Communications and Network, 2013, 5, 43-47
doi:10.4236/cn.2013.52B008 Published Online May 2013 (http://www.scirp.org/journal/cn)
An Iterative Maximum Likelihood Synchronization
Method for OFDM System
Chi-Min Li, Wei-Tse Sun
Department of Communications, Naviga tion and Control Engineer ing, National Taiwan Ocean University, Keelung, Chinese Taipei
Received 2013
ABSTRACT
OFDM system divides a wideband transmission bandwidth into several overlapped narrowband subcarriers to avoid the
serious frequency selectiv e fading problem. However, if Ti ming Offset (TO) and Carrier Frequency Offset (CFO) exist
between the Transmitter (Tx) and Receiver (Rx), OFDM system will suffer the Inter-Symbol Interference (ISI) and In-
ter-Carrier Interference (ICI) that degrade the system performance dramatically. In this paper, we propose an iterative
maximum likelihood method for synchronization. We also adopt the overlap concept to reduce the plateau problem.
Simulation results sh ow that the proposed meth od can predict the time delay and frequ ency offset correctly even under
the multipath fading scenarios.
Keywords: Synchronization; Maximum Likelihood; OFDM
1. Introduction
Orthogonal Frequency Division Multiplexing (OFDM)
has been widely adopted in the current wireless commu-
nication systems, such as IEEE 802.11 a/g, and Long
Term Evolution (LTE). To correctly accomplish symbol
detection, an OFDM system has to establish the time and
frequency synchronization [1]-[7] firstly to avoid the Bit
Error Rate (BER) degradatio n. Poor synchronization will
destroy the orthogonal property of the received signal
and system will suffer the Inter-Symbol Interference (ISI)
and Inter-Carrier Interference (ICI). Therefore, synchro-
nization is a very important issue for the transmission,
especially for the OFDM system.
In [3], authors proposed a Maximum Likelihood (ML)
method via adopting the correlation properties of the
Cyclic Prefix (CP) with the copied portion of an OFDM
symbol to achieve the time and frequency synchroniza-
tion. Its performance depends notably on the length of
the inserted CP. The longer of the length of CP, the better
of the synchronization can be obtained. Based on this
behavior, if we use a preamble with two repetitions to
conduct the synchronization, the ML method should
perform well. However, a serious plateau problem arises
that causes the Receiver (Rx) can not predict the arrival
time precisely. This problem comes from the inherent
correlation properties of the preamble with two or multi-
ple repetitions. Many methods have been proposed to
reduce the plateau problem [2,4].
In this paper, we propose an iterative ML synchroniza-
tion method that adopts the overlap concept to reduce the
plateau problem. This paper is organized as follows. In
Section II, the conventional ML synchronization method
and the proposed iterative method will be described in
detail. Simulation results of these methods are compared
in Section III. Finally, some conclusions for this paper
are given in Section IV.
2. Method Descriptions
In OFDM system, Transmitter (Tx) can insert a CP at the
Guard Interval (GI) of the OFDM signal to avoid the ISI.
Assume the received signal can be expressed as
2/
() ()()
jnN
rn xnewn

  (1)
where

is the time delay,

is the normalized fre-
quency offset and is the Additive White Gau ssian
Noise (AWGN). ()wn
Consider an observation interval with 2N+L samples
(Figure 1), the interval contains a complete OFDM sym-
bol of N+L samples, where N is the number of subcarrier
and L is the length of the CP.




,1,...,-1 ',1,...,-1ILINNN     L
Figure 1. ML synchronization method.

