# Transceiver Design for Dual-Hop Nonregenerative MIMO-OFDM Relay Systems Under Channel Uncertainties

**ABSTRACT** In this paper, linear transceiver design for dual-hop nonregenerative [amplify-and-forward (AF)] MIMO-OFDM systems under channel estimation errors is investigated. Second order moments of channel estimation errors in the two hops are first deduced. Then based on the Bayesian framework, joint design of linear forwarding matrix at the relay and equalizer at the destination under channel estimation errors is proposed to minimize the total mean-square-error (MSE) of the output signal at the destination. The optimal designs for both correlated and uncorrelated channel estimation errors are considered. The relationship with existing algorithms is also disclosed. Moreover, this design is extended to the joint design involving source precoder design. Simulation results show that the proposed design outperforms the design based on estimated channel state information only.

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**ABSTRACT:**In this paper, we investigate the resource allocation for decode-and-forward (DF) relay-assisted SC-FDMA systems. Considering the subcarrier adjacency restriction of SC-FDMA, we present an optimal algorithm which refers to the set partitioning problem. In order to reduce the computational complexity of the optimal resource allocation, we also propose a suboptimal algorithm which is based on the greedy heuristic thinking. Simulation results show that the spectral efficiency of the optimal algorithm is much higher than that of the round robin algorithm. It is also shown that the greedy algorithm, which has much lower complexity, performs quite close to the optimal algorithm.Communications and Networking in China (CHINACOM), 2012 7th International ICST Conference on; 01/2012 -
##### Conference Paper: Robust transceiver optimization for non-regenerative MIMO relay systems

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**ABSTRACT:**In this paper, we design a robust relay transceiver for non-regenerative MIMO relay system where imperfect channel state information (CSI) is available at the relay node. The CSI errors of relay to destination link at the relay node are assumed to be norm bounded due to the quantization and feedback. And the worst-case design approach is consdiered. The robust optimization aims to minimize worst-case sum mean-square errors (MSE) with the power constraints at the relay under channel uncertainty. An iterative algorithm is proposed for the optimization. Further, the subproblems in the iteration are proven to be equivalent to semidefinite programming representations with efficient solutions. Simulation results verify the robustness of the proposed MIMO relay transceivers when compared to the non-robust design.Wireless Personal Multimedia Communications (WPMC), 2012 15th International Symposium on; 01/2012 -
##### Conference Paper: Pilot transmission scheme and robust filter design for non-regenerative multi-pair two-way relaying

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**ABSTRACT:**In this paper, a pilot transmission scheme and a robust self-interference aware relay transceive filter for multipair two-way relaying are introduced. The bidirectional pairwise communications of multiple single-antenna nodes are simultaneously performed via an intermediate non-regenerative multi-antenna relay station and a pilot transmission scheme is proposed to obtain channel state information at the relay station and at the nodes. It is assumed that the nodes can use the available channel state information to subtract the back-propagated self-interference and the cases of perfect and imperfect self-interference cancellation are investigated. The relay station performs linear signal processing based on the estimated channels and a robust relay transceive filter approach is introduced which utilizes the fact that the nodes can perform self-interference cancellation. The proposed pilot transmission scheme requires less resources for channel estimation than conventional schemes and the proposed robust filter design increases the achievable sum rate in case of imperfect channel state information.Personal Indoor and Mobile Radio Communications (PIMRC), 2012 IEEE 23rd International Symposium on; 01/2012

Page 1

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 12, DECEMBER 20106325

Transceiver Design for Dual-Hop Nonregenerative

MIMO-OFDM Relay Systems Under

Channel Uncertainties

Chengwen Xing, Shaodan Ma, Yik-Chung Wu, and Tung-Sang Ng, Fellow, IEEE

Abstract—In this paper, linear transceiver design for dual-hop

nonregenerative [amplify-and-forward (AF)] MIMO-OFDM sys-

tems under channelestimationerrorsis investigated.Second order

moments of channel estimation errors in the two hops are first

deduced. Then based on the Bayesian framework, joint design of

linear forwarding matrix at the relay and equalizer at the destina-

tion under channel estimation errors is proposed to minimize the

total mean-square-error (MSE) of the output signal at the desti-

nation. The optimal designs for both correlated and uncorrelated

channelestimationerrorsareconsidered.Therelationshipwithex-

istingalgorithmsisalsodisclosed.Moreover,thisdesignisextended

to thejointdesign involvingsource precoder design.Simulation re-

sults show that the proposed design outperforms the design based

on estimated channel state information only.

IndexTerms—Amplify-and-forward(AF),equalizer,forwarding

matrix, minimum mean-square-error (MMSE).

I. INTRODUCTION

I

be one of the essential parts for future communication systems

(e.g., LTE, IMT-Adanced, Winner Project). In dual-hop coop-

erative communication, relay nodes receive signal transmitted

from a source and then forward it to the destination [1], [2].

Roughly speaking, there are three different relay strategies:

decode-and-forward (DF), compress-and-forward (CF), and

amplify-and-forward (AF). Among them, AF strategy is the

most preferable for practical systems due to its low complexity

[3]–[7].

On the other hand, for wideband communication, multiple-

input multiple-output (MIMO) orthogonal-frequency-division-

multiplexing (OFDM) has gained a lot of attention in both in-

dustrial and academic communities, due to its high spectral ef-

ficiency, spatial diversity and multiplexing gains [8]–[11]. The

combination of AF and MIMO-OFDM becomes an attractive

Nordertoenhancethecoverageofbasestationsandquality

of wireless links, dual-hop relaying is being considered to

ManuscriptreceivedMarch15,2010;acceptedAugust15,2010.Dateofpub-

lication August 30, 2010; date of current version November 17, 2010. The as-

sociate editor coordinating the review of this manuscript and approving it for

publication was Prof. Xiqi Gao.

C. Xing is with the School of Information and Electronics, Beijing Instistute

of Technology, Beijing, China (e-mail: cwxing@eee.hku.hk).

S. Ma, Y.-C. Wu, and T.-S. Ng are with the Department of Electrical and

Electronic Engineering, The University of Hong Kong, Hong Kong (e-mail:

sdma@eee.hku.hk; ycwu@eee.hku.hk; tsng@eee.hku.hk).

Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2010.2070797

option for enabling high-speed wireless multi-media services

[12].

