MULTIPLE-INPUT MULTIPLE-OUTPUT (MIMO) is. Spatial Degrees of Freedom of Large Distributed MIMO Systems and Wireless Ad Hoc Networks

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1 22 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 3, NO. 2, FEBRUARY 23 Spatial Degrees of Freeom of Large Distribute MIMO Systems an Wireless A Ho Networks Ayfer Özgür, Member, IEEE, OlivierLévêque, Member, IEEE, an Davi Tse, Member, IEEE Abstrat We onsier a large istribute MIMO system where wireless users with single transmit an reeive antenna ooperate in lusters to form istribute transmit an reeive antenna arrays. We haraterize how the apaity of the istribute MIMO transmission sales with the number of ooperating users, the area of the lusters an the separation between them, in a line-of-sight propagation environment. We use this result to answer the following question: an istribute MIMO provie signifiant apaity gain over traitional multi-hop in large aho networks with n soure-estination pairs ranomly istribute over an area A? Two iametrially opposite answers [24] an [26] have emerge in the urrent literature. We show that neither of these two results are universal an their valiity epens on the relation between the number of users n an A/, whih we ientify as the spatial egrees of freeom in the network. is the arrier wavelength. When A/ n, there are n egrees of freeom in the network an istribute MIMO with hierarhial ooperation an ahieve a apaity saling linearly in n as in [24], while apaity of multihop sales only as n. On the other han, when A/ n as in [26], there are only n egrees of freeom in the network an they an be reaily ahieve by multihop. Our results also reveal a thir regime where n A/ n. Here, the number of egrees of freeom are smaller than n but larger than what an be ahieve by multi-hop. We onstrut saling optimal arhitetures for this intermeiate regime. Inex Terms Spatial egrees of freeom, large sale MIMO, multi-user MIMO, virtual MIMO, istribute MIMO, wireless a ho networks, hierarhial ooperation, linear apaity saling. I. INTRODUCTION MULTIPLE-INPUT MULTIPLE-OUTPUT (MIMO) is one of the key tehnologies to ahieve high-ata rates in urrent wireless systems. Both the transmitter an the reeiver are equippe with multiple antennas, whih allows to spatially multiplex several streams of ata an transmit them simultaneously. When the sattering environment is rih enough to allow reeive antennas to separate out signals from ifferent transmit antennas, MIMO hannels offer large Manusript reeive 24 January 22; revise 3 June 22. The work of Ayfer Özgür was supporte in part by the European Researh Counil grant NOWIRE ERC-29-StG-2437 while she was with EPFL, Switzerlan. An initial version of this paper was presente at the Information Theory an Appliations Workshop, San Diego, January 2. A. Özgür is with Stanfor University, Pakar Eletrial Engineering, 35 Serra Mall, Room 25 Stanfor, California , USA ( aozgur@stanfor.eu). O. Lévêque is with the Eole Polytehnique Féérale e Lausanne, Faulté Informatique et Communiations, Builing INR, Station 4, CH - 5 Lausanne, Switzerlan ( olivier.leveque@epfl.h). D. Tse is with the University of California at Berkeley, Department of EECS, Berkeley, CA 9472, USA, ( tse@ees.berkeley.eu). Digital Objet Ientifier.9/JSAC apaity gains: the apaity of suh a MIMO hannel with M transmit an reeive antennas is proportional to M. This was establishe in the groun-breaking papers of Fohini- Gans [] an Telatar [2] whih assume an i.i.. faing moel for hannels between ifferent antenna pairs. However, when there is insuffiient sattering in the environment or when antennas are ensely pake together in small han-hel evies, the hannels between ifferent antenna pairs beome orrelate. Suh physial onstraints prevent MIMO apaity from inreasing inefinitely with M. The impat of orrelate faing [3], [4], [5], [6], [7], the sattering environment [8], [9] an antenna oupling [], [] on MIMO apaity was stuie by a large boy of follow-up researh, whih also provie a haraterization of the inherent spatial egrees of egrees of freeom in a MIMO system as a funtion of the area an the geometry of the antenna arrays an the angular sprea of the physial environment [2], [3], [4], [5]. Reently, istribute MIMO ommuniation arise as a promising tool to ahieve large performane gains in wireless networks, similar to those provie by onventional MIMO in the point-to-point wireless hannel. Here, wireless users with a single transmit an reeive antenna ooperate in lusters to form istribute transmit an reeive antenna arrays. A ommon assumption in the performane analysis of istribute MIMO systems is that hannels between ifferent pairs of noes are subjet to inepenent faing [6], [24]. Analogous to the point-to-point ase, this leas to apaity gains linear in the number of noes M ontaine in the transmit an reeive lusters. In a istribute setting, noes (or antennas) are typially muh farther apart from eah other as ompare to lassial MIMO, so an i.i.. faing moel may seem appropriate. However, the number of noes 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 inefinitely with M an orrelations between pairwise hannels are expete to limit performane when M is large. In this paper, we provie a rigorous lower boun on the saling of the apaity of a large istribute MIMO system with the area of the transmit an reeive lusters, the separation between the lusters an the number of noes M ontaine in eah luster assuming a line-of-sight propagation moel an a ranom istribution of noes over the luster areas. We show that the apaity of istribute MIMO systems sales at least as /3/$3. 23 IEEE min } M, when / min M, / } when ()

2 ÖZGÜR et al.: SPATIAL DEGREES OF FREEDOM OF LARGE DISTRIBUTED MIMO SYSTEMS AND WIRELESS AD HOC NETWORKS 23 where is the arrier wavelength. This result ientifies an / as the spatial egrees of freeom in the istribute MIMO hannel in the two orresponing regimes. The apaity of the hannel sales linearly in M when the physial hannel has more than M spatial egrees of freeom. Fortunately, this an be often the ase for atual networks. Consier for example two lusters of imensions m m separate by a istane of m. When ommuniation takes plae aroun 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 egrees of freeom for all users. When the istane between the two lusters is km, is still. The spatial egrees of freeom are expete to be even larger in sattering environments. The istribute MIMO hannel, an therefore its analysis, iffers from the lassial MIMO hannel in a ouple of ways. First, the istanes between ifferent pairs of users an be signifiantly ifferent in the istribute ase, whih results in heterogeneous hannel gains itate by the geometry of the network. Seon, while in lassial MIMO, the separation between the antenna arrays is typially muh larger than the length (or the iameter) of the arrays, for istribute systems these two imensions an be omparable. Our analysis takes into aount these new aspets brought by the istribute nature of the problem. Mathematially, our approah iffers signifiantly from existing results stuying the spatial egrees of freeom of lassial MIMO hannels. While suh results (for example, see [2], [4], [5]) are base on stuying the singular values of the ontinuous propagation operator uner approximations for the regime where, our analysis is mathematially rigorous an is base on ranom matrix analysis. The haraterization of the saling of the apaity of istribute MIMO systems allows us to ientify the number of spatial egrees of freeom in large wireless a ho networks, an to reonile some seemingly ontraiting results in the urrent literature on saling laws for wireless networks. The stuy of the asymptoti regime where the number of users in a wireless network is large was initiate by the seminal paper [7] of Gupta an Kumar an reeive signifiant attention in the literature [8], [9], [2], [2], [22], [23]. Gupta an Kumar showe that the