Broadcast Channels with Cooperating Decoders

Transcription

1 To appear in the IEEE Transations on Information Theory, Deember Broadast Channels with Cooperating Deoders Ron Dabora Sergio D. Servetto arxiv:s/ v3 [s.it] 1 Nov 2006 Abstrat We onsider the problem of ommuniating over the general disrete memoryless broadast hannel BC) with partially ooperating reeivers. In our setup, reeivers are able to exhange messages over noiseless onferene links of finite apaities, prior to deoding the messages sent from the transmitter. In this paper we formulate the general problem of broadast with ooperation. We first find the apaity region for the ase where the BC is physially degraded. Then, we give ahievability results for the general broadast hannel, for both the two independent messages ase and the single ommon message ase. Index Terms Broadast hannels, ooperative broadast, relay hannels, hannel apaity, network information theory. A. Motivation I. INTRODUCTION In the lassi broadast senario the reeivers deode their messages independently of eah other. However, the inreasing interest in networking motivates the onsideration of broadast senarios in whih eah node in the network, besides deoding its own information, tries to help other nodes in deoding. This problem omes up naturally in sensor networks, where a transmitter external to the sensor network wants to download data into the network, e.g., to onfigure the sensor array. The onept of ooperation among reeivers is also relevant to general ad-ho networks, sine suh ooperation provides a method for inreasing the rates without inreasing the spetrum alloation. Therefore, this motivates the study of the effet of reeiver ooperation on the rates for the broadast hannel. B. The Disrete Memoryless Broadast Channel DMBC) The broadast hannel was introdued by Cover in [1]. Following this initial work, Bergmans proved an ahievability result for the degraded BC, [2], and also a partial onverse that holds only for the Gaussian broadast hannel [3]; in [4] Gallager established a onverse that holds for any disrete memoryless degraded broadast hannel. In [5] El-Gamal generalized the apaity result for the degraded broadast hannel to the more apable ase, and in [6] and [7] he showed that feedbak does not inrease the apaity region for the physially degraded ase. Several other lasses of broadast hannels were studied in the following years. For example, the sum and produt of two degraded broadast The authors are with the Shool of Eletrial and Computer Engineering, Cornell University, Ithaa, NY. URL: Parts of this work were presented at the International Symposium on Information Theory, Chiago, 2004 and the International Symposium on Information Theory, Adelaide, Australia, Work supported by the National Siene Foundation, under awards CCR CAREER), CCR , and ANR hannels were onsidered in [8], and in [9], [10] and [11] the deterministi broadast hannel was analyzed. For the general broadast hannel, Cover derived an ahievable rate region for the ase of two independent senders in [12]. In [13] Korner and Marton onsidered the apaity of general broadast hannels with degraded message sets. The best ahievable region and the best upper bound for the two independent senders ase were derived by Marton in [14], and a simple proof of Marton s ahievable region appeared later in [15]. Another upper bound for the general broadast hannel, the so-alled degraded, same-marginals DSM) bound, was presented in [16]. This bound is weaker than the upper bound in [14] but stronger than Sato s upper bound previously presented in [17]. We note, however, that while Marton s upper bound is the strongest, it is valid only for the two-reeiver ase, while Sato s bound and the DSM bound an be extended to more than two reeivers. The effet of feedbak on the apaity of the Gaussian broadast hannel was studied in [18] and [19], and in [20] the ase of orrelated soures was onsidered. A survey on the topi, with extensive referenes to previous work, an be found in [21]. In reent years the Multiple-Input-Multiple-Output MIMO) Gaussian broadast hannel has attrated a lot of attention. Initially, the sum-rate apaity was haraterized in [22], [23], [24], [25], and finally, in [26] the apaity region was obtained. None of the early work on the DMBC onsidered diret ooperation between the reeivers. In the ooperative broad- W 1 W 2 Enoder X n py 1,y 2 x) Broadast Channel Y n 1 Y n 2 Reeiver 1 C 12 R x1 R x2 C 21 Reeiver 2 Fig. 1. Broadast hannel with two private messages and ooperating reeivers. ast senario, a single transmitter sends two messages to two reeivers enoded in a single hannel odeword X n, where the supersript n denotes the length of a vetor. Eah of the reeivers gets a noisy version of the odeword, Y1 n at R x1 and Y2 n at R x2. After reeption, the reeivers exhange messages over noiseless onferene links of finite apaities C 12 and C 21, as depited in Figure 1. The onferene messages are, in general, funtions of Y1 n at R x1), Y2 n at R x2), and the previous messages reeived from the other deoder. After onferening, eah reeiver deodes its own message. We note that in a reent work, [27], the authors onsider the problem of interative deoding of a single broadast message over the independent broadast hannel by a group W ^ 1 W ^ 2

2 2 of ooperating users. In our work we extend this senario to the general hannel and also onsider the two independent senders ase. C. Cooperative Broadast: A Combination of Broadasting and Relaying The senario in whih one transeiver helps a seond transeiver in deoding a message is learly a relay senario. Hene, ooperative broadast an be viewed as a generalization of the broadast and relay senarios into a hybrid broadast/relay system, whih better desribes future ommuniation networks. Senarios of this type have attrated onsiderable attention reently both from the pratial and the theoretial aspets. From the pratial aspet, new protools are proposed for the ollaborative broadast senario. For example in [28] the authors present a protool for ollaborative deision making involving broadasting and relaying. From the theoretial aspet, there is a onsiderable effort invested in haraterizing the apaity of an entire network. This work started with [29] and reent results appear in [30] and the following work [31], [32] and [33]. This work fouses on the Gaussian ase. A omplementing approah for studying the performane of a network is to ombine the basi building bloks of a network, namely multiple aess, relaying and broadasting and study the apaity of these ombinations. The reent work on relaying fouses on extending the single relay results derived in [34] to the MIMO ase see for example [35]) and to the multiple level ase [36], [37]. Another reent result was introdued in [38] where joint deoding was applied to the ombined deode-and-forward and estimate-and-forward sheme of [34, theorem 7]. A third approah for studying the performane of an entire network is the network oding approah sparked by the work of [39], whih fouses on enoding at the nodes for maximizing the network throughput, separately from the hannel oding. In this paper we fous on the ombination of broadast and relay. A relevant work in this ontext is [40], in whih the apaity of a lass of independent relay