IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 12, DECEMBER 2010 6325
Transceiver Design for Dual-Hop Nonregenerative
MIMO-OFDM Relay Systems Under
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
to thejointdesign involvingsource precoder design.Simulation re-
sults show that the proposed design outperforms the design based
on estimated channel state information only.
matrix, minimum mean-square-error (MMSE).
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 , .
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
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 –. The
combination of AF and MIMO-OFDM becomes an attractive
of wireless links, dual-hop relaying is being considered to
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: email@example.com).
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:
firstname.lastname@example.org; email@example.com; firstname.lastname@example.org).
Color versions of one or more of the figures in this paper are available online
Digital Object Identifier 10.1109/TSP.2010.2070797
option for enabling high-speed wireless multi-media services
In the last decade, linear transceiver design for various
systems has been extensively investigated because of its low
implementation complexity and satisfactory performance ,
(MMSE) is one of the most important and frequently used cri-
teria –. For example, for point-to-point MIMO and
MIMO-OFDM systems, linear MMSE transceiver design has
been discussed in details in –. Linear MMSE trans-
ceiver design for multiuser MIMO systems has been considered
in , . For single carrier AF MIMO relay systems, linear
MMSE forwarding matrix at the relay and equalizer at the
destination are joint designed in . Furthermore, the linear
MMSE transceiver design for dual hop MIMO-OFDM relay
systems based on prefect channel state information (CSI) is
proposed in .
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 ,  and
point-to-point MIMO-OFDM systems .
In this paper, we take a step further and consider the linear
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
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
6326 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 12, DECEMBER 2010
proposed algorithms provide an obvious advantage in terms of
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 . 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.
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
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
denotes the all zero matrix. The notation
Hermitian square root of the positive semidefinite matrix
represents the expectation operation. The
stacks the columns of the matrix
single vector. The symbol
denotes theblock diagonalmatrix with
, , and
identity matrix, while
is a Hermitian matrix.
represents Kronecker product.
. The notation
and as 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
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
at the relay on
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
is the data vector transmitted by the source with co-
, under a power con-
is the maximum transmit power. Then the resulting signal is
transmitted to the destination. The received data
tination on the
at the des-
where the symbol
subcarrier at the second hop with zero mean and covari-
. In order to guarantee the trans-
can be recovered at the destination, it is assumed
,, andare greater than or equal to
received at the relay and the signal
the destination in frequency domain can be compactly written
is the additive Gaussian noise vector on
Notice that in general the matrix
trix. This corresponds to mixing the data from different sub-
carriers at the relay, and is referred as subcarrier cooperative
AF MIMO-OFDM systems . It is obvious that when the
tems needs very high complexity. On other hand, it has been
shown in  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
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
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
. Therefore, we focus on time domain channel estimation.
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
hop channel estimation.
From the received signal model in frequency domain given
by (3), the corresponding time domain signal is
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
where the matrices
are defined as
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 the tap of the multi-path MIMO
where the element
is expressed as
Based on the signal model in (7), the linear minimum-mean-
square-error (LMMSE) channel estimate is given by 
with the corresponding MSE
for channel covariance matrix. For uncorrelated channel
channel tap .
On the other hand, the channel in frequency domain and time
domain has the following relationship1:
is the prior information
is the variance of
is computed according to (13), we have
In case there is no prior information on
sign uninformative prior to
proach infinity . In this case,
channel estimator (11) and estimation MSE (12) reduce to that
of maximum likelihood (ML) estimation [25, p. 179].
, we can as-
, that is, ap-
, and then the
block diagonal elements from
is the matrix taken from the following
Furthermore, based on (15), for an arbitrary square matrix
is proved in Appendix B that
1This relationship holds for both perfect CSI and estimated CSI.
6328 IEEE 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
we have the following property:
main. Furthermore, as the two channels are estimated indepen-
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 destination to minimize
where the expectation is taken with respect to
, , and.2Since
dent, the MSE expression (20) can be written as
, andare indepen-
andare independent, the first term of
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 .
