Linear Precoding and Equalization for Network MIMO with Partial Cooperation Saeed Kaviani, Student Member, IEEE, Osvaldo Simeone, Senior Member, IEEE, Witold A

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 5, JUNE 2012 2083 Linear Precoding and Equalization for Network MIMO With Partial Cooperation Saeed Kaviani, Student Member, IEEE, Osvaldo Simeone, Senior Member, IEEE, Witold A. Krzymien,´ Senior Member, IEEE, and Shlomo Shamai (Shitz), Fellow, IEEE

Abstract—A cellular multiple-input–multiple-output (MIMO) network-level interference management appears to be of fun- downlink system is studied, in which each base station (BS) trans- damental importance to overcome this limitation and harness mits to some of the users so that each user receives its intended the gains of MIMO technology. Confirming this point, multicell signal from a subset of the BSs. This scenario is referred to as network MIMO with partial cooperation since only a subset cooperation, which is also known as network MIMO, has been of the BSs is able to coordinate their transmission toward any shown to significantly improve the system performance [3]. user. The focus of this paper is on the optimization of linear Network MIMO involves cooperative transmission by multi- beamforming strategies at the BSs and at the users for network ple base stations (BSs) to each user. Depending on the extent of MIMO with partial cooperation. Individual power constraints at multicell cooperation, network MIMO reduces to a number of the BSs are enforced, along with constraints on the number of streams per user. It is first shown that the system is equivalent to scenarios, ranging from a MIMO broadcast channel (BC) [4] in a MIMO interference channel with generalized linear constraints case of full cooperation among all BSs to a MIMO interference (MIMO-IFC-GC). The problems of maximizing the sum rate (SR) channel [5], [6] in case no cooperation of the BSs is allowed. and minimizing the weighted sum mean square error (WSMSE) In general, network MIMO allows cooperation only among a of the data estimates are nonconvex, and suboptimal solutions cluster of BSs for transmission to a certain user [7], [8] (see with reasonable complexity need to be devised. Based on this, suboptimal techniques that aim at maximizing the SR for the also references in [3]). MIMO-IFC-GC are reviewed from recent literature and extended In this paper, we consider a MIMO interference channel with to the MIMO-IFC-GC where necessary. Novel designs that aim at partial cooperation of the BSs. It is noted that all BSs cooperat- minimizing the WSMSE are then proposed. Extensive numerical ing for transmission to a certain user have to be informed about simulations are provided to compare the performance of the con- the message (i.e., the bit string) intended for the user. This can sidered schemes for realistic cellular systems. be realized using the backhaul links among the BSs and the Index Terms—Cooperative communication, linear precoding, central switching unit. We focus on the sum-rate maximization multicell processing, network MIMO, partial cooperation. (SRM) and the minimization of weighted sum-MSE (WSMSE) under per-BS power constraints and constraints on the number I. INTRODUCTION of streams per user. Moreover, although nonlinear processing NTERFERENCE is known to be a major obstacle in techniques such as vector precoding [9], [10] may generally be I realizing the spectral efficiency increase promised by useful, we focus on more practical linear processing techniques. multiple-antenna techniques in wireless systems. Indeed, Both the SRM and WSMSE minimization (WSMMSE) prob- multiple-input–multiple-output (MIMO) capacity gains are lems are nonconvex [11], and thus suboptimal design strategies severely degraded and limited in cellular environments due of reasonable complexity are called for. to the deleterious effect of interference [1], [2]. Therefore, The contributions of this paper are as follows.

1) It is ﬁrst shown in Section II-A that network MIMO with Manuscript received June 24, 2011; revised November 14, 2011; accepted partial BS cooperation, i.e., with partial message knowl- January 5, 2012. Date of publication February 11, 2012; date of current version edge, is equivalent to a MIMO interference channel, in June 12, 2012. This paper was presented in part at the IEEE Global Commu- nication Conference, Houston, TX, December 2011. The work of S. Kaviani which each transmitter knows the message of only one and W. A. Krzymien´ was supported in part by TRLabs, by the Rohit Sharma user under generalized linear constraints, which we refer Professorship, and by the Natural Sciences and Engineering Research Council to as MIMO-IFC-GC. of Canada. The work of O. Simeone was supported by the U.S. National Science Foundation under Grant CCF-0914899. The work of S. Shamai was supported 2) We review the available suboptimal techniques that have by the Israel Science Foundation. The review of this paper was coordinated by been proposed for the SRM problem [12]–[14] and ex- Prof. H.-F. Lu. tend them to the MIMO-IFC-GC scenario where neces- S. Kaviani and W. A. Krzymien´ are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6A 2G4, sary in Section V. Since these techniques are generally Canada, and also with TRLabs, Edmonton, AB T5K 2M5, Canada (e-mail: unable to enforce constraints on the number of streams, [email protected]; [email protected]). we also review and generalize techniques that are based O. Simeone is with the Center for Wireless Communications and Signal Processing Research, New Jersey Institute of Technology, Newark, NJ 07102 on the idea of interference alignment [5] and are able to USA (e-mail: [email protected]). impose such constraints. S. Shamai (Shitz) is with the Department of Electrical Engineer- 3) Then, we propose two novel suboptimal solutions for ing, Technion–Israel Institute of Technology, Haifa 32000, Israel (e-mail: [email protected]). the WSMMSE problem in Section VI under arbitrary Digital Object Identiﬁer 10.1109/TVT.2012.2187710 constraints on the number of streams. It is noted that the

We now detail the signal model for the channel at hand, which is referred to as MIMO interference channel with partial T ∈ Cdk message sharing. Deﬁne as uk =[uk,1,...,uk,dk ] the dk × 1 complex vector representing the dk ≤ min(Mknt,nr) independent information streams intended for user k.Weas- sume that uk ∼CN(0, I). The data streams uk are known to all the BSs in the set Mk. In particular, if m ∈Mk,themth n ×d BS precodes vector uk via a matrix Bk,m ∈ C t k so that the n signal x˜m ∈ C t sent by the mth BS can be expressed as x˜m = Bk,mUk. (1)

k∈Km Imposing a per-BS power constraint, the following constraint must be then satisﬁed: E 2 E H x˜m =tr x˜mx˜m H ≤ Fig. 1. Downlink model with partial BS cooperation or equivalently partial = tr Bk,mBk,m Pm message knowledge. k∈Km m = 1,...,M (2) WSMMSE problem without such constraints would be

trivial, as it would result in zero minimum mean square where Pm is the power constraint of the mth BS. error (MMSE) and no stream transmitted. The proposed The signal received at the kth user can be written as solutions are based on a novel insight into the single-user M MMSE problem with multiple linear constraints, which is y = H x˜ + n˜ (3a) discussed in Section IV. k k,m m k m=1 4) Finally, extensive numerical simulations are provided in Section VII to compare the performance of the proposed = Hk,mBk,mUk + Hk,jBl,jUl + n˜k

schemes in realistic cellular systems. m∈Mk l= k j∈Ml (3b) Linear MMSE precoding and equalization techniques proposed in this paper were discussed briefly in [15]. The detailed nr ×nt where Hk,m ∈ C is the channel matrix between the mth analysis and discussion (including the proofs to the lemmas) are BS and kth user, and nk is additive complex Gaussian noise included in this paper. Additionally, we have also reviewed and 1 n˜k ∼CN(0, I). We assume ideal channel state information extended available solutions to the SRM problem. Furthermore, (CSI) at all nodes. In (3b), we have distinguished between we have included discussions of the complexity and overhead the first term, which represents useful signal, the second term, of the proposed techniques and previously available (and/or which accounts for interference, and the noise. extended) solutions. Notation: We denote the positive definite matrices as A 0. [·]+ denotes max(·, 0). Capital bold letters represent ma- A. Equivalence With MIMO-IFC-GC trices, and small bold letters represent vectors. We denote We now show that the MIMO interference channel with · T the transpose operator with ( ) and conjugate (Hermitian) partial message sharing and per-BS power constraints previ- · H −(1/2) transpose with ( ) . A represents the inverse square of ously described is equivalent to a specific MIMO interference positive definite matrix A. channel with individual message knowledge and generalized linear constraints, which we refer to as MIMO-IFC-GC. Definition 1: MIMO-IFC-GC: The MIMO-IFC-GC consists II. SYSTEM MODEL AND PRELIMINARIES of K transmitters and K receiver, where the kth transmitter Consider the MIMO downlink system shown in Fig. 1, with has mt,k antennas and the kth receiver has mr,k antennas. The M BSs forming a set M and K users forming a set K. Each BS received signal at the kth receiver is is equipped with nt transmit antennas, and each mobile user employs nr receive antennas. The mth BS is provided with the yk = Hk,kxk + Hk,lxl + nk (4) messages of its assigned users set Km ⊆K. In other words, the l=k kth user receives its message from a subset of Mk BSs Mk ⊆ where n ∼CN(0, I), the inputs are x ∈ Cmt,k , and the M. Notice that if K contains one user for each transmitter k k m channel matrix between the lth transmitter and the kth receiver m and Mk = 1, then the model at hand reduces to a standard