hannels with noiseless relay is derived. Note that the ase of noiseless relay is also related to the Wyner-Ziv problem [41]. This relationship will be highlighted in the sequel. Lastly, we note that a reent work, [42], presented an ahievability result for the general DMBC with a single wireless ooperation hannel from one reeiver to the seond reeiver. This ahievable rate region is shown to be the apaity region for the physially degraded broadast/relay hannel. D. Main Contributions and Organization In the following we summarize the main ontributions of this work. We initially study a speial ase of the general setup formulated in Setion I-B: the ase of the physially degraded broadast hannel. Although the physially degraded BC is of little pratial interest, it is useful in developing the oding onept for the general BC with ooperation. For the physially degraded BC, we present both an ahievability result and a onverse. Together, these two results give the apaity region for this setup. Furthermore, this new region is shown to be a strit enlargement of the lassial region without ooperation [21]. Next, we give an ahievability result for the general BC with ooperating reeivers. This region is also greater, in general, than the lassi ahievable region given in [14] for the broadast hannel. We also onsider the ase where a single ommon message is transmitted to both reeivers. We onsider two different ooperation strategies and derive the ahievable rates for eah of them. We also derive an upper bound on the ahievable rates for this senario. Here we provide results that expliitly link the available ooperation apaity to the inrease in the rate of information. Lastly, we show that for a speial ase of the general BC, namely when one hannel is distintly better than the other, the upper and lower bounds oinide, resulting in the apaity for that ase. The rest of this paper is strutured as follows: in setion II we define the mathematial framework. In setion III we analyze the physially degraded BC, and derive the apaity region for that ase, and in setion IV we present an ahievability result for the general broadast hannel with ooperating reeivers. Next, setion V presents ahievability results and an upper bound on the rates for the ase where only a single ommon message is transmitted. Conluding remarks are provided in setion VI. II. DEFINITIONS AND NOTATIONS First, a word about notation: in the following we use H ) to denote the entropy of a disrete random variable RV), and I ; ) to denote the mutual information between two disrete random variables, as defined in [43, Ch. 2]. We denote random variables with apital letters X, Y, et., and vetors with boldfae letters, e.g., x, y. We denote by A ǫ n) X) the weakly typial set for the possibly vetor) random variable X, see [43, Ch. 3] for the definition of A ǫ n) X). When referring to a typial set we may omit the random variables from the notation, when these variables are lear from the ontext. We denote the ardinality of the finite set A with A. We use X to denote the disrete and finite) range of X. Finally, we denote the probability distribution of the RV X over X with px) and the onditional distribution of X given Y with px y). Definition 1: A disrete broadast hannel is a hannel with disrete input alphabet X, two disrete output alphabets, Y 1 and Y 2, and a probability transition funtion, py 1, y 2 x). We denote this hannel by the triplet X, py 1, y 2 x), Y 1 Y 2 ). Definition 2: A memoryless broadast hannel is a broadast hannel for whih the probability transition funtion of a sequene of n symbols is given by py n 1, y n 2 x n ) = n i=1 py 1,i, y 2,i x i ), where y n k = y k,1, y k,2,..., y k,n ), k 1, 2, and x n = x 1, x 2,..., x n ). We shall assume the hannel to be disrete and memoryless.

3 3 Definition 3: The physially degraded broadast hannel is a broadast hannel in whih the probability transition funtion an be deomposed as py 1, y 2 x) = py 1 x)py 2 y 1 ). Hene, for the physially degraded BC we have that X Y 1 Y 2 form a Markov hain. Definition 4: An R 12, R 21 )-onferene between R x1 and R x2 is defined by two onferene message sets W 12 = 1, 2,..., 2 nr 12, W21 = 1, 2,..., 2 nr21, and two mapping funtions, h 12 and h 21 whih map the reeived sequene of n symbols and the onferene messages at one reeiver into a message transmitted to the other reeiver: h 12 : Y n 1 W 21 W 12, h 21 : Y n 2 W 12 W 21. We note that this is not the most general definition of a onferene, see for example [44], [45] for a more general form. In this paper we onsider only onferenes in whih eah reeiver sends at most one message to the other reeiver. Note that there are ases where a single onferene message is enough to ahieve apaity: for example, in setion III a single onferene step ahieves apaity for the physially degraded broadast hannel, and in [45] a single onferene step ahieves apaity for the disrete memoryless multiple aess hannel ounterpart of the setup disussed here. Definition 5: A C 12, C 21 )-admissible onferene is a onferene for whih R 12 C 12 and R 21 C 21. Definition 6: A 2 nr1, 2 nr2 ), n, C12, C 21 ) ) ode for the broadast hannel with ooperating reeivers having onferene links of apaities C 12 and C 21 between them, onsists of two sets of integers W 1 = 1, 2,..., 2 nr1, W2 = 1, 2,..., 2 nr 2, alled message sets, an enoding funtion f : W 1 W 2 X n, a C 12, C 21 )-admissible onferene and two deoding funtions h 12 : Y n 1 W 21 W 12, h 21 : Y n 2 W 12 W 21, g 1 : W 21 Y n 1 W 1, 1) g 2 : W 12 Y n 2 W 2. 2) Definition 7: The average probability of error is defined as the probability that the deoded message pair is different from the transmitted message pair: P n) e = Pr g 1 W 21, Y n 1 ) W 1 or g 2 W 12, Y n 2 ) W 2). We also define the average probability of error for eah reeiver as: P n) e1 P n) e2 = Prg 1 W 21, Y n 1 ) W 1), 3) = Prg 2 W 12, Y n 2 ) W 2 ), 4) where we assume transmission of n symbols for eah odeword. By the union bound we have that max P n) e1, P n) e2 P e n) P n) e1 + P n) n) e2. Hene, P e 0 implies that both P n) e1 0 and P n) e2 0, and when both individual error probabilities go to zero then P e n) goes to zero as well. In the analysis that follows, we assume that user 1 and user 2 selet their respetive messages W 1 and W 2 independently and uniformly over their respetive message sets. Definition 8: A rate pair R 1, R 2 ) is said to be ahievable, if there exists a sequene of ) 2 nr1, 2 nr2, n, C12, C 21 ) ) odes with P e n) 0 as n. Obviously, this is satisfied if both P n) e1 0 and P n) e2 0 as n inreases. Definition 9: The apaity region for the disrete memoryless broadast hannel with ooperating reeivers is the onvex hull of all ahievable rates. III. CAPACITY REGION FOR THE PHYSICALLY DEGRADED BROADCAST CHANNEL WITH COOPERATING RECEIVERS We onsider the physially degraded broadast hannel with three independent messages: a private message to eah reeiver and a ommon message to both. We note that for the physially degraded hannel, following the argument in [43, theorem ], we an inorporate a ommon rate to both reeivers by replaing R 2, the private rate to the bad reeiver, obtained for the two private messages ase with R 0 +R 2, where R 0 denotes the rate of the ommon information. Without ooperation, the apaity region for the physially degraded BC X Y 1 Y 2 given in [43, theorem ], is the onvex hull of all the rate triplets R 0, R 1, R 2 ) that satisfy R 1 IX; Y 1 U), 5) R 0 + R 2 IU; Y 2 ), 6) for some joint distribution pu)px u)py 1 x)py 2 y 1 ), where U min X, Y 1, Y 2. 