Multiuser MIMO Downlink Transmission Schemes over Time-Varying Channels

TSINGHUA SCIENCE AND TECHNOLOGY I S S N 1 0 0 7 - 02 1 4 0 5 / 1 8 pp 7 6 9 - 7 7 7 Volume 13, Number 6, December 2008

Multiuser MIMO Downlink Transmission Schemes over Time-Varying Channels* WANG Hongmei (ฆ‫܃‬ਜ), XU Xibin (༘๰μ), ZHAO Ming (ვ ੜ), ZHOU Shidong (ᄻಷՊ), YAO Yan (ྍ ཮)** Tsinghua National Laboratory for Information Science and Technology, Department of Electronic Engineering, Tsinghua University, Beijing 100084, China Abstract: The performance of multiuser multiple-input-multiple-output (MIMO) downlink systems with block diagonalization (BD) depends on the accuracy of the channel state information (CSI) available at the transmitter and the receiver. In time-varying channels, the CSI available at the transmitter (CSIT) is always outdated due to an inherent time delay between the uplink channel estimation and the downlink data transmission in time division duplexing (TDD) systems. This leads to a drastic degradation of system capacity. This paper first analyzes the effect of the outdated CSIT on multiuser MIMO downlink systems using the BD method and then proposes two linear processing methods, BD precoding with user selection and scheduling at the transmitter and total minimum mean squared error (MMSE) decoding at the receiver (TBDUSS-RTMMSE) and BD precoding at the transmitter with partial MMSE decoding at the receiver (TBD-RPMMSE), to mitigate the interference among data streams and users. Analysis and simulation results show that these methods can effectively reduce the impairment of the outdated CSIT to increase the system sum capacity in a suitable time delay region of the CSIT. Key words: multiuser multiple-input-multiple-output (MIMO) downlink system; time-varying wireless channel; outdated channel state information; minimum mean squared error (MMSE)

Introduction Multiuser multiple-input-multiple-output (MIMO) systems have gradually grasped more research interest[1], since with multiple transmitting antennas at the transmitter, no extra time slots or frequency resources are needed to support multiple users. Recent theoretic results[2-6] have indicated that the sum capacity of multiuser MIMO downlink channels can be achieved using Received: 2007-07-20; revised: 2007-12-16

* Supported by the National High-Tech Research and Development (863) Program of China (No. 2006AA01Z282) and the TsinghuaQualcomm Project

** To whom correspondence should be addressed. E-mail: [email protected]; Tel: 86-10-62781396

dirty-paper coding (DPC)[7] at the transmitter. However, DPC is difficult to implement in practical systems due to the high computational burden of the successive encoding and decoding. A reduced complexity scheme named zero-forcing DPC (ZFDPC) is proposed[2,8]. It uses QR decomposition combined with DPC at the transmitter to optimize the sum capacity and provides performance very close to the Sato bound[9], but its complexity is still very high. A low-complexity linear suboptimal strategy zero-forcing beamforming (ZFBF) scheme is proposed[10] and can serve multiple users each equipped with a single receiving antenna at a time like DPC. When users have multiple antennas, block diagonalization (BD)[11,12] is an efficient method to optimize the sum capacity. The previous work is based on the assumption that

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Tsinghua Science and Technology, December 2008, 13(6): 769-777

the channel state information (CSI) is perfectly known at the transmitter. However, in many situations, this condition is not reasonable, especially when the channel varies rapidly. In this situation, the CSI available at the transmitter (CSIT) is always outdated due to the inherent delay between the uplink estimation of the CSI and the downlink data transmission in time division duplexing (TDD) systems. This will lead to the precoding matrices not matching with the real propagation channel, resulting in interference among data streams and users and drastic degradation in system capacity. Other studies[13,14] have just investigated the effect of the outdated CSIT on the capacity of the multiuser MIMO system with the BD method, but did not propose a solution to combat its impairment. To mitigate the interference, a Hadamard method[15] is proposed for multiuser MISO systems, where users have a single antenna. For multiuser MIMO systems, a linear processing method[16] is proposed, but it has two drawbacks. The first one is that when the CSIT time delay is small, the equivalent channel matrix of other users cannot be estimated accurately at each user. The second one is that with the increase of the CSIT time delay, the interference that one user suffers from other users also increases, and this method[16] cannot effectively suppress interference. Then with the increase of the CSIT time delay, the system performance degrades rapidly. This paper analyzes the effect of the outdated CSIT on multiuser MIMO systems using BD, gives the upper bound of the interference that one user suffers from other users with respect to the CSIT time delay, and then proposes two linear processing methods to mitigate the interference. The first method selects and schedules users according to the CSIT time delay and uses the BD method to calculate the precoding matrices for all the selected users at the transmitter. Then the transmitter sends sets of orthonormal pilot signals through the precoding matrices so that each user can estimate the product of its channel matrix and all the precoding matrices. Each user calculates a total minimum mean square error (MMSE) decoding matrix that is used to mitigate the interference and improve the system performance. This method is called the TBDUSS-RTMMSE method and is suitable for relatively long delays. The second method uses the BD method to calculate the precoding matrices for all users at the transmitter, and the partial MMSE method to