apaity of multihop ooperation sales as n with inreasing number of user n in the network. In this traitional ommuniation arhiteture, pakets are route from eah soure to its estination along a path where intermeiate users at as relays. Eah relay eoes the pakets sent from the previous relay an forwars them to the next. A n saling for the total apaity implies that the rate per user ereases as / n with inreasing system size n. Can more sophistiate ooperation between users signifiantly inrease the apaity of large wireless networks? Two iametrially opposite answers have emerge in the reent literature: ) Capaity an be signifiantly improve when users form istribute MIMO arrays via a hierarhial ooperation arhiteture [24]. In regimes where power is not a limiting fator [25], the apaity an sale almost linearly with n implying a onstant rate per user. 2) The saling of the apaity is upper boune by n ue to the spatial onstraints impose by the physial hannel [26]. Nearest-neighbor multi-hop alreay ahieves this saling an more sophistiate ooperation is useless. The key ifferene between these two results is their assumptions for the hannel moel between pairwise noes. [24] assumes that the phases of the hannel gains an be moele as uniformly istribute ranom variables, inepenent aross ifferent pairs of noes in the network. [26], 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 [26] 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. What is the way to reonile these two sets of results? We answer this question in the seon part of the paper builing on the result in (). We show that uner the physial hannel moel of [26], the istribute MIMO base hierarhial ooperation arhiteture in [24] ahieves a apaity saling as }} n, A max min n,. (2) in a network of n soure-estination pairs uniformly istribute over an area A an ommuniating aroun a arrier wavelength. The saling of the apaity epens on how n ompares to A/, whih an be interprete as the spatial egrees of freeom available in the network. The two earlier results an be reovere as two speial ases of this new result: ) When A/ n, the apaity sales linearly in n. In this regime, there are suffiient spatial egrees of freeom for all the n users in the network an they an be exploite by istribute MIMO ommuniation. The i.i.. faing assumption aross ifferent noe pairs in [24] leas to n egrees of freeom an therefore [24] inherently assumes that the network operates in this regime. 2) When A/ n, the apaity sales as n. In this regime, the spatial egrees of freeom available in the network are as few as n, an therefore they an be reaily ahieve by multihop. By assuming that the ensity of noes is fixe as the number of noes n grows, [26] assumes that the number of spatial egrees of freeom A/ is proportional to n. Therefore, [26] inherently assumes that the network operates in this regime. Therefore, neither of the two onlusions in [24] an [26] that more sophistiate ooperation an provie signifiant apaity gains or is useless are universal. They orrespon to two ifferent operating regimes of large wireless networks. (2) larifies the onitions for a network to be in either of these two regimes. Inee, (2) also unovers a thir regime where the network is partially limite in spatial egrees of freeom. When n A/ n, the number of spatial egrees of freeom is smaller than n, so the spatial limitation is felt, but larger than what an be ahieve by simple multi-hopping. (Multi-hop ahieves n saling inepenent of A/.) We show that either a moifiation of the hierarhial ooperation

3 24 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 3, NO. 2, FEBRUARY 23 sheme in [24] or a version of the MIMO-multihop sheme in [25] an ahieve the A/ available egrees of freeom an therefore the optimal saling of the apaity in this regime. The main iea behin the first moifiation is to allow only a subset N of the soure-estination pairs to ommuniate at a time using the hierarhial ooperation sheme, an then take turns among ifferent subsets. N = A/, the number of spatial egrees of freeom in the network, so eah subset orrespons to a ilute network whih is not limite in spatial egrees of freeom. The iea behin the seon sheme is to form lusters of an intermeiate size an hop aross several lusters to reah estinations where eah hop is performe via istribute MIMO transmissions. The luster size is hosen ritially to ensure linear saling for the istribute MIMO transmissions. Traitionally, the literature on saling laws for wireless networks seeks the saling of the apaity with the number of noes n when all systems parameters are ouple with n in a speifi way. One ommon approah is to assume that the area sales linearly in n, while all other parameters remain fixe, as in [26]. As we have alreay seen, this immeiately implies that the spatial egrees of freeom in the network A/ are proportional to n. 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. For atual networks, there an be a huge ifferene between A/ an n.takean example of a network serving n = users on a ampus of km 2, operating at 3 GHz: A/ =, 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 are inee an not as given by (2). For the lassial MIMO hannel, it is now well unerstoo that there are a number of qualitatively ifferent regimes. When the antennas are separate suffiiently apart, the apaity inreases linearly in n. However for ense antenna arrays the apaity is limite by physial onstraints an annot inrease linearly in n. To obtain an analogous unerstaning of the operating regimes of large wireless networks, we avoate in this paper a shift of the large networks researh agena from seeking a single universal saling law, where parameters of the network are ouple to 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. We have introue this approah in [25] where we eouple the number of noes an the amount of power available. The urrent paper follows the spirit of [25], both mathematially an philosophially, 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. D T D R Fig.. Two square lusters of area separate by istane. A version of this problem has been stuie in an inepenent onurrent work [3], however both the formulation of the problem an the propose arhitetures iffer from the urrent paper: in [3], the network area A is taken either fixe or proportional to n, while the arrier wavelength sales own to zero with inreasing n. The arhiteture in [3] is obtaine by iluting the istribute MIMO transmissions at eah level of the hierarhy in [24], as oppose to iluting the whole network as we o here. In the urrent paper, we also show that the same performanes an be alternatively ahieve with a MIMO-multihop strategy. II. SPATIAL DEGREES OF FREEDOM OF DISTRIBUTED MIMO SYSTEMS A. Moel We onsier a istribute MIMO transmission between two square lusters of area separate by istane (see Figure ), with eah luster ontaining M noes istribute uniformly at ranom over the area. Eah noe is equippe with one antenna, oriente in the iretion perpeniular to the plane, with a given power buget P. We assume that ommuniation takes plae over a flat hannel with banwith W an that the signal reeive by noe i at the RX luster at time-slot m is given by y i [m] = M h ik x k [m]+z i [m] k= where x k [m] is the signal sent by noe k at the TX luster at time m, z i [m] is aitive white irularly symmetri Gaussian noise (AWGN) of power spetral ensity N /2 Watts/Hz. In a line-of-sight environment, the omplex baseban-equivalent hannel gain h ik between transmit noe k an reeive noe i at time m is given by h ik = G exp(2πjr ik/) (3) r ik where is the arrier wavelength, r ik is the istane between noe i an noe k an G is Friis onstant given by ( ) 2 G = G t G r 4π with G t an G r being the transmit an reeive antenna gains, respetively. Finally, we assume full hannel state information at all the noes, whih is a reasonable assumption here, as the hannel oeffiients only epen on the noe positions an these o not vary over time.

