Joint Transmitter-Receiver Design for the Downlink Multiuser Spatial Multiplexing MIMO System

Joint T ransmitter -Recei v er Design for the Do wnlink Multiuser Spatial Multiple xing MIMO System Pengfei Ma, W enbo W ang, Xiaochuan Zhao and Kan Zheng W ireless Signal Proce ssing and Network Lab, Ke y Laborato ry of Universal W irele ss Commu nication Ministry of Ed ucation, Beijing University of Posts an d T elecommu nications, Beijing, China Abstract —In the multiuser spatial multiplexing multiple-in put multiple-outp ut (MIMO) system, the join t transmitter-r eceiver (Tx-Rx) design is i n vestigated to minimize th e weighted su m power under the post-processing signal-to-in terference-and-noise ratio (post-SINR) c onstraints f or all subchannels. Firstly , we show that the u plink- downlink duality is equivalent to th e Lagrangian duality in the optimization problems. Then, an iterati ve algo rithm fo r th e j oint Tx-Rx design is pro posed according to th e above result. S imulation results sh ow that the algorithm can not only satisfy the post-SINR constraints, but a lso easily adjust the power distribution among the u sers b y ch anging the weights accordingly . So th at the transmitti ng power to the edge users in a cell can be decreased effectively to alleviate t he adjacent cell interference wi thout performance penalty . Index T erms —spatial mu ltiplexing, M IMO, p ower allocation, Lagrangian duality . I . I N T RO D U C T I O N Spatial mu ltiplexing for the mu ltiple-inpu t mu ltiple-outp ut (MIMO) systems, employing multip le transmit and receive antennas, ha s been recogn ized as an effecti ve way to improve the spectral e fficiency o f the wirele ss link [1]. Mo re recently , the multiuser schem es have been in vestigated for the spatial multiplexing MIMO systems. This pape r focuses on the down- link multiuser sche mes in which each user can not c oopera te with th e others th us suffers from the interf erence from th em. Mainly , there are two kinds of multiuser schemes. One is the precod er o r the tr ansmit b eamform ing, such as the dirty-p aper coding ( DPC) [2] an d th e zero-forcing (ZF) [3], e tc., which mitigates the m ultiuser interfere nce on ly by pr ocessing at th e transmitter . The other is the join t transmitter-receiver (Tx -Rx) design, such as the nullspace- directed SVD (Nu- SVD) [4] an d the minimum total mean squared error (TMMSE) [5], etc. In general, the for mer possesses lower com plexity but more perfor mance penalty . With the great development of sign al processors, the latter g radually dr aws m ore attentio n. For the joint Tx-Rx de sign, the schemes p ropo sed in [4][5] minimize mean squ ared error (MM SE), or maximize the cap acity under the tra nsmit p ower constraint. Whereas on some occasions, such as th e multimed ia communication , it is req uired to minimize the total transmit power wh ile guaran tee the qua lity of service (QoS). [6 ][7] investigate the beamfor ming an d the power alloc ation po licy wh en all users are subjected to a set of post-pro cessing signal-to-interf erence- and-no ise ratio (po st-SINR) constrain s in the up link SIMO and the downlink MISO. [8][9] extend this work to the downlink MIMO and the MIMO network , howev e r the MIM O systems discussed in [8][ 9] are assumed that there is on ly one sub stream betwee n each pair of the transm itter and re- ceiv er . In other words, only the mu ltiuser in terferen ce ap pears in the so-called diversity MIMO system in [8][9]. For the multiuser spatial multip lexing MIMO system, however , bo th the multiuser in terferen ce between in dividual users an d self- interferen ce be tween individual sub streams of a user should be mitigated . For the downlink, the transmit beamform ing affects the interferen ce signatur e of all re