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December 2013, 20(6): 49–54
www.sciencedirect.com/science/journal/10058885 http://jcupt.xsw.bupt.cn
The Journal of China
Universities of Posts and
Telecommunications
Frequency offset estimation based on peak power ratio in LTE system
WANG Jun1
(�), GUAN Bao2
, LIU Shou-yin1
, XIE Wen-wu1
1. Department of Electronics and Communication Engineering, Central China Normal University, Wuhan 430079, China
2. Department of LTE, Wuhan Hongxin Telecom. Technologies CO., LTD, Wuhan 43009, China
Abstract
Orthogonal frequency division multiplexing (OFDM) is one of the key techniques for long term evolution (LTE) system.
Frequency offset estimation of OFDM is an essential issue. Especially in the high-speed environment, the frequency offset
will become large. Based on the features of LTE uplink physical random access channel (PRACH), this paper proposes a
new frequency offset algorithm by using peak power ratio to enlarge the range of frequency offset estimation. According to
the relation between frequency offset and the power delay profile (PDP), the ratio of the peak power of the PDP at the main
window to that at the negative window or positive window is utilized to estimate frequency offset. Simulation results show
that the new proposed algorithm extends the estimation range of frequency offset from 1 000 Hz to 1 250 Hz. Meanwhile
the accuracy of frequency offset estimation is almost not lost. Particularly in low signal noise ratio (SNR), the new
algorithm has lower mean square error (MSE) compared with traditional phase differential algorithm.
Keywords frequency offset estimation, LTE, peak power ration, PDP
1 Introduction
LTE is an attractive technique for high data rate
transmission. It has flexible frequency spectrum
application strategy, variable bandwidth and simplified
lower network cost to adapt the diverse service
requirement. OFDM is one of the key techniques for LTE
system. Similar to the OFDM, the single carrier frequency
division multiplexing access (SC-FDMA) technique [1]
used in LTE uplink channels is typically sensitive to
frequency offset. Therefore the frequency offset estimation
is essential for LTE system. The training sequence is very
important for reliable frequency offset estimation in near
Shannon limit coding system [2].
Phase differential algorithms [3–7] are the traditional
frequency offset estimation method based on training
sequence. In Ref. [3], the maximum likelihood (ML) is
used to estimate frequency offset with estimation range
limited to ±1/2 inter-carrier spacing. Ref. [4] uses one
unique symbol with a repetition within half a symbol
Received date: 05-06-2013
Corresponding author: LIU Shou-yin, E-mail: [email protected]
DOI: 10.1016/S1005-8885(13)60108-9
period, and allows a large acquisition range for the carrier
frequency offset. However, this algorithm [4] need to
design two-symbol training sequence, which will usually
be placed at the start of the frame, then it is not suit
applicable for LTE system. The algorithms in Refs. [5–7]
can release the limitation between accuracy and estimation
range, but demands high SNR for good performance. In
addition, all of these papers [3–7] are based on the phase
differential algorithms.
In this paper, a new frequency offset estimation
algorithm based on peak power ratio is proposed, which
uses the characteristics of Zadoff-Chu (ZC) sequence for
physical random access channel (PRACH). First we
analyze the relation between the frequency offset and PDP,
where PDP is the power distribution derived from the cross
correlation between received preamble sequence and the
cyclic shift of original ZC sequence. And then the
one-to-one map table between peak power ratio value and
frequency offset is obtained. Secondly we judge the sign of
frequency offset by comparing the peak power of the PDP
at negative window and that at positive window. Finally,
the frequency offset can be obtained by calculating the
ratio of the peak power of the PDP at main window to that
50 The Journal of China Universities of Posts and Telecommunications 2013
at negative window or positive window and looking up the
one-to-one map table.
Simulation results show that the new algorithm extends
the estimation range of frequency offset from 1 000 Hz to
1 250 Hz. Meanwhile the accuracy of frequency offset
estimation is almost not losing. Particularly in low SNR,
the new algorithm has lower MSE compared with
traditional phase differential algorithm.