4 ÖZGÜR et al.: SPATIAL DEGREES OF FREEDOM OF LARGE DISTRIBUTED MIMO SYSTEMS AND WIRELESS AD HOC NETWORKS 25 B. Main Result Our goal in this setion is to provie a lower boun on the spatial egrees of freeom of the system esribe above. It an be inferre from [26] that the spatial egrees of freeom of suh a istribute MIMO system are upperboune (up to logarithmi fators) by / as gets large, irrespetive of the istane. tually, the spatial egrees of freeom erease as inreases, beause of the reution of the aperture of the MIMO system. We prove below that the spatial egrees of freeom of the system are at least min M, /} if / min M, / } if (again, up to logarithmi fators). Notie that in the seon regime, the obtaine lower boun mathes the upper boun foun in [26]. Finally, notie that if /, then the system has learly at least one egree of freeom. Let us now state our main result. Theorem 2.: Let /, an let the noes in the transmit luster D T perform inepenent signaling with power P eah, suh that the long-istane SNR between these two lusters efine as SNR() =M GP N W 2 (4) is greater than or equal to B. Then there exists a onstant K>inepenent of M,, an suh that the apaity C MIMO of the istribute MIMO hannel from the transmit luster D T to the reeive luster D R is lowerboune by } / K min M, if A/ log( /) C M K min M, if } / log( /) with high probability as M gets large. The rest of this setion is evote to the proof of this result, whih is mae of three ingreients. The first key ingreient provies a lower boun on the MIMO hannel apaity average over the ranom noe positions in the first regime where /. Lemma 2.: If /, then the expete apaity E(C MIMO ), average over the ranom noe positions, satisfies } / E(C MIMO ) K min M, (5) log( /) Next, we show that the apaity of the istribute MIMO hannel with given ranom noe positions is lose to its expete value with high probability. The proof relies on lassial onentration arguments an is relegate to the en of the present setion. Lemma 2.2: In general, if M noes partiipate to the MIMO transmission, then for all ε>, there exists K> suh that C MIMO E(C MIMO ) KM /2+ε with high probability as M gets large. Finally, we show that the stuy of the seon regime ( ) an be brought bak to the ase = by simply reuing the set of transmitting an reeiving noes by a fator 2, so that the interluster istane beomes of the same orer as their raius. Again, the proof is relegate to the en of the present setion. Lemma 2.3: If, then the spatial egrees of freeom of the system are of the same orer as when =. Combining these three lemmas yiels the result given in Theorem 2.: Assume first /. Then by Lemma 2., E(C MIMO ) K min / } / M, log( /) If M log(/), then Lemma 2.2 allows to onlue that C MIMO itself is with high probability at least of orer M.Ifon the other han M> / log(/), then it shoul be notie that in this ase, it is useless to have all the M noes partiipating to the MIMO transmission. Only M = / log(a suffie. /) Applying then the onentration result replaing M by M allows to onlue. Finally, Lemma 2.3 shows that for all, the egrees of freeom of the system are lowerboune by } C MIMO K / min M, log( /) with high probability for some other onstant K >. This onlues the proof. In the sequel, we provie the proof of Lemma 2.. The proofs of Lemmas 2.2 an 2.3 are relegate to the en of the present setion. Proof of Lemma 2.. For notational onveniene, we start by efining f ik = = G h ik = r ik exp(2πjr ik /) x k w i exp(2πj x k w i /) (6) 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. Notie that r ik ( /), so (+2 2 /) f ik, (7) where = (+2 2) an the first inequality follows from the fat that. Remembering the efinition of SNR() given (4), we obtain the following expression for the average apaity of the istribute MIMO hannel (where we reall here that H, F are the matries with entries h ik, f ik, respetively): E(C MIMO )=E ( log et ( I + )) P N W HH = E ( log et ( I + SNR() FF /M )) = M E (log ( + SNR() ))

5 26 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 3, NO. 2, FEBRUARY 23 Fig. 2. xm D T f im x k f lm f ik f lk S = E(f ik flk f lm fim ), i l, k m where is an eigenvalue of FF /M pike uniformly at ranom. By applying now the Paley-Zygmun inequality, stating that for a non-negative ranom variable X, (E(X) t)2 P(X t) E(X 2, <t<e(x) ) we obtain that for <t<e(): w l w i D R E(C MIMO ) M log ( + SNR() t) P( > t) From (7), we further obtain M log ( + SNR() t) E() = M 2 E ( tr(ff ) ) = M 2 (E() t)2 E( 2 ) M E( f ik 2 ) 2. i,k= E( 2 )= M 3 E(tr(FF FF )) = M M 3 E(f ik flkf lm fim) 2+ M 3 i,k,l,m= M i,k,l,m= i l,k m E(f ik f lk f lmf im ) 2+MS where S = E(f ik flk f lm fim ), with i l an k m (notie that S oes not epen on the speifi hoie of i l an k m). See Figure 2. Choosing then t = 2 /2, we obtain E(C MIMO ) (M 4 /4) log ( +SNR() 2 /2) 2+MS K min M, } S for a onstant K> inepenent of M an S, provie that SNR() B, whih was our initial assumption (4). 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.. an irularly-symmetri, S =, so the apaity is of orer M. On the other han, if we onsier the line-of-sight hannel moel (6) in the senario where noes are plae on a single straight line, then a simple omputation shows that S is of orer, so that the apaity is also of orer in this ase. The problem we are looking at lies between these two extreme ases. Our aim in the following is to show the Stritly speaking, this only shows that the lower boun on the apaity is of orer, but in this ase, the matrix F an be shown to be essentially rank one, so the atual apaity is inee of orer. following lemma, whih allows to onlue the proof of Lemma 2.. Lemma 2.4: If /, then there exists a onstant K>inepenent of, an, suh that ( ) ( ) S K log. (8) We now give a proof iea for Lemma 2.4. Let us first expliitly write the expression for S. We have S = E(f ik flk f lm fim ) = A 4 x k x m w i w l ρe 2πjΔ/ D T D T D R D R (9) where Δ= x k w i x k w l + x m w l x m w i () an 4 ρ = () x k w i x k w l x m w l x m w i We first erive the result (8) by approximating the internoe istanes in the regime. This