Duaity, Poite Water-fiing, and Optimization for MIMO B-MAC Interference Networks and itree Networks 1 An Liu, Youjian Eugene) Liu, Haige Xiang, Wu Luo arxiv:1004.2484v3 [cs.it] 4 Feb 2014 Abstract This paper gives the ong sought network version of water-fiing named as poite water-fiing. Unike in singe-user MIMO channes, where no one uses genera purpose optimization agorithms in pace of the simpe and optima water-fiing for transmitter optimization, the traditiona water-fiing is generay far from optima in networks as simpe as MIMO mutiaccess channes MAC) and broadcast channes BC), where steepest ascent agorithms have been used except for the sum-rate optimization. This is changed by the poite water-fiing that is 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 more genera cass of MIMO B-MAC interference networks, which is a combination of mutipe interfering broadcast channes, from the transmitter point of view, and mutiaccess channes, from the receiver point of view, incuding MAC, BC, interference channes, X networks, and most practica wireess networks as specia case. It is poite because it strikes an optima baance between reducing interference to others and maximizing a ink s own rate. Empoying it, the reated optimizations can be vasty simpified by taking advantage of the structure of the probems. Deepy connected to the poite water-fiing, the rate duaity is extended to the forward and reverse inks of the B-MAC networks. As a demonstration, weighted sum-rate maximization agorithms based on poite water-fiing and duaity with superior performance and ow compexity are designed for B-MAC networks and are anayzed for Interference Tree itree) Networks, a sub-cass of the B-MAC networks that possesses promising properties for further information theoretic study. The work was supported in part by NSFC Grant No.60972008, and in part by US-NSF Grant CCF-0728955 and ECCS- 0725915. An Liu Emai: wendao@pku.edu.cn), Haige Xiang, and Wu Luo are with the State Key Laboratory of Advanced Optica Communication Systems & Networks, Schoo of EECS, Peking University, China. Youjian Eugene) Liu is with the Department of Eectrica, Computer, and Energy Engineering, University of Coorado at Bouder, USA. The corresponding author is Wu Luo.
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 2 Index Terms Water-fiing, Duaity, MIMO, Interference Channe, One-hop Network, Transmitter Optimization, Network Information Theory
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 3 I. INTRODUCTION The transmitter optimization of networks is an important but hard probem due to nonconvexity and the ack of understanding of the optima input structure that can be taken advantage of. This paper provides a poite water-fiing input structure that is optima for MIMO mutiaccess channes MAC), broadcast channes BC), and a set of achievabe regions of a more genera cass of MIMO B-MAC interference networks, paving the way for efficient optimization agorithms. A. System Setup The paper considers the optimization of genera one-hop mutipe-input mutipe-output MIMO) interference networks, where each muti-antenna transmitter may send independent data to mutipe receivers and each muti-antenna receiver may coect independent data from mutipe transmitters. Consequenty, the network is a combination of mutipe interfering broadcast channes BC), from the transmitter point of view, and mutiaccess channes MAC), from the receiver point of view, and thus is named the Broadcast-Mutiaccess Channe B-MAC), or the B-MAC Interference) Network. It incudes BC, MAC, interference channes [1] [3], X channes [4], [5], X networks [6], and most practica communication networks, such as ceuar networks, wireess LAN, cognitive radio networks, and digita subscriber ine DSL), as specia cases. Therefore, optimization of such networks has both theoretica and practica impact. We consider a set of achievabe rate regions with the foowing assumptions. 1) The input distribution is circuary symmetric compex Gaussian 1 ; 2) The interference among the inks are specified by any binary couping matrices of zero and one that can be reaized by some transmit and receive schemes. That is, after canceation, the interference of a signa to a ink is the product of an eement of the couping matrix, a channe matrix, and the signa; 3) Each signa is decoded at no more than one receiver. The reasons for considering the above setting are as foows. 1) The setting is optima in terms of capacity region for MAC and BC and aows us to give the optima input structure, poite water-fiing structure, for a boundary points, which had been an open probem since the discovery of the water-fiing structure of the sum-rate optima point [7] [9]; 2) The setting incudes a wide range of interference management techniques as specia cases, such as a) spatia interference reduction through beamforming matrices without 1 The circuary symmetric assumption wi be reaxed in a future work.
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 4 interference canceation, which becomes the spatia interference aignment at high SNR [4], [10] [12]; b) some combination of interference canceation using dirty paper coding DPC) [13] at transmitters and MMSE pus successive interference canceation SIC) at receivers 2 ; c) transmitter cooperation, where a transmitter cance another transmitter s signa using DPC when another transmitter s signa is avaiabe through, e.g., an optica ink between them; 3) The scenario that each signa is decoded at no more than one receiver is usefu in practice where ow compexity is desired and a receiver does not know the transmission schemes of interference from undesired transmitters; 4) Treating interference as noise is optima in the weak interference regime [14] [19] or is asymptoticay optima at high SNR in terms of degree of freedom for some cases, such as MIMO MAC, MIMO BC, two-user MIMO interference channe [20], and some of the MIMO X channes [5]; 5) Limiting to the above setting enabes us to make progress towards the optima soution. The resuts and insight serve as a stepping stone to more sophisticated design. For exampe, the extension to Han-Kobayashi scheme [2], [21], [22], where a signa may be decoded and canceed at more than one receiver is discussed in Remark 5 and in future work at Section VI. B. Singe-user Sefish) Water-fiing The optimization of B-MAC networks has been hindered by that beyond the fu soution in the 1998 semina work [23] on singe antenna MAC, itte is known about the Pareto optima input structure of the achievabe region. Thus, genera purpose convex programming is empoyed to find the boundary points of a MAC/BC capacity region [24]. But it is desirabe to expoit the structure of the probem to design more efficient agorithms that wi work we for both convex and non-convex optimization of networks. For sum-rate maximization in MIMO MAC, a water-fiing structure is discovered in [7]. For individua power constraints, it resuts in a simpe iterative water-fiing agorithm [7]. Using duaity, the approach is modified for sum-rate maximization for BC [8], [9]. The above approach works because the sum-rate happens to ook ike a singe-user rate and thus, cannot be used for the weighted sum-rate maximization. The approach in [8] is generaized for weighted sum-rate maximization with a singe antenna at receivers [25], where the generaized water-fiing structure no onger has a simpe water-fiing 2 Certain combinations of DPC and SIC may resut in partia canceation and thus, are not incuded.
