Sum Capacity of Multiuser MIMO Broadcast Channels with Block Diagonalization

Transcription

1 Sum Capacty of Multuser MIMO Broadcast Channels wth Block Dagonalzaton Zukang Shen, Runhua Chen, Jeffrey G. Andrews, Robert W. Heath, Jr., and Bran L. Evans The Unversty of Texas at Austn, Austn, Texas 7872 Emal: {shen, rhchen, jandrews, rheath, Abstract The sum capacty of a Gaussan broadcast MIMO channel can be acheved wth Drty aper Codng DC). Deployng DC n real-tme systems s, however, mpractcal. Block Dagonalzaton ) s an alternatve precodng technque for downlnk multuser MIMO systems, whch can elmnate nteruser nterference at each recever, at the expense of suboptmal sum capacty vs. DC. In ths paper, we study the sum capacty loss of for a fxed channel. We show that ) f the user channels are orthogonal to each other, then acheves the complete sum capacty; and 2) f the user channels le n a common row vector space, then the gan of DC over can be bounded by the mnmum of the number of transmt and receve antennas and the number of users. We also compare the ergodc sum capacty of DC wth that of n a Raylegh fadng channel. Smulatons show that can acheve a sgnfcant part of the total throughput of DC. An upper bound on the ergodc sum capacty gan of DC over s derved, whch can be evaluated wth a few numercal ntegratons. Wth ths bound, we can easly estmate how far away s from beng optmal n terms of ergodc sum capacty, whch s useful n drectng practcal system desgns. I. INTRODUCTION Multple-nput-multple-output MIMO) systems can sgnfcantly ncrease the spectral effcency by explotng the spatal degree of freedom created by multple antennas. It has been shown that the pont-to-pont MIMO channel capacty scales lnearly wth the mnmum number of transmt and receve antennas n Raylegh fadng channels ]. For Gaussan MIMO broadcast channels BC), t was conjectured n 2]3] and recently proven n 4] that Drty aper Codng DC) 5] can acheve the capacty regon. A dualty relatonshp for the MIMO broadcast DC capacty regon to the MIMO multple access channel MAC) capacty regon s shown n 3]6]. The sum capacty n a multuser MIMO broadcast channel s defned as the maxmum aggregaton of all the users data rates. Although the sum capacty of a Gaussan MIMO BC channel can be acheved wth DC 3], deployng DC n real systems s very complcated and mpractcal. An alternatve low-complexty precodng technque s Block Dagonalzaton ) 0] 3], whch s an extenson of the zero-forcng precodng technque for downlnk multuser MIMO systems. Wth, each user s precodng matrx les n the null space of all other users channels. Hence f the channel matrces of all users are perfectly known at the transmtter, then there s no nterference at every recever, renderng a smple recever structure. Furthermore, the transmtter s complexty s much lower for than DC. On the other hand, s nferor n terms of sum capacty to DC, snce the users sgnal covarance matrces are generally not optmal for sum capacty. The sum capacty gan of DC vs. TDMA has been studed n 5]. For a fxed channel, t has been proven that the gan of DC over TDMA s bounded by the mnmum of the number of users and the number of transmt antennas, for dfferent user number, antenna settng and SNR. Furthermore, t has been shown n 6] that the ergodc sum capacty of scales the same as DC n the number of users for Raylegh fadng channels. In ths paper, we focus on the sum capacty gan of DC over. We defne s sum capacty to be the maxmum total throughput over