This paper consists of five sections. Sect. 2 describes the
phase differential algorithm [3] simply. Sect. 3 explains the
new frequency offset estimation algorithm based on peak
power ratio in detail. In Sect. 4 simulation results are
presented. Finally, a conclusion is drawn in Sect. 5.
2 Phase differential algorithm
The phase differential algorithm proposed in Ref. [3] is
usually applied in LTE system. The training sequence in
the algorithm consists of two repetition symbols. Assume
x1(n) and x
2(n) are the symbols transmitted repeatedly in
time domain with length L, the delay between two symbols
is Nd samples, in addition, frequency offset is Δf
c. Thus, the
received sequences 1
( )r n and 2
( )r n can be expressed
as:
[ ]c s
j 2π
1 1 1( ) ( )e ( ); 0,1, , 1
f nT
r n x n n n L
ϕ η− Δ +Δ= + = −… (1)
[ ]c d s
j 2π ( )
2 2 2( ) ( )e ( ); 0,1, , 1
f n N T
r n x n n n L
ϕ η− Δ + +Δ= + = −…
(2)
where Δφ is an unknown random phase with uniform
probability density in [0, 2π), 1
( )nη and 2
( )nη denote
the additive white Gaussian noise (AWGN) with zero
mean and variance 2
n
σ . Hence, the function of the
correlation between r1(n) and r
2(n) is obtained as:
[ ]
[ ]
c sc d s
c d s
1 1
* *
l 1 2 d
0 0
1 1
j 2πj2π * *
1 2 1 2
0 0
1 1
j 2π ( )* *
1 2 1 2
0 0
( ) ( ) ( ) ( )
e ( ) ( ) [ ( )e ( )]
[ ( ) ( )e ] [ ( ) ( )]
L L
n n
L L
f nTf N T
n n
L L
f n N T
n n
R r n r n r n r n N
x n x n x n n
n x n n n
ϕ
ϕ
η
η η η
− −
= =
− −− Δ +ΔΔ
= =
− −Δ + +Δ
= =
= = + =
+ +
+ =
∑ ∑
∑ ∑
∑ ∑
c d s
1
j2π *
1 2
0
e ( ) ( )
L
f N T
n
x n x n η−
Δ
=
+∑ (3)
where
[ ]
[ ] ]
c s
c d s
1 1
j 2π * *
1 2 1 2
0 0
1
*j 2π ( )
1 2
0
= [ ( )e ( )] [ ( ) ( )
[ ( ) ( )].e
L L
f nT
n n
L
f n N T
n
x n n n x n
n n
ϕ
ϕ
η η η
η η
− −− Δ +Δ
= =
−Δ + +Δ
=
+ ⋅
+
∑ ∑
∑
If we ignore the impact of the noise η, Eq. (3) can be
approximately expressed as:
c d s
1
2j2π
l 1
0
e ( )
L
f N T
n
R x n
−Δ
=
= ∑ (4)
Then, the frequency offset can be derived from the
phase of Rl, which is given by:
l
c
d s
arg
2π
R
f
N T
Δ = (5)
In LTE uplink, two demodulation reference
signals (DMRS) symbols in physical uplink shared
channel (PUSCH) are usually used to estimate frequency
offset. The time interval of two symbols is 0.5 ms, that is
d s
0.5 msN T = . As l
arg( ) ( π, π)R ∈ − , the estimation range
is:
c
1 000 Hz 1 000 Hzf− < Δ < (6)
We can see that the estimation range of phase
differential algorithm is restricted by the time interval and
estimation accuracy of the l
arg( )R .
3 The new frequency offset estimation algorithm
In LTE uplink, PRACH is responsible for random access.
