Capacity Gain From Two-Transmitter and Two-Receiver Cooperation

Transcription

1 3822 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 10, OCTOBER 2007 Caacity Gain From Two-Transmitter and Two-Receiver Cooeration Chris T. K. Ng, Student Member, IEEE, Nihar Jindal, Member, IEEE, Andrea J. Goldsmith, Fellow, IEEE, and Urbashi Mitra, Fellow, IEEE Abstract Caacity imrovement from transmitter and receiver cooeration is investigated in a two-transmitter, two-receiver network with hase fading and full channel state information (CSI) available at all terminals. The transmitters cooerate by first exchanging messages over an orthogonal transmitter cooeration channel, then encoding jointly with dirty-aer coding. The receivers cooerate by using Wyner Ziv comress-and-forward over an analogous orthogonal receiver cooeration channel. To account for the cost of cooeration, the allocation of network ower and bandwidth among the data and cooeration channels is studied. It is shown that transmitter cooeration outerforms receiver cooeration and imroves caacity over noncooerative transmission under most oerating conditions when the cooeration channel is strong. However, a weak cooeration channel limits the transmitter cooeration rate; in this case, receiver cooeration is more advantageous. Transmitter-and-receiver cooeration offers sizable additional caacity gain over transmitter-only cooeration at low signal-to-noise ratio (SNR), whereas at high SNR transmitter cooeration alone catures most of the cooerative caacity imrovement. Index Terms Caacity, transmitter and receiver cooeration, dirtyaer coding, Wyner Ziv comress-and-forward, ower and bandwidth allocation, wireless ad hoc network. I. INTRODUCTION In a wireless ad hoc network, each node can communicate with any other node over the wireless medium. Hence, grous of nodes may cooerate among one another to jointly encode or decode the transmission signals. In this corresondence, we consider a scenario where there are two clustered transmitters and two clustered receivers, with each transmitter intending to send an indeendent message to a different receiver. We assume the channels between the transmitter and receiver clusters are under quasi-static hase fading, and all terminals have erfect channel state information (CSI). When the clustered nodes do not cooerate, the four-node network is an interference channel [1]: its caacity remains an oen roblem in information theory. We study this roblem from a different ersective and ask the question: How much does cooeration increase the set of achievable data rates? However, we do not allow cooeration to occur for free. We assume the nodes in Manuscrit received August 16, 2006; revised Aril 26, This work was suorted by the U.S. Army under MURI Award W911NF , the ONR under Award N , DARPA s ITMANET rogram under Grant TFIND, a grant from Intel, and the National Science Foundation ITR under Grant CCF The material in this corresondence was resented in art at the IEEE International Symosium on Information Theory, Chicago, IL, June/July 2004, and at the IEEE Information Theory Worksho, San Antonio, TX, October C. T. K. Ng and A. J. Goldsmith are with the Deartment of Electrical Engineering, Stanford University, Stanford, CA 94305, USA ( ngctk@wsl. stanford.edu; andrea@wsl.stanford.edu). N. Jindal is with the Deartment of Electrical and Comuter Engineering, University of Minnesota, Minneaolis, MN 55455, USA ( nihar@umn. edu). U. Mitra is with the Deartment of Electrical Engineering, University of Southern California, Los Angeles, CA , USA ( ubli@usc. edu). Communicated by G. Kramer, Guest Editor for the Secial Issue on Relaying and Cooeration. Digital Object Identifier /TIT a cluster cooerate by exchanging messages over an orthogonal cooeration channel which requires some fraction of the available network ower and bandwidth. To cature the cost of cooeration, we lace a system-wide ower constraint on the network, and examine different bandwidth allocation assumtions for the data and cooeration channels. The notion of cooerative communication has been studied in several recent works. In [2], [3], the authors showed