Communication Tradeoffs in Wireless Networks Serj Haddad, Olivier Le´veˆque Information Theory Laboratory School of Computer and Communication Sciences EPFL. 1015 Lausanne, Switzerland Abstract We characterize the maximum achievable broadcast rate in a wireless network under various fading assumptions. Our result exhibits a duality between the performance achieved in this context by collaborative beamforming strategies and the number of degrees of freedom available in the network. Index Terms wireless networks, broadcast capacity, low SNR communications, beamforming strategies, random matrices 6 1 I. INTRODUCTION 0 Thereisa vastbodyofliteratureonthesubjectofmultiple-unicastcommunicationsinadhocwirelessnetworks.Becauseof 2 the inherent broadcast nature of wireless signals, managing the interference between the multiple source-destination pairs is a n keyissueandhasledtovariousinterestingproposals[1],[2],[3],[4],[5],[6],[7],[8].Insomeoftheseworks,itappearedthat a J the model considered for the fading environment may substantially impact the performance of the proposed communication 4 schemes(see[9]).Inparticular,thechanneldiversity,bothspatialandtemporal,turnsouttobeakeyparameterfortheanalysis 2 of the various schemes. In the present paper, we address an a priori much easier scenario (previously considered in [10]). Instead of every source ] nodewillingto communicateeachto adifferentdestinationnode,we considerthebroadcastscenario,whereeachsourcenode T wishes to send some piece of information to all the other nodes in the network. This situation is to be encountered e.g. when I . control signals carrying channel state information should be broadcasted to the whole network. In this context, the broadcast s c nature of the wireless medium can only help relaying communications, so that the situation seems simpler to handle, if not [ trivial. What we show in the following is that even in this simpler scenario, the optimal communication performance highly 1 depends on the nature of the wireless medium. The conclusions we draw put again channel diversity to the forefront. But v whereas diversity was beneficial for establishing multiple parallel communication channels in the multiple-unicast scenario, it 5 turns out that in the presentcase, diversity is on the contrarydetrimentalto a properbroadcasting of information.A duality is 0 further established between the number of degrees of freedom available for multi-party communicationsand the beamforming 4 gain of broadcast transmissions, which allows for a better dissemination of information. At one end, in a rich scattering 6 0 environment, degrees of freedom are prominent, while beamforming is practically infeasible. At the other end, degrees of . freedom become a scarce resource, while high beamforming gains can be achieved via collaborative transmissions. 1 0 Our analysis relies on the simplistic line-of-sight fading model for signal attenuation over distance, where signal amplitude 6 attenuation is inversely proportionalto distance and phase shifts are also proportionalto distance. Yet, this model, along with 1 another parameter characterizing the sparsity of the network, allows to capture the different regimes mentioned above and to : v characterize the performance trade-offs. In addition, we would like to highlight here that despite the simplicity of the model, i the mathematical analysis needed to establish the result on the maximum achievable broadcast rate in the network requires a X preciseandcarefulstudyofthespectralnormofunconventionalrandommatrices,rarelystudiedinthemathematicalliterature. r a II. MODEL There are n nodes uniformly and independently distributed in a square of area A = nν, ν > 0. Every node wants to broadcast a different message to the whole network, and all nodes want to communicate at a common per user data rate r n bits/s/Hz. We denote byR =nr the resultingaggregatedata rate andwill often referto it simply as“broadcastrate” in the n n sequel. The broadcast capacity of the network, denoted