And the random access channel (RACH) preamble is
constructed by ZC sequences and their cyclic shifted
versions. In addition, the uth root ZC sequence is defined
by:
( ) ZC
π ( 1)
j
ZCe ; 0 1
un n
N
ux n n N
+−= −� � (7)
where NZC
denotes the length of ZC sequence, which is
equal to 839 [1]. For the uth root ZC sequence, the
preambles with zero correlation zones of length NCS
are
defined by:
[ ], ZC( ) ( )mod
u v u v
x n x n C N= + (8)
Furthermore�the cyclic shift [1] is given by:
ZC
CS CS
CS
CS
RA
start shift CSRA
shift
RA RA RA
shift group shift
; 0,1,..., 1, 0,
for unrestricted sets
0; 0, for unrestricted sets
( mod ) ;
0,1,..., 1,
v
N
vN v N
N
N
C
v
d v n N
n
v n n n
⎢ ⎥
= − ≠⎢ ⎥
⎣ ⎦
==
⎢ ⎥
+⎢ ⎥
⎣ ⎦
= + −
for restricted sets
⎧
⎪
⎪
⎪
⎪
⎪⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪⎩
(9)
where the unrestricted sets are chosen in normal speed
mode, and the restricted sets are chosen in high speed
mode. In normal speed mode, the cyclic shift of preambles
is equal to NCS
, and one root ZC sequence can generate
Issue 6 WANG Jun, et al. / Frequency offset estimation based on peak power ratio in LTE system 51
ZC CS
N N preambles. In high speed mode, the cyclic shift
of preambles is not equal to NCS
, and one root ZC sequence
can generate RA RA RA
shift group shiftn n n+ preambles. In this paper, in
order to estimate the frequency offset in high speed
environment we focus on the high speed mode of PRACH.
Denoting that ( ) ( )u
r n x n= is the original sequence of
RACH preamble, and fΔ is the frequency offset in Hz.
In addition, ωΔ denotes the phase to frequency offset
fΔ . Hence, the received RACH preamble with frequency
offset ˆ( )r n is given by
( )j j
1 1ˆ( ) ( )e ( ) e ( )
n n
u
r n r n n x n n
ω ωψ ψ− Δ − Δ= + = + (10)
where s
2π f fωΔ = Δ , 1
( )nψ denotes the AWGN with
zero mean and variance 2
n
σ , and s
f denotes the sampling
rate of the RACH preamble, which can be expressed as
s sym1f T= (
symT denotes the sampling period of the
PRACH symbol). Therefore, Eq. (10) can be expressed as:
SEQ
sym
sym SEQ
j2π
j2π
1ˆ( ) ( )e ( )= ( )e
fT
T n
fT n T
u ur n x n n x nψ
Δ−
− Δ= + +
SEQ ZCj2π /
1 1 ( )= ( )e ( )
fT n N
un x n nψ ψ− Δ + (11)
where ZC SEQ sym
N T T= , and SEQT denotes the length of
preamble sequence. PDP is the cross correlation between
the receiver preamble sequence and the cyclic shift of
original ZC sequence, which can be calculated by
[ ]ZC
2
1
*
ZC
0
ˆ( ) ( ) ( )mod
N
u
n
P l r n x n l N
−
=
= +∑ (12)
where ZC
0,1,2,..., 1l N= − and ( )P l denotes the PDP.
With Eq. (7) and Eq. (11), when frequency offset fΔ is
SEQ1 T , we can get:
ZC ZC
ZC ZC
ZC ZC
π ( 1)
j
j2π /
1
( 1/ )( 1/ 1) 2 1/ 1
jπ
j2π /
1
( 1/ )( 1/ 1) 2 1/ 1 2
jπ jπ
1
ˆ( ) e e ( )
e e ( )=
e e ( )
un n
N n N
u n u n u n u
N n N
u n u n u n u n
N N
r n n
n
n
ψ
ψ
ψ
+−−
+ + + − − −−−
+ + + + + −−
= + =
+
+ =
ZC
jπ(1/ 1) /
1 ( 1/ )e ( )
u N
u
x n u nψ++ +
(13)
It can be seen that the cyclic shift period of preamble
sequence is ( )ZC
1 modu
d u N= . In order to let u
d be an
integer, we can get ( )ZCZC
mod1u
d NmN u= +⎡ ⎤⎣ ⎦
, where
m is the smallest positive integer. From Eq. (13), when
frequency offset is SEQ
1 T , the cyclic shift period of
preamble sequence is u
d , and the cross correlation peak
power between received sequence ˆ( )r n and original
sequence ( )r n will occur at u
d .
