ( + ) where the lat equality follow from [8, ]. One can obtain PDF of RV X, pecified in Theorem by taking the derivative of F X (), (4), with repect to a apple f X () e S P( + )K ( p P( + )) (6) + S p P( + )K ( p P( + )) where P and S +. Fig. 3 illutrate f X (), derived in (6) for variou value of and auming that r rd. It i clear that for larger value of (i.e. lower per hop) the PDF curve move left, toward the PDF() ai, conequently, when!, the entire power of the PDF will hrink at the point +, which implie that communicating via relay i impoible. III. OUTAGE ANALYSIS Outage probability i a common tandard criterion, characterizing the performance of communication ytem operating in fading environment. By definition, (e.g. [9], []), outage probability, p out, i the probability that the intantaneou receiver output fall below a certain threhold which i pecified according the epected performance (e.g. BER, capacity, etc.) of the ytem. In order to evaluate the outage behaviour of the ytem illutrated in Fig., we conider two cenario: we firt invetigate outage behaviour of AF relaying ytem when there i not a direct link available between the ource and the detination, then the problem i further invetigated when a direct link between the ource and the detination i available. In the following, let define N and P N P. Suppoe that input ignal vector i i.i.d. circularly ymmetric comple Gauian with identity covariance matri, i.e. E( H )I. Moreover we aume that the perfect Channel State Information (CSI) of all the link, i.e. h r,h d and h rd, are available at the detination but only h r i known for the relay. P out R.5 R R With S D link 3 4 P/N(dB) Fig. 4. Outage probability for variou R. r rd d A. Without Source to Detination Link Let aume that the detination terminal doe not hear the ource tranmiion, which can be e.g due to occurrence of deep fading in S-D channel. Therefore the ource communicate with the detination only via the relay. Therefore, the mutual information, auming (3), i I log + h r h rd h r + h rd bp/hz (7) + where the factor reflect the fact that information i conveyed to the detination in two time lot. The outage probability i defined a p out (R, ) P log ( + h r h rd h r + h rd + ) apple R h r h rd P h r + h rd + apple R (8) The probability pecified in (8) wa already derived in (4). By ubtituting r R in (4), p out (R, ) i p out (R, ) e ( r+ rd)rp r rdr(r + ) K p r rdr(r + ) (9) Fig. 4 (olid line) illutrate the theoretical outage probability for the non S-D link cenario, for variou value of tranmiion rate when r rd. B. With Source to Detination Link Suppoe that the detination hear both the ource and the relay tranmiion and employ MRC technique for combining them. Therefore the receiver output at the detination can be written a hr h rd tot h r + h rd + + h d () 8

4 , Theory 4, Numeric 5.6, Theory 5.6, Numeric C 3 With S D link P outage 3 Without S D link 3 4 P/N (db) P/N (db) Fig. 5. -outage capacity for variou value. r rd d Fig. 6. Outage probability, theoretical and numerical reult, R.5. Therefore, the outage probability i h r h rd p out (R, ) P ( h r + h rd + {z } X + h d apple R {z } ) () Y where the RV X i ditributed according (6) and RV Y i eponentially ditributed with parameter d. Then, p out in () can be written a p out (R, ) P (X + Y apple r) F X (r) e de e f Y (y) f X ()dyd e d f X ()d e d F X ()d () d e e ( r+ d)p rd r rd( + ) K p r rd( + ) d where the fourth equality i obtained uing integration by part algorithm. The outage probability in (), or equivalently, the tatitic of the receiver output at the detination, become more challenging to obtain. In fact the integral appeared in () i non-trivial and eem not to have general cloed form olution in the form of known mathematical function. However, we develop ome theoretical cloed form olution in ection IV-B. We alo reort to the numerical integration method for olving the problem.we employ Gau-Kronrod quadrature adaptive integration algorithm [] to obtain p out from (). Fig. 4 (dahed line) preent the outage probability for variou value of and uing Gau-Kronrod algorithm. The theoretical, cloed form reult will follow in ection IV-B. Fig. 5 illutrate the outage capacity for the cenario eplained in III-A (olid line) and III-B (dahed line). IV. ASYMPTOTIC OUTAGE BEHAVIOUR We will evaluate both the cenario eplained in III-A and III-B at high and low regime. A. Low For the relay cenario eplained in III-A: the outage probability i given in (9). When!(i.e.! ), K ( )!, which mean that the p out!. That i intuitively true, becaue at low (!), the relay, almot, only forward the receiver noie toward the detination. For the relay cenario eplained in (III-B): the outage probability i given in (). At low regime, h r h rd h r + h rd + be written a! a!. Then, outage probability can p out (R, ) P( h d apple R ) e (3) where r R.Therefore, the outage probability of a AF relaying ytem ubject to Rayleigh fading i equivalent to the outage probability of the direct ource to detination tranmiion. Then, the -outage capacity can be calculated from P( h d < C ). (4) Auming lim log( + ), and ome manipulation,! (4) will be implified to lim!! C bp/hz. (5) d From MIMO theory it i well known that the optimum tranmiion trategy at low i to allocate all the available power to the bet antenna (the antenna correponding to the tronget eigenmode of MIMO channel). Likewie, ince the relay terminal doe not improve the capacity at low, the bet policy i to allocate all available power to the ource tranmiion, intead of plitting it between the ource 9

