Design Rules for Efficient Scheduling of Packet Data on Multiple Antenna Downlink

Transcription

1 Desgn Rules for Effcent Schedulng of acet Data on Multple Antenna Downln Davd J. Mazzarese and Wtold A. rzyme Unversty of Alberta / TRLabs Edmonton, Alberta, Canada E-mal: djm@ ece.ualberta.ca / wa@ece.ualberta.ca ABSTRACT We provde a novel analyss of the optmal number of users that should be allocated power n order to acheve the sumcapacty of the MIMO broadcast channel, as well as the optmal power allocaton and the optmal transmtter covarance matrces n the asymptotcally hgh power regon. We study cases where recevers are equpped wth a sngle or wth multple antennas, and pont out the fundamental dfferences between these systems. Ths analyss s then appled to multple users schedulng algorthms for throughput maxmzaton, wth the addtonal goal of provdng low-complexty solutons. We also provde a dscusson of the smlartes and dfferences wth receve antenna selecton algorthms and MIMO channels wth cochannel nterference. I. ITRODUCTIO MIMO systems are envsoned to enable hgh-speed data transmsson on the downln of fourth generaton wreless networs. When the base staton s equpped wth multple transmt antennas, a scheduler taes advantage of the latency allowed wth pacet data communcaton and provdes ncreased throughput by means of multuser dversty. Recent nformaton theoretc advances have proved that the nature of the MIMO broadcast channel (BC) requres to revst schedulng algorthms to be able to transmt to more than one user at a tme n each fadng state n order to acheve sum-capacty [][2][3][4]. The very frst attempts at combnng the advantages of MIMO systems wth multuser dversty used sngle-user schedulng strateges. However these cannot acheve spatal multplexng gan f the recevers have a sngle antenna, and they can even suffer from channel hardenng f no care s taen n the explotaton of spatal dversty jontly wth multuser dversty [5]. We revst schedulng algorthm strateges when the base staton s equpped wth multple transmt antennas. Our analyss provdes gudelnes for the desgn of nearoptmal low-complexty schedulng algorthms for maxmum throughput wth optmal sgnalng,.e. drtypaper codng [6]. We thus obtan desgn crtera for schedulng algorthms applcable wth sub-optmal transmsson schemes, such as transmtter channel nverson. We tacle the problem by explorng the followng two fundamental questons. What s the optmal number of actve users n any gven fadng state or channel realzaton? What are the optmal power allocaton and transmt covarance matrces? umercal methods exst that allow to obtan the optmal covarance matrces, but a more precse understandng of the underlyng soluton s stll requred. The underlyng objectve of throughput maxmzaton wth multple antennas s to acheve the maxmum spatal multplexng gan avalable n MIMO systems. Hence, the nature of the problem largely depends on the number of antennas at each moble user s recever. The scope of ths paper s lmted to throughput maxmzaton. However n cellular systems farness must be ensured to prevent some users, typcally those close to the base staton, to hold all the resources. The study presented here s applcable to groups of users wth the same statstcal channel condtons over a short perod of tme, n whch case farness n terms of throughput and delay wll be provded statstcally. One scenaro one could thn of s to group users s such a way and apply throughput maxmzaton wthn the groups, whle farness s provded by some other means between groups of users. roportonally-far schedulng has been consdered n a broader context [9], but due to the assumed short-term statstcal equvalence of all user channels, t s not useful n ths study. The remander of the paper s organzed as follows. In Secton II we present the channel model. We brefly revew recent advances on the MIMO BC n Secton III. We provde a novel analyss of the optmal number of actve users and ther power allocaton n the hgh power regon n Secton IV. We apply our analyss to the desgn of low-complexty maxmum-throughput schedulng algorthms n Secton V. Conclusons are gven n Secton VI. II. SYSTEM MODEL We consder the downln of a cellular system, n whch the base staton has transmt antennas. There are users n the sector served by the base staton. User s equpped wth M receve antennas. We call ths channel the (,M,) MIMO BC. The complex channel gans are assumed to be ndependent between users and between antenna elements. The channel remans constant n each tme slot, and changes randomly from slot to slot (quasstatc fadng). The channel between the base staton and user s descrbed by a matrx H of sze M by. The elements of H represent small-scale fadng and they are modeled as..d. complex Gaussan random varables wth

