Power Adaptation in Space-Time Block Code
Myeongsook Seo and Sang Wu Kim
Department of Electrical Engineering
Korea Advanced Institute of Science and Technology
373-1 Kusong-dong, Yusong-gu, Taejon 305-701, Korea
Abstract- Transmission power adaptations are considered
in a space-time block code (STBC) system utilizing feedback
channel gain informations. We investigate the optimal power
adaptation scheme that minimizes the average BER subject
a fixed average transmission power constraint. We present
the power gain provided by the optimal adaptation scheme
over the conventional STBC without power adaptation. The
power gain provided is found to be 13.8dB at BER of
in a Rayleigh fading channel. Also, we analyze the performance of STBC system with optimal truncated power adaptation which compensates for fading above a certain fade
depth: below the cutoff level data transmission is suspended
(i.e. outage condition is declared). Finally, we propose a simple adaptation scheme for STBC systems based on two-bits
feedback information that provides a power gain of 4.3dB at
.
BER of
I. I NTRODUCTION
Recently, transmit diversity has drawn a lot of interest
because of its simplicity of implementation and feasibility
of having multiple antennas at the base station. One attractive approach is space-time block code (STBC), proposed by Alamouti [1], which can achieve full diversity
without bandwidth expansion. STBC system can achieve
the diversity order of for transmit antennas and one
receiver antenna. The reason is that STBC maintains orthogonality between the antennas and it can avoid selfinterference in flat fading channels. Furthermore, since
the receiver can process the received signal using simple
linear processing, the complexity is low. This transmit diversity is currently considered in the WCDMA system [2].
In this paper, we consider the transmission power adaptation in space-time block code (STBC) systems, where
the transmitter is equipped with two antennas and the receiver is equipped with one antenna. We assume that
channel gain informations at each antenna are obtained
at the receiver and then transported over a dedicated feedback link to the transmitter for power adaptations. We
investigate the optimal power adaptation scheme that minimizes the average BER under a fixed average transmission power constraint. We show that the optimal adaptation provides a power gain of 13.8dB at BER of over conventional STBC without power adaptation. Then
we analyze the STBC system employing optimal trun-
cated power adaptation scheme that minimizes the average BER. We show that the optimal truncated power adaptation provides a power gain of 15dB at BER of and
outage probability of .
The optimal adaptation requires a complete channel
gain information, and thus presents the theoretical performance limit. In practice, the feedback channel information may be limited. So, we propose a simple adaptation
scheme based on two-bits feedback information (one bit
for each antenna). We show that the proposed scheme
provides a power gain of 4.3dB at BER of over the
conventional STBC without power adaptation.
An attempt was made in [3] to adapt STBC according to
the available channel information, where a predetermined
orthogonal STBC was adapted by means of a linear transformation under a fixed instantaneous transmission power
constraint.
The remainder of this paper is organized as follows.
Section II describes the system model. In Section III,
we investigate the optimal adaptation scheme that minimizes the average BER. In Section IV, we propose a simple adaptation scheme based on two-bits feedback information. In Section V, numerical results are presented. Finally, conclusions are made in Section VI.
II. S YSTEM
MODEL
We consider the downlink of a mobile communication
system where the base station is equipped with two transmit antennas, and the mobile is equipped with one received antenna. The channel gain between the transmit
antenna one and the receiver is denoted by and be
tween the transmit antenna two and the receiver is denoted
by . We assume that and are constant across two
consecutive bits, and they are independent. We assume
a single user and BPSK signaling in a frequency nonselective Rayleigh fading channel. Fig. 1 shows the model
of proposed STBC system, wherein the gains (weights)
, at each antenna is adapted based on the
feedback channel information and . The traditional
STBC system without power control corresponds to a spe
cial case of
.
The received signals and at time and !#" ,
respectively, where " is the symbol duration, are given
0-7803-7206-9/01/$17.00 © 2001 IEEE
3188
This indicates that the optimal strategy is to use only
one antenna providing the larger gain. Then, it follows
from (6) that the average BER is given by
by
"
! "
" !
!
! 7 8 ;:=?
