On the Rate Duality of MIMO Interference Channel and its Application to Sum Rate Maximization

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1 On the Rate Duality of MIMO Interference Channel and its Application to Sum Rate Maximization An Liu 1, Youjian Liu 2, Haige Xiang 1 and Wu Luo 1 1 State Key Laboratory of Advanced Optical Communication Systems & Networs, School of EECS, Peing University 2 Department of Electrical and Computer Engineering, University of Colorado at Boulder 1 Corresponding Author: luow@pu.edu.cn Abstract In this paper, we establish a rate duality between the forward and reverse lins of MIMO interference channel, where the reverse lins are obtained by exchanging the roles of transmitters and receivers in the forward lins, and the corresponding channel matrices are conjugate transpose of the forward channel matrices. Since the capacity region for general interference channel is unnown, we show that the forward and reverse lins have the same achievable rate region by treating interference as noise under some sum power constraint. The explicit expression of the corresponding input covariance matrix transformation is provided. We discuss the connection between the proposed transformation and the MAC-BC transformations in the previous wors. As an application, a duality based iterative algorithm is proposed to maximize the sum rate of MIMO interference channel under sum power constraint. We also extend the algorithm to individual power constraint. The proposed algorithms are shown to be effective by simulation. Index Terms Multiple-Input Multiple-Output (MIMO), Interference Channel, Duality, Resource Allocation I. INTRODUCTION Multiple-Input Multiple-Output (MIMO) systems promise high spectral efficiency [1]. They have generated interest in the research on the capacity region for multi-user MIMO channels such as MIMO multiple-access channel (MAC), MIMO broadcast channel (BC) and MIMO interference channel (IFC). The duality between MIMO MAC and MIMO BC capacity region is established in [2], [3]. 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 [4] [6]. Based on the SINR duality, a general rate duality is presented in [7] between the MIMO MAC and the MIMO BC, which is applicable to systems with and without nonlinear interference cancellation. On the other hand, the capacity region of general IFC is still an open problem. However, there are already many wors investigating the resource allocation for MIMO IFC from practical considerations. Most of these wors assume that the receiver treats the interference as noise, i.e., not allowing multiuser encoding/decoding techniques. This assumption is reasonable in many applications where the complexity of the receiver must be ept low, or the receivers do not now the coding and modulation schemes used by the interfering transmitter. In [8], a centralized global approach and a distributed This wor was supported in parts by NSF Grants CCF , ECCS iterative approach based on the gradient projection method are proposed to maximize the sum rate. These algorithms are shown to converge to some local optimum. The most popular distributed approach is the game theoretical approach, where the resource allocation problem is formulated as a noncooperative game. A mathematical framewor was provided in [9] to derive a unified set of sufficient conditions guaranteeing the uniqueness of the Nash equilibrium (NE) and the global convergence of waterfilling based asynchronous distributed algorithms. Unfortunately, NE is observed to be inefficient in term of the sum rate [8]. The authors in [10] extended the SINR duality between MIMO MAC and MIMO BC to any MIMO networ with linear beamformers by using a generic duality theory in linear programming. Then the networ duality is used to solve the joint MIMO beamforming and power control problem. Recently, closed form interference alignment schemes are proposed in [11] to approach and analysis the capacity of the interference networs 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 nowledge. Motivated by the duality between MIMO MAC and MIMO BC capacity region, it is natural to as that whether similar duality also holds for MIMO IFC. Since the capacity region of MIMO IFC is in general unnown, this wor establishes the duality of the achievable rate region for MIMO IFC under the following assumptions: The receiver treats the interference from others as noise and does not attempt to decode the messages from others. All transmitters and receivers now perfect channel state information of all lins. The corresponding input covariance matrix transformation is also provided. Although we focus on MIMO IFC in this paper, the proposed rate duality and transformation can also be applied to any MIMO networ [13]. We show that the transformation in [7] is equivalent to a special case of the proposed transformation. For the MIMO MAC and the MIMO BC, the proposed transformation is also equivalent to the one developed in [2] at the boundary point of the capacity region. However, at the inner point of the capacity region, they are different because the proposed transformation eeps the sum transmit power unchanged while not decreasing the achievable rate, and the transformation in [2] eeps the achievable rate the same while not increasing the sum transmit power. The

