Research Collection. Mismatched decoding for the relay channel. Conference Paper. ETH Library. Author(s): Hucher, Charlotte; Sadeghi, Parastoo

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1 Researh Colletion Conferene Paper Mismathed deoding for the relay hannel Author(s): Huher, Charlotte; Sadeghi, Parastoo Publiation Date: 2010 Permanent Link: Rights / Liense: In Copyright - Non-Commerial Use Permitted This page was generated automatially upon download from the ETH Zurih Researh Colletion. For more information please onsult the Terms of use. ETH Library

2 Mismathed Deoding for the Relay Channel Charlotte Huher and Parastoo Sadeghi The Australian National University, Canberra, ACT, 0200, Australia Abstrat We onsider a disrete memoryless relay hannel, both relay and destination may have an inorret estimation of the hannel. This estimation error is modeled with mismathed deoders. In this paper, we provide a lower-bound on the mismath apaity of the relay hannel. Moreover, we prove that this lower-bound is indeed the exat apaity for the degraded relay hannel when random oding is used. I. INTRODUCTION The deoding method that minimizes the error probability is the maximum-likelihood (ML) deoder. However, it annot always be implemented in pratie beause of some hannel estimation errors or hardware limitations. An alternative deoder an then be a mismathed one, based on a different metri. The theoretial performane of mismathed deoding has been studied sine the 1980 s when Csiszàr and Körner in [1], and Hui in [2] both provided a lower-bound on the ahievable apaity in a point-to-point ommuniation hannel. In [3], the authors proved that this lower-bound is the exat apaity when random oding is used. The mismath apaity of multiple-aess hannels has also been haraterized in [4]. There is inreasing evidene that future wireless ommuniations will be based not on point-to-point transmission anymore, but on ooperation between the nodes in a network (see [5],[6]). The simplest model of a ooperative network is the relay hannel for whih apaity bounds have been derived in 1979 by Cover and El Gamal in [7]. In this paper, we onsider a disrete memoryless relay hannel with mismathed deoders at both reeivers (the relay and the destination). We provide a lower-bound on the mismath apaity of suh a hannel and prove that it is indeed the exat apaity of the mismathed degraded relay hannel when random oding is used. II. THE RELAY CHANNEL AND MISMATCHED DECODER We onsider a disrete memoryless relay hannel onsisting of one soure, one relay and one destination. We use the same setup as in [7]: The soure broadasts a signal x 1 X 1 whih is reeived by both the relay and the destination. The relay transmits a signal x 2 X 2 whih is reeived by the destination. Reeived signals at relay and destination are denoted by y 1 Y 1 and y Y respetively (see Figure 1). The hannel is modeled by a set of probability distributions p(y 1,y x 1,x 2 ). We onsider three mismathed deoders