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 5 interpretation. Directy appying singe-user water-fiing to networks is referred to as sefish water-fiing here. It is we known to be far from optima [26] [28] because it does not contro interference to others. Based on sefish water-fiing, game-theoretic, distributed, and iterative agorithms have been we studied for DSL [26], [29] [33], for MIMO interference channes, e.g., [34] [37], and for mutiaccess channes, e.g., [28]. The agorithms converge ony under stringent conditions and the performance is not near optima. The importance of controing interference to others has been recognized in iterature, e.g., [11], [38] [40]. But a systematic, genera, and optima method has not been found. In interference pricing method, each user maximizes its own utiity minus the interference cost determined by the interference prices. With a proper choice of the interference price which can be reverse engineered from the KKT conditions, the interference pricing based method can find a stationary point of the sum utiity maximization probem. Severa monotonicay convergent interference pricing agorithms have been proposed in [41] [43] for the SISO/MISO interference channes, and the MIMO interference channe with singe data stream transmission. Except for the SISO case, a these agorithms update each user s beam sequentiay and exchange interference prices after each update 3. 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 is optima. What is the optima input structure for a boundary points of the MAC/BC capacity region? Does it resembe water-fiing? Can it be used to design agorithms with much ower compexity and better performance than genera purpose optimization agorithms. More ambitiousy, one can ask the above questions for the B-MAC networks with respect to the achievabe region defined above. What is the optima method to contro interference to others? Can we decompose a B-MAC network to mutipe equivaent singe-user channes so that the distributed) optimization can be made easy? 3 In contrast, the poite water-fiing based agorithms presented in this paper take advantage of the structure of the optima input, reative to the achievabe regions, take care of interference to others using duaity, and consider mutipe data streams and various interference canceation.
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 6 This paper gives an optima network version of water-fiing for the above probems. The optimaity is reative to the capacity regions of MAB/BC and the achievabe regions of the genera B-MAC networks. C. SINR and Rate Duaity In this paper, we extend the MAC-BC duaity to the forward and reverse inks of the B-MAC networks. The duaity is deepy connected to the new water-fiing structure and can be used to design efficient iterative agorithms. The extension is a simpe consequence of the signato-interference-pus-noise power ratio SINR) duaity, which states that if a set of SINRs is achievabe in the forward inks, then the same SINRs can be achieved in the reverse inks when the set of transmit and receive beamforming vectors are fixed and the roes of transmit and receive beams are exchanged. Thus, the optimization of the transmit vectors is equivaent to the optimization of the receive vectors in the reverse inks, which has ower compexity. The SINR duaity between MAC and BC was found in [44]. Aternating optimization based on it has been empoyed in [45] [51]. In [52], the SINR duaity is extended to any one-hop MIMO networks with inear beamformers. The MAC-BC rate duaity has been estabished in [53] [56]. The forward ink BC and reverse ink MAC have the same capacity region and the dua input covariance matrices can be cacuated by different transformations. In [54], the covariance transformation is derived based on the fipped channe and achieves the same rates as the forward inks with equa or ess power. The transformation cannot be cacuated sequentiay in genera B-MAC networks, uness there is no oops in the interference graphs as discussed ater. The MAC-BC duaity can aso be derived from the SINR duaity as shown in [55], where a different covariance transformation is cacuated from the MMSE receive beams and the SINR duaity. Such a transformation achieves equa or higher rates than the forward inks under the same sum power and can be easiy generaized to B-MAC networks as foowed in this paper. The above MAC-BC duaity assumes sum power constraint. It can be generaized to a singe inear constraint using minimax duaity and SINR duaity [57], [58]. Efficient interior point methods have been appied to sove optimizations of MAC or BC with mutipe inear constraints in [59], [60]. Expoiting the poite water-fiing structure is expected to produce more efficient agorithms that aso work for the more genera B-MAC networks as discussed ater.
