MMSE-Based User Selection Algorithms for Multiuser ... - IEEE Xplore

MMSE-Based User Selection Algorithms for Multiuser Linear Precoding Ngo.c-D˜ung Ðào and Yong Sun Toshiba Research Europe Ltd., Telecommunications Research Lab 32 Queen Square, Bristol, BS1 4ND, England Telephone: +44-117-906 0700, Fax: +44-117-906 0701, Email: {ngoc.dao, yong.sun}@toshiba-trel.com

Abstract—User selection for multiuser linear precoding in broadcast channel is investigated in this paper. Assuming complete knowledge of channel state information at the transmitter, new user selection algorithms are introduced using minimum mean square error criterion. A novel iterative user selection is proposed offering a flexible performance-complexity tradeoff. New algorithms are also developed for orthogonal frequency division multiple access systems. Simulation results show that the proposed methods outperform well-known algorithms, which select users based on users’ orthogonality or sum rate bound. Index Terms—Multiuser precoding, user selection, OFDMA.

I. I NTRODUCTION We consider a downlink broadcast channel of wireless systems where a base station with multiple transmit antennas sends several data streams to uncoordinated users simultaneously. In this scenario, assuming complete channel state information is available at the transmitter, dirty paper coding (DPC) techniques can be employed to pre-cancel the interuser interference [1]. Since DPC has high complexity, low complexity multiuser (MU) precoding methods have been proposed, for example linear [2], [3] and nonlinear [4], [5] precoding. Those precoders have been optimized with different criteria, for example zero-forcing (ZF) [2], [3] and minimum mean-square error (MMSE) [2]. These linear precoders have also been investigated in a context of coordinated base stations [6]. While the interuser interference is nulled out in ZF precoders, the MMSE precoders allow an amount of interuser interference such that total distortion of recovered signals is minimized. Therefore, MMSE-based precoders perform better than ZF-based precoders. We will thus study in particular the linear MMSE precoder designed in [2] (see also [3]). In wireless communication systems, the number of active users is often larger than the number of users that can be served at the same time. Therefore user scheduling is an important task. In general, the scheduler could try to provide communication links for users with different quality of services (QoS) requirements. In this paper, we limit our study to a scheduler, which greedily selects users to maximize the system capacity. A number of user selection methods with different cost metrics have been proposed. For example, in [7], the authors suggest to select the users with largest channel power and maximal orthogonality among selected users for linear ZF precoding. Another method is proposed in [8] to maximize the upper bound on the users’ sum rate. Recently, a user

selection method to minimize MMSE is presented in [9], in which the uplink-downlink duality is exploited. This method requires certain optimization tools, which might be complex for practical hardware implementations. In this paper, we will present new methods to select users for linear MMSE precoding, which are also applicable for linear ZF precoding. First we show that the user selection for the linear MMSE precoder [2] in point-to-multipoint MIMO communications can be translated into the problem of receive antenna selection for linear MMSE receivers in point-to-point MIMO communications. Thus, the incremental and decremental receive antenna selection algorithms, for example in [10], can be extended for the user selection. The decremental search gives better performance than the incremental search. But when the number of users is large, the complexity of the former is much higher than the latter. We thus propose a novel search strategy, which select users iteratively. The iterative selection method offers a flexible performance-complexity tradeoff. We also extend the newly proposed methods to assign users in resource blocks (RB) of orthogonal frequency division multiple access (OFDMA) systems. Simulation results show that our user selection algorithms outperform well-known algorithms, which select users based on the user orthogonality [7] and sum rate upper bound [8]. The proposed algorithms can nearly approach the performance of the optimal exhaustive search, but with much less complexity. II. S YSTEM M ODEL Assume that the channel exhibits frequency-flat fading. The base station has M transmit antennas, there are totally Kt single-antenna users of which K (K ≤ M ) users are to be selected. users have the same unit noise power. Let  T All T T be the channel matrix, where H = h1 h2 . . . hT K superscript T denotes matrix transpose operation, and hk is the row channel vector of user k (k = 1, . . . , K). Assume that base station has full knowledge of H. Now we briefly review the structure of the linear MMSE precoder and its MSE [2], [3]. The transmit signal is precoded by a matrix P as x = Ps, where s is the data vector of all users. Note that the elements of s have the same unit power. The received signals of all users is   Etr Etr Hx + n = HPs + n (1) z= K K

