A Conjugate Operation for Mitigating Intercarrier Interference of OFDM Systems

Hen-Geul Yeh Yuan-Kwei Chang Electrical Engineering Department EECS Department California State University, Long Beach University of California, Irvine Long Beach, CA 90840-8303 Irvine, CA 92697 E-mail: heyeh@csulb.edu E-mail: ykchang1107@yahoo.com

Abstract In order to mitigate the intercarrier interference (ICI), a two-path algorithm is developed for orthogonal frequency division multiplexing (OFDM) systems. The first path is a regular OFDM system for transmission and receiving. The second path requires a conjugate operation of the 1st block of data to form the 2nd block of data for transmission at the transmitter. The received 2nd block data will be conjugated at the receiver and then demodulated. The demodulated results of the 1st path are combined with that of the 2nd path with equal weights to form the final detected data symbols. This conjugate cancellation (CC) scheme provides (1) a high signal to ICI power ratio in the presence of small frequency offsets (33 dB higher than that of linear self-cancellation [1-2] algorithms, at %1.0=fT of subcarrier frequency spacing); (2) backward compatibility with the existing OFDM system; (3) significantly better BER performance in both additive white Gaussian noise (AWGN) and frequency selective fading channels.

I. INTRODUCTION

Lately, researchers in universities and manufacturers in industry show great interest in orthogonal frequency division multiplexing (OFDM) systems. OFDM has already been accepted for the new wireless local area network standards as IEEE 802.11. OFDM is very robust to wireless channel impairments (e.g. multi-path propagation and frequency-selective fading). The receiver design is simplified due to less complex equalization. The OFDM system has high spectral efficiency due to overlapping subcarrier spectra. However, one of the major disadvantages of such a multi-carrier modulated system is the sensitivity of its performance to synchronization error, such as frequency or phase offsets. The frequency offset can result from a Doppler shift due to a mobile environment, as well as from a carrier frequency synchronization error. Such frequency offsets cause a loss of the carriers

orthogonality, and hence create intercarrier interference (ICI). Currently, three different approaches for mitigating ICI have been proposed including: ICI self-cancellation [1-2], frequency-domain equalization [3], and time-domain windowing scheme [4]. Frequency offset estimation techniques using training sequence such as pilot symbols are proposed in [5-6]. This study focuses on the ICI cancellation scheme and its effect on OFDM communication systems in AWGN and fading channels, assuming that the synchronization, including phase, frequency, and timing has done by using repeated preamble sequence, but the ICI may still exist due to the frequency offset estimation error or unexpected Doppler velocity.

This paper is organized as follows. The OFDM

system in additive white Gaussian noise (AWGN) channel is discussed in Section II. The math model of the transceiver is described in Section III, along with a discussion of the weighting function of the data symbol due to frequency offset. Section IV presents a CC scheme and the corresponding architecture of the transceiver. Simulation results are discussed in Section V. Conclusions are given in Section VI.

II. OFDM SYSTEMS A. Regular OFDM Transmitter

A conventional OFDM modulation is employed at the transmitter. The baseband transmitted signal kx at the output of the IFFT can be written as

1,...,2,1,021

0

===

Nkedxnk

NjN

nnk

(1)

where nd is the data symbol, and

1...,,1,0,2

= Nkenk

Nj

, represent the corresponding orthogonal frequencies of N subcarriers. Clearly, a group of N different data symbols is mapped onto N subcarriers via the IFFT processor. Note that the IFFT

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will have TOFDM seconds to complete its operation. The duration TOFDM for an OFDM symbol is NTs, where Ts is the data symbol time duration. For simplicity, T is used to replace TOFDM in the following discussion.

B. Regular Receiver Baseband Processing At the receiver, the OFDM signal is mixed with a local oscillator signal. Assuming it is f above the carrier frequency of the received OFDM signal due to frequency estimation error or Doppler velocity, the baseband FFT demodulator output is then given by

1,...,2,1,0121

0==

=

NnerNdnk

NjN

kkn

(2)

where kfTk

Nj

kk wexr +=2

represents the received signal at the input to the FFT processor, kw is the

AWGN, and nd is the output of the FFT processor.

The 1...,,1,0,2

=

Nke

fTkN

j

, represents the corresponding frequency offset of the received signal at the sampling instants, and fT is the frequency offset to subcarrier frequency spacing ratio.

