A Scalable Limited Feedback Design for Network MIMO Using Per-Cell

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single cell SDMA scheme with limited feedback has been considered extensively in the literatures [8]–[11].

However, these results cannot be applied to the multi- ¡ ¤

cell scenario with limited feedback capturing the dynamic ¦§ ¨

number of cooperating BSs and the heterogeneous path

© loss effects. ¢ £ ¥ One conventional limited feedback design, namely the Givens rotation [19]–[21], could potentially address the above challenges. Using Givens rotation, a unitary matrix (channel Fig. 1. An sample network MIMO system with (nT ,N,nR,K) = , , , . The 3 BSs collaboratively serve the 3 MSs via multi-cell SDMA direction) is decomposed into products of Givens matrices. (4 3 2 6) Each Givens matrix contains two Givens parameters, which are quantized using scalar quantizer and fed back to the summarize the main results in Section VI. BSs. As a result, it offers the flexibility because when the Notations: Matrices and vectors are denoted with boldface number of cooperating BSs changes, the number of Givens uppercase and lowercase letters, respectively; A† and tr A matrices also changes accordingly. Hence, the same scalar denote the conjugate transpose and trace of matrix A, respec-{ } quantizer can be used to quantize unitary matrices of time- 2 tively; IL represents the L L identity matrix; CN(µ, σ ) varying dimensions. However, the issue of Givens rotation denotes the circularly symmetric× complex Gaussian distribu- approach is the poor feedback efficiency due to scalar (or 2 tion, with mean µ and variance σ ; C and R+ denote the set two-dimensional vector) quantization. In this paper, we shall of complex numbers and positive real numbers, respectively. propose a novel scalable limited feedback mechanism using per-cell product codebooks3 to address the dynamic MIMO II. SYSTEM MODEL configurations and heterogeneous path loss effects, along with a low-complexity realtime feedback indices selection algo- A. Network MIMO Channel Model rithm. In the proposed feedback scheme, the product codebook Consider a network MIMO system (nT ,N,nR,K) with (defined in Section III-B) that is used for CSI feedback in the N BSs serving K active MSs in the downlink direction, as network MIMO systems is simply the Cartesian product of shown in Fig. 1. The N BSs are inter-connected via high- N per-cell product codebooks, with N denoting the number speed backhauls, and collaboratively serve the K MSs through of cooperating BSs. Cartesian product operation allows for a the standard SDMA scheme. Without loss of generality, we single per-cell product codebook to be used irrespective of the assume that each BS has nT antennas and each MS has nR number of cooperating BSs. We shall show that the proposed antennas4. Assume limited feedback based block diagonaliza- per-cell product codebook based limited feedback mechanism tion (LF-BD) [7], [10] is employed for SDMA transmission can asymptotically achieve the same performance as the joint- in the network MIMO system, the downlink signal model can cell codebook approach. We derive an asymptotic per-user be written as, throughput loss due to limited feedback with per-cell product K codebooks. Based on the results, we show that when the num- yk = HkWk Pkdk + Hk Wj Pj dj +zk,(1) sum ber of per-user feedback-bits Bk is NnT nR log2(ρgk ) , p j=1X,j6=k p O c c the system operates in a noise-limited regime with per-user sum nRρgk residual CCI throughput scaling as nR log . On the other 2 NnT | {z } O where y CnR×1 is the received signal vector at the kth hand, when the number of per-user feedback-bits Bk does k ∈CnR×NnT not scale with the system SNR ρ, the system operates in an MS; Hk denotes the aggregate CSI matrix of th ∈ CNnT ×nR th interference limited regime with per user throughput scaling the k MS; Wk is the precoder for the k † ∈ NPmax nRBk MS, with WkWk = InR ; Pk = Kn InR is the power as 2 . Numerical results show while the proposed c R O (NnT ) allocation matrix for the kth MS, with P representing scheme is flexible to accommodate dynamic number of co- max c c nR×1 the maximal transmit power of one BS; dk C is the operating BS, it has significant performance gains over the th∈ reference baselines (e.g. Givens rotation approach). transmitted symbol vector intended for the k MS, satisfying E † CnR×1 The rest of this paper is organized as follows. We introduce dkdk = InR ; zk denotes the noise vector, with n o ∈ the network MIMO system model and codebook-based CSI CN 2 E † 2 i.i.d. (0, σ ) entries, i.e., zkzk = σ InR . In this paper, feedback model in Section II. The proposed per-cell product n o it is assumed that KnR NnT . codebook based limited feedback framework is introduced in For precoder design, LF-BD≤ imposes the following condi- Section III. Asymptotic performance analysis of the proposed tions on Wk to eliminate CCI at the transmitter side [7], [10], scheme is elaborated in Section IV. We present the numerical results along with discussions in Section V. Finally, we HckWj = 0n ×L, k = j; k, j =1, 2, ,K, (2) R ∀ 6 ··· 3”Per-cell product codebook” refers to the codebook that is designed with 4Althoughb c we assume that the cooperating BSs and active MSs have the single BS antenna configuration; while ”joint-cell codebook” refers to the homogeneous antenna configurations (i.e., every BS has nT transmit antennas, codebook that is designed with the aggregate antenna configuration of the and every MS has nR receive antennas.), the proposed per-cell limited N cooperating BSs. In other words, in joint-cell codebook design, the N feedback mechanism can be directly extended to the cases with heterogeneous cooperating BSs are treated as one aggregate BS, which is called super-BS antenna configurations (i.e., different BSs and/or different MSs have different in the paper. number of antennas.). 3

