A General Rate Duality of the MIMO Multiple Access Channel and the MIMO Broadcast Channel

A General Rate Dualit y of the MIMO Multiple Access Channel and the MIMO Broa dcast Channel Raphae l Hunger and Michael Joham Associate Institute for Signal Processing, T echnische Universit ¨ at M ¨ unchen, 80290 Mu nich, Germa ny T elephon e: +49 8 9 28 9-28 508, Fax: +49 8 9 28 9-285 04, Email: hunger@t um.de Abstract — W e present a general rate d uality between th e multiple access channel (M A C) and the broadcast channel (BC) which is applicable t o systems wi th and without nonlinear interference cancellation. Different to the state-of-the-art rate duality with int erference subtraction from V ishwanath et al., the proposed duality is fil ter -based i nstead of cov ariance-based and exploits the arising unitary degree of fr eedom to decorrelate ev ery poin t-to-point link. Th eref or e, it allows f or noncooperativ e stream-wise d ecoding which reduces complexity and latency . Moreo ver , the co n version from one domain to the other does not exhib it any d ependencies d uring its computation makin g it accessible to a parallel implementation instead of a serial one. W e addi tionally derive a rate du ality for systems with multi- antenna ter minals when linear fi ltering w ithout interference (pre- )subtraction is applied and the different streams of a single user are n ot treated as self-interference. Both dualities are based on a framework alre ady app lied to a mean-square-erro r d uality between the M A C and the BC. Thanks t o this nove l rate duality , any rate-based optimization with linear fi ltering in t he BC can now be handled in the dual MA C wh ere t he arising expressions lead to more efficient algorithmic solutions than in the BC due to the alignment of the channel and precoder indices. I . I N T R O D U C T I O N In the p ast few y ears, dualities were successfully employed as the linking element between the multiple a ccess channel (MA C) and the broadcast chan nel (BC). Thank s to various versions of dualities, ma ny region s of the MAC and the BC were classified to b e identical under a sum-power con straint. First, the signal-to -interfer ence-an d-noise-ratio (SINR) regions under sing le-stream transmission per user were shown to be identical in [ 1], [2]. Second, the mean-square- error (MSE) regions of the MA C and the BC coincide which has be en proven by m eans of th e SINR d uality in [3] an d later in [4] o r directly in [ 5], [6]. An d th ird, th e rate regions of the MA C a nd th e BC under Gau ssian signaling and nonlinear interferen ce cancellatio n have recently been shown to be the same, see [7] fo r the single-a ntenna case, [8] for the multi- antenna case, an d [9] for th e coincid ence o f the dirty- paper coding rate region and th e c apacity r egion. A stream-wise duality with power constraints on su bsets of an tennas which holds for the optimu m filter s of a quality-o f-service power minimization was presented in [10] fo r systems with and without n onlinear inter ference cancellation . Du e to its stream- wise n ature, conversion fro m one domain to the d ual is com- plicated since it is not clear how to allocate the SINRs to the users in case of multi-an tenna terminals. B esides the capa bility of proving co ngrue ncy of two regions, d ualities also deliver explicit conv ersion for mulas how to switch from on e domain to the other . In case o f th e rate d uality in [ 8], (ar bitrary) optimu m receive filter s generating suf ficient statistics are assum ed both in the MAC and in the BC. Given