9
# < >A@9B
! #
" DCFEG9H
IKJ C L C
(8)
(1)
where the pdf of - is given by
$ +
where is the average transmission power and
and
G9H IKJ C MN# OQP SRTVUXW P SRTVUXW are information bits at time and ! " , and and
(9)
are complex random variables representing noise and
interference.
The combiner is designed to keep the orthogonality and
MN# of STBC so as to detect separately without self (10)
%'& $ ( interference. The combiner generates outputs and ,
MY# where
In what follows, we will assume
.
7 8 that
9
Now, we want to minimize
in (8) by adapting ac
! cording to C , subject to
(2)
%'& ( Z:=# < [ G H IKJ C L C (11)
Then, it follows from (1) and (2) that
" !
!
" ! !
!
Using calculus of variations [4], we can show that the optimal is the solution of
"!
$ !
$ (4)
$
Adfe
d
T
aW
bC
"Oc
#
(12)
b gC "Oc # g h
#
gC
"Oc
(13)
i is the Lambert W function [5], defined as
where d
where
(3) which yields
Thus, the received SNR is given by
"
#
P ]R \_^4`
KjFPk + l -
j (14)
(5) Thus, the average BER with the optimal power adaptation
can be obtained from (8) and (13) as
III. STBC WITH C OMPLETE
C HANNEL I NFORMATION
7 8
9
m:n<
u rt
#?>poqs
td
@ b wv
gC
"
#yx
C
i| D
QP SR P SR L
Ez{
In this section, we assume that the transmitter is provided
with complete feedback channel informations, and , where b is the solution of
and adapts the power based on them. We consider adapt#
g C " and , subject to %'&
!
(
, such that
ing the
:=# < b
#yx E
'
%
&
(
@
v "
C d
the average BER is minimized. If we let
!
,
then
i} OQP SR P SR L C "!*)
"
#
"
#
!
-
(15)
(16)
Fig. 2 is a plot of the optimum weighting factor ver- . We find that the optisus the channel gain C of
(6) mum is decreasing with $ C ,+ indicating that deep fades
$,+
should be compensated by transmitting at a higher power.
0/2143
This may require a large average transmit power and may
where -.
$ +
$ $ and the equality is satisfied cause a large interference to other users. Note that the opwhen
timal adaptation requires channel gain informations and , since - ~/2143 if
6
5
$ +
. In contrast, the
$
$ +
(7)
required feedback informations in [3] are and .
otherwise.
$,+
3189
where b
Optimal Truncated Power Adaptation
In this subsection, we investigate the optimal truncated
power adaptation in a space-time block code system. The
transmitter sends data only when the channel
gain - is
#
$ +
greater than a given threshold level . Otherwise, data
transmission is suspended (i.e. outage is declared). Thus,
#
the data transmission suffers an outage probability of
is the solution of
%'& (
#
gC
"
:=< b
" C d @ v #yx
W
i} OQP SR P SR L C E
(24)
Fig. 3 is a plot of the optimum weighting
factor ver# C
sus the
.
# : # W G H IKJ C L C
# channel gain of $ + - for
As increases, the optimal weighting factor increases,
i.e. received SNR in (18) increases. As consequence, we
(17)
DQP W can expect a reduction of BER at the expense of an inThis type of power control has been analyzed in [6] [7]. creased outage probability. For comparison purpose, we
also include the weighting factor for channel inversion
The power adaptation
# without
# truncation corresponds to a ( c C ), where the factor 0.72 has been chosen to
special case of
X
4 .
! satisfy (11).
In order to maintain a constant data rate
c " cP independent of the outage, the bit duraIV. STBC WITH L IMITED C HANNEL
#
#
tion during
$ + - should be set to " .