7) Next, onsider ooperation between reeivers over the physially degraded BC. First note that for this ase, the link from R x2 to R x1 does not ontribute to inreasing the rates due to ooperation, and that only the link from R x1 to R x2 does. This is due to the data proessing inequality see [43, theorem 2.8.1]): sine X Y 1 Y 2 form a Markov hain, any information about X ontained in Y 2 will also be ontained in Y 1, and thus onferening annot help: IX; Y 1, Y 2 ) = IX; Y 1 ) + IX; Y 2 Y 1 ) = IX; Y 1 ). = 0 For the rest of this setion then, we shall onsider only a ommuniation link from the good reeiver R x1, to the bad reeiver R x2 i.e. we set C 21 = 0). This implies that W 21 is a onstant and we an thus omit it from the analysis. We begin with a statement of the theorem: Theorem 1: The apaity region for sending independent information over the disrete memoryless physially degraded broadast hannel X Y 1 Y 2, with ooperating reeivers having a noiseless onferene link of apaity C 12, as defined in Setion II, is the onvex hull of all rate triplets R 0, R 1, R 2 ) that satisfy R 1 IX; Y 1 U), 8) R 0 + R 2 min IU; Y 1 ), IU; Y 2 ) + C 12 ), 9)

4 4 for some joint distribution pu)px u)py 1, y 2 x), where the auxiliary random variable U has ardinality bounded by U min X, Y 1. We note that this result presented in [46] was simultaneously derived in [42] for the ase of a wireless relay. A. Ahievability Proof In this setion, we show that the rate triplets of theorem 1 are indeed ahievable. We will show that the region defined by 8) and 9) with R 0 = 0 is ahievable. Inorporating R 0 > 0 easily follows as explained earlier. 1) Overview of Coding Strategy: The oding strategy is a ombination of a broadast ode as an outer ode used to split the rate between R x1 and R x2, and an inner ode for R x2, using the ode onstrution for the physially degraded relay hannel, desribed in [34, theorem 1]. We first generate odewords U n for R x2, aording to the relay hannel ode onstrution. Then, the odewords for R x2 are used as loud enters for the odewords transmitted to R x1 whih are also the output to the hannel). Upon reeption, R x1 deodes both its own message and the message for R x2, and then uses the relay ode seletion to selet the message relayed to R x2. R x2 uses its reeived signal, Y2 n, to generate a list of possible Un andidates, and then uses the information from R x1 to resolve for the orret odeword. 2) Details of Coding Strategy: a) Code Generation: 1) Consider first the set of M R = 2 nc12 relay messages. These are the messages that the relay R x1 transmits to R x2 through the noiseless finite apaity onferene link between the two reeivers. Index these messages by s, where s 1, 2,..., M R. Next, fix pu) and px u). 2) For eah index s [1, M R ], generate 2 nr2 onditionally independent odewords uw 2 s) n i=1 pu i), where w 2 1, 2,..., 2 nr2. 3) For eah odeword uw 2 s) generate 2 nr1 onditionally independent odewords xw 1, w 2 s) xw 1 uw 2 s)) n i=1 px i u i w 2 s)), where w 1 1, 2,..., 2 nr 1. 4) Randomly partition the message set for R x2, 1, 2,..., 2 nr 2, into MR sets S 1, S 2,..., S MR, by independently and uniformly assigning to eah message an index in [1, M R ]. b) Enoding Proedure: Consider transmission of B bloks, eah blok transmitted using n hannel symbols. Here we use nb symbol transmissions to transmit B 1 message pairs w 1,i, w 2,i ) [ 1, 2 nr1 ] [ 1, 2 nr 2 ], i = 1, 2,..., B 1. As B we have that the rate R 1, R 2 ) B 1 B R 1, R 2 ). Hene, any rate pair ahievable without bloking an be approahed arbitrarily lose with bloking as well. Let w 1,i and w 2,i be the messages intended for R x1 and R x2 respetively, at the i th blok, and also assume that w 2,i 1 S si. R x1 has an estimate ŵ 2,i 1 of the message sent to R x2 at blok i 1. Let ŵ 2,i 1 S ŝi. At the i th blok the transmitter outputs the odeword xw 1,i, w 2,i s i ), and R x1 sends the index ŝ i to R x2 through the noiseless onferene link. ) Deoding Proedure: Assume first that up to the end of the i 1) th blok there was no deoding error. Hene, at the end of the i 1) th blok, R x1 knows w 1,1, w 1,2,..., w 1,i 1 ), w 2,1, w 2,2,..., w 2,i 1 ) and s 1, s 2,..., s i ), and R x2 knows w 2,1, w 2,2,..., w 2,i 2 ) and s 1, s 2,..., s i 1 ). The deoding at blok i proeeds as follows: 1) R x1 knows s i from w 2,i 1. Hene, R x1 determines uniquely ŵ 1,i, ŵ 2,i ) s.t. uŵ2,i s i ),xŵ 1,i, ŵ 2,i s i ),y 1 i) ) A n) ǫ. If there is none or there is more than one, an error is delared. 2) R x2 reeives s i from R x1. From knowledge of s i 1 and y 2 i 1), R x2 forms a list of possible messages, Li 1) = w 2 : y 2 i 1),uw 2 s i 1 )) A n) ǫ. Now, R x2 uses s i to find a unique ŵ 2,i 1 S si Li 1). If there is none or there is more than one, an error is delared. 3) Analysis of the Probability of Error: The ahievable rate to R x2 an be proved using the same tehnique as in [34, theorem 1]. For the ease of desription assume that R x1 is onneted via an orthogonal hannel to R x2 and let X denote the hannel input from R x1 and Y the orresponding hannel output to R x2. Thus, R x2 has ombined input Y 2, Y ). The overall transition matrix is given by py 1, y 2, y x, x ) = py 1, y 2 x)py x ). 10) Additionally, we selet the transition matrix py x ) and the input and output alphabets X, Y suh that the apaity of the orthogonal hannel X Y is C 12. An example for suh a seletion is letting X = Y = 0, 1,..., 2 C12 1, where is denotes the eil funtion. Letting [a] denotes the integer part of the real number a, we set the hannel transition funtion to be 1 α, Y py X ) = = X α, Y = mod X + 2 [C12], 2 C12 ), with α seleted suh that HY X ) = C 12 C 12. The apaity of this hannel is C 12 and is ahieved by letting px 1 ) =, x X. This setup is equivalent to the 2 C 12 original setup desribed in setion I-B. Now onsider the rate to R x2. The Markov hain U X Y 1, Y 2 ) ombined with the ondition in 10) implies the following probability distribution funtion p.d.f.) pu, y 1, y 2, y, x ) = py 1, y 2 u)py x )pu, x ). Now, applying [34, theorem 1], with pu, x ) = pu)px ), we have that see also [32]) R 2 min IU, X ; Y 2, Y ), IU; Y 1 X ) = min IU, X ; Y ) + IU, X ; Y 2 Y ), IU; Y 1 ) = min IX ; Y ) + IU; Y X ) + IU; Y 2 Y ) +IX ; Y 2 Y, U), IU; Y 1 ) = min C 12 + IU; Y 2 ), IU; Y 1 ). Next, onsider the rate to R x1. From the proof of [34, theorem 1] we have that R x1 deodes W 2. Therefore, R x1 an now use suessive deoding similar to the deoding at R x1 in [43, Ch ], whih imply that the ahievable rate to R x1 is given

