J. Electromagnetic Analysis & Applications, 2009, 1: 118-123
doi:10.4236/jemaa.2009.12019 Published Online June 2009 (www.SciRP.org/journal/jemaa)
Copyright © 2009 SciRes JEMAA
1
ICI Performance Analysis for All Phase OFDM Systems
Renhua Ge1, Shanlin Sun1,2
1Guilin College of Aerospace Technology, Guilin, 541004, China; 2Tianjin University, Tianjin, 300072, China.
Email: grh1962@163.com
Received January 26th, 2009; revised March 2nd, 2009; accepted March 10th, 2009.
ABSTRACT
Orthogonal frequency division multiplexing (OFDM) is a strong candidate for the next generation wireless communica-
tion. But the frequency offset between the local oscillators at the transmitter and receiver causes a single frequency
offset in the signal, while a time-varying channel can cause a spread of frequency offsets known as the Doppler spread.
Frequency offsets ruin the orthogonal of OFDM sub-carriers and cause inter-carrier interference (ICI), therefore,
quickly diminishing the performance of the system. A novel all phase OFDM (AP-OFDM) system is established. APFFT
is introduced for the first time to overcome ICI aroused by carrier frequency offset (CFO) in OFDM systems. This
scheme makes use of APFFT in time domain and zero inserting in frequency domain to reduce the amount of ICI gener-
ated as a result of frequency offset, with little additional computational complexity. At the same time, the proposed sys-
tem has zero phase error. It is proved to be correct and effective in mathematics. The simulation results indicate that
AP-OFDM system has a better performance than conventional OFDM system.
Keywords: All Phase Orthogonal Frequency-division Multiplexing (APOFDM), Carrier Frequency Offset (CFO), In-
ter-carrier Interference (ICI)
1. Introduction
Orthogonal frequency-division multiplexing (OFDM)
communication systems require precise frequency syn-
chronization [1-4], since otherwise inter-carrier interfer-
ence (ICI) will occur. Currently, three different ap-
proaches for reducing ICI have been developed in the
literature. One is to estimate and remove the frequency
offset [3-7]. While many methods exist that can estimate
and remove the frequency offset quite accurately, they
often have considerable computational complexity. An-
other approach is to use signal processing and/or coding
to reduce the sensitivity of the OFDM system to the fre-
quency offset [8]. These methods can either be used as
low complexity alternatives to fine frequency-offset es-
timation techniques or they can be used together with a
somewhat accurate oscillator. Windowing has been used
in [9] and [10] to reduce the ICI created as a result of
frequency offset. A simple and effective method known
as ICI self-cancellation scheme [11,12] has been pro-
posed by Zhao and Haggman. Other frequency-domain
coding methods have been proposed in [13,14] that do
not reduce the data rate. However, these methods produce
less reduction in ICI. A new self-cancellation scheme has
been proposed by Alireza, Seyedi and Saulnier [15]. This
method has better performance with more computational
complexity.
This paper concentrates on the further development of
the ICI cancellation method and APFFT is proposed for
ICI cancellation. The advantages of APFFT are zero
phase error and less ICI. It improves the OFDM system's
ability to resist frequency offset with the cost of limited
additional computational complexity.
2. The Model of APOFDM System
In conventional FFT system, the spectrum of the receiver
signal can expressed as [16].


