Linear Capaity Saling in Wireless Networks: Beyon Physial Limits? Ayfer Özgür, Olivier Lévêque EPFL, Switzerlan {ayfer.ozgur, olivier.leveque}@epfl.h Davi Tse University of California at Berkeley tse@ees.berkeley.eu Abstrat We investigate the role of ooperation in wireless networks subjet to a spatial egrees of freeom limitation. To aress the worst ase senario, we onsier a free-spae lineof-sight type environment with no sattering an no faing. We ientify three qualitatively ifferent operating regimes that are etermine by how the area of the network A, normalize with respet to the wavelength λ, ompares to the number of users n. In networks with A/λ n, the limitation in spatial egrees of freeom oes not allow to ahieve a apaity saling better than n an this performane an be reaily ahieve by multihopping. This result has been reently shown in [7]. However, for networks with A/λ > n, the number of available egrees of freeom is minn, A/λ), larger that what an be ahieve by multi-hopping. We show that the optimal apaity saling in this regime is ahieve by hierarhial ooperation. In partiular, in networks with A/λ > n, hierarhial ooperation an ahieve linear saling. I. INTRODUCTION Multi-hop is the ommuniation arhiteture of urrent wireless networks suh as mesh or a ho networks. Pakets are sent from eah soure to its estination via multiple relay noes. Eah relay eoes the pakets sent from the previous relay an forwars them to the next relay. Can more sophistiate ooperation between noes signifiantly inrease apaity of suh networks? This is an important question onerning future ommuniation arhitetures for suh networks, an information theory has been brought to bear to try to she some light on this question. Aopting the saling law formulation of Gupta an Kumar [], muh fous has been on the asymptoti regime where the number of noes is large. Two iametrially opposite answers have emerge: ) Capaity an be signifiantly improve when noes form istribute MIMO arrays via an intelligent ooperation arhiteture [3], [4]. The total egrees of freeom in the network is n, the number of noes, an in regimes where power is not a limiting fator, the apaity an sale almost linearly with n. 2) The total egrees of freeom in the network is not n but is atually upper boune by n ue to the spatial onstraints impose by the physial hannel [7]. Nearestneighbor multi-hop is optimal to ahieve this saling []. This is no mathematial ontraition between these two sets of results. They are base on two ifferent hannel moels. The key ifferene is the assumption on the phases of the hannel gains between the noes. [3], [4] assume that the phases are uniform an inepenent aross the ifferent hannel gains. [7], on the other han, starts from physial priniples an regars the phases as funtions of the loations of the noes. While the physial hannel moel use in [7] is more funamental, the i.i.. phase moel is also wiely aepte in wireless ommuniation engineering, partiularly for noes in far fiel from eah other. Is there a way to reonile the two sets of results? A eeper look at [7] provies a lue. The spatial egrees of freeom limitation in [7] is atually itate by the iameter of the network rather than the number of noes. More preisely, the spatial egrees of freeom in the network are limite by A/λ, where A is the area of the network an λ is the arrier frequeny. This number an be heuristially thought of as an upper boun to the total egrees of freeom in the network as a whole an puts a limitation on the maximum possible ooperation gain. The onlusion that the apaity sales like n omes from the assumption that the ensity of noes is fixe as the number of noes n grows, so that A/λ is proportional to n. But for atual networks, there an be a huge ifferene between A/λ an n. Take an example of a network serving n =, users on a ampus of km 2, operating at 3 GHz: A/λ =, while n is only, two orers of magnitue smaller. So while multi-hop an ahieve a total throughput of the orer of bits/s/hz, there is still a lot of potential for ooperation gain, sine the spatial egrees of freeom upper boun is,. So the ultimate ooperation gain is limite by A/λ, while multi-hop performane epens on the number of noes n only an not on A/λ. But the number of noes an the area are two inepenent parameters of a network, eah of whih an take on a wie range of values. To yiel a omplete piture of whether ooperation an help, the key is to remove the artifiial oupling between these two parameters an analyze the apaity in terms of the two parameters separately. This is the goal of the present paper. We fous on a physial hannel moel similar to that use in [7], but with only a line-of-sight hannel between eah pair of noes, a ase in whih spatial limitation is expete to be the most severe. Our main result is that in the regime when n an A/λ are both large, the apaity of the network is approximately ) n, A max minn, λ ). )
oringly, the optimal operation of the network falls into three ifferent operating regimes: ) A/λ n: The number of spatial egrees of freeom is too small, ooperation is useless an nearest neighbor multi-hopping is optimal. 2) A/λ > n: The number of spatial egrees of freeom is n, ooperation is very useful, an the optimal performane an be ahieve by the same hierarhial ooperation sheme introue in [4]. Spatial egree of freeom limitation oes not ome into play an the performane is as though the phases are i.i.. uniform aross the noes. 3) n A/λ n: The number of egrees of freeom is smaller than n, so the spatial limitation is felt, but larger than what an be ahieve by simple multi-hopping. A moifiation of the hierarhial ooperation sheme ahieves optimal saling in this regime. Regime ) is essentially the onlusion of [7]; regime 2) is essentially the onlusion of [4] in the ase when power is not a limiting fator). Thus, the valiity of the results in these papers is not universal but epens on the relationship between n an A/λ. The upper boun of A/λ on the spatial egrees of freeom of the network is alreay establishe by [7]. The main tehnial ontributions of the present paper are two-fole: ) we show that there are atually minn, A/λ) spatial egrees of freeom available in the physial hannel moel when A/λ n; 2) we show that hierarhial ooperation an ahieve these egrees of freeom. Both mathematially an philosophially, the present paper follows the same spirit of [5]. [5] avoates a shift of the large networks researh agena from seeking a single universal saling law, where the number of noes n sales with all systems parameters ouple with n in a speifi way, to seeking a multi-parameter family of saling laws, where the key parameters are eouple an many ifferent limits with respet to these parameters are taken. A single saling law with a partiular oupling between parameters is often arbitrary an too restritive to over the wie ranges that the multiple parameters of the network an take on. The speifi parameters that were eouple in [5] were the number of noes an the amount of power available. The urrent paper follows the approah of [5], but fouses on the number of noes an the area of the network, while assuming there is a suffiient amount of power available that it is not limiting performane. A future goal of this researh program is to investigate the epenene of the apaity on the number of noes, the area of the network an the amount of power all together. II. MODEL There are n noes with transmitting an reeiving apabilities that are uniformly an inepenently istribute in a retangle of area A A. Eah noe has an average transmit power buget of P Watts an the network is alloate a total banwith of W Hertz aroun a arrier frequeny of f, f W. Every noe is both a soure an a estination for some traffi request. The soures an estinations are ranomly paire up one-to-one into n soure-estination pairs without any onsieration on noe loations. Eah soure wants to ommuniate to its estination at the same rate R bits/s/hz. The aggregate throughput of the system is T = nr. We assume that ommuniation takes plae in free-spae line of sight type environment an the omplex basebanequivalent hannel gain between noe i an noe k is given by H ik = G ej2πr ik/λ r ik 2) where r ik is the istane between the noes i an k an λ is the arrier wavelength. Note that the loations of the users are rawn ranomly but remain fixe over the uration of the ommuniation. Therefore for a given realization of the network, the hannel oeffiients in 2) are eterministi. The parameter G is given by the Friis formula, G = G Tx G Rx λ 2 6π 2, 3) where G Tx an G Rx are the transmitter an reeiver antenna gains respetively. The isrete-time omplex baseban signal reeive by noe i at time m is given by n Y i [m] = H ik X k [m] + Z i [m] 4) k=, k i where X k [m] is the signal sent by noe k at time m subjet to an average power onstraint E X k 2 ) P/W an Z i [m] is omplex white irularly symmetri Gaussian noise of variane N. The moel in 2), 3) orrespons to free-spae propagation. It is equivalent to the moel in Setion IV of [7] but with no satterers. We onsier the ase of no satterers sine the spatial egrees of freeom limitation is expete to be most severe in this ase. It has been shown in [5] that a wireless aho network is power-limite when the long-range SNR in the network is smaller than B an the long range SNR has been ientifie as GP SNR l := n N W A). 5) α For the urrent ase α = 2, whih implies that SNR l = SNR s, where SNR s is the SNR in a point-to-point transmission over the typial nearest neighbor istane in the network. See also [6].) In the present paper, our goal is to onentrate on the effet of the spatial egrees of freeom limitation on the apaity of wireless aho networks. To be able to solely onentrate on this fator, we assume there is no power limitation in our network. Formally, we assume that P an W are suh that SNR l > B, 6) for every A an n. For the urrent ase of α = 2, the onition an be equivalently state as SNR s > B. When this onition fails to hol, the network beomes power limite an the behavior of the apaity as well as optimal operation an be signifiantly ifferent.
D T A D R A Fig.. Two square lusters of area separate by a istane. III. MAIN RESULT The main result of [7] is to show that uner the network an the hannel moel esribe in the previous setion with the aitional restrition A = n, the apaity of the wireless network is upper boune by T K n log n) 2, with high probability, where K > is a onstant inepenent of n. Coupling the area of the network a priori with the number of noes in the network is restritive an oes not allow to eue the nature of the limitation impose here. A relatively straightforwar generalization of the analysis in [7] gives the following result. Let us efine the normalize area of the network with respet to the wavelength λ as, A := A λ 2. Uner the network an hannel moel esribe in the previous setion, the apaity of the wireless network is upper boune by { K T min n log n) 2, ) ) 2 A log A if A > n K n log n) 2 if A n with high probability where K > is a onstant inepenent of n an A. For A n, this result says that the maximum ahievable apaity is of orer n, whih is ahievable by a simple multi-hopping sheme []. For A > n, the ahievability remains an open issue so far. The following theorem is the main ontribution of the present paper. Theorem 3.: Consier the network an the hannel moel esribe in the previous setion an assume A > n, the total throughput ahieve by hierarhial ooperation is lower boune by, T K 2 minn, ) ε A ) with high probability, for any ε > an a onstant K 2 > inepenent of n an A. The theorem an be interprete as follows: When A > n 2, hierarhial ooperation an ahieve an aggregate throughput T > K 2 n ε for any ε >. When A < n 2, hierarhial ooperation an ahieve an aggregate throughput T > K 2 A /2+ε. Note that this throughput is larger than n, when A > n. With probability as n. IV. HIERARCHICAL COOPERATION IN LOS ENVIRONMENTS The proof of Theorem 3. relies on the following lemma whih establishes a lower boun on the apaity of a MIMO transmission between two lusters of noes. For notational simpliity, in the sequel we assume that all the istanes in the network are normalize with respet to the arrier wavelength λ. Note that when the istanes r ik are expresse in wavelength units, the hannel moel in 2), 3) takes the simplifie form, H ik = G ej2πr ik r ik, G = G Tx G Rx 6π 2. 7) Lemma 4.: Consier two square lusters of area separate by a istane see Figure ), with eah luster ontaining M noes istribute uniformly at ranom over. Let, an the noes in the transmit luster D T perform inepenent signalling with power P /M suh that GP > B. 8) N W2 Then the apaity of the MIMO hannel from the transmit luster D T to the reeive luster D R is lower boune by C MIMO = log et I + P )) N W M HH ) / K 3 min M, log /) with high probabilityfor some onstant K 3 > inepenent of M, an. The lemma is the analog of Lemma 4.3 in [4] whih lower bouns the apaity of a MIMO transmission between two lusters of noes uner the i.i.. phase moel. With i.i.. phases, the apaity of the MIMO transmission sales linearly in M. The onition 8) ensures that the MIMO transmission is not power limite. For the LOS ase, we have the aitional / log/) term, whih orrespons to the spatial egrees of freeom between the two lusters. When this term is smaller than M, the apaity of the MIMO transmission is not any more linear in M. This in turn egraes the performane of the hierarhial ooperation sheme whih is base on suh MIMO transmissions. The apaity of a MIMO transmission between two lusters uner the urrent LOS hannel moel has been investigate earlier in [8]. The result state in Theorem of [8] is equivalent to Lemma 4. above. However, the proof of Theorem in [8] is base on an approximation whih is not fully justifie. Through private ommuniation, we have been informe of a follow-up work [9] by the same authors, whih inepenently from our urrent paper, also investigates the performane of the hierarhial ooperation sheme uner the LOS hannel moel. Next, we investigate the performane of the hierarhial ooperation sheme an show how Lemma 4. allows to prove the result in Theorem 3.. The ore of the proof is the following reursion lemma.
Lemma 4.2: Consier a network of n noes uniformly istribute over an area A > n an the available power P per noe satisfies 6). Assume that there exists a ommuniation sheme for this network that ahieves an aggregate throughput T K 4 minn, A ) b with high probability for some b < an a onstant K 4 > inepenent of n an A. Then, we an onstrut another sheme for this network that ahieves a higher aggregate throughput T K 5 minn, A ) 2 b ε with high probability for any ε > an a onstant K 5 > inepenent of n an A. As soon as we have a sheme to start with, Lemma 4.2 an be applie reursively, yieling a sheme that ahieves higher throughput at eah step of the reursion. Note that 2 b > b for b <. We first show that a simple timesharing strategy between the soure-estination pairs TDMA) satisfies the onitions of the lemma with b =. Note that with TDMA, eah soure noe transmits only a fration /n of the total time of ommuniation. Hene when ative, eah soure noe an transmit with elevate power np an still satisfy its average power onstraint P. This yiels an SNR larger than SNR l in 5) for eah transmission, hene a onstant rate. Therefore, the aggregate throughput ahieve by TDMA is onstant inepenent of n an A. Starting with TDMA, b =, an applying Lemma 4.2 reursively h times, we get a hierarhial sheme that ahieves an aggregate throughput of orer minn, A ) h h+ ε for any ε >. Therefore given any ε >, we an hoose ε = ε/2 h an h suh that h+ ε/2 an we a get a sheme that ahieves the performane in Theorem 3.. Proof of Lemma 4.2: We will prove the lemma by onentrating separately on the two ases A > n 2 an n < A n 2. In the first ase, we provie a brief overview of the three-phase sheme from Lemma 3. in [4] an verify that it ahieves the same performane in [4] uner the urrent eterministi phase moel. The reaer shoul refer to [4] for a preise analysis. For the ase n < A n 2, a moifiation of the sheme is require to ahieve the performane given in Lemma 4.2. A. A > n 2 Let us ivie the network into square lusters of area. Eah luster ontains approximately M = A n noes. A partiular soure noe s sens M bits to its estination noe in three steps: S) Noe s first istributes its M bits among the M noes in its luster, one bit for eah noe; S2) These noes together an then form a istribute transmit antenna array, sening the M bits simultaneously to the estination luster where lies; S3) Eah noe in the estination luster observes the MIMO transmission in the previous phase; it quantizes eah observation to Q bits, with a fixe Q, an ships them to, whih an then o joint MIMO proessing of all the quantize observations an eoe the M transmitte bits from s. From the network point of view, all soure-estination pairs have to eventually aomplish these three steps. Step 2 is longrange ommuniation an only one soure-estination pair an operate at a time. Steps an 3 involve loal ommuniation an an be parallelize aross lusters. Sine there are M soure noes in every luster, this gives a total traffi of exhanging MM ) M 2 bits insie eah luster in phase. We an hanle this traffi by setting up M sub-phases, an assigning M pairs in eah sub-phase to ommuniate their bit. The traffi to be hanle at eah sub-phase is similar to our original network ommuniation problem with n users on an area A, but now instea, we have M users on area. We hanle this traffi using the ommuniation sheme given in Lemma 4.2. Note that if this sheme ahieves an aggregate throughput K 4 minn, A ) b in the network of n noes an area A, it will ahieve an aggregate rate K 4 minm, ) b insie the lusters of M noes an area. 2 This an be verifie by heking that the lusters of M noes an area satisfy the onitions of the lemma. We have > M for the lusters if A > n for the original network an SNR l M, ) = M GP = SNR l > B N W if P satisfies 6). Moreover when A > n 2, we have > M 2, so the performane of the sheme is K 4 minm, ) b = M b. The traffi in the thir phase is hanle similarly to the first phase. Then, we nee: M 2 b /K 4 time slots to omplete phase all over the network; We hanle the traffi in M subphases, eah subphase is omplete in M b /K 4 time-slots. n/k 3 time-slots to omplete the suessive MIMO transmissions in the seon phase, if the istribute MIMO transmissions between any two lusters an ahieve a rate of K 3 M bits/time-slot; We perform one MIMO transmission for eah of the n soure-estination pairs in the network. QM 2 b /K 3 K 4 time slots to omplete phase 3 all over the network; The traffi in the thir phase is symmetrial to the traffi in the first phase, but larger by a fator of Q/K 3. This fator omes from the fat that eah MIMO transmission lasts /K 3 time slots, an eah of the orresponing /K 3 observations is quantize to Q bits. In [4], it is shown that eah estination noe is able to eoe the transmitte bits from its soure noe from the M quantize signals it gathers by the en of Phase 3. Thus, the aggregate throughput ahieve by the sheme an be alulate as follows: eah soure noe is able to transmit M bits to its estination noe, hene nm bits in total are elivere 2 We ignore the performane loss ue to inter-luster interferene sine it oes not hange the saling law. The reaer is referre to [6] for etails.
to their estinations in M 2 b /K 4 + n/k 3 + QM 2 b /K 3 K 4 time slots, yieling an aggregate throughput of nm M 2 b /K 4 + n/k 3 + QM 2 b bits/time-slot. /K 3 K 4 Choosing M = n 2 b to maximize this expression yiels an aggregate throughput T = K 5 n 2 b for a onstant K5 >. Note that this throughput an only be ahieve if the MIMO transmissions in phase 2 ahieve a rate linear in M. The rate of the MIMO transmissions are lowerboune in Theorem 4. for the eterministi phase moel uner ertain onitions. The luster areas an the separation between the lusters shoul satisfy the onition an the users shoul transmit with power satisfying onition 8). It is easy to verify that. Note that is always true unless the ommuniating lusters are neighbors. 3 Let us verify that the power onition 8) for the MIMO transmission an be satisfie uner the average power onstraint P per noe satisfying 6). In the seon phase, the MIMO transmissions between lusters are performe suessively an eah noe in the network transmits only M/n of the time. Therefore when ative, eah noe an transmit with elevate power np/m an still satisfy its average power onstraint P. Observe that if P = np, the onition 8) is satisfie given 6) an the fat that < A. Therefore, Theorem 4. lowerbouns the rate of the MIMO transmissions in the seon phase. The lower boun is linear A in M if / loga M. If /) > n 2, using = MA n an A, we obtain for suffiiently large M, / log /) M A /n logm A /n) M ε for any ε >. The ε is introue to ompensate for the logarithmi term an in turn yiels an n ε egraation in the overall throughput as state in Lemma 4.2. This onlues the proof of the lemma for networks with A > n 2. B. n < A n 2 In the ase n < A n 2, the proof of the lemma iffers from the earlier ase A > n 2 in two aspets. When n < A n 2, the MIMO transmissions between the lusters are limite in spatial egrees of freeom. More preisely, in Theorem 4., the performane is lower boune by the seon / log/) term an it is not anymore linear in M. This fat requires a moifiation in the operation of this phase. The seon ifferene is the following: We have seen that when A > n 2 for the original network, we have > M 2 for the smaller lusters. In other wors, when the network is not spatial egrees of freeom limite at the largest sale, it is not spatial egrees of freeom limite at any sale. In the urrent ase, when n < A n 2, the network is limite in spatial egrees of freeom at the largest sale, but the smaller lusters may or may not be spatial egrees of freeom limite. 3 The speial ase of neighboring lusters is exlue from the urrent isussion an an be hanle separately as in [4]. More preisely, for a luster of smaller size, we an either have M < M 2 or > M 2. This fat requires a more areful analysis. In partiular, we separately onsier the two ases n < A n 24 b) 5 2b an n 24 b) 5 2b < A n 2. ) n 24 b) 5 2b < A n 2 : As before, we ivie the network into lusters of area that ontain M = n /A noes an the goal again is to aomplish steps S-S2-S3 for every soure-estination pair in the network. We hoose the luster size in the following partiular way, M = n 2 2 b A 22 b). 9) This is a vali hoie in the sense that M < n, in partiular M < n 2 b given the onition A n 2 for the network. The onition n 24 b) 5 2b < A ensures that > M 2. Therefore as before, the sheme given in the hypothesis of Lemma 4.2 ahieves an aggregate throughput K 4 minm, ) b = M b when use insie the lusters of area an number of noes M. We use this sheme to hanle the traffi insie the lusters in phases an 3 as before. In the seon phase, the MIMO transmissions ahieve a rate A / log /) / log / A ). Sine this rate is not linear in M, the MIMO transmissions for eah soure-estination pair an not be omplete in onstant number of time-slots as before. In orer for these MIMO transmissions of low rate not to result in too many MIMO observations in the thir phase, we introue the following moifiation to step S2). Let M = / A log / A ). ) We ranomly ivie the M noes in the soure luster to M/M groups eah ontaining M noes. We o the same ivision also in the estination luster. We ranomly assoiate one-to-one the M/M groups in the soure luster with the M/M groups in the estination luster. The earlier M M MIMO transmission between the soure an the estination luster is now ivie into M/M suessive MIMO transmissions, eah of size M M. In eah of these M M MIMO transmissions, a group of M noes in the soure luster are simultaneously transmitting their bits to their orresponing group in the estination luster. Note that these M M MIMO transmissions are not limite in spatial egrees of freeom, preisely ue to our hoie for M in ). We will later verify that these M M MIMO transmissions ahieve a rate K 3 M. If this is the ase, we nee: M 2 b /K 4 time slots to omplete phase all over the network; n M/M /K 3 time-slots to omplete the suessive MIMO transmissions in the seon phase, if the istribute M M MIMO transmissions between any two groups an ahieve a rate of K 3 M bits/time-slot;
QM 2 b /K 3 K 4 time slots to omplete phase 3 all over the network; Note that although eah luster reeives M M/M MIMO transmissions in total, M/M MIMO transmissions per eah estination noe in the luster, eah noe has one MIMO observation of uration /K 3 time-slots for eah of the other noes. The moifiation in the seon phase is preisely mae to ensure this fat. Thus, the aggregate throughput ahieve by the sheme is given by nm M 2 b /K 4 + nm/m K 3 + QM 2 b /K 3 K 4 ) bits per time-slot. It an be verifie that for the hoie of the luster size in 9), we have M 2 b = nm /. A The three terms in the enominator of ) are orer-wise equal or in other wors, 9) is the luster size that maximizes the throughput expression in ). This yiels an aggregate throughput T = K 5 M = K 5 A A ε = K 5 n b 2 b A b 22 b) A ε, for a onstant K 5 > an for any ε >, whih is introue to ompensate for the logarithmi term in ). It an be verifie that when A n 2 the above throughput, T K 5 A ) 2 b ε whih is the performane laime in the lemma. It remains to verify that we an ahieve a rate K 3 M in the M M MIMO transmissions between the two lusters of area. Note that sine the M noes in eah group are hosen ranomly among the M noes in eah luster, without any onsieration on noe loations, they are uniformly an inepenently istribute over the area. It an be reaily verifie that the onition in Theorem 4. is satisfie. It remains to verify that we an transmit with power P /M suh that P satisfies 8). Note that ue to the extra time ivision between the M/M istint groups in eah luster, eah noe is transmitting in only M /M of the total transmission time of the luster. On the other han, ue to the time sharing between the lusters in the seon phase, eah luster is only ative in a fration M/n of the total ompletion time of the phase. Therefore uring the M M MIMO transmissions, the noes in the transmit group an transmit with elevate power np/m an still satisfy their average power onstraint P. This, in turn, means that they an satisfy the power requirement 8) in Theorem 4.. 2) n < A n 24 b) 5 2b : In this ase, we hoose the luster area as = A 3 4 b. 2) For this hoie, the urrent onition n < A n 24 b) 5 2b on the network gives M < M 2. This implies that, the sheme given in the hypothesis of Lemma 4.2 an now ahieve an aggregate throughput K 4 minm, ) b = ) b when use insie the lusters of area an number of noes M. Applying exatly the sheme in the earlier ase ), we now get an aggregate throughput nm. M 2 A b/2 /K 4 + nm/m K 3 + QM 2 A b/2 /K 3 K 4 The three terms in the enominator of this expression are orer-wise equal for the luster area given in 2). Therefore, the throughput ahieve is given by T = K 5 M = K 5 A 2+b 24 b) A ε K 5 A ) 2 b ε, for a onstant K 5 > an any ε >. The last inequality follows from the fat that b <. Combining the onlusions of Setions IV-A an IV-B above ompletes the proof of Lemma 4.2. APPENDIX A PROOF OF LEMMA 4. Lemma 4. will be proven in two steps. We first lower boun the expete apaity of the MIMO hannel over ranom noe positions an then show that for a ranom realization of the noe positions, the apaity of the orresponing MIMO hannel is not that ifferent from its expete value. We formally state these two results in the following lemmas. Lemma A.: The expete apaity C MIMO of the MIMO hannel in Lemma 4. is lower boune by EC MIMO ) = Elog et I + P /M)HH ) ) ) / K 3 min M,, log /) for a onstant K 3 >, where the expetation is taken over the inepenent an uniform istribution of noe positions over the transmit an reeive omains of area A). Lemma A.2: Let s = min M,, for any t > / log/) P C MIMO EC MIMO ) > t) e 2t2 s. Choosing t = s /2+ε2, ε 2 >, the probability in the seon lemma ereases to zero for inreasing s. This implies that the eviations of C MIMO from EC MIMO ) are, at most, of the orer of s. Therefore ombining the results of these two lemmas yiels the result given in Lemma 4.. In the sequel, we prove Lemma A.. The proof of Lemma A.2 losely follows the proof of Proposition 5.2 in [2] an is skippe ue to spae limitations. Proof of Lemma A.: For notational onveniene, we start by efining f ik = r ik e j 2πr ik = x k w i ej 2π x k w i 3) where r ik enotes the istane between the noes k D T an i D R loate at positions x k an w i respetively. Note that r ik + 2 2 /), an therefore + 2 2 /) f ik, 4)
where := +2 2) an the first inequality follows from the fat that. The first ingreient of the proof of Lemma A. is the Paley- Zygmun inequality use in [4] to prove Lemma 4.3. We have EC MIMO ) = E log et I + P )) N W M HH = E log et I + GP )) N W 2 M FF = M E log + GP )) N W 2 λ M log + GP ) N W 2 t Pλ > t ) for any t >, where λ is an eigenvalue of /M)FF pike uniformly at ranom. By Paley-Zygmun s inequality, if < t < Eλ), we have EC MIMO ) M log + GP ) Eλ) t) 2 N W 2 t Eλ 2 ) Given 3), we have Eλ) = M 2 E trff ) ) = M 2 M E f ik 2 ) 2. i,k= Eλ 2 ) = M 3 EtrFF FF )) = M M 3 Ef ik flk f lmfim ) 2 + M 3 i,k,l,m= M i,k,l,m= i l,k m Ef ik f lk f lmf im ) 2 + M S where the last inequality follows from the upper boun in 4). S = Ef aa fba f bb fab ) where a, b are two ifferent inies notie that S oes not epen on the speifi hoie of a an b). See Figure 2. Choosing then t = 2 /2, we obtain EC MIMO ) M 4 /4) log + GP 2 ) 2N W 2 2 + M S K 3 min M, ) S for a onstant K 3 > inepenent of M an S if GP N W 2 > B. The quantity S, whih takes values between an, itates therefore the apaity saling. In the ase where the hannel matrix entries f ik are i.i.. phases, S =, so the apaity EC MIMO ) is of orer M. At the other en, if we onsier the LOS hannel moel in 3) in the senario where noes are plae on a single straight line, then a simple omputation shows that S =, so that EC MIMO ) is of orer in this ase, we know that the matrix F is also rank one, so the lower boun mathes the upper boun on the apaity, up to a log M term). The problem we are looking at lies between these two Ay x xb D T h ab xa h bb haa h ba w b wa D R Fig. 2. S = Ef aa f ba f bb f ab ) Ax Fig. 3. Coorinate system. Aw extremes. Our aim in the following is to show that if both A an grow large an, then there exists K 3 > inepenent of an, suh that ) S K 3 log. 5) This implies that ) / EC MIMO ) K 3 min M, log /) w Az whih ompletes the proof. The rest of the setion is evote to proving 5). Let us first expliitly write the expression for S. We have S = Ef aa fba f bb fab) = A 4 x a x b w a w b ρ e j 2π 6) D T D T D R D R where = x a w a x a w b + x b w b x b w a, 7) ρ = x a w a x a w b x b w b x b w a ). 8) We first erive the result 5) by approximating the istane in 3) in the regime. This approximate analysis aptures most of the intuitions for the preise erivation whih is given afterwars. Consier two noes at positions x = x, y) D T an w = + w, z) D R, where x, y, w, z [, ] see Figure 3). Using the assumption that, we obtain x w = + x + w)) 2 + y z) 2 + x + w) + y z)2 2 whih in turn implies = x a w a x a w b + x b w b x b w a 2 y a z a ) 2 y a z b ) 2 + y b z b ) 2 y b z a ) 2 ) = y b y a )z b z a )
Next, let us also make the approximation that ρ in 8): this is atually assuming that the spatial egrees of freeom between the two lusters are mainly etermine by the phases of the hannel oeffiients an not so muh by the amplitues. We will see below that this intuition is orret. These two suessive approximations lea to the following expression for S: S S = j 2π y a y b z a z b e y b y a) z b z a) = 2 j2π y a y b z a z b e y b y a) z b z a), y a where the seon equation follows from the symmetry of the integran. Note that this expression oes not epen on the horizontal positions of the noes. This an be interprete as follows. Provie the above approximation is vali, the MIMO apaity saling between two lusters of M noes separate by a istane is the same, be the noes uniformly istribute on two squares of area or on two parallel vertial) lines of length. This result is of interest in itself an an be proven rigorously. We show below that the above integral is inee of orer /. Let us ompute the first integral, whih yiels 2πj z b e y b y a)z b z a) = j2π y b y a ) This implies that j 2π z b e 2π e j y b y a)z b z a) z b = y b y a)z b z a) z b = K 6 y b y a for a onstant K 6 inepenent of an. We an ivie the integration over y a an y b into two parts, = j2π y a y b z a z b e y b y a)z b z a) y a ya+ε 3) ε3 ) y a y b + y a y a z a y a+ε 3 j2π z b e y b y a)z b z a), for any < ε 3 <. The first term an be simply boune by ε 3, whih yiels the following upper boun for S S 2ε 3 + 2K 6 ε3 2ε + 2K 6 log/ε 3 ) y a So hoosing ε 3 = /, we finally obtain S S K 3 y a+ε 3 y b y b y a log /) for a onstant K 3 inepenent of an. We will next prove 5) without making use of the above approximations.. Proof of Inequality 5): We start again with the expression for S in 6). Note that ue to the symmetry of an ρ in w a an w b, we an upper boun 6) as S 4 A 4 x a x b D T D T DR w ej 2π xa w x b w ) x a w x b w Expressing this upper boun more expliitly in the oorinate system in Figure 3, we obtain the following upper boun for S, x a y a x b y b w z ej 2π g 2 a,bw,z) G a,b w, z) 9) where an g a,b w, z) = + x a + w)) 2 + y a z) 2 + x b + w)) 2 + y b z) 2. G a,b w, z) = 2 + x a + w)) 2 + y a z) 2 + x b + w)) 2 + y b z) 2. Let us first fous on the integral insie the square in 9). The key iea behin the next steps of the proof is ontaine in the following two lemmas. Lemma A.3: Let g : [, ] R be a C 2 funtion suh that g z) > for all z [, ] an g hanges sign at most twie on [, ] say e.g. g z) in [z, z + ] an g z) outsie). Let also G : [, ] R be a C funtion suh that Gz) 2 > an G z) hanges sign at most twie on [, ]. Then 2πgz) ej z Gz) 4. π 2 Lemma A.4: Let g : [, ] R be a C 2 funtion suh that there exists z [, ] an > with g z) z z for all z [, ] an g hanges sign at most twie on [, ]. Let also G : [, ] R be a C funtion suh that Gz) 2 > an G z) hanges sign at most twie on [, ]. Then 2π gz) ej z 4 Gz). π 2 The proof of Lemma A.3 is relegate to Appenix B. The proof of Lemma A.4 follows the same lines an is omitte ue to spae limitations. Let now ε 3 > an let us ivie the integration omain x a, x b, y a, y b ) [, ] 4 in 9) into three subomains see Figure 4): U = { y a y b } /) x b x a ε 3 { < y a y b } /) x b x a < ε 3 U 2 = U 3 = { y a y b } /) x b x a Consier first the integral over U. It an be verifie from the expression 23) for the first orer partial erivative of g a,b with 2
ε A/ U 3 U 2 x b U x a Fig. 4. Domains of integration: the relative positions of the points x a an x b etermine in whih omain one is U on the figure). respet to z given in Appenix B that if x a, x b, y a, y b ) U, then ) g a,b w, z) z K 7 y b y a x b x a for a onstant K 7 > inepenent of an. Notie next that G a,b y, z). It an further be heke that both 2 g a,b z w, z) an G a,b 2 z w, z) hange sign at most twie on the interval z [, ] for w fixe). Therefore, applying Lemma A.3, we onlue that w z ej 2π g a,bw,z) G a,b z) U U 2 U 3 D T w z ej 2π g a,bw,z) G a,b y, z) K 8 y b y a /) x b x a Sine we know that this integral is also less than, this in turn implies w z ej 2π g 2 a,bw,z) G a,b w, z) x a x b y a y b U K 8 = K 8 log/ε 3 ) U x a x b y a y b y b y a /) x b x a Seon, it is easy to hek that x a x b y a y b w z ej 2π g a,bw,z) U 2 G a,b w, z) 2 2ε 3. The integral over the thir omain of integration U 3 is more eliate. Notie first that the obvious boun x a x b y a y b w z ej 2π g 2 a,bw,z) U 3 G a,b w, z) 2 allows to obtain S K 8 log/ε 3 ) + 2ε 3 + 2 whih an be mae smaller than K 3 / ) log /) by hoosing ε 3 = / when A 3/4 as / / in this ase). For the remainer of the proof, let us therefore assume that A 3/4. As before, we fous on the integral insie the square in the following term x a x b y a y b w z ej2π g 2 a,bw,z) U 3 G a,b w, z). 2) Let us start by onsiering the simplest ase where the points x a an x b are loate on the same horizontal line, i.e. y a = y b. In this ase, the seon term in the expression 23) for g a,b z w, z) beomes zero, so we eue the following lower boun: g a,b z w, z) K 9 A 3/2 2 x b x a z y a This, together with the above mentione properties of the funtions g a,b an G a,b, allows us to apply Lemma A.4 so as to obtain w z ej2π g a,bw,z) G a,b w, z) K A 3/4 xb x a for a onstant K > inepenent of an. A slight generalization of this argument see Appenix B for etails) shows that not only when y a = y b but for any x a, x b, y a, y b ) U 3, we have w z ej2π g a,bw,z) G a,b w, z) K K A 3/4 A 3/4 x b x a ) 2 + y b y a ) 2 ) /4 2) xb x a Sine we also know that the above integral is less than, we further obtain w z ej2π g 2 a,bw,z) G a,b w, z) min K 2 ) A 3/2 x b x a, For any < η <, we an now upper boun 2) as x a x b y a y b w z ej2π g 2 a,bw,z) U 3 G a,b w, z) U 3 { x b x a < η} 2 + K x a x b y a y b x b x a U 3 { x b x a η} 2 2η + K implying that A 3/2 A 3/2 log/η) = 2η + K log/η) S K 8 log/ε 3 ) + 2ε 3 + 2η + K log/η) Choosing finally ε 3 = η = / allows to onlue that S K / ) log /) also in the ase where A 3/4.
APPENDIX B TECHNICAL DETAILS Proof of Lemma A.3. By the integration by parts formula, we obtain z ej2πgz) = Gz) 2πgz)) ej = j 2πg z)gz) z j 2πg z) j 2πg z)gz) e2πjgz) z g z)gz) + g z)g z) j 2πg z)gz)) 2 e j 2πgz) whih in turn yiels the upper boun z ej2πgz) Gz) 2π g ) G) + g ) G) g ) z) + z g z)) 2 Gz) + G z) z g z)gz)) 2. By the assumptions mae in the lemma, we have g z) z g z)) 2 Gz) 2 z z g z) g z)) 2 z+ = z g z) 2 g z)) 2 + z g z) z g z)) 2 ) z g z) z + g z)) 2 ) = 2 g ) g ) + 2 g z ) 2 g. z + ) So g z) z g z)) 2 Gz) 6. 2 We obtain in a similar manner that z G z) g z)gz)) 2 6 2 Combining all the bouns, we finally get z ej2πgz) Gz) 4. π 2 Expression for the first orer erivative of g a,b w, z): It an be verifie that g a,b w, z) = xb / + x + w) x / + x + w) 2 + y a z) 2 + yb y a x a y z) y / + x b + w) 2 + y z) 2 22) So the expression for the first orer partial erivative of g a,b w, z) with respet to z is given by g a,b z w, z) = xb z y a )/ + x + w) x x a / + x + w) 2 + z y a ) 2) 3/2 + yb / + x b + w) 2 y / + x b + w) 2 + z y) 2) 23) 3/2 y a x b x a Fig. 5. Tilte referene frame. Proof of equation 2): In orer to prove 2), we nee to make a hange of oorinate system, replaing w, z) by w, z ), where w is now in the iretion of the vetor x a x b an z is perpeniular to it see Figure 5 ). In this new oorinate system, the integral reas w gdr z ej 2πg a,bw,z ) G a,b w, z ) where g a,b w, z ), G a,b w, z ) have the same form as g a,b w, z), G a,b w, z), but now, the omain of integration D R is a tilte square, as iniate on the Figure 5. Using then the same argument as in the ase y a = y b, we onlue that w gdr z ej 2πg a,bw,z ) G a,b w, z ) K A 3/4 z gd R x b x a. Notiing finally that x b x a = x b x a ) 2 + y b y a ) 2 allows to onlue 2). REFERENCES [] P. Gupta an P. R. Kumar, The Capaity of Wireless Networks, IEEE Trans. on Information Theory 42 2), pp.388-44, 2. [2] A. Özgür, O. Lévêque, E. Preissmann, Saling laws for one an twoimensional ranom wireless networks in the low attenuation regime, IEEE Trans. on Information Theory 53 ), 27, 3573-3585. [3] S. Aeron, V. Saligrama, Wireless A ho Networks: Strategies an Saling Laws for the Fixe SNR Regime, IEEE Trans. on Information Theory 53 6), 27, 244-259. [4] A. Özgür, O. Lévêque, D. Tse, Hierarhial Cooperation hieves Optimal Capaity Saling in A-Ho Networks, IEEE Trans. on Information Theory 53 ), pp.3549-3572, 27. [5] A. Özgür, R. Johari, O. Lévêque, D. Tse, Information Theoreti Operating Regimes of Large Wireless Networks, IEEE Trans. on Information Theory 56 ), pp.427-437, 2. [6] A. Özgür Ayin. Funamental limits an optimal operation in large wireless networks. PhD thesis, Lausanne, 29. [7] M. Franeshetti, M.D. Migliore, P. Minero, The apaity of wireless networks: information-theoreti an physial limits, preprint, 27. [8] S.-H. Lee an S.-Y. Chung, Effet of hannel orrelation on the apaity saling in wireless networks, in Pro. IEEE International Symposium on Information Theory, Toronto, Canaa, July 28. [9] S.-H. Lee an S.-Y. Chung, On the apaity saling of wireless a ho networks: Effet of finite wavelength, submitte to IEEE International Symposium on Information Theory ISIT 2). w