5 5 by R 1 IX; Y 1 U). Combining both bounds we get the rate onstraints of theorem 1. B. Converse Proof In this setion we prove that for P e n) 0, the rates must satisfy the onstraints in theorem 1. First, note that for the ase of the physially degraded broadast hannel with ooperating reeivers we have the following Markov hain: X n Y1 n W 12 Y1 n ), Y 2 n ). 11) Considering the definition of the deoders in 1) and 2), and the definition of the probability of error for eah of the reeivers in 3) and 4), we have from Fano s inequality [43, Ch. 2.11]) that HW 1 Y1 n n) ) P e1 log 2 nδp n) HW 2 Y n 2, W 12 Y n 1 )) P n) e2 log 2 2 nr 1 1 ) + hp n) e1 )12) e1 ), nδp n) e2 ), 2 nr 2 1 ) + hp n) e2 )13) where hp) is the entropy of a Bernoulli RV with parameter P. Note that when P n) e1 0 then δp n) e1 ) 0 and when P n) e2 0 then δp n) e2 ) 0. Now, for R x1 we have that nr 1 = HW 1 ) = IW 1 ; Y n 1 ) + HW 1 Y n 1 ). Applying inequality 12), and then proeeding as in [4] we get the bound on R 1 as nr 1 n k=1 IX k ; Y 1,k U k ) + nδp n) e1 ), where U k Y 1,1, Y 1,2,..., Y 1,k 1, W 2 ). For R x2 we an write nr 2 = HW 2 ) a) IW 2 ; Y2 n, W 12Y1 n n) )) + nδp e2 ) 14) = IW 2 ; Y2 n ) + IW 2; W 12 Y1 n ) Y 2 n n) ) + nδp e2 ), where the inequality in a) is due to 13). Proeeding as in [4], we bound IW 2 ; Y2 n) n k=1 IU k; Y 2,k ). Next, we bound IW 2 ; W 12 Y1 n) Y 2 n ) as follows: IW 12 Y n 1 ); W 2 Y n 2 ) HW 12 Y n 1 ) Y n 2 ) HW 12 Y n 1 )) nc 12, 15) where the first inequality follows from the definition of mutual information, the seond is due to removing the onditioning and the third is due to the admissibility of the onferene. Combining both bounds we get that nr 2 n k=1 IU k ; Y 2,k ) + nc 12 + nδp n) e2 ). 16) The bound on R 2 an be developed in an alternative way. Begin with 14): nr 2 IW 2 ; Y2 n, W 12Y n n) )) + nδp a) 1 e2 ) IW 2 ; Y2 n, Y 1 n n) ) + nδp e2 n ) = IW 2 ; Y 1,k, Y 2,k Y1 k 1, Y2 k 1 ) + nδp n) e2 ),17) k=1 where a) follows from the fat that W 1, W 2 ) Y1 n, Y 2 n) W 12, Y2 n ) is a Markov relation and from the data proessing inequality. Next, we an write IW 2 ; Y 1,k, Y 2,k Y1 k 1, Y2 k 1 ) a) = IW 2 ; Y 1,k Y1 k 1, Y2 k 1 ) = HY 1,k Y1 k 1, Y2 k 1 ) HY 1,k Y1 k 1, Y2 k 1, W 2 ) b) HY 1,k ) HY 1,k Y1 k 1, Y2 k 1, W 2 ) ) = HY 1,k ) HY 1,k Y1 k 1, W 2 ) = IY 1,k ; Y1 k 1, W 2 ) = IY 1,k ; U k ), 18) where the equality in a) is due to the physial degradedness and memorylessness of the hannel, b) is due to removing the onditioning, and ) is beause the Markov hain makes Y 1,k independent of Y2 k 1 given Y1 k 1. Plugging this into 17), we get a seond bound on R 2 : nr 2 n k=1 IU k ; Y 1,k ) + nδp n) e2 ). Colleting the three bounds we have: R 1 1 n IX k ; Y 1,k U k ) + δp n) e1 ), 19) n R 2 1 n R 2 1 n k=1 n k=1 n k=1 IU k ; Y 2,k ) + C 12 + δp n) e2 ), 20) IU k ; Y 1,k ) + δp n) e2 ). 21) Using the standard time-sharing argument as in [43, Ch. 14.3], we an write the averages in 19) - 21) by introduing an appropriate time sharing variable, with ardinality upper bounded by 4. Therefore, if P n) e1 0 and P n) e2 0 as n, the onvex hull of this region an be shown to be equivalent to the onvex hull of the region defined by R 1 IX; Y 1 U), 22) R 2 IU; Y 2 ) + C 12, 23) R 2 IU; Y 1 ). 24) Finally, the bound on the ardinality of U follows from the same arguments as in the onverse for the non-ooperative ase in [4]. Note however, that Y 2 is absent from the minimization on the ardinality f. equation 7) for the nonooperative ase). The reason is that even when Y 2 = 1, information to R x2 represented by the random variable U), an be sent through the onferene link between the two reeivers.

6 6 C. Disussion To illustrate the impliations of theorem 1, onsider the physially degraded binary symmetri broadast hannel BSBC) depited in figure 2. For this hannel, theorem 1 U X Y 1 Y 2 p U p 1 Fig. 2. The physially degraded BSBC. p U, p 1 and p 2 are the transition probabilities at the left, middle and right segments respetively. implies that U = 2. Due to the symmetry of the hannel, the probability distribution of U whih maximizes the rates, is a symmetri binary distribution, PrU = 0) = PrU = 1) = 1 2. The resulting apaity region for this ase is depited in figure 3 for the ase where R 0 = 0. In the figure, the bottom line dash) is the non-ooperative apaity region, and the top line dash-dot) is the maximum possible sum rate, whih requires that C 12 hp 12 ) hp 1 ), where p 2 hp) = p log 2 p) 1 p)log 2 1 p), p 12 = p 1 1 p 2 ) + p 2 1 p 1 ). This maximum sum-rate of IX; Y 1 ) is obtained by summing the rate to R x1 given by 22) and the maximum possible rate for R x2 given by 24), and using the Markov hain relation U X Y 1. The middle line solid) is the apaity region for IX;Y 1) IX;Y )+C 2 12 IX;Y 2) R 2 C 12 R 1 IX;Y 1) Fig. 3. The apaity region for the physially degraded BSBC. Top, middle and bottom lines orrespond to maximum possible ooperation, partial ooperation and no-ooperation senarios respetively. the partial ooperation ase where 0 < C 12 < hp 12 ) hp 1 ). As an be seen from this example, the apaity region derived in this setion is stritly larger than the apaity region for the non-ooperation ase. Indeed, summing the onstraints on R 0, R 1 and R 2 without ooperation equations 5), 6)), results in a maximum ahievable sum-rate of R 0 + R 1 + R 2 IX; Y 1 ) IU; Y 1 ) IU; Y 2 )), 25) where the seond term is always positive due to the Markov hain U X Y 1 Y 2 assuming the degrading hannel is non-invertible 1 ). In this setup, the maximum possible sumrate, IX; Y 1 ), is ahieved only when U is a onstant, and thus no information is sent to R x2. When R 0 + R 2 > 0, beause of the relationship R 0 + R 2 IU; Y 2 ) < IU; Y 1 ), we annot ahieve the maximum sum-rate of IX; Y 1 ) to R x1. However, summing 23) or 24) with 22), results in a maximum ahievable sum-rate with ooperating reeivers of R 0 + R 1 + R 2 IX; Y 1 ) + min 0, C 12 IU; Y 1 ) IU; Y 2 )).26) Comparing this to non-ooperative sum-rate given by 25), it is lear that ooperation allows a net inrease in the sum-rate, by at most C 12. IV. ACHIEVABLE RATES FOR THE GENERAL BROADCAST CHANNEL WITH COOPERATING RECEIVERS For the lassi general BC senario, the best ahievability result was derived by Marton in [14]. This result states that for the general BC, any rate pair R 1, R 2 ) satisfying R 1 IU; Y 1 ), 27) R 2 IV ; Y 2 ), 28) R 1 + R 2 IU; Y 1 ) + IV ; Y 2 ) IU; V ), 29) for some joint distribution pu, v, x, y 1, y 2 ) = pu, v, x)py 1, y 2 x), is ahievable. We note that Marton s largest region ontains three auxiliary RVs, W, U, V ), where W represents information deoded by both reeivers. Here we use a simplified version, where W is set to a onstant. We now onsider ooperation between the reeivers. We begin with a statement of the theorem: Theorem 2: Let X, py 1, y 2 x), Y 1 Y 2 ) be any disrete memoryless broadast hannel, with ooperating reeivers having noiseless onferene links of finite apaities C 12 and C 21, as defined in Setion II. Then, for sending independent information, any rate pair R 1, R 2 ) satisfying subjet to, where, R 1 RU), R 2 RV ), R 1 + R 2 RU) + RV ) IU; V ), C 21 IÛ; Y 2) IÛ; Y 1), 30) C 12 IˆV ; Y 1 ) IˆV ; Y 2 ), 31) RU) = IU; Y 1, Û), 32) RV ) = IV ; Y 2, ˆV ), 33) for some joint distribution pu, v, x, y 1, y 2, û, ˆv) = pu, v, x)py 1, y 2 x)pû y 2 )pˆv y 1 ), is ahievable, with u U, v V, û Û, ˆv ˆV, Û Y and ˆV Y In the next subsetions we provide the proof of this theorem. 1 It an be shown that IU; Y 1 ) IU; Y 2 ) = 0 for the degraded hannel setup implies that if R 0 + R 2 > 0 then HY 1 Y 2 ) = 0, i.e. the hannel from R x1 to R x2 is invertible. Under these irumstanes, this setup an be replaed by an equivalent setup in whih both reeivers get Y 1, but suh a degenerate setup is not interesting.

