Distributed MIMO Network Optimization Based on Duality and Local Message Passing

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1 Forty-Seventh Annual Allerton Conference Allerton House, UIUC, Illinois, USA September 30 - October 2, 2009 Distributed MIMO Network Optimization Based on Duality and Local Message Passing An Liu 1, Ashutosh Sabharwal 2, Youjian Liu 3, Haige Xiang 1, Wu Luo 1 1 State Key Laboratory of Advanced Optical Communication Systems & Networks, School of EECS, Peking University 2 Department of ECE, Rice University 3 Department of Electrical and Computer Engineering, University of Colorado at Boulder 1 Corresponding Author: luow@pku.edu.cn Abstract In a communication network, it is often impractical for each node to learn the global channel knowledge (network connectivity and channel state information of each link). In this paper, we address distributed rate optimization for Time-Division Duplex (TDD) Multiple-Input Multiple-Output (MIMO) networks when part of the local channel knowledge is learned via message passing between each transmitter and its intended receivers. The distributed optimization algorithm is based on a rate duality and the corresponding input covariance matrix transformation between the forward and reverse links of TDD MIMO networks under the assumption of global channel knowledge. Noting that the key information required by the proposed transformation is the interference-plus-noise covariance matrix, we propose a local covariance matrix transformation such that each node can distributedly optimize its input covariance matrix by only exchanging interferenceplus-noise covariance matrix locally. It is observed from the simulation that the proposed algorithm achieves a performance close to the one with global channel knowledge and outperforms the existing distributed algorithms. Index Terms Multiple-Input Multiple-Output (MIMO) network, Local message Passing, Duality, Rate optimization I. INTRODUCTION The capacity of Multiple-Input Multiple-Output (MIMO) network heavily depends on the available channel knowledge [1], [2]. Most work on the capacity analysis of MIMO networks assume global channel knowledge of the whole network. It is possible for multiple access channel (MAC) and broadcast channel (BC) that the central node learns the global channel knowledge. However, in distributed MIMO networks, nodes often do not know the global channel knowledge. Inspired by belief propagation algorithm commonly used in LDPC decoding, a message passing algorithm is proposed in [3] for nodes to incrementally learn the channel knowledge (network connectivity and channel state information of each link). The message is chosen to be the channel state information (CSI) of the connected links and the CSI propagated from other links. However, for a MIMO network with time varying CSI and/or a moderate or large number of nodes, the cost is high to propagate CSI through all connected links. Therefore in this paper, we study a different kind of messages, each of The work was supported in part by NSFC Grant No , and in part by US-NSF Grant CCF and ECCS /09/$ IEEE 795 which is the current interference-plus-noise covariance matrix of a desired link, for time division duplex (TDD) MIMO networks. Instead of propagating the matrices along the network, the messages are between the transmitters and their desired receivers and are not accumulative. This reduces the load of message passing especially when the number of the desired links with active data transmission is small compared to the number of the interfering links. The distributed rate optimization algorithm relies on a rate duality between the forward and reverse links of TDD networks. There are already a number of works on the duality of MIMO networks. The duality between MIMO MAC and MIMO BC capacity region is established in [4], [5]. The signal-to-interference-and-noise power ratio (SINR) duality between MIMO MAC and MIMO BC has also been established and used to perform resource allocation [6] [8]. Based on the SINR duality, a general rate duality and the corresponding input covariance matrix transformation is presented in [9] between the MAC and the BC, which is applicable to systems with and without nonlinear interference cancellation. The transformation in [9] is elegant in the sense that the capacity can be achieved without intra-user successive interference cancellation. A significant progress was achieved in [10], where the SINR duality between MAC and BC is extended to any MIMO network with linear beamformers by the duality of linear programming. Then the network duality is used to solve the joint MIMO beamforming and power control problem. It also presented a distributed version of the algorithm tailored for a cellular downlink. Most SINR duality based works above focus on beamforming, where the transmit signal dimension of each link is fixed as one. Although in some works, multiple beams are used for each link, the objective is usually to optimize the SINR of each beam rather than the rate of each link. Few works consider the optimization of the number of beams and resource allocation over the beams of each link such that the rate of each link is optimized. Recently, closed form interference alignment schemes are proposed in [11] to optimize degree of freedom at high SNR. In [12], the duality of the interference alignment is established and used to design distributed interference alignment achieving nearly the same performance as the one with global channel knowledge. However, how to choose the