2 zero mean and unt varance. The AWG varance at each receve antenna s normalzed to one. The transmtter has a total power constrant at each channel use. We assume that the transmtter and the moble users perfectly now all channel complex fadng gans. We defne the channel matrx H = [ H T H T ] T when M = for all users. The absence of path loss and shadow fadng maes ths model applcable for groups of users wth approxmately the same SR averaged over small-scale fadng over several tme slots. When shadow fadng and path loss are accounted for, more elaborate schedulng strateges must be appled, such as proportonally-far schedulng. As shown n [9], the nsghts obtaned n the analyss of throughput maxmzaton wth the smple channel model consdered here are stll relevant n the analyss of systems assumng more complete channel models. III. BACGROUD O THE MIMO BROADCAST CHAEL The MIMO BC s a degraded broadcast channel. It was recently proved that drty-paper codng acheves the capacty regon of that channel [4]. Drty-paper codng s a theoretcal random codng technque for nterference cancellaton at the transmtter of a non-causally nown nterference source [6]. In partcular, the sum-capacty of the MIMO BC s achevable by drty-paper codng wth successve encodng at the transmtter and optmal transmt covarance matrces. These optmal matrces can be obtaned numercally through effcent algorthms [7]. The problem s n general solved on the dual sum-power MIMO multple-access channel (MAC) where t s convex. The optmal covarance matrces obtaned for the sumpower MIMO MAC can then be transformed to gve the optmal covarance matrces on the MIMO BC [3], such that the rate vectors acheved on both channels are the same. The transmt covarance matrces for the MIMO BC depend on the chosen encodng order. The power allocated to a gven user s equal to the trace of ts transmt covarance matrx, and there s no conservaton of the power allocaton between the MIMO MAC and the MIMO BC. Thus t s not suffcent to study the optmal power allocaton on the sum-power MIMO MAC. However, a user that s allocated no power on ether channel s also allocated no power on the dual channel. Thus, the optmal number of actve users remans the same. We brefly revew the MAC to BC transformatons [3]. Successve decodng s used on the MIMO MAC. The decodng order s the followng: user s decoded frst, user 2 s decoded second, and so on untl user s decoded last. The same rate vector s acheved on the MIMO BC usng the covarance matrces obtaned wth the MAC to BC transformatons when user s encoded last, user 2 s encoded second to last, and so on wth user beng encoded frst. The optmal covarance matrx of sze M M of user on the MIMO MAC s. It does not depend on the decodng order chosen on the MAC. The rate vector achevable on the MAC s determned by the decodng order. The reversed encodng order must be used on the BC n order to acheve the same rate vector, wth the optmal covarance matrx of sze of user on the MIMO BC gven by = B FG A A G F B, () where the sngular value decomposton of the effectve channel s 2 2 B H A = F G. (2) F and G are untary matrces, and s a dagonal matrx wth nonnegatve man dagonal entres. Let I be the dentty matrx of sze. A and B represent the nterference experenced by user on the BC and on the MAC respectvely: A = I + H H (3) ( = ) M B I H H =+ = +. (4) IV. ASYMTOTICALLY OTIMAL UMBER OF ACTIVE USERS AD OWER ALLOCATIO A. Sngle-antenna recevers Ths secton s devoted to studyng the asymptotcally optmal power allocaton requred to acheve the sumcapacty of the (,,) MIMO BC n the lmt when the total transmt power becomes large. On the dual sumpower MIMO MAC the covarance matrx of user s a scalar p. Defne p r lm =. (5) In order to prove our results, we mae the followng assumptons: There are at least as many users as transmt antennas:. At least users are allocated a non-vanshng fracton of the total transmt power on the dual sum-power MIMO MAC. We number these users such that r > for =,, +. These assumptons are reasonable. As long as there are at least as many users as transmt antennas, t s only possble to explot the dmensons avalable n the MIMO channels by allocatng a non-vanshng fracton of the total transmt power to at least users n the hgh power regon. In fact, t s even possble that more than users are allocated a non-vanshng fracton of the total transmt power n order to acheve the sum-capacty of the dual sum-power MIMO MAC [8]. After MAC to BC transformatons of the MAC power allocaton we obtan the optmal BC transmt covarance matrces,,. We can prove that [9]: Tr lm = f (6) Tr ( ) lm > r f > (7) Therefore only users are allocated a non-vanshng fracton of the total transmt power at the sum-capacty n