calculate the decoding matrix at each user, which is called the TBD-RPMMSE and is suitable for small delays. Simulation results demonstrate that these two methods can effectively reduce the impairment of the outdated CSIT and increase the system capacity in the suitable CSIT time delay region.

1 Multiuser MIMO Downlink System with BD Consider a multiuser MIMO downlink system with M transmitting antennas at the transmitter and K users, each equipped with N k (k 1! K ) receiving

antennas, as shown in Fig. 1. The notation MU [ M  N1 ! N K ] is used to represent such a system.

Let bk represent the Lk u 1 transmitted data symbol vector for the k-th user, where Lk is the number of parallel data streams simultaneously transmitted to the user. This data symbol vector is first passed through the precoding matrix, which is characterized by an M u Lk nonzero matrix Tk . Then the precoded signals for all users are added together and launched into the MIMO channels by the M transmitting antennas.

Fig. 1 Multiuser MIMO downlink system model using the BD method

Assume that the channels between the various transmitting and receiving antennas are flat fading and independent of each other. Denote the MIMO channel for the k-th user as an N k u M matrix H k , where each element in the i-th row and j-th column, [ H k ]i  j , represents the channel gain from the j-th transmitting antenna of the transmitter to the i-th receiving antenna of the k-th user. Also assume a rich scattering environment so that all the entries of H k are independently identically distributed (i.i.d.) complex Gaussian random variables with zero mean and variance V h2 , and H k is full rank with probability one. Then the

WANG Hongmei (ฆ‫܃‬ਜ) et alġMultiuser MIMO Downlink Transmission Schemes over ...

received signals at the k-th user can be expressed as an N k u 1 vector, yk

where nk is an N k u 1 noise vector whose elements are i.i.d. zero mean white complex Gaussian random variables with variance V n2 . Interference is inevitable since all users occupy the same time and frequency resource. However, whenever the number of antennas meets the constraint M > K 1i z k

Ni  k 1! K } , interference can be thor-

oughly removed simply by appropriately choosing the precoding matrix Tk , assuming that the perfect CSI of all users is available at the transmitter and receiver. The BD scheme[11,12] was developed to mitigate the interference by designing the precoding matrix Tk for the k-th user lying within the nullspace of all the other Gk ( H1H , H 2H ,!, users’ channels. Denoting H kH1  H kH1 … H KH ) H , where the superscript

H

represents conjugate transpose. The k-th user is free from multiuser interference as long as Tk lies within the nullspace of Gk , i.e., GkTk

0 , where 0 stands

for a zero matrix. Then, the N k u 1 received signal vector of the k-th user becomes yk H k Tk bk  nk

(2)

which is equivalent to an interference-free single-user MIMO system. The conventional way to find one basis of the nullspace of Gk is through using the singular value decomposition (SVD) of Gk , Gk

Uk[Ȉ

0][Vk(1) Vk(0) ]H

(3)

where the column vectors of Vk(0) form an orthonormal basis of the nullspace of Gk . The precoding matrix can then be chosen as a linear combination of the basis vectors, (4) Tk Vk(0) Ak Pk where Pk is an Lk u Lk non-negative real diagonal matrix, whose diagonal elements represent the power allocated to the data streams of the k-th user, and Ak is the precoding matrix for the equivalent interference-free single-user MIMO system given in Eq. (2). Here, Tk (k 1! K ) satisfy the total power constraint