7 7 A. Overview of Coding Strategy As in the ahievability part of theorem 1, the proposed ode is a hybrid broadast-relay ode. Here, we ombine the relay ode onstrution of [34, theorem 6] and the broadast ode onstrution of [15]. The fat that in these two theorems the hannel enoding and the relay operation are performed independently, allows to easily ombine them into a hybrid oding sheme. The enoder generates broadast odewords, eah seleted from a odebook onstruted similarly to the onstrution of [15]. This odebook splits the rate between the two users. Next, eah relay R x1 ats as a relay for R x2 and vie-versa) generates its odebook aording to the onstrution of [34, theorem 6]. In the deoding step, using the reeived signal Y1 n at R x1 and Y2 n at R x2 ), eah reeiver generates a list of the possible transmitted relay messages and uses the onferene message from the next time interval to resolve for the relay massage. Then, eah reeiver uses the deoded relay message and its reeived hannel output to deode its own message. B. Enoding at the Transmitter 1) Let ǫ > 0 and n 1 be given. Fix pu, v, x), pû y 2 ) and pˆv y 1 ), and let δ > 0 be a positive number, whose seletion is desribed in the next item. Let A n) δ U) denote the set of strongly typial i.i.d. sequenes of length n, u U n, as defined in [43, Ch. 13.6]. Let A n) δ V ) denote the set of strongly typial i.i.d. sequenes of length n, v V n. Let S n) [U]δ denote the set of all sequenes u A n) δ U), suh that A n) δ V u) is nonempty as defined in [47, orollary 5.11], and similarly define S n) [V ]δ for the sequenes v A n) δ V ). 2) Selet 2 nru) ǫ) strongly typial sequenes u in an i.i.d. manner, aording to the probability 1,u S n) p u) = S n) [U]δ [U]δ 0, otherwise. ] Label these sequenes by uk), k [1, 2 nru) ǫ). Selet 2 nrv ) ǫ) strongly typial sequenes v in an i.i.d. manner, aording to the probability 1,v S n) p v) = S n) [V ]δ [V ]δ 0, otherwise. Label these sequenes by vl), l [ 1, 2 nrv ) ǫ)]. Note that from [47, orollary 5.11] we have that S n) [U]δ 1 δ)2nhu) ψ), where ψ 0 as n and δ 0, so for any ǫ > 0 we an always find 0 < δ ǫ suh that for n large enough we obtain S n) [U]δ > 2 niu;y1,û) ǫ) and S n) [V ]δ > 2nIV ;Y2,ˆV ) ǫ). 3) [ Define the ells B i = i 1)2 nru) R 1 ǫ) + 1, i2 nru) R1 ǫ)], i [ ] 1, 2 nr1. This is a partition of the u sequenes into 2 nr1 sets. Define the ells C j = [ j 1)2 nrv ) R2 ǫ) + 1, j2 nrv ) R2 ǫ)], j [ ] 1, 2 nr2, whih form a partition of the v sequenes into 2 nr2 sets. 4) For every pair of integers w 1, w 2 ) [ ] 1, 2 nr1 [ ] 1, 2 nr 2, define the set Dw1,w 2 = uk),vl)) : k B w1, l C w2, uk),vl)) A n) ǫ U, V ). Here, A n) ǫ U, V ) denotes the strongly typial set for the random variables U and V as defined in [43, Ch. 13.6]. In the following we may omit the random variables when referring to the strongly typial set, when these variables are lear from the ontext. We now have the following slightly modified) lemma from [15]: Lemma 1: For any 2-D ell B i C j, ǫ > 0, and n large enough, we have that Pr D ij = 0) ǫ, provided that R 1 + R 2 < RU) + RV ) IU; V ) 2ǫ ǫ 1, 34) where ǫ 1 0 as ǫ 0 and n. Proof: The proof of this lemma is obtained by diret appliation of the tehnique used to prove [15, Lemma in pg. 121], and therefore will not be repeated here. 5) For eah message pair w 1, w 2 ), selet one pair uk w1,w 2 ),vl w1,w 2 )) D w1,w 2. For eah of the seleted pairs one pair for eah message pair), generate a odeword aording to xw 1, w 2 ) n i=1 p x i u i k w1,w 2 ), v i l w1,w 2 )). 6) To transmit the message pair w 1, w 2 ) the transmitter outputs xw 1, w 2 ). C. Enoding the Relay Messages Consider first the relay enoding at R x2, whih ats as a relay for R x1. 1) R x2 -relay has a set of 2 nc21 relay messages indexed by s [ ] 1, 2 nc21. For eah index s, generate 2 nr i.i.d. sequenes û, eah with probability pû) = n i=1 pû i), pû) = X,Y 1,Y 2 pû y 2 )py 1, y 2 x)px), and px) = U,V pu, v, x). Label these odewords ûz s ), s [ ] 1, 2 nc 21, z [1, 2 nr ]. 2) Randomly and uniformly partition the message set [1, 2 nr ] into 2 nc21 sets S s, s [ ] 1, 2 nc21. 3) Enoding: Assume that after reeiving y 2 i 1) we have at R x2 that ûz i 1 s i 1 ),y 2i 1) ) A n) ǫ, and that z i 1 S s s i 1 is known from the previous i transmission of z i 2 ). Then, at the i th transmission interval the relay transmits the index s i to R x1. Relay enoding at R x1 is performed in a symmetri manner to the relay enoding at R x2. The orresponding variables for R x1 are S s and ˆvz s ), s [ ] 1, 2 nc12, z [1, 2 nr ]. D. Deoding the Relay Messages at the Relays Consider deoding the relay message at R x2. The relay deoder at R x2 uses its hannel input y 2 i), and its previously deoded s i to generate the relay message z i as follows: upon reeiving y 2 i), the relay R x2 deides that the message z i was reeived at time i if û z i s i ),y 2i)) A n) ǫ. Following the argument in [34, theorem 6] see also the proof in [43, Ch.

