Int. J. Communications, Network and System Sciences, 2009, 2, 852-856
doi:10.4236/ijcns.2009.29099 Published Online December 2009 (http://www.SciRP.org/journal/ijcns/).
Copyright © 2009 SciRes. IJCNS
Adaptive Channel Estimation in OFDM System Using
Cyclic Prefix (Kalman Filter Approach)
P. V. NAGANJANEYULU1, K. SATYA PRASAD2
1Department of ECE, Guntur Engineering College, Guntur, India
2JNTU, Kakinada, India
E-mail: {pvnaganjaneyulu, prasad_kodati}@yahoo.co.in
Received August 31, 2009; revised October 1, 2009; accepted October 29, 2009
ABSTRACT
OFDM is a promising technique for high data rate transmission and the channel estimation is very important
for implementation of OFDM. In this paper, cyclic prefix (CP) can be used as a source of channel informa-
tion which is originally used to reduce inter symbol interference (ISI). Based on this CP observation, we pro-
pose two cross coupled dual Kalman filters to track the channel variations without additional training se-
quences. One Kalman filter AR parameter estimation and another for fading channel estimation.
Keywords: Cyclic Prefix, Kalman Filter
1. Introduction
In OFDM systems, due to user mobility, each carrier is
subject to Doppler shifts resulting in time-varying fading.
Thus, the estimation of the fading process over each car-
rier is essential to achieve coherent symbol detection at
the receiver. In that case, training sequence/pilot aided
techniques and blind techniques are two basic families
for channel estimation. Training based methods require
the transmission of explicit pilot sequences followed
by suitable filtering. This paper focuses on estimation
of fading wireless channels for OFDM, using the ideas
of Cyclic Prefix (CP) based estimation and adaptive
filtering.
The time-varying fading channels are usually mod-
elled as zero-mean wide-sense stationary circular com-
plex Gaussian processes, whose stochastic properties
depend on the maximum Doppler frequency denoted by
fd. According to the Jakes model [1], the theoretical
Power Spectrum Density (PSD) of the fading process, is
band-limited. Moreover, it exhibits twin peaks at ± fd.
The fading wireless channel statistics can be directly
estimated by means of the Least Mean Square (LMS)
and the Recursive Least Square (RLS) algorithms as in
[2]. Alternatively, Kalman filtering algorithm combined
with an Autoregressive (AR) model to describe the time
evolution of the fading processes and it provides superior
performance over the LMS and RLS based channel esti-
mators in [3]. In addition, when the AR model parame-
ters are unknown, dual filtering algorithms are used to
estimate the fading channels.
In this paper, for the channel estimation of OFDM, a
system model and architecture over fading channels are
presented. In the next section a CP based model and the
different channel estimation algorithms Kalman and
Dual-Kalman are discussed. The performance results are
discussed in the next section, finally simulation results
are presented.
1.1. Existing Methods for Channel Estimation
Different Channel Estimation methods are proposed
based on training sequence, blind and semi-blind. In
practice we either assume the channel is invariant and
use the initial training to get the channel estimation are
periodically employ training sequence to trap the channel
variations. These will cause performance loss or increase
the overhead of the system. So, we present that the CP in
OFDM which is used to reduce ISI and normally dis-
carded at the receiver can be viewed as a training se-
quence for channel estimation. In paper [3], channel es-
timation is proposed by two Kalman filters based on
noisy data as training sequence. In this paper, we present
two cross coupled Kalman by using CP as training se-
quence and their performances are compared.
2. System Model
In the following, we consider a low to moderate Doppler
environment, which allows for a block fading (quasi-
static) channel assumption. This implies that the channel
P. V. NAGANJANEYULU ET AL. 853
n
Noise
n
z
CP
Figure 1. Block schematic of the OFDM system.
tap variations within OFDM symbol duration are negli-
gible, and hence we may define an L×1 channel tap vec-
tor for each OFDM symbol as

(0)(1). . . (1)T
nnnn
hh hLh (1)
where hn(l) is the lth channel tap for the nth OFDM sym-
bol.
The classical Doppler spectrum for each of the L
channel taps is approximated by an independent AR-2
process [4].
For the lth channel tap at nth OFDM symbol, we have
112 2
()()() ()
nnn
hlah l ahl vl

 (2)
where a1 and a2 are the AR-2 coefficients are defined in
[5] and vn(l) is the modelling noise for lth tap at symbol n.
2.1. OFDM Architecture over Fading Channel
We consider an OFDM system as in Figure 1 with N data
subcarriers. Input data are buffered, converted to a paral-
lel stream and modulated to i.i.d. equi-probable symbols
Xn(k), where Xn(k) denotes the kth symbol of the nth
OFDM symbol. Each symbol mapped to some complex
constellation points, X
n(k), k=0,1,. . ,N-1 at each n. The
modulation is implemented by N-point inverse discrete
Fourier transform (IDFT) for the symbol vector
(0) (1). . . (1)T
nnn n
XX XNX (3)
is
1
2/
0
1
()(), 0m
N
jmkN
nn
k
-1
x
mgi XkeN
N
 