that cooeration enlarges the achievable rate region in a channel with two cooerative transmitters and a single receiver. Under a similar channel model, a nonorthogonal cooerative transmission scheme was resented in [4]. In [5], the transmitters forward arity bits of the detected symbols, instead of the entire message, to achieve cooeration diversity. A channel with two cooerative transmitters and two noncooerative receivers was considered in terms of diversity for fading channels in [6]. It was shown that orthogonal cooerative rotocols can achieve full satial diversity order. A similar channel configuration without fading was analyzed in [7] with dirty-aer coding transmitter cooeration. Achievable rate regions and caacity bounds for channels with transmitter and/or receiver cooeration were also resented in [8] [15]. In this work, we examine the imrovement in sum caacity from transmitter cooeration, receiver cooeration, and transmitterand-receiver cooeration. For transmitter cooeration, we consider dirty-aer coding (DPC), which is caacity-achieving for multile-antenna Gaussian broadcast channels [16]. For receiver cooeration, we consider Wyner Ziv comress-and-forward, which in relay channels is shown to be near-otimal when the cooerating node is close to the receiver [12], [13]. The DPC transmitter cooeration scheme was resented in [17], [18], with amlify-and-forward receiver cooeration in [17]. Our work differs from revious research in this area in that i) we consider receiver cooeration together with transmitter cooeration, and ii) we characterize the cooeration cost in terms of ower as well as bandwidth allocation in the network. The remainder of the corresondence is organized as follows. Section II resents the system model. In Section III, we consider the gain in caacity from transmitter cooeration, receiver cooeration, and transmitter-and-receiver cooeration. Numerical results of the cooeration rates, in comarison to corresonding multile-antenna channel caacity uer bounds, are resented in Section IV. Section V concludes the corresondence. II. SYSTEM MODEL Consider an ad hoc network with two clustered transmitters and two clustered receivers as shown in Fig. 1. We assume the nodes within a cluster are close together, but the distance between the transmitter and receiver clusters is large. The channel gains are denoted by h 1 ;...;h 4 2. To gain intuition on the otential benefits of cooeration, we consider a simle model where the channels exerience quasi-static hase fading [12]: the channels have unit magnitude with indeendent and identically distributed (i.i.d.) random hase uniform between 0 and 2. Hence, h i = e j ;i = 1;...; 4, where i U[0; 2] and i is fixed after its realization. We assume all nodes have erfect CSI, and the transmitters are able to adat to the realization of the channels. There are three orthogonal communication channels: the data channel between the transmitter and receiver clusters, the cooeration channel between the transmitters, and the cooeration channel between the receivers. In the data channel, Transmitter 1 wishes to send to Receiver 1 at rate R 1, and likewise Transmitter 2 to Receiver 2 at rate R 2. In this corresondence, we investigate the caacity imrovement in the sum rate R 1 + R 2 from cooeration. Let x [x 1 x 2] T /$ IEEE

2 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 10, OCTOBER Moreover, we consider two scenarios on the allocation of bandwidth between the data channel and the cooeration channels: Under bandwidth assumtion 1), we assume dedicated orthogonal channels for cooeration, and each channel has a bandwidth of 1 Hz (i.e., B t = B = B r =1). Under bandwidth assumtion 2), however, there is a single 1-Hz channel to be divided into three different bands to imlement the cooerative schemes. We thus allocate B t;b and B r such that B t + B + B r =1. Bandwidth assumtion 1) is alicable when the shortrange cooeration communications take lace in searate bands which may be satially reused across all cooerating nodes in the system, and hence the bandwidth cost for a articular cooerating air can be neglected. In contrast, bandwidth assumtion 2) is