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where Etr is the total transmit power, n is the additive Gaussian noise vector of all users, each element of n has zero mean and unit variance. The precoding matrix is given as −1 H  H P = β HH H + αIM

(2)

where α = K/Etr , superscript H stands for matrix transpose H conjugate. Let  F = H H + αIM , the power normalization Etr factor β = tr(F−2 , where tr(X) denotes the trace of HH H) matrix X. The MSE of the linear MMSE precoder is 

−1  . (3) ψ = tr α−1 HHH + IK The linear ZF precoder can be obtained by removing the term αIM in (2) [2]. At high signal-to-noise ratio (SNR), the MSE of ZF precoder converges to that of MMSE precoder, which can be obtained by removing the identity matrix IK in (3). Loosely speaking, it could be interpreted that the MMSE precoder has a more general formulation than that of ZF precoder. Thus the solutions of user selection for the linear MMSE precoder could be well extended to the linear ZF precoder, with marginal modifications. Next, we derive the user selection algorithms with the linear MMSE precoding in frequency-flat channels.

Consider another quantity φ as  −1 φ = tr α−1 HH H + IM ⎧ k ⎪  1 ⎪ ⎪ ⎪ , for k ≤ M M − k + ⎪ −1 ⎨ α δi2 + 1 i=1 = M ⎪  ⎪ 1 ⎪ ⎪ , for k > M ⎪ −1 δ 2 + 1 ⎩ α i i=1 = M − k + ψ.

(5)

Thus, instead of finding the user set with minimal MSE ψl , we can equivalently find the user set that minimizes φ. The format of φ in the first line of (5) is mathematically convenient to derive low-complexity user selection algorithms, because the size of the matrix in the right hand side of (5) is always M × M , which is independent of the number of users k being processed. Going back to the problem of interest, let Ω = {1, 2, . . . , Kt } be the set of indices of all users, S ⊆ Ω be a subset of Ω containing the indices of any K users, and Hs be the associated channel matrix of these K users. Then the set ω of K users to be selected satisfies

−1 H ω = arg min tr α−1 Hs Hs + IM S

−1 H = arg min tr Hs Hs + αIM . (6) S

III. U SER S ELECTION FOR L INEAR P RECODING A. Non-Iterative User Selection Algorithms We will derive the selection algorithms, in which the MSE in (3) is minimized. A brute-force  t  exhaustive search for all to minimize MSE in (3) combinations sets of users K K could be prohibitively complex for a large number of user Kt . Thus we will seek for low-complexity algorithms. Let us start by deriving a more convenient form of (3) to enable lowcomplexity user selection. Assume that H is the channel matrix of some arbitrary k users. The matrix H can be decomposed by a singular value decomposition as H = UΔVH , where U and V are unitary matrices, and Δ is a diagonal matrix containing singular values δi (i = 1, 2, . . . , n) of H. Note that during the selection process, k may be larger or smaller than K. If k ≤ M , then n = k; otherwise, n = M . The MSE metric in (3) becomes

−1

−1 = tr α−1 UDDH UH + Ik ψ = tr α−1 HHH + Ik  

−1 H −1 H = tr U α DD + Ik U ⎧ k ⎪  1 ⎪ ⎪ ⎪ , for k ≤ M ⎪ −1 δ 2 + 1 ⎨ α i i=1 (4) = M ⎪  ⎪ 1 ⎪ ⎪k − M + , for k > M . ⎪ ⎩ α−1 δi2 + 1 i=1

The above optimization problem can be solved efficiently using the matrix inversion lemma, which is in fact similar to the receive antenna selection algorithms in [10, Algorithm IV]. This algorithm decrementally selects some receiver antennas for MMSE receiver in point-to-point MIMO. Thus we have actually translated the problem of user selection in pointto-multipoint MIMO downlink into the problem of antenna selection in point-to-point MIMO downlink. Now let H be the channel matrix of all users. If hi is the channel vector of a deselected user i, the channel matrix of remaining users (without user i) is HKt −1 . We have H H HH Kt −1 HKt −1 = H H − hi hi .

