Channel aware schedulng for broadcast MIMO systems wth orthogonal lnear precodng and farness constrants G Prmolevo, O Smeone and U Spagnoln Dp d Elettronca e Informazone, Poltecnco d Mlano Pzza L da Vnc, 3 I-133 Mlano (Italy) Abstract In the downlnk of a broadcast fadng channel, the base staton can captalze on multuser dversty through channel aware schedulng In MIMO systems, the desgn of the scheduler has to take nto account the processng performed at the trasmtter and the recevers In ths work, we consder channel aware schedulng for orthogonal lnear precodng at the base staton that guarantees nterference free recepton for each scheduled users The problem s set n a novel mathematcal framework and a schedulng algorthm s proposed that s shown by smulaton to guarantee superor performance as compared to know technques Moreover, farness constrants nspred by the proportonal far crteron are ntroduced n the schedulng process n order to guarantee the desred long term farness propertes I INTRODUCTION In the downlnk of a broadcast fadng channel, the base staton can captalze on multuser dversty provded by ndependent fadng realzatons across dfferent users Channel aware schedulng s a technque that allows to acheve ths goal by approprately tmng transmsson to a subset of one or more users n each avalable tme (code/frequency) slot Schedulng s performed accordng to the knowledge of the channel state nformaton avalable at the scheduler wth the general goal of grantng transmsson to the users that have nstantaneous channel near the peak [1] If base staton and users are equpped wth a sngle antenna, t has been shown that transmsson to the user wth the strongest channel s a strategy that acheves channel capacty [] owever, f the base staton s equpped wth an antenna array, more users can be served smultaneously n the same tme slot In partcular, f the base staton has n T antennas, up to n T users can be allocated n the same tme slot wth controlled nterference In ths case, transmsson to a sngle user s not the optmal soluton and the desgn of the scheduler becomes more complcated dependng on the beamformng and power allocaton strategy [3] [4] In MIMO systems (e, antenna array at both base staton and termnals), the scheduler can leverage on another degree of freedom snce each user can be assgned to multple spatal channels [5] In fact, f each user has n R recevng antennas (n R n T ), the base staton can grant up to n R spatal channels (out of the avalable n T ) to any user [6] The desgn of the scheduler has to take nto account the processng performed at the transmtter (eg, lnear precodng and power allocaton) and the recevers (eg, lnear equalzer) Schedulng wth lnear processng at the base staton that smply assocates each spatal channel wth a transmttng antenna and lnear nterfaces at the recevers has been consdered n [5] (zero forcng equalzer) and [7] (MMSE equalzer) In ths paper, we consder channel aware schedulng for x 1 x K M 1 M K 1 K n 1 y 1 n K y K B 1 B K ~ y1 y~ K equalzaton decodng equalzaton decodng Fg 1 Block dagram of a broadcast channel wth lnear nterfaces at the transmtter (base staton) and recevers (users) orthogonal lnear precodng at the base staton and lnear zero-forcng equalzers at the recevers (Orthogonal Space Dvson Multple Access, OSDMA [8]) Usng the OSDMA transmttng/recevng strategy, the spatal channels ntended for a gven user do not nterfere nether wth the sgnal destned to other users nor among themselves Channel aware schedulng for OSDMA has been frst studed n [8] We set the problem n a novel mathematcal framework and propose a schedulng algorthm that s shown by smulaton to guarantee superor performance n terms of sum capacty Moreover, farness constrants nspred by the