2 established rate duality of MIMO IFC opens the way to design efficient rate optimization algorithms. We show an application of sum rate maximization in this paper. Note that although the proposed algorithm shares similar elements as the ones in [10], the rate optimization problem solved in this paper is in general more difficult than the SINR optimization problem in [10]. We find the optimal dimension of the transmit signal and show that uniform power allocation over these dimensions is optimal. We also consider the individual power constraint. In [13], we extend the results of this paper to any MIMO networ. The properties of the proposed transformation are studied and used to derive some interesting theoretical results. We also design algorithms to optimize the input covariance matrices for MIMO networs with individual rate constraint. In [14], the proposed algorithm is extended to a distributed rate optimization algorithm based on local channel nowledge learned by local message passing. The rest of the paper is organized as follows. Section II describes the system model. Section III establishes the rate duality of MIMO IFC. Iterative algorithms based on the established duality is proposed to maximize the sum rate under both sum and individual power constraint in Section IV. We verify the results by numerical simulations in Section V and give the conclusion in Section VI. II. SYSTEM MODEL We first define an achievable rate region for MIMO IFC. Then we show that precoding combined with successive decoding can achieve any point in the achievable rate region. Finally we summarize the corresponding SINR duality. A. Definition of the Achievable Rate Region Assume K MIMO lins share the same physical wireless medium. Each transmitter and receiver are equipped with L T and L R antennas respectively 1. The transmission signals of all lins interfere with each other. The receive signal for the th lin can be expressed as y = H, x + H l, x l + w, l where x C LT 1 is the transmit signal of the th lin, H l, C LR LT is the channel matrix from the l th transmitter to the th receiver, and w C LR 1 is the Gaussian noise vector with zero mean and unit covariance matrix. Denote { the} input covariance matrix of the th lin as Σ = E x x C LT LT. The interference-plus-noise of the th lin is K l=1,l H l,x l + w, whose covariance matrix is Ω = I + H l, Σ l H l,. (1) Define l=1,l Σ 1:K = {Σ, =1,,K}. (2) 1 We assume different lins have the same numbers of transmit antennas and the same number of receive antennas in this section only for simplicity of notations. The duality results in this paper also hold for arbitrary number of antennas. And treating interference as noise, the mutual information of the th lin is given by a function of Σ 1:K as follow [15] I (Σ 1:K )=log I + H, Σ H, Ω 1. (3) Now, we define the achievable rate region to be the set of all mutual information that can be achieved under the sum power constraint P T : { R (P T ) r C Σ K 1 : (4) K 1:K: =1 Tr(Σ ) P T r I (Σ 1:K ), 1 K} R K +. Similarly, denote the input covariance matrix of the th lin in the reverse lins as ˆΣ C LR LR. Then the interference plus noise covariance matrix is ˆΩ = I + l=1,l H,l ˆΣ l H,l. (5) Treating interference as noise, the mutual information of the th lin in the reverse lins is ) Î (ˆΣ1:K = log I + H ˆΣ, H ˆΩ 1,. where ˆΣ 1:K = {ˆΣ, =1 K}. Finally, the achievable rate region of the reverse lins is defined as { ˆR (P T ) r C K 1 ˆΣ : (6) K 1:K: =1 Tr(ˆΣ ) P T ) } r (ˆΣ1:K Î, 1 K R K +. The outer boundary of R is called the Pareto boundary. At any point of the Pareto boundary, it is impossible to improve one of the rates, without simultaneously decreasing at least one of the other rates. Note that R is in general not convex. If the available degrees of freedom are allowed to be split, e.g., timesharing between two (or more) transmission schemes Σ 1:K and Σ 1:K is allowed, a larger rate region R may be achieved, where R denote the convex hull of R. In this paper, we will focus on R. The region R can be easily drawn from R. B. Precoding with Successive Decoding Any point in the region R (P T ) can be achieved by precoding with successive decoding as follows. Decompose the input covariance matrices as Σ = T P T, =1,,K, where T = [t,1,, t,m ] C LT 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 ran of Σ 2. Note that the decomposition is not unique. Let T = T P 1 2, then any T = T V also satisfies the decomposition Σ = T (, T ) where V C M M is a unitary matrix. In this way, the signal x is divided into M 2 Here, we assume M is no less than the maximum ran of all input covariance matrices so that all precoding matrices can have the same dimension for the simplicity of notations.