In the last decade, linear transceiver design for various

systems has been extensively investigated because of its low

implementation complexity and satisfactory performance [8],

[13].Forlineartransceiverdesign,minimummean-square-error

(MMSE) is one of the most important and frequently used cri-

teria [14]–[20]. For example, for point-to-point MIMO and

MIMO-OFDM systems, linear MMSE transceiver design has

been discussed in details in [14]–[16]. Linear MMSE trans-

ceiver design for multiuser MIMO systems has been considered

in [17], [18]. For single carrier AF MIMO relay systems, linear

MMSE forwarding matrix at the relay and equalizer at the

destination are joint designed in [19]. Furthermore, the linear

MMSE transceiver design for dual hop MIMO-OFDM relay

systems based on prefect channel state information (CSI) is

proposed in [20].

In alltheaboveworks, CSI is assumed tobe perfectly known.

Unfortunately, in practical systems, CSI must be estimated and

channel estimation errors are inevitable. When channel esti-

mation errors exist, in general, two classes of designs can be

employed: min-max and stochastic designs. If the distribu-

tions of channel estimation errors are known to be unbounded,

stochasticdesign is preferred.Stochasticdesign includes proba-

bility-based design and Bayesian design. In this paper, we focus

on Bayesian design, in which an averaged mean-square-error

(MSE) performance is considered. Recently, Bayesian linear

MMSE transceiver design under channel uncertainties has been

addressed for point-to-point MIMO systems [22], [23] and

point-to-point MIMO-OFDM systems [24].

In this paper, we take a step further and consider the linear

MMSEtransceiverdesignfordual-hopAFMIMO-OFDMrelay

systems without the direct link. For channel estimation in the

two hops, both the linear minimum mean square error and max-

imum likelihood estimators are derived, based on which the

secondordermomentsofchannelestimationerrorsarededuced.

Using the Bayesian framework, channel estimation errors are

taken into account in the transceiver design criterion. Then a

general closed-form solution for the optimal relay forwarding

matrix and destination equalizer is proposed. Both the uncorre-

lated and correlated channel estimation errors are considered.

The relationship between the proposed algorithm and several

existing designs is revealed. Furthermore, the proposed closed-

form solution is further extended to an iterative algorithm for

joint design of source precoder, relay forwarding matrix and

destination equalizer. Simulation results demonstrate that the

1053-587X/$26.00 © 2010 IEEE

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6326 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 12, DECEMBER 2010

proposed algorithms provide an obvious advantage in terms of

datamean-square-error(MSE)comparedtothealgorithmbased

on estimated CSI only.

We want to highlight that the solution proposed in this

paper can be directly extended to the problem minimizing the

weighted MSE. Various objective metrics such as capacity

maximization and minimizing maximum MSE can be trans-

formed to a weighted MSE problem with different weighting

matrices [14]. For clearness of presentation, we only consider

a sum MSE minimization problem. On the other hand, mini-

mizing the transmit power with a QoS requirement is a different

perspective for transceiver design. Formulating and solving this

problem is out of the scope of this paper.

Thispaperisorganizedasfollows.Systemmodelispresented

in Section II. Channel estimators and the corresponding covari-

ance of channel estimation errors are derived in Section III. The

optimization problem for transceiver design is formulated in

Section IV. In Section V, the general optimal closed-form so-

lution for the relay forwarding matrix and destination equalizer

design problemis proposed. Theproposed closed-formsolution

isfurtherextendedtoaniterativealgorithmtoincludethedesign

of sourceprecoderinSection VI.Simulationresults are givenin

Section VII and finally, conclusions are drawn in Section VIII.

The following notations are used throughout this paper.

Boldface lowercase letters denote vectors, while boldface up-

percase letters denote matrices. The notations

denote the transpose, Hermitian, and conjugate of the matrix

, respectively, and is the trace of the matrix

symbol

denotes the

denotes theall zero matrix. The notation

Hermitian square root of the positive semidefinite matrix

such that

and

The symbol

represents the expectation operation. The

operation

stacks the columns of the matrix

single vector. The symbol

The symbol

means

denotes theblock diagonalmatrix with

elements.

,, and

. The

identity matrix, while

is the

,

is a Hermitian matrix.

into a

represents Kronecker product.

. The notation

andas the diagonal

II. SYSTEM MODEL

In this paper, we consider a dual-hop AF MIMO-OFDM re-

laying cooperative communication system, which consists of

one source with

antennas, one relay with

tennas and

transmit antennas, and one destination with

antennas, as shown in Fig. 1. At the first hop, the source trans-

mits data to the relay, and the received signal

the

subcarrier is

receive an-

at the relay on

(1)

where

variancematrix

can be an arbitrary covariance matrix. The matrix

MIMO channel between the source and relay on the

rier. The symbol

is the additive Gaussian noise with zero

mean and covariance matrix

carrier. At the relay, for each subcarrrier, the received signal

is multiplied by a forwarding matrix

straint

is the data vector transmitted by the source with co-

onthesubcarrier,and

is the

subcar-

on thesub-

, under a power con-

whereand

is the maximum transmit power. Then the resulting signal is

transmitted to the destination. The received data

tination on the

subcarrier is

at the des-

(2)

where the symbol

the

subcarrier at the second hop with zero mean and covari-

ance matrix

. In order to guarantee the trans-

mitted data

can be recovered at the destination, it is assumed

that

,, andare greater than or equal to

The signal

received at the relay and the signal

the destination in frequency domain can be compactly written

as

is the additive Gaussian noise vector on

[6].

received at

(3)

(4)

where

(5a)

(5b)

(5c)

(5d)

(5e)

(5f)

Notice that in general the matrix

trary

trix. This corresponds to mixing the data from different sub-

carriers at the relay, and is referred as subcarrier cooperative

AF MIMO-OFDM systems [20]. It is obvious that when the

numberofsubcarrier

islarge,transceiverdesignforsuchsys-

tems needs very high complexity. On other hand, it has been

shown in [20] that the low-complexity subcarrier independent

AF MIMO-OFDM systems [i.e., the system considered in (3)

and (4)] only have a slight performance loss in terms of total

datamean-square-error(MSE)comparedtothesubcarriercoop-

erative AF MIMO-OFDM systems. Therefore, in this paper, we

focus on the more practical subcarrier independent AF MIMO-

OFDM relay systems.

in (4) can be an arbi-

matrix instead of a block diagonal ma-

III. CHANNEL ESTIMATION ERROR MODELING

In practical systems, channel state information (CSI) is un-

known and must be estimated. Here, we consider estimating the

channels based on training sequence. Furthermore, the two fre-

quency-selectiveMIMO channelsbetween thesource and relay,

and that between the relay and destination are estimated inde-

pendently.In thispaper, thesource-relay channel is estimatedat

the relay, while the relay-destination channel is estimated at the

destination.Theneachchannelestimationproblemisa standard

point-to-point MIMO-OFDM channel estimation.