proportonal far crteron [9] [1] are ntroduced n the schedulng process n order to guarantee the desred long term farness propertes II SIGNAL MODEL AND PROBLEM FORMULATION The broadcast channel wth lnear nterfaces at the transmtter and recevers s depcted n fg 1 Let K be the set of K avalable users The th user s equpped wth an antenna array of n R, elements and the base staton wth n T antennas The subset of K(t) users that are served by the base staton wthn the tth tme slot s denoted as K(t) K and ts element ndexed by k =1,,, K(t) The scheduler allocates d k (t) n R,k spatal channels to the kth user so that all the avalable n T spatal channels are used: K(t) X d k (t) =n T (1) k=1 The sgnal ntended for the kth user, collected n the d k (t) 1 vector x k (t) s lnearly precoded by the n T d k (t) matrx M k (t) Followng the conventonal notaton (see, eg, [6]) and referrng to fg 1, the sgnal receved by the kth user across ts n R,k recevng antennas wthn the tth tme slot can -783-8939-5/5/$ (C) 5 IEEE
be wrtten as the n R,k 1 vector y k (t) y k (t) = k (t)m k (t)x k (t)+ X k (t)m (t)x (t)+n k (t) 6=k K(t) () where k (t) s the n R,k n T channel matrx of the kth user and n k (t) s the zero mean addtve Gaussan nose wth E[n k (t)n k (t) ]=σ ni nr,k The receved sgnal y k (t) les n a n R,k -dmensonal lnear space owever, only d k (t) n R,k spatal channels are assgned to the kth user Therefore, the useful part of the receved sgnal spans a d k (t)-dmensonal subspace that we refer to as recevng subspace In order to account for ths, at the recever, the n R,k 1 receved sgnal y k (t) s pre-fltered by the d k (t) n R,k matrx B k (t) ỹ k (t) = B k (t) y k (t) = k (t)m k (t)x k (t)+ + X k (t)m (t)x (t)+ñ k (t), (3) 6=k K(t) wherewehavedefned the d k (t) n T equvalent channel k (t) =B k (t) k (t) and ñ k (t) =B k (t) n k (t) In order to smplfy the analyss and wthout lmtng the generalty of the approach, we assume B k (t) B k (t) =I dk (t), (4) so that E[ñ k (t)ñ k (t) ]=σ ni dk (t) The range space of B k (t) corresponds to the recevng subspace for the kth termnal As a last step, equalzaton and detecton s performed on ỹ k (t) In ths work, we assume that the channel matrces k (t) are known to the transmtter and recevers, eg, by transmsson of plot symbols and feedback of the channel state nformaton from the recevers to the base staton [11] An analyss of the effect of mperfect channel state nformaton and feedback delays s proposed n [1] In order to smplfy the notaton, n the followng the temporal dependence on t s omtted A Problem formulaton In prncple, we would lke to fnd the subset of users K and the set of precodng matrces M ={M } K and pre-flterng matrces B = {B } K so that the sum capacty s maxmzed under a total power constrant (recall also constrants (1) and (4)): {B, M} =argmax B,M st C (B, M) tr(m M ) P, where C (B, M) s the lnk capacty for the th user [6] wth C (B, M) =log I d +R 1 ( M M R = σ ni d + X k6= k K (5a) (5b) ) (6) M k M k (7) In (6)-(7) the assumpton of Gaussan codebooks wth E[x x ] = I d s mpled Moreover, as a result of the optmzaton problem (5) the th user belongs to the set of actve users K f d > or equvalently (M, B ) are not empty matrces Soluton of the optmzaton problem (5) s not known, even for the case of gven sets K and B In [8], an algorthm s proposed for obtanng an approxmate soluton based on the addtonal constrant of zero nter-user nterference and the separate computaton of precodng and schedulng as explaned n Sec III The treatment s amed at settng the results of [8] n the dscussed mathematcal framework and revew the man concepts A novel approxmate soluton of (5) s then proposed n Sec IV The algorthm s stll based on the ncluson of the zero nter-user nterference constrant