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Int. J. Electron. Commun. (AEÜ) 63 (2009) 502–505

www.elsevier.de/aeue

LETTER

JointtransceivervectorprecodingbasedonGMDmethodfor

MIMOsystems

Feng Liu∗, Lingge Jiang, Chen He

Department of Electronic Engineering, Shanghai Jiao Tong University, Mailbox 1164, Dongchuan Road 800#, Shanghai 200240, China

Received 2 February 2007; accepted 18 February 2008

Abstract

We heuristically propose a joint transceiver vector precoding (VP) based on geometric mean decomposition (GMD) method

for multiple input multiple output (MIMO) systems. In fact, it is the generalization and improvement of the GMD transceiver

with Tomlinson-Harashima precoding (THP). We first propose the GMD-VP design with zero forcing (ZF) criterion, then

exploit the extended channel matrix to obtain the improved regularized scheme. Simulation results show that the proposed

schemes obtain performance improvement and complexity reduction.

? 2008 Elsevier GmbH. All rights reserved.

Keywords: Vector precoding; GMD method; MIMO systems

1. Introduction

Vector precoding (VP) is a transmitter side technique that

generalizes the Tomlinson-Harashima precoding (THP). In

[1], the authors propose a zero forcing (ZF) VP (ZF-VP) and

an improved regularized VP (Reg.-VP) for multiple input

multiple output (MIMO) systems. In [2], the authors pro-

pose a minimum mean square error (MMSE) VP (MMSE-

VP) that outperforms the ZF-VP and Reg.-VP. The sphere

encoder is utilized to find the optimal perturbation vector.

Motivated by the excellent property of the recently pro-

posed geometric mean decomposition (GMD) [3] and its

ZF transceiver design with THP [4], we heuristically con-

sider the joint transceiver VP design based on the GMD

method, which is the generalization of the GMD-THP. We

propose two schemes: the ZF-GMD-VP and the improved

Reg.-GMD-VP. Although setting the linear receive filter to

the unitary matrix obtained by the GMD of the channel is not

optimal, simulation results show that this joint transceiver

∗Corresponding author.

E-mail address: fengliu@sjtu.edu.cn (F. Liu).

1434-8411/$-see front matter ? 2008 Elsevier GmbH. All rights reserved.

doi:10.1016/j.aeue.2008.02.013

design can achieve performance improvement and complex-

ity reduction.

2. System model

We consider a MIMO system with MTtransmit and MR

(?MT) receive antennas. There is a total power constraint

PTfor the transmitted symbols. Perfect channel state infor-

mation is assumed at both transmitter and receiver. Since

we only have the real-valued sphere encoding algorithm on

hand, we use an equivalent real-valued signal model [5] ex-

pressed as

y = Hx + n,

where y is the 2MR×1 received symbols, x is the 2MT×1

transmitted symbols, n is the 2MR×1 independent identical

distributed (IID) real Gaussian noise with zero mean and

variance ?2

variance real-valued Gaussian channel.

VP schemes generate transmitted symbols x as

(1)

n, and H is the 2MR× 2MTIID zero-mean unit-

x = ?Fd = ?F(s + p),(2)

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F. Liu et al. / Int. J. Electron. Commun. (AEÜ) 63 (2009) 502–505503

where F is the 2MT× 2MRprecoding matrix, s is the data

vector with variance ?2

from the lattice ?Z2MRwith positive scalar ?, d is the mix

of the data and perturbation symbols, and ? is a power con-

straint factor satisfying ?x?2= PT. At the receiver, p is re-

moved by the component-wise symmetric modulo operation

which is defined by

s, p is the perturbation vector chosen

mods?(b) = b − ??(b + ?/2)/??.

Themodulation

model is the A point amplitude shift keying (ASK)

{±1/2,±3/2,...,±(A−1)/2} corresponding to the square

M(=A2) point quadrature amplitude modulation (QAM)

constellation for the complex-valued model. For this con-

stellation the variance of data symbols is ?2

and the scalar ? is set to be ? = A.

constellationforthereal-valued

s= (M − 1)/12

3. Proposed schemes

3.1. ZF-GMD-VP

The channel matrix can be decomposed by the GMD

method [3] as

H = QRPT,

where Q is a 2MR×2MRunitary matrix, P is a 2MT×2MT

unitary matrix, R is a 2MR× 2MT upper triangular ma-

trix with diagonal elements equal to the geometric mean of

the positive singular values of H. Based on (3), our joint

transceiver partially equalizes the channel by QTat the re-

ceiver, which obtains

ˆd =1

where?isusedforautomaticgaincontrol.Atthetransmitter,

we see the equivalent channel as

˜H = QTH = RPT.

The precoding matrix of ZF approach is chosen to be the

pseudo-inverse of˜H:

FZF=˜H†= PR†.

Thus, the data symbol can be easily detected by the

symmetric modulo operation to eliminate the perturbation

vector as

ˆ s = mods?(ˆd)

= mods?

?1

Here we use the fact that mods?(s) = s and mods?(p) = 0.

We can see that the modulo operation reduces the effect

of noise and the factor ? controls the noise power. So we

maximize ? to achieve maximal received signal to noise

(3)

?QTy, (4)

(5)

(6)

?

s + p +1

?QTn

?

?

= s + mods?

?QTn

. (7)

ratio (SNR). Since x = ?PR†(s + p), by the transmit power

constraint we have

?

? =

PT

?PR†(s + p)?2=

Therefore, maximizing ? is equivalent to minimizing

?R†(s + p)?2. Now the optimal perturbation vector of

ZF-GMD-VP can be found by the sphere encoder

?

PT

?R†(s + p)?2.(8)

popt

ZF= arg min

p∈?Z2MR

?R†(s + p)?2.(9)

3.2. Reg.-GMD-VP

In [1], the regularized approach of the VP technique is

also provided. However, we find the original Reg.-VP is out-

performed by the ZF-VP. As shown by Hochwald et al. [1],

the optimal factor ? can be found with the aide of simula-

tion to improve the performance by providing some trade-

off between residual interference and noise amplification.

From literatures (such as [6]) we know that the regularized

approach, which is also the MMSE approach for some sce-

narios, can be simply expressed with the corresponding ZF

approach based on the extended channel matrix (here de-

fined as He= [H ?n/?sI]). Therefore, the Reg.-GMD-VP

can be easily developed as the ZF-GMD-VP. Due to space

limitation, we only give a simple deduction. Now, the GMD

method is applied to obtain

He= QeRePT

where Qeis a 2MR× 2MRunitary matrix, Peis a 2(MT+

MR) × 2(MT+ MR) unitary matrix, and Re is a 2MR×

2(MT+ MR) matrix whose 2MTleft columns construct an

upper triangular matrix Re1and 2MRright columns are all

zero vectors. Thus we rewrite Reas

e,(10)

Re= [Re1

and rewrite Peas

?Pe11

where Pe11has a dimension of 2MT× 2MR.

We use QT

eat the receiver as the ZF approach for partial

equalization. The pseudo-inverse of the equivalent extended

channel matrix is

0]

Pe=

Pe12

Pe22

Pe21

?

,

˜H†

e= (QT

=

eHe)†= (RePT

?Pe11

?Pe11

?Pe11R†

e)†= PeR†

[Re1

??R†

e

Pe12

Pe22

Pe12

Pe22

?

Pe21

?

0]†

=

Pe21

e1

0

?

=

e1

Pe21R†

e1

.(11)

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504F. Liu et al. / Int. J. Electron. Commun. (AEÜ) 63 (2009) 502–505

The upper part of (11) is corresponding to the regularized

processing of the original channel matrix. Thus we obtain

the precoding matrix of Reg.-GMD-VP as

FReg.= Pe11R†

and the perturbation vector is obtained by

e1

(12)

popt

Reg.= arg min

p∈?Z2MR

= arg min

p∈?Z2MR

= arg min

p∈?Z2MR

?PeR†

e(s + p)?2

?R†

e(s + p)?2

?R†

e1(s + p)?2. (13)

We remark that MR?MT is not necessary for the Reg.-

GMD-VP.

4. Simulation results

4.1. Performance comparison

We measure the performance by the uncoded bit error

rate (BER). The simulation results employing 16 QAM for

MT=MR=4 and MT=MR=10 MIMO systems are shown

in Figs. 1 and 2, respectively. Note, the regularized VP with

optimal factor is labeled as Reg.-VP (optimal).

We can see the Reg.-GMD-VP performs much better than

the ZF-GMD-VP as the Reg.-VP (optimal) outperforms

the ZF-VP. The ZF-GMD-VP and the Reg.-GMD-VP out-

perform the corresponding ZF and regularized GMD-THP,

respectively. The Reg.-GMD-VP even shows very close per-

formance to the enhanced UCD-THP [7]. For (4,4) MIMO

systems, Fig. 1 shows that compared with the ZF-VP and

Reg.-VP, the proposed schemes achieve improvement gain

in high SNR region. Interestingly, the Reg.-GMD-VP even

810 12 14

SNR (dB)

16 182022

10−7

10−6

10−5

10−4

10−3

10−2

10−1

BER

ZF−GMD−THP

Reg.−GMD−THP

UCD−THP

ZF−VP

Reg.−VP

Reg.−VP (optimal)

MMSE−VP

ZF−GMD−VP

Reg.−GMD−VP

Fig. 1. Averaged uncoded BER comparison for (4,4) MIMO sys-

tems with 16 QAM.

8 10121416 1820

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

SNR (dB)

BER

ZF−GMD−THP

Reg.−GMD−THP

UCD−THP

ZF−VP

Reg.−VP

Reg.−VP (optimal)

MMSE−VP

ZF−GMD−VP

Reg.−GMD−VP

Fig. 2. Averaged uncoded BER comparison for (10,10) MIMO

systems with 16 QAM.

−20

−100 102030 40

101

102

103

104

105

106

SNR (dB)

Average number of visited nodes of

sphere encoder

ZF−VP

Reg.−VP

Reg.−VP (optimal)

MMSE−VP

ZF−GMD−VP

Reg.−GMD−VP

(4.4) MIMO

(10.10) MIMO

Fig. 3. Complexity comparison of VP schemes for MIMO systems

with 16 QAM.

outperforms the MMSE-VP up to 1dB SNR gain in high

SNR region. Unfortunately, such gain disappears for sys-

tems significantly larger than 4 antennas. We can see that

there is some performance loss for the Reg.-GMD-VP com-

pared with the MMSE-VP from Fig. 2. Although the ZF-

VP outperforms the ZF-GMD-VP, the Reg.-GMD-VP is still

superior to the Reg.-VP (optimal). Furthermore, simulation

showstheproposedReg.-GMD-VPhasthebestperformance

among different factors (i.e. ?), thus optimal factor selection

is not needed in contrast with the Reg.-VP.

4.2. Complexity comparison

Fig. 3 shows the complexity (measured by the averaged

number of visited nodes of sphere encoder) comparison of

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F. Liu et al. / Int. J. Electron. Commun. (AEÜ) 63 (2009) 502–505 505

VP schemes for (4,4) and (10,10) MIMO systems with

16 QAM. The ZF-GMD-VP and Reg.-GMD-VP show the

lowest and almost fixed complexity for all SNR, while the

complexity of ZF-VP is the highest with small variation and

the complexity of MMSE-VP increases from that of our

schemes to that of ZF-VP as SNR increases. Interestingly,

the complexity of conventional regularized VP schemes de-

creases first and then increases as SNR increases. The com-

plexity of ZF-GMD-VP and Reg.-GMD-VP is nearly the

same and is only about 30 and 6 percent of that of ZF-VP

for (4,4) and (10,10) MIMO systems, respectively. Thus

the complexity of our schemes seemly give the lower bound

of that of conventional transmitter side VP. Compared with

the MMSE-VP, the proposed schemes also have significant

complexity reduction in the practical SNR region.

5. Conclusion

We improved the GMD-THP transceiver by the VP tech-

nique. The Reg.-GMD-VP shows very close performance

to the UCD-THP and even outperforms the MMSE-VP for

(4,4) MIMO systems. The proposed schemes have great

complexity reduction compared with the conventional VP.

Acknowledgement

This research is supported by the Science and Tech-

nology Committee of Shanghai Municipality (under Grant

no. 06DZ15013) and the National Natural Science Founda-

tion of China (under Grant no. 60772100).

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