Page 1
An Iterative Decoding Technique for Cooperative STBCOFDM Systems with
Multiple Carrier Frequency Offsets
Yeong Jun KimP*o
P*
P, Kyung Soo Woo*
P ChungAng University, P **
P, Hyun Il YooP*
PElectronics and Telecommunications Research Institute (ETRI)
yscho@cau.ac.kr
Abstract
P, Heesoo LeeP**
P, Hyun Kyu ChungP**
P, and Yong Soo ChoP*
P
P
In this papper, an iterative decoding technique for cooperative spacetime block coded orthogonal
frequency division multiplexing (STBCOFDM) systems is proposed with the aim of mitigating the
intercarrier interference (ICI) caused by multiple carrier frequency offsets (CFOs) in cooperative
transmission. The proposed iterative decoding technique is shown to mitigate the noise enhancement
effect on STBCOFDM signals caused by multiple CFOs, and is effective in reducing ICI, especially for
systems having a large FFT size and large multiple CFOs.
Introduction I.
Mobile multihop relay (MMR) networks have
garnered significant attention as an extension to the
conventional singlehop cellular network by combining a
fixed cellular infrastructure with multihop relaying
technology [1]. The cooperative MMR has also received
a great deal of attention recently since it can solve the
problem of lack of space for attaching multiple antennas
on a station as well as mitigate the decrease of diversity
gain in multiple antennas [2]. When the cooperative
MMR is applied to orthogonal frequency division
multiplexing (OFDM) cellular systems for performance
enhancement, the effect of carrier frequency offset (CFO)
on OFDMbased cooperative MMR systems should be
investigated since the OFDM system is quite sensitive to
CFO due to the narrowness of subcarrier spacing.
In an OFDMbased cooperative MMR system with a
space time block code (STBC), the coded signals are
transmitted from distributed relay stations (RSs). In the
received signal at the mobile terminal (MT), there may
exist multiple CFOs that are not identical for different
transmitters. Even if the distributed stations are perfectly
synchronized, multiple CFOs may occur due to different
Doppler shifts when the MT moves. In the case where
RS1 and RS2 are in a cooperative mode, the CFO
between RS1 and MT may increase when the MT moves
toward RS1, whereas the CFO between RS2 and MT
decreases. In the event that the received signal suffers
from multiple CFOs, the CFOs cannot be reduced
completely by merely adjusting the carrier frequency of
the receiver at the MT. One method of overcoming the
problem of multiple CFOs is to estimate the CFOs using
preambles or pilots in the downlink and to inform the
distributed RSs of the corresponding CFO estimates
through the uplink. However, this feedback process will
increase the overhead in the uplink. Another way of
mitigating the multiple CFOs without increasing uplink
overhead is to cancel the intercarrier interference (ICI) at
the MT.
In a cooperative STBCOFDM system, joint ICI
mitigation and STBC decoding at the MT is possible by
modeling the STBCOFDM system with multiple CFOs
in the frequency domain and then canceling the ICI by
solving linear equations [3]. A frequencydomain
equalization technique for distributed STBCOFDM
systems with multiple CFOs was proposed in an effort to
mitigate ICI and ISI at the MT by using the simplified
maximumlikelihood (SML) method [4]. In both
techniques, only partial matrices in the channel model
including ICI are used for ICI cancellation to reduce
computational complexity, as in the case of ICI
cancellation for OFDM systems with a single antenna [5].
However, the performances of these techniques are
degraded significantly as CFOs increase, especially for
moving MTs in cooperative STBCOFDM systems. In
this paper, an iterative decoding technique for
cooperative STBCOFDM systems with multiple CFOs
is proposed with the aim of increasing the range of CFO
for ICI mitigation under reduced computational
complexity. It is also shown that, in the cooperative
STBCOFDM system with multiple CFOs, performance
loss may occur due to the phase difference between two
received STBC signals with different CFOs, where the
phase change between two adjacent symbols for each
received STBC signal increases linearly as its CFO
increases. In this paper, we analyze the effect of multiple
CFOs on STBCOFDM signals in the cooperative MMR
and propose an iterative decoding technique that
mitigates the effect of multiple CFOs on the cooperative
STBCOFDM system while attaining an increased
convergence range.