3 the hgh power regon on the MIMO BC. Furthermore we also prove the followng property of the optmal BC covarance matrces of these users. Let the ran-one optmal covarance matrx of user be: = tr vv, (8) where v s the transmt beamformng vector of user and π = tr. We prove that for a gven j > [9]: lm Hv j = for all such that < < j. (9) Ths result tells us that asymptotcally n the hgh power regon on the (,,) MIMO BC, the optmal transmt beamformng vector of user, who s among the frst users to be encoded, becomes asymptotcally orthogonal to the channel matrces of all other users that are allocated an asymptotcally non-vanshng fracton of the total transmt power and that are encoded pror to user. As a consequence of the above property on the (,,) MIMO BC, we can express the asymptotc rates acheved by the users n the hgh power regon and we can solve for the frst-order asymptotc value of the sum-capacty n a smple way: Fg.. (a) Fracton of power allocated to each user on the MIMO MAC User (b) Fracton of power allocated to each user on the MIMO BC User (c) Rates acheved by each user on the MIMO BC User Total power n reference to the nose level (db) BC BC sum = IM + H H = =. () lm C lm R log The optmzaton over the covarance matrces wth the orthogonalty constrant can be reformulated as: ( π HvvH ) BC lm Csum max log + π,, π v,, v = (3,,8) MIMO BC optmal power allocaton and users rates wth encodng order 8 to. subject to π = and π =, =,, and =,, : Hv j =, j > and v =. () Solvng the above problem leads to the QR decomposton of the channel matrx H = RQ and waterfllng power allocaton. Let the (,) dagonal element of the upper trangular matrx R be r, then BC 2 lm Csum log + 2 log + log ( r ) (2) n= rnn = As a frst order approxmaton, the asymptotcally optmal power allocaton on the MIMO BC, gven an arbtrary encodng order, s unform over the users that are allocated a non-vanshng fracton of the total power n the hgh power regon. Our proof s vald for any. The asymptotc optmalty of unform power allocaton n the hgh power regon had been shown for = 2 n [] where the transmsson strategy used a QR decomposton and drty-paper codng. Here we prove that not only QR decomposton wth drty-paper codng s asymptotcally optmal, but t s the frst-order asymptotcally optmal procedure to acheve the sum-capacty of the (,,) MIMO BC n the hgh power regon for any. As a consequence of the orthogonalty property we deduced that unform power allocaton s optmal on the (,,) MIMO BC n the hgh power regon. We can thus drectly prove that the asymptotcally optmal power allocaton on the dual sum-power MIMO MAC s also unform for any, whch was prevously nown only for large []. We now show some numercal results to llustrate our fndngs. We consder a fxed channel realzaton and we let the total transmt power ncrease to very large values. We observe the fracton of the total power allocated to each user both on the dual MIMO MAC and on the MIMO BC. We consder one realzaton of the (3,,8) MIMO BC. Fgure (a) shows the optmal power allocaton on the MIMO MAC as a functon of the total transmt power n reference to the nose level. The power allocated to each user s normalzed to the total transmt power. The power allocaton s ndependent of the encodng order. Only users {,2,5,6} are allocated a non-vanshng fracton of the total transmt power n the hgh power regon on the MIMO MAC. Snce there are 3 base staton antennas, after MAC to BC transformatons wth encodng order 8 to, only users {2,5,6} are stll allocated a non-vanshng fracton of the total transmt power n the hgh power regon on the MIMO BC as shown n Fgure (b). The rate acheved by user for the set of optmal covarance matrces,, and the encodng order to s gven by: R BC ( j ) ( j< ) I + H H = log I + H H M j M j (3) These rates are shown n Fgure (c). User acheves a constant non-zero rate at hgh power, whch becomes asymptotcally neglgble wth respect to the sum-capacty. B. Multple-antenna recevers Wth antennas at each recever only onedmensonal channels are allocated a non-vanshng fracton of the total transmt power on the MIMO BC when the total transmt power goes to nfnty. These onedmensonal channels all belong to the same user, as long as that user s allocated a non-vanshng fracton of the total transmt power on the dual MIMO MAC and ts MAC covarance matrx s full ran asymptotcally, and that user s encoded frst by drty-paper codng. Thus we

