Communications and Network, 2010, 2, 145-151
doi:10.4236/cn.2010.23022 Published Online August 2010 (http://www.SciRP.org/journal/cn)
Copyright © 2010 SciRes. CN
Downlink MBER Transmit Beamforming Design Based on
Uplink MBER Receive Beamforming for TDD-SDMA
Induced MIMO Systems
Sheng Chen, Lie-Liang Yang
School of Electronics and Computer Sciences, University of Southampton, Southampton, UK
E-mail: {sqc, lly}@ecs.soton.ac.uk
Received January 10, 2010; revised April 26, 2010; accepted May 30, 2010
Abstract
The downlink minimum bit error rate (MBER) transmit beamforming is directly designed based on the
uplink MBER receive beamforming solution for time division duplex (TDD) space-division multiple-access
(SDMA) induced multiple-input multiple-output (MIMO) systems, where the base station (BS) is equipped
with multiple antennas to support multiple single-antenna mobile terminals (MTs). It is shown that the dual
relationship between multiuser detection and multiuser transmission can be extended to the rank-deficient
system where the number of users supported is more than the number of transmit antennas available at the
BS, if the MBER design is adopted. The proposed MBER transmit beamforming scheme is capable of
achieving better performance over the standard minimum mean square error transmit beamforming solution
with the support of low-complexity and high power-efficient MTs, particularly for rank-deficient
TDD-SDMA MIMO systems. The robustness of the proposed MBER transmit beamforming design to the
downlink and uplink noise or channel mismatch is investigated using simulation.
Keywords: Minimum Bit Error Rate, Time Division Duplex, Multiple-Input Multiple-Output, Transmit
Beamforming, Receive Beamforming, Space-Division Multiple-Access
1. Introduction
Motivated by the demand for increasing throughput in
wireless communication, antenna array assisted spatial
processing techniques [1-7] have been developed in
order to further improve the achievable spectral
efficiency. In the uplink, the base station (BS) has the
capacity to implement sophisticated receive (Rx)
beamforming schemes to separate multiple user signals
transmitted by mobile terminals (MTs). This provides a
practical means of realising multiuser detection (MUD)
for space-division multiple-access (SDMA) induced
multiple-input multiple-output (MIMO) systems. Tradi-
tionally, adaptive Rx beamforming is based on the mini-
mum mean square error (MMSE) design [2,5,6,8,],
which requires that the number of users supported is no
more than the number of receive antenna elements. If
this condition is not met, the system becomes rank-
deficient. Recently, adaptive minimum bit error rate
(MBER) Rx beamforming design has been proposed
[9-12], which outperforms the adaptive MMSE Rx
beamforming, particularly in hostile rank-deficient
systems.
In the downlink with non-cooperative MTs at the
receive end, the mobile users are unable to perform
sophisticated cooperative MUD. If the downlink's
channel state information (CSI) is known at the BS, the
BS can carry out transmit (Tx) preprocessing, leading to
multiuser transmission (MUT). The assumption that the
downlink channel impulse response (CIR) is known at
the BS is valid for time division duplex (TDD) systems
due to the channel reciprocity. However, for frequency
division duplex systems, where the uplink and downlink
channels are expected to be different, CIR feedback from
the MT's receivers to the BS transmitter is necessary [13].
Many research efforts have been made to discover the
equivalent relationship between the MUD and MUT
[14-19]. Notably, Yang [20] has derived the exact
equivalency between the MUD and MUT for TDD
systems under the condition that the number of antennas
at the BS is no less than the number of MTs supported1.
1All the existing works, such as [14-20], consider the designs of MUD
and/or MUT using second-order statistics based criteria, which implies
that the MIMO s
y
stem must have full r ank.
S. CHEN ET AL.
146
According to the results of [20], the MUT can be
obtained directly from MUD. Since the BS has to
implement MUD, it can readily implement MUT based
on its uplink MUD solution with no extra computational
complexity cost. This is very attractive as this strategy
enables the employment of low-complexity and high
power-efficient MTs to achieve good downlink per-
formance.
In general, the BS can design MUT when the
downlink CSI is available. The Tx beamforming design
based on the MMSE criterion is popular owing to its
appealing simplicity [13,21]. Since the bit error rate
(BER) is the ultimate system performance indicator,
research interests in MBER based Tx beamforming
techniques have inten sified recently [22-25]. Th is MBER
based MUT design invokes a constrained nonlinear optimi-
sation [22-25], which is typically solved using the
iterative gradient-based optimisation algorithm known as
the sequential quadratic programming (SQP) [26].
However, the computational complexity of the SQP
based MBER MUT solution can be excessive and may
become impractical for high-rate systems [23]. This
contribution adopts a very different approach to design
the MBER Tx beamforming for TDD-SDMA induced
MIMO systems, which does not suffer the above men-
tioned difficulty of high complexity. We prove that
Yang's results [20] are more general and the exact
equivalency between the MUD and MUT is not
restricted only to second-order statistics based designs.
Therefore, we can apply the results of [20] to implement
the MBER Tx beamforming scheme directly based on
the MBER Rx beamforming solution already available at
the BS with no computational cost at all, even for
rank-deficient systems where the number of antennas at
the BS is less than the number of MTs supported. The
robustness of the proposed scheme is also investigated
when the downlink and uplink noise powers or channels
mismatch.
2. Multiuser Beamforming System
The TDD-SDMA induced MIMO system considered is
depicted in Figure 1, where the BS employs
antennas to support L
K
single-antenna MTs. When the
uplink is considered, the received signal vector
at the BS is given by
,1 ,2,
=[ ]
UUU
xx xxT
UL
=1
=n= n
K
UUkk
k
s
xHsh U
(1)
where the channel matrix is given by
LK
12
=[ ]
K
Hhhh, =
kk
Figure 1. Schematic diagram of the TDD-SDMA induced
MIMO system employing transmit and receive beamfor-
mings at the BS. The system employs
L
antennas at the
BS to support
K
single-antenna MTs.
dditive white Gaussian noise (AWGN) vaector with
2
[nn] = 2
H
UU UL
E
I, and
L
I represents the LL
ithout thloss of generalit
assume the binary phase shift keying (BPSK) modulation.
The result however can be extended to modulation
schemes with multiple bits per symbol by adopting the
minimum symbol-error-rate design [9]. The MUD at the
BS consists of a bank of Rx beamformers
,=u, 1,
H
Ukk U
ykKx
identity matrix. We y, we
(2)
where is the Rx beamformer's
uk
an
weight vector for
user kd
denotes the Hermitian operator. The
decisi variae vector ,1 ,2,
=[ ]
T
UUU UK
yy yy for
the
on bl
K
transmitted symbols ca
== n
HHH
UU U
yUxUHsU
n be expressed as
(3)
with the LK
Rx beamforming coeff
y (4)
The real part of is a sufficient s
he BS employs the Tx beamforming for the
do
icient matrix
expressed b
12
=[u uu].
K
U
U
y tatistics for detecting
s.
T
wnlink transmission to the
K
MTs, with the LK
transmission preprocessing matx
12
=[d dd]D
ri
(5)
where is the precoder's coef
,
K
k
dficient vector for
preprocessing the symbol k
s
to be transmitted to the
kth MT. Note that we u the same notation s to
esent the downlink symbol vector, witout
distinction from the uplink symbol vector for the purpose
of notational simplification. Due to the reciprocity of the
downlink and uplink channels, the received signal vector
,1 ,2,
=[ ]
T
DDD DK
yy yy, received by the
se
repr h
K
MTs, is
k
A
hg with k
A
and
denoting the channel coefficient and the steering vector
for the kth user, respectively,
k
g
12
=[ T
]
K
s
ss
K
s
he uplink c
hannel
contains
the data symbols transmitted by the MTs to the
K
BS, and nU denotes t
expressed as
,=n
T
D
D
yHDs (6)
where is the downlink AW
n
D
GN vector with
2
[nn2] =
H
D
DD
E
IK
and denotes the transpose
T
Copyright © 2010 SciRes. CN
S. CHEN ET AL.147
r. Thoperatoe real part of the decision variable ,
D
k
y is
detecting the symbol k
used by the th MT for
k
s
transmitted from the BS to the kth MT.
Under the condition LK, there exists an exact
equivalency between the Tx beamforming preprocessi
matrix D and the Rx beamfoing weng
rm ight matrix U
expressed by [2 0]
*
=,
DU (7)
where *denotes thejugate ope
} conrator,
12
,Λ,,={
K
diag

