Communications and Network, 2013, 5, 666-670
http://dx.doi.org/10.4236/cn.2013.53B2119 Published Online September 2013 (http://www.scirp.org/journal/cn)
Channel Estimation for SCM-OFDM Systems by Modified
Kalman Filter*
Tao Peng1, Yue Xiao1, Shaoqian Li1, Huaqiang Shu2, Eric Pierre Simon2
1Nation Key Lab of Sci. and Techno. On Commun., University of Electronic Science and Technology of China, Chengdu, China
2IEMN Lab, TELICE Group University of Lille, Lille, France
Received July 2013
ABSTRACT
In this paper, the problem of channel estimation for superposition coded modulation-orthogonal frequency division
multiplexing (SCM-OFDM) systems over frequency selective channels is investigated. Assuming that the path delays
are known, a new channel estimator based on modified Kalman filter algorithms is introduced for the estimation of the
multipath Rayleigh channel complex gains (CG). In the simulation, the mean square error (MSE) and bit-error-rate
(BER) performances are given to verify the effectiveness of the Kalman estimation algorithms for SCM-OFDM sys-
tems.
Keywords: Superposition Coded Modulation (SCM); Orthogonal Frequency Division Multiplexing (OFDM); Channel
Estimation; Kalman Filter
1. Introduction
As a kind of non-orthogonal multiple access scheme,
interleave division multiple access (IDMA) was devel-
oped by Ping et al. [1,2], in which random interleavers
were employed as the only means of user separation. In
general, IDMA outperforms conventional code division
multiple access (CDMA) in terms of power and band-
width efficiency. The key innovation of IDMA is the
introduction of low-rate channel coding, chip-level inter-
leaving and low-complexity multiu s e r detection.
Motivated by the concept of IDMA, superposition
coded modulation (SCM) partitions the data to multi layer,
where each layer is treated by a user equivalently. The
low-rate encoder for all layers is typically identical, and
the interleaver of every layer is distinct, which is used to
combat the inter-layer interference. SCM has several ad-
vantages over conventional coded modulation schemes
such as trellis-coded modulation (TCM) and bit-inter -
leaved coded-modulation with iterative decoding (BICM-
ID). First, the transmitted signal of SCM can be approx-
imated as a Gaussian variable according to the central
limit theorem. Second, for adaptive modulation, the rate
adaptation can be simply realized in SCM by adjusting
the number of layers. Furthermore, a low-cost chip-by-
chip iterative detection a lgorithm can be adopted in SCM,
where the complexity is independent of the number of
layers [3,4]. Furthermore, SCM can be combined with
orthogonal frequency division multiplexing (OFDM) to
combat the frequency selective fading and improve the
throughput [5].
For signal detection in the receiver, reliable channel
information is needed. There have only been few litera-
tures regarding channel estimation for IDMA and SCM
systems. For example, least square (LS) and minimum
mean square error (MMSE) algorithms are employed to
estimate channel response of IDMA systems [6,7]. How-
ever, these estimation algorithms perform per-user chan-
nel estimation using pilot symbols in the frequency do-
main, which lead to poor estimation performance.
To alleviate this problem, channel estimation for SCM-
OFDM systems is investig ated in this paper. The estima-
tion of physical channel parameters includes estimating
multipath delays and multipath complex gains (CGs). It
is well known that the path delays are quasi-invariant
over several OFDM blocks whereas the CGs may change
significantly even within one OFDM block. Therefore, the
delays are assumed to be perfectly estimated and only the
problem of CGs estimation is considered in this paper.
Due to the excellent estimation performance of Kalman
*
This work was supported by the Foundation Project of National
Key
Laboratory of Science and Technology on Communications under
Grant 9140C020404120C0201, National High
-
Tech R&D Program
of China (
“863” Project under Grant number 2011AA01A105), N
a-
ti
onal Grand Special Science and
Technology Project of China under
Grant No. 2010ZX03006
-002-
02, and the Fundamental Research
Funds for the Central Universities.

