capacity scaling of wireless ad hoc networks shannon meets maxwell.bak

1702 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 3, MARCH 2012

Capacity Scaling of Wireless Ad Hoc Networks:Shannon Meets Maxwell

Si-Hyeon Lee, Student Member, IEEE, and Sae-Young Chung, Senior Member, IEEE

Abstract—In this paper, we characterize the information-the-oretic capacity scaling of wireless ad hoc networks withrandomly distributed nodes. By using an exact channel modelfrom Maxwell’s equations, we successfully resolve the conflictin the literature between the linear capacity scaling by Özgüret al. and the degrees of freedom limit given as the ratio of thenetwork diameter and the wavelength by Franceschetti et al.In dense networks where the network area is fixed, the capacityscaling is given as the minimum of and the degrees of freedomlimit � to within an arbitrarily small exponent. In extendednetworks where the network area is linear in , the capacityscaling is given as the minimum of and the degrees of freedomlimit � to within an arbitrarily small exponent. Hence, werecover the linear capacity scaling by Özgür et al. if � � ��in dense networks and if � � � �� in extended networks.Otherwise, the capacity scaling is given as the degrees of freedomlimit characterized by Franceschetti et al. For achievability, amodified hierarchical cooperation is proposed based on a lowerbound on the capacity of multiple-input multiple-output channelbetween two node clusters using our channel model.

Index Terms—Capacity scaling, channel correlation, cooperativemultiple-input multiple-output (MIMO), degrees of freedom, hi-erarchical cooperation (HC), Maxwell’s equations, physical limit,wireless ad hoc networks.

I. INTRODUCTION

P IONEERED by Gupta and Kumar in [1], the capacityscaling in wireless ad hoc networks has been actively

studied over the last decade. In this research, we consideruniformly and independently distributed nodes in a unit area(dense network) or an area of (extended network), each ofwhich wanting to communicate to a random destination atthe same rate of . The goal is to find out the maximallyachievable scaling of the aggregate throughputwith . In their seminal paper [1], Gupta and Kumar showedthat throughput scaling higher than cannot be achievedif each node treats interference as noise and that the multihop

Manuscript received February 05, 2010; revised August 29, 2011; acceptedSeptember 09, 2011. Date of current version February 29, 2012. The materialin this paper was presented in part at the 2008 IEEE International Symposiumon Information Theory and at the 2010 IEEE International Symposium on In-formation Theory. This work was supported in part by the Defense AcquisitionProgram Administration and the Agency for Defense Development under Con-tract UD060048AD.

The authors are with the Department of Electrical Engineering, Korea Ad-vanced Institute of Science and Technology (KAIST), Daejeon 305-701, Korea(e-mail: [email protected]; [email protected]).

Communicated by M. Franceschetti, Associate Editor for CommunicationNetworks.

Digital Object Identifier 10.1109/TIT.2011.2177741

scheme can achieve .1 This gap was closed in [3],where it was shown that the multihop via percolation theory canachieve . To information theorists, a natural question iswhat the information-theoretic capacity scaling is without suchunderlying physical-layer assumptions.

The information-theoretic capacity scaling is highly depen-dent on the channel model. Furthermore, it is important to use arealistic channel model to get results that are closer to reality. Inwireless networks in line-of-sight (LOS) environments, wherethe spatial locations of nodes are fixed with sufficiently large in-ternode separation compared to the wavelength, the baseband-equivalent channel response between two nodes and is givenas

(1)

from Maxwell’s equations where , is the distancebetween nodes and , denotes the wavelength where is

the speed of light and is the carrier frequency, andby Friis’ formula where is the product of the transmit andreceive antenna gains.

Recently, Özgür et al. characterized the information-theo-retic capacity scaling in [4]. Instead of using the exact channelmodel (1) with a distance-dependent phase, however, they as-sumed that the baseband-equivalent channel response betweentwo nodes and is given as

(2)

where is independent and identically distributed (i.i.d.).For this channel model, the capacity scaling is shown to bearbitrarily close to linear in both dense and extended networks,which means that each source can communicate to its des-tination as if there were no interference. A key componentto achieve such a scaling is the cooperative multiple-inputmultiple-output (MIMO) transmission between two node clus-ters whose sizes are comparable to that of the network. If thepenalty to form such a virtual MIMO is negligible, the classicalMIMO results [5], [6] under the i.i.d. channel phase assumptionmake the linear throughput scaling possible. Such an overheadis indeed shown to be arbitrarily small by using hierarchicalcooperation (HC). In the HC scheme, each cluster forms avirtual antenna array using MIMO transmissions between

1In this paper, we use the following asymptotic notations [2]. 1) �� if �� as� tends to infinity for some constant �. 2) �� if � �� as� tends to infinity for some constants� and � . 3) �� if �� as� tends to infinity for someconstant �.

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LEE AND CHUNG: CAPACITY SCALING OF WIRELESS AD HOC NETWORKS 1703

small-scale clusters inside it. Similarly, each small-scale clusterforms a virtual antenna array by MIMO transmissions betweeneven smaller clusters inside it. This builds up a hierarchy anda plain time division multiple access (TDMA) is performed atthe bottom hierarchy.