transmit cov ariance matrices in the MAC are converted to transmit covariance matrices in th e d ual BC. Dep endenc ies during these tran sformation s prevent a parallel proc essing and force a serial implementation. In addition, the r eceiv ed data streams have to be d ecoded jointly wh ich en tails a high computatio nal complexity . Our co ntribution in this p aper is twofold. First, we p resent a novel rate duality for systems with non linear interferenc e cancellation. One of the key steps in volved is th e change fro m the cov ariance matrices to the t ransmit filters by which we ga in an isometry as degree of f reedom. This degree o f freedo m is then used to d ecorrelate e very p oint-to- point link thus making a fast parallel stream-wise d ecoding p ossible. As the stream s of a sing le user now do n ot interfer e with each other, we can employ an SINR duality in th e style o f ou r MSE dua lity in [5], [6]. Th erein, the tran smit filters in the dual dom ain are scaled rece iv ers of the pr imal domain and the rec eiv e filters are scaled transmitters of the pr imal domain. W e en d up with a system of linear eq uations to determ ine these scaling factor s. Our second contr ibution is a ra te duality for linear filterin g applicable to mu lti-antenna ter minals whe re different streams of a user a re not tr eated as self-in terferenc e. Up to now , such a duality did not exist and h itherto existing dualities for linear filtering treat dif ferent streams of a user as virtual users contributing interfer ence to the user und er conside ration, see [1], [2], [1 1] for example. In general, the maximum po ssible rate canno t b e obtaine d whe n a duality based on virtual users is applied. Th e und erlying framework for the p roposed linear duality is similar to the pr oposed nonlinear duality presented in the following. Key observation is again th e fact that deco rrelation allows for a stream-wise decoding which also a chieves the rate that is p ossible un der joint deco ding. I I . S Y S T E M M O D E L T wo sy stems are considered , n amely the MAC wh ere K multi-anten na users send their data to a co mmon base station which is equipped with N antenna s, an d the BC where the signal flow is rev ersed, i.e ., the base station serves the users. In the f ormer case the tran smission between the k th user an d the base station is d escribed by the ch annel matrix H k ∈ C N × r k with r k denoting the number of transmit antennas at u ser k . The BC link, howe ver , is characterized by th e Hermitian channel m atrix H H k . User k multip lexes L k data streams. I f in terferenc e c ancellation is applied in th e MAC, we assume for the sake of rea dability that the decod ing order is chosen such that user 1 is decode d last, whereas the re versed encodin g o rder is c hosen in the BC, i.e., user 1 is precod ed first. For different sortings, the user s hav e to be relab eled correspo ndingly . Un der these assumptions, the rate of user k in the MA C with non linear interference cancellation reads as [12] R MAC k = log 2   σ 2 η I N + P ℓ ≤ k H ℓ Q ℓ H H ℓ     σ 2 η I N + P ℓk S ℓ H k   , (2) where S ℓ ∈ C N × N is the BC tran smit covariance matrix o f user ℓ . I f only linea r filtering withou t interferen ce sub traction is applied , user k experien ces interference from all othe r users. I I I . R A T E D U A L I T Y F O R S Y S T E M S U T I L I Z I N G I N T E R F E R E N C E S U B T R AC T I O N A. Ben efits of th e Ra te Duality wit h Interfer en ce Cancellation Besides the ability to show congrue ncy between the two capacity r