Channel Estimation for Space-Time Block Coded OFDM Systems
using Few Received Symbols
Biling Zhang
1
, Yipu Yuan
2
, Jung-Lang Yu
3
, Wei-Ting Hsu
3
and Yu-Jie Huang
3
1
School of Network Education, Beijing Univ. of Posts and Telecommunications, Beijing, 100876, P.R. China
2
College of Physics & Information Engineering, Quanzhou Normal Univ., Quanzhou, 362000, P.R. China
3
Department of Electrical Engineering, Fu Jen Catholic Univ., New Taipei City, 24205, Taiwan
Keywords: Space-Time Block Code, Multiple-Input Multiple-Output, Orthogonal Frequency Division Multiplexing,
Subspace Channel Estimation, Forward-backward Method, Noise Prewhitening Technique.
Abstract: A novel subspace channel estimation is proposed for space-time block coded (STBC) multiple-input
multiple–output (MIMO) orthogonal frequency division multiplexing (OFDM) systems. Considering the
zero-padding technique, the signal model of the STBC-OFDM system is first studied. The major drawback
of the subspace method is the slow convergence which requires a large number of received symbols to
compute the noise subspace. The repetition index scheme is developed here to increase the number of
equivalent received symbols. Using the space-time coding property, the forward-backward method is
presented to improve the convergence speed further. Moreover, the repetition index scheme transforms the
white noise into non-white one. The noise prewhitening technique is proposed to reduce the non-white noise
effect. Computer simulations show the effectiveness of the proposed forward-backward method and noise
prewhitening technique.
1 INTRODUCTION
Orthogonal frequency division multiplexing
(OFDM) techniques have been extensively
developed in the modern wireless communication
systems. The OFDM system utilizes the guard
interval such as cyclic prefix (CP) or zero-padding
(ZP) to avoid the inter-symbol interference (ISI) and
converts the frequency-selective fading channel into
a group of narrowband flat-fading channels with the
help of discrete Fourier transform (DFT) property.
The long term evolution (LTE)-advanced standard
adopts the OFDM technique in the fourth-generation
(4G) system to achieve a minimum peak data rate
and spectral efficiency requirements (Dahlman,
2011). The OFDM with index modulation is further
investigated for the next generation communication
systems (Wen, 2016). Moreover, the multiple-input
multiple-output (MIMO) system is proposed to
increase the channel spectral and energy efficiency
(Lu, 2014). The space-time coding technique offers
both spatial diversity and coding gains (Jafarkhani,
2005). The space-time trellis coding (STTC) was
introduced by Tarokh et al. At the same time,
Alamouti presented the space-time block coding
(STBC) scheme which uses two antennas at the
transmitter and a simple maximum likelihood
decoding algorithm at the receiver.
The channel estimation is necessarily performed
before the receiver design in the coherent
communication system. The pilot-based and blind-
based channel estimations are the two most popular
methods. The spectral efficiency of the pilot-based
channel estimation is lower because of the insertion
of periodic pilots. On the other hand, the blind
subspace method, which requires a large number of
received symbols to compute the noise and/or signal
subspaces, is converged slowly but no extra
bandwidth is needed.
In this paper we propose a fast convergence
subspace technique to estimate the channel
coefficients of the STBC ZP-OFDM systems. To
improve the convergence speed of the blind method,
the forward-backward averaging technique is
presented in (Yu, 2009) for the MIMO ZP-OFDM
system with Alamouti STBC. Using the circular
property of a channel matrix, the cyclic repetition
method (CRM) is investigated in (Zhang, 2014) for
both the CP-OFDM and the ZP-OFDM systems with
STBC coding. The repetition index scheme (RIS)
Zhang, B., Yuan, Y., Yu, J-L., Hsu, W-T. and Huang, Y-J.
Channel Estimation for Space-Time Block Coded OFDM Systems using Few Received Symbols.
DOI: 10.5220/0006381700870092
In Proceedings of the 7th International Joint Conference on Pervasive and Embedded Computing and Communication Systems (PECCS 2017), pages 87-92
ISBN: 978-989-758-266-0
Copyright © 2017 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
87