III. ANALYSIS The ICI resulting from the carrier frequency offset is analyzed by using both mathematically in discrete domain and graphically with continuous curves in this section. Following the similar approach in [2], expressions are derived for each demodulated subcarrier at the receiver in terms of each transmitted subcarrier and N complex weighting functions. Without loss of generality, the noise kw in the received signal is ignored in the following discussion. Substituting (1) into (2) and after some manipulation, it can be shown that

1,...,2,1,011

0

)(21

0==

=

+

=

NnedN

dN

k

fTnmN

kjN

mmn

(3)

Taking the advantage of the properties of geometric series, it can be derived as

1,...,2,1,0

1

11

1

0

)(2

)(21

0

==

=

=

+

+

=

Nnud

e

edN

d

nm

N

mm

fTnmN

j

fTnmjN

mmn

(4)

where

)(2

)(2

1

11fTnm

Nj

fTnmj

nm

e

eN

u+

+

=

(5)

Clearly, the complex weighting functions 110 ...,,, Nuuu indicate the contribution of each of the

N data symbols md to the FFT output nd . Those terms that mn, they represent the cross talk from the undesired data symbols. The weighting function of the

( ) ( )kn rFFTdIFFT pair, the transmitter-receiver operation, is a periodic function with a period equal to N. If the normalized frequency offset fT equals zero, then nd is exactly equal to nd . With some manipulation, equation (5) can be rewritten as

))(sin(

))(sin(1))(1(

fTnmN

fTnmN

eufTnmN

Nj

nm

+

+=

+

(6)

Equation (6) consists of a rotation factor ))(1( fTnmN

Nj

e+

and the Dirichlet function

))(sin(

))(sin(1

fTnmN

fTnmN +

+

. To illustrate a complete cycle

at 8=N , 0=n , and 0=fT , the range of m is set from zero to eight in Fig. 1. Note that there are small approximately anti-symmetrical regions around the zero-crossing points at the integer indexes m = 1, 2, 6 and 7 for the real weighting function. Similarly, there are small approximately anti-symmetrical regions around the zero-crossing points at the integer indexes m = 2, 3, 4, 5 and 6 for the imaginary weighting function. These small approximately anti-symmetrical regions around the zero-crossing points at the majority integer indexes are very useful for ICI cancellation as explained in the following sections.

0-7803-8521-7/04/$20.00 (C) 2004 IEEE39660-7803-8521-7/04/$20.00 2004 IEEE

0 1 2 3 4 5 6 7 8-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Normalized index m

We

ight

ing

fun

ctio

n of

re

gula

r O

FDM

Real

Imag

0 1 2 3 4 5 6 7 8-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Normalized index m

We

ight

ing

fun

ctio

n of

re

gula

r O

FDM

Real

Imag

Fig. 1. The weighting function.

III. CONJUGATE ALGORITHM

To mitigate the impact of ICI, a new conjugate algorithm is developed. The basic idea is to add a simple circuitry that provides weighting factors with opposite polarities. To do this, this new algorithm should produce the same weighting curves as that of Fig. 1. But these curves shift to the right (instead of left) by fT when the frequency offset fT is greater than zero. This shift-to-right operation changes the weighting functions from zero to negative or positive values in the reversed direction of that of the shift-to-left operation. The combined weighting functions (regular and conjugate algorithms) will provide a significantly smaller weighting function on undesired data symbols while maintaining the same weighting function on the desired data symbol for small frequency offsets as illustrated in the next three subsections. A. The Conjugate Algorithm At the transmitter, this algorithm requires a conjugate operation on the IFFT output as defined in (7):

1,..,0)(21

0

*

*21

0

'==

=

=

=

Nkededx nkNjNn

n

nkN

jN

nnk

(7)

where nd is the data symbol, and

1...,,1,0,2

=

Nkenk

Nj

, represent the corresponding orthogonal frequencies of N subcarriers. Note that in order to demodulate the original signal xk and the conjugate signal xk separately, xk needs to be transmitted independently. This can be achieved by using frequency division multiplexing (FDM), or time division

multiplexing (TDM), or code division multiplexing (CDM), or dual antennas. At the receiver, the conjugate algorithm requires a conjugate operation on the received signal first, and then performs the FFT operation as defined in (8):

( ) 1,...,2,1,01 210

*''==

=

NnerNdnk

NjN

kkn

(8)

where '2

''k

fTkN

j

kk wexr +=

represents the received

signal, 'kw is the independent AWGN, and 'nd is the

output of the FFT processor. The term

1...,,1,0,2

=

Nke

fTkN

j

, represents the corresponding frequency offset of the received signal at the sampling instants. Without loss of generality, the noise 'kw is assumed to be zero in the following discussion. Substituting (7) into (8) and after some manipulation, it can be shown that

1,...,2,1,011

0

)(21

0

'==

=

=

NnedN

dN

k

fTnmN

kjN

mmn

(9).