C.-M. LI, W.-T. Sun
44
Define intervals
I
and '
I
as in Figure 1, the re-
ceived signal has the following correlation prop erties [3]:
22
2
*2/ 2/*
*2/ *
2
[()*( )]
()( )()()
()()()()
, 0
, ,
0 , e ls e
jmN jkN
jkN
xw
j
x
Erkrk m
ExkxkmeExkewkm
Exk mewkEwkwk m
m
emNkI
 


 










 










(2)
where 2
2[()]
xExk
  is the energy of the transmitted
signal and 2
2[()]
wEwk
  denotes the energy of the
AWGN.
Let the Log-Likelihood Function (,)

  be defined
as


**
1
22
(,)
() ()cos2() ()
() ()
2
L
k
rkr kNrkr kN
rkrk N
















 (3)
where 1
SNR
SNR
  is a constant related to the Sig-
nal-to-Noise Ratio (SNR) and denotes the phase.
The ML method estimates the time delay and frequency
offset as Equation(5) and Equation(6) respectively.


(,)
ML
122
*
max(,)
ˆ
max( ,())
()( )()( )
2
L
k
rkrk Nrkrk N






 










(4)

ML
122
*
ˆ
argmax( )()( )()
2
L
k
rkrk Nrkrk N












(5)
ML
ML
ˆ1*
ML ˆ
1
ˆ
()()
2
L
k
rkr kN





 
 (6)
In the ML merthod, the performance of the estimated
time delay and frequency offset depend heavily on the
length of the inserted CP. Table 1 lists th e parameters for
the ML method in [3] under different lengths of CP. Fig-
ures 2 and 3 are the corresponding time and frequency
simulation results. According to the results, we can note
that if we increase the leng th of the CP, the Mean Square
Errors (MSE) for both the time and frequency offsets can
be improved.
Based on above observ ation , if we use a prea mble w ith
two repetitions to conduct the synchronization, the ML
method should perform well. A simple extension for the
ML method can be depicted in Fi gur e 4.
Table 1. Simulation parameters for the ML method.
Modulation Type QPSK
Number of Subcarriers , N 256
Cyclic Prefix Length , L 4、6、8
16、32
Delay T ime(sample) , 128
Normalized Frequency Offset , 0.4
Channel Model AWGN
Monte Carlo 10000times
0246810 12 1416 18 20
10
-2
10
0
10
2
10
4
Time Offset Estimation
SNR(dB)
MSE
ML Method,

=128,

=0.4,L=4
ML Method,

=128,

=0.4,L=6
ML Method,

=128,

=0.4,L=8
ML Method,

=128,

=0.4,L=16
ML Method,

=128,

=0.4,L=32
Figure 2. MSE of the time delay.
0246810 12 14 16 1820
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Frequency Offset Estimation
SNR(dB )
MSE
ML Method,=128,=0.4,L=4
ML Method,=128,=0.4,L=6
ML Method,=128,=0.4,L=8
ML Method,=128,=0.4,L=16
ML Method,=128,=0.4,L=32
Figure 3. MSE of the frequency offset.
'
,1,. ..,1, ,1,. .. ,1
222
NNN
II   N

 




  
Figure 4. Preamble with two repetitions.

C.-M. LI, W. –T. Sun 45
In Figure 4, we construct a preamble with two repeti-
tions to conduct the ML estimation. The ML method
calculates the correlation between the defined intervals I
and I’. However, a serious plateau problem will arise.
This problem comes from the inherent correlation prop-
erties of the preamble with two or multiple repetitions.
The plateau problem can be easily solved if we calculate
the cross-correlation in an overlap manner. Figure 5 is
an example that the intervals I and I’ are overlapped. In
this case, the length of the overlap region is exactly to L.
In the simulation part, we will demonstrate that with the
help of the overlap region, the plateau problem can be
efficiently reduced. The corresponding time delay and
frequency offset estimations can be formulated as in
Equation(7) and Equation(8).
overlap
2
22
12
22
*
ˆ
argmax( )()( )()
22 2
NL
k
NN
rkr krkrk









 