II. An Iterative Decoding Technique for
Cooperative STBCOFDM Systems
with Multiple CFOs
1
Page 2
2
In a cooperative STBCOFDM system, STBCcoded
signals are transmitted from distributed relay stations,
RS1 and RS2. In the received signal at the MT, the CFO
for the link between RS1 and MT is usually different
from that between RS2 and MT. Assuming that RS1 and
RS2 have one transmit antenna and the MT has one
receive antenna, the received OFDM signals with
multiple CFOs can be expressed in the frequency domain
as follows:
Y = CX + W (1)
where
[]
[]
1
s m
*
221
1
0
1
0
221
01
2 ,
R M
s
+
2
R M
s
= (0)( 1)
( )=
m
( )
m
( )
m
( )=
m
( ) ( )
k
X
( )
m
( ) ( )
m
X
( ) ( )
k
X
( )
m
= (0)( 1)
( )=
m
( )
m
( )
m
(0)( 1)
( )
k
( )
k
T
TT
T
ss
N
∑
m
k
N
∑
mm
k
k m
≠
T
T
T
T
ss
T
TT
N
mmm
R M
s
m
N
YY
k
mk
N
XX
N
CC
+
−
=
−
=
+
−
⎡
⎣
⎤
⎦
−
⎡
⎣
⎤
⎦
=
⎡
⎢
⎣
⎤
⎥
⎦
−
⎡⎤
⎦
=⎣
=−
=
YYY
Y
YC + W
C+C + W
XXX
X
CCC
CCC
C
?
?
?
?
2
21
,
*
m
*
2 1,2 1,
*
221
( )
k
( )
k
( )
k
( )=
m
( )
m
( )
m
R M
m
m
T
ss
CC
W
⎡
⎣
W
+
+
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
−
⎤
⎦
W
where
2,( )
s i m
+
2( )(2 )/
(2
)
)
( )/2(/
j
j
cp
j
cp
Ck
j N N
+
s i
+
N
Aeif k m
+
s i
j k m
−
N j
e
N N
+
N
Beif
πε
π πε
⎧
⎪⎪⎨
⎪
⎪⎩
=
=
−
k m
≠
(1)/
−
2
( 1)/
−
2
sin(
sin(
)
/
( )
k
)
sin(
k
)
( )
k
.
sin( ()/)
j
j
j
j
jnN
s i
+
j
j
j
jnN
s i
+
j
AHe
NN
πε
−
BHe
NmN
πε
πε
πε
πε
πε
=
=
+
Here,
index of the subcarrier, respectively. Also,
RTB1
MT link and RS2MT link, respectively, and
the normalized CFO for the signal through link
Ncp, and s are the size of the FFT, the length of the
cyclic prefix (CP), and the index of the STBCencoded
codeword, respectively.
frequency response of the
(2)
si
+
th symbol period for link j . The complex
number denotes the ICI term affecting the
2
s i m
C
+
k
and denote the target index and adjacent
denotes
TBM or RTB2
TBM, where RTB1
TBM and RTB2
TBM represent the RS1
ε denotes
m
k
j
j
. ,
j
N
2
k
( )
k
th subcarrier at the
j
s i
+
H
represents the

th subcarrier due to the signal of the th subcarrier at
the
different from
,( )
j
m
k
(2)
si
+
th symbol period for link
. When k is equal to m ,
denotes the frequency response of the
Here, the matrix of size (2
C
to as an ICI channel matrix.
If the conventional linear combining technique is
applied for decoding the STBCcoded signals, the
performance will be significantly degraded due to
multiple CFOs. Given an ICI channel matrix, the
transmitted sequence can be decoded by multiplying (1)
with the inversion of . This technique is referred to as
ZF ICI cancellation technique in this paper. Here, it is
assumed that channels and CFOs are already estimated
by preambles and/or pilots, and are known to the MT.
Although the ZF ICI cancellation technique can mitigate
the ICI caused by multiple CFOs, it cannot be practically
implemented due to high computational complexity,
especially for a large . Assuming that multiple CFOs
are small, the ZF ICI cancellation technique can be
simplified by calculating the inverse matrix of a partial
matrix instead of inversing the full ICI channel matrix.