¦

K k 1

trace(Tk TkH )  PT , where PT is the total

power. The N k u 1 received signal vector of the k-th

H kVk(0) Ak Pk bk  nk

yk

(5)

Then, yk is passed through an Lk u N k decoding

(1)

i 1

max{¦ i

user becomes

K

H k ¦ Ti bi  nk

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matrix Rk , and the overall transmission relationship between the transmitter and the k-th user becomes yˆ k Rk H kVk(0) Ak Pk bk  nˆ k

(6)

[12]

where nˆ k Rk nk . The results have shown that the maximum capacity for a multiuser MIMO downlink system when using the BD method can be achieved if Ak and Rk are calculated using the SVD of H kVk(0) and the total transmitted power is allocated across all transmitted data streams for all users with the water-filling scheme[17]. The SVD of H kVk(0) is H V (0) Uˆ Ȉ AH (7) k

k

k

k

k

where Ȉ k is a diagonal matrix, Rk

Uˆ k . Then Eq.

(6) becomes yˆ k

where Ȉˆ k

Ȉˆ k bk  nˆ k

(8)

Ȉ k Pk , which is also a diagonal matrix.

The multiuser MIMO downlink channel can then be divided into

¦

K k 1

Lk parallel interference-free ei-

genmode subchannels.

2

Effect of Outdated CSIT

From Section 1, the maximum sum capacity of the multiuser MIMO downlink system using the BD method can be achieved if perfect CSI is known at the transmitter. However, in time-varying channels, the CSIT, which is used to calculate the precoding matrices, is always outdated due to the inherent time delay between the uplink estimation of the CSI and the downlink data transmission in TDD systems. For the k-th user (k 1! K ) , let H k t Td denote its outk

dated CSIT, where t represents the time of the downlink data transmission and Tdk is the time delay between the uplink channel estimation and the downlink data transmission for the k-th user. For simplicity, assume that the time delays of all users are the same and represented as Td . Then, the precoding matrix for the k-th user is Tk t Td , which lies within the nullspace of Gk t Td . Here, Gk t Td

( H1Ht Td  H 2Ht Td !

H kH1t Td  H kH1t Td … H KHt Td )H . The SVD of Gk t Td is Gk t Td

U k t Td [ Ȉ k t Td

H 0][Vk(1) Vk(0)  t  Td  t  Td ]

(9)

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Then,

Tk t Td

Vk(0) Pk t Td , where Pk t Td is t Td Ak t Td

H k t

U k H k t T  1 _ U k _2 ǻk

(11)

d

the power allocation matrix, Ak t Td is the precoding

where ǻk is also a complex Gaussian matrix and

matrix for the equivalent interference-free single-user MIMO system in Eq. (2) according to the outdated CSIT, and Ak t Td is equal to the right singular matrix

ǻk ~ CN(0 Nk uM  V h2 MI Nk uNk ) which is independent of

of the product H k t Td and Vk(0) t Td . Let H k t denote the channel matrix of the k-th user when data streams are transmitted to it. Then, the N k u 1 received signal vector of the k-th user becomes K

yk t

i 1

H k tTk t Td bk t  H k t

¦T

i 1i z k

b  nk t

i t Td i t

(10)

From the previous analysis, Tit Td (i 1 2! K ) generally do not match with the real time channels due to the fact that the SVD used in calculating the precoding matrices is a nonlinear function with nonunique outputs, and small variations of the channels can, therefore, result in major shifts of the precoding and produce large errors[18]. In this case, the transmitted data streams are not mapped correctly to the eigenmodes and are degraded by noise and interference between each other. The remainder of this section analyzes the interference that one user suffers from other users with respect to the CSIT time delay. First highlight two specific assumptions in the channel model: (1) Assume that the channels between different transmitting and receiving antennas are flat fading and independent. With the assumption of a rich scattering environment, the k-th user’s channel H k can be modeled as a complex Gaussian matrix whose entries are i.i.d zero-mean complex Gaussian variables with a common variance V h2 . That is H k ~ CN(0 Nk uM 

