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Efficient Transmit Antenna Selection for Multiuser MIMO Systems with Block Diagonalization Runhua Chen, Jeffrey G. Andrews, and Robert W. Heath, Jr. Wireless Networking and Communications Group Department of Electrical and Computer Engineering The University of Texas at Austin, Austin, Texas 78712-0240 Email:{rhchen, jandrews, rheath}@ece.utexas.edu

Abstract— Block diagonalization is a precoding technique for multiuser MIMO systems that pre-cancels inter-user interference at the transmitter side. When there are a large number of base station antennas but a limited number of RF amplifiers, the system performance can be significantly improved by switching a subset of antennas to the RF chains and exploiting antenna selection diversity. The optimal antenna subset can be obtained by exhaustively searching over all possible antenna combinations. This brute-force search, however, is prohibitively complicated and impractical. To reduce the complexity, in this paper we propose several low-complexity suboptimal transmit selection algorithms that minimize a symbol error rate (SER) upper bound or maximize a capacity lower bound. Simulation results show that our proposed algorithms perform very close to the optimal exhaustive search, while the complexity is much lower.

I. I NTRODUCTION A downlink multiuser multi-input multi-output (MUMIMO) communication system involves a base transceiver station (BTS) transmitting to multiple mobile stations (MS) simultaneously over the same frequency band. Compared to conventional TDMA systems, MU-MIMO substantially increases the system capacity by multiplexing users in the spatial domain [1] and reduces the latency for each user. The capacity optimal approach for MU-MIMO systems requires dirty paper coding (DPC) [2]-[5], which however is impractical in commercial wireless systems due to its high complexity associated with the successive encoding process. A more practical MU-MIMO broadcast transmission technique is block diagonalization (BD) precoding, which pre-multiplies each user’s signal by a precoding matrix at the BTS. When channel state information (CSI) is available at the transmitter, precoding can be used with a large number of BTS antennas to reduce or to eliminate co-channel interference (CCI). The number of BTS antennas (RF amplifiers) is required to be larger than the total number of receive antennas at mobile users to completely eliminate CCI [6]-[9]. When there are a large number of BTS antennas but a limited number of RF amplifiers, the system performance can be greatly improved by selecting a subset of transmit antennas with good channel conditions and switching them to the RF chains. As antenna elements are much cheaper than RF amplifiers, antenna selection is able to efficiently exploit the spatial selectivity introduced by additional antennas This work is supported by AT&T Laboratories, Inc., Austin, Texas.

and improve the system diversity gain without significantly increasing the equipment cost. Antenna selection for single user MIMO systems has been extensively studied, where the selection can be applied at both transmitter and receiver ends, in an effort to maximize the capacity or minimize the error rate [10]-[16]. Relatively fewer results are available for antenna selection in multiuser MIMO systems. In [9], transmit antenna selection for MU-MIMO system under the block diagonalization signaling is studied, where two selection algorithms are proposed to optimize the system symbol error rate (SER) and sum rate capacity. Brute-force search is required to find the optimum antenna subset [9]. This approach, however, is highly complicated due to the huge number of possible antenna subsets, especially as the number of BTS antennas becomes large. As a result, a low-complexity algorithm is critical to efficiently exploit the diversity gain promised by MU-MIMO antenna selection, while keeping the selection complexity low. In this paper we propose several sub-optimal antenna selection algorithms to reduce the computational complexity. Following a greedy selection mechanism, our proposed algorithms eliminate one antenna from the BTS antenna array at a time, which causes the minimum performance degradation compared to a system where this antenna is not removed. This greedy procedure is repeated until the desired antenna subset is reached. Two algorithms are proposed in this paper with different selection metrics. The first algorithm aims to minimize the post-processing SER lower bound, while the second Frobenius norm based algorithm maximizes a sum rate capacity lower bound. The computational complexity of our proposed algorithms is analyzed and shown to be much smaller than the exhaustive search. Simulation results demonstrate that our algorithms perform very close to the optimum exhaustive search methods. II. S YSTEM M ODEL Consider the MU-MIMO system illustrated in Fig. 1 with MT RF chains and MT > MT antennas at the BTS, and K mobile users where the kth user has MR,k receive antennas, k = 1, 2, . . . , K. A narrow-band flat-fading channel is assumed, which is satisfied if orthogonal frequency division multiplexing (OFDM) is used. A low mobility environment is considered which is the common application scenario for multiuser MIMO systems. The channel from the

3499 1930-529X/07/$25.00 © 2007 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2007 proceedings.