has been used in the channel estimation of the
single-input single–output ZP-OFDM system (Pan,
2013). We extend the RIS method to the STBC ZP-
OFDM systems here. Two novel techniques
including noise prewhitening and forward-backward
techniques are developed to enhance the
performance of RIS channel estimation. Simulation
results will verify that the proposed techniques
effectively improve the performance of the subspace
channel estimation.
Notations are defined as follows. Vectors and
matrices are denoted by boldface lower and upper
case letters, respectively; superscripts of (.)*,(.)T
and (.)H, denote the complex conjugate, transpose,
and conjugate transpose, respectively; E{.} denotes
the statistical expectation; I
n
denotes an n×n identity
matrix;

stands for Kronecker product; 0
m
×
n
denotes a m×n matrix with all zero entries.
|| ||
F

denotes the matrix or vector Frobenius norm. Let
[(0) ( 1)]
TT T
MVV V
be a Ma×b matrix and
each submatrix V(n) be a a×b matrix,
,
()
MS
V T
and
,
()
MT
V H
denote a (M+S-1)a×Sb block Toeplitz
matrix and a Ta × (M – T + 1)b block Hankel matrix,
respectively
,
(0)
()
(1) (0)
(1)
MS
M
M










V0
V
VV
0V


T
,
(0)(1) ()
(1) (2)
()
(1) () ( 1)
MT
M
T
TQ M









VV V
VV
V
VV V




H
2 SYSTEM MODEL
Fig. 1 shows a K-user uplink STBC MIMO-OFDM
system with J receive antennas. Let N be the number
of subcarriers,
() () ()
(0) ( 1)
T
kk k
ii i
ssN



s 
be the
OFDM symbol of the k-th user at time i, and
()k
i
s
and
()k
i
s
be the coded symbols transmitted through
the 1st and 2nd antennas, respectively. With two
consecutive OFDM symbols
()
2
k
i
s
and
()
21
k
i

s
, the
modified Alomouti’s STBC is given by (Zhang,
2014)
() () () ()*
221 2 21
() () () ()*
221 21 2
kk k k
ii i Ni
kk k k
ii i Ni



 



 

 

 
ss s Js
ss s Js
where
112
[]
NNN

J ωω ω ω
and
i
ω
is the i-th column of I
N
.
Let
()k
i
x
,
()k
i
x
and
()k
i
x
be the inverse discrete Fourier
transforms (IDFT) of
()k
i
s
,
()k
i
s
and
()k
i
s
, respectively.
After the IDFT operations, we have
() () () ()*
221 2 21
() () () ()*
221 21 2
kk k k
ii i i
kk k k
ii i i



 



 

 

 
xx x x
xx x x
. Let H
i
(n)∊C
JK
be the
channel coefficients from the i-th transmit antenna
of all users to the receive antennas,
ZP
(0) ( )
T
TT
ii i
L





HH H
, L and L
ZP
be the lengths
of channel impulse response (CIR) and ZP,
respectively. Without loss of generality, we assume
L
ZP
=L. Collecting the signals among all the receive
antennas yields
(1) ( )
() () ()
T
J
ii i
nynyn



y 
and
(0) ( 1)
T
TT
ii i
P





yy y
where P=N+L. Then the
two consecutive signal vectors y
2i
and y
2i+1
are given
respectively by
21,121,2212
() ()
iLN iLN i i
 yHxHxw TT
(1)
**
2 1 1, 2 2 1, 1 2 1 2 1
() ()
iLN iLN i i


 yHxHxw TT
(2)
where
i
w
is the additive white Gaussian noise
(AWGN) vector,
(0) ( 1)
T
TT
ii i
N