I NFORMATION
Also, to make an average transmission# power of
, the
-
should be set to
transmission# power during
c . As a result,$ + the received SNR, during
#
$ + - , is given by
In this section, we consider the situation where the chan
# "!
nel information at the transmitter is limited. We assume
" $ + # # i D
that the transmitter knows only whether the channel gain
D
"
$ for antenna is above a threshold . The required feed# $ + - (18) back
information in this case is two bits (one bit for each
antenna). The weighting factor is adapted as
Then, it follows from (18) that the average BER is given
by
6
5 (25)
$ ,
7 8 ;
9
: <
> @ B
A
W
" DCFEn
#
G9H IKJ G9H IKJ
C L C (19)
W
where the conditional pdf of
$ + - given $ + -
given by
#
is
$
for some constants and , subject to %'& ! ( . This is illustrated in Fig. 5. Then, it follows from (4)
and (25) the average BER is given by
#
- C
7 8
9
$ +
,
$,+ W
#
DQ
SRP
PSR C
(20)
#
D
7
Now, we want to minimize 98 in (19) by adapting according to C , subject to
G H IKJ G H IKJ
C :=< [
G H IKJ C
%'& ( ;
L C
W
Using calculus of variations, the optimal
given by
dfe
for C
b gC "Oc # h
#
gC
"Oc
:=< > oqs
u rt
# p
td
D
W
C i| DQP SR P SR L @ b v
#
gC
#
"
x
:# : 6
# >A@ B
#
I
R P S
R L C L C
i?P S
m:
!
< : #
"
#
: < : < >A@ B
!
"
>A@ B
I
R P S
R L C L C
i?P S
(21)
#
"
C
!
C
C
!
!
C E
C E
C E
I R P SR L C L C (26)
i?P S
)
P ! T , we can show that
Using the inequality > 3 (22)
) #" I
97 8
#" (27)
!
is
It follows from (19), (20) and (22) that the average BER
with the optimal truncated power adaptation is given by
7 8
9
where
#" I
Ez{
#"
(23)
3190
.
.
:# P
: < P
$
&%
'`
$
&%
W
'`
` " I R S )(R C
L
W
` " RS )(R C
L (28)
7
We want to minimize the upper bound on 98
ject to
g
, sub- the total power on one antenna providing a higher channel gain and controlling the transmission power on that
antenna according to the Lambert W function of the chan%'& ! ( (29) nel gain. The optimal adaptation is found to provide a
power gain of 13.8dB at BER of over the convenIt follows from (25) that this constraint can be restated as tional STBC without power adaptation.
The optimal strategy requires a large transmit power to
! (30)
D
compensate for deep fades. In order to avoid the large
transmit power (thereby avoiding a strong interference to
where
other users in multi-user systems), we considered the opZ: <
.
P SR L C (31) timal truncated power adaptation, wherein the transmit
ter sends data only when
# channel gain $ + - is greater
. Otherwise, data transmission
than
a
threshold
level
Since the upper bound is minimized by minimizing " I !
is suspended and an outage is declared. The power gain
#" , the power adaptation problem is characterized by
over the conventional STBC without# power adaptation in
creases as the outage probability increases. The opti#
"
#
"
Min
" I "
I !
(32) mal truncated power control is found to provide a power
gain of 15dB over the conventional STBC without power
subject to D
! .
adaptation at BER of and outage probability of
. These optimal strategies require a complete channel
gain information, and thus presents the theoretical perV. N UMERICAL R ESULTS
formance limit.
#
In practice, the feedback channel information may be
Fig. 4 is a plot of average BER versus 9 "Oc
with
limited.
So, we proposed a simple adaptation scheme
optimal power adaptation. We find that the STBC with opbased
on
two-bits
feedback information (one bit for each
timal power adaptation provides a power gain of 13.8dB
antenna):
The
transmitter
needs to know only whether the
over the conventional STBC without power adaptation at
BER of . When the optimal truncated power adapta- channel gain $ for antenna is above a threshold. This
tion is applied to STBC, the power gain over the conven- adaptation scheme is found to provide a power gain of
tional STBC without power adaptation increases
at the ex- 4.3dB over the conventional STBC without power adap#
pense of increased outage probability . Figure 4 shows tation at BER of .
that the optimal truncated power adaptation provides a
power gain of 15dB over the conventional STBC without
power
adaptation at BER of and outage probability References
# of . We also find that the power gain provided
[1] S. M. Alamouti, “A simple transmit diversity techby the optimal power adaptation in (13) over the channel
nique for wireless communications,” IEEE Jounal
inversion is 0.3dB at BER of .
on Selected Areas in Communications, Vol. 16, No.