MIMO interference channel. Moreover, when all transmitters 1 K K In case the noise is not uncorrelated across the antennas, each user can al- cooperate in transmitting to all the users, i.e., m = for all ways whiten it as a linear preprocessing step. Therefore, a spatially uncorrelated m ∈Mor equivalently Mk = M, then we have a MIMO BC. noise can be assumed without loss of generality. KAVIANI et al.: LINEAR PRECODING AND EQUALIZATION FOR NETWORK MIMO WITH PARTIAL COOPERATION 2085

m ×m is Hk,l ∈ C r,k t,k . The data vector intended for user k which is referred to as the mean square error (MSE) matrix (see d is uk ∈ C k , with dk ≤ min(mt,k,mr,k) and uk ∼CN(0, I). [17] for a review). The name comes from the fact that that the m ×d The precoding matrix for user k is deﬁned as Bk ∈ C t,k k , jth term on the main diagonal of Ek is the MSE so that xk = Bkuk. Inputs xk have to satisfy M generalized 2 linear constraints MSEk,j = E |uˆk,j − uk,j| (10)

K K on the estimation of the kth user’s jth data stream uk,j.Using E H H ≤ tr Φk,m xkxk = tr Φk,mBkBk Pm (5) the definition of MIMO-IFC-GC, it is easy to see that the MSE- k=1 k=1 matrix can be written as a function of the equalization matrix { }K m ×m Ak and all the transmit matrices Bk k=1 as for given weight matrices Φk,m ∈ C t,k t,k and m = 1,...,M. The weight matrices are such that matrices H H H − H M Ek = Ak HkBkBk Hk,kAk Ak Hk,kBk m=1 Φk,m are positive definite for all k = 1,...,K. − H H H We remark that the positive definiteness of matrices Bk Hk,kAk + Ak ΩkAk + Ik (11) M m=1 Φk,m guarantees that the system is not allowed to transmit infinite power in any direction [16]. where Ωk is the covariance matrix that accounts for noise and Lemma 1: Let (l)k be the lth BS in subset Mk of BSs that interference at user k, i.e., know user k’s message. The MIMO interference channel with H H partial message sharing (and per-transmitter power constraints) Ωk = I + Hk,lBlBl Hk,l. (12) is equivalent to a MIMO-IFC-GC. This equivalent MIMO- l= k IFC-GC is defined with mt,k = Mknt, mr,k = nr, channel matrices, i.e., III. PROBLEM DEFINITION AND PRELIMINARIES Hk,l = H ···H (6) k,(1)l k,(Ml)l In this paper, we consider the optimization of the sum of some specific functions f (E ) of the MSE-matrices E of beamforming matrices k k k all users k = 1,...,Kfor the MIMO-IFC-GC. Specifically, we T address the following constrained optimization problem: B = BT ···BT (7) k k,(1)k k,(Mk)k K minimize fk(Ek) and weight matrices Φk,m being all zero, except that their lth Bk,Ak ∀k × k=1 nt nt submatrix on the main diagonal is Int ,ifm =(l)k.(If k/∈K , then Φ = 0). We emphasize that the definition of K m k,m H ≤ MIMO-IFC-GC and this equivalence rely on the assumption of subject to tr Φk,mBkBk Pm,m= 1,...,M linear processing at the transmitters. k=1 Proof: The proof follows by inspection. Notice that ma- (13) M trices m=1 Φk,m are positive definite by construction. Given the generality of the MIMO-IFC-GC, which includes where the optimization is over all transmit beamforming matri- the scenario of interest of the MIMO interference channel with ces Bk and equalization matrices Ak. Specifically, we focus on partial message sharing as per the aforementioned Lemma, in the WSMSE functions the following, we focus on the MIMO-IFC-GC as previously dk defined and return to the cellular application in Section VII. It fk(Ek)=tr{WkEk} = wkjMSEkj (14) is noted that a model that subsumes the MIMO-IFC-GC has j=1 been studied in [16], as discussed here. d ×d with given diagonal weight matrices Wk ∈ C k k , where the main diagonal of W is given by [w ,...,w ] with B. Linear Receivers and MSE k k,1 k,dk nonnegative weights wkj ≥ 0. With cost function (14), we refer In this paper, we focus on the performance of the MIMO- to the problem (13) as the WSMMSE problem. IFC-GC under linear processing at the receivers. Therefore, the Of more direct interest for communications systems is the kth receiver estimates the intended vector uk using the receive maximization of the SR. This is obtained from (13) by selecting × ∈ Cdk mr,k processing (or equalization) matrix Ak as the SR functions

H uˆk = Ak yk. (8) fk(Ek) = log |Ek|. (15)

The most common performance measures, such as weighted With cost function (15), problem (13) is referred to as the SRM sum rate (SR) or bit error rate, can be derived from the estima- problem. In fact, from information-theoretic considerations, it tion error covariance matrix for each user k can be seen that (15) is the maximum achievable rate (in bits per channel use) for the kth user, where the signals of the other H Ek = E (uˆk − uk)(uˆk − uk) (9) users are treated as noise (see, e.g., [17]). 2086 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 5, JUNE 2012