c r
desi ne
a
d
is for achieving the transmit
power int, and U or D is assumed to have
been based on some second-order statistic
criterion. The exact relationship (7) is valid for
22
=
onst
g
D
U
. A simple scheme to implement the transmit
power constraint is to set =1/
kk
u for 1kK
relationship (7) is easy to understand. Under
the condition of LK, H
Und H)
have the same full rank and the same second-order
statistic properties, given (7).
For rank-deficient systems where <LK, H
UH
and T
HD
no longer ha ve the full rank . Inde ed, both the
[20]. The
H of (3) a
M
TD
of
ormi
(6
ngMSE Rx beamforming and MMSE Tf
turn out to be deficient in this case, exhibiting a high
BER floor. However, the MBER Rx beamforming
scheme [10,12] has been shown to consistently
outperform the MMSE design and is capable of
operating in rank-deficient systems where the number of
MTs is more than the number of receive antennas at the
BS. We will show in the next section that the relationship
(7) is not restricted only to second-order statistics based
designs. Specifically, with the MBER MUD and MUT
designs, (7) is valid and, moreover, the restriction
LK is no longer required. This enables us to use (7)
directly for designing the MBER Tx beamforming based
on the available MBER Rx beamforming solution even
for rank-deficient systems.
3. MBER Receive and Transmit
x beam
Beamforming
The BER of detecting k
s
using th
mforming with the
e up
be weior can
lin
be show
k Rx
n aght vectk
u
to be [10, 12]
() ()
,()[ ]
1
()=,
NqHq
skk
Rx kk
sgn s
PQ