2 number of beams for each link such that the interference alignment is always feasible is still an open problem. In this paper, we establish a duality of an achievable region for general one-hop MIMO networks, where the achievable region is defined as the set of all rates that can be achieved by nonlinear interference cancellation at both the transmitters (dirty paper coding) and receivers (successive interference cancellation), under sum power constraint. The corresponding input covariance matrix transformation is also provided. In [13], we show that, at the boundary point of the capacity region for the special case of MAC and BC, the proposed transformation gives the same solution as the one developed in [4]. However, at the inner point of the capacity region, they are different because the proposed transformation keeps the sum transmit power unchanged while not decreasing the achievable rate, and the transformation in [4] keeps the achievable rate the same while not increasing the sum transmit power. We also show that the transformation in [9] can be directly derived from the proposed transformation. A common disadvantage of these transformations are the requirement of the global channel knowledge. We modify the proposed transformation such that each node can distributedly calculate the input covariance matrix by only exchanging interference-plus-noise covariance matrix locally. A distributed rate optimization algorithm is proposed based on the modified input covariance matrix transformation.the main advantages of the proposed algorithm over the distributed algorithms in [10] and [12] is that the number of data streams, or number of beams, of a user is automatically optimized during the iterations and a better performance can be achieved as shown by the simulation results. The rest of the paper is organized as follows. Section II defines an achievable region of MIMO networks by precoding with nonlinear interference cancellation. Section III presents the distributed rate optimization algorithm based on the proposed rate duality and local message passing. The performance of the proposed algorithm is verified in section IV and the conclusion is given in section V. II. SYSTEM MODEL We first define an achievable region for general one-hop MIMO network, where each transmitter may send independent information to multiple receivers.then we summarize the corresponding SINR duality. A. Definition of the Achievable Region Consider a MIMO network comprised of K transmitters and L receivers, where the k th transmitter sends independent information to a subset of the L receivers denoted as Sk T. For convenience, denote Sl R as the set of transmitters sending useful information to the l th receiver. All transmitters and receivers are equipped with L T and L R antennas respectively 1. The received signal at the l th receiver can be expressed as 1 We assume different transmitters/receivers have the same numbers of transmit/receive antennas in this section only for simplicity of notations. The results in this paper also hold for arbitrary number of antennas. 796 y l = K H k k =1 l S T k,l x k,l + w l, (1) where for l Sk T, x C L T 1 and H C L R L T respectively are the transmit signal and the channel matrix from the k th transmitter to the l th receiver, and w l C L R 1 is the Gaussian noise vector with zero mean and unit covariance matrix. Now we will define an achievable region by applying dirty paper coding (DPC) at the transmitters and successive decoding at the receivers. At the k th transmitter, the encoding order is denoted by πk e(l), l ST k, which means that x is the πk e(l)th one to be encoded at the k th transmitter. For the l th receiver, the decoding order is denoted by πl d(k), k SR l, which means that x is the πl d(k)th one to be decoded at the l th receiver. For simplicity of notations, we let Sk T = {1,, L}, k = 1,, K and Sl R = {1,, K}, l = 1,, L in the rest of the paper. All results also hold for arbitrary Sk T and SR l. Now define,l Φk for any 1 k, k K and 1 l,l L such that if x k,l causes interference to x, Φ k,l = 1, otherwise, Φ k,l = 0. 0 if k = k and πk e (l) > πe k ) (l Φ k,l 0 if l = l and πl d = (k) > πd l ) (k (2) 0 if k = k and l = l 1 otherwise 1 k,k K,1 l,l L, { } Let Σ = E x x C L T L T be the covariance matrix of x. The interference-plus-noise of the signal x is K L,l k =1 l =1 Φk H k,l x k,l + w l, whose covariance matrix is Define Ω = I + K k =1 l =1 L Φ k,l H k,l Σ k,l H k,l. (3) Σ 1:KL = {Σ, k = 1,, K,l = 1,, L} (4) as the set of input covariance matrices. Then the mutual information (rate) corresponding to x is given by a function of Σ 1:KL [14] I (Σ 1:KL ) = log I + H Σ H Ω 1. (5) Let π = { π e k (.), πd l (.), k = 1 K, l = 1 L} denote the set of encoding and decoding orders. Then for a fixed π, we define an achievable region to be the set of all mutual information that can be achieved under the sum power constraint P T : R π (P T ) Σ 1:KL : K k=1 { r R KL L l=1 Tr(Σ + : (6) ) P T r I (Σ 1:KL ),1 k K,1 l L}.