4 .5.5 proved [9] that asymptotcally n the hgh power regon, only one user s allocated a non-vanshng fracton of the total transmt power. The latter two assumptons can always be satsfed as long as at least one user has a fullran channel matrx, whch occurs almost certanly n a rch scatterng envronment. Let lm tr (a) Fracton of power allocated to each user on the MIMO MAC User 5 5 (b) Fracton of power allocated to each user on the MIMO BC 5 5 (c) Rates acheved by each user on the MIMO BC 2 5 Fg. 2. User 4 User 4 User User Total power n reference to the nose level (db) (4,4,4) MIMO BC optmal power allocaton and users rates wth encodng order 4 to. = r, where r s a constant. Then we can prove that [9]: lm = I (4) lm tr ( j ) = for j =,,. (5) The optmal power allocaton on the BC s very dfferent than that on the MAC after MAC to BC transformatons. Even though several users could be allocated a non-vanshng fracton of the total transmt power on the MIMO MAC, only one wll be allocated a non-vanshng fracton of the total transmt power on the MIMO BC. However, one must be careful n concludng that transmttng to only one user s suffcent to acheve the sum-capacty. We can only say that the rato of the rate acheved by user to the sum-capacty tends to one as the total transmt power goes to nfnty, but the convergence s slow due to the logarthmc growth of the sum-capacty wth power. Moreover, smulatons show that the asymptotc result only occurs at very large values of the total transmt power. As the power s large and ncreases, but as t s stll below the threshold where only one user s allocated power on all ts egenmodes, several onedmensonal channels are allocated power such that ths power ncreases wth the total transmt power untl the total transmt power reaches the threshold, and these onedmensonal channels belong to more than one user. Beyond that threshold, all the addtonal power s allocated to only one user. We consder a realzaton of the (4,4,4) MIMO BC. Fgure 2 shows the optmal power allocaton on the MIMO MAC, and on the MIMO BC wth encodng order 4 to, as well as the users rates, as a functon of the total transmt power n reference to the nose level. All four users are user user 2 user 3 user non zero egenvalue maxmum of 4 non zero egenvalues 5 5 Total power n reference to the nose level (db) Fg. 3. allocated a non-vanshng fracton of the total transmt power on the MIMO MAC, but after MAC to BC transformatons only user 4 wll be allocated a nonvanshng fracton of the total transmt power n the hgh power regon. We notce that at 5 db, the rates of users 2 and 3, whch reman constant, are not neglgble compared to the sum-capacty. They wll only become neglgble at much hgher values of the total transmt power. We can tae a closer loo at the power allocaton by observng the rato of the egenvalues of the optmal covarance matrces to the transmt power. After MAC to BC transformatons wth the encodng order 4 to, we see n Fgure 3 that the power allocaton progressvely shfts from egenmodes of users, 2 and 3 to all of the fourth user s egenmodes n the hgh power regon. In general when users are equpped wth dfferent numbers of receve antennas, only one-dmensonal channels are allocated a non-vanshng fracton of the total transmt power n the hgh power regon, and these onedmensonal channels belong to the J users that are encoded frst by drty-paper codng such that these users are allocated a non-vanshng fracton of the total transmt power on the dual sum-power MIMO MAC and M M. (6) = J+ 2 = J+ maxmum of 2 non zero egenvalues maxmum of 2 non zero egenvalues (4,4,4) MIMO BC optmal power allocaton per egenmode wth encodng order 4 to. The proofs of the above results le n the ncremental asymptotc ran of the nterference matrces B I that