approximation, mae alreay by various authors [2], [9], [4], [5] in ifferent ontexts, allows us to erive an upper boun on S, an orresponingly, a lower boun on the spatial egrees of freeom. In aition, we provie in the Appenix a rigorous erivation of the lower boun, whih oes not make use of the approximation. As far as we know, this erivation is new. Consier two noes at positions x =( x, y) D T an w =( + w, z) D R,wherex, 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 Δ= x k w i x k w l + x m w l x m w i 2 ((y k z i ) 2 (y k z l ) 2 +(y m z l ) 2 (y m z i ) 2 ) = (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 egrees of freeom between the two lusters are mainly etermine by the phases of the hannel oeffiients an not so muh by the amplitues. In the Appenix, we show that this intuition is orret. These two suessive approximations lea to the following approximation for S: S S = y k y m z i =2 y k y m z i y k 2πj z l e 2πj z l e (ym y k)(z l z i) (ym y k)(z l z i)

6 ÖZGÜR et al.: SPATIAL DEGREES OF FREEDOM OF LARGE DISTRIBUTED MIMO SYSTEMS AND WIRELESS AD HOC NETWORKS 27 Ay x w Az (where we reuse here the notation aopte in the proof of Lemma 2.) an be seen as a funtion of the noe positions x k an w i : C MIMO = g M (x,...,x M ; w,...,w M ) Fig. 3. Ax Coorinate system. Aw 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 spatial egrees of freeom between two lusters of M noes separate by a istane are the same, be the noes uniformly istribute on two squares of area or on two parallel (vertial) lines 2 of length. We show below that the above integral is of orer /. Let us ompute the first integral, whih yiels 2πj z l e (ym y k)(z l z i) = 2πj (y m y k ) e 2πj z l = (ym y k)(z l z i) z l = This implies that 2πj z l e (ym y k)(z l z i) K y m y k for a onstant K>inepenent of, an. We an ivie the integration over y k an y m into two parts, so 2πj y k y m z i z l e (ym y k)(z l z i) y k ( (yk +ε) ε ) = y k y m + y k y m y k y k +ε 2πj z i z l e (ym y k)(z l z i) for any <ε<. Thefirst term an be simply boune by ε, whih yiels the following upper boun on S S 2ε +2K ε y k y m y m y k 2ε +2K log ( ε ) y k +ε So hoosing ε = /,wefinally obtain ( ) ( ) S S K log for a onstant K>inepenent of, an. Proof of Lemma 2.2. First observe that the apaity of the istribute MIMO hannel C MIMO =loget(i + SNR() FF /M ) 2 This result an be prove rigorously; atually, the rigorous argument following for the two squares applies equally likely to the ase of two parallel lines As we will see in the following, the apaity oes not vary signifiantly with the noe positions, whih allows us to apply the following (simplifie version of the) theorem by MDiarmi [28]. Theorem 2.2: Let x,...,x M be a family of i.i.. ranom variables istribute in a boune region A R 2,anlet f M : R 2M 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 an x,...,x k, x k,...,x M A. Then for all t>, P ( f M (x,...,x M ) E(f M (x,...,x M )) >t) ) 2exp ( 2t2 M 2 M In orer to apply the above theorem (replaing f M by g M ), we nee to upperboun the ifferenes g M (x,...,x k,...,x M ; w,...,w M ) g M (x,...,x k,...x M ; w,...,w M ) an 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 onsier the first ase here. Fix k M. Notie first that moifying the vetor x k only moifies a single olumn of the matrix F.Letus efine F as being the matrix F with olumn k remove (so F is an M (M ) matrix). Beause of what was just observe, F F is a rank one matrix, so using the interlaing property of the singular values of F an F [29, Theorem 7.3.9], we obtain that for all j M, j j an j j+ where... M are the eigenvalues of FF /M an... M are the eigenvalues of F F /M. Remember now that g M (x,...,x M ; w,...,w M ) =loget(i + SNR() FF /M )= Defining M log( + SNR() j ) j= g M = loget(i + SNR() F F /M ) = M log( + SNR() j ) j=

7 28 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 3, NO. 2, FEBRUARY 23 /2 /2 Fig. 4. The istane between the two highlighte sublusters is equal to +. an applying the above inequalities on the eigenvalues, we see that g M (x,...,x M ; w,...,w M ) g M log(+snr() ) It an be easily seen M. Besies, we are intereste in the regime where the growth of SNR() 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 ) 2K log M This finally shows, via MDiarmi s theorem, that gm P( (x,...,x M ; w,...,w M ) ) E(g M (x,...,x M ; w,...,w M )) >t ( t 2 ) 2exp 2MK 2 (log M) 2 whih gives the result, by setting t = M /2+ε. Proof of Lemma 2.3. The proof of this lemma is base on the following simple observation. When,thetwo lusters are lose to eah other, as illustrate on Figure 4. Consier now the two square sublusters of size 2 2 that are the most separate horizontally. These two sublusters are now separate by a istane +, whih is of the same orer as. On the other han, both the area an the number of noes in these sublusters remain of the same orer as in the original lusters. More preisely, the new area is /4 an orresponingly, the number of noes in eah subluster is aroun M/4 with high probability. By letting only the noes in these two sublusters partiipate to the MIMO transmission, we therefore see that the same orer of spatial egrees of freeom an be ahieve as when =. III. AD HOC WIRELESS NETWORKS An a ho network is a olletion of wireless users whih an self-organize an ommuniate among themselves without the help of any fixe infrastruture. Optimal ooperation arhitetures for suh networks have reeive muh fous in the asymptoti regime where the number of users is large. Consier an a ho network where n soure-estination pairs are inepenently an uniformly istribute over an area A. Eah soure noe wants to ommuniate to its orresponing estination at the same rate R bits/s/hz. For this network, we have evelope a hierarhial ooperation arhiteture in [24] that maintains a onstant per-pair ommuniation rate R even when the network serves a growing number of users, provie that the hannels between pairwise users are subjet to i.i.. faing. More preisely, when the phases of the hannel oeffiients an be moele as i.i.. ranom variables inepenent of the noe loations, hierarhial