towards an information theory of large networks: an ...towards an information theory of large...

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 8, AUGUST 2003 1877

Towards an Information Theory of Large Networks:An Achievable Rate RegionPiyush Gupta, Member, IEEE,and P. R. Kumar, Fellow, IEEE

Abstract—We study communication networks of arbitrary sizeand topology and communicating over a general vector discretememoryless channel (DMC). We propose an information-theoreticconstructive scheme for obtaining an achievable rate region in suchnetworks. Many well-known capacity-defining achievable rate re-gions can be derived as special cases of the proposed scheme. Afew such examples are the physically degraded and reversely de-graded relay channels, the Gaussian multiple-access channel, andthe Gaussian broadcast channel. The proposed scheme also leadsto inner bounds for the multicast and allcast capacities.

Applying the proposed scheme to a specific wireless network ofnodes located in a region of unit area, we show that atransport ca-pacityof �( ) bit-meters per second (bit-meters/s) is feasible in acertain family of networks, as compared to the best possible trans-port capacity of �( ) bit-meters/s in [16], where the receivercapabilities were limited. Even though the improvement is shownfor a specific class of networks, a clear implication is that designingand employing more sophisticated multiuser coding schemes canprovide sizable gains in at least some large wireless networks.

Index Terms—Discrete memoryless channels (DMCs), Gaussianchannels, multiuser communications, network information theory,relay networks, transport capacity, wireless networks.

I. INTRODUCTION

T HE last few decades have seen a tremendous growth inwireless communication. The most popular examples are

cellular voice and data networks and satellite communicationsystems. These and other similar applications have moti-vated researchers to extend Shannon’s information theory for asingle-user channel to some that involve communication amongmultiple users. A few such examples are the multiple-accesschannel, the broadcast channel, and the interference channel.The exact capacity region is, however, known in the mostgeneral case only for the multiple-access channel, while thebroadcast capacity region is known only for few specific chan-nels, like the additive white Gaussian noise (AWGN) channel

Manuscript received September 11, 2001; revised April 25, 2003. This paperis based on work supported in part by the U.S. Army Research Office underContracts DAAD19-00-1-0466 and DAAD19-01010-465, DARPA under Con-tracts N00014-01-1-0576 and F33615-01-C-1905, and the Office of Naval Re-search under Contract N00014-99-1-0696. Any opinions, findings, and conclu-sions are those of the authors and do not necessarily reflect the views of theabove agencies.The material in this paper was presented in part at the IEEEInternational Symposium on Information Theory, Washington, DC, June 2001,and at the IMA Workshop on Wireless Networks, Minneapolis, MN, August2001.

P. Gupta is with Bell Laboratories, Lucent Technologies, Murray Hill, NJ07974 USA (e-mail: [email protected]).

P. R. Kumar is with the Department of Electrical and Computer Engineering,and the Coordinated Science Laboratory, University of Illinois at Urbana-Cham-paign, Urbana, IL 61801 USA (e-mail: [email protected]).

Communicated by P. Narayan, Associate Editor for Shannon Theory.Digital Object Identifier 10.1109/TIT.2003.814480

and the deterministic channel [7], and even fewer results areavailable for the interference channel [23]. It should be furthernoted that the above applications as well as the channel modelsused for analyzing them involve mainly single-hop wirelesscommunication.

Recently, there has been considerable interest in anotherclass of wireless networks, namely,ad hocwireless networks ormultihop wireless networks. These networks consist of a groupof nodes that communicate with each other over a wirelesschannel without any centralized control. Examples of suchnetworks are in coordinating an emergency rescue operation,networking mobile users of portable-yet-powerful computingdevices (laptops, PDAs, smart phones) on a campus using,for instance, IEEE 802.11 wireless local-area network (LAN)technology [20], sensor networks [3], automated transportationsystems [10],Bluetooth [17], and HomeRF [24]. Ad hocwireless networks differ from conventional cellular networks inthat all links are wireless and there is no centralized control. Asevery node may not be in direct communication range of everyother node, nodes inad hocnetworks cooperate in routing eachother’s data packets. Lack of centralized control and possiblenode mobility give rise to a number of challenging design andperformance evaluation issues in such wireless networks, manyof which have no counterparts in the cellular networks or in thewired networks, like the Internet.

An important performance analysis issue is to determinewhat the traffic-carrying capacity of such multihop wirelessnetworks is. An attempt to address this issue was made in [16]under certain models of communications motivated by currenttechnology. For this, an important notion of thetransportcapacityof a network was introduced. Consider a network of

nodes. Suppose the network hassource–destination pairs,communicating independent information over the network.Let be a vector of feasible rates for thesource–destination pairs and be the distance between the

th source and its destination. Then, , wherethe supremum is taken over all feasible rate vectors, is definedas thetransport capacityof the network. In other words, thetransport capacity of a network is the maximum bit–distanceproduct that can be transported by the entire network per secondand is measured in bit-meters per second (bit-meters/s), wherea bit transported over a distance of 1 m toward its destination iscounted as 1 bit-meter. It was shown in [16], that under somemodels of noninterference motivated by current technology,the transport capacity of a network of nodes located in aregion of unit area is bit-meters/s, even assuming thenode locations, traffic patterns, and the range/power of eachtransmission, are all optimally chosen. (A specific network

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1878 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 8, AUGUST 2003

construction that achieves a transport capacity ofbit-meters/s was also given.) If the network’s transport capacitywere to be equitably divided among all nodes, then eachnode could only obtain bit-meters/s. An implication ofthis scaling law is that anad hocwireless network furnishes anaverage throughput to each user for nonvanishingly far awaydestinations that diminishes to zero as the number of nodesincreases in the network. This suggests that only smalladhoc networks or networks supporting mainly nearest neighborcommunications are feasible with current technology.