3 Now consider the reverse links of the above channel, where the receivers and the transmitters are exchanged, and the channel matrices are conjugate transpose of the corresponding forward channel matrices. For the k th receiver, the decoding order ˆπ k d(.) is set as the inverse order of πe k (.), i.e., ˆπ k d(l) = πe k (L l + 1), l = 1,, L. For the lth transmitter, the encoding order ˆπ l e (.) is set as the inverse order of πl d(.), i.e., ˆπe l (k) = πd l (K k + 1), k = 1,, K. Denote ˆx C LR 1 as the signal from the l th transmitter to the k th receiver in the reverse links. Let ˆΣ { } = E ˆx ˆx C L R L R be the input covariance matrix of ˆx. Then the interference-plus-noise covariance matrix corresponding to ˆx is ˆΩ = I + L K l =1 k =1 Φ k,l H ˆΣ k,l H. Define } ˆΣ 1:KL = {ˆΣ, k = 1,, K,l = 1,, L. Then the mutual information (rate) corresponding to ˆx is given by [14] ) Î (ˆΣ1:KL = log I + H ˆΣ H ˆΩ 1. Let ˆπ = {ˆπ e l (.), ˆπd k (.), l = 1 L,k = 1 K} denote the set of encoding and decoding orders for the reverse links. Then for a fixed ˆπ, the achievable region for the reverse links is defined as the set of all mutual information that can be achieved under the sum power constraint P T : ˆRˆπ (P T ) ˆΣ 1:KL : K k=1 ˆr Î {ˆr R KL L l=1 Tr(ˆΣ + : (7) ) P T (ˆΣ1:KL ),1 k K,1 l L }. Note that if the duality holds for the rate regions defined in (6) and (7), i.e., R π (P T ) = ˆRˆπ (P T ), then it also holds for the larger achievable regions defined by a convex closure of the union of the rate regions corresponding to all possible encoding and decoding orders as follows, R(P T ) = convex closure π R π (P T ) ˆR(P T ) = convex closure ˆπ ˆRˆπ (P T ), i.e., R(P T ) = ˆR(P T ). Therefore in this paper, we will focus on the R π (P T ), the region R(P T ) can be easily drawn from R π (P T ). B. Precoding with Nonlinear Interference Cancellation Any point in the region R π (P T ) can be achieved by precoding with nonlinear interference cancellation as follows. Decompose the input covariance matrices as Σ = T P T, k = 1,, K,l = 1,, L, where T = [t,1,,t,m ] C L T M, t,m = 1, m is the normalized precoding matrix, P = diag (p ) C M M is the positive diagonal power allocation matrix, p = [p,1,, p,m ] C M 1 is the power allocation vector, and M is the number of streams which is no less than the 797 rank of Σ 2. Note that the decomposition is not unique. Let T = T P 1 2, then any T = T V also satisfy the decomposition Σ = T ), ( T where V C M M is a unitary matrix. In this way, the signal x is divided into M streams. When decoding x, after subtracting the already decoded signals, successive decoding is again used to decode the M streams of x. Without loss of generality, we assume the intra-signal decoding order is π M (m) = m, m = 1,, M, i.e., the first stream is decoded first and the M th stream is decoded last. The receive vector for the m th stream of x is obtained by MMSE criteria as r,m (8) =α,m ( M i=m+1 H t,i t,i H + Ω ) 1 H t,m, where t,i = p,i t,i, and α,m is chosen such that r,m = 1. This is called MMSE-SIC (MMSE receiver combined with Successive Interference Cancellation) receiver in this paper. For convenience, define R = [r,1,,r,m ], (9) R = {R kl, k = 1,, K,l = 1,, L}, (10) p = [p 11,,p 1L,,p K1,,p KL ] T, (11) T = {T, k = 1,, K,l = 1,, L}. (12) where R C LR M is the normalized MMSE-SIC receive matrix for x, p is the overall power allocation vector, T and R are the sets of precoding and MMSE-SIC receive matrices respectively. Define a non-negative coupling matrix Ψ R KLM KLM + being a function of T,R, where the element at the ((k 1)LM + (l 1)M + m) th row and ( (k 1)LM + (l 1)M + m ) th column of Ψ is Ψ k,l,m,m 0 k = k, l = l and m m = r,m H t,m 2 k = k, l = l and m < m r,m H k,l t 2 k,l,m otherwise Φ k,l (13) where Φ k,l is defined in (2). The coupling matrix Ψ describes the cross-talk between different streams [6]. Then the SINR and the rate of the m th stream of x can be expressed as a function of T, R and p γ,m (T,R,p) = 1 + r p,m,m H 2 t,m K L M k =1l =1 m =1 p k,l,m Ψk,l,m,m (14) r,m (T,R,p) = log (1 + γ,m (T,R,p)), (15) 2 Here, we assume M is no less than the maximum rank of all input covariance matrices so that all precoding matrices can have the same dimension for the simplicity of notations.