but, dfferently from [8], t performs ontly precodng and schedulng III REVIEW OF MIMO-OSDMA WIT LSV SCEDULING Accordng to the approxmate soluton of (5) proposed n [8], at frst the schedulng step s performed Ths amounts to select the subset K and the correspondng K matrces B k Recall that the choce of B k mples the allocaton of d k spatal channel to the kth user and the correspondng recevng subspace Then, the desgn of the precodng matrces M k s carred out wth the addtonal constrant of grantng nter-user nterference free transmsson (MIMO-OSDMA) A Largest Sngular Value (LSV) channel aware schedulng In [8], selecton of the subset K and of the correspondng K matrces {B k } K k=1 s performed so as to set as actve the spatal channels correspondng to the largest sngular values of matrces { } K To elaborate, let λ,, =1,,r = rank( ) be the non-zero sngular values of channel matrx gathered n the dagonal matrx Λ and (u,, v, ) the correspondng left and rght sngular vectors collected by columns n matrces U and V respectvely: = U Λ V The LSV algorthm selects the n T largest sngular values of the set {λ, =1,, K, =1,,r } and bulds matrces B k wth the correspondng left sngular vectors v, Ths algorthm can equvalently be stated as the soluton of the followng optmzaton problem: fnd the set B = {B } K so that (recall also constrants (1) and (4)): B =argmax B B, (8) Notce that an user belongs to K f the correspondng number of assgned channel d s not zero, or equvalently B s not empty B MIMO Orthogonal Space Dvson Multple Access (OS- DMA) Gven the output of the schedulng algorthm (e, the set K and matrces {B k } K k=1 ), the precodng matrces M k are derved by maxmzng the sum capacty (5) wth an addtonal zero nterference constrant among the users Notce that the -783-8939-5/5/$ (C) 5 IEEE
zero-nterference assumpton s capacty achevng n the hgh sgnal-to-nose rato regme where the nterference plays a maor role n defnng the system performance In partcular, the followng optmzaton problem s solved: KX M =argmax C (B, M) (9a) M KX st tr(m k M k ) P, (9b) k=1 M = f 6=, (9c) Constrant (9c) ensures zero-nterference among the K actve users Therefore, the capacty for the th user can be wrtten as (6) wth R = σ ni d The resultng precodng matrx M k has the followng form [8] M k = Θ k P k, (1) where Θ k s a n T d k matrx s selected as Θ k = V kṽk (11) where V k s a n T d k bass of the d k dmensonal null space of the n T (n T d k ) matrx k wth the sngular value decomposton k = [ 1 k 1 k K] = = Ū Ū Λ V V, (1) ths guarantees the zero nter-user nterference constrant (9c) Matrx Ṽ k s the d k d k range space of k, thus ensurng zero nterference among all the spatal channels Fnally, P k s a d k d k dagonal matrx that defnes power allocaton over the spatal channels selected accordng to the mult-user waterfllng crteron IV MIMO-OSDMA WIT SVS SCEDULING The approxmate soluton of the problem (9) proposed by [8] suffers from degraded performance (as t wll be shown n Sec VI) manly because the precodng matrces M k and the preflterng matrces B k (and the assocated set K) are optmzed separately ere we propose a ont optmzaton that approxmates problem (9) as follows ) The zero nterference constrant (9c) s mposed, thus obtanng a MIMO- OSDMA system as n [8] As explaned n Sec III-B, the resultng precodng matrces have the form (1)-(11) ) The obectve functon C (B, M) s approxmated by ts frst term of the Taylor expanson: C (B, M) 1/σ n B M The latter approxmaton s expected to hold at suffcently low sgnal-to-nose ratos The resultng optmzaton problem reads (recall also constrants (1) and (4) and defne V = { V } K ): {B, V } =argmax B, V N (B, V ), (13a) st V =,6=, (13b) wherewedefned N (B, V )= B V The obectve functon (13a) s