Polite Water-filling for the Boundary of the Capacity/Achievable Regions of MIMO MAC/BC/Interference Networks

Transcription

1 2011 IEEE Internationa Symposium on Information Theory Proceedings Poite Water-fiing for the Boundary of the Capacity/Achievabe Regions of MIMO MAC/BC/Interference Networks An iu 1, Youjian iu 2, Haige Xiang 1, Wu uo 1 1 State Key aboratory of Advanced Optica Communication Systems & Networks, Schoo of EECS, Peking University 2 Department of Eectrica and Computer Engineering, University of Coorado at Bouder Abstract We found a network version of water-fiing, named poite water-fiing, that optima for a boundary points of the capacity regions of MAC and BC and for a boundary points of a set of achievabe regions of a genera cass of interference networks, named MIMO B-MAC networks that incude BC, MAC, interference channes, X networks, and most practica networks as specia cases. It poite because it strikes an optima baance between reducing interference to others and maximizing a ink s own rate. Unike in singe-user MIMO channes, where the optima input covariance can be soved by the water-fiing, the traditiona water-fiing far from optima in networks. Thus, genera purpose optimization agorithms have been used for networks but have high compexity and do not work we for non-convex cases. Together with our duaity resut, the poite water-fiing can be used to design highy efficient ow-compexity iterative centraized/dtributed agorithms for the optimization of input covariance matrices, incuding both power and beamforming matrices, because it takes the advantage of the structure of the probems. References to the resuting agorithms that outperform the state-of-the-art by a wide margin are provided. Index Terms Poite Water-fiing, MIMO Interference Network, Transmitter Optimization, Network Information Theory I. INTRODUCTION We soved the open probem of finding the right form of water-fiing that optima for a boundary points of the capacity regions of MIMO mutiaccess channes (MAC), broadcast channes (BC). Furthermore, it optima for a boundary points of a set of achievabe regions of some genera onehop MIMO interference networks, where each transmitter may send independent data to mutipe receivers and each receiver may coect independent data from mutipe transmitters. Such a network a combination of mutipe interfering BCs, from the transmitter point of view, and MACs, from the receiver point of view, and named the B-MAC network. It incudes BC, MAC, interference channes, X networks, and most practica communication networks as specia cases. The poite waterfiing expected to resut in an update of most reated network optimization agorithms. We assume Gaussian input and that each signa decoded by no more than one receiver. The setting optima for MIMO MAC/BC and incudes most practica interference management techniques as specia cases, such as 1) spatia interference reduction through beamforming matrices without interference canceation, which automaticay becomes spatia interference The work was supported in part by NSFC Grant No , and in part by US-NSF Grant CCF and ECCS aignment at high SNR [1]; 2) some combination of interference canceation using dirty paper coding (DPC) [2] at transmitters and MMSE pus successive interference canceation (SIC) at receivers; 3) transmitter cooperation, where a transmitter cance another transmitter s signa using DPC when another transmitter s signa avaiabe through, e.g., an optica ink between them. The extension to Han-Kobayashi scheme [3], where a signa may be decoded and canceed at more than one receiver aso dcussed in Remark 2 and in future work at Section V. The optimization of B-MAC networks has been hindered by that itte known about the Pareto optima input structure of the capacity/achievabe regions. Iterative traditiona waterfiing can ony be empoyed for finding the sum-capacity point of the MIMO MAC/BC [4], [5]. Thus, genera purpose optimization agorithms such as steepest ascent has been used to find other boundary points of a MAC/BC capacity region [6]. But it desirabe to expoit the structure of the probem to design more efficient agorithms that wi work we for both convex optimization of BC/MAC and non-convex optimization of more genera interference networks. Directy appying singe-user water-fiing to networks referred to as sefh water-fiing here. It