ooperation ahieves an aggregate throughput T = nr that sales linearly in n. Current ommuniation arhitetures for a ho networks an not provie salable performane. The traitional approah is to forwar information from the soure noe to the estination by following a path, with intermeiate noes on the path ating as relays. The aggregate throughput of this multi-hop arhiteture sales as n with inreasing network size n [7]. Distribute MIMO is key to the linear saling of the arhiteture in [24]. Hierarhial ooperation allows noes to effiiently organize in lusters an establish the n ommuniations in the network via istribute MIMO transmissions between large lusters. Uner i.i.. faing, the apaity of these istribute MIMO transmissions are linear in the the number of noes M ontaine in the transmit an reeive lusters. In the earlier setion, however, we have seen that the area of the lusters an the separation between them poses a limit on the linear saling of istribute MIMO. In this setion, we evaluate the performane of hierarhial ooperation uner the physial moel in (3) an show that it an still ahieve linear saling but provie that n. When A <n, we present two moifiations of the hierarhial ooperation arhiteture that ahieve the optimal saling. Our result is formally summarize in the following theorem. Theorem 3.: Consier a wireless network of n noes istribute uniformly at ranom over a square area A suh that A/ n. Assume that eah noe is subjet to an average power onstraint P an the network is alloate a total banwith of W. The hannels between pairwise users are governe by the LOS moel in (3). Let us efine the SNR of the network as GP SNR = n (2) N WA an assume that SNR B. For any ε>, there exists a onstant K>inepenent of n, an 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 boun on the best ahievable apaity saling uner the moel in (3) is evelope in [26] for the ase where the ensity of the noes remains fixe as the number of noes n grows. In this ase, [26] shows that the apaity of the wireless network is upperboune by T K n (log n) 2, A

8 ÖZGÜR et al.: SPATIAL DEGREES OF FREEDOM OF LARGE DISTRIBUTED MIMO SYSTEMS AND WIRELESS AD HOC NETWORKS 29 with high probability, where K > is a onstant inepenent of n. The above result an at first lea to the onlusion that the best saling ahievable in wireless networks is n an therefore multi-hop is saling optimal. However, a eeper look reveals that the onlusion that the apaity sales like n omes from the assumption that the ensity of noes is fixe, so that A/ is proportional to n. A relatively straightforwar generalization of the analysis in [26] gives the following refine upper boun on apaity: K n (log n) 2 if A n ( T K A log ) 2 A if n> A > n (3) K n (log n) 2 n if A 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,whihis ahievable by a simple multihop. For A/ > n, the ahievability of the upper boun was an open problem an is now establishe in Theorem 3. when SNR B. This leas to the onlusion that in the regime when n an A/ are both large an SNR B, the apaity of the network is approximately }} n, A max min n, oringly, the optimal operation of the network falls into three ifferent operating regimes: ) A/ n: The number of spatial egrees of freeom is too small, more sophistiate ooperation is useless an multihop is optimal. 2) A/ n: The number of spatial egrees of freeom is n, the optimal performane an be ahieve by the same hierarhial ooperation sheme introue in [24]. Spatial egree of freeom limitation oes not ome into play an the performane is as though phases were i.i.. uniform aross noe pairs. 3) n< A/ < 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 the optimal saling in this regime. The SNR in (2) an be ientifie as the typial SNR between nearest neighbor noes in the network uner the hannel moel (3). Note that in a ranom network of n noes istribute over an area A, the typial separation between nearest neighbor pairs is given by A/n. The onition (2) ensures that these hannels are in the high-snr regime. Note that hannels between pairs further away an be in low-snr. Ientifying optimal ooperation arhitetures for networks with SNR B uner the physial hannel moel remains an open problem. Optimal arhitetures for suh networks have been ientifie in [25] uner the i.i.. faing moel. A. Optimal ooperation in networks with limite spatial egrees of freeom Capitalizing on the result of Theorem 2., 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 [24] sales linearly in n uner the LOS moel of (3). (B) When n< A/ < n, we show that a ilute hierarhial ooperation arhiteture ahieves the saling in Theorem 3.. Here, only a ranomly hosen subset of the soure-estination pairs operate at a time while remaining noes stay silent. This reates a ilute network for whih ase () hols so the network oes not experiene any limitation in spatial egrees of freeom. 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 hybri arhiteture ombining istribute MIMO with multi-hop, introue in [25]. Here, noes form MIMO lusters of an intermeiate size an information is route from one luster to the next via suessive istribute MIMO transmissions between ajaent lusters. The luster size is ritially hosen at the largest possible sale that allows for linear saling of the istribute MIMO transmissions. When n = A/, this luster size is a single noe an the hybri arhiteture reues to pure multi-hop. When A/ = n, the luster size is as large as n, an the arhiteture reues to pure hierarhial ooperation. The ifferene between the two strategies in (B) an (C) arises when we moify the hannel moel to h ik = G ej2πr ik/ (4) r α/2 ik where α is the power path loss exponent etermining how fast signal power eays with istane in the environment. Although not physial, this hannel moel provies a simple way to apture the impat of larger path loss attenuation ue to multiple propagation paths, an at the same time it preserves the spatial orrelation between hannels by keeping the epenene of the phases to the geometri struture of the network. It has been shown in [26] that multiple paths o not hange the saling of the number of spatial egrees of freeom in a large network. Therefore it suffies to onentrate on a LOS moel for the phases. However, multiple paths an have signifiant impat on the power. For example, with an aitional reflete path from the groun plane the power path loss over istane r inreases from r 2 to r 4. Uner this new moel, the power onition for ahieving linear apaity for istribute MIMO beomes SNR() =M GP N W α B in Theorem 2.. oringly, the ilute hierarhial ooperation arhiteture in (B) ahieves the performane in Theorem 3. when GP SNR l = n N W ( A) B α SNR l is efine as the long-range SNR of a network in [25]. This quantity an be ientifie as n times the reeive SNR in a point-to-point transmission over the largest sale in the network, the iameter A. Theextran omes from the network