. When the LMMSE
For the inner expectation, the following equation holds:
where based on (17) the matrix
is defined as
Applying (23) and the corresponding result for
(22), the first term of
where the matrix
is defined as
Similarly, the second term of
in (21) can be simplified
Based on (25) and (27), the
(21) equals to
Notice that the matrix
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
XING et al.: DUAL-HOP NONREGENERATIVE MIMO-OFDM RELAY SYSTEMS6329
minimizes the total MSE of the output data at the destination
can be formulated as the following optimization problem:
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
optimization problem (31) is rewritten as
power over the
The Lagrangian function of the optimization problem (32) is
Differentiating (33) with respect to
ting the corresponding results to zero, the Karush-Kuhn-Tucker
(KKT) conditions of the optimization problem (32) are given
, and , and set-
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 . Based on these facts, the KKT conditions are
the necessary conditions.3From KKT conditions, we can derive
thefollowingtwo usefulpropertieswhichcanhelpus tofindthe
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
Furthermore, the corresponding satisfies
Property 2: Define the matrices
based on eigenvalue decomposition (EVD) and singular
value decomposition (SVD) as
,,, , and
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 equalizer must be in the form
and arranged in
Right multiplying both sidesof (34a) with
plying both sides of (34b) with
(41), the first two KKT conditions become
are the first
are to be determined. The matrix
and , respec-
is the first
and left multi-
, and making use of (40) and
3Notice that the solution ?
? ? also satisfies the KKT conditions, but this solution is meaningless
as no signal can be transmitted .
? ??? ? ?
? ? and ?
? ??? ?
6330 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 12, DECEMBER 2010
paper, we consider AF MIMO-OFDM relay systems, the ma-
and canbe ofarbitrary dimensioninsteadofthe
square matrices considered in point-to-point systems , .
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:
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
Combining Property 2 and Property 3, and following the ar-
gument in , it can be concluded that the optimal solution
and is unique. Now, substituting (44) and (45)
into (42) and (43), and noticing that all matrices are diagonal,
andcan be easily solved to be
andare twodiagonal matrices
where the matrices
are the principal sub-
and , and
. The matrices
, 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-
are the first,
, , and are un-
Substituting (44)–(47) into (48) and (49), it can be straightfor-
wardly shown that
and can be expressed as functions of
,,, and are defined as
all the results in this section, we have the following summary.
Summary: The optimal forwarding matrix
is a diagonal selection matrix with diagonal elements
andgiven by (50)–(52).
From the above summary, it is obvious that the problem of
finding optimal forwarding matrix and equalizer reduces to
, and it can be solved based on (51) and the
following two constraints [i.e., (34f) and (36)]
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-
,. 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.
the derived solution reduces to the optimal solution proposed in
XING et al.: DUAL-HOP NONREGENERATIVE MIMO-OFDM RELAY SYSTEMS6331
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 . 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
inatedby one matrixinversion of
three matrix multiplications and one EVD in (38), one matrix
, 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 . In (39), the matrix inversion,
matrix multiplications and SVD operation costs
With the diagonal structures of
matrix multiplications in (53) and (54) have complexities
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
. 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 , –:
Then according to (14), we have
. Similarly, for the second hop, we also have
where the specific form of
Putting (60) into the left-hand side of (38), the expression
can be easily derived based on
and comparing with the right-hand side of (38), we have
Substituting (62) into (51),
With (63) and the facts that
, can be straightforwardly computed to be
is the principal submatrix of.
B. Correlated Channel Estimation Error
Due to limited length of training sequence,
not be possible to achieve . 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-
. From (38), it can
and cannot be
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 , , the eigenvalue spread of
SPA is a good approximation. With SPA, the left-hand side of
is the maximum eigenvalue of. Ap-
is small, and
6332IEEE 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
can be calculated by (64) but with
VI. EXTENSION TO THE JOINT DESIGN INVOLVING
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
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
. Then by using the substitutions
and into the first line of (21), and fol-
that the data MSE at destination in the
in (68) as all correlations are
the data MSE with source precoder is to use the substitutions
the optimization problem of joint transceiver design is formu-
, and , in
to the three design variables, and there is no closed-form solu-
tion. However, when
’s are fixed, the solution for
is the maximum transmit power at the source. In gen-
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
MSE (69) is rewritten as
. On the other hand, when ’s and ’s are
’sand ’sare fixed,thedata
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
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.