alicable when satial reuse is not considered. III. CAPACITY GAIN FROM COOPERATION Fig. 1. System model of a network with two clustered transmitters and two clustered receivers. denote the transmit signals, and y [y 1 y 2] T 2 2 denote the corresonding received signals. In matrix form, the data channel can be written as y = Hx + n 1 n 2 ; H h 1 h 2 h 3 h 4 (1) where n 1 ;n 2 CN(0; 1) are i.i.d. zero-mean circularly symmetric comlex Gaussian (ZMCSCG) white noise with unit variance. Let B denote the bandwidth of the data channel, and P 1 E[jx 1 j 2 ];P 2 E[jx 2 j 2 ] denote the transmission ower of Transmitter 1, Transmitter 2, resectively; the exectation is taken over reeated channel uses. There is also a static, additive white Gaussian noise (AWGN) cooeration channel between the two transmitters with channel gain G. As we assume the cooerating nodes are close together, the case of interest is when G is large. We assume the two transmitters can simultaneously transmit and receive on this full-dulex cooeration channel. Let x 0 1;x be the transmit signals, and y1;y the received signals, then the cooeration channel is described by y 0 1 = Gx n 3; y 0 2 = Gx n 4 (2) where n 3;n 4 CN(0; 1) are i.i.d. unit-variance ZMCSCG noise. Let B t denote the transmitter cooeration channel bandwidth, and P t E[jx 0 1j 2 + jx 0 2j 2 ] denote the cooeration transmission ower. Between the two receivers, there is an analogous full-dulex static AWGN cooeration channel, with channel gain equal to G. Let x 00 1 ;x be the signals sent on this channel, and y 00 1 ;y be the received signals. The receiver cooeration channel is then defined by y 00 1 = Gx n 5 ; y 00 2 = Gx n 6 (3) where n 5 ;n 6 CN(0; 1) are again i.i.d. unit-variance ZMCSCG noise. Let B r denote the receiver cooeration channel bandwidth, and P r E[jx 00 1j 2 + jx 00 2 j 2 ] denote the receiver cooeration transmission ower. To cature the system-wide cost of transmitter and receiver cooeration, we consider a total network ower constraint P on the data and cooeration transmissions P 1 + P 2 + P t + P r P: (4) A. Transmitter Cooeration We first consider transmitter cooeration in our network model assuming noncooerating receivers (i.e., P r = 0;B r = 0). In the transmitter cooeration scheme, the transmitters first fully exchange their intended messages over the orthogonal cooeration channel, after which the network becomes equivalent to a multile-antenna broadcast channel (BC) with a two-antenna transmitter where f 1 ;f 2 are the rows of H y 1 = f 1 x + n 1 ; y 2 = f 2 x + n 2 (5) f 1 [h 1 h 2 ]; f 2 [h 3 h 4 ]: (6) The transmitters then jointly encode both messages using DPC, which is caacity-achieving for the multile-antenna Gaussian BC [16]. Causality is not violated since we can offset the transmitter cooeration and DPC communication by one block. The sum caacity achieved by DPC in the multile-antenna BC is equal to the sum caacity of its dual multile-access channel (MAC) [19], [20] R DPC = B log I + f H 1 (P 1=B)f 1 + f H 2 (P 2=B)f 2 (7) = B log(1 + 2(P 1 + P 2 )=B +2P 1 P 2 =B 2 ) (8) where log is base 2 and 1 0 cos( ). Note that R DPC is symmetric and concave in P 1;P 2, thus it is maximized at P 3 1 = P 3 2 =(P 0 P t )=2. By symmetry each transmitter uses ower P t=2 to exchange messages in the cooeration channel, which suorts the cooeration sum rate R t =2B t log(1 + GP t=(2b t)): (9) To ensure each transmitter reliably decodes the other s message, we need R t R DPC ; hence, the transmitter cooeration sum rate is R TX = max min(r t;r DPC ): (10) B ;B;0P P Note that R t is increasing in P t while for R DPC it is decreasing, the otimal Pt 3 is thus achieved at R t = R DPC. Under bandwidth assumtion 1) with B t = B =1, the otimal ower allocation is P 3 t = 2( D 0 G 0 P 0 2) ; if G 2 6=2 G (11) P (P +4) 2(G + P +2) ; else where D 4(G +1)+G 2 (2P +1)+GP (2 + GP=2). Under bandwidth assumtion 2) with B t + B =1;Pt 3 is found by equating R t and R DPC for given B t;b, which is numerically comuted as it involves solving equations with non-integer owers. The otimal bandwidth allocation Bt 3 ;B 3 are found through numerical one-dimensional otimization [21].