8 8 13.6]), there exists suh z i with probability that is arbitrarily lose to one as long as ) R I Û; Y2, 35) and n is suffiiently large. Relay deoding at R x1 is done in a symmetri manner to the relay deoding at R x2. E. Deoding at the Reeivers We first find the rate onstraint for deoding at R x1. R x1 deodes its message w 1,i 1 based on its hannel input y 1 i 1) and the relay indies s i and s i 1 : 1) From knowledge of s i 1 and y 1i 1), R x1 alulates the set L 1 i 1) suh that L 1 i 1) = z [1, 2 nr ] : û z s i 1),y1 i 1) ) A n) ǫ. 2) At the time interval of the i th odeword, R x1 reeives the relayed s i. Sine s i is seleted from a set of 2 nc21 possible messages, it an be transmitted over the noiseless onferene link without error. 3) R x1 now hooses ẑ i 1 as the relay message at time i 1 if and only if there exists a unique ẑ i 1 S s L1 i i 1). Again, following the reasoning in [34, theorem 6], this an be done with an arbitrarily small probability of error as long as R IÛ; Y 1) + C 21, 36) and n is large enough. Combining this with inequality 35) we get the onstraint on the relay information rate: C 21 IÛ; Y 2) IÛ; Y 1). 37) This expression is similar to the Wyner-Ziv expression for the rate required to transmit Y 2 to reeiver R x1 up to a given distortion, determined by pû y 2 ) and a deoder. Here the performane of the deoder are implied in the mutual information IU; Y 1, Û). The ompressed Y2 n is then used by R x1 to assist in deoding W 1. 4) Lastly, R x1 deodes w 1,i 1 or, equivalently uk w1,i 1,w 2,i 1 )) by hoosing ukŵ1,i 1,ŵ 2,i 1 ) suh that ukŵ1,i 1,ŵ 2,i 1),y 1 i 1),û ẑ i 1 i 1)) s A n) ǫ. From the point-to-point hannel oding theorem see [15]) we have that ŵ 1,i 1 = w 1,i 1 with probability that is arbitrarily lose to one, as long as z i 1 was orretly deoded at R x1 and ) R 1 RU) I U; Y 1, Û, 38) for suffiiently large n. Combining this with equation 37) yields the rate onstraint on R 1 : R 1 RU), 39) as long as C 21 IÛ; Y 2) IÛ; Y 1). 40) Using symmetri arguments to those presented for deoding at R x1 we find the rate onstraint for R x2 to be R 2 RV ), 41) as long as C 12 IˆV ; Y 1 ) IˆV ; Y 2 ). 42) Combining equations 34), 39), 40), 41) and 42), gives the onditions in theorem 2. F. Error Events In the sheme desribed above we have to aount for the following error events for deoding w 1,i 1, w 2,i 1 ): 1) Enoding at the transmitter fails: E D,i = D w1,i 1,w 2,i 1 = 0. 2) Joint typiality deoding fails: E 0,i = ukw1,i 1,w 2,i 1 ),vl w1,i 1,w 2,i 1 ), xw 1,i 1, w 2,i 1 ),y 1 i 1),y 2 i 1) ) / A n) ǫ 3) Deoding at the relays fails: E 1,i = E 112,i, z [1, 2 nr ] s.t. E 11,i = û z s i 1),y2 i 1) ) A n) ǫ, E 12,i = z [1, 2 nr ] s.t. ) ˆv z s i 1,y1 i 1) ) A n) ǫ. 4) Deoding the relay message at the reeivers fails: E 2,i = E 222,i, where E 21,i = E 21,i E 21,i and E 22,i = E 22,i E 22,i, E 21,i = z i 1 / S s L1 i 1), i E 21,i z = z i 1 s.t. z S s L1 i 1), i E 22,i z = i 1 / S s L2 i 1), i z z i 1 s.t. z S s L2 i 1), i L 2 i 1) z [1, 2 nr ] : ) ˆv z s i 1,y2 i 1) ) A n) ǫ. 5) Final deoding at the reeivers fails: E 3,i = E 31,i E32,i, where, E 31,i = ukw1,i 1,w 2,i 1),y 1 i 1), ûz i 1 s i 1 )) / A n) ǫ w 1 w 1,i 1 s.t. ukw1,w 2 ),y 1 i 1),ûz i 1 s i 1 )) A n) ǫ, E 32,i = vlw1,i 1,w 2,i 1),y 2 i 1), ˆvz i 1 s i 1 )) / A n) ǫ w 2 w 2,i 1 s.t. vlw1,w 2 ),y 2 i 1), ˆvz i 1 s i 1 )) A n) ǫ. We now bound the probability of the error events at time E 22,i = i. Note that at time i both R x1 and R x2 share the same s i 1 and s i 1 irrespetive whether the deoding at the relays was orret at time i 1. Hene, a deoding error at time i 1 does not affet the deoding at time i. Now, from lemma 1 it follows that by taking n large enough the probability of E D,i an be made arbitrarily small, as long as 34) is satisfied. Additionally, by taking n large enough, the probability PrE 0,i E D,i ) an be made arbitrarily small by the properties of strongly typial sequenes, see [43, lemma ]. The probability PrE 1,i ) an be made arbitrarily small as long as 40) and 42) are satisfied, as explained is setion IV-D. Next, the Markov lemma [50, lemma 4.2] and the Markov hains Y 1 Y 2 Û and Y 2 Y 1 ˆV, imply that PrE 21,i 0,i ) and PrE 22,i 0,i ) an be made arbitrarily small by taking n large enough, and.

