An Analytical Expression of the Probability of Error for Relaying with Decode-and-forward

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1 An Analytical Expression of the Probability of Error for Relaying with Decoe-an-forwar Alexanre Graell i Amat an Ingmar Lan Department of Electronics, Institut TELECOM-TELECOM Bretagne, Brest, France alexanre.graell@telecom-bretagne.eu Institute for Telecommunications Research, University of South Australia, Aelaie, Australia ingmar.lan@unisa.eu.au Abstract For a three-noe relay network that employs binary linear coes an the ecoe-an-forwar strategy, we erive analytic upper an lower bouns for the probability of error at the estination. The bouns are base on union-boun techniques an the input-output weight enumerator of the encoer. As an application, we use this boun to optimise the ecoer at the estination. Our approach is verifie by simulation results. I. INTRODUCTION In a three-noe relay network, a source communicates to a estination over a wireless channel, assiste by a cooperating relay (see Fig. 1). Such relay channels were introuce by van er Meulen [1], an their capacities were stuie in etail by Cover an El Gamal []. Among the various cooperation schemes propose an investigate [3, 4], ecoe-an-forwar is one of the most practical. In this scheme, the relay ecoes the ata transmitte by the source, an forwars the estimate to the estination. The error rate of ecoe-an-forwar in uncoe systems has been analize in [5, 6]. For this analysis an also for the ecoing algorithms at the estination, the source-to-relay-toestination channel may be moelle as an equivalent onehop link with an equivalent signal-to-noise ratio (SNR) [7, 8]. The value of this equivalent SNR has been investigate in [7, 9]. The ecoing threshols of coe systems with iterative ecoing at the estination has been analyse with EXIT charts [8, 10]. Publishe results for coe systems have some major shortcomings. First, ecoing errors at the relay introuce memory in the virtual source-to-relay-to-estination channel (in contrast to the often applie assumption); there are no analytical methos available to analyse how this affects the probability of error at the estination. Secon, error floors with respect to the source-to-estination SNR have been observe [11], but no theoretical explanations are available. An thir, current methos to etermine the equivalent SNR for coe systems o not consier that the source-to-relay-to-estination channel has memory. In the present paper we evelop novel analytical bouns for the probability of error, taking into account ecoing errors at the relay. The upper boun is base on a union boun approach This work has been supporte in parts by the Australian Research Council uner the ARC Discovery Grant DP an by NEWCOM++. s γ sr x s ch s-r y sr γ s ch s- r x r ch r- y s y r Fig. 1. Relay network consisting of source s, relay r an estination, an the three channels (ch) in between. an the lower boun on the ominant term, both uner the mil an reasonable assumption that the SNR of the sourceto-relay is not too low. As an example application of these bouns, we optimise the equivalent SNR such that the error rate is minimise. The outline of the paper is as follows. In Section III we efine the system moel, the relaying strategy, an the ecoing strategy at the estination. Section IV eals with the boun on the probability of error. Section V escribes the optimization of the relay-to-estination SNR assume by the estination. Numerical results are presente in Section VI. Conclusions an an outlook to future work are provie in Section VII II. NOTATION Throughout the paper, we write vectors in bolface letters, an the i-th element of a vector a as a i. The Hamming weight of a vector a is enote by w(a), an the Hamming istance between two vectors a an b is enote by (a, b); for convenience we may simply speak of weight an istance. The support of a vector a is enote by S(a) = {i : a i 0}, an its complement by S c (a) = {i : a i = 0}. Notice that S(a) = w(a). The BPSK moulate symbol of a bit x i {0, 1} is written as x i { 1, +1}, an we use the BPSK mapping 0 +1 an 1 1. The signal-to-noise ratio of an AWGN channel is enote by γ = E s /N 0, where E s is the receive signal energy an N 0 is the single-sie noise power ensity. We use the complementary error function erfc(z) = / π e s s. z