3 streams. Successive decoding is used to decode the M streams of x. Without loss of generality, we assume the decoding order is π =1,,M, i.e., the first stream is decoded first and the M th stream is decoded last. After subtracting the first (m 1) th stream of x, the remaining signal passes through a receive vector (r,m ) C 1 LR to generate the estimated data for the m th stream of x, where r,m is obtained by MMSE criteria as [13] r,m =α,m ( M i=m+1 H, p,i t,i t,i H, +Ω ) 1 H, t,m, where 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 (7) R =[r,1,, r,m ], (8) R = {R, =1,,K}, (9) p =[p 1,, p K ] T, (10) T = {T, =1,,K}. (11) 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 optimal receive matrices respectively. Define a non-negative coupling matrix Ψ R+ KM KM being a function of T, R [4] Ψ (T, R) ( 1)M+m,(l 1)M+n 0 = l and m n = r,m H 2 l,t l,n otherwise 1, l K, 1 m, n M, (12) where Ψ i,j denote the element of the i th row and j th column of Ψ. For simplicity, we will use Ψ l,n,m to denote Ψ ( 1)M+m,(l 1)M+n, which describes the cross-tal from the n th stream of the l th lintothem th stream of the th lin. Then the SINR and the rate of the m th stream in the th lin can be expressed as a function of T, R and p r p,m,m H 2,t,m γ,m (T, R, p) = 1+ K M l=1 n=1 p l,nψ l,n,,m (13) r,m (T, R, p) =log (1 + γ,m (T, R, p)), (14) The sum rate of all M streams of x is M R MS (T, R, p) = r,m (T, R, p), (15) m=1 For MMSE-SIC receiver, the following fact holds [16]. Fact 1: For a given set of input covariance matrices Σ 1:K and any decomposition: Σ = T P T, =1,,K,the MMSE-SIC receiver described above can achieve the mutual information of MIMO IFC, i.e., R MS (T, R, p) =I (Σ 1:K ), where R MS (T, R, p) and I (Σ 1:K ) are defined in (15) and (3) respectively. Further more, the achieved rate (T, R, p) is the same for different decoding orders π. R MS Similarly, in the reverse lins, we use R = [r,1,, r,m ] C LR M and T = [t,1,, t,m ] C M LT as the normalized precoding and receive matrix respectively. We change the power allocation vector to q =[q,1,,q,m ] T. And the decoding order is denoted as ˆπ = M,, 1, which has an inverse order of π. Then for the reverse lins, 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) = 1+ K M l=1 n=1 q l,nψ,m, (16) l,n ˆr,m (R, T, q) =log (1 + ˆγ,m (R, T, q)), (17) where q = [ q T 1,, q T K] 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 decoding order π =1,,M/ˆπ = M,, 1 as described above in the forward/reverse lins. C. SINR Duality for MIMO IFC { For given set of SINR values γ,m }, 0, 1 K, 1 m M precoding and receive matrices {T, R}, define {( D = diag γ 0,m/ ) r,m H 2,t,m, 1 K, 1 m M }, where D R+ KM KM3. Then the SINR duality for MIMO IFC is stated in the following Lemma. Lemma 1: Let γ,m 0 = γ,m (T, R, p), 1 K, 1 m M, where γ,m (T, R, p) is the SINRs achieved by the transmission and reception strategy {T, R, p} with p 1 = P T in the forward lins. Then the same SINRs can be achieved in the reverse lins using the transmission and reception strategy {R, T, q}, i.e., ˆγ,m (R, T, q) =γ,m 0, 1 K, 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 (18) The SINR duality for any wireless networ is proved in [10] when there is no successive decoding. With successive decoding, the only difference is that some elements of the coupling matrix Ψ is set to zero because of successive decoding. However, the proof in [10] did not rely on the specific value of Ψ as long as it is non-negative. Therefore, the same proof can be used to prove lemma 1. In the next section, we will use the SINR duality to derive the duality of the achievable rate region for the MIMO IFC. 3 Note that γ 0,m / r,m H,t,m 2 is the (( 1)M + m) th diagonal element of D.