all possble user sets. Hence the TDMA sum capacty s automatcally ncorporated n s sum capacty defnton. Therefore, the general bound on the gan of DC vs. TDMA apples to the gan of DC vs.. We show that for a fxed channel ) f user channels are orthogonal to each other, then acheves the same sum capacty as DC; 2) f user channels le n the same subspace, then the gan of DC over can be reduced to the mnmum of the number of transmt and receve antennas and the number of users. Furthermore, the ergodc sum capacty of DC s compared to that of n a Raylegh fadng channel. We show that acheves a sgnfcant part of the DC sum capacty for low and hgh SNR regmes, or when the number of transmt antennas s much larger than the sum of all users receve antennas. An upper bound on the ergodc sum capacty gan of DC over s derved. The proposed upper bound on the gan can be evaluated wth a few numercal ntegratons, hence provdng an easy way to compare the performance of vs. DC wthout performng the tme-consumng Monte Carlo smulatons. II. SYSTEM MODEL AND BACKGROUND In a K-user downlnk multuser MIMO system, we denote the number of transmt antennas at the base staton as N t and the number of receve antennas for the jth user as,j. The transmtted symbol of user j s denoted as a N j,j )- dmensonal vector x j, whch s multpled by a N t N j precodng matrx T j and sent to the basestaton antenna array. The receved sgnal y j for user j can be represented as y j = H j T j x j + k=,k j H j T k x k + v j ) where v j denotes the Addtve Gaussan Whte Nose AWGN)

2 vector for user j wth varance Ev j v j ] = σ2 I, where ) denotes the matrx conjugate transpose. Matrx H j C,j N t denotes the channel transfer matrx from the basestaton to the jth user, wth each entry followng an..d. complex Gaussan dstrbuton CN 0, ). For analytcal smplcty, we assume that rankh j ) = mn,j, N t ) for all users. It s also assumed that the channels H j experenced by dfferent users are ndependent. The key dea of block dagonalzaton s to precode each user s data x j wth the precodng matrx T j UN t, N j ), such that H T j = 0 for all j and, j K, 2) where Un, k) represents the class of n k untary matrces,.e. the collecton of vectors u,..., u k ) where u C n for all, and the k-tuple u,..., u k ) s orthonormal. Hence wth precodng matrces T j, the receved sgnal for user j can be smplfed to y j = H j T j x j + v j. 3) Let H j = H T H T j H T j+ H T K] T, where ) T denotes the matrx transpose. To satsfy the constrant n 2), T j shall be n the null space of H j. Let Ñ j denote the rank of H j. Let the sngular value decomposton of H j be H j = Ũ j Λj Ṽ j Ṽ0 j], where Ṽ j contans the frst Ñ j rght sngular vectors and Ṽ 0 j contans the last N t Ñj) rght sngular vectors of H j. The columns n Ṽ 0 j form a bass set n the null space of H j, and hence T j can be any rotated verson of Ṽ 0 j. Note that T j satsfyng 2) does not always exst. The suffcent condton for the exstence of such matrces s N t K,j, as shown n 0]. In the rest of the paper, we assume that every user has the same number of receve antennas,.e. {,k } K k= =. III. VS. DC: SUM CAACITY FOR FIXED CHANNELS Consder a set of fxed channels for a multuser MIMO system, where K = {, 2,, K} denote the set of all users, and A be a subset of K. Let H j = H j T j denote the effectve channel after precodng for user j A. The total throughput acheved wth appled to the user set A wth total power can be expressed as C H A,, σ 2 ) = max {Q j : Q j 0, TrQ j ) } j A j A I + σ 2 H jq j H j 4) where Q j = Ex j x j ] s user j s nput covarance matrx of sze N j N j and TrA) denotes the trace of matrx A. Let A be the set contanng all A,.e. A = {A, A 2, }. The sum capacty of s defned as the maxmum total throughput of over all possble user sets,.e. C H,,K,, σ 2 ) A A C H A,, σ 2 ). 