The i.i.d. phase assumption in (2) makes the throughput anal-ysis easier in [4], but such an artificial assumption can lead toresults contradicting the physics. Recently, the linear capacityscaling in [4] turned out to be contradictory to the physical limiton degrees of freedom (DoF) when is not sufficiently small.In [7], Franceschetti et al. showed, using Maxwell’s equations,that DoF in extended networks is limited by the ratio of the net-work diameter and . By rescaling the network size, the DoFlimit becomes in dense networks. This is a fundamentallimitation independent of power attenuation and fading models.Hence, the linear capacity scaling in [4] is in fact not attainablefor and in dense and extendednetworks, respectively. The cause of such a conflict is the i.i.d.channel phase assumption in [4] that ignores the channel corre-lation due to the distance-dependent channel phase.

Two contradictory results [4], [7] highlight the importanceof exact channel models based on Maxwell’s equations. Thus,the ultimate goal would be the characterization of the infor-mation-theoretic capacity scaling of wireless ad hoc networksfrom Maxwell’s equations without any artificial assumptions.In this paper, we accomplish this goal by characterizing theinformation-theoretic capacity scaling of wireless ad hoc net-works using an exact channel model from Maxwell’s equationsin LOS environments. In dense networks, we establish the ca-pacity scaling given as to within an arbitrarilysmall exponent. Hence, the capacity scaling is linear in if

. Otherwise, the capacity scaling is given as theDoF limit characterized by Franceschetti et al. In extendednetworks, the capacity scaling is given as to withinan arbitrarily small exponent. Hence, the capacity scaling islinear in if and is given as the DoF limitcharacterized by Franceschetti et al. otherwise.

Since the converse is straightforward from the previous worksin [4] and [7], our main contribution is to show the achievability.We note that under the far-field assumption, i.e., is much

smaller than the internode separation , where denotes

the network diameter; the DoF limit is, in general, higherthan the throughput scaling of the multihop via percolationtheory in [3]. For achievability, we modify the HC scheme in [4]according to an achievable MIMO rate between two node clus-ters. We show that the capacity of the MIMO channel betweentwo node clusters is at least proportional to the minimum of thenumber of nodes in the cluster and the product of the ratio ofthe cluster diameter and and the angular spread between theclusters. In our modified HC scheme, only a subset of nodes in acluster performs the MIMO transmission such that the numberof participating nodes is proportional to the achievable MIMOrate, whereas all nodes in the cluster participate in the MIMOtransmission in the HC scheme of [4].

The organization of this paper is as follows. In Section II,the system model is presented. In Section III, we present the

main theorems on the capacity scaling and their implications. InSection IV, a modified HC scheme is constructed according toan achievable MIMO rate between node clusters. We concludethis paper in Section V.

The following notations will be used in this paper.denotes the circularly symmetric complex Gaussian randomvector with zero mean and covariance matrix of . anddenote the set of real numbers and the set of natural numbers,respectively. and denote the expectation and conjugatetranspose, respectively. denotes the modulo- operation.

denotes the positive part of , i.e.

ifif

For two integers and such that , denotes theset . For a set , denotes the cardinality ofthe set. The logarithm function is base 2 unless otherwisespecified.

II. SYSTEM MODEL

There are uniformly and independently distributed nodesin a square of unit area (called a dense network) or a squareof area (called an extended network). It is assumed that thenode locations are fixed for the duration of the communication.Each node has an average transmit power constraint of andthe network is allocated a total bandwidth around the car-rier frequency . The wavelength is assumed tobe much smaller than the average separation distance betweenneighbor nodes given as and for dense and ex-tended networks, respectively. Furthermore, we assume a verymild lower bound on such that for an arbitrarilylarge constant . We assume that is a monotonicallynonincreasing function of . This corresponds to using highercarrier frequencies to handle more traffic due to the increasednumber of nodes. Every node is a source and a destination si-multaneously, and the source–destination pairs are determinedrandomly. Every source wants to communicate to its destinationat the same rate of . The aggregate throughputof the network is given as .

We consider the LOS environment, i.e., no multipath fading.2

From Maxwell’s equations in far fields, the discrete-time base-band-equivalent channel gain between nodes and at timeis given as

(3)

where , is the distance between nodes andat time , and by Friis’ formula, where is the

product of the transmit and receive antenna gains.3 Note that ifis fixed, vanishes as tends to zero. In extended networks,

however, we assume that is a constant since we can increaseproportional to without increasing the physical size of

2Our analysis can be extended to cases where there is multipath fading.However, we believe that having a finite number of paths would not affect thethroughput scaling laws.

3A channel model with a path-loss exponent larger than two is considered inAppendix D.



the antennas beyond a small fraction of the internode separa-tion.4 In dense networks, it is proper to assume that the nodesize is upper-bounded by for some constant since thenetwork area is now fixed. Hence, is assumed to befor dense networks because we can make proportional to

.5

The discrete-time baseband-equivalent output at nodeat time is given as

where is the discrete-time baseband-equivalent input atnode at time and is the additive Gaussian noise

at node at time . The channel state informationis available only at the receivers. From now on, we will omit thetime index for notational convenience.

