Distributed Precoding for Multicell MIMO Networks

Yingquan Zou1,2, Chunguo Li1 and Luxi Yang1, Member, IEEE 1School of Information Science and Engineering, Southeast University, Nanjing, JiangSu, China

2 School of Electronic and Information Engineering, Nanjing University of Information Engineering, Nanjing, JiangSu, China Email: zouyingquan0309@gmail.com, cgli, lxyang@seu.edu.cn

Abstract—Distributed precoder for multicell multi-input multi-output (MIMO) networks is designed in this paper, which is implemented in a distributed manner instead of the central manner. Considering the interference can be transformed into the useful contribution, the criterion of maximizing the signal to leakage plus noise ratio (SLNR) is first introduced for the distributed precoder design. Moreover, in order to further reduce the channel state information feedback overhead, another precoder in terms of matching filter (MF) is then proposed for the distributed precoder design in multicell MIMO systems. Computer simulations have shown that the proposed precoding schemes outperform the existing method in terms of the achievable rate and the cumulative distribution function of the whole multicell MIMO system.

Keywords—Multi-input multi-output (MIMO), multicell, cooperation, precoding.

I. INTRODUCTION Most of the existing investigations on the downlink of cellular system mainly focus on single cell systems, where the co-channel interference (CCI) from the other cells is usually treated as the background noise, leading to an interference-limited system. In order to solve the CCI problem, the frequency reuse and sub-carrier allocation methods are traditionally the main two techniques, where the spectral efficiency is quite low as a result of the lost spatial freedom. Fortunately, coordinated techniques for multi-cell multi-input multi-output (MIMO) communications have drawn amount of interest in recent years due to the fast improvement of processing capability and the backhaul capacity of base transceiver station (BTS). Investigations on multi-cell cooperative communications show that multi-cell coordination techniques can provide great multi-user diversity gain [1]-[3].

In multi-cell coordination systems, the interference can be transformed into useful signals by properly designing the coordinated precoder for multiple base stations with the full or partial channel state information (CSI) among different BTS. In [4], dirty paper coding (DPC) was first introduced to the multi-cell coordination system, where all of BTS and users are equipped with single antenna. It is shown that DPC is an optimal precoding scheme [5], but it is only a theoretical conclusion that cannot be applied in practical engineering. Some joint precoders are proposed for the coordinated multicell system subject to individual power constraints

instead of the total power constraint [6]. However, , there are many issues for the joint precoders in realistic cellular systems such as the computational complexity, the channel state information (CSI) feedback overhead, and the synchronization, leading to a strong motivation to develop coordinated precoder in a distributed fashion. In order to reduce the complexity and the CSI feedback overhead, a clustered BTS coordination strategy is proposed in [7][8], where the coordination among different clusters is limited. However, there are still a great mount of CSI feedback overhead among different clusters and among different base stations within the same cluster.

In this paper, a multi-cell downlink precoding scheme is first proposed, which is based on the criteria of the signal to leakage plus noise ratio (SLNR) maximization. The key feature of the proposed scheme lies that it avoids the data exchanges among different BTSs and determines the coordinated precoder in a distributed manner. In order to further decease the CSI feedback overhead, another multi-cell downlink precoding scheme based on the criteria of matching filter is presented, where neither the CSI nor the data is shared among BTSs. The rest of the paper is organized as follows: Section II describes the underlying system model. Section III proposes the precoding scheme based on SLNR, which is applicable to the multiple receiving antennas scenario. Section IV designes another precoding scheme based on the criteria of matching filter. Section V conducts the simulations of the two proposed schemes. Section VI concludes this paper.

II. SYSTEM MODEL Considering a cellular MIMO system composed of three

BTSs (or cells), where the antenna number of each BTS and each user is, respectively, Nt and Nr. Each BTS serves only one user. The frequency reuse factor of the system is one. Fig. 1 depicts the system model, where the solid and the dashed lines denote the useful signal and the interference, respectively.