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 7 D. Contributions The foowing is an overview of the contributions of this paper. Duaity: As a simpe consequence of the SINR duaity, we show that the forward and reverse inks of B-MAC networks have the same achievabe rate regions in Section III-A. The dua input covariance matrices are obtained from a transformation based on the MMSE fitering and the SINR duaity. Unike the covariance transformation in [54], which achieves the same rate as the forward inks but with equa or ess power and cannot be cacuated sequentiay for genera B-MAC networks, the transformation in this paper achieves equa or arger rates with the same power and can be cacuated easiy for genera B-MAC networks. We show that the two transformation coincide at the Pareto rate points. Poite Water-fiing: The ong sought network version of water-fiing for a the Pareto optima input of the achievabe rate region of the B-MAC networks is found in Section III-B. Different from the traditiona sefish water-fiing, the poite water-fiing strikes an optima baance between reducing interference to others and maximizing a ink s own rate in a beautifuy symmetric form. Conceptuay, it tes us that the optima method to contro the interference to others is through optima pre-whitening of the channe. It offers an eegant method to decompose a network into mutipe equivaent singe-user channes and thus, paves the way for designing/improving ow-compexity centraized or distributed/gametheoretic agorithms for reated optimization probems, e.g., those in [4], [11], [26] [28], [34], [48], [61]. The poite water-fiing has the foowing structure. Consider ink with channe matrix H, and Pareto optima input covariance matrix Σ. The equivaent input 1/2 covariance matrix Q ˆΩ Σ ˆΩ1/2 is found to be the water-fiing of the pre- and postwhitened equivaent channe H = Ω 1/2 H ˆΩ 1/2,, where Ω is the interference-pus-noise covariance of the forward ink, used to avoid interference from others. The physica meaning of ˆΩ, used to contro interference to others, is two foded. In terms of the reverse ink, it is the interference-pus-noise covariance resuted from the optima dua input. In terms of the forward ink, it is the Lagrangian penaty for causing interference to other inks. The deep connection to duaity is that the optima Lagrange mutipiers work out beautifuy to be the optima dua reverse ink powers. Extension to Singe Linear Constraint: We show that a resuts in the paper, incuding
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 8 duaity, poite water-fiing structure, and agorithms, can be generaized from the case of a sum power constraint and white noise to the case of a singe inear constraint and coored noise in Section III-C. As discussed in Section VI, the singe inear constraint resut can be extended to hande mutipe inear constraints, which arise in individua power constraints, per-antenna power constraints, and interference reduction to primary users in cognitive radios [57], [59], [60]. Weighted Sum-Rate Maximization: In Section IV, highy efficient weighted sum-rate maximization agorithms are designed to iustrate the power of the poite water-fiing. itree Networks: The optimization for B-MAC networks is not convex in genera. To anayze the agorithms and to provide a monotonicay converging agorithm, we introduce the Interference Tree itree) Networks, in Section IV-B. For a fixed interference canceation scheme, itree networks are B-MAC networks that do not have any directiona oops in the interference graphs. It appears to be a ogica extension of MAC and BC. An approach to making progress in network information theory is to study specia cases such as deterministic channes [21], [62] and degree of freedom [5], [21], [63] in order to gain insight. itree networks ooks promising in this sense. The rest of the paper is organized as foows. Section II defines the achievabe rate region and summarizes the preiminaries. Section III presents the theoretica resuts on the duaity and poite water-fiing. As an appication, poite water-fiing is appied to weighted sum-rate maximization in Section IV, where itree networks is introduced for the optimaity anaysis. The performance of the agorithms is verified by simuation in Section V. The resuts of the paper provides insight into more genera probems. They are discussed in Section VI aong with the concusions. II. SYSTEM MODEL AND PRELIMINARIES We first define the achievabe rate region, foowed by a summary of the SINR duaity in [52]. We use for L 2 norm and 1 for L 1 norm. The compex gradient of a rea function wi be used extensivey and is defined as foows. Define fz) : C M N R. The extension of the resuts in [64] gives the gradient of fz) over Z as ) dfz) Z f, dz
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 9 where dfz) dz and z i,j = x i,j + jy i,j, CM N is defined as dfz) dz = f z i,j = 1 f 2 x i,j j f 2 f z 1,1. f z M,1 f z 1,N. f z M,N, y i,j, i, j. If Z is Hermitian, it can be proved that the above formua can be used without change by treating the entries in Z as independent variabes. A. Definition of the Achievabe Rate Region We consider the genera one-hop MIMO interference networks named B-MAC networks, where each transmitter may have independent data for different receivers and each receiver may want independent data from different transmitters. There are L data inks. Assume the set of physica transmitter abes is T = {TX 1, TX 2, TX 3,...} and the set of physica receiver abes is 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 with two inks, 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 L T and L R respectivey. The received signa at R is L y = H,k x k + w, 1) k=1 where x k C L T 1 k is the transmit signa of ink k and is assumed to be circuary symmetric compex Gaussian CSCG); H,k C L R L Tk is the channe matrix between T k and R ; and w C L R 1 is a CSCG noise vector with zero mean and identity covariance matrix. To hande a wide range of interference canceation, we define a couping matrix Φ {0, 1} L L as a resut of some interference canceation scheme. It specifies whether interference is competey canceed or treated as noise: if x k, after interference canceation, sti causes interference to x, Φ,k = 1 and otherwise, Φ,k = 0. Remark 1. The couping matrices vaid for the resuts of this paper are those for which there exists a transmission and receiving scheme such that each signa is decoded and possiby canceed) at no more than one receiver, because in the achievabe rate region defined ater, a rate is
