Simplified transmitter design for MIMO systems with channel uncertainty

The Journal of China Universities of Posts and Telecommunications April 2009, 16(2): 20–23 www.sciencedirect.com/science/journal/10058885

www.buptjournal.cn/xben

Simplified transmitter design for MIMO systems with channel uncertainty DU Juan1, 2 ( ), KANG Gui-xia1, ZHANG Ping1 1. Key Laboratory of Universal Wireless Communications, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing 100876, China 2. School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, China

Abstract

This article investigates transmitter design in Rayleigh fading multiple input multiple output (MIMO) channels with spatial correlation when there are channel uncertainties caused by a combined effect of channel estimation error and limited feedback. To overcome the high computational complexity of the optimal transmit power allocation, a simple and suboptimal allocation is proposed by exploiting the transmission constraint and differentiating a bound based on Jensen inequality on the channel capacity. The simulation results show that the mutual information corresponding to the proposed power allocation closely approaches the channel capacity corresponding to the optimal one and meanwhile the computational complexity is greatly reduced. Keywords MIMO, capacity, channel state information (CSI), channel estimation error, covariance feedback, mutual information, power allocation

1

Introduction 

MIMO techniques are expected to be widely used in future wireless communications to meet the increasing demand for capacity. By exploiting the CSI, the transmitted data can be adapted to the spatial characteristics of the channel, which results in significant capacity gain over conventional single-antenna systems [1–2]. The transmitter can obtain the CSI in two ways, based on the uplink-downlink reciprocity by using feedback link. However, there are errors in the channel estimation, which can arise from the estimation variance because of noise and the quantization error in the feedback channel. In addition, the feedback of perfect (instantaneous) CSI from the receiver to the transmitter appears to be impractical since the instantaneous CSI becomes outdated at the transmitter and the overhead associated with the instantaneous feedback can be excessive. Thus, the CSI available at the transmitter is generally imperfect in practical systems. If imperfect CSI is not taken into account in the transmitter design, the system performance will be degraded. The design of algorithms exploiting imperfect CSI is still an open problem. Recent articles have studied this topic in Received date: 12-07-2008 Corresponding author: DU Juan, E-mail: [email protected] DOI: 10.1016/S1005-8885(08)60195-8

terms of maximizing the ergodic capacity [3–9]. The capacity and the optimal transmission scheme for multiple input single output (MISO) system with channel statistics (channel mean or covariance, which changes more slowly than the instantaneous CSI in a fading channel environment) feedback were analyzed in Ref. [3]. In Refs. [4–6], the results were extended to MIMO systems with correlated antennas under covariance feedback. The effect of channel estimation error on transmitter design was studied in Ref. [7], and channel estimation error and channel covariance feedback were simultaneously considered in Ref. [8]. An adaptive MIMO transmission approach in Ref. [9] was designed for spatially correlated channels to offer substantial capacity improvements. Transmitter design is characterized by its input covariance matrix, including transmitting independent symbols along the corresponding directions specified by the eigenvectors of the input covariance matrix, with the corresponding eigenvalues specifying the powers allocated in each direction. With channel covariance feedback, the optimal input covariance matrix that maximizes the ergodic capacity has the same eigenvectors as the transmit correlation matrix [4–5], whereas the corresponding eigenvalues have no closed-form solutions so that it is not easy to solve them. In general, an exhaustive search over the whole valid domain [3,5] or numerical optimization [6] is needed to find the optimal power

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DU Juan, et al. / Simplified transmitter design for MIMO systems with channel uncertainty

allocation, which costs time and computational resource, especially the optimization is nonlinear in Ref. [8]. Hence, with channel covariance feedback and channel estimation error, it is necessary to find an approximation of the optimal power allocation. Based on the Jensen inequality and the transmit power constraint, this article discusses a simple, suboptimal power allocation.

2 Modeling and overview

21

expectation operator. Then, the ergodic capacity is given by Refs. [7–8]. C max I  Q Q :tr  Q İP

ˆ ˆ ª º HQH max H « log 2 I  2 » 2 Q:tr  Q İP V n  V E tr  Rt Q »¼ «¬ ˆ ˆ ª º HQH max H « log 2 I  » 2 Q:tr  Q İP 1  V E tr  Rt Q ¼» ¬«

(6)

where I  Q is the mutual information with Q , and H is

We consider a wireless communications system using t transmitting antennas and r receiving antennas. The channel is assumed to be flat with a linear model in which the received r u 1 vector y depends on the transmitted t u 1 vector x

the expectation operation over channel samples. Let the eigendecomposition of Q be:

according to: y Hx  n

where U Q is a t u t unitary matrix, and ȁQ is a t u t

(1)

Q U Q ȁQU Q

(7)

where the entries of r u 1 noise vector n are independently distributed Gaussian noise samples with variance V n2 1 , and

diagonal matrix. Yoo [8] has proved that the optimal Q has the same eigenvectors as Rt , which means U Q U t . Then,

H is the r u t channel matrix modeled as in Ref. [10]. H H w Rt1/ 2 (2)