(7)

Making use of the matrix inversion lemma [11] for a positive definite matrix A and a vector a, (7) becomes −1  −1 H  −1 = A + Aa 1 − aH Aa a A. (8) A − aaH By replacing A = (HH H + αIM )−1 and a = hH i , we get −1  H HKt −1 HKt −1 + αIM   H −1 = A + AhH hi A. (9) i 1 − hi Ahi Hence, once A is known, the left hand side of (9) can be computed without explicite matrix inversion. Let hi A = gi , (9) can be simplified as  H −1 −1 H  HKt −1 HKt −1 + αIM = A+ 1 − gi hH gi gi . (10) i

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TABLE I A LGORITHM I-A – D ECREMENTAL U SER S ELECTION 1

Input H, α, K, Ω = {1, 2, . . . , Kt }

2 3 4

Initialization: compute A = HH H + αIM for l = 1 : Kt − K compute gi = hi A ∀i ∈ Ω

5 6 7 8 9





j = arg min 1 − gi hH i i∈Ω

−1



TABLE II A LGORITHM I-B – I NCREMENTAL U SER S ELECTION

−1

gi 2

−1

update A ← A + 1 − gj hH j Ω = Ω\j end for Output selected user indices ω = Ω

gjH gj

Consequently, −1  = tr (A) + tr HH Kt −1 HKt −1 + αIM

1 2

Input H, α, K, Ω = {1, 2, . . . , Kt }, ω = ∅ Initialization: compute A = α−1 IM and j = arg max hi 2

3 4

ω = ω ∪ j, Ω = Ω\j, gj = hj A for l = 1 : K − 1

i∈Ω



−1

5 6

update A ← A − 1 + gj hH j compute gi = hi A ∀i ∈ Ω

7

j = arg max 1 + gi hH i

8 9 10

Ω = Ω\j, ω = ω ∪ j end Output: index set of selected user ω



−1

i∈Ω

gjH gj

gi 2

TABLE III A LGORITHM I-C – O NE -L OOP I TERATIVE U SER S ELECTION

gi 2 . (11) 1 − gi hH i

1 2 3

Input H, α, K, number of iterations r Select K random users, their index set ω and channel matrix HK Ω = {1, 2, . . . , Kt } \ω

Thus, the deselected user j is the one that makes the left hand side of (10) minimized.

4 5 6

Initialization: compute A = HH K HK + αIM for l = 1 : r compute gi = hi A∀i ∈ Ω

j = arg min i∈Ω

gi 2 . 1 − gi hH i

(12)

One by one users are deselected until there are K users left, which are the selected users. After dropping one user, the  −1 H matrix A can be updated as A ← A + 1 − gi hH gi gi . i The pseudocode for the decremental user selection algorithm is summarized in the Table I. Similar to the increment receive antenna selection approach, we can give another incremental user selection algorithm. The process starts with an empty set of users. Then one by one users will be added to the selected list. Note that incremental antenna selection is not suitable for the MMSE receiver as the number of receive antenna must be at least the same as the number of the transmit antennas to achieve a good performance [10]. Again, the matrix inversion lemma can be applied for incremental user selection. The only change in (8) is the reverse of addition and minus operators, −1  −1 H  −1 = A − Aa 1 + aH Aa a A. (13) A + aaH At the beginning of the incremental user selection, let A = α−1 IM , a = hH i , Ω = {1, 2, . . . , Kt }, gi = hi A. The first user is selected such that gi 2 2 = arg max hi  . (14) j = arg max i∈Ω 1 + gi hH i∈Ω i A is updated as A ← A − −1 matrix  Then Hthe H gj gj . The second user can be selected simi1 + gj hj larly. We provide the pseudocode of incremental user selection algorithm in Table II. We have shown that the problem of user selection in MU-MIMO can be translated into a similar problem of receive antenna selection in single user MIMO communications. Some necessary modifications have been made to employ the conventional decremental and incremental search strategies in solving our user selection problem. Next, we present a completely new strategy, which iteratively selects users rather than the above decremental and incremental user selections.