2 III. SYSTEM MODEL We consier the wireless relay channel epicte in Fig. 1: source s communicates with estination with the help of relay r, which uses the ecoe-an-forwar strategy. The source-to-estination channel, the source-to-relay channel an the relay-to-estination channel are moelle as binary input AWGN channels with signal-to-noise ratio (SNR) γ s, γ sr an, respectively, an BPSK moulation. The source employs a binary linear coe C {0, 1} N of length N an rate R; an an encoer E mapping user ata u {0, 1} K to coewors x s C. The user ata is assume to be uniformly istribute, an thus also the coewors. A coewor x s C is transmitte over a wireless channel. Due to the broacast nature of the wireless channel both the estination an the relay receive a noisy observation of x s, enote by y s an y sr, respectively. The relay ecoes y sr an generates the estimate x r C of the transmitte coewor x s. It then cooperates with the source by forwaring x r to the estination. We assume that the source an the relay transmit through orthogonal channels. Note that the relay may not be able to ecoe y sr correctly, an therefore x r an x s may iffer. Base on the two noisy observations y s an y r, the estination estimates the coewor x s that was transmitte by the source; this estimate is enote by ˆx s. A true maximum-likelihoo (ML) estimation woul take into account that the relay may make erroneous ecisions an the corresponing statistics. Inee, the relay introuces errors with memory an thus this shoul be consiere by the estination. As this is by far too complex for practical implementations, we use the following ecoer (cf. [8, 9, 11]). The source-to-estination SNR γ s is assume to be known by the estination (Assumption 1). The observation y r is assume to be the output of a (virtual) memoryless AWGN channel with input x s an SNR (Assumption ). Assumption 1 an are commonly use, though not explicitly state [7 9]. Base on this moel, the estination computes the L- values L s,i = 4γ s y s,i, L r,i = 4γ ry r,i, (1) i = 1,,...,N, where y i enotes the i-th element of vector y. For the analysis in Section IV, we will nee the conitional istributions of these L-values. Given x s,i = ±1, where x s,i {±1} enotes the BPSK moulate symbol of the bit x s,i {0, 1} (see above), L s,i has mean ±4γ s an variance 8γ s. Similarly, given x r,i = ±1, L r,i has mean ±4 an variance 8 /. Using above L-values, the ecoing rule is N ˆx s = argmax x i (L s,i + L r,i ), () x C i=1 Note that this ecoing rule is optimal if Assumption 1 an Assumption hol. While Assumption 1 is reasonable, Assumption cannot be true as ecoe-an-forwar introuces errors with memory. If the estination has some knowlege of the source-to-relay channel, it can exploit this information by properly weighing the relaye information by γ r. In general, the optimum value of for above ecoer is a function of all three SNRs γ sr,, an γ s. In the remainer of this paper, we analyse the probability of error at the ecoer an the effect of the value of. IV. A BOUND ON THE ERROR PROBABILITY The error events at the relay an at the estination are efine by e r := {x r x s }, e := {ˆx s x s }, (3) respectively. The complement of e r is enote by ē r. We also efine the bit error event at the estination by e b := {ˆx s,i x s,i for any i}. (4) For the analysis we assume without loss of generality that the all-zero coewor was transmitte, i.e. x s = 0. The error event e can then equivalently be written as { } e (L s,i +L r,i ) < 0 for any x C, x 0, (5) i S(x) where S(x) enotes the support of x (see above). Let A w, be the input-output weight enumerator (IOWE) of encoer E, giving the number of coewors of weight generate by input weight w; let further A = K i=1 A w, be the weight enumerator (WE) of encoer E, giving the number of coewors of weight. Also, enote by min the minimum istance of coe C. A. A boun on the frame error probability The probability of error at the estination can be written as p(e) = p(e e r )p(e r ) + p(e ē r )p(ē r ), (6) where we istinguish between the two cases where the relay makes an error an where it oes not. Using the union boun, the probability of error at the relay can be upper boune by p(e r ) 1 N ( ) A erfc γsr. (7) = min The probability of no error at the relay is upper-boune by p(ē r ) 1. We will now analyse the two conitional probabilities of error in (6). Consier first the case that no error occurs at the relay, i.e. the term p(e ē r ). The relay network of Fig. 1 is then equivalent to a system where x s is transmitte over two inepenent parallel channels with SNR γ s an. Let b be a nonzero coewor of C with support S(b) (as efine above). Splitting up the error event in (5) an taking the union boun, we obtain the upper boun p(e ē r ) = p(e x s = 0,x r = 0) p (L s,i + L r,i ) < 0 x s = 0,x r = 0 b C i S(b) = 1 N A erfc (γ s + γ r. (8) = min γ s + γ r