Eq. (7) can be rewritten as: Q U t ȁQU t

where H w is a r u t matrix with zero-mean unit-variance of independent identically distributed (i.i.d.) complex Gaussian entries. Here, Rt1/ 2 is the unique square-root of Rt , such that

Rt1/ 2 Rt1/ 2

Rt ,

represents the t u t

Rt

transmit correlation matrix, which is complex Hermitian positive semidefinite and can be captured at the receiver by estimating long-term channel statistics, thus we can regard it as a constant. Furthermore, Rt can be eigen-decomposed as:

Rt

U t ȁU t t

where (<)

(3) denotes the conjugate transpose, U t is a t u t

unitary matrix, and ȁt is a t u t diagonal matrix. The minimum mean square error (MMSE) estimation of H w is Hˆ w , which is performed at the receiver, and the MMSE

estimation error matrix E w is given by E w

H w  Hˆ w ,

where Hˆ w and E w are uncorrelated and their entries are i.i.d. with zero-mean and variances 1  V E2

and V E2 ,

respectively. Thus, the channel matrix can be written as: H Hˆ  E w Rt1/ 2

(4)

where Hˆ Hˆ w Rt1/ 2

(5)

We define the total channel estimation error matrix E H  Hˆ , which is given by E E w Rt1/ 2 . We assume that the power constraint at the transmitter is tr(Q )İP , where tr(<) denotes trace operation, Q is the input covariance matrix

Q H ( xx ) , and

H

is the

(8)

Substituting Eqs. (3) and (8) into Eq. (6), we obtain that: C max I  ȁQ 

ȁQ :tr ȁQ İP

ª º ˆ ȁ U Hˆ HU t Q t » max H «log 2 I 

2 ȁQ :tr  ȁQ İP « 1  V E tr U t ȁU t t U t ȁQU t » ¬ ¼ ª ˆ ȁ U Hˆ º HU t Q t » max H «log 2 I  ȁQ :tr  ȁQ İP « 1  V E2 tr  ȁt ȁQ » (9) ¬ ¼ From the above, we know that there is no closed-form of ȁQ . In addition, Eq. (9) is not concave in ȁQ , which makes the optimization more difficult, and thus obtaining the optimal solution ȁQ opt usually requires considerable time and computational resource. If there is no CSI at the transmitter, the capacity is maximized when we adopt equal power allocation on each transmitting antenna [2]: P Qequ I (10) t with eigenvalue matrix ȁQ equ ( P t ) I . The corresponding mutual information is given by: ª º 1 ˆ ˆ » HQ I  Qequ H « log 2 I  equ H 2 1  V E tr  Rt Qequ « » ¬ ¼

(11)

3 Simplified transmitter design This section discusses a simple suboptimal ȁQ . From Eq. (3), by substituting Eq. (5) into Eq. (9), we obtain:

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The Journal of China Universities of Posts and Telecommunications

C

max 

ȁQ :tr ȁQ İP

I  ȁQ

ª Hˆ R1/ 2U ȁ U R1/ 2 Hˆ w* º » max H «log 2 I  w t 2t Q t t ȁQ :tr  ȁQ İP « 1  V E tr  ȁt ȁQ » ¬ ¼

1/ 2 ª ˆ H U ȁ U t U t ȁQU t U t ȁt1/ 2U t Hˆ w* º » max H «log 2 I  w t t ȁQ :tr  ȁQ İP « 1  V E2 tr  ȁt ȁQ » ¬ ¼ * ª Hˆ w ȁt ȁQ Hˆ w º » max H «log 2 I  ȁQ :tr  ȁQ İP « 1  V E2 tr  ȁt ȁQ » ¬ ¼ (12) Assume that ȁt ȁQ D and tr  ȁt ȁQ q , then the Jensen

bounded on I  ȁQ in Eq. (12) is: ª

H «log 2 I  ¬«

log 2 I 

Hˆ w DHˆ w 1  V E2 q r 1  V

2 E

1V q 2 E



D

t

¦ log i 1

2

1



r 1  V

2 E

1V q 2 E

From ȁQ

(19)

ȁt1 D and Eq. (19), the suboptimal ȁQ is thus

given by: ȁQ sub

P ȁt1 tr  ȁt1

(20)

Then, the suboptimal input covariance Q is: Qsub U t ȁQ sub U t

(21)

With Qsub , we get the corresponding mutual information: ˆ ˆ ª º HQ sub H I  Qsub H «log 2 I  » 2 1  V E tr  Rt Qsub ¼» ¬«

(22)

which satisfies I  Qsub İC , where the ergodic capacity C

H Hˆ Hˆ w º D » İlog 2 I  1  V E2 q ¼»

w

d

corresponds to the optimal power allocation.