3 3824 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 10, OCTOBER 2007 B. Receiver Cooeration Next we consider receiver cooeration in our network model without transmitter cooeration (i.e., P t =0;B t =0). When the receivers cooerate, there is no advantage in using a decode-and-forward scheme since, due to channel symmetry, each receiver decodes just as well as its cooerating node does. Instead, each receiver emloys comressand-forward to send a comressed reresentation of its own observation to the other receiver through the orthogonal cooeration channel. The comression of the undecoded signal is realized using Wyner Ziv source coding [22], which exloits as side information the correlation between the observed signals of the receivers. Comress-and-forward is shown to be near-otimal when the cooerating node is close to the receiver in relay channels [12], [13]. Suose Receiver 2 sends its observation to Receiver 1 via comress-and-forward over the cooeration channel. Then Receiver 1 is equivalent to a two-antenna receiver that observes the signals [y 1 y 2 + z 2 ] T, where z 2 CN(0; ^N 2 ) is the comression noise indeendent from y 1 ;y 2. The variance of the comression noise is given in [8], [12] ^N 2 = H(6 x =B)HH + f 2 (6 x=b)f H 2 + f 1 (6 x=b)f H (2 R =B 0 1) f 1 (6 x=b)f H 1 +1 (12) where 6 x E[xx ] is the covariance matrix of the transmit signals, and R r2 is the rate at which Receiver 2 comresses its observation with Wyner Ziv source coding. Likewise, Receiver 1 follows similar comress-and-forward rocedures to send its observation to Receiver 2 at rate R r1 with comression noise z 1 that has variance ^N 1 given by ^N 1 = H(6 x =B)H H + f 2 (6 x =B)f H 2 + f 1 (6 x =B)f H 1 +1 (2 R =B 0 1) f 2 (6 x =B)f H 2 +1 : (13) In the absence of transmitter cooeration (i.e., 6 x is diagonal), note that the comression noise variance is symmetric in Receiver 1 and Receiver 2. Suose Receiver 1 and Receiver 2 use ower P r 1 ;Pr 2, resectively, to erform comress-and-forward, then the Wyner Ziv source coding rate is given by the caacity of the receiver cooeration channel R ri = B r log(1 + GP ri =B r )); i =1; 2: (14) When each receiver has a noisy version of the other s signal, the network is equivalent to a multile-antenna interference channel where each receiver has two antennas with ~y 1 = ~ h 1 x 1 + ~ h 2 x 2 +[n 1 ~n 2 ] T (15) ~y 2 = ~ h 3x 1 + ~ h 4x 2 +[~n 1 n 2] T (16) ~h 1 [h 1 2 h 3 ] T ; ~ h2 [h 2 2 h 4 ] T (17) ~h 3 [ 1h 1 h 3] T ; ~ h4 [ 1h 2 h 4] T (18) ~y i for i =1; 2 is the aggregate signal from recetion and cooeration at Receiver i; ~n i i.i.d. CN(0; 1), and i 1=(1 + ^N i ) is the degradation in antenna gain due to the comression noise. We assume equal ower allocation P r1 = P r2 = P r=2 between the receivers, which results in the symmetric comress noise variance ^N 1 = ^N 2 ^N 2P 1 P 2 =B 2 +2(P 1 + P 2 )=B +1 = ([1 + GP r =(2B r )] B =B 0 1)((P 1 + P 2 )=B +1) : (19) Equal ower allocation achieves the saddle oint that satisfies the strong interference condition [23], under which each receiver decodes the messages from both transmitters and symmetric allocation of the receiver cooeration ower is otimal. The sum caacity of the interference channel is R IC =min B log I + ~ h 1 (P 1 =B) ~ h H 1 + ~ h 2 (P 2 =B) ~ h H 2 ; B log I + ~ h 3 (P 1 =B) ~ h H 3 + ~ h 4 (P 2 =B) ~ h H 4 (20) = B log(1 + (1 + )(P 1 + P 2 )=B +2P 1 P 2 =B 2 ) (21) where 1 = 2. The interference channel sum caacity is symmetric in P 1 ;P 2, but not concave. As R IC does not have a structure that lends readily to analytical maximization, the receiver cooeration sum rate is found through numerical exhaustive search over the ower and bandwidth allocation variables R RX =max B;B R IC; subject to: P r P 0 P 1 0 P 2 : (22) C. Transmitter and Receiver Cooeration The cooeration schemes described in the revious sections can be combined by having the transmitters exchange their messages and then erform DPC, while the receivers cooerate using comress-and-forward. Let C tx (G; P t ;B t ) denote the rate region suorted by the transmitter cooeration channel, with (R 1 ;R 2 ) 2C