9 2 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 3, NO. 2, FEBRUARY 23 effet, it reflets the potential power gain ue to ooperation over the global sale. For the multi-hop MIMO arhiteture in (3), the power requirement is given by GP SNR( )=M N W ( ) α B where SNR( ) is the analog of SNR l but for a luster of area ontaining M = n/a noes. In partiular, we will hoose M in the sequel suh that M = A 2 n.when n< A/ < n, wehave M n. It is easy to verify that when α > 2, SNR( ) SNR l (see (3.2) in [25]), therefore the seon onition is less stringent than the first. When α =2,SNR( )=SNR l = SNR in (2). Therefore, for the LOS moel, the two arhitetures are equivalent. A etaile isussion on the relevane of the SNR parameters above in networks with i.i.. faing is provie in [27]. The below isussion assumes that the reaer is familiar with the hierarhial ooperation an the MIMO multihop arhitetures an their performane analysis. A etaile esription of these strategies an be also foun in [27].. Hierarhial ooperation when A/ n In this regime, the upper boun in (3) allows for throughput saling linear in n. Potentially hierarhial ooperation an ahieve arbitrarily lose to linear saling. One nees 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 noes in the lusters. This is easy to verify: onsier a MIMO transmission between two lusters of area an size M = n/a. The separation between these two lusters is upperboune by the iameter of the network A. Therefore A A A = n M Therefore when A/ n, M, so by Eq. (5), istribute MIMO transmissions operate with full egrees of freeom (up to a logarithmi fator), just like in the ase of i.i.. phases. To ompensate for the logarithmi fator, we argue that / log( /) M ε for any ε> an suffiiently large M. This in turn implies that the apaity of the istribute MIMO transmissions sale as M ε. The erease by M ε is apture in the n ε egraation in the overall throughput in Theorem 2.. We also nee to verify that the istribute MIMO transmissions have suffiient power as require in onition (4). In the hierarhial ooperation arhiteture, the MIMO transmission between lusters of area an size M take plae insie a larger luster of area A an size M = A n/a in the next level of the hierarhy. Therefore the separation between the TX an RX lusters is upper boune by A.During the MIMO transmissions eah noe transmits with elevate power P m = M P M. This is beause of the time-ivision between MIMO transmissions from ifferent lusters. Eah noe transmits only a fration M/M of the time, therefore it an transmit with elevate power M P M an still satisfy the average transmit power onstraint P. See [27] for etails. Therefore, the SNR for the MIMO transmissions is given by SNR() =M GP m N WA = M GP N WA = n GP N WA B where the last inequality is the power onition in Theorem 3.. Therefore, MIMO transmissions at eah level of the hierarhy have full egrees of freeom an suffiient power. Hierarhial ooperation ahieves an aggregate throughput saling arbitrarily lose to linear in n in this ase. 2. 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 oul outperform multihopping strategies, whose asymptoti performane n is stritly suboptimal ompare to the upper boun A/. A iret appliation of the hierarhial ooperation sheme fails to improve on multi-hop in this ase, but it turns out that a simple aaptation of the sheme to this spatially limite situation ahieves the optimal saling. The iea is the following: organize the ommuniation of the n soure-estination pairs into n/n sessions, eah involving N soure-estination pairs, where N = A/. It is possible to hoose here the noes in a way suh that eah group of N noes statistially oupies the total area of the network. This way, no group of N noes onsiere alone feels the spatial limitation, as for this ilute network N = A/ an we are in the ase A/ N above. The sessions operate suessively an the traffi in eah session is hanle using hierarhial ooperation where only the N hosen noes are involve. The rest of the noes remain silent. Sine noes are ative only a fration of N/n of the total time, when ative they an transmit with elevate power P m = np/n an still satisfy their iniviual power onstraint P. Therefore, for the ilute network of N noes in eah session, the SNR is 3 N GP m N WA = n GP N WA B Therefore, the ilute network is neither power nor spaelimite an hierarhial ooperation ahieves aggregate throughput of orer N ε =( A/) ε for any fixe ε>. With time-ivision aross ifferent groups of noes, the same throughput is ahievable in the whole network. 3. MIMO multi-hop when n A/ < n Consier the MIMO multi-hop strategy esribe in Setion 3.3 of [27]. On the global sale, this hybri arhiteture is similar to multi-hop. The pakets of eah soure-estination pair are transferre by hopping from one luster to the next. At eah hop, the pakets are eoe an then re-enoe for the next hop. The arhiteture iffers from multi-hop by the fat that eah hop is performe via istribute MIMO transmissions assiste by hierarhial ooperation. Let us hoose the luster size M suh that / = M where = AM/n is the A 2 n area of the luster. This leas to M =. This hoie of the luster size ensures that the lusters are not limite in spatial egrees of freeom. Therefore, the apaity of 3 With the moel in (4), this onition beomes SNR l B.