By following a similar derivation as that of Section III, one obtains the weighting functions for the FFT output:

))(sin(

))(sin(1))(1(

fTnmN

fTnmN

evfTnmN

Nj

nm

=

(10)

Equation (10) is similar to (6), but the sign of the frequency offset term, fT , is changed from positive to negative. This weighting function of the

( )( ) ( )*'* )( kn rFFTdIFFT pair, conjugate transmitter receiver operation, is identical to the weighting function of (6) at 0=fT . On the other hand, the frequency offset 0>fT , will result a shift to the right operation on the weighting function of (10) as opposed to a shift to the left of (6). B. Conjugate Cancellation Scheme Assuming that both outputs of a regular

( ) ( )kn rFFTdIFFT pair OFDM system and a conjugate ( )( ) ( )*'* )( kn rFFTdIFFT pair OFDM system can be combined without interfering each other at the receiver by using a division multiplexing technique, the final detected symbol is then chosen as

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the averaged detected symbols of the regular OFDM receiver and the conjugate algorithm as follows:

1,...,2,1,0)(21 '"

=+= Nnddd nnn . (11)

This is the CC scheme. In fact, it is a simple version of time space coding if dual antennas are employed [7]. The signal to ICI power ratio (SIR) of the CC algorithm, as a function of frequency offsets, is plotted in Fig. 2. The SIR of a regular OFDM system is independent to N. But the SIR of the CC algorithm is a function of N for large frequency offsets. For a small frequency offset, the SIR of the CC algorithm is about the same for different N. It shows that the SIR of the CC algorithm is about 17 dB higher than that of the regular algorithm at 4% frequency offset. On the other hands, the SIR of the CC algorithm is smaller than that of the regular OFDM algorithm at 25.0>fT . Fig. 3 depicts the SIR for four different systems: regular OFDM, self-cancellation schemes with constant and linear components of ICI [1-2], and this CC scheme at N = 16. It is shown that this CC algorithm has a highest SIR than others when frequency offsets are small. A single antenna architecture for the transmitter and receiver is described in Figs. 4(a) and (b), respectively. A CDM, TMD, or FDM is required at the parallel transmitter to make parallel transmission. A de-multiplexing circuit is required at the receiver in order to perform parallel receiving operations accordingly. A 2N-point FFT is applied for both path data for efficient computation.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-20

-10

0

10

20

30

40

50

60

Normalized Frequency index

SICI

ra

tio (d

B), Re

gula

r, C

onj p

aral

lel C

om

bined

Regular

N=8

N=128

N=16 N=32

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-20

-10

0

10

20

30

40

50

60

Normalized Frequency index

SICI

ra

tio (d

B), Re

gula

r, C

onj p

aral

lel C

om

bined

Regular

N=8

N=128

N=16 N=32

Fig. 2. SIR of the CC algorithm vs. frequency offsets.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

10

20

30

40

50

60

70

80

90

100

110

Normalized Frequency index

SICI

ratio

(dB

), Reg

ula

r, N=

16, Co

nj, c

onst

and

line

ar

CC

Regular

Linear Self-Cancellation

Constant Self-Cancellation

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

10

20

30

40

50

60

70

80

90

100

110

Normalized Frequency index

SICI

ratio

(dB

), Reg

ula

r, N=

16, Co

nj, c

onst

and

line

ar

CC

Regular

Linear Self-Cancellation

Constant Self-Cancellation

Fig. 3. SIR of four algorithms vs. frequency offsets.

S/PN-Point

IFFT

P/Snd

f1

( )*

1st path

2nd path

TDM

Modulation(BPSK,QPSK,

16QAM,64QAN)

S/PN-Point

IFFT

P/Snd

f1

( )*

1st path

2nd path

TDM

Modulation(BPSK,QPSK,

16QAM,64QAN)

Fig. 4(a). The parallel transmitter.