using the metris q sr (x 1,x 2,y 1 ), q rd (x 2,y) and q sd (x 1,x 2,y), the subsripts sr, rd and sd stand for the soure-relay, relay-destination and soure-destination links, respetively. w enoder x 1 y 1 Fig. 1. relay deoder/enoder p(y 1,y x 1,x 2 ) x 2 Relay hannel y deoder The following setup is used to prove the ahievability of the lower-bound on the mismath apaity derived in this paper. We onsider the transmission of B bloks of length n. In eah blok i {1,...,B, a message w i {1,...,2 nr is transmitted from the soure. Let us partition the set {1,...,2 nr into 2 nr0 independent subsets denoted by S s,s {1,...,2 nr0, suh that any w i {1,...,2 nr belongs to a unique subset S si. The message is then oded as x 1 (w i s i ) X1 n and x 2(s i ) X2 n at the soure and relay, respetively. Random oding: The hoie of the set C = {x 1 (..),x 2 (.) of odewords is random: 2 nr0 iid odewords in X2 n are first generated aording to the probability distribution p(x 2 )= n i=1 p(x 2i) and indexed by s {1,...,2 nr0 : x 2 (s); for eah x 2 (s), 2 nr iid odewords in X1 n are generated aording to the probability distribution p(x 1 x 2 (s)) and indexed by w {1,...,2 nr : x 1 (w s). Two transmission steps: Let us assume that (i 2) bloks have already been sent. Thus the relay has already deoded w i 2 and s i 1, the destination has deoded w i 3 and s i 2. In order to derive a lower bound on the apaity of the mismathed relay hannel, we hoose to use threshold deoders as in [2], [4]. Indeed, it an be proven that the deoding error probability of an ML mismathed deoder (whih is implemented in pratie) is upper-bounded by the deoding error probability of the onsidered threshold deoder. In blok (i 1), the soure and relay transmit x 1 (w i 1 s i 1 ) and x 2 (s i 1 ), respetively; the relay and destination reeive y 1 (i 1) and y(i 1). The relay is able to detet w i 1 as the unique w suh that (x 1 (w s i 1 ), x 2 (s i 1 ), y 1 (i 1)) is jointly typial and q sr (x 1 (w s i 1 ), x 2 (s i 1 ), y 1 (i 1)) is larger than a threshold to be defined later. The relay is thus able to determine s i suh that w i 1 S si. The soure is obviously also aware of s i. In blok i, the soure and relay transmit x 1 (w i s i ) and x 2 (s i ), respetively; the relay and destination reeive y 1 (i) and y(i). The destination an detet s i as the unique s suh that (x 2 (s i ), y(i)) is jointly typial and q rd (x 2 (s i ), y(i)) is larger than some threshold. It is then able to detet w 25

3 w i 1 as the unique w suh that w S si L(y(i 1)), L(y(i 1)) is the set of all w {1,...,2 nr suh that (x 1 (w s i 1 ), x 2 (s i 1 ), y(i 1)) is jointly typial and q sd (x 1 (w s i 1 ), x 2 (s i 1 ), y(i 1)) is larger than some threshold. Notation: Let E p (q(x)) denote the expeted value of q(x) w.r.t. the probability distribution p(x). LetI f (X; Y ) denote the usual mutual information between X and Y w.r.t. the probability distribution f(x, y). For a probability distribution p(x) on a finite set X and a onstant δ > 0, let Np(x) δ denote the set of all probability distributions on X that are within δ of p(x): Np(x) δ = {f P(X ): x X, f(x) p(x) δ. LetT p(x) δ be the set of all sequenes in X n whose type is in N p(x) δ : Tp(x) {x δ = X n : f x Np(x) δ,f x (x) is the number of elements of the sequene x that are equal to x, normalized by the sequene length n. In the following, we drop the subsript arguments when the ontext is lear enough and write f(x) Np δ and (x, y) Tp δ. III. MAIN RESULTS Theorem 1: (Ahievability) Consider a disrete memoryless relay hannel desribed by the probability distribution p(y 1,y x 1,x 2 ). The apaity C M obtained using the mismathed deoders q sr (x 1,x 2,y 1 ), q rd (x 2,y) and q sd (x 1,x 2,y) is lower-bounded by: C LM = max I LM(p(x 1,x 2 )) (1) p(x 1,x 2) I LM (p(x 1,x 2 )) = min{i sr (p(x 1,x 2 )),I rd (p(x 2 )) + I sd (p(x 1,x 2 )) (2) I sr (p(x 1,x 2 )) = min I f (X 1 ; Y 1 X 2 )+I f (X 1 ; X 2 ) (3) I rd (p(x 2 )) = min I f (X 2 ; Y ) f D rd (4) I sd (p(x 1,x 2 )) = min I f (X 1 ; Y X 2 )+I f (X 1 ; X 2 ) f D sd (5) and D sr ={f(x 1,x 2,y 1 ):f(x 1 )=p(x 1 ),f(x 2,y 1 )=p(x 2,y 1 ), E f (q sr (x 1,x 2,y 1 )) E p (q sr (x 1,x 2,y 1 )) (6) D rd ={f(x 2,y):f(x 2 )=p(x 2 ),f(y) =p(y), E f (q rd (x 2,y)) E p (q rd (x 2,y)) (7) D sd ={f(x 1,x 2,y):f(x 1 )=p(x 1 ),f(x 2,y)=p(x 2,y), E f (q sd (x 1,x 2,y)) E p (q sd (x 1,x 2,y)). (8) Let now suppose that there exist three probability distributions ˆf i, i {sr, rd, sd suh that E ˆfi (q i ) >E p (q i ) with strit inequality. Theorem 2: (Converse) With the above assumption and for a degraded relay hannel, if for some input distribution p(x 1,x 2 ),therater>c LM, then the average probability of error, averaged over all random odebooks drawn aording to p(x 1,x 2 ), approahes one as the blok length tends to infinity. IV. PROOF OF THEOREM 1 A. Upper-bounding the error probability In order to prove the ahievability of this lower-bound, we onsider the four possible error events desribed in [7] adapted to the use of a threshold deoder. For eah blok i, these possible error events are: E 0i : (x 1 (w i s i ), x 2 (s i ), y 1 (i), y(i)) is not jointly typial; E 1i : there exists w w i suh that (x 1 ( w s i ), x 2 (s i ), y 1 (i)) is jointly typial and q sr (x 1 ( w s i ), x 2 (s i ), y 1 (i)) is larger than some threshold; E 2i : there exists s s i suh that (x 2 ( s), y(i)) is jointly typial and q rd (x 2 ( s), y(i)) is larger than some threshold; E 3i = E 3i E 3i E 3i : w i 1 / S si L(y(i 1)) E 3i : there exists w w i 1 suh that w S si L(y(i 1)). Let F i be the deoding error event in blok i. Let us assume that no error has ourred till blok i 1. Thus, the deoding error probability in blok i is given by: { 3 k 1 3 p e (i) = Pr E ki Fi 1 Eli p ek (i). k=0 l=0 k=0 1) Probability of error event E 0i : By Sanov s theorem [8, Theorem ], this probability is exponentially small in n. There exists ψ>0 suh that p e0 (i) < 2 nψ. 2) Probability of error event E 1i : An error ours if there exists