Remark 1: Consider an iterative algorithm where, at each constraints on the number of streams d1 but is otherwise iteration, a WSMMSE problem is solved with the weight ma- convex and, in this special case, can be efficiently solved [17]. trices Wk assumed to be nondiagonal and selected based on The global optimal solution for the single-user problem with the previous MSE-matrix Ek. This algorithm can approximate multiple linear power constraint (and a rank constraint) is still the solution of (13) for any general cost function fk(Ek).This unknown [21]. The WSMMSE problem is trivial without rank was first pointed out in [18] for the weighted SRM problem in a constraint, as previously explained, and is nonconvex. Here, we MIMO BC and then in [19] for the single-antenna interference first review a key result in [17] and [22] that shows that, with channel and a general utility function and has been general- K = 1 and a single constraint (M = 1), the solution of the ized to a MIMO (broadcast) interference channel in [20] with WSMMSE problem can be, however, efficiently found. We then conventional power constraints. It is not difficult to see that discuss that, with multiple constraints (M>1), this is not the this result also extends to the MIMO-IFC-GC, which is not case, and a solution of the WSMMSE problem, even with K = subsumed in the model of [20] due to the generalized linear 1, must be found through some complex global optimization constraints. We explicitly state this conclusion here. strategies. One such technique was recently proposed in [21] Lemma 2 [20]: For strictly concave utility functions fk(·) based on a sophisticated gradient approach. At the end of this for all k, the global optimal solution of problem (13) and the section, we then propose a computationally and conceptually solution of simpler solution based on a novel result (see Lemma 5) that our numerical results have shown to have excellent performance. K This will be then leveraged in Section VI-B to propose a novel minimize {tr {WkEk}−tr {Wkgk(Wk)} Bk,Ak,Wk ∀k solution for the general multiuser case. k=1 To elaborate, consider a scenario where the noise-plus- +f (g (W ))} k k k interference matrix Ωk (12) is fixed and given (i.e., not subject K to optimization). Now, we solve the WSMMSE problem with H ≤ subject to tr Φk,mBkBk Pm,m= 1,...,M K = 1 for specified weight matrices W and Φm. For the rest k=1 of this section, we drop the index k = 1 from all quantities for (16) simplicity of notation. We proceed by solving the problem at hand: first with respect to A for fixed B and then with respect to where gk(·) is the inverse function of the ∇fk(·), are the same. B without loss of optimality. The first optimization, over A,is Consequently, to find an approximate solution of (13), at each easily seen to be a convex problem (without constraints) whose step, matrices Wk for k = 1,...,K are updated by solving solution is given by the minimum MSE equalization matrix (16) with respect to Wk only (i.e., we keep (Ak, Bk) ∀k − H H 1 unchanged in this step). Then, using the obtained matrices Wk, A = HBB H + Ω HB. (18) for k = 1,...,K, the problem (16) reduces to a WSMMSE Plugging (18) in the MSE matrix (11), we obtain problem with respect to matrices Ak and Bk for k = 1,...,K − (i.e., matrices Wk are kept fixed). This results in the iterative H H −1 1 algorithm, which is discussed in Remark 1 and leads to a E = I + B H Ω HB . (19) suboptimal solution of (13). In the special case of the SRM problem, we have f (E ) = log |E | and g (W )=W−1,in We now need to optimize over B the following problem: k k k k k k which problem (16) is then equivalent to the problem −1 minimize tr W I + BHHHΩ−1HB B K K { }− | | H minimize tr WkEk log Wk subject to tr ΦmBB ≤ Pm,m= 1,...,M. (20) Bk,Ak,Wk ∀k k=1 k=1 K Consider first the single-constraint problem, i.e., M = 1. H ≤ The global optimal solution for single-user WSMMSE problem subject to tr Φk,mBkBk Pm,m= 1,...,M. k=1 with M = 1 is given in [21] and [22] and reported here. Recall that, according to Definition 1, matrix Φ is positive definite. (17) 1 Lemma 3 [22]: The optimal solution of the WSMMSE The optimization problem (17) can be solved in an iterative problem with K = 1 and a single trace constraint (M = 1) is fashion, where, at each iteration, the weights are selected as given by − 1 − 1 Wk = Ek , and then, the WSMMSE problem is solved with 2 B = Φ1 UΣ (21) respect to matrices (Ak, Bk) for k = 1,...,K. × where U ∈ Cmt d is the matrix of the eigenvectors of matrix −(1/2) −1 H −(1/2) IV. SINGLE-USER CASE (K = 1) Φ1 HΩ H Φ1 corresponding to its largest eigenvalues γ1 ≥···≥γd, and Σ is a diagonal matrix with diagonal The WSMMSE and SRM problems are nonconvex, and √ terms pi defined as thus, global optimization is generally prohibitive. In this sec- + tion, we address the case of a single user (K = 1). In particular, wi 1 pi = − (22) the SRM problem with K = 1 is nonconvex if one includes μγi γi KAVIANI et al.: LINEAR PRECODING AND EQUALIZATION FOR NETWORK MIMO WITH PARTIAL COOPERATION 2087 with μ ≥ 0 being the “waterfilling” level chosen to satisfy the Proof: The proof is given in the Appendix. H single power constraint tr{Φ1BB } = P1. Lemma 5 suggests that, to solve the single-user multiple- Proof: Introducing the “effective” precoding matrix B¯ = constraint problem, under some technical conditions, one can 1/2 ¯ −(1/2) minimize instead the dual problem on the right-hand side of Φ1 B and “effective” channel matrix H = HΦ1 ,the problem is equivalent to that discussed in , Theorem 1[22]. (24). Lemma 3 showed that this is always possible with a In the case of multiple constraints, the approach used in single constraint. The conditions in Lemma 5 hold in most Lemma 3 cannot be leveraged. Here, we propose a simple but cases where the power constraints for the optimal solution are effective approach, which is based on the following considera- satisfied with equality. While this may not be always the case, in tions summarized in the following two lemmas: practice, e.g., if the power constraints represent per-BS power Lemma 4: The precoding matrix (21) and (22) for a given constraints as in the original formulation of Section II, this fixed μ>0 minimizes the Lagrangian function condition can be shown to hold [23]. Inspired by Lemma 5, here we propose an iterative approach −1 − 1 − − 1 L ¯ ¯ H 2 H 1 2 ¯ to the solution of the WSMMSE problem with K = 1 that (B; μ)=tr W I + B Φ1 H Ω HΦ1 B is based on solving the dual problem maxλ≥0 minB L(B; λ). L + μ tr B¯ B¯ H (23) Specifically, to maximize infB (B; λ) over λ 0, in the proposed algorithm, the auxiliary variables λ is updated at the where B¯ is the effective precoding matrix previously defined. jth iteration via a subgradient update given by [16] Proof: We first note that (23) is the Lagrangian function of λ(j) = λ(j−1) + δ P − tr Φ BBH ∀m (28) the single-user single-constraint problem solved in Lemma 2. m m m m Then, we prove (23) by contradiction. Assume that the min- to attempt to satisfy the power constraints. Having fixed the imum of the Lagrangian function is attained at where the (j) L vector λ , problem minB(B, λ) reduces to minimizing corresponding E is not diagonal. Then, one can always (j) (23) with Φ = Φ(λ(j))= λ Φ and μ = 1. This can find a unitary matrix Q ∈ Cd×d such that the matrix B¯ ∗ = 1 m m m ∗ H be done using Lemma 3 so that, from (21) and (22), at BQ¯ diagonalizes E since, with B¯ ,wehaveE = Q (I + − the jth iteration, B(j) is obtained as Φ(λ(j)) (1/2)U(j)Σ(j), ¯ H −(1/2) H −1 −(1/2) ¯ −1 { } B Φ1 H Ω HΦ1 B) Q [22]. Function tr WE where U(j) is the matrix of eigenvectors of matrix ¯ ∗ is Schur concave, and therefore, the matrix B does not de- Φ(λ(j))−(1/2)HHHΦ(λ(j))−(1/2) corresponding to its largest crease function tr{WE} with respect to B¯ , whereas B¯ B¯ H = (j) eigenvalues γ1 ≥···≥γdand Σ is a diagonal matrix with B¯ ∗B¯ ∗H. This implies that the minimum of tr{WE} is reached √ − + when the MSE matrix is diagonalized. Therefore, we can set the diagonal terms pi = [ (wi/γi) (1/γi)] . without loss of generality B¯ = UΣ, where U is defined as in Lemma 3 and Σ is diagonal with nonnegative elements on the V. S UM RATE MAXIMIZATION main diagonal. Substituting this form of B¯ into the Lagrangian function, we obtain a convex problem in the diagonal elements The SRM problem for a number of users K>1 is non- of Σ, whose solution can be easily shown to be given by (22) convex, even when removing the constraints on the number of for the given μ. This concludes the proof. streams per user. The general problem in fact remains noncon- Lemma 5: Let p be the optimal value of the single-user vex and is nonprobabilistic-hard [24]. Therefore, since finding WSMMSE problem with multiple constraints (K = 1,M ≥ the global optimal has prohibitive complexity, one needs to 1).Wehave resort to suboptimal solutions with reasonable complexity. In this section, we review several suboptimal solutions to the SRM p ≥ max inf L(B; λ) (24) problem that have been proposed in the literature. Since some λ≥0 B of these techniques were originally proposed for a scenario where that does not subsume the considered MIMO-IFC-GC, we also propose the necessary modifications required for application −1 L(B; λ)=tr W I + BHHHΩHB to the MIMO-IFC-GC. Note that these techniques perform an optimization over the transmit covariance matrices by relaxing M the rank constraint due to the number of users per streams (see H + λm tr ΦmBB − Pm (25) discussion here). Therefore, we also review and modify when m=1 necessary a different class of algorithms that solve problems re- is the Lagrangian function of the single-user WSMMSE prob- lated to SRM but are able to enforce constraints on the number of transmitting streams per user. The WSMMSE problem does lem at hand, and λ =(λ1,...,λM ). Moreover, if there exists an optimal solution B˜ achieving p that, together with a strictly not seem to have been addressed previously for the MIMO-IFC- positive Lagrange multiplier λ˜ > 0, satisfies the conditions GC and will be studied in the next section.