xx x
,
(1) ( )
() () ()
T
J
ii i
nxnxn





x 
. Let
(0) ( 1)
T
TT
ii i
P





yy y
,
221
() () ()
T
TH
iii
mmm






yyy
,
(0) ( 1)
T
TT
ii i
N





xx x
,
221
() () ()
T
TT
iii
mmm






xxx
,
[(0) ()]
TTT
L
HH H
,
and
12
**
21
() ()
()
() ()
ll
l
ll








HH
H
HH
. Then integrating (1)
and (2) with reshuffling the order of sequences
yields
i
y
as follows,
1,
()
iLN i i


y
Hx w T
(3)
Based on (3), the authors in (Zhang, 2014)
developed a subspace method to estimate the
channel coefficients. In the following, we will
propose a fast convergence subspace method based
on the repetition index scheme.
SPCS 2017 - International Conference on Signal Processing and Communication Systems
88

3 PROPOSED METHODS
The repetition index scheme has the following
property (Pan, 2013)
1,
21 2 1
21 2 1
1,
()
()
ii
LNq
Jq Kq
Jq Kq
LNq
ii




 

 
 
 

 
 
yx
H
00
00
H
yx
T
T
(4)
With the repetition index property in (4), we define a
composite signal matrix




,1,1,,
() ()
P
Qi LNQ NQi PQ i

y
Hx w TT T T
,
(5)
where the parameter Q is referred as the repetition
index. We can see that there are Q times equivalent
signal vectors generated from
i
y
with identical
channel matrix
1, 1
()
LNQ
H T
. Assuming that the
CIR remains unchanged during the transmission of
N
S
STBC OFDM blocks
, 1,...,
iS
iNy
, we can
stack the received signals as
()
,1 , 2( 1)
() ()
1, 1
[() ()]
( )
s
Ss
ss
N
QPQ PQNJPQQN
NN
LNQ Q Q




Yyy
HX W
 TT
T
(6)
where
()
,1 , 2( 1)
[() ()]
s
Ss
N
QNQ NQNKNQQN
Xxx TT
and
()
,1 , 2(1)
[() ( )]
s
Ss
N
QPQ PQNJPQQN
Www TT
.
Since
1, 1
()
LNQ
H T
is a
2 ( 1) 2 ( 1)JP Q KN Q 
block Toeplitz
matrix, it is of full column rank if
JK
. Besides, a
necessary condition for
()
s
N
Q
X
having full row rank is
2( 1)/
s
NKNQ Q
. If neglecting the channel
noise in (6) with the above two assumptions, we can
derive the noise subspace, U
n
, from the eigenvalue
decomposition (EVD) of the correlation matrix
()()
ss
NNH
QQ
YY
(Yu, 2009), (Zhang, 2014), (Pan, 2013).
The noise subspace U
n
∊C
2J(P+Q-1) 
,  =2J(P+Q-1)-
2K(N+Q-1), is orthogonal complement to the
channel matrix
1, 1
()
LNQ
H T
such that
1, 1
()
H
nLNQ
UH0 T
. Let u
g
be the g-th column of U
n
.
Then
1, 1
()
H
gLNQ
uH0 T
can be represented by
1, 1
()
H
LPQ g
uH 0 H
(7)
When the channel noise is considered, the estimated
noise subspace will be deviated from the true one
and the homogeneous equation in (7) might be
solved by the least square method. Let H=[h
1
…
h
2K
]. The channel estimation can be computed by the
following optimization problem
2
2
1, 1
1
11
2
1
1
ˆ
arg min ( )
arg min
k
k
K
H
LPQ g k
gk
K
H
kk
k










h
h
Huh
h Ψ h
H
(8)
where
1, 1 1, 1
1
() ()
H
L
PQ g L PQ g
g

 