V MI N 2 h

k u Nk

) , where 0 Nk uM is an N k u M zero matrix,

and I Nk u Nk is an N k u N k identity matrix. (2) Assume that the channels are wide-sense stationary uncorrelated scattering (WSSUS) Rayleigh fading channels based on the Jakes model[19]. H k t and H k t Td are then correlated realizations of the channel

distribution. Then, the relationship between H k t Td and H k t can be written as Ref. [20],

relation of the i.i.d time-varying channel coefficients, defined as U k E[[ H k t ]i  j [ H k t Td ] i  j ]  V h2 . Here, the superscript * denotes the conjugate. Under the assumption of Jakes channel model[19], the channel coefficient U k J 0 (2ʌf dk Td ) , where J 0 (˜) is the zeroth-order Bessel function of the first kind and f dk is the maxi-

H k t ¦ Ti t Td bi t  nk t K

H k t Td and H k t , and U k is the common time-cor-

mum Doppler frequent shift of the k-th user determined by the carrier frequency as well as the mobile speed. Here, f d k represents the normalized time delay of the k-th user. Substituting Eq. (11) into Eq. (10), yk t

§

·

©

¹

H k tTk t Td bk t  ¨¨ U k H k t Td  1 _ U k _2 ǻk ¸¸ < K

¦

i 1i z k

Ti t Td bi t  nk t

H k tTk t Td bk t  1 _ U k _2 ǻk

K

¦T

i 1i z k

b  nk t

i t Td i t

H k tVk(0) Pk t Td bk t  1  _ U k _2 ǻk < t Td Ak t Td K

¦

i 1i z k

Vi (0) Pi t Td bi t  nk t t Td Ai t Td

(12)

The second term in Eq. (12) is the interference that the k-th user suffers from other users. Then the interference is bounded by the following theorem: Theorem 1 The upper bound of the interference the k-th user suffers from other users is PI - (1 _ U k _2 ) PT N kV h2 (13)

Proof The interference that the k-th user suffers from other users is PI

trace{E[( 1 _ U k _2 ǻk ( 1 _ U k _2 ǻk

Since

K

¦V

j 1 j z k

K

¦V

i 1i z k

(0) j t Td

Ai t Td (i 1! K )

(0) i t Td

Ai t Td Pi t Td bit ) <

A j t Td Pj t Td b j t )H ]}

is

an

unitary

(14)

matrix,

ǻkVi (0) and ǻkVi (0) t Td Ai t Td t Td have the same distribution[21]. Assume that all transmitted data symbols are independent and energy normalized. Then, ­ I  i j; (15) E[bi t b Hj t ] ® ¯ 0 i z j

WANG Hongmei (ฆ‫܃‬ਜ) et alġMultiuser MIMO Downlink Transmission Schemes over ...

The interference can then be written as K ­° ª§ · PI (1 _ U k _2 )trace ® E «¨ ǻk ¦ Vi (0) Pi t Td ¸ < t Td ¹ °¯ ¬© i 1izk H K § · º ½° (0) ǻ V P ¨ k ¦ i t Td i t Td ¸ » ¾ © i 1i zk ¹ ¼» ¿° (1 _ U k _2 )

K

¦ E{trace[( ǻ V k

i 1i z k

(0) i t Td

Pi t Td ) <

( ǻkVi (0) Pi t Td ) H ]} t  Td Recalling the fact that trace(YZ )

(16) trace( ZY ) , then

Eq. (16) becomes

PI

(1 _ U k _2 )

K

¦ E{trace[( ǻ V k

i 1i z k

(0) i t Td

K

H (0) (1 _ U k _2 ) ¦ E[trace( Pi t Td Vi(0)H Pi t Td )] t Td ǻk ǻkVi t Td i 1i z k K

H (0) (1 _ U k _2 ) ¦ trace{[ Pi t Td Vi (0)H Pi t Td ]} t Td E ( ǻk ǻk )Vi t Td i 1i z k

(17)

PIk i

N kV h2 I M u M into Eq. (17),

(0) (1 _ U k _2 ) N kV h2 trace[( Pi t Td Vi (0)H Pit Td )] t Td Vi t Td

Let Bi t Td

(0) (0) Vi (0)H t Td Vi t Td . Since Vi t Td

the nullspace of Gi t Td , then Bi t Td

PI

(1 _ U k _2 ) N kV h2

(18) is the basis of

I Li u Li .