as C , is defined as the maximum achievable aggregate data rate R . n n We assume that communication takes place over a flat fading channel with bandwidth W and that the signal Y [m] received j by the j-th node at time m is given by Y [m]= h X [m]+Z [m], j jk k j k X∈T where is the set of transmitting nodes, X [m] is the signal sent at time m by node k and Z [m] is additive white circularly k j T symmetric Gaussian noise (AWGN) of power spectral density N /2 Watts/Hz. We also assume a common average power 0 budget per node of P Watts, which implies that the signal X sent by node k is subject to an average power constraint k 1 E(X 2) P. In line-of-sight environment, the complex baseband-equivalentchannel gain h between transmit node k and k jk | | ≤ receive node j is given by exp(2πir /λ) h =√G jk , (1) jk r jk where G is Friis’ constant, λ is the carrier wavelength, and r is the distance between node k and node j. Let us finally jk define GP SNR = n1 ν, s − N W 0 which is the SNR available for a communication between two nodes at distance nν−21 in the network. We focus in the following on the low SNR regime, by which we mean that SNR = n γ for some constant γ > 0. This s − means that the power available at each node does not allow for a constant rate direct communication with a neighbor. This couldbethecasee.g.,inasensornetworkwithlowbatterynodes,orinasparsenetwork(largeν)withlongdistancesbetween neighboring nodes. In order to simplify notation, we choose new measurement units such that λ = 1 and G/(N W) = 1 in these units. This 0 allows us to write in particular that SNR =n1 νP. s − III. MAIN RESULT Before stating our main contribution, let us recall what is known for the multiple-unicast scenario [11]. In this case, the aggregated network throughput scales as1 nSNR if (A/λ2) n2 s ≥ T (√A/λ)SNR if n (A/λ2) n2 . n  s ∼ ≤ ≤ √nSNRs if 1 A/λ2 n ≤ ≤ Such an aggregate throughput is achieved bya hierarchical coooperative strategy involving network-wide distributed MIMO transmissionsinthefirsttwocases,whileasimplemulti-hoppingstrategyachievestheperformanceclaimedinthethirdregime. We therefore see that the wider the area is, the more degrees of freedom are available for communication in the network. The case where A n2 (corresponding to a sparse network of density O(1/n)) models the case where the phase shifts are ∼ large enough to ensure sufficient channel diversity and full degrees of freedom of MIMO transmissions. On the contrary, in the regime where A n (corresponding to a network of constant density), and even though this may seem surprising at first ∼ sight, phase shifts do not allow for efficient MIMO transmissions, so that multi-hopping becomes the best way to transfer information across the network. A totally different scenario awaits us in the broadcast case. Our main result is the following: the aggregate broadcast rate scales as min SNR ,1 if (A/λ2) n2 s { } ≥ R n n ∼min SNR ,1 if 1 (A/λ2) n2 s  √A/λ ≤ ≤ (cid:26)(cid:18) (cid:19) (cid:27) and is achieved by a simple broadcast transmission in the first case and by a multi-stage beamforming strategy in the second  case. The performance is further capped at 1, which means that such beamforming gains can only be obtained at low SNR. We see here for a sparse network of density O(1/n) (regime where A n2), no particular beamforming gain can be ∼ obtained, while the beamforming gain increases as the network gets denser and denser. Let us mention here that the result where the network is of constant density (A n) has been previously established in [12]. ∼ A final observation shows the duality of the two previous results: in the regime where A/λ2 n (that is, for networks of ≥ constant density or sparser) and at low SNR, we have T R n n =n SNR SNR s s which captures the fact that high beamforming gains can only be obtained at the expense of a reduced number of degrees of freedom (or reciprocally). 