5) It has been proven that the sum capacty of a multuser Gaussan broadcast channel s acheved wth drty paper codng 3]. Wth the dualty results n 3], the DC sum capacty can be expressed as C D C H,,K,, σ 2 ) = max I + σ 2 {S j : S j 0, K TrS j ) } H j S j H j where S j of sze s the sgnal covarance matrx for user j n the dual multple access channel. In ths secton, we are nterested n the gan of DC over n terms of sum capacty. Anaous to 5], we defne the rato of DC to as 6) GH,,K,, σ 2 ) C D CH,,K,, σ 2 ) C H,,K,, σ 2 ). 7) The gan s obvously dependent on the channel realzatons {H k } K k=, the total power, and nose varance. In the next theorem, we gve a bound on GH,,K,, σ 2 ) that s vald for any {H k } K k=,, and σ2. Theorem : The sum capacty gan of DC over s lower bounded by and upper bounded by the mnmum of N t and K,.e. GH,,K,, σ 2 ) mn{n t, K} 8) roof: Theorem 3 n 5] states that where C D C H,,K,, σ 2 ) C T DMA H,,K,, σ 2 ) mn{n t, K} 9) C T DMA H,,K,, σ 2) = max max k {Q k :Q k 0,TrQ k ) } I + σ 2 H kq k H k 0) s the maxmum sngle user capacty among all users. The defnton of the sum capacty for ndcates that C H,,K,, σ 2 ) C T DMA H,,K,, σ 2 ). ) Further, snce DC s optmal for sum capacty, we have C D C H,,K,, σ 2 ) C H,,K,, σ 2 ). 2) Combnng 9), ), and 2) completes the proof. The above bound can be tghtened n two specal cases. Lemma : Assume N t and K Nt. If {H k } K k= are mutually orthogonal H H 2 H K,.e. H H j = 0 for j, then C D C H,,K,, σ 2 ) = C H,,K,, σ 2 ). roof: lease see appendx I. Lemma shows when the user channels are mutually orthogonal, user cooperaton s not necessary to acheve the sum capacty because all users do not nterfere wth each other. Interestngly, can also acheve the same capacty n ths case. Ths s dfferent from the TDMA scheme n 5]

3 where, even f the users are mutually orthogonal, t s not possble to acheve the same sum capacty as DC. Actually, the gan of DC over TDMA can stll be at the maxmum,.e. mn{n t, K}, when the users are mutually orthogonal. The next Lemma shows a bound on the gan of DC over when all user channels are n the same vector subspace. Lemma 2: Assume N t. If the row vector spaces of all user channels are the same,.e. spanh ) = spanh 2 ) = = spanh K ), whch s denoted as W, then GH,,K,, σ 2 ) mn{, K}. roof: lease see appendx II. IV. VS. DC: ERGODIC SUM CAACITY IN RAYLEIGH FADING CHANNELS In ths secton, we analyze the ergodc capacty of a multuser MIMO system wth block dagonalzaton vs. DC. Let H j = H j T j of sze N j ) be the effectve channel for user j after precodng. Assumng that H j are ndependent for dfferent j and the elements n H j are..d. complex Gaussan random varables, we have the followng theorem on the probablty densty functon of H j. Theorem 2: In a downlnk MIMO system wth Block Dagonalzaton appled to a fxed set of users, f the MIMO channel for each user s modeled as..d. complex Gaussan, then the effectve channel after precodng s also an..d. complex Gaussan matrx. roof : Snce H j = H j T j and H j s..d. complex Gaussan, then H j condtoned on T j s also complex Gaussan and ndependent of T j. Hence H j s ndependent of T j. Theorem 