CP of length
gi is appended to form the transmitted vector as
(0)(1) . . . (1)()(1). . . (1)T
nnnnnnn
xxxgixgixgixgi N
'
x
(4)
where
()(), 01
nn
x
mxNmmgi 
The received symbol corrupted by fading channel and
AWGN becomes
1
0
()()()(), 0m1
L
nnnn
l
y
mhlxmlzm NgiL

(5)
where n is the OFDM symbol index,
zn(m) is an AWGN sample with zero mean and vari-
ance σ2 at instant m in the nth OFDM symbol.
Demodulation involves removing the cyclic prefix and
taking N-point DFT of the received vector to get

(0) Y(1). . . Y(1)T
nn nn
YNY (6)
In frequency domain, we have over each subcarrier
()()()()
nnnn
YkXkH kZ k
(7)
where Hn(k) is the channel frequency response at subcar-
rier k given by
1
2/
0
1
()(), 01
L
jlkN
nn
l
Hkhlek N
N
 
(8)
and zn(k) is the noise on subcarrier of nOFDM
symbol i.e.,
th
kth
1
2/
0
1
()(), 01
N
jmkN
nn
m
Zkzmek N
N
 
(9)
At the receiver, the channel estimator is followed by
frequency domain equalizer. After equalization, the es-
timated symbol at the kth symbol becomes
()()()()
ˆ() ˆˆˆ
()()()
nnnn
n
nn
YkXkH kZ k
Xk
n
H
kHkH
 
k
(10)
where is the estimate of
ˆ()
n
Hk ()
n
H
k
)
defined in Equation
(8). The estimated symbolsˆ(
n
X
kare then demapped to out-
put bits.
3. CP Based Channel Estimation Techniques
This section describes the use of various adaptive filter-
ing algorithms in CP based frame work for channel esti-
mation in OFDM systems. From Equation (5), we know
that
()(0)() (1)(1)...(1)(1)(
nn nnn
ymhxmhxmhLxmLzm)
 (11)
Gathering the received samples of the nth received
OFDM symbol for time instants 0mgi –1, we obtain a
gi×1 vector
,(0) (1) (2)...(1)T
nCPnn nn
yyy ygiy (12)
n
y
Channel
S/P
Training/decision directed
'
n
x
Data out
Channel
Estimation
DFT
Channel
Equalization
QPSK
Demodulation
Data in P/S
QPSK Modu-
lation IDFT
Copyright © 2009 SciRes. IJCNS
P. V. NAGANJANEYULU ET AL.
854
which is the CP of the received OFDM symbol, and
,(0) (1) (2)...(1)T
nCPnn nn
zzz zgiz (13)
is the gi×1 vector of AWGN samples affecting the CP
part of the nth received OFDM symbol.
3.1. Kalman Filtering (KF) Algorithm
When operating in a non-stationary environment, Kal-
man filter [6] is known to yield an optimal solution to the
linear filter problem. This subsection describes the ap-
plication of KF to the channel estimation problem in
OFDM. For this purpose, the system is formulated as a
state-space model, with unknown channel taps compris-
ing the state of the system. We assume that the state Sn,
to be estimated at OFDM symbol index n, comprises of
channel taps at two consecutive OFDM symbols [7].
121
T
nnn
L
shh (14)
From Equation (1) and Equation (2) we have
1
(0)(1). . . (1)T
nnn n
L
hhhL
h

11111
(0)(1). . . (1)T
nnnn
L
hh hL

h
and
1122nn n
aa
n
hh hv
(15)
From above equations we get
12
21
nn
LL L
n
nL1Ln
aa

 


 


 
hh
0I v
hIIh (16)
We observe that Equation (16) provides the basis for
forming the process equation as
1nnn
sBs v
(17)
Here, transition matrix
222
LL L
L1L
L
L
aa



0I
BII (18)
0L×L denotes the
L
×L matrix of all zeros and IL is the
L
×L identity matrix.
Process noise vector

1L 21
(0)(1)...(1) T
nnnn
L
vv vL
v0 (19)
where vn(l) is the modelling noise as in (2)
From Equation (11), we have
( )(0)( )(1)(1)...(1)(1)( )
nn nnn
ymhxmhxmhLxmLzm 
11
1
(0)(0) (1) . . . (1)
(1)(1) (0) (1) . . .
(1)
nnnn
nn nn
n
yx xNgixNgiL
yx xxNgi
ygi