Hs
u (8)
=1q
skU
N


u
u
where e usual Gaussian error function
()Q is th, []
denotes tal part, he re=2
K
s
N
ate transmit sy
is the number o
equipritimmbol v
f all th
ectors, s
e
()q obable leg
for 1
s
qN
, and ()q
k
s
denotes the k th element f
()q
s. The MBER solutik is then defined as
MBER, ,
=arg( ).
min
kRxkk
k
P
u
uu ( )
The optimisation (9) can be solved using a
ient-based numerical optisation algorithm [1
o
for
9
2].
BER is iario a ti
als norlise e beamforming
on
nv
way
u
im
ant t
ma
gra
No
k
u
d
te that the
, and one canposi
th ve scaling of
weight vector to a unit length, yielding =1
k
u, which
significantly reduces the computational complexity of
imisation. This is also useful for directly implementing
the Tx beamforming design from the Rx beamforming
design using the relationship (7), as the scalin matrix
Λ can be chosen as the identity matrix in this case.
Adaptive MBER Rx beamforming can be achieved using
opt
tha
-mu
g
the stochastic gradient-based algorithm known as the
least bit error rate [10, 12]. It is clear from (8) and (9)
t the optimal MBER Rx beamforming solution to
k
u is self-centred without regarding the effect on the
other users. This type of optimisation is referred to as the
egocentric-optimisation (E-optimisation) and the resul
ting solution is known to be egocentric-optomum (E-opti
m) [20]. Th e average BER over a ll the
K
users for
the Rx beamforming with the beamforming weight
matrix U is obviously given by
,
=1
() ()
()= ( )=
()[
1
RxRx k k
k
NqHq
Ksk
PP
K
sgn s
Q

Uu
uH (10)
1K
=1 =1
]
,
k
kq
skU
KN


s
u
and the MBER Rx beamforming solution
U
ply given by
MBER =arg( )
min Rx
P
U
U (11)
is simMBERMBER,1 MBER,2MBER,
=[ ]
K
Uuu u.
u forSince all the Rx beamforming vectors
MBER,k
1kK
are optimum in some sen
matrix MBER
U is overall optimum
se (E-optimum), the
Rx beamforming
(O-optimum) [20].
With the precoder coefficient matrix D, the BER of
detecting k
s by the kth MT is
() ()
,=1
Tx kq
sD
Q
N