00 /)()1(
/sin
sin1



jNkNj ee
Nk
k
N
kX
 (1)
where
is the normalization frequency offset. Thus,
the amplitude spectrum is:

Nk
k
N
kX /sin
sin1
)(

(2)
And the phase spectrum is:
01 )(
1


 k
N
N (3)
Frequency offset can introduce ICI and phase noise
and diminish the performance of the system Figure 1
ICI Performance Analysis for All Phase OFDM Systems 119
Copyright © 2009 SciRes JEMAA
shows the block of APOFDM. Figure 2 is the N-all phase
FFT circuit [16,17].
Assuming that X = {X(k), k = 0, 1, …, N-1} is the
complex digital sequence in frequency domain and
  
N
kn
j
N
Nk
ekX
N
kXIFFTnx
2
12/
12/
1

 . After zero
interval interpolation, we obtain sequence

,0,1,2,...,2 1XXkk N



. and
,0,2,,2(1
2
0,1,3,5,, 21
k
Xk N
Xk
kN
 



)
.
The number of the point for the new sequence is 2N,
so 2N-IFFT should be implemented.
12
0
2
)(
2
1
)(
N
k
kn
N
WkX
N
nx
1
0
2
2
)2(
2
1N
m
mn
N
WmX
N, n=0,1,,2N-1 (4)
 


12
0
)(2
2
2
1N
k
Nnk
N
WkX
N
Nnx

N
N
N
m
mn
NWWmX
N
2
2
1
0
2
2
2
2
1

 
nxWmX
N
N
m
mn
N

1
0
2
2
2
2
1, n=0,1,,2N-1
(5)
The purpose of zero interval interpolation in frequency
is to obtain the copy of N subcarriers of OFDM systems
and keep the number of subcarriers unchangeable.
Therefore, the former N points are the copy of the last N
points in time domain. These 2N points are indispensable
to APFFT. That is to say, in receiver, we only need two
times clock sample compared with FFT system to get
APFFT sequence in time domain.
To digital sequence with 2N points, the first point x(0)
is removed and the remaining sequence can be expressed
as ).12(,)2(),1(),(),1(,)2(),1(

NxNxNxNxNxxx 
In this digital sequence, we can obtain N sub-sequences
and each of them has N points including x(N). Then,
every sub-sequence with a recycling style is moved. Fi-
nally, the point x(N) is moved to the first place and then
Figure 1. The block of APOFDM system
Figure 2. The block of N-step APFFT
120 ICI Performance Analysis for All Phase OFDM Systems
get the others N sun-sequences with N points. Making
them aligned and adding them, a new N points sequence
is obtained, which is called all-phase digital sequence.
For instance, N=4 and 2N-1=7, we can get the se-
quence . This sequence
can be divided into 4 sub-sequences.
)7(),6(),5(),4(),3(),2(),1( xxxxxxx
1st section: , 2nd section:

)4(),3(),2(),1( xxxx

)5(),4(),3(),2(xxxx
),5(),4(xx
),2(),1(xx
, 3rd section: ,
4th section: . After cycle moving,
we obtain: , ,
)6(),5(),4(),3( xxxx
)4(),3(),2(),5( xxxx
)7(),6(xx

)4(),3( xx

)4(),3(),6(),5(xxxx and
)4(),7(),6(),5( xxxx. We
add them together with parallelism principle and normal-
ize them. The AP digital sequence can be expressed as

)4(4
4
1
,)7()3(3
4
1
,)6(2)2(2
4
1
,)5(3)1(
4
1xxxxxxx .
Now, we analyzes the connection between
kX
 
and
. According to the character of discrete Fourier trans-
form, is the FFT of sequence

kX

kX

Nxx,1 x,...,2
cycle moving in time domain. Thus
 
N
ki
j
ekXkX
2
, i = 0, 1, , N - 1 (6)
where (k =0,1,…N-1). The
spectrum of APFFT is the sum of , thus:

kX

1
0
/2
N
n
Nknj
enx

kX
  

1
0
/2
1
0
11 N
i
Nkij
N
i
AP eiX
N
iX
N
kX
(7)
Supposing that

)
2
(0
0
N
nk
j
enx, is a single fre-
quency complex signal, thus,
 
N
ki
j
N
m
N
km
j
N
i
j
AP eemxe
N
kX

2
1
0
2
1
0
2][
10
 

  1
0
2
1
0
/2
2
000
1N
m
mkkj
N
i
Nikkjj eee
N



Nkkj
kkj
Nkkj
kkj
j
e
e
e
e
e
N/2
2
/2
2
20
0
0
0
0
1
1
1
11


 
 