ceiv ers, wh ereas the receive beamfor ming only affects th at of th e correspon ding user . [7][8] construct a du al system, called the virtual u plink, a nd in dicate that th e vir tual u plink can obtain the same post-SINR a s the primary downlink. Moreover, the receive beamfo rming matrix of the virtual uplink is iden tical with the transmit beamforming matrix of the pr imary downlink. The design of th e downlink, therefor e, can resor t to the virtu al uplink . In this pap er , w e extend the d uality der iv e d for M IMO network in [9] to th e multiuser spatial multiplexing MIMO system. Accord ing to the up link-downlink duality , we pro pose a joint Tx- Rx schem e to minimize the weig hted su m power under the post-SINR constraints of all the subch annels. Notation : Boldface u pper-case letters denote matrices, and boldface lower-case letters deno te colum n vectors. tr ( · ) , ( · ) ∗ , ( · ) H , || · || 2 and || · || F denote trace, conju gate, con jugate tra ns- position, Euclidian n orm and Frobenius n orm, respectively . diag ( x ) de notes a diag onal matrix with diagon al elements drawn f rom the vector x . [ · ] i,j , [ · ] : ,j denote the ( i , j ) -th element and j - th colum n of a matrix, respectively . I I . S Y S T E M M O D E L W e co nsider a base station (BS) with M anten nas and K m obile stations (MS’ s) each h aving N i ( i = 1 , . . . , K ) antennas. T here are L i ( i = 1 , . . . , K ) substreams between BS and M S i ( i = 1 , . . . , K ) , that is to say , BS transmits L i symbols to MS i simultaneou sly . Th e signal recovered by M S k can be w ritten as y DL k = A H k H k K X i =1 B i diag ( √ p i ) x i + A H k n k (1) where y DL k ∈ C L k × 1 is th e recovered signal vector . x i ∈ C L i × 1 ( i = 1 , . . . , K ) is th e transmitted signal vector from BS to MS i with zero-m ean and norma lized covariance matrix I . p i ∈ R L i × 1 denotes the power vector allocated to MS i . A lin- ear p ost-filter A k ∈ C N k × L k is used to recover an estimation of the transmitted signal vector x k . T he MI MO c hannel from BS to MS k is den oted as H k ∈ C N k × M , an d assumed flat faded. Hence, its eleme nts are th e com plex chann el gain s, and they are ind epende ntly identically distrib uted (i.i.d.) zero-mean complex Gaussian rand om variables with the unity variance. Moreover , the p erfect chann el state information are assumed av ailable at both transmitter and receiver via some way , for example, chan nel measuremen t at rec eiv er and fast fe edback to the tran smitter for the fr equency di vision du plex (FDD) systems, or inv o king the chann el rec iprocity in time division duplex (TDD) systems. B i ∈ C M × L i is used to weig ht x i and transform it into a M × 1 vector . n k ∈ C N k × 1 is the noise vector with the c orrelation matrix R n = σ 2 n I . For simplicity , in th e sequel we assume L 1 = . . . = L K = L . W e design the A k , B k and p k ( k = 1 , . . . , K ) in (1) to minimize th e weigh ted sum power und er the post-SINR con- straints, which can be de noted as the f ollowing optimization problem . min p , A k , B k w T p s.t. S I N R DL k,j ≥ γ k,j ( k = 1 , . . . , K, j = 1 , . . . , L ) (2) where p = [ p T 1 , . . . , p T K ] T and w ∈ R K L × 1 is th e weight vector . w affects the power distribution am ong users, an d its value is determ ined b y various factors, su ch as the position s of users in a cell an d the inter ference en vironme nt of the neighbo ring cells. γ k,j is the gi ven po st-SINR goal for the MS k ’ s j -th substream. I I I . T H E P R O O F O F U P L I N K - D O W N L I N K D UA L I T Y If A k = [ a k, 1 , . . ., a k,L ] , B k = [ b k, 1 , . . ., b k,L ] , p k = [ p k, 1 , . . ., p k,L ] T , (1 ) can be r ewritten into y DL k =    a H k, 1 H k b k, 1 √ p k, 1 . . . a