How Much Does Transmit Correlation Affect the Sum-Rate Scaling of MIMO Gaussian Broadcast Channels?

Transcription

1 562 IEEE TRANSACTIONS ON COUNICATIONS, VOL. 57, NO. 2, FEBRUARY 2009 How uch Does Transmt Correlaton Affect the Sum-Rate Scalng of IO Gaussan Broadcast Channels? Tareq Y. Al-Naffour, asoud Sharf, and Babak Hassb, Senor ember, IEEE Abstract Ths paper consders the effect of spatal correlaton between transmt antennas on the sum-rate capacty of the IO Gaussan broadcast channel.e., downlnk of a cellular system. Specfcally, for a system wth a large number of users n, we analyze the scalng laws of the sum-rate for the drty paper codng and for dfferent types of beamformng transmsson schemes. When the channel s..d., t has been shown that for large n, the sum rate s equal to log log n + log P + o where s the number of transmt antennas, P s the average sgnal to nose rato, and o refers to terms that go to zero as n. When the channel exhbts some spatal correlaton wth a covarance matrx R non-sngular wth trr, we prove that the sum rate of drty paper codng s log log n + log P +logdetr +o. We further show that the sum-rate of varous beamformng schemes acheves log log n + log P + log c + o where c depends on thetypeofbeamformng.wecannfactcomputec for random beamformng proposed n [] and more generally, for random beamformng wth precodng n whch beams are pre-multpled by a fxed matrx. Smulaton results are presented at the end of the paper. Index Terms Broadcast channel, channel state nformaton, mult-user dversty, transmt correlaton, wreless communcatons. I. INTRODUCTION ULTIPLE nput multple output IO communcaton has been the focus of a lot of research whch bascally demonstrated that the capacty of a pont to pont IO lnk ncreases lnearly wth the number of transmt and receve antennas. Research focus has shfted recently to the role of multple antennas n multuser systems, especally Paper approved by A. H. Banhashem, the Edtor for Codng and Communcaton Theory of the IEEE Communcatons Socety. anuscrpt receved January 3, 2006; revsed arch, 2007 and February 28, The work of T. Y. Al-Naffour has been supported by Kng Fahd Unversty of Petroleum and nerals, Dhahran, Saud Araba. The work of. Sharf and B. Hassb s supported n part by the NSF under grant no. CCR and CCR , by the Davd and Lucy Packard Foundaton, and by Caltech Lee Center for Advanced Networkng. The work of T. Y. Al-Naffour was supported by Unversty Project no. IN070342, Deanshp of Scentfc Research, Kng Fahd Unversty of Petroleum and nerals, Dhahran, Saud Araba. T. Y. Al-Naffour s wth the Department of Electrcal Engneerng, Kng Fahd Unversty of Petroleum and nerals, P.O. Box 083, Dhahran, 326, Saud Araba e-mal: naffour@kfupm.edu.sa.. Sharf s wth the Department of Electrcal and Computer Engneerng, Boston Unversty, 8 Sant arys Street, Boston, A 0225 e-mal: sharf@bu.edu. B. Hassb s wth the the Department of Electrcal Engneerng, Calforna Insttute of Technology, 200 E. Calforna Boulevard, Pasadena, CA 925 e-mal: hassb@systems.caltech.edu. Dgtal Object Identfer 0.09/TCO /09$25.00 c 2009 IEEE broadcast scenaros.e., one to many communcaton as downlnk schedulng s the major bottleneck for future broadband wreless networks. An overvew of the research on ths problem can be found n [3], [6], [6]. In these scenaros, when multple users are present, one s usually nterested n quantfyng the maxmum possble sum rate to all users and 2 devsng computatonally effcent algorthms for capturng most of ths rate [3]. The frst queston was settled recently by usng a technque smlar to wrtng on drty paper and hence known as drty paper codng DPC. Whle DPC solves the broadcast problem optmally, t s computatonally expensve and requres a great deal of feedback as the transmtter needs perfect channel state nformaton for all users [3]. There has been ncreased nterest recently to devse smple technques that utlze multuser dversty and acheve a sumrate close to the sum-rate capacty of the IO broadcast channel see, e.g., [], [5] [8], [2], [5]. The scheme proposed n [], known as opportunstc multple random beamformng or concsely random beamformng, has been proved to asymptotcally maxmze the sum-rate or throughput of the downlnk of sngle antenna cellular systems by transmttng to the users wth the best channel condtons for a gven set of random beams. The gan of ths and other beamformng schemes can be attrbuted to multuser dversty each user experences a dfferent channel and therefore the transmtter can explot ths varaton and choose the users that have the best channel condtons. Clearly, the multuser gan would be specally magnfed when the channels between the transmtter and the users are changng ndependently. In ths paper we focus on a mult-antenna downlnk channel n the presence of correlaton between transmt antennas. Ths correlaton s caused by local scatterers around the base staton or the fact that the transmt antennas n the base staton are not spaced far enough to create ndependent channels. The overrdng queston then s to analyze the effect of ths correlaton on the sum-rate of DPC and varous beamformng schedulng technques. Specfcally, we consder three varatons of random beamformng, namely, random beamformng wth channel whtenng, beamformng wth general precodng, and determnstc beamformng. In the frst, the transmtter spatally whtens the channel and then uses random beamformng. In random beamformng wth precodng, the transmtter employs a more general precodng matrx. In both of these transmsson