Fig. 6 is a plot of the optimal weighting
# factors 8, pp. 1451-1458, October 1998.
9
and and the threshold versus
. We find
"Oc
that the optimal threshold decreases with the transmis[2] 3GPP, “Technical Specification 25.214, Physical
sion power . Also, the optimal increases with 9 ,
layer procedures(FDD),” June 2000.
whereas the optimal is almost invariant to 9 .#
. The [3] G. Jongren and B. Ottersten, “Combining transmit
Fig. 7 is a plot of average BER versus 9 "Oc
antenna weights and orthogonal space-time block
adaptive STBC scheme with limited feedback information
(two-bits feedback) provides a power gain of 4.3dB over
codes by utilizing side information,” in Proc. 33th
the conventional STBC without power adaptation at BER
Asilomar Conference on Signals, Systems and Comof .
puters, Oct. 1999.
VI. C ONCLUSIONS
[4] R. Courant and D. Hilbert, Methods of Mathematical
Physics, Vol. I, New York:Interscience, 1953.
We considered transmission power adaptations in a
space-time block code (STBC) system. We investigated
the optimal adaptation scheme that minimizes the average BER under a fixed average transmission power constraint. We presented the power gain provided by the optimal adaptation scheme over the conventional STBC without power adaptation. The optimal strategy is to allocate
[5] R. M. Corless et al., “On the Lambert W function,”
Advances in Computational Mathematics, Vol. 5,
pp.329-359, 1996.
[6] Sang Wu Kim and Andrea J. Goldsmith, “Truncated
power control in code-division multiple access communications,” IEEE Trans. Veh. Technol., Vol. 49,
No. 3, pp. 965-972, May 2000.
3191
[7] A. J. Goldsmith and S. G. Chua, “Variable-data
variable-power MQAM for fading channels,” IEEE
Trans. on Commun., Vol. 45, pp. 1218-1230, Oct.
1997.
h1 = α1e jθ1
b2 b1 STBC
− b2 b1
Encoder b1 b2
PTTw1
PTTw2
r2 , r1
h2 = α2e
Power adaptation
jθ 2
Combiner
h1
h2
Channel
estimator
b̂1
b̂2
h1 , h2
(b) Receiver
(a) Transmitter
Figure 1: Block diagram of the proposed system.
7
PTT/N0= 2dB
PTT/N0= 4dB
PTT/N0= 8dB
PTT/N0= 10dB
PTT/N0= 12dB
6
5
w
4
3
2
1
0
0
1
2
3
4
g
Figure 2: The optimal weighting factor C versus channel gain C for complete feedback information.
3192
10
PTT/N0=10dB
P0=0
8
-2
P0=10
-1
P0=10
Channel inversion (w=0.72/g)
w
6
4
5.0
5.0
4.5
4.5
4.0
2
4.0
q1
q2
3.5
3.5
q1 , q2
0
0.0
0.5
1.0
1.5
2.0
g
Figure 3: The optimal weighting factor C versus channel gain C for complete feedback information when truncated power adaptation is applied.
10
-1
10
-2
2.5
Threshold η
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0.0
0.0
0
5
10
15
20
PTT/N0[dB]
Figure 6: The optimal weighting
factors , and the
#
.
threshold versus 9 "Oc
0
10
3.0
2.5
STBC without power adaptation
STBC with channel inversion (w=0.72/g)
STBC with optimal power adaptation when P0=0
-2
STBC with optimal power adaptation when P0=10
-1
Average BER
STBC with optimal power adaptation when P0=10
10
-3
1x10
-4
1x10
-5
10
-6
10
-7
10
-8
10
-9
5
10
15
20
25
PTT/N0[dB]
#
Figure 4: Average BER versus 9 "Oc
with complete
channel information.
10
1
10
0
No diversity
STBC without power adaptation
STBC with two-bits feedback adaptation
STBC with channel inversion
STBC with optimal power adaptation when P0=0
Average BER
-2
10
-1
10
-2
10
-3
1x10
-4
1x10
-5
10
-6
STBC with optimal power adaptation when P0=10
-1
STBC with optimal power adaptation when P0=10
wi
q1
5
10
15
20
25
PTT/N0[dB]
q2
Figure 7: Average BER versus 9 "Oc
η
Threshold η
3.0
g
Figure 5: Weighting factors , versus channel gain C
with two-bits feedback information.
3193
#
.