4 The sum rate of all M streams of x is r s (T,R,p) = M r,m (T,R,p), (16) m=1 For MMSE-SIC receiver, the following fact holds [15]. Fact 1: For a given set of input covariance matrices Σ 1:KL and any decomposition: Σ = T P T, k = 1,, K,l = 1,, L, the sum rate in (16) achieved by the MMSE-SIC receiver equals to the mutual information in (5), i.e., r s (T,R,p) = I (Σ 1:KL ). Further more, the achieved rate r s is the same for different intra-signal decoding orders π M. Similarly, in the reverse links, we use R and T as the normalized precoding and receive matrix respectively. We change the power allocation vector to q = [q,1,, q,m ] T. The intra-signal decoding order is denoted as ˆπ M (m) = M m + 1, m = 1,, M, which has an inverse order of π M. Then for the reverse links, the SINR and the rate of the m th stream of ˆx can be expressed as t q,m,m H r,m 2 ˆγ,m (R,T,q) = K L M 1 + q k,l,m Ψ,m k,l,m k =1l =1 m =1 (17) ˆr,m (R,T,q) = log (1 + ˆγ,m (R,T,q)), (18) where q = [ q T 1,1,,q T 1,L,,qT K,1,,qT K,L] T is the overall power allocation vector. For simplicity, we will use {T,R,p} /{R,T,q} to denote the transmission and reception strategy with encoding and decoding orders {π, π M } / {ˆπ, ˆπ M } as described above in the forward/reverse links. C. SINR Duality for MIMO network { For given set of SINR values γ0 = γ,m }, 0,1 k K,1 l L,1 m M precoding and receive matrices {T,R}, define a diagonal matrix D R+ KLM KLM, where for any 1 k K, 1 l L, and 1 m M, the [(k 1)LM + (l 1)M + m] th diagonal element of D is defined as D,m = γ 0,m r,m H 2. (19) t,m Then the SINR duality for MIMO network is stated in the following Lemma. Lemma 1: Set the elements of γ 0 as the SINRs achieved by the transmission and reception strategy {T,R,p} with p 1 = P T in the forward links, i.e., let γ 0,m = γ,m (T,R,p),1 k K,1 l L,1 m M. Then the same SINRs can be achieved in the reverse links using {R,T,q}, i.e., ˆγ,m (R,T,q) = γ 0,m,1 k 798 K,1 l L,1 m M, where the power allocation q satisfies q 1 = P T and is given by q = ( D 1 Ψ T (T,R) ) 1 1. (20) The SINR duality for any wireless network is proved in [10] without nonlinear interference cancellation. With nonlinear interference cancellation, the only difference is that some elements of the coupling matrix Ψ is set to zero because of the interference cancellation. However, the proof in [10] did not rely on the specific value of Ψ as long as it is nonnegative. Therefore, the same proof holds for lemma 1. The duality of the achievable region for any MIMO network can be easily derived from the SINR duality. III. DISTRIBUTED MIMO NETWORK OPTIMIZATION We first present the rate duality and the corresponding input covariance matrix transformation for any MIMO network. Then we propose a distributed rate optimization algorithm based on a distributed version of the proposed transformation using local message passing. A. Rate Duality The proofs of all the following results are given in [13]. The rate duality is obtained by showing that for any set of rates achievable in the forward links, a set of equal or larger rates is achievable in the reverse links with the same sum power constraint, and vice versa. The key is the input covariance matrix transformation stated in the following lemma. Lemma 2: For any set of input covariance matrices Σ 1:KL satisfying the sum power constraint P T and achieving some rate point r in the region R π (P T ), the transformation that gives the input covariance matrices ˆΣ 1:KL satisfying the same sum power constraint P T and achieving a rate point ˆr r in the region ˆRˆπ (P T ) is given by ˆΣ = R diag (q )R,k = 1,, K,l = 1,, L, (21) where R, k = 1,, K,l = 1,, L is the MMSE- SIC receive matrices defined in (9) corresponding to some decomposition of the input covariance matrix: Σ = T P T, k = 1,, K,l = 1,, L, and q is given by (20). Proof: Suppose Σ 1:KL achieving some rate point r in the region R π (P T ) with sum power constraint P T. According to fact 1, the transmission and reception strategy {T,R,p} with p 1 = P T can achieve the same rate point r, where T, R and p are defined in (12), (10) and (11) respectively. Then it follows from lemma 1 that {R,T,q} with q 1 = P T can also achieve the same rate point r in the reverse links, where q is given by (20). Note that T may not be the set of MMSE-SIC receive matrices for the reverse links corresponding to the transmission scheme R and q.