T. PENG ET AL.
667
filter [8], a new channel estimator based on modified
Kalman filter algorithms is obtain ed to estimate the mul-
tipath Rayleigh channel CGs. In the final, simulation re-
sults verify that the Kalman estimation algorithms can
greatly improve the mean square error (MSE) performance
compared to the LS and MMSE algorithms, and can
achieve almost th e same bit-error-rate (BER) performance
as the ideal channel estimation.
The rest of this paper is organized as follows. Section
II introduces the system model of SCM-OFDM systems
adopted in this paper. In Section III, channel model and
Kalman channel estimation algorithms are presented for
SCM-OFDM systems. Simulation results are provided in
Section IV to demonstrate the effectiveness of the esti-
mation algorithms. The conclusions are drawn in Section
V.
2. System Model
2.1. Transmitter of a SCM-OFDM System
The superposition coded modulation scheme with K lay-
ers is shown in Figure 1. A binary data sequence d is
first serial-to-parallel conver ted i nto K subsequences. Then
the data of each layer is encoded, interleaved and mod-
ulated independently. Finally, all the data of K layers are
linearly superimposed to transmission. For layer-k, the
data sequence dk is first encoded by a low-rate encoder,
resulting in a coded sequence ck of length N. Then the
coded sequence ck is interleaved by a distinct chip-level
interleaver πk to produce a permutated sequence
k
c
.
After interleaving, the randomly sequence
k
c
is mod-
ulated to Xk by binary phase shift keying (BPSK). The
output signal is a linear superposition of K independently
coded symbols
( )( )
1
1,0 .
K
k
k
XmNX mm
=
=≤≤−
∑
(1)
Then the superposition signal is fed into an inverse fast
Fourier transform (IFFT) modulator. After IFFT and cyc-
lic prefix (CP) insertion, the time domain transmitted
signal can be represented as
( )()
12
0
, 1
1
Njmn N
mg
xn XmnN
NNe
π
−
=
≤− ≤=−
∑
(2)
where N is the total number of subcarriers and Ng is the
length of CP.
2.2. Receiver of a SCM-OFDM System
Since each layer of SCM systems can be treated as one
user of multi-user systems, the suboptimal iterative de-
tection [2] can be applied in SCM systems. The receiver
of SCM-OFDM is composed by a multi-layer detector
(MLD) and K decoders (DECs), which are applied to
solve inter-layer interference and the coding constraint
separately. The receiver performs the iterative processes
to update the extrinsic information between ESE and
DECs. After the last iteration, the DECs produce hard
decisions towards information bits. The receiver of SCM-
OFDM is illustrated in Figure 2.
The transmitted signals will pass through the multipath
fading channels and be corrupted by additive white Gaus-
sian noise (AWGN). At the receiver, after FFT processing
and CP removal, the frequency domain received signal is
given by
() ()( )( )
()( )( )
1
,
K
k
k
RmXmHm Wm
HmXmWm
=
=+
=+
∑
(3)
where H(m) is the channel frequency response at the mth
subcarrier, and W(m) is the AWGN with zero mean and
variance σ2.
To detect the mth chip in the kth layer, the received
signal can be rewritten as
Figure 1. Transm it t er s t ructure of a SC M system.
S/P
Channel
Encoder interleaver Modulation
1
π
1
d
1
c
1
X
1
c
+
Channel
Encoder interleaver Modulation
k
π
k
d
k
c
k
c
Channel
Encoder interleaver Modulation
K
π
K
d
K
c
K
c
d
X
k
X
K
X






