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A New Carrier Frequency Offset EstimationUsing CP-ICA Scheme in OFDM Systems

Jong-Deuk KimDept. of Electronics Engineering, University of InCheon

177 Dohwa-Dong, Nam-Gu,InCheon 402-749, Korea [email protected]

Youn-Shik ByunDept. of Electronics Engineering, University of InCheon

177 Dohwa-Dong, Nam-Gu,InCheon 402-749, Korea [email protected]

Abstract — a well-known problem of OFDM is its sensitivity tocarrier frequency offset which leads to inter-carrier interference(ICI) in the OFDM symbol. This ICI causes severe degradation of the BER performance of the OFDM receiver. In this paper, wepropose a new ICI cancellation algorithm which estimatesfrequency offset at the time-domain by using Cyclic Prefix -Independent Component Analysis (CP-ICA) method to the

received subcarriers phase rotation. This algorithm is based on astatistical blind estimation method, which mainly utilizes theeigenvalue decomposition (EVD), rotating phasor and the 4th cumulants.

Keywords-OFDM;Frequency Offset; ICI; CP-ICA

I. I NTRODUCTION

RTHOGNAL Frequency Division Multiplexing( OFDM )is emerging as the preferred modulation scheme in

modern high data rate wireless communication systems. Awell known problem of OFDM, however, is its sensitivity tofrequency offset between the transmitted and received signal,which may be caused by Doppler shift in the channel, or bythe difference between the transmitter and receiver localoscillator frequencies. The carrier frequency offset causes lossof orthogonality between subcarriers , and the signalstransmitted on each carrier are not independent of each other,thus leading to the ICI [1, 2]. Therefore, a synchronization

problem of carrier frequency is very crucial for possibleapplication of OFDM in high data rate transmission wirelesscommunications. Many researchers have proposed variousmethods to combat the ICI in OFDM systems. The existingapproaches that have been developed to reduce ICI can becategorized as using pilot method, blind scheme using Cyclic

Prefix and the ICI self-cancellation (SC) scheme [3]. Mostexisting frequency offset estimation techniques in OFDMsystems require pilot symbols [4, 5, 6, 7]. However, the use of

pilot symbols reduces the bandwidth efficiency because the pilots occupy some part of the useful bandwidth. Therefore, a blind approach scheme for frequency offset estimation wasintroduced [8]. However, the algorithm relies on virtualsubcarriers, which also lowers the bandwidth efficiency. For more realistic synchronization, some methods have been

proposed for blind estimation of timing offset as well as

frequency offset [9, 10, 11, 12]. These algorithms mainlyutilize the cyclic prefix, and some methods [11] require theOFDM signal to be cyclostationary. In addition, statisticalseparation approaches have also been explored to do a blindsource separation [13, 14, 15].

In this paper, we propose a blind method to estimate

frequency offset in OFDM systems. This method is based onCyclic Prefix– Independent Component Analysis (CP-ICA)approach that statistically estimates the frequency offset andcorrects the offset using the estimated value at the receiver. The

paper is organized as follows : Section Ⅱ describes thestandard system model and the problem of ICI in OFDM.Section Ⅲ describes the independent component analysisscheme and the proposed CP-ICA methods in this paper.Section Ⅳ provides a simulation result that tests the BER

performance and also compares it with the SC, ML and CP-ML[3, 5, 10]. Finally, concluding remarks are presented insection Ⅴ.

II. SYSTEM MODEL AND I NTER -C ARRIER I NTERFERENCEPROBLEM IN OFDM SYSTEMS

In OFDM systems, the input bit streams are multiplexedinto N symbol streams, each with symbol period T s, and eachsymbol stream is used to modulate parallel, synchronoussubcarriers [16]. The subcarriers are spaced by 1/NT s infrequency, and thus they are orthogonal over the interval ( 0,T s). A typical discrete-time baseband OFDM transceiver system is shown in Figure 1. First, a serial-to-parallel (S/P)converter groups the stream of input bits from the sourceencoder into groups of log 2 Ð bits, where Ð is the alphabet of size of the digital modulation scheme employed on each

subcarrier. A total of N such symbols, X l (m) , are created.Then, the N symbols are mapped to bins of an inverse fastFourier transform (IFFT). These IFFT bins correspond to theorthogonal subcarriers in the OFDM symbols.

21

0

1( ) ( )

n m N i

N l l

m

x n X m e N

π −

=

= ∑ (1)

where X l (m) is the baseband l -symbols on m -th subcarrier.

O

1525-3511/07/$25.00 ©2007 IEEE

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2007 proceedings.