S/P 2N-PointFFT

P/S

f1

De-mux

( )*

N Even Bins

1st path

2nd path

De-modulation

ndS/P 2N-

PointFFT

P/S

f1

De-mux

( )*

N Even Bins

1st path

2nd path

De-modulation

nd

Fig. 4(b). The parallel receiver.

IV. SIMULATION RESULTS

The signal processing for the transmitter and receiver

described in Figs. 4(a) and 4(b), respectively, was simulated in AWGN channel. For fair comparison, each branch at the transmitter of the CC system is at half of the original signal power of the regular OFDM system. At N = 16, the BER performance of 16-QAM for a regular OFDM and a CC OFDM in AWGN channel is provided in Fig. 5. At 04.0=fT , it shows that the performance of the CC algorithm is better than that of the regular algorithm by 14 dB at BER =10-6.

0-7803-8521-7/04/$20.00 (C) 2004 IEEE39680-7803-8521-7/04/$20.00 2004 IEEE

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 4 8 12 16 20 24 28

Eb/No dB

BER

of 1

6-Q

AM

OFD

M in

AW

GN Theo-AWGN

Reg-0.02CC-0.02Reg-0.04CC-0.04Reg-0.06CC-0.06

Fig. 5. BER performance in AWGN channel.

A 3-ray Rummler fading model [8] with frequency offset is employed. A 25% guard time is applied to the DQPSK OFDM signal with N=1024. Frequency-domain differential coding is applied in both systems in order to avoid channel response estimation. The data rate is 20 Mbps. This channel model is composed of three propagation paths: a line-of-sight (LOS) path, and two specular reflections (2nd and 3rd rays) with Rayleigh distribution. The 2nd ray has a signal strength that is 83% of that in the LOS path with a 50 ns delay. The amplitude of the 3rd ray is 2% of that of the LOS ray with a 150 ns delay. Fig. 6 shows that the BER performance of the CC system is significantly better than that of regular system when the frequency offset becoming a dominate factor, especially at high SNR.

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

4 8 12 16 20 24 28 32 36 40

Eb/No (dB)

BER

of D

QPS

K O

FDM

Reg-0.01CC-0.01Reg-0.04CC-0.04Reg-0.1CC-0.1

Fig. 6. BER performance in fading channel.

V. CONCLUSIONS

By using the CC scheme, the sensitivity of the OFDM system to ICI is reduced. The key feature of this new scheme is that it provides a much better SIR over the regular OFDM system in the presence of small frequency offset. This CC scheme provides an excellent BER performance for small frequency offset over a regular OFDM system in AWGN channel and 3-ray Rummler fading channels. Although the bandwidth efficiency is reduced to half due to twice transmission, it can be compensated by using larger signal alphabet sizes. This CC scheme can also be considered as a simple space coding with dual antennas as in [7].

REFERENCES

[1] Y. Zhao and S-G. Haggman, Intercarrier interference self-cancellation scheme for OFDM mobile communication systems, IEEE Trans. Commun., vol. 49, no. 7, pp. 1185-1191, July 2001.

[2] J. Armstrong, Analysis of new and existing methods of reducing intercarrier interference due to carrier frequency offset in OFDM, IEEE Trans. Commun., vol. 47, no. 3, March 1999, pp. 365-369.

[3] N. A. Dhahi et al., Optimum finite-length equalization for multicarrier transceivers, IEEE Trans. Commun., vol. 44, no. 1, pp. 56-64, Jan. 1996.

[4] C. Muschallik, Improving an OFDM reception using an adaptive Nyquist windowing, IEEE Trans. Consumer Electron., vol. 42, pp. 259-269, Aug. 1996.

[5] P. H. Moose, A technique for orthogonal frequency division multiplexing frequency offset correction, IEEE Trans. Commun., vol. 42, no. 10, Oct. 1994, pp. 2908-2914.

[6] T. M. Schmidl and D. C. Cox, Robust frequency and timing synchronization for OFDM, IEEE Trans. Commun., vol. 45, no. 12, pp1613-1621, Dec. 1997.

[7] S. M. Alamouti, A simple transmit diversity techniquefor wireless communications, IEEE J. Select. Areas Commun., vol. 16, no. 10, pp. 1451-1458, Oct. 1998.

[8] M. Rice, A. Davis, C. Bettweiser, A wideband channel model for aeronautical telemetry, IEEE Trans. Aerospace and Electronics Systems, vol. 40, pp. 57-69, Jan. 2004.

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