III. MAIN RESULT

We first present a lower and an upper bound on the capacityscaling for dense networks in Theorems 1 and 2, respectively.In Theorems 3 and 4, we present a lower and an upper bound onthe capacity scaling for extended networks, respectively.6

Theorem 1: Consider a network of nodes on a unit area,in which source–destination pairs are assigned arbitrarily.For any , a scheme exists that achieves an aggregatethroughput

with high probability,7 where is a positive constant inde-pendent of both and .

The aggregate throughput scaling in Theorem 1 can beachieved by the modified HC scheme constructed in Section IV.Note that Theorem 1 holds even if source–destination pairingis arbitrary.

In the following theorem, we show an upper bound on thethroughput scaling. If the source–destination pairs can be de-termined according to the node locations, then an aggregatethroughput scaling of would be achievable for any byletting each of the source–destination pairs be nearest neigh-bors. Therefore, for the upper bound on the capacity scaling,we limit our interest to random source–destination pairing.

4For each node, we can deploy �� antennas vertically that form anantenna array of length ��, which gives a vertical beamforming gain of�� . Hence, the product of the transmit and receive beamforming gainscan be �� .

5In dense networks, we can vertically deploy �� antennas foreach node that form an antenna array of length �� .

6We note that a similar result was also independently shown in [8] based onthe same channel model as in [9] at the same time this paper was submitted. Inthis paper, we derive a lower bound on the MIMO transmission between twonode clusters without any artificial assumptions, which is the key ingredientin the achievability, whereas the work in [8] assumed that interfering signalsfrom other transmitting nodes in the network to the MIMO transmission areindependent. In addition, the effect of � on � is considered in this paper, butnot in [8].

7With probability approaching 1 as � tends to infinity.

Theorem 2: Consider a network of nodes on a unit area, inwhich source–destination pairs are assigned randomly. Theaggregate throughput in the network is upper-bounded as

(4)

with high probability, where is a positive constant indepen-dent of both and .

The first term in the minimum in (4) is the DoF limit shownin [7].8 The second term in the minimum in (4) is obtained fromthe fact that the transmission rate from a source to its destinationis upper-bounded by the capacity of the single-input multiple-output (SIMO) channel between the source and the remainingnodes in the network (see, e.g., [4, Th. 3.1]).

Theorems 1 and 2 establish the capacity scaling in dense net-works to within an arbitrarily small exponent. To see the effectof on the capacity scaling, let for . Note thatthe condition is needed for the far-field approximation tohold. If , the capacity scaling is arbitrarily close to linear.If , the capacity scaling is given as the DoF limit.

Now, we give an achievable aggregate throughput scaling inextended networks.

Theorem 3: Consider a network of nodes on an area ,in which source–destination pairs are assigned arbitrarily.For any , a scheme exists that achieves an aggregatethroughput


The aggregate throughput scaling in Theorem 3 can beachieved by the modified HC scheme in Section IV.

For random source–destination pairing, the following the-orem shows an upper bound on the capacity scaling whose ex-ponent is arbitrarily close to that of the lower bound in Theorem3.

Theorem 4: Consider a network of nodes on an area , inwhich source–destination pairs are assigned randomly. Theaggregate throughput in the network is upper-bounded as

(5)


The first term in the minimum in (5) is the DoF limit shownin [7], and the second term in the minimum in (5) is obtainedsimilarly as the derivation of the second term in the minimumin (4).

Similarly, as in dense networks, let for to seehow affects the capacity scaling in extended networks. Notethat means that is a constant, regardless of . If ,the capacity scaling is arbitrarily close to linear. If ,the capacity scaling is given as the DoF limit.

8In [7], the DoF limit in extended networks is studied and it is shown to bedetermined by the ratio of the network diameter and the wavelength. In densenetworks, the DoF limit can be obtained by rescaling the network size.



Remark 1: The exponent signifies the increase of tohandle more traffic as increases. For example, considera network with an area of 0.01 with and

. Then, the DoF limit is an order of100, and hence, the network is not DoF limited. Now, assumethat the network size grows to an area of 1 with .If , i.e., the carrier frequency remains the same, then thenetwork becomes DoF limited since the DoF limit is an orderof 1000. Now, if , i.e., the carrier frequency is increasedto 3 GHz , then the network is not DoF limitedsince the DoF limit is now an order of 10 000.

IV. MODIFIED HC

In this section, Theorems 1 and 3 are proved by constructing amodified HC scheme. Let us first consider a cooperative MIMObetween two node clusters, which is the key to the construc-tion of the modified HC scheme. Consider independently anduniformly distributed nodes in each of two horizontally alignedsquare areas with side length and distance betweenthe centers, as shown in Fig. 1. Let and denote theleft and right clusters of nodes in Fig. 1, respectively. The

-by- cooperative MIMO channel from to is givenas

(6)

where is the -by-1 received vector at , is the -by-channel matrix from (3), is the -by-1 transmitted vectorfrom , is the -by-1 external interference vector with co-variance matrix , and is the -by-1 additive Gaussian noise

vector . Let and .The following theorem presents an achievable MIMO rate from

to .910 The proof is in Appendix A.