M T' Tx x1,1 user 1

Precoder

...

S/P

x1, 2

T1, p

x1, N1

s1,1 s1, 2 s1,3

M R ,1

MS 1

RF 1

s1, M T RF 2 s2,1 s 2, 2

x 2,1 x 2, 2

T2, p

x 2, N 2

s

...

TK , p

selection

RF RF 3

3

Switching

s K ,1 s K ,2 s K ,3 s K ,M T

channel

RF M T

p

Antenna Selection Module

selection

MS 2

propagation

... ... ...

Precoder

x K ,NK

,

... ...

... ... S/P

x K ,2

1

s 2, M T

x K ,1

user K

M R,2

Precoder s2,3

...

S/P

... ...

user 2

M R, K

p

MS K channel feedback

Fig. 1. Block diagram of the MU-MIMO system with BD precoding: perfect feedback is assumed with {Hk }K k=1 exactly known at the transmitter for precoder design.

BTS to the kth MS is given as Hk ∈ CMR,k ×MT , where (i,j) Hk denotes the channel fading coefficient from the jth transmit antenna to the ith receive antenna of user k. It is assumed that both the BTS and MSs experience sufficient local scattering, therefore Hk has full rank (i.e. rank (Hk ) = min (MR,k , MT )) with probability one. In addition, it is asK sumed that {Hk }k=1 are independent due to users’ different geographical locations, thereforethe composite channel matrix H HH · · · HH has full rank. The channel H = HH 1 2 K is assumed perfectly known at the base station, satisfied by channel reciprocity in TDD system or feedback in FDD systems. At a particular time instant, the MR,k -dimensional data vector for user k is denoted as xk = [xk,1 , xk,2 , . . . , xk,MR,k ]T , where Qk = E{xk xH k } is the input covariance matrix and K P = k=1 trace (Qk ) is the sum transmit power. Data vector xk is then pre-multiplied with a MT ×MR,k precoding matrix Tk and sent to the BTS antenna array. The received signal at the kth MS is expressed as yk = Hk

K

Tj xj + nk ,

(1)

j=1

where nk ∈ CMR,k ×1 is the AWGN noise with zero mean and variance No . The principal idea of BD is to find the unitary precoding matrix Tk ∈ U (MT , MR,k ) such that Hk Tj = 0,

1 ≤ k = j ≤ K,

(2)

where U(n, k) denotes the set of n × k unitary matrices with orthonormal columns. If (2) is satisfied, co-channel interference is completely eliminated and user k observes a single-user MIMO link whose effective channel is Hk Tk . † ˜ k = H† · · · , H† , H† · · · , H† denote the Let H 1

k−1

k+1

K

aggregate interference channel observed by user k. To satisfy the zero-interference constraint (2), precoder Tk should lie in

˜ k , thus constraint (2) can be rewritten as the null space of H ˜ k Tk = 0, H

1 ≤ k ≤ K.

(3)