Here, the partial matrix is defined as a sparse matrix
consisting of diagonal elements and a few elements
adjacent to the diagonal elements, ignoring other
elements.
Next, an iterative decoding technique is described. A
full ICI matrix is considered so that the CFO range for
ICI cancellation can be increased, where the ICI terms
are cancelled gradually through iterations [6]. The
received OFDM signals with multiple CFOs can be
expressed in the frequency domain as
(
=
Y = CX + W
if is
th subcarrier.
N will be referred
j
k
m
2,( )
s i m
+
j
Ck
m
x2 )
N
C
N
)
diag ICI
+
CC X + W (2)
where
[]
?
,0,1
0
k
1
k
(0)
−
( 1)
T
T
ICI
T
ICI N
−
ICI diag
diag
(0)
N
diagN
+
−
−
(
⎡⎤
⎦
=− =⎣
=
CCCCC
CCC
?
?
[]
,
.
( 1) 1)(
ICI kkkkk
N
=
CCC0CC
?
1)
−
r
The decoded signal after the
th iteration is given by
r
( ) ( )
1
ˆr
X
diag
−
=
CY (3)
where
( )
r
( 1)
0
ˆ
0.
r
ICI
if r
if r
−
=
>
⎧⎪
⎪ ⎩
=⎨
−
Y
Y
YCX
The decoded signal after the Rth iteration is given by
1
( )
R
11
0
1
11
1
ˆ
( 1) (
−
)
( 1) (
−
)
R
∑
rr
diagICI diag
r
R
∑
rr
diag diagICI diag
r
−
−−
=
−
−−
=
=
=+
XCCCY
CYCCCY
1
−
(4)
where
Page 3
11
01
(0)( 1)
diagN
diagN
−−
−
⎡
⎣
⎤
⎦
=−
CC
matrix inversion, associated with
C
?
1
−
.
Note that a (2
cooperative STBC decoding, needs to be performed
times for the calculation of
. The convergence
condition for the iterative decoding technique is given by
the following inequality [6]:
x2)
N
1
diag
−
C
1
max
1
diag ICI
λ
−
=<
CC
(5)
where
max
.
λ
denotes the maximum eigenvalue of
1
diag
Since STBC decoding is performed on the unit of two
symbols, there exists a phase change between two
adjacent symbols in the received signal when CFO is
present. However, in the case of the conventional STBC,
where two transmit antennas are placed on a single
transmitter, the phase change between two adjacent
symbols due to CFO can be assumed to be identical for
each path, resulting in no phase difference between two
simultaneously received signals. On the other hand, in
the case of the cooperative STBC, where two transmit
antennas are placed on different stations, the CFO for
RTB1
resulting in a phase difference between two received
signals. The phase difference increases linearly as the
difference between the CFOs or time index increases.
The degree of the phase difference caused by multiple
CFOs is associated with the effect of noise enhancement
when ZF ICI cancellation techniques are used. The effect
of noise enhancement due to phase difference in the
process of STBC decoding can be explained by the
following equation:
1
( ) ( )
mm
⎡
−
⎢
=
⎢
−
⎣
ICI
−
CC
TBM link is usually different from that for RTB2
TBM link,
R M
1
1
s
2
s
R M
2
2
s
1
s
θ
*
2
*
2
212
θ
*
1
212
( )
m
( )
m e
( )
m
1
H
det( ( ))
( )
m
( )
m e
( )
m
j
R MR M
s
j
R M
+
R M
s
W
HH
m
W
⎣
HH
−
−
+
−
+
⎤
⎥
⎥⎦
−
⎡
⎢
⎤
⎥
⎦
HW
1
s
1
s
2
s
2
s
.
22
θθ
**
2121
θ
**
221221
( )
m
( )
m
( )
m
( )
m e
( )
m e
det( ( ))
H
( )
m H
( )
m e
( )
m H
( )
m e
R M
s
R M
s
jj
R M
s
+
R M
s
j
R MR MR M R M
+
HH
HH
mHH
−−
+
−
+
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
=
−
= −−
H
(6)
where
12
R M
2
R M
1
21
R M
1
R M
2
θ j
−
Here, and denote phase changes for RTB1
TBM
link and RTB2
( )
m
H
TBM link due to multiple CFOs, respectively.
denotes the channel matrix for (2
signals, where
2
( )
s i
Hm
+
represents the frequency
response of the th subcarrier at the
m
period for link . Also,
of the matrix. In (6), the effects of ICI other than the
phase difference are ignored for notational simplicity.