For point-to-point MIMO-OFDM systems, channels can be

estimated in either frequency domain or time domain. The ad-

vantage of time domain over frequency domain channel esti-

mation is that there are much fewer parameters to be estimated

[25]. Therefore, we focus on time domain channel estimation.

Becausethechannelsinthetwohopsareseparatelyestimatedin

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XING et al.: DUAL-HOP NONREGENERATIVE MIMO-OFDM RELAY SYSTEMS6327

Fig. 1. AF MIMO-OFDM relaying diagram.

time domain, we will present the first hop channel estimation as

anexampleandthesameprocedurecanbeappliedtothesecond

hop channel estimation.

From the received signal model in frequency domain given

by (3), the corresponding time domain signal is

(6)

where

matrix with dimension

matrix, it is proved in Appendix A that (6) can be rewritten as

is the normalized discrete-Fourier-transform (DFT)

. Based on the properties of DFT

(7)

where the matrices

are defined as

(8)

It is obvious that

channel between the source and relay in the time domain and

is the length of the multi-path channel. The data matrix

a block circular matrix as

is thetap of the multi-path MIMO

is

...

...

...

...

...

...

...

...

(9)

where the element

is expressed as

(10)

Based on the signal model in (7), the linear minimum-mean-

square-error (LMMSE) channel estimate is given by [25]

(11)

with the corresponding MSE

(12)

where

for channel covariance matrix. For uncorrelated channel

taps,

, where

the

channel tap [24].

On the other hand, the channel in frequency domain and time

domain has the following relationship1:

is the prior information

and

is the variance of

(13)

where

channel estimate

isthefirstcolumnsof

is computed according to (13), we have

.Ifthefrequencydomain

(14)

where

In case there is no prior information on

sign uninformative prior to

proach infinity [26]. In this case,

channel estimator (11) and estimation MSE (12) reduce to that

of maximum likelihood (ML) estimation [25, p. 179].

Taking the

(14) gives

.

, we can as-

, that is,ap-

, and then the

block diagonal elements from

(15)

where

partition of

is thematrix taken from the following

...

...

...

(16)

Furthermore, based on (15), for an arbitrary square matrix

is proved in Appendix B that

, it

(17)

1This relationship holds for both perfect CSI and estimated CSI.

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6328IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 12, DECEMBER 2010

A similar result holds for the second hop. In particular, de-

noting the relationship between the true value and estimate of

the second hop channel as

(18)

we have the following property:

(19)

where

main. Furthermore, as the two channels are estimated indepen-

dently,

andare independent.

is the length of the second hop channel in time do-

IV. TRANSCEIVER DESIGN PROBLEM FORMULATION

At the destination, a linear equalizer

subcarrier to detect the transmitted data

problem is how to design the linear forwarding matrix

relay and the linear equalizer

the MSE of the received data at the destination:

is adopted for each

(see Fig. 1). The

at the

at the destination to minimize

(20)

where the expectation is taken with respect to

,, and.2Since

dent, the MSE expression (20) can be written as

,,

,andare indepen-

(21)

Because

andare independent, the first term of

is

2In this paper, the MSE is in fact an average of the traditional MSE over all

possible channel estimation errors ??

channel estimator is adopted, it is equivalent to the conditional MSE corre-

sponding to the partial CSI case defined in [27].

and ??

. When the LMMSE

(22)

For the inner expectation, the following equation holds:

(23)

where based on (17) the matrix

is defined as

(24)

Applying (23) and the corresponding result for

(22), the first term of

to

becomes

(25)

where the matrix

is defined as

(26)

Similarly, the second term of

in (21) can be simplified

as

(27)

Based on (25) and (27), the

(21) equals to

(28)

where

(29)

(30)

Notice that the matrix

ceive signal

Subject to the transmit power constraint at the relay, the joint

design of relay forwarding matrix and destination equalizer that

is the correlation matrix of the re-

subcarrier at the relay.on the

Page 5

XING et al.: DUAL-HOP NONREGENERATIVE MIMO-OFDM RELAY SYSTEMS 6329

minimizes the total MSE of the output data at the destination

can be formulated as the following optimization problem:

(31)

Remark 1: In this paper, the relay estimates the source-relay

channel and the destination estimates the relay-destination

channel. The forwarding matrix

designed at the relay. Therefore, the estimated second hop CSI

should be fed back from destination to relay. However, when

channel is varying slowly, and the channel estimation feedback

occurs infrequently, the errors in feedback can be negligible.

and equalizer are

V. PROPOSED CLOSED-FORM SOLUTION FOR

In this section, we will derive a closed-form solution for the

optimizationproblem (31).Inordertofacilitatetheanalysis,the

optimization problem (31) is rewritten as

’S AND

’S

(32)

withthephysicalmeaningof

power over the

The Lagrangian function of the optimization problem (32) is

beingthemaximumallocated

subcarrier.

(33)

wherethepositivescalars

Differentiating (33) with respect to

ting the corresponding results to zero, the Karush-Kuhn-Tucker

(KKT) conditions of the optimization problem (32) are given

by [28]

andaretheLagrangemultipliers.

,and , and set-

(34a)

(34b)

(34c)

(34d)

(34e)

(34f)

(34g)

(34h)

Itisobviousthattheobjectivefunctionandconstraintsof(32)

are continuously differentiable. Furthermore, it is easy to see

that solutions of the optimization problem (32) satisfy the regu-

larity condition, i.e., Abadie constraint qualification (ACQ), be-

cause linear independence constraint qualification (LICQ) can

be proved [29]. Based on these facts, the KKT conditions are

the necessary conditions.3From KKT conditions, we can derive

thefollowingtwo usefulpropertieswhichcanhelpus tofindthe

optimal solution.

Property 1: It is proved in Appendix C that for any

isfying the KKT conditions (34a)–(34e), the power constraints

(34g) and (34h) must occur on the boundaries

sat-

(35)

(36)

Furthermore, the correspondingsatisfies

(37)

Property 2: Define the matrices

based on eigenvalue decomposition (EVD) and singular

value decomposition (SVD) as

,,,, and

(38)

(39)

with elements of the diagonal matrix

decreasing order. Then with KKT conditions (34a) and (34b), it

is proved in Appendix D that the optimal forwarding matrix

and equalizermust be in the form

andarranged in

(40)

(41)

where

and

tively, and

columns of

Right multiplying both sidesof (34a) with

plying both sides of (34b) with

(41), the first two KKT conditions become

and

are the first

are to be determined. The matrix

columns of

. Similarly,

, and

and, respec-

is the first

.

and left multi-

, and making use of (40) and

(42)

(43)

3Notice that the solution ?