˜ k as Denote the singular value decomposition (SVD) of H † (1) (0) ˜k Λ ˜ ˜ ,V ˜k = U ˜ k, 0 ˜ ˜k = where L V H ˜ k k Lk ×(MT −L k) K ˜ k, Λ ˜k = MR,j , MT is the rank of H min j=1,j=k ˜k × L ˜ k diagonal matrix condiag λ1,k , . . . , λL˜ k ,k is the L (1) ˜ ˜ k right contains the first L taining the singular values, V k (0) ˜ ˜ singular vectors, and Vk contains the last MT − Lk singular ˜ (0) form a null space basis of H ˜ k. vectors. The columns of V k As a result, the columns of any Tk satisfying (3) should be ˜ (0) . To guarantee that linear combinations of the columns of V k the null space is not empty, a necessary condition for perfect interference cancelation with BD is specified as follows [6][9]. Lemma 1: To ensure that the null space is not empty and the interference can be perfectly cancelled bymultiuser precoding, K 2 a necessary condition for BD is MT ≥ k=1 MR,k . In the remainder of this paper we assume that all MSs have the same number of receive antennas MR . We note that our proposed work can be easily generalized to the case with different numbers of antennas per user, but this is not discussed here for notational brevity. III. M ULTIUSER A NTENNA S ELECTION A multiuser transmit antenna selection technique was proposed in [9], which works for a system where the number of RF units MT is exactly the minimum required for supporting multiuser downlink precoding, i.e., MT = KMR . It can also be applied when there are a large number of users and a MT users are selected, so the constraint MT = subset of M R KMR is met. The BTS has MT > MT BTS antennas, more than strictly required for interference cancelation. With the additional spatial selectivity introduced by excess antennas, the key idea of antenna selection is to choose a subset of MT antennas with good channel conditions and switch them to the RF chains. By transmitting over the “preferred” antennas, system performance in terms of SER and capacity can be greatly improved thanks to transmit selection diversity. This performance enhancement comes at a relatively low cost because antenna elements are much cheaper than RF amplifiers. Even though only a subset of antennas are used, simulation show that it achieves the same diversity performance as a full system where all MT antennas are simultaneously used. The optimal antenna selection can be performed by exhausMT tively searching over the possible CM antenna subsets under T n denotes m choosing a given performance metric, where Cm n. Two exhaustive search algorithms have been proposed in [9] to optimize the SER or the sum rate capacity, which will be briefly introduced in this section. In the following, the selected antenna subset is denoted as A ⊆ {1, 2, . . . , MT } which contains the indexes of the selected antennas. We use P to denote the set of all possible antenna subsets where MT card (P) = CM . For a particular antenna subset A, the T

K

corresponding channel matrices are given as {Hk,A }k=1 .


A. SER-based Exhaustive Antenna Selection With a linear ZF or MMSE receiver at each mobile terminal, it was shown in [9][16] that the post-processing SNR of the kth user is lower bounded by λmin (Hk,A Tk,A ), where λmin denotes the minimum singular value and Hk,A Tk,A denotes the effective channel of user k after precoding, given a certain BTS antenna subset A. Given this result, the SER-based MT exhaustive search chooses from CM possible antenna subsets T to maximize the system SER upper bound. Algorithm 1: SER-based Exhaustive Search - To optimize the SER lower bound, the optimal antenna subset is selected as (4) Aopt = arg max min λmin (Hk,A Tk,A ). A∈P k=1,...,K

B. Capacity-based Exhaustive Antenna Selection Antenna selection can also be implemented by to optimize the sum rate capacity after precoding. Algorithm 2: Capacity-based Exhaustive Search - The optimum antenna subset is selected as Aopt

=

arg max A∈P

MR K



 log2 1 +

k=1 i=1

γ−

No |λk,i |2



+

M R No

 |λk,i |2  , (5)

where diag (λk,1 , . . . , λk,MR ) are the singular values of Hk,A Tk,A , 0 x≤0 , (6) (x)+ = x x>0 and γ is the waterfilling threshold determined by the sum power constraint P =

MR K

γ−

k=1 i=1

No |λk,i |2

.

(7)

+

IV. L OW-C OMPLEXITY A NTENNA S ELECTION One issue with the exhaustive search is the computational MT antenna combinations complexity. Because a total of CM T need to be searched over, the computational complexity MT grows linearly with CM . This complexity easily becomes inT tractably high for any commercial wireless network, especially as MT becomes large. To reduce the computational complexity and exploit transmit antenna selection as a practical diversity scheme, two low-complexity algorithms are proposed in this section. The first algorithm aims to minimize an upper bound of the postdecoding SER, and the second algorithm aims to maximize a lower bound of the sum rate capacity represented by the aggregate Frobenius norm of the effective channel. Both algorithms follows a greedy antenna selection approach, where one BTS antenna is deactivated in each iteration, until the number of remaining antennas is equal to the number of RF units.