3 The last line is obtaine by the following consierations: Given x s = 0 an x r = 0, L s,i an L r,i have positive mean values. Therefore, i S(b) (L s,i+l r,i ) is Gaussian with mean 4(γ s + ) an variance 8(γ s + /) (see Section III); notice that = S(b). Consier now the case that an error occurs at the relay, i.e. the term p(e e r ). This probability can be written as p(e e r ) = p(e e r,x s = 0) = = p(e x s = 0,x r = a)p(x r = a e r,x s = 0) (9) a C a =0 (Notice that p(x r = 0 e r,x s = 0) = 0 an thus a = 0 may as well be inclue in the summation.) The computation of (9) is cumbersome. To simplify the computation, we make the following assumption: whenever the relay makes an error, it ecoes to a coewor that has minimum istance to the transmitte coewor (Assumption 3). Notice that this assumption is justifie if γ sr is high, which is require for the ecoe-an-forwar scheme to be useful. Define C min = {x C : w(x) = min } the set of minimum weight coewors. Then, assuming x s = 0 as before, we have x r = 0 (if the relay ecoes correctly) or x r C min (if the relay makes an error). Corresponingly, (9) can be re-written as p(e e r,x s = 0) = = p(e x s = 0,x r = a)p(x r = a e r,x s = 0) = p(e x s = 0,x r = a), (10) for any a C min. Here we use that p(e x s = 0,x r = a) takes the same value for all a C min, as the coe is linear; an that p(x r = a e r,x s = 0) = 1 by Assumption 3. Similar to the case of no error at the relay, we split up the error event in (5) an take the union boun: p(e x s = 0,x r = a) p (L s,i + L r,i ) < 0 x s = 0,x r = a, (11) b C i S(b) }{{} p (ˆx s = b x s = 0,x r = a) for any a C min. Each term in the sum enotes a pairwise error probability p ( ), namely the probability that the estination ecies for b instea of 0 if x s = 0 an x r = a were transmitte. We will now write these probabilities using the complementary error function by etermining the mean an the variance of the sum of the L-values, similar to the (8). Since x s = 0, all values L s,i have positive mean. Since x r = a, the values L r,i have positive mean for i S(b) S c (a), an they have negative mean for i S(b) S(a); S c (a) enotes the complement set of S(a). Using the mean values of these L-values (see Section III) an the number of terms in each set, the mean value of the overall sum can be written as S(b) 4γ s + S(b) S c (a) 4γ r S(b) S(a) 4γ r = S(b) 4(γ s + γ r ) S(b) S(a) 4γ r = 4(γ s + γ r) 8w(a b)γ r, (1) where a b enotes the element-wise prouct of the two vectors; notice that S(b) S(a) = w(a b). The variance of the overall sum is 8(γ s +γ r /), as in the case where the relay ecoes error free. Using this mean an variance, the pairwise error probability can be written as p (ˆx s = b x s = 0,x r = a) = = 1 erfc (γ s + γ r ) w(a b)γ r (γ. (13) s + γ r For a = 0, we obtain the terms of the sum in (8). For w(a b) = 0, the argument in (13) is maximum, an thus the error probability is minimum. On the other han, with increasing w(a b) (increasing overlap of a an b), the argument ecreases an thus the error probability increases. The overlap between a an b is maximum when b = a. In this case we have that w(a b) = min. If b ( a, however, ) the maximum value of w(a b) is given by min min, ; we call this value w m (b). Corresponingly, the pairwise error probability can be upper boune by p (ˆx s = b x s = 0,x r = a) 1 erfc (γ s + ) w m (b)γ r (γ, (14) s + γ r where w m(b) = min if b = a, an w m(b) = w m (b) otherwise. Finally, using (14), (11) an (10) in (9), p(e e r ) can be upper boune by p(e e r ) p (ˆx s = b x s = 0,x r = a) b C b C 1 erfc (γ s + ) w m(b)γ r (γ = s + γ r = 1 erfc min (γ s γ r ) ( min γ s + γ r ) N A erfc (γ s + γ r ) w m(b)γ r = min (γ, (15) s + γ r