? ? also satisfies the KKT conditions, but this solution is meaningless

as no signal can be transmitted [14].

? ??? ? ?

? ? and ?

? ??? ?

?

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6330IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 12, DECEMBER 2010

where thematrix

Similarly,

paper, we consider AF MIMO-OFDM relay systems, the ma-

trices

and canbe ofarbitrary dimensioninsteadofthe

square matrices considered in point-to-point systems [14], [22].

Then, the solutions satisfying KKT conditions and obtained by

solving (42) and (43) are not unique. To identify the optimal so-

lution, we need an additional information which is presented in

the following Property 3.

Property 3: Putting the results of Property 1 and Property

2 into the optimization problem (32), based on majorization

theory, it is proved in Appendix E that the optimal

have the following diagonal structure:

isthe principalsubmatrixof

principalsubmatrixof

.

isthe .Inthis

and

(44)

(45)

where

tobedetermined,and

is obtained by applying majorization theory to the original op-

timization problem. It is also a necessary condition for the op-

timal solution, and contains different information from that of

Property 2.

Combining Property 2 and Property 3, and following the ar-

gument in [14], it can be concluded that the optimal solution

of

andis unique. Now, substituting (44) and (45)

into (42) and (43), and noticing that all matrices are diagonal,

andcan be easily solved to be

and are twodiagonal matrices

.NoticethatProperty3

(46)

(47)

where the matrices

matrices of

and

with dimension

are the principal sub-

and , and

. The matrices

columns of

,

and

, respectively. From (46) and (47), it can be seen

that the optimal solutions are variants of water-filling solution.

Furthermore, the eigen channels of two hops are paired based

on the best-to-best criterion at the relay.

In the general solution (46), (47),

known. However, notice that from (35) and (37) in Property 1,

the optimal forwarding matrix and equalizer should simultane-

ously satisfy

are the first,

and

,, andare un-

(48)

(49)

Substituting (44)–(47) into (48) and (49), it can be straightfor-

wardly shown that

andcan be expressed as functions of

(50)

(51)

where

,,, andare defined as

(52a)

(52b)

(52c)

(52d)

and

being1or0,andservestoreplacetheoperation‘ ’.Combining

all the results in this section, we have the following summary.

Summary: The optimal forwarding matrix

izer

are

is a diagonal selection matrix with diagonal elements

and equal-

(53)

(54)

where

(55)

(56)

with

andgiven by (50)–(52).

From the above summary, it is obvious that the problem of

finding optimal forwarding matrix and equalizer reduces to

computing

, and it can be solved based on (51) and the

following two constraints [i.e., (34f) and (36)]

(57)

(58)

In the following subsections, we will discuss how to compute

.

Remark 2: When both channels in the two hops are flat-

fading channels, theconsidered system reducesto single-carrier

AF MIMO relay system. Note that for single-carrier systems no

power allocation has to be calculated since only one carrier ex-

ists, i.e.,

,. In this case, the proposed closed-

form solution is exactly the optimal solution for the transceiver

design under channel estimation errors in flat-fading channel.

Furthermore,whentheCSIinthetwohopsareperfectlyknown,

the derived solution reduces to the optimal solution proposed in

[19].

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XING et al.: DUAL-HOP NONREGENERATIVE MIMO-OFDM RELAY SYSTEMS6331

Remark3: Noticethatwhenthesource-relaylinkisnoiseless

and the first hop channel is an identity matrix, the closed-form

solution can be simplified to the optimal linear MMSE trans-

ceiver under channel uncertainties for point-to-point MIMO-

OFDM systems [24]. Moreover, if single carrier transmission

is employed, the closed-form solution further reduces to the op-

timal point-to-point MIMO LMMSE transceiver under channel

uncertainties [22].

Remark4: Thecomplexityoftheproposedalgorithmisdom-

inatedby one matrixinversion of

three matrix multiplications and one EVD in (38), one matrix

inversion of

, two matrix multiplications and one

SVD in (39), four matrix multiplications in (53), four matrix

multiplications in (54), and two water-filling computations

in (55) and (56). Note that the matrix inversions in (53) and

(54) are the same as those in (38) and (39) and therefore

their computations could be saved. Specifically, in (38), the

matrix inversion, matrix multiplications and EVD operation

have complexities of

, respectively [30]. In (39), the matrix inversion,

matrix multiplications and SVD operation costs

, and

With the diagonal structures of

matrix multiplications in (53) and (54) have complexities

of

,

, and

,

, respectively.

and, the

and

, respectively.

On the other hand, the complexities for the two water-filling

computations in (55) and (56) are

AF MIMO-OFDM system with

of the proposed transceiver design is approximately upper

bounded by

, where

. As a result, for the

subcarriers, the complexity

.

A. Uncorrelated Channel Estimation Error

When the channel estimation errors are uncorrelated (for ex-

ample, by using training sequences that are white in both time

and space dimensions), the following condition must be satis-

fied [10], [31]–[33]:

(59)

Then according to (14), we have

. Similarly, for the second hop, we also have

(60)

where the specific form of

(26).

Putting (60) into the left-hand side of (38), the expression

becomes

can be easily derived based on

(61)

Applying eigen-decomposition

and comparing with the right-hand side of (38), we have

(62)

Substituting (62) into (51),

reduces to

(63)

where

With (63) and the facts that

, can be straightforwardly computed to be

is theprincipal submatrix of.

and

(64)

where

equals

(65)

B. Correlated Channel Estimation Error

Due to limited length of training sequence,

not be possible to achieve [31]. In this case, the channel esti-

mation errors are correlated, and

be seen that the relationship between

expressed in a closed-form. Then the solution for

be directly obtained. Here, we employ the spectral approxima-

tion (SPA)

may

. From (38), it can

andcannot be

cannot

(66)

Forspectralapproximation,

where

plying (66) totheMSE formulationin (28),it is obviousthatthe

resultant expression forms an upper-bound to the original MSE.