4 III. RATE DUALITY OF MIMO IFC The proofs of all the following results are given in [13]. First, we have the following theorem which establishes the duality of the achievable rate region between the forward and reverse lins of the MIMO IFC. Theorem 1: When treating the interference from other transmitters as noise, the forward and reverse lins of the MIMO IFC described in section II have the same achievable rate region under the same sum power constraint P T, i.e., R (P T )= ˆR (P T ), where R (P T ) and ˆR (P T ) are defined in (4) and (6) respectively. The proof of this is obtained by showing that for any set of rates achievable in the forward lins, a set of equal or larger rates is achievable in the reverse lins with the same sum power constraint, and vice versa. The ey point is to find an input covariance transformation from the forward/reverse lins to the reverse/forward lins so that a set of nondecreasing rates is achieved in the reverse/forward lins with the same sum power constraint. Suppose Σ 1:K is a set of input covariance matrices achieving some rate point r in the region R (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 (11), (9) and (10) respectively. Then it follows from lemma 1 that the transmission and reception strategy {R, T, q} can also achieve the same rate point r in the reverse lins, where q is given by (18). Note that T may not be the set of MMSE-SIC receive matrices for the reverse lins corresponding to the transmission scheme R and q.ifweuse the optimal receiver in the reverse lins, we may achieve a set of rates even larger than that in the forward lins. Now we summarize the transformation in the following lemma. Lemma 2: For any set of input covariance matrices Σ 1:K satisfying the sum power constraint P T and achieving some rate point r in the region R (P T ), the transformation that gives the input covariance matrices ˆΣ 1:K satisfying the same sum power constraint P T and achieving a rate point ˆr r in the region ˆR (P T ) is given by ˆΣ = R diag (q ) R, =1,,K, (19) where R, = 1,,K is the MMSE-SIC receive matrices for the reverse lins defined in (8) corresponding to some decomposition of the input covariance matrix: Σ = T P T, =1,,K, and q is given by (18). Note that for different decompositions of the input covariance matrices, R and q are different in general. Hence the transformation in (19) may not be unique. However, for the transformation at the Pareto boundary point of the achievable rate region, we have stronger results as stated in the following theorem. Theorem 2: For a given set of optimal input covariance matrices Σ 1:K achieving some Pareto boundary point of the achievable rate region, the solution of the transformation in (19) satisfies the following matrix equations ˆΩ 1 H ˆΣ, H, =Σ H, Ω 1 H,, 1 K (20) Further more, if the number of optimal solutions achieving the corresponding Pareto boundary point is finite in either reverse lins or forward lins, then the transformation is unique, i.e., for all decompositions of Σ 1:K in the forward lins, it will be transformed to the same ˆΣ 1:K in the reverse lins, and vice versa. As shown by [13], the transformation in (19) can be applied to any MIMO networ. The following proposition states the connection between the transformation in (19) and that for the MIMO MAC and the MIMO BC in [7]. Proposition 1: Under a special decomposition of the input covariance matrix Σ = T P T such that T H, Ω 1 H,T is diagonal matrix, the normalized MMSE-SIC receive vector given by (7) equals to the normalized MMSE receive vector, i.e., r,m = ( M 1 α,m i=m+1 H, t,i t,i H, ) +Ω H, t,m = α,m Ω 1 H,t,m. The existence of this special decomposition is shown in [7]. Under this special decomposition, the transformation in (19) becomes equivalent to the one in [7]. As shown in [7], the advantage of using this special decomposition is that a parallel stream-wise decoding is possible without intra-signal interference cancellation. Now we derive the connection between the proposed transformation and the one in [2]. Proposition 2: At the Pareto boundary point, the solution of the transformation in (19) satisfies the following equations ˆΣ =Ω 1/2 F G 1/2 ˆΩ Σ 1/2 ˆΩ G F Ω 1/2 (21) =1 K, where Ω 1/2 H ˆΩ 1/2 is decomposed using the SVD as Ω 1/2 H ˆΩ 1/2 = F Δ G, where Δ is a square and diagonal matrix. Proposition 2 indicates that the transformation equations for the MIMO MAC and the MIMO BC in [2] also hold for the MIMO IFC at the Pareto boundary point. In MIMO MAC and MIMO BC, the equations in (21) can be solved one by one iteratively. However, for MIMO IFC, they have to be solved jointly, which is very difficult. The transformation in (19) provides an explicit expression of the solution for the equations in (21). IV. AN APPLICATION: MAXIMIZING THE SUM RATE The problem of maximizing the sum rate is formulated as follow max I (Σ 1:K ),s.t. Tr (Σ ) P T, (22) Σ 1:K =1 =1 where I (Σ 1:K ) is defined in (3). As shown in section II-B, problem (22) is equivalent to the following problem M max r,m (T, R, p),s.t. p 1 P T, (23) T,R,p =1 m=1