If
2
( )
s i
Hm
+
is timeinvariant for the period of a STBC
codeword and the average power of the channel is
identical for both links, the average noise power in the
decoded signal can be derived as
STBC
1
R M
θ
2
R M
θ
x2)
j
(2)
si
+
th symbol
denotes the determinant
j
det( ) i
j
{}
{}{}
1
s
2
s
2
1
22
2
m
2
m
22
( )
m
( )
m
( )
m
( )
m
R MR M
E
E H E H
DD
σσ
−
=+
HW
(7)
where
{}
2
2
2
() ( )
m
j
j
ms
P E H
=
{}
12
121
s
2
s
44
44
R MR M
22
()()
2cos(θθ) ( )
m
( )
m
R M
m
P
R M
m
P
R MR M
D
EHH
=+
++
where
subcarrier. Note that
when ,
N
respectively. If
corresponds to noise enhancement of 3dB or SNR loss of
3dB.
In order to reduce the effect of noise enhancement, ML
decoding , instead of ZF decoding, can be applied to (3)
in an iterative decoding technique. The ML decoding part
of the th subcarrier at the
m
ˆ
( )(
argmin
X m
where
( )
( )
m
Y
Y
2
m
σ denotes the noise variance at the m th
+
, and
ε
are 64, 0.1, and 0.1,
is equal to
()
m
P
will become
12
R M R M
(θθ)
/ 2
π
1 R M
ε
2
R M
, this result
1
4
R M
2
4
(
R M
m
P
)
th iteration is given by
r
( )
r
( )
r
( )
( )
m
( ) ( ) )
m
X
diag
m
=−
XYC
m
f r
i
(8)
( 1)
,2 ,21
( )
m
0
ˆ
( )
m
0.
r
T
r
T
ICI
T
ICI
)
i
mm
if r
−
+
=
⎧⎪
⎪
⎩
=⎨
⎡
⎣
⎤
⎦
−>
YCCX
Here,
,2
ICIm i +
C
is the (2
m
+ th column vector in
ICI
C
.
Computational complexities for ICI cancellation
techniques in the cooperative STBCOFDM transmission
are summarized in XTable IX where ,
diag
N
partial
N
, and
denote the size of , the size of a partial matrix,
and the size of constellation. In the STBC cooperative
case, is equal to 2. The number of multiplications
diag
N
required for the ZF ICI cancellation technique using a
full ICI channel matrix is proportional to
for the simplified ZFICI cancellation technique is
proportional to
(
partial
NN
−
number of multiplications required for the iterative
decoding technique using ZF or ML is proportional to
. Note that when is large, the computational
complexity for the iterative decoding technique is much
smaller than that for the ZF ICI cancellation technique.
III. Simulation
In this section, performances of ICI cancellation
techniques for cooperative STBCOFDM systems with
multiple CFOs are evaluated. The FFT size, number of
samples in CP, modulation order, and channel model
, and that
c
N
( )
diagk
C
1)
3
N
/2
+
and
3
partial
N
. The
2
NN
3
Page 4
used for the simulation are 64, 16, 16QAM, and ITUR
Table I. Computational complexities for ICI cancellation techniques
ICI cancellation technique
ZF ICI cancellation technique
using full ICI channel matrix
Simplified ZF ICI cancellation
technique
Number of multiplications
3
((2 ) )
NO
3
partial
(( /2 1)
+
)
partial
O NNN
−
Iterative decoding technique
using ZF
23
diag
( (2 ) )
O R
?????
(( /2 1)
+
)
matrix product in
the ICI generation
matrix inversion in
ZF decoding for diagonal terms
diag
N O N
???????????
NN
+−
Iterative decoding technique
using ML
22
diag
( (2 ) )
O R
?????
((/2 1)
+
)
matrix product in
the ICI generation
matrix inversion in
ML decoding for diagonal terms
diag
N
c diag
N O N
?????? ? ??????? ?