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 10 determined by one equivaent channe. Future extension to the Han-Kobayashi scheme where a common message may be decoded by mutipe receivers is discussed in Remark 5. We give exampes of vaid couping matrices. For a BC MAC) empoying DPC SIC) where the th ink is the th one to be encoded decoded), the couping matrix is given by Φ,k = 0, k and Φ,k = 1, k >. Fig. 1 iustrates a B-MAC network empoying DPC and SIC. The first receiver abeed by R 1 /R 2 is the intended receiver for x 1 and x 2 and the second receiver abeed by R 3 is the intended receiver for x 3. Therefore, x 1 and x 2 is ony decoded at R 1 /R 2 and x 3 is ony decoded at R 3. The foowing Φ a, Φ b, Φ c, Φ d are vaid couping matrices for ink 1, 2, 3 under the corresponding encoding and decoding orders: a. x 3 is encoded after x 2 and x 2 is decoded after x 1 ; b. x 2 is encoded after x 3 and x 2 is decoded after x 1 ; c. x 3 is encoded after x 2 and x 1 is decoded after x 2 ; d. There is no interference canceation. 0 1 1 0 1 1 Φ a = 0 0 1, Φb = 0 0 0, 1 0 0 1 1 0 0 0 1 0 1 1 Φ c = 1 0 1, Φd = 1 0 1. 1 0 0 1 1 0 We give the detais on how to obtain the couping matrix Φ c. At R 1 /R 2, because x 1 is decoded after x 2, we have Φ c 1,2 = 0 and Φ c 2,1 = 1. The interference between x 1 and x 3 can not be canceed by DPC or SIC and thus Φ c 1,3 = 1 and Φ c 3,1 = 1. Note that R 3 is not aowed to decode x 2. The interference from x 2 to x 3 is canceed by DPC at the transmitter T 2 /T 3 rather than by SIC at the receiver R 3. Therefore, we have Φ c 2,3 = 1 and Φ c 3,2 = 0. Remark 2. Some combination of encoding and decoding orders for DPC and SIC may resut in partia interference canceation that can not be described by the couping matrix of 0 s and 1 s. For exampe, in Fig. 1, if x 2 is encoded after x 3 and x 1 is decoded after x 2, the receiver R 1 /R 2 can decode the data embedded in x 2 but can ony reconstruct a signa that is a function of both x 2 and x 3 due to the subtety in DPC, resuting in partia canceation [4], [65]. One may empoy the resuts in this paper to obtain the inner/outer bounds of the achieved rates using pessimistic/optimistic couping matrices. For exampe, a pessimistic couping matrix can assume none of x 2 and x 3 is canceed, and an optimistic couping matrix can assume both x 2 and x 3
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 11 Figure 1. Exampe of a B-MAC network are canceed. It is an interesting future research to design coding techniques to fuy cance the interference from both x 2 and x 3 to x 1. The achievabe rate region in this paper refers to the foowing. Note that Φ, = 0 by definition. The interference-pus-noise of the th ink is L k=1 Φ,kH,k x k + w, whose covariance matrix is L Ω = I + Φ,k H,k Σ k H,k, 2) k=1 where Σ k is the covariance matrix of x k. We denote a the covariance matrices as Σ 1:L = Σ 1, Σ 2,..., Σ L ). Then the mutua information rate) of ink is given by [66] I Σ 1:L, 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 is defined as R Φ P T ) Σ 1:L : L =1 TrΣ ) P T r I Σ 1:L, Φ), 1 L}. { r R L + : 4) Assuming the sum power constraint is the necessary first step for more compicated cases. It has its own theoretica vaue and the resut can be used for individua power constraints and the more genera mutipe inear constraints as discussed in Section III-C and VI. Definition 2. If [I Σ 1:L, Φ)] =1:L is a Pareto rate point of R Φ P T ), the input covariance matrices Σ 1:L are said to be Pareto optima.
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 12 A bigger achievabe rate region can be defined by the convex cosure of Φ Ξ R Φ P T ), where Ξ is a set of vaid couping matrices. For exampe, if DPC and/or SIC are empoyed, Ξ can be a set of vaid couping matrices corresponding to various encoding and/or decoding orders. In this paper, we focus on a fixed couping matrix Φ, which is the basis to study the bigger regions. The optima couping matrix Φ, or equivaenty, the optima encoding and/or decoding order of the weighted sum-rate maximization probem is partiay characterized in Section IV-A. The couping matrix setup impies successive decoding and canceation. If there is no interference canceation at the transmitters, empoying joint decoding at each receiver does not enarge the achievabe regions that we consider, where one message is decoded at no more than one receiver. The reason is as foows. For B-MAC networks with any given set of input covariance matrices, after isoating a MAC at a receiver from the network, the time sharing of different decoding orders achieves the same region as that of joint decoding [67]. And this is achieved without changing the interference to others because the transmission covariance matrices are not changed. The performance gain of joint decoding over that of SIC in genera cases wi be studied in future works. The above definition ensures the foowing. Theorem 1. For each boundary point r of R Φ P T ), there is no r R Φ P T ) satisfying r r and k, s.t. r k > r k, i.e., a boundary points of R Φ P T ) are strong Pareto optimums. The proof is given in Appendix A. In Section III, we estabish a rate duaity between the achievabe rate regions of the forward and reverse inks. 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 is the transpose of that for the forward inks. We use the notation ˆ to denote the terms in the reverse inks. The interference-pus-noise covariance matrix of reverse ink is L ˆΩ = I + Φ k, H ˆΣ k, k H k,. 5) k=1 ) And the rate of reverse ink is given by Î ˆΣ1:L, Φ T = og I + H ˆΣ, H ˆΩ 1,. For a fixed Φ T, the achievabe rate region for the reverse inks is defined as:
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 13 ˆR Φ T P T ) ˆΣ 1:L : L =1 Tr ˆΣ ) P T ˆr Î {ˆr R L + : 6) ˆΣ1:L, Φ T ), 1 L }. B. SINR Duaity for MIMO B-MAC Networks The above achievabe rate region can be achieved by the foowing spatia mutipexing scheme. Definition 3. The Decomposition of a MIMO Link to Mutipe SISO Data Streams is defined as, for ink and any M RankΣ ), where the physica meaning of M is the number of SISO data streams for ink, finding a precoding matrix T = [ p,1 t,1,..., p,m t,m ] satisfying Σ = T T = M m=1 p,m t,m t,m, 7) where t,m C L T 1 is a transmit vector with t,m = 1; and p = [p 1,1,..., p 1,M1,..., p L,1,..., p L,ML ] T are the transmit powers. Note that the precoding matrix aways exists but is not unique because T = T V with unitary V C M M aso gives the same covariance matrix in 7). Without oss of generaity, we assume the intra-signa decoding order is that the m th stream is the m th to be decoded and canceed. The receive vector r,m C L R 1 for the m th stream of ink is obtained by the MMSE fitering as M r,m = α,m i=m+1 H, p,i t,i t,i H, + Ω ) 1 H, t,m, 8) where α,m is chosen such that r,m = 1. This is referred to as MMSE-SIC receiver. For each stream, one can cacuate its SINR and rate. For convenience, define the coections of transmit and receive vectors as T = [t,m ] m=1,...,m,=1,...,l, 9) R = [r,m ] m=1,...,m,=1,...,l. 10)
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 14 Define the R M M + [48] as 1 ) th i=1 M k 1 th i + m row and i=1 M i + n) coumn of a cross-tak matrix Ψ T, R) 0 k = and m n, Ψ k,n,m = r,m H 2,t,n k =, and m < n, r Φ,k,m H 2,kt k,n otherwise. Then the SINR and the rate for the m th stream of ink can be expressed as γ,m T, R, p) = p,m r,m H,t,m 2 1 + L M k p k,n Ψ k,n,m k=1 n=1 11), 12) r,m T, R, p) = og 1 + γ,m T, R, p)). 13) Such decomposition of data to streams with MMSE-SIC receiver is information ossess. Fact 1. The mutua information in 3) is achieved by the MMSE-SIC receiver [68], i.e., it is equa to the sum-rate of a streams of ink : r s M m=1 r,m T, R, p) 14) = I Σ 1:L, Φ). In the reverse inks, we can obtain SINRs using R as transmit vectors and T as receive vectors. The transmit powers is denoted as q = [q 1,1,..., q 1,M1,..., q L,1,..., q L,ML ] T. The decoding order of the streams within a ink is the opposite to that of the forward ink, i.e., the m th stream is the m th ast to be decoded and canceed. Then the SINR for the m th stream of reverse ink is ˆγ,m R, T, q) = q,m t 1 +,m H, r,m 2 L M k q k,n Ψ,m k,n k=1 n=1. 15) For simpicity, we wi use {T, R, p} {R, T, q}) to denote the transmission and reception strategy described above in the forward reverse) inks.