5 where r,m (T, R, p) is defined in (14). First we discuss on the optimal number of streams M and the optimal power allocation. The following proposition is useful. Proposition 3: For any input covariance matrix Σ and any integer M no less than the ran of Σ, we can always find a decomposition Σ = T 0 P 0 T 0 such that T 0 C LT M and the corresponding power allocation matrix P 0 = Tr (Σ) I/M R+ M M, i.e., uniform power allocation over the streams of the same lin is optimal. The proof is given in [13]. Suppose the ran of the optimal input covariance matrix Σ for x is M. Then from proposition 3, as long as the number of streams M for x is no less than M, we will not lose any optimality. Indeed, if we choose M = M, the algorithms have the lowest complexity. However, we may not be able to now M before actually solving problem (22). For single user MIMO systems, we only need M min (L T,L R ) active data streams to achieve the capacity. For MIMO IFC, more active data streams will cause more interference between lins. Thus set M = min (L T,L R ) is sufficient to fully utilize the spatial diversity of the multiantenna system [1]. Proposition 3 also states that uniform power allocation over the streams of the same lin will not lose any optimality, which is useful when designing algorithm for the individual power constraint. To solve problem (23), we use similar algorithm as that proposed for MIMO BC in [6]. For fixed T, p, the optimal receive vector for each data stream is decoupled, i.e., the SINR of the m th data stream of x,l is maximized individually by MMSE receiver in (7). However, for fixed R, p, the transmit vectors of all data streams have to be optimized jointly. The SINR duality in lemma 1 implies that the optimal transmit vectors in the forward lins can be found by optimizing the receive vectors at the reverse lins. This leads to an alternating algorithm, which switches the optimization between the forward lins and reverse lins. When switching to the reverse lins, the power allocation q is first optimized. Then for fixed R, q, the optimal receive vector for each data stream is decoupled in the reverse lin, and is given individually by MMSE-SIC receiver ( m 1 t,m = β,m i=1 H, q,ir,i r,i H, + ˆΩ ) 1 H, r,m (24) where ˆΩ is defined in (5) with ˆΣ l = M i=1 q l,ir l,i r l,i and β,m is chosen such that t,m =1. For fixed T, R, the optimal power allocation q in the reverse lins is given by solving the following problem [17] min q K M =1 m=1 1 1+ˆγ,m (R, T, q), s.t. q 1 P T, (25) where ˆγ,m (R, T, q) is defined in (16). Problem (25) is a signomial programming (SP) problem and has been studied in [17]. It is not convex and the optimal solution is still unnown. However, there exist algorithms to obtain the local optimum [17]. In high SINR region, problem (25) can be approximated Table I ALGORITHM A: ALGORITHM FOR SOLVING PROBLEM (22) Choose p (0) and T (0).Setn 0. While not converge do 1. Update in the forward lins a) Compute R (n+1) from p (n) and T (n) using (7); b) Compute D and Ψ from T (n) and R (n+1) ; c) Compute q (n) = ( D 1 Ψ T ) 1 1; 2. Update in the reverse lins a) Using q (n) as the initial value, obtain q (n+1) by solving the SP problem in (25), or the GP problem in (26) for high SINR region. b) Compute T (n+1) from q (n+1) and R (n+1) using (24); c) Compute D and Ψ from T (n+1) and R (n+1) ; d) Compute p (n+1) = ( D 1 Ψ ) 1 1; n n +1; End by the following geometric programming (GP) problem K M 1 min q ˆγ,m (R, T, q), s.t. q 1 P T, (26) =1 m=1 GP can be converted to a convex problem and then the globally optimal solution can be obtained by efficient algorithms [17]. The proposed rate maximization algorithm is summarized in Table I and is called algorithm A in this paper. The proof of the convergence for algorithm A is similar to the one in [6]. Following the same arguments as in [6], it is easy to show that the sum rate is monotonically increasing from one iteration to the next, and is obviously bounded due to the sum power constraint at the transmitters. Thus the proposed algorithm converges to a local optimum when i. Now suppose each user subject to an individual power constraint, e.g., Tr (Σ ) P 0, =1,,. Then the above algorithm can be extended to the case of individual power constraint under the following minor modifications. Delete step 2-a) and in step 2-d) p is given by uniform power allocation, i.e., p 0,m = P 0 /M, =1,,K,m=1,,M. This modified algorithm is called algorithm B in this paper. Remar 1: Note that algorithm B tries to increase the