2 ndcates that f s appled to a fxed set of users, the ergodc capacty of user j can be easly evaluated wth the egenvalue dstrbuton of H j H j ] 8]. where λ 2 j,n are nth unordered egenvalues of H j H j and H j s of sze N t ) ). Inequalty a) holds because the RHS assumes all users are smultaneously transmttng for all channel realzatons. Inequalty holds because the RHS assumes equal power s allocated to every non-zero egenmodes. Equalty c) holds because λ j, has the same dstrbuton for j =, 2,,. For notatonal smplcty, we denote α = λ 2, and N = N t ). Wth Theorem 2 and ], the dstrbuton of α can be expressed as p N, α ) = where ϕ k+ α ) = B. An Upper Bound on the Ergodc Sum Capacty of DC A. A Lower Bound on Ergodc Sum Capacty wth It s well known that the sum capacty of a K-user broadcast Let A = {,, } be a subset{ of users,.e. A K, channel wth DC s upper bounded f the recevers are N for =,, I where I = mn K, t allowed to cooperate 2]3]. Let H = H }. Wth T H T 2 H T K] T, { } and N {N t, K } and M = mn{n t, KN R }, then mn K, Nt and the elements n {H} K k= are generated accordng to an..d. complex Gaussan dstrbuton, we have E C D C H,,K,, σ 2 ) ] E I + ] σ 2 HQH E C H A,, σ 2 ) ] a) E I + σ 2 H jq j H M j = E + )] m σ 2 λ2 m = ME + )] σ 2 α m= Nr = E + ] j,n ) Γ0 α σ 2 λ2 j,n M σ 2 p N,M α )dα 8) n= σ Nr E + ] 2 /Γ 0 N n= r σ 2 λ2 C coop H,,K,, σ 2 ) 9) j,n where λ 2 m s mth unordered egenvalue of H H and α = E + ] λ 2 N r σ 2 λ2 ; p N,M α ) s the dstrbuton for α, whch s gven by j, 4) wth and N replaced by M and espectvely. c) = E + ] The parameter Γ 0 s optmzed so that the ergodc sum σ 2 λ2 capacty s maxmzed wth the average power constrant,, 3).e. M ) C H A,, σ 2 σ 2 /Γ 0 Γ 0 σ2 α p N,M α)dα =. Detals on the ) nequalty 8) can be found n 7]. m= k! k + N )! for k = 0,,, m, and ϕ m α ) 2 α N e α 4) ] /2 L N Nr k α ) 5) L n m k x) = k! ex m n dk x e x dx k x n m+k). 6) Hence 3) can be evaluated wth a numercal ntegraton. Now we can lower bound the ergodc sum capacty wth by E C H,,K,, σ 2 ) ] max I C H A,, σ 2 ). 7) It s mportant to{ note that } n order to evaluate the lower bound, N up to I = mn K, t numercal ntegratons need to be carred out because of the maxmzaton n the RHS of 7).

4 Ergodc Sum Capacty bts/s/hz) DC a) SNR db) Ergodc Sum Capacty Gan: DC vs SNR db) Bounds n 20) Sum Capacty Bts/s/Hz) DC a) K Ergodc Sum Capacty Gan: DC vs Bounds n 20) K Fg.. Ergodc sum capacty of DC vs. n Raylegh fadng channels. N t = 0, = 2, K = 5. Fg. 3. Ergodc sum capacty of DC vs. n Raylegh fadng channels. N t = 0, = 2, SNR = 20 db. Ergodc Sum Capacty bts/s/hz) DC a) N t Ergodc Sum Capacty Gan: DC vs Bounds n 20) N t Fg. 2. Ergodc sum capacty of DC vs. n Raylegh fadng channels. = 2, K = 3, SNR = 20 db. C. An Upper Bound on the Ergodc Capacty of DC vs. From the above two sectons, we can upper bound the ergodc sum capacty gan of DC over as E C D C H,,K,, σ 2 ) ] E C H,,K,, σ 2 )] C cooph,,k,, σ 2 ) max C H A,, σ 2 ). 