A. SER-based Low-Complexity Selection The first proposed low-complexity algorithm aims to minimize the SER upper bound of the system. The key idea is to greedily reduce the number of selected antennas, until the number of remaining active BTS antennas is equal to MT . At the beginning, all transmit antennas are active. Then in each iteration, for each transmit antenna we compare the post-decoding SNR lower bound assuming it is removed from the BTS array. The antenna that maximizes the SNR lower bound is selected and removed, which intuitively causes the minimum SER performance degradation. This procedure is repeated where a single antenna is removed at a time, until the optimum MT antennas are finally selected. The proposed algorithm is detailed as follows. Algorithm 3: SER-based Low-Complexity Search 1) Stage s = 0: Let all MT BTS antennas be active, and K feedback the channel {Hk }k=1 to the BTS. Let A = {1, 2, . . . , MT } denote the set of active BTS antennas, and let S = φ denote the set of inactive antennas. 2) Stage s = s + 1: a) For every antenna i ∈ A, temporarily deactivate it by setting A˜ = A − {i}, and calculate the postdecoding SNR lower bound by (8) λi,min = min λmin Hk,A˜Tk,A˜ , k=1,...,K

where Hk,A˜ is the channel matrix of user k associ˜ and T ˜ ated with the active BTS antenna set A, k,A is the corresponding precoding matrix. b) Find the antenna that maximizes the SNR lower bound (9) iopt = arg max λi,min . i∈A

c) Deactivate antenna iopt by letting A = A − {iopt }. 3) If s < (MT − MT ), go to stage (2). Else, exit the iteration. This algorithm needs to perform MT −MT iterations, where no more than MT − s + 1 antenna need to be searched over in the sth iteration. As a result, the size of search space is upper bounded by MT −MT

MT − MT , 2 s=1 (10) which is greatly simplified than the exhaustive search method MT where CM possible combinations have to be searched over. (MT − s + 1) = (MT + MT + 1)

T

B. Norm-based Low-Complexity Algorithm The second low-complexity antenna selection algorithm is proposed where the performance metric is the aggregate Frobenius norm of the effective channel, denoted as N=

K k=1

P Hk,p Tk,p 2F . KMR

(11)

The Frobenius norm is chosen as the selection metric because it is closely related to the eigenvalues of the effective channel


0

10

after precoding, and provides a tight lower bound on the sum capacity represented by [17]

−1

10

−2

10

SER

C

1 log det I + Hk,A Tk,A Qk,A T†k,A H†k,A = No k=1 K P ≥ log 1 + Hk,A Tk,A 2F KMR No k=1 K P 2 > log 1 + (12) Hk,A Tk,A F . KMR No K

−4

10

k=1

As a result, we can maximize the sum capacity lower bound (12) by maximizing the aggregate Frobenius norm. Algorithm 4: Norm-based Low-Complexity Search 1) Stage s = 0: Let all MT BTS antennas be active, and K collect the channel {Hk }k=1 at the BTS. Let A = {1, 2, . . . , MT } denote the set of active BTS antennas, and let S = φ denote the set of inactive antennas. 2) Stage s = s + 1: a) For every antenna i ∈ A, temporarily deactivate it by setting A˜ = A − {i},. Use the channel Hk,A˜ to calculate the precoding matrices Tk,A˜. Calculate the aggregate Frobenius norm as K k=1

Hk,A˜Tk,A˜2F

iopt = arg max Ni i∈A

(14)

c) Deactivate antenna iopt by letting A = A − {iopt }. 3) If s < (MT − MT ), go to stage (2). Else, exit the iteration. Similar to the SER-based low-complexity algorithm, the norm-based algorithm follows a greedy selection approach. In each iteration, the BTS selects the optimal antenna that generates the maximum Frobenius norm if this antenna is deactivated. The algorithm terminates when the number of active transmit antennas reaches MT . The search size is MT −MT

−5

−6

10

MT − MT , 2 s=1 (15) which is much less complicated than the exhaustive search MT over CM antenna subsets. (MT − s + 1) = (MT + MT + 1)

T

V. N UMERICAL R ESULTS Simulation results are presented in this section to demonstrate the performance of our proposed low-complexity antenna selection algorithms. It is assumed that each user experience an i.i.d Rayleigh fading channel with zero mean and variance 1, and the channels of all users are independent.

