OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 19 — Sep. 23, 2013
  • pp: 22847–22853
« Show journal navigation

A performance investigation of correlation-based and pilot-tone-assisted frequency offset compensation method for CO-OFDM

Shengjiao Cao, Changyuan Yu, and Pooi-Yuen Kam  »View Author Affiliations


Optics Express, Vol. 21, Issue 19, pp. 22847-22853 (2013)
http://dx.doi.org/10.1364/OE.21.022847


View Full Text Article

Acrobat PDF (1056 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We carry out a comprehensive analysis to examine the performance of our recently proposed correlation-based and pilot-tone-assisted frequency offset compensation method in coherent optical OFDM system. The frequency offset is divided into two parts: fraction part and integer part relative to the channel spacing. Our frequency offset scheme includes the correlation-based Schmidl algorithm for fraction part estimation as well as pilot-tone-assisted method for integer part estimation. In this paper, we analytically derive the error variance of fraction part estimation methods in the presence of laser phase noise using different correlation-based algorithms: Schmidl, Cox and Cyclic Prefix based. This analytical expression is given for the first time in the literature. Furthermore, we give a full derivation for the pilot-tone-assisted integer part estimation method using the OFDM model.

© 2013 OSA

1. Introduction

Orthogonal frequency-division multiplexing (OFDM) has attracted much research interest due to its dispersion tolerance, ease of frequency domain equalization and high spectral efficiency. To the contrary, OFDM is more sensitive to frequency offset (FO), due to its longer symbol duration, which is N (number of subcarriers) times that of a single carrier system. This calls for accuracy in frequency recovery hundreds or thousands of times greater than that in a single carrier system with the same bit rate [1

1. T. Pollet, M. Van Bladel, and M. Moeneclaey, “BER sensitivity of OFDM systems to carrier frequency offset and wiener phase noise,” IEEE Trans. Commun. 43(2), 191–193 (1995). [CrossRef]

]. The presence of FO would cause loss of orthogonality between subcarriers and thus degrade the system performance. Frequency offset compensation (FOC), therefore, is one of the most critical functions to implement in OFDM systems.

Various methods have been employed for FO estimation and compensation in both wireless [2

2. T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997). [CrossRef]

4

4. J. van de Beek, M. Sandell, and P. O. Borjesson, “ML estimation of time and frequency offset in OFDM systems,” IEEE Trans. Signal Process. 45(7), 1800–1805 (1997). [CrossRef]

] and optical domain [5

5. F. Buchali, R. Dischler, M. Mayrock, X. Xiao, and Y. Tang, “Improved frequency offset correction in coherent optical OFDM systems,” in IEEE Proc. ECOC 2008, paper Mo.4.D.4.

8

8. S. Cao, S. Zhang, C. Yu, and P.-Y. Kam, “Full-range pilot-assisted frequency offset estimation for OFDM systems,” in Proceedings of OFC/NFOEC 2013, paper JW2A.53.

]. The frequency offset can be divided into a fraction and an integer part of the carrier spacing (f0) and estimated separately. In [8

8. S. Cao, S. Zhang, C. Yu, and P.-Y. Kam, “Full-range pilot-assisted frequency offset estimation for OFDM systems,” in Proceedings of OFC/NFOEC 2013, paper JW2A.53.

], we proposed to use a correlation-based method for estimating the fraction part and a pilot-tone-assisted method for the integer part. The fraction part estimation methods are either based on repeated training symbols including Schmidl [2

2. T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997). [CrossRef]

] (which is the method employed in [8

8. S. Cao, S. Zhang, C. Yu, and P.-Y. Kam, “Full-range pilot-assisted frequency offset estimation for OFDM systems,” in Proceedings of OFC/NFOEC 2013, paper JW2A.53.

]) and Moose [3

3. P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun. 42(10), 2908–2914 (1994). [CrossRef]

], or cyclic prefic (CP) [4

4. J. van de Beek, M. Sandell, and P. O. Borjesson, “ML estimation of time and frequency offset in OFDM systems,” IEEE Trans. Signal Process. 45(7), 1800–1805 (1997). [CrossRef]

]. Here, we will analytically derive the estimation variance in the presence of laser phase noise for these methods. To our knowledge, there is no other work on this topic in the literature. The analytical expressions are confirmed through simulation. CP method is found to be the most robust to laser phase noise among the three. Furthermore, we investigate the performance of correlation-based methods under different amount of chromatic dispersion. Schmidl and Moose estimators are robust to dispersion while the CP estimator degrades severely in the presence of dispersion. The pilot-tone assisted integer part estimation method is given in [8

8. S. Cao, S. Zhang, C. Yu, and P.-Y. Kam, “Full-range pilot-assisted frequency offset estimation for OFDM systems,” in Proceedings of OFC/NFOEC 2013, paper JW2A.53.

] without solid proof. In this paper, we will give a full derivation of the method based the OFDM model. Its performance will be further investigated through both analysis and simulation.

The reminder of the paper is organized as follows. In Section 2, a model incorporating frequency offset, laser phase noise, channel distortion and additive white Gaussian noise is derived for CO-OFDM systems. Subsequently, correlation-based fraction part estimation methods are investigated in the presence of chromatic dispersion and laser phase noise in section 3, where analytical expressions and simulation results of error variances are given. In section 4, the pilot-tone assisted integer part estimation method is presented with a more detailed derivation from the system model. In Section 5 we draw the conclusions.

2. System Model

The discrete time domain samples of an OFDM signal are obtained by taking DFT transform from Xki, the frequency-domain complex modulation symbol associated to the k-th subcarrier and i-th OFDM symbol:
xNi+n=(1/N)k=0N1Xkiej2πkn/N,
(1)
where N is the DFT size. Assuming perfect time synchronization, the received signal sampled at {0