egions, the decisive reason for u tilizing the r ate duality is that all rate expr essions are concave fun ctions of the transmit covariance matr ices in the MA C but not in the BC. Moreover , the optimal sorting of the users can easily be obtain ed in the MAC. As a consequen ce, many rate- based maxim izations can be solved with efficient a lgorithms conv erging to the glo bal optim um in the MAC and afterwards conv erted to th e BC by m eans of the duality conv ersion formu las. B. Sta te-of-the- Art Duality By means o f the MAC-to-BC conversion, we illustrate th e state-of-the- art rate d uality from [8]. Both in the MA C and in the BC, all rate expressions de pend only on the transmit covariance matrices and not on the ma trix valued receive filters since they are imp licitly assumed to gen erate suffi cient statistics. Based on the se statistics, the L k data streams o f u ser k have to be decoded jointly . Gi ven a set of transmit cov ariance matrices { Q k } in the MAC which fu lfills a total transmit power constraint and obtain s a rate tu ple R MAC 1 , . . . , R MAC K under the assumption o f optimum r eceiv e filters, the du ality in [8] genera tes a set of transmit covariance matrices { S k } for the BC that fulfills the same to tal transmit power con- straint and ach iev es th e same rate tu ple R BC 1 , . . . , R BC K . In th e BC, o ptimum receivers yield ing sufficient statistics are ag ain required and all streams of every individual user have to b e decoded jointly as well. T wo key methods utilized are the effective channe l an d the flip ped channel idea. The f ormer one imp lies that th e capacity of a point-to-point M IMO system with channel matrix H subject to an a dditive Gaussian distortio n (noise plus indepen dent interfer ence) with cov ariance matr ix X eq uals the cap acity of a point-to -point system with effective channel matrix L − 1 H sub ject to additiv e Gaussian distortion with identity covariance matrix if X = LL H . Given an ar bitrary effecti ve chan nel of a point-to -point sy stem, a system with reversed signal flow and He rmitian effective channel ( flip ped channel ) has th e same capacity [ 13]. Ac cording to (1), the rate of u ser k in the MAC can be exp ressed as R MAC k = log 2   I N + X − 1 k H k Q k H H k   , (3) with th e sub stitution X k = σ 2 η I N + P k − 1 ℓ =1 H ℓ Q ℓ H H ℓ . Intro- ducing the Cholesky decom position X k = L k L H k , applyin g the determ inant equality | I a + AB | = | I b + B A | f or arb itrary A and B of ap propr iate dimensions, and inserting tw o identity matrices I r k = F − 1 k F k = F H k F − H k , ( 3) ca n be expressed as R MAC k = log 2   I N + L − 1 k H k F − 1 k F k Q k F H k F − H k H H k L − H k   . Now , L − 1 k H k F − 1 k can be regarded as the e ffecti ve chann el for the c ovariance m atrix F k Q k F H k . How F k must be cho sen will be c larified below . Flipping th e channel, o utcomes in [8] ensure the existence of a covariance m atrix Z k ∈ C N × N with R MAC k = log 2   I r k + F − H k H H k L − H k Z k L − 1 k H k F − 1 k   , tr( Z k ) ≤ tr( F k Q k F H k ) . (4) The ra te of user k in the BC is ( cf. E q. 2) R BC k = log 2   I r k + Y − 1 k H H k S k H k   = log 2   I r k + F − H k H H k S k H k F − 1 k   , (5) with the substitution Y k = σ 2 η I r k + P K ℓ = k +1 H H k S ℓ H k = F H k F k . Equality betwe en R MAC k in (4) an d R BC k in (5) h olds, if S k = L − H k Z k L − 1 k . (6) Implicitly , Z k depend s on F k as will be sh own soon. Thus, S k depend s o n Y k which itself is a function of all S ℓ with ℓ > k . These dep endencies