th where Hk is the quantized aggregated CSI matrix of the k between the quantization source V and the codedwords in the MS (see equation (12)). entire codebook ϕ. Mathematically, b In the network MIMO system, the aggregate CSI matrix H k ∗ seen by the kth MS can be partitioned into N submatrices, J = argmin dc (VJ , V) . (8) VJ ∈ϕ i.e., The following assumptions are made through the rest of Hk = [Hk, Hk, Hk,N ] , k =1, 2, ,K, (3) 1 2 ··· ∀ ··· this paper. Firstly, all the MSs have perfect channel state nR×nT information between the cooperative BSs and themselves. where Hk,n C denotes the CSI matrix between the nth BS and the∈ kth MS, which is commonly modeled as [22], Secondly, the large-scale fading component Gk is known at all the BSs and MSs within the network. Thirdly, each MS (w) Hk,n = √gk,nsk,nH , k =1, 2, ,K; n =1, 2, ,N, shall feed back the CSI knowledge to the cooperative BSs k,n ∀ ··· ··· (4) through zero-delay error-free feedback links. In addition, flat block-fading MIMO channel is assumed. th th where gk,n R denotes the path loss from the n BS to k ∈ + MS; sk,n R+ denotes the lognormal shadowing component; (w) ∈ III. SCALABLE LIMITED FEEDBACK MECHANISM BASED H CnR×nT denotes a random matrix with i.i.d. CN(0, 1) k,n ON PER-CELL PRODUCT CODEBOOK entries,∈ i.e., the Rayleigh fading component. th Moreover, the aggregate CSI matrix Hk of the k MS can In the joint-cell codebook based CSI feedback schemes [14], be factorized as, [15], the cooperating BSs are treated as one composite BS (super BS), and the codebooks are designed for the aggregate (w) Hk = H Gk, k =1, 2, ,K, (5) k ∀ ··· CSI matrix Hk in a standard way. Moreover, authors of [15] (w) has also considered the heterogeneous path loss effects and CnR×NnT where Hk denotes a random matrix with i.i.d. ∈ Nn ×Nn the effects of increasing cluster size with a variable number CN(0, 1) entries; G R T T represents the large-scale k + of cooperating BSs. While the joint-cell codebook approaches fading component, which∈ is given by, provide some preliminary solutions for network MIMO systems, two important issues, namely the dynamic number of Gk = diag √gk,1sk,11n √gk,2sk,21n T T ··· cooperating BSs and the heterogeneous path loss effects, are ignored. In this section, we shall introduce a scalable √gk,N sk,N 1n , (6) T limited feedback design based on per-cell product codebook to accommodate the dynamic MIMO configurations and deal 1×nT where 1n equals to [1 1 1] R , and diag(a) with the heterogeneous path loss effects. Moreover, we shall T ··· ∈ + denotes a diagonal matrix with diagonal entries given by the formulate the realtime feedback indices determination (at the elements of vector a. mobile) as a combinatorial search problem, and propose a low-complexity solution. B. Codebook-based Limited Feedback Model For downlink transmission with the LF-BD scheme, the A. Scalable Per-cell Product Codebook based Limited Feed- essential information (i.e., users channel subspaces) required at back Design the BSs side can be obtained through the codebook-based CSI In order to accommodate the dynamics of cooperating BSs feedback scheme. In a general codebook-based CSI feedback and deal with the heterogeneous path loss effects, we propose 5 framework, the design metric for minimizing the residual CCI (w) to quantize span Hk with per-cell product codebooks, is chordal distance [7], [10], which is defined as, n o rather than quantizing span Hk directly, where span A 1 † † denotes the row space of matrix{ } A. The proposed per-cell{ } dc (V1, V2)= V1V1 V2V2 , (7) √2 − F product codebook based CSI quantization procedure involves

NnT ×nR 6 both MS side processing and BS side processing, i.e., code- where V1, V2 C are two orthonormal matrices ; ∈ words indices generation at the MS side and quantized aggre- and A F denotes the Frobenius norm of matrix A. Withoutk k loss of generality, we assume the BS and MS share gate CSI matrix reconstruction at the BS side. Denote the N the same CSI quantization codebook, denoted by ϕ. The J th per-cell product codebooks as ϕ1, ϕ2, , ϕN , where ϕn = nT ×nR † ··· Bk,n V V C , V V = In , with cardinality 2 , and codeword in the codebook ϕ is given by VJ and the cardinality ∈ R B the total number of per-user feedback-bits Bk is given by of the codebook ϕ is 2 , with B representing the number N of feedback-bits. In the conventional single-cell MIMO com- Bk = n=1 Bk,n. Note that the N per-cell product codebooks munications, the codeword index J ∗ reported by the MS is couldP be identical. The proposed scalable limited feedback generated through the minimization of the chordal distance processing at the MS and the BS sides is summarized below: Feedback Indices Generation (MS side): take the kth MS 5For limited feedback based SDMA transmission, a widely adopted design as an example. The feedback indices generation is a mapping metric for CSI feedback schemes is minimizing the inter-user interference, (w) from the normalized aggregated CSI matrix Hk to the rather than maximizing the desired signal power [7], [10]. ∗ ∗ ∗ 6 feedback indices J , J , , J corresponding to In this paper, an orthonormal matrix refers to a matrix with orthonormal k,1 k,2 ··· k,N columns. the N per-cell product codebooks ϕ1, ϕ2, , ϕN . ··· 4

• Normalization: To handle the heterogeneous path loss Remark 2 (Generalization of Common Cooperating BSs): effects, we first normalize the aggregate CSI matrix Hk Without loss of generality, we have assumed that the K MSs by the large-scale fading component Gk, i.e., have the same set of cooperating BSs. The above per-cell product codebook limited feedback framework can also be H(w) H G−1. (9) k = k k applied directly to the case where each MS has a different • Decomposition: Apply the standard singular value de- active cooperating BSs set. For example, suppose MS-1 has composition (SVD) [23] to the normalized CSI matrix BS-1 and BS-2 as its active set and MS-2 has BS-2 and BS-3 (w) as its active set. This can be accommodated by our framework Hk , we have, by considering a common active set of BS-1, BS-2, BS-3 for † (w) (w) (w) (w) ˜ (w) both MS-1 and MS-2 and setting g g . Hk = Uk Sk 0nR×NnT −nR Vk Vk 1,3 = 2,1 =0 h i h i † As a summary, the proposed per-cell product codebook (w) (w) (w) = Uk Sk Vk . (10) based limited feedback mechanism has the following advan- h i tages. (w) • Realtime Feedback Indices Selection: Vk is then • The proposed scheme relies on per-cell product code- mapped into N per-cell product codebooks with indices books, which are designed offline based on single-BS J ∗ , J ∗ , , J ∗ , and these codewords indices k,1 k,2 ··· k,N MIMO configurations. The proposed scheme is scalable are fed back to the BSs via an error-free feedback w.r.t. any number of cooperating BSs. link. The realtime indices generation is formulated as a • Standard precoder codebooks (such as the Grassmannian combinatorial optimization in Section III-B. codebook, Lloyd’s codebook, etc.) can be used in the Aggregate CSI Matrix Reconstruction and SDMA Precoder proposed framework because the heterogeneous path loss Computation (BS side): After collecting all the indices from issue is handled realtime in equation (9) and (12). the K MSs, the BSs try to reconstruct the CSI matrices and • We could further exploit the special structure of N compute the SDMA precoder from the N codeword indices per-cell product codebooks in the proposed framework J ∗ , J ∗ , , J ∗ . to derive a low complexity feedback indices selection k,1 k,2 ··· k,N ∗ ∗ ∗ algorithm. • Reconstruction: The indices J ,J , ,J k,1 k,2 ··· k,N from the kth MS are used to constructn a quantized versiono (w) (w) B. Problem Formulation of Vk (denoted as Vk ), i.e., † The feedback indices generation at the MS side is non- (w) 1 b † † † Vk = VJ∗ VJ∗ VJ∗ . (11) trivial, since it involves combinatorial search over the N √N k,1 k,2 ··· k,N h i per-cell product codebooks. In the following, we shall first b (w) • Denormalization: Vk is used to construct the quan- define the aggregate codeword for the N per-cell product tized aggregate CSI matrix H of the kth MS, which is codebooks and then formulate the feedback indices generation b k given by, problem as a chordal distance minimization problem between b † the aggregate-codeword and the quantization source. w ( ) nT ×nR Hk = V Gk, k =1, 2, ,K. (12) Definition 1 (aggregate-codeword): Let VJ C k ∀ ··· k,n ∈ h i denote the Jk,n-th codeword in codebook ϕn, an aggregate- • SDMAb Precoderb Computation: Apply the standard ¯ codeword V (Jk,1,Jk,2, ,Jk,N ) is defined to be, SVD to the quantized interference channel H−k seen by ··· th † the k MS, we have, ¯ 1 † † † b V (Jk,1,Jk,2, ,Jk,N )= VJ VJ VJ , † ··· √N h k,1 k,2 ··· k,N i H = U S V V , (13) (15) −k −k −k −k −k b b b b be where V ϕ , n =1, 2, ,N. where H is given by, Jk,n n −k By the definition∈ ∀ of the aggregate-codeword,··· the feedback † th ˆ b ˆ † ˆ † ˆ † ˆ † ˆ † indices at k MS side can be determined through the follow- H−k = H1 H2 Hk−1 Hk+1 HK . (14) h ··· ··· i ing optimization problem. The SVD operation (13) generates an orthonormal basis Problem 1 (Optimal Feedback Indices Generation):

