mmse vector precoding

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0165-1684/$ - se

doi:10.1016/j.si

$This resear

Committee of

06DZ15013).�CorrespondE-mail addr

Signal Processing 87 (2007) 2823–2833

www.elsevier.com/locate/sigpro

MMSE vector precoding with joint transmitter and receiverdesign for MIMO systems$

Feng Liu�, Lingge Jiang, Chen He

Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

Received 23 November 2006; received in revised form 28 May 2007; accepted 29 May 2007

Available online 7 June 2007

Abstract

The transmitter side (Tx) vector precoding (VP) shows excellent performance. However, joint transmitter and receiver

(Tx–Rx) design might achieve further improvement. In this paper, we propose a novel joint Tx–Rx VP for multiple input

multiple output (MIMO) systems. A unitary matrix is used at the receiver to partially equalize the channel. We design the

joint Tx–Rx VP with the minimum mean square error (MMSE) criterion and deduce the general closed-form solution.

Then we apply several methods including the singular value decomposition, QR decomposition and geometric mean

decomposition (GMD) to specify the general joint Tx–Rx VP design. Moreover, by using the extended channel, we achieve

an improved scheme to obtain further performance gain. Simulation results show that the specification with GMD method

outperforms the other specifications and the Tx MMSE VP. The improved joint Tx–Rx MMSE VP with GMD is superior

to other MMSE VP schemes.

r 2007 Elsevier B.V. All rights reserved.

Keywords: MIMO systems; MMSE vector precoding; Joint transmitter and receiver design; Geometric mean decomposition

1. Introduction

It has been shown that the multiple input multipleoutput (MIMO) systems achieve high capacity andbetter performance than the single-antenna systems[1–3]. If the channel state information (CSI) isavailable at the transmitter, precoding techniquecan be exploited to improve the system performance.

The ‘‘vector perturbation technique’’ [4,5] is anew transmitter side (Tx) technique which gener-

e front matter r 2007 Elsevier B.V. All rights reserved

gpro.2007.05.025

ch is supported by the Science and Technology

Shanghai Municipality (under Grant No.

ing author. Tel.: +8621 34204578 603.

ess: [email protected] (F. Liu).

alizes the Tomlinson–Harashima precoding (THP)[6] for MIMO systems. It is also called as vectorprecoding (VP) by the following researchers. Thekey idea is to add a perturbation component to thedata symbols, which shapes the symbols to satisfy aspecial goal. As in the THP, the perturbationsymbol in VP is an integer scaled by a factorand can be eliminated by a modulo operation atthe receiver. In contrast with the THP, theperturbation symbol in VP can be purposivelydesigned. In [4,5] Tx zero forcing (ZF) VP and animproved regularized VP are proposed whichminimizes the transmit energy. In [7] Tx minimummean square error (MMSE) VP is designed tominimize the mean square error (MSE) instead ofthe transmit energy and achieves better performance

.

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dx.doi.org/10.1016/j.sigpro.2007.05.025

mailto:[email protected]

ARTICLE IN PRESSF. Liu et al. / Signal Processing 87 (2007) 2823–28332824

than the Tx ZF VP and the Tx regularized VP.All these Tx VP schemes significantly outperformconventional THP. Without the cooperation amongthe receiver antennas, these Tx VP schemes areadaptable to broadcast channels and point-to-pointchannel.

However, joint transmitter and receiver (Tx–Rx)design for VP might achieve further performanceimprovement. In this paper, we consider a simplescenario where only a unitary matrix is used at thereceiver. This design adapts to the point-to-pointchannel. With MMSE criterion, we deduce a closed-form solution of the joint Tx–Rx VP design. Wefind it generalizes the Tx MMSE VP. Then somespecifications of the general joint Tx–Rx VP areconsidered: by singular value decomposition (SVD),by QR decomposition (QRD), and by geometricmean decomposition (GMD) [8]. Then we proposean improved version of the joint Tx–Rx VP by usingthe extended channel. We compare the performanceof these specifications by computer simulations andfind that the specifications by GMD methodsignificantly outperform the Tx VP and the speci-fications by SVD and QRD methods. The improvedjoint Tx–Rx MMSE GMD VP is shown to besuperior to the other VP schemes and achievefurther diversity gain.