∇ L =0 (26) B A. Soft Interference Nulling H tr ΦmB˜ B˜ = Pm ∀m (27) A solution to the SRM problem for the MIMO-IFC-GC was proposed in [12]. In this technique, the optimization is over × H ∈ Cmt,k mt,k then (24) holds with equality. all transmit covariance matrices Σk = BkBk . 2088 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 5, JUNE 2012

The constraints on the number of streams would impose a It turns out that such linearized problem can be solved using rank constraint on Σk as rank(Σk)=dk. Here and in all the semidefinite programming (SDP). Specifically, denoting with following reviewed techniques, unless stated otherwise, such Ω(j) matrix (12) corresponding to the solution B(j) at the previ- k k rank constraints are relaxed by assuming that the number of (j) (j) (j)H H ous iteration j, i.e., Ω = I + Hk,lB B H ,the transmitting data streams is equal to the transmitting antennas k l=k l l k,l SDP problem to be solved to find the solutions B(j+1) for the to that user, i.e., dk = mt,k. From (15) and (18), we can rewrite k the (negative) SR as (j + 1)th iteration is given by

K K K K H (j) log |Ek| = − log Ωk + Hk,kΣkH + log |Ωk| { } k,k minimize tr Yk + tr Ck Σk Yk,Σk ∀k k=1 k=1 k=1 k=1 (29) K { }≤ where Ωk is defined in (12). Notice that it is often convenient to subject to tr Φk,mΣk Pm,m= 1,...,M k=1 work with the covariance matrices instead of the beamforming ⎡ ⎤ 1 matrices Bk since this change of variables may render the H (j) (j) (j) 2 optimization problem convex as, for instance, when minimizing ⎢ Hk,kΣkHk,k + Ωk Wk Ωk ⎥ ⎣ 1 ⎦0 the first term only in (29). It can then be seen that the SRM 2 W(j)Ω(j) Y problem is, however, nonconvex due to the presence of the k k k − log |Ωk| term, which is indeed a concave function of the and Σk 0,k= 1,...,K matrices Σk. An approximate solution is then be found in [12] via an where iterative scheme, whereby, at each (j + 1)th iteration, given the previous solution Σ(j), the nonconvex term − log |Ω | is k k (j) (j) H (j)−1 approximated using a first-order Taylor expansion as Wk = I + Hk,kΣk Hk,kΩk (32) −1 − | |− (j) log Ωk log Ωk (j) H (j) H (j) Ck = Hi,k I + Hi,lΣl Hi,l Wi −1 i= k l − tr Ω(j) H Σ − Σ(j) HH (30) k k,l l l k,l −1 l= k × H Σ(j)HH I + H Σ(j)HH H i i i i,l l i,l i,k (j) (j) H l where Ωk = I + l= k Hk,lΣk Hk,l. Since the resulting problem (33) K Y minimize − log |Ωk + Hk,kΣkHk,k| and k is an auxiliary optimization variable, which is Σk,k=1,...,K (j) k=1 defined using the Schur complement as Yk = WkΩk H (j) −1 −1 (Hk,kΣkH + Ω ) to convert the original optimization (j) H k,k k + tr Ωk Hk,lΣlHk,l problem to an SDP problem [13]. The derivation requires minor l= k modifications with respect to [13] and is therefore not detailed. K The scheme is referred to as “SDP relaxation” in the following: subject to tr {Φk,mΣk}≤Pm,m= 1,...,M (31) See [13] for further details about the algorithm. k=1 is convex, a solution can be efficiently found. Following the C. PWF original reference [12], we refer to this scheme as “soft interference nulling.” See [12] for further details about the algorithm. In [16], the (weighted) SRM problem for a general model that includes the MIMO-IFC-GC is studied. We review the approach here for completeness. Two algorithms are pro- B. SDP Relaxation posed, whose basic idea is to iteratively search for a solution A related approach is taken in [13] for the SRM problem2 for of the Karush–Kuhn–Tucker (KKT) conditions [11] for the a MIMO-IFC with regular per-transmitter, rather than gener- (weighted) SRM problem. Notice that, since the problem is alized, power constraints. Similarly to the previous technique, nonconvex, being a solution of the KKT conditions is only the optimization is over the transmit covariance matrices and necessary (as proved in [16]) but not sufficient to guarantee under the relaxed rank constraints. In particular, the authors first global optimality. It is shown in [16] that any solution Σk, k = approximate the problem by using the approach in [18]. Then, 1,...,K of the KKT conditions must have a specific structure an iterative solution is proposed by linearizing a nonconvex that is referred to as “polite waterfilling (PWF),” which is term similar to soft interference nulling, as previously reviewed. reviewed here for the SRM problem. Lemma 6 [16]: For a given set of Lagrange multipliers λ = 2 More generally, the reference studies the weighted SRM problem. (μλ1,...,μλM ), where μ>0 and λi ≥ 0fori = 1,...,M KAVIANI et al.: LINEAR PRECODING AND EQUALIZATION FOR NETWORK MIMO WITH PARTIAL COOPERATION 2089 associated with the M power constraints in (13), define the Remark 2: Other notable algorithms designed to solve the covariance matrices SRM problem for the special case of a MIMO-BC with generalized constraints are [25] and [26]. As explained in [16], these M schemes are not easily generalized to the scenario at hand where Ωˆ = λ Φ + HH Σˆ H (34) k m k,m j,k j j,k the cost function is not convex. As such, they will not be further m=1 j= k studied here. with − D. Leakage Minimization ˆ 1 −1 − H 1 Σk = Ω Ωk + Hk,kΣkHk,k . (35) μ k While the techniques previously discussed do not enforce constraints on the number of stream per users, here we extend An optimal solution Σk, k = 1,...,K, of the SRM problem a technique previously proposed in [27] that aims at aligning must have the PWF form interference through minimizing the interference leakage and − 1 − 1 is able to enforce the desired rank constraints. It is known ˆ 2 H ˆ 2 Σk = Ωk VkPkVk Ωk (36) that this approach solves the SRM problem for high signal- to-noise ratio (SNR). In this algorithm, it is assumed that the where the columns of V are the right singular vectors of the k power budget is divided equally between the data streams, −(1/2) ˆ −(1/2) “pre- and postwhitened channel matrix” Ωk Hk,kΩk and the precoding matrix of user k from BS m is given as with (12) for k = 1,...,K, and Pk is a diagonal matrix with Bk,m = (Pm/Kmdk)B¯ k,m, where B¯ k,m is an nt × dk ma- diagonal elements pk,i. The powers pk,i must satisfy ¯ H ¯ trix of orthonormal columns (i.e., Bk,mBk,m = I). The equal- + ization matrices are also assumed to have orthonormal columns 1 1 H pk,i = − (37) (i.e., Ak Ak = I). Hence, there is no interstream interference μ γk,i for each user. Total interference leakage at user k is given by √ where γk,i is the ith singular value of the whitened matrix H I = tr Ak QkAk (40) −(1/2) −(1/2) ˆ ≥ k Ωk Hk,kΩk . Parameter μ 0 is selected to satisfy the constraint where Q = (P /K d )H B¯ B¯ H H H . k j= k m∈Mj m m j k,m k,m k,m j,m M K M To minimize the interference leakage, the equalization matrix λm tr {Φk,mΣk}≤ λm (38) Ak for user k can be obtained as Ak = vdk (Qk), where m=1 k=1 m=1 vdk (A) represents a matrix with columns given by the eigenvectors corresponding to the d smallest eigenvalues of which is implied by the constraints of the original problem (13). k A. Now, for fixed matrices A , the cost function (40) can be Moreover, parameters λ ≥ 0 are to be chosen to satisfy each k i rewritten as individual constraint in (13). To obtain a solution Σk, k = 1,...,K, according to PWF ¯ H ˆ ¯ I = tr Bk,mQk,mBk,m (41) form as described in Lemma 6, [16] proposes to use the inter- ∈M ˆ k m k pretation of Ωk in (34) as the interference plus noise covariance ˆ H H 4 matrix and Σk in (35) as the transmit covariance matrix both at where Qˆ k,m = ∈K (Pm/Kmdk)H AjA Hj,m. 3 j=k,j m j,m j the “dual” system. Minimizing over the matrices Bk leads to choosing Based on this observation, the algorithm proposed in [16] ¯ ˆ Bk,m = vdk (Qk,m). The algorithm iterates until convergence. works as follows: At each jth iteration, the first one calculates We refer to this scheme as “min leakage” in the following. (j) the covariance matrices Σk in the original system using the PWF solution of Lemma 6; then, one calculates the matrices ˆ (j) E. Max-SINR Σk using again PWF in the dual system as previously explained. Finally, at the end of each jth iteration, one updates Another algorithm called “max-SINR” has been proposed in the Lagrange multipliers as [27], which is based on the maximization of SINR, rather than directly on the SR. This algorithm is also able to enforce rank K (j) constraints. The max-SINR algorithm assumes equal power tr Φk,mΣk (j+1) (j) k=1 allocated to the data streams and attempts at maximizing the λm = λm (39) Pm SINR for each stream by selecting the receive filters. Then, it exchanges the role of transmitter and receiver to obtain the thus forcing the solution to satisfy the constraints of the SRM transmit precoding matrices, which maximizes the max-SINR. problem (13). For details on the algorithm, we refer to [16]. This iterates until convergence. A modification of this algorithm is given in [28] by maximizing the ratio of the average signal 3In the “dual” system, the role of transmitters and receivers is switched, i.e., the kth transmitter in the original system becomes the kth receiver in the “dual” 4In the original work [27], which is proposed for the interference channels, system. The channel matrix between the kth transmitter and the lth receiver in the algorithm iteratively exchanges the role of transmitters and receivers to H the dual system is given by Hl,k. update the precoding and equalization matrices similarly. 2090 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 5, JUNE 2012