1uptologarithmic factors 2 s Fig.1. √n √nnetworkdivided intoclusters ofsizeM = nν/4 nν/2.Twoclusters ofsizeM placed onthesamehorizontal lineandseparated by × 2c1 × 4 distance d= nν4/2 pairupandstartback-and-forth beamforming. Thevertical separation between adjacent cluster pairsisc2nν/4+ǫ. IV. BROADCASTING STRATEGIESIN DIFFERENT REGIMES FirstnotethatundertheLOSmodel(1)andtheassumptionsmadeinthe SectionII, asimple timedivisionschemeachieves a broadcast (aggregate) rate R of order min(SNR ,1). Indeed, a rate of order 1 is obviously achieved at high SNR2. At n s low SNR (i.e. when SNR n γ for some γ > 0), each node can spare power while the others are transmitting, so as to s − ∼ compensateforthepathlossoforder1/nν betweenthesourcenodeandothernodeslocatedatdistanceatmost√2nν,leading to a broadcast rate of order R log(1+nP/nν) n1 νP =SNR . n − s ∼ ∼ In the following,we will see that, at low SNR, while the described simple TDMA based broadcastscheme is order-optimal for A n2, it is not optimal for sparse networks with area A < n2 (ν < 2) (for simplicity, as stated in Section II, we take ≥ λ=1). On theotherhand,the back-and-forthbeamformingscheme,presentedin [13], provesto be order-optimalforA n2. ≤ Asdescribedin[13],theback-and-forthbeamformingschemeinvolvessourcenodestakingturnstobroadcasttheirmessages. Eachtransmissionisfollowedbyaseriesofnetwork-wideback-and-forthtransmissionsthatreinforcethestrengthofthesignal, sothatattheend,everynodeisabletodecodethemessagesentfromthesource.Thereasonwhyback-and-forthtransmissions are useful for small area networks/dense networks is that in line-of-sight environment, nodes are able to (partly) align the transmitted signals so as to create a significant beamforming gain for each transmission (whereas this would not be the case in high scattering environment/sparse networks with i.i.d. fading coefficients). In short, the back-and-forth beamforming scheme is split into two phases: Phase 1. Broadcast Transmission. The source node broadcasts its message to the whole network. All the nodes receive a noisy version of the signal, which remains undecoded. This phase only requires one time slot. Phase 2. Back-and-ForthBeamformingwithTime Division.Uponreceivingthesignalfromthebroadcastingnode,nodes start multiple back-and-forthbeamformingtransmissionsbetween the two halvesof the networkto enhancethe strength of the signal. Although this simple scheme probably achieves the optimal performance claimed in Theorem IV.1 below, we lack the analytical tools to prove it. For this reason, we propose a time-division strategy, where clusters of size M = nν/4 nν/2 and 2c1 × 4 separated by horizontal distance d = nν/2 pair up for the back-and-forth transmissions. During each transmission, there are 4 Θ nν/4 ǫ cluster pairs operating in parallel, so Θ(n1 ǫ) nodes are communicating in total. The number of rounds needed − − to serve all nodes must therefore be Θ(nǫ). (cid:0) (cid:1) After each transmission, the signal received by a node in a given cluster is the sum of the signals coming from the facing cluster, of those coming from other clusters, and of the noise. We assume a sufficiently large vertical distance c nν/4+ǫ 2 separating any two cluster pairs. We show below that the broadcast rate between the operating clusters is Θ(n2−32νP) = Θ(n1−ν2SNRs). Since we only need Θ(nǫ) number of rounds to serve all clusters, phase 2 requires Θ(n−2+32ν+ǫP−1) time slots. As such,back-and-forthbeamformingachievesa broadcastrate ofΘ(n2−32ν−ǫP)=Θ(n1−ν2−ǫSNRs) bitspertimeslot. In view of the described scheme, we are able to state the following result. 