5 Therefore, the set of rates achieved by the input covariance matrices in (21) in the reverse links must be larger or equal to r. Theorem 1: For any MIMO network, the achievable regions of the forward and reverse links defined in (6) and (7) respectively are the same under the same sum constraint P T, i.e., R π (P T ) = ˆRˆπ (P T ). Note that for different decompositions of the input covariance matrices, R and q are different in general. Hence the transformation in (21) may not be unique. However, for the transformation at the boundary point of the achievable region, we have stronger results as stated in the following theorem. Theorem 2: For a given set of input covariance matrices Σ 1:KL achieving some Pareto point, the solution of the covariance matrix transformation in (21) satisfies the following matrix equations ˆΩ 1 H ˆΣ H = Σ H Ω 1 H (22) 1 k K,1 l L. Further more, the covariance matrix transformation in (21) is unique, i.e., for all decompositions of Σ 1:KL in the forward links, it will be transformed to the same ˆΣ 1:kL in the reverse links, and vice versa. The transformation equations in (22) are simple and natural. At the boundary point, the covariance transformation will give the same set of rates for the forward and reverse links. Hence we have log I + H ˆΣ H ˆΩ 1 = log I + H Σ H Ω 1 log I + ˆΩ 1 H ˆΣ H = log I + Σ H Ω 1. (23) H The theorem shows that not only the determinant but also the corresponding matrices are equal, i.e., ˆΩ 1 H ˆΣ H = Σ H Ω 1 H. B. Distributed Rate Optimization 1) Problem Formulation: The problem is formulated as a multi-objective optimization problem which simultaneously optimizes the rates of all links for fixed encoding and decoding orders subject to the individual power constraint as follow. max {I (Σ 1:KL ),k = 1,, K,l = 1,, L} (24) Σ 1:KL s.t. Σ 0, Tr (Σ ) P, 0 k = 1,, K,l = 1,, L, The solution to problem (24) is a set of Pareto points of R π (P T ). Pareto solutions are those for which improvement in one rate can only occur with the worsening of at least one other rate. To solve problem (24), we propose a distributed adaptive algorithm which tries to increase the rates of all links in each updating of the input covariance matrices until one of the Pareto points is achieved ) Assumptions on the distributed MIMO Network Model: Assume ideal TDD MIMO network, i.e., the channel matrices in the reverse links are conjugate transpose of the corresponding forward channel matrices. Assume block fading channel and each block is divided into several rounds, where one round consists of a half round of transmission in the forward links (forward round) followed by a half round of transmission in the reverse links (reverse round). The following local channel state information (CSI) is assumed for the link corresponding to x (ˆx ). Both the transmitter and the receiver have perfect channel state information about H. The interference-plus-noise covariance matrix Ω (ˆΩ ) is supposed to be perfectly estimated by the l th (k th ) receiver in the forward (reverse) round. A local message passing strategy is applied where the transmitter learns the interference-plus-noise covariance matrix at the previous round from the corresponding receiver. 3) Benefit of the Duality with Global knowledge: Note that when we iteratively apply the proposed covariance matrix transformation (21) 3 from forward (reverse) to reverse (forward) links back and forth, the rate of each link is monotonically increasing from one iteration to the next. Since the rate is obviously bounded due to the sum power constraint, the rate of each link converges to some local optimum. This method can be used to achieve some Pareto boundary point of the achievable region if it converges to the global optimum. The gain in each iteration is due to the following two operations. MMSE improves the SINRs in both the forward and reverse links (only with local CSI) When switching from the forward links to the reverse links, the SINRs are kept unchanged by solving the following SINR balancing problem 4 (with global channel knowledge) [6] max min ˆγ,m (R,T,q) q γ,m 0, s.t. q 1 P T, 1 k K, 1 l L, 1 m M (25) Note that the power allocation in (20) is actually the unique solution of the SINR balancing problem (25). By the above two operations, the duality converts the optimization of the transmit vectors to the optimization of receive vectors, which can be solved much more easily by using MMSE receiving. However, solving (25) requires global channel knowledge. Therefore, we propose a heuristic algorithm to solve (24) based on a distributed version of the transformation in (21). 4) Operation at each Node: Here we only give the operation for x at the l th receive node in the forward links. The 3 The transformation from the reverse links to forward links is obtained by exchanging the roles of the transmit and the receive vectors in the forward links, replacing the channel matrices by their conjugate transpose, and replacing the coupling matrix by its transpose Ψ T. 4 The SINRs are kept unchanged similarly when switching from the reverse links to the forward links.