we known to be far from optima [7], [8] because it does not contro interference to others. Based on sefh water-fiing, game-theoretic, dtributed, and iterative agorithms have been we studied for DS [9], [10], for MIMO interference channes, e.g., [11], and for mutiaccess channes, e.g., [8]. The agorithms converge ony under stringent conditions and the performance not near optima. The importance of controing interference to others has been recognized in iterature, e.g., [1], [12]. But a systematic, genera, and optima method has not been found. Consequenty, the foowing probems have been open. In singe-user MIMO channes, no one uses genera purpose optimization agorithms for transmitter optimization because the simpe water-fiing optima. Is there a network version of water-fiing that optima for a boundary points of the MAC/BC capacity region so that it can be used to design ower compexity and better performance optimization agorithms compared with the genera purpose ones. More ambitiousy, one can ask the above questions for some B-MAC networks with respect to some achievabe regions. Furthermore, what the optima method to contro interference to others? Can we decompose a B /11/$ IEEE 2015

2 MAC network to mutipe equivaent singe-user channes so that the (dtributed) optimization can be made easy? The main contribution of th paper to give a network version of water-fiing for the above probems. The rest of the paper organized as foows. Section II defines the achievabe rate region and summarizes the preiminaries on rate duaity. Section III presents the theoretica resuts on poite water-fiing. Together with our duaity resut [13], the poite water-fiing can be used to design highy efficient owcompexity iterative centraized/dtributed agorithms for the optimization of input covariance matrices, incuding both power and beamforming matrices. The agorithms and numeric resuts are given in the references in Section IV. The concusions are given in Section V. II. SYSTEM MODE AND PREIMINARIES Consider a generaized one-hop MIMO interference network named a MIMO B-MAC network [13], [14], where each transmitter may have independent data for different receivers and each receiver may want independent data from different transmitters. There are data inks. Assume the set of physica transmitter abes T = {TX 1, TX 2, TX 3,...} and the set of physica receiver abes R = {RX 1, RX 2, RX 3,...}. Define transmitter T of ink as a mapping from to ink s physica transmitter abe in T. Define receiver R as a mapping from to ink s physica receiver abe in R. For exampe, in a 2-user MAC, the sets are T = {TX 1, TX 2 }, R = {RX 1 }. And the mappings coud be T 1 = TX 1, T 2 = TX 2, R 1 = RX 1, R 2 = RX 1. The numbers of antennas at T and R are T and respectivey. The received signa at R R y = H,k x k + w, (1) where x k C T 1 k the transmit signa of ink k and assumed to be circuary symmetric compex Gaussian (CSCG); H,k C R Tk the channe matrix between T k and R ; and w C R 1 a CSCG noe vector with with zero mean and identity covariance matrix. To hande a wide range of interference canceation, define a couping matrix Φ {0, 1} + as a function of the interference canceation scheme [13]. It specifies whether interference competey canceed or treated as noe: if x k, after interference canceation, sti causes interference to x, Φ,k = 1 and otherwe, Φ,k = 0. The couping matrices vaid for the resuts of th paper are those for which there exts a transceiver scheme such that each signa decoded (and possiby canceed) by no more than one receiver. For exampe, in a BC (MAC) empoying DPC (SIC) where the th ink the th one to be encoded (decoded), the couping matrix given by Φ,k = 0, k and Φ,k = 1, k >. For genera B-MAC networks, certain combinations of DPC and SIC are aso aowed as dcussed in [13]. The achievabe regions in th paper refer to the foowing. Note that Φ, = 0 by definition. The interference-pus-noe covariance of ink Ω = I + Φ,k H,k Σ k H,k, (2) where Σ k the covariance matrix of x k. We denote a the covariance matrices as Σ 1: = (Σ 1,..., Σ ). Then the achievabe rate of ink given by [15] I (Σ 1:, Φ) = og I + H, Σ H, Ω 1. (3) Definition 1: The Achievabe Rate Region with a fixed couping matrix Φ and sum power constraint P T defined as { R Φ (P T ) r R + : (4) Σ 1: : =1 Tr(Σ ) P T r I (Σ 1:, Φ), 1 }. Definition 2: If [I (Σ 1:, Φ)] =1: a Pareto