Notation: Throughout the paper, we will denotevectors and matrices by lower and upper case boldletters, respectively. We use E( � ), ( � )H, ( � )�1, andtr( � ) for expectation, conjugate transposition, theinverse, and the trace of a matrix, respectively. Theidentity matrix is denoted by I. The norm of avector or matrix is denoted as || � ||.

The rest of this paper is organized as follows.Section 2 introduces the system model. Subse-quently, we propose the joint Tx–Rx MMSE VPin Section 3. Simulation results are provided inSection 4. Finally, Section 5 concludes the paper.

2. System model

We consider a MIMO system with MT transmitand MR receive antennas in a frequency flat fadingchannel. A block of data symbols with length NB istransmitted at each channel instance. There is atotal power constraint PT for the transmitted block.Throughout the paper, we assume that perfect CSIis known at both the transmitter and receiver.

The signal model is expressed as

y½n� ¼ Hx½n� þ n½n�; n ¼ 1; 2; � � � ;NB, (1)

where y[n] is the nth MR� 1 symbol vector amongthe received block, x[n] the nth MT� 1 symbolvector among the transmitted block, n[n] thecorresponding nth MR� 1 vector of independentadditive white Gaussian noise with zero-mean andcovariance Rn ¼ s2nI, and H the MR�MT indepen-dent zero-mean unit-variance complex Gaussianchannel. We note that the more general frequency-selective channel can be represented by a spatial–temporal channel with a larger dimensionality.Hence, (1) is rather general.

VP schemes generate transmitted symbols x[n]as a function f (which will be derived later) of themix of data symbols s[n] and perturbation symbolsp[n] as

x½n� ¼ f d½n�ð Þ; n ¼ 1; 2; � � � ;NB, (2)

where d½n�9s½n� þ p½n� is the nth MR� 1 mixedsymbol vector. The perturbation vector p[n] ischosen from an integer lattice aZMR with an integerscalar a. At the receiver, a modulo operation(mod a) is used to remove the perturbation vectorp[n]. If square M-point Gray coded quadratureamplitude modulation (QAM) constellation

�1

2; � � � ;�

ffiffiffiffiffiffiMp� 1

2

� �2

with variance s2s ¼ ðM � 1Þ=6 is used, the scalar isset to be a ¼

ffiffiffiffiffiffiMp

.In this paper, we consider a simple scenario for

joint Tx–Rx design by using a unitary matrix AH atthe receiver to partially equalize the channel as

d½n� ¼ gAHy½n�; n ¼ 1; 2; � � � ;NB, (3)

where g is a power control factor. Finally, thesymbol is estimated after the modulo operation

s½n� ¼ d½n� mod a; n ¼ 1; 2; � � � ;NB. (4)

The proposed joint Tx–Rx VP design is demon-strated by Fig. 1.

3. Proposed schemes

In this section, we consider the problem of jointTx–Rx VP design with the MMSE criterion for theabove model. We first deduce the generalizedsolution of the problem. Then specializations basedon different methods are given. And then animproved version of the joint Tx–Rx VP designis proposed with the extended channel. Finally,we provide the performance analysis and somediscussion.

ARTICLE IN PRESS

s[n]( )f ⋅

x[n]d[n]H gAH

n[n]

ˆ [n]d ˆ[n]s

Sphere

encoding

mod

MRpopt[n]∈�

Fig. 1. Demonstration of the proposed joint Tx–Rx VP design for MIMO systems.

F. Liu et al. / Signal Processing 87 (2007) 2823–2833 2825

3.1. Problem formulation and solution

Our goal is to find the optimal solution todetermine the perturbation vector p[n] AND thefunction f (i.e. the relationship between x[n] andd[n]). The power control factor g can also beincluded into this question. We choose the MMSEcriterion to design the joint Tx–Rx VP. Themeasurement is the total MSE of the differencebetween the mixed symbol block and its estimation,which is defined as