With Eq. (12) and Eq. (13), the peak power of the PDP
at 0l = is given by:
ZC
ZC
sym
21
*
0
0
21
*j2π
1
0
ˆ( ) ( ) ( )
( )( )e ( )
N
l u
n
N
fT n
uu
n
P f r n x n
x nx n nψ
−
==
−− Δ
=
Δ = =
⎡ ⎤ =+⎣ ⎦
∑
∑
ZC
2
1
SEQ *
1
0 ZC
2π
exp j ( ) ( )
N
u
n
n fT
x n n
N
ψ−
=
⎛ ⎞Δ⎛ ⎞
− +⎜ ⎟⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
∑ (14)
If we ignore the impact of noise 1
( )nψ , Eq. (14) can be
expressed as:
( )
( )
ZC
2
1
SEQ
0
0 ZC
2
SEQ
SEQ
ZC
2
ZC
SEQ SEQ
ZC
SEQ
ZC
2π
( ) exp j
1 exp j2π
2π
1 exp j
1
sin π exp jπ
π
sin
N
N
l
n
n fT
P f
N
fT
fT
N
N
fT fT
N
fT
−
==
Δ⎛ ⎞
Δ = − =⎜ ⎟
⎝ ⎠
− − Δ=
Δ⎛ ⎞
− −⎜ ⎟
⎝ ⎠
⎛ ⎞−Δ − Δ⎜ ⎟
⎝ ⎠ =Δ
∑
( )2
SEQ
SEQ
ZC
sin π
π
sin
fT
fT
N
ΔΔ
(15)
Because ( )ZCZC
mod1u
d NmN u= +⎡ ⎤⎣ ⎦
, the peak power
of the PDP at u
l d= ± is given by:
( )
( )
( )
zc
ZC
21
*
ZC
0
1
SEQ
0 ZC
2
2
ZC
SEQ
SEQ
ZC
ˆ( ) ( )mod
2π
exp j
2
exp jπ
sin π( 1)
π
( 1)sin
u
N
l d u u
n
N
n
u u u
P f x r nn d N
n fT
N
d nd d
u
N
fT
fT
N
−
=±=
−
=
Δ = =±⎡ ⎤⎣ ⎦
Δ⎛ ⎞
− ⋅⎜ ⎟
⎝ ⎠
⎛ ⎞± ±⎜ ⎟− =⎜ ⎟
⎝ ⎠
Δ ±
⎛ Δ ±
∑
∑
SEQ
2
ZC
ZC ZC
exp jπ( 1
1 ( 1)
exp jπ =u
fT
N d
N N
⎛
Δ ±⎜
⎞⎝
⎜ ⎟
⎝ ⎠
⎞ ⎛ ⎞− ± +⎟ ⎜ ⎟
⎠ ⎝ ⎠
52 The Journal of China Universities of Posts and Telecommunications 2013
( )2
SEQ
SEQ
ZC
sin π( 1)
π
( 1)sin
fT
fT
N
Δ ±
⎛ ⎞Δ ±⎜ ⎟
⎝ ⎠
(16)
Fig. 1 shows the curves of 0
( )l
P f= Δ , ( )u
l d
P f=− Δ and
( )u
l d
P f= Δ . Each of these curves is an approximately sinc
distribution. 0
( )l
P f= Δ decreases with the increasing of
the absolute value of the frequency offset when
SEQ0 abs( ) 1f T� � . In addition, it decreases to zero when
SEQ1f TΔ = ± . ( )
u
l d
P f=− Δ and ( )u
l d
P f= Δ are the left and
right curve of 0
( )l
P f= Δ , which increase with the
increasing of the absolute value of the frequency offset
when SEQ
0 abs( ) 1f T� � and the sign of frequency
offset is negative or positive, and reach to maximum value
when SEQ
1f TΔ = − and SEQ
1f TΔ = respectively.