(7)
overlap
overlap
overlap
ˆ1
2*
ˆ
1() ()
2
ˆ
NL
k
N
rkr k






 
 (8)
The main difference between the Equations 5-6 with
Equations 7-8 is the range for the cross-correlation cal-
culation. The overlap concept can ease the plateau prob-
lem of synchronization if a preamble contains multiple
repetitions. Besides, the performance of the synchron iza-
tion can be further improved if we adopt the iteration
operation for the preamble. For example, if the Part-A of
the preamble in Figure 5 can be further constructed with
two repetitions as illustrated in Figure 6. The preamble
wilt multiple repetitions can be easily generated if we
adopt the Constant Pilot Padding Method (CPPM) [9]
proposed previously. Thereafter, we can conduct the fol-
lowing iterative steps to increase the synchronization
precision for both the time and frequency.
Step.1 Calculate the ML estimation for the intervals
(
I
and '
I
) with N/2 separation in Figrue 7. In this
case, the intervals
I
and '
I
contain the overlap region
with length equals to L.
L
'
22222 222
,1,...,1, ,1,...,1
222
NNN
ILI 



 LN



Figure 5. Overlap cross-correlation calculation.
Figure 6. Preamble for iterative operations.
 
I
I’

CP B B B B
N/4 N/4 N/4 N/4
N/2+L N/2 +L
Figure 7. ML calculation for interval I and I’.
 
I’

CP B B B B
N/4 N/4 N/4 N/4
3N/2
3N/2
L
I
Figure 8. ML calculation for interval I and I’
Step 2 Calculate the ML estimation for the intervals
(
I
and '
I
) with N/4 separation in Figure 8. In this
case, the intervals
I
and '
I
contain the overlap region
with length equals to N/2.
Step 3 Calculate the estimated time and frequency off-
sets via the Equations 9-10. We can have more precise
synchronization estimation which will be shown at the
simulations in Section III.
1
2
1
Iterative
31
4
2
3
11
24
12
ˆarg max{()
()
()()
2
N
L
k
N
L
kL
NN
LL
kkL
θk
k
kk









 

 

 














 





 









(9)
Iterrative
Iterrative
ˆ1
2
Iterative 1
ˆ
1
ˆ()
N
L
k
k




 

  (10)
where
*
1
*
2
()()()
2
()() ()
4
N
krkrk
N
krkrk





C.-M. LI, W.-T. Sun
46
2
2
1
2
2
2
() ()()
2
() ()()
4
N
krk rk
N
krk rk




(11)
3. Computer Simulations
Table 2 lists the parameters to evaluate the proposed over-
lap concept. The delay time offset is 128 in this simula-
tion. Figrue 9 is the Log-Likelihood function calculation
if the two-repetition preamble conducts only non- over-
lapped cross correlation calculation, L/2-overlapped cal-
culation and the L-overlapped cross correlation calcula-
tion respectively. Results show that if the leng th of over-
lap equals to or greater than the length of CP, it can avoid
the plateau problem efficiently. Figures 10 and 11 are
the corresponding time and frequency offsets estimation
for different SNR scenarios. With the help of the over-
lapped cross-correlation calculation, the proposed con-
cept can predict the offsets correctly.
Table 3 lists the parameters to evaluate the proposed
iterative method under the multipath fading ch anne l. Path
attenuations are according to the Vehicular-A [8] channel
model. Figures 12 and 13 are the estimated time and
frequency offsets. Results show with the proposed itera-
tive method, the synchronization precision can be further
improved for both the AWGN and multipath fading chan-
nels.
Table 2. Simulation parameters.
Modulation Type QPSK
Number of Subcarriers 256
Cyclic Prefix Length 32
Delay Time(sample) 128
Normalized Frequency Offset 0.4
Channel Model AWGN
Monte Carlo 10000times
050100150 200 250300
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
Time Offset Estimation
Time(sample)
Log-Likelihood Function of Time O ffset
X: 129
Y: 0.01301
Non-Overlapped ML Met hod
L/2-Overlapped ML Met hod
L-Overlapped ML Method
Figure 9. Log-Likelihood function calculation.
0 2 4 6 81012 14 16 1820
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
Time Offset Estimation
SNR(dB)
MSE
Non-Overlapped ML Method
L/2-Overlapped ML Method
L-Overlapped ML Method
Figure 10. MSE of the timing offset estimation.
0246810 12141618 20
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Frequency Offset Est imation
SNR(dB)
MSE
Non-Overlapped ML Method
L/2-Overlapped ML Method
L-Overlapped ML Method
Figure 11. MSE of the frequency offset estimation.
Table 3. Simulation parameters.
Modulation Type QPSK
Number of Subcarriers 256
Cyclic Prefix Length 32
Delay Time(sample) 128 132 136
Path Gain(dB) 0 -1 -9
Normalized Frequency Offset0.4
Channel Model AWGN 3Paths Rayleigh Fading Channel
Monte Carlo 10000times
0246810 1214 16 18 20
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
Time Offset Estimation
SNR(dB )
MSE
L-Overlapped ML Method，AWG N
L-Overlapped ML Method，3Paths
It er at ive M L M et ho d，AWGN
It er at ive M L M et ho d，3Paths
Figure 12. MSE of the time offset estimation.