2 AL-NAFFOURI et al.: HOW UCH DOES TRANSIT CORRELATION AFFECT THE SU-RATE SCALING OF IO GAUSSIAN BROADCAST CHANNELS? 563 schemes, the transmtted sgnal needs to be scaled properly to mantan the average power constrant. Fnally, n determnstc beamformng, as ts names suggests, we use a fxed beamformer for all channel uses n place of the randomly varyng one. When the number of users s large and there s no correlaton, the sum rate for DPC and random beamformng asymptotcally concde see Lemma and [], [2] R log log n + log P + o where n s the number of users, s the number of transmt antennas, and P s the average sgnal to nose rato, and o represents terms that go to zero as n. It turns out that ths s not case for the channel wth transmt correlaton. In ths case, the sum-rate can be wrtten as log log n + log P + log c + o 2 where the constant c whch refers to the sum-rate loss due to correlaton depends on the schedulng scheme and the egenvalues of the covarance matrx R. The results of ths paper strongly depend on assumng that the users have a common correlaton matrx R. Ths s essental to make the users s channels and hence SINR s d. Otherwse, t would be very dffcult to use extreme value theory to perform the scalng analyss. Fortunately, our analyss extends to the case when the users experence dfferent path loss.e. when the users correlaton matrces are the same modulo some multplcatve constant. The paper s organzed as follows. After ntroducng the channel model n the next secton, we revew n Secton III the dfferent schedulng schemes studed n ths paper. We obtan the scalng law of the sum-rate for DPC and random beamformng schemes n Secton IV and V, respectvely. Secton V-B, whch s the heart of the paper, s devoted to dervng the scalng law of random beamformng n a spatally correlated envronment. We use ths result to derve the scalng laws for random beamformng wth precodng and for determnstc beamformng. We conclude the paper wth smulatons and conclusons. II. CHANNEL ODEL AND PROBLE FORULATION In ths paper we consder a mult-antenna Gaussan broadcast channel wth n recevers equpped wth one antenna and a transmtter base staton wth antennas. Let St be the vector of the transmt symbols at tme slot t, andlet Y t be the receved sgnal at the th recever. We can then wrte the receved sgnal at the th user as Y t PH St+W,,...,n, 3 where W s the addtve nose whch s complex Gaussan wth zero mean and unt varance, CN0,. oreover, St s the transmt symbol satsfyng the power constrant E{S S}.HereP denotes the average transmt power or One could argue that when users have dfferent correlatons matrces, we wll contnue to observe multuser rchness smlar to that we observe n the whte case. equvalently the average SNR consderng the normalzaton of the varances for channel and nose. The channel H s a complex channel vector, known perfectly to the recever, and dstrbuted as CN0,R. The covarance matrx R s a measure of the spatal correlaton and s assumed to be non-sngular wth trr. 2 We also assume that H follows a block fadng model,.e., t remans constant durng a coherence nterval T and vares ndependently from one such nterval to the next. We fnally note that the channel s dentcally dstrbuted across users but s ndependent from one user to another. Denotng the average rate of the the user by R over all the channel realzatons, we are nterested n analyzng the behavor of the sum-rate,.e., n R, of downlnk for large n. In the followng secton, we revew the schedulng schemes that wll be consdered n ths paper. III. REVIEW OF TRANSISSION SCHEES IN THE DOWNLINK A. Drty Paper Codng DPC The capacty regon of the mult-antenna broadcast channel s acheved by drty paper codng when full channel state nformaton CSI s avalable to the transmtter and users. Intutvely, f the transmtter knows the channels of all users, t can use DPC to pre-subtract the nterference for each user whle preservng the average power constrant [3]. ore precsely, the sum rate capacty, R DPC, can be wrtten as see [2] and the references theren, { } R DPC E max {P log det + H P H,...,P n, P P } 4 In a system wth a large number of users n, andforfxed and P, t has been shown that the sum-rate of DPC behaves as n, R DPC log log n + log P + o, 5 when there s no spatal correlaton,.e., R I [], [2]. Scalng of the sum rate capacty has also been nvestgated for other regons of n,, andp see [8] [0] for detals. There are two major drawbacks of ths scheme. Frst, t s very computatonally complex, both at the recevers and transmtter. oreover, t requres full CSI feedback from all actve users to the transmtter of the base staton ths feedback requrement ncreases wth the number of antennas and users and wth the decrease of the coherence tme of the system. B. Random Beamformng Gven these drawbacks of DPC, research has focused on devsng algorthms for multuser broadcast channels that have less computatonal complexty and/or less feedback and stll acheve most of the sum-rate promsed by DPC such as 2 We assume that the spatal correlaton s nvarant across users. Ths assumpton s realstc because ths s effectvely the transmt correlaton among antennas at the base staton. In the case when R s rank defcent, the results of ths paper apply wth replaced by the rank of the autocorrelaton matrx and wth the SNR kept fxed at P/.