10 ÖZGÜR et al.: SPATIAL DEGREES OF FREEDOM OF LARGE DISTRIBUTED MIMO SYSTEMS AND WIRELESS AD HOC NETWORKS 2 istribute MIMO transmissions at eah hop sales linearly in M provie that there is suffiient power. Sine the istribute MIMO transmissions at eah hop take plae over a istane, the power onition in (4) yiels SNR() =M GP B N W This is equivalent to SNR B in Theorem 3. 4, as M/ = n/a. When the apaity of the istribute MIMO transmissions at eah hop sale linearly in M, the aggregate throughput of the MIMO multi-hop arhiteture is given by nm /2 ε in Eq. (3.3) of [27]. Plugging our hoie M = 2 n,we obtain an aggregate throughput saling as ( A/) ε. IV. ACKNOWLEDGMENT We woul like to thank Mar Desgroseilliers for helping with the preparation of the present paper. APPENDIX Rigorous proof of Lemma 2.4. We now prove equation (8) without making use of approximations. We start again with expression (9) for S. Notie that ue to the symmetry of Δ an ρ in w i an w l, we an upper boun (9) as S 4 A 4 x k x m D T D T w DR e2πj ( x 2 k w x m w ) x k w x m w Expressing this upper boun more expliitly in the oorinate system shown on Figure 3, we obtain: S x k y k x m y m w z e2πjg 2 G k,m (w, z) (5) where ( g k,m (w, z) = ( + (x k + w)) 2 + (y k z) 2 ( + ) (x m + w)) 2 + (y m z) 2 / an G k,m (w, z) = ( 2 + (x k + w)) 2 + (y k z) 2 ( + (x m + w)) 2 + (y m z) 2 Let us first fous on the integral insie the square in (5). The key iea behin the next steps of the proof is ontaine in the following two lemmas. Lemma A.: 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 4 With the moel in (4), this onition beomes SNR() B. A G(z) 2 > an G (z) hanges sign at most twie on [, ]. Then z e2πjg(z) G(z) 4. π 2 Proof: By the integration by parts formula, we obtain z e2πjg(z) = G(z) 2πjg (z)g(z) = e2πjg(z)) z 2πjg (z) 2πjg (z)g(z) e2πjg(z) z g (z)g(z)+g (z)g (z) 2πj(g (z)g(z)) 2 e 2πjg(z) whih in turn yiels the upper boun ( z e2πjg(z) G(z) 2π g () G() + g () G() g ) (z) + z (g (z)) 2 G(z) + G (z) z g (z)(g(z)) 2 By the assumptions mae in the lemma, we have g (z) z (g (z)) 2 G(z) z g (z) 2 (g (z)) 2 z g (z) ( = z 2 (g (z)) 2 + ) z g (z) z + (g (z)) 2 ( = 2 g () g () + 2 z+ z z g (z) (g (z)) 2 ) g (z ) 2 g (z + ) So g (z) z (g (z)) 2 G(z) 6. 2 We obtain in a similar manner that G (z) z g (z)(g(z)) Combining all the bouns, we finally get z e2πjg(z) G(z) 4 π 2 Lemma A.2: 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 G(z) 2 > an G (z) hanges sign at most twie on [, ]. Then z e2πjg(z) G(z) 4. π 2 Proof: The proof follows the steps of the previous lemma. In orer to highlight the ifferenes an for the sake of reaability, we fous here on the simple ase where G(w, z). For any ε>, wehave z+ε z e j 2πg(z) = z e j 2πg(z) + z ε [,z ε] [z +ε,] z j 2πg (z) j 2πg (z) ej 2πg(z)

11 22 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 3, NO. 2, FEBRUARY 23 ε U 3 U 2 x m U x k Fig. 5. Domains of integration: the relative positions of the points x k an x m etermine in whih omain one is (U on the figure). Note that the first term an be simply upperboune by 2ε. The seon term an be boune by the integration by parts formula, whih yiels. z j 2πg (z) 2πg(z) j 2πg ej (z) [,z ε] [z +ε,] 2πg(z) z ej ε = 2πjg (z) 2πg(z) ej + 2πjg (z) + z +ε z ε + U z z +ε U 2 U 3 D T g (z) 2πg(z) 2πj(g ej (z)) 2 z g (z) 2πg(z) 2πj(g ej (z)) 2 Using the assumptions on g an g in the lemma an following similar steps to the proof of Lemma A., we obtain z exp(2πjg(z)) 2ε+ ( 2 2π g () + 2 g (z ε) + 2 g (z + ε) + 2 ) g () 2ε + 8 2π ε 2 Choosing ε = π yiels the esire result z exp(2πjg(z)) 8 π whih ompletes the proof. Let now ε>an let us ivie the integration omain (x k,x m,y k,y m ) [, ] 4 in (5) into three subomains (see Figure 5): U = y m y k ( } /) x m x k ε U 2 = < y m y k ( } /) x m x k <ε U 3 = y m y k ( } /) x m x k Consier first the integral over U. One an hek that g k,m (w, z) = + xm x k ym y k (/ + x + w) x (/ + x + w) 2 +(y k z) 2 (y z) y (/ + x m + w) 2 +(y z) 2 So the first orer partial erivative 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 )(/ + x + w) x ( (/ + x + w) 2 +(z y k ) 2) 3/2 y k (/ + x m + w) 2 y ( (/ + x m + w) 2 +(z y) 2) 3/2 (6) From this expression, we eue that if (x k,x m,y k,y m ) U,then g k,m (w, z) z K A ( ) y m y k x m x k for a onstant K > inepenent of, an. Notie next that G k,m (y, z). It an further be heke that both 2 g k,m z (w, z) an G k,m 2 z (w, z) hange sign at most twie on the interval z [, ] (for w fixe). Therefore, applying Lemma A., we onlue that w z e2πjg G k,m (z) w z e2πjg G k,m (y, z) K y m y k ( /) x m x k Sine we know that this integral is also less than, thisin turn implies w z e2πjg 2 G k,m (w, z) x k x m y k y m U K = K log x k x m y k y m U y m y k ( /) x m x k ( ) ε Seon, it is easy to hek that x k x m y k y m w z e2πjg 2 U 2 G k,m (w, z) 2ε The integral over the thir omain of integration U 3 is more eliate. Notie first that the obvious boun x k x m y k y m w z e2πjg 2 U 3 G k,m (w, z) 2 allows to obtain S K ( ) log +2ε +2 ε ( ) whih an be mae smaller than K log ( ) by hoosing ε = when A3/4 (as in this ase). For the remainer of the proof, let us therefore assume that 3/4 /. As before, we fous on the integral insie the square in the following expression x k x m y k y m w z e2πjg 2 U 3 G k,m (w, z) (7)