rate point of R Φ (P T ), the input covariance matrices Σ 1: are said to be Pareto optima. In Section III, we characterize the Pareto optima input covariance based on a rate duaity between the forward and reverse inks of a MIMO B-MAC network. The reverse inks are obtained by reversing the directions of the forward inks and repacing the channe matrices by their conjugate transposes. The couping matrix for the reverse inks the transpose of that for the forward inks. We use the notation ˆ to denote the corresponding terms in the reverse inks. The interference-pusnoe covariance of reverse ink ˆΩ = I + Φ k, H ˆΣ k, k H k,. (5) And the rate of reverse ink given by Î ˆΣ1:, Φ T = og I + H ˆΣ, H ˆΩ 1,. The achievabe rate region of the reverse inks can be defined simiary as in (4) and denoted by ˆR Φ T (P T ). We review the rate duaity in [13]. The key a covariance transformation from the forward input covariance matrices to the reverse ones, which cacuated as foows. Step 1: Decompose the signa of each ink to M streams as Σ = M m=1 p,mt,m t,m,, where t,m C T 1 a transmit vector with t,m = 1 and p = [p 1,1,..., p 1,M1,..., p,1,..., p,m ] T are the transmit powers. Step 2: Each receiver empoys SIC and the inear MMSE fiter (MMSE-SIC). Without oss of generaity, we assume the intra-signa decoding order that the m th stream the m th one to be decoded. For each stream, compute the receive vector by MMSE agorithm: ( M 1 r,m = α,m H, p,i t,i t,i H, ) + Ω H, t,m, i=m+1 where α,m chosen such that r,m = 1. Then the achieved SINR γ,m can be easiy cacuated. Note that the above decomposition of data to streams with MMSE-SIC receiver information ossess, i.e., I (Σ 1:, Φ) = M m=1 og (1 + γ,m). (6) 2016

3 Step 3: In the reverse inks, use {r,m } as transmit vectors and {t,m } as receive vectors. The decoding order of the streams within a ink the opposite to that of the forward ink, i.e., the m th stream the m th ast to be decoded and canceed. Appy the SINR duaity [16] to obtain the reverse transmit powers q = [q 1,1,..., q 1,M1,..., q,1,..., q,m ] T which achieve the same SINRs {γ,m } as in the forward inks. Step 4: The Covariance Transformation from Σ 1: to ˆΣ 1: ˆΣ = M m=1 q,m r,m r,m, = 1,...,. (7) Then, the rate duaity in [13] restated in the foowing theorem. Theorem 1: For any Σ 1: satfying =1 Tr (Σ ) P T and achieving a rate point r in the forward inks, its covariance transformation ˆΣ 1: achieves a rate point ˆr r in the reverse inks and satfies =1 Tr ˆΣ = =1 Tr (Σ ) P T. Therefore, the achievabe rate regions of the forward and reverse inks are the same under the same sum power constraint, i.e., R Φ (P T ) = ˆR Φ T (P T ). III. MAIN RESUTS In th section, the structure and properties of the Pareto optima input are characterized. We first show that the Pareto optima input covariance matrices have a poite water-fiing structure defined beow. It generaizes the we known singeuser water-fiing soution to networks. Definition 3: Given input covariance matrices Σ 1:, obtain its covariance transformation ˆΣ 1: by (7) and cacuate the interference-pus-noe covariance matrices Ω 1: and ˆΩ 1:. For each ink, pre- and post- whiten the channe H, to produce an equivaent singe-user channe H = Ω 1/2 H ˆΩ 1/2,. Define 1/2 Q ˆΩ Σ ˆΩ1/2 as the equivaent input covariance matrix of the ink. The input covariance matrix Σ said to possess a poite water-fiing structure if Q a water-fiing over H, i.e., Q = G D G, (8) D = ( ν I 2 ) +, where ν 0 the poite water-fiing eve; the equivaent channe H s thin singuar vaue decomposition (SVD) H = F G, where F C R N, G C T N, R N N ++ ; and N = Rank (H, ). Σ 1: said to possess the poite water-fiing structure if a Σ s do. For B-MAC, the poite water-fiing structure proved by observing that the Pareto optima input covariance matrix for each ink the soution of some singe-user optimization probem. We use the notation to denote the Pareto optima variabes. Without oss of generaity, assume that Σ = M m=1 p,m t,m t,m, = 1,...,, where p,m > 0,, m, achieves a Pareto rate point [Ĩ > 0] and ˆΣ = =1,..., M m=1 q,m r,m r,m, = 1,..., are the corresponding covariance transformation. The corresponding interference-pusnoe covariance matrices are denoted as Ω and ˆΩ. Then it can be proved by contradiction that Σ the soution of the foowing singe-user optimization probem for ink, where the transmsion and reception schemes