20) I Notce that the upper bound n 20) { s a functon } of N t,, K,, and σ 2. Furthermore, mn K, Nt + numercal ntegratons are necessary to evaluate the bound n 20). V. NUMERICAL RESULTS In ths secton, we provde some numercal demonstratons of the gan of DC over. Fg. a) shows the ergodc sum capacty of DC vs. under dfferent SNRs, wth N t = 0, = 2, and K = 5. In the low SNR regme, acheves almost the same sum capacty as DC because beamformng to the user wth the best channel egenvalue s asymptotcally optmal for sum capacty n low SNRs. As SNR goes to nfnty, the sum capacty of both DC and ncrease wth the same slope because both and DC explot the maxmum number of egenmodes among the users. Fg. shows the gan of DC over from the curves n Fg. a), as well as the bound on the gan evaluated from 20). As SNR ncreases, the bound from 20) gets tghter to the results from Monte Carlo smulatons. For low SNR, the bound n 20) s loose manly because ) the lower bound on assume equal power allocaton to all non-zero egenvalues; 2) the cooperatve upper bound on DC s also loose n low SNR. Fg. 2 a) shows the ergodc sum capacty of DC vs. for dfferent N t, wth = 2, K = 3, and SNR = 20 db. As the number of transmt antenna ncreases, the sum capacty of gets closer to the sum capacty of DC. Fg. 2 shows the gan of DC over from the curves n Fg. 2 a). It s observed that the bound from 20) s very tght for N t 8, wth the specfed, K, and SNR. Fg. 3 a) shows the ergodc sum capacty of DC vs. for dfferent numbers of users, wth N t = 0, = 2, and SNR = 20 db. For small numbers of users, acheves almost the same sum capacty as DC. As the number of users ncreases, DC exhbts hgher performance than. Fg. 3 show the gan of DC over from the curves on Fg. 3 a). For small numbers of users, the bound from 20) s very tght compared to the smulatons. For larger numbers of users, the bound from 20) loosens. The man reason s that the lower bound on the sum capacty of n 7) only consders users 5 and hence the effect of multuser dversty s not reflected n 7). In summary, acheves a sgnfcant part of the sum capacty for low and hgh regmes, or when N t K. The bound n 20) s tght for medan to hgh SNRs or when K Nt. AENDIX I ROOF OF LEMMA roof: Let the SVD of H be H = U Λ V 2) where U s of sze and U U = I; Λ = dag{λ,, λ,2,, λ,nr } s a dagonal matrx of sze ; and V s of sze N t and V V = I. Furthermore, H H V = V Λ 2. For j, V j V = Λ 2 j ) V j H j H j H H V Λ 2 ) = 0 because H j H = 0. Let H = H H 2 H K], then the SVD of H can be expressed as H = UΛV, where U = bdag{u, U 2,, U K } s an untary block dagonal matrx of sze K K ; Λ = bdag{λ, Λ 2,, Λ K } s a dagonal matrx of sze K K ; and V = V V 2 V K ] s of sze N t K and V V = I. The capacty of the pont-to-pont MIMO channel H can be regarded as an upper bound on the sum capacty of the broadcast channel because user cooperaton s allowed wth H. Hence C D C H,,K,, σ 2 ) C coop H,,K,, σ 2 ) 22) = + ),n σ 2 λ2,n = n= 23)

5 where,n s the power allocated to user s nth egenmode and,n s obtaned by the water-fllng algorthm wth total power constrant,n =. = n= On the other hand, snce V j V = 0 for j, we have H j V = 0 for j. Thus we can set T j = V j to satsfy the null constrant n 2). Notce the effectve channel H j = H j V j has the same sngular values as H j. Hence C H,,K,, σ 2 ) max = {Q j : Q j 0, TrQ j ) } j K I + σ 2 H jq j H j 24) j K = n= + ),n σ 2 λ2,n 25). Wth 23), 25), and the fact that C D C C, we have C D C = C as the condtons n Lemma 3 are satsfed. AENDIX II ROOF OF LEMMA 2 roof: Let E = e e 2 e ] be a bass n W, whch s the row vector space spanned by {H } K k=, where e s of sze N t. Hence EE = I. Let the SVD of H be H = U Λ V. There exsts a untary matrx R of sze such that V = R E. Then H = U Λ R E. Denote H W) = U Λ R, t s easy to see that H W) has the same sngular values of H. Hence C D C H,,K,, σ 2 ) {S j: S K j 0, TrS j) } {S j : S j 0, K TrS j ) } {S j : S j 0, K TrS j ) } I + σ 2 H j S j H j I + I + σ 2 E H W) σ 2 H W) ) S j H W) ) S j H W) E = C D C H W),,K,, σ2 ). 