K { (j) (j)H}≤ power to the interference plus noise power (SINR-like) term. and the power constraints k=1 tr Φk,mBk Bk Pm However, these techniques are only given for standard MIMO for all m. (j) interference channels and not for MIMO-IFC-GC. Once the matrices Bk are obtained using the results in Lemma 7, the iterative procedure continues with the (j + 1)th iteration. We refer to this scheme as extended MMSE-IA VI. MEAN SQUARE ERROR MINIMIZATION (eMMSE-IA). In this section, we propose two suboptimal techniques to Remark 3: The algorithm previously proposed reduces to solve the WSMMSE problem. We recall that, with the WS- that introduced in [19] in the special case of per-transmitter power constraints and single-antenna receivers. It is noted that, MMSE problem, enforcing the constraint on dk is necessary to avoid trivial solutions. Performance comparison among all in such case, problem (42) can be solved in a distributed the considered schemes will be provided in Section VII for a fashion so that each transmitter k can calculate its matrix multicell system with network MIMO. (more precisely vector, given the single antenna at the receivers) independently from the other transmitters. In the MIMO-IFC- GC, the power constraints couple the solutions of the different A. MMSE Interference Alignment users and thus make a distributed approach infeasible. A technique referred to as MMSE interference alignment (MMSE-IA) was presented in [19] for an interference channel B. Diagonalized MMSE with per-transmitter power constraints and where each receiver We now propose an iterative optimization strategy inspired is endowed with a single antenna. Here, we extend the approach by the single-user algorithm that we put forth in Section IV. to the MIMO-IFC-GC. At the (j + 1)th iteration, given the matrices obtained at the The idea is to approximate the solution of the WS- previous iteration, we proceed as follows: The WSMSE (14) MMSE problem by optimizing the precoding matrices B , k with the definition of MSE-matrices (11) is a convex function followed by the equalization matrices A and iterating k in each A and B when (B , A ) ∀j = k are fixed. Neverthe- the procedure. Specifically, arbitrarily initialize B . Then, k k j j k less, it is not jointly convex in terms of both (A , B ). Inspired at each iteration j, the following hold: 1) For each user k k by Lemma 5 for the corresponding single-user problem, we k, evaluate the equalization matrices using the MMSE propose a (suboptimal) solution based on the solution of the (j) (j−1) (j−1)H H solution (18), obtaining A =(Hk,kB B H + k k k k,k dual problem for calculation of (Ak, Bk). To this end, we first − − − Ω(j 1))−1H B(j 1), where, from (12), we have Ω(j 1) = obtain A as (18). Then, we simplify the Lagrangian function k k,k k k k (j−1) (j−1)H H (j) I + Hk,lB B H . 2) Given matrices A , with respect to Bk by removing the terms independent of Bk. l=k l l k,l k H H the WSMMSE problem becomes Specifically, by defining Υk = l= k Hl,kAlWlAl Hl,k,we have that the Lagrangian function at hand is given by K (j) − −1 minimize tr WkEk L(B ; λ)=tr W I + BHHH Ω 1H B Bk,k=1,...,K k k k k,k k k,k k k=1 ! " K H H H +tr ΥkBkBk +tr λmΦk,m BkBk . (46) subject to tr Φk,mBkB ≤ Pm ∀m ∈M (42) k m k=1 This Lagrangian function for user k is the same as the La- where E(j) is (11) with A(j) in place of A . Fixing the k k k grangian function (25) of single-user WSMMSE problem when (j) ∀ equalization matrices Ak k, this problem is convex in Bk Φ(λ) is replaced with Fk(λ)=Υk + λmΦk,m.Matrix and can be solved by enforcing the KKT conditions. Therefore, Fk(λ) is nonsingular, and therefore, using the same argument (j) matrices Bk for the jth iteration can be obtained as follows: as in the proof of Lemma 5, for a given Lagrange multipliers λ (j) and given other users’ transmission strategies (Al, Bl) ∀l = k, Lemma 7: For given equalization matrices Ak , a solution (j) the optimal transmit precoding matrix can be obtained as Bk , k = 1,...,K, of the WSMMSE problem is given by 1 −1 − K Bk = Fk(λ) 2 UkΣk (47) B(j) = HH A(j)W A(j)HH + μ Φ k l,k l l l l,k m k,m × ∈ Cmt,k dk l=1 m where Uk is the eigenvectors of −(1/2) H −1 −(1/2) Fk(λ) Hk,kΩk Hk,kFk(λ) corresponding to × HH A(j)W (43) its largest eigenvalues γ ≥···≥γ , and Σ is diagonal k,k k k k,1 √ k,dk k matrices with the elements pk,i given by where μm are Lagrangian multipliers satisfying w 1 + ≥ p = k,i − (48) μm 0 (44) k,i γ γ k,i k,i K (j) (j)H − with λ 0 being the Lagrangian multipliers satisfy the μm tr Φk,mBk Bk Pm = 0 (45) k=1 power constraints. Since this scheme diagonalizes the MSE KAVIANI et al.: LINEAR PRECODING AND EQUALIZATION FOR NETWORK MIMO WITH PARTIAL COOPERATION 2091 matrices defined in (9), it is referred to as diagonalized MMSE SRM problem is studied by decomposing the multiuser problem (DMMSE). into single-user problems for each user. Each single-user prob- To summarize, the proposed algorithm at each iteration j lem is a standard single-user SRM problem with an additional (j) interference power constraint. The approach used in [14]–[16] evaluates the transmit precoding matrices Bk , given other − − (j 1) (j 1) assumes that the number of transmitted streams is equal to nr. users’ transmission strategies (Al , Bl ) using (47) and (48), updates the equalization matrices using the MMSE solu- In this paper, we address WSMMSE problem and allow for an ≤ tion (18), and updates the λ via a subgradient update arbitrary number of streams (dk nr). Remark 6: Our algorithms consists of an inner loop, which K solves the WSMMSE problem, and an outer loop, which is the (j+1) (j) − H λm = λm + δ Pm tr Φk,mBkBk (49) subgradient algorithm to update λ. The subgradient algorithm k=1 in the outer loop is convergent (with a proper selection of the step sizes [30]) due to the fact that the dual function to satisfy the power constraints. infB L(B; λ) is a concave function with respect to λ [11]. The Remark 4: In this paper, we assume perfect knowledge of inner loops of the proposed algorithms in this paper (i.e., eMM- CSI. Therefore, each transmitter and receiver has sufficient in- SEIA and DMMSE) are convergent since the objective function formation to calculate the resulting precoders and equalizers by decreases at each iteration. A discussion of the convergence for running the proposed algorithms. Under this assumption, which a special case of the eMMSEIA algorithm can be found in [19]. is common to other reviewed works such as [12] and [13], no However, the original problem is nonconvex, and our solutions exchange of precoder and equalizer vectors is required between are only local minima. Nevertheless, the DMMSE algorithm is the transmitters and receivers. Nevertheless, in practice, the shown to converge to a local minimum with better performance CSI may only be locally available, in the sense that trans- compared to the previously known schemes in Section VII. mitter k knows channel matrices Hl,k, for all l = 1,...,K, whereas receiver k is aware of channel matrices H , for all k,l VII. NUMERICAL RESULTS l = 1,...,K. The proposed DMMSE and the reviewed PWF [14], [16] algorithms require, aside from the local CSI, that We consider a hexagonal cellular system where each BS is the transmitter k also has available the interference-plus-noise equipped with nt transmit antennas and each user has nr re- covariance matrix, Ωk and the current equalization matrices Al ceive antennas. The users are uniformly located at random. Two for all l = 1,...,K to update the precoder for user k. Hence, tiers of surrounding cells are considered as interference for each to enable DMMSE and PWF with local CSI, exchange of cluster. We consider the worst-case scenario for the intercluster the equalizer matrices is needed between the nodes. Similarly, interference, which will be the condition that interfering BSs the proposed eMMSEIA and the min leakage and Max-SINR transmit at the full allowed power [7], [8], [31], [32]. We define algorithms [27] require the transmitters to know the equalizing the cooperation factor κ as a number of BSs cooperating on matrices Al for l = 1,...,K at each iteration, in addition to transmission to each user. The κ BSs are assigned to each user, the local CSI. Moreover, each receiver must know the current so that the corresponding channel norms (or, alternatively, the precoders Bl for all l = 1,...,K. Therefore, the overhead for corresponding received SNRs) are the largest. the proposed eMMSEIA and the min leakage and Max-SINR The propagation channel between each BS’s transmit an- algorithms involves the exchange of equalizer and precoder tennas and the mobile user’s receive antenna is characterized matrices between the transmitters and the receivers. However, by path loss, shadowing, and Rayleigh fading. The path-loss −β these latter algorithms can also be adapted using the bidi- component is proportional to dkm, where dkm denotes distance rectional optimization process proposed in [29]. This process from BS m to mobile user k and β = 3.8 is the path-loss involves bidirectional training, followed by data transmission. exponent. The channel from the transmit antenna t of the BS In the forward direction, the training sequences are sent using b at the receive antenna r of the kth user is given by [7] # the current precoders. Then, at the user receivers, the equalizers $ % −β are updated to minimize the least-square error cost function. (r,t) (r,t) (t) dk,b Hk,b = αk,b γ0ρk,bA Θk,b (50) In the backward training phase, the current equalizers are used d0 to send the training sequences, and the precoders are updated (r,t) ∼CN (dBm) accordingly. Finally, the soft interference nulling (SIN) [12] where αk,b (0, 1) represents Rayleigh fading, ρk,b is and SDP relaxation [13] techniques are applied in a centralized the lognormal shadow fading between the bth BS and the kth fashion (rather than by updating the transmitter and receiver for user with a standard deviation of 8 dB, and d0 = 1 km is the cell each user at each iteration), and they require centralized full radius. γ0 is the interference-free SNR at the cell boundary. We knowledge of all channel matrices. consider one user randomly located per cell for the numerical Remark 5: In [13], the SRM problem for a MIMO-IFC results. with regular per-transmitter, rather than generalized, power When sectorization is employed, the transmit antennas are constraints is addressed. The problem is addressed by solving equally divided among the sectors of a cell. Each transmit an SDP problem at each iteration. Moreover, the optimization antenna has a parabolic beam pattern as a function of the is over the transmit covariance matrices and under the relaxed direction of the user from the broadside direction of the antenna. rank constraint. This enforces a constraint on the number of (For more details, see [7] and [33].) The antenna gain is a transmitted streams per user. In [14]–[16], study the (weighted) function of the direction of the user k from the broadside 2092 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 5, JUNE 2012