A natural question is whether the above constriction on theaverage throughput provided to each node with the size of thenetwork is a limitation of current technology, or whether itcan be overcome by employing more sophisticated multiusercoding schemes. In other words, what are the fundamentalinformation-theoretic limits on the traffic-carrying capacity ofmultihop wireless networks? This paper aims at taking a stepin addressing this challenging question. As mentioned earlier,the exact information-theoretic capacity region is not knownfor even simple networks like a general broadcast channel or aninterference channel. Thus, here we will only attempt to obtainan achievable rate region in general networks. Even though theobtained rate region will not be capacity defining in general, itwill nevertheless include many well-known capacity-achievingrate regions as special cases. A few such examples are thephysically degraded and reversely degraded relay channels, theGaussian multiple-access channel, and the Gaussian broadcastchannel. Furthermore, the proposed scheme will achieve atransport capacity of bit-meters/s in a specific wirelessnetwork of nodes located in a unit-area region, which will bemotivated by the multiple-input multiple-output architecture.

Consider a set of nodes . Letdenote the signal transmitted by nodein the th transmis-

sion, and be the signal received by nodein the th trans-mission. Let the range space of be , and the range spaceof be . Nodes in communicate with each other overa general -user vector discrete memoryless channel (V-DMC)that is specified by , thejoint probability that is receivedin the th transmission given thatwas transmitted.

Suppose has source–destination pairsWe propose a constructive scheme to deter-

mine an information-theoretic achievable rate regionin . As a communication

strategy, consider afeedforward flowgraph for each– pair , in . (One can subsequently

optimize over the choice of all such feedforward flowgraphs.)In each such flowgraph , nodes in are groupedinto disjoint level sets as follows:

, , and, for , nodes inwill receive information only from nodes in levels for

, while they will send information only to nodes inlevels for . Thus, will onlyhave arcs of the form , for , .Finally, let .

To illustrate the main idea, let us first consider the case of, i.e., when has a single source–destination pair .

Let be a flowgraph for as above. Let denotethe signal transmitted by , th node in level , in oneusage of the channel, and

where denotes the number of nodes in level . Also, letdenote the signal received by . Our main result is the

following.

Main Result: Rate is achievable for source–des-tination pair in a network of nodes communi-cating over a V-DMC whenever there exists a flowgraph

, , and a joint probability distributioncorresponding to , such

that, for some : ,satisfies

(1)

for . Above

and is theconditional mutual informationbetweenand given .

Later, in Theorem 3.1, we will apply the preceding resultto obtain achievable rates for the AWGN channels with andwithout fading. We will also extend the procedure in Section IVto obtain an achievable rate region for a network with mul-tiple source–destination pairs communicating over the Gaussianchannels.

The basic idea behind our main result, which will be madeprecise in the sequel, is as follows. As nodes in cannotrelay any information from to , they constantly transmitsome symbol that facilitates transmissions by nodes in

. Now if in each levelevery node can receive informa-tion at rate from the component of its received signal

that is correlated with ,the signal vectors transmitted by the nodes in levels

, then level will be able tosimultaneously send common information at rate to everynode in each , , whenever the followingare true.

• In level , every node canreceive information at rate from the component of itsreceived signal that is correlated with , thesignal vector sent by the nodes in (first term withinminimum in (1)).

• In each level , every nodecan obtain additional information

GUPTA AND KUMAR: TOWARDS AN INFORMATION THEORY OF LARGE NETWORKS 1879

Fig. 1. A two-level relay channel.

at rate from the component of its receivedsignal that is correlated with (minimum in thesecond term in (1)).

It should be noted that in the above scheme the informationsent by source at rate reaches not only destinationbutalso all the nodes in the intermediate levels .Hence, the scheme also leads to an inner bound on themulticastcapacityfrom source to a set of destination nodes, for any

.We next illustrate the main result on atwo-level relaychannel.

Example: Two-Level Relay Channel

Consider a network of four nodes: source, destination, and two relays: and . Consider a flowgraph

for such that , , , and; see Fig. 1. Let and denote, respectively,

the signal transmitted and the signal received by the node in. Then, a rate is achievable from to

with vanishingly small probability of error whenever there existrates , , and probability distribution suchthat the following hold (which again will be made precise in thesequel).

1) Relay can receive information at rate from thetransmission of source. This is achievable if

(2)

2) Relay can receive information at rate from relay, and an additional information at rate from

the transmission of source. To achieve this, employssuccessive cancellationdecoding. Thus, it first decodesinformation at rate from the signal component of itsreceived signal that is correlated to the signal trans-mitted by relay , , treating the uncorrelated compo-nent of the transmission ofas noise. It then cancels fromits received signal the component correlated to anddecodes additional information at rate fromthe remaining component of that is correlated with thesignal transmitted by, . This is feasible if

(3)

3) Destination can receive information at rate fromrelay , and additional information at rate

from relay and at rate from the transmis-sion of source . Again, employing successive cancella-tion decoding, this is achievable if

(4)

For each , collecting the bounds on rate in(2)–(4), we obtain the expression given by (1) for the two-levelrelay channel.

We next identify some special cases of our general model thathave been considered in the literature.

When has only three nodes—, , and one additional node—the resultant network is commonly referred to as arelay

channel, which was introduced in [22]. The capacities of twospecial types of relay channels—thephysically degradedandreversely degradedrelay channels—were obtained in [8]. Theachievable rate given by our result for a relay channel is exactlythe maximum of those in [8, Theorems 1 and 2]. Specifically,two flowgraphs are possible for a relay channel.