(12)
for 1kK
()[ ]
qT
skk
sgn s

hDs
1
()= Nq
P
D
. The avrage BER o
K
ever all the users
eamforming with the precodeweight
then given by
for
matrix
the Tx b
D is r's
,
=1
() ()
)= ()=
()[ ]
1
Tx k
k
NqTq
Ks
P
K
sgn s
Q

DD
hDs (13)
=1 =1
1
(
.
K
Tx
kk
kq
sD
P
KN

Copyright © 2010 SciRes. CN
S. CHEN ET AL.
148
The MBER Tx beamforming solution for can be
obtained by minimising subject to given
transmit power constraint (14)
r opt
g the
computatio
Unlike the E-optimum of the MBER
the optimal MBER soultion to a precoden
D
the()
Tx
PD
MBER=arg( )
min Tx
P
D
DD
s.t.transmit power constraint ismet
This constrained nonlineaimisation problem for
example can be solved usinSQP algorithm [22, 23,
25], which is however
nally expensive.
Rx beamforming,
r's colum
vector k
d is not self-centred, as it is clear from (12) to
(14) that an optimal solution to k
d of user k not only
maximises the kth user's performance but also pays
attention on mitigating its effect on the other 1K
users. This type of optimisation is referred to as the
altruistic-optimisation (A-optimisation) and the resulting
solution is known to be altruistic-optimum (A-optimum)
[20]. Denote M RMBER,1MBER,2MBER,
=[ ]
k
Ddd d. Since
all the columns MBER,k
d for 1kK are optimu
some sense (A-optimum), the precoding matrix MBER
D
is also O-optimum [20].
BE
m in
applying the computationally ex
ectly derive it as
(15)
cost at all, provided that the
more general. We
start by examining the probabilit
Instread ofpensive
SQP algorithm to find the MBER Tx beamforming
solution, we can dir
*
MBER MBER
=,DU
with no computational
relation (7) is not restricted to second-order statistics
based designs and it is also valid for the MBER design.
We now show that (7) is indeed much
y density functions
(PDFs) of []
D
y and []
u
y, the real part of
D
y in
(6) and the real part of U
y in (3), respectively. Without
loss of generality, we assume that =1
H
kk
uu , 1kK
,
as the BER is invariant to the length of k
u [12]. Denote
=
R
I
jH, wHH ith th part =[]
RHHthe
imaginary part =[
IHH and 2=1j. Similarly , let
=
e real
]
,
R
I
jUU U and =
UU U
R
I
jnn n
[]=( )().
TTT T
URRII RUIU
. Then
R
I
UyUHUHsUn n
(16)
[]
U
y, denoted as ( )
UK
pw , is
obviously Gause mean
[[ ])
U I
y s
The PDF of
sian with th
(17)
and the covariance matrix
H
On the other hand, with the notation
12
, ,,w w
]=(TT
RRI
EUHUH
22
12
[]
U
H
UU

uu 2
1
222
21
2
12
[[ ]]=
[ ]
[] [
[] []
U
K
H
UUU
UK UKU
Cov

 