of other inks are fixed as the Pareto optima scheme: { p, t, r, k }. min Tr (Σ ) (9) Σ 0 s.t. og I + H, Σ H 1, Ω ( Ĩ ) Tr Σ A () Tr Σ A () (10) m = 1,..., M k, k = 1,...,, k, where A () = Φ k,h k, r r H k,; and Tr Σ A () the interference from ink to the m th stream of( ink k and ) constrained not to exceed the optima vaue Tr Σ A (). The constraints force the rates to be the Pareto point whie the power of ink minimized. The agrangian of probem (9) (λ, ν, Θ, Σ ) M k =Tr (Σ (A (λ) Θ)) k + ν Ĩ ν og I + H, Σ H, m=1 λ Tr Σ A () 1 Ω, (11) where the dua variabes ν R + and λ = k [λ ] m=1,...,mk,k R M k 1 + are associated with the rate constraint and the interference constraints in (10) respectivey; A (λ) I + Mk k m=1 λ A () a function of λ; Θ the matrix dua variabes associated with the positive semidefiniteness constraint on Σ. Because probem (9) convex, the duaity gap zero [17] and Σ minimizes the agrangian (11) with optima dua variabes ν and λ ] = [ λ, i.e., it the soution of m=1,...,m k,k ) min Tr Σ A ( λ ν og I + H, Σ H 1 Σ 0, Ω. (12) Note that in probem (12), the constant terms in the agrangian (11) have been deeted, and the term Tr (Σ Θ) expicity handed by adding the constraint Σ 0. The foowing theorem proved in Appendix B of [13] states ] that the physica meaning of the optima dua variabes [ λ exacty the optima power aocation in the reverse k inks. Theorem 2: The optima dua variabes of probem (9) are given by λ = q, m = 1,..., M k, k (13) q,m p,m (1 + γ,m ) r,m H 2, t,m ν =, m (14) γ 2,m where γ,m the SINR of the m th stream of ink achieved by { p, t, r }. Therefore, ) ) A ( λ = A ([ q ] k = ˆΩ. The poite water-fiing structure shown by a singeuser-channe view using the above resuts. et H = 2017

4 Ω 1/2 H ˆΩ 1/2, ˆΩ 1/2 and Q = Σ ˆΩ1/2. Since ˆΩ nonsinguar, probem (12) equivaent to a singe-user optimization probem min Tr (Q ) ν og I + H Q H Q 0, (15) Σ ˆΩ1/2 of which ˆΩ1/2 an optima soution. Since the optima soution to probem (15) unique and given by the waterfiing over H [5], [15], the foowing theorem proved. Theorem 3: For each, perform the thin SVD as H = F G. At a Pareto rate point, the input covariance matrix Σ must have a poite water-fiing structure, i.e., the equivaent input covariance matrix Q ˆΩ1/2 Σ ˆΩ1/2 satfies Q = G D G, (16) D = ( ν I 2 ) +. Simiary, the corresponding ˆΣ produces ˆQ 1/2 Ω ˆΣ Ω1/2, which satfies ˆQ = F D F. (17) Remark 1: The insight given by the above proof that restricting interference to other inks can be achieved by prewhitening the channe with reverse ink interference-pus-noe covariance matrix. And thus, the B-MAC can be converted to virtuay independent equivaent channes H, = 1,...,. The restriction of interference achieved in two steps. First, in H 1/2 = Ω H ˆΩ 1/2,, the mutipication of ˆΩ 1/2 reduces the channe gain in the interfering directions so that in Q, ess power wi be fied in these directions. Second, in Σ = ˆΩ 1/2 Q ˆΩ 1/2, the power to the interfering directions further reduced by the mutipication of ˆΩ 1/2. Remark 2: The agrangian interpretation of ˆΩ makes it possibe to extend the duaity and poite water-fiing to Han- Kobayashi transmsion scheme. Canceing the interference requires the interference power to be greater than a threshod rather than ess than it. Therefore, some agrange mutipiers are negative in A (λ). If we sti interpret the agrange mutipier as reverse ink power, we must introduce the concept of negative power for the duaity and poite water-fiing for Han-Kobayashi scheme. The matrix ˆΩ ikey remains positive definite. Otherwe, the soution to probem (12) has infinite power, which suggests there no feasibe power aocation to satfy a the constraints. Theorem 3 says that at the Pareto rate point, it necessary that Σ 1: and ˆΣ 1: have the poite water-fiing structure. The foowing theorem states that Σ having the poite waterfiing structure suffices for ˆΣ to have the poite water-fiing structure even at a non-pareto rate point. Th enabes the optimization of the network part by part. A emma needed for the proof and reveas more insight to the duaity. Athough the covariance transformation preserves tota power such that =1 Tr (Σ ) = ( =1 Tr ˆΣ ), in genera, Tr (Σ ) = Tr ˆΣ not true. Surpringy, Tr (Q ) = Tr ˆQ, true as stated in the foowing emma proved in Appendix C of [13]. emma 1: et ˆΣ1: be the covariance transformation of Σ 1:. Define