3 564 IEEE TRANSACTIONS ON COUNICATIONS, VOL. 57, NO. 2, FEBRUARY 2009 random beamformng [5] and zero forcng [7] see also [5], []. A random beamformng scheme was proposed n [] where the transmtter sends multple n fact random orthonormal beams chosen to users wth the best sgnal to nterference rato SINR. In ths scheme the only feedback requred from each user s the SINR of the best beam and the correspondng ndex. Specfcally, the transmtter chooses random orthonormal beam vectors φ m of sze generated accordng to an sotropc dstrbuton. Now these beams are used to transmt the symbols s t,s 2 t,...,s t by constructng the transmtted vector St φ m ts m t, t,...,t 6 m After T channel uses, the transmtter ndependently chooses another set of orthogonal vectors {φ m } and constructs the sgnal vector accordng to 6 and so on. From now on and for smplcty, we wll drop the tme ndex t. The sgnal Y at the th recever s gven by Y PH S + W 7 P H φ m s m + W,,...,n8 m where ESS I snce the s s are assumed to be dentcal and ndependently assgned to dfferent users. The th recever uses ts knowledge of the effectve channel gan H φ m, somethng that can be arranged by tranng, to calculate SINR s, one for each transmtted beam H φ m 2 SINR,m P + k m H φ k, m,...,. 2 9 Each recever then feeds back ts maxmum SINR,.e. max SINR,m, along wth the maxmzng ndex m. Thereafter, the transmtter assgns s m to the user wth the hghest m correspondng SINR,.e. max SINR,m. If we do the above n schedulng, the throughput for large n can be wrtten as [2] 3, R RBF Elog + max SINR,m + o 0 n where the term o accounts for the small probablty that user may be the strongest user for more than one sgnal s m []. To further quantfy 0, [] used the fact that the SINR,m s are d over and employed extreme value theory [20] to argue that max SINR,m behaves lke P n log n and hence concluded that the sum rate capacty scales as n, meanng that the sum-rate of random beamformng behaves the same as that of DPC for large number of users. C. Other Beamformng Schemes The scalng result apples for d channels. As such, we derve n Secton V the scalng law of ths scheme for 3 The proof follows from the fact the when n s large the maxmum SINR and the th maxmum SINR behave qute smlarly. correlated channels. Alternatvely, gven ths correlaton, we consder the followng beamformng schemes. Random beamformng wth channel whtenng In the presence of correlaton, one can frst whten the channel and then use random beamformng schedulng. In ths case, and nstead of usng Φ as the beamformng matrx 4, we would use αr /2 Φ where α s a constant to make sure that the transmt symbol has an average power of. The scalng of ths scheme would follow drectly from the scalng of random beamformng over d channels see Secton V-A. Random beamformng wth general precodng ore generally, we can precode wth a general matrx αa /2 before beamformng,.e. we use αa /2 Φ to transmt the nformaton symbols. The scalng of ths scheme follows drectly from the scalng of random beamformng over correlated channels and so s consdered n Sectons V-B and V-D. We go one step further and show how to compute the sum-rate when the beamformng matrx s premultpled by the full rank matrx A. Determnstc beamformng Fnally, by fxng the beamformng matrx Φ, we obtan determnstc beamformng, a scheme analyzed by Park and Park [4] for the two antenna case and whch we further analyze n Secton V-C. As we mentoned above, and as we shall soon see, all these schemes have scalng smlar to the d case wth a penalty term log c where c s a constant that depends only on the schedulng scheme and the correlaton matrx R. IV. EFFECT OF TRANSIT CORRELATION ON THE SU-RATE OF DPC In ths secton, we derve the scalng laws of DPC for correlated channels. As mentoned earler, drty paper codng acheves the sum-rate capacty of the mult-antenna broadcast channel. The sum-rate capacty s gven by 4 and ts behavor when n s large s gven by 5 for d channels. It turns out that when the number of users s large, the sum-rate capacty wll be decreased by a constant whch depends on the covarance matrx of the channel. It should be mentoned that throughout the paper, we assume R s fxed and non-sngular wth trr. The next theorem proves ths statement. The proof s along the same lne as the proof for the..d. case as shown n [] wth the only dfference that the lower bound rather than beng acheved wth random beamformng s acheved wth a specal type of determnstc beamformng 5.Wefrst gve the lower bound n the followng lemma. Lemma : Consder a Gaussan broadcast channel wth a channel covarance matrx R whch s non-sngular wth trr. Let there be one transmtter wth antennas and n users wth sngle antennas that have access to the 4 Note that Φ s an orthonormal matrx composed of the beam column vectors φ,...,φ. 5 It should be stressed that the optmalty here s n the asymptotc of large number of users.