K

¦ trace[ P

i 1 i z k

i  t  Td

]

(19)

Assuming that the total transmitted power is PT , K

¦ trace[ P

i 1 i z k

i  t  Td

] - PT

(20)

Thus, Eq. (13) is obtained. ƶ Theorem 1 indicates that when the normalized delay f dk Td (k 1! K ) is small, the common time-correlation Uk is large and the interference that the k-th user suffers from other users is small. With increasing f dk Td , Uk decreases, and then the interference that the k-th user suffers from other users increases, resulting in the decrease of the signal-interference-noise ratio (SINR) and the degradation of the system capacity.

3

estimation of the CSI and the downlink data transmission in TDD systems. However, each user can perfectly acquire its real time CSI. Then the impairment of the outdated CSIT on the multiuser MIMO downlink system using BD method can be mitigated at each user by processing the received signals. Based on the analysis in Section 2, this section proposes two linear processing schemes, TBDUSS-RTMMSE and TBDRPMMSE, to mitigate the impairment of the outdated CSIT.

3.1 TBDUSS-RTMMSE When the transmitter is transmitting data streams to users, orthonormal pilot signals are first sent through the precoding matrices Ti t Td (i 1! K ) generated

Pi t Td ) H <

( ǻkVi (0) Pi t Td )]} t Td

Substituting E[ ǻkH ǻk ]

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Proposed Transmission Schemes

In time-varying channels, the CSIT is inevitably outdated due to the inherent time delay between the uplink

by the transmitter from the outdated CSIT. If each user knows the pilot signals, each user can estimate the product of its channel matrix and the precoding matrices of all users. Then, H k t Tit Td for i 1! K can be estimated for the k-th user. H k tTi t Td  i 1! k  1 k  1,!, K can only be estimated accurately when the normalized delay f dk Td (k 1! K ) is relatively large

because when the normalized delay is relatively large, the common time-correlation, U k , is relatively small, and then the interference that the k-th user suffers from other users is relatively large. Since the interference is also the transmitted power of other users distributed onto the k-th user, when f dk Td is also relatively large, the transmitted power for other users received at the k-th user is relatively large. These matrices can be used to calculate a correction matrix based on the MMSE criteria to compensate the impairment of the outdated CSIT[16], but the method cannot effectively suppress the interference when the time delay is relatively long, so with the increase of the CSIT time delay, the system performance degrades rapidly. From Theorem 1, it can be seen that with the increase of f dk Td , the interference that the k-th user suffers from other users increases, resulting in the decrease of the SINRs of the data streams and the degradation of the system capacity. Reducing the number of users can reduce the proportion of the transmitted power allocated to other users and can effectively mitigate the interference and improve the SINRs of the

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774

data streams. Although reducing the number of users may also reduce the total number of data streams transmitted, when f dk Td is relatively large, reducing the number of users can effectively increase the SINRs of the data streams. There is a trade-off between the SINRs of the data streams and the total number of users. Accurate determination of the number of users according to f dk Td can effectively improve the system sum capacity. To ensure fairness among users, a round-robin (RR) scheme is used to select users. Assume that Ks users are selected, H k t Tit Td for i 1! K s can be estimated for the k-th user. Let

H ki t H k t Ti t Td . Rewrite the N k u 1 received signal vector of the k-th user as

yk t

Ks

H k t ¦ Ti t Td bi t  nk t

(21)

i 1

Then, a correction matrix C k t can be derived based on the following MMSE criteria to compensate the impairment of the outdated CSIT: ­min trace( E || C k t yk t  bk t ||22 ) ° Ck t (22) ® °¯st trace(|| C k t ||22 ) 1 Then, 1

§ Ks · C k t H ¨ ¦ H ki t H kiHt  V n2 I Nk u Nk ¸ (23) ©i1 ¹ The decoded data symbol vector for the k-th user is Ks § · (24) bˆ k t C k t ¨ H k t ¦ Tit Td bit  nk t ¸ i 1 © ¹ This method can effectively reduce the interference among data streams and users due to the outdated CSIT when the time delay is relatively long. Fewer users are selected with longer delays. H kk  t