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 15 The achievabe SINR regions of the forward and reverse inks are the same. Define the achievabe SINR regions T Φ P T ) and ˆT Φ T P T ) as the set of a SINRs that can be achieved under the sum power constraint P T in the forward and reverse inks respectivey. For given set of SINR vaues γ 0 = [ ] γ,m 0, define a diagona matrix D T, R, m=1,...,m,=1,...,l γ0 ) R M M + being a function of T, R and γ 0, where the 1 i=1 M i + m) th diagona eement is given by D 1 i=1 M i+m, 1 i=1 M i+m = γ0,m/ We restate the SINR duaity, e.g. [52], as foows. r,m H 2,t,m. 16) Lemma 1. If a set of SINRs γ 0 is achieved by the transmission and reception strategy {T, R, p} with p 1 = P T in the forward inks, then γ 0 is aso achievabe in the reverse inks with {R, T, q}, where the reverse transmit power vector q satisfies q 1 = P T and is given by And thus, one has T Φ P T ) = ˆT Φ T P T ). q = D 1 T, R, γ 0) Ψ T T, R) ) 1 1. 17) III. THEORY In this section, we first estabish a rate duaity as a simpe consequence of the SINR duaity in [52]. Then the poite water-fiing structure and properties of the Pareto optima input are characterized. Finay, we discuss the extension from the sum power constraint and white noise to a singe inear constraint and coored noise. A. A Rate Duaity for B-MAC Networks The rate duaity is a simpe consequence of the SINR duaity. Theorem 2. The achievabe rate regions of the forward and reverse inks of a B-MAC network defined in 4) and 6) respectivey are the same, i.e., R Φ P T ) = ˆR Φ T P T ). Proof: For any rate point r in the region R Φ P T ) achieved by the input covariance matrices Σ 1:L, the covariance transformation defined beow can be used to obtain ˆΣ 1:L such that a reverse ink rate point ˆr r under the same sum power constraint P T can be achieved, according to Lemma 2. The same is true for the reverse inks. Therefore, we have R Φ P T ) = ˆR Φ T P T ).
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 16 Definition 4. Let Σ = M m=1 p,mt,m t,m, = 1,..., L be a decomposition of Σ 1:L. Compute the MMSE-SIC receive vectors R from 8) and the reverse transmit powers q from 17). The Covariance Transformation from Σ 1:L to ˆΣ 1:L is defined as ˆΣ = M m=1 q,m r,m r,m, = 1,..., L. 18) Lemma 2. For any input covariance matrices Σ 1:L satisfying the sum power constraint P T and achieving a rate point r R Φ P T ), its covariance transformation ˆΣ 1:L achieves a rate point ˆr r in the reverse inks under the same sum power constraint. Proof: According to fact 1, r is achieved by the transmission and reception strategy {T, R, p} with p 1 = L =1 Tr Σ ). It foows from Lemma 1 that {R, T, q} with q 1 = L =1 Tr Σ ) can aso achieve the same rate point r in the reverse inks. Because T may not be the optima MMSE-SIC receive vectors, the reverse ink rates may be improved with a better receiver to obtain ˆr r. The foowing coroary foows immediatey from Lemma 2 and Theorem 2. Coroary 1. For any input covariance matrices Σ 1:L achieving a Pareto rate point, its covariance transformation ˆΣ 1:L achieves the same Pareto rate point in the reverse inks. The foowing makes connection of the covariance transformation 18) to the existing ones. First, it is essentiay the same as the MAC-BC transformations in [57] and [69]. The difference is that we do not specify the decomposition of Σ in 7), whie two particuar decompositions are used in [57] and [69]. Second, the MAC-BC transformation in [54] is equivaent to the covariance transformation in 18) at the Pareto rate point. Theorem 3. For any input covariance matrices Σ 1:L achieving a Pareto rate point, its covariance transformation ˆΣ 1:L produced by 18) is the same as that produced by the MAC-BC transformation in [54], i.e., ˆΣ = Ω 1/2 F G ˆΩ 1/2 Σ ˆΩ1/2 G F Ω 1/2, = 1,..., L, 19) where F C L R N, G C L T N are obtained by the thin singuar vaue decomposition SVD): Ω 1/2 H ˆΩ 1/2, = F G ; and N is the rank of H,.