H k, 1 H k b k,L √ p k,L . . . . . . . . . a H k,L H k b k, 1 √ p k, 1 . . . a H k,L H k b k,L √ p k,L    x k + A H k H k K X i =1 ,i 6 = k B i diag ( √ p i ) x i + A H k n k (3) The diag onal elements of the first par t in th e right- hand side (RHS) of (3 ) d enote th e useful signals, an d the non -diago nal elements d enote th e self-interferen ce. T he med ial and the last parts in th e RHS of (3 ) deno te the multiuser interfer ence a nd the no ise, respectively . Moreover, the post-SINR of the MS k ’ s j -th sub stream can be den ote as S I N R DL k,j = a H k,j R s,DL k,j a k,j a H k,j R I + n,D L k,j a k,j R s,DL k,j = p k,j H k b k,j b H k,j H H k R I + n,D L k,j = L X i =1 ,i 6 = j p k,i H k b k,i b H k,i H H k + K X m =1 ,m 6 = k H k B m diag ( p m ) B H m H H k + σ 2 n I (4) If x m = [ x m, 1 , . . ., x m,L ] , y DL k = [ y k, 1 , . . ., y k,L ] , the link power gain between x m,n and y k,j can be den oted as [ φ k,j ] m,n = || a H k,j H k b m,n || 2 2 (5) then (4 ) can b e rewritten into S I N R DL k,j = p k,j [ φ k,j ] k,j L P i =1 ,i 6 = j p k,i [ φ k,j ] k,i + K P m =1 ,m 6 = k L P n =1 p m,n [ φ k,j ] m,n + σ 2 n || a k,j || 2 2 (6) By sub stituting (6 ) into the co nstraint ine quality of (2), we obtain c T k,j p + σ 2 n || a k,j || 2 2 ≤ 0 ( k = 1 , . . . , K, j = 1 , . . . , L ) (7) where the m -th elem ent of c k,j ∈ R K L × 1 is [ c k,j ] m = ( − [ φ k,j ] k,j γ k,j m = ( k − 1) L + j [ φ k,j ] ⌈ m L ⌉ ,m − ( ⌈ m L ⌉ − 1) L m 6 = ( k − 1) L + j (8) where  m L  round s m L to the nearest integer greater than o r equal to m L . Write (7) into the matrix form , we ob tain Cp + d ≤ 0 (9) where C ∈ R K L × K L and d ∈ R K L × 1 are C = [ c 1 , 1 , . . ., c 1 ,L , . . ., c K, 1 , . . ., c K,L ] T d = σ 2 n  || a 1 , 1 || 2 2 , .., || a 1 ,L || 2 2 , .., || a K, 1 || 2 2 , .., || a K,L || 2 2  T (10) So, (2) is equiv alent to the f ollowing op timization pro blem min p k , A k , B k w T p s.t. Cp + d ≤ 0 , p ≥ 0 (11) Subsequen tly , to obtain the Lagrang ian duality of ( 11) [9], we divide the so lving pro cess of (1 1) into two steps similar with [10]. First, a ssuming A k and B k ( k = 1 , . . . , K ) are fixed, the Lagrang ian fu nction of (11) is L ( p , λ , µ ) = w T p + λ T ( Cp + d ) − µ T p (12 ) where λ ≥ 0 , µ ≥ 0 are th e Lagra ngian multiplier s associated with the ineq uality con straints. Then the Lagrang ian duality o f (11) is max λ,µ min p L ( p , λ , µ ) s.t. λ ≥ 0 , µ ≥ 0 (13) According to the Slater’ s condition , (11) is equiv a lent to (1 3). Since the gradie nt of the Lagrangian function (12) with respect to p vanishs at op timal points, we obtain w T − µ T = − λ T C . Substituting it into (12), we obtain min p L ( p , λ , µ ) = d T λ . Moreover , as λ ≥ 0 and µ ≥ 0 , (13) can be rewritten to max λ d T λ s.t. C T λ + w ≥ 0 λ ≥ 0 (14) Similar with (6)-( 9), substitute (1 0) into ( 14), we obtain max λ d T λ s.t. S I N R U L k,j ≤ γ k,j ( k = 1 , . . . , K, j = 1 , . . . , L ) (15) where S I N R U L k,j = b H k,j R s,U L k,j b k,j b H k,j R I + n,U L k,j b k,j R s,U L k,j = λ k,j H k a k,j a H k,j H H k R I + n,U L k,j = L X i =1 ,i 6 = j λ k,i H k a k,i a H k,i H H k + K X m =1 ,m 6 = k H m A m diag ( λ m ) A H m H H m + [ w ] ( k − 1) L + j I (16) where λ = [ λ T 1 , . . . , λ T K ] . Furth ermore , S I N R U L k,j is the post- SINR of MS k ’ s j -th sub stream in th e vir tual uplin k y U L k = B H k K X i =1 H i A i diag ( p λ i ) x i + B H k √ w k (17) (14) max imizes the weigh ted sum power und er the ma ximum post-SINR con straints, howe ver, it has no physical meanin g [9]. But it can be sh own that (14) is equi valent to the following optimization pr oblem min λ d T λ s.t. S I N R U L k,j ≥ γ k,j ( k = 1 , . . . , K, j = 1 , . . . , L ) (18) Theor em 1 : At the op timal point, the post-SINR con