9 9 PrE 21,i i,1 ) and PrE 22,i i,1 ) an be made arbitrarily small by taking n large enough as long as 40) and 42) are satisfied. Finally, PrE 3 2,i E 0,i E D,i ) and PrE 32,i E 2,i E 0,i E D,i ) an be made arbitrarily small by taking n large enough by the Markov lemma and the hains U, Y 1 Y 2 Û and V, Y 2 Y 1 ˆV, and as long as 39) and 41) are satisfied. This onludes the proof of theorem 2. G. An Upper Bound Proposition 1: Assume the broadast hannel setup of theorem 2. Then, for sending independent information, any ahievable rate pair R 1, R 2 ) must satisfy R 1 IX; Y 1 ) + C 21, R 2 IX; Y 2 ) + C 12, R 1 + R 2 IX; Y 1, Y 2 ), for some distribution px) on X. Proof: The proof uses the ut-set bound [43, theorem ]. First we define an equivalent system by introduing two orthogonal hannels X 2 Y 1 from R x2 to R x1 and X 1 Y 2 from R x1 to R x2. The joint probability distribution funtion then beomes p y 1, y 1 ), y 2, y 2 ) x, x 1, x 2 ) = py 1, y 2 x)py 1 x 2 )py 2 x 1 ), where the signal reeived at R x1 is Y 1, Y 1 ) and the signal reeived at R x2 is Y 2, Y 2). As in the proof in setion III-A.3, we selet X 1, X 2, Y 1, Y 2, px 1 ), px 2 ), py 1 x 2 ) and py 2 x 1) suh that the apaities of the hannels X 2 Y 1 and X 1 Y 2 are C 21 and C 12 respetively. Additionally, the odewords for the onferene transmissions are determined independently from the soure odebook so we set px, x 1, x 2 ) = px)px 1 )px 2 ). Now, from the ut-set bound, letting the transmitter and R x2 form one group and R x1 the seond group, we have R 1 IX, X 2; Y 1, Y 1 X 1) = IX 2 ; Y 1, Y 1 X 1 ) + IX; Y 1, Y 1 X 1, X 2 ) = IX 2; Y 1 X 1) + IX 2; Y 1 X 1, Y 1) + IX; Y 1 X 1, X 2 ) + IX; Y 1 X 1, X 2, Y 1 ) = IX 2; Y 1) + IX; Y 1 ) = C 21 + IX; Y 1 ), where IX 2 ; Y 1 X 1, Y 1 ) = IX; Y 1 X 1, X 2 ) = 0 follows from diret appliation of the distribution funtion. Similarly we obtain the rate onstraint on R 2. Lastly, for the sum-rate onsider the transmitter in one group and the reeivers in the seond. Then, the ut-set bound results in R 1 + R 2 IX; Y 1, Y 2, Y 1, Y 2 X 1, X 2) = IX; Y 1, Y 2 X 1, X 2 ) + IX; Y 1, Y 2 X 1, X 2, Y 1, Y 2 ) = IX; Y 1, Y 2 ), yielding the last onstraint in the proposition. H. Remarks Comment 4.1: Observing the rate onstraints in theorem 2 we an see that when 30) and 31) are satisfied then the ooperative rates are greater than the non-ooperative rates due to the generally) positive terms adding to IU; Y 1 ) and IV ; Y 2 ). Comment 4.2: We note that although we present a single letter haraterization of the rates, we are not able to apply standard ardinality bounding tehniques suh as those used in [48] or [49] for bounding U and V. The method of [48] annot be applied sine it relies on the fat that the auxiliary random variables are independent, whih is not the ase here. The method of [49] annot be applied as explained in the omment for theorem 2 in [20]. The ardinality bounds on Û and ˆV are trivial sine they are transmitted over noiseless links. Comment 4.3: The relay strategies an be divided into two general lasses. The first lass is referred to as deode-andforward DAF). In this strategy, the relay first deodes the message intended for the destination and then generates a relay message based on the deoded information. The seond lass is referred to as estimate-and-forward EAF). In this lass the relay does not deode the message intended for the destination but transmits an estimate of its hannel input to the destination. For the physially degraded BC we used DAF, based on [34, theorem 1], to derive theorem 1, and for the general BC we used the EAF sheme of [34, theorem 6], to derive theorem 2. Of ourse, one an also ombine both strategies and perform partial deoding at eah reeiver of the other reeiver s message before onferening, following [34, theorem 7]. This ombination will, in general, result in an inreased ahievable rate region. I. Speial Cases 1) No Cooperation: C 12 = C 21 = 0: Consider first ooperation from R x2 to R x1. Setting C 21 = 0 in theorem 2 implies that HÛ Y 1) = HÛ Y 2). 43) From equation 32), the onstraint on R 1 an be written in the form Now we find IU; Û Y 1): R 1 IU; Y 1 ) + IU; Û Y 1). IU; Û Y 1) = HÛ Y 1) HÛ Y 1, U) a) = HÛ Y 2) HÛ Y 1, U) b) = HÛ Y 2, Y 1, U) HÛ Y 1, U) 44) = IÛ; Y 2 Y 1, U). where a) is due to 43), and b) is due to the Markov hain U U, V ) X Y 1, Y 2 ) Y 2 Û, whih implies that given Y 2, Û is independent of Y 1 and U. Now, sine mutual information is non-negative, we onlude that IU; Û Y 1) = 0. Hene, the rate onstraint on R 1 beomes R 1 IU; Y 1 ).

10 10 Similarly, the maximum rate R 2 is given by IV ; Y 2 ), and in onlusion when C 12 = C 21 = 0 we resort bak to the rate region without ooperation derived in [14] with a onstant W ). 2) Full Cooperation: C 12 = HY 1 Y 2 ), C 21 = HY 2 Y 1 ): When C 12 = HY 1 Y 2 ), we get from 31) that HY 1 Y 2 ) = C 12 IˆV ; Y 1 ) IˆV ; Y 2 ) = HˆV Y 2 ) HˆV Y 1 ), whih is satisfied when ˆV = Y 1. Plugging this into 33), we get that when full ooperation from R x1 to R x2 is available, the rate onstraint for R x2 beomes R 2 IV ; Y 2, Y 1 ). Using the same reasoning we onlude that when full ooperation from R x2 to R x1 is available, the rate onstraint for R x1 beomes R 1 IU; Y 1, Y 2 ). 3) Partial Cooperation: When 0 < C 12 < HY 1 Y 2 ) and 0 < C 21 < HY 2 Y 1 ), we get that C 21 HÛ Y 1) HÛ Y 2) HÛ Y 1) C 21 + HÛ Y 2). 45) Hene, the ahievable rate to R x1 is upper bounded by R 1 IU; Y 1, Û) = IU; Y 1 ) + IU; Û Y 1) = IU; Y 1 ) + HÛ Y 1) HÛ U, Y 1) a) IU; Y 1 ) + HÛ Y 2) HÛ U, Y 1) + C 21 b) = IU; Y 1 ) + HÛ Y 2, Y 1, U) HÛ U, Y 1) + C 21 R 1 IU; Y 1 ) + C 21 IÛ; Y 2 U, Y 1 ). 46) where a) is due to 45) and b) follow from the same reasoning leading to equation 44). Similarly, R 2 IV ; Y 2 ) + C 12 IˆV ; Y 1 V, Y 2 ). Note that there exist negative terms IÛ; Y 2 U, Y 1 ) and IˆV ; Y 1 V, Y 2 ) in the ahievable rate upper bounds. This an be explained as follows: the mutual information IÛ; Y 2 U, Y 1 ) an be onsidered as a type of anillary information that Û ontains, sine this information is ontained in Û while U and Y 1 are already known - therefore, this information is a noise part of Y 2 whih does not inlude any helpful information for deoding U at R x1. Thus, for ooperating in the optimal way, Û has to be a type of suffiient and omplete ooperation information. evaluate the ahievable region. For the single ommon message ase, we are able to derive results for partial ooperation without auxiliary variables, whih make this region expliitly omputable. This senario is depited in figure 4. W Enoder X n py 1,y 2 x) Broadast Channel Y n 1 Y n 2 Reeiver 1 R x1 R x2 Reeiver 2 W ^ C 12 C 21 ^ W Fig. 4. The single message broadast hannel with ooperating reeivers. Ŵ and Ŵ are the estimates of W at R x1 and R x2 respetively. For this senario we need to speialize the definitions of a ode and the average probability of error as follows: A 2 nr, n, C 12, C 21 ) ) ode for sending a ommon message over the broadast hannel with ooperating reeivers having onferene links of apaities C 12 and C 21 between them, is defined in a similar manner to definition 6 with W 1, W 2 and W 1 W 2 all replaed with W = 1, 2,..., 2 nr. The average probability of error is defined similarly to definition 7 with W 1 and W 2 replaed with W. The apaity for the non-ooperative single message senario is given in [5] by C = sup min IX; Y 1 ), IX; Y 2 ) ). 