Ψ uu HH
. The
solution of the optimization problem in (8) is the
eigenvectors of  corresponding to the first 2K
smallest eigenvalues. Because of the essence of
blind subspace method, the values of
ˆ
H
in (8) differ
those of H from a 2K2K ambiguity matrix. For
simplicity, we denote the method in (8) as ‘ZP-RIS’.
4 FORWARD-BACKWARD
TECHNIQUE
The forward-backward technique is developed to
increase the equivalent received symbols from (1)
and (2) in this section. We first stack
221
and
ii

yy in
an alternative way
*
1, 1 1, 2
21 21
21
**
* *
*
1, 2 1, 1
2 2
2
() ()
() ()
LN LN
i i
i
LN LN
i i
i

 


















HH
yw
x
HH
yw
x
TT
TT
(9)
The symbol in (9) can be treated as another STBC
OFDM received signal. Let
(0) ( 1)
T
TT
ii i
P





yy y
,
21 2
() () - ()
T
TH
iii
mmm






yy y
,
(0) ( 1)
T
TT
ii i
N





xx x
,
21 2
() () ()
T
HH
iii
mmm






xxx
. Then reshuffling the
order of sequences in (9) yields
i
y
as follows,
1,
()
iLN i i

yHxw T
(10)
The vector in (10) is referred as the backward STBC
symbol. Applying the repetition index technique
onto the backward STBC symbol yields


 
,1,1,,
()
P
Qi LNQ NQi PQ i
yHxw TT TT
(11)
From (11), we obtain a signal matrix similar to (6)
()
,1 , 2(1)
() ()
1, 1
[() ()]
( )
s
Ss
ss
N
Q PQ PQN JPQQN
NN
LNQ Q Q




Yyy
HX W
 TT
T
(12)
Channel Estimation for Space-Time Block Coded OFDM Systems using Few Received Symbols
89

Using (6) and (12), the forward-backward technique
generates the signal matrix by
() () () () ()
1, 1
()
sss ss
NNN NN
QQQLNQQQ
 


YYY HXW T
(13)
where
() () ()
[ ]
sss
NNN
QQQ
XXX. A necessary condition
for
()
s
N
Q
X
having full row rank is
(1)/
s
NKNQ Q
. That means the forward-
backward technique can converge faster than the
methods using
() ()
or
s
s
NN
QQ
YY
, respectively. The
channel estimation using the forward-backward
technique is called ‘ZP-RIS-FBM’.
5 NOISE PREWHITENING
TECHNIQUE
The noise effect on the repetition index technique is
discussed in this section. We observe that the
repetition index technique makes the noise matrix

,
P
Qi
w T
nonwhite even though the noise vector
i
w is white, i.e.,


2
2
H
ij wijPJ
E  ww I . Using the
definition of the block Toeplitz matrix, the
correlation matrix of


,
P
Qi
w T
is calculated by
 


2
,,
2
2
(| | 1) '
[ (1, 2, , 1, , , , 1, , 2, 1) ]
H
PQ i PQ i w pre
w J
PQ s
E
diag R R R R





ww R
I


TT
(14)
where

min ,RPQ
. Since R
pre
is a diagonal
matrix, it can be decomposed as
R
pre
=LL
H
where L
is given by
2
(| | 1) '
(1,, 1, ,, , 1,,1)
J
PQ s
diag R R R R

 LI

(15)
From (15), we define the prewhitening matrix as
L
-1
and a prewhitening signal matrix


() 1
,,
()
L
P
Qi PQi

zL z TT
for z
i
is a proper size signal
vector. Then signal matrix in (5) can be prewhitened
as
 
() () ()
,1,1,,
() ()
LL L
P
Qi LNQ NQi PQ i
yHxw TT TT
(16)
where
() 1
1, 1 1, 1
() ()
L
LNQ LNQ

