Channel Estimation for SCM-OFDM Systems by Modified Kalman Filter

doi:10.4236/cn.2013.53B2119

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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

Email: [email protected]

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

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

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

After interleaving, the randomly sequence

is mod-

ulated to Xk by binary phase shift keying (BPSK). The

output signal is a linear superposition of K independently

coded symbols

( )( )

1,0 .

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

( )()

, 1

Njmn N

xn XmnN

NNe

−

≤− ≤=−

∑

(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

() ()( )( )

()( )( )

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

Channel

Encoder interleaver Modulation

Channel

Encoder interleaver Modulation



OFDM

T. PENG ET AL.

668

Figure 2. Receiver structure of a SCM system.

()( )()()( )( )

()()( )

' 1,'

k kk

MLI

RmHmX mHmXmWm

HmX mm

= ≠

=++

∑

(4)

where

( )

MLI

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

approximated as a Gaussian random variable with mean

( )

MLI

and variance

( )

MLI

Var m

. Based on

the definition of extrinsic LLR, the output of ESE detec-

tor is calculated by

( )

( )( )( )

( )

MLI

ESE kMLI

Rm Em

eX mHmVar m

−

(5)

where

( )

MLI

and

( )

MLI

Var m

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)

( )

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

R RRN= −





, 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

[ ]

( )

,' 10

n nn

j jq

nn lq

eqT e

πτ π

−

= =







∑∑

(9)

where

is the lth path delay normalized by the sam-

pling time, Ts is the sampling time, and

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

[ ]

( )

LjN

s ls

nn l

nT e

πτ

−

∑

(10)

Let us define the

NL×

matrix FH and the

LN×

matrix α as follows,

[ ]

( )()()

( )

0 ,,...,1

ll ls

T NT

αα α



= −





(11)

[ ]

j nN

πτ

−

=FH

(12)

Then substituting (11) and (12) in (10) yields

s=H FHα

(13)

DEC

deinterleaver

−

MLD

interleaver



( )

1ESE

( )

1DEC

P/S FFT

DEC

deinterleaver

−

interleaver

( )

ESE K

()

DEC K

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:

( )

0,2 ,1

14, 2

besseljf Talgorithm

f Talgorithm

πσ





=−





( )

diag= −US P

( )

,zerosL L=P

( )

app

diag N=KF

2) Recursive Estimation: for symbol index i = 1,2,…,

Prediction step:

 

11ii i−−

=Sαα

ii −

−

= +P PU

( )

' '2

iaa ap

ii ii

eye N

−−

= +K PKKPK

Updata step:

 

( )

11i iiii

ip a

−−

=+−KR Kαα α

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)

−7.219

−4.219

0.4

−6.219

−10.219

3.2

−12.219

4.6

−14.219

Table 2. Multipath channel model B.

Path number

Average (dB)

Delay (Ts)

−1

0.6

−9

1.4

−10

2.2

−15

3.5

−20

Figure 3. MSE performances of the Kalman estimation algo-

rithms for SCM-OFDM systems in ch a nn el m o del A.

-2

-1

-3

-5 0 5 10 15 20

SNR(dB)

MSE

LS algorithm

MSSE algorithm

Kalman algorithm1

Kalman algorithm2

-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

-2

-1

-3

SNR(dB)

BER

LS algorithm

MSSE algorithm

Kalman algorithm1

Kalman algorithm2

-4

-6-4-202 468 10 1214

Ideal algorithm