5 and the relay. The channel capacity will be doubled if the ource terminal tranmit with full power. (i.e. P r and P P ) lim!! C d bp/hz. (6) From inpection of (5) and (6) it i clear that plitting total available power between the ource and the relay i not optimum at very low. In fact at very low regime, the bet policy i to avoid cooperation. However, the minimum intantaneou which one can benefit cooperation i out of cope of the paper. B. High For the relay cenario eplained in III-A: the outage probability at high from (9) i ( r+ rd)r p out (R, ) r p r rde K p r rdr (7) where r R. The equation in (7) i derived by ubtituting in (9) and auming that r i arbitrary value depending on R and. Furthermore, uppoe that r!, i.e. low tranmiion rate [3]. Then, uing [,.3.], K ( p r rdr) p. Therefore, (7) will be further r rdr implified to p out (R, ) lim r!! e ( r+ rd)r (8) ( r + rd )r ( r + rd )( R ). (9) Note that (8) i equivalent to the CDF of an eponential RV with parameter r + rd. Conequently, at high and low tranmiion rate regime, the S-R-D link can be modelled by an eponential random variable with parameter r + rd. Interetingly, thi reult i reported in [3] too, in which obtain the ame reult taking a totally different approach. The eponent in (9) prove that the relay path at high provide full diverity order. For the relay cenario eplained in III-B:, the outage probability will be calculated from () when! p p out (R, ) e d () r rd e e ( r+ rd d) K ( p r rd)d A eplained before, the integral in () i non-trivial but for the cae when ( r + rd d ) p r rd, the cloed form olution i available a follow p out (R, ) ( + 3 )e () d 3 r e ( r+ rd)r (K ( r) K ( r)) where ( r + rd d ) p r rd. In order to obtain () from (), we employ [,.43.9] in which e K ()d () e r r + + (K (r) K + (r)) + ( + ) + Fig. 6 how the theoretical outage probability obtained uing () and alo uing numerical Gau-Kronrod adaptive algorithm in () for variou value of and wide regime. The perfect merge of numerical and theoretical curve confirm the correctne of the calculation. Moreover, when r! (i.e. low and moderate tranmiion rate and high ), () can be further implified to p out (R, ) lim r!! ( r + rd )e d ( r + rd )r r + rd d de ( r+ r)r ) d ( r + rd )( R ) (3) where we again apply the aumption that K ( p r rdr) p in () and eploit erie epanion of r rdr e r at r!. The eponent in (3) how the diverity order where one of the order i provided by S-D link and the other one by the S-R-D link. V. CONCLUSION An equivalent S-R-D channel model i derived for AF relaying protocol. Uing the channel model, the outage probability (or equivalently outage capacity) i derived for the S-D-R link. the problem i then etended to a more complicated cenario where a direct link between the ource and the detination i available. Theoretical reult on outage probability are derived. The aymptotic (high and low ) behaviour of the communication ytem of conideration are invetigated. The comparion of theoretical and numerical reult confirm the correctne of calculation. REFERENCES [] I. E. Telatar, Capacity of multi-antenna gauian channel, Telecommunication, European Tranaction on, vol., pp , No/Dec 999. [] E. C. van der Meulen, Three-terminal communication channel, Advance in Applied Probability, vol. 3, p., 97. [3] J. Laneman, D. Te, and G. Wornell, Cooperative diverity in wirele network: Efficient protocol and outage behavior, Information Theory, IEEE Tranaction on, vol. 5, pp , dec. 4. [4] M. M. Molu and N. Goertz, A tudy on relaying oft information with error prone relay, pp , ept.. [5] M. O. Hana and M. S. Alouini, End-to-end performance of tranmiion ytem with relay over rayleigh fading channel, Wirele Communication, IEEE Tranaction on, vol., November 3. [6] I. Krikidi, J. Thompon, S. McLaughlin, and N. Goertz, Amplifyand-forward with partial relay election, IEEE Communication Letter, vol., April 8. [7] A. S. Avetimehr and D. N. C. Te, Outage capacity of the fading relay channel in the low nr regime, Information Theory, IEEE Tranaction on, vol. 53, April 7. [8] I. Gradhteyn and I. Ryzhik, Table of Integral, Serie, and Product. Academic, 7th ed., 7. [9] D. TSE and P. Viwanath, Fundamental of Wirele Communication. Cambridge Pre, May 5. [] A. Goldmith, Wirele Communication. Cambridge Pre, September 5. [] W. H. Pre, S. Teukolky, W. T. Vetterling, and B. P. Flannery, Numerical Recipie: The Art of Scientific Computing. Cambridge Pre, 3 ed., September 7. [] F. W. J. Oliver, D. W. Lozier, R. F. Boivert, and C. W. Clark, NIST Handbook of Mathematical Function. Cambridge Pre, July.