The signal received at user k can be expressed as

, ( ) , ( )1,

K

k k b k k k b j j kj j k= ≠

= + +∑y H x H x n (1)

This work was supported by the National Basic Research Program of China under Grant 2007CB310603, the National Natural Science Foundation of China under Grant 60902012, the National Science and Technology Major Project of China under Grant 2009ZX03003-004, the Ph.D. Programs Foundation of Ministry of Education of China under Grant 20090092120013.

978-1-4244-7555-1/10/$26.00 ©2010 IEEE

Figure 1 Schematic of three cells MIMO network

k k k=x W s (2) where t kN L

k×∈W C is the precoding matrix, ks ( 1kL × ) denotes

the symbol transmitted to user k with Hk k =s s IE , nk is the

whit additive Gaussian noise (AWGN). The transmit power of the kth BTS is

( ) ( ) ( )H H H Hk k k k k k k k kTr Tr Tr P= = =x x W s s W W WE E (3)

where ( )Tr ⋅ and ( )H⋅ , respectively, denotes the trace and the conjugate transpose of the matrix, Pk is the transmit power for the kth BTS.

On the one hand, an upper bound of the underling system can be obtained if the CCI is neglected as

, ( ) , ( )2

0

log detH H

k b k k k k b kupkR

N

⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

H W W HI (4)

where N0 is the covariance of the AWGN noise. On the other hand, a lower bound of sum rate can be obtained if the CCI is treated as the noise as

, ( ) , ( )2

0 , ( ) , ( )1,

log detH H

k b k k k k b kdbk K

H Hk b k k k k b k

j j k

RN

= ≠

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟= +⎜ ⎟⎜ ⎟+⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠∑

H W W HI

H W W H (5)

With the underling system model, the goal of this paper is to determine the coordinated precoder Wk in a distributed manner, where no data exchange is needed among different BTSs.

III. DESIGN OF DISTRIBUTED PRECODER BASED ON SLNR CRITERION

In this section, a distributed precoder is proposed for coordinated BTSs, where each BTS determines its own precoder based on its local CSI without any other BTSs’ CSI. The existing criterion of the signal to interference plus noise ratio (SINR) can not be used in our system since all of the SINRs are related to each other, indicating that the SINR expressions cannot be decoupled into simple expressions. Thus, many issues such as CSI feedback overhead and the computational complexity are involved in the criterion of SINR for multicell systems. In order to avoid the high computational complexity and the CSI feedback overhead, the criterion of the signal to leakage plus noise ratio (SLNR) maximization proposed for multi-user MIMO (MU-MIMO) in [9] is introduced to the multicell MIMO networks for the

coordinated precoder design, where the precoder is determined in a distributed manner with quite low computational complexity.

A. Coordinated Precoder Design for Single-Antenna Users When each BTS has multiple antennas and each user has

only single antenna, there is only one data stream for each link, where the signal power received at user k can be expressed as

( )2

, kk b kh w , the interference power for user k is

( )2

,1,

K

uk b uu u k= ≠∑ h w , and the interference power resulted by user k

is ( )2

,1,

K

ku b ku u k= ≠∑ h w , where . denotes 2 norm, kw is the

precoding vector of the kth BTS. Note that since kW in (2) becomes a column vector here. Thus, the SINR and the SLNR for user k can be described as

( )2

, ( )1 2 22

, ( )1,

, ,..., k b k kk K K

k k b u uu u k

SINRσ

= ≠

=+ ∑

h ww w w

h w (6)

( )2

, ( )

22, ( )

1,

k b k kk k K

k u b k ku u k

SLNRσ

= ≠

=+ ∑

h ww

h w (7)

where the SINR of user k denotes ratio of received useful signal power to the sum of received interference power from other co-channels; while SLNR depicts ratio of received useful signal power to the total interference for other users in the co-channel and interference power. The core idea of SLNR is maximizing the useful signal power for each user and minimizing the interference to other users in the co-channel, in other words, maximizing the ratio of the useful signal power to the total interferences for other users in the co-channel and interference power.