NkkjNkkj
kkjkkj
NkkjNkkj
kkjkkj
j
ee
ee
ee
ee
e
N////2 00
00
00
00
0
1










Nkk
kk
e
N
j
/sin
sin
1
0
2
0
2
2
0
(8)
Thus, the normalized spectrum of APFFT is
 

Nk
k
e
N
kX j
ap /sin
sin1
2
2
2
0


(9)
Having taken into account all the combinations of N
points sequence including x(N), the plus phase error and
the minus of x(N) offset each other, thus the phase error
is zero.
Figure 3 is the spectrum of the same signal with FFT
and APFFT respectively. ICI caused by CFO has more
difference between them. The ICI of FFT is higher than
that of APFFT.
Figure 4 is the phase and amplitude spectrum of
OFDM and APOFDM systems with frequency offset
0.05. From Equation (1), (9), Figure 3 and Figure 4, we
can indicate that, with APFFT, the phase error of the re-
ceiver signal is 0 and the normalized ICI power is the
square of that of FFT signal. Therefore, its sensitivity to
Figure 3. Signal spectrum of APFFT and FFT
Figure 4. Phase and amplitude spectrum of signal s=exp
(j*(w*t*20/N+20*pi/180)) with APFFT and FFT, frequency
offset is 0.05
Copyright © 2009 SciRes JEMAA
ICI Performance Analysis for All Phase OFDM Systems 121
Copyright © 2009 SciRes JEMAA
1
ICI caused by CFO is lower and APFFT is more suitable
for OFDM system.
3. Performance Analysis for AP-OFDM Sys-
tems with CFO
The change of performance duo to CFO is more inter-
ested in the research content. In this section we use a
method similar to the one used in [16] to calculate the
exact BER of the proposed system in the presence of a
frequency shift. We assume that the system operates over
an AWGN channel and that the data symbols are quater-
nary phase-shift keying (QPSK) modulated. Similar
analysis is possible if a 64-QAM modulation is used [16].
We assume an ideal additive white Gaussian noise
(AWGN) channel. The CFO does not change during one
OFDM symbol. The sampled signal for the kth
sub-channel after the receiver fast Fourier transform
processing can be written as:
1
0
0,
,0,1,,
N
kk lklk
llk
yXS SXnkN

 
(10)
where
1,13,15,17,
3,33,35,37,
5,53,55,57,
7,73,75,77
k
j
jjj
j
jj
Xj
j
jjj
j
jj
 


 


 
 
j
denotes
the transmitted symbol for the kth sub-carrier, nk is a
complex Gaussian noise sample (with its real and imagi-
nary components being independent and identically dis-
tributed with variance ), and N is the number of
sub-carriers. The second term in (12) is the ICI caused by
the CFO. The sequence (ICI coefficients) depends
on the CFO and is given by
2
k
S
 

0
2
/sin
sin1


j
APk e
Nk
k
N
AkXS
 (11)
If the ICI assumed to be a Gaussian- distributed ran-
dom variables (RV) with a zero mean, both the BER and
SER can readily be computed for carious modulation
formats. The approximate error rates then can be ex-
pressed in terms of the function and effective SNR
for the kth sub-carrier as
)(Q
2
2
0
1APICIS
S
eff
S

(12)
where 2
2/

ks XE which is the SNR for the kth
sub-carrier in the absence of a CFO. The variance of the
signal constellation 2
k
XE will be independent of if all
sub-carriers use the same modulation format, which is the
normal case. The variance of the ICI on the kth sub-car-
rier can be given by

1
,0
2
2
N
kll
klAPICI S
(13)
Compared with FFT-OFDM systems [16], the variance
of the ICI of APFFT-OFDM systems can be expressed by