4 AL-NAFFOURI et al.: HOW UCH DOES TRANSIT CORRELATION AFFECT THE SU-RATE SCALING OF IO GAUSSIAN BROADCAST CHANNELS? 565 CSI and the transmtter knows the CSI perfectly. We assume the transmtter uses the determnstc beamformng matrx Φ U where U s the untary matrx consstng of the egenvectors of R. Then for large n, the sum-rate of ths schedulng s R BF D log log n+ log P + log detr+o. Proof: See Secton V-C for the proof. Clearly s a lower bound for the sum-rate capacty. In the next theorem we show that s ndeed an upper bound for the sum-rate as well. Theorem : Consder a Gaussan broadcast channel wth an autocorrelaton matrx R defned n Lemma. Let there be one transmtter wth antennas and n users wth sngle antennas that have access to the CSI. Assume further that the transmtter knows the CSI perfectly. The sum-rate capacty whch s acheved by DPC scales lke R DPC log log n + log P + log det R + o, 2 for large n. Proof: Lemma mples that the rght hand sde of 2 s achevable. All we need to prove the theorem s to show that the sum-rate of DPC can not be larger than 2. We use the sum rate capacty expresson gven n 4 to obtan an upper bound for the sum-rate. To ths end, defne H H w R 2, where H w s N0,I. Wth ths decomposton, the sum-rate capacty can be wrtten as R DPC E max { log det R + {P,...,P n, P P } } Hw P H w detr 3 Now usng the geometrc-arthmetc mean nequalty, deta we obtan tra tr Hw P H w max trh w H w max H w 2 P to replace the log det wth an upper bound log det R + Hw P H w log trr + trhw P H w log trr +max H w 2 P Snce H w 2 s χ 2 2 dstrbuted, wth hgh probablty, the maxmum max H w 2 behaves lke log n + Olog log n. Thus, trr R DPC log + P log n + Olog log n + P log det R + o 4 or usng the fact that for large n log trr + P log n + Olog log n log P +loglogn +log trr P log log n + O log n P log log n +loglogn + O, 5 log n we can further smplfy 4 and obtan R DPC log log n + log P + log det R + o whch s the desred upper bound. Ths completes the proof of the theorem. V. EFFECT OF TRANSIT CORRELATION ON RANDO BEAFORING The determnstc beamformng scheme of Lemma asymptotcally acheves the DPC sum-rate. However t has the drawback that, unless the H s change very rapdly over dfferent channel uses, t wll often transmt to a fxed set of users. To make the schedulng more short-term far, t s useful to further randomze the user selecton by random beamformng see [], [5] for more detals. In ths secton, we analyze the effect of correlaton on the sum-rate of random beamformng. We start by the smplest case n whch the beamformng matrx s multpled by R /2 n order to whten the channel. We then turn our attenton to the random beamformng scheme and fnally use t to deduce the sum rates of determnstc beamformng and beamformng wth general precodng. A. Random Beamformng wth Channel Whtenng To whten the channel, we multply all the beams wth γr /2 where γ s a normalzaton factor. The transmt symbol s therefore equal to St γr /2 φ m ts m t 6 m We choose γ to satsfy the power constrant that the transmt symbol average power s bounded by unty, E{γS R S} γe{trsr S } γe{trr S S} γtrr ES S} γ trr 7 Thus, the constrant E{γS R S} mples that γ. We can therefore wrte the SINR as trr SINR,m H R /2 φ m 2 Pγ + k m H 8 R /2 φ k 2 H wφ m 2 Pγ + k m Hw φ,m,..., 9 2 k

5 566 IEEE TRANSACTIONS ON COUNICATIONS, VOL. 57, NO. 2, FEBRUARY 2009 where H w H R /2 has covarance of I and therefore has..d. Gaussan entres wth zero mean and unt varance. Therefore we can apply the random beamformng result of [] to obtan the sum rate of random beamformng wth channel whtenng. Ths s summarzed n the followng Theorem. Theorem 2: Consder a Gaussan broadcast channel wth a channel covarance matrx R defned n Lemma. Let there be one transmtter wth antennas and n users wth sngle antennas that have access to the CSI. If the transmtter knows the channel autocorrelaton perfectly, then the sum rate capacty for random beam formng wth channel whtenng denoted by R BF W sgvenby R BF W log log n + log P log trr + o 20 for suffcently large n. When the the channel s..d, Theorem 2 reduces to the already known result of []. It s also worth mentonng that 20 s less than the sum-rate acheved by DPC n 2. B. Sum-Rate of Random Beamformng In ths secton, we study the effect of transmt correlaton on random beam-formng. To do ths, we need to derve the CDF and pdf of the SINR defned n 9. The sum rate capacty of random beamformng s gven by 0. Now consder the expectaton n 0. The averagng here s done over H and Φ n the followng order, E log + max SINR,m n } E Φ {E H s Φ log + max SINR,m Φ n 2.e., we evaluate the expectaton by frst condtonng on Φ and calculatng the expectaton over H and we subsequently average over Φ. The advantage of dong so s that Φ s common among all users and so, by condtonng over Φ, all the SINR s, SINR,m,...,SINR n,m reman d. Ths n turn allows us to evaluate max SINR,m usng extreme value n theory provded we can evaluate the CDF and pdf of the SINR. It turns out that the man challenge les n calculatng the CDF. When the channel s d, calculatng the CDF s straghtforward as the SINR numerator and denomnator are ndependent []. Ths ceases to be the case n the presence of correlaton and n evaluatng the CDF, we use a contour ntegral representaton of the unt step and fnd the CDF usng the Gaussan ntegral. Once the CDF s avalable, we appeal to results n extreme value theory to obtan the behavor of max SINR,m when n s large and proceed to calculate the n expectaton n 2. Wth the scalng law for random beamformng at hand, t becomes straghtforward to obtan the scalng laws of random beamformng wth precodng and of determnstc beamformng. Dstrbuton of SINR, Gven Φ : We frst obtan the complementary CDF of SINR,m defned n 9 by defnng the auxlary varable S as S x ρ + H + xφ m φ m xih 22 Here ρ P just to smplfy the notaton and where the beamformng matrx Φ s gven and H s an vector wth Gaussan entres and wth covarance matrx R. We can wrte the probablty that SINR,m >xas, P SINR, >xp S >0 P H usdh π e HR H usdh 23 detr where us s the unt-step functon. To evaluate P S >0, we can vew S as a weghted sum of correlated Gaussan random varables and employ one of varous technques that have been suggested n the lterature. Unfortunately, the expressons we get nvolve recursons and nfnte sums and hence