−1

7

J1 = arg max 1 + gi hH i

8

Ω=

9 10

update A ← A − 1 + gJ1 hH J1 compute gi = hi A∀i ∈ ω

11

J2 = arg min 1 − gi hi

12

Ω=Ω∪

13 14 15

i∈Ω Ω\J1 , ω =

i∈ω J2 , ω

gi 2

ω ∪ J1





 H −1

−1

gJH gJ1 1

gi 2

= ω\J2



−1

update A ← A + 1 − gJ2 hH J2 end for Output: index set of selected user ω

−1

gJH gJ2 2

B. Iterative User Selection Algorithms At the beginning, a set ω, which we call selected set, of K random users is selected. The other users are in the set Ω = {1, 2, . . . , Kt } \ω, which we call remaining set. Each iteration is processed as follows. In the first half of an iteration, one user is added to the selected set from the remaining set so that the MSE of the new selected set with K + 1 users is minimal. In the second half of the iteration, a user in selected set of K + 1 users is dropped such that the remaining K users have smallest MSE. Several iterations can be made to further refine the MSE of selected users. Each add/drop operation can also utilize the matrix inversion lemma as implemented in the decremental and incremental user selection algorithms. We call this algorithm one-loop iterative user selection. The pseudocode of this algorithm is provided in Table III. There are several ways to improve the performance of the above one-loop iterative user selection. In the following, we give an illustrative example. If in one iteration the added and dropped users are the same, the selection process will finish since no further improvements can be achieved. In this case, we may randomly select another set of K users and run a secondary iterative search. Finally, the two iterative searches may give two different user sets, the set with smaller MSE will be selected. We may give some constraints to limit the complexity of the algorithm. For example, the total

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TABLE IV U SER S ELECTION A LGORITHMS FOR F REQUENCY F LAT C HANNELS Algorithm I-A: Decremental selection I-B: Incremental selection I-C: One-loop iterative selection I-D: Twoloop iterative selection

Description All users are initially selected. Then one-by-one user is dropped. Strongest user is initially selected. Then one-by-one user is added. A set of random user is initially selected. Next one user is added to the selected set; then one user is dropped. These add/drop processes are iteratively repeated. Similar to one-loop iterative user selection algorithm, but the user with largest channel norm is included in the initial user set. If in one iteration, the added and dropped users are the same, a secondary iterative loop is performed. Another random set of users is selected, not to include all the users have been selected in the primary iterative loop. Finally, the MSE of the primary and secondary iterative searches are compared. The user set with smaller MSE is selected

number of iterations of the primary and secondary search is fixed. We call the modified algorithm two-loop iterative user selection. Additionally, the user with largest channel norm can be selected at the beginning to reduce the randomness of the initial user set. The pseudocode of the two-loop iterative algorithm is omitted for brevity. Instead, we summarize and compare the developed noniterative and iterative algorithms in Table IV. C. Complexity Comparison The complexity of the proposed algorithms includes the complexities in three stages: initialization, user search, and matrix update. While the initialization and matrix update stages have fixed complexity, the user search stage has varying complexity since each search strategy involves different number of searched users. The search time and power consumption are proportional to the number of searched users. In the following, we use the number of searched users to define a measure of complexity. First we consider the complexity of the incremental user search. When searching for the k-th user (1 ≤ k ≤ K), the number of users to be checked is Kt − k + 1. Thus to search for K users, the total number of searched users is Cinc = K(2Kt − K + 1)/2.