straints in (15) and (18) are activ e. And the solu tions of (15) and (18) are identica l. Pr oof : W ith out any loss of the gener ality , we assum e S I N R U L k,j < γ k,j . From (1 6), we can find λ k,j contribute to th e n umerato r o f S I N R U L k,j and th e d enomina tor of S I N R U L m,n ( m 6 = k , n 6 = j ) . In oth er words, S I N R U L k,j is a monoto ne inc reasing functio n of λ k,j , wh ile S I N R U L m,n ( m 6 = k , n 6 = j ) is a mono tone decreasing functio n of λ k,j . So increasing λ k,j until S I N R U L k,j = γ k,j , we obtain a larger d T λ without breakin g any po st-SINR constrain t. Likewise, if S I N R U L k,j > γ k,j , decr easing λ k,j until S I N R U L k,j = γ k,j , a smaller d T λ is obtained. As a result, the co nstraints o f ( 15) and (18) beco me a linea r equ ations C T λ + w = 0 , and its solution is λ ∗ = − ( C T ) − 1 w . Similar with Theorem 1 , at the op timal poin t of (1 1) p ∗ = − C − 1 d . Summarize the ab ove statement, we obtain the following conclusion . Theor em 2 : In th e d ownlink multiuser spa tial mu ltiplexing MIMO system, if the tr ansmit and receive b eamfor ming matri- ces are B k and A H k ( k = 1 , . . . , K ) , respecti vely , a s long as the following con ditions are satisfied, th e downlink optimization problem (2) is eq uiv alen t to the virtual uplink op timization problem (1 8). 1) I n the virtual uplink, the transmit and receive b eamform - ing matric es are A k and B H k ( k = 1 , . . . , K ) , respectively . 2) In the virtu al up link pr oblem ( 18), the we ight vector w is the noise p ower vector . 3) In the v irtual up link prob lem (18), the noise p ower v ector d is the weight vector . When A k and B k ( k = 1 , . . . , K ) ar e no t fixed, (18) is a joint optim ization prob lem denoted as min λ, A k , B k d T λ s.t. S I N R U L k,j ≥ γ k,j ( k = 1 , . . . , K, j = 1 , . . . , L ) (19) Theor em 3 : I f th e no ise power vector in the virtual up link is th e weigh t vector w , the joint o ptimization pro blem (2) is equivalent to (1 9). At the optima l point, the beam formin g matrices of th e virtu al uplink and the primal downlink are common . Pr oof : Let ( B ∗ k , A ∗ k , p ∗ ) ( k = 1 , . . . , K ) be the g lobal minimum of (2). Accor ding to Th eor em 2 , ( 19) h as the solution ( A ∗ k , B ∗ k , λ ∗ ) k = 1 , . . . , K . Mo reover , this solutio n is defin itely the global min imum. Other wise, a b etter solu tion of (2) would b e fo und by ap plying Theo r em 2 again. So the v irtual up link and the primal downlink have th e co mmon beamfor ming matr ices. The weight vector w decides whe ther A k and B k are used to streng then the useful signals or alleviate the interf erence to other users. Whe n a user ’ s weight tur ns high er , its transmit power will d ecrease. I n this occasio n, it benefits to ap ply the beamfor ming to incr ease the signal ga in, a s the inter ference to other users is much less im portant. On the other hand, once the weight gets lower , th e beamf ormer sho uld tr y to sup press interferen ce to other s [9]. T o m itigate the adjacent cell in terferen ce, we can increa se the weights of edge u sers in a ce ll, which w ould induce the declining o f th e tran smit power f rom the BS to them. In order to ho ld the po st-SINR und er th is circ umstance, obviously , th e beamfor ming matrices would b e used to boost up th e signa l gain. I V . T H E J O I N T T X - R X B E A M F O R M I N G S C H E M E It is rather difficult to solve the joint optimization pro blem (2) directly . Howe ver , it is easy to o btain A k ( k = 1 , . . . , K ) in the pr imal d ownlink, and so d oes B k ( k = 1 , . . . , K ) in th e virtual up link. Mo reover , it is p roved in the previous section that the prima l downlink is eq uiv alen t to the virtual uplink, and