It is seen from (7) that the SLNR for each user depends only on their own precoding vector. Therefore, the optimum solution to the coordinated precoder can be obtained distributedly based on maximizing SLNR, where the optimization problem can be established as

22, ( )

22, ( )

1,

arg max , s.t. k

k b k koptk k kK

k u b k ku u k

Pσ

= ≠

= =+ ∑

w

h ww w

h w (8)

(8) can be equivalently rewritten as

, ( )

, ( ) , ( )2

, ( )1,

arg maxk

k b u

H Hk k b k k b k kopt

k KH Hkk k b u k

u u kkPσ

= ≠

=⎛ ⎞

+⎜ ⎟⎝ ⎠

∑w

w h h ww

Iw h h w (9)

where , ( ) , ( )H

k k b k k b kA h h and , ( )

2

, ( )1,

k b k

KHk

k k b ku u kkP

σ= ≠

= + ∑IB h h . Thus,

the optimization problem can be rewritten as: 2arg max , s.t.

k

Hopt k k kk k kH

k k k

P= =w

w A ww ww B w

(10)

The above optimization problem (10) actually is a generalized Rayleigh ratio quotient problem, whose optimum solution is the generalized eigenvector corresponding to the largest eigenvalue. If kB is invertible, then optimal solution is obtained as

max 1( )optk k k k kP ξ −=w B A (11)

where max 1( )k k kξ −B A denotes the energy-normalized eigenvector corresponding to the maximum eigenvalue by conducting singular value decomposition (SVD) on 1

k k−B A .

B. Coordinated Precoder Design for Multi-Antenna Users When each BTS and each user have multiple antennas, each

user receives multiple data streams. Thus, the corresponding SINR and SLNR of user k can be expressed as follows,

( ) ( )( )

, ( )

, ( )

, ( )

1 22

, ( )1,

, ,..., k b k

k b u

H Hk k b k k

k K KH H

r k u k b u uu u k

TrSINR

N Trσ= ≠

=+ ∑

W H H WW W W

W H H W(12)

( ) ( )( )

, ( )

, ( )

, ( )

2, ( )

1,

k b k

u b k

H Hk k b k k

k k KH H

r k k u b k ku u k

TrSLNR

N Trσ= ≠

=+ ∑

W H H WW

W H H W (13)

where Wk is the precoding matrix of the kth BTS. It is seen from (13) that it is very difficult to design Wk. Fortunately, the criterion of SLNR maximization as in (12) is proposed for the coordinated precoder design. The mathematical problem is established as follows

( )arg max , s.t. k k

k

opt H kk k k k k L L

k

PSLNRL ×= =

WW W W W I (14)

where kL is the number of data streams for user k . (14) can be further expressed as

( )

( ), ( )

, ( )

, ( )

2

, ( )1,

argmax k b k

k

u b k

H Hk k b k kopt

k KH Hr k

k u b k ku u kk

Tr

NTrPσ

= ≠

=⎛ ⎞⎛ ⎞

+⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠∑

W

W H H WW

W I H H W(15)

where , ( ) , ( )H

k k b k k b kA H H and

( ), ( )

2

, ( )1,

k b u

KHr k

k k b uu u kk

NPσ

= ≠

= + ∑B I H H . The above optimization

problem is still hard to solve. Thus, a lower bound of SLNR is derived as

( )( ) 1,...,

mink

H Hk k k k k k

k HH l Lk k kk k k

TrSLNR

Tr == ≥

W A W w A ww B wW B W

(16)

Based on the lower bound of SLNR, a modified problem can be obtained as

1,...,arg max min , s.t.

k kkk

Hopt Hk k k kk k k L LHl L

k k k k

PL ×=

= =w

w A ww W W Iw B w

(17)

whose optimum solution can be expressed as

1( )opt kk l k k

k

PL

ξ −=w B A ， 1,..., kl L∈ (18)

where 1( )l k kξ −B A is the energy-normalized eigenvector corresponding to the largest eigenvalue of 1

k k−B A .

IV. COORDINATED PRECODER DESIGN BASED ON MATCHING FILTER CRITERION

Section III presents a coordinated precoding scheme based on maximizing SLNR criterion for the single-antenna user and multi-antenna user, respectively. Although the proposed schemes need no data exchanges among different BTSs, it still needs to share CSI among BTSs. Considering this phenomenon, a novel precoding matrix based on the matching filter (MF) is proposed in this section.