Notice that when the training sequences are close to white se-

quence [35], [36], the eigenvalue spread of

SPA is a good approximation. With SPA, the left-hand side of

(38) becomes

isreplacedby,

is the maximum eigenvalue of. Ap-

is small, and

(67)

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6332 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 12, DECEMBER 2010

Comparing (67) to (61), it is obvious that the problem becomes

exactly the same as that discussed for uncorrelated channel es-

timation errors. Therefore, the allocated power to the

carrier

can be calculated by (64) but with

.

sub-

replaced by

VI. EXTENSION TO THE JOINT DESIGN INVOLVING

SOURCE PRECODER

Notice that the design in the previous section is suitable for

scenarios where the source has fixed precoder. For example,

the source precoder can be set to

or space-time block coding matrix for increasing diversity. On

the other hand, if source precoder, relay forwarding matrix and

destination equalizer are jointly designed, we can proceeds as

follows. First, with a source precoder

the system model in (2) is rewritten as

for full spatial multiplexing

before transmission,

(68)

It can be seen that (68) is the same as (2) except

is in the place of. Furthermore, without loss of gener-

ality, we can assume

represented by

. Then by using the substitutions

andinto the first line of (21), and fol-

lowingthesamederivationinSectionIV,itcanbeeasilyproved

that the data MSE at destination in the

in (68) as all correlations are

subcarrier is

(69)

where

(70)

Comparing(28)to(69),itcanbeseenthatanotherwaytoobtain

the data MSE with source precoder is to use the substitutions

,

(28).

Withtheadditionalpowerconstraintforthesourceprecoders,

the optimization problem of joint transceiver design is formu-

lated as

, and, in

(71)

where

eral,theoptimizationproblem(71)isnonconvexwithrespective

to the three design variables, and there is no closed-form solu-

tion. However, when

’s are fixed, the solution for

’scanbedirectlyobtainedfromresultsgivenby(46)and(47)

is the maximum transmit power at the source. In gen-

’s and

with substitutions

and

fixed, the optimization problem (71) is convex with respect to

’s. Therefore, an iterative algorithm can be employed for

joint design of source precoder, relay forwarding matrix and

destination equalizer.

Inordertosolve

’swhen

MSE (69) is rewritten as

,,

. On the other hand, when’s and’s are

’sand’sare fixed,thedata

(72)

with

(73)

(74)

In (73), we have used the spectral approximation

, so that the objective function for designing

’s is consistent with that of

there is no correlation in the second hop channel estimation

error,

Notice that the data MSE (72) is equivalent to the following

expression involving Frobenius norm

’s and ’s. However, if

and there is no approximation.

(75)

Furthermore, the two power constraints in the optimization

problem (71) can also be reformulated into expressions in-

volving Frobenius norm

(76)

(77)

where

(78)

Because the last term

be neglected, and the optimization problem (71) with respective

to

’s can be formulated as the following second-order conic

programming (SOCP) problem [see (79) at the bottom of the

in (72) is independent of’s, it can

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XING et al.: DUAL-HOP NONREGENERATIVE MIMO-OFDM RELAY SYSTEMS6333

page].Thisproblemcanbeefficientlysolvedbyusinginterpoint

polynomial algorithms [28].

When

’s are fixed, the proposed solutions for

’s in the previous section are the optimal solution for the

corresponding optimization problem. On the other hand, when

’s and’s are fixed, the solution for

the SOCP problem is also the optimal solution. It means that

the objective function of joint transceiver design monotonically

decreases at each iteration, and the proposed iterative algorithm

converges.

’s and

’s obtained from

VII. SIMULATION RESULTS AND DISCUSSIONS

In this section, we investigate the performance of the

proposed algorithms. For the purpose of comparison, the

algorithm based on estimated channel only (without taking

the estimation errors into account) is also simulated. An AF

MIMO-OFDM relay system where the source, relay and

destination are equipped with same number of antennas,

is considered. The number of

subcarriers

channels in both hops is

is generated according to the HIPERLAN/2 standard [10].

The signal-to-noise ratio (SNR) of the first hop is defined as

, and is fixed as 30 dB. At the source, on

each subcarrier, four independent data streams are transmitted,

and QPSK is used as the modulation scheme. The SNR at the

second hop is defined as

MSE is referred to total simulated MSE over all subcarriers

normalized by

. Each point in the following figures is an av-

erage of 10 000 realizations. In order to solve SOCP problems,

the widely used optimization Matlab toolbox CVX is adopted

[39].

Based on the definition of

matrix. In the following, only the effect of spatial correlation in

training sequence is demonstrated, and the training is white in

time domain. In this case,

canbewrittenas

is set to be 64, and the length of the multipath

. The channel impulse response

. In the figures,

in (9),is a block circular

is a block diagonal matrix, and

,whereis

the spatial correlation matrix of the training sequence. Further-

more, the widely used exponential correlation model is adopted

todenotethespatialcorrelation[22],[23],andthereforewehave

(80)

Itisassumedthatthesametrainingsequenceisusedforchannel

estimation in the two hops. Based on the definition of

in (24) and (26), and together with (80), we have

and

(81)

where

estimation errors and

process.

First, we investigate the performance of the proposed algo-

rithm with fixed source precoder

(81). Fig. 2 shows the MSE of the received signal at the destina-

tion with different

. It can be seen that the performance of the

proposed algorithm is always better than that of the algorithm

based on estimated CSI only, as long as

more, the performance improvement of the proposed algorithm

over the algorithm based on only estimated CSI enlarges when

increases.

Fig. 3 shows the MSE of the output data at the destination for

both proposed algorithm and the algorithm based on estimated

CSI only with fixed source precoder

ferent

. It can be seen that although performance degradation

is observed for both algorithms when

algorithm shows a significant improvement over the algorithm

based on estimated CSI only. Furthermore, as

bestdataMSEperformance,itdemonstratesthatwhitesequence

is preferred in channel estimation.

can be viewed as the variance of channel

is SNR during channel estimation

and whenin

is not zero. Further-

and with dif-

increases, the proposed

gives the

...

(79)

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6334 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 12, DECEMBER 2010

Fig.2. MSEofreceivedsignalatthedestinationfordifferent? when? ? ???

and with ?

? ? .

Fig. 3. MSE of received signal at the destination for different ? when

? ???

? ?? dB and with ?

? ? .

Fig. 4 shows the bit error rates (BER) of the output data at

the destination for different

, when

that the BER performance is consistent with MSE performance

in Fig. 2.