Theorem 5: The capacity of the cooperative MIMOchannel from to is lower-bounded as

for any with high probability as tends to infinity,where and are positive constants independent of , ,

, and and is given as

(7)

9A more general version of Theorem 5 was shown previously in [9, Th. 1],where multiple antennas per node were assumed.� in Theorem 5 correspondsto the fourth term in the minimum in [9, eq. (4)], which was obtained based onan approximation and, therefore, differs slightly from � . In deriving Theorem5, however, no approximation is used, and therefore, the result is now exact.

10For the simplicity of presentation, � and � are assumed to be horizon-tally aligned. However, the proof of Theorem 5 in Appendix A can be easilyextended to cases where � and � are not horizontally aligned, which willresult in the same conclusion as in Theorem 5.

We have the following corollaries for certain classes of .

Corollary 1: If there is no external interference, i.e., ,the capacity of the cooperative MIMO channel fromto is lower-bounded as

with high probability as tends to infinity, where andare positive constants independent of , , , and .

Proof: We choose in Theorem 5, e.g., .

Corollary 2: If and, where and , the

capacity of the cooperative MIMO channel from tois lower-bounded as

with high probability as tends to infinity, where andare positive constants independent of , , , and .

Proof: We choose in Theorem 5.

Note that matches the DoF limit predicted in [7] and [10]given as the product of the normalized cluster diameter andthe angular spread between the clusters.

In Sections IV-A and IV-B, Theorems 1 and 3 for dense andextended networks, respectively, are proved by constructing amodified HC scheme.

A. Dense Network

Let us construct the modified HC scheme for dense networksconsisting of hierarchy levels. For an area of , there are anorder of nodes with high probability.11 For simplicity, weassume that there are exactly nodes in our description ofthe scheme, but our results hold without such an assumption.Consider a -tuple such that

and for all and a -tuplesuch that for all

. For , let and . Considera hierarchical structure of the network such that the network isdivided into square areas of , each of those square areas isagain divided into smaller square areas of , and so on, i.e.,at the th hierarchy level for , each square area ofis divided into smaller square areas of .

Let for denote the achievablethroughput when a cluster of nodes operates as a networkhaving its own source–destination pairs in an arbitrarymanner. The following lemma gives as a function of

for .12

Lemma 1: Fix . Consider a cluster of nodes. If,for any two clusters and of nodes inside the cluster of

nodes, a rate of is achievable with high probability forthe MIMO communication from randomly chosen nodes

11See [4, Lemma 4.1] for the proof.12Since � �� for � � �� has a recursive form, it also depends on

� � � � � � � � � � � � � .



Fig. 1. Cooperative MIMO between � and � with side length � and dis-tance � � �� between the centers.

in cluster to randomly chosen nodes in cluster whenother nodes in the cluster of nodes are silent, we have

(8)with high probability, where is a positive constant indepen-dent of both and .

Proof: We construct a scheme for the cluster of nodeswhen it operates as a network having its own source–des-tination pairs in an arbitrary manner. From now on, a clusterindicates a cluster of nodes inside the cluster of nodesunless otherwise specified. We randomly assign the indices

to nodes in each cluster and let forand denote the set

of nodes in cluster .The scheme consists of three phases. Let us first explain the

scheme briefly from the perspective of source in cluster andits destination in cluster . In the first phase, source in cluster

distributes its message to . In the second phase, per-forms MIMO transmission to . In the last phase, destination

in cluster collects quantized MIMO observations fromand decodes the message. The detailed operation in each phaseis as follows.

1) Phase 1: Each cluster operates in parallel according to the9-TDMA scheme in [4] illustrated in Fig. 2. Source incluster distributes its message to , i.e., the messageof is split into subblocks and each node inreceives one subblock. For a cluster, this can be done bysetting up subphases, where source–destina-tion pairs in each of the subphases are assigned as follows:in subphase ,

is the set of source–destination pairs. Be-cause is achievable for a network ofnodes, time slots are needed for eachsubphase. Since there are subphases in each TDMAslot, Phase 1 needs a total oftime slots.

2) Phase 2: We perform successive MIMO transmissions forall source–destination pairs, i.e., MIMO transmission from

to for source in cluster and destination

Fig. 2. Big square and small squares represent a cluster of � nodes and theclusters of � nodes inside it, respectively. In Phases 1 and 3, the clusters of� nodes operate in parallel according to the following 9-TDMA scheme: thetotal time of the phase is divided into nine TDMA slots, and, in the �th TDMAslot for � � �� , clusters marked with � operate simultaneously while theother clusters are silent.

in cluster . Since a rate of is assumed to be achiev-able for each MIMO transmission, time slots areneeded for each source–destination pair. Since we havesource–destination pairs, a total of time slotsare needed for Phase 2. After Phase 2, each node quan-tizes the MIMO observations at a fixed rate subblocksper time slot.13

3) Phase 3: Each cluster operates in parallel according to the9-TDMA scheme in [4] depicted in Fig. 2. Destination

in cluster collects the quantized observations of theMIMO transmission intended for it from and, then,decodes the message. Note that each quantized MIMO ob-servation consists of subblocks. By setting up

subphases for a cluster similarly as in Phase 1, wheresource–destination pairs are assigned in each of the

subphases, a total oftime slots are needed for Phase 3.