 

 





(0)
(1)
(1) (2) . . . ()(1)
n
n
nnn n
h
h
xgixgixgi LhL





 


()
n

z
m
where
01mgi
We observe from above that following provides the
basis for forming measurement equation as
,nCPn nnCP

,
y
As z (20)
where the measurement matrix n
Ain Equation (20) is
formed from the matrix An by augmenting it with a null
matrix as
L2
ngi n
g
iL


A0 A (21)
Here A
n is a gi×L matrix of transmitted symbols that
determine the CP of the received OFDM symbol.
11
1
(0) (1) . . . (1)
(1) (0) (1) . . .
nn n
nn n
n
xx Ngix NgiL
xx xNgi

 

A
(1) (2) . . . ()
nn n
g
iL
xgixgixgi L

 

Considering that the CP appended to an OFDM sym-
bol is a replication of the last gi values of that symbol,
we may write An in terms of transmitted CP value as,
11 1
11
(0) (1) (2) . . . (1)
(1) (0) (1) . . . (2)
(
nn nn
nnnn
n
n
xxgixgi xgiL
xxxgixgiL
x
 

 

A 
1) (2) . . . ()
nn
g
iL
gix gix giL
 
An has gi rows corresponding to gi consecutive time
instants of the CP. The L elements of each row are the
transmitted symbol values affecting the received CP
value at that instant. This matrix structure assumes that
the CP length is at least equal to the number of taps in the
channel impulse response, i.e. no inter block interference.
The measurement noise vector Zn,CP, in Equation (20),
comprises the gi×l vector of AWGN samples affecting
the cyclic prefix part of the OFDM symbol.
We observe that Equation (17) and Equation (20) pro-
vide the basis for forming the process equation and
measurement equation, respectively for the state space
model, as follows
1nnn
sBs v
,nCPn nnCP

,
y
As z (22)
A Kalman filter is employed to estimate the unknown
Copyright © 2009 SciRes. IJCNS
P. V. NAGANJANEYULU ET AL. 855
Q
state of the system. Cyclic prefix of the received OFDM
symbol yn,CP is given as input observation to Kalman
algorithm, the following estimation equations are given
by [3]
|12 21|11
[] H
nnL Ln n 
BBPP (23)
1, 1
ˆ
[]
nginCP nn

αyAs
(24)
|1 2
[] H
ngigin nnn
CAAPQ
(25)
1
2|1
[] H
nLginnn n

KAPC
(26)
21 1
ˆˆ
[]
nLn nn
sBsKα (27)
1
ˆˆ
[]
nLn
R
hs
, 2
[ ]
L
LLLL
R
0I (28)
22 2|1
[] []
nLLLnnnn
IKAPP
(29)
where Kn is the 2L×gi Kalman gain matrix , is the
state estimate at the nth OFDM symbol, Q1 and Q
2 are
the covariance matrices of vn and Zn,CP respectively,
Pn|n-1 is the priori covariance matrix of estimation error ,
and Pn is the current covariance matrix of estimation
error. When the channel taps are modelled as a zero
mean random process, the algorithm is initialized with an
all-zero state vector. Besides this, the assumption of un-
correlated scattering (US) causes the different channel
taps to be i.i.d., and the error covariance matrix is ini-
tialized as an identity matrix.
ˆn
s
00 2
ˆ1
L
ss0
00000
ˆˆ
()()
H
n
E


ssssIP
The receiver operates in training and decision directed
modes. In training mode the known transmitted CP
(xn,CP ) and CP part of the received OFDM symbol (yn,CP)
form the input to the above Kalman filter algorithm, and
get the channel estimation Hn(k), we get
()
ˆ() ˆ()
n
n
n
Yk
Xk
H
k
(30)
In decision directed mode the receiver uses the esti-
mated channel vector from the previous OFDM symbol
to demodulate the received symbol and generate an esti-
mate of transmitted CP (). Here the transmitted
CP part can be estimated by previous estimated channel
i.e.,
,
ˆ()
nCP
Xk
,
,
1
()
ˆ() ˆ()
nCP
nCP
n
Yk
Xk
Hk
(31)
This estimated CP and CP of the received OFDM
symbol (yn,CP) helps to estimate the channel.
The equations from (23) to (29) can be carried out by
providing the AR parameters that are involved in the
transition matrix B and the driving process variances are
available. In case, these are unknown, for estimating
these parameters Dual-Kalman filtering technique is
used.
3.2. Dual-Kalman Filtering Algorithm
To estimate the AR parameters θn from the estimated
fading process ĥn, Equation (28) is firstly represented as
an AR-2 model to express the estimated fading process
as a function of θn (AR parameter vector).
1
12
2
ˆˆˆ
nnn
a
a