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Figure 1. Baseband OFDM system

At the receiver, the signal is converted back to a discrete N point sequence yl (n) , corresponding to each subcarrier. Thisdiscrete signal is demodulated using the N point fast Fourier transform (FFT) operation at the receiver. The demodulatedsymbol stream is given by

21

0

( ) ( ) ( )kn N i

N l l l

n

Y k y n e W k π − −

=

= +∑ (2)

where W l (k) corresponds to the FFT of the samples of w l (n) , which is the Additive White Gaussian Noise (AWGN)introduced in the channel. Multipath distortion can also cause

Inter-symbol Interference (ISI) when adjacent symbolsoverlap with each other. This is prevented in OFDM system

by the insertion of a cyclic prefix between successive OFDMsymbols. This CP is discarded at the receiver to cancel out ISI.It is due to the robustness of OFDM to ISI and multipathdistortion that it has been considered for various wirelessapplication and standards. The main disadvantage of OFDM,however, is its susceptibility to small differences in frequencyat the transmitter and receiver, normally referred to asfrequency offset. The frequency offset can be caused byDoppler shift due to relative motion between the transmitter and receiver, or by the difference between the transmitter andreceiver local oscillator frequencies. In this paper, the carrier frequency offset is modeled as a multiplicative factor introduced in the channel, as shown in Figure 2.

Figure 2. Carrier Frequency Offset Model

The received signal is given by

2( ) ( ) ( )

ni N l l l y n x n e w n

π ε = + (3)

where ε(= f offset / ∆ f) is the normalized frequency offset, f offset is the frequency difference between the transmitted andreceiver frequencies and ∆ f is the subcarrier spacing. w l (n) isthe AWGN introduced in the channel. The effect of thisfrequency offset on the received symbol stream can beunderstood by considering the received symbol Y l (k) on the k-th subcarrier.

12 /

0

2 ( )1 1

0 0

1

0,

( ) ( ) , 0,1, ...., 1

1( ) ( )

( ) (0) ( ) ( ) ( )

N i kn N

l l n

n m k N N i N

l l m n

N

l l l l l m m k

Y k y n e k N

X m e W k N

X k I X m I m k W k

π

π ε

−−

=

+ −− −

= =

−

= ≠

= = −

= +

= + − +

∑

∑ ∑

∑

(4)

The ICI components are the interfering signals transmittedon subcarriers other than the k-th subcarrier. The complexcoefficients are given by

1(1 )( )1 sin( ( ))( )

s in( ( ) / )

i m k N

l

m k I m k e

N m k N

π ε π ε π ε

− + −+ −− =

+ −(5)

The first term in (4) denotes the useful component. Thetransmitted symbol X l (k) is rotated and attenuated. The secondterm is the complex coefficients for the ICI components in thereceived signal. For sufficiently large N , ICI can be modeled asa Gaussian random process with zero mean variance 2

I σ by the

central limit theorem. Therefore, we can estimate the frequencyoffsets from the estimate of phase rotation.

III. FREQUENCY OFFSET ESTIMATION USING CP-ICA

To rigorously define Independent Component Analysis(ICA), we can use a statistical “latent variables” model [13-15]. We have an observable random vector ( ) y n .

1 1

0 0

( ) ( ) ( ) , 0,1,...., 1 I I

ij jn jni j

y n a x n w n n N − −

= =

= + = −∑∑ (6)

y(n) is assumed to be linear mixtures of some mutuallystatistically independent variables a ij . Mathematically, Y isgenerated according to Y=AX . The matrix A is mixing matrix.In order to estimate unmixing matrix ζ = A -1., whitening

processing is needed. One popular method for whitening is touse the eigenvalue decomposition of covariance matrix.

[ , ] ( { })T T v v v v v E D eig YY E D E = Ε = (7)

where E v is the orthogonal matrix of eigenvectors of ( { })T eig E YY and D v is the diagonal matrix of its eigenvalues.

The whiteness or uncorrelatedness is a necessary condition for

the stronger statistically independent condition [13 - 14].(1/2) , ( )T

v v vO E D E Y OY O AX W RX W −= = = + = + (8)

where O is whitening matrix and after whitening, newvector Y is white. i.e., its components are uncorrelated andtheir variance equal unity. The whitening transforms themixing matrix A into a new rotating matrix R (= O A ) . We

just need to find an orthogonal transformation U to makecomponents of X = U Y .


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( ) ( )ˆ( ) ( ) ( )

UY U R X W U OAX OW

UO AX UO W UO AX W X W

= + = +

= + = + = +

(9)

Therefore, ζ can be constructed as ζ = UO = A -1.In order to mitigate the ICI in OFDM systems, we will have

to estimate the carrier frequency offset (CFO). The proposedCFO estimation using CP-ICA performs as follows: iψ (i =1,2)

are linear mixtures of statistically independent each other bythe frequency offset and AWGN.