OFDM
x

T. PENG ET AL.
668
Figure 2. Receiver structure of a SCM system.
()( )()()( )( )
()()( )
'
' 1,'
K
kk
k kk
MLI
kk
RmHmX mHmXmWm
HmX mm ζ
= ≠
=++
=+
∑
(4)
where
( )
MLI
k
m ζ
is the total interference term with re-
spect to layer-k on subcarrier-m, consisted of both in-
ter-layer interference and noise.
According to the central limit theorem,
( )
MLI
k
m ζ
is
approximated as a Gaussian random variable with mean
( )
( )
MLI
k
Em ζ
and variance
( )
( )
MLI
k
Var m ζ
. Based on
the definition of extrinsic LLR, the output of ESE detec-
tor is calculated by
( ) ( )
( )( )( )
( )
( )
( )
2
MLI
k
ESE kMLI
k
Rm Em
eX mHmVar m
ζ
ζ
−
=
(5)
where
( )
( )
MLI
k
Em ζ
and
( )
( )
MLI
k
Var m ζ
can be ob-
tained by the mean an d va riance of Xk(m).
At each iteration, by the priori LLRs feedback from
the DECs, the mean and variance of Xk(m) can be calcu-
lated as follows
( )
( )
( )
( )
( )
tanh2 ,
kDEC k
EXme Xm=
(6)
( )
( )
( )
( )
( )
2
1.
kk
VarXm EXm= −
(7)
In the above detection algorith m, the channel informa-
tion will be utilized at every iterative step. And the sys-
tem performance can be enhanced by improving the ac-
curacy of the channel estimation. Therefore, channel pa-
rameter estimation is an important task for SCM-OFDM
systems.
3. Channel Estimat ion B ased on Modified
Kalman Filter
3.1. Channel Model
The Rayleigh fading channel model with Jakes’s Doppler
spectrum is the most accepted random model to represent
temporal variations of the equivalent baseband channel
CG in wireless communication. Thus, this channel model
is considered in this paper with a delay spread L. The
normalized Doppler frequency is denoted as fDT, where
fD represents the maximum Doppler frequency shift and
T is an OFDM symbol period.
After transmission over a multipath Rayleigh channel,
the received OFDM symbol is given in the frequency
domain (after removing CP and taking FFT) by a matrix
form as
= +RHX W
(8)
where
()( )()
0,1 ,...,1
T
R RRN= − 

R
, X and W are
defined in a similar way as R.
The
NN×
channel matrix H can be written in terms
of physical channel parameters as
[ ] ( )
''
1
22
,' 10
1 l
n nn
LN
j jq
NN
ls
nn lq
eqT e
N
πτ π
α
−
−
−
= =

=


∑∑
H
(9)
where
l
τ
is the lth path delay normalized by the sam-
pling time, Ts is the sampling time, and
l
α
is the lth
path CGs.
When the channel is assumed to be time invariant
within a block, the channel matrix H can be simplified to
a diagonal matrix denoted as
[ ]
( )
2
,1
l
n
LjN
s ls
nn l
nT e
πτ
α
−
=
=.
∑
H
(10)
Let us define the
NL×
matrix FH and the
LN×
matrix α as follows,
[ ] ( )()() ( )
,
0 ,,...,1
ll ls
l
T NT αα α

= −

α

(11)
[ ]
2
,
l
j nN
nl
e
πτ −
=FH
(12)
Then substituting (11) and (12) in (10) yields
s=H FHα
(13)
DEC
deinterleaver
1
1
π
−
1
d
MLD
interleaver
1
π
d
R



( )
1ESE
eX
( )
1DEC
eX
P/S FFT
r
DEC
deinterleaver
1
K
π
−
K
d
interleaver
K
π
( )
ESE K
eX
()
DEC K
eX

T. PENG ET AL.
669
After obtained the path delays and CGs, the channel
matrix can be calculated by (13). Note that α will be
time-varying actually if the normalized Doppler fre-
quency fDT is larger. In this paper, α is assumed to be
fixed when the channel is time-invariant. And the delays
are assumed to be perfectly estimated and only the prob-
lem of CGs estimation is considered.
3.2. Modified Kalman Filter Algorithms
In this subsection, the problem of channel estimation for
SCM-OFDM systems over time invariant channels is con-
sidered. Assuming that the path delays are known, a new
channel estimator based on modified Kalman filter algo-
rithms is obtained to estimate the channel CGs. In this
estimation algorithm, each CG within one OFDM sym-
bol can be estimated by using a Kalman filter recursively.
For clarity, the modified Kalman filter algorithms are
summarized in the following, where two different Kal-
man initialization methods are given as algorithm 1 and
algorithm 2.
1) Initialization:
( )
( )
( )
13
42
0,2 ,1
14, 2
d
d
besseljf Talgorithm
f Talgorithm
π
πσ