 
HL H TT
. We can see
that






() () 2
,, 2(1)
LLH
P
QiPQ i wPQJ
E 

wwI TT .
Consequently, the noise subspace calculated from


()
,
L
P
Qi
y T
will get lower perturbations than that from


,
P
Qi
y T
. Let
(,) ()
1
s
s
NL N
QQ

YLY
and
()
,
L
g
u
1, ,g


 be the eigenvectors corresponding to the
first α smallest eigenvalues of
(,)(,)
ss
NL NLH
QQ
YY
. Thus
we have
() ()
1, 1
()
LH L
gLNQ


uH T
() 1
1, 1
()=
LH
gLNQ


uL H0 T because those eigenvectors
are orthogonal to the channel matrix
()
1, 1
()
L
LNQ

H T
.
The channel estimate can be obtained from (7) and
(8) by letting
()
H
L
g
g

uLu
. Similar method can be
applied to
()
s
N
Q
Y
and
()
s
N
Q
Y to get improvements in
channel estimation.
6 SIMULATION RESULTS
Computer simulations are given to demonstrate the
superiority of the proposed subspace channel
estimation for STBC ZP-OFDM systems. The
number of subcarrier is
N = 32 or 64, the CP length
is L = 8, the ZP length is L
zp
= 12, 16-QAM
modulation schemes is applied, and the channel
impulse responses is assumed to be i.i.d. complex
Gaussian random variable with zero mean and unit
variance. The signal-to-noise ratio (SNR) is defined
as
2
2
2
2
[]
(1)
[]





ii
s
F
n
i
F
E
NK L
SNR
P
E
yw
w
(17)
The performance metric of the channel estimation is
the normalized mean-square error (NMSE) is given
by
2
2
1
ˆ
|| ( ) ( ) ||
1
|| ( ) ||




m
N
F
i
mF
ii
NMSE
Ni
HH
H
(18)
where
N
m
is the number of Monte-Carlo trials, H(i)
and
ˆ
()
iH
are the true channel of the i-th trial and the
estimated channel after ambiguity correction,
respectively. The notations ‘ZP-2009’ and ‘ZP-
2009-FBM’ denote the channel estimation method
developed in (Yu, 2009), ‘ZP-CRM’ and ‘ZP-CFBM’
are used to indicate the fast subspace channel
estimation in (Zhang, 2014), and ‘PreW’ represents
the noise prewhitening technique. It is noted that the
numbers of equivalent STBC-OFDM symbols are
N
S
,
2N
S
, NN
S
and 2NN
S
for ‘ZP-2009’, ‘ZP-2009-FBM’,
‘ZP-CRM’ and ‘ZP-CFBM’, and
QN
S
and 2QN
S
for
‘ZP-RIS’ and ‘ZP-RIS-FBM’, respectively
Fig. 2 shows the NMSE versus the input SNR
when
N
s
= 100, N=32 and Q=8. Since
i
y is a 2JN1
vector, the minimum number of STBC-OFDM
SPCS 2017 - International Conference on Signal Processing and Communication Systems
90

