PIERS Proceedings, Cambridge, USA, July 2–6, 2008
166
An Efficient BER Analysis of OFDM Systems with ICI Conjugate
Cancellation Method
Vivek K. Dwivedi and G. Singh
Department of Electronics and Communication Engineering
Jaypee University of Information Technology, Solan 173 215, India
Abstract— In this paper, we have presented bit error rate (BER) analysis of the orthogonal
frequency division multiplexing (OFDM) communication systems with the concept of conjugate
cancellation scheme. In the first part of our work, an effort has been made to illustrate the mathematical derivation for BER of the OFDM system with frequency offset. Later, we have derived
a formula for the BER analysis of OFDM system with the conjugate cancellation scheme. Bit
error rate of the proposed conjugate cancellation scheme is better than the ICI self-cancellation.
We also discusses carrier to interference ratio (CIR) and compared this proposed results with the
others reported results.
1. INTRODUCTION
The next generation broadband multimedia communication systems will integrate various function
and application in a system. This system support large data rates with sufficient robustness to
radio channel impairments, requires careful choosing of modulation technique. The suitable choice
seems to OFDM which is special case of multi-carrier communication system, where single data
stream is transmitted over number of lower sub-carrier. OFDM communication systems can be seen
as either a modulation or multiplexing technique. One main region to use OFDM is to increase
the robustness against frequency selective fading or narrow band interference. This communication
system is broadly considered as an effective approach for the future high speed wireless multimedia
communication systems. The basic principle of OFDM is to split the high-data stream into number
of lower rate data streams which are transmitted simultaneously over number of subcarriers. High
spectral efficiency and multipath immunity are two major features of the OFDM technique.
However, OFDM system is very sensitive to carrier frequency offset between transmitter and a
receiver, which destroys the orthogonality between sub-carriers and creates inter carrier interference
(ICI). The reduction of the signal amplitude and introduction of the ICI are two destructive effects
caused by carrier frequency offset in OFDM systems. So there is a need to reduce ICI. Recently
several methods have been proposed and developed to estimate and adjustment of the effects of ICI.
Among the all ICI cancellation schemes, the ICI-self cancellation scheme is a simple way for ICI
reduction. The main idea is to modulate one data symbol onto the next sub-carrier with predefined
inversed weighting coefficient “−1”. By doing so the ICI signals generated within a group can be self
cancelled each other. Several methods [1–6] have been developed for bit error rate (BER) analysis
and ICI self cancellation of OFDM system. Kang et al. [1] and Zhao and Haggman [2] discuss the
self cancellation scheme for OFDM to reduce the effects of frequency offset error. In [3] BER upper
bound of OFDM system is analyzed without ICI self cancellation and [4] BER of OFDM system
is analyzed using self cancellation but this method is less accurate. Yeh et al. [7] discussed ICI
mitigation using conjugate cancellation but they have not provided any mathematical analysis of
ICI. In [8, 9] the inter-carrier interference (ICI) by Doppler effects in time domain in orthogonal
frequency division multiplexing systems is observed.
In this paper, we have presented BER analysis of the OFDM system with concept of conjugate cancellation scheme. In the first part of our work an effort has been made to illustrate the
mathematical derivation for BER of OFDM system with frequency offset. Later, we have derived a
formula for BER analysis of OFDM system with the conjugate cancellation scheme. The organization of the paper as follows. The Section 2 discusses the system model of the OFDM communication
systems. The Section 3 is concern with the conjugate cancellation scheme for ICI. The Section 4
discusses the results and finally, Section 5 concludes the work.