NnT ×nR Finding out N codewords indices, that are denoted as of the right null space of H−k, i.e., V−k C . ∈ ∗ ∗ ∗ Jk,1, Jk,2, , Jk,N , in the N per-cell product codebooks We can set Wk = V−k, which satisﬁese equation (2). ··· b b ϕ1, ϕ2, , ϕN respectively, such that the chordal distance Remark 1 (Ways of Sendinge back Feedback Indices): between··· the aggregate-codeword V¯ (J ,J , ,J ) and th c b ∗ k,1 k,2 k,N When the k MS sends back the feedback indices J , (w) ··· k,1 the quantization source V is minimized. Mathematically, J ∗ , ,J ∗ , it could the feedback index J ∗ the nth BS, k k,2 ··· k,N k,n the feedback indices generation problem can be modeled as n = 1, 2, ,N ; or it could send all the feedback indices the following optimization problem, ∀J ∗ , J ∗ ,··· , J ∗ to its nearest BS; or it could simply k,1 k,2 ··· k,N broadcast all the feedback indices J ∗ , J ∗ , , J ∗ , ¯ (w) k,1 k,2 k,N min dc V (Jk,1,Jk,2, ,Jk,N ) , Vk (16) ··· J ,J , ··· ,J which will be received by all the N cooperating BSs thanks k,1 k,2 k,N ··· subject to VJ ϕn, n =1, 2, ,N. (17) to the broadcast nature of the wireless media. k,n ∈ ∀ ··· 5

In the above proposed per-cell product codebook Algorithm 1 Low-complexity Indices Selection Algorithm based limited feedback scheme, the aggregate codeword (ISA) ¯ V (Jk,1,Jk,2, ,Jk,N ) can be thought as a codeword in the • Step 1: Find Localized Centroids: ··· th product codebook ϕP er = ϕ1 ϕ2 ϕn, i.e., At the k ( k = 1, 2, ,K) MS, the standard SVD ∀ ··· (w) ⊗ ⊗···⊗ operation is first applied to the CSI matrix H , i.e., ¯ k,n V (Jk,1,Jk,2, ,Jk,N ) ϕ = ϕ1 ϕ2 ϕn,(18) ··· ∈ ⊗ ⊗···⊗ (w) (w) (w) (w) (w) H = U S 0n ×(n −n ) V V where VJ ϕn, n =1, 2, ,N. The product codebook k,n k,n k,n R T R k,n k,n k,n ∈ ∀ ··· h i h i ϕ is important because it allows for a single codebook to be = U(w)S(w)V(w). e (20) designed, and the real codebook that is used for CSI feedback k,n k,n k,n (w) is simply the Cartesian product of N single cell codebooks Then, the nT (nT nR) orthonormal basis V of the N k,n × − (w) ϕn n=1. { } row space of the CSI matrix Hk,n is set as the centroid of Remark 3 (Backward Compatibility): When N equals 1, (w) i.e., the single-BS scenario, Problem 1 will degenerate to the the sub-codebook ϕ¯ Vk,n ,δn . In this step, the standard 2 conventional feedback index generation problem. SVD operation is with the time complexity of nT n . O R • Step 2: Construct Sub-codebooks: After finding out the centroid V(w) of the sub-codebook C. Low-complexity Solution k,n ϕ V(w),δ , the kth k , , ,K MS proceed to Problem 1 belongs to the standard combinatorial search ¯ k,n n ( =1 2 ) ∀ ··· problem [24] and the optimal solution requires exhaustive select all the codewords in codebook ϕn that are within (w) search over the N per-cell product codebooks, which has the neighborhood δn of the centroid Vk,n , to construct (w) exponential complexity w.r.t. the number of feedback-bits Bk. sub-codebook ϕ¯ Vk,n ,δn . Specifically, However, in the practical communication systems, a MS may not be able to support such complicated operations. In order to (w) If V ϕn and dc V, Vk,n <δn, address this issue, we shall propose a low-complexity search- ∈ (w) ing algorithm, which exploit the per-cell product codebook then V ϕ¯ V ,δn , n =1, 2, ,N, ∈ k,n ∀ ··· structure and decomposes the searching process over the N where the time complexity of this step is of the order codebooks into the searching over N sub-codebooks with N Bk,n reduced size. For illustration purpose, we shall first give the n=1 2 . O definition of sub-codebook. • StepP 3: Indices Selection with Sub-codebooks: th Definition 2 (Sub-codebook): A sub-codebook The k ( k = 1, 2, ,K) MS then searches the ∀ ··· (w) feedback indices within the restricted sub-codebooks ϕ¯ V ,δn , is defined as a collection of codewords k,n ϕ V(w),δ to solve Problem 1. Specifically, in the original codebook ϕ , which lies in the neighborhood ¯ k,n n n n on=1 (w) the kth MS tries to find out the feedback indices of δn of the quantization source Vk,n . Mathematically, we 7 J ∗ , J ∗ , , J ∗ through solving the following have , k,1 k,2 ··· k,N noptimization problem witho exhaustive search: (w) (w) ϕ¯ V ,δn , V V ϕn; dc V, V <δn .(19) k,n ∈ k,n ¯ (w) n o min dc(V (Jk,1,Jk,2, ,Jk,N ) , Vk ) J ,J ,··· ,J Based on Definition 2, we can propose our low-complexity k,1 k,2 k,N ··· searching algorithm as well as complexity analysis in the (w) subject to VJk,n ϕ¯ Vk,n ,δn , (21) following (see Algorithm 1 below). ∈ where the time complexity of this step is of the order Remark 4 (Performance-Complexity Tradeoff of δn): In N (w) (w) the above algorithm, the value of δn can be utilized to of n=1 ϕ¯ Vk,n ,δn ,with ϕ¯ Vk,n ,δn de- O tradeoff the average quantization distortion performance Q (w) noting the cardinality of the sub-codebook ϕ¯ Vk,n ,δ n , and the computational complexity. In particular, when which depends on δn and Bk,n. δ1 = δ2 = = δN = √nR, the above algorithm reduces to the··· traditional exhaustive search algorithm. As long as δn √nR, n = 1, 2, ,N, then B N (w) ≤ k,n∀ ··· (nT ,N,nR,K) and the per-user feedback-bits Bk. In order to Pn=1 Bk n=1 ϕ¯ Vk,n ,δn 2 = 2 , which is the ≤ have tractable analysis so as to obtain design insights, we shall timeQ complexity of the exhaustive search method for solving N analyze the performance of the proposed limited feedback Problem 1 with the original codebooks ϕn . { }n=1 design using random codebooks [10], [18].