1

2

1

1

1

2

[2 ( / 4)]

:

:,

: ( / 4)

( ) :

( )

( )

n L

N

n N L

l

N

l thl

cyclic prefix parts

useful data parts

L CP length N

y n received l symbols

y n

y n

ψ

ψ ψ

ψ

ψ

−

=−

−

= −

×=

=

=

=

∑

∑(10)

Step 1. Polar Decomposition

112

[ , ] ( )( )

,

T

T

E D e i g l e n g t h

P E D E O P

ψ ψ ψ

−

=

= =

(11)

where E is eigenvectors, D is eigenvalues, P is whitener matrix, and O is pseudo-inverse of P .

Step 2. Centering and Whitening

( ){ },r r Oψ ψ ψ ψ = − Ε ϒ = ⋅ (12)

where r ψ is centering, ϒ is statistically independentmatrix and {}Ε ⋅ denotes the expectation operator.

Step 3. 4 th cumulants of independent components [15]

0 1 2 3

0 2 1 3

0 3 1 2

{( )}, {( )}

[ , ] [ , ]

[ , ] [ , ]

[ , ] ( [ , ])

T H yy yy

yy yy yy

yy yy

yy yy

C

conj

τ τ τ τ

τ τ τ τ

τ τ τ τ

ℜ = Ε ϒϒ ℑ = Ε ϒϒ

= ℜ −ℜ ℜ

−ℜ ℜ

− ℑ ℑ

(13)

where T is transpose, H is Hermitian, C is the 4 th

cumulants and iτ is delay factor.

Step 4. Extraction important eigenvector from step 3

1 1

1

1,

[ , ] ( )

a r g m a x ( { } ) , {1, 2 , . . . , }

a rg m a x ( { } )i

j j i

E D e ig C

i d ia g D n

j d i a g D∈

∈ ≠

== =

=

(14)

where i is 1 st and j is 2nd index of max eigenvalues.

Step 5. Buffering as respects important eigenvector

1 1[ ] 11 2

2 2[ ] 1

( ), [ ], 2

( )M K

M K

m m E i B m m M K

m m E j×

×

= == = =

= =(15)

where B is buffer with respect to max eigenvectors matrix.

Step 6. Rotating to the Step 5

1 1 1 1

1 2

1 2

2 2 2 2

m ax ( ( )) min( ( )) m ax( ( )) min( ( ))(1, ) (1, )

( ,1) ( ,1)

cos sin

sin cos

1 0[ , ] , ( )

0T T

E i E i E j E j g m K m K

m M m M

c s R

s c

E D eig G E g g E R

θ θ θ θ

− − =

= = − −

= = ⋅

(16)

where g is real and imaginary axis separation to the maxeigenvectors(= B), R is basis rotating matrix, G is rotatedmatrix of independent components.

Step 7. Extraction max eigenvector from the G matrix andobtains basis rotating matrix ( = R).

( )( )

3 3

3

[ , ] ( )

arg max( { 3}) , {1, 2, ... , }

(2) 1 (3)1 (1)( ), ,

2 2 2

k

E D eig G

k diag D n

E k c sc

θ θ θ θ

∈

=

= =

− −= = + =

⋅

(17)

where k is index of max eigenvalues and Ө is maxeigenvector.

Step 8. Update basis rotating matrix.

( ) 1 0, ,

0 1u b b u

c con j s R U U U R

s c

− = = = ⋅

(18)

where R u is updated basis rotating matrix, U b is basisUnitary, U is updated new rotating unitary matrix.

Step 9. Comparison for the optimal rotated matrix

If | s | > Th [→ Go Step 5. updating buffer]

T u B R B= ⋅ (19)

Else if | s | < Th [→ Obtain the optimal rotated matrix]

[ ] ( )M K U Oζ × = ⋅ (20)

End


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where Th is decision threshold (=1 / No. of subcarriers) andζ is the optimal rotated square matrix.

Step 10. Frequency Offset Estimation

( )1

11

6

:( )

: ( 1)tan ( ) / 2 ,

( 10 ), 1 ,

th

M

th

M

M M

M rowvector

M rowvector conj

if end

ζ

ζ ε ζ ζ π

ε ε ε −

−−

−

−= ⋅

< − = +

∑

(21)

IV. SIMULATION R ESULTS

In order to compare the three different ICI cancellationschemes, we will assume that timing synchronization is

perfect for all methods at the receiver. BER curves were usedto evaluate the performance of each scheme. The OFDMtransceiver system was implemented as specified by Figure 1.Frequency offset was introduced as the phase rotation as given

by using (3). In this simulation, we choose 16-QAM and 64-QAM of modulation schemes and the AWGN channel isassumed. The OFDM system parameters chosen are asfollows: the number of subcarriers N = 256, the number of CPlength L= N/4, and normalized frequency offsets ε = {0.032,0.49, 0.65, 0.98}. The simulation is carried out over 1000OFDM symbols for each SNR value. The BER curve for standard OFDM without ICI cancellation shows in Figure 3.

