A Downlink Decomposition Transmit Pre-processing Technique for ...

A Downlink Decomposition Transmit Pre-processing Technique for Multi-user MIMO Systems Ruly Lai-U Choi, and Ross D. Murch Department of Electrical & Electronic Engineering The Hong Kong University of Science & Technology Clear Water Bay, Kowloon, Hong Kong Email: [email protected] and [email protected]

ABSTRACT In recent years, wireless multiple-input multiple-output (MIMO) systems with multiple antennas employed at both the transmitter and receiver have gained attention because of their promising performance improvement. However, to date, there is still a lack of solution for the downlink of multi-user MIMO systems. In this paper, we introduce a transmit pre-processing technique at the transmitter for the downlink of multi-user MIMO systems. It decomposes the multi-user MIMO downlink channel into multiple parallel independent single-user MIMO downlink channels. Some key properties of this technique are figured out, such as linearly increasing capacity and high bandwidth efficiency. Simulation results are also provided and these results demonstrate the potential of our technique in terms of performance and capacity. I. INTRODUCTION In recent years, wireless multiple-input multiple-output (MIMO) systems with multiple antennas employed at both the transmitter and receiver have gained attention because of their promising improvement in terms of performance and bandwidth efficiency [1]. Several techniques have been proposed [2]-[5] such as VBLAST (Vertical Bell Laboratories Layered SpaceTime), MLD (maximum likelihood detection), and singular value decomposition (SVD) based techniques. However, in the downlink these techniques are only suitable for single-user MIMO operation since they assume that the system includes a single receiver with multiple receive antennas and joint signal detection. In this paper, we introduce a MIMO transmit preprocessing technique at the transmitter for the downlink of multi-user MIMO systems. The technique is based on decomposing a multi-user MIMO downlink channel into parallel independent single-user MIMO downlink channels. Once the multi-user channels are decomposed, any single-user MIMO technique can be applied in the usual way to each user. Previously, there has been only limited work on multi-user MIMO systems for the downlink. Examples include [6] and [7], where multi-user MIMO systems are considered and the processing at both of the receivers

and transmitter is assumed to be linear. The work in [6] attempts to find the antenna weights for the transmitter and receivers jointly by maximizing the spectral efficiency for a multi-user MIMO TDMA system and [7] tries to find the antenna weights jointly by maximizing the signal to interference-plus-noise ratio (SINR) for a multi-user MIMO CDMA system. However, the solution for the antenna weights in [6] are not guaranteed to exist since the proposed iterative algorithm is not guaranteed to converge. On the other hand, the solution for the antenna weights in [7] is sub-optimal. Another problem is that particular linear receiver structures are assumed in both of the systems and these impose certain restrictions on the systems. Our work is different in that we introduce a transmit pre-processing technique at the transmitter of the base station (BS) for the downlink of multi-user MIMO systems for decomposing a multi-user MIMO downlink channel into multiple parallel independent single-user MIMO downlink channels. Therefore, all the techniques for single-user MIMO systems, such as V-BLAST, MLD, and joint transmit and receive MIMO processing (e.g. SVD based techniques), can be applied for each user of the multi-user MIMO systems. Some key properties of this decomposition are figured out, such as linearly increasing capacity and high bandwidth efficiency. The structure of this paper is as follows. In Section II, the system model of a multi-user MIMO system is introduced, while the problem formulation and the solution are provided in Section III. Then, Section IV gives some discussions of the key properties and some simulation results are provided in Section V. Finally, Section VI concludes our work. II. SYSTEM MODEL The configuration of our proposed MIMO system is shown in Figure 1, where M antennas are located at the base station (BS) and N k antennas are located at the k-th mobile station (MS). In total, there are K MSs or users in the system. At the BS, the data are processed before transmission, which we refer to as transmit preprocessing, and then launched into the MIMO channel. Let b(k ) represent the Lk × 1 transmit data symbol vector for user k , where Lk is the number of data symbols transmitted simultaneously for user k ( k = 1,..., K ). This

data symbol vector passes through a transmit precoder, which is characterized by the precoding matrix T(k ) , a M × Lk matrix that takes in Lk nonzero values and outputs M terms. Each of the M output terms is transmitted by each of the M transmit antennas. We assume that the channel is flat fading and denote the MIMO channel to user k as H (k ) , which is a N k × M matrix. Its (i, j ) -th element is the complex gain from j-th transmit antenna at the BS to the i-th receive antenna at MS k . Also, its elements are independently identically distributed (i.i.d.) zero mean complex Gaussian random variables with unity variance (real and imaginary part variance of 0.5). At the receiver of user k , N k receive antennas are used to receive the Lk data symbols and the received signals can be written by a vector of length N k , which is given by K

r ( k ) = H ( k ) ∑ T (i ) b ( i ) + n ( k )