6 operations at other nodes are similar. The node performs a distributed version of the transformation in (21) as follows. 1. In the forward round, estimate the interference-plusnoise matrix and calculate the normalized MMSE-SIC receive matrix R in (21) with local CSI. 2. In the reverse round, use R as the normalized precoding matrix. Solve the optimal solution q of the following local SINR balancing problem to obtain the power allocation q. where max γ 0,m = γ,m = min ˆγ,m q 1 m M γ,m 0, s.t. q 1 P 0 (26) p,m r,m H t,m 2 r,m Ω r,m + M m =1 is the SINRs achieved in the forward links, and ˆγ,m = q,m r,m H t,m 2 t,mˆω t,m + p,m Ψ,m,m M q,m Ψ,m m =1,m (27) (28) is the SINRs achieved in the reverse links. Note that γ,m can be calculated at the l th receiver in the forward links using the above local CSI. However when switching to the reverse links, neither the l th transmitter nor the k th receiver knows ˆΩ before actual transmission. In the algorithm, ˆΩ is replaced by the interference-plus-noise covariance matrix at the previous round. Let Ψ be an M M matrix where the element [ at the m th row and m th column is Ψ,m,m. Let ˆσ = t,1ˆω t,1,,t ˆΩ,M t,m ]. Define the following matrices γ,m 0 D = diag r,m H 2, 1 m M t,m [ Λ = D Ψ T 1 P1 T D 0 Ψ T D ˆσ 1 P1 T D ˆσ 0 ],, (29) where D R+ M M, 1 = [1,,1] T R+ M 1, Λ R (M+1) (M+1) +. Then the optimal solution of problem (26) is obtained by solving the following eigensystem [6] Λ q ext = ˆλ q ext with [ q ext ] = 1, (30) M+1 where ˆλ is the maximum eigenvalue; q ext = [ ( ] T T q),1 is the corresponding dominant eigenvector of Λ, which is scaled so that its last component equals one; q is the optimal solution of problem (26). Then the covariance matrix transformation at the node using only local CSI is ˆΣ = R diag ( q ) R. (31) Note that the transformation in (31) satisfies the individual power constraint. 5) Summary of the Distributed Optimization Algorithm: Within each block, for each signal x (ˆx ), the input covariance matrix is updated iteratively using the local covariance matrix transformation in (31) in a round by round fashion as follows. 1. Let M = min (L T, L R ) and i = 1. In the first forward round, the covariance matrices Σ (1) is randomly generated at ( the k ) th transmitter, where the rank of Σ (1) is M and Tr Σ (1) = P In the i th reverse round, the l th transmitter calculates (i) the input covariance matrices ˆΣ using the transformation in (31) with the local CSI: Σ (i), H, Ω (i) and ˆΩ (i 1), where Ω (i) is the interference-plus-noise covariance matrix estimated by the l th receiver in the i th forward round, ˆΩ (i 1) is the interference-plus-noise covariance matrix estimated by the k th receiver in the (i 1) th reverse round and is learned in the i th forward round (see step 3). For i = 1, we let ˆΩ (0) = I. Then the l th transmitter sends Ω (i) to the k th receiver. 3. In the (i + 1) th forward round, the k th transmitter calculates the input covariance matrices Σ (i+1) using the transformation similar to (31) 5 (i) with the local CSI: ˆΣ, H, ˆΩ (i) and Ω(i), where Ω(i) is learned in the ith reverse round (see step 2). Then the k th transmitter sends ˆΩ (i) to the lth receiver Let i = i + 1 and enter the next round. Keep updating Σ (i) (i) and ˆΣ until it converges. Remark 1: We do not explicitly optimize the number of data streams M for each signal. It is observed from the simulation that as long as M is larger that the rank of the optimal input covariance matrix, the rank of Σ (i) will automatically converge to the optimal value 6. This is an advantage over the algorithms in [12], whose performance heavily depends on the choice of M, which is in general difficult to optimize. Remark 2: On the i th round, the interference-plus-noise covariance matrix is a function of the CSI within 2i hops away. Therefore, each node incrementally and implicitly learns the network state information round after round. Remark 3: The receiver can calculate the transmission strategy by exchanging the interference-plus-noise covariance matrix with the transmitter. Therefore, it is more efficient than exchanging transmission strategy. 5 As already explained in footnote 3. 6 This is related to the fact that for any input covariance matrix Σ and any integer M no less than the rank of Σ, we can always find a decomposition Σ = TPT such that T C L T M and P C M M.