2Wecoarselyapproximate logP by1here! 3 Theorem IV.1. For any ǫ>0, 0<ν <2, and P =O(n−2+32ν), the following broadcast rate Rn =Ω n2−32ν−ǫP =Ω n1−ν2−ǫSNRs is achievablewith high probability3 in the network(cid:16). As a conse(cid:17)quence(cid:0),when P =Ω((cid:1)n−2+32ν), a broadcastrate Rn =Ω(n−ǫ) is achievable with high probability. Before proceeding with the proof of the theorem, the following lemma provides an upper bound on the probability that the number of nodes inside each cluster deviates from its mean by a large factor. The proof is provided in the Appendix. Lemma IV.2. Let us consider a cluster of area M with M =nβ for some ν 1<β <ν. The number of nodes inside each cluster is then between ((1 δ)Mn1 ν, (1+δ)Mn1 ν) with probability large−r than 1 nν exp( ∆(δ)Mn1 ν) where ∆(δ) − − − − M − − is independent of n and satisfies ∆(δ)>0 for δ >0. As shownin Fig. 1, two clustersof size M = nν/4 nν/2 placed onthe same horizontalline andseparatedbydistanced= 2c1 × 4 nν/2 forma cluster pair. Duringthe back-and-forthbeamformingphase,there are manycluster pairsoperatingsimultaneously. 4 Given that the cluster width is nν/4 and the vertical separation between adjacent cluster pairs is c nν/4+ǫ, there are 2c1 2 nν/2 N = =Θ nν/4 ǫ C nν/4 +c nν/4+ǫ − 2c1 2 (cid:16) (cid:17) cluster pairs operating at the same time. Let and denote the receiving and the transmitting clusters of the i-th cluster i i R T pair, respectively. Twokeyingredientsforanalyzingthemulti-stageback-and-forthbeamformingschemearegiveninLemmaIV.3andLemma IV.4. The proofs are presented in the Appendix. Lemma IV.3. The maximum beamforming gain between the two clusters of the i-th cluster pair can be achieved by using a compensation of the phase shifts at the transmit side which is proportional to the horizontal positions of the nodes. More precisely, there exist a constant c > 0 (remember that c is inversely proportional to the width of cluster i) and a constant 1 1 K >0 such that the magnitude of the received signal at node j is lower bounded with high probability by 1 i ∈R exp(2πi(r x )) Mn1 ν jk k − − K , 1 (cid:12) rjk (cid:12)≥ d (cid:12)(cid:12)kX∈Ti (cid:12)(cid:12) where x denotes the horizontal position (cid:12)of node k. (cid:12) k (cid:12) (cid:12) Lemma IV.4. For every constantK >0, there exists a sufficiently large separating constantc >0 such that the magnitude 2 2 of interfering signals from the simultaneously operating cluster pairs at node j is upper bounded with high probability i ∈R by NC exp(2πi(r x )) Mn1 ν jk k − − K logn. (cid:12)(cid:12) rjk (cid:12)(cid:12)≤ 2 dnǫ (cid:12)(cid:12)Xll==1i kX∈Tl (cid:12)(cid:12) (cid:12) 6 (cid:12) Proof of Theorem IV.1: The firs(cid:12)t phase of the scheme results(cid:12)in noisy observationsof the message X at all nodes, which (cid:12) (cid:12) are given by Y(0) = SNR X +Z(0), k k k where E(X 2)=E(Z(0) 2)=1 and SNR is the signalp-to-noise ratio of the signal Y(0) received at the k-th node. In what | | | k | k k follows, we drop the index k from SNR and only write SNR = min SNR . Note that it does not make a difference at k k k { } which side of the cluster pairs the back-and-forthbeamformingstarts or ends. Hence, assume the left-handside clusters ignite the scheme by amplifying and forwarding the noisy observations of X to the right-hand side clusters. The signal received at node j is given by i ∈R NC exp(2πi(r x )) Y(1) = jk − k AY(0)+Z(1) (2) j r k j jk Xl=1kX∈Tl 3thatis,withprobability atleast1−O(cid:0)n1p(cid:1)asn→∞,wheretheexponent pisaslargeaswewant. 4 where A is the amplification factor (to be calculated later) and Z(1) is additive white Gaussian noise of variance Θ(1). We j start by applying Lemma IV.3 and Lemma IV.4 to lower bound NC exp(2πi(r x )) exp(2πi(r x )) NC exp(2πi(r x )) jk k jk k jk k − − − (cid:12) (cid:12) (cid:12) rjk (cid:12)≥(cid:12) rjk (cid:12)−(cid:12) rjk (cid:12) (cid:12)(cid:12)Xl=1kX∈Tl (cid:12)(cid:12) (cid:12)(cid:12)kX∈Ti (cid:12)(cid:12) (cid:12)(cid:12)Xll==1i kX∈Tl (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) logn Mn1−ν(cid:12)(cid:12) (cid:12)(cid:12) 6 Mn1−ν (cid:12)(cid:12) ≥ K1−K2 nǫ d =(cid:12)Θ d . (cid:12) (cid:18) (cid:19) (cid:18) (cid:19) For the sake of clarity, we can therefore approximate4 the expression in (2) as follows NC exp(2πi(r x )) NC exp(2πi(r x )) (1) jk k jk k (0) (1) Y = − A SNR X + − AZ +Z j r k r k j jk jk Xl=1kX∈Tl p Xl=1kX∈Tl AMn1 ν A√N Mn1 ν − √SNRX + C − Z(0)+Z(1) ≃ d d j AMn1 ν AMn1 ν N = − √SNRX + − C Z(0)+Z(1), d d Mn1 ν j r − where d NC exp(2πi(r x )) Z(0) = jk − k Z(0). √NCMn1−ν Xl=1kX∈Tl rjk k Note that E(Z(0) 2)=Θ(1). Repeating the same process t times in a back-and-forthmanner results in a final signal at node | | j in the left or the right cluster (depending on whether t is odd or even) that is given by i ∈R AMn1 ν t AMn1 ν t N Y(k) = − √SNRX + − C Z(0) j d d Mn1 ν (cid:18) (cid:19) (cid:18) (cid:19) r − +...