When source precoder design is considered, the proposed al-

gorithm is an iterative algorithm. Fig. 5 shows the convergence

behavioroftheproposediterativealgorithmwithdifferentinitial

values of

. In the figure, the suboptimal solution as the initial

value for

refers to the solution given in [24] based on the first

hop CSI. It can be seen that the proposed algorithm with sub-

optimal solution as initial value has a faster convergence speed

than that with identity matrix as the initial value.

Fig. 6 compares the data MSEs of the proposed iterative

algorithm under channel uncertainties and the iterative algo-

rithm based on estimated CSI only in [20]. Similar to the case

. It can be seen

Fig. 4. BER of received data at the destination for different ? when ? ? ???

and with ?

? ? .

Fig. 5. Convergence behavior of the proposed iterative algorithm when ? ?

??? and ?

? ????.

with fixed source precoder, the proposed joint design algorithm

taking into account the channel estimation uncertainties per-

forms better than the algorithm based on estimated CSI only.

Finally, Fig. 7 illustrates the data MSE of the iterative trans-

ceiver design algorithm based on estimated CSI only [20] and

the proposed algorithms with source precoder jointly designed

or simply set to

. It can be seen that when CSI is per-

fectly known

, the algorithms with source precoder

design performs better than that by setting precoder

On the other hand, when

rithm with simple precoder

algorithm based on estimated CSI only with source precoder

design. Furthermore, when the channel estimation errors in-

creases, the performance gap between the proposed algorithms

with and without source precoder design decreases. Notice that

.

, even the proposed algo-

performs better than the

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XING et al.: DUAL-HOP NONREGENERATIVE MIMO-OFDM RELAY SYSTEMS 6335

Fig. 6. MSE of received signal at the destination for different ? when ? ?

???.

Fig. 7. MSE of received data at the destination for different ? , when ? ? ???

and ? ???

? ?? dB.

the algorithm without source precoder design has a much lower

complexity, thus it represents a promising tradeoff in terms of

complexity and performance.

VIII. CONCLUSION

In this paper, linear transceiver design was addressed for AF

MIMO-OFDM relaying systems with channel estimation errors

based on MMSE criterion. The linear channel estimators and

the corresponding MSE expressions were first derived. Then a

general solution for optimal relay forwarding matrix and desti-

nationequalizer wasproposed. When thechannelestimationer-

rorsareuncorrelated,theoptimalsolutionisinclosed-form,and

it includes several existing transceiver design results as special

cases. Furthermore, the design was extended to the case where

source precoder design is involved. Simulation results showed

that the proposed algorithms offer significant performance im-

provements over the algorithms based on estimated CSI only.

APPENDIX A

PROOF OF (7)

BasedonthecharacteristicsofDFToperation,thematrix

definedin(6)isa

(82) at the bottom of the page, whose element

(8).Itis obviousthat

is the

channels between the source and relay in the time domain and

is the length of the multi-path channel.

Ontheotherhand,basedonthedefinitionof

the relationship between

and

blockcirculantmatrixgivenby

is defined in

tapofthemulti-pathMIMO

in(6),wehave

which is given by (83).

(83)

From (82) and (83), by straightforward computation, the signal

model given in (6) can be reformulated as

(84)

where the matrix

is defined in (9).

... ... ...............

(82)

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6336IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 12, DECEMBER 2010

APPENDIX B

PROOF OF (17)

For the expectation of the following product

(85)

where

patible dimension to

and are tworandom matrices with com-

element of, theis

(86)

If the two random matrices

andsatisfy

(87)

where

trix, then we have the equality

is a matrix whileis ama-

. Asandare scalars, (86)

can be further written as

(88)

Finally, writing (88) back to matrix form, we have [37]

(89)

Notice that this conclusion is independent of the ma-

trix variate distributions of

mined by their second order moments. Putting

and, but only deter-

,and

, into (89), we have (17).

APPENDIX C

PROOF OF PROPERTY 1

Right multiplying both sides of (34a) with

equality holds

, the following

(90)

Left multiplying (34b) with

, we have

(91)

After taking the traces of both sides of (90) and (91) and with

the fact that the traces of their right-hand sides are equivalent,

i.e.,

we directly have

(92)

By the property of trace operator

and (92) reduces to

(93)

On the other hand, based on the definition of

can be also expressed as

in (30),

(94)

Comparing (93) with (94), it can be concluded that

(95)

Putting (95) into (34c), we have

. As, it is straightforward that

(96)

Furthermore, based on the fact

and taking summation of both sides of (96), the following

equation holds:

(97)

Putting (97) into (34e), we have

(98)

and it follows that

(99)

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XING et al.: DUAL-HOP NONREGENERATIVE MIMO-OFDM RELAY SYSTEMS6337

Sincefortheoptimalequalizer

, it can be concluded that

fied, we must have

,

. In order to have (34c) satis-

(100)

Furthermore, as

, based on (34e), it is also concluded that

(101)

Finally, (96) constitutes the second part of the Property 1.

APPENDIX D

PROOF OF PROPERTY 2

Defining a full rank Hermitian matrix

, then for an arbitrary

written as

matrix, it can be

(102)

wherethe innermatrix

.

equals to

Putting (102) into (34a), and with the following definitions

[the same as the definitions in (38) and (39)]:

(103)

(104)

the equalizer

can be reformulated as

(105)

where the second equality is due to the matrix inversion lemma.

Putting (96) from Appendix C into (34b), after multiplying

both sides of (34b) with, we have

(106)

Then substituting

have

in (102) and in (105) into (106), we

(107)

Since

their ranks by

concluded that

andarerectangulardiagonalmatrices(denoting

andrespectively), based on (107), it can be

has the following form

(108)

where

thermore, putting (108) into the definition of

have

is of dimension and to be determined. Fur-

in (105), we

(109)

where

stituting (108) and (109) into (102) and (105), it can be con-

cluded that

is of dimension, and to be determined. Sub-

(110)

(111)

where

(112)

and

is theprincipal submatrix of.