don t lend themselves to further mathematcal manpulatons. Instead, we use the followng representaton of the unt step functon [4] us 2π e jω+βs dω 24 jω + β whch s vald for any β > 0. Ths frees 23 from the constrant on S and, as we shall see, allows us to compute 23 n closed form. Usng 24, we can express 23 as P S >0 2π + dω detr jω + β dh e jω+βs HR H Usng the defnton of S n 22, we get P S >0 2π detr dω e jω+β x ρ jω + β dh e H RH 25 dω e jω+β ρ 2π detr jω + β det R 26 where R R + xjω + βi + xjω + βφ m φ m 27 a Evaluatng the roots of R: Now to evaluate the ntegral wth respect to ω, we need to fnd the roots of det R wth respect to ω. To ths end, note that det R detu Λ U +jω + βxi + xφ m φ m detλ +jω + βxi + xφ m φ m 28 detλ det Adetjω + βi A 29 where U Λ U represents the egenvalue decomposton of R Δ, φ m Uφm, and A +xλ /2 φ m φ mλ /2 xλ 30

6 AL-NAFFOURI et al.: HOW UCH DOES TRANSIT CORRELATION AFFECT THE SU-RATE SCALING OF IO GAUSSIAN BROADCAST CHANNELS? 567 Now detλ det A detxi + xφ m φ m x x + x x because xi + xφ m φ m has x as an egenvalue wth multplcty andanegenvalueatx + x φ m 2. We can thus wrte det R x detjω + βi A Now consder the equaton detjω + βi A 0 3 The roots of ths equaton, wth respect to jω+β, are / A where A s an egenvalue of the matrx A. Snce A s Hermtan and nonsngular, these egenvalues are real and nonzero. To fnd these egenvalues, decompose A as where A A + A 2 A +xλ /2 φ m φ mλ /2 and A 2 xλ The matrx A has only one nonzero egenvalue, + xφ m Λφ m. The egenvalues of A 2 are xλ Λ xλ Λ xλ Λ where λ Λ λ 2 Λ λ Λ are the dagonal elements of Λ ordered 6. The second largest egenvalue of A thus satsfes [3] λ A { λ A +λ A 2 32 λ A +λ A 2 { 0 xλ φ 33 mλφ m xλ 2 Ths means that λ A xλ < 0. So the second largest egenvalue s negatve. The largest egenvalue, however, s postve otherwse A would be negatve defnte or sngular, nether of whch s the case. Ths means that 3 has exactly one postve root λ λ A Henceforth, we drop the dependence upon the matrx A as t s understood. From above, we can express R as det R x jω + β λ jω + β 6 In general, the egenvalues of a sze matrx K are wrtten as λ K λ 2 K λ K. We wll drop the dependence on K for notatonal convenence whenever t s understood. b Dervng the CDF of SINR: Wth the above factorzaton of det R, we can proceed to evaluate the probablty P λ >0 n 26 and hence the CDF of the SINR can be wrtten as, P S >0 x 2π detr e jω+β x ρ jω + βjω + β λ jω + β dω 34 Usng partal fracton expanson, we can wrte jω + βjω + β λ jω + β ζ ζ jω + β + λ jω + β + ζ 0 35 jω + β The term ζ jω+β λ s the only one that contrbutes to the ntegral n 34 the other terms ntegrate to zero snce the poles are outsde the contour of ntegraton, and so we only need to calculate ζ ζ jω + β λ jω + β λ jω+β λ and P S >0 ζ e jω+β x ρ 2π detr x λ jω + β dω ζ x detr x e ρ λ 38 Ths represents the probablty P SINR,m >x. Thus, the CDF of the SINR s gven by ζ x F x detr x e ρ λ Or, upon replacng ζ by ts value obtaned n 37, F x detr λ λ x x λ e ρ λ 39 We would lke to emphasze that the egenvalues of A,, are functons of x. 2 Probablty Densty Functon of SINR: To fnd the pdf df x of the SINR, we smply evaluate the dervatve dx. To do ths, we frst need to fnd the dervatve of the egenvalues dλ dx. So let q be the egenvector assocated wth. Then, we can wrte q 2 A q Λ/2 φ m φ m x k m φ k φ k Λ/2 q where we used the notaton q 2 A q Aq. We can use ths to show that d dx q 2 B 40

7 568 IEEE TRANSACTIONS ON COUNICATIONS, VOL. 57, NO. 2, FEBRUARY 2009 where B Λ /2 φ m φ m IΛ/2. We can n turn use ths result to show that d λ λ λ2 q 2 C λ2 q 2 C dx x λ x 2 λ 2 4 where C Λ /2 φ m φ m Λ/2. From 39 4, we can show that the SINR pdf s gven by { ρ fx detr e ρ q 2 C λ x λ q 2 B λ x λ } λ 2 q 2 C λ2 q 2 C x λ 42 3 Scalng Law of the axmum SINR: Lemma 2: Let F x denote the CDF of SINR,m gven by 39 and let fx denote the assocated pdf gven by 42. Then F x ρ lm x fx φ m 2 Λ Proof: See Appendx A for the proof. Note that n the absence of spatal correlaton, ΛI, and the above lmt reduces to F x lm ρ x fx φ m ρ 2 whch s the scalng obtaned n []. Usng extreme value theory [7] [9], and the lemma above, we know that max SINR,m behaves lke n ρ log n. Upon substtutng ths n 2 and notng that φ m 2 Λ the φ s are dentcally dstrbuted, we can wrte R RBF P E φm log + log n + olog log n + o φ m 2 Λ m P E φm log log n + o 43 φ m 2 Λ m log log n + log P + E φ m log φ m 2 Λ +o. 44 It thus remans to calculate the expectaton n 44 for whch we need to derve the CDF of. φ m 2 Λ 4 Calculatng the CDF of : φ 2 Λ Lemma 3: The CDF of y s gven by φ 2 Λ Gx Pr φ 2 <x Λ η x λ u Λ λ x Λ where η j λ j Λ Λ. Proof: See Appendx B for the proof. 5 Calculatng the Sum-Rate: Now all we need to do to calculate the sum-rate n 44 s to compute E log φ 2 Λ where the dstrbuton of s gven n Lemma 3. We φ 2 Λ employ ntegraton by parts and use the CDF to calculate the expectaton as follows λ Λ E log Gylogy φ 2 λ Λ Λ λ Λ λ Λ Gy y dy Gλ Λ logλ Λ λ Λ logλ Λ λ Λ y + λλ η y y dy λ Λ logλ Λ + logλ Λ + η k0 λ Λ λ Λ Gy y dy λλ η y y dy λλ λ Λ k dy logλ Λ λλ λ Λ k y + k+ λ Λ y k+ k dy logλ Λ + η log λ λ k k +2 k η log λ y k+2 Λ λ Λ Therefore the sum-rate of beamformng can be wrtten as, R RBF log log n + log P + log λ Λ+ η log λ λ k k +2 k { } Λ k+2 λ Λ k+2 + o. 45