Therefore, it can be shown that the solution to (2) is equivalent to the solution of

T( k )

  H (1)T( k ) = 0    M    H ( k −1)T( k ) = 0    , for k = 1,..., K = arg ( k +1) T ( k ) = 0  H 0 < trace ( T ( k ) T ( k ) ) ≤ Pk  H   M    H ( K ) T ( k ) = 0   

(3) ) be the nth column of T (k ) , since Letting t (k n (k ) ( i ) ( k ) H T = 0 implies that t n is in the null space or (i ) kernel of H , we can simplify (3) further as the solution of

(1)

t (nk ) ∈

i =1

where the noise n (k ) is an N k ×1 vector, whose elements are i.i.d. zero mean complex Gaussian random variables with variance σ 2 (real and imaginary part variance of 0.5σ 2 ). Throughout this paper, we denote a K -user system with M transmit antennas at the BS and N k antennas at the k-th MS as a ( M , [ N 1 , N 2 , L , N K ]) system and we will refer to a single-user system with M transmit antennas at the BS and N antennas at the MS as a ( M , N ) system. III. PROBLEM FORMULATION By using the system model introduced in Section II, our primary objective is to select the non-zero K precoding matrices, (T (1) , T ( 2) ,..., T ( K ) ) , for the K users such that at the receiver of each MS there is no interference from the other K − 1 users. Mathematically, the problem statement can be expressed as

(

T (1) , T ( 2) ,..., T ( K )

)=

  (1) K (i ) (i )   H ∑T b = 0  i =1,i ≠1     ( 2) K (i ) (i )  T b = 0  (2)  H ∑ arg i =1,i ≠ 2   H 0< trace( T ( k ) T ( k ) ) ≤ Pk , k =1, 2,... K  M     ( K ) K (i ) (i )  ∑ T b = 0  H i i K 1 , = ≠   where trace(.) represents the trace operation and the constraint states that the transmit power of each user is K limited. It should be noted that H ( k ) ∑ T(i )b (i ) represents i =1, i ≠ k

the interference to user k due to other K − 1 users and therefore, our primary objective function in (2) nulls all interference for each user. Since {b(i ) , i = 1,..., K } are arbitrary data vectors, H (k )

K

∑ T (i ) b (i ) = 0 implies

i =1,i ≠ k

H ( k ) T (i ) = 0

for

i≠k .

K

I ker(H (i ) )

(4)

i =1,i ≠ k

where n = 1,..., L k , ker(X) denotes the null space or kernel of X , and I represents the intersection of the subspaces. Notice that the precoding matrix T (k ) should be a nonzero matrix, otherwise, no signal is transmitted. To guarantee the existence of a nonzero precoding matrix, it is easy to show that the necessary condition for the existence of the solution to (3) is as follows: The number of the transmit antennas is larger than the sum of the number of receive antennas of any K − 1 users. Mathematically, it can be expressed as M > max{

K

∑ N i , k = 1,2,..., K }

.

(5)

i =1,i ≠ k

{

}

Under this necessary condition, let v 1( k ) , v (2k ) ,..., v (nkk ) be an orthogonal normal basis of the subspace

K

I ker(H (i ) ) ,

i =1,i ≠ k

where n k is the dimension of this subspace. This orthogonal normal basis can be found by computing the orthogonal normal basis of the subspace

[

ker( H (1)

T

L H ( k −1)

T

H ( k +1)

T

L H(K )

] ) . Hence, it can

T T

be shown that the dimension of the subspace, n k , is nk = M −

K

∑ Ni

(6)

i =1,i ≠ k

with probability one. By letting

[

V ( k ) = v1( k )

]

v(2k ) L v (nkk ) ,

(7)

from (4), we can write a solution to (3) as T (k ) = V (k ) A (k )

(8)

where A (k ) is a nonzero n k × L k matrix, which can be designed alone by some criteria or can be jointly designed with the structure of the receiver. Also, note H

that trace( A ( k ) A ( k ) ) ≤ Pk because V (k ) is orthogonal H

normal and trace(T ( k ) T ( k ) ) ≤ Pk .

Furthermore, by substituting (8) into (1), we can obtain r ( k ) = H ( k ) T ( k ) b ( k ) + n ( k ) = H ( k ) V ( k ) A ( k ) b ( k ) + n ( k ) .(9)