IV. ASYMPTOTIC PERFORMANCE ANALYSIS A. Asymptotic Optimality In this section, we shall quantify the asymptotic per- We start with comparing the asymptotic performance of the formance of the proposed per-cell product codebook based proposed per-cell product codebook based limited feedback limited feedback mechanism w.r.t. the system conﬁgurations scheme and the joint-cell codebook approach. Denote ϕJoint 7In this paper, we use the chordal distance as the distance measure. Other and ΦJoint as a random joint-cell codebook and the collec- distance metrics can also be applied here. tion of all possible random joint-cell codebooks, respectively. 6

Similarly, denote ϕP er and ΦP er as a random product per-cell B. Effect of Limited Feedback product codebook and the collection of all possible random Within the framework of limit feedback study, the through- product per-cell product codebooks, respectively, where a put loss due to CSI quantization is a common performance random product per-cell product codebook is defined as the measure [8]–[11]. In this paper, we extend the concept into Cartesian product of N random per-cell product codebooks, network MIMO configuration and define per-user throughput ¯ i.e., ϕP er = ϕ1 ϕ2 ϕN . Let Dk (ΦJoint) and loss RLoss as follows. ¯ ⊗ ⊗···⊗ k Dk (ΦP er) denote the average quantization distortion aver- Definition 3 (Per-user Throughput Loss): The per-user aged over all possible random joint-cell codebooks and random Loss th throughput loss Rk (w.r.t. the k MS) is defined as the per-cell product codebooks, respectively, which are defined as, throughput gap between the global CSI8 (GCSI) case and the proposed per-cell product codebook based limited feedback D¯ Φ , k Joint design, i.e., (w) (w) E 2 Loss CSIT LF min dc V, Vk Hk ; ϕJoint ΦJoint ,(22) R = R R , k =1, 2, ,K, (26) V∈ϕJoint ∈ k k k − ∀ ···

D¯ k ΦP er , where CSIT is the abbreviation for Channel State Information at the Transmitter Side and LF is the abbreviation for Limited E 2 (w) (w) min dc V, Vk Hk ; ϕJoint ΦP er . (23) Feedback. Moreover, V∈ϕP er ∈ CSIT 1 † † In order to show the efficiency of the proposed per-cell R = E log det In + HkWkPkW H (27), k 2 R σ2 k k product codebook based limited feedback scheme, we shall establish the asymptotic optimality of the proposed limited LF E 2 Rk = log2 det InR + σ InR + feedback design w.r.t. the joint-cell codebook approach and K −1 summarize the main results in the following Lemma. † † † † Hk Wj Pj W H HkWkPkW H ,(28) Lemma 1 (Asymptotic Optimality): For sufficiently large j k k k j=1X,j6=k nT and finite N, we have: c c c c (w) with W denoting the percoder for the kth MS designed based (I) The NnT nR orthonormal basis Vk of the row-space k × (w) on GCSI. Note that the residual CCI term in equation (28) is of the nR NnT normalized CSI matrix Hk has the same structure× as the aggregate-codeword defined in (15) due to limited feedback effects, and the expectation operation almost surely (i.e., with probability 1); is taken over Rayleigh fading and the lognormal shadowing (II) The proposed per-cell product codebook based limited effect, as well as the random per-cell product codebooks (for LF feedback scheme and the joint-cell codebook approach Rk only). achieve the same average quantization distortion, i.e., The following theorem quantifies the asymptotic per- user throughput loss of the proposed limited feedback D¯ k (ΦP er)= D¯ k (ΦJoint) , k =1, 2, ,K. (24) scheme w.r.t. the network configuration (n ,N,n ,K) , ∀ ··· T R the per-user feedback-bits Bk and the path loss geometry gk,1,gk,2, ,gk,N , which consists of all the path losses ··· th Proof: Please refer to Appendix A for the proof. between the N cooperating BSs and the k MS, and gk,n { } By virtue of Lemma 1, we can derive the average quanti- has been normalized to the weakest path so that gk,n 1. zation distortion associated with the random per-cell product Theorem 1 (Asymptotic Per-user Throughput Loss):≥In the codebooks, which is summarized in the following lemma. network MIMO system with the proposed per-cell product Lemma 2 (Average Quantization Distortion): For codebook based limited feedback scheme, the asymptotic per- sufficiently large Bk, nT , and small nR, the average user throughput loss is given by, quantization distortion associated with the random per-cell Bk Loss − n Nn −n sum product codebooks is given by, R = nR log 2 R( T R) ρg , (29) k O 2 k B − k P n (Nn −n ) max sum D¯k (Φper) nR2 R T R . (25) where ρ = σ2 is termed as system SNR; gk is defined to ≈ N be n=1 gk,n. Proof: Please refer to Appendix B for the proof. PProof: Please refer to Appendix C for the proof. Remark 5 (Average Quantization Distortion): In Lemma 2 As direct consequences of Theorem 1, we have the following the average quantization distortion is associated with quantiz- corollaries. (w) ing the row-space of the normalized CSI matrix Hk , which Corollary 1 (Scaling Law for the Noise-Limited Regime): consists of CN(0, 1) entries. As a result, the expression given For the per-cell product codebook based feedback scheme, in (25) is the same as equation (8) given in reference [10], the per-user feedback-bits Bk required to bound the per-user which is in fact first proved in reference [18]. throughput loss within a constant ε shall scale according to In the rest of this section, we shall derive the asymptotic the following expression, performance of the proposed limited feedback design based sum Bk nR(NnT nR) log ρg c(ε), (30) on the above two lemmas, and study the effects of limited ≈ − 2 k − feedback and the advantage of macrodiversity provided by BS 8In this paper, global CSI means that all the cooperating BSs has perfect cooperation. CSI knowledge of the whole network. 7