1) , , and . For a given jointdistribution , we have

Hence,

2) , , and . For andprobability distribution , we have

Choosing the maximum of the suprema of the above two over, and and , respectively, our main

result states that a rateis achievable in a relay channel when-ever

which is exactly the maximum of the achievable rates given in[8, Theorems 1 and 2]. Hence, our result is a generalization ofthe achievable rate for the one-relay channel in [8, Theorems 1and 2] to networks having more than one relay node.

A network having four nodes—, , and two relays: and—with being , is analyzed in [25]

when the network is communicating over the Gaussian channel.In this case, one of the achievable rates obtained there is a spe-cial case of our result.

Later, in Section IV, we will extend our main result to net-works having multiple source–destination pairs. We will also


then discuss how the capacity-defining achievable rate regionsof the Gaussian multiple-access and broadcast channels can beobtained as special cases of our constructive scheme.

The rest of the paper is organized as follows. In Section II,we obtain an achievable rate for a source–destination pair in anetwork of nodes communicating over a general vector discretememoryless channel. We apply this result in Section III to de-termine an achievable rate over the AWGN channels with andwithout fading. In Section IV, we extend the procedure given inthe previous section to obtain an achievable rate-vector regionfor networks with multiple source–destination pairs. We applythis result in Section V to arrive at a better achievable boundon the transport capacity of wireless networks than in [16]. InSection VI, we discuss a specific wireless network ofnodeslocated in a region of unit area in which the proposed schemeachieves a transport capacity of bit-meters/s.

II. A CHIEVABLE RATE FOR A SOURCE–DESTINATION PAIR

OVER V-DMC

Consider a set of nodes communicatingover a general V-DMC specified by

the joint probability that node receives , in thtransmission, , given that

was transmitted.Suppose that has a single source–destination pair .

In order to specify when a rate is achievable from sourcetodestination , we next provide a few definitions.

Definition 2.1: A -code for the network withsource–destination pair consists of the following ele-ments.

1) An index set of integers .

2) An encoding function for the source,yielding codewords , each oflength . The set of codewords is referred to as codebook

.

3) Encoding functions

s.t.

4) A decoding function .

The above definition, and those to follow, are straightforwardgeneralizations of the ones for the single-relay channel analyzedin [8].

If the message is sent, let

Fig. 2. A feedforward flowgraphG(s; d).

denote the conditional probability of error in the message de-coded by destination. Then, themaximal probability of error

for the -code is defined as

Therateof a -code is bits per transmission.

Definition 2.2: A rate is said to beachievable forsource–destination pair in if there exists a sequenceof -codes with such that the maximalprobability of error as .

Given the set of nodes with source–destination pair ,we construct afeedforward flowgraph as follows. As acommunication strategy, let us group nodes ininto disjointlevel sets such that the followinghold.

1) , .

2) For , nodes in will receive informationonly from nodes in levels for , whilethey will send information only to nodes in levelsfor . Thus, will only havearcs of the form , for .

3) .1

Fig. 2 illustrates such a feedforward flowgraph. Let be thenumber of nodes in for , and be thenumber of nodes in . We will use to indicate the thnode at level , according to some arbitrary but fixed num-bering of nodes in .

We will denote by the signal transmitted by nodein one usage of the channel, and by

the signal vector transmitted by all the nodes in. Similarly,let

1Nodes inL are those nodes inN which will not participate in “relaying”information froms to d (but they may “facilitate” the relaying process by con-stantly transmitting symbols independent ofx that “open” the channel for othertransmissions).


denote the signal vector transmitted by the nodes in. Let

Finally, let denote the signal received by node .Then an achievable rate in network of arbitrary size and

topology and having a single source–destination pair isas follows.

Theorem 2.1:A rate from source to destination in net-work of nodes communicating over a V-DMC is achievablewhenever

where, for some

satisfies

(5)

for .Proof: See Appendix II.

III. A CHIEVABLE RATE OVER GAUSSIAN CHANNELS

We next obtain an achievable rate for a single source–desti-nation pair in a network of nodes communicatingover the AWGN channels with and without fading.

For , let be the signal transmitted by nodeinthe th transmission. Then, the signal received by nodeat theend of the th transmission is given by

(6)

where is the channel gain from nodeto node and

is a sequence of independent and identically distributed (i.i.d.)Gaussian random variables with meanand variance , for

. An example of the channel gain term is to modelthe signal power path loss from nodeto , i.e.,

(7)

where is the distance between nodesand and is thesignal power path loss exponent. This is the physical model

studied, for instance, in [16]. For this model of the channel gain,, , and will all be real-valued random variables.

Here, we will also consider a more general model that, inaddition to the signal-power path loss, also incorporates channelvariation due to Rayleigh flat fading [6], [28]. Specifically, thechannel gain from node to node is given by

(8)

where : is an i.i.d. complex Gaussiancollection with .2 Furthermore, each use ofthe channel involves independent . Under this channelmodel, , , and will all be complex-valued randomvariables, with noise .

As is by now a standard practice [6], we will analyze theGaussian fading channel under two scenarios concerning theavailability of the channel state information at various nodes inthe network:

1) CSIR:For each , the channel gains :will be known to the receiving node.

2) CSIRT:The channel gains :will be known to each node in .3

Let denote the constraint on the average transmissionpower available to node, i.e., needs to satisfy

(9)

where the expectation is also over under CSIRT if nodevaries its transmit power with the channel state information.

For the above network channel settings, we next obtain anachievable rate from a given source nodeto a given destinationnode in . For this, recall the definitions of the flowgraph

and the associated variables from Section II. In addition,let and denote, respectively, thechannel gain and the distance from node to node .Also, for the sake of simplicity in notation, let us relabel thetransmission power constraint and the noise power at nodeas and , respectively.

An achievable rate from to in is obtained as follows.Each node assigns a fraction

of its power to aid level in transmittingits information, for each . This is done in a “co-herent” way so that the square roots of the powers received fromdifferent nodes in aid of the transmission of the same level addup at each node. Let denote the total “coherent”power received by node from the transmissions of level

as well as those aiding them (made precise in the sequel).Then, the following holds.