+ AMn1−ν t−s NC Z(s)+...+Z(t), d Mn1 ν j (cid:18) (cid:19) r − where d NC exp(2πi(r x )) Z(s) = jk − k Z(s). √NCMn1−ν Xb=1kX∈Tb rjk k Note againthatE(Z(s) 2)=Θ(1), and Z(t) is additivewhite Gaussian noise of varianceΘ(1). Finally,note that LemmaIV.4 | | j ensures an upper bound on the beamforming gain of the noise signals, i.e., NC exp(2πi(r x )) exp(2πi(r x )) NC exp(2πi(r x )) logn Mn1 ν jk k jk k jk k − − − + − 1+K . (cid:12)(cid:12)(cid:12)Xl=1kX∈Tl rjk (cid:12)(cid:12)(cid:12)≤(cid:12)(cid:12)(cid:12)kX∈Ti rjk (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)Xll==1i kX∈Tl rjk (cid:12)(cid:12)(cid:12)(cid:12)≤(cid:18) 2 nǫ (cid:19) d (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 6 (cid:12) (n(cid:12)otice indeed that the first term(cid:12) in(cid:12)the middle expressionis t(cid:12)rivia(cid:12)lly upperboundedby Mn1 ν(cid:12)/d, as it containsM terms, all (cid:12) −(cid:12) less than 1/d). Now, we want the power of the signal to be of order 1, that is: AMn1 ν t 2 AMn1 ν 2t E − √SNRX = − SNR=Θ(1) (3)  (cid:18) d (cid:19) !  (cid:18) d (cid:19)  d  1 A=Θ SNR−2t . ⇒ Mn1 ν (cid:18) − (cid:19) Since at each round of TDMA cycle there are Θ N Mn1 ν =Θ n1 ǫ nodes transmitting, then every node will be active C − − Θ NCMn1−ν fraction of the time. As such, the amplification factor is given by n (cid:0) (cid:1) (cid:0) (cid:1) (cid:16) (cid:17) nν A=Θ τP , N M (cid:18)r C (cid:19) 4We make this approximation to lighten the notation and make the exposition clear, but needless to say, the whole analysis goes through without the approximation; itjustbecomesbarely readable. 5 where τ is the number of time slots between two consecutive transmissions, i.e. every τ time slots we have one transmission. Therefore, we have d nν 1 A=Θ SNR−2t =Θ τP Mn1 ν N M (cid:18) − (cid:19) (cid:18)r C (cid:19) 1 d 2 τ =O n−ǫSNR−1/t . ⇒ P (cid:18)Mn1−ν(cid:19) ! We can pick the number of back-and-forthtransmissions t sufficiently large to ensure that SNR−1t =O(nǫ), which results in 2 1 d 1 τ =O =O . P (cid:18)Mn1−ν(cid:19) ! (cid:18)n2−32νP(cid:19) Moreover, the noise power is given by t−1E AMn1−ν t−s NC Z(s) 2 +E Z(t) 2 tE AMn1−ν t NC Z(0) 2 +1 Xs=0  (cid:18) d (cid:19) rMn1−ν !  (cid:18)(cid:16) j (cid:17) (cid:19)≤  (cid:18) d (cid:19) rMn1−ν !    AMn1 ν 2t N  − C t +1 ≤ d Mn1 ν (cid:18) (cid:19) − (a) t+1=Θ(1), ≤ where (a) is true if and only if SNR = Ω MNn1C−ν = Ω(nν/2−1−ǫ) (check eq. (3)), which is true: Distance separating any two nodes in the network is as most √2nν, which implies that the SNR of the received signal at all the nodes in the network (cid:0) (cid:1) is Ω(nτP/nν)=Ω nν/2 1 . − Given that the required τ =O 1 , we can see that for P =O(n3ν/2 2) the broadcastrate between simultaneously (cid:0) (cid:1) n2−3ν/2P − operating clusters is Ω(n2 3ν/2P). Finally, applying TDMA of n = Θ(nǫ) steps ensures that X is successfully − (cid:0) (cid:1) NCMn1−ν decoded at all nodes and the broadcast rate R =Ω n2 3ν/2 ǫP . This completes the proof of the theorem. n − − (cid:0) (cid:1) V. OPTIMALITYOF THESCHEME We start with the general upper bound already established in [13] on the broadcast capacity of wireless networks at low SNR, which applies to a general fading matrix H. Theorem V.1. Let us consider a network of n nodes and let H be the n n matrix with h =0 on the diagonaland h = jj jk × the fading coefficient between node j and node k in the network. The broadcast capacity of such a network with n nodes is then upper bounded by C P H 2 n ≤ k k where P is the power available per node and H is the spectral norm (i.e. the largest singular value) of H. k k We now aim to specialize Theorem V.1 to line-of-sight fading, where the matrix