Fig. 3. Per-cell SR for a MIMO-IFC-GC with M = 5andκ = 1, 2, 3, 5, Fig. 2. Per-cell SR for a MIMO-IFC-GC with M = 3andκ = 2. nt = 4, nr = dk = 2, and two users per cell. direction of the tth transmit antenna of the bth BS denoted by (t) ∈ − Θk,b [ π,π], Θ3dB is the half-power angle, and As is the sidelobe gain. The antenna gain is given as [33] ⎛ ⎞ 2 Θ(t) (t) − ⎝ k,b ⎠ A Θk,b = min 12 ,As . (51) dB Θ3dB

For the three six-sector cells, As = 20, 23 dB, and Θ3dB= (70π)/180, (35π)/180, respectively [7], [33], [34]. When there is no sectorization, we set A = 1. We first compare different algorithms (for the solution of the SRM problem) without enforcing rank constraints on SIN, PWF, SDP relaxation and setting dk = min(mt,k,mr,k)=nr for the eMMSEIA and DMMSE algorithms. To solve the SRM problem, the weight matrices in the eMMSEIA and DMMSE −1 algorithms are updated at each iteration as Wk = Ek using Fig. 4. Per-cell sum rate of the schemes that can support dk < the current MSE-matrix Ek. Fig. 2 compares the per-cell SR min(mt,k,mr,k ) for dk = 1, nt = 4, nr = 2, M = 3, and κ = 2. of the algorithms discussed in this paper for a cluster with M = 3 cells and a cooperation factor κ = 2. The results show was not designed to handle rank constraints. We have adopted the PWF algorithm to support dk < min(mt,k,mr,k) by using that our proposed DMMSE algorithm outperforms other tech- − ˆ (1/2) H −(1/2) niques, whereas the PWF algorithm [14], [16] has a similar a thin SVD of Ωk Hk,kΩk when computing (36). performance. Our proposed eMMSEIA scheme converges to a In Fig. 5, we vary the size of cluster M, showing also the poorer local optimum value compared with these two schemes. advantages of coordinating transmission over larger clusters, The soft interference nulling (SIN) [12] and SDP relaxation even when the number of cooperating BSs κ is fixed. Recall that [13] algorithms, which use the approximation of the nonconvex M represents the set of BSs whose transmission is coordinated, terms in the objective function, perform worse in this example. but only κ BSs cooperate for transmission to a given user. As In Fig. 3, we evaluate the effect of partial cooperation for an example, for a cluster size of M = 7, a cooperation factor of the DMMSE, eMMSEIA, and PWF algorithms in a cluster of κ = 4 performs almost as well as the full cooperation scenario size M = 5, where each BS is equipped with nt = 4 transmit with κ = 7. Moreover, the performance gains with respect to antennas, each user employs nr = 2 receive antennas, and the noncooperative case κ = 1 are evident. We also show the two users are randomly dropped in each cell. Recall that the performance with a cluster containing a single cell, i.e., M = 1. cooperation factor κ represents the number of BSs cooperating This highlights the performance gains attained, even in the in transmission to each user. It can be seen that, as κ increases, absence of message sharing among the BSs (i.e., κ = 1) due the performance improves with diminishing returns as κ grows to the coordination of the BSs within the cluster. large. Moreover, the relative performance of the algorithms Finally, the effect of sectorization is studied in Fig. 6, where confirms the considerations previously. nt = 6 transmit antennas at each BS are divided equally into In Fig. 4, we compare again the performance of the schemes S = 1, 3, 6 sectors. Each cell contains six users, each equipped considered in Fig. 3 but with a stricter requirement on the with nr = 2 receive antennas. The users are randomly located number of streams, namely, dk = 1. It can be seen that the at the distance of 2/3d0 from its BS. For a given channel proposed DMMSE tends to perform better than PWF, which realization, the DMMSE algorithm is used to obtain the per-cell KAVIANI et al.: LINEAR PRECODING AND EQUALIZATION FOR NETWORK MIMO WITH PARTIAL COOPERATION 2093