they have the com mon beamf orming matrices A k and Fix p and B k Fix A k and B k Fix A k and B k Fix λ and A k ✲ ❄ ✛ ✻ Optimize A k Optimize λ Optimize B k Optimize p Fig. 1. Block diagram of the joint Tx-Rx beamforming scheme B k ( k = 1 , . . . , K ) . Ther efore, we d ivide the solving proce ss into fou r steps shown as Fig. 1. When p and B k ( k = 1 , . . . , K ) are fixed, optimize A k ( k = 1 , . . . , K ) to maximize S I N R DL k,j ( k = 1 , . . . , K, j = 1 , . . . , L ) . Then ob serving (4), it is a generalized Rayleigh quotient pro blem, and its solutio n is a k,j = ˜ a k,j / || ˜ a k,j || 2 ˜ a k,j = ξ max ( R s,DL k,j , R I + n,D L k,j ) (20) where ξ max ( X , Y ) is the do minant gen eralized e igenv ector of the matr ix p air ( X , Y ) . When λ an d A k ( k = 1 , . . . , K ) are fixed, in the sam e way , B k ( k = 1 , . . . , K ) ca n be obtain ed by b k,j = ˜ b k,j / || ˜ b k,j || 2 ˜ b k,j = ξ max ( R s,U L k,j , R I + n,U L k,j ) (21) The pro posed algo rithm is summ arized in the fo llowing. Initialize B (0) k ( k = 1 , . . . , K ) and p (0) randomly. Set the noise vector of the virtual uplink to w . n = 0 1) Update in the primal downlink . a) Calculate A ( n +1) k ( k = 1 , . . . , K ) from B ( n ) k ( k = 1 , . . . , K ) and p ( n ) using (4)(20). b) Calculate C ( n ) from B ( n ) k ( k = 1 , . . . , K ) and A ( n +1) k ( k = 1 , . . . , K ) using (5)(8)(10 ). c) Solve λ ( n ) = − (( C ( n ) ) T ) − 1 w 2) Update in the virtual up link. a) Calculate B ( n +1) k ( k = 1 , . . . , K ) from A ( n +1) k ( k = 1 , . . . , K ) and λ ( n ) using (16)(21). b) Calculate C ( n +1) from B ( n +1) k ( k = 1 , . . . , K ) and A ( n +1) k ( k = 1 , . . . , K ) using (5)(8)(10 ). c) Solve p ( n +1) = − ( C ( n +1) ) − 1 d n = n + 1 3) Repeat 1) and 2) u ntil K P k =1 || A ( n ) k − A ( n +1) k || F + K P k =1 || B ( n ) k − B ( n +1) k || F ≤ ε . In the simulation, we set ε = 0 . 0001 . By iteration, ( A ( n +1) k , B ( n +1) k , p ( n +1) )( k = 1 , . . . , K ) co n- verges to the optimal solution to th e optimizatio n pro blem (2). Once any element in p ( n +1) is n egativ e, which in dicates the post-SINR g oals γ k,j ( k = 1 , . . . , K, j = 1 , . . . , L ) can - 4 - 2 0 2 4 6 8 1 0 1 2 1 4 1 6 - 5 0 5 1 0 1 5 2 0 2 5 3 0 3 5 W e i g h t e d su m p o w e r ( d B ) [ dB ] K = 4 K = 3 K = 2 Fig. 2. T otal transmit po wer versus SINR goal γ , when K = 2 , 3 , 4 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 - 5 0 5 1 0 1 5 2 0 2 5 3 0 w e i g h t e d su m p o w e r ( d B ) ( dB ) K = 8 K = 7 K = 6 K = 5 Fig. 3. T otal transmit po wer versus SINR goal γ , when K = 5 , 6 , 7 , 8 . not be attained , γ k,j should be d ecreased to r elax the po st- SINR con straints. When w = 1 , the proposed algorithm is similar with the one in [11]. V . S I M U L A T I O N R E S U LT In this section, we assume that a BS with 8 antenn as ( M = 8 ) is comm unicating with K M S’ s each with 2 antennas, ( N 1 = . . . = N K = 2 ) . Also we assume th at the numb er o f sub streams of each MS is the same, equal to 2 , ( L 1 = . . . = L K = 2) . QPSK is employe d in th e simulatio n and no forward error coding is considered. The p ost-SINR goals f or all substreams ar e γ ( γ k,j = γ ) . Addition ally we assume MS 1 is an edge user in a cell, accordin g to the pre vio us section, a high er weight shou ld b e assigned to it to mitigate the ad jacent cell interferenc e. Th us, the weight vector is set to w = [ w , w, 1 , . . . , 1] T , where w is the we ight c orrespo nding to the two substre ams of MS 1 and w > 1 . Fig. 2 ,3 plot the curves of the total transmit power P k,j p k,j versus the p ost-SINR g oal γ , when w = 5 . I n Fig. 2, K = 2 , 3 , 4 , the system config uration satisfies M ≥ K L , 0 5 1 0 1 5 2 0 6 8 1 0 1 2 1 4 1 6 1 8 1 8 2 0 2 2 2 