=−


S
( )
( )
2
1
v
diag= −US P
( )
0
,zerosL L=P
( )
app
diag N=KF
2) Recursive Estimation: for symbol index i = 1,2,…,
Prediction step:
 
11ii i−−
=Sαα
1
1i ii − −
= +P PU
( )
( )
' '2
11
iaa ap
ii ii
eye N σ
−−
= +K PKKPK
Updata step:
 
( )
'
11i iiii
ip a
−−
=+−KR Kαα α
11
i ia
ii ii−−
= −P PKKP
In the above modified Kalman filter algorithms, the
estimation for the channel CG of each OFDM symbol is
performed continuously and recursively. With the increas-
ing number of estimation, the estimation performance
will become more and more accurate.
4. Simulation Results
In this section, computer simulations are carried out to
verify the effectiveness of the Kalman estimation algo-
rithms for SCM-OFDM systems. Simulations are per-
formed for a SCM-OFDM system with four layers, em-
ploying BPSK modulation. System parameters are as
follows, N = 128 subcarriers, Ng = N/8, Np = N/8 pilots,
and 1/Ts = 2 MHz. The channels are assumed to be
time-invariant, thus the normalized Doppler frequency is
selected as fDT = 0.0001. Two Rayleigh channel model
[9,10] are chosen as shown in Tables 1 and 2.
Figures 3 and 4 show the MSE performances of the
Kalman channel estimation algorithms compared to LS
and MMSE estimation algorithms in channel model A
and model B. It can be found that the performances of
the Kalman algorithms are significantly better than that
of the LS and MMSE algorithms. And the performance
gaps between the Kalman algorithms and others become
greater with increasing SNR. Moreover, the Kalman al-
gorithm 2 can achieve more excellent estimation perfor-
mance than Kalman algorithm 1 in both channel models.
Figure 5 depicts the BER performances of the Kalman
channel estimation algorithms for data detection com-
pared to LS and MMSE estimation algorithms. In this
figure, the ideal estimation algorithm is included, where
the exact channel information is perfectly known in the
receiver. From this figure, it can be observed that the
Kalman algorithms can enhance the BER performance
obviously rather than the LS and MMSE algorithms.
Meanwhile, the performance of the Kalman algorithm 2
is almost the same as that of the ideal algorithm.
Table 1. Multipath channel model A.
Path number
Average (dB)
Delay (Ts)
0
−7.219
0
1
−4.219
0.4
2
−6.219
1
3
−10.219
3.2
4
−12.219
4.6
5
−14.219
10
Table 2. Multipath channel model B.
Path number
Average (dB)
Delay (Ts)
0
0
0
1
−1
0.6
2
−9
1.4
3
−10
2.2
4
−15
3.5
5
−20
5
Figure 3. MSE performances of the Kalman estimation algo-
rithms for SCM-OFDM systems in ch a nn el m o del A.
10
0
10
-2
10
-1
10
1
10
-3
-5 0 5 10 15 20
SNR(dB)
MSE
LS algorithm
MSSE algorithm
Kalman algorithm1
Kalman algorithm2
10
-4