H is given by 0 if j =k h = exp(2πir ) (4) jk  jk if j =k  rjk 6 The rest of the section is devoted to proving the proposition below which, together with Theorem V.1, shows the asymptotic  optimalityoftheback-and-forthbeamformingschemeforsmallareanetworks/densenetworks(0<ν <2)andtheasymptotic optimality of the simple TDMA based broadcast scheme for high scattering environment/sparse networks (ν 2) at low ≥ SNR and under LOS fading. Proposition V.2. Let H be the n n matrix given by (4). For every ε>0, there exists a constant c>0 such that × H 2 n2−32ν+ǫP =n1−ν2+ǫ SNRs if 0<ν <2 k k ≤(n1−ν+ǫP =nǫ SNRs if ν 2 ≥ with high probability as n gets large. Analyzing directly the asymptotic behavior of H reveals itself difficult. We therefore decompose our proof into simpler k k subproblems. The first building block of the proof is the following Lemma, which can be viewed as a generalization of the classical Gersˇgorin discs’ inequality. 6 Fig.2. √n×√nnetworksplitintoK clusters andnumberedinorder.Assuch,Rj ={j−√K−1,j−√K,j−√K+1,j−1,j,j+1,j+√K− 1,j+√K,j+√K+1 ,whichrepresents thecentersquarecontaining theclusterj andits8neighbors (markedinshades). } Lemma V.3. Let B be an n n matrix decomposed into blocks B , j,k =1,...,K, each of size M M, with n=KM. jk × × Then K K B max max B , max B jk kj k k≤ (1 j K k k 1 j K k k) ≤ ≤ k=1 ≤ ≤ k=1 X X The proof of this Lemma is relegated to the Appendix.The second buildingblock of this proofis the followinglemma, the proof of which is also given in the Appendix. Lemma V.4. Let H be the m m channel matrix between two square clusters of m nodes distributed uniformly at random, × each of area A=mν, ν >0, then b m2+ǫ m1+ǫ H 2 max , k k ≤ Ad d2 (cid:26) (cid:27) b m2+ǫ if 0<ν <2 Ad ≤(max mA2+dǫ,md12+ǫ if ν ≥2 n o with high probability as m gets large, where 2√A d A denotes the distance between the centers of the two clusters. ≤ ≤ Proof of Proposition V.2: First we consider the case where ν 2. The strategy for the proof is now the following: ≥ in order to bound H , we divide the matrix into smaller blocks, apply Lemma V.3 and Lemma V.4 in order to bound the k k off-diagonalterms H . For the diagonalterms H , we reapply Lemma V.3 and proceed in a recursive manner,until we jk jj k k k k reach small size blocks for which a loose estimate is sufficient to conclude. Note that a network with area A = nν has a density of n1 ν. This means that a cluster of area A = m nν 1 contains 0 − 1 1 − m nodes with high probability. Let us therefore decompose the network into K square clusters of area m nν 1 with m 1 1 1 − 1 nodes each. Without loss of generality, we assume each cluster has exactly m nodes and K =n/m =A /A . By Lemma 1 1 1 0 1 V.3, we obtain K1 K1 H max max H , max H (5) jk kj k k≤ (1≤j≤K1k=1k k 1≤j≤K1k=1k k) X X where the n n matrix H is decomposedinto blocks H , j,k =1,...,K , with H denoting the m m channelmatrix jk 1 jk 1 1 × × between cluster number j and cluster number k in the network. Let us also denote by d the corresponding inter-cluster jk distance, measured from the centers of these clusters. Based on Lemma V.4, we obtain H 2 max m21+ǫ ,m11+ǫ (=a) m11+ǫ k jkk ≤ (A1djk d2jk ) d2jk with high probability as m , where (a) follows from the fact that A /m = nν 1 nν/2 d k, since ν 2 1 1 1 − j (equivalently, m1 1 ). → ∞ ≥ ≥ ≥ A1 ≤ djk 7 Let us now fix j 1,...,K and define R = 1 k K : d < 2√A and S = 1 k K : d 2√A 1 j 1 jk 1 j 1 jk 1 ∈ { } { ≤ ≤ } { ≤ ≤ ≥ } (see Fig. 2). By the above inequality, we obtain K1 H H +√nǫ √m1 jk jk k k≤ k k d jk Xk=1 kX∈Rj kX∈Sj with high probability as m gets large. Observe that as there are 8t clusters or less at distance t√A from cluster j, so we 1 1 obtain √m √K1 √m K m 1 1 1 1 8t =O . kX∈Sj djk ≤ Xt=2 t√A1 r A1 ! There remains to upper bound the sum over R . Observe that this sum contains at most 9 terms: namely the term k = j j and the 8 terms corresponding to the 8 neighboring clusters of cluster j. It should then be observed that for each k R , j ∈ H H(R ) , where H(R ) is the 9m 9m matrix made of the 9 9 blocks H such that j ,j R . Finally, k jkk ≤ k j k j 1× 1 × j1,j2 1 2 ∈ j this leads to K1 K m H 9 H(R ) +8√nǫ 1 1 jk j k k≤ k k A k=1 r 1 X Using the symmetry of this bound and (5), we obtain K m H 9 max H(R ) +8√nǫ 1 1 (6) j k k≤ 1≤j≤K1k k r A1 A key observation is now the following: For all 1 j K , the 9M 9M matrix H(R ) has exactly the same structure 1 j ≤ ≤ × as the original matrix H. Therefore, without loss of generality, let us assume H = max H(R ) = H(R ) . Finally, to bound H , the same technique may be reused. This leads to the fokllow1king recurs1i≤vej≤sKol1uktion. j k k 1 k 1 k k K m H =O H +√nǫ 1 1 1 k k k k r A1 ! K m K m =O H +√nǫ 2 2 +√nǫ 1 1 2 k k r A2 r A1 ! l K m =O H +√nǫ t t l k k t=1r At ! X l =O H +√nǫ√n1 ν K , k lk − t! t=1 Xp where m denotes the number of nodes in a square cluster of area A . Moreover, K = A /A = m /m denotes the i i i i 1 i i 1 i number of square clusters of area A and m nodes in a square network of area A con−taining m − nodes (note that i i i 1 i 1 A = A = nν and m = n). Finally, H denotes the norm of the channel matrix of−the network with−square area A and 0 0 i i k k m nodes. i Note thatwe havea trivialboundon H . Applyforthis the slightly modifiedversionof the classical Gersˇgorininequality i k k (which is nothing but the statement of Lemma V.3 applied to the case M =1): ml ml ml 1 H max max (H ) , max (H ) = max l l jk l jk k k≤ (1≤j≤mlk=1| | 1≤j≤mlk=1| |) 1≤j≤ml k=1 rjk X X Xk6=j For any 1 j m , it holds with high probability that for c large enough, l ≤ ≤ ml 1 √ml ct logn Xkk=6=1j rjk ≤ Xt=1 tnν−21 =O(cid:16)√mln1−2ν logn(cid:17)=O(cid:16)n1−νpAllogn(cid:17), where the first inequality comesfromthe fact that at a distance tnν−21 there are at most ct clusters of area nν−1 with at most logn nodes each. This implies that H =O √nǫn1 ν√A for any ǫ>0. Therefore, we have l − l k k (cid:0) (cid:1) l A kHk=O √nǫn1−ν Al+√nǫ√n1−ν sAtt1!. p Xt=1 − 8 Upon optimizing over the Ai’s, we get Ai =nν−l+i1. Note that Ai is a decreasing function of i and A0 =nν. As such, for ν 2, we get the desired result ≥ H =O n1−2νnǫ+2(l1+1) , k k (cid:16) (cid:17) where for any ǫ >ǫ, we can pick l large enough so as ǫ <ǫ+ 1 (notice that ǫ and ǫ can be as small as we want). ′ ′ 2(l+1) ′ For 0 < ν < 2, we will take the following approach: We notice that a dense network can be seen as a superposition of sparse networks. In other words, we will look at a network with n nodes uniformly and independently distributed over the area nν, as the superposition of n1 ν/2 networkswith m=nν/2 nodes uniformlyand independentlydistributed over the area − nν =m2. Again, by Lemma V.3, we obtain n1−ν/2 n1−ν/2 H max max H , max H k k≤ 1 j n1−ν/2 k jkk 1 j n1−ν/2 k kjk  ≤ ≤ Xk=1 ≤ ≤ kX=1  wherethen n matrixH isdecomposedintoblocksHjk,j,k =1,...,n1−ν/2,with Hjk denotingthe m m channelmatrix × × between sparse network number j and sparse network number k. Since each of these sparse networks has area m2 with m nodes, we can apply the upper bound we got for ν =2, and j,k =1,...,n1 ν/2, obtain − ∀ Hjk =O m−21+ǫ =O n−ν4+2ǫ , k k which results in (cid:16) (cid:17) (cid:0) (cid:1) H =O n1−34ν+2ǫ . k k (cid:16) (cid:17) This finally proves Proposition V.2. VI. CONCLUSION In this work, we characterize the broadcast capacity of a wireless network at low SNR in line-of-sight environment and under various assumptions regarding the network density. The result exhibits a dichotomy between sparse networks, where nodecollaborationcan hardlyhelp enhancingcommunicationrates, andconstantdensitynetworks,wheresignificantgainscan be obtained via collaborative beamforming. VII. ACKNOWLEDGMENT S. Haddad’s work is supported by Swiss NSF Grant Nr. 200020-156669. APPENDIX Proof of Lemma IV.2: The number of nodes in a given cluster is the sum of n independently and identically distributed Bernoulli random variables B , with (B =1)=M/nν. Hence i i P n P Bi (1+δ)Mn1−ν ≥ ! i=1 X n =P exp s B exp(s(1+δ)Mn1 ν) i − !≥ ! i=1 X En(exp(sB ))exp( s(1+δ)Mn1 ν) 1 − ≤ − M M n = exp(s)+1 exp( s(1+δ)Mn1−ν) nν − nν − (cid:18) (cid:19) exp( Mn1 ν(s(1+δ) exp(s)+1))=exp( Mn1 ν∆ (δ)) − − + ≤ − − − where ∆ (δ) = (1+δ)log(1+δ) δ by choosing s = log(1+δ). The proof of the lower bound follows similarly by + − considering the random variables B . The conclusion follows from the union bound. i − Proof of Lemma IV.3: We present lower and upper boundson the distance r separating a receiving node j and a jk i ∈R transmitting node k . Denote by x , x , y , and y the horizontaland the verticalpositionsof nodesj and k, respectively i j k j k ∈T (as shown in Fig. 3). An easy lower bound on r is jk r x +x +d jk k j ≥ 9 Fig.3. Coordinate system. On the other hand, using the inequality √1+x 1+ x, we obtain ≤ 2 r = (x +x +d)2+(y y )2 jk k j j k − q (y y )2 j k =(x +x +d) 1+ − k j s (xk+xj +d)2 (y y )2 1 j k x +x +d+ − x +x +d+ . ≤ k j 2d ≤ k j 2c2 1 Therefore, 1 0 r x x d . ≤ jk − k− j − ≤ 2c2 1 After bounding r , we can proceed to the proof of the lemma as follows: jk exp(2πi(r x )) exp(2πi(r x x d)) jk k jk k j − = − − − (cid:12) rjk (cid:12) (cid:12) rjk (cid:12) (cid:12)(cid:12)kX∈Ti (cid:12)(cid:12) (cid:12)(cid:12)kX∈Ti (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) exp(2πi(r x x d(cid:12))) (cid:12) (cid:12) (cid:12) jk− k− j −(cid:12) ≥ℜ rjk ! kX∈Ti cos cπ21 K Mn1−ν, ≥ r(cid:16) (cid:17) ≥ 1 d jk kX∈Ti when the constant c is chosen sufficiently large so that cos π >0. 1 c2 1 Proof of Lemma IV.4: There are NC clusters transmittin(cid:16)g s(cid:17)imultaneously. Except for the horizontally adjacent cluster of a given cluster pair (i-th cluster pair), all the rest of the transmitting clusters are considered as interfering clusters (there are N 1 of these). With high probability,each cluster contains Θ(Mn1 ν) nodes. For the sake of clarity, we assume here that C − − every cluster contains exactly Mn1 ν nodes, but the argument holds in the general case. In this lemma, we upper bound the − magnitude of interfering signals from the simultaneously interfering clusters at node j as follows i ∈R NC exp(2πi(r x )) NC cos(2π(r x )) NC sin(2π(r x )) jk k jk k jk k − − + − (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) rjk (cid:12)≤(cid:12) rjk (cid:12) (cid:12) rjk (cid:12) (cid:12)(cid:12)Xll==1i kX∈Tl (cid:12)(cid:12) (cid:12)(cid:12)Xll==1i kX∈Tl (cid:12)(cid:12) (cid:12)(cid:12)Xll==1i kX∈Tl (cid:12)(cid:12) (cid:12) 6 (cid:12) (cid:12) 6 (cid:12) (cid:12) 6 (cid:12) We onlyupperb(cid:12)oundthe firstterm(cosineter(cid:12)ms)(cid:12)in theequationaboveasfol(cid:12)lows(cid:12) (we canupperboundthese(cid:12)condterm(sine (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) terms) in exactly the same fashion): NC cos(2π(r x )) NC cos(2π(r x )) cos(2π(r x )) NC cos(2π(r x )) jk − k jk − k E jk − k + E jk − k (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(cid:12)(cid:12)Xll==1i kX∈Tl rjk (cid:12)(cid:12)(cid:12)≤(cid:12)(cid:12)(cid:12)Xll==1i kX∈Tl rjk − (cid:18) rjk (cid:19)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)Xll==1i kX∈Tl (cid:18) rjk (cid:19)(cid:12)(cid:12)(cid:12) (cid:12) 6 (cid:12) (cid:12) 6 (cid:12) (cid:12) 6 (cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)NC cos(2π(r x )) cos(2π(r x )) (cid:12)(cid:12) (cid:12)(cid:12)NC cos(2π(r x )) (cid:12)(cid:12) jk − k E jk − k + E jk − k (cid:12) (cid:12) ≤(cid:12)(cid:12)(cid:12)Xll==1i kX∈Tl rjk − (cid:18) rjk (cid:19)(cid:12)(cid:12)(cid:12) Xll==1i (cid:12)(cid:12)(cid:12)kX∈Tl (cid:18) rjk (cid:19)(cid:12)(cid:12)(cid:12) (cid:12) 6 (cid:12) 6 (cid:12) (cid:12) (cid:12)(cid:12)NC cos(2π(r x )) cos(2π(r x )) (cid:12)(cid:12) NC (cid:12) cos(2π(r x )) (cid:12) jk − k E jk − k + E jk − k ≤(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Xll=6=1i kX∈Tl rjk − (cid:18) rjk (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Xll=6=1i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)kX∈Tl′ (cid:18) rjk (7)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 10

Communication Tradeoffs in Wireless Networks

Similar Communication Tradeoffs in Wireless Networks