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 17 Proof: The proof reies on Theorem 5 in Section III-B, which states that at the Pareto rate point, both Σ 1:L and its covariance transformation ˆΣ 1:L satisfy the poite water-fiing structure as in 28) and 29) respectivey. Then we have Ω 1/2 ˆΣ Ω 1/2 = F D F = F G G D G G F = F G 1/2 ˆΩ Σ ˆΩ1/2 G F. where the first and ast equations foow from 28) and 29) respectivey. Remark 3. The above two transformations are different at the inner point. The transformation in 18) keeps the sum transmit power unchanged whie not decreasing the achievabe rate, whie the MAC-BC transformation in [54] keeps the achievabe rate the same whie not increasing the sum transmit power. Furthermore, the extension of MAC-BC transformation in [54] to B-MAC networks can ony be sequentiay cacuated for itree networks defined in Section IV-B. B. Characterization of the Pareto Optima Input We show that the Pareto optima input covariance matrices have a poite water-fiing structure defined beow. It generaizes the we known singe-user water-fiing soution to networks. Definition 5. Given input covariance matrices Σ 1:L, obtain its covariance transformation ˆΣ 1:L by 18) and cacuate the interference-pus-noise covariance matrices Ω 1:L and ˆΩ 1:L. For each ink, pre- and post- whiten the channe H, H = Ω 1/2 H ˆΩ 1/2,. Define Q 1/2 ˆΩ Σ ˆΩ1/2 to produce an equivaent singe-user channe as the equivaent input covariance matrix of the ink. The input covariance matrix Σ is said to possess a poite water-fiing structure if Q is a water-fiing over H, i.e., Q = G D G, 20) D = ) ν I 2 +, where ν 0 is the poite water-fiing eve; the equivaent channe H s thin singuar vaue decomposition SVD) is H = F G, where F C L R N, G C L T N, R N N ++ ; and N = Rank H, ). Σ 1:L is said to possess the poite water-fiing structure if a Σ s do.
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 18 For B-MAC, the poite water-fiing structure is proved by observing that the Pareto optima input covariance matrix for each ink is 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,..., L, where p,m > 0,, m, achieves a Pareto rate point [Ĩ > 0] and ˆΣ = M m=1 q,m r,m r,m, = 1,..., L are the corresponding covariance =1,...,L transformation. The corresponding interference-pus-noise covariance matrices are denoted as Ω and ˆΩ. Then it can be proved by contradiction that Σ is the soution of the foowing singe-user optimization probem for ink, where the transmission and reception schemes of other inks are fixed as the Pareto optima scheme, { p k,m, t k,m, r k,m, k }. s.t. min Tr Σ ) 21) Σ 0 og I + H, Σ H 1, Ω Ĩ ) ) Tr Σ A ) k,m Tr Σ A ) k,m 22) m = 1,..., M k, k = 1,..., L, k, ) where A ) k,m = Φ k,h k, r k,m r k,m H k,; and Tr Σ A ) k,m is the interference from ink to the m th ) stream of ink k and is constrained not to exceed the optima vaue Tr Σ A ) k,m. The constraints force the rates to be the Pareto point whie the power of ink is minimized. The Lagrangian of probem 21) is L λ, ν, Θ, Σ ) M k =Tr Σ A λ) Θ)) k + ν Ĩ ν og I + H, Σ H, m=1 ) λ k,m Tr Σ A ) k,m 1 Ω, 23) k M k 1 where the dua variabes ν R + and λ = [λ k,m ] m=1,...,mk,k R+ are associated with the rate constraint and the interference constraints in 22) respectivey; A λ) I + Mk k m=1 λ k,ma ) k,m is a function of λ; Θ is the matrix dua variabes associated with the positive semidefiniteness constraint on Σ. Because probem 21) is convex, the duaity gap is zero [70] and Σ minimizes the Lagrangian 23) with optima dua variabes ν and λ = [ λk,m ] m=1,...,m k,k, i.e., it is the soution of )) min Tr Σ A λ ν og I + H, Σ H, Σ 0 Ω 1. 24)
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 19 Note that in probem 24), the constant terms in the Lagrangian 23) have been deeted, and the term Tr Σ Θ) is expicity handed by adding the constraint Σ 0. The foowing theorem proved in Appendix B states that the physica meaning of the optima dua variabes [ λk,m ] k is exacty the optima power aocation in the reverse inks. Theorem 4. The optima dua variabes of probem 21) are given by λ k,m = q k,m, m = 1,..., M k, k 25) q,m p,m 1 + γ,m ) r,m H 2, t,m ν =, m 26) where γ,m is the SINR of the m th stream of ink achieved by { p k,m, t k,m, r k,m }. Therefore, ) ) A λ = A [ q k,m ] k = ˆΩ. γ 2,m The poite water-fiing structure is shown by a singe-user-channe view using the above resuts. Let H = 1/2 Ω H ˆΩ 1/2, and Q = ˆΩ 1/2 Σ ˆΩ1/2. Since ˆΩ is non-singuar, probem 24) is equivaent to a singe-user optimization probem min Tr Q ) ν og I + H Q H, 27) Q 0 of which ˆΩ1/2 Σ ˆΩ1/2 is an optima soution. Since the optima soution to probem 27) is unique and is given by the water-fiing over H [8], [66], the foowing theorem is proved. Theorem 5. 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 satisfies Q = G D G, 28) D = ν ) I 2 +. Simiary, the corresponding ˆΣ produces ˆQ Ω 1/2 ˆΣ Ω1/2, which satisfies ˆQ = F D F. 29) Remark 4. The insight given by the above proof is that restricting interference to other inks can be achieved by pre-whitening the channe with reverse ink interference-pus-noise covariance