NNN
+−
Ped A, respectively.
RTB2
assumed that the MT is located between RS1 and RS2,
signifying that the received signal powers and
propagation delays for both links are identical.
XFig. 1X shows the BER performances of the ZF ICI
cancellation technique when a full ICI channel matrix
and a partial matrix are used. From this figure, we can
see that the ZF ICI cancellation technique with a full ICI
channel matrix has the same slope as the analytic result
(Alamouti) without CFO. The performance loss of about
3dB in the ZF ICI cancellation technique is attributed to
the noise enhancement effect due to the phase difference
in the process of STBC decoding. Note that when the ZF
ICI cancellation technique with a partial matrix is used,
an error floor occurs for most cases. Better performance
is obtained with more terms in the partial matrix.
XFig. 2X shows BER performances of iterative decoding
techniques with ZF and ML for cooperative STBC
OFDM systems with multiple CFOs. Note that the BER
performance of the iterative decoding technique with ZF
after three iterations (Iterative ZF (3)) approaches that of
the ZF ICI cancellation technique with a full ICI channel
matrix (ZF ICI (Full)), shown in
decoding technique with ML (Iterative ML (3)) shows
roughly 3dB better performance at a BER of
iterative decoding with ZF (Iterative ZF (3)), due to
mitigation of the noise enhancement effect. From XFig. 1X
and XFig. 2X, we can see that the performance is improved
in the following ascending order: ZF ICI (Partial 10),
Iterative ZF (3), ZF ICI (Full), Iterative ML (3).
Computation complexities required for ICI cancellation
techniques can be easily obtained from
the FFT size is equal to 64, the number of multiplications
required for ZF ICI (Full), ZF ICI (Partial 10), Iterative
ZF (3), and Iterative ML (3) are in the order of 2M, 60K,
50K, and, 115K, respectively. When the FFT size is equal
to 1024, the number of multiplications is in the order of
8.6G, 1M, 13M, and 14M, respectively. The complexity
reduction gain of the iterative decoding technique over
the ZFICI cancellation technique (Full) becomes
significant as the FFT size increases. For the parameters
set for this simulation,
max
λ
convergence condition in (5) for iterative decoding with
ZF.
1
R M
ε
for RTB1
TBM link and
2
R M
ε
for
TBM link are set to 0.1 and 0.1, respectively. It is
XFig. 1X. The iterative
than
XTable IX. When
4
10
is 0.76, satisfying the
05 10 1520 2530
10
6
10
5
10
4
10
3
10
2
10
1
10
0
Ch=PedA, v=3Km, 16QAM, CFO R1M/R2M=0.1/0.1, ZF ICI cancellation technique
Analytic(Alamouti)
ZF ICI (Full)
ZF ICI (Full)
ZF ICI (Partial 2)
ZF ICI (Partial 6)
ZF ICI (Partial 10)
Fig. 1. BER performances of ZF ICI cancellation
techniques for cooperative STBCOFDM systems with
multiple CFOs
05 1015 2025 30
10
6
10
5
10
4
10
3
10
2
10
1
10
0
Ch=PedA, v=3Km, 16QAM, CFO R1M/R2M=0.1/0.1, iterative decoding techniques
Analytic(Alamoutic)
Iterative ML (1)
Iterative ML (2)
Iterative ML (3)
Iterative ZF (1)
Iterative ZF (2)
Iterative ZF (3)
Fig. 2. BER performances of iterative decoding
techniques for cooperative STBCOFDM systems with
multiple CFOs
IV. Conclusion
In this paper, it was shown that the proposed iterative
decoding technique is effective in reducing ICI caused by
cooperative STBCOFDM systems with multiple CFOs
when the system has a large FFT size and large multiple
4
Page 5
CFOs. It was also shown that the proposed iterative
decoding technique with ML not only achieves the
lowest BER performance at the reduced complexity, but
also mitigates the noise enhancement effect on STBC
signals caused by multiple CFOs.
Acknowledgement
This research is supported by ETRI, Ubiquitous
Computing and Network (UCN) Project, MIC 21st
Century Frontier R&D Program in Korea.
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