7 Improvement by local SINR balancing Improvement by MMSE receiving 9 Improvement by equal power allocation Improvement by MMSE receiving The average rate of User 2 (bit/channel use) The average rate of User 2 (bit/channel use) The average rate of User 1 (bit/channel use) The average rate of User 1 (bit/channel use) Figure 1. Improvements of the proposed algorithm after each round for a 2 user interference channel with L T = L R = 3 Figure 2. Improvements of Max-SINR after each round for a 2 user interference channel with L T = L R = 3 Remark 4: Although block fading channel is assumed in the paper, this algorithm can also be used to track the time varying channel. IV. SIMULATION RESULTS In this section, numerical simulations are used to verify the performance of the proposed algorithm. We assume block fading channels and each channel matrix H has zero-mean i.i.d. Gaussian entries with unit variance. Each active data transmission is assumed to subject to the same individual power constraint P T /N s, where N s is the number of active data transmissions and P T is the total transmit power. Each simulation is done over 1000 randomly generated channel realizations. We compare the performance of the proposed algorithm with the following algorithms. The algorithm based on the proposed covariance matrix transformation with global CSI in [13] is called centralized algorithm. The IWFA algorithm is the iterative waterfilling in [16]. Max-SINR is the Algorithm 2 proposed in [12]. The MMK scheme is the joint scheme proposed in [17]. Both the proposed algorithm and the Max-SINR use MMSE receiver to improve the SINRs by switching between the forward and reverse links back and forth. The main difference is that when switching to the reverse (forward) links, the proposed algorithm tries to keep the SINR improvement by local SINR balancing. But the Max-SINR use equal power allocation over the streams of the same link, which is likely to decrease the SINR improvement. Fig. 1 plots the improvement by MMSE receiving and local SINR balancing of the proposed algorithm after each round for a 2 user interference channel. The local SINR balancing actually improves instead of preserving the average rates of all users in the first few rounds. Fig. 2 plots the rate improvement by MMSE receiver and equal power allocation of Max-SINR after each round. It is observed that equal power allocation decreases the average rates of all users in the first few rounds. Average Sum Rate (bit/channel use) Proposed Alg. with M = min(lt,lr) Max SINR with M = Mopt Max SINR with M = min(lt,lr) Number of Rounds Figure 3. Convergence behaviors of the proposed algorithm and Max-SINR for a 3 user interference channel with L T = L R = 3 The convergence behaviors of the proposed algorithm and Max-SINR are displayed in Fig. 3 for a 3 user interference channel. The total transmission power is 10dB. In the figure, M = M opt means that the number of data streams is optimized for each link by exhaustive search, and M = min (L T, L R ) means that the number of data streams is fixed as min (L T, L R ). It is observed that the proposed algorithm converges faster than Max-SINR and most gain comes from the first few rounds. In Fig. 4, we compare the sum rate achieved by different algorithms for a 3 user interference channel. It can be seen that the sum rate achieved by the proposed algorithm is close to that achieved by the centralized algorithm, which is higher than that achieved by all other algorithms. And the performance of Max-SINR heavily depends on the choice of M, which is difficult to optimize. Note that the performance 801