require that S k has to b e compu ted before S k − 1 and consequently , one has to start with the compu tation of S K followed by S K − 1 , . . . , S 1 . It r emains to de termine the ma trices Z k ∀ k . Introd ucing the reduced sin gular-value-deco mposition (rSVD) L − 1 k H k F − 1 k = U k D k V H k ∈ C N × r k (7) with the two (sub -)unitar y matrice s U k ∈ C N × rank( H k ) and V k ∈ C r k × rank( H k ) , th e matrix Z k reads as Z k = U k V H k · F k Q k F H k · V k U H k . (8) The pro of for th e sum -power conservation can be fou nd in [8] . From the MAC-to-BC con version, it can be concluded that ev ery rate tuple in the MAC can also be achieved in the du al BC. Conversely , the transfor mation from the BC to th e MAC which f ollows from the same framework, states that every rate tuple in the BC c an also be achieved in the MA C. He nce, the duality of these two domains is proven and as a con sequence, their capacity regions are con gruen t. Sum ming up, the state- of- the-art rate duality including in terference cancellation is serial in two senses: First, it requires a serial implemen tation o f the covariance matrix con version due to the depend encies o f S k on S ℓ with ℓ > k . Second , the application of the duality r equires that the d ifferent streams associated to a user are deco ded jointly o r , at the b est, in a serial fashion. C. Pr oposed F ilter -Based Duality The previously describ ed state- of-the- art rate duality is mainly deduced fr om info rmation theor etic consid erations, where op timum receiv ers generate sufficient statistics and capacity is achieved via joint decoding with inter- an d intra- user successive interference cancellation . Approac hing fr om a signal processing po int of view enables us to d erive a novel intuitive du ality of low complexity . Switching from arbitrary su fficient statistics gener ating optimum r eceivers to MMSE r eceivers, we are a ble to express all rates in ter ms of error covariance matrices, which in turn only depend on the tran smit covariance matrices, i.e ., on the outer prod uct of the pre coding filters. The remaining degree o f freedo m is a u nitary rotation and we u tilize th is isom etry in ord er to decorre late e very single point-to -point li nk. Doing so, the er ror covariance m atrix becom es diagona l and capacity is achieved with separate stream -wise decoding making intra-user inter- ference c ancellation su perfluou s. The fact th at stream- wise en- coding/d ecoding achiev es capacity has already been observed in [2], [ 14]. The re, howe ver , intra-user successive d ecoding must be app lied and all stream s ar e decoded one by one. As a ll rates c an n ow be expressed as function s of the SINRs of the individual streams, we apply a lo w-comp lexity SINR duality in the style of our MSE duality in [6], [5]. In a nutshell, the scaled MMSE receivers a re used as p recode rs in the dual domain and scaled precod ing filters serve a s the receiv e filters in th e dual domain. This dua l domain f eatures the same SINR values as the o riginal one an d theref ore achieves the same user rates. In th e following, we give an elaborate deriv ation of the MA C-to-BC conv ersion. 