� p½n�;x½n�; gð Þ9XNB

n¼1

E d½n� � d½n�� 2n o

, (5)

which can be computed as

�ðp½n�;x½n�; gÞ ¼XNB

n¼1

E gAHðHx½n� þ n½n�Þ � d½n�

�� 2n o

¼XNB

n¼1

E gAHðHx½n� þ n½n�Þ � d½n�

� �Hn� gAH

ðHx½n� þ n½n�Þ � d½n��

¼XNB

n¼1

g2xH½n�HHHx½n�

� gdH½n�AHHx½n� þ g2trðRnÞ

�gxH½n�HHAd½n� þ dH½n�d½n��. ð6Þ

Now the optimizing problem reads as

p½n�; x½n�; g �

¼ argmin � p½n�;x½n�; gð Þ

s:t:XNB

n¼1

xH½n�x½n� ¼ PT. ð7Þ

Theorem 1. The solution of the problem (7) is

given by

popt½n� ¼ argminp½n�2aZMR

LA s½n� þ p½n�ð Þ�� 2,

n ¼ 1; 2; � � � ;NB, ð8Þ

x½n� ¼ g�1ðHHHþ xIÞ�1HHAd½n�

¼ g�1HHðHHH þ xIÞ�1Ad½n�; n ¼ 1; 2; � � � ;NB,

ð9Þ

g ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

PT

XNB

n¼1

dH½n�AHH HHH þ xI ��2

HHAd½n�

vuut

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

PT

XNB

n¼1

dH½n�AHðHHH þ xIÞ�1HHHðHHH þ xIÞ�1Ad½n�

vuut ,

ð10Þ

where x ¼ NB trðRnÞ=PT, and we use the Cholesky

factorization ðHHH þ xIÞ�1 ¼ LHL to obtain the

triangular matrix L. The minimal MSE obtained by

the above optimal solution is given by

� ¼ xXNB

n¼1

dH½n�AHLHLAd½n�

¼ xXNB

n¼1

LA s½n� þ p½n�ð Þ�� 2. ð11Þ

Proof. : see Appendix A.On comparing the above results with the Tx

MMSE VP [7], we can see if we set A ¼ I, the abovejoint Tx–Rx design turns into the Tx MMSE VP.Therefore, our joint Tx–Rx MMSE VP generalizesthe Tx MMSE VP. Furthermore, if the goal is tominimize the power control factor g, it turns intothe regularized VP discussed in [4,5], which isobviously inferior to the MMSE design.

3.2. Specifications of the general solution

In the former subsection, the matrix A at thereceiver can be any unitary. Thus, it gives a generalform of the joint Tx–Rx MMSE VP with a unitarymatrix at the receiver. However, it is difficult todetermine the optimal unitary matrix A. Here wespecify the general form by some existing methodsfor extra performance improvement. We consider


the SVD, QRD, and GMD [8] methods. Thesemethods decompose the channel H into the follow-ing general form:

H ¼ QRPH, (12)

where Q and P are unitary matrices. For the SVDmethod, R is a real diagonal matrix with singularvalues of H. For the QRD method, P ¼ I, and R isan upper triangular matrix. For the GMD method,R is a real upper triangular matrix with diagonalelements all equal to the geometric mean of thepositive singular values of H. We note the TxMMSE VP can be viewed as a special case of theabove general form where Q ¼ I, R ¼ RH

0 , andP ¼ Q0 using the QRD: HH ¼ Q0R0.

We choose the unitary matrix A ¼ Q to partiallyequalize the channel. Now, the joint Tx–Rx MMSEVP can be specified as

x½n� ¼ g�1PðRHRþ xIÞ�1RHd½n�

¼ g�1PRHðRRH þ xIÞ�1d½n�; n ¼ 1; 2; � � � ;NB,

ð13Þ

g ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

PT

XNB

n¼1

dH½n�ðRRH þ xIÞ�1RRHðRRH þ xIÞ�1d½n�

vuut

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

PT

XNB

n¼1

RHðRRH þ xIÞ�1d½n�� 2

vuut , ð14Þ

� ¼ xXNB

n¼1

dH½n�ðRRH þ xIÞ�1d½n�. (15)

By the Cholesky factorization

ðRRH þ xIÞ�1 ¼ BHB (16)

we can further simplify Eq. (15) as

� ¼ xXNB

n¼1

dH½n�BHBd½n� ¼ xXNB

n¼1

B s½n� þ p½n�ð Þ�� 2.

(17)

In order to minimize the total MSE, eachindependent item in Eq. (17) needs to be minimized.Now the optimal perturbation vector is chosen by


B s½n� þ p½n�ð Þ�� 2; n ¼ 1; 2; � � � ;NB,

(18)

which can be resolved by the closest-point searchalgorithm (sphere encoding).