the equivaent covariance matrices Q ˆΩ 1/2 Σ ˆΩ1/2 and ˆQ Ω 1/2 ˆΣ Ω 1/2. The power of the forward and reverse ink equivaent ( covariance ) matrices of each ink equa, i.e., Tr (Q ) = Tr ˆQ,. Theorem 4: For a given Σ 1: and its covariance transformation ˆΣ 1:, if any Σ has the poite water-fiing structure, so does ˆΣ, i.e., ˆQ Ω 1/2 ˆΣ Ω 1/2 given by water-fiing over 1/2 the reverse equivaent channe H ˆΩ H, Ω 1/2 as in (17). Proof: Because water-fiing uniquey achieves the singeuser MIMO channe capacity [15], Q achieves the capacity of H. Since the capacities of H and H are the same under the same power constraint [15], ˆQ achieves the capacity of H with the same power by emma 1 and Theorem 1. Therefore, ˆQ a water-fiing over H. Decompose a MIMO probem to SISO probems using beams as in step 1 of the covariance transformation can often reduce the compexity. Note that the decomposition Σ = M m=1 p,mt,m t,m, not unique. For different decompositions, {r,m } and {q,m } are different, and thus the covariance transformation may aso be different. But if the input covariance matrices have the poite water-fiing structure, its covariance transformation unique and has an expicit matrix expression. Theorem 5: For any input covariance matrices Σ 1: satfying the poite water-fiing structure, its covariance transformation (7) unique, i.e., for a decompositions of Σ 1:, it wi be transformed to the same dua input ˆΣ1:, and vice versa. Furthermore, the dua input ˆΣ 1: satfies the foowing ˆΩ 1 H, ˆΣ H, = Σ H, Ω 1 H,, = 1,...,, (18) and can be expicity expressed as ( H, Σ H, + Ω ˆΣ = ν (Ω 1 where ν the poite water-fiing eve in (8). The proof given in [13]. ) 1 ), = 1,..., (19) IV. APPICATIONS AND FUTURE WORKS The resuts of th paper provide a stepping stone for soving many important probems. Some appications and future works are ted beow. Weighted Sum-Rate Maximization, and Other Optimization Probems: In [13], the weighted sum-rate maximization under a singe inear constraint used to iustrate the superiority of poite water-fiing. The owest compexity per iteration of the agorithms in [13] inear with respect to the tota number of data inks. Severa times faster convergence speed and severa orders of magnitude higher accuracy than the state-of-the-art achieved. Mutipe inear Constraints, Cognitive Radio, and Other Optimization Probems: Assuming the sum power constraint the necessary first step for more compicated cases. It has its own theoretica vaue and the resut can be easiy extended to a singe inear constraint and 2018

5 the more genera mutipe inear constraints as shown in [13] and [18]. Thus, the poite water-fiing can be empoyed to optimize B-MAC networks with mutipe inear constraints, such as individua power constraints, per antenna power constraints and interference constraints in cognitive radio. In [18], the Poite water-fiing expoited to design high efficiency and ow compexity agorithms for weighted sum-rate maximization under mutipe inear constraints in MIMO B-MAC networks. Again, severa times faster convergence speed and orders of magnitude higher accuracy than the state-of-the-art are demonstrated by numerica exampes. Other optimization probems incude ones with quaity of service requirement, such as those studied in our recent work [14]. Dtributed Optimization and Finite Rate Message Passing: In practice, dtributed/game-theoretic optimization agorithms with partia channe state information are desirabe [14]. The insight from poite water-fiing we suited as it turns the probem into singe-user optimization probems under the infuence of interference from and to others as summarized by the covariance matrices Ω and ˆΩ. Partia or fu knowedge of these covariance matrices can be obtained from reverse ink transmsion or piot training in time divion dupex (TDD) systems [14], or from message passing among the nodes in frequency divion dupex (FDD) systems [19]. The observation in [13], [14] that very few iterations suffices for the poite water-fiing based agorithms to achieve most of the gain makes the message passing approach meaningfu. In practice, the message passing further imited to finite rate. The singe-user view of the poite water-fiing makes it convenient to extend the resuts on the singe-user finite rate feedback in [20] to B-MAC networks. Extension to Han-Kobayashi Transmsion Scheme: Han- Kobayashi scheme