rates of all users in each iteration until one of the Pareto points is achieved. Ideally if algorithm B actually converges to the global optimum, it will achieve one of the Pareto boundary points of the achievable rate region R (P T ) corresponding to the individual power constraint P 0, =1,,K. V. NUMERICAL RESULTS We evaluate the performance in this section. We assume bloc fading channels and the channel matrices are given by H,l = g,l H w, where H w has zero-mean i.i.d. Gaussian entries with unit variance. For performance evaluation, we define a term called pseudo global optimum which is defined by the maximum of several local optimums obtained by running the algorithm for several times with different randomly generated initial values. In all simulations, the numbers of users and antennas are set as K =2and L T =4,L R =2 respectively. And the initial transmit vectors and power allocation are given by waterfilling over the channel matrix of each lin. We also compare the proposed algorithms with

6 Average Sum Rate (bit/channel use) Figure 1. 0dB 10dB Pseudo Global Optimum Alg. A with SP Alg. A with GP Alg. B IWFA Interference Strength g c (db) Sum Rate vs. g c for a Two-users Symmetric System the iterative waterfilling (IWFA) in [9]. For algorithm B and IWFA, we assume each user subject to equal individual power constraint, i.e., Tr (Σ ) P T /K, =1,,K, where P T is the total power. First, we consider symmetric systems, i.e., g, = g d, and g,l = g c, l. Fig. 1 plots the sum rate achieved by different algorithms vs. g c with g d =0dB for total power P T =0dB and 10dB respectively. The pseudo global optimum is also given for comparison. In all simulations, algorithm A with SP achieves nearly the same sum rate as the pseudo global optimum. The performance of algorithm A with GP and algorithm B is close to that of algorithm A with SP, which is much higher than that of IWFA. We also evaluate the performance for asymmetric systems, where g l,, l are generated randomly with uniform distribution between -5 db and 10 db. We let P T =3dB in the simulation. The average sum rate for algorithm A with SP, with GP, algorithm B and IWFA is , , and (bits/channel use) respectively, while the pseudo global optimum is (bits/channel use). [3] 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 [4] M. Schubert and H. Boche, Solution of the multiuser downlin beamforming problem with individual SINR constraints, IEEE Transactions on Vehicular Technology, vol. 53, no. 1, pp , Jan [5] S. Shi, M. Schubert, and H. Boche, Downlin MMSE transceiver optimization for multiuser MIMO systems: Duality and sum-mse minimization, IEEE Transactions on Signal Processing, vol. 55, no. 11, pp , Nov [6] G. Zheng, T.-S. Ng, and K.-K. Wong, Joint power control and beamforming for sum-rate maximization in multiuser MIMO downlin channels, IEEE GLOBECOM 06., pp. 1 5, Dec [7] 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 [8] S. Ye and R. Blum, Optimized signaling for mimo interference systems with feedbac, IEEE Transactions on Signal Processing, vol. 51, no. 11, pp , Nov [9] 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 [10] B. Song, R. Cruz, and B. Rao, Networ duality for multiuser MIMO beamforming networs 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 -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 networs 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 networs, in preparation. [14] A. Liu, A. Sabharwal, Y. Liu, H. Xiang, and W. Luo, Distributed MIMO networ optimization based on local message passing and duality, submitted to the 47th Allerton Conf. on Communication, Control, and Computing, Monticello, Illinois, USA, [15] T. M. Cover and J. A. Thomas, Elements of Information Theory. New Yor: John Wiley and Sons, [16] M. Varanasi and T. Guess, Optimum decision feedbac 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 [17] M. Chiang, C. W. Tan, D. Palomar, D. O Neill, and D. Julian, Power control by geometric programming, IEEE Transactions on Wireless Communications, vol. 6, no. 7, pp , July VI. CONCLUSION In this paper, we show that similar duality as observed between MIMO MAC and MIMO BC also holds for the forward and reverse lins of MIMO interference channel. The corresponding covariance matrix transformation and the connection to the existing transformations for MIMO MAC and MIMO BC are discussed. These results are obtained by a natural extension of the SINR duality in [10]. To illustrate the application of the duality, iterative algorithms are proposed to solve the problem of sum rate maximization under both sum and individual power constraint. REFERENCES [1] I. E. Telatar, Capacity of multi-antenna gaussian channels, Europ. Trans. Telecommu., vol. 10, pp , Nov./Dec [2] 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 , 2003.