ε 3 n where c(ε)= nR(NnT nR) log2 2 R 1 . Joint−cell codebook, 12 bits − − baseline 1 Proof: Setting right hand side (RHS) of equation (49) in 2.8 Per−cell codebook, 12 bits Givens rotation, 24 bits Appendix C to ε, and solving for Bk will result in equation Givens rotation, 12 bits 2.6 (30) directly. proposed scheme Corollary 2 (Scaling Law for the Interference-Limited Regime): 2.4 For the proposed per-cell product codebook limited feedback 2.2 scheme, if per-user feedback-bits (i.e., Bk) does not scale baseline 2 with system SNR ρ, then for sufﬁciently large Pmax, the 2 LF per-user throughput Rk tends to a constant and scales according to, 1.8 baseline 3 n B ln 2 1.6 RLF = R k , k =1, 2, ,K.(31) k O (Nn n ) Nn ∀ ···

T R T Per−user Ergodic Capacity (bits per channel−use) − 1.4 Proof: Please refer to Appendix D for the proof.

Remark 6 (Effects of Heterogeneous Path Losses): Note 6 8 10 12 14 16 18 20 that, the results given in Theorem 1 and Corollary 1 above Interference−free SNR (dB) are similar to those results stated in Theorem 1 and Theorem Fig. 2. Performance comparison of the proposed per-cell product codebook 2 of reference [10]. The major difference is the path loss based limited feedback design with joint-cell codebook and Givens rotation sum approaches, in a network MIMO system (nT ,N,nR,K) = (4, 3, 2, 6). In effect term gk , which results from the different path losses th the proposed scheme, baseline 1 and baseline 3, we adopt 4 bits per BS for from the N cooperating BSs to the k MS. CSI feedback; while in baseline 2, we adopt 8 bits per BS for CSI feedback. Remark 7 (Scaling Laws for the Proposed Limited Feedback DesWhileign): the proposed per-cell product codebook limited feedback scheme can In the noise-limited regime, the minimum number of feedback- achieve 95% ∼ 97% of the performance of joint-cell codebook approach, it performs much better than the Givens rotation approach. bits Bk required to maintain a bounded per-user throughput loss, shall scale w.r.t. the number of cooperating BSs according to, model speciﬁed in [25] is used, i.e., PL(dB) = 130.19 +

sum 37.6 log10(d(km)), with 8 dB lognormal shadowing effects. Bk = NnT nR log ρg . (32) O 2 k Users are assumed to be uniformly distributed within the cooperating cells. We use interference-free SNR to represent Moreover, the residual CCI term in equation (28) is negligi- the receiving SNR at the cell edge of a single-cell single-MS ble for the noise-limited case. Following the proof of Theorem scenario. 1, we can show that in the noise-limited regime, the achievable per-user throughput of the proposed limited feedback scheme scales as, A. Performance of the Proposed Limited Feedback Design Fig. 2 illustrates the per-user average throughput versus n ρgsum RLF = n log R k . (33) interference-free SNR. The simulation results show that, in k R 2 Nn O T the practical settings, the proposed per-cell product codebook On the other hand, in the interference-limited regime, the based limited feedback scheme can achieve 95% 97% achievable per-user throughput of the proposed per-cell prod- the performance of the joint-cell codebook approach.∼ More- uct codebook based limited feedback scheme shall scale as, over, the proposed scheme achieves much better throughput n B compared with Baseline 2 (Givens rotation approach with RLF R k . (34) k = 2 doubled number of bits for limited feedback) and Baseline O (NnT ) 3 (Givens rotation approach). This is because the Givens V. NUMERICAL RESULTS AND DISCUSSIONS rotation approach has quite low feedback efficiency due to two-dimensional vector quantization, compared with matrix In this section, we shall study the performance of the quantization in the codebook-based approach. proposed per-cell product codebook limited feedback scheme and verify the analytical results via simulations. We shall first compare the proposed per-cell product codebook limited feed- B. Performance-complexity Tradeoff of the Low-complexity back scheme with several baseline schemes. Baseline 1: joint- ISA cell codebook approach; Baseline 2 and 3: Givens rotation In this section, we study the performance-complexity trade- approach with different number of feedback-bits. In the Givens off of the low-complexity ISA. Here, performance is mea- rotation approach, the two Givens parameters of a Givens sured in terms of per-user ergodic capacity, and complexity matrix are quantized with a two-dimensional vector quan- is measured in terms of the number of codewords that are tizer [19]–[21]. Then, we proceed to study the performance- searched for generating the codewords indices. As stated in complexity tradeoff of the proposed low-complexity feedback section III-C, the parameters δn (n =1, 2, ,N) determine indices selection algorithm, as well as the analytical results the tradeoff between performance and complexity··· of algorithm in Section IV. In the simulations, two-dimensional hexagonal 1. Fig. 3 shows the performance-complexity tradeoff with cellular model is considered, with a cell-radius of 300 meters different choices of δn, where we set δn to the same value and the carrier frequency is set to be 2 GHz. The path loss for all n. The complexity numbers shown in the figure denote 8

0 10 10

96% Perfect CSIT 9 Scaling Feedback−bits

−1 Fixed Feedback−bits 10 8

92% 7

−2 10 6 86% per−cell codebook feedback 5 −3 10

Normalized Complexity δ = 1.0 4 n 24 bits δ = 0.9 n −4 3 10 δ = 0.8 Per−user Ergodic Capacity (bits per channel−use) n δ = 0 n 2 10 12 14 16 18 20 22 24

−5 Interference−free SNR (dB) 10 6 9 12 15 18 21 24 Number of Feedback−bits Fig. 4. Per-user ergodic capacity versus interference-free SNR, for perfect CSIT case, per-cell product codebook feedback with scaling feedback-bits Fig. 3. Performance-complexity tradeoff of the proposed low-complexity and fixed feedback-bits, with a system configuration (nT ,N,nR,K) = feedback indices selection algorithm (algorithm 1), with different choices (8, 3, 2, 12). For the scaling feedback-bits case, the per-user feedback-bits of δn for a network MIMO system (nT ,N,nR,K) = (4, 3, 2, 6). The scale according to equation (30), and we set ε = 1. Specifically, the feedback- complexity numbers shown in the figure are relative complex defined in bits used in the scaling feedback-bits case are [24 30 36 42 48 57 63 72], equation (35). Setting δn = 1, 0.9, 0.8 can achieve 96%, 92%, 86% the corresponding to the 8 SNR values respectively. A constant gap of about 0.5 performance of exhaustive search, with 10%, 1%, 0.1% the complexity of (bits per channel use) between the scaling feedback and the perfect CSIT exhaustive search, respectively. Larger δn gives better performance at the cost case is observed. For the fixed feedback-bits case, the system is interference- of higher complexity, and vice versa. The curve with δn = 0 is for reference. limited.