2. SYSTEM MODEL
Figure 1 shows a typical discrete-time base-band equivalent model of an OFDM system. Input
binary serial data stream is encoded using suitable modulation technique (M — QAM, BPSK, and
QPSK). The further symbols are transferred in the serial-to-parallel converter (S/P) in this stage
Progress In Electromagnetics Research Symposium, Cambridge, USA, July 2–6, 2008
167
duration of bits is increased. First part of parallel bit stream is subjected to IFFT block and second
part is also subjected to FFT block. The modulated symbols are serialized using a parallel-to-serial
converter (P/S). Now guard band addition is done because at the receiver one OFDM symbol is over
lapped with the other symbol due to multipath distortion [1, 2, 7]. To eliminate the problem of inter
symbol interference a guard time inserted between two symbols, duration of guard interval should
be greater than maximum delay spread. Guard time consists no signal at all. Before transmission
first part of data is sent as it is and conjugate is taken for second data. In next block digital signal
is converted to analog via the digital-to-analog converter (D/A) before being sent down to the
channel. At the receiver side, guard interval is removed and the received symbol is converted from
analog to digital using the analog-to-digital converter (A/D). In next process first part of data is
transferred as it is in serial to parallel converter and conjugate is taken for second part of data
before transferring to the serial to parallel converter and then first and second part of data are sent
in FFT block. After FFT and IFFT block data is sent for parallel to serial (P/S) conversion and
then for demodulation. ICI cancellation is done after demodulation using diversity combiner.
Figure 1: System model for communication systems.
3. CONJUGATE ICI CANCELLATION SCHEME
Input data bits are encoded by using suitable modulation technique like (QPSK or QAM) and
output of his block is Xk . IFFT out put at the transmitter is:
xn =
K
1 X
Xk e2πjnk/N
N
k=−K
where n = 0, 1, 2, . . . , N − 1, and N ≥ 2K + 1 where K is number of sub carries N is the period
of IFFT. At received sequence after passing through the channel can be expressed as:
" K
#
X
1
yn =
Xk Hk e2πjn(k+ε)/N + wn
N
k=−K
where n = 0, 1, 2, . . . N − 1, where Hk is channel transfer function at the frequency of kth sub
carrier, ε is relative frequency offset of channel, wn is Additive White Gaussian Noise (AWGN).
Out put of DFT demodulator can be expressed as:
( " K
#
)
N
−1
X
X
1
2πjn(k+ε)/N
Yk =
Hk X k e
+ wn e−2πjkn/N
N
n=0
n=−K
(1)
jπε(N −1)
/N + I
+
W
Yk = (Xk Hk ) {(sin πε) /N sin (πε/N )} e
k
| {zk } |{z}
II
III
PIERS Proceedings, Cambridge, USA, July 2–6, 2008
168
The first component is the modulation value Xk is modified by channel transfer function. This
component experiences an amplitude reduction and phase shift due to the frequency offset. Second
term is ICI term, which arises due to frequency mismatch of oscillator transmitter and receiver.
After some manipulation second term ICI can be expressed as:
Ik =
N
−1
X
l=0
l6=k
Ik ≈
N
−1
X
l+ε−k
1
Xl Hl (sin π(l + ε − k)/ sin π ((l + ε − k)/N )) × ejπ(N −1)( N )
N
(2)
Xl Hl (sin π(l + ε − k)/π(l + ε − k))
l=0
Third is Additive White Gaussian Noise in frequency domain which can be expressed as
Wk =
N
−1
X
wn e−2πkn/N
n=0
Now we will analyze the second part of data before transmission we will take conjugate of the
original signal:
!∗
Ã
K
K
X
1 X ∗ −2πjnk/N
1
=
Xk e2πjnk/N
Xk e
x0n =
N
N
k=−K
k=−K
n = 0, 1, . . . N − 1. At the receiver a conjugate algorithm is requires a conjugate operation on
received signal first and FFT is performed:
#
" K
X
1
X ∗k Hk e2πjn(−k+ε)/N + wn
yn0 =
N
k=−K
where n = 0, 1, 2, . . . N − 1. Out put of DFT demodulator can be expressed as:
( " K
#
)∗
N
−1
X
X
1
2πjn(−k+ε)/N
0
Hk X ∗k e
+ Wn e−2πjkn/N
Yk =
N
n=0
k=−K
(
)
−jπε(N −1)
(sin
πε)
/N + I 0 + W 0
¡ πε ¢ e
Yk0 = (Xk Hk )
k
k
|{z}
|{z}
N sin N
II
(3)
III
ICI term can be expressed as:
N
−1
X
Wk0 =
l=0
N
−1
X
1
Xl Hl
N
Ã
sin π(l − ε − k)
¡ l−ε−k ¢
=
sin
π
N
l=0, l6=k
¶
µ
N
−1
X
sin π(l − ε − k)
Ik0 ≈
Xl Hl
π(l − ε − k)
Ik0
!