amenable to an effcent numercal optmzatonandwllbeshownnsecvitoyeldrelevantadvantages as compared to the separate optmzaton proposed n [8] Notce that n order to smplfy the soluton of the problem, the remanng term of the precodng matrces (1), Ṽ k and P k, are assumed to be computed accordng to Sec III-B, thus guaranteeng zero nterference among dfferent streams and the enforcement of the total power constrant A Successve Vector Selecton (SVS) channel aware schedulng Problem (13) can be effcently solved by a greedy approach as detaled n the followng The dea s to select at each step the spatal channel that yelds the largest ncrease of the obectve functon (13a) Let us denote wth the superscrpt (n) the quanttes of nterest as computed at the nth teraton At each teraton a spatal channel (out of the n T avalable) s allocated to a specfc user so that a total number of n T teratons are needed We are nterested n updatng the recevng subspaces B (ntalzed as B () equaltoanempty matrx) and the transmttng subspaces V, or equvalently ts orthogonal complement V (see (1), ntalzaton: V () = I nt ) Let u be a possble canddate vector to be ncluded n the recevng subspace B (n) of user at the nth teraton ( =1,, K) As a result of the choce of u at the nth teraton, the obectve functon (13a) modfes as (droppng the functonal dependence on B, V for smplcty of notaton) N (n) (u )= N (n 1) + N (n) (u ) (14) Among all the possble vectors u for all users =1,, K, the vector u s selected so as to maxmze the ncrease of (u )Inthefollowng,the computaton of N (n) (u ) s carred out To elaborate, we need to defne for each user a bass U (n) that spans the range space of the channel matrx that at the nth teraton has not be assgned to any recevng subspace the obectve functon P K N (n) Formally, t s: span{u (n) } = span{u } null{b (n) } Therefore, the correspondng ntalzaton s U () = U At the nth teraton we have P K (n) ˆd = n and the possble canddate vectors to be ncluded n the recevng subspace of the th user are lnear combnatons of the columns of U (n) : u = U (n) a, wth ka k =1 (15) Wth the selecton of (15), the correspondng recevng subspace s updated as B (n) =[B (n 1) u ] whle ts transmt subspace remans unchanged, V (n) = V (n 1), snce no new constrant (9c) s mposed upon t It follows that N (n) (u )= u V (n) (16) Then, let v = u be the vector correspondng to u on the transmtter sde The choce of u for user results n an -783-8939-5/5/$ (C) 5 IEEE
addtonal zero-nterference constrant for any user 6= (see (13)), that leads to h V (n) = V (n 1) w, (17) where w s the proecton of v over V (n 1), scaled to unt length w =( V (n 1) V (n 1) v )/ V (n 1) V (n 1) v (18) V (n) s updated as well, so that span( V (n) ) = null( V (n) ) Ths step can be performed, eg, by updatng the QR decomposton of (17) [13] On the other hand, nothng changes at the recever sde of the th user, B (n) = B (n 1) Itseasyto show that N (n) (u )= B (n) w 6= (19) To sum up, from (16) and (19) the ncrease of obectve functon (13) due to the choce of vector u at the nth teraton s N (n) (u )= u V (n) X B (n) w 6= () The frst term n () accounts for the ncreased useful power receved by user on the newly assgned spatal channel, whereas the other terms represent the power loss suffered from the other users from not beng allowed to transmt over w anymore Recallng (18) and (15), functon () can be easly recognzed to be a sum of Raylegh quotents n terms of vector a Whle the maxmzaton of a sngle Raylegh quotent s analytcally feasble snce t corresponds to the soluton of a generalzed egenvalue problem, maxmzng a sum of Raylegh quotents s much more dffcult and costly ere, we resort to a sub-optmal approach, by restrctng a to be a column of an dentty matrx, whch