0

2

4

6 8 10 12 14 SNR per branch per receive antenna (dB)

16

18

20

Fig. 2. SER performance of exhaustive and low-complexity antenna selection with 2 users, 2 receive antennas and 2 data substreams per user, using 4QAM modulation and MMSE receiver per MS. 0

10

−1

10

(13)

b) Find the antenna that maximizes the aggregate Frobenius norm

SER−based exhaustive SER−based low−complexity no additional TX 1 additional TX 2 additional TX 4 additional TX 6 additional TX 8 additional TX 10 additional TX

10

−2

10

SER

Ni =

−3

10

−3

10

−4

10

SER−based exhaustive SER−based low−complexity no additional TX 1 additional TX 2 additional TX 4 additional TX

−5

10

−6

10

0

2

4


16

18

20

Fig. 3. SER performance of exhaustive and low-complexity antenna selection with 2 users, 4 receive antennas and 4 data substreams per user, using 4QAM modulation and MMSE receiver per MS

A. SER Performance Fig. 2 plots the average SER of the exhaustive and the proposed SER-based low-complexity antenna selection algorithms, with 2 users, 2 receive antennas per user, QPSK modulation and MMSE receiver. We find that the performance of the low-complexity algorithm is almost identical to that of the exhaustive search method, especially when the number of extra transmit antennas is fewer than 4. Similarly, Fig. 3 compares the SER results with 2 users, 4 receive antennas per user, and 4QAM modulation. Again, the low-complexity algorithm performs almost the same as the exhaustive search method, with substantially lower complexity.


30

25

20 sum capacity

is expected that the difference between the brute-force and the proposed suboptimal algorithms will be increased. This is an independent problem and hence is not discussed in this paper.

Capacity−based exhaustive Norm−based low−complexity no additional TX 1 additional TX 2 additional TX 4 additional TX 6 additional TX 8 additional TX 10 additional TX

VI. C ONCLUSIONS

15

10

5

0

0

2

4

6 8 10 12 14 SNR per branch per receive antenna(dB)

16

18

20

Two low-complexity transmit antenna selection algorithms for downlink MU-MIMO systems with block diagonalization are proposed in this paper. The goal is to select a subset of BTS antennas to maximize the sum-rate capacity, or to minimize the post-decoding SER. The optimal brute-force search method obtains the optimal antenna subset, however its complexity MT grows linearly with CM . Following the greedy selection T mechanism, the proposed algorithms remove one antenna at each iteration that causes the minimum performance degradation, therefore the complexity is much lower. Simulation results show that the proposed algorithms perform very close to the optimal exhaustive search.

Fig. 4. Sum rate capacity of exhaustive and low-complexity antenna selection with 2 users, 2 receive antennas per user. 45 Capacity−based exhaustive Norm−based low−complexity no additional TX 1 additional TX 2 additional TX 4 additional TX

40

35

sum capacity

30

25

20

15

10

5

0

2

4


16

18

20

Fig. 5. Sum rate capacity of exhaustive and low-complexity antenna selection with 2 users, 4 receive antennas per user.

B. Capacity Performance The sum rate capacity of the exhaustive and the Frobenius norm based low-complexity algorithms is compared in Fig. 4, with 2 users, 2 receive antennas per user. Again, it is confirmed that the capacity performance of these two algorithms are very close to each other. The capacity with 2 users, 4 receive antennas is compared in Fig. 5. It is found that the Frobenius norm based low-complexity algorithm achieves about 95% of the capacity of the capacity-based exhaustive search, with much lower complexity. When there are a large number of users in the system, the base station cannot support all users at the same time, therefore a subset of users should be selected at each time instant to meet the antenna constraint. Under this condition, it

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