4 i

(13)

where di is the (i,i)th element of the diagonal matrix D . Under the constraint tr( D) q , for each fixed q, by using the Lagrange multiplier method, it is easy to find the optimal di , which maximizes the right hand of Eq. (13): q (14) ; i 1,2,..., t t Substituting Eq. (14) into the transmitter power constraint tr  ȁt1 D İP , then we have (q t ) tr  ȁt1 İP , which means:

di  q

qİ

P ; i 1,2,..., t tr  ȁt1

di

2009

tP tr  ȁt1

(15)

Numerical results

In this section, the performance of the above suboptimal allocation is illustrated for a 4 u 4 MIMO channel for V E2 0.05 and the correlation factor D 0.9 in the exponential model Rt(i , j )

D i  j ; i, j 1, 2, 3, 4; D İ1 . This

model is widely used for its simplicity. However, the results in this article still hold for general Rt as long as it is symmetric and full-rank. The optimal allocation and the equal power allocation are also included for comparison. Fig. 1 shows the mutual information corresponding to the optimal, suboptimal, and equal power allocation, with the normalized eigenvalues ȁQ opt , ȁQ sub , and ȁQ equ depicted in Fig. 2.

On the other hand, substituting Eq. (14) into the right hand of Eq. (13), we can obtain: t r 1  V E2 q  ˜ log 1 ¦ 2 (16) 1  V E2 q t i 1 We differentiate Eq. (16) with respect to q: r 1  V E2 d 1 t dq

ln 2

¦ 1V i 1



2 E

q ª¬t 1  V E2 q  rq 1  V E2 º¼

(17)

which suggests that Eq. (16) increases when more q is assigned. Thus, the equality holds in Eq. (15): tP q (18) tr  ȁt1 Substituting Eq. (18) into Eq. (14), the diagonal element of D is:

Fig. 1 Mutual information for a 4 u 4 MIMO channel with different power allocation for V E2 0.05 and D 0.9

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DU Juan, et al. / Simplified transmitter design for MIMO systems with channel uncertainty

23

then a one-dimensional search is needed. The Newton method requires calculating the gradient, the Hessian matrix and function value at each optimization step, and these values are approximated by taking the sample mean over a lot of channel samples (we use 106 samples in this article) due to the expectation operator in Eq. (9). However, ȁQ sub can be obtained by Eq. (20) directly without the Newton method and the one-dimensional search. Hence, the computational complexity, resource and time of ȁQ sub are greatly reduced compared with that of ȁQ opt . Fig. 2 Different power allocation for 4 u 4 MIMO channel for V E2 0.05 and D 0.9

The mutual information of suboptimal power allocation I  Qsub is a tight bound to the ergodic capacity C at all signal to noise ratio (SNR). Any minor difference between them occurs only at high SNR when the effect of channel estimation error dominates that of the Gaussian noise ( tr  ȁt ȁQ !! V n2 1 ), and the mutual information converges to I  Qsub

P of

H ªlog 2 I  1 V E2 Hˆ w Hˆ w º , which equals to

¬ ¼ that of Rayleigh fading MIMO channels with no spatial correlation at the transmitter (no covariance feedback), and thus small difference in power allocation occurs. When there are only covariance feedback are considered, the ergodic capacity converges to the mutual information with equal power allocation at high SNR in Refs. [6,8]. When channel covariance feedback and channel estimation error are simultaneously considered here, the ergodic capacity is greatly larger than the mutual information with equal power allocation, and the gap between them increases with the increase of the SNR. Fig. 2 depicts the four normalized eigenvalues of each power allocation for a 4 u 4 MIMO channel. At high SNR, ȁQ sub is more closely approaching ȁQ opt than at other levels

of SNR, which is different as the corresponding mutual information shown in Fig. 1. Hence, the channel capacity that is not concave in the eigenvalues of input covariance matrix is further verified. The power allocation ȁQ equ is greatly different from ȁQ opt and thus has a poor performance. In the following, we compare the computational complexity of C and I  Qsub , which is determined by ȁQ opt and ȁQ sub . Eq. (9) demonstrates that C is not concave in ȁQ due to the presence of tr  ȁt ȁQ in the dominator;

furthermore, ȁQ has no closed-form solution. To obtain the optimal solution, the Newton method is first adopted, and

5

Conclusions

In this article, we have investigated the more practical case with spatially correlated Rayleigh fading with channel estimation error at the receiver and channel covariance feedback for MIMO systems. The optimal transmitter design is studied. It is shown that the optimal power allocation is not easy to be obtained because of the stochastic nature of the channel and generally much time and computational resource are required. A simple and suboptimal power allocation is derived, which is based on the Jensen inequality and the transmission power constraint. The results are shown by numerical simulations. The performance and complexity of the proposed power allocation are compared with the optimal one, and the efficiency of the proposed power allocation is clearly demonstrated. Acknowledgements

This work was supported by the National Natural Science Foundation of China (60502038), the Hi-Tech Research and Development Program of China (2006AA01Z272, 2006AA01Z283), Beijing New Star Program of Science and Technology, China (2007A046), and 111 Project of Ministry of Education (MOE) of China (B07005).

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Acknowledgements This work was supported by the National Natural Science Foundation of China (60432040, 60772021), the Research Fund for the Doctoral Program of Higher Education (20060013008), University IT Research Center Project (INHA UWB-ITRC), Korea, and the Project iCHIP financed by Italian Ministry of Foreign Affairs, the Innovation Fund for Graduate Student of Bupt, China.

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