47) px) In the following we onsider two ooperation shemes, referred to as a single-step sheme and a two-step sheme. These shemes are desribed in figure 5. In the single-step sheme, after reeption eah reeiver generates a single ooperation message based on its hannel input. In the two-step sheme, after reeption one reeiver generates a ooperation message based only on its hannel input, as in the previous ase, but the seond reeiver generates its ooperation message only after deoding whih is done with the help of the onferene message from the first reeiver). In both ases eah reeiver generates a single onferene message, however in the singlestep onferene the emphasis is on low delay, while in the two-step onferene we sarifie delay in order to gain rate. V. THE GENERAL BROADCAST CHANNEL WITH A SINGLE COMMON MESSAGE We now onsider the ase where only a single message, rather than two independent messages, is transmitted to both reeivers. The main motivation for onsidering this ase is that in the two independent messages ase it is diffiult to speify an expliit ooperation sheme, and we therefore have to represent ooperation through auxiliary random variables. Hene, we annot identify diretly the gain from ooperation, exept in the ase of full ooperation, and we also annot y 1 i) y 2 i) Time i R x1 R x2 Reeption W 21 y 2 i)) Time i+1 Single Step Conferene Conferening y 1 i) W 12 y 1 i)) y 2 i) Time i R x1 R x2 Reeption Time i+1 W 21 y 2 i)) Conferening step 1 Two-Step Conferene Time i+2 W 12 y 1 i),wi)) ^ Conferening step 2 Fig. 5. Shemati desription of the single-step and the two-step onferene shemes.

11 11 A. Deoding with a Single-Step Cooperation In this setion we onstrain both deoders to output their deoded messages after a onferene that onsists of a single message from eah reeiver, based only on its reeived hannel input. For this ase, we an speialize the derivation of theorem 2 and get the following ahievable rate for the broadast hannel with partially ooperating reeivers: Theorem 3: Let X, py 1, y 2 x), Y 1 Y 2 ) be any disrete memoryless broadast hannel, with ooperating reeivers having noiseless onferene links of finite apaities C 12 and C 21, as defined in setion II. Then, for sending a ommon message to both reeivers, any rate R satisfying subjet to R sup px) min IX; Y 1, Û), IX; Y 2, ˆV ), C 21 IÛ; Y 2) IÛ; Y 1), C 12 IˆV ; Y 1 ) IˆV ; Y 2 ), for some joint distribution px, y 1, y 2, û, ˆv) = px)py 1, y 2 x)pû y 2 )pˆv y 1 ) is ahievable, with Û Y and ˆV Y The proof of theorem 3 follows the same lines of the proof of theorem 2 and will not be repeated here. We next show how we an inrease the rates by introduing the two-step onferene. B. Deoding with a Two-Step Cooperation We onsider a two-step onferene: at the first step only one reeiver deodes the message. The seond reeiver deodes after the seond step. Therefore, after the first reeiver deodes the message, relaying to the seond reeiver redues to the deode-and-forward relay situation of [34, theorem 1]. The rates ahievable with a two step onferene are given in the following theorem: Theorem 4: Assume the broadast hannel setup of theorem 3. Then, for sending a ommon message to both reeivers, any rate R satisfying R sup px) R 12 px)) min [ ] max R 12 px)), R 21 px)) IX; Y 1 ) + C 21, IX; Y 2 ) IˆV ; Y 1 Y 2, X) R 21 px)) min IX; Y 2 ) + C 12, + min C 12, HˆV Y 2 ) HˆV Y 1 ) )), IX; Y 1 ) IÛ; Y 2 Y 1, X) + min C 21, HÛ Y 1) HÛ Y 2) )), for some joint distribution px, y 1, y 2, û, ˆv) = px)py 1, y 2 x)pû y 2 )pˆv y 1 ) is ahievable, with Û Y and ˆV Y 1 + 1, and with the appropriate C 12 IˆV ; Y 1 Y 2, X) or C 21 IÛ; Y 2 Y 1, X) the one used for the first ooperation step). Proof: 1) Overview of Coding Strategy: The sheme desribed in theorem 3 uses a single-step onferene for both deoders. However, if we let one reeiver use a two-step onferene, then that reeiver, instead of using onferene information derived from the raw input of the other reeiver, an use information generated by the seond reeiver after it already deoded the message. This onferene information is less noisy, and thus the rate to the first reeiver an be inreased. To put this in more onrete terms, assume that at time i+1, R x1 sends to R x2 the index s i+1 of the partition into whih its relay message at time i, denoted zˆv,i, belongs. In appendix B we show that R x2 an deode the message w 0,i with an arbitrarily small probability of error as long as and R IX; Y 2 ) IˆV ; Y 1 Y 2, X) ) + min C 12, HˆV Y 2 ) HˆV Y 1 ), 48) C 12 IˆV ; Y 1 Y 2, X). 49) We now introdue the following modifiations to the sheme used in theorem 3: 2) Relay Sets Generation at R x2 : R x2 partitions the message set W into 2 nc21 subsets in a uniform and independent manner. Denote these subsets with S s, s [ ] 1, 2 nc21. 3) Relay Enoding at R x2 : R x2 has an estimate ŵ 0,i of the message w 0,i. Now, R x2 looks for the partition into whih ŵ 0,i belongs and sends the index of this partition, denoted s i+2, to R x1 at time i ) Deoding at R x1 : Upon reeption of y 1 i), R x1 generates the set L 1 i) = w W : xw),y 1 i)) A n) ǫ X, Y 1 ). At time i + 2, upon reeption of s i+2, R x1 looks for an index w suh that w L 1 i) S s. If a unique suh w exists then R x1 sets i+2 ŵ 0,i = w, otherwise an error is delared. 5) Bounding the Probability of Error: Using the proof tehnique in [34, theorem 1], it an be easily shown that assuming orret deoding at R x2, then any rate R IX; Y 1 ) + C 21 is ahievable to R x1. Combining the bounds derived above, we onlude that with a two-step onferene at R x1, any rate satisfying R min C 12 IˆV ; Y 1 Y 2, X), IX; Y 1 ) + C 21, IX; Y 2 ) IˆV ; Y 1 Y 2, X) + min C 12, HˆV Y 2 ) HˆV Y 1 ) )), ia ahievable. Repeating the same derivation when R x2 uses a two-step onferene, and ombining with the previous ase proves theorem 4. Setting Û = Y 2, ˆV = Y1 in theorem 4 we obtain the following ahievable region: