Spatial Degrees of Freedom of Large Distributed MIMO Systems and Wireless Ad hoc Networks

Transcription

1 Spatial Degrees of Freedom of Large Distributed MIMO Systems and Wireless Ad ho Networks Ayfer Özgür Stanford University Olivier Lévêque EPFL, Switzerland David Tse University of California at Berkeley Abstrat We onsider a large distributed MIMO system where wireless users with single transmit and reeive antenna ooperate in lusters to form distributed transmit and reeive antenna arrays. We haraterize how the apaity of the distributed MIMO transmission sales with the number of ooperating users, the area of the lusters and the separation between them, in a line-of-sight propagation environment. We use this result to answer the following question: an distributed MIMO provide signifiant apaity gain over traditional multi-hop in large adho networks with n soure-destination pairs randomly distributed over an area A? Two diametrially opposite answers [4] and [6] have emerged in the urrent literature. We show that neither of these two results are universal and their validity depends on the relation between the number of users n and A/, whih we identify as the spatial degrees of freedom in the network. is the arrier wavelength. When A/ n, there are n degrees of freedom in the network and distributed MIMO with hierarhial ooperation an ahieve a apaity saling linearly in n as in [4], while apaity of multihop sales only as n. On the other hand, when A/ n as in [6], there are only n degrees of freedom in the network and they an be readily ahieved by multihop. Our results also reveal a third regime where n A/ n. Here, the number of degrees of freedom are smaller than n but larger than what an be ahieved by multi-hop. We onstrut saling optimal arhitetures for this intermediate regime. I. INTRODUCTION Multiple-input multiple-output MIMO) is one of the key tehnologies to ahieve high-data rates in urrent wireless systems. Both the transmitter and the reeiver are equipped with multiple antennas, whih allows to spatially multiplex several streams of data and transmit them simultaneously. When the sattering environment is rih enough to allow reeive antennas to separate out signals from different transmit antennas, MIMO hannels offer large apaity gains: the apaity of suh a MIMO hannel with M transmit and reeive antennas is proportional to M. This was established in the groundbreaking papers of Fohini-Gans [] and Telatar [] whih assumed an i.i.d. fading model for hannels between different antenna pairs. However, when there is insuffiient sattering in the environment or when antennas are densely paked together in small hand-held devies, the hannels between different antenna pairs beome orrelated. Suh physial onstraints prevent MIMO apaity from inreasing indefinitely with M. The impat of orrelated fading [3], [4], [5], [6], [7], the sattering environment [8], [9] and antenna oupling [], [] on MIMO apaity was studied by a large body of follow-up researh, whih also provided a haraterization of the inherent spatial degrees of degrees of freedom in a MIMO system as a funtion of the area and the geometry of the antenna arrays and the angular spread of the physial environment [], [3], [4], [5]. Reently, distributed MIMO ommuniation arised as a promising tool to ahieve large performane gains in wireless networks, similar to those provided by onventional MIMO in the point-to-point wireless hannel. Here, wireless users with a single transmit and reeive antenna ooperate in lusters to form distributed transmit and reeive antenna arrays. A ommon assumption in the performane analysis of distributed MIMO systems is that hannels between different pairs of nodes are subjet to independent fading [6], [4]. Analogous to the point-to-point ase, this leads to apaity gains linear in the number of nodes M ontained in the transmit and reeive lusters. In a distributed setting, nodes or antennas) are typially muh farther apart from eah other as ompared to lassial MIMO, so an i.i.d. fading model may seem appropriate. However, the number of nodes M partiipating to the transmission an be also muh larger in this ase sine there are no physial paking onstraints like in MIMO. Capaity an not sale indefinitely with M and orrelations between pairwise hannels are expeted to limit performane when M is large. In this paper, we provide a rigorous lower bound on the saling of the apaity of a large distributed MIMO system with the area of the transmit and reeive lusters, the separation between the lusters d and the number of nodes M ontained in eah luster assuming a line-of-sight propagation model and a random distribution of nodes over the luster areas. We show that the apaity of distributed MIMO systems sales at least as min } M, d when d / min M, / } when d ) where is the arrier wavelength. This result identifies d and / as the spatial degrees of freedom in the distributed MIMO hannel in the two orresponding regimes. The apaity of the hannel sales linearly in M when the physial hannel has more than M spatial degrees of freedom. Fortunately, this an be often the ase for atual networks. Consider for example two lusters of area m separated by a distane of m. When ommuniation takes plae around

2 a arrier frequeny of 3 GHz, / =. As long as there are less than users in eah luster, the line-of-sight hannel has suffiient spatial degrees of freedom for all users. d is When the distane between the two lusters is km, still. The spatial degrees of freedom are expeted to be even larger in sattering environments. The distributed MIMO hannel, and therefore its analysis, differs from the lassial MIMO hannel in a ouple of ways. First, the distanes between different pairs of users an be signifiantly different in the distributed ase, whih results in heterogeneous hannel gains ditated by the geometry of the network. Seond, while in lassial MIMO, the separation between the antenna arrays d is typially muh larger than the length or the diameter) of the arrays, for distributed systems these two dimensions an be omparable. Our analysis takes into aount these new aspets brought by the distributed nature of the problem. Mathematially, our approah differs signifiantly from existing results studying the spatial degrees of freedom of lassial MIMO hannels. While suh results for example, see [], [4], [5]) are based on studying the singular values of the ontinuous propagation operator under approximations for the regime where d, our analysis is mathematially rigorous and is based on random matrix analysis. The haraterization of the saling of the apaity of distributed MIMO systems allows us to identify the number of spatial degrees of freedom in large wireless ad ho networks, and to reonile some seemingly ontraditing results in the urrent literature on saling laws for wireless networks. The study of the asymptoti regime where the number of users in a wireless network is large was initiated by the seminal paper [7] of Gupta and Kumar and reeived signifiant attention in the literature [8], [9], [], [], [], [3]. Gupta and Kumar showed that the apaity of multihop ooperation sales as n with inreasing number of user n in the network. In this traditional ommuniation arhiteture, pakets are routed from eah soure to its destination along a path where intermediate users at as relays. Eah relay deodes the pakets sent from the previous relay and forwards them to the next. A n saling for the total apaity implies that the rate per user dereases as / n with inreasing system size n. Can more sophistiated ooperation between users signifiantly inrease the apaity of large wireless networks? Two diametrially opposite answers have emerged in the reent literature: ) Capaity an be signifiantly improved when users form distributed MIMO arrays via a hierarhial ooperation arhiteture [4]. In regimes where power is not a limiting fator [5], the apaity an sale almost linearly with n implying a onstant rate per user. ) The saling of the apaity is upper bounded by n due to the spatial onstraints imposed by the physial hannel [6]. Nearest-neighbor multi-hop already ahieves this saling and more sophistiated ooperation is useless. The key differene between these two results is their assumptions for the hannel model between pairwise nodes. [4] assumes that the phases of the hannel gains an be modeled as uniformly distributed random variables, independent aross different pairs of nodes in the network. [6], on the other hand, starts from physial priniples and regards the phases as funtions of the loations of the nodes. While the physial hannel model used in [6] is more fundamental, the i.i.d. phase model is also widely aepted in wireless ommuniation engineering, partiularly for nodes in far field from eah other. What is the way to reonile these two sets of results? We answer this question in the seond part of the paper building on the result in ). We show that under the physial hannel model of [6], the distributed MIMO based hierarhial ooperation arhiteture in [4] ahieves a apaity saling as }} n,min A max n,. ) in a network of n soure-destination pairs uniformly distributed over an area A and ommuniating around a arrier wavelength. The saling of the apaity depends on how n ompares to A/, whih an be interpreted as the spatial degrees of freedom available in the network. The two earlier results an be reovered as two speial ases of this new result: ) When A/ n, the apaity sales linearly in n. In this regime, there are suffiient spatial degrees of freedom for all the n users in the network and they an be exploited by distributed MIMO ommuniation. The i.i.d. fading assumption aross different node pairs in [4] leads to n degrees of freedom and therefore [4] inherently assumes that the network operates in this regime. ) When A/ n, the apaity sales as n. In this regime, the spatial degrees of freedom available in the network are as few as n, and therefore they an be readily ahieved by multihop. By assuming that the density of nodes is fixed as the number of nodes n grows, [6] assumes that the number of spatial degrees of freedom A/ is proportional to n. Therefore, [6] inherently assumes that the network operates in this regime. Therefore, neither of the two onlusions in [4] and [6] that more sophistiated ooperation an provide signifiant apaity gains or is useless are universal. They orrespond to two different operating regimes of large wireless networks. ) larifies the onditions for a network to be in either of these two regimes. Indeed, ) also unovers a third regime where the network is partially limited in spatial degrees of freedom. When n A/ n, the number of spatial degrees of freedom is smaller than n, so the spatial limitation is felt, but larger than what an be ahieved by simple multi-hopping. Multi-hop ahieves n saling independent of A/.) We show that either a modifiation of the hierarhial ooperation sheme in [4] or a version of the MIMO-multihop sheme in [5] an ahieve the A/ available degrees of freedom and therefore the optimal saling of the apaity in this regime. The main idea behind the first modifiation is to allow only

3 a subset N of the soure-destination pairs to ommuniate at a time using the hierarhial ooperation sheme, and then take turns among different subsets. N = A/, the number of spatial degrees of freedom in the network, so eah subset orresponds to a diluted network whih is not limited in spatial degrees of freedom. The idea behind the seond sheme is to form lusters of an intermediate size and hop aross several lusters to reah destinations where eah hop is performed via distributed MIMO transmissions. The luster size is hosen ritially to ensure linear saling for the distributed MIMO transmissions. Traditionally, the literature on saling laws for wireless networks seeks the saling of the apaity with the number of nodes n when all systems parameters are oupled with n in a speifi way. One ommon approah is to assume that the area sales linearly in n, while all other parameters remain fixed, as in [6]. As we have already seen, this immediately implies that the spatial degrees of freedom in the network A/ are proportional to n. But the number of nodes and the area are two independent parameters of a network, eah of whih an take on a wide range of values. For atual networks, there an be a huge differene between A/ and n. Take an example of a network serving n = users on a ampus of km, operating at 3 GHz: A/ =, while n is only, two orders of magnitude smaller. So while multi-hop an ahieve a total throughput of the order of bits/s/hz, there is still a lot of potential for ooperation gain, sine the spatial degrees of freedom are indeed and not as given by ). For the lassial MIMO hannel, it is now well understood that there are a number of qualitatively different regimes. When the antennas are separated suffiiently apart, the apaity inreases linearly in n. However for dense antenna arrays the apaity is limited by physial onstraints and annot inrease linearly in n. To obtain an analogous understanding of the operating regimes of large wireless networks, we advoate in this paper a shift of the large networks researh agenda from seeking a single universal saling law, where parameters of the network are oupled to n in a speifi way, to seeking a multi-parameter family of saling laws, where the key parameters are deoupled and many different limits with respet to these parameters are taken. A single saling law with a partiular oupling between parameters is often arbitrary and too restritive to over the wide ranges that the multiple parameters of the network an take on. We have introdued this approah in [5] where we deoupled the number of nodes and the amount of power available. The urrent paper follows the spirit of [5], both mathematially and philosophially, but fouses on the number of nodes and 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 dependene of the apaity on the number of nodes, the area of the network and the amount of power all together. A version of this problem has been studied in an indepen- D T d D R Fig.. Two square lusters of area separated by distane d. dent onurrent work [3], however both the formulation of the problem and the proposed arhitetures differ from the urrent paper: in [3], the network area A is taken either fixed or proportional to n, while the arrier wavelength sales down to zero with inreasing n. The arhiteture in [3] is obatained by diluting the distributed MIMO transmissions at eah level of the hierarhy in [4], as opposed to diluting the whole network as we do here. In the urrent paper, we also show that the same performanes an be alternatively ahieved with a MIMO-multihop strategy. A. Model II. SPATIAL DEGREES OF FREEDOM OF DISTRIBUTED MIMO SYSTEMS We onsider a distributed MIMO transmission between two square lusters of area separated by distane d see Figure ), with eah luster ontaining M nodes distributed uniformly at random over the area. Eah node is equipped with one antenna, oriented in the diretion perpendiular to the plane, with a given power budget P. We assume that ommuniation takes plae over a flat hannel with bandwidth W and that the signal reeived by node i at the RX luster at time-slot m is given by M y i [m] = h ik x k [m] + z i [m] k= where x k [m] is the signal sent by node k at the TX luster at time m, z i [m] is additive white irularly symmetri Gaussian noise AWGN) of power spetral density N / Watts/Hz. In a line-of-sight environment, the omplex baseband-equivalent hannel gain h ik between transmit node k and reeive node i at time m is given by h ik = G expπjr ik/) 3) r ik where is the arrier wavelength, r ik is the distane between node i and node k and G is Friis onstant given by G = G t G r 4π with G t and G r being the transmit and reeive antenna gains, respetively. Finally, we assume full hannel state information at all the nodes, whih is a reasonable assumption here, as the hannel oeffiients only depend on the node positions and these do not vary over time. )

4 B. Main Result Our goal in this setion is to provide a lower bound on the spatial degrees of freedom of the system desribed above. It an be inferred from [6] that the spatial degrees of freedom of suh a distributed MIMO system are upperbounded up to logarithmi fators) by / as gets large, irrespetive of the distane d. tually, the spatial degrees of freedom derease as d inreases, beause of the redution of the aperture of the MIMO system. We prove below that the spatial degrees of freedom of the system are at least min M, /d} if d / min M, / } if d again, up to logarithmi fators). Notie that in the seond regime, the obtained lower bound mathes the upper bound found in [6]. Finally, notie that if d /, then the system has learly at least one degree of freedom. Let us now state our main result. Theorem.: Let d /, and let the nodes in the transmit luster D T perform independent signaling with power P eah, suh that the long-distane SNR between these two lusters defined as SNRd) = M GP N Wd 4) is greater than or equal to db. Then there exists a onstant K > independent of M,, and d suh that the apaity C MIMO of the distributed MIMO hannel from the transmit luster D T to the reeive luster D R is lowerbounded by } /d K min M, if d A/ log /d) C M } / K min M, log if d /) with high probability as M gets large. The rest of this setion is devoted to the proof of this result, whih is made of three ingredients. The first key ingredient provides a lower bound on the MIMO hannel apaity averaged over the random node positions in the first regime where d /. Lemma.: If d /, then the expeted apaity EC MIMO ), averaged over the random node positions, satisfies } /d EC MIMO ) K min M, 5) log /d) Next, we show that the apaity of the distributed MIMO hannel with given random node positions is lose to its expeted value with high probability. The proof relies on lassial onentration arguments and is relegated to the end of the present setion. Lemma.: In general, if M nodes partiipate to the MIMO transmission, then for all ε >, there exists K > suh that C MIMO EC MIMO ) K M /+ε with high probability as M gets large. Finally, we show that the study of the seond regime d ) an be brought bak to the ase d = by simply reduing the set of transmitting and reeiving nodes by a fator, so that the interluster distane beomes of the same order as their radius. Again, the proof is relegated to the end of the present setion. Lemma.3: If d, then the spatial degrees of freedom of the system are of the same order as when d =. Combining these three lemmas yields the result given in Theorem.: Assume first d /. Then by Lemma., EC MIMO ) K min /d M, } /d log /d) If M log/d), then Lemma. allows to onlude that C MIMO itself is with high probability at least of order M. If on the other hand M > /d log/d), then it should be notied that in this ase, it is useless to have all the M nodes partiipating to the MIMO transmission. Only M = /d loga suffie. /d) Applying then the onentration result replaing M by M allows to onlude. Finally, Lemma.3 shows that for all d, the degrees of freedom of the system are lowerbounded by C MIMO K min M, } / log /) with high probability for some other onstant K >. This onludes the proof. In the sequel, we provide the proof of Lemma.. The proofs of Lemmas. and.3 are relegated to the end of the present setion. Proof of Lemma.. For notational onveniene, we start by defining f ik = d h ik = d expπjr ik /) G r ik d = x k w i expπj x k w i /) 6) where r ik denotes the distane between the nodes k D T and i D R loated at positions x k and w i, respetively. Notie that d r ik d + /d), so + /d) f ik, 7) where = + ) and the first inequality follows from the fat that d. Remembering the definition of SNRd) given 4), we obtain the following expression for the average apaity of the distributed MIMO hannel where we reall here that H, F are the matries with entries h ik, f ik, respetively): EC MIMO ) = E log det I + )) P N W HH = E log det I + SNRd)FF /M )) = M Elog + SNRd)))

5 xm D T f im x k Fig.. f lm f ik f lk w l w i D R S = Ef ik flk f lm fim ), i l, k m where is an eigenvalue of FF /M piked uniformly at random. By applying now the Paley-Zygmund inequality, stating that for a non-negative random variable X, PX t) we obtain that for < t < E): EX) t) EX, < t < EX) ) EC MIMO ) M log + SNRd)t) P > t ) From 7), we further obtain M log + SNRd)t) E) = M E trff ) ) = M E) t) E ) M E f ik ). i,k= E ) = M 3 EtrFF FF )) = M M 3 Ef ik flkf lm fim) + M 3 i,k,l,m= M i,k,l,m= i l,k m Ef ik f lkf lm f im) + M S where S = Ef ik flk f lm fim ), with i l and k m notie that S does not depend on the speifi hoie of i l and k m). See Figure. Choosing then t = /, we obtain EC MIMO ) M 4 /4) log + SNRd) / ) + M S K min M, } S for a onstant K > independent of M and S, provided that SNRd) db, whih was our initial assumption 4). The quantity S, whih takes values between and, ditates therefore the apaity saling. In the ase where the hannel matrix entries f ik are i.i.d. and irularly-symmetri, S =, so the apaity is of order M. On the other hand, if we onsider the line-of-sight hannel model 6) in the senario where nodes are plaed on a single straight line, then a simple omputation shows that S is of order, so that the apaity is also of order in this ase. The problem we are looking at lies between these two extreme ases. Our aim in the following is to show the following lemma, whih allows to onlude the proof of Lemma.. Lemma.4: If d /, then there exists a onstant K > independent of, and d, suh that ) d S K log d ). 8) We now give a proof idea for Lemma.4. Let us first expliitly write the expression for S. We have S = Ef ik flk f lm fim) = dx k dx m dw i dw l ρe πj / D T D T D R D R where A 4 = x k w i x k w l + x m w l x m w i ) and ρ = d 4 x k w i x k w l x m w l x m w i 9) ) We first derive the result 8) by approximating the inter-node distanes in the regime d. This approximation, made already by various authors [], [9], [4], [5] in different ontexts, allows us to derive an upper bound on S, and orrespondingly, a lower bound on the spatial degrees of freedom. In addition, we provide in the Appendix a rigorous derivation of the lower bound, whih does not make use of the approximation. As far as we know, this derivation is new. Consider two nodes at positions x = x, y) D T and w = d + w, z) D R, where x,y,w,z [,] see Figure 3). Using the assumption that d, we obtain x w = d + x + w)) + y z) d + x + w) + y z) d whih in turn implies = x k w i x k w l + x m w l x m w i d y k z i ) y k z l ) + y m z l ) y m z i ) ) = d y m y k )z l z i ) Next, let us also make the approximation that ρ in ): this is atually assuming that the spatial degrees of freedom between the two lusters are mainly determined by the phases of the hannel oeffiients and not so muh by the amplitudes. In the Appendix, we show that this intuition is orret. Stritly speaking, this only shows that the lower bound on the apaity is of order, but in this ase, the matrix F an be shown to be essentially rank one, so the atual apaity is indeed of order.

6 Ay x Ax d Fig. 3. Coordinate system. Aw w Az These two suessive approximations lead to the following approximation for S: S S = dy k dy m dz i = dy k dy m dz i y k πj dz l e πj dz l e d ym y k)z l z i) d ym y k)z l z i) where the seond equation follows from the symmetry of the integrand. Note that this expression does not depend on the horizontal positions of the nodes. This an be interpreted as follows. Provided the above approximation is valid, the spatial degrees of freedom between two lusters of M nodes separated by a distane d are the same, be the nodes uniformly distributed on two squares of area or on two parallel vertial) lines of length. We show below that the above integral is of order d/. Let us ompute the first integral, whih yields πj dz l e d ym y k)z l z i) d = πj y m y k ) e πj This implies that πj dz l e d ym y k) z l z i) z l = d ym y k)z l z i) K d y m y k z l = for a onstant K > independent of, and d. We an divide the integration over y k and y m into two parts, so πj dy k dy m dz i dz l e d ym y k)z l z i) y k yk +ε) ε ) = dy k dy m + dy k dy m y k y k +ε dz i πj dz l e d ym y k)z l z i) for any < ε <. The first term an be simply bounded by This result an be proved rigorously; atually, the rigorous argument following for the two squares applies equally likely to the ase of two parallel lines ε, whih yields the following upper bound on S S ε + K d ε ε + K d log ε dy k ) y k +ε So hoosing ε = d/, we finally obtain ) ) d S S K log d dy m y m y k for a onstant K > independent of, and d. Proof of Lemma.. First observe that the apaity of the distributed MIMO hannel C MIMO = log deti + SNRd)FF /M) where we reuse here the notation adopted in the proof of Lemma.) an be seen as a funtion of the node positions x k and w i : C MIMO = g M x,...,x M ;w,...,w M ) As we will see in the following, the apaity does not vary signifiantly with the node positions, whih allows us to apply the following simplified version of the) theorem by MDiarmid [8]. Theorem.: Let x,...,x M be a family of i.i.d. random variables distributed in a bounded region A R, and let f M : R M R be a measurable funtion suh that there is a onstant M with f M x,...,x k,...,x M ) f M x,...,x k,...x M ) M for all k M and x,...,x k,x k,...,x M A. Then for all t >, P f M x,...,x M ) Ef M x,...,x M )) > t) ) exp t M M In order to apply the above theorem replaing f M by g M ), we need to upperbound the differenes g M x,...,x k,...,x M ;w,...,w M ) g M x,...,x k,...x M ;w,...,w M ) and g M x,...,x M ;w,...,w k,...,w M ) g M x,...,x M ;w,...,w k,...,w M ) As the problem is symmetri, we only onsider the first ase here. Fix k M. Notie first that modifying the vetor x k only modifies a single olumn of the matrix F. Let us define F as being the matrix F with olumn k removed so F is an M M ) matrix). Beause of what was just observed, F F is a rank one matrix, so using the interlaing property

7 of the singular values of F and F [9, Theorem 7.3.9], we obtain that for all j M, / / j j and j j+ where... M are the eigenvalues of FF /M and... M are the eigenvalues of F F /M. Remember now that g M x,...,x M ;w,...,w M ) = log deti + SNRd)FF /M) = Defining M log + SNRd) j ) j= g M = log deti + SNRd) F F /M) = M log + SNRd) j ) j= and applying the above inequalities on the eigenvalues, we see that g M x,...,x M ;w,...,w M ) g M log+snrd) ) It an be easily seen M. Besides, we are interested in the regime where the growth of SNRd) is no more than polynomial in M it is atually onstant in the ase of interest), so for all ε >, there exists a onstant K > suh that g M x,...,x M ;w,...,w M ) g M K log M so by the triangle inequality, g M x,...,x k,...,x M ;w,...,w M ) g M x,...,x k,...x M ;w,...,w M ) K log M This finally shows, via MDiarmid s theorem, that gm P x,...,x M ;w,...,w M ) ) Eg M x,...,x M ;w,...,w M )) > t t ) exp MK log M) whih gives the result, by setting t = M /+ε. Proof of Lemma.3. The proof of this lemma is based on the following simple observation. When d, the two lusters are lose to eah other, as illustrated on Figure 4. Consider now the two square sublusters of size that are the most separated horizontally. These two sublusters are now separated by a distane d+, whih is of the same order as. On the other hand, both the area and the number of nodes in these sublusters remain of the same order as in the original lusters. More preisely, the new area is /4 and orrespondingly, the number of nodes in eah subluster is around M/4 with high probability. By letting only the nodes in these two sublusters partiipate to the MIMO transmission, we therefore see that the same order of spatial degrees of freedom an be ahieved as when d =. d Fig. 4. The distane between the two highlighted sublusters is equal to d +. III. AD HOC WIRELESS NETWORKS An ad ho network is a olletion of wireless users whih an self-organize and ommuniate among themselves without the help of any fixed infrastruture. Optimal ooperation arhitetures for suh networks have reeived muh fous in the asymptoti regime where the number of users is large. Consider an ad ho network where n soure-destination pairs are independently and uniformly distributed over an area A. Eah soure node wants to ommuniate to its orresponding destination at the same rate R bits/s/hz. For this network, we have developed a hierarhial ooperation arhiteture in [4] that maintains a onstant per-pair ommuniation rate R even when the network serves a growing number of users, provided that the hannels between pairwise users are subjet to i.i.d. fading. More preisely, when the phases of the hannel oeffiients an be modeled as i.i.d. random variables independent of the node loations, hierarhial ooperation ahieves an aggregate throughput T = nr that sales linearly in n. Current ommuniation arhitetures for ad ho networks an not provide salable performane. The traditional approah is to forward information from the soure node to the destination by following a path, with intermediate nodes on the path ating as relays. The aggregate throughput of this multi-hop arhiteture sales as n with inreasing network size n [7]. Distributed MIMO is key to the linear saling of the arhiteture in [4]. Hierarhial ooperation allows nodes to effiiently organize in lusters and establish the n ommuniations in the network via distributed MIMO transmissions between large lusters. Under i.i.d. fading, the apaity of these distributed MIMO transmissions are linear in the the number of nodes M ontained in the transmit and reeive lusters. In the earlier setion, however, we have seen that the area of the lusters and the separation between them poses a limit on the linear saling of distributed MIMO. In this setion, we evaluate the performane of hierarhial ooperation under the physial model in 3) and show that it an still ahieve A linear saling but provided that n. When A < n, we present two modifiations of the hierarhial ooperation arhiteture that ahieve the optimal saling. Our result is formally summarized in the following theorem. Theorem 3.: Consider a wireless network of n nodes

8 distributed uniformly at random over a square area A suh that A/ n. Assume that eah node is subjet to an average power onstraint P and the network is alloated a total bandwidth of W. The hannels between pairwise users are governed by the LOS model in 3). Let us define the SNR of the network as GP SNR = n ) N W A and assume that SNR db. For any ε >, there exists a onstant K > independent of n, and A suh that an aggregate throughput T K min n, } ε A/ is ahievable with high probability as n gets large. We present the arhitetures that ahieve this performane in the next setion. An upper bound on the best ahievable apaity saling under the model in 3) is developed in [6] for the ase where the density of the nodes remains fixed as the number of nodes n grows. In this ase, [6] shows that the apaity of the wireless network is upperbounded by T K n log n), with high probability, where K > is a onstant independent of n. The above result an at first lead to the onlusion that the best saling ahievable in wireless networks is n and therefore multi-hop is saling optimal. However, a deeper look reveals that the onlusion that the apaity sales like n omes from the assumption that the density of nodes is fixed, so that A/ is proportional to n. A relatively straightforward generalization of the analysis in [6] gives the following refined upper bound on apaity: K nlog n) if A n T K A log ) A if n > A > n 3) K nlog n) n if A with high probability, where K > is a onstant independent of n, and A. For A/ n, this result says that the maximum ahievable apaity is of order n, whih is ahievable by a simple multihop. For A/ > n, the ahievability of the upper bound was an open problem and is now established in Theorem 3. when SNR db. This leads to the onlusion that in the regime when n and A/ are both large and SNR db, the apaity of the network is approximately n,min max n, }} A ordingly, the optimal operation of the network falls into three different operating regimes: ) A/ n: The number of spatial degrees of freedom is too small, more sophistiated ooperation is useless and multihop is optimal. ) A/ n: The number of spatial degrees of freedom is n, the optimal performane an be ahieved by the same hierarhial ooperation sheme introdued in [4]. Spatial degree of freedom limitation does not ome into play and the performane is as though phases were i.i.d. uniform aross node pairs. 3) n < A/ < n: The number of degrees of freedom is smaller than n, so the spatial limitation is felt, but larger than what an be ahieved by simple multi-hopping. A modifiation of the hierarhial ooperation sheme ahieves the optimal saling in this regime. The SNR in ) an be identified as the typial SNR between nearest neighbor nodes in the network under the hannel model 3). Note that in a random network of n nodes distributed over an area A, the typial separation between nearest neighbor pairs is given by A/n. The ondition ) ensures that these hannels are in the high-snr regime. Note that hannels between pairs further away an be in low-snr. Identifying optimal ooperation arhitetures for networks with SNR db under the physial hannel model remains an open problem. Optimal arhitetures for suh networks have been identified in [5] under the i.i.d. fading model. A. Optimal ooperation in networks with limited spatial degrees of freedom Capitalizing on the result of Theorem., in this setion we prove Theorem 3. in three steps: A) When A/ n, we verify that the performane of the hierarhial ooperation arhiteture in [4] sales linearly in n under the LOS model of 3). B) When n < A/ < n, we show that a diluted hierarhial ooperation arhiteture ahieves the saling in Theorem 3.. Here, only a randomly hosen subset of the soure-destination pairs operate at a time while remaining nodes stay silent. This reates a diluted network for whih ase ) holds so the network does not experiene any limitation in spatial degrees of freedom. Different subsets take turns to operate. C) When n < A/ < n, an alternative way to ahieve the saling in Theorem 3. is to use a hybrid arhiteture ombining distributed MIMO with multi-hop, introdued in [5]. Here, nodes form MIMO lusters of an intermediate size and information is routed from one luster to the next via suessive distributed MIMO transmissions between adjaent lusters. The luster size is ritially hosen at the largest possible sale that allows for linear saling of the distributed MIMO transmissions. When n = A/, this luster size is a single node and the hybrid arhiteture redues to pure multi-hop. When A/ = n, the luster size is as large as n, and the arhiteture redues to pure hierarhial ooperation.

9 The differene between the two strategies in B) and C) arises when we modify the hannel model to h ik = G ejπr ik/ r α/ ik 4) where α is the power path loss exponent determining how fast signal power deays with distane in the environment. Although not physial, this hannel model provides a simple way to apture the impat of larger path loss attenuation due to multiple propagation paths, and at the same time it preserves the spatial orrelation between hannels by keeping the dependene of the phases to the geometri struture of the network. It has been shown in [6] that multiple paths do not hange the saling of the number of spatial degrees of freedom in a large network. Therefore it suffies to onentrate on a LOS model for the phases. However, multiple paths an have signifiant impat on the power. For example, with an additional refleted path from the ground plane the power path loss over distane r inreases from r to r 4. Under this new model, the power ondition for ahieving linear apaity for distributed MIMO beomes SNRd) = M GP N Wd α db in Theorem.. ordingly, the diluted hierarhial ooperation arhiteture in B) ahieves the performane in Theorem 3. when SNR l = n GP N W A) α db SNR l is defined as the long-range SNR of a network in [5]. This quantity an be identified as n times the reeived SNR in a point-to-point transmission over the largest sale in the network, the diameter A. The extra n omes from the network effet, it reflets the potential power gain due to ooperation over the global sale. For the multi-hop MIMO arhiteture in 3), the power requirement is given by SNR ) = M GP N W ) α db where SNR ) is the analog of SNR l but for a luster of area ontaining M = n/a nodes. In partiular, we will hoose M in the sequel suh that M = A n. When n < A/ < n, we have M n. It is easy to verify that when α >, SNR ) SNR l see 3.) in [5]), therefore the seond ondition is less stringent than the first. When α =, SNR ) = SNR l = SNR in ). Therefore, for the LOS model, the two arhitetures are equivalent. A detailed disussion on the relevane of the SNR parameters above in networks with i.i.d. fading is provided in [7]. The below disussion assumes that the reader is familiar with the hierarhial ooperation and the MIMO multihop arhitetures and their performane analysis. A detailed desription of these strategies an be also found in [7].. Hierarhial ooperation when A/ n In this regime, the upper bound in 3) allows for throughput saling linear in n. Potentially hierarhial ooperation an ahieve arbitrarily lose to linear saling. One needs to hek however that the MIMO transmissions taking plae at all levels of the sheme are fully effiient, i.e. have apaity saling linearly in the number of nodes in the lusters. This is easy to verify: onsider a MIMO transmission between two lusters of area and size M = n/a. The separation d between these two lusters is upperbounded by the diameter of the network A. Therefore d A A A = n M Therefore when A/ n, M, so by Eq. 5), d distributed MIMO transmissions operate with full degrees of freedom up to a logarithmi fator), just like in the ase of i.i.d. phases. To ompensate for the logarithmi fator, we argue that /d log /d) M ε for any ε > and suffiiently large M. This in turn implies that the apaity of the distributed MIMO transmissions sale as M ε. The derease by M ε is aptured in the n ε degradation in the overall throughput in Theorem.. We also need to verify that the distributed MIMO transmissions have suffiient power as required in ondition 4). In the hierarhial ooperation arhiteture, the MIMO transmission between lusters of area and size M take plae inside a larger luster of area A and size M = A n/a in the next level of the hierarhy. Therefore the separation between the TX and RX lusters is upper bounded by A. During the MIMO transmissions eah node transmits with elevated power P m = M P M. This is beause of the time-division between MIMO transmissions from different lusters. Eah node transmits only a fration M/M of the time, therefore it an transmit with elevated power M P M and still satisfy the average transmit power onstraint P. See [7] for details. Therefore, the SNR for the MIMO transmissions is given by SNRd) = M GP m N WA = M GP N WA = n GP N WA db where the last inequality is the power ondition in Theorem 3.. Therefore, MIMO transmissions at eah level of the hierarhy have full degrees of freedom and suffiient power. Hierarhial ooperation ahieves an aggregate throughput saling arbitrarily lose to linear in n in this ase.. Hierarhial ooperation when n A/ < n In this regime, equation 3) shows that a linear throughput saling is not ahievable by any means. Nevertheless, the question remains whether one ould outperform multihopping strategies, whose asymptoti performane n is stritly suboptimal ompared to the upper bound A/. A diret appliation of the hierarhial ooperation sheme fails to improve on multi-hop in this ase, but it turns out that a simple adaptation of the sheme to this spatially limited situation ahieves the optimal saling.

10 The idea is the following: organize the ommuniation of the n soure-destination pairs into n/n sessions, eah involving N soure-destination pairs, where N = A/. It is possible to hoose here the nodes in a way suh that eah group of N nodes statistially oupies the total area of the network. This way, no group of N nodes onsidered alone feels the spatial limitation, as for this diluted network N = A/ and we are in the ase A/ N above. The sessions operate suessively and the traffi in eah session is handled using hierarhial ooperation where only the N hosen nodes are involved. The rest of the nodes remain silent. Sine nodes are ative only a fration of N/n of the total time, when ative they an transmit with elevated power P m = np/n and still satisfy their individual power onstraint P. Therefore, for the diluted network of N nodes in eah session, the SNR is 3 N GP m N WA = n GP N WA db Therefore, the diluted network is neither power nor spaelimited and hierarhial ooperation ahieves aggregate throughput of order N ε = A/) ε for any fixed ε >. With time-division aross different groups of nodes, the same throughput is ahievable in the whole network. 3. MIMO multi-hop when n A/ < n Consider the MIMO multi-hop strategy desribed in Setion 3.3 of [7]. On the global sale, this hybrid arhiteture is similar to multi-hop. The pakets of eah soure-destination pair are transferred by hopping from one luster to the next. At eah hop, the pakets are deoded and then re-enoded for the next hop. The arhiteture differs from multi-hop by the fat that eah hop is performed via distributed MIMO transmissions assisted by hierarhial ooperation. Let us hoose the luster size M suh that / = M where = AM/n is the A n area of the luster. This leads to M =. This hoie of the luster size ensures that the lusters are not limited in spatial degrees of freedom. Therefore, the apaity of distributed MIMO transmissions at eah hop sales linearly in M provided that there is suffiient power. Sine the distributed MIMO transmissions at eah hop take plae over a distane, the power ondition in 4) yields SNRd) = M GP N W db This is equivalent to SNR db in Theorem 3. 4, as M/ = n/a. When the apaity of the distributed MIMO transmissions at eah hop sale linearly in M, the aggregate throughput of the MIMO multi-hop arhiteture is given by nm / ε in Eq. 3.3) of [7]. Plugging our hoie M = A n, we obtain an aggregate throughput saling as A/) ε. 3 With the model in 4), this ondition beomes SNR l db. 4 With the model in 4), this ondition beomes SNR) db. IV. ACKNOWLEDGMENT We would like to thank Mar Desgroseilliers for helping with the preparation of the present paper. APPENDIX Rigorous proof of Lemma.4. We now prove equation 8) without making use of approximations. We start again with expression 9) for S. Notie that due to the symmetry of and ρ in w i and w l, we an upper bound 9) as S d4 A 4 dx k dx m D T D T dw DR eπj x k w x m w ) x k w x m w Expressing this upper bound more expliitly in the oordinate system shown on Figure 3, we obtain: S dx k dy k dx m dy m dw dz eπjg k,mw,z) G k,m w,z) where g k,m w,z) = d + x k + w)) + y k z) d + ) x m + w)) + y m z) / and 5) G k,m w,z) = d d + x k + w)) + y k z) d + x m + w)) + y m z) Let us first fous on the integral inside the square in 5). The key idea behind the next steps of the proof is ontained in the following two lemmas. Lemma.: Let g : [,] R be a C funtion suh that g z) > for all z [,] and g hanges sign at most twie on [,] say e.g. g z) in [z,z + ] and g z) outside). Let also G : [,] R be a C funtion suh that Gz) > and G z) hanges sign at most twie on [,]. Then dz eπjgz) Gz) 4. π Proof: By the integration by parts formula, we obtain dz eπjgz) = Gz) πjg z)gz) = eπjgz)) dz πjg z) πjg z)gz) eπjgz) dz g z)gz) + g z)g z) πjg z)gz)) e πjgz)

11 whih in turn yields the upper bound dz eπjgz) Gz) π g ) G) + g ) G) g ) z) + dz g z)) Gz) + G z) dz g z)gz)) By the assumptions made in the lemma, we have So g z) dz g z)) Gz) z dz g z) g z)) z+ = dz g z) g z)) + dz g z) z g z)) ) dz g z) z + g z)) ) = g ) g ) + g z ) g z + ) dz g z) g z)) Gz) 6. We obtain in a similar manner that formula, whih yields. dz j πg z) [,z ε] [z +ε,] j πg z) πgz) z ej ε z ε = πjg z) + dz πgz) ej + πjg z) + dz z +ε z +ε ej πgz) g z) πgz) πjg ej z)) g z) πgz) πjg ej z)) Using the assumptions on g and g in the lemma and following similar steps to the proof of Lemma., we obtain dz expπjgz)) ε + π g ) + g z ε) + g z + ε) + ) g ) ε + 8 π ε Choosing ε = π yields the desired result dz expπjgz)) 8 π whih ompletes the proof. dz G z) g z)gz)) 6 Combining all the bounds, we finally get dz eπjgz) Gz) 4 π ε d U 3 U x m U x k U U U 3 D T Lemma.: Let g : [,] R be a C funtion suh that there exists z [,] and > with g z) z z for all z [,] and g hanges sign at most twie on [,]. Let also G : [,] R be a C funtion suh that Gz) > and G z) hanges sign at most twie on [,]. Then dz eπjgz) Gz) 4. π Proof: The proof follows the steps of the previous lemma. In order to highlight the differenes and for the sake of readability, we fous here on the simple ase where Gw,z). For any ε >, we have dz e j πgz) = + z+ε z ε [,z ε] [z +ε,] dz e j πgz) dz j πg z) j πg z) ej πgz) Note that the first term an be simply upperbounded by ε. The seond term an be bounded by the integration by parts Fig. 5. Domains of integration: the relative positions of the points x k and x m determine in whih domain one is U on the figure). Let now ε > and let us divide the integration domain x k,x m,y k,y m ) [,] 4 in 5) into three subdomains see Figure 5): U = y m y k } /d) x m x k ε U = < y m y k } /d) x m x k < ε U 3 = y m y k } /d) x m x k Consider first the integral over U. One an hek that g k,m w,z) = xm + x k ym y k d/ + x + w)dx d/ + x + w) + y k z) y z)dy d/ + x m + w) + y z)

12 So the first order partial derivative of g k,m w,z) with respet to z is given by g k,m z w,z) = xm + x k ym z y k )d/ + x + w) dx d/ + x + w) + z y k ) ) 3/ y k d/ + x m + w) dy d/ + x m + w) + z y) ) 3/ 6) From this expression, we dedue that if x k,x m,y k,y m ) U, then g k,m z w,z) K A ) y m y k d d x m x k for a onstant K > independent of, and d. Notie next that G k,m y,z). It an further be heked that both g k,m z w,z) and G k,m z w,z) hange sign at most twie on the interval z [,] for w fixed). Therefore, applying Lemma., we onlude that dw dz eπjg k,mw,z) G k,m z) dw dz eπjg k,mw,z) G k,m y,z) K d y m y k /d) x m x k Sine we know that this integral is also less than, this in turn implies dw dz eπjg k,mw,z) G k,m w,z) U K d = K d log U y m y k /d) x m x k ) ε Seond, it is easy to hek that dw U dz eπjg k,mw,z) G k,m w,z) ε The integral over the third domain of integration U 3 is more deliate. Notie first that the obvious bound dw dz eπjg k,mw,z) U 3 G k,m w,z) d allows to obtain S K d log whih an be made smaller than K log ) by hoosing ε = d when A3/4 ase). ) + ε + ε d ) d d d as d d in this For the remainder of the proof, let us therefore assume that d A 3/4 /. As before, we fous on the integral inside the square in the following expression dw dz eπjg k,mw,z) U 3 G k,m w,z) 7) Let us start by onsidering the simplest ase where the points x k and x m are loated on the same horizontal line, i.e. y k = y m. In this ase, the seond term in the expression 6) for g k,m z w, z) beomes zero, so we dedue the following lower bound: g k,m z w,z) K A3/ d x m x k z y k This, together with the above mentioned properties of the funtions g k,m and G k,m, allows us to apply Lemma. so as to obtain dw dz eπjg k,mw,z) G k,m w,z) d K A 3/4 xm x k for a onstant K > independent of, and d. A slight generalization of this argument see below for details) shows that not only when y k = y m but for any x k,x m,y k,y m ) U 3, we have dw dz eπjg k,mw,z) G k,m w,z) d K A 3/4 d K A 3/4 x m x k ) + y m y k ) ) /4 xm x k A 3/ 8) Sine we also know that the above integral is less than, we further obtain dw dz eπjg k,mw,z) G k,m w,z) } min K d x m x k, For any < η <, we an now upper bound 7) as follows: dw dz eπjg k,mw,z) U 3 G k,m w,z) U 3 x m x k < η} d + K U 3 x m x k η} A 3/ x m x k d ) η + K log = η + K d ) log d η η implying that S K d log A 3/ ) + ε + η + K d log ε ) η Choosing finally ) ε = η = d/ allows to onlude that d S K log d) also in the ase where d A 3/4 /.

13 x m x k Fig. 6. Tilted referene frame. Proof of equation 8). In order to prove 8), we need to make a hange of oordinate system, replaing w,z) by w,z ), where w is now in the diretion of the vetor x k x m and z is perpendiular to it see Figure 6). In this new oordinate system, the integral reads dw dz eπjg k,mw,z ) DR G k,m w,z ) where g k,m w,z ), G k,m w,z ) have the same form as g k,m w,z), G k,m w,z), but now, the domain of integration D R is a tilted square, as indiated on the Figure 6. Using then the same argument as in the ase y k = y m, we onlude that dw dz eπjg k,mw,z ) d DR G k,m w,z ) K A 3/4 z DR x m x k Notiing finally that x m x k = x m x k ) + y m y k ) allows to onlude that 8) holds. REFERENCES [] G. J. Foshini, Layered Spae-Time Arhiteture For Wireless Communiation in a Fading Environment when Using Multi-Element Antennas, AT&T Bell Labs Teh. Journal ), 4 59, 996. [] E. 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