the data estimates. Specifically, we have proposed an exten- sion of the recently introduced MMSE interference alignment strategy and a novel strategy termed diagonalized MSE matrix (DMMSE). Our proposed strategies support transmission of any arbitrary number of data streams per user. Extensive numerical results show that the DMMSE outperforms most previously proposed techniques and performs just as well as the best known strategy. Moreover, our results bring insight into the advantages of partial cooperation and sectorization and the impact of the size of the cooperating cluster of BSs and sectorization. We have concluded with a brief discussion on the complexity of the algorithms. Due to the difficulty of complete complexity analysis, particularly in terms of speed of convergence, we have presented a discussion based on our simulation experiments. The PWF algorithm converges in almost the same number Fig. 5. Per-cell SR of the proposed DMMSE scheme for cluster sizes M = of iterations as the DMMSE algorithm. The complexity per 1, 3, 7 versus the cooperation factor κ, with nt = nr = 2, SNR = 20 dB, and iteration of PWF and DMMSE is also almost the same as a single user per cell. O 2 O 3 K( (κntnr)+ (nr)) (required for the thin SVD operation). However, the PWF algorithm contains additional operations (matrix inversion and SVD) to obtain the precoding matrices from the calculated transmit covariance matrices.5 In addition, the PWF algorithm includes a waterfilling algorithm within its inner loop, which is not required in the DMMSE algorithm. The eMMSEIA algorithm has lower complexity per iteration O 3 (i.e., K (nr)) than the PWF and DMMSE algorithms since its complexity is due to a matrix inversion per iteration per user. However, eMMSEIA converges in a larger number of iterations than DMMSE and PWF. The complexity per iteration for the SDP relaxation is higher than that for the SIN algorithm. (This is because of the extra auxiliary positive semidefinite matrix variable Y introduced in the SDP relaxation algorithm). The SIN algorithm also converges in a smaller number of iterations than the SDP relaxation algorithm.

Fig. 6. CDF of the per-cell sum rates achieved by DMMSE for S = 1, 3, 6 sectors per cell, M = 1, 3, 7 coordinated clusters, and κ = 1, 2, 3 cooperation APPENDIX factors, with γ0 = 20 dB, nt = 6, and nr = 2. The circles represent the mean PROOF OF LEMMA 5 values of the per-cell SRs. The inequality (24) follows from weak Lagrangian duality. sum rate. The cumulative distribution functions (CDFs) of per- We now prove the second part of the statement. Recognizing cell sum rates are computed using a large number of channel now that tr{WE} with (19) is a Schur-concave function of the realizations. The gains of sectorization and cooperation are diagonal elements of (19),6 it can be argued that the minimum compared. For example, the system with coordination of seven is attained when E is diagonalized as we did for Lemma 4. cells and κ = 3 cooperation factor and without sectorization Defining R = HHΩ−1H, we can conclude that BHRB must performs better than the sectorized system with S = 6 and with also be diagonal in this search domain. Now, assume that no coordination between the BSs. an optimal solution of the single-user WSMMSE problem is denoted as B˜ . Without loss of generality, we can assume that this solution diagonalizes the MSE matrices. The necessity of VIII. CONCLUSION the KKT conditions can be proved as in [16] and in special cases In this paper, we have studied a MIMO interference channel such as the MIMO interference channel with partial message with partial cooperation at the BSs and per-BS power con- sharing of Section II; it also follows from linear independence straints. We have shown that the channel at hand is equiva- constraint qualification conditions [30]. lent to a MIMO interference channel under generalized linear constraints (MIMO-IFC-GC). Focusing on linear transmission 5This can be performed together with finding the MMSE receive matrices. 6 strategies, we have reviewed some of the available techniques A Schur-concave function f(x) of vector x =(x1,...,xd) is such that j j f(x) ≤ f(x) if x majorizes x, i.e., if x ≥ x for all j = for the maximization of the SR and extended them to the i=1 [i] i=1 [i] MIMO-IFC-GC when necessary. Moreover, we have proposed 1,...,d,wherex[i] (and x[i]) represents the vector sorted in decreasing order, ≥···≥ ≥···≥ two novel strategies for minimization of the weighted MSE on i.e., x[1] x[d] (and x[1] x[d]). 2094 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 5, JUNE 2012

Hence, there exists a Lagrange multiplier vector λ˜, which, Thus, for the given Lagrange multiplier λ˜, which, together with together with B˜ , satisﬁes the KKT conditions of the WSMMSE B˜ , satisfying the KKT conditions of (20), B˜ must satisfy (55) problem (20) [18], [30]. As stated in the Lemma, we consider and (59). If all power constraints are satisﬁed with equality by the case that λ˜m are also strictly positive (i.e., λ˜m > 0 for all this solution, then (55) and (59) also solve the single constraint m). Simplifying the KKT condition (26), we have7 problem

M − −1 ∇ L − ˜ ˜ ˜ ˜ ˜ minimize tr W I + BHHHΩ 1HB B = RBEWE + λmΦmB = 0. (52) B m=1 M ˜ H ≤ ˜ Left-multiplying (52) by B˜ H gives us subject to tr Φ(λ)BB λmPm. (60) m=1 ˜ H ˜ ˜ ˜ ˜ H ˜ ˜ B RBEWE = B λmΦm B. (53) The solution of this problem is given in Lemma 3 as m − 1 H B(λ˜)=Φ(λ˜) 2 UΣ (61) SinceB˜ RB˜ and, correspondingly, E˜ are diagonal matrices, ˜ H ˜ ˜ B ( m λmΦm)Bmust also be diagonal. For simplicity, we M where U consists of d eigenvectors of Φ(λ˜)−(1/2)RΦ introduce Φ(λ˜)= λ˜mΦm. Since λ˜m > 0 for every m; m=1 ˜ −(1/2) therefore, Φ(λ˜) is a nonsingular matrix. This can be easily (λ) corresponding to its largest eigenvalues,√ and Σ is a diagonal matrix with the diagonal elements of pi, which is verified due to the structure of Φm. Hence, we can write B˜ HΦ(λ˜)B˜ = Δ˜ , where Δ˜ ∈ Cd×d is a diagonal matrix. There- given by fore, we can write + wi − 1 Φ(λ˜)1/2B˜ = U˜ Σ˜ (54) pk,i = (62) μγi(λ˜) γi(λ˜) × where U˜ ∈ Cmt d consists of orthonormal columns (i.e., ˜ H ˜ ˜ d×d for a waterfilling value of μ ≥ 0, which satisfies the power U U), and√ Σ ∈ C is a diagonal matrix with the diagonal constraint terms of p˜i. Hence, we can write ˜ ˜ −1/2 ˜ ˜ H B = Φ(λ) UΣ. (55) tr Φ(λ˜)B(λ˜)B(λ˜) ≤ λ˜mPm. (63) m Replacing the structure of B˜ given in (55), we can write