Notice that the multi-user MIMO system denoted by (1) has been decoupled to K parallel single-user MIMO systems. A close observation of (9) shows that we can think of the equivalent single-user MIMO channel of user k as H ( k ) V ( k ) and the equivalent transmit processing can be represented as A (k ) . As shown in Figure 2, the multi-user MIMO channel is decomposed to K parallel single-user MIMO channels. We refer to the decomposing process in (9) as Multi-user MIMO Decomposition and we refer to a multi-user MIMO system applying this decomposition as MU-MIMO system. IV. KEY PROPERTIES The key properties of the Multi-user MIMO Decomposition discussed in Section III are as follows: 1. It decomposes a multi-user MIMO downlink channel into K parallel independent single-user MIMO channels (See Figure 2). Therefore, all the techniques for single-user MIMO systems, such as V-BLAST, MLD, and joint transmit and receive MIMO processing (e.g. SVD based techniques), can be applied for each user of the multi-user MIMO systems. 2. Each equivalent single-user MIMO channel has the same Gaussian properties as a normal MIMO channel. As shown in Figure 2, the equivalent channel of user k can be given by H ( k ) V ( k ) , whose dimension is N k × n k . Since the elements of H (k ) are i.i.d. zero mean complex Gaussian random valuables with unity variance and V (k ) is orthogonal normal, the elements of H ( k ) V ( k ) are also i.i.d. zero mean complex Gaussian random valuables with unity variance. Hence, the equivalent system for user k after the Multi-user MIMO Decomposition is a system with n k transmit antennas and N k receive antennas. 3. Increasing the number of transmit antennas of the multi-user system by one increases the number of transmit antennas of each single-user by one. That is, increasing the value of M by one increases the value of n k by one for all users ( k = 1,..., K ). This can be observed from the relationship (6) discussed in Section III. 4. The capacity of this multi-user MIMO system increases linearly with the number of transmit antennas. For example, a K -user MU-MIMO system with 2 K transmit antennas at the BS and 2 receive antennas at each MS is equivalent to K parallel (2,2) systems. When the number of transmit antennas increases, the number of users can increase without affecting of capacity of the original users. Therefore, the capacity increases linearly with the number of transmit antennas.

5. The total number of simultaneous data streams for all K the users, ∑ Li , can be larger than the number of i =1

transmit antennas. For example, a 3-user system with 7 transmit antennas at the BS and 2 receiver antennas at each MS can be decomposed to 3 parallel (3,2) systems. We can transmit 3 data streams for each user ( L1 = L2 = L3 = 3 ) and therefore, the total number of simultaneous data streams for all the users is 9, which is larger than 7, the original number of transmit antennas in the multi-user system. Notice that a MLD receiver has to be used at each user to retrieve the three data streams. V. SIMULATION RESULTS In this section, the MU-MIMO System introduced in the previous sections is investigated by computer simulation. In the simulation, 4-QAM modulation is utilized. The flat fading MIMO channel, whose elements are i.i.d. zero mean complex Gaussian random variables with variance one, is fixed for 100 symbols and more than 10,000 independent channels are used to obtain each BER simulation. Throughout this section, we consider a K user system with M transmit antennas at the BS and N receive antennas at each MS ( N1 = N 2 = L = N K = N ), and we will refer to it as a ( M , [ 1 N ,42 N ,L , N ]) system. In 43 K

order to satisfy the necessary condition for the existence of a nonzero precoding matrix solution, we assume M > ( K − 1) N . Also, we assume that the number of data streams equals to for each user L ( L1 = L 2 = L = L K = L ). We denote a single-user system with M transmit antennas at the BS and N receive antennas at each MS as a ( M , N ) system. Figure 3 provides sample performance comparison between our MU-MIMO systems and single-user MIMO systems. Receive antennas N=2 are used for each user. Three different cases are compared: L = 1 with SVD based technique (i.e. the data stream is transmitted through the channel with the largest singular value), L = 2 with MMSE receiver at each user, and L = 2 with ML receiver at each user. It can be observed that the performance of our 3-user MU-MIMO system, (6,[2,2,2]) configuration, is similar to that of the singleuser system, (2,2) configuration, for all cases. This is consistent with the result in Section IV, in which we show that (6,[2,2,2]) MU-MIMO system is equivalent to a (n k ,2) configuration single-user system and n k = 2 from (6). We also deduced in Section IV that a K -user MU-MIMO system with 2 K transmit antennas at the BS and 2 receive antennas at each MS is equivalent to K parallel (2,2) systems. These results reveal that the capacity of the MU-MIMO system increases as the number of the transmit antennas increases. Since multiple-input single-input (MISO) system is a special case of a MIMO system, our approach is applicable to multi-user MISO systems (we refer to it as MU-MISO system). In Figure 4, we provide performance