w w i suh that the metri q sr (x 1 ( w s i ), x 2 (s i ), y 1 (i)) is greater than the threshold Υ δ sr =min p(x 1,x 2,y 1 )q sr (x 1,x 2,y 1 ), δ (x 1,x 2,y 1) X 1 X 2 Y 1 δ is a small positive number. The probability of error event E 1i is thus given by p e1 (i) =Pr{ w w i, x 1 ( w s i ) T δ p, q sr (x 1 ( w s i ), x 2 (s i ), y 1 (i)) Υ δ sr F i 1. Using Sanov s theorem, we obtain the upper-bound R sr δ = p e1 (i) (2 nr 1)(n +1) X1 X2 Y1 2 n R δ sr (n +1) X1 X2 Y1 2 n( R δ sr R), min f(x 1,x 2,y) D δ sr D(f(x 1,x 2,y 1 ) p(x 1 )p(x 2,y 1 )) (10) Dsr δ = {f(x 1,x 2,y 1 ):f(x 1 ) Np δ,f(x 2,y 1 ) Np δ, E f (q sr (x 1,x 2,y 1 )) Υ δ sr, (11) with D(..) denoting the KL divergene [8, equation (2.26)]. Thus, if R < R sr, δ the probability of error event E 1i is exponentially small in n: p e1 (i) < 2 nψ for some ψ>0. (9) 26

4 3) Probability of error event E 2i : Using the threshold Υ δ rd =min p(x 2,y)q rd (x 2,y), (12) δ (x 2,y) X 2 Y we an write the probability of error event E 2i as p e2 (i) =Pr { s s i, x 2 ( s) Tp δ,q rd (x 2 ( s), y(i)) Υ δ rd Fi 1 (n +1) X2 Y 2 n( R δ rd R0), the upper-bound is obtained using Sanov s theorem with R rd δ = min D(f(x 2,y) p(x 2 )p(y)) (13) f(x 2,y) D δ rd Drd δ = {f(x 2,y):f(x 2 ) Np δ,f(y) Np δ, E f (q rd (x 2,y)) Υ δ rd. (14) If R 0 < R rd δ,thenp e2(i) < 2 nψ for some ψ>0. 4) Probability of error event E 3i : Error event E 3i an be deomposed into two different events E 3i and E 3i. If we assume that the previous transmission was orretly reeived at destination, then w i 1 L(y(i 1)). Moreover, the fat that the error event E 2i does not our implies that w i 1 Sŝi = S si. Thus the first term of the deomposition has a probability zero and we only need to onsider E p e3 (i) =Pr { w w i 1, w S si L(y(i 1)) Fi 1 E Pr { w S si Fi 1 w w i 1, w L(y(i 1)) E { L(y(i 1)) 2 nr0 Fi 1. denotes the ardinality of the onsidered set. Let 1, x 1 ( w s i 1 ) Tp δ, ϕ( w y) = q sd (x 1 ( w s i 1 ), x 2 (s i 1 ), y(i 1)) Υ δ sd 0, otherwise. the threshold is defined by Υ δ sd =min p(x 1,x 2,y)q sd (x 1,x 2,y). δ (x 1,x 2,y) X 1 X 2 Y (15) Using Sanov s theorem, we an upper-bound the expeted ardinality of L(y(i 1)) given that w w i 1 R sd δ = E { L(y(i 1)) w w i 1 F = E ϕ( w y) F w w i 1 i 1 i 1 (2 nr 1)(n +1) X1 X2 Y 2 n R δ sd (n +1) X1 X2 Y 2 n( R δ sd R), min f(x 1,x 2,y) D δ sd 3i. D(f(x 1,x 2,y) p(x 1 )p(x 2,y)) (16) Dsd δ = {f(x 1,x 2,y):f(x 1 ) Np δ,f(x 2,y) Np δ, E f (q sd (x 1,x 2,y)) Υ δ sd. (17) The error probability is then upper-bounded by p e3 (i) (n +1) X1 X2 Y 2 n( R δ sd R) 2 nr0 (n +1) X1 X2 Y 2 n( R δ sd +R0 R). Replaing R 0 by the onstraint previously omputed p e3 (i) (n +1) X1 X2 Y 2 n( R δ sd + R δ rd R). If R< R δ sd + R δ rd,thenp e3(i) < 2 nψ for some ψ>0. B. Existene of a random ode If R < R sr δ and R < R rd δ + R sd δ, then the total error probability is exponentially small in n: p e < 4B 2 nψ. Thus, as n tends to infinity, the probability of finding a set of odewords C respeting p e (C) < 4B 2 