ZF-THP Combined with Receive Beamforming for Multi-user MIMO Downlinks

LI Xinmin1,2 1. College of Communication and Information engineering

Xi’an University of Science and Technology Xi’an, 710054, China yekailee@sina.com

BAI Baoming2 2．State Key Lab. of Integrated Service Networks

Xidian University Xi’an, 710071, China

Abstract—In this paper, a method of Tomlinson-Harashima precoding (THP) combined with receive beamforming for the multi-user MIMO broadcast channel is proposed. When the base station and user terminals all have multiple antennas, each user’s receiver maximizes the system sum rate with receive beamforming, and the base station use THP based on zero forcing criteria to cancel multiuser interferences (MUI). Simulation results show that the proposed method, THP-BF, has better BER performance than traditional THP and some linear precoding methods such as block diagonalization (BD) and zero forcing (ZF). The sum rate of ZF-THP at asymptotically high SNR is computed, and the computation results are consistent with simulation results. Simulation results reveal that ZF-THP has higher sum rate than linear precoding methods (such as ZF and BD) due to its more degrees of freedom of MIMO channels, and that THP-BF also has higher sum rate than traditional THP at low and moderate SNR.

Keywords-MIMO, receive beamforming, Tomlinson-Harashima precoding

I. INTRODUCTION

Multiple-input multiple-output (MIMO) wireless systems have recently received considerable attentions due to its potential high spectral efficiency, and will be one of the key techniques in future mobile communication systems. In the future mobile communication systems, both base station and mobile terminals have multiple antennas to transmit or receive radio signals. In order to cancel multi-user interferences (MUI) and achieve high system capacity, precoding techniques are uses the multi-antenna downlinks.

Precoder designs for the downlink can be divided into two categories: linear and nonlinear techniques. Linear precoding include block diagonalization (BD) [1], zero-forcing (ZF) [2] and orthogonal space division multiplexing (OSDM) [3]. These methods are easy to realize, but they achieve lower sum rates than dirty paper coding (DPC) [4]-[5] because they reduce the degrees of freedom of channels between the base station and users’ terminals. A regularized block diagonalization (RBD) technique is proposed [6], which significantly improves sum rate and diversity order, because it balances the MUI suppression achieved by reducing the overlap of the spaces spanned of different users and MIMO processing gain which requires users use as much as possible the available subspaces.

Nonlinear methods include Tomlinson-Harashima precoding (THP), nested linear/lattice codes [7] and vector perturbation (VP) [8]. THP was initially proposed for the

equalization of intersymbol interference and can be readily extended to MIMO downlink channels [9]. It can be designed with zero-forcing (ZF) criterion or minimum mean square error (MMSE) criterion [9][10], which is a suboptimal realization of DPC with lower complexity and some capacity loss compared with ideal DPC [11]. So several problems about THP are studied in recent years, such as ordering, scheduling methods and system performance analysis with limited feedback [12]-[14].

In this paper, a method of Tomlinson-Harashima precoding (THP) combined with a space multiplexing structure[15] for multi-user MIMO downlinks is proposed, where the base station eliminates the multi-user interferences (MUI) with THP, and each user combines the signals from its multi-antenna with receive beamforming respectively. Different from [15] where ideal ZF-DPC is used, our proposed method uses THP to cancel MUI, so it has lower complexity than ideal DPC does. Besides, unlike [4] [7] [9], our method can work for the case of multi-antenna user terminals. The analytical and simulation results show it has better performance and higher sum rate than traditional THP and linear ZF or BD method.

This paper is organized as follows. In Section II we describe the multi-user MIMO downlink channel and the system model that we proposed. In Section III we discuss the design criteria of our proposed system and its performance analysis is presented in Section IV. Simulation Results are given in Section V. Finally, we give a short conclusion in Section VI.

Notations: Boldface letters denote matrix-vector quantities. The operation ( )tr ⋅ and ( )H⋅ represents the trace and the Hermitian transpose of a matrix, respectively. The function det( )⋅ represents the determination of a matrix, and I is a identity matrix.

II. SYSTEM MODEL

Consider the downlink transmission from a base station with N antennas to K users, and each user equipping with M receive antennas ( N KM≥ ). Figure 1 shows the corresponding block diagram of the system, where U is the signal vector (size 1N × ) composed by all users’ signals and N-KM zeros filled. Each user’s signal ku (size 1M × ) is taken from a square M-QAM signaling constellation Φ on the odd-integer grid (E.g., for M=16, { } , , { 1, 3}I Q I Qa ja a aΦ = + ∈ ± ± ). B is the linear preequalization matrix, which is a upper

This work was partially supported by the NSFC under Grant U0635003 and France Telecom R&D Beijing Company

The 1st International Conference on Information Science and Engineering (ICISE2009)

978-0-7695-3887-7/09/$26.00 ©2009 IEEE 2779

triangular with size of KM N× . F is the linear precoding matrix (size N N× ). Hk represents the kth user’s MIMO channel (size M N× ), which has i.i.d. entries distributed according to CN(0,1). kp represents the kth user’s linear receiver (or receive beamformer) (size M M× ), while kn is the AWGN at the receiver k. All the MOD operations constrain input signal to lie within the boundary region of Φ, resulting in the output signals with small power.

We assume the BS knows the channel state information (CSI) of all users, so we design the precoding matrix F, preequalization matrix B and the receive beamformer kp jointly to achieve maximum sum rate and eliminate MUI.

III. THP COMBINING WITH RECEIVE BEAMFORMING All signals before modulo-operation can be computed as

1 2[ ... ]H H H H H HK= = +Z z z z GP HFU GP n (1)

where P is block diagonal matrix(size KM KM× ): 1 2( , , ..., )Kdiag=P p p p

and H is the channel matrix (size KM N× ),

1 2[ ... ]H H H HK=H H H H

Starting with QR-type decomposition of equivalent channel:

Heq = =H P H RQ (2)

where R is the upper triangular matrix with size KM N× , and its diagonal elements are

kkr (k=1,2,…KM). Q is a unitary

matrix with size N N× . According to THP, we define 1 1 1

11 22diag( , , , )KM KMr r r− − −×=G , the preequalization matrix

B=GR and the precoding matrix F=QH. By using ZF-THP, the multi-user interferences and the multi-antenna interferences can be cancelled, which means

1( )H − =GP H FB I So the multi-user MIMO downlink channel is decomposed

into KM parallel independent AWGN channels and each user has M independent data streams.

Neglecting the precoding loss and assuming each antenna with equal power, the signal-to-noise ratio of each channel is

2

k kkSNR r snr= ⋅

where kkr is the diagonal entries in R corresponding to the kth user, and snr is the ratio of the average power of user signal

2xσ to the covariance of Gaussian noise 2

nσ . Then the sum rate of all these parallel channels is computed as follows

2

21

log (1 )KM

kki

C r snr=

= + ⋅∑ (3)

For high SNR, the sum rate can be approximately 2

21

log ( )KM

kkk

C r snr=

≈ ⋅∑

2 2log log (det( )HK M snr= + R R

2 21

log log ( det( ))K

H H

k k k kk

KM snr=

= + ∏ p H H p (4)

Assume that each user only knows its own channel state information, and no collaboration between users, we mean that

1H

2H

KH

2n

Kn

1P

2P

KP ×

×

×1

g

2g

Kg

1n

U

1y

2y

Ky

1

^

U

2

^

U

KU^

1z

2z

Kz

Figure 1. Block diagram of MU-MIMO downlink

the preprocessing matrix P is unitary block diagonal and kp is solely determined by kH . To maximize the system sum rate,

kp must be the set of eigenvectors associated with the M maximum eigenvalues of the corresponding channel matrix

HkkHH .

Compared with traditional THP, in our proposed method (called THP-BF), receive beamforming design is based on the maximization of the sum rate while precoder design is based on zero forcing THP (ZF-THP). In fact if P=I, this method is the traditional THP. It is also important to mention that each user receiver just needs to know its own channel to perform the optimization, and the preprocessing matrix kg is diagonal elements of G, which can be transmitted to each user’s receiver with less broadcast data. Because THP is of very reasonable complexity, THP-BF can be very feasible.

IV. SUM RATE ANALYSIS FOR HIGH SNR

Our proposed THP-BF and traditional THP are both based on zero forcing THP (ZF-THP), which both use QR decomposition of the channel matrix, and the sum rate is determined by the diagonal elements of upper triangular matrix R (see Eq.3). Theorem 1: Suppose the channel matrix H has i.i.d. entries, and each entry of H has a normalized zero-mean complex Gaussian distribution. Then the sum rate of ZF-THP system is given as follows

2

2 221

n

σlog log ( 1)

σ

KM

u

THPk

C KM e N kψ=

= + ⋅ − +∑ (5)

where ( )ψ ⋅ is the Euler’s digamma function, which is given by

1

1

1 1( ) ( 1) (1)1

m

lm m

m lψ ψ ψ

−

== − + = +

− ∑ (6)

and (1) 0.577215ψ = − .

Proof: the complex Gaussian H=RQ is the QR-type decomposition where Q is a unitary and R=[ ijr ] is a lower

triangular matrix. The random variables 2kkr (k=1,2,…,KM)

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from R are statistically independent and 2kkr has central chi-

square distribution with 2(N-k+1) degrees of freedom, which is equivalent to a complex Wishart matrix with N-k+1 degrees of freedom, i.e. ),1(1 IW +− kN . Using the property of Wishart Matrix:

1[log det( ( 1, ))] ( 1)

eE W N k N kψ− + = − +I

so substituting the equation above into Eq.3, the sum rate of ZF-THP system is achieved as Eq.5. ■

In order to cancel MUI, linear precoding techniques choose the equivalent channel matrix of each user’s receiver orthogonal to other users’, so they reduce degrees of freedom of MIMO channel. When linear BD method is used, for high SNR the sum rate is determined by central Wishart matrix with N-(K-1)M degrees of freedom, which can be computed by [5]

2 1

2 220

n

σlog log ( ( 1) )σ

M

u

BDn

C KM K e N K M nψ−

=

= + ⋅ − − −∑ (7)

When linear ZF method is used, for high SNR the sum rate of ZF is computed by [5]

2

2 22

n

σlog log ( 1)

σu

ZFC KM e KM N KMψ= + ⋅ ⋅ − + (8)

It is noted that in practice, the sum rate of ZF-THP is smaller than Eq.5 [11], due to several losses such as shaping loss in ZF-THP. And our proposed THP-BF has the same sum rate as traditional ZF-THP for high SNR, because the equivalent channel

eqH also has a normalized zero-mean

complex Gaussian distribution.

V. SIMULATION RESULTS The sum rate differences between ZF-THP and BD or ZF are

shown in Fig.2, where K=3, M equals 1 or 2, and M are increased. It is apparent that ZF-THP has higher sum rate than ZF and BD, because ZF-THP remains more degrees of freedom of MIMO channel than such linear precoding techniques (BD or ZF).

Fig.3 shows the variation of the sum rate with SNR, where N=6, K=3, M=2, SNR= 2

n2 /σσ x

and simulate via 10000 complex Gaussian channel matrices. From Fig.3, for moderate and low SNR, our proposed THP-BF has more sum rate than traditional THP. When SNR gets much large, they have same sum rate, because they are both based on ZF-THP.

The comparison of theoretical computation and simulation results of the sum rates of THP-BF, ZF-THP, BD and ZF with N=6, K=3, M=2, SNR=30dB is shown in Table 1, where theoretical computation results is achieved from Eq.5, Eq.7 and Eq.8, and simulation results from Fig.3. We can see that theoretical computation results are consistent with simulation results, which verifies theoretical analysis and computation true.

Fig.4 shows the variation of the average bit error rate (BER) with SNR when N=6, K=3, M=2, SNR= 2 2

nσ / σ

u. The channel

model used in the simulation is a quasi-static flat Rayleigh fading channel and changes with each transmission. 16QAM modulation is assumed and Grey mapping is used. Fig.4 shows

THP-BF has better BER performance than traditional THP and some linear precoding techniques such as BD and ZF.

Table 1 comparison of theoretical computation and simulation results of the sum rates with N=6, K=3, M=2, SNR=30dB

Techniques ZF-THP/THP-BF BD ZF

theoretical results(bit/s/Hz) 67.31 59.16 53.22

simulation results(bit/s/Hz) 67.35 59.13 54.80

6 7 8 9 10 11 12 13 14 150

2

4

6

8

10

12

14

N

Cap

aciti

es D

iffer

ence

s(bi

t/s/

Hz)

ZF-THP vs. ZF(M=1)

ZF-THP vs. BD(M=1)ZF-THP vs. ZF(M=2)

ZF-THP vs. BD(M=2)

Figure 2. Capacity differences between ZF-THP and BD or ZF

0 5 10 15 20 25 300

10

20

30

40

50

60

70

SNR(dB)

Sum

Rat

e(bi

t/s/

Hz)

THP-BF

THP

BD

ZF

Figure 3. Comparison of the sum rates

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0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

SNR(dB)

Bit

Err

or R

ate

ZF

BD-MMSETHP

THP-BF

Figure 4. Average BER vs. SNR

VI. CONCLUSION AND FUTURE RESEARCH In this paper, when the base station and user terminals all

have multiple antennas, a method of Tomlinson-Harashima precoding combined with receive beamforming for the multi-user MIMO broadcast channel is proposed. Compared with traditional THP and some linear precoding techniques such as BD and ZF, our proposed THP-BF has better BER performance and higher sum rate. Due to some sum rate loss in THP[11], we will take into consider nested linear/lattice codes combined with receive beamforming to increase sum rate. In the future research, optimal ordering of all users in THP-BF is also necessary to study.

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