comparison results for MU-MISO systems and single user MISO systems. We can see that the performance of (3,[1,1,1]) configuration is similar to that of (1,1) configuration, the performance of (4,[1,1,1]) configuration is similar to that of (2,1) configuration, and the performance of (5,[1,1,1]) configuration is similar to that of (3,1) configuration. These results are consistent with the analysis in Section IV that increasing the number of transmit antennas of the multi-user system by one increases the number of transmit antennas of each single-user by one. A close observation to Figure 3 and Figure 4 reveals the flexibility of our method. Multiple data streams can be transmitted simultaneously for each user. In Figure 3, the (6,[2,2,2]) MU-MIMO system when L = 1 outperforms that when L = 2 . The (7,[2,2,2]) MUMIMO system outperforms the (6,[2,2,2]) MU-MIMO system when L = 1 . Moreover, the ML receiver provides better performance than the MMSE receiver when L = 2 . In Figure 4, we can see that the (4,[1,1,1]) MU-MISO system when L = 1 provides better performance than that when L = 2 , and the (4,[1,1,1]) MU-MISO system outperforms the (3,[1,1,1]) MU-MISO system when L = 1 . Therefore, we can trade off between the data rate and the performance. Moreover, we can increase the number of transmit antennas to improve the performance and also the structure of the receiver can assist improving the performance. VI. CONCLUSIONS In this paper, we have introduced a transmit preprocessing technique at the transmitter for the downlink of multi-user MIMO systems. It decomposes the multiuser MIMO downlink channel into parallel independent single-user MIMO downlink channels. Used together with this decomposition technique, all the previous proposed MIMO processing techniques, which are only suitable for the downlink of single user MIMO systems,

are applicable in the downlink of multi-user MIMO systems. Some key properties are figured out. Simulation results are also provided and these results reveal the potential of our technique in terms of performance and capacity. REFERECES [1] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas," Wireless Personal Commun., vol. 6, no. 3, pp. 311-335, Mar. 1998. [2] G. D. Golden, G. J. Foschini, R. A. Valenzuela, and P. W. Wolniansky, “Detection algorithm and initial laboratory results using V-BLAST space-time communication architecture,” IEE Electronics Letters, vol. 35, no. 1, pp. 14-16, Jan. 1999. [3] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criterion and code construction,” IEEE Transactions on Information Theory, vol. 44, no. 2, pp. 744-765, March 1998. [4] H. Sampath, and A. J. Paulraj, “Joint transmit and receive optimization for high data rate wireless communications using multiple antennas,” Thirtythird Asilomar Conference on Signals, Systems, and Computers, 1999, vol. 1, pp. 215-219. [5] K. K. Wong, R. D. Murch, and K. B. Letaief, “Optimizing time and space MIMO antenna system for frequency selective fading channels,” IEEE Journal on Selected Areas in Communications, vol. 19, pp. 1395-1407, July 2001. [6] K. K. Wong, R. D. Murch, R. S. Cheng, and K. B. Letaief, “Optimizing the spectral efficiency of multiuser MIMO smart antenna systems,” WCNC2000 IEEE, vol. 1, pp. 426-430. [7] R. L. Choi, K. B. Letaief, and R. D. Murch, “MIMO CDMA antenna systems,” ICC2000 IEEE, vol. 2, pp. 990 –994.

1

ˆ (1) Rx 1 b

N

1 b (1)

b

b

( 2)

(K )

T(1) T

M

( 2)

N1

2 O

M

1

T

M

M

Base Station (BS)

ˆ ( 2) Rx 2 b

N

N2

M (K )

MS 1 (User 1)

1 N

NK

MS 2 (User 2)

M ˆ (K ) Rx K b MS K (User K)

Figure 1. System configuration of a multi-user MIMO system

n (1) b (1)

A (1)

H (1) V (1)

n ( 2) b ( 2)

A

( 2)

H

( 2)

V

A

bˆ ( 2)

Rx 2

MS 2 (User 2)

n

(K )

bˆ ( K )

Rx K

H (K )V (K )

(K )

MS 1 (User 1)

( 2)

M b (K )

bˆ (1)

Rx 1

MS K (User K)

Figure 2. MU-MIMO Decomposition: decomposing a multi-user MIMO channel into parallel single-user MIMO channels 0

10

(2,2), L=1, SVD (6,[2,2,2]), L=1, SVD (2,2), L=2, MMSE receiver (6,[2,2,2]), L=2, MMSE receiver (2,2), L=2, ML receiver (6,[2,2,2]), L=2, ML receiver (3,2), L=1, SVD (7,[2,2,2]), L=1, SVD

−1

Average BER

10

−2

10

−3

10

−4

10

0

5

10 E /N per receive branch (in dB) b

15

0

Figure 3. Performance comparison between single-user MIMO systems and MU-MIMO systems 0

10

(1,1), L=1 (3,[1,1,1]), L=1 (2,1), L=1, SVD (4,[1,1,1]), L=1, SVD (3,1), L=1, SVD (5,[1,1,1]), L=1, SVD (2,1), L=2, ML receiver (4,[1,1,1]), L=2, ML receiver

−1

Average BER

10

−2

10

−3

10

−4

10

0

5

10 Eb/N0 per receive branch (in dB)

15

Figure 4. Performance comparison between single user MISO systems and MU-MISO systems