3.3. Improved joint Tx– Rx MMSE VP with the

extended channel

The specification schemes in the previous subsec-tion is based on the SVD/QRD/GMD methods ofchannel H. From the literature, e.g. [9], we knowusing the extended channel He9½H sn=ssI� forprecoding can further improve the performance.Here, we develop an improved version of jointTx–Rx MMSE VP based on the extended channel.

Now the mentioned methods decompose theextended channel He as

He ¼ QeRePHe . (19)

The following steps are done as in subsection 3.2withQ,R, P replaced byQe,Re, Pe. We note that sinceHe has a dimension expansion, dimension reduction isneeded for correct matrix computation. Thus, only theproper upper-left parts of Qe, Re, Pe are used in theimproved joint Tx–Rx MMSE VP schemes.

3.4. Performance analysis

For the specifications with SVD, QRD, and GMDmethods, we provide some brief discussion. The SVDmethod gives a diagonal matrix R, while the QRDand GMD methods give a triangular matrix R. Sincethe sphere encoding considers interference cancella-tion to find the optimal perturbation vector, weknow from Eq. (18) that the sphere encoding cannotbenefit from a diagonal channel B. Therefore, theQRD and GMD specifications obviously outperformthe SVD specification. Moreover, the GMD methodgives a triangular matrix with equal diagonalelements, which eliminates the imbalance amongthe subchannel gains. Thus, the GMD specification issuperior to the QRD specification.

The authors of [8] propose two schemes (namedGMD and UCD) based on the GMD method forMIMO systems [10,11]. In fact, the GMD scheme [10]is a ZF THP based on the GMD of channel H, whilethe UCD scheme [11] is a MMSE THP with water-filling power allocation based on the GMD of theextended channel He. In [11], the authors prove thatthe GMD scheme has a diversity of order dGMD ¼

ðM �mþ 1Þm and the UCD scheme has a diversityof order dUCD ¼Mm, where M ¼ maxðMR;MTÞ

and m ¼ minðMR;MTÞ. The authors also point outthat the water filling power allocation does not helpimprove diversity gains. Therefore, we can expect ourjoint Tx–Rx MMSE VP with GMD has equaldiversity order as that of the GMD scheme and the

ARTICLE IN PRESSF. Liu et al. / Signal Processing 87 (2007) 2823–2833 2827

improved joint Tx–RxMMSE VP with GMD has thesame diversity order as that of the UCD scheme.

Since the exact performance is difficult to analyze,we will use computer simulations to compare theseschemes in the next section.

3.5. Further remarks

In [12], the authors propose a joint precodingmatrix and perturbation vector design. From (12),we know that it can be regarded as a special case ofour design. Let Q ¼ I and the QRD: HH

¼ Q0R0,similar processing can be done. The difference is theoptimization of [12] includes the QRD-M algo-rithm. Compared with [12], our design even doesnot constrain the precoding matrix to be linear,although the linear structure is proved optimal. Wepoint out the QRD-M algorithm discussed in [12]can be further considered in our design to reduce thecomplexity of the sphere encoding, which is not thefocus of this paper.

4. Simulation results

The performance is measured in terms of theuncoded bit error rate (BER) over the equivalent

0 2 4 6 8 10 12 1

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N

BE

R

Tx MMSE VP

Tx-Rx MMSE VP by GMD

Improved Tx-Rx MMSE VP

Tx-Rx MMSE VP by SVD


Tx-Rx MMSE VP by QRD


Fig. 2. Average uncoded BER performance comparison for the specifica

4-QAM for (2,2) MIMO systems.

received SNR which is defined as: Eb=N0 ¼

MRs2s=Rms2n, where Rm is the modulation rate (i.e.bits per modulated symbol).

4.1. Performance comparison for the different

specification schemes

We show the performance of the differentspecifications based on different methods in Figs.2 and 3 for a 2-by-2 MIMO system with 4-QAMand 16-QAM, respectively. We see that the specifi-cations with SVD and QRD methods performworse than the Tx MMSE VP. The specificationwith GMD method significantly outperformsthe Tx MMSE VP. In detail, there is about 2 and3 dB SNR gain for the Tx–Rx MMSE VP by GMDin comparison with the Tx MMSE VP atBER ¼ 10�5 for 4-QAM and 16-QAM, respec-tively. We attribute this to the property of GMDmethod, which eliminates the imbalance betweenthe gains of subchannels. By using the extendedchannel, the improved Tx–Rx MMSE VP by GMDhas further processing gain of about 4 dB comparedwith the Tx MMSE VP. Therefore, we only considerthe specification with GMD in the followingsimulation.

4 16 18 20 22 24 26 28

0 (dB)

by GMD

by SVD

by QRD

tions of the joint Tx–Rx MMSE VP with different methods using

ARTICLE IN PRESS

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Eb/N

0 (dB)

10-5

10-4

10-3

10-2

10-1

100

BE

R

Tx MMSE VP


Improved Tx-Rx MMSE VP by GMD

Tx-Rx MMSE VP by SVD

Improved Tx-Rx MMSE VP by SVD

Tx-Rx MMSE VP by QRD

Improved Tx-Rx MMSE VP by QRD

Fig. 3. Average uncoded BER performance comparison for the specifications of the joint Tx–Rx MMSE VP with different methods using

16-QAM for (2,2) MIMO systems.

F. Liu et al. / Signal Processing 87 (2007) 2823–28332828

4.2. The performance comparison for the GMD-

based schemes

In this subsection, we compare the performanceof the GMD-based schemes, including the GMD,UCD, the joint Tx–Rx MMSE VP by GMD, andthe improved joint Tx–Rx MMSE VP by GMD. Wealso provide the Tx ZF VP in [5] and Tx MMSE VP[7] for comparison. Since the Tx regularized VP [5]is inferior to the Tx MMSE VP, we omit it in oursimulation.

Figs. 4 and 5 show the performance comparisonfor a 2-by-2 MIMO system with 4-QAM and16-QAM, respectively. We see the GMD schemeonly outperforms the Tx ZF VP, while the UCDscheme is superior to all others. The joint Tx–RxMMSE VP by GMD significantly outperforms theGMD scheme. Compared with the UCD scheme,the improved joint Tx–Rx MMSE VP by GMDshows little performance loss. As analyzed in theabove section, our joint Tx–Rx MMSE VP designdoes not consider power allocation, which leads tosome performance loss in comparison with theUCD scheme.

We give more simulation data to compare therelated schemes. In Figs. 6 and 7, the performance is

compared for a 3-by-3 MIMO system with 4-QAMand 16-QAM, respectively. Similar results hold asthe 2-by-2 MIMO systems. We see the specificationwith GMD still outperforms the Tx MMSE VP.The improved joint Tx–Rx MMSE VP by GMDhas about 2 dB SNR gain compared with the TxMMSE VP.

Furthermore, the slopes of these curves in highSNR region indicating the diversity order verify ouranalysis about the diversity order. From Figs. 4–7,we see the diversity orders of the GMD and jointTx–Rx MMSE VP with GMD schemes are equal,while the UCD and improved joint Tx–Rx MMSEVP by GMD schemes share the same diversityorder. Moreover, the Tx MMSE VP shows the samediversity order as the GMD, while the Tx ZF VPobtains low diversity order.

4.3. Explanation for the performance improvement of

the GMD-based schemes

The performance of VP schemes is difficult toanalyze theoretically. In this subsection, we try toexplain the performance improvement of theproposed GMD-based VP schemes by simulation.As mentioned before, the GMD method eliminates

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0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Eb/N

0 (dB)

GMD

UCD

Tx ZF VP

Tx MMSE VP


Improved Tx-Rx MMSE VP by GMD10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

BE

R

Fig. 4. Average uncoded BER performance comparison for different schemes using 4-QAM for (2,2) MIMO systems.

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Eb/N

0 (dB)

GMD

UCD

Tx ZF VP

Tx MMSE VP



10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

BE

R



the difference of subchannel gains, which affects thesearching of the perturbation vectors. Looking atthe triangular matrices used in Eqs. (8) and (18) for

sphere encoding, we find that the difference of thediagonal elements of this triangular matrix playsan important role in searching the perturbation

ARTICLE IN PRESS

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Eb/N

0 (dB)

GMD

UCD

Tx ZF VP

Tx MMSE VP



10-5

10-6

10-4

10-3

10-2

10-1

100

BE

R


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Eb/N

0 (dB)

GMD

UCD

Tx ZF VP

Tx MMSE VP



10-5

10-6

10-7

10-4

10-3

10-2

10-1

100

BE

R


F. Liu et al. / Signal Processing 87 (2007) 2823–28332830

vectors. With computer simulation, we can see itexplicitly. Fig. 8 shows the average variance of thediagonal elements of the triangular matrix used forsphere encoding with 4-QAM, where A, B, and C

represent the Tx MMSE VP, the Tx–Rx MMSE VPby GMD, and the improved Tx–Rx MMSE VP byGMD, respectively. We compare the (2,2), (5,5),and (10,10) MIMO systems. From the left part of

ARTICLE IN PRESS

0 10 20 30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Eb/N

0 (dB)

Variance

0 10 20 30

10-5

10-4

10-3

10-2

10-1

100

101

Eb/N

0 (dB)

Variance

A. (2,2)

B. (2,2)

C. (2,2)

A. (5,5)

B. (5,5)

C. (5,5)

A. (10,10)

B. (10,10)

C. (10,10)

Fig. 8. Comparison of average variance of the diagonal elements of the triangular matrix used for sphere encoding with 4-QAM. A

represents the Tx MMSE VP scheme, B represents the Tx–Rx MMSE VP by GMD scheme, and C represents the improved Tx–Rx MMSE

VP by GMD scheme. (2,2), (5,5), and (10,10) MIMO systems are considered. Left: linear Y-axis; right: logarithmic Y-axis.


Fig. 8 (using linear scale for Y-axis) we can see thatthe average variance of Tx MMSE VP significantlyincreases with the SNR, while the GMD-based VPschemes keep a very low level of average variance.Bigger variance means higher imbalance among thediagonal elements. This helps explain why theGMD-based VP schemes significantly outperformthe Tx MMSE VP in high SNR region. However, ifthe antenna number increases, the average varianceof Tx MMSE VP reduces. This can explain why theperformance improvement of the GMD-based VPschemes decreases as the number of antennasincreases compared with the Tx MMSE VP.Furthermore, we change the Y-axis into logarithmicscale to see it in detail as the right part of Fig. 8. Wefind that the average variance of the improvedTx–Rx MMSE VP by GMD is the smallest amongthe three schemes, which explains why the improvedTx–Rx MMSE VP by GMD has a superiorperformance.

5. Conclusion

We presented the joint transmitter and receiverdesign of VP with MMSE criterion for MIMO

systems, while the receiver is multiplied with aunitary matrix. This joint Tx–Rx VP MMSE designgeneralizes the transmitter side MMSE VP. Wededuced a closed-form solution with any unitarymatrix at the receiver. Then we used severalmethods to specify the general solution. We foundthe specification with GMD method improves theperformance due to the elimination of gain imbal-ance among subchannels of MIMO systems incomparison with the Tx MMSE VP. Moreover,we exploited the extended channel to obtain animproved version of the joint Tx–Rx MMSE VP.The improved joint Tx–Rx MMSE VP with GMDachieves further diversity gain and processing gain.Although our design still has little performance losscompared with the UCD scheme, it significantlyoutperforms the Tx MMSE VP. The simulationresults verified our design and analysis.

Acknowledgment

The authors are grateful to the anonymousreviewers for their helpful suggestions for improvingthe submitted manuscript.


Appendix A. Proof of Theorem 1:

Proof. We use the Lagrangian multiplier method tosolve the question. The cost function is

L p½n�; x½n�; g; lð Þ ¼ � p½n�;x½n�; gð Þ

þ lXNB

n¼1

xH½n�x½n� � PT

!, ð20Þ

where l is the Lagrangian multiplier to be deter-mined. By setting the derivatives of Lðp½n�;x½n�; g; lÞto zeros with respect to x[n], g, and l, respectively,we get

qL

qx½n�¼ 0) g2xH½n�HHH� gdH½n�AHHþ lxH½n� ¼ 0,

(21)

qL

qg¼ 0)

XNB

n¼1

2gxH½n�HHHx½n� � dH½n�AHHx½n�

þ2g trðRnÞ � xH½n�HHAd½n��¼ 0, ð22Þ

qL

ql¼ 0)

XNB

n¼1

xH½n�x½n� � PT ¼ 0. (23)

Rearrange Eq. (21) to obtain

x½n� ¼ g�1 HHHþlg2

I

� �1HHAd½n�

¼ g�1HH HHH þlg2

I

� �1Ad½n�, ð24Þ

� ¼XNB

n¼1

dH½n�AHðHHH þ xIÞ�1HHHHHHðHHH þ xIÞ�1Ad½n� þ dH½n�d½n�

�dH½n�AHHHHðHHH þ xIÞ�1Ad½n� � dH½n�AHðHHH þ xIÞ�1HHHAd½n�

þxdH½n�AHðHHH þ xIÞ�1HHHðHHH þ xIÞ�1Ad½n�

8>><>>:

9>>=>>;

¼XNB

n¼1

dH½n�AH

ðHHH þ xIÞ�1HHHHHH HHH þ xI ��1

þ I

�HHHðHHH þ xIÞ�1 � ðHHH þ xIÞ�1HHH

þxðHHH þ xIÞ�1HHHðHHH þ xIÞ�1

8>>><>>>:

9>>>=>>>;Ad½n�

¼ xXNB

n¼1

dH½n�AHðHHH þ xIÞ�1Ad½n�, ð30Þ

and

gxH½n�HHHx½n� � dH½n�AHHx½n� þlgxH½n�x½n� ¼ 0,

gxH½n�HHHx½n� � xH½n�HHAd½n� þlgxH½n�x½n� ¼ 0.

ð25Þ

From Eq. (24) we prove that the optimal relation-ship between x[n] and d[n] is linear. Bring Eq. (25)into Eq. (22):

XNB

n¼1

�2lgxH½n�x½n� þ 2g trðRnÞ

� �¼ 0. (26)

Then we get

lg2¼

NB � trðRnÞPNB

n¼1

xH½n�x½n�

¼NB � trðRnÞ

PT9x. (27)

Thus Eq. (24) can be written as

x½n� ¼ g�1ðHHHþ xIÞ�1HHAd½n�

¼ g�1HHðHHH þ xIÞ�1Ad½n�. ð28Þ

Now by taking Eq. (28) into Eq. (23), we get

g ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

PT

XNB

n¼1

dH½n�AHHðHHH þ xIÞ�2HHAd½n�

vuut

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

PT

XNB

n¼1

dH½n�AHðHHH þ xIÞ�1HHHðHHH þ xIÞ�1Ad½n�

vuut .

ð29Þ

With Eqs. (28) and (29), the MSE expression (6) canbe computed

where the last step uses the fact

ðHHH þ xIÞ�1HHHHHHðHHH þ xIÞ�1

� Iþ I�HHHðHHH þ xIÞ�1

þ I� ðHHH þ xIÞ�1HHH

ARTICLE IN PRESSF. Liu et al. / Signal Processing 87 (2007) 2823–2833 2833

þ xðHHH þ xIÞ�1HHHðHHH þ xIÞ�1

¼ ðHHH þ xIÞ�1HHHHHHðHHH þ xIÞ�1

� Iþ 2xðHHH þ xIÞ�1


¼ ðHHH þ xIÞ�1HHHHHH��ðHHH þ xIÞ þ 2xI

�ðHHH þ xIÞ�1


¼ ðHHH þ xIÞ�1HHH � I� �

HHH þ xI �� ðHHH þ xIÞ�1 þ xðHHH þ xIÞ�1

�HHHðHHH þ xIÞ�1

¼ x �ðHHH þ xIÞ�1HHH þ I �

ðHHH þ xIÞ�1


¼ xðHHH þ xIÞ�1, ð31Þ

where the matrix inversion lemma is applied for.Then we use the Cholesky factorization

ðHHH þ xIÞ�1 ¼ LHL, (32)

so that the MSE expression (30) can be furthersimplified:

� ¼ xXNB

n¼1

dH½n�AHLHLAd½n�

¼ xXNB

n¼1

LA s½n� þ p½n�ð Þ�� 2. ð33Þ

From Eq. (33), the optimal perturbation vectorminimizing the MSE can be effectively found by theclosest-point search algorithm (sphere encoding) as


LA s½n� þ p½n�ð Þ�� 2,

n ¼ 1; 2; � � � ;NB. ð34Þ

Thus, we complete the proof. &

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