are allocated nfnte power n the successve encodng process. Ths s smlar n nature to the problem of cochannel nterference n MIMO systems []. However on the MIMO BC there s addtonal cooperaton at the transmtter that allows to cope wth ths nterference, so that even f users are equpped wth one receve antenna t s stll possble to allocate nfnte power to users, and that other users acheve a constant rate. V. -USER SCHEDULIG ALGORITHMS As seen n [8] t s necessary and suffcent to transmt to users at a tme wth any number of receve antennas per user n the medum power regon n order to lose only a margnal amount of spectral effcency relatve to the sumcapacty. It mght also be requred to transmt to no more

5 Rato of spectral effcency to the sum capacty SUR schedulng SUR (+) schedulng Gorohov Algorthm I Gorohov Algorthm III umber of users.95.9 (b).85 SUR schedulng SUR (+) schedulng Gorohov Algorthm I.8 Gorohov Algorthm III Random user schedulng umber of users Fg. 4. Rato of the spectral effcency wth several -user schedulng algorthms and drty-paper codng to the sum-capacty. Total power n reference to the nose level: (a) db, (b) db. than users at a tme due to constrants of lnear spatal multplexng schemes or for complexty reducton. Optmal -user schedulng by exhaustve search of groups of users among ncurs a combnatoral complexty, whch maes t mpractcal wth a large number of users and even a moderate number of transmt antennas. We consder the followng schedulng algorthms to select users n each tme slot: Random -user schedulng: users are randomly selected. Sngle-User Rates (SUR-) schedulng: the users wth the largest ndvdual capactes are selected. SUR-(+) schedulng: frst select the + users wth the largest ndvdual capactes, then the users that offer the largest sum-capacty by exhaustve search. Gorohov s receve antenna selecton algorthms [2]: by treatng each user as a dfferent antenna n a sngle recever wth multple antennas, Gorohov s lowcomplexty algorthms allow to select receve antennas out of and to lmt the capacty loss. Wth 4 transmt antennas and 5 users wth a sngle receve antenna each, random -user schedulng acheves only 75% of the sum-capacty at db as shown n Fgure 4.b. It performs even worse at db. It s unable to explot multuser dversty. Other strateges such as round-robn schedulng that also do not explot multuser dversty perform poorly. Schedulng users ndependently of one another s also not a good soluton as seen wth SUR- schedulng. The performance s mproved wth SUR-(+) schedulng wth an ncrease n complexty. As a consequence of the asymptotc analyss we can conclude that the (,) open-loop cooperatve MIMO capacty s a good approxmaton of the sum-capacty of the (,,) MIMO BC n the hgh power regon. Hence, receve antenna selecton algorthms, such as proposed n [2], are applcable for maxmum-throughput jont -user schedulng on the (,,) MIMO BC n the hgh power regon. Moreover, due to ther nherent nterferenceavodance propertes they also perform well n the low and medum power regons as seen n Fgure 4. (a) VI. COCLUSIO We have shown that n the hgh power regon of the MIMO broadcast channel wth transmt antennas, t s asymptotcally optmal to allocate a non-vanshng fracton of the total transmt power to users f users are equpped wth sngle receve antennas, or to only one user f users are equpped wth receve antennas. In the sngle receve antenna case an addtonal orthogonalty property was proved as a frst order approxmaton of the sum-capacty, whch showed that the channel between the users that are allocated a non-vanshng fracton of the total power s completely orthogonalzed by the jont acton of drtypaper codng and optmal beamformng. We observed that n the medum power regon t s best to transmt to users smultaneously n each fadng state to approach the sumcapacty. As a consequence of ths fact and of a hgh power approxmaton of the sum-capacty, receve antenna selecton algorthms provde an effcent way of schedulng users wth low-complexty and near-optmal throughput. ACOWLEDGMET The authors gratefully acnowledge fundng for ths wor provded by TRLabs, Roht Sharma rofessorshp and the atural Scences and Engneerng Research Councl (SERC) of Canada. REFERECES [] G. Care and S. Shama, On the achevable throughput of a multantenna Gaussan broadcast channel, IEEE Trans. on Inform. Theory, vol. 49, no. 7, pp , July 23. 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