(15)

For decremental search, when searching for the k-th user, the number of users to be searched is also Kt −k+1. However, the number of users to be deselected is Kt − K. Hence the total number of searched users in decremental selection is Cdec = (Kt − K)(Kt + K + 1)/2.

(16)

On the other hand, there are two searches in each iteration of the iterative algorithm. In the first and second halves of each iteration, Kt − K users (in the remaining list) and K + 1 users in the selected list are searched. If r iterations are run, the total number of searched users is Cite = r(Kt + 1).

(17)

TABLE V C OMPLEXITY C OMPARISON FOR U SER S ELECTION A LGORITHMS Algorithm Incremental Decremental Iterative

Kt = 8, K = 4 26 26 9r

Kt = 12, K = 4 42 68 17r

Kt = 16, K = 4 58 126 21n

Hence the complexity of incremental and iterative strategies is linear in the total number of users Kt , while the complexity of decremental search is squared in the total number of user Kt . We provide numerical examples of the complexities of the incremetal, decremental, and iterative user selection algorithms in Table VIII. Obviously, the iterative algorithm offers flexible complexity search compared with the incremental and decremental ones. We have investigated the user selection algorithms for frequency-flat channels. Next, we will extend these algorithms for user selection for frequency-selective channels with OFDMA transmission. IV. U SER S ELECTION IN OFDMA S YSTEMS In OFDMA systems, the users’ data is transmitted in resource blocks (RBs). Each RB consists of N consecutive subcarriers and T OFDM symbols [12]. The number of subcarriers N in each RB are often selected so that the channels of subcarriers are highly correlated. The number of OFDM symbols T is designed so that the channel variation during a transmission of a RB is negligible. Therefore, we can use one user’s channel vector of a subcarrier in each RB of for user selection purpose. The developed user selection algorithms can be easily applied to select users in one RB. We now consider a more complex scenario, where users can be jointly assigned to two or more RBs to exploit the frequency diversity of the OFDM channel. For illustration, we study the joint user selection for two RBs, RB-1 and RB-2. To simplify our presentation, we assume that the users to be selected in RB-1 and RB-2 are K1 and K2 , respectively, such that Kt = K1 + K2 (K1 , K2 ≤ M ). Let H1 and H2 be channel matrices of all users in RB-1 and RB-2, respectively. The first algorithm for two-RB user selection is a combination of the decremental and incremental algorithms. In particular, initially Kt − 1 users are allocated in RB-1, only the strongest user (with largest norm of channel vector) is pre-assigned in RB-2. Then, one user is moved from RB-1 to RB-2 such that the sum of MSE of two RBs is minimal. This moving process is repeated K2 − 1 times, until the numbers of users in each RB meet the requirements. We provide the pseudocode of this algorithm in Table VI. Next we present another user selection for two RBs based on the one-loop iterative algorithm. At the beginning, users are randomly divided into two sets Ω1 and Ω2 , each with K1 and K2 users. For each iteration, in the first half, one user is moved from set Ω1 to set Ω2 such that the sum MSE of two sets is minimized; in the second half, one user is removed from set Ω2 back to set Ω1 such that the sum MSE of two

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.

TABLE VI A LGORITHM II-A – D ECREMENTAL -I NCREMENTAL U SER S ELECTION 1 2

Initialization: input H1 , H2 , α, Kt , K1 , K2 Ω0 = {1, 2, . . . , Kt }

3

 2 Ω2 = arg max h2,k 

4

Ω1 = Ω0 \Ω2

5 6 7

A1 = H H , A2 = HH Ω1 HΩ1 + αIM Ω2 HΩ2 + αIM for l = 1 : K2 − 1 compute gi,k = hi,k A−1 for all k ∈ Ω1 and i = 1, 2 i