The coordinated precoder Wk can be composed of two items as shown below

k k k=W G Ω (19) where the diagonal matrix kΩ denotes the powers of all data streams for the kth BST, and Gk is the preprocessing matrix of the kth BTS. About the design of Gk, the existing method is the criterion of zero forcing (ZF) [8], namely,

( ) 1H Hk k k k

−=G H H H (20)

Another existing method is based on the criterion of the minimization of mean squared error (MMSE), namely,

1

0H Hk k k k

k

NP

−⎛ ⎞

= +⎜ ⎟⎝ ⎠

G H H H I (21)

where kP is the transmit power of the kth BTS. Now, we propose a novel criterion of matching filter (MF) for the design of Gk as

Hk k=G H (22)

After accomplishing the design of Gk, the second item kΩ in (19) can be obtained by the water-filling method. In fact, the design of kΩ can be achieved for two different SINR regimes consisted of the high and the low SINR scenarios. In the case of high SINR, the powers for multiple data streams can be allocated equally, namely,

kk

t

PN

=Ω I . (23)

In the case of low SINR, kΩ can be determined as

k k=Ω Λ , (24) where kΛ is a diagonal matrix, whose diagonal elements are the eigenvalues of kH . To this end, the coordinated preocder Wk has been designed.

V. SIMULATION RESULTS In this section, the performances of the proposed

coordination schemes are simulated by the Monte Carlo method, where the number of Monte Carlo cycles is 10000. Assume that there are three cells as shown in Fig. 1. The existing coordinated precoders in terms of MMSE and ZF [8] are also plotted for the comparison. Fig. 2 shows the sum rate versus the signal to noise ratio (SNR) realized by the proposed precoders and the existing

precoders [8], where the antenna number of each BTS and that of each user is 4 and 2. It is seen from Fig. 2 that the proposed MF coordinated precoder is the best one. And the proposed SLNR coordinated precoder also outperforms the existing precoders based on ZF and MMSE. Fig. 3 and Fig. 4 show the cumulative distribution function (CDF) versus the achievable rate realized by the proposed schemes and the existing schemes, where the antenna number of each BTS and that of each user in Fig. 3 is 2 and 1 yet the antenna configuration is correspondingly 4 and 2 in Fig. 4. From these two figures, it is seen that the proposed coordinated precoders based on SLNR and MF always outperform the existing schemes based on MMSE and ZF.

VI. CONCLUSION This paper investigates the coordinated preocder design for

the MU-MIMO multi-cell networks, where two novel precoders, respectively, based on SLNR and MF are proposed. Numerical simulations confirm that the two proposed schemes outperform the existing coordinated schemes in terms of achievable rate and cumulative distribution function. The key feature of the two proposed precoders is that no data exchange is needed among different cells, where these two precoders are applicable to any antenna configurations of both the BTSs and the users. Thus, the proposed precoders greatly reduce the CSI feedback overhead and the computational complexity.

REFERENCES

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[9] M. Sadek , A. Tarighat, and A. H. Sayed, “A leakage-based precoding scheme for downlink multi-user MIMO channels,” IEEE Trans. Wireless Commun., vol. 6, no. 5, pp. 1711-1721, 2007.

-10 -5 0 5 10 15 200.5

1

1.5

2

2.5

3

3.5

SNR (dB)

Sum

Rat

e(bp

s/H

z)

ZFMMSE

MFMSLNR with full CSI

Figure 2 Comparison of sum rate and SNR with Nt=4 and Nr=2.

0 0.5 1 1.5 2 2.5 3 3.5 40.4

0.5

0.6

0.7

0.8

0.9

1

Capacity(bps/Hz)

CD

FZFMMSE

MFMSLNR with full CSI

Figure 3 Cumulative distribution function of system sum rate with Nt=2 and Nr=1.

0 0.5 1 1.5 2 2.5 3 3.5 40.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Capacity(bps/Hz)

CD

F

ZFMMSEMFMSLNR with full CSI

Figure 4 Cumulative probability distribution of system sum rate with Nt=4

and Nr=2.