8 AL-NAFFOURI et al.: HOW UCH DOES TRANSIT CORRELATION AFFECT THE SU-RATE SCALING OF IO GAUSSIAN BROADCAST CHANNELS? 569 C. Sum-Rate of Determnstc Beamformng Here we consder the case where the beamformng matrx Φ s fxed over all channel uses. In ths case, we can use the same analyss as we done n the case of random beamformng wth the only excepton that we do not need to take expectaton over the beamformng matrx. Therefore, we may wrte the sum-rate for the determnstc beamformng matrx Φ as, R BF D log log n + log P + log + o φ U Λ Uφ 46 where U Λ U s the egenvalue decomposton of the correlaton matrx R. One nterestng specal case would be the case where the Uφ s are the columns of the dentty matrx. In ths case, the beamformng matrx s n fact equal to U and the argument n the logarthm would therefore reduce to λ m. Thus, when n s large, the sum-rate s gven by R BF D log log n + log P + log det R + o. 47 Keepng n mnd that the egenvalues of Λ are such that Λ, t s clear that the geometrc mean of s would be less than. Eq. 47 n fact proves Lemma. It should be also mentoned that ths result s obtaned n [4] for 2. Sum rate loss DPC Determnstc RBF RBF RBF wth whtenng Correlaton factor Fg.. Sum-rate loss versus the correlaton factor α for a system wth 2and n 00. Sum rate D. Sum-Rate of Random Beamformng wth Precodng We can consder a generalzaton of the random beamformng by usng precodng. In ths scheme the new beamformng matrx s γa /2 Φ where A s a postve defnte matrx and γ s just a normalzaton factor to adjust the transmt power. Agan smlar to Secton V-B, we can state that γ has to be tra. less than In order to analyze the sum-rate, we can follow along the same lne as what we dd for the analyss of the random beamformng wth the only excepton that the covarance matrx of the channel s replaced wth R A /2 RA /2. Therefore the same result holds for ths case wth the new covarance matrx R. Here s the man result. Corollary : Consderng the random beamformng schedulng wth beamformng matrx γa /2 Φ where Φ s a random untary matrx, the sum-rate of ths scheme can be wrtten as R BF Prec log log n + log P + E log + trλ P φ U Λ Uφ + o. 48 for large n, whereu Λ U represents the egenvalue decomposton of R. 6 Determnstc RBF RBF RBF wth channel whtenng Correlaton factor Fg. 2. Sum-rate versus the correlaton factor α for a system wth 2, P 0,andn 00. VI. SIULATION RESULTS In ths secton we present the smulaton results for the sumrate of beamformng schemes and DPC. In the frst example, we consder a system wth two transmt antennas,.e., 2, and 00 users. The covarance matrx s assumed to be lke [ ] α F 49 α where α s the correlaton. Fg. shows the sum-rate loss relatve to the case of no correlaton versus the correlaton coeffcent α for DPC, RBF and RBF wth whtenng. It s clear that RBF outperforms the one wth channel whtenng. Fg. 2 also shows the actual sum-rate for such a settng for RBF and RBF wth whtenng. Fg. 3 shows the sum-rate loss for the there antenna case 3where the covarance matrx

9 570 IEEE TRANSACTIONS ON COUNICATIONS, VOL. 57, NO. 2, FEBRUARY 2009 Sum rate loss DPC Determnstc RBF RBF RBF wth whtenng Correlaton factor Fg. 3. Sum-rate loss versus the correlaton factor α for a system wth 3and n 00. Sum rate RBF wthout correlaton Determnstc RBF RBF RBF wth channel whtenng Number of users Fg. 4. Sum-rate versus the number of users n a system wth 2, P 0, and α 0.5 s now gven by F α α2 α α 50 α 2 α where α s changng from 0 to 0.8. In Fg. 4, we show the sum-rate versus the number of users n system wth 2, α 0.5, P 0for beamformng scheme and t s compared to the case of havng no correlaton. VII. CONCLUSION Ths paper consders the effect of spatal correlaton on varous multuser schedulng schemes for IO broadcast channels. Specfcally, we consdered drty paper codng and varous random, determnstc, and channel whtenng beamformng schemes. When the channel s..d. and for large number of users, the sum rate of all these technques exhbts the same scalng, namely, as log log n + log P + o where n s the number of users, s the number of transmt antennas and P s the average SNR. In the presence of a correlaton between transmt antennas, the channel matrx has a covarance matrx R whch s assumed to be non-sngular and trr. In ths case, the sum-rate of DPC and beamformng schemes wll be dfferent. It turns out that n these case, the sum-rate can be wrtten as log log n + log P + log c + o where c < s a constant that only depends on the schedulng scheme and the covarance matrx R. ForDPC,c s just the geometrc mean of the egenvalues of R. We further obtan c for dfferent beamformng schemes; For example, for the case of beamformng wth channel whtenng, c wll be equal to the harmonc mean of the egenvalues of R. Itsworth mentonng, numercal results suggest that sum-rate of random beamformng outperforms that of the random beamformng wth channel whtenng 7. APPENDIX A: PROOF OF LEA 2 From 39 and 42, we can wrte F x fx λ q 2 C ρ λ q 2 B λ 2 q 2 C λ2 q 2 C x λ 5 To evaluate the lmt of ths expresson, we need to nvestgate the behavor of the egenvalues and egenvectors of A as x. Now from the bound 33, we deduce that lm for all m x We now have to evaluate the behavor of the maxmum of egenvalue as x tends to nfnty. Ths s done by usng the Raylegh quotent for the maxmum egenvalue as, λ max u u Au 2 max u u Λ /2 φ m φ mλ /2 xλ /2 φ φ u 52 2 m The vector u that maxmzes λ s the assocated egenvector. Snce any vector u of dmenson can be wrtten as u α Λ /2 φ, we can wrte Au 2 as Au 2 u Au u α m Λ /2 φ m x α Λ /2 φ m α 2 m x m α 2 53 where we used the fact that the φ s are orthonormal vectors. Now as x tends to nfnty, Au 2 couldgoto and s maxmzed when m α2 s equal to zero.e., α 0for m and as a result α m. We have thus proved φ m 2Λ that lm q lm u Λ /2 φ m 54 x x φ m 2Λ 7 Channel whtenng s lke zero forcng n that t takes care of the worst egenvalue and thus would result n a bg waste of power.