H − 1 − 1 On the other hand, summing up the KKT conditions λ˜m(Pm − ˜ H ˜ ˜ ˜ H ˜ 2 ˜ 2 ˜ ˜ B RB = Σ U Φ(λ) RΦ(λ) UΣ = D. (56) H tr{ΦmBB })=0 for all m, we obtain that Thus, we can conclude from the preceding equation that U˜ must ! " −(1/2) −(1/2) contain the eigenvectors of Φ(λ˜) RΦ(λ˜) . H tr λ˜mΦm B˜ B˜ = λ˜mPm. (64) Now, plugging (55) into (26) and left-multiplying it with m m Φ−(1/2), we get −1 −1 If we set μ = 1 and comparing (59) and (62), we can conclude ˜ ˜ ˜ ˜ 2 ˜ ˜ 2 ˜ ΓΣ I + ΓΣ W I + ΓΣ = Σ (57) that p˜i = pi ∀i, which, together with a comparison of (61) and (55), we can conclude that B(λ˜)=B˜ and the μ = 1isthe ˜ ˜ ˜ where Γ˜(λ) = diag[γ1(λ) ···γd(λ)] is a diagonal matrix optimal Lagrange multiplier of the single-constraint WSMMSE with the diagonal terms of the d largest eigenvalues of problem (60). Following Lemma 4, this precoding matrix is also Φ(λ˜)−(1/2)RΦ(λ˜)−(1/2). Since all the matrices are diagonal, a result of minimization of the Lagrangian function (23) when ˜ (57) reduces to the scalar equations μ = 1 and Φ1 = Φ(λ), which means w γ (λ˜) i i = 1. (58) p =infL(B; λ˜). (65) 2 B 1 +˜piγi(λ˜) On the other hand, we have Solving these equations gives us the optimal p˜i given by + max inf L(B; λ) ≥ inf L(B; λ˜). (66) wi 1 λ≥0 B B p˜i = − . (59) γi(λ˜) γi(λ˜) which in concert with (24) and (65) results in

0 7 ∇ { H } p =infL(B; λ˜) = max inf L(B; λ) (67) We use a differentiation rule Xtr AX B = BA and ≥ −1 −1 −1 B λ 0 B ∇Xtr{Y } = −Y (∇XY)Y . For the complex gradient operator, each matrix and its conjugate transpose are treated as independent variables [35]. thus concluding the proof. KAVIANI et al.: LINEAR PRECODING AND EQUALIZATION FOR NETWORK MIMO WITH PARTIAL COOPERATION 2095

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Lee, “Multi-cell MIMO the President’s Doctoral Prize of Distinction (2009 and 2010), and the Queen downlink with cell cooperation and fair scheduling: A large-system limit Elizabeth II Graduate Student Scholarship (2011). 2096 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 5, JUNE 2012

Osvaldo Simeone (SM’12) received the M.Sc. de- Shlomo Shamai (Shitz) (S’80–M’82–SM’89–F’94) gree (with honors) and the Ph.D. degree in informa- received the B.Sc., M.Sc., and Ph.D. degrees in elec- tion engineering from Politecnico di Milano, Milan, trical engineering from Technion—Israel Institute of Italy, in 2001 and 2005, respectively. Technology, Haifa, Israel, in 1975, 1981, and 1986, He is currently with the Center for Wireless Com- respectively. munications and Signal Processing Research, New During 1975–1985, he was with the Communi- Jersey Institute of Technology, Newark, where he is cations Research Labs as a Senior Research Engi- an Assistant Professor. His current research inter- neer. Since 1986, he has been with the Department ests include the cross-layer analysis and design of of Electrical Engineering, Technion—Israel Institute wireless networks, with emphasis on information- of Technology, where he is currently the William theoretic, signal processing, and queuing aspects. Fondiller Professor of Telecommunications. His re- Specific topics of interest include cognitive radio, cooperative communications, search interests include information theory and statistical communications. rate-distortion theory, ad hoc, sensor, mesh and hybrid networks, distributed Dr. Shamai (Shitz) has served as Associate Editor for Shannon Theory estimation, and synchronization. of the IEEE TRANSACTIONS ON INFORMATION THEORY and has also Dr. Simeone co-received of Best Paper Awards at the IEEE Workshop on served on the Board of Governors of the Information Theory Society. He is Signal Processing Advances in Wireless Communications in 2007 and the IEEE a member of the Executive Editorial Board of the IEEE TRANSACTIONS ON Wireless Rural and Emergency Communications, also in 2007. He currently INFORMATION THEORY. He received the 2011 Claude E. Shannon Award, the serves as an Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS. 1999 van der Pol Gold Medal of the Union Radio Scientifique Internationale, the 2003 and the 2004 Joint IT/COM Societies Paper Award, the 2007 IEEE Information Theory Society Paper Award, the 2009 European Commission FP7, Witold A. Krzymien´ (M’79–SM’93) received the Network of Excellence in Wireless COMmunications (NEWCOM++)Best M.Sc. (Eng.) and Ph.D. degrees in electrical engi- Paper Award, the 2010 Thomson Reuters Award for International Excellence neering from Poznan´ University of Technology, in Scientific Research, the 1985 Alon Grant for distinguished young scientists, Poznan,´ Poland, in 1970 and 1978, respectively. the 2000 Technion Henry Taub Prize for Excellence in Research, and he was a Since April 1986, he has been with the Department corecipient of the 2000 IEEE Donald G. Fink Prize Paper Award. of Electrical and Computer Engineering, Univer- sity of Alberta, Edmonton, AB, Canada, where he currently holds the endowed Rohit Sharma Profes- sorship in Communications and Signal Processing. In 1986, he was one of the key research program architects of the newly launched TRLabs, which is Canada’s largest industry–university–government precompetitive research consortium in the Information and Communication Technology area that is headquartered in Edmonton. His research activity has been closely tied to the consortium ever since. Over the years, he has also done collaborative research with Nortel Networks, Ericsson Wireless Communications, German Aerospace Centre (DLR-Oberpfaffenhofen), TELUS Communications, Huawei Tech- nologies, and the University of Padova, Padua, Italy. He held visiting research appointments with the Twente University of Technology, Enschede, The Netherlands, from 1980 to 1982; Bell-Northern Research, Montréal, QC, Canada, from 1993 to 1994; Ericsson Wireless Communications, San Diego, CA, in 2000; Nortel Networks Harlow Laboratories, Harlow, U.K, in 2001; and the Department of Information Engineering, University of Padova, in 2005. His current research interests include multiuser multiple-input–multiple- output (MIMO) and MIMO orthogonal frequency-division multiplexing systems and multihop relaying and network coordination for broadband cellular applications. Dr. Krzymien´ is a Fellow of the Engineering Institute of Canada, and a licensed Professional Engineer in the Provinces of Alberta and Ontario, Canada. He is an Associate Editor for the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY and a member of the Editorial Board of Wireless Personal Communications (Springer). From 1999 to 2005, he was the Chairman of Commission C (Radio Communication Systems and Signal Processing) of the Canadian National Committee of the Union Radio Scientifique Internationale (URSI), and from 2000 to 2003, he was the Editor for Spread Spectrum and Multi-Carrier Systems of the IEEE TRANSACTIONS ON COMMUNICATIONS. He received a Polish National Award of Excellence for his Ph.D. thesis, the 1991/1992 A.H. Reeves Premium Award from the Institution of Electrical Engineers (U.K.) for a paper published in the IEE Proceedings—Part I,and the Best Paper Award at the IEEE WIRELESS COMMUNICATIONS AND NETWORKING CONFERENCE in April 2008.