 1
0
1
0
22
2
2ˆ
var
1
ˆ
N
n
N
n
ICI
AP
ICI
AP N
n
N
nN
N
nX
N

 1
0
22
3
2
2
ˆN
n
ICI nNnN
N
 


 2
1
2
6
121
ˆ3
3
2NN
N
NNN
N
N
ICI
22
2
2
ˆ
3
1
ˆ
3
12
ICIICI
N
N

(14)
Equation (14) shows that, after APFFT, the ICI noise
to every sub-carrier is the third of that in FFT system.
Using the method in [16], the effect caused by CFO can
be expressed with the losing of SNR.

S
D

2
3
1
10ln
10
(15)
4. APOFDM Performance Test with Simulation
4.1 The Design Based SIR
According to the method of ICI cancellation in [15], we
choose N=64, L=2, M=1 and SIRmin=25dB to design a
system and compare its performance with FFT system,
Zhao-Haggman (L=2) system and A-Seyedi system. Fig-
ure 5 shows that the performance of APFFT is better than
FFT and Zhao-Haggman (L=2) systems in low frequency
offset, and poorer in high frequency offset. We will only
be interested in frequency shifts in the range 5.0
.
4.2 The Design Based BER
As SIR can not represent the system performance, and
the result in Figure 6 obtained before frequency com-
pensation, thus the APFFT-OFDM system SIR can not
necessarily better than the other existing systems. The
model method in [12] is used and all phase compensa-
tion is adopted, BER performances are compared in Fig-
ure 6. The parameters are same to the above. Assuming
that channel is pilot and the pilot frequency account for
20% of the symbol. The CFO is 0.5. From Figure 6,
when BER=10-4, APFFT is better than Zhao-Haggman
system and A. Seyedi system 1.1 dB and 0.8 dB respec-
tively. This is because of the zero phase error of APFFT
with CFO. Equation (14), Figure 3 and Figure 4 can
prove it to be correct. That is to say, the simulation and
the theory are consistent.
122 ICI Performance Analysis for All Phase OFDM Systems
Figure 5. The correction between SIR and normalization
frequency offset in ideal AWGN
Figure 6. Calculated and simulated BER for the FFT system,
Zhao-Haggman system, A. Seyedi system and APFFT sys-
tem over an AWGN channel with perfect equalization
Figure 7. BER of FFT, Zhao-Haggman, A. Seyedi and
APFFT-OFDM system over a frequency-selective Channel
4.3 BER of Frequency-Selective System
APFFT can not estimate the CFO but can reduce the sen-
sitivity to CFO. Thus, it is true that APFFT is more suited
to the OFDM system. In a frequency-selective channel, 4
sub-paths are chosen to establish channel model and
every sub-path attenuation coefficient accord with
Rayleigh distribution. Channel model and Power spectral
density are the same with [18]. The cyclic prefix is TCP =
T/8. Figure 7 shows that, when channel is perfect, the
lowest BER varies considerably in all the schemes we
have mentioned above. The BER of APOFDM is best.
5. Conclusions
The amplitude of FFT has the character of function sinc
in frequency domain. Moreover, the amplitude spectrum
of APFFT is the square of function sinc. At the same time,
the phase of FFT can change with CFO but APFFT can
not. The plus phase error and minus phase error balance
out each other, thus, the phase error of APFFT is zero. So,
APFFT is more suitable for OFDM system. Compared
with conventional OFDM system, APFFT reduce the
sensitivity of system to CFO. In this paper, AP-OFDM
system is established, the character of ICI cancellation in
OFDM system is proved in mathematics, and the meth-
ods in [15] are used to analyze the performance of
APOFDM system. Through Monte Carlo simulations, we
have shown that the proposed system has better per-
formance in the presence of an oscillator frequency offset
or when ICI is created as a result of channel fading. Sig-
nificantly larger gains are achieved when the equalization
process is imperfect.
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