Furthermore, the two power constraints in the optimization
problem (71) can also be reformulated into expressions in-
volving Frobenius norm
Because the last term
be neglected, and the optimization problem (71) with respective
’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
XING et al.: DUAL-HOP NONREGENERATIVE MIMO-OFDM RELAY SYSTEMS6333
polynomial algorithms .
’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
’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
channels in both hops is
is generated according to the HIPERLAN/2 standard .
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
. 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
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,
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
the spatial correlation matrix of the training sequence. Further-
more, the widely used exponential correlation model is adopted
estimation in the two hops. Based on the definition of
in (24) and (26), and together with (80), we have
estimation errors and
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
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
. 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
is preferred in channel estimation.
can be viewed as the variance of channel
is SNR during channel estimation
is not zero. Further-
and with dif-
increases, the proposed
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
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
. In the figure, the suboptimal solution as the initial
refers to the solution given in  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 . 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  and
the proposed algorithms with source precoder jointly designed
or simply set to
. It can be seen that when CSI is per-
, 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
XING et al.: DUAL-HOP NONREGENERATIVE MIMO-OFDM RELAY SYSTEMS6335
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.
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-
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.
PROOF OF (7)
(82) at the bottom of the page, whose element
channels between the source and relay in the time domain and
is the length of the multi-path channel.
the relationship between
is defined in
which is given by (83).
From (82) and (83), by straightforward computation, the signal
model given in (6) can be reformulated as
where the matrix
is defined in (9).
............ ...... ...
6336 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 12, DECEMBER 2010
PROOF OF (17)
For the expectation of the following product
patible dimension to
andare two random matrices with com-
element of, the is
If the two random matrices
trix, then we have the equality
is a matrix while is a ma-
. As and are scalars, (86)
can be further written as
Finally, writing (88) back to matrix form, we have 
Notice that this conclusion is independent of the ma-
trix variate distributions of
mined by their second order moments. Putting
and, but only deter-
, into (89), we have (17).
PROOF OF PROPERTY 1
Right multiplying both sides of (34a) with
, the following
Left multiplying (34b) with
, we have
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,
we directly have
By the property of trace operator
and (92) reduces to
On the other hand, based on the definition of
can be also expressed as
Comparing (93) with (94), it can be concluded that
Putting (95) into (34c), we have
. As , it is straightforward that
Furthermore, based on the fact
and taking summation of both sides of (96), the following
Putting (97) into (34e), we have
and it follows that
XING et al.: DUAL-HOP NONREGENERATIVE MIMO-OFDM RELAY SYSTEMS6337
, it can be concluded that
fied, we must have
. In order to have (34c) satis-
, based on (34e), it is also concluded that
Finally, (96) constitutes the second part of the Property 1.
PROOF OF PROPERTY 2
Defining a full rank Hermitian matrix
, then for an arbitrary
matrix , it can be
Putting (102) into (34a), and with the following definitions
[the same as the definitions in (38) and (39)]:
can be reformulated as
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
in (102) andin (105) into (106), we
their ranks by
and respectively), based on (107), it can be
has the following form
thermore, putting (108) into the definition of
is of dimension and to be determined. Fur-
in (105), we
stituting (108) and (109) into (102) and (105), it can be con-
is of dimension , and to be determined. Sub-
is the principal submatrix of.
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
On the other hand, substituting (111) into (96) in Appendix C,
Comparing (113) and (114), it follows that
6338 IEEE 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
on (115) and (116), the optimization problem (32) becomes as
is a constant part independent of. Therefore, based
For any given
be decoupled into a collection of the following suboptimization
, then the optimization problem (117) can
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,
diagonal elements in decreasing order. The objective function
can be rewritten as
. Together with
are in de-
is a diagonal matrix with the
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,
Putting (120) into the definition of
structure of the optimal
is a diagonal matrix to be determined,
in (112), the
is given by
is also a diagonal matrix.
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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
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
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
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.