y
uu
uu u
uu uu

2]
.
H
K
u (18)
22HH

=
D
DD
R
I
j
nn n
and given ,
.
*
=DU
[]=( )
TT
DRRIID
R

y
HUHU sn
The PDF of
(19)
[]
D
y, denoted as 12
(, ,,)
D
K
pww w, is
Gaussian with the mean
s (20)
nce
[[ ]]=(TT
DR
and the covaria matrix
)
R
II
EyHUHU
2
[[ ]]=.
D
DK
Cov
yI
by the
(21)
The BER (10) is determined
K
marginal
PDFs of []
U
y, ,()
Uk
pw
k
nl
ar
forhich are
Gausd are speci
eleovariance
gin
1kK
fied
ments of t
al PDFs
, w
he c
of
sian distributed aby the mean
vector (17) and the diagona
matrix (18). The K m[]
D
y,
,()
D
kk
pw for 1kK
, are als
of the co
o G
atri
aussian distributed
and are specified by the mean vector ( and the
diagonal evariance m (21). These
two sets of the marginal PDFs are almost ``identical'',
given 22
=
UD
20)
xlements
. The difference is that ,()
Uk k
pw only
depends on the kcolumn vector of U while
,()
th
D
kk
pw depends on the entire matrix *
U, leading to
the BER expressions (10) and (13). We quote the
following result from [20].
Proposition 1 An E-optimum solution in a MUD is
equivalent an A-optimum solu the
corresponding MUT
et MBER
U be the MBER Rxeamforming
solution of (11). That is, the column vectors of MBER
U
are E-optimum in the c
totion in
.
Now l B
ontext of the MBER Rx
linear aK use The
ngles of arrival (departure) for the four users were
Beamforming. Then *
MBER
U is the MBER Tx
Beamforming solution of (1 4), and the co lumn vectors of
*
MBER are Aum in the context of the MBER Tx
Beamforming. Note that we do not require LK
4. Simulation Study
Full-rank system. The BS employed a four-element
ntenna array with half-wavelength element
spacing to support four single-antenna BPSrs.
U-optim
.
a 2
,
plink
were
16, 15 and 30, respectively, and the u
elffor the four users chann coeficients
= 0.70710.7071
k
Aj
, 14k
. The full uplink CSI
was assumed to be known at the BS and was used by the
BS to design the uplink Rx beamforming. Figure 2
compares the average BER performance, defined in (,
nk MM and MBER Rx beamforming
schemes. The BS then directly implemented the down-
10)
of the upliSE
Copyright © 2010 SciRes. CN
S. CHEN ET AL.149
Figure 2. Average BER performance comparison of the
uplink Rx and downlink Tx beamforming schemes with
both the MMSE and MBER designs for the TDD-SDMA
induced MIMO system which consists of a four-antenna BS
to support four single-antenna BPSK MTs.
link Tx beamforming from the resulting uplink Rx
beamforming according to (7). Assuming the exact
reciprocity of the uplink and downlink channels as well
as 22
=
UD
, the average BERs, as defined in (13), of
both the MMSE and MBER Tx beamforming schemes so
esigned are also plotted in Figure 2, in comparison with
h that of
d
the average BERs of the MBER and MMSE Rx
beamforming designs. As expected, the performance of
the proposed downlink Tx beamforming design agreed
witthe uplink Rx beamforming design.
The robustness of the proposed Tx beamforming
design was next investigated when the downlink and
uplink noise powers or channels were mismatched. In the
case of noise mismatching, the downlink noise power
was 3 dB more than the uplink noise power. The average
BERs of the MBER and MMSE Tx beamforming desi-
gns under this uplink and downlink noise power
mismatching are plotted in Figure 3, in comparison with
the case of equal uplink and downlink noise powers. It
can be seen that the 3 dB noise-power mismatching had
little influence on the performance of the MBER Tx
beamforming scheme but it had some influence on the
performance of the MMSE Tx beamforming design. This
was expected as the MMSE design is explicitly influen-
ced by the noise power while the BER calculation is
relatively insensitive to the noise variance estimate used.
In the case of channel mismatching, the uplink channel
coefficients were = 0.70710.7071
k
Aj
for 14k
,
but the downlink channel coefficients were
=0.6 0.8
k
A
j for 14k. The average BER
performance of the MBER and MMSE Tx beamforming
designs under this uplink and downlink channel mis-
matching are plotted in Figure 4, in comparison with the
Figure 3. Average BER performance of the downlink Tx
beamforming schemes with both the MMSE and MBER
designs for the TDD-SDMA induced MIMO system which
consists of a fou r-antenna BS to support four single-antenna
BPSK MTs, when the dow nlink and uplink noise powers
mismatch.
Figure 4. Average BER performance of the downlink Tx
beamforming schemes with both the MMSE and MBER