1) Deriva tion: Assuming tha t ev ery MA C covariance ma- trix Q k = T k T H k is generated by the pr ecoder T k ∈ C r k × L k , the symbol estimate of user k in the MA C is ˆ s k = G k h H k T k s k + X ℓ>k H ℓ T ℓ s ℓ + X ℓ k is removed by succ essi ve interferen ce cancellation, the MMSE receiver for user k is G k = T H k H H k  X ℓ ≤ k H ℓ T ℓ T H ℓ H H ℓ + σ 2 η I N  − 1 . (9) Using (9 ) an d the matrix -inv ersion lemm a, the MMSE error covariance m atrix C k = E[( s k − ˆ s k )( s k − ˆ s k ) H ] r eads as C k = I L k − G k H k T k =  I L k + T H k H H k X − 1 k H k T k  − 1 , ( 10) with X k = σ 2 η I N + P k − 1 ℓ =1 H ℓ T ℓ T H ℓ H H ℓ . The rate of user k can be expre ssed in te rms of its error covariance matrix R MAC k = log 2 | C − 1 k | = − log 2 | C k | , (11) cf. (3). Note that the rate of user k is in variant to a unitar y matrix W k multiplied f rom th e r ight han d side to T k yielding T ′ k = T k W k . Mo reover , th e rate expressions of other user s only dep end on the transmit covariance matrices an d not on the filters th emselves theref ore b eing also inv ariant to this isometry . Last but not lea st, the transm it power tr( Q k ) = tr( T k T H k ) = tr( T ′ k T ′ H k ) is inv ariant under this isometry W k . Although W k does not influ ence the interfere nce cov ariance matrix experienced by any other user, it c an be used as a spatial d ecorrelatio n filter for every poin t-to-po int link wh ich in co njunction with th e MMSE receiver G ′ k = W H k G k diagona lizes the error-cov ariance ma trix C k . T o this end, W k must be ch osen as the eigenbasis of G k H k T k which is also the eigenbasis of T H k H H k X − 1 k H k T k . Du e to the d ecorrelatio n, all point-to- point lin ks from the u sers to the ba se station achieve capacity without intra-user successiv e interference c ancellation thus making sep arate stream decoding p ossible. This way , the rate o f u ser k can be expressed as the sum of the ind ividual streams’ rates, i.e., R MAC k = P L k i =1 R MAC k,i , wh ere R MAC k,i = log 2 (1 + SINR MAC k,i ) . Let t ′ k,i be the i th column of T ′ k and g ′ T k,i be the i th row of G ′ k , th en th e g eneral SINR definition in th e M A C SINR MAC k,i = | g ′ T k,i H k t ′ k,i | 2 g ′ T k,i  X k + P m 6 = i H k t ′ k,m t ′ H k,m H H k  g ′∗ k,i (12) reduces fo r th e special c hoice o f th e de correlation filter W k to SINR MAC k,i = | g ′ T k,i H k t ′ k,i | 2 σ 2 η k g ′ k,i k 2 2 + P ℓk P L ℓ m =1 | g ′ T ℓ,m H k t ′ k,i | 2 α 2 ℓ,m . Equating SINR BC k,i with the MA C SINR from (1 3), we get α 2 k,i h σ 2 η k g ′ k,i k 2 2 + X ℓk L ℓ X m =1 α 2 ℓ,m | g ′ T ℓ,m H k t ′ k,i | 2 = σ 2 η k t ′ k,i k 2 2 , (17) which n eeds to hold for all user s k and all streams i ∈ { 1 , . . . , L k } thu s ge nerating the system of lin ear eq uations M ·  α 2 1 , 1 , . . . , α 2 K,L K  T = σ 2 η  k t ′ 1 , 1 k 2 2 , . . . , k t ′ K,L K k 2 2  T (18) with the P K k =1 L k × P K k =1 L k block upper trian gular matr ix M =    M 1 , 1 · · · M 1 ,K 0 . . . . . . 0 0 M K,K    . (19) The o ff-diagonal blocks with a < b rea d as (cf. Eq. 1 7) M a,b = − ( G ′ b H a T ′ a ) H ⊙ ( G ′ b H a T ′ a ) T ∈ R L a × L b (20) with the Had amar d produ ct ⊙ , an d M a,a is diag onal with [ M a,a ] i,i = σ 2 η k g ′ a,i k 2 2 − X ℓ 0 , M is colum n diagon ally d ominant. So, M is an M-matrix such that its inverse exists with nonnegative entries [15] yielding valid solutions α 2 k,i ≥ 0 . Because of the block upper triang ular structure o f M we can quick ly so lve for α 2 1 , 1 , . . . , α 2 K,L K via b ack-sub stitution, in particular since the d iagonal blo cks M k,k are d iagonal matrices. Note that a rank-d eficient prec oder T m manifests in zer o columns an d zero rows in M which hav e to be removed before inversion. The respective α 2 m, · and k t ′ m, · k 2 2 in (18) also have to be removed, and fin ally , p m, · = 0 and b m, · = 0 m ust be cho sen. Summing u p the rows of (18), we ob tain K X k =1 L k X i =1 α 2 k,i k g ′ k,i k 2 2 | {z } k p k,i k 2 2 σ 2 η = σ 2 η K X k =1 L k X i =1 k t ′ k,i k 2 2 , (22) stating that the dua l BC co nsumes the same power as th e MA C. Thus, the same or la rger (if MMSE receivers are ch osen for B 1 , . . . , B K ) rates can b e achieved in the d ual BC as in the primal MAC under the same transmit power co nstraint. The re verse directio n o f th e duality transforming BC filters to the MA C can be handled with the same fram ew ork. Due to its similarity , we skip its d eriv ation. Fr om this d irection of the duality , it fo llows tha t th e BC rate region is a subset o f the MA C cap acity region. In com bination with th e fo rmer r esult of the MAC-to-BC conversion stating that the MA C capacity region is a sub set of the BC rate region, the following theorem becomes evident with the aid of [ 9] ( cf. [8]): Theorem III.1: The capa city r e g ions o f the MAC a nd the BC ar e con gruent under a sum-power co nstraint. As a c onsequen ce, any optimizatio n in the BC can be solved in the MA C, which offers concave rate expressions suitable for efficient globally conver gent algo rithms. Since both capa city regions are co ngru ent, we optimize over the same region and therefor e, do not introd uce any subop timality at this poin t. Having found the solution in th e MAC we can c onv ert it back to the BC by mean s of the duality . Optim ality in on e domain translates itself to optimality in the other d omain. Th e main advantage of the propo sed filter-based du ality c ompared to the state-of-the- art d uality in [ 8] is that b oth the con version and the decodin g in the dual domain can be parallelized and need not be app lied serially a s in [8]. Th e com putation of the tran smit and receive filters feature s no depend encies and the decod ing process d oes n ot require in tra-user interferen ce cancellation or intra-user join t deco ding of the streams, all streams of a user can be decod ed indep endently in p arallel. 2) Algorithmic Implementatio n: Given ar bitrary precod ing filters T k ∀ k in the MA C, MMSE receivers G k are first computed v ia ( 9) for all k , see Lin e 2 in Alg. 1. The decor- relation filter W k is cho sen as the eigenbasis of G k H k T k and afterwards, the transmit and receive filters are ad apted, see Lines 3 an d 4. Ther eby , a p arallel stream-wise deco ding is possible with out intra -user interfere nce cancellation. Having set u p the linear system o f eq uations in (1 8) whic h en sures the conservation of the SINRs in th e BC, the pr ecoders P k and receivers B k are com puted with (1 6), cf. Line 8 . I V . R A T E D UA L I T Y F O R S Y S T E M S W I T H O U T I N T E R F E R E N C E S U B T R A C T I O N In case of linear filtering , i. e., when nonlinear inter-user interferen ce c ancellation is not ap plied, user k experien ces interferen ce from all other users ℓ 6 = k . Up to now , a rate duality for the linear case without interfer ence subtraction does not exist in the literature wh en mu lti-antenna terminals are in volved a nd different streams sha ll not be tr eated as self- interferen ce. By jointly decodin g the streams in the MA C, user k can achieve the rate R MAC k = log 2    I N +  X ℓ 6 = k H ℓ Q ℓ H H ℓ + σ 2 η I N  − 1 H k Q k H H k    = − log 2   I N − X − 1 H k Q k H H k   , (23) with the sub stitution X = σ 2 η I N + P K ℓ =1 H ℓ Q ℓ H H ℓ . In co n- trast to systems with in terference ca ncellation describ ed in th e previous section, th is matr ix is commo n to MMSE rece iv ers G k = T H k H H k X − 1 (24) for all users k and