ejπ(N −1)(
l−ε−k
N
)
(4)
wn∗ e−2πkn/N
n=0
Out put of receiver after using conjugate cancellation scheme is:
Yk00 = (Yk + Yk0 )/2
(5)
By putting the values in Equation (5) from Equations (1) and (3) we will get three terms Ist term
is desired signal at out of the receiver is:
C(k) = [(Xk Hk ) (sin πε) /πε]
Progress In Electromagnetics Research Symposium, Cambridge, USA, July 2–6, 2008
169
Second term is ICI component at output of the receiver after conjugate cancellation scheme will
be:
Ik00 = (Ik + Ik0 )/2
Third term is Additive White Gaussian Noise at the OFDM receiver out put is:
Wk00 = (Wk + Wk0 )/2
00
In order to evaluate the statistical properties
h
i of ICI after conjugate cancellation
h
i assume E(Ik ) = 0
and assuming average channel gain E |Hl |2 = |H|2 is constant and E |Xl |2 = |X|2
h¯ ¯ i
2
2
σIC
= E ¯Ik00 ¯ = |X|2 |H|2 (sin πεε)2 × .2195
Bit error rate of QPSK modulated OFDM system is given in [10]:
´
³p
BER = 1/2 ∗ Q
Es /N0
Bit error rate of QPSK OFDM system after conjugate inter carrier interference cancellation is given
by:
s
½
¾
(sin πε) 2
2
2
BER ≤1/2 ∗ Q |X| |H|
/(N0 + |X|2 |H|2 (sin πε ε)2 × .2195)
πε
v
Ã
!
u
2
2
u |X|2 |H|2 ½ (sin πε) ¾2
|X|
|H|
2
/ 1+
(sin πε ε) × .2195
=1/2 ∗ Qt
N0
πε
N0
s
½
¾ µ
¶
Eb (sin πε) 2
Eb
2
BER =1/2 ∗ Q
/ 1+
(sin πε × ε) × .2195
N0
πε
N0
4. RESULTS AND DISCUSSION
For simulation modulation is QPSK N = 64, guard interval = 7, Fig. 2 shows compression of
BER between self-cancellation method and conjugate cancellation method for frequency offset 0.1
and 0.2. In this frequency offset 0.2 by using self cancellation method BER greater than 10−1 at
SNR = 0 dB and BER less than 10−4 at SNR = 10 dB and for proposed method BER is 10−1 for
0 dB SNR and BER is just less than 10−5 at 10 dB SNR. But for normalized frequency offset 0.1
10
10
0
Frequency offset 0.2
-1
-2
BER
10
-3
10
Without compensation
Without compensation
With ICI self cancellation
With ICI self Cancellation
With proposed scheme
With proposed scheme
-4
10
Frequency offset 0.1
-5
10
-6
10
0
1
2
3
4
5
6
7
8
9
10
Eb /No (in dB)
Figure 2: Comparison between proposed conjugate ICI cancellation schemes with self-cancellation scheme.
PIERS Proceedings, Cambridge, USA, July 2–6, 2008
170
for self cancellation method BER at 0 dB SNR is greater than 10−1 and BER is less than 10−5 at
10 dB SNR but for conjugate cancellation method BER 10−1 at 0 dB SNR and greater than 10−5
at 10 dB SNR. So proposed conjugate method is better than the ICI self-cancellation method as
discussed in [2].
Figure 3 shows compression of carrier to interference ratio among different methods like [2],
standard OFDM system and proposed scheme for normalized frequency offset 0.25 and our result
is comparable with [11]. Carrier to interference ratio (CIR) of proposed scheme is close to that of
the conventional OFDM systems. It means that the effect of ICI distortion in the proposed scheme
is close to the one in the conventional OFDM systems.
Figure 3: Carrier to interference ratio (CIR) versus normalized frequency offset.
5. CONCLUSION
In this paper, we have suggested a simple ICI cancellation scheme to reduce the frequency offset
sensitivity of the OFDM system which is based upon conjugate cancellation scheme. In this scheme
two sequences are transmitted in each data symbol. First sequence is original received sequence
and another sequence is conjugate of the original sequence. Thus the transmitted two sequences
are conjugate of each other rather than adjacent sub-carriers with opposite polarities in order to
cancel ICI. It gives better bit error rate than [2], [4] and [11].
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