cances more interference and especiay beneficia when the interfering channe gain arge. But its optimization sti an open probem. As dcussed in Remark 2 of Section III, the agrangian interpretation of the reverse ink interference-pus-noe covariance matrix for the poite water-fiing makes it possibe to extend the duaity and poite water-fiing to Han- Kobayashi transmsion scheme so as to hep understand the optima input structure and design ow compexity optimization agorithms. The approach in th paper may aso be usefu in muti-hop networks, reay channes, and ad-hoc networks. V. CONCUSION Th paper soves the probem of finding the right form of water-fiing that optima for capacity regions/achievabe regions of genera MIMO interference networks named B- MAC networks with Gaussian input and any vaid couping matrix. The main contribution the dcovery that a the Pareto optima input has a poite water-fiing structure, which optimay baances between reducing interference to others and maximizing a ink s own rate. Th network version of water-fiing provides insight into understanding interference in networks and an eegant method to decompose a network to mutipe equivaent singe-user inks. It can be empoyed to design/improve most reated network optimization agorithms. REFERENCES [1] K. Gomadam, V. Cadambe, and S. Jafar, Approaching the capacity of wireess networks through dtributed interference aignment, IEEE GOBECOM 08, pp. 1 6, Dec [2] M. Costa, Writing on dirty paper (corresp.), IEEE Trans. Info. Theory, vo. 29, no. 3, pp , [3] T. S. Han and K. Kobayashi, A new achievabe rate region for the interference channe, IEEE Trans. Inform. Th., vo. 27, no. 1, pp , Jan [4] W. Yu, W. Rhee, S. Boyd, and J. Cioffi, Iterative water-fiing for Gaussian vector mutipe-access channes, IEEE Trans. Info. Theory, vo. 50, no. 1, pp , [5] N. Jinda, W. Rhee, S. Vhwanath, S. A. Jafar, and A. Godsmith, Sum power iterative water-fiing for muti-antenna Gaussian broadcast channes, IEEE Trans. Info. Theory, vo. 51, no. 4, pp , Apr [6] H. Vwanathan, S. Venkatesan, and H. Huang, Downink capacity evauation of ceuar networks with known-interference canceation, IEEE J. Seect. Areas Commun., vo. 21, no. 5, pp , June [7] O. Popescu and C. Rose, Water fiing may not good neighbors make, in Proceedings of GOBECOM 2003, vo. 3, 2003, pp [8]. ai and H. E Gama, The water-fiing game in fading mutipe-access channes, IEEE Transactions on Information Theory, vo. 54, no. 5, pp , [9] W. Yu, G. Gin, and J. Cioffi, Dtributed mutiuser power contro for digita subscriber ines, IEEE J. Seect. Areas Commun., vo. 20, no. 5, pp , [10] G. Scutari, D. Paomar, and S. Barbarossa, Asynchronous iterative waterfiing for gaussian frequency-seective interference channes, IEEE Transactions on Information Theory, vo. 54, no. 7, pp , juy [11], The MIMO iterative waterfiing agorithm, IEEE Trans. on Signa Processing, vo. 57, pp , May [12] W. Yu, Mutiuser water-fiing in the presence of crosstak, Information Theory and Appications Workshop, San Diego, CA, U.S.A, pp , feb [13] A. iu, Y. iu, H. Xiang, and W. uo, Duaity, poite water-fiing, and optimization for MIMO B-MAC interference networks and itree networks, submitted to IEEE Trans. Info. Theory, Apr. 2010; reved Oct [Onine]. Avaiabe: [14], MIMO B-MAC interference network optimization under rate constraints by poite water-fiing and duaity, IEEE Trans. Signa Processing, vo. 59, no. 1, pp , Jan [Onine]. Avaiabe: [15] E. Teatar, Capacity of muti-antenna gaussian channes, Europ. Trans. Teecommu., vo. 10, pp , Nov./Dec [16] B. Song, R. Cruz, and B. Rao, Network duaity for mutiuser MIMO beamforming networks and appications, IEEE Trans. Commun., vo. 55, no. 3, pp , March [17] S. Boyd and. Vandenberghe, Convex Optimization. Cambridge University Press, [18] A. iu, Y. iu, H. Xiang, and W. uo, Technica report: Poite water-fiing for weighted sum-rate maximization in MIMO B-MAC networks under mutipe inear constraints, Peking University and University of Coorado at Bouder Joint Technica Report, Dec [Onine]. Avaiabe: iue/pubications/index.htm [19] V. Aggarwa, Y. iu, and A. Sabharwa, Sum-capacity of interference channes with a oca view, submitted to IEEE Trans. Inf. Theory, Oct., [20] D. Wei,. Youjian, and B. Rider, Quantization bounds on grassmann manifods and appications to mimo communications, IEEE Trans. Info. Theory, vo. 54, no. 3, pp ,