nψ tends to one. Let C be suh a set of odewords of length n. Its average error probability is lower than 4B 2 nψ.throwingaway the worst half of the odewords, we end up with a set of odewords C of length n 2 whose maximum error probability is lower than 2 4B 2 nψ, whih tends to zero, and whose rate is R 1 n whih tends to R. C. Letting δ tend to zero We note that lim δ 0 Υδ sr = E p (q sr (x 1,x 2,y 1 )). Thus, the set Dsr δ beomes D sr ={f(x 1,x 2,y 1 ):f(x 1 )=p(x 1 ),f(x 2,y 1 )=p(x 2,y 1 ), E f (q sr (x 1,x 2,y 1 )) E p (q sr (x 1,x 2,y 1 )) and with these new onstraints on the probability distribution D(f(x 1,x 2,y 1 ) p(x 1 )p(x 2,y 1 )) = I f (X 1 ; Y 1 X 2 )+I f (X 1 ; X 2 ). Thus the first onstraint of the rate beomes I sr (p(x 1,x 2 )) min I f (X 1 ; Y 1 X 2 )+I f (X 1 ; X 2 ). (18) In the same way, we find the final expressions of I rd (p(x 1,x 2 )) and I sd (p(x 1,x 2 )). V. PROOF OF THEOREM 2 The proof of Theorem 2 is in essene similar to the one of [4, Theorem 3]. A. Deoding at relay Let us assume that R> min I f (X 1 ; Y 1 X 2 )+I f (X 1 ; X 2 ). (19) Let f be the probability distribution that ahieves (19). Let f =(1 ɛ)f + ɛ ˆf sr. We reall that ˆfsr is a probability distribution that respets E ˆfsr (q sr ) > E p (q sr ). Then, for suffiiently small ɛ, R>I f (X 1 ; Y 1 X 2 )+I f (X 1 ; X 2 ) (20) E f (q sr ) >E p (q sr ). (21) 27

5 Using (20), (21) and the ontinuity of the divergene, we an find Δ > 0, ɛ>0 and a neighborhood U of f(x 1,x 2,y 1 ) suh that for all f U and p (x 2,y 1 ) N ɛ p,wehave R>D(f(x 1 x 2,y 1 ) p(x 1 ) p (x 2,y 1 )) + Δ (22) E f (q sr ) >E p (q sr )+Δ, (23) D(...) is defined in [4]. Let V be a suffiiently small neighborhood of p(x 1,x 2,y 1 ), suh that for every p (x 1,x 2,y 1 ) V,thenp (x 2,y 1 ) N ɛ p and E p (q sr ) <E p (q sr )+Δ. (24) Let us assume that the true message in blok i is w i and that the triple (x 1 (w i s i ), x 2 (s i ), y 1 (i)) has empirial type in V. If there exists another message w w i suh that the empirial type of (x 1 ( w s i ), x 2 (s i ), y 1 (i)) is in U, then a deoding error ours. Let W ( w) take the value 1 if the empirial type of (x 1 ( w s i ), x 2 (s i ), y 1 (i)) is in U and 0 otherwise. The expetation of W = w w i W ( w) given y 1 is E(W y 1 )=(2 nr 1)π 0. =2 nr π 0, (25) =. denotes the behavior of the expression when n and π 0 = E(W ( w ) y 1 ) = Pr{W ( w ) = 1 y 1 with w being a random message different from w i. For two different messages w and w,theeventsw( w) and W ( w ) are independent. Thus the variane of W given y 1 is Var(W y 1 )= w w i Var(W ( w) y 1 ). Sine W ( w) an only take the values 0 and 1, we an upperbound Var(W ( w) y 1 ) E(W ( w) y 1 ). Thus Var(W y 1 ) w w i E(W ( w) y 1 ). =2 nr π 0. (26) Using (25), (26) and the fat that for any random variable X, Pr(X =0) Var(X) E(X) 2, we an write Pr(W =0 y 1 ) 2nR π 0 (2 nr π 0 ) 2 =2 nr 1 π 0. (27) Using the seond part of Sanov s theorem, we obtain the asymptoti behavior π 0. = 2 n R sr, Rsr = min f U D(f(x 