8 Average sum Rate (bit/channel use) Centralized Alg. Proposed Alg. with M = min(lt,lr) Max SINR with M = Mopt IWFA Max SINR with M = min(lt,lr) Total transmit power (db) Figure 4. Performance comparison for a 3 user interference channel with L T = L R = 3 Average sum Rate (bit/channel use) Centralized Alg. Proposed Alg. with M = min(lt,lr) Max SINR with M = Mopt MMK IWFA Max SINR with M = min(lt,lr) Total transmit power (db) Figure 5. Performance comparison for a two user X channel with L T = L R = 3 gain over Max-SINR is due to the power control by local SINR balancing. To make fair comparison, the same nonlinear interference cancellation is also used in Max-SINR. Fig. 5 compares the sum rate for a 2 user X channel [17]. Again, similar results as in Fig. 4 is observed. V. CONCLUSION In this paper, an efficient distributed rate optimization algorithm is proposed for MIMO TDD network by combining the ideas of local message passing and the duality. The local message passing allows nodes to incrementally learn the network state information. The proposed rate duality and the corresponding input covariance matrix transformation are obtained by a simple extension of the SINR duality in [10]. Although the proposed transformation needs global channel knowledge, we modify it such that each node can do the transformation locally with channel knowledge learned from 802 local message passing. Then the modified transformation is used to design distributed algorithm which tries to increase the rates of all links in each update of the input covariance matrices until one of the Pareto points is achieved. The proposed algorithm is shown by simulation to outperform the existing distributed algorithms. REFERENCES [1] I. E. Telatar, Capacity of multi-antenna gaussian channels, Europ. Trans. Telecommu., vol. 10, pp , Nov./Dec [2] A. Goldsmith, S. Jafar, N. Jindal, and S. Vishwanath, Capacity limits of MIMO channels, IEEE J. Select. Areas Commun., vol. 21, no. 5, pp , [3] V. Aggarwal, Y. Liu, and A. Sabharwal, Sum-capacity of interference channels with a local view, submitted to IEEE Trans. Inf. Theory, Oct., [4] S. Vishwanath, N. Jindal, and A. Goldsmith, Duality, achievable rates, and sum-rate capacity of Gaussian MIMO broadcast channels, IEEE Trans. Info. Theory, vol. 49, no. 10, pp , [5] H. Weingarten, Y. Steinberg, and S. S. (Shitz), The capacity region of the Gaussian MIMO broadcast channel, in IEEE International Symposium on Information Theory (ISIT 04), Chicago, IL, USA, June 2004, p [6] M. Schubert and H. Boche, Solution of the multiuser downlink beamforming problem with individual SINR constraints, IEEE Transactions on Vehicular Technology, vol. 53, no. 1, pp , Jan [7] S. Shi, M. Schubert, and H. Boche, Downlink MMSE transceiver optimization for multiuser MIMO systems: Duality and sum-mse minimization, IEEE Transactions on Signal Processing, vol. 55, no. 11, pp , Nov [8] G. Zheng, T.-S. Ng, and K.-K. Wong, Joint power control and beamforming for sum-rate maximization in multiuser MIMO downlink channels, IEEE GLOBECOM 06., pp. 1 5, Dec [9] R. Hunger and M. Joham, A general rate duality of the MIMO multiple access channel and the MIMO broadcast channel, IEEE GLOBECOM 2008, pp. 1 5, Dec [10] B. Song, R. Cruz, and B. Rao, Network duality for multiuser MIMO beamforming networks and applications, IEEE Transactions on Communications, vol. 55, no. 3, pp , March [11] V. R. Cadambe and S. A. Jafar, Interference alignment and degrees of freedom of the k-user interference channel, IEEE Transactions on Information Theory, vol. 54, no. 8, pp , [12] K. Gomadam, V. Cadambe, and S. Jafar, Approaching the capacity of wireless networks through distributed interference alignment, IEEE GLOBECOM 08, pp. 1 6, Dec [13] A. Liu, Y. Liu, H. Xiang, and W. Luo, Rate duality and optimization of input covariance matrices for MIMO networks, in preparation. [14] I. Telatar, Capacity of multi-antenna Gaussian channels, Euro. Trans. Telecommun., vol. 10, pp , [15] M. Varanasi and T. Guess, Optimum decision feedback multiuser equalization with successive decoding achieves the total capacity of the gaussian multiple-access channel, in Proc. Thirty-First Asilomar Conference on Signals, Systems and Computers, vol. 2, 1997, pp [16] G. Scutari, D. Palomar, and S. Barbarossa, Competitive design of multiuser MIMO systems based on game theory: A unified view, IEEE Journal on Selected Areas in Communications, vol. 26, no. 7, pp , September [17] M. Maddah-Ali, A. Motahari, and A. Khandani, Communication over MIMO X channels: Interference alignment, decomposition, and performance analysis, IEEE Transactions on Information Theory, vol. 54, no. 8, pp , Aug