Fig. 1 The curves of 0
( )l
P f= Δ and ( )u
l d
P f=± Δ
As analyzed above, we can utilize the ratio of 0
( )l
P f= Δ
and ( )u
l d
P f=± Δ to estimate the frequency offset, and the
range of frequency offset estimation is SEQ
1 T f− < Δ <
SEQ1 T , while
SEQT is equal to 0.8 ms in PRACH.
Therefore the theoretical estimation range of the proposed
algorithm is 1 250 1 250f− < Δ < , which is larger than
that of the phase differential algorithm [3]. The ratio of
0
( )l
P f= Δ and ( )u
l d
P f=± Δ can be given by:
( )
( )
2
SEQ
SEQ
ZC0
2
SEQ
SEQ
ZC
sin π
π
sin
( )
( )
sin π( 1)
π
( 1)sin
u
l
l d
fT
fT
NP f
R
P f
fT
fT
N
=
=±
ΔΔ
Δ= =
ΔΔ ±
⎡ ⎤Δ ±⎢ ⎥
⎣ ⎦
(17)
For convenience, the one-to-one map function between
peak power ratio and frequency offset can be listed after
calculating by Eq. (17). A sample of the one-to-one map is
shown in Table 1.
In our proposed algorithm, the PDP could be used to
perform the judgment of frequency offset, which is shown
in Fig. 2. As shown in Fig. 2, the three windows (include
negative window, main window and positive window) with
length Ncs
are set at l = − du, l = 0, and l = d
u respectively.
The simulation results show that the peak power of the
PDP moves from main window to positive window in case
of positive frequency offset and from main window to
negative window in case of negative frequency offset.
Therefore, the sign of frequency offset can be determined
by comparing the peak power of negative window with
that of positive window. In addition, when peak power of
negative window is larger than that of positive window, the
sign is negative. Otherwise, the sign is positive.
As analyzed above, the main steps of the proposed
frequency offset estimation algorithm are as follows�
Step 1 Calculate the PDP by using Eq. (12).
Step 2 Find the peak power of the PDP calculated by
Step 1 at positive window and negative window, denoted
as u
l d
P=′ and u
l d
P=−′ respectively, and compare the peak
power between u
l d
P=′ with u
l d
P=−′ to determine the sign
is negative or positive.
Step 3 Find the peak power of the PDP calculated by
Step 1 at main window, denoted as 0l
P=′ .
Step 4 Calculate the peak power ratio value R′ =
0
max( , )u u
l l d l d
P P P= = =−′ ′ ′ , and determine the frequency offset
value by looking up Table 1.
Table 1���
�The one-to-one map function between peak power
ratio and frequency offset
Frequency
offset fΔ /Hz
Peak power
ratio R
Frequency
offset fΔ /Hz
Peak power
ratio R
1 1 560 000 900 0.151 2
2 389 370 901 0.150 0
3 172 780 902 0.148 8
� � � �
400 4.515 6 1 247 5.78×10-6
401 4.482 6 1 248 2.56×10-6
402 4.449 8 1 249 6.41×10-7
� � 1 250 0
700 0.617 3
701 0.613 4
702 0.609 4
� �
Issue 6 WANG Jun, et al. / Frequency offset estimation based on peak power ratio in LTE system 53
Fig. 2 PDP
4 Simulation results
In this section, we compare the performance of our
proposed algorithm with phase differential algorithm [3].