APPENDIX E

PROOF OF PROPERTY 3

Taking the trace of both sides of (42) and (43), and noticing

that the resultant two equations are the same, it is obvious that

(113)

On the other hand, substituting (111) into (96) in Appendix C,

we have

(114)

Comparing (113) and (114), it follows that

(115)

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6338IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 12, DECEMBER 2010

For the objective function in the optimization problem (32),

substituting (40) and (41) into the MSE expression in (28), the

MSE on the

subcarrier can be written as

(116)

where

on (115) and (116), the optimization problem (32) becomes as

is a constant part independent of . Therefore, based

(117)

For any given

be decoupled into a collection of the following suboptimization

problems:

, then the optimization problem (117) can

(118)

where the constant part

positive semidefinite Hermitian matrices

is neglected. For any two

and , we have

denotes the, where

largest eigenvalue of the matrix

the fact that elements of the diagonal matrix

creasing order, the objective function of (118) is minimized,

when

diagonal elements in decreasing order. The objective function

can be rewritten as

[38]. Together with

are in de-

is a diagonal matrix with the

(119)

where

onal elements of the matrix

It follows that

3.H.3]. Then, based on [15, Theorem 1], the optimal

the following structure:

denotes the vector which consists of the main diag-

.

is a Schur-concave function of[38,

has

(120)

where

and

Putting (120) into the definition of

structure of the optimal

is adiagonal matrix to be determined,

.

in (112), the

is given by

(121)

where

is also adiagonal matrix.

REFERENCES

[1] A. Scaglione, D. L. Goeckel, and J. N. Laneman, “Cooperative com-

munications in mobile ad hoc networks,” IEEE Signal Process. Mag.,

pp. 18–29, Sep. 2006.

[2] J.N.Laneman,D.N.C.Tse,andG.W.Wornell,“Cooperativediversity

in wireless networks: Efficient protocols and outage behavior,” IEEE

Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004.

[3] O.Munoz-Medina,J.Vidal,andA.Agustin,“Lineartransceiverdesign

in nonregenerative relays with channel state information,” IEEE Trans.

Signal Process., vol. 55, no. 6, pp. 2953–2604, Jun. 2007.

[4] X. Tang and Y. Hua, “Optimal design of non-regenerative MIMO

wireless relays,” IEEE Trans. Wireless Commun., vol. 6, no. 4, pp.

1398–1407, Apr. 2007.

[5] C.-B. Chae, T. W. Tang, R. W. Health, and S.-Y. Cho, “MIMO re-

laying with linear processing for multiuser transmission in fixed relay

networks,” IEEE Trans. Signal Process., vol. 56, no. 2, pp. 727–738,

Feb. 2008.

[6] A. S. Behbahani, R. Merched, and A. J. Eltawil, “Optimizations of a

MIMO relay network,” IEEE Trans. Signal Process., vol. 56, no. 10,

pt. 2, pp. 5062–5073, Oct. 2008.

[7] B. Khoshnevis, W. Yu, and R. Adve, “Grassmannian beamforming for

MIMO amplify-and-forward relaying,” IEEE J. Sel. Areas Commun.,

vol. 26, no. 8, pp. 1397–1407, Oct. 2008.

[8] D.TseandP.Viswanath,FundamentalsofWirelessCommunication.

Cambridge , U.K.: Cambridge Univ. Press, 2005.

[9] S. Ma and T.-S. Ng, “Time domain signal detection based on

second-order statistics for MIMO-OFDM system,” IEEE Trans. Signal

Process., vol. 55, no. 3, pp. 1150–1158, Mar. 2007.

[10] J. Chen, Y.-C. Wu, S. Ma, and T.-S. Ng, “Joint CFO and channel esti-

mation for multiuser MIMO-OFDM systems with optimal training se-

quences,” IEEE Trans. Signal Process., vol. 56, no. 8, pp. 4008–4019,

Aug. 2008.

[11] F. Verde, D. Darsena, and A. Scaglione, “Cooperative randomized

MIMO-OFDM downlinkfor multicell networks:Design andanalysis,”

IEEE Trans. Signal Process., vol. 58, no. 1, pp. 384–402, Jan. 2010.

[12] I.Hammerstrom and A. Wittneben,“Power allocation schemes for am-

plify-and-forward MIMO-OFDM relay links,” IEEE Trans. Wireless

Commun., vol. 6, no. 8, pp. 2798–2802, Aug. 2007.

[13] H. Bolcskei, D. Gesbert, C. B. Papadias, and A.-J. Van Der Veen,

Space-Time Wireless Systems.

Press, 2006.

[14] H. Sampath, P. Stoica, and A. Paulraj, “Generalized linear precoder

and decoder design for MIMO channels using the weighted MMSE

criterion,” IEEE Trans. Commun., vol. 49, no. 12, pp. 2198–2206, Dec.

2001.

[15] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, “Joint robust Tx-Rx

beamformingdesignformulticarrierMIMOchannels:Aunifiedframe-

work for convex optimization,” IEEE Trans. Signal Process., vol. 51,

no. 9, pp. 2381–2399, Sep. 2003.

[16] M. Joham, W. Utschick, and J. A. Nossek, “Linear transmit processing

in MIMO communications systems,” IEEETrans. Signal Process., vol.

53, no. 8, pp. 2700–2712, Aug. 2005.

[17] S. Serbetli and A. Yener, “Transceiver optimization for mutiuser

MIMO systems,” IEEE Trans. Signal Process., vol. 52, no. 1, pp.

214–226, Jan. 2004.

[18] Z. Q. Luo, T. N. Davidson, G. B. Giannakis, and K. M. Wong, “Trans-

ceiver optimization for block-based multiple access through ISI chan-

nels,” IEEE Trans. Signal Process., vol. 52, no. 4, pp. 1037–1052, Apr.

2004.

[19] W. Guan and H. Luo, “Joint MMSE transceiver design in non-regen-

erative MIMO relay systems,” IEEE Commun. Lett., vol. 12, no. 7, pp.

517–519, Jul. 2008.

Cambridge, U.K.: Cambridge Univ.

Page 15

XING et al.: DUAL-HOP NONREGENERATIVE MIMO-OFDM RELAY SYSTEMS6339

[20] Y. Rong, X. Tang, and T. Hua, “A unified framework for optimizing

linear non-regenerative multicarrier MIMO relay communication sys-

tems,” IEEE Trans. Signal Process., vol. 57, no. 12, pp. 4837–4851,

Dec. 2009.

[21] L. Musavian, M. R. Nakhi, M. Dohler, and A. H. Aghvami, “Effect of

channel uncertianty on the mutual information of MIMO fading chan-

nels,” IEEE Trans. Veh. Technol., vol. 56, no. 5, pp. 2798–2806, Sep.

2007.

[22] M. Ding and S. D. Blostein, “MIMO minimum total MSE transceiver

design with imperfect CSI at both ends,” IEEE Trans. Signal Process.,

vol. 57, no. 3, pp. 1141–1150, Mar. 2009.

[23] X. Zhang, D. P. Palomar, and B. Ottersten, “Statistically robust design

of linear MIMO transceiver,” IEEE Trans. Signal Process., vol. 56, no.