2A random vector � 2 is said to be complex Gaussian if(Re(�); Im(�)) 2 is Gaussian.### � N (��; ��) denotes that### isa complex circularly symmetric Gaussian random vector with mean�� andcovariance matrix��.

3Alternatively, one can consider a block-fading channel model where thefading coefficients remain fixed over a block of transmissions and take i.i.d.values from block to block [6], [18], [26]. If the length of the blocks over whichthe fading coefficients remain constant is sufficiently long, the above scenarioholds by learning the channel gains implicitly or explicitly.


Theorem 3.1:i) In network of nodes communicating over the AWGN

channel without fading given by (6) and (7), rateis achievablefrom source to destination whenever

(10)where, for some

satisfies

(11)

for . Above

(12)

and

(13)

ii) For the AWGN channel with fading given by (6) and (8)and the channel state information known only to the receivingnodes (CSIR), an achievable ratefrom to in is alsogiven by (10) and (11), except that now

(14)

and

(15)

where the expectation is taken over the variation in the respec-tive channel gain coefficients.

iii) For the AWGN channel with fading and CSIRT (i.e., thechannel state information is known to each node in ), a rate

is achievable from to whenever there exist a flowgraphand complex transmission gains

such that, for each ,

(16)

and, for some

satisfies (11) with

(17)

and

(18)

Proof: See Appendix III.

As stated earlier, in the above scheme, information fromsource at rate not only reaches destination but alsoreaches all the nodes in intermediate levels .Hence, the above scheme also leads to an achievable bound onthe multicast capacity. Nevertheless, it may sometimes be moreprofitable to send different information over different paths, orin general over different subnets. In the sequel, we will givean approach that will allow splitting the rate of informationtransfer from to over different flowgraphs between them.The approach will be a special case of a network with multiplesource–destination pairs, discussed next.

IV. NETWORKSWITH MULTIPLE SOURCE–DESTINATION PAIRS

Consider a set of nodes commu-nicating over the AWGN channel with or without fading, de-scribed in the previous section. Supposehas source–des-tination pairs . As for thesingle source–destination pair case, we next provide a few defi-nitions to specify when a rate vector for the source–destinationpairs in is said to be achievable.

Definition 4.1: A -code for con-sists of

1) index sets of integers ;

2) encoding functions :, where if ,

otherwise;


3) decoding functions : , ,where if , otherwise.

Corresponding to each source–destination pairis a message , which is assumed to be indepen-

dently and uniformly distributed over its range. Then the prob-ability of error that the message is not decoded correctly isgiven by

and the maximal probability of error for the source–destinationpair is given by

Definition 4.2: A rate vector issaid to be achievable if there exists a sequence of

-codes with, such that as .

Next, recall the definition of flowgraph from Sec-tion II. For each , let be a flowgraph forsource-destination pair . Suppose that haslevels . Let de-note the index of the level to which nodebelongs in ,where denotes that does not participate inrelaying information from to . Let denote the set of allsource–destination pairs for which nodedoes relay informa-tion, i.e.,

Then, an achievable rate-vector region infor source–destina-tion pairs is as follows.

Theorem 4.1:An achievable rate-vector regionin networkof nodes communicating over a Gaussian channel and havingsource–destination pairs

is given by the closure of the convex hull of the set of all ratevectors such that, for some flowgraphs

: , intermediate level rates

and random variable with probability distribution :, for some finite , the following constraints are satisfied

for each node :

for each and , and

for each and . Above, for each ,: are some

successive decoding orderings satisfying

and

for some transmission gains

with , for some power allocations

satisfying , or and

Proof: See Appendix IV.

It is worth noting that the achievable-rate regions given byTheorem 4.1 for the -user Gaussian multiple-access channeland the -user Gaussian broadcast channel are exactly the ca-pacity-defining achievable rate regions given for these channelsin [1], [21] as well as [4], [14], respectively.

We next extend Theorems 3.1 and 4.1 so as to allow dif-ferent information to be sent over different flowgraphs betweena source–destination pair . The basic idea is to colocatevirtual source–destination pairs at the same pair of nodesand

in the above scheme. We illustrate this with an example. Con-sider a network with a single source–destination pair .Suppose that sends information at rate over path, ormore generally, feedforward flowgraph, , .Then we replace by ,where and are located at and , respectively, for each

. Now the rate partition is achievablefor if belongs to the rate regiongiven by Theorem 4.1 forwith flowgraphs . Generalization ofthe approach for the case where there are multiple source–des-tination pairs is straightforward.

A limitation of the above approach is that it does not usenet-work coding[2]. That is, since intermediate nodes handle in-formation sent by different virtual source–destination pairs in-dependently, they do not perform coding of bits received overdifferent paths. It is shown in [2] that such coding strictly in-creases the achievable rate in multicasting, i.e., when the sameinformation is to be sent from a source to multiple destinations.Generalizing the network coding scheme of [2] for a network ofindependent DMCs to the more general channel model that weconsider is a challenging problem.

V. ACHIEVABLE BOUND ON TRANSPORTCAPACITY OF

WIRELESSNETWORKS

We next apply Theorem 4.1 to obtain an achievable bound onthe transport capacity of arbitrary wireless networks (cf. Sec-tion I). Even though the achievable bound is not in an explicitform, it nevertheless improves upon that given in [16] for net-works with limited receiver capabilities. In fact, in the next sec-tion we will construct a specific wireless network ofnodeslocated in a unit-area region where the proposed scheme will


achieve a transport capacity of bit-meters/s, as comparedto the best possible transport capacity of bit-meters/s in[16].