In total

time slots are needed to transport messages, i.e.,subblocks. Hence, the constructed scheme yields an aggregatethroughput of (9), which proves Lemma 1.

In the earlier explanation of the scheme, we focused on themodified operation from the scheme in [4] and the resultingscaling law of the throughput. The readers should refer to [4]for a more detailed description of the scheme. However, takingthose details into account does not change the throughputscaling.

13From [4, Appendix II], a strategy exists for each node to encode the ob-servation of an MIMO transmission at a fixed rate � such that the resultant� -by-� quantized MIMO channel has the same multiplexing gain asthe original MIMO channel.

(9)



The modified HC scheme is constructed recursively using thescheme in the proof of Lemma 1 for the original network of

nodes and using the multihop via percolation theory [3] forclusters of nodes at the bottom hierarchy. Now, let us showan achievable throughput scaling using the modified HC schemewith hierarchy levels. Note that throughput achieved by themodified HC scheme depends on the choice ofand . First, we choose as for

, where

For the modified HC scheme with the aforementioned choice of, the following lemma shows that a rate of

is achievable for the MIMO transmis-sions in Phase 2 at the th hierarchy level for .

Lemma 2: In Phase 2 at the th hierarchy level of the modifiedHC scheme for , a rate of isachievable for the MIMO transmissions between clusters ofnodes.

Proof: Fix . In Phase 2 at the th hierarchy level,we let each transmitting cluster of nodes use a randomlygenerated Gaussian code according to , where

(10)

This satisfies the average power constraint of per node be-cause each node participates in the MIMO transmission forfraction of time in Phase 2.

Consider the MIMO transmission from cluster ofnodes to cluster of nodes inside cluster of nodes inPhase 2 at the th hierarchy level of the modified HC scheme.To prove that the capacity of the MIMO channel from to

is at least linear in , we use Corollary 2. Byadopting the notations for Corollary 2, let and denote theside length of and and the distance between the centers,respectively, and let be given as (7). The MIMO transmis-sion from to is interfered by the MIMO transmissionby nodes in each cluster of nodes that operates simulta-neously with . Let denote the set of clusters of nodesthat operate simultaneously with . Then, can be split intosubgroups according to their distance to such that the th sub-group contains or less clusters of nodes and thedistance between the centers of and each cluster inis greater than or equal to for , as illus-trated in Fig. 3. The number of such subgroups can be simplybounded by . Let denote the number of clusters of

nodes in . Then, the MIMO channel from tois given as (6), in which is substituted for and the in-terference is given as , where is the

-by- channel matrix from to andis the -by-1 transmitted vector from .

Fig. 3. For cluster � , clusters that operate simultaneously with � according tothe 9-TDMA scheme are represented as shaded. The set of shaded clusters withdots represents � ��, and the set of shaded clusters with slash lines represents� ��.

Now, let us show that the covariance matrix of satisfiesthe conditions in Corollary 2 for and . Let

. Then, we have

and

where is from the following lemma, which is a direct con-sequence of the matrix trace inequality in [11].

Lemma 3: If ’s are positive semidefinite matrices, then.

By applying similar bounding techniques as those forand in Appendix A to and

, we can show

and

with high probability, where is given as



Because and , we have

and

Hence, the conditions in Corollary 2 are satisfied forand . From Corollary 2 for and , thecapacity of the MIMO channel from to is lower-bounded as

for some constant with high probability, where is fromthe choice of in (10). Furthermore, becauseand , we have .

Hence, we have , which provesLemma 2.

Now, by substituting and for andin (8), where is a positive constant independent

of both and , we have the recursive form of forfor the modified HC scheme given as

(11)

where is a positive constant independent of both and .The following lemma gives an achievable throughput scaling

using the modified HC scheme with hierarchy levels when wechoose that maximizes (11) for .The proof is at the end of this section.

Lemma 4: In dense networks, the modified HC scheme withhierarchy levels achieves

with high probability, where is a positive constant indepen-dent of both and ,

ifif

for someif

and

for , where for.The following corollary is obtained straightforwardly from

Lemma 4.

Corollary 3: In dense networks, the modified HC scheme withhierarchy levels achieves


Now, we are ready to prove Theorem 1.Proof of Theorem 1: Fix . Let be the smallest

integer such that and let be the smallest integer suchthat and . Let usdefine functions and for foras

Fix . We will show that is larger thanfor all . Let us first show that .

is given as

If , we have

If , we get

Thus, we conclude that . Now, we are readyto show for all . Note that .For

For ,

Hence, we prove thatfor all . By letting , we equiva-lently prove that the achievable rate of the modified HC schemewith hierarchy levels in Corollary 3 is lower-bounded as



Now, we have

where is because . Hence, Theorem 1 isproved.