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 20 matrix. And thus, the B-MAC can be converted to virtuay independent equivaent channes H, = 1,..., L. The restriction of interference is achieved in two steps. First, in H = Ω 1/2 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 power to the interfering directions is further reduced by the mutipication of ˆΩ 1/2. Remark 5. The Lagrangian interpretation of ˆΩ Q ˆΩ 1/2, the makes it possibe to extend the duaity and poite water-fiing to Han-Kobayashi transmission scheme. Canceing the interference requires the interference power to be greater than a threshod rather than ess than it. Therefore, some Lagrange mutipiers are negative in A λ). If we sti interpret the Lagrange mutipier as reverse ink power, we must introduce the concept of negative power for the duaity and poite waterfiing for Han-Kobayashi scheme. The matrix ˆΩ ikey remains positive definite. Otherwise, the soution to probem 24) has infinite power, which suggests there is no feasibe power aocation to satisfy a the constraints. Theorem 5 says that at the Pareto rate point, it is necessary that Σ 1:L and ˆΣ 1:L have the poite water-fiing structure. The foowing theorem states that Σ having the poite water-fiing structure suffices for ˆΣ to have the poite water-fiing structure even at a non-pareto rate point. This enabes the optimization of the network part by part. A emma is needed for the proof and reveas more insight to the duaity. Athough the covariance transformation preserves tota power such that L =1 Tr Σ ) = ) L =1 Tr ˆΣ ), in genera, Tr Σ ) = Tr ˆΣ is not true. Surprisingy, ) Tr Q ) = Tr ˆQ, is true as stated in the foowing emma proved in Appendix C. Lemma 3. Let ˆΣ 1:L be the covariance transformation of Σ 1:L. Define the equivaent covariance matrices Q 1/2 ˆΩ Σ ˆΩ1/2 and ˆQ Ω 1/2 ˆΣ Ω 1/2 equivaent covariance matrices of each ink is equa, i.e., Tr Q ) = Tr. The power of the forward and reverse ink ) ˆQ,. Theorem 6. For a given Σ 1:L and its covariance transformation ˆΣ 1:L, if any Σ has the poite water-fiing structure, so does ˆΣ, i.e., ˆQ Ω 1/2 ˆΣ Ω 1/2 is given by water-fiing over the reverse equivaent channe H 1/2 ˆΩ H, Ω 1/2 as in 29). Proof: Because water-fiing uniquey achieves the singe-user MIMO channe capacity [66], Q achieves the capacity of H. Since the capacities of H and H are the same under the same power constraint [66], ˆQ achieves the capacity of H with the same power by Lemma 3 and
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 21 Lemma 2. Therefore, ˆQ is a water-fiing over H. Decompose a MIMO probem to SISO probems can often reduce the compexity. 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 is unique and has an expicit matrix expression. Theorem 7. For any input covariance matrices Σ 1:L satisfying the poite water-fiing structure, its covariance transformation 18) is unique, i.e., for a decompositions of Σ 1:L, it wi be transformed to the same dua input ˆΣ1:L, and vice versa. Furthermore, the dua input ˆΣ1:L satisfies the foowing matrix equation and can be expicity expressed as ˆΩ 1 H, ˆΣ H, = Σ H, Ω 1 H,, = 1,..., L, 30) ˆΣ = ν Ω 1 where ν is the poite water-fiing eve in 20). ) ) 1 H, Σ H, + Ω, = 1,..., L 31) The proof is given in Appendix D. The matrix equation in 30) is natura. At the Pareto rate point, the covariance transformation wi give the same rates for the forward and reverse inks. Hence we have og I + H ˆΣ, H ˆΩ 1, = og I + H, Σ H, Ω 1 1 og I + ˆΩ H ˆΣ I, H, = og + Σ H, Ω 1 H,. 32) Theorem 7 shows that not ony the determinant but aso the corresponding matrices are equa, i.e., ˆΩ 1 H ˆΣ, H, = Σ H, Ω 1 H,. C. Extension to a Singe Linear Constraint and Coored Noise So far we have assumed sum power constraint L =1 Tr Σ ) P T and white noise for simpicity. But individua power constraints is more common in a network, which can be handed by the foowing observation. A resuts in this paper can be directy appied in a arger cass of probems with a singe inear constraint ) L =1 Tr Σ Ŵ P T and/or coored [ ] noise with covariance E w w = W, where Ŵ and W are assumed to be non-singuar 4 4 A singuar constraint or noise covariance matrix may resut in infinite power or infinite capacity.
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 22 and Hermitian. The singe inear constraint appears in Lagrangian functions for mutipe inear constraints, which arise in cases of individua power constraints, per-antenna power constraints, interference reduction in cognitive radios, etc. [57], [58]. Combined with a Lagrange mutipier update, the agorithms in this paper can be generaized to sove the cases of mutipe inear constraints. For a singe inear constraint and coored noise, we denote ) L ) [H,k ], Tr Σ Ŵ P T, [W ], 33) =1 as a network where the channe matrices is given by [H,k ]; the input covariance matrices must satisfy the inear constraint ) L =1 Tr Σ Ŵ P T ; and the covariance matrix of the noise at the receiver of ink is given by W. The extension is faciitated by the foowing emma which can be proved by variabe substitutions. Lemma 4. The achievabe rate region of the network 33) is the same as the achievabe rate region of the network with sum power constraint and white noise ) [ ] L ) W 1/2 H,k Ŵ 1/2 k, Tr Σ P T, [I]. 34) If Σ 1:L achieves certain rates and satisfies the sum power constraint in network 34), Σ 1:L obtained by Σ = Ŵ 1/2 Σ Ŵ 1/2 in network 33) and vice versa. =1, achieves the same rates and satisfies the inear constraint The above impies that the dua of coored noise in the forward ink is a inear constraint in the reverse ink and the dua of the inear constraint in the forward ink is coored noise in the reverse ink as stated in the foowing theorem. Theorem 8. The dua of the network 33) is the network [ ] L ) ] ) H k,, Tr ˆΣ W P T, [Ŵ, 35) =1 in the sense that 1) both of them have the same achievabe rate region; 2) If Σ 1:L achieves certain rates and satisfies the inear constraint in network 33), its covariance transformation ˆΣ 1:L achieves equa or arger rates, satisfies the dua inear constraint in network 35), and ) L =1 Tr ˆΣ W = ) L =1 Tr Σ Ŵ P T.