10 AL-NAFFOURI et al.: HOW UCH DOES TRANSIT CORRELATION AFFECT THE SU-RATE SCALING OF IO GAUSSIAN BROADCAST CHANNELS? 57 and lm x λ φ m 2 Λ Usng the above, t s easy to verfy that and lm x λ 2 q 2 C x λ 0 Γ e jω+β 4π +2 dω dω 2 e jω2 jω + β det xjω + β Λ jω 2 I Now use partal fracton expanson to show that det xjω + β Λ jω 2 I 56 lm λ 2 q 2 C x x λ lm q 2 C x x λ 0 From 54 and the defnng expresson of B, we also deduce that lm x q 2 B φ m 2 Λ φ mφ m φ m Iφ m 0 Thus, the only nonzero lmt n the denomnator of 5 s and ρ q m 2 C λ F x lm x fx λ2 ρ q m 2 C ρ φ m 2 Λ 55 APPENDIX B: PROOF OF LEA 3 Consder the nequalty y φ 2 >x Λ whch can be equvalently wrtten as x φ 2 Λ > 0. As we dd to derve the SINR CDF above, we use the unt-step representaton u x φ 2 Λ e x φ 2 Λ jω +β dω 2π jω + β Now the pdf of φ s pφ Γ π δ φ 2 Alternatvely, followng the approach of [4], we can use an ntegral representaton for the Drac delta pφ Γ π dω 2 e jω2 φ 2 2π So the probablty p φ 2 Λ >xp x φ 2 Λ > 0 s gven by p φ 2 Λ >x Γ 4π +2 dω dω 2 x jω + β x λ jω Λ + β jω 2 where η j λ j Λ. We thus have Λ p φ 2 >x Λ Γ 2π η x λ jω 57 Λ + β jω 2 Γ e jω+β 4π 2 x dω jω + β η dω 2 x λ jω e jω 2 58 Λ + β jω 2 e jω+β x Λ x η dω jω + β 59 or after some straght-forward calculatons, p φ 2 >x Λ η x Λ u x Λ Alternatvely, the CDF, Gx p φ 2 Λ Gx <x s gven by η x u x Λ Λ whch completes the proof of the Lemma. dφ ejω+β x φ 2 Λ e jω2 φ 2 jω + β Γ 4π +2 e jω+β dω jω + β dω 2 e jω2 dφe φ xjω +Λ jω 2Iφ ACKNOWLEDGENT T. Y. Al-Naffour would lke to acknowledge Kng Fahd Unversty of Petroleum and nerals, Saud Araba for ts support of ths work. The authors would lke to thank Amr F. Dana at Calforna Insttute of Technology for numerous dscussons and hs very useful comments. We would also lke to thank aralle Fakhreddne and ohammed Eltayeb for ther help wth the smulatons.

11 572 IEEE TRANSACTIONS ON COUNICATIONS, VOL. 57, NO. 2, FEBRUARY 2009 REFERENCES []. Sharf and B. Hassb On the capacty of IO broadcast channel wth partal sde nformaton, IEEE Trans. Inform. Theory, vol. 5, no. 2, pp , Feb [2] A. F. Dana,. Sharf, A. Vakl, and B. Hassb, Dfferentated rate schedulng for the down-lnk of cellular systems, to be publshed. [3] Q. H. Spencer, C. B. Peel, A. L. Swndlehurst, and. Haardt, An ntroducton to the mult-user IO downlnk, IEEE Commun. ag., vol. 42, no. 0, pp , Oct [4] B. Hassb and T. L. arzetta, ultple-antennas and sotropcally random untary nputs: The receved sgnal densty n closed form, IEEE Trans. Inform. Theory, vol. 48, no. 6, pp , June [5] B. Hochwald, B. Peel, and A. L. Swndlehurst, A vector perturbaton technque for near capacty multantenna multuser communcaton part II: perturbaton, IEEE Trans. Commun., vol. 53, no. 3, pp , ar [6] G. Care and S. Shama, On the achevable throughput of a multantenna Gaussan broadcast channel, IEEE Trans. Inform. Theory, vol. 49, no. 7, pp , July [7] T. Yoo and A. Goldsmth, On the optmalty of mult-antenna broadcast schedulng usng zero-forcng beamformng, IEEE J. Select. Areas Commun., vol. 24, no. 3, pp , ar [8] H. Vswanathan and S. Venkatesan, Asymptotcs of sum rate for drty paper codng and beamformng n multple antenna broadcast channels, n Proc. 4 st Annual Allerton Conf., [9] B. Hochwald and S. Vswanath, Space tme multple access: lnear growth n the sum rate, n Proc. 40 th Annual Allerton Conf., [0] N. Jndal and A. Goldsmth, Drty-paper codng versus TDA for IO Broadcast channels, IEEE Trans. Inform. Theory, vol. 5, no. 5, pp , ay []. Kountours and D. Gesbert, Robust mult-user opportunstc beamformng for sparse networks, n Proc. IEEE Workshop Sgnal Processng Advances Wreless Commun., SPAWC, 2005, pp [2] H. Wengarten, Y. Stenberg, and S. Shama, The capacty regon of the Gaussan multple-nput multple-output broadcast channel, IEEE Trans. Inform. Theory, vol. 52, no. 9, pp , Sept [3] H. Lütkepohl, Handbook of atrces. New York: John Wley & Sons, 996. [4] D. Park and S. Y. Park, Effect of transmt antenna correlaton on multuser dversty, n Proc. IEEE ISIT, 2005, pp [5] P. Vswanath, D. N. Tse, and R. Laroa, Opportunstc beamformng usng dump antennas, IEEE Trans. Inform. Theory, vol. 48, no. 6, pp , June [6] A. Goldsmth, S. A. Jafar, N. Jndal, and S. Vshwanath, Capacty lmts of IO channels, IEEE J. Select. Areas. Commun., vol. 2, no. 5, pp , June [7]. R. Leadbetter, Extreme value theory under