8 9 10 11 12 13

k∈Ω0



−1

j = arg min

g1,k 2

1−g1,k hH



−1

g2,k 2

−

1+g2,k hH k∈Ω1 1,k 2,k −1 H update A1 ← A1 + 1 − g1,j hH g1,j g1,j 1,j −1 H update A2 ← A2 − 1 + g2,j hH g2,j g2,j 2,j Ω1 = Ω1 \j, Ω2 = Ω2 ∪ j

 

 

1 2

input H1 , H2 , α, Kt , K1 , K2 , number of iterations r Ω0 = {1, 2, . . . , Kt }, Ω1 = {1, 2, . . . , K1 } , Ω2 = Ω0 \Ω1

3 4 5

A1 = H H , A2 = HH Ω1 HΩ1 + αIM Ω2 HΩ2 + αIM for l = 1 : r compute gi,k = hi,k A−1 for all k ∈ Ω1 and i = 1, 2 i

7



−1

J1 = arg min



g1,k 2

H k∈Ω1 1−g1,k h1,k

−

−1

g2,k 2

1+g

hH

2,k 2,k  −1 H update A1 ← A1 + 1 − g1,J1 hH g1,J g1,J1 1,J 1 1  −1 H H

8 9 10

update A2 ← A2 − 1 + g2,J1 h2,J g2,J g2,J1 1 1 Ω1 = Ω1 \J1 , Ω2 = Ω2 ∪ J1 −1 compute gi,k = hi,k Ai for all k ∈ Ω2 and i = 1, 2

11

J2 = arg min −

12 13 14 15 16

Algorithm Description II-A Combination of decremental and incremental algorithms developed for one RB user selection. II-B Generate two random sets of users. In each iteration, first move one user from set 1 to set 2, and then remove one user from set 2 to set 1. II-C Similar to Algorithm II-B. If the indices of added and dropped users are the same, in the next iteration, one user is not moved from set 1 to set 2, but moved from set 2 to set 1, then one user is removed from set 1 to set 2. II-D Similar to Algorithm II-C, but there are two separate runs, different in the direction, that a user is moved in the first half of the first iteration: In the 1st run, a user is moved from set 1 to set 2. In the 2nd run, a user is moved from set 2 to set 1.

end for Output: two sets of user indices Ω1 and Ω2

TABLE VII A LGORITHM II-B – O NE -L OOP I TERATIVE U SER S ELECTION FOR 2 RB S

6

TABLE VIII S UMMARY OF U SER S ELECTION A LGORITHMS FOR T WO RB S

g1,k 2

1+g1,k hH

+

g2,k 2

1−g2,k hH 2,k −1 H update A1 ← A1 − 1 + g1,J2 hH g1,J g1,J2 1,J2 2 −1 H update A2 ← A2 + 1 − g2,J2 hH g2,J g2,J2 2,J2 2 Ω1 = Ω1 ∪ J2 , Ω2 = Ω2 \J2 k∈Ω2

 

1,k

 

end for Output: two sets of user indices Ω1 and Ω2

sets is minimized. The pseudocode for the one-loop iterative user selection for two RBs, which we call Algorithm II-B, is given in Table 7. It may be possible that the users indices J1 and J2 are the same (c.f. lines 6 and 11 in Table VII). We thus can extend the idea of two-loop iteration for one-RB (Algorithm I-D) to the two-RB user selection problem. In this case, in the primary loop, one user is moved from set Ω1 to set Ω2 , and then one user is moved from set Ω2 to set Ω1 . In the secondary loop, users are moved in the reverse direction: first from set Ω2 to set Ω1 , then from set Ω1 to set Ω2 . The pseudocode for the two-loop iteration for two RBs are not presented for brevity. This algorithm is referred to as Algorithm II-C. Algorithm II-C can also be developed in some other ways; one of its extension is discussed in the following. Note that Algorithm II-C starts by moving one user from set Ω1 to set Ω2 . We can also reversely move one user from set Ω2 to set Ω1 . Therefore, we can develop another algorithm, named Algorithm II-D, in which Algorithm II-C is run twice. In the first run, a user is moved from set Ω1 to set Ω2 in the first