tx (G; P t ;B t ) iff (2 R =B 0 1)B t =G +(2 R =B 0 1)B t =G P t (23) which follows from the caacity of AWGN channels. Suose Receiver 1 uses ower P r1 to comress-and-forward to Receiver 2 with comression noise z 1 2CN(0; ^N 1 ), and in the oosite direction Receiver 2 uses ower P r2 with comression noise z 2 2 CN(0; ^N 2), where ^N 2 ; ^N 1 are as given in (12), (13). When both transmitters know the intended transmit messages, and each receiver has a noisy version of the other s signal, the network is equivalent to a multile-antenna BC with a two-antenna transmitter and two two-antenna receivers ~y 1 = ~ H 1 x +[n 1 ~n 2] T (24) where ~y 2 = ~ H 2 x +[~n 1 n 2 ] T (25) ~H 1 h 1 h 2 2 h 3 2 h 4 ; 2 1=(1 + ^N 2) (26) ~H 2 1h 1 1h 2 h 3 h 4 ; 1 1=(1 + ^N 1 ): (27) Suose the transmit signals intended for Receiver 1, Receiver 2 have covariance matrices 6 x1 ; 6 x2, resectively. Note that 6 x = 6 x1 + 6 x2 since DPC yields statistically indeendent transmit signals. Let DPC encode order (1) denote encoding for Receiver 1 first, then for Receiver 2; the corresonding DPC rates are R (1) 1;DPC = B log I + H ~ 1(6 x=b) H ~ H 1 I + ~ H 1 (6 x2 =B) ~ H H 1 (28) R (1) 2;DPC = B log I + ~ H 2 (6 x2 =B) ~ H H 2 : (29) Under encode order (2), when DPC encoding is erformed for Receiver 2 first followed by Receiver 1, the rates are R (2) 1;DPC = B log I + ~ H 1 (6 x1 =B) ~ H H 1 (30) R (2) 2;DPC = B log I + H ~ 2 (6 x =B) H ~ H 2 I + ~ H 2 (6 x1 =B) ~ H H 2 : (31)

4 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 10, OCTOBER The transmitter-and-receiver cooeration sum rate is given by the following otimization roblem: R TX0RX = max B ;B;B R1 + R2 (32) subject to: (R 1 ;R 2 ) 2C tx (G; P t ;B t ) (33) (R 1 ;R 2 ) 2 R (i) 1;DPC ;R(i) 2;DPC ;i=1; 2 (34) Tr(6 x1 +6 x2) P 0 P t 0 P r1 0 Pr2: (35) In general, the ower and bandwidth allocated for transmitter cooeration and receiver cooeration need to be otimized jointly. However, since the search sace is large, we consider a subotimal allocation scheme. We assume a fixed comression noise target ( ^N 1; ^N 2) that is suorted by receiver cooeration, with which the equivalent multile-antenna BC matrices are ~ H 1 ; ~ H 2 as given in (26), (27). Then we find the otimal transmitter inut distributions 6 3 x1; 6 3 x2 that maximize the DPC BC sum rate for ~ H 1 ; ~ H 2 using the sum ower iterative water-filling algorithm [24]. In the end, we verify through bisection search that the total ower required to achieve the DPC rates and to suort ( ^N 1 ; ^N 2 ) is feasible under the network ower constraint. The transmitter-and-receiver cooeration sum rate is then found through numerical exhaustive search over ( ^N 1; ^N 2) and the bandwidth allocation variables. In the numerical results, the subotimal allocation scheme is able to achieve rates that aroach the multile-inut multile-outut (MIMO) caacity uer bound as G increases. Fig. 2. Cooeration sum rates under bandwidth assumtion 1) at P = 0 db. IV. NUMERICAL RESULTS In this section, we resent the caacity gain achieved by the cooeration schemes at different signal-to-noise ratios (SNRs) under different bandwidth assumtions. The cooeration rates are comared against the baseline of noncooerative sum caacity. With neither transmitter nor receiver cooeration, the network is a Gaussian interference channel under strong interference, and the noncooerative sum caacity is 2 2 C NC =min log(1 + jh 1j P1 + jh 2j P2); log(1 + jh 3j 2 P1 + jh 4j 2 P2) (36) = log(1 + P ): (37) The cooeration rates are also comared to the caacity of corresonding multile-antenna channels as if the cooerating nodes were colocated and connected via a wire. With such colocated transmitters, the channel becomes a multile-antenna BC with a two-antenna transmitter. The BC sum caacity is given by the sum caacity of its dual MAC C BC = max log I + f H 1 P 1f 1 + f H 2 P 2f 2 (38) P +P P = log(1 + 2P + P 2 =2) (39) where the last equality follows from the sum caacity being symmetric and concave in P 1 ;P 2 : thus, it is maximized at P1 3 = P2 3 = P=2. With similarly colocated receivers, the channel becomes a multile-antenna