Consider nodes located in a disk of unit area. Letbe the location of node in , for . For notationalconvenience, let and .Also, let denote a rate vector in thenetwork, where denotes the average rate of informationtransfer (in bits per transmission) from nodeto node .

Suppose that the bandwidth available to the network is

hertz. Also, suppose that is the noise spectral densityat node . First consider the case when the channel has nofading. Let be the set of all achievable rate vectorsinTheorem 4.1 for the node-location vector, the power-con-straint vector , and the noise-power vector

. Now fix a rate vector

Employing by now standard characterization of transformingrates from bits per transmission to bits per second for aband-limited AWGN channel without fading (see, for instance,[9] and references therein), we obtain that bits persecond can be sent from nodeto node , for each ,with vanishingly small probability of error. Thus, the totalbit-meters transported by the network per second is

For the AWGN channel with fading, the above continues tohold except that, since inputs and outputs are now all complex,the rate from node to node in bits per second is given by

. Hence, we obtain the following result.

Theorem 5.1:The transport capacity of a wireless network ofnodes communicating over a Gaussian channel of bandwidthhertz is lower-bounded by

- (19)

where is the achievable rate region given by Theorem4.1 for the node-location vector with the spatial separationbetween nodesand being , the transmit-power constraintvector , and the noise-power vector

, and is (resp., ) when the channel hasfading (resp., no fading).

We next construct a specific wireless network ofnodeswhere a transport capacity of bit-meters/s is achieved.

VI. FEASIBILITY OF TRANSPORTCAPACITY OF

BIT-METERS/S

Our constructive example is motivated by the multipletransmit–receive antenna architecture [11], [12], [29], and thebasic idea is as follows. Consider a network ofnodes, where

source nodes (for notational simplicity, letbe even) areplaced “close” to the origin and destination nodes are placed“close” to ; see Fig. 3. Now since the source nodes areclose to each other, they can exchange among themselvesall the information that they have to send to their respective

Fig. 3. Placement of nodes.

destinations in a “short” time. Similarly, the destinationnodes will be able to exchange among themselves all theinformation that they will receive from the source nodesin a “short” time. Thus, source nodes act as transmitantennas and destination nodes act as receive antennas.Under appropriate conditions on the channel characteristicsand specific choices of flowgraphs and power allocations, weobtain that bits/s can be sent from the source nodes to thedestination nodes. Since the source–destination pairs aremeters apart, we obtain that bit-meters/s is feasible. Wemake the construction precise in the following.

Consider a set of nodes communicating over the AWGNchannel with fading given by (6) and (8). Here, we considerCSIRT, i.e., the channel gains :are known to each node in . Suppose the bandwidth availableto the network is hertz. Further, suppose the average poweravailable to each node in for transmission is and the noisepower at each node is. For the sake of brevity, we discuss herethe case where has a single source–destination pair ; theextension to the multiple source–destination case is straightfor-ward. The placement of nodes in is as given in Fig. 3. Sourcenode is placed at the origin, while destination nodeis placedat . Additionally, nodes are uniformly placed, to-gether with , on the circle centered at and of radius ,where

(20)

The remaining nodes are similarly placed on the circlecentered at and of radius .4 Now information fromsource is sent to destinationover a set of flowgraphs

, each having levels. Flowgraph has levelconsisting of all nodes close to and level havingonly node from among the nodes close to, for some arbitrarybut fixed ordering of the nodes; see Fig. 4. Letbe the rate atwhich destination receives information over . Thus, the cu-mulative rate of information flow from to is .

We next apply Theorem 4.1 to obtain the feasibility ofbits per transmission. Sinceand are 1 m apart, The-

orem 5.1 would then allow us to show that a transport capacityof bit-meters/s is feasible. In order to use The-orem 4.1 in the current setup, we treat information flow fromto

over different flowgraphs as information transfer betweencolocated virtual source–destination pairs (cf. Section IV). Withthis we next give specific choices of various parameters that willallow the rate-constraint conditions in Theorem 4.1 to hold for

and, thus, ; since Theorem 4.1 actuallyoptimizes over these parameters, the feasible rate it will obtain

4Note that nodes inN are located within the unit-area disk centered at( ; 0).


Fig. 4. Flowgraphs.

can only be better. We start by specifying how the transmis-sions are scheduled in the network. Letbe a random vari-able that takes values in with probability distribution:

and , for .Now, when , for , only the nodes in leveltransmit, while, for , only node transmits. Then,the rate-constraint conditions in Theorem 4.1 are satisfied forinformation flow at rate from to when the following hold.

1) Source can send information at rate to nodes in .This is achievable with high probability if

where is the channel gain from sourceto theth node in level . This is feasible if

(21)

2) Nodes in can simultaneously send independent infor-mation at rate to node in , for .To achieve this, nodes in performzero-forcing beam-forming [6]. Let denote the matrix of channel gainsfrom the nodes in to those in . Then, almost surely

has full rank, as the channel fading coefficients be-tween all pairs of nodes in and are statistically in-dependent. Thus, exists almost surely. Now, when

, the nodes in transmit the signal vector, where for the

zero-probability event that does not have full rank,while otherwise

where . Thenand thus,

which, by near symmetry, ensures that the constraint onthe average transmission power at the nodes inis sat-isfied. Furthermore, when , the signal vector re-ceived by the nodes in is

a.s.