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 23 Proof: Appy Lemma 4 to networks 33) and 35) to produce a network and its dua with sum power constraint and white noise. Then the resut foows from Lemma 2. Remark 6. For MIMO BC, the duaity resut here reduces to that in [57]. IV. ALGORITHMS In this section, we use the weighted sum-rate maximization to iustrate the benefit of poite water-fiing. We first present the simper case of a sub-cass of B-MAC networks, the interference tree itree) networks, with a concave objective function. Then the agorithm is modified for the genera B-MAC networks. Readers who are interested in agorithm impementation ony may directy go to Section IV-D and read Tabe II for Agorithm PT and Tabe III for Agorithm PP. Agorithm P and P1 are more of theoretica vaue. A. The Optimization Probem and the Choice of the Couping Matrices We consider the foowing weighted sum-rate maximization probem WSRMP) with a fixed couping matrix Φ, WSRMP: gφ) = where w 0 is the weight for ink. max f Σ 1,..., Σ L, Φ) 36) Σ 1:L s.t. f Σ 1,..., Σ L, Φ) L w I Σ 1:L, Φ), =1 Σ 0,, L Tr Σ ) P T, We first take a detour to give a partia characterization of the optima choice of Φ, or equivaenty, the optima encoding order in DPC and decoding order in SIC for B-MAC networks. It is in genera a difficut probem because the encoding and decoding orders need to be optimized jointy. However, for each Pseudo BC transmitter/pseudo MAC receiver in any B-MAC network, the optima encoding/decoding order is easiy determined by the weights and is consistent with the optima order of an individua BC or MAC, as proved in Theorem 9 beow. =1
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 24 Definition 6. In a B-MAC network, a transmitter with a set of associated inks, whose indices forms a set L B, is said to be a Pseudo BC transmitter if the inks in L B either a interfere with a ink k or none of them interfere with the ink k, k L C B, i.e., the coumns of the couping matrix Φ indexed by L B, excuding rows indexed by L B, are the same. A receiver with a set of associated inks, whose indices forms a set L M, is said to be a Pseudo MAC receiver if the inks in L M are either a interfered by a ink k or none of them is interfered by the ink k, k L C M, i.e., the rows of the couping matrix Φ indexed by L M, excuding coumns indexed by L M, are the same. For exampe, if L B, ink is the ast one to be decoded at its receiver, then the corresponding transmitter is a pseudo BC transmitter. If L M, ink is the first one to be encoded at its transmitter, then the corresponding receiver is a pseudo MAC receiver. Theorem 9. In a B-MAC network empoying DPC and SIC with the optima encoding and decoding order π of the foowing probem max π g Φπ)), 37) if there exists a pseudo BC transmitter pseudo MAC receiver), its ink with the n th smaest) weight is the n th one to be encoded decoded). argest Proof: It is proved by isoating the inks of a Pseudo BC transmitter or a Pseudo MAC receiver from the network to form an individua BC or MAC. In the optima soution of 37), {Σ : L B } and π are aso the optima input and encoding order that maximizes the weighted sum-rate of a BC with fixed interference from inks in L C B and under mutipe inear constraints that the interference to inks in L C B must not exceed the optima vaues. The resut on BC with mutipe inear constraints in [71] impies that the optima encoding order is as stated in the theorem. For a pseudo MAC receiver, using simiar method and generaizing the resut in Section III-C to mutipe inear constraints gives the desired decoding order. We briefy discuss the impact of the choice of the couping matrix Φ. The achievabe rate region with any vaid couping matrix is outer and inner bounded by the capacity region and the achievabe rate region without any interference canceation respectivey. In some cases, such as MIMO MAC, MIMO BC, two-user MIMO interference channe [20], and some MIMO X channes [5], the achievabe rate without interference canceation is optima in terms of degree
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, APR. 2010, REVISED MAR. 2011 25 of freedoms DOF) at high SNR, and thus the choice of the couping matrix Φ has no impact on the DOF. For genera cases, it is difficut to anayze the performance of different couping matrices. But the optima encoding/decoding order for each Pseudo BC transmitter/pseudo MAC receiver in Theorem 9 can be used to improve any given orders in a B-MAC network. A the rest of the paper is for a fixed couping matrix Φ and the argument Φ in f Σ 1,..., Σ L, Φ) is omitted. We consider centraized agorithms with goba channe knowedge. Distributed agorithms can be deveoped based on them as discussed in Section VI. B. itree Networks itree networks appears to be a natura extension of MAC and BC. We define it beow. Definition 7. A B-MAC network with a fixed couping matrix is caed an Interference Tree itree) Network if after interference canceation, the inks can be indexed such that any ink is not interfered by the inks with smaer indices. Definition 8. In an Interference Graph, each node represents a ink. A directiona edge from node i to node j means that ink i causes interference to ink j. Remark 7. An itree network is reated to but different from a network whose channe gains has a tree topoogy. A network with tree topoogy is an itree network ony if the interference canceation order is proper. For exampe, a MAC, which has a tree topoogy, is not an itree network if the successive decoding is not empoyed at the receiver. On the other hand, a network with oops may sti be an itree network. We give such an exampe in Fig. 2 where DPC and SIC are empoyed. With encoding/decoding order A, where the signa x 2 is decoded after x 1 and the signa x 3 is encoded after x 2, each ink {2, 3, 4} is not interfered by the first 1 inks. Therefore, the network in Fig. 2 is sti an itree network even though it has a oop of nonzero channe gains. However, for encoding/decoding order B, SIC is not empoyed at R 1 /R 2, and x 2 is encoded after x 3 at T 2 /T 3. The network in Fig. 2 is no onger an itree network because the interference graph has directiona oops. itree networks can be equivaenty defined using their interference graphs. Lemma 5. A B-MAC network with a fixed couping matrix is an itree network if and ony if after interference canceation, its interference graph does not have any directiona oop.