designs for the TDD-SDMA induced MIMO system which
consists of a four-antenna BS to support four single-antenna
BPSK MTs, when the downlink and uplink channels
mismatch.
SI.
case of the exact reciprocity of uplink and downlink
channels. From Figure 4, it can be seen that this
imperfect CSI had little influence on the MMSE Tx
beamforming scheme. It is also seen that the MBER Tx
beamforming design was not overly sensitive to the
mperfect CiRank-deficient system. The BS again employed a
four-element linear antenna array with half-wavelength
element spacing but the number of single-antenna BPSK
users was increased to six. The angles of arrival (depar-
ture) for the six users were 2
, 15, 10, 30, 25
Copyright © 2010 SciRes. CN
S. CHEN ET AL.
150
Figure 5. Average BER performance comparison of the
uplink Rx and downlink Tx beamforming schemes with
both the MMSE and MBER designs for the TDD-SDMA
induced MIMO system which consists of a four-antenna BS
to support six single-antenna BPSK MTs.
and respectively, and the uplink channel co-effi-
36,
cients for the six users were = 0.70710.7071
k
Aj,
16k. Given the exact reciprocity of the uplink and
downlink channels as well as 22
=
UD
, the average
BERs of both the MBER and MMSE Tx beamforming
esigns, implemented directly from the corresponding
eam
e average BERs of th
rank-defi d simila
d
Rx bforming schemes according to (7), are plotted in
Figure 5, in comparison with the
MBER and MMSE Rx beamforming designs. For this
cient system, both the MMSE Rx and Tx
beamforming solutions exhibiter high BER
floors while both the MBER Rx and Tx beamforming
solutions achieved similarly adequate performance.
The robustness of the proposed Tx beamforming
design was then investigated again under the scenarios of
mismatched downlink and uplink noise powers or
channels. In the case of the downlink channel having 3
dB more noise power than the uplink channel, the
average BERs of the MBER and MMSE Tx
beamforming designs, implemented directly from the
corresponding Rx beamforming solutions according to
the relation (7), are plotted in Figure 6, in comparison
with the case of 22
=
UD
. It can be seen that the 3 dB
noise-power mismatching had little influence on
performance. In the case of channel mismatching, the
uplink channel coefficients were = 0.70710.7071
k
Aj
for 16
k, but the downlink channel coefficients
were =0.6 0.8
k
A
j for 16k. The average BER
performance of th and MMSE Tx beamforming
designs under this uplink and downlink channel
mismatching are plotted in Figure 7, in comparison with
the case of an identical uplink and downlink channel.
Figure 6. Average BER performance of the downlink Tx
beamforming schemes with both the MMSE and MBER
designs for the TDD-SDMA induced MIMO system which
consists of a four-antenna BS to support six single-antenna
BPSK MTs, when the downlink and uplink noise powers
mismatch.
Figure 7. Average BER performance of the downlink Tx
beamforming schemes with both the MMSE and MBER
designs for the TDD-SDMA induced MIMO system which
consists of a four-antenna BS to support six single-antenna
BPSK MTs, when the downlink and uplink channels
mismatch.
From Figure 7, it can be seen that the MBER Tx
nk MBER transmit beamforming solution has
where the number of MTs supported is
beamforming design was not overly sensitive to the
imperfect CSI.
5. Conclusions
he downliT
been derived directly based on the uplink MBER receive
beamforming design for TDD-SDMA induced MIMO
systems. It has been shown that even for rank-deficient
TDD systems,
e MBER
Copyright © 2010 SciRes. CN
S. CHEN ET AL.
Copyright © 2010 SciRes. CN
151
me than the number of transmit antennas available at
lent relationship between the MUD
still valid, if the MBER
ime Wireless Communications,” Cambridge
ss, Cambridge, 2003.
. Viswanath, “Fundamentals of Wireless
munications, Vol. 5, 1998, pp. 23-27.
“Adap-
[10]
ro c e e d i n g s o f International
or
the BS, the equiva
and the corresponding MUT is
design is adopted. The proposed MBER transmit beam-
forming design imposes no computational cost at all at
the BS and is capable of achieving good downlink BER
performance with the support of low-complexity and
high power-efficient MTs. The robustness of this transmit
beamforming scheme to the downlink and uplink noise
or channel mismatching has been investigated by
simulation.
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