3.2 TBD-RPMMSE When f dk Td is relatively small, U k is large and then the interference from other users is relatively small. Also, the transmitted power for other users received at the k-th user is relatively small. So at the k-th user, the product of H k t and Tit Td (i 1! k  1 k  1! K ) cannot be accurately estimated, so only H kk t H k t Tk t Td is estimated, which is an N k u Lk matrix. Then, this matrix is used to calculate a correction matrix C k t based on the MMSE criteria to reduce the

the impairment of the outdated CSIT, 1

(25) ( H kkH t H kk t  V n2 I ) H kkH t Then, the decoded data symbol vector at the k-th user becomes K § · bˆ k t C k t ¨ H k t ¦ Ti t Td bi t  nk t ¸ i 1 © ¹ K § · C k t H k tTk t Td bk t  C k t ¨ 1 _ U k _2 ǻk ¦ Ti t Td bit  nk t ¸ i 1i z k © ¹ (26) This method can effectively improve the system capacity when f d k Td is relatively small because the in-

C k t

terference from other users is relatively small and can be looked on as noise. This method can effectively reduce the interference between the data streams of the k-th user itself and can also overcome the first drawback of the method proposed by Zhang and Niu[16].

4 Simulation Results To demonstrate the effectiveness of our proposed methods, TBDUSS-RTMMSE and TBD-RPMMSE, we compare the sum capacities of the multiuser MIMO downlink system using these two methods with those of the multiuser MIMO downlink system using the BD method with perfect CSIT, the method proposed by Zhang and Niu[16], one stream scheme, and open loop system. These systems are named the TBDUSSRTMMSE system, the TBD-RPMMSE system, the perfect CSIT system, the TBD-RMMSE system, the one stream system, and the open loop system. The MU[6,2,2,2] system was used for the simulations. Assume that the channels between different transmitting and receiving antennas are independent and flat fading and that the elements of all the user’s MIMO channel matrices are i.i.d. complex Gaussian random variables with zero mean and unit variance. The Jakes fading model[19] was used to simulate the time correlation of the channel fading process. All input data symbols were treated as independent Gaussian signals. The total transmitted power was PT 1 . The power allocation scheme for the first five systems was the water-filling method and for the open loop system was the equal power allocation method.

4.1 System sum capacity for various fdTd Figure 2 shows the system sum capacity for various

WANG Hongmei (ฆ‫܃‬ਜ) et alġMultiuser MIMO Downlink Transmission Schemes over ...

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f dTd when SNR = 20 dB. For small f dTd , e.g., when

f dTd 004, the sum capacity of the TBDUSS-RTMMSE

f dTd  002 , the sum capacity of the proposed

system with 2 users is 1.26 bits/(s·Hz) higher than that of the TBD-RPMMSE system, 2.0 bits/(s·Hz) higher than that of the TBD-RMMSE system, 3.4 bits/(s·Hz) higher than that of the TBD-RMMSE system, and 10.5 bits/(s·Hz) higher than that of the open loop system. With the further increase of f dTd , e.g., when

TBD-RPMMSE system is a bit higher than that of the TBD-RMMSE system and much higher than those of the one stream system and open loop system. When f dTd is very small, the sum capacities of the proposed TBD-RPMMSE system and the TBD-RMMSE system converge towards that of the perfect CSI system. This is to be expected since when the channel variation is slower, the transmitter has more accurate CSI, and then the interference between the data streams and users is smaller. For the proposed TBDUSS-RTMMSE method with one user and one stream scheme, the system sum capacities are degraded because when f dTd is relatively small, the TBD-RPMMSE system and the TBD-RMMSE system can sustain relatively more data streams, while the TBDUSS-RTMMSE system and one stream system have limitations on the total number of transmitted data streams which limits the system sum capacity. The system capacity of the open loop system is the least.

f dTd ! 0.08 , the TBD-RMMSE system, the one stream

system, the TBD-RPMMSE system, and the TBDUSSRTMMSE system with 2 users all deteriorate due to the large CSIT delay. The TBDUSS-RTMMSE system with 1 user has a larger sum capacity than the other four systems because with the further increase of f dTd , reducing the number of users effectively increases the SINR of the data streams, and thus the system capacity. When f dTd is relatively large, the other four systems can sustain relatively less data streams. When f dTd 01, the sum capacity of the TBDUSS-RTMMSE system with 1 user is 4.7 bits/(s·Hz) higher than that of the TBD-RMMSE system, 2.7 bits/(s·Hz) higher than that of the one stream system, 5.6 bits/(s·Hz) higher than that of the TBD-RPMMSE system and 2.4 bits/(s·Hz) higher than that of the TBDUSS-RTMMSE system with 2 users. Further increasing f dTd , the sum capacities of the former four systems are close to and then lower than that of the open loop system, and the sum capacity of the TBDUSS-RTMMSE system when selecting 1 user is close to that of the open loop system. Figure 2 shows that the proposed method, TBDRPMMSE, is suitable for small f dTd and the proposed method, TBDUSS-RTMMSE, is suitable for relatively large f dTd . For the TBDUSS-RTMMSE