translates to restrctng our search of the optmal u to the columns of U (n) Ths approach has been proved by smulaton to yeld performance very close to the optmum soluton V SVS ALGORITM WIT PROPORTIONAL FAIRNESS CONSTRAINTS The algorthms dscussed so far am to maxmze the system throughput If the users are unbalanced, wth some of them experencng strongly attenuated channels, t s expected that the algorthms wll result n an unfar sharng of system resources that mght preclude communcaton to some users (see also Sec VI) Smlarly to the proportonal far crteron [1], a schedulng procedure that acheves over a long term an approprate balance between sum capacty and farness among users can be defned by modfyng (13) as follows (here we explct for convenence the tme dependence): {B(t), V (t)} =argmax B, V log(e[n (B, V )]), (1a) st (t) V (t) =,6= (1b) where E[ ] refers to the long term average over tme Followng the analyss of [9] and the consderatons n Sec IV, t can be shown that a procedure that (approxmately) converges to the soluton of (1) can be obtaned by mplementng the SVS algorthm on normalzed channel matrces N (t) = (t) () α (t) wth α (t) = µ1 1tc ˆα (t)+ 1 t c N (B(t 1), V (t 1)), (3) where parameter t c rules the memory of the algorthm Parameter α (t) measures the channel power that each user has been allowed to use wthn a wndow of t c tme slots The ratonale of the algorthm s that f a gven user has been gnored by the schedulng procedure n the consdered tme wndow, the matrx scalng () wll force the SVS algorthm to allocate resources to t The crteron (1) s a farness constrants on the channel norms: ts mplcaton on the channel rates s not obvous and wll be nvestgated n the next Secton by numercal smulatons VI NUMERICAL SIMULATIONS The performance of the proposed SVS algorthm s compared wth the LSV algorthm [8] by Monte Carlo smulatons We consder K = 4 users, where each user has the same number of recevng antennas n R = whle the base staton s equpped wth n T =4antennas Where not stated otherwse, the channels are assumed to be subect to dentcally dstrbuted Ralegh fadng, vec( ) CN, I nt n R As reference performance, a random user selecton algorthm s consdered that chooses randomly a set K of users such that (1) s satsfed On ths subset, orthogonal precodng s appled as detaled n Sec III-B Moreover, the performance of a n T n T sngle user MIMO lnk s evaluated n order to set a reference level for the sum capacty of the multuser system The ergodc sum capacty s plotted versus the sgnal to nose rato P/σ n n fg The proposed SVS algorthm yelds a gan of about 4dB as compared to the LSV algorthm, whose performance are, n ths case, smlar to random users selecton As explaned n Sec IV, the advantage of SVS s due to the ont computaton of the transmttng and recevng subspaces Fg 3 shows sum capacty versus outage probablty for P/σ n =1dB It can be seen that the slope of the outage probablty for SVS s comparable to that of a sngle user channel, provng the ablty of the SVS algorthm to approprately explot the dversty of the broadcast channel Let us now consder unbalanced users n order to valdate the performance of the SVS algorthm wth farness constrants To be specfc, the channels are assumed to be selected so that vec( ) CN,β I nt n R, where β1 = db, β = 5dB, β 3 = 1dB and β 4 = db The performance of the SVS algorthm s evaluated wth and wthout farness constrants (t c = ) The results are summarzed n fg 4 n terms of ergodc sum capacty and ndvdual -783-8939-5/5/$ (C) 5 IEEE