MAC with a two-antenna receiver, the sum caacity of which is given by C MAC = max log I + h 1 P 1 h1 H + h 2 P 2 h2 H (40) P +P P = log(1 + 2P + P 2 =2); (41) where h 1;h 2 are the columns of H h 1 [h 1 h 3 ] T ; h 2 [h 2 h 4 ] T (42) and (41) follows again from the symmetry and concavity of P 1 ;P 2. Note that the MAC in (40) is not the dual channel of the BC in (38); Fig. 3. Cooeration sum rates under bandwidth assumtion 1) at P = 10 db. nonetheless, they evaluate to the same sum caacity: C BC = C MAC. Last, with colocated transmitters and colocated receivers, the channel becomes a MIMO channel where the transmitter and the receiver each have two antennas. The caacity of the MIMO channel is C MIMO = max log ji + H6 x H H j (43) Tr(6 )P where the otimal inut covariance matrix 6x 3 is found by water-filling over the eigenvalues of the channel [25]. Figs. 2 and 3 show the cooeration rates at SNRs of P = 0 and 10 db, resectively, under bandwidth assumtion 1) where the cooeration channels occuy searate dedicated bands. Figs. 4 and 5 show the cooeration rates at 0 and 10 db under bandwidth assumtion 2) where the network bandwidth is allocated among the data and cooeration channels. The exected rates are comuted via Monte Carlo simulation over random channel realizations. Under bandwidth assumtion 2), as the network bandwidth needs to be divided among the data

5 3826 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 10, OCTOBER 2007 bounded by C BC and C MAC, butr TX-RX continues to imrove and aroaches the MIMO channel caacity C MIMO. As SNR increases, however, the additional caacity imrovement from transmitter-and-receiver cooeration over transmitter-only cooeration becomes insignificant, as C BC tends to C MIMO in the limit of high SNR [26]. Fig. 4. Cooeration sum rates under bandwidth assumtion 2) at P = 0 db. V. CONCLUSION We have studied the caacity imrovement in the sum rate from DPC transmitter cooeration, Wyner Ziv comress-and-forward receiver cooeration, as well as transmitter-and-receiver cooeration when the cooerating nodes form a cluster in a two-transmitter, two-receiver network with hase fading and full channel state information available at all terminals. To account for the cost of cooeration, we imosed a system-wide transmission ower constraint, and considered the allocation of ower and bandwidth among the data and cooeration channels. It was shown that transmitter cooeration outerforms receiver cooeration and imroves caacity over noncooerative transmission under most oerating conditions when the cooeration channel is strong. However, when the cooeration channel is very weak, it becomes a bottleneck and transmitter cooeration undererforms noncooerative transmission; in this case, receiver cooeration, which always erforms at least as well as noncooeration, is more advantageous. Transmitter-and-receiver cooeration offers sizable additional caacity gain over transmitter-only cooeration at low SNR, whereas at high SNR, transmitter cooeration alone catures most of the cooerative caacity imrovement. We considered a simle model where the channels between the transmitter and receiver clusters are under hase fading to gain intuition on the otential benefits of cooeration. When the channels are under Rayleigh fading, for examle, ower and bandwidth allocation become less tractable since we cannot exloit the symmetry in the channels; however, the DPC and comress-and-forward cooeration schemes are still alicable and we exect comarable cooerative caacity gains can be realized. We assumed erfect CSI is available at all terminals; the system model is alicable in slow-fading scenarios when the channels can be tracked accurately. The CSI assumtion is critical: without CSI we cannot erform DPC or Wyner Ziv comression effectively and we exect the benefits of cooeration to be considerably diminished. Fig. 5. Cooeration sum rates under bandwidth assumtion 2) at P = 10 db. and cooeration channels, it takes a higher cooeration channel gain G to achieve cooeration rates comarable to those under bandwidth assumtion 1); nevertheless, under both bandwidth