C.-M. LI, W. –T. Sun
47
[2] Z. S. Pang and X. M. Li, “A Novel Synchronization Al-
gorithm for OFDM System Based on Training Sequence
Added Scramble Code,” IEEE Communications Tech-
nology and Applications (ICCTA), 2009
0246810121416 18 20
10
-6
10
-5
10
-4
10
-3
Frequenc y Offset Es t imation
SNR(dB)
MSE
L-Overlapped ML Method，AWGN
L-Overlapped ML Method，3Paths
Iterative ML Method，AWGN
Iterative ML Method，3Paths
[3] J. J. van de Beek, M. Sandell and P. O. Borjesson, “ML
Estimation of Time and Frequency Offset in OFDM Sys-
tems,” IEEE Transactions on Signal Processing, Vol. 45,
No.7, 1997, pp.1800-1805. doi;10.1109/78.599949
[4] H. Minn, M. Zeng and V. K. Bhargava, “On Timing Off-
set Estimation for OFDM Systems,” IEEE Communica-
tions on Letters, Vol. 4, No. 7, 2000, pp. 242-244.
doi;10.1109/4234.852929
[5] S. D. Ma, X. Y. Pan, G. H. Yang and T. S. Ng, “Blind
Symbol Synchronization Based on Cyclic Prefix for
OFDM Systems,” IEEE Transactions on Vehicular Tech-
nology ,Vol. 58, No. 4, 2009, pp. 1746-1751.
doi;10.1109/TVT.2008.2004031
Figure 13. MSE of the frequency offset estimation.
4. Conclusions [6] Z. Q. He, Y. Han and X. Fu, “Frequency Synchronization
Algorithm Based on PN Sequence,”4th International
Conference on Wireless Communications, Networking
and Mobile Computing, 2008.
In this paper, we propose an iterative ML method for
synchronization. We adopt the overlap concept to reduce
the plateau problem. Actually, any form of preamble
contains a two-repetition or multiple-repetition structure
can conduct the proposed iterative method easily. Simu-
lation results show that the proposed method can predict
the time delay and frequency offset correctly even under
the multipath fading scenarios.
[7] H. Hwang and H. Park, “A Timing Synchronization De-
sign In OFDM Systems,” IEEE International Workshop
on Satellite and Space Communications, 2008,
pp.290-293. doi;10.1109/IWSSC.2008.4656815
[8] 3GPP TS 25.101 V5.12.0, “Technical Specification
Group Radio Access Network User Equipment (UE) Ra-
dio Transmission and Reception (FDD),” 2004.
REFERENCES [9] C. M. Li, H. C. Chen, P. J. Wang, J. C. Wu and I. T.
Tang, ”Timing Synchronization and Channel Estimation
of a Constant Pilot Padding OFDM System,” The 14th
Asia-Pacific Conference on Communications (APCC2008)
Tokyo, Japa n.
[1] T. M. Schmidl and D. C. Cox, “Robust Frequency and
Timing Synchronization for OFDM,” IEEE Transactions
on Communication, Vol. 45, No. 12, 1997, pp.1613-1621.
doi:10.1109/26.650240