th erefor e ha s to be co mputed on ly once. Applying G k , u ser k experien ces the erro r covariance matrix C k = I L k − T H k H H k X − 1 H k T k , (25) which is again decorr elated by the isometry W k since the rate R MAC k = − log 2 | C k | is again in variant under this uni- tary degree of freedo m. Choosin g W k as the eigenb asis o f T H k H H k X − 1 H k T k , we adapt the receiv e filter G ′ k = W H k G k and the transmit filter T ′ k = T k W k . Due to the decorr elation, the er ror covariance matr ix W H k C k W k is diagon alized and all L k streams of user k can be decode d separately yield ing the rate R MAC , lin k = P L k i =1 R MAC , lin k,i , with th e rate R MAC , lin k,i = log 2 (1 + SINR MAC , lin k,i ) (26) of u ser k ’ s stream i . Its SINR now reads as SINR MAC , lin k,i = | g ′ T k,i H k t ′ k,i | 2 σ 2 η k g ′ k,i k 2 2 + P ℓ 6 = k P L ℓ m =1 | g ′ T k,i H ℓ t ′ ℓ,m | 2 . W e apply the same r ule for finding the pr ecoding and receive filters P k and B k of user k in the BC as we do in case of inter- ference cancellation, i.e., p k,i = α k,i g ′∗ k,i and b k,i = α − 1 k,i t ′∗ k,i , see ( 16). Wi th these transforma tions, the BC SINR reads as SINR BC , lin k,i = α 2 k,i | g ′ T k,i H k t ′ k,i | 2 σ 2 η k t ′ k,i k 2 2 + P ℓ 6 = k P L ℓ m =1 | g ′ T ℓ,m H k t ′ k,i | 2 α 2 ℓ,m . Equating the BC an d MAC SINRs yields the system of linear equations ( 18), where the matr ix M is not block u pper triangular as in (19), since inter-user interference cancellation is no t a pplied: M =    M 1 , 1 · · · M 1 ,K . . . . . . . . . M K, 1 · · · M K,K    . (27) For this reason, (18) is solved v ia LU-factorization [16, Sec- tion 3.2 .5] and for ward-backward substitution . The diago nal blocks o f M are d iagonal m atrices with diago nal entries [ M a,a ] i,i = σ 2 η k g ′ a,i k 2 2 − X ℓ 6 = a L ℓ X m =1 [ M ℓ,a ] m,i , (28) such th at M is again an M-matrix satisfyin g th e power conservation eq uation (22). W ith slight modifications, Alg. 1 can be used to perfor m the MA C-to-BC conversion with out nonlinear inter-user inter ference cancellation. In Line 2, G k must be compute d according to ( 24), a nd in L ine 7, the ma trix M fo llows from (27), ( 20), and (28). Again , th e converse direction o f the du ality und erlies the same framew ork and completes the p roof o f the duality in ca se o f linear filterin g without in ter-user interference ca ncellation: Theorem IV .1: The MIMO MAC and the MIMO BC sha r e the sa me rate r e gion u nder linea r filtering an d a sum-p ower constraint bo th for separate and joint de -/encodin g of each user’ s da ta str eams. This n ovel r ate duality f or systems witho ut in terferen ce can- cellation allows us to con vert any rate-b ased optimization from the BC to the MAC with out loss o f optimality . An immediate benefit is that we can switch fr om the rate expression R MAC , in terferenc e k = − log 2 Y i  I L k − T H k H H k X − 1 H k T k  i,i with separate stream decoding and hence self -interfere nce to the one in (23) with jo int stream dec oding R MAC , lin k = − log 2   I L k − T H k H H k X − 1 H k T k   , which is always larger than or eq ual to R MAC , in terferenc e k . Moreover , the channel and precoder indices are aligned in Algorithm 1 Novel stream-wise MAC-to-BC conversion. 