weak mxng condtons, Studes n Probablty Theory, AA Studes n athematcs, pp. 46 0, 978. [8] N. T. Uzgoren, The asymptotc development of the dstrbuton of the extreme values of a sample, Studes n athematcs and echancs Presented to Rchard von ses. New York: Academc Press, pp , 954. [9] H. A. Davd, Order Statstcs. New York: Wley, 970. [20]. R. Leadbetter and H. Rootzen, Extremal theory for stochastc processes, The Annals of Probablty, vol. 6, pp , 988. Tareq Al-Naffour obtaned hs.s. degree n electrcal engneerng from Georga Insttute of Technology, Atlanta, n 998, and hs Ph.D. n Electrcal Engneerng from Stanford Unversty, CA n He was a vstng scholar at the Calforna Insttute of Technology, Pasadena, CA Jan Aug, 2005 and durng the summer of He s currently a Fulbrght scholar at the Unversty of Southern Calforna. He has held nternshp postons at NEC Research Labs, Tokyo, Japan n 998; Adaptve Systems Lab, Unversty of Calforna at Los Angeles n 999; Natonal Semconductor, Santa Clara, CA, n 200 and 2002; and Beceem Communcatons, Santa Clara, CA, n Dr Al- Naffour s currently an Assstant Professor at the Electrcal Engneerng Department, Kng Fahd Unversty of Petroleum and nerals, Saud Araba. Hs research nterests le n the areas of adaptve and statstcal sgnal processng and ther applcatons to wreless communcatons and n multuser nformaton theory. He has over 50 publcatons n journal and conference Proceedngs. Dr Al- Naffour s the recpent of a 200 best student paper award at the IEEE- EURASIP Workshop on Nonlnear Sgnal and Image Processng NSIP 200 for hs work on adaptve flterng analyss. asoud Sharf S 99 receved hs Ph.D. n Electrcal Engneerng 2005 from the Calforna Insttute of Technology. In 2005, he was a post-doctoral scholar n the EE department at Caltech. Snce January 2006, he has been an assstant Professor at Boston Unversty. Dr. Sharf was awarded the C. H. Wlts Prze n 2006 for best doctoral thess n Electrcal Engneerng at Caltech. He s a member of the Center for Informaton and Systems Engneerng at Boston Unversty. Hs research nterests nclude ad-hoc and sensor networks, multple-user multpleantenna communcaton channels, crosslayer desgn for wreless networks, and mult-user nformaton theory. Hs recent research has focused on collaboratve communcaton schemes n ad-hoc and sensor networks and the capacty of multple antenna broadcast channels. Babak Hassb was born n Tehran, Iran, n 967. He receved the B.S. degree from the Unversty of Tehran n 989, and the.s. and Ph.D. degrees from Stanford Unversty n 993 and 996, respectvely, all n electrcal engneerng. He has been wth the Calforna Insttute of Technology snce January 200, where he s currently Professor and Executve Offcer of Electrcal Engneerng. From October 996 to October 998 he was a research assocate at the Informaton Systems Laboratory, Stanford Unversty, and from November 998 to December 2000 he was a ember of the Techncal Staff n the athematcal Scences Research Center at Bell Laboratores, urray Hll, NJ. He has also held short-term appontments at Rcoh Calforna Research Center, the Indan Insttute of Scence, and Lnkopng Unversty, Sweden. Hs research nterests nclude wreless communcatons and networks, robust estmaton and control, adaptve sgnal processng, and lnear algebra. He s the coauthor of the books both wth A.H. Sayed and T. Kalath Indefnte Quadratc Estmaton and Control: A Unfed Approach to H 2 and H Theores New York: SIA, 999 and Lnear Estmaton Englewood Clffs, NJ: Prentce Hall, He s a recpent of an Alborz Foundaton Fellowshp, the 999 O. Hugo Schuck best paper award of the Amercan Automatc Control Councl wth H. Hnd and S.P. Boyd, the 2002 Natonal Scence Foundaton Career Award, the 2002 Okawa Foundaton Research Grant for Informaton and Telecommuncatons, the 2003 Davd and Luclle Packard Fellowshp for Scence and Engneerng and the 2003 Presdental Early Career Award for Scentsts and Engneers PECASE, and was a partcpant n the 2004 Natonal Academy of Engneerng Fronters n Engneerng program. He has been a Guest Edtor for the EEE TRANSACTIONS ON INFORA- TION THEORY specal ssue on space-tme transmsson, recepton, codng and sgnal processng, was an Assocate Edtor for Communcatons of the IEEE TRANSACTIONS ON INFORATION THEORY durng , and s currently an Edtor for the Journal Foundatons and Trends n Informaton and Communcaton.