Ergodc sum Capacty [bt/s/z] 5 15 1 5 Random selecton LSV schedulng [8] SVS schedulng Sngle User 4 6 8 1 1 14 16 18 P/σ n [db] Ergodc capacty [bt/s/z] 8 7 6 5 4 3 1 C 1 C C 3 Fg Ergodc sum capacty versus P/σ n (K =4,n T =4,n R =) C 4 SVS schedulng Far SVS schedulng sum capacty user capacty 1 3 4 5 t 1 8 Random selecton LSV schedulng [8] SVS schedulng Sngle User Fg 4 Ergodc sum capacty and ndvdual capactes versus tme t for the SVS algorthm wth and wthout farness constrants (unbalanced users, K =4,n T =4,n R =,P/σ n =1dB) Outage probablty 6 4 based on the ont optmzaton of precodng and schedulng has been proposed and ts performance proved by smulaton to be superor to know technques n terms of sum capacty Moreover, the ntroducton of proportonal farness constrants has been dscussed and a modfcaton of the algorthm that yelds desrable long term farness propertes has been proposed and valdated through smulaton 5 1 15 Ergodc sum capacty [bt/s/z] Fg 3 Sum capacty versus outage probablty (K =4,n T =4,n R =, P/σ n =1dB) ergodc capacty versus tme t The total throughput loss ncreases as the farness constrants are mposed by the scalng algorthm dscussed n Sec V and converges to approxmately bt/s/z Ths decrease of the sum capacty translates n a more far sharng of resources as proved by the ndvdual channel capactes Notce that the channel capactes for t = correspond to the performance of the SVS algorthm wth no farness constrants Even though the farness constrant (1) was mposed on the channel norms, rather than on the ndvdual rates as n [1], the smulaton results ndcate that the proportonal far crteron s very closely followed by the channel capactes as well In fact, the user capactes approxmately converge to the dashed lnes n fg 4 that denote the ndvdual capactes as obtaned by sharng the long term sum capacty accordng to the proportonal far crteron (e, n proporton to the sngle user capactes E[log I d +P/K ] [1]) VII CONCLUSION The problem of channel aware schedulng for broadcast MIMO channels wth orthogonal lnear precodng and lnear nterfaces at the recevers has been nvestgated An algorthm REFERENCES [1] R Knopp and P umblet, Informaton capacty and power control n sngle cell multuser communcatons, Proc IEEE Int Computer Conf, pp 331-335, June 1995 [] D N C Tse, Optmal power allocaton over parallel Gaussan channels, Proc IEEE ISIT, p 7, 1997 [3] Vswanathan and K Kumaran, "Rate schedulng n multple antenna downlnk wreless systems," Proc Allerton Conference, 1 [4] I Koutsopoulos, T Ren and L Tassulas, "The mpact of space dvson multplexng on resource allocaton: a unfed vew," Proc IEEE Infocom, pp 533-543, 3 [5] RWeathJr,MAry,AJPaulra, MultuserdverstyforMIMO wreless systems wth lnear recevers, Proc Aslomar Conf Sgnals, Systems and Computers, pp 1194-1199, Nov 1 [6] AGoldsmth,SAJafar,NJndalandSVshwanath,"Capactylmts of MIMO channels," IEEE J Select Areas Commun, vol 1, no 5, pp 684-7, June 3 [7] O Shn and K Bok, Antenna-asssted round robn schedulng for MIMO cellular systems, IEEE Comm Letters, vol 7, no 3, pp 19-111, March 3 [8] Q Spencer, A Lee Swndlehurst and M aardt, "Zero-forcng methods for downlnk spatal multplexng n multuser MIMO channels", IEEE Trans Sgnal Processng, vol 5, no, pp 461-471, Feb 4 [9] J Kushner and P Whtng, "Convergence of proportonal-far sharng algorthms under general condtons", IEEE Trans Wreless Commun, vol 3, no 4, pp 15-159, July 4 [1] P Vswanath, D N C Tse, R Laroa, Opportunstc beamformng usng dumb antennas, IEEE Trans Inform Theory, vol 48, no 6, pp 177-194, June [11] T Marzetta and B ochwald, "Fast transfer of channel state nformaton n wreless systems," submtted to IEEE Trans Sgnal Processng [also avalable on http://marsbell-labscom/] [1] G Prmolevo, O Smeone and U Spagnoln, "Effects of mperfect channel state nformaton on the capacty of broadcast OSDMA-MIMO systems", Proc IEEE SPAWC, 4 [13] G Golub and C F van Loan, Matrx Computatons, Johns opkns, 1996, 3rd edton -783-8939-5/5/$ (C) 5 IEEE