1.3 the relative complexity with respect to the exhaustive search 1.2 Simulation method with original codebooks, i.e., Analytical 1.1 Relative complexity 1 No. of codewords searched with Algorithm 1 (n , N, n , K) = (8, 2, 2, 8) , T R No. of codewords searched with exhaustive search 0.9 N (w) 0.8 n=1 ϕ¯ Vk,n ,δn (n , N, n , K) = (8, 3, 2, 12) = . (35) T R Q Bk 0.7 2 As shown in the figure, with δn = 0.9, the low-complexity 0.6 ISA can achieve the performance of the exhaustive search 92% 0.5 approach, with only 1% complexity of the exhaustive search. Per−MS Ergodic Capacity (bits per channel−use) 0.4 With larger δn, algorithm 1 can achieve better performance at (n , N, n , K) = (8, 4, 2, 16) T R the cost of higher computational complexity, and vice versa. 1 2 3 4 5 6 7 8 9 10 Per−BS feedback−bits (B /N) k C. Verification of the Analytical Expressions Fig. 5. Per-user throughput in the interference-limited regime with different In this section, we shall compare the analytical results stated system configurations (nT ,N,nR,K). The y-axis is the per-user ergodic in Corollary 1 and Corollary 2 with numerical results, and capacity (in bits per channel use), and the x-axis is the per-BS feedback-bits. It is observed that the per-user throughput scale linearly with the feedback- demonstrate the validity of those analytical studies. Fig. 4 bits, as stated in Corollary 2. The simulation results match the analytical shows the simulation results for GCSI case, per-cell product results stated in Corollary 2. codebook limited feedback with scaling feedback-bits and fixed feedback-bits. For the per-cell product codebook based limited feedback with scaling feedback-bits, we set ε = 1 VI. CONCLUSIONS and let the feedback-bits scale according to equation (30). Compared with GCSI case, about 0.5 (bits per channel use) In this paper, we have proposed a scalable per-cell product throughput loss is achieved with scaling-feedback. codebook based limited feedback framework for network Fig. 5 shows the performance of the proposed per-cell prod- MIMO systems, along with a low-complexity feedback indices uct codebook based limited feedback scheme in interference selection algorithm. We have shown that the proposed limited limited regime, with different number of feedback-bits per feedback design can asymptotically achieve the same perfor- BS (i.e., Bk ). We can see that the per-user ergodic capacity mance as the joint-cell codebook approach. When the number N sum of per-user feedback-bits scales as NnT nR log (ρg ) , scales linearly w.r.t. feedback-bits per BS under all system O 2 k the proposed scheme operates in a noise-limited regime with configurations (nT ,N,nR,K). The simulation results shown sum nRρgk in Fig. 5 match the analytical results stated in Corollary 2. a per-user throughput scaling as nR log2 . On O NnT 9 the other hand, when the number of per-user feedback-bits aggregate-codeword defined in (15) almost surely (i.e., with does not scale with system SNR, the system operates in probability 1), which proves the first statement of Lemma 3. an interference-limited regime with a per-user throughput Note that, the first statement of Lemma 1 in turn suggests nRBk scaling as 2 . The numerical results show that the that the codewords in the joint-cell codebook shall have the O (NnT ) proposed scheme can achieve similar performance as the joint- same structure as the aggregate-codeword defined in (15) cell codebook approach and performs much better than the almost surely for sufficiently large nT and finite N. As a Givens rotation approach in practical settings. One interesting result, the second statement of Lemma 1 can be derived from direction for further study is to consider adaptive feedback-bits the definition of average distortion associated with joint-cell allocation based on users path loss geometry. codebooks and per-cell product codebooks given in equation (22) and (23), respectively. Therefore, the expected distortion APPENDIX A (i.e., average distortion) associated with the random per-cell PROOF OF LEMMA 1 product codebooks shall be the same as the expected distortion (i.e., average distortion) with random joint-cell codebooks. For ease of elaboration, we first introduce the following intermediate lemma. Lemma 3: Consider a random vector h CNnT with APPENDIX B CN 2 ˜ h ∈ PROOF OF LEMMA 2 i.i.d. (0, σ ) entries, and let h = khk denote the direction of h. Partitioning h˜ into N sub-vectors, i.e., h˜ = By virtue of Lemma 1, for large Bk, the average quanti- † zation distortion associated with the random per-cell product † † † nT h˜ h˜ h˜ , where hñ C , n = 1, 2, ,N, 1 2 ··· N ∈ ∀ ··· codebooks can be approximated as [18], weh then have, i 1 Γ 1 Bk ¯ α − α − α ˜† ˜ 1 Dk (Φper) β 2 + o(1) (42) Pr lim hnhn = =1, n =1, 2, ,N.(36) ≈ α nT →∞ n o N ∀ ··· where α = n (Nn n ); β = 1 nR (NnT −i)! ; Γ (x) R T R α! i=1 (nR−i)! th ˜ − Proof: Denote h˜i as the i element of h. Since h is a random denotes the Gamma function; the o(1)Qterm can be ignored 2 Γ 1 vector with i.i.d. CN(0, σ ) entries, all h˜i (i =1, 2, ,NnT ) ( α ) ··· when Bk is large or nR is small [18]. Since limα→∞ α = are identically distributed and satisfy, 1, for large α (which is true in network MIMO system), we Γ 1 † NnT 1 ( α ) Var h˜ h˜ , i , , ,Nn(37), have α 1. i i = 2 − =1 2 T ≈ (NnT ) (NnT + 1) ∀ ··· Substituting Stirling’s approximation for factorial, we get n o − nR 1 † † β (NnT ) 2 and, where Var h˜ h˜ denotes the variance of h˜ h˜ . Moreover, NnRnT i i i i ≈ (nR) n o † 1 it is straightforward to get that E h˜ h˜ = , and 1 nR (a) n n N − α β − nR, (43) n o nR 1 ˜† ˜ N−1 ≈ 2NnRnT ≈ Var h hn = 2 , n =1, 2, ,N. (NnT ) n N (NnT +1) ∀ ··· n o − ˜† ˜ 1 nR 1 Define ψt = hnhn N >ǫ , where ǫ denotes any where (a) is because of that lim (Nn ) 2NnRnT = 1 n − o nT →∞ T given positive number. Using Chebyshev’s inequality, we get, and Nn is usually very large for network MIMO system. T Therefore, the approximation of the average quantization † 1 N 1 Pr ψt = Pr h˜ hñ >ǫ − ,(38) { } n − N ≤ N 2 (Nt + 1) ǫ2 distortion associated with the random per-cell product codebooks D¯k (Φper) can be further simplified as D¯ k (Φper) B which implies that [26, page 37], − k ≈ n (Nn −n ) nR2 R T R .