4 where A = A 1 for = min, an A = A otherwise; notice that the first term correspons to the case where b = a. This completes the computation of an upper boun on the probability of error in (6). The corresponing lower boun is obtaine by consiering only the terms with minimum istance (with the corresponing multiplicities) in (7), (8) an (15). B. A boun on the bit error probability The bit error probability p(e b ) can be boune in a similar way to the frame error probability. We first write it as p(e b ) = p(e b e r )p(e r ) + p(e b ē r )p(ē r ). (16) We then use the upper boun on p(e r ) as given in (7), p(ē r ) 1, an p(e b ē r ) 1 N erfc (γ s + γ r ) γ s + γ, (17) r A (b) = min with the bit multiplicity A (b) = K w=1 w K A w,. Assuming again that the relay ecoes on a coewor at minimum istance (Assumption 3), we have p(e b e r ) = p(e b e r,x s = 0) = = p(e b x s = 0,x r = a)p(x r = a e r,x s = 0) = 1 A p(e b x s = 0,x r = a). (18) Note that in contrast to the frame error probability, the probability p(e b x s = 0,x r = a) epens on a an not only on its weight w(a), as several input weights may lea to a coewor of minimum weight. However, we can consier the worst case, which is given by the maximum weight of information wors that generate a coewor of minimum weight. We enote the corresponing bit multiplicity by A (b),max min. Then, using the same approach as for the frame error probability, we obtain the upper boun p(e b e r ) = 1 p(e b x s = 0,x r = a) A a C min 1 A(b),max min + 1 N A (b) = min erfc min (γ s γ r ) ( min γ s + γ r ) + erfc (γ s + ) w m(b)γ r (γ, (19) s + γ r where A (b) = A (b) A(b),max min for = min, an A (b) = A (b) otherwise. Note that by using A (b),max min, this upper boun oes not epen on a. V. OPTIMIZATION OF Since a true ML ecoer is far too complex, a common approach in the literature is to moel the source-to-relayto-estination channel as a virtual memoryless channel with SNR γ eq, where γ eq epens on γ sr an [7, 9]. At the estination is then set to γ eq. In general, however, is also a function of γ s. Moreover, ue to ecoing at the relay, the noise on this virtual channel is not linear, not Gaussian, an not memoryless. Therefore the ecoer is suboptimal, an the performance epens on the value chosen for γ r. In this paper, we propose to optimize γ r as a function of the triplet (γ s, γ sr, ). Notice that in the ecoing rule (), the value of γ r is use to trae-off the reliability of the sourceto-estination channel, epening on γ s, to the reliability of the virtual source-to-relay-to-estination channel, epening on γ sr an. We use the lower boun on the error probability, erive above, to perform this optimization. In particular, for each triplet (γ s, γ sr, ), we etermine numerically the value of which minimizes the lower boun on the error probability; we enote this optimum value by γ opt. VI. NUMERICAL RESULTS In this Section, we evaluate the tightness of the boun on the error probability erive in Section IV by comparing the bouns to simulation results. For the examples in this Section, we have chosen a rate-1/ 4-states convolutional encoer with generator polynomials (1, 5/7) in octal form, an we use the information wor length K = 64. We consier two ifferent scenarios. In Scenario 1, the SNRs γ sr an are fixe, an γ s varies. In Scenario, γ sr an vary with γ s as: γ sr = γ s g sr, = γ s g r, (0) where the gains g sr = ( s / sr ) α an g r = ( s / r ) α are ue to shorter istances [1]. The values s, sr an r enote the istances between source an estination, source an relay, an relay an estination, respectively; usually α 6; here we assume α =. In Fig. we plot the bouns on the frame error rate (ashe curves with empty markers) together with the simulation results (soli curves with soli markers) for Scenario 1 as a function of γs b (in B), where γb s = γ s/r enotes the SNR per information bit. We assume the SNRs γsr b = γr b = 5 B. When two ashe curves with the same markers are plotte, the lower curves correspon to the lower boun, while the upper curves correspon to the upper boun. Several values for γ r are consiere. In all cases the bouns match very well with the simulation results. For high γs b the simulation results are on the upper bouns. Due to the use of the union boun, the upper boun iverges for low SNRs. On the other han the lower boun gives also a goo approximation of the performance. Note that for γsr b = 5 B, the probability of error at the relay is p(e r ) , an thus not negligible. Very goo matching is also observe for higher error probabilities at the relay.