1 x 2,y 1 ) p(x 1 ) p (x 2,y 1 )). Using (22), we an then lower-bound π 0 2 n(r Δ) and onlude that the probability of no deoding error tends to 0 when n : Pr(W =0 y 1 ) 2 nr 2 n(r Δ) =2 nδ. (28) B. Deoding at destination The seond inequality an be dealt in two separate parts. Indeed, we have shown in the diret part that R 0 min I f (X 2 ; Y ) f D rd (29) R R 0 + min I f (X 1 ; Y X 2 )+I f (X 1 ; X 2 ). f D sd (30) We thus have to show that if one of these inequalities is reversed, an error ours with asymptoti probability one. This an be done using the same reasoning as in previous subsetion. VI. MATCHED DECODING CASE In the mathed deoding ase, i.e. q sr (x 1,x 2,y 1 ) = log p(y 1 x 1,x 2 ), q rd (x 2,y) = logp(y x 2 ) and q sd (x 1,x 2,y) = logp(y x 1,x 2 ), the apaity oinides with the one of degraded relay omputed by Cover and El Gamal in [7]. Indeed, for any distribution f D sr,wehave I f (X 1 ; Y 1 X 2 )+I f (X 1 ; X 2 ) I f (X 1 ; Y 1 X 2 ) = H(Y 1 X 2 ) H f (Y 1 X 1,X 2 ) (31) H(Y 1 X 2 )+ f(x 1,x 2,y 1 )logp(y 1 x 1,x 2 ) (32) x 1,x 2,y 1 H(Y 1 X 2 )+ p(x 1,x 2,y 1 )logp(y 1 x 1,x 2 ) (33) x 1,x 2,y 1 = I(X 1 ; Y 1 X 2 ), (31) holds beause f(x 2,y 1 )=p(x 2,y 1 ), (32) follows from the non-negativity of the divergene and (33) is obtained using E f (log p(y 1 x 1,x 2 )) E(log p(y 1 x 1,x 2 )). Moreover, by hoosing f(x 1,x 2,y 1 ) = p(x 1 )p(x 2 )p(y 1 x 1,x 2 ) D sr, I f (X 1 ; X 2 ) = 0 and I f (X 1 ; Y 1 X 2 )+I f (X 1 ; X 2 )=I(X 1 ; Y 1 X 2 ). The bound is ahievable, so I sr (p(x 1,x 2 )) = I(X 1 ; Y 1 X 2 ). In the same way, we an prove that I rd (p(x 1,x 2 )) = I(X 2 ; Y ) and I sd (p(x 1,x 2 )) = I(X 1 ; Y X 2 ). Finally, in the mathed ase, the following rate is ahievable R =min{i(x 1 ; Y 1 X 2 ),I(X 2 ; Y )+I(X 1 ; Y X 2 ) (34) =min{i(x 1 ; Y 1 X 2 ),I(X 1,X 2 ; Y ). (35) ACKNOWLEDGEMENT This work was supported under Australian Researh Counil s Disovery Projets funding sheme (projet no. DP ). The authors would also like to thank Prof. Lapidoth for useful disussions. REFERENCES [1] I. Csiszàr and J. Körner, Graph deomposition: A new key to oding theorems, IEEE Trans. Inform. Theory, vol. 27, no. 1, pp. 5 12, Jan [2] J. Hui, Fundamental issues of multiple aessing, Ph.D. dissertation, M.I.T., 1983, hap. IV. [3] N. Merhav, G. Kaplan, A. Lapidoth, and S. Shamai Shitz, On information rates for mismathed deoders, IEEE Trans. Inform. Theory, vol. 40, no. 6, pp , Nov [4] A. Lapidoth, Mismathed deoding and the multiple-aess hannel, IEEE Trans. Inform. Theory, vol. 42, no. 5, pp , Sep [5] A. Sendonaris, E. Erkip, and B. Aazhang, User Cooperation Diversity. Part I and II, IEEE Trans. Commun., vol. 51, no. 11, pp , Nov [6] A. Nosratinia and A. Hedayat, Cooperative Communiation in Wireless Networks, IEEE Commun. Mag., vol. 42, no. 10, pp , Ot [7] T. Cover and A. Gamal, Capaity theorems for the relay hannel, IEEE Trans. Inform. Theory, vol. 25, no. 5, pp , Sep [8] T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed. Wiley-Intersiene,