The simulation model is built based on 3GPP protocol
standards [1] and we use the four channels AWGN,
extended pedestrian A (EPA), extended vehicular A (EVA)
and Extended Typical Urban (ETU) [8] for simulation.
Table 2 and Table 3 show the system parameters used in
the simulation for our proposed algorithm and phase
differential algorithm, respectively.
Table 2 Simulation parameters for the proposed algorithm
Parameter Value
System bandwidth 20 MHz
Carrier frequency 2.1 GHz
Channel model AWGN,EPA,EVA,ETU
Link PRACH
Preamble format 3
PRACH Configuration Index 51
Logical root sequence number 384
zero correlation zones of length 0
Table 3 Simulation parameters for phase differential algorithm
Parameter Value
System bandwidth 20 MHz
Carrier frequency 2.1 GHz
Modulation QPSK
Channel model AWGN,EPA,EVA,ETU
Link PUSCH
PUSCH RB number 50
These figures from Fig. 3 to Fig. 6 compare the MSE of
the two algorithms at different SNR when frequency offset
is 700 Hz in AWGN, EPA, EVA and ETU channels,
separately. The MSE of the two algorithms are very close
when SNR>0 dB. Additionally, the MSE of two algorithms
increase with the decreasing of the SNR.
When SNR<0 dB, we can see that in most cases our
proposed algorithm has lower MSE than that of phase
differential algorithm. Moreover, our proposed algorithm
provides gains about 3 dB in comparison with phase
differential algorithm when SNR<0 dB and 700 HzfΔ = .
Fig. 3 Performance of two algorithms in AWGN channel
(Frequency offset=700 Hz)
Fig. 4 Performance of two algorithms in EPA channel
(Frequency offset=700 Hz)
Fig. 5 Performance of two algorithms in EVA channel
(Frequency offset=700 Hz)
The estimation accuracies of our proposed algorithm are
discussed in different frequency offsets in AWGN channel.
Fig. 7 shows the MSE of our proposed algorithm in different
frequency offsets with SNR=10 dB. Our proposed algorithm
widens estimation range from 1 000 Hz to 1 250 Hz. In
addition, when the frequency offset is beyond 400 Hz, our
proposed algorithm can achieve the best performance.
Fig. 8 shows the estimated frequency offset in different
frequency offsets for different SNR. It depicts that our
54 The Journal of China Universities of Posts and Telecommunications 2013
proposed algorithm can get better performance when the
SNR � − 12 dB and the frequency offset is beyond
400 Hz.
Fig. 6 Performance of two algorithms in ETU channel
(Frequency offset=700 Hz)
Fig. 7 MSE in different frequency offsets (SNR=10 dB)
Fig. 8 Estimated frequency offset in different frequency
offsets for different SNR
5 Conclusions
In this paper, we proposed a new frequency offset
estimation algorithm based on peak power ratio in LTE
uplink. Moreover, our proposed algorithm uses the ratio of
the peak power of the PDP at the main window to that at
the negative window or positive window to estimate
frequency offset, according to the characteristics of cyclic
shift of ZC sequence caused by frequency offset.
Through numerical simulations, we can see that our
proposed algorithm can extend the estimation range of
frequency offset. Meanwhile the accuracy of frequency
offset estimation is almost not lost. Particularly in low
SNR, it has lower MSE compared with traditional phase
differential algorithm. As in the case of high-speed
movement (such as high-speed railway environment), the
frequency shift generates by the Doppler frequency shift
and oscillator of the base station and the terminal may be
greater than 1 000 Hz, in this case the conventional
method can’t estimate the frequency offset accurately.
Moreover, the proposed method can improve the estimates
range from 1 000 Hz to 1 250 Hz. Therefore, our proposed
method can be combined with the traditional methods to
improve the estimate range and accuracy in the practical
application.
Acknowledgements
This work was supported by the National Natural Science
Foundation of China (60572117), the Scientific Research Foundation
for the returned Overseas Chinese scholars, State Education Ministry.
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