Thus, rates are simultaneously feasiblefrom nodes in to node in , respectively, with highprobability if

(22)

Now, recall that and the entries of areindependent complex Gaussian with mean zero and vari-ance close to unity for large (since is given by (20),

, for large ). Thus,is nearly distributed as , which has mean

and variance (see[15, Theorem 3.3.28] and its generalization to complexvectors, for instance, in [19]). Furthermore, it is easy toshow that, for and any

Hence, (22) holds if

Therefore, rate is feasible from to with high prob-ability if

(23)

3) Finally, destination should be able to successfully de-code all the information that is being sent over flowgraphs

. This is achievable with high probability if

where the last step follows from (20). Thus, rateisfeasible with high probability if

(24)

Hence, from (21)–(24), we determine that a rate of bitsper transmission is achievable fromto . Since and are 1m apart, we obtain from Theorem 5.1 that a transport capacityof bit-meters/s is feasible.

The above improvement in the transport capacity is for a spe-cific class of networks. Whether such a favorable scaling lawfor the transport capacity holds for a network with randomly lo-cated nodes is a challenging open question.

VII. CONCLUSION

We have constructed an information-theoretic scheme forobtaining an achievable rate region in a network of arbitrary


size and topology. Many well-known capacity-defining achiev-able rate results can be derived as special cases of our scheme.A few such examples are the physically degraded and reverselydegraded relay channels, the Gaussian multiple-access channel,and the Gaussian broadcast channel. The proposed scheme alsoleads to a better achievable bound on the transport capacity ofwireless networks than previously obtained. In fact, the schemeallows a feasible transport capacity of bit-meters/s ina specific wireless network of nodes located in a region ofunit area, as compared to the best possible transport capacityof bit-meters/s shown earlier [16] for less sophisticatedreceiver operation. Even though the improvement is shownfor a specific class of networks, a clear implication is that onemay obtain sizable gains by designing and employing moresophisticated multiuser coding schemes in at least some largewireless networks. Quantifying the gains in a general networkis an important subject for future work.

APPENDIX ISOME DEFINITIONS AND BACKGROUND RESULTS

In this appendix, we recall some standard definitions and re-sults from the literature.

Let denote a finite collection of discreterandom variables with some fixed joint distribution

Let be a subset of . Consider indepen-dent copies of , . Let denote the vector

. Thus, for

Then, by the law of large numbers

where the convergence takes place simultaneously with proba-bility for all subsets, .

Next recall the definition of jointly -typical sequences [9]:The set of -typical -sequences

is defined by

Let denote the restriction ofto the coordinates of . For example, if

We recall the following well-known fact (see, for instance,[9]).

Lemma 1.1:For any , for sufficiently large ,

i) , .

ii) .

iii) , wheredenotes the cardinality of the set .

To bound the probability of error, we will use anotherwell-known result on the probability that conditionally inde-pendent sequences are jointly typical. Let andbe three subsets of . Letbe such that and are conditionally independentgiven but otherwise share the same pairwise marginalsof as shown below. Then the probability ofjoint typicality of is as follows (see [9]).

Lemma 1.2:Let

and

Then, for such that

We next discuss the proof of Theorem 2.1, which obtains anachievable rate in a network of nodes communicating over ageneral V-DMC and having a single source–destination pair.

APPENDIX IIPROOF OFTHEOREM 2.1

The proof is a generalization of the one given in [8] for thesingle-relay channel. As Theorem 2.1 deals with a network ofarbitrary size and topology, its proof discussed next is ratherinvolved. A prior study of the much simpler case in [8] may aidin understanding this proof.

The basic approach of the proof is similar to [8]. Relay nodesat a particular level in a given flowgraph performsuccessivecancellationto decode the information transmitted by the nodesin the previous levels, including the source node. They then em-ploy random conditional codebooksand random partitionstoencode their own information as well as aid the transmissionsby the nodes in the subsequent levels, in order to jointly relayinformation from the source node to the nodes in the levels evenfurther down, including the destination node. Complication inthe proof arises due tomultiple levels of relaying, with eachlevel possibly having multiple nodes, all of which need to per-form many levels of decoding and encoding jointly in order to


achieve the rate given in Theorem 2.1. This requires that therandom codebooks, the random partitions, and the encoding anddecoding procedures for various nodes in the network are care-fully chosen. We make this precise in the following.

Fix a flowgraph . For this flowgraph, fixand a probability mass function for

Consider a rate satisfying (5) for some

As in [8], we useblock Markov encodingto achieve rate .Consider blocks of transmission, each ofsymbols. A se-quence of messages

will be sent from source to destination in transmissions.As , for fixed and , the rate is arbitrarilyclose to .

Generation of Random Codebooks

In each -block , node in levelof uses the same random codebook, denoted by

, to determine the codeword to transmit in block, for. Now, all the random codebooks

employed by nodes in a particular level , denoted by

are generated according to a specificjoint conditional distribu-tion, as follows.

• First construct the codebooks for nodes in levelby generating at random i.i.d. -sequences

drawn according to

• Next construct the codebooks for as follows.Consider a node at . It selects a codeword to transmitthat depends not only on the index of its own level, de-noted by , but also on the index transmitted bythe higher level . Specifically, is obtained asfollows. For each , generate

conditionally independent-sequences, denotedby , , drawn ac-cording to

• Next, proceeding recursively, and supposing thathave been generated, we use the

following procedure to construct the codebooksfor . As at , a node at level selects acodeword to transmit that depends not only on the

index but also on the indexestransmitted by higher levels. This way the nodes at level

not only send the index of their own level but also“aid” the transmissions of the nodes in higher levels

. More specifically, let de-note the vector , then is obtainedas follows. For each

generate conditionally independent -sequences,, , drawn according to

Random Partitions

We next construct a nested set of random partitionsof . These nested random partitions

will allow us to recursively use the random binning argument,introduced in [27] and used in [8] for the single-relay channel,to show that is achievable.

Let . Having obtained

we generate a random partition

of by randomly and independently assigningto according to a uniform distribution over

the indexes . That is, all the indexes in theset are made members of the set .