hhhw
n
n
(32)
12 1
12
221
12
12
ˆˆˆ
(0)(0) (0)
ˆˆˆ
(1)(1) (1)
ˆˆˆ
(1)(1) (1)
nnn
nnn
n
nnn
LL
hhh a
hhh
a
hLh LhL



















w

nn
rHθw (33)
where rn is the estimated channel vector, θn is the AR
parameter vector defines as .
12
[ ]
T
naaθ
and wn is the L×1 noise vector as in Equation (19).
When the channel is assumed to be stationary, the AR
parameters are time-invariant and satisfy the following
relationship
1nn
θθ
(34)
As Equation (33) and Equation (34) define a state-
space representation for the estimation of the AR pa-
rameters, a second Kalman filter can be used to recur-
sively estimate θn as follows [3]
/1 1/1
22
[]
nnn n


PP
(35)
1
ˆ
[] [
nLnn
αrHθ1
]
(36)
/1 3
[]
nnn
H
LL

CHHPQ
n
C
(37)
/1
1
2
[]
nnn
H
L

KHP (38)
21 1
ˆˆ
[] nn
nn

θθKα (39)
//
22 2
[] []
nnn nn

1
IKHPP
(40)
where Q3 is the covariance matrix of the wn , the error
covariance matrix and the initial AR parameter vector
are defined as
002
ˆ
θθ01
2
0/0 0000
ˆˆ
()()
H
E



θθθθ IP
3.3. Noise parameters estimation
Apart from estimating the AR parameters, we also need
to estimate the noise parameters for the fading channel
Copyright © 2009 SciRes. IJCNS
P. V. NAGANJANEYULU ET AL.
Copyright © 2009 SciRes. IJCNS
856
4. Results and Conclusions
In this analysis we compare CP based dual Kalman and
Kalman. At SNR of 10db, BER of Kalmanis 10-0.8 where
as for dual Kalman; this value is 10-1.81. For Overall SNR
is concerned, dual Kalmangives better performance and
doesn’t require any additional training sequence like
training bits and noise. (Figure 2)
5. References
[1] R. Steele, “Mobile radio communications,” New York:
IEEE Press, 1992.
[2] X. W. Wang and K. J. R. Liu, “Adaptive channel estimation
using cyclic prefix in multicarrier system,” IEEE Commu-
nication Letters Magazines, Vol. 3, pp. 291–293, October
1999.
Figure 2. BER v/s SNR for Kalman and Dual Kalman.
environment i.e., variance of v
n and Zn,CP. This can be
done by using the error covariance matrices. From Equa-
tion (23) and Equation (29) we can write the noise vari-
ances recursively as
[3] A. Jamoos, D. Labarre, E. Grivel, and Najim, “Two cross
coupled Kalman filters for joint estimation of MC-DS-
CDMA fading channels and their corresponding autore-
gressive parameters,” Proceedings of EUSIPCO, Antalya,
Turkey, September 4–8, 2005.
22 1|1
[]
H
HH
nLLnn nnnnn

 BBKααKLPP (41)
[4] M. K. Tsatsanis, G. B. Giannakis, and G. Zhou, “Estimation
and equalization of fading channels with random coeffi-
cients,” Signal Process, Vol. 53, pp. 211–229, 1996.
11
11
ˆˆ
()( 1)T
n
n
Qn Qn
nn

D
LD (42)
12
1 0 . . . . . 0
L
D [5] A. Jamoos, J. Grolleau, and E. Grivel, “Kalman vs H algo-
rithms for MC-DS-CDMA channel estimation with or with-
out a priori AR modelling,” IEEE Multicarrier Spread
Spectrum, Springer Verlag, pp. 427–436, 2007.
|1
[]
H
H
ngigin nn nnn
Mαα AAP
(43)
221
11
ˆˆ
()( 1)T
n
n
Qn Qn
nn
M[6] I. R. Petersen and A. V. Savkin, “Robust Kalman filtering
for signals and systems with large uncertainties,” Boston,
MA: Birkhäuser, 1999.
1
D
D
(44)
11
1 0 . . . . . 0
g
i
D, [7] Y. Li, L. J. Cimini, and N. R. Sollenberger, “Robust chan-
nel estimation for OFDM systems with rapid dispersive
fading channels,” IEEE Transactions on Communications,
Vol. 46, pp. 902–915, July 1998.
where and are the estimated variances of
the process noise vn and modelling noise Zn,CP respec-
tively.
1
ˆ()Qn 2
ˆ()Qn