assumtions the relative erformance of transmitter and receiver cooeration follows similar trends. When G is small, the transmitter cooeration rate is imaired by the rovision that each transmitter decodes the message of the other, which becomes a erformance burden under a weak cooeration channel. Receiver cooeration, on the other hand, always erforms better than or as well as noncooerative transmission, since the comress-and-forward rates adat to the channel conditions. When G is higher than aroximately 5 db, however, the receiver cooeration rate begins to trail behind that of transmitter cooeration. When G is large, both R TX and R RX aroach the multile-antenna channel caacity C BC and C MAC. 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Berry, Member, IEEE Abstract In cooerative relaying, multile nodes cooerate to forward a acket within a network. To date, such schemes have been rimarily investigated at the hysical layer with the focus on communication of a single end-to-end flow. This aer considers cooerative relay networks with multile stochastically varying flows, which may be queued within the network. Throughut otimal network control olicies are studied that take into account queue dynamics to jointly otimize routing, scheduling and resource allocation. To this end, a generalization of the Maximum Differential Backlog algorithm is given, which takes into account the cooerative gains in the network. Several structural characteristics of this olicy are discussed for the secial case of arallel relay networks. Index Terms Backressure algorithm, cooerative relaying, wireless networks, wireless resource allocation. I. INTRODUCTION Given stochastically varying traffic, there is a growing body of work on throughut otimal control schemes for wireless networks that jointly address issues such as routing, scheduling and hysical-layer resource allocation, e.g., [1] [6]. By throughut otimal we mean that a control scheme stabilizes all the queues within the network whenever it is ossible to do so. In other words, such a scheme stabilizes the network for any rate in the network s stability region. Many of these schemes utilize some version of a maximum differential backlog (MDB) olicy (also sometimes called the Backressure Algorithm) [4], which has the desirable roerty of requiring no a riori knowledge of the traffic statistics. A feature of all the above models is that each acket is forwarded along a single route of oint-to-oint links. At any time a acket resides at a single location in the network, and the resources needed for the next transmission do not deend on the revious transmissions of the acket. Recently, there has been much interest in various cooerative relaying techniques (e.g., [7] [12]) that do not satisfy these assumtions. With such techniques, multile nodes cooerate to relay a acket. For examle, consider the four node arallel relay network from [7], in Fig. 1. Suose that node 1 has traffic to send to node 4. The arrows in Fig. 1 indicate the feasible links for this traffic using traditional oint-to-oint forwarding. 1 If node 1 broadcasts the same acket to both nodes 2 and 3, then these nodes can cooeratively forward this acket to node 4 by, for examle, forming a distributed antenna array. In /$ IEEE Manuscrit received Setember 2, 2006; revised May 13, This research was suorted in art by NSF under Grants CCR and CCR , by ARO under Grant DAAD , and by DARPA under Grant W911NF The material in this corresondence was resented in art at the IEEE International Symosium on Information Theory, Nice, France, June E. Yeh is with the Deartment of Electrical Engineering, Yale University, New Haven, CT USA ( edmund.yeh@yale.edu). R. Berry is with the Deartment of Electrical Engineering and Comuter Science (EECS), Northwestern University, Evanston, IL USA ( rberry@eecs.northwestern.edu). Communicated by M. Franceschetti, Guest Editor for the Secial Issue on Relaying and Cooeration. Digital Object Identifier /TIT For simlicity, we assume that node 1 cannot directly transmit to node 4, e.g., the direct link may be of too oor a quality to be feasible.