4 2 6 2 8 3 0 t h e t r a n sm i t p o w e r o f u se r 1 ( d B ) w t h e t r a n s m i t p o w e r o f u s e r 1 t o t a l t r a n s m i t p o w e r t o t a l t r a n sm i t p o w e r ( d B ) Fig. 4. When K = 4 , γ = 10 dB, the transmit powe r of MS 1 and total po wer versus the weight w . the multiuser in terferen ce, thus, can be effecti vely sup pressed throug h the beamfor ming [4]. Under the circ umstance, i ncreas- ing the transmit power of any user has nearly no effect to the p ost-SINR of othe r u sers. Th erefor e, all the substream s can attain r elativ ely high po st-SINR. In Fig. 3, K = 5 , 6 , 7 , 8 , M ≥ K L does not h old any more. Consequently , th e m ultiuser interferen ce can not be effecti vely mitigated, wh ich mean s any enhancem ent in the transmit power of a ny user is very likely to deterio rate the post-SINR of other users. As shown in Fig . 3, with the user number increa sing, the av ailable post-SINR of each user is dec reased. When K = 8 , o nly 0 dB post- SINR can be attained . In these two figur es, the total transmit power increases with the numb er of users and the p ost-SINR goal γ . Especially when K = 7 , 8 and γ ≥ 0 dB in Fig. 3, du e to the residual m ultiuser in terferenc e, the slopes of the curves are much steeper than that in Fig . 2 where the multiuser interferen ce is negligible. And the steeper the cur ves are, the more power would b e p aid f or the u nit increase o f the post- SINR of each user . Fig. 4 shows the cu rves of the tran smit power P j p 1 ,j of MS 1 and the total tr ansmit power P k,j p k,j versus th e weight w , when K = 4 and γ = 1 0 dB. The left vertical ax is is correspo nding to the tra nsmit power of MS 1 and the right on e is to the total tran smit po wer . Obviously , as the w is increasing, the transmit power of MS 1 is decreasin g wh ile the total power is in creasing, b ecause the o ptimization ob ject is to minimize the weig hted sum power P k,j w k,j p k,j . Mo reover , when w changin g from 1 to 20 , th e tran smit power of MS 1 decreases almost 1 0 dB, h owe ver the to tal power incr eases on ly abo ut 1 dB, whic h demonstrates that the p roposed algorith m ada pts the power allocation policy very ef f ectiv ely with negligible penalty on per forman ce. V I . C O N C L U S I O N In this pap er , we in vestigate the join t Tx-Rx design for the downlink multiuser spatial m ultiplexing MIMO system. W e show , first, the uplin k-downlink duality has the following characteristics: 1) In both of the prim al downlink an d the virtual uplink , th e substreams can attain the sam e post-SINR goal; 2) The b eamfor ming matr ices are commo n in both of the prima l downlink and the virtual u plink. Based on the du ality , a joint Tx-Rx bea mform ing scheme is propo sed. Simu lation r esults demo nstrate that the scheme can not only satisfy the po st-SINR constraints which g uarantee the pe rforma nce of the comm unication link s, but also easily adjust the power distribution amon g users by changing the weights co rrespon dingly , which can be used to d iminish the power of th e edge users in a ce ll to alleviate the adjacent ce ll interferen ce. R E F E R E N C E S [1] I. T elatar , ”Capacity of m ulti-an tenna G aussian channels” , Eur . T rans. T elecommun , vol. 10, no. 6, pp.585-595, Nov . /Dec. 1999. [2] Q. Caire and S. Shamai, ”On the achie vabl e throughp ut of a multiant enna gaussian broadc ast channel”, IEEE T rans. Inform. theory , vol. 49, no. 7, pp.1691-1706, July 2003. [3] Q. Spencer , A. Swindlehurst and M. Haardt, ”Zero-forcing methods for do wnlink spa tial m ultiple xing in mult iuser mimo channels”, IEEE T rans. Signal Pr ocessing , vol . 52, no. 2, pp.461- 471, Feb. 2004. [4] Z.G. Pan,K.K. W ong and T . S. 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