B. Extended Network

In extended networks, both and the distance betweennodes is increased by a factor of as compared to those indense networks. Hence, for the same transmit power, the re-ceived power at each node remains the same as in dense net-works. By rescaling the space, let us consider an extended net-work as an equivalent dense network on a unit area but with thewavelength reduced to . Since the wavelength is given as

in the equivalent dense network, Theorem 3 is proved.Proof of Lemma 4: First, consider the case of

. Since , this implies . In thiscase, is , and hence, the recursive form of in(11) becomes

(12)

for all . Note that by using themultihop via percolation theory [3] for the cooperation for the

clusters of nodes.14 By choosing that maxi-

mizes (12) for , is ob-

tained. Because and , Lemma 4 is provedfor the case of .

Next, consider the case of for some. Let us first assume that is for

and is for . For the choice of

that maximizes (11) under this assumption,we will show that the range of where the assumption is validis the same as the range of corresponding toin Lemma 4.

14In [3], a path-loss exponent larger than two is considered and a multihop viapercolation theory is shown to achieve��

��. For the path-loss exponent equal

to two, however, it achieves�� due to the interference power proportionalto ��.

Since is assumed to be for , weobtain

(13)

For , is assumed to be , and

hence, the recursive form of in (11) is given as

(14)

Let us assume that for has

the form of for some positive

constants , , and independent of both and. Then, the recursive formulas and

are obtained by choosing as

that maximizes (14) for . Using the conditionsand from (13), and

for are given as

Because , for is given as (15).Now, the range of that makes the assumption, i.e., is

for and is for ,

valid is given asifif .

(16)By using and from (15), we can show that therange of in (16) is the same as the range of correspondingto in Lemma 4. Hence, we prove that for

, the modified HC scheme with levels achieves

(15)



Since and , Lemma 4 is provedfor the case of .

V. CONCLUSION

We characterized the information-theoretic capacity scalingof wireless ad hoc networks from Maxwell’s equations withoutany artificial assumptions. The capacity scaling is given as theminimum of the number of nodes and the DoF limit given asthe ratio of the network diameter and the wavelength. Accord-ingly, a network becomes DoF-limited if in densenetworks and in extended networks. Our resultsindicate that the linear throughput scaling in [4] that was shownunder the i.i.d. channel phase assumption is indeed achievableto within an arbitrarily small exponent in the non DoF-limitedregime. In the DoF-limited regime, the DoF limit characterizedby Franceschetti et al. in [7] that generally has higher scalingthan that of the multihop scheme can be achieved to within anarbitrarily small exponent by using the modified HC scheme.

We also considered a channel model with a path-loss ex-ponent larger than two. In dense networks, the throughputscaling using the modified HC scheme for remains thesame as when . However, the throughput scaling using themodified HC scheme is decreased for in extended net-works due to the power limitation. This suggests, as a furtherwork, an upper bound considering both the DoF limitation dueto the channel correlation and the power limitation due to thepower attenuation over the distance.

APPENDIX APROOF OF THEOREM 5

The capacity of the MIMO channel from to islower-bounded as

(17)

for any , where is , is chosen uni-formly among the eigenvalues , of

, is chosen uniformly among the eigenvalues ,of , and is chosen uniformly among the eigen-

values , of . is from choosing theinput as , is because assuming Gaussian interferenceminimizes the mutual information for given noise and interfer-ence covariance matrices [12], [13], is because the geometricmean is upper-bounded by the arithmetic mean, is because

, is from the Paley–Zygmund inequality[4], [14], and is from Lemma 3.

Note that and . To get a lower boundon (17), we need a lower bound on and an upper bound on

. Let . Then, , where

. Note that constants and exist independentof and such that for all , .First, is given as

Since , we have .Next, is upper-bounded as

where ,

, and

.Note that ’s for all and

follow an identical distribution, but theyare not necessarily independent of each other. Nevertheless,

strongly converges to



as the following lemma shows, where the expectation is overuniform node distributions.

Lemma 5: The sample mean

strongly converges to . That is

The proof of the aforementioned lemma is given inAppendix B. Furthermore, the following lemma gives anupper bound on , which is proved in Appendix C.

Lemma 6: .From Lemmas 5 and 6, we have

with high probability astends to infinity.

Now, by using the bounds on and , Theorem 5 isproved.

APPENDIX BPROOF OF LEMMA 5

Let us first present a theorem on the strong convergence ofthe sample mean of a sequence of not necessarily independentrandom variables. The proof is in [15].

Theorem 6: Let be a sequence of not necessarilyindependent complex-valued random variables, each of whichfollows an identical probability density function such that

and and are bounded. Suppose that

(18)

Then, the strong law of large numbers holds for , i.e.

Now, let us prove Lemma 5 using Theorem 6. For , letand denote the collections of random variables given as

and

ifotherwise.

Note that , , and. Let for be the th random vari-

able in with an arbitrary ordering. We construct a sequenceof random variables as follows: for , we let

denote the random variable , where is the

integer satisfying .