Fig. 2

System sum capacity for various fdTd with

SNR=20 dB, M = 6 , N k = 2 , K = 3

With the increase of

f dTd , e.g., when 002 

f dTd  008 , the sum capacities of the TBD-RMMSE

system and the TBD-RPMMSE system deteriorate rapidly due to the increased interference, e.g., when f dTd 004 , their sum capacities are 7.1 and 7.0 bits/(s·Hz) less than that of the perfect CSI system. Although the sum capacity of TBDUSS-RPMMSE system with 2 users deteriorates, it is higher than the TBD-RPMMSE system, the TBD-RMMSE system, one stream system, and open loop system, e.g., when

method, when f dTd is large, reducing the number of users can improve the system sum capacity. Thus, for their specified f dTd ranges, they both can effectively improve the system sum capacity.

4.2 System sum capacity for various SNR Figure 3 shows the sum capacity versus signal-noiseratio (SNR) when f dTd 002 . The system sum capacity of the proposed TBD-RPMMSE system is slightly higher than that of the TBD-RMMSE system, and much higher than those of the one stream system and the open loop system, e.g., at SNR=25 dB, the sum

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Tsinghua Science and Technology, December 2008, 13(6): 769-777

capacity of the TBD-RPMMSE system is 0.1 bits/(s·Hz) higher than that of the TBD-RMMSE system, 5.0 bits/(s·Hz) higher than that of the one stream system, and 14.6 bits/(s·Hz) higher than that of the open loop system.

Figure 5 shows the sum capacity for various SNR when f dTd 01 . The sum capacities of the TBDRMMSE and one stream scheme systems reach their ceilings for large SNR. The sum capacity of the proposed TBDUSS-RTMMSE system with 1 user increases nearly linearly as the SNR. For all SNR, the proposed method TBDUSS-RTMMSE with 1 user outperforms the TBD-RMMSE system and one stream system, e.g., at SNR=25 dB, the sum capacity of the TBDUSS-RTMMSE with 1 user is 7.6 bits/(s·Hz) higher than that of the TBD-RMMSE system and 4.6 bits/(s·Hz) higher than that of the one stream system. Over most of all the region of SNR, the sum capacity of the TBDUSS-RTMMSE with 1 user is higher than that of the open loop system.

Fig. 3 System sum capacity for various SNR with fdTd 0.02 , M = 6 , N k = 2 , K = 3

Figure 4 shows the sum capacity for various SNR when f dTd 004 . The system sum capacity of the proposed method, TBDUSS-RTMMSE with 2 users is higher than those of the TBD-RMMSE system, one stream system, and open loop system, e.g., at SNR= 25 dB. The sum capacity of the proposed TBDUSSRTMMSE system with 2 users is 1.7 bits/(s·Hz) higher than that of the TBD-RMMSE system, 3.1 bits/(s·Hz) higher than that of the one stream system, and 9.6 bits/(s·Hz) higher than that of the open loop system.

Fig. 5 System sum capacity for various SNR with fdTd 0.1 , M = 6 , N k = 2 , K = 3

5

Fig. 4 System sum capacity for various SNR with fdTd 0.04 , M = 6 , N k = 2 , K = 3

Conclusions

This paper investigated the impairment of the outdated CSIT on the multiuser MIMO downlink systems using the BD method. Two linear processing methods, TBDUSS-RTMMSE and TBD-RPMMSE, are proposed to reduce the impact of the outdated CSIT. Analysis and simulation results demonstrate that both methods effectively reduce the impairment and increase the system sum capacity for relatively large and relatively small time delays. Their system sum capacities are higher than those of the TBD-RMMSE system, the one stream system, and the open loop system in the reasonable CSIT time delays.

WANG Hongmei (ฆ‫܃‬ਜ) et alġMultiuser MIMO Downlink Transmission Schemes over ...

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