12 ÖZGÜR et al.: SPATIAL DEGREES OF FREEDOM OF LARGE DISTRIBUTED MIMO SYSTEMS AND WIRELESS AD HOC NETWORKS 23 z x m x k D R w Fig. 6. Tilte referene frame. Let us start by onsiering the simplest ase where the points x k an x m are loate on the same horizontal line, i.e. y k = y m. In this ase, the seon term in the expression (6) for g k,m z (w, z) beomes zero, so we eue the following lower boun: g k,m (w, z) z K A3/2 2 x m x k z y k This, together with the above mentione properties of the funtions g k,m an G k,m, allows us to apply Lemma A.2 so as to obtain w z e2πjg G k,m (w, z) K A 3/4 xm x k for a onstant K>inepenent of, an. A slight generalization of this argument (see below for etails) shows that not only when y k = y m but for any (x k,x m,y k,y m ) U 3,wehave w z e2πjg G k,m (w, z) K A 3/4 K A 3/4 ((x m x k ) 2 +(y m y k ) 2 ) /4 xm x k A 3/2 (8) Sine we also know that the above integral is less than, we further obtain w z e2πjg 2 G k,m (w, z) } min K 2 x m x k, For any <η<, we an now upper boun (7) as follows: x k x m y k y m w z e2πjg 2 U 3 G k,m (w, z) U 3 x m x k <η} 2 + K x k x m y k y m U 3 x m x k η} A 3/2 x m x k 2 ( ) 2η + K log =2η + K ( ) log η η implying that S K log A 3/2 ( ) +2ε +2η + K log ε ( ) η Choosing( finally ) ε = η = / allows to onlue that S K log ( ) also in the ase where 3/4 /. Proof of equation (8). In orer to prove (8), we nee to make a hange of oorinate system, replaing (w, z) by (w,z ),wherew is now in the iretion of the vetor x k x m an z is perpeniular to it (see Figure 6). In this new oorinate system, the integral reas w DR z e2πjg k,m(w,z ) 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 omain of integration D R is a tilte square, as iniate on the Figure 6. Using then the same argument as in the ase y k = y m, we onlue that w DR z e2πjg k,m(w,z ) G k,m (w,z ) K A 3/4 x m x k Notiing finally that x m x k = (x m x k ) 2 +(y m y k ) 2 allows to onlue that (8) hols. REFERENCES [] G. J. Foshini, Layere Spae-Time Arhiteture For Wireless Communiation in a Faing Environment when Using Multi-Element Antennas, AT&T Bell Labs Teh. Journal (2), 4 59, 996. [2] E. Telatar, Capaity of Multi-Antenna Gaussian Channels, European Trans. on Teleommuniations (6), , November 999. [3] D. Shiu, G. J. Foshini, M. J. Gans, an J. M. Kahn, Faing orrelation an its effet on the apaity of multi-element antenna systems, IEEE Trans. Commun., 48(3), pp.52-53, 2. [4] S. L. Loyka, Channel apaity of MIMO arhiteture using the exponential orrelation matrix, IEEE Commun. Lett., vol. 5, pp , 2 [5] H. Bölskei, D. Gesbert an A. J. Paulraj, On the apaity of OFDMbase spatial multiplexing systems, IEEE Trans. Commun., 5 (2), pp , 22. [6] D. Gesbert, H. Bölskei, D. A. Gore an A. J. Paulraj, Outoor MIMO wireless hannels: moels an performane preition, IEEE Trans. Commun., 5 (2), pp , 22. [7] K. Liu, V. Raghavan an A. Sayee, Capaity Saling an Spetral Effiieny in Wieban Correlate MIMO Channels, IEEE Trans. Inf. Theory, pp , Ot. 23. [8] G. G. Raleigh an J. M. Cioffi, Spatio-temporal oing for wireless ommuniation, IEEE Trans. Commun. 46(3), pp , 998. [9] A. M. Sayee, Deonstruting multi-antenna faing hannels, IEEE Trans. Signal Proess., vol. 5, pp , Ot. 22. [] R. Janaswamy, Effet of element mutual oupling on the apaity of fixe length linear arrays, IEEE Antennas Wireless Propagat. Lett., vol., pp. 576, 22.

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Inf. Theory, 58 (3), 72 75, Marh 22. Ayfer Özgür reeive her B.S. egrees in eletrial engineering an physis from Mile East Tehnial University, Turkey, in 2 an the M.S. egree in ommuniations from the same university in 24. From 2 to 24, she worke as harware engineer for the Defense Inustries Development Institute in Turkey. She reeive her Ph.D. egree in 29 from the Information Proessing Group at EPFL, Switzerlan. Her issertation won the 2 EPFL Best Ph.D. Thesis Awar. In 2 an 2, she was a post-otoral sholar with the Algorithmi Researh in Network Information Group at EPFL. Dr. Özgür is urrently an Assistant Professor in the Eletrial Engineering Department at Stanfor University. Her researh interests inlue network ommuniation, wireless systems, an information an oing theory. Olivier Lévêque was born in Switzerlan in 97. He reeive the engineer physiist iploma from the Swiss Feeral Institute of Tehnology-Lausanne (EPFL), Switzerlan, in 995 an omplete the Ph.D. egree in the Mathematis Department of EPFL in 2. He is now with the Laboratory of Information Theory at EPFL. His researh interests inlue stohasti alulus, ranom matries, wireless ommuniations an information theory. Davi N. C. Tse (M 96 SM 97 F 9) reeive the B.A.S. egree in systems esign engineering from the University of Waterloo,Waterloo, ON, Canaa, in 989, an the M.S. an Ph.D. egrees in eletrial engineering from the Massahusetts Institute of Tehnology, Cambrige, in 99 an 994, respetively. From 994 to 995, he was a postotoral member of tehnial staff at AT & T Bell Laboratories. Sine 995, he has been with the Department of Eletrial Engineering an Computer Sienes, University of California at Berkeley, where he is urrently a Professor. Dr. Tse reeive a 967 NSERC 4-year grauate fellowship from the government of Canaa in 989, a NSF CAREER awar in 998, the Best Paper Awars at the Infoom 998 an Infoom 2 onferenes, the Erlang Prize in 2 from the INFORMS Applie Probability Soiety, the IEEE Communiations an Information Theory Soiety Joint Paper Awar in 2, the Information Theory Soiety Paper Awar in 23, an the 29 Freerik Emmons Terman Awar from the Amerian Soiety for Engineering Euation. He has given plenary talks at international onferenes suh as ICASSP in 26, MobiCom in 27, CISS in 28, an ISIT in 29. He was the Tehnial Program o-hair of the International Symposium on Information Theory in 24 an was an Assoiate Eitor of the IEEE Trans. Inf. Theory from 2 to 23. He is a oauthor, with P. Viswanath, of the text Funamentals of Wireless Communiation, whih has been use in over 6 institutions aroun the worl.