26) Snce the sze of H W) for =, 2,, K) s, anaous to Theorem n 5], we can obtan C D C H,,K,, σ 2 ) = C D C H W),,K,, σ2 + ) σ 2 λ2 max 27) where λ max λ,n where λ,n s the th user s K, n nth sngular value. On the other hand, f spanh ) = spanh 2 ) = = spanh K ) and only one user s supported wth, we have C H,,K,, σ 2 ) = C T DMA H,,K,, σ 2 )28) + ) σ 2 λ2 max 29) Then we can mmedately obtan by Theorem 3 n 5]. GH,,K,, σ 2 ) mn{, K} 30) REFERENCES ] I. E. Telatar, Capacty of Mult-Antenna Gaussan channels, European Trans. on Telecommuncatons, vol. 0, no. 6, pp , Nov./Dec ] G. Care and S. Shama, On the Achevable Throughput of a Multantenna Gaussan Broadcast Channel, IEEE Trans. on Informaton Theory, vol. 49, No. 7, pp , Jul ] S. Vshwanath, N. Jndal, and A. Goldsmth, Dualty, Achevable Rates, and Sum-Rate Capacty of Gaussan MIMO Broadcast Channels, IEEE Trans. on Informaton Theory, vol. 49, No. 0, pp , Oct ] H. Wengarten, Y. Stenberg, and S. Shama Shtz), The Capacty Regon of the Gaussan MIMO Broadcast channel, n roc. IEEE Int. Symposum on Informaton Theory, pp. 74, Jun ] M. Costa, Wrtng on Drty aper, IEEE Trans. on Informaton Theory, vol. 29, no. 3, pp , May ]. Vswanath and D. N. C. Tse, Sum Capacty of the Vector Gaussan Broadcast Channel and Uplnk-Downlnk Dualty, IEEE Trans. on Informaton Theory, vol. 49, No. 8, pp Aug ] W. Yu and J. M. Coff, Sum Capacty of Gaussan Vector Broadcast Channels, IEEE Trans. on Informaton Theory, vol. 50, no. 9, pp , Sep ] W. Yu, W. Rhee, S. Boyd, and J. M. Coff, Iteratve Water-Fllng for Gaussan Vector Multple-Access Channels, IEEE Trans. on Informaton Theory, vol. 50, no., pp , Jan ] N. Jndal, W. Rhee, S. Vshwanath, S. A. Jafar, and A. Goldsmth, Sum ower Iteratve Water-fllng for Mult-Antenna Gaussan Broadcast Channels, IEEE Trans. on Informaton Theory, vol. 5, no. 4, pp , Apr ] Q. H. Spencer, A. L. Swndlehurst, and M. Haardt, Zero-Forcng Methods for Downlnk Spatal Multplexng n Multuser MIMO Channels, IEEE Trans. on Sgnal rocessng, vol. 52, no. 2, pp , Feb ] L. U. Cho and R. D. Murch, A Transmt reprocessng Technque for Multuser MIMO Systems Usng a Decomposton Approach, IEEE Trans. on Wreless Communcatons, vol. 3, no., pp , Jan ] K. K. Wong, R. D. Murch, K. B. Letaef, A Jont-Channel Dagonalzaton for Multuser MIMO Antenna Systems, IEEE Trans. on Wreless Communcatons, vol. 2, no. 4, pp , Jul ] R. Chen, J. G. Andrews, and R. W. Heath. Jr., Multuser Space-Tme Block Coded MIMO System wth Downlnk recodng n roc. IEEE Int. Conf. on Communcatons, vol. 5, pp , Jun ] R. Chen, J. G. Andrews, and R. W. Heath. Jr., Transmt Selecton Dversty for Multuser Spatal Multplexng Systems, n roc. IEEE Global Communcatons Conf., vol. 4, pp , Dec ] N. Jndal and A. Goldsmth, Drty aper Codng vs. TDMA for MIMO Broadcast Channels, IEEE Trans. on Informaton Theory, vol. 5, no. 5, pp , May ] T. Yoo and A. J. Goldsmth, Optmalty of Zero-Forcng Beamformng wth Multuser Dversty, n IEEE Int. Conf. on Communcatons, vol., pp , May ] Z. Shen, R. W. Heath. Jr., J. G. Andrews, and B. L. Evans, Comparson of Space-Tme Water-fllng and Spatal Water-fllng for MIMO Fadng Channels, n roc. IEEE Global Communcatons Conf., vol., pp , Dec ] R. J. Murhead, Aspects of Multvarate Statstcal Theory, John Wley & Sons, Inc ] A. Edelman, Egenvalue and Condton Numbers of Random Matrces, h.d. thess, MIT, May ] T. M. Cover and J. A. Thomas, Elements of Informaton Theory, John Wley & Sons, Inc. 99.