Figure 3. BER performance without ICI cancellation of a standard OFDMsystem

In the case of larger alphabet sizes and larger frequencyoffset, OFDM systems do deteriorate the performance toogreatly. Therefore, we can conclude that larger alphabet sizesand larger frequency offsets are more sensitive to ICI.

Figure 4 and Figure 5 provide comparisons of the performance of the SC, ML, CP-ML and CP-ICA schemes for different alphabet sizes and different values of the frequencyoffset in 0.0< ε <0.5 . In the presence of very small frequencyoffset and small alphabet size, self-cancellation [3] gives the

best results. However, for larger alphabet sizes and larger frequency offset such as 64-QAM and frequency offset of 0.49, SC and ML [5] method does offer a serious decrease in

performance. But the CP-ML [10] and CP-ICA method givesthe best overall results.

Figure 4. BER Performance with ICI Cancellation, ε = 0.032


Figure 6 and Figure 7 show comparisons of the performanceof the SC, ML, CP-ML and CP-ICA schemes for differentalphabet sizes and different frequency offsets in 0.5< ε <1.0 .We can observe the BER error floor. The SC, ML and CP-MLmethods are very hardly depredated BER. Since thesetechniques do not completely cancel the ICI from adjacentsubcarriers and the effect of this residual ICI increases for larger alphabet sizes and frequency offset values. However, byusing the proposed CP-ICA method for high normalizedfrequency offset, we can compensate the frequency driftcaused by the difference between oscillators in the transmitter and the receiver. This is attributed to the fact that the CP-ICAmethod estimate the frequency offset very accurately andcancel the frequency offset. The proposed CP-ICA methodhave an extremely performance overall in 0.0< ε <1.0 . It givesa significant boost to performance.



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Frequency offset estimation for different values ε ={0.032,0.49, 0.65, 0.98} and 64-QAM subcarrier modulation is shownin Figure 8. This figure shows that using CP-ICA method canestimate the frequency offsets very accurately overall.

Figure 8. Frequency Offset Estimation for different values ε = {0.032, 0.49,0.65, 0.98}, 64-QAM

V. CONCLUSION

Inter-carrier interference which results from the frequencyoffset degrades the performance of the OFDM systems. Theself-cancellation (SC), the maximum likelihood (ML) and CP-ML estimation techniques were proposed in previous

publications [3, 5, 10] for mitigation of the ICI.The SC [3] does not require very complex hardware or

software for implementation. However, it is not bandwidthefficient as there is a redundancy of 2 for each carrier. The ML[5] method also introduces the same level of redundancy but

provides better BER performance, since it accurately estimatesthe very small frequency offset. Its implementation is morecomplex than the SC method. On the other hand, the CP-ML

[10] method does not reduce bandwidth efficiency as thefrequency offset can be estimated from the Cyclic Prefix ineach OFDM frame. Also, it performed the best of the threemethods in 0.0< ε <0.5 . However, this method performs atfrequency domain for estimation of the frequency offset.

In this paper, we propose using CP-ICA scheme for estimation of frequency offsets in OFDM systems. Thismethod is a blind scheme and performs more efficiently attime-domain than at frequency domain. Since our scheme doesnot need any pilot symbol in estimation, we can expectenhanced bandwidth efficiency in OFDM systems. We derive

the frequency offset estimation technique by using ICAanalyzing the phase rotation of the OFDM symbols. The carrier frequency offset is very accurately estimated using our technique based on Cyclic Prefix-Independent ComponentAnalysis. While the proposed CP-ICA method has littleweakness that complexity is a little heavy and the number of symbols used is enough needed, simulation results show thatthe proposed frequency offset estimator is more accurate thanthe other estimators in 0.0< ε <1.0 .

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[3] Y. Zhao and S. Häggman, “Intercarrier interference self-cancellationscheme for OFDM mobile communication systems,” IEEE Trans. onCommunications , vol. 49, no. 7, pp. 1185 – 1191, July 2001.

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[6] U. Lambrette, M. Speth and H. Meyr, “OFDM burst frequencysynchronization by single carrier training data, ” IEEE Communications

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[9] M. Luise and R. Reggiannini, “Carrier acquisition and tracking for OFDM systems,” IEEE Trans. on Communications, vol. 44, no. 11, pp.1590-1598, November 1996.

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[12] J. H. Gunther, H. Liu and A. L. Swindlehurst, “A new approach for symbol frame synchronization and carrier frequency estimation inOFDM communication,” Proc. of The IEEE Int. Conf. on Acoustics,Speech and Signal Processing, pp.2725-2728, Phoenix, AZ, March1999.

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