Pr  ψt Pr ψt 0 as nT , (39) ≤ { } → → ∞ APPENDIX C t≥[nT  tX≥nT PROOF OF THEOREM 1   which proves Lemma 3 [26, pape 34-35]. Here is the proof of Theorem 1. We next proceed to prove the first statement of Lemma 1. (w) Nn ×n (a) We first partition V C T R into N sub-matrices, i.e., Loss k R E log det In ∈ k ≤ 2 R † † † † K (w) (w) (w) (w) 1 † † Vk = Vk,1 Vk,2 Vk,N , (40) + H W P W H h i h i ··· h i σ2 k j j j k j=1X,j6=k (w) CnT ×nR c c where Vk,n , n =1, 2, ,N. Then, as a direct (b) 1 (w) † ∈ ∀ ··· = E log det In + V Gk consequence of Lemma 3, 2 R σ2 k † K (w) (w) 1 2 (w) (w) Pr lim Vk,n Vk,n = InR =1, (41) † nT →∞ N Wj Pj Wj GkV S h i k k j=1X,j6=k h i c c (c) Equation (41) suggests that when nT is sufficiently large, 1 (w) † (w) log det In + E V Gk the orthonormal basis V shall have the same structure as the ≤ 2 R σ2 k k 10

K 2 (5) † (w) (w) where E denotes expectation taken over the distribution Wj Pj Wj GkVk Sk , of Zk, and (e) is given in Appendix B of [10] with D¯ = j=1X,j6=k h i Bk − − c c n 2 nR(NnT nR) . where (a) and (c) are obtained following the approaches in R (w) • Step 6: Finally, substituting equation (48) into (44), we [10]; (b) follows by substituting equation (10) for Hk . † get, (w) K † (w) Let Fk = E V Gk Wj Pj W GkV ¯ sum k j=1,j6=k j k pγ˜k pD (K 1) gk Fk = D¯In = − In . (49) 2 P R R c c 2 NnT N S(w) V(w) G W S(w) k , and note that k , k, j and k are Therefore, the asymptotic per-user throughput loss due to h i mutually independent, the expectation canc be carried out step limited feedback is given by, by step. Bk Loss − − sum (w) nR(NnT nR) Rk = nR log2 2 ρgk . (50) • Step 1: substituting the decomposition of Vk , i.e., O (w) (w) (w) Vk = Vk XkYk + Vk Zk (see Lemma 1 of [10]). APPENDIX D K b † e (w) † (w) PROOF OF COROLLARY 2 Fk = E  V Gk Wj Pj W GkV  k j k The proof of Corollary 2 can be summarized as follows. h i j=1X,j6=k  c c 2 K (1) (w)  E S IFL (a) † † k R E log det Hk Wj W H  h i k ≈ 2 j k †  Xj=1  (d) E † (w)  c c  = NnT Zk Vk Gk  K  † † E log det Hk WjW H  K e − 2 j k † (w)  j=1X,j6=k  Wj Pj W GkV Zk , (44)  c c  j k j=1X,j6=k  K  c c e (b) † † log det E Hk Wj W H  2 ≈ 2 j k where (d) is because of that E(1) S(w) =  Xj=1  k  c c  h i  K  NnT InR (see Appendix B of [10]) and we have used † † log det E Hk WjW H  the LF-BD conditions (2). − 2 j k  j=1X,j6=k  • Step 2: calculating the expectation of  c c  K † Bk W P W . Let  n Nn −n  j=1,j6=k j j j (c) 2 R( T R) = nR log 1+  P c c O 2 K 1 K − Q E(2) W P W†    k =  j j j  , (45) (d) Bk ln 2 j=1X,j6=k  = , (51) O (NnT nR) (K 1) c c − − where E(2) denotes expectation taken over the distribution where (a) is because the noise term is negligible compared of Wj. When the number of active users is large and the with the CCI term; (b) holds in the orderwise sense; (c) users are randomly distributed, we can safely conclude follows the same arguments as in the proof of Theorem 1; (d) B c nR(K−1) NPmax k that Qk pINnT , with p = . Bk n (Nn −n ) NnT KnR is because is quite small and 2 R T R ≈ (w) nR(NnT −nR) • V Bk ln 2 ≈ Step 3: calculating expectation over k . 1+ . nR(NnT −nR) † (w) (w) pne Rγk (K 1) E(3) V G Q G V I k k k k k = 2− (46)nR , REFERENCES h i (NnT ) e e [1] M. Karakayali, G. Foschini, and R. Valenzuela, “Network coordination where E(3) denotes expectation taken over the dis- for spectrally efﬁcient communications in cellular systems,” IEEE Wire- (w) less Commun. Mag., vol. 13, no. 4, pp. 56–61, Aug. 2006. tribution of V (isotropic distribution), and γk = k [2] G. Foschini, K. Karakayali, and R. Valenzuela, “Coordinating multiple n N g s . T n=1 k,ne k,n antenna cellular networks to achieve enormous spectral efﬁciency,” IEE • StepP 4: calculating expectation over lognormal- Proceedings of Communications, vol. 153, no. 4, pp. 548–555, Aug. shadowing. Let 2006. [3] J. Andrews, W. Choi, and R. Heath, “Overcoming interference in spatial (4) sum multiplexing MIMO cellular networks,” IEEE Wireless Commun. Mag., γ˜k = E γk = nT g , (47) { } k vol. 14, no. 6, pp. 95–104, Dec. 2007. E(4) [4] S. Jing, D. N. C. Tse, J. B. Soriaga, J. Hou, J. E. Smee, and R. Padovani, where denotes expectation taken over lognormal- “Multicell downlink capacity with coordinated processing,” EURASIP J. shadowing. Wirel. Commun. Netw., vol. 2008, no. 5, pp. 1–19, Jan. 2008. • Step 5: calculating expectation over quantization error [5] L.-U. Choi and R. D. Murch, “A transmit pre-processing technique for multi-user MIMO systems using a decomposition approach,” IEEE Zk. We have, Trans. Wireless Commun., vol. 3, no. 1, pp. 20–24, Jan. 2004. [6] Q. Spencer, A. Swindlehurst, and M. Haardt, “Zero-forcing methods n γ˜ (e) pγ˜ D¯ (K 1) R k E(5) † k for downlink spatial multiplexing in multiuser MIMO channels,” IEEE 2 p ZkZk = −2 InR , (48) (NnT ) n o (NnT ) Trans. Signal Process., vol. 52, no. 2, pp. 461–471, Feb. 2004. 11