iteration of the primary (outer) loop, while in the second run, a user is moved from set Ω2 to set Ω1 . We summarize the user selection algorithms for two RBs in Table VIII. V. S IMULATION R ESULTS The uncoded bit error rate (BER) performance of the proposed user selection algorithms will be presented and also compared with the algorithms based on users’ orthogonality [7] and sum rate [8] in frequency-flat channels. The simulated MU-MIMO system has M = 4 transmit antenna base station, serving K = 4 out of Kt = 16 users, all users employ 4QAM signal. The channels between any transmit-receive antennas are uncorrelated and have the same Rayleigh distribution with unit power. Performances of the linear MMSE precoder with different user selection algorithms are shown in Fig. 1. It is clear that our three proposed MMSE-based algorithms outperform the orthogonality-based and capacity-based algorithms; the iterative algorithm with r = 4 iterations yields about 1 dB gain over the orthogonality-based algorithm at a BER of 10−4 . The performances of the three MMSE-based algorithms also verify the performance-complexity tradeoffs (c.f. Table V). We also present performances of the proposed algorithms for ZF precoding in Fig. 2. Our algorithms again give smaller BER than the orthogonality and capacity-based algorithms. It is worthwhile to note that although MMSE precoder performs better than ZF precoder without user selection, their performance gap becomes smaller when user selection is possible. Performances of the algorithms for joint user selection in two RBs of an OFDMA system are also presented for an OFDM system with 1024 subcarriers. There are 64 RBs, each RB has 16 subcarriers and spreads over 20 OFDM symbols. There are Kt = 8 users to be assigned for two neighbor RBs, each RB thus serves 4 users. The uncorrelated channel is examined, where each OFDM channel is generated according to the power and delay profiles of the modified wideband ITU-T Pedestrian-B channel model recommended for the IEEE 802.16m system evaluation methodology [12]. From simulation results, the iterative algorithm with 6 iterations nearly achieves the optimal performance of highly complicated

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.

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random user selection capacity−based orthogonality−based MMSE−based incremental MMSE−based iterative, 4 iterations MMSE−based decremetal MMSE−based exhaustive 6

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Random user selection Combined incremental−decremental II−C, 4 iterations II−D, 4 iterations II−D, 6 iterations Exhaustive

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Fig. 1. Performances of user selection algorithms with linear MMSE precoding, 4 transmit antennas, 16 users choose 4, users employ 4QAM signal.

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Fig. 3. Performances of user selection algorithms for 2 neighbor RBs, K1 = K2 = 4, OFDMA system with 1024 subcarriers, 4QAM for all users.

algorithms can be extended in several ways, e.g. for user selection with nonlinear precoding.

−1

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ACKNOWLEDGMENT −2

Uncoded BER

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The authors would like to thank for fruitful discussions with their colleagues at Toshiba Research Europe Ltd. and the support and approval of its directors.

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random user selection capacity−based orthogonality−based MMSE−based incremental MMSE−based iterative, 4 iterations MMSE−based decremetal MMSE−based exhaustive

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Fig. 2. Performances of user selection algorithms with linear ZF precoding, 4 transmit antennas, 16 users choose 4, users employ 4QAM signal.

exhaustive search, it gains about 1.4 dB over the combined incremental-decremental algorithm at a BER of 10−3 . VI. C ONCLUSION This paper presented new user selection algorithms for MU-MIMO linear precoding. We have shown that, based on MMSE criterion, the user selection in MU-MIMO precoding can be mathematically transformed into the problem of receive antenna selection for MMSE receivers. Thus, two algorithms, namely incremental and decremental search, have been extended from existing MMSE-based antenna selection algorithms. Importantly, we propose a new iterative user selection strategy, which offers a flexible performance-complexity tradeoff, compared to the conventional decremental and incremental search approaches. Simulation results show that our proposed algorithms perform better than well-known algorithms using user orthgonality and capacity bound criteria. The presented

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