Let us show that satisfies the conditions in The-orem 6. First, it is easy to show that and arebounded, i.e., and . Next, theleft-hand side term of the inequality in (18) is written as

Consider two random variablesand in . If

and , andare independent of each other, and hence,

. Otherwise,. Using this, let us get an upper bound on

for each as follows.1) for some : In this case,

is . For each random variable in ,random variables in are independent of

. Thus, we have

for some positive constant .2) for some : Let

and . Then, we get



Fig. 4. Two nodes � and � in� and two nodes � and � in � . � and � denote�� and ��, respectively.

for some positive constant .

Let . Now, we have

which is finite. Hence, from Theorem 6

which concludes the proof of Lemma 5.

APPENDIX CPROOF OF LEMMA 6

Consider two uniformly and independently distributed nodesand in and two uniformly and independently distributed

nodes and in . Consider a cartesian coordinate systemwhose origin is at the bottom left corner of . Let

, , , and de-note the coordinates of nodes , , , and , respectively. Let

and for random variables and denote thesupport of the probability density function and the sup-port of the conditional probability density function , re-spectively. Let denote the set of suchthat the line through and intersects , and let denote

. Let and let denote thelength of where the length of an interval

is defined as .15 Let and let , , , anddenote when is fixed at , , ,

and , respectively. Let .See Fig. 4.

15Here, we follow the convention that �� is the counterclockwise anglefrom � to � and � �� .

Now, we are ready to prove Lemma 6. is upper-bounded as

Furthermore, both andare upper-bounded by

if

if(19)

for some positive constants and , where .The upper bound (19) is derived at the end of this appendix.

Using the aforementioned upper bounds, isupper-bounded separately for the cases of and

. If , we have

for some positive constant . If , we have

for some positive constant .



Because , is upper-bounded as

if

if .

Finally, is upper-bounded as follows:

for some positive constants , , and . Since, Lemma 6 is proved.

Now it remains to show the upper bound (19). The upperbound (19) is obtained by using the following lemma, whoseproof is at the end of this appendix.

Lemma 7: Let be a periodic Lebesgue-integrable func-tion on with period that satisfiesand . Let be a nonnegativeand Lebesgue-integrable function on . Consider an in-terval and constants and . If a partition

of exists for finite such thatand is monotone on each

interval for , we have

where is such thatfor all .

To boundgiven as (20) using Lemma 7, we first show that

consists of a finite number of intervals suchthat is monotone for each. Becauseand have a one-to-one relationship, we have

where

We can easily show that has at most two crit-ical points from its derivative with respect to and that

can be split into at most four intervalssuch that is a constant for each. Hence,

can be split into at most six intervals suchthat is monotone for each, implyingthat can also be split into at most six intervalssuch that is monotone for each. Because

and , we have

(21)

from Lemma 7, where is such that

for all .In the same way

(22)

We bound separately for the cases ofand . Without loss of

generality, assume that . First, consider thecase of . Note that is decreasingin and is increasing in

. For the case of ,

(20)



,and hence, we have . Therefore, we have

where is when for given ,, . We have the following bounds on :

(23)

for some positive constants and . The upper boundholds since and the lower bound is obtained as

where is the angle between the line through and and thehorizontal line crossing and is because .From (23), we have

for some positive constant . Using this bound in (21) and(22), the upper bound (19) for the case of isobtained.

Now consider the case of . The followinglemma gives a lower bound on for the case of ,whose proof is given at the end of this appendix.

Lemma 8: When is given, is lower-boundedby .

From the aforementioned lemma

and hence

Using this bound in (21) and (22), the upper bound (19) for thecase of is proved.

Proof of Lemma 7: It can be easily shown that for any intervalon which is monotonically increasing, we have

(24)

and for any interval on which is monotonicallydecreasing

(25)

From (24) and (25), Lemma 7 is directly obtained.

Proof of Lemma 8: Assume that is given. Then,is included in either or . is given

as follows:

Fix . The derivative of with respect to has the

form of a rational polynomial , where is positivefor every and is a cubic function of witha positive cubic coefficient whose roots are given as (26).

Since when , which vio-lates the assumption , contains at most oneroot of . Because the cubic coefficient of is posi-tive, is maximized when is 0 or , and hence,

is minimized when is 0 or .In a similar way, we can show that is minimized when

is or for fixed . Thus, is lower-bounded by .

APPENDIX DEXTENSION TO A PATH-LOSS EXPONENT LARGER THAN TWO

In this appendix, we consider the channel model with a path-loss exponent larger than two, i.e., the discrete-time baseband-equivalent channel gain (3) between nodes and at time ischanged to

(27)

with the path-loss exponent . For , this channelmodel approximates the channel when there are a direct pathand a reflected path off the ground plane between transmit andreceive antennas with a sufficiently large horizontal distance.For and , however, the channel model (27) is not a

(26)



direct consequence of Maxwell’s equations, and hence, the DoFlimit characterized in [7] is not valid for this channel model.