T. PENG ET AL.
670
Figure 4. MSE performances of the Kalman estimation algo-
rithms for SCM-OFDM systems i n channel model B.
Figure 5. BER performances of the Kalman estimation algo-
rithms for SCM-OFDM systems i n channel model A.
5. Conclusion
In this paper, channel estimation for SCM-OFDM sys-
tems is investigated over frequency selective channels.
Compared to LS and MMSE algorithms, the Kalman
algorithms can significantly improve the MSE and BER
performances. More sp ecifically, the Kalman algorithm 2
can achieve the performance very close to the ideal algo-
rithm.
REFERENCES
[1] P. Li, L. H. Liu, K. Y. Wu and W. K. Leung, “A Unified
Approach to Multiuser Detection and Space-Time Coding
with Low Complexity and Near ly Optimal Pe rformance,”
Proceedings of 40th Allerton Conference, 2002, pp. 170-
179.
[2] P. Li, L. H. Liu, K. Y. Wu and W. K. Leung, “Interleave
division multiple access,” IEEE Transaction on Wireless
Communication, Vol. 5, No. 4, 2006, pp. 938-947.
[3] J. Tong and P. Li, “Iterative Decoding of Superposition
Coding,” Proce edings of 4th International Symposium on
Turbo Codes, 2006.
[4] J. Tong, P. Li and X. Ma, “Superposition Coded Modula-
tion with Peak-Power Limitation,” IEEE Transactions on
Information Theory, Vol. 55, No. 6, 2009, pp. 2562-2576.
http://dx.doi.org/10.1109/TIT.2009.2018224
[5] J. Tong, P. Li, Z. H. Zhang and W. K. Bhargava, “Itera-
tive Soft Compensation for OFDM Systems with Clip-
ping and Superposition Coded Modulation,” IEEE Tran-
sactions on Communications, Vol. 58, No. 10, 2010, pp.
2861-2870.
http://dx.doi.org/10.1109/TCOMM.2010.083110.09296
[6] H. Schoeneich and P. A. Hoeher, “Iterative Pilot-Layer
Aided Channel Estimation with Emphasis on Interlea-
ve-Division Multiple Access Systems,” EURASIP Jour-
nal on Advances in Signal Processing, Vol. 2006, 2006,
pp. 1-15. http://dx.doi.org/10.1155/ASP/2006/81729
[7] S. Suyama, H. Suzuki, K. Fukawa and L. Zhang, “Itera-
tive Multiuser Detection with Soft Decision-Directed
Channel Estimation for MC-IDMA and Performance
Comparison with Chi p-Interleaved MC-CDMA,” IEICE
IEEE Transactions on Communications, Vol. E92-B, No.
5, 2009, pp. 1495-1503.
[8] S. Ghandour-Haidar, L. Ros, J.M. Brossier, “On the Use
of First-Order Autoregressive Modeling for Rayleigh Flat
Fading Channel Estimation with Kalman Filter,” Signal
Processing, Vol. 92, No. 2, 2012, pp. 601-606.
http://dx.doi.org/10.1016/j.sigpro.2011.08.014
[9] E. P. Simon, L. Ros, H. Hijazi and M. Ghogho, “Joint
Carrier Frequency Offset and Channel Estimation for
OFDM Systems via the EM Algorithm in the Presence of
Very High Mobility,” IEEE Transaction on Signal Pro-
cess, Vol. 60, No. 2, 2012, pp. 754-765.
http://dx.doi.org/10.1109/TSP.2011.2174053
[10] E. P. Simon, L. Ros, H. Hijazi, et al., “Joint Carrier Fre-
quency Offset and Fast Time-Varying Channel Estima-
tion for MIMO-OFDM Systems,” IEEE Transactions on
Vehicular Technology, Vol. 60, No. 3, 2011, pp. 955-965.
http://dx.doi.org/10.1109/TVT.2011.2104970
100
10-2
10-1
101
10-3
-5 0 5101520
SNR(dB)
MSE
LS algorithm
MSSE algorithm
Kalman algorithm1
Kalman algorithm2
10-4
10
0
10
-2
10
-1
10
-3
SNR(dB)
BER
LS algorithm
MSSE algorithm
Kalman algorithm1
Kalman algorithm2
10
-4
-6-4-202 468 10 1214
Ideal algorithm