[7] C. WANG, Adaptive downlink multi-user MIMO wireless systems. PhD Vincent Lau (SM’04) obtained B.Eng (Distinc- dissertation, Hong Kong University of Science and Technology, Aug. tion 1st Hons) from the University of Hong Kong 2007. (1989-1992) and Ph.D. from Cambridge University [8] N. Jindal, “MIMO broadcast channels with finite-rate feedback,” IEEE (1995-1997). He was with HK Telecom (PCCW) as Trans. Inf. Theory, vol. 52, no. 11, pp. 5045–5060, Nov. 2006. PLACE system engineer from 1992-1995 and Bell Labs - [9] T. Yoo, N. Jindal, and A. Goldsmith, “Multi-antenna downlink channels PHOTO Lucent Technologies as member of technical staff with limited feedback and user selection,” IEEE J. Sel. Areas Commun., HERE from 1997-2003. He then joined the Department of vol. 25, no. 7, pp. 1478–1491, 2007. ECE, Hong Kong University of Science and Tech- [10] N. Ravindran and N. Jindal, “Limited feedback-based block diagonal- nology (HKUST) as Associate Professor. His cur- ization for the MIMO broadcast channel,” IEEE J. Sel. Areas Commun., rent research interests include the robust and delay- vol. 26, no. 8, pp. 1473–1482, Oct. 2008. sensitive cross-layer scheduling of MIMO/OFDM [11] K. Huang, J. Andrews, and R. Heath, “Performance of orthogonal wireless systems with imperfect channel state information, cooperative and beamforming for SDMA with limited feedback,” IEEE Trans. Veh. cognitive communications, dynamic spectrum access as well as stochastic Technol., vol. 58, no. 1, pp. 152–164, Jan. 2009. approximation and Markov Decision Process. [12] M. Trivellato, H. Huang, and F. Boccardi, “Antenna combining and codebook design for the MIMO broadcast channel with limited feedback,” in Asilomar Conference on Signals, Systems and Computers, 2007, Pacific Grove, USA, nov. 2007, pp. 302 –308. [13] M. Trivellato, F. Boccardi, and H. Huang, “On transceiver design and channel quantization for downlink multiuser MIMO systems with limited feedback,” IEEE J. Sel. Areas Commun., vol. 26, no. 8, pp. 1494–1504, 2008. [14] J. H. Kim, W. Zirwas, and M. Haardt, “Efficient feedback via subspace- based channel quantization for distributed cooperative antenna systems with temporally correlated channels,” EURASIP J. Adv. Signal Process, vol. 2008, no. 2, pp. 1–13, Jan. 2008. [15] L. Thiele, M. Schellmann, T. Wirth, and V. Jungnickel, “Cooperative Multi-User MIMO based on Reduced Feedback in Downlink OFDM Systems,” in 42nd Asilomar Conference on Signals, Systems and Com- puters, Monterey, USA, Oct. 2008. [16] D. J. Love and R. W. H. Jr., “Limited feedback unitary precoding for spatial multiplexing systems,” IEEE Trans. Inf. Theory, vol. 51, no. 7, pp. 2967–2976, Aug. 2005. [17] B. Mondal, S. Dutta, and R. W. Heath, “Quantization on the Grassmann manifold,” IEEE Trans. Signal Process., vol. 55, no. 8, pp. 4208–4216, Aug. 2008. [18] W. Dai, Y. Liu, and B. Rider, “Quantization bounds on Grassmann manifolds and and applications to MIMO communications,” IEEE Trans. Inf. Theory, vol. 54, no. 3, pp. 1108–1123, Mar. 2008. [19] J. C. Roh and B. Rao, “An efficient feedback method for MIMO systems with slowly time-varying channels,” vol. 2, 2004, pp. 760–764 Vol.2. [20] M. A. Sadrabadi, A. K. Khandani, and F. Lahouti, “Channel feedback quantization for high data rate MIMO systems,” IEEE Trans. Wireless Yi Long received his Ph.D. degree from Tsinghua Commun., vol. 5, no. 12, pp. 3335–3338, Dec. 2006. University, Beijing, China in 2008. From 2008 to [21] H. Long, W. Wang, H. Zhao, and K. Zheng, “Precoding vector distribu- 2010 he worked on cellular cooperative communication under spatial correlated channel and nonuniform codebook design,” tions. He is now an engineer at Huawei Technology in IEEE ICC’08, May 2008, pp. 4506–4510. PLACE Co. Ltd., Shenzhen, China. His research interests [22] H. Zhang and H. Dai, “Cochannel interference mitigation and coop- PHOTO include interference management, MIMO, and iter- erative processing in downlink multicell multiuser MIMO networks,” HERE ative algorithms. EURASIP J. Wirel. Commun. Netw., vol. 2004, no. 2, 2004. [23] D. Tse and P. Viswanath, Fundamentals Of Wireless Communication. Cambridge University Press, 2005. [24] C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity. Dover Publications, 1998. [25] IEEE 802.16m evaluation methodology document. IEEE 802.16m- 08/004r4. [Online]. Available: http://www.ieee802.org/16/tgm/ [26] P. K. Sen and J. M. Singer, Large Sample Methods in Statistics: An Introduction with Applications. Chapman & Hall, New York, 1993.

Yong Cheng (S’09) received the B.Eng. (1st honors) from Zhejiang University (2002-2006), Hangzhou, China, and the MPhil in Electronic and Computer Engineering from the Hong Kong University of PLACE Science and Technology (HKUST) (2008-2010), PHOTO Hong Kong. His research interests include limited HERE feedback design for MIMO/MISO systems, resource control and optimization in wireless networks, cooperative MIMO systems, as well as relay networks. 0 10

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−1 10

−2 10 Per−user Outage Probability (P

−3 10 1 2 3 4 5 6 7 Number of Cooperating BSs (N)