5 γ' r=γr γ' r=γeq γ' r=3b γ' r=γeq γ' r=γopt 10 - γ' r=γopt FER BER s/ sr=3/, s/ r= s/ sr=3, s/ r=3/ γ b s Fig.. FER bouns (ashe curves with empty markers) an simulation results (soli curves with soli markers) for the relay network in Fig. 1. Scenario 1 with γsr b = 5 B an γr b = 5 B. The worst performance is obtaine if we set =, as no information about the source-to-relay channel is exploite. Slightly better results are achieve if we moel the source-to-relay-to-estination channel as a virtual memoryless AWGN channel with SNR γ eq. However, performance can significantly be improve if other values for are use. For instance, if we set = 3 B a gain of 1 B is achieve at FER=10 5. The optimization of γ r using the proceure escribe in Section V yiels the best results. A gain of.4 B is achieve at FER=10 5 with respect to the curve with γ r = γ eq. We remark that the optimal value γ opt of γ r ecreases with increasing values of γ s. In Fig. 3 we plot the lower bouns on the bit error rate (ashe curves with empty markers) together with the simulation results (soli curves with soli markers) for Scenario as a function of γs b. Two cases are consiere: in case 1, the relay is closer to the source than to the estination, an therefore we use s / sr = 3 an s / r = 3/; in case, the relay is closer to the estination than to the source, an therefore we use s / sr = 3/ an s / r = 3. For both cases, the curve for γ r = γ eq an the one for = are inistinguishable, therefore only one is plotte. Inee, it turns out that γ eq is very close to. For both cases the bouns are tight. Note that the lower boun is also tight when the probability of error at the relay p(e r ) is high. For instance, for γs b = 0 B an case, we have p(e r) 5 10, an the lower boun preicts still well the actual performance. For case 1, γ opt is almost ientical to γ eq an, thereby no ifference is observe in the plotte curves. On the other han for case the optimization of accoring to the proceure escribe in the previous Section yiels a gain of aroun 0.5 B. For other values of s / sr an s / r, we observe a similar behaviour. VII. CONCLUSIONS AND FUTURE WORK In the present paper we have evelope a new analytical metho to upper an lower boun the error rate of relaying γ b s Fig. 3. BER bouns (ashe curves with empty markers) an simulation results (soli curves with soli markers) for the relay network in Fig. 1. Scenario, case 1 (relay closer to source) an case (relay closer to estination). with ecoe-an-forwar. The bouns are base on unionboun approaches an the weight enumerators of the coe. As an application we have consiere the optimisation of the equivalent SNR of the source-to-relay-to-estination link at the estination. In future work we will exten our bouning approach to schemes with re-encoing at the relay, which leas to istribute turbo-like coes, an to scenarios with multiple users an network coing at the relay. REFERENCES [1] E. C. van er Meulen, Three-terminal communication channels, Av. Appl. Prob., vol. 3, pp , [] T. Cover an A. El Gamal, Capacity theorems for the relay channel, IEEE Trans. Inform. Theory, vol. 5, no. 5, pp , [3] J. N. Laneman, D. N. C. Tse, an G. W. Wornell, Cooperative iversity in wireless networks: Efficient protocols an outage behavior, IEEE Trans. Inform. Theory, vol. 50, pp , Dec [4] G. Kramer, I. Marić, an R. D. Yates, Cooperative communications, FNT in Networking, vol. 1, no. 3-4, pp , 006. [5] A. Ribeiro, X. Cai, an G. Giannakis, Symbol error probabilities for general cooperative links, IEEE Trans. Wireless Commun., vol. 4, no. 3, pp , 005. [6] P. Anghel an M. Kaveh, Exact symbol error probability of a cooperative network in a rayleigh-faing environment, IEEE Trans. Wireless Commun., vol. 3, no. 5, pp , 004. [7] T. Wang, A. Cano, G. Giannakis, an J. Laneman, High-performance cooperative emoulation with ecoe-an-forwar relays, IEEE Trans. Commun., Jan [8] S. Yang an R. Koetter, Network coing over a noisy relay : a belief propagation approach, in Proc. IEEE Int. Symp. Inf. Theory (ISIT), pp , 007. [9] H. Sneessens an L. Vanenorpe, Soft ecoe an forwar improves cooperative communications, Proc. IEEE Int. Workshop on Comput. Avances in Multi-Sensor Aaptive Processing, Jan [10] C. Hausl, F. Schreckenbach, an I. Oikonomiis, Iterative network an channel ecoing on a tanner graph, Proc. 44th Annual Allerton Conf. on Commun., Control, an Computing, Jan [11] G. Zeitler, R. Koetter, G. Bauch, an J. Wimer, Design of network coing functions in multihop relay networks, Proc. Int. Symp. on Turbo Coes an Rel. Topics (ISTC), 008. [1] M. D. Yacoub, Founations of Mobile Raio Engineering. CRC Press, 1993.