1: for k = 1 : K do 2: G k ← T H k H H k  P ℓ ≤ k H ℓ T ℓ T H ℓ H H ℓ + σ 2 η I N  − 1 3: W k ← eig enbasis ( G k H k T k ) decorrelation matrix 4: G ′ k ← W H k G k and T ′ k ← T k W k decorr elate 5: end for 6: set up M with (19) – (2 1), remove zero column s/rows 7: solve fo r α 2 1 , 1 , . . . , α 2 K,L K via (1 8) 8: p k,i = α k,i g ′∗ k,i and b k,i = 1 α k,i t ′∗ k,i ∀ k , ∀ i the MA C, see (23), whe reas they are n’t in the BC. A lthough (weighted) sum-rate maximization remains a noncon cav e max - imization in the MAC , th e aforem entioned indices alignm ent allows for simpler exp ressions an d reduced- complexity algo- rithms. Last but no t least, MA C p recoder s are charac terized by o nly P K k =1 r 2 k variables instead o f N P K k =1 r k in the BC. Summing u p, solving r ate b ased o ptimizations with linear filtering in the MA C and app lying the proposed duality is more efficient than solv ing the prob lem in th e BC. R E F E R E N C E S [1] H. Boche and M. Schubert, “A General Duality Theory for Uplink and Do wnlink Beamforming, ” in Proc . IEEE V ehicul ar T echn. Conf. (V TC) fall, V ancouver , Canada , September 2002, pp. 87 – 91 vol.1. [2] P . Vi swana th and D. Tse, “Sum Capacit y of the V ector Gaussian Broadca st Channel and U plink-Do wnlink Duality , ” IEEE T rans. Inf. Theory , vol. 49, no. 8, pp. 1912–1921, August 2003. [3] S. Shi an d M. Sch ubert, “MMSE Transmit Optimiza tion for Multi user Multia ntenna Systems, ” in Pr oc. IEEE Internat. Conf. on Acoustics, Speec h, and Signal Pro c. (ICASSP), Philadelphia, USA , Mar . 200 5. [4] A. Khachan, A. T enen baum, and R. S. Adve, “Linear Processi ng for the Do wnlink in Multiu ser MIMO Systems with Multi ple Data Streams, ” in IEEE Internat ional Conf. on Comm. , June 2006, vol. 9, pp. 4113–4118. [5] A. Mezgha ni, M. Joham, R. Hunger , and W . Utschick, “Transc ei ve r Design for Multi-User MIMO Systems, ” in Pr oc. ITG/IEEE WSA 2006 , March 2006. [6] R. Hunger , M. Joham, and W . Utschick, “On the MSE -Dualit y of the Broadca st Channel and the Multiple Access Channe l, ” Submitte d to IEEE T ransact ions on Signal Proc essing . [7] N. Jindal, S. V ishwan ath, and A. Goldsmith, “On the Duality of Gaussian Multiple -Access and Broadca st Channel s, ” IEEE T ra ns. Inf. Theory , vol. 50, no. 5, pp. 768–783, May 2004. [8] S. V ishwa nath, N. Jindal, and A. Goldsmith, “Duality , Achie v able Rates, and Sum-Rate Capaci ty of MIMO Broadc ast Channels, ” IEEE T r ans. Inf. Theory , vol. 49, pp. 2658–2668, October 2003. [9] H. W eingarten, Y . Steinber g, and S. S. Shamai, “The Capacity Region of the Gaussian Multiple-Inpu t Multiple-Out put Broadcast Channel, ” IEEE T rans. Inf. Theory , vol. 52, no. 9, pp. 3936–3964, 2006. [10] W . Y u and T . Lan, “Transmit ter Optimizat ion for the Multi-Antenna Do wnlink Wit h Per-Ante nna Powe r Constrain ts, ” IEE E T rans. Signal Pr ocess. , vol. 55, no. 6-1, pp. 2646–2660, June 2007. [11] B. Song, R. L. Cruz , and B. D. Rao , “Network Dualit y for Mult iuser MIMO Be amforming Net works an d Appl icati ons, ” IEEE T rans. Com- mun. , vol. 55, no. 3, pp. 618–630, 2007. [12] W . Y u, W . Rhee , S. Boyd, and J.M. Cioffi, “Iterat i ve W ater -Filli ng for Gaussian V ector Mult iple-Ac cess Channels, ” IEEE T rans. Inf. The ory , vol. 50, no. 1, pp. 145–152, January 2004. [13] E. T elatar , “Capacity of Multi-Ante nna Gaussian Channels, ” Europea n T ransactions on T elecommun icatio ns , vol. 10, pp. 585–595, 1999. [14] D. Tse and P . V iswanath, “On the Capacit y of the Multipl e Antenna Broadca st Channel , ” DIMACS W orkshop. American Math. Soc. 2003 , vol. 62, pp. 87–105, October 2002. [15] A. Berman and R. J. Plemmons, Nonne gative Matrices in the Mathe- matical Sciences , Academic Press, 1979. [16] G. H. Golub and C.F . V an Loan, Matrix Computatio ns , Johns Hopkins Uni ver sity Press, 2nd edition, 1991.