Now, let us present throughput scalings using the modifiedHC scheme constructed in Section IV for the channel modelin (27). In dense networks, we can get the same throughputscaling in Theorem 1. In extended networks, the throughputscaling using the modified HC scheme is decreased because thenetwork becomes power-limited. For the same transmit power,the received power at each node in extended networks is de-creased by a factor of as compared to the dense net-work. By rescaling the space, an extended network can be con-sidered as an equivalent dense network on a unit area but withthe average power constraint per node reduced to in-stead of and the wavelength reduced to instead of .Note that the average power constraint per node isless than . As the bursty modification of the HC scheme in[4], we use the bursty version of the modified HC scheme, i.e.,we use the modified HC scheme with operating power for

fraction of the time and keep silent for the remainingfraction of the time. This satisfies the average power constraintper node and yields an aggregate throughput scalingof .

REFERENCES

[1] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEETrans. Inf. Theory, vol. 46, no. 2, pp. 388–404, Mar. 2000.

[2] D. E. Knuth, “Big omicron and big omega and big theta,” ACM SIGACTNews, vol. 8, pp. 18–24, Apr.–Jun. 1976.

[3] M. Franceschetti, O. Dousse, D. N. C. Tse, and P. Thiran, “Closing thegap in the capacity of wireless networks via percolation theory,” IEEETrans. Inf. Theory, vol. 53, no. 3, pp. 1009–1018, Mar. 2007.

[4] A. Özgür, O. Lévêque, and D. N. C. Tse, “Hierarchical cooperationachieves optimal capacity scaling in ad hoc networks,” IEEE Trans.Inf. Theory, vol. 53, no. 10, pp. 3549–3572, Oct. 2007.

[5] G. J. Foschini, “Layered space-time architecture for wireless commu-nication in fading environments when using multi-element antennas,”Bell Labs Tech. J., vol. 1, pp. 41–59, Apr. 1996.

[6] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur.Trans. Telecommun., vol. 10, pp. 585–595, Nov. 1999.

[7] M. Franceschetti, M. D. Migliore, and P. Minero, “The capacity ofwireless networks: Information-theoretic and physical limits,” IEEETrans. Inf. Theory, vol. 55, no. 8, pp. 3413–3424, Aug. 2009.

[8] A. Özgür, O. Lévêque, and D. N. C. Tse, “Linear capacity scaling inwireless networks: Beyond physical limits?,” in Proc. Inf. Theory Appl.,San Diego, USA, Feb. 2010, pp. 1–10.

[9] S.-H. Lee and S.-Y. Chung, “Effect of channel correlation on the ca-pacity scaling in wireless networks,” in Proc. IEEE Int. Symp. Inf.Theory, Toronto, Canada, Jul. 2008, pp. 1128–1132.

[10] A. Poon, R. Brodersen, and D. N. C. Tse, “Degrees of freedom inmultiple-antenna channels: A signal space approach,” IEEE Trans. Inf.Theory, vol. 51, no. 2, pp. 523–536, Feb. 2005.

[11] X. Yang, X. Yang, and K. L. Teo, “A matrix trace inequality,” J. Math.Anal. Appl., vol. 263, pp. 327–333, 2001.

[12] S. Ihara, “On the capacity of channels with additive non-Gaussiannoise,” Inf. Control, vol. 37, no. 1, pp. 34–39, 1978.

[13] S. N. Diggavi and T. M. Cover, “The worst additive noise under acovariance constraint,” IEEE Trans. Inf. Theory, vol. 47, no. 7, pp.3072–3081, Nov. 2001.

[14] J. P. Kahane, Some Random Series of Functions. Cambridge, U.K.:Cambridge Univ. Press, 1985.

[15] R. Lyons, “Strong laws of large numbers for weakly correlated randomvariables,” Michigan Math. J., vol. 35, pp. 353–359, 1988.

Si-Hyeon Lee (S’08) received the B.S. degree, summa cum laude, in electricalengineering from the Korea Advanced Institute of Science and Technology(KAIST), Daejeon, South Korea, in 2007. She is currently pursuing the Ph.D.degree in electrical engineering at KAIST. Her research interests includenetwork information theory and wireless communication systems.

Sae-Young Chung (S’89–M’00–SM’07) received the B.S. (summa cum laude)and M.S. degrees in electrical engineering from Seoul National University,Seoul, South Korea, in 1990 and 1992, respectively, and the Ph.D. degree inelectrical engineering and computer science from the Massachusetts Instituteof Technology, Cambridge, MA, USA, in 2000. From September 2000 toDecember 2004, he was with Airvana, Inc., Chelmsford, MA, USA. SinceJanuary 2005, he has been with the Department of Electrical Engineering,Korea Advanced Institute of Science and Technology, Daejeon, South Korea,where he is currently a KAIST Chair Professor. He has served as an AssociateEditor of the IEEE TRANSACTIONS ON COMMUNICATIONS since 2009. He is theTechnical Program Co-Chair of the 2014 IEEE International Symposium onInformation Theory. His research interests include network information theory,coding theory, and wireless communications.


capacity scaling of wireless ad hoc networks shannon meets maxwell.bak

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