symbols required for subspace channel estimation is
192. Thus only the method ‘ZP-2009’ is failed to
work but ‘ZP-2009-FBM’ works very well. The
CRM and CFBM using the overlap-and-add (OLA)
scheme slowly reduce the NMSE and obtain a
higher NMSE than the ‘ZP-2009-FBM’ as the SNR
increases. The proposed RIS and RIS-FBM methods
derive much lower NMSE than the compared ones.
The effect of the prewhitening technique is not
recognizable for
Q=8 since the nonwhite noise in
(14) is close to white. We further examine the
NMSE when
N=64, Q=16 and N
s
= 100 in Fig. 3. In
this scenario, both the ‘ZP-2009’ and ‘ZP-2009-
FBM’ are failed to work. The effect of nonwhite
noise is evident for the ‘ZP-RIS’ method and the
prewhitening technique alleviates the effect
considerably. The proposed channel estimations still
outperform the compared ones.
Fig. 4 shows the NMSE versus the number of
STBC-OFDM symbols when
SNR=20dB, N=32 and
Q=8. Both the CRM and the CFBM methods reduce
the NMSEs very slowly when
N
s
increases. The ‘ZP-
2009’ and the ‘ZP-2009-FBM’ decrease the NMSEs
very sharply after
N
s
>180 and N
s
>90, respectively.
On the other hand, the NMSEs of the proposed RIS
and RIS-FBM methods have dropped quickly when
N
s
>20 and N
s
>10, respectively. Fig. 5 shows the
NMSE versus the number of STBC-OFDM symbols
when
SNR=20dB, N=64 and Q=16. We can see that
the proposed methods outperform the other ones.
When the input SNR is small, the nonwhite noise
drastically influences the performance of the
subspace channel estimation and prewhitening
technique improves this effectively.
7 CONCLUSIONS
In this paper, we proposed a RIS method to enhance
the convergence of the performance of subspace
channel estimation in STBC ZP-OFDM systems.
Based on the STBC property, a FBM method
generating twice equivalent STBC OFDM symbols
as the RIS is presented. The RIS method turns the
white noise into nonwhite. The prewhitening
technique is developed to alleviate the nonwhite
effect. Computer Simulations showed the proposed
methods reduced the NMSEs quickly with few
STBC OFDM symbols.
ACKNOWLEDGEMENTS
This work was supported by the National Natural
Science Foundation of China under Grants No.
61501041, the Ministry of Science and Technology,
Taiwan under Grants No. MOST-104-2221-E-030-
004-MY2, the Open Foundation of State Key
Laboratory under Grants No. ISN16-08, the Special
Foundation for Young Scientists of Quanzhou
Normal University of China under Grants No.
201330 and Fujian Province Education Department
under Grants JA13267.
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(1)
i
y
(1) (1)
221
(1) (1)
221
ii
ii








xx
xx
(1) (1)
221
(1) (1)
221
ii
ii








ss
ss
(1)
2
(1)
21
i
i

s
s
() ()
221
() ()
221
KK
ii
KK
ii








xx
xx
() ()
221
() ()
221
KK
ii
KK
ii








ss
ss
()
2
()
21
K
i
K
i

s
s
H
()J
i
y
(2)
i
y
()
ˆ
k
i
s
Figure 1: STBC-ZP-OFDM systems.
Channel Estimation for Space-Time Block Coded OFDM Systems using Few Received Symbols
91

Figure 2: NMSE versus the input SNR when N=32, Q=8,
16-QAM modulation is used,
N
S
=100 STBC OFDM, K =
2,
J = 3, and N
m
= 1000.
Figure 3: Scenario is the same as that in Fig. 2 except for
N=64, Q=16.
Figure 4: NMSEs versus the number of STBC-OFDM
symbols when
N=32, Q=8, 16-QAM modulation is used,
input SNR=20dB,
K = 2, J = 3, and N
m
= 1000.
Figure 5: Scenario is the same as that in Fig. 4 except for
N=64, Q=16.
0 5 10 15 20 25 30 35 40
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
10
1
SNR(dB)
NMSE
ZP−2009
ZP−2009−FBM
ZP−CRM
ZP−CFBM
ZP−RIS
ZP−RIS−PreW
ZP−RIS−FBM−PreW
0 5 10 15 20 25 30 35 40
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
10
1
SNR(dB)
NMSE
ZP−2009
ZP−2009−FBM
ZP−CRM
ZP−CFBM
ZP−RIS
ZP−RIS−PreW
ZP−RIS−FBM−PreW
10
1
10
2
10
3
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
10
1
Number of STBC−OFDM Symbols
NMSE
ZP−2009
ZP−2009−FBM
ZP−CRM
ZP−CFBM
ZP−RIS
ZP−RIS−PreW
ZP−RIS−FBM−PreW
10
1
10
2
10
3
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
10
1
Number of STBC−OFDM Symbols
NMSE
ZP−2009
ZP−2009−FBM
ZP−CRM
ZP−CFBM
ZP−RIS
ZP−RIS−PreW
ZP−RIS−FBM−PreW
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