Encoding

The above random codebooks and nested random partitionsare employed to do encoding as follows. Letbe the new index to be sent from sourcein block , fordestination . As indicated during the codebook generationstep, node at level transmits in block a codewordfrom its codebook that depends not only on its estimateof , the index that has to send, but also on its estimatesof , the indexes that higher levels

send in block .To make the above precise, we will use the following notation.

1) Estimate by node of the index thatlevel needs to send in block.

2) “ -step forward” estimate by node ofthe index to be sent by level in block .

This estimate is obtained by using “set inclusion,”defined in the sequel, at the end of block , for

.

3) “ -step backward” estimate by node ofthe index sent by level in block . This


estimate is made by using joint typicality at the endof block , for (details will be givenin the sequel).

In order to initialize the encoding procedure, as well as thedecoding procedure described in the sequel

are independently and uniformly chosen from , and aremade known to all the nodes

before the beginning of block.Now the indexes

are obtained as follows:

• , i.e., the source node simply chooses thesymbol ;

• for , is the index of thecell in partition which contains (the cell in

indexed by ). How the node obtainsthe one-step backward estimate at the end ofblock will be specified in the sequel.

Next, node obtains the forward estimates ,based on , using set inclusion, as follows. For each

, estimates as , whereis such that . Note that

depends only on , which, in turn, depends only on. Thus, the estimate can be made by node

at the end of block .Let

Then, having obtained the estimates above, which one shouldnote can be made at the end of block , node transmitsin block the codeword .

Decoding

The decoding will be defined in an iterative fashion. Supposethat at the end of block , node has estimates

Decoding at node at the end of block then proceeds asfollows:

• first obtains the one-step backward estimateof , the index sent by level in block , asfollows. Using the forward estimates

and upon receiving the signal , node estimatesiff there exists auniqueindex

such that , where

-

(25)

In the usual fashion of information theory, an error eventcaptures situations where there does not exist any such es-timate , or there is more than one such. This contin-gency will be taken into account in the probability of errorcalculations in the sequel.

• Next iteratively decodes the indexes sent by levels, as follows. Having obtained

backward estimates

node estimates as the index sentby in block iff there exists aunique

such that

where

are jointly -typical (26)

As above, an error occurs when such a uniquedoes notexist.

Probability of Error Calculations

Let be the index to be sent from source in block ,for destination . Define the “ideal” indexes to be sentby levels as follows:

• ;• for , , where is such

that .For the random coding scheme described above, we will de-

clare an error in block if the following event occurs:

(27)

where

1) is the event that :, , , : ,

are not jointly -typical,

2) is the event that node fails to obtainone-step backward estimate at the end of block

that matches the ideal symbol , i.e., either


or there exists such that;

3) for , is the event that nodefails to obtain -step backward estimateat the end of block that matches the ideal symbol

, i.e., either

or there exists such that

Now let

(28)

Thus, is the event that all the decoding steps at allnodes till the end of block are error free.

We will now show that is small. This willlead to a recursive proof that is small. We beginas follows.

Lemma 2.1:For sufficiently large

(29)

Proof: The event implies that the followinghold:

(30)

for each , , , .Furthermore

are jointly -typical, for each .Now is a forward estimate that depends only on

, which in turn depends only on made atthe end of block . Hence, depends only

on the estimates : made by the end of block.

Hence,

Moreover, by (30), we get that

since the dependence of on is in the same wayas depends on .

Thus,

are not jointly -typical

Hence, from Lemma 1.1 i), we get that

when is sufficiently large. This completes the proof of Lem-ma 2.1.

Next let

.

Hence, is the event that all the decodingsteps conducted at at the end of block for layers

are error free, as well as thejoint typicality property holds.

We first show that the conditional probability of error in ob-taining the one-step backward estimate

is small.

Lemma 2.2: If

then for sufficiently large

Proof: Recall from (25) that is the set of allsuch that

are jointly -typical. Let ,where

• is the event that , and• is the event that there exists such

that .As in Lemma 2.1, the event implies that

is the event that

are not jointly -typical.Also, implies that is the event

that

are jointly -typical. In particular, the eventimplies that

are jointly -typical.Thus,

(31)


Hence,

are jointly -typical

where uses (31), is from the definition of ,follows from an argument similar to Lemma 2.1, andholds due to Lemma 1.2 since and

are conditionally independent given

for each .Thus, if


which completes the proof of Lemma 2.2.

We next show that the conditional probability of error in ob-taining the -step backward estimate

can also be made small for each .

Lemma 2.3: If


for .Proof: Recall that is the set of all

such that

are jointly -typical. Let ,where

• is the event that

and

• is the event that there existssuch that

Now

(32)

We next evaluate the two terms on the right-hand side of (32)separately. For the first term, arguing along lines similar to (31),we get that

(33)

Now consider the second term in (32)

(34)

where follows from the definitions of andand an argument similar to Lemma 2.1, and the

equality follows from the definition of .Substituting (33) and (34) in (32), we get

(35)

Hence, we obtain the bound given at the top of the fol-lowing page, where uses (35), is from the defini-tion of , follows from an argument similarto Lemma 2.1, and follows from the fact that each

is independently and uniformly assignedto , , and from Lemma 1.2, as,for each and

are conditionally in-dependent given

Thus, if


and the proof of Lemma 2.3 is complete.

Now we bound the probability of error over theblocks ofsize sent from source to destination , for rate satisfying(5). Let


-

-

be the sequence of indexes transmitted by sourcein -blocks. We assume the indexes are i.i.d. random variablesuniformly distributed on . Let

be the sequence decoded at destinationat the end of block ,and let

Then the probability of error is

where and follow from the definition of in (27),uses the definition of , and uses Lemmas

2.1–2.3 and that is the total number of nodes in .Now, using the standard random coding arguments, we de-

duce that there exists a selection of codebooks


for every node in each level of such that

By throwing away the worse half of the inand reindexing them, we get that

for each . Thus, for and suffi-ciently large, the maximal probability of error , forrates

where satisfies (5). First letting , then ,and, finally, , we see that is achievable. Thiscompletes the proof of Theorem 2.1.

APPENDIX IIIPROOF OFTHEOREM 3.1

First consider the AWGN channel without fading (i.e., thechannel gains are given by (7)). Fix a flowgraph .For this flowgraph, let , and let ,

, , be such that

for each

Next, let , be i.i.d. real-valued Gaussian ran-dom variables with mean zero and covariance one. Let

(36)

where

Now codebooks for nodes in are generated as follows.Given a rate satisfying (11) for some

generate i.i.d. -variate normal random vectors ,, , having mean zero and covari-

ance . Then, , the codebook for node , is given by

where

(37)

The codewords so generated satisfy the power constraint (9)with high probability, when is large enough. With these code-books, the encoding and decoding procedures described in theproof of Theorem 2.1, and employing the standard discretizationand quantization arguments used for computing the capacityof general memoryless channels (i.e., channels with not neces-sarily finite input and output alphabets; see [13, Ch. 7]), we de-duce that the rate given by Theorem 2.1 is achievable with

arbitrarily small probability of error whenever (5) holds. Now,from (6), (7), and (36), we obtain

where is given by (13) and is the total powerreceived by node for . From (6), (7), and (36), wedetermine that is as given in (12).

Next consider the AWGN channel with fading and CSIR. Letand , , ,

be as above. Then, the signals transmitted bynodes and the codewords are as in (36) and (37), respectively,except that and are now all complexcircularly symmetric Gaussian. Then, as argued in [29], sincethe channel gains

are known to the receiving node , the channel outputfor node is . Thus, asabove, we obtain that the rate in Theorem 2.1 is achievablewhenever (5) holds with replaced by . Now

where uses the fact that, for , and areindependent and follows from (6) and (8), withand as given in (14) and (15), respectively.

Finally, consider the AWGN channel with fading and CSIRT.The key conceptual difference in obtaining an achievable regionhere as compared to the CSIR case arises due to the fact that thechannel state information is now also available to the transmit-ting nodes. Consequently, each transmitting node can now dopower allocation for aiding various levels in the flowgraph de-pending on the current channel state. More specifically, considera flowgraph as above. Then, node transmits

where the complex gains

satisfy (16) in order to meet the power constraint at , and: , as above, are i.i.d. with .

The rest of the argument is now similar to the CSIR case.


APPENDIX IVPROOF OFTHEOREM 4.1

The basic idea for showing that a rate vectoris achievable in is as follows. As

in the single source–destination pair case, block Markovencoding is used. A codebook for each nodeis generatedby applying the random-codebook generation procedure ofAppendix III to , for each . Now, in eachblock , node transmits a weighted sum over of thecodewords it would have transmitted if were the onlysource–destination pair in , with the weights chosen so asnot to violate the constraint on the total power available fortransmission to node. Furthermore, a “scheduling” randomvariable is introduced, which allows nodes to allocatedifferent fractions of their available transmission power atdifferent transmissions to various levels of the flowgraphs thatthey belong. At the end of the transmission of each block,each node performssuccessive cancellationto iterativelydecode indexes for the flowgraphs it belongs.

We now expand on the above. Given ,let, for each , and

be such that

Also, given the probability distribution : , generatea -sequence , where areidentically and independently drawn according to .

Codebook Generation:The codebook for each nodeinis generated as follows. For each that the “scheduling”random variable can take, each nodeallocates a fraction

of its available power to aid the transmission of levelin , for each , . Now, given

generate i.i.d. Gaussian random variables

having mean zero and variance one. Then the codebook for node, , is given by

where

Above

for the AWGN channel without fading or with fading and CSIR,and for the AWGN channel

with fading and CSIRT. The codewords so generated satisfy thepower constraint of nodewith high probability when

and is large enough.Random Partitions:For each , obtain nested

random partitions : of usingthe random-partitioning procedure given in Appendix II.

Encoding: In -block , each node transmits the codeword, where, for each

forward estimates : are obtainedby applying the encoding procedure given in Appendix II to

.Decoding: Each node decodes usingsuccessive cancellation

[5], [9], [30]. Thus, each node defines an order in which itsuccessively decodes the signals received from different levelsof the different flowgraphs that it belongs to. While decodinga particular level of a particular flowgraph, the signal compo-nents corresponding to already decoded levels are removed fromthe received signal and the signal components corresponding tolevels with higher order act as noise.

More specifically, for each , letdenote the step in which nodedecodes level

of flowgraph , where denotes thatthe information transmitted by levelof flowgraphis known to without any decoding operation (for instance,

, , ). A restriction on

is that, for each , the order it assigns to decodingthe signals transmitted by different levels within flowgraph

should correspond to the decoding order describedin Appendix II, i.e., for each

Then, from straightforward extensions of the probability of errorcalculations for the single source–destination pair case given inAppendix II and the -user multiple-access channel in [9, Sec.14.3], we deduce that each nodewill be able to decode all therequired indexes with arbitrarily small probability of error if, foreach , : satisfy

for , and

for each , where is the total powerreceived by nodecorresponding to the signal component trans-


mitted by level of flowgraph for when the sched-uling random variable takes value , and it is given by

Finally, by employing time sharing over the rate vectors, that are achievable using the above en-

coding–decoding procedure, we obtain a convex achievablerate region. This completes the proof of Theorem 4.1.

ACKNOWLEDGMENT

The authors wish to thank the Associate Editor and the anony-mous reviewers for their many helpful comments. They alsowish to thank Bertrand Hochwald and Gerhard Kramer for someinsightful discussions.

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