8, pp. 3678–3689, Aug. 2008.

[24] F. Rey, M. Lamarca, and G. Vazquez, “Robust power allocation algo-

rithms for MIMO OFDM systems with imperfect CSI,” IEEE Trans.

Signal Process., vol. 53, no. 3, pp. 1070–1085, Mar. 2005.

[25] E. G. Larsson and P. Stoica, Space-Time Block Coding for Wireless

Communications. Cambridge , U.K.: Cambridge Univ. Press, 2003.

[26] C. P. Robert, The Bayesian Choice.

[27] F. A. Dietrich, P. Breun, and W. Wolfgang, “Robust Tomlinson-Ha-

rashima precoding for the wireless broadcast channel,” IEEE Trans.

Signal Process., vol. 55, no. 2, pp. 631–644, Feb. 2007.

[28] S. Boyd and L. Vandenberghe, Convex Optimization.

U.K.: Cambridge Univ. Press, 2004.

[29] D. P. Bertsekas, A. Nedic, and A. E. Ozdaglar, Convex Analysis and

Optimization.New York: Athena Scientific, 2003.

[30] R. A. Horn and C. R. Johnson, Matrix Analysis.

Cambridge Univ. Press, 1985.

[31] P. Stoica and O. Besson, “Training sequence deisgn for frequency

offset and frequency-selective channel estimation,” IEEE Trans.

Commun., vol. 51, no. 11, pp. 1910–1917, Nov. 2003.

[32] M. Ghogho and A. Swami, “Training design for multipath channel and

frequency-offset estimation in MIMO systems,” IEEE Trans. Signal

Process., vol. 54, no. 10, pp. 3957–3965, Oct. 2006.

[33] H. Minn and N. Al-Dhahir, “Optimal training signals for MIMO

OFDM channel estimation,” IEEE Trans. Wireless Commun., vol. 5,

no. 5, pp. 1158–1168, May 2006.

[34] A. Beck, A. Ben-Tal, and Y. C. Eldar, “Robust mean-squared error es-

timation of multiple signals in linear systems affected by model and

noise uncertainties,” Math. Program., vol. 107, pp. 155–187, 2006,

Springer.

[35] J. G. Andrews, A. Ghosh, and R. Muhamed, Fundamentals of

WiMAX. Englewood Cliffs, NJ: Prentice-Hall, 2007.

[36] F. Ohrtman and K. Roeder, Wi-Fi Handbook: Building 802.11 b Wire-

less Networks.New York: McGraw-Hill, 2003.

[37] S. Kay, Fundamental of Statistical Signal Processing: Estimation

Theory. Englewood Cliffs, NJ: Prentice-Hall, 1993.

[38] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and

Its Applications.New York: Academic, 1979.

[39] M. Grant, S. Boyd, and Y. Y. Ye, CVX: Matlab Software for Dis-

ciplined Convex Programming ver. V.1.0RC3, Feb. 2007 [Online].

Available: http://www.stanford.edu/boyd/cvx/

New York: Springer , 2001.

Cambridge,

Cambridge, U.K.:

ChengwenXingreceivedtheB.Eng.degreefromXi-

dian University, Xi’an, China, in 2005 and the Ph.D.

degree in electrical and electronic engineering from

the University of Hong Kong (HKU), Hong Kong, in

2010.

Since September 2010, he has been with the

School of Information and Electronics, Beijing

Institute of Technology (BIT), Beijing, China, where

he is currently a Lecturer. His current research

interests include statistical signal processing, convex

optimization, multivariate statistics, optimization,

matrix analysis and cooperative communication systems.

Shaodan Ma received the B.Sc. (Eng.) and the

M.Eng.Sc. degrees from NanKai University, Tianjin,

China, in 1999 and 2002, respectively, all in elec-

trical engineering, and the Ph.D. degree in electrical

and electronic engineering from The University of

Hong Kong (HKU), Hong Kong, in 2006.

Since 2006, she has been with the Department of

Electrical and Electronic Engineering, HKU, as a

Postdoctoral Fellow. Her research interests include

wireless communication systems, spread spectrum

techniques, MIMO systems, OFDM technique, and

digital signal processing.

Yik-Chung Wu received the B.Eng. (EEE) degree in

1998 and the M.Phil. degree in 2001 from The Uni-

versity of Hong Kong (HKU), Hong Kong. After re-

ceiving the M.S. degree, he was a Research Assistant

with the same university. He received the Ph.D. de-

gree in 2005 from Texas A&M University, College

Station.

During his study at Texas A&M University, he

was fully supported by the prestigious Croucher

Foundation scholarship. From August 2005 to

August 2006, he was with the Thomson Corporate

Research, Princeton, NJ, as a Member of Technical Staff. Since September

2006, he has been with HKU as an Assistant Professor. His research interests

are in general area of signal processing and communication systems, and in

particular, receiver algorithm design, synchronization techniques, channel

estimation, and equalization.

Dr. Wu was a TPC member for IEEE VTC Fall 2005, Globecom 2006, 2008,

ICC2007,and2008.He iscurrentlyservingasan AssociateEditorfor theIEEE

COMMUNICATIONS LETTERS.

Tung-Sang Ng (S’74–M’78–SM’90–F’03) received

the B.Sc. (Eng.) degree from The University of

Hong Kong (HKU), Hong Kong, in 1972, and the

M.Eng.Sc. and Ph.D. degrees from the University of

Newcastle,Australia, in 1974 and 1977, respectively,

all in electrical engineering.

He worked for BHP Steel International and The

University of Wollongong, Australia after graduation

for 14 years before returned to HKU in 1991, where

he was Professor and Chair of Electronic Engi-

neering. He was Head of Department of Electrical

and Electronic Engineering from 2000 to 2003 and Dean of Engineering from

2003 to 2007. His current research interests include wireless communication

systems, spread spectrum techniques, CDMA, and digital signal processing.

He has published more than 300 international journal and conference papers.

Dr. Ng was the General Chair of ISCAS’97 and the VP-Region 10 of IEEE

CAS Society in 1999 and 2000. He was an Executive Committee Member and a

Board Member of the IEE Informatics Divisional Board (1999–2001) and was

an ordinary member of IEE Council (1999–2001). He was awarded the Hon-

orary Doctor of Engineering Degree by the University of Newcastle in 1997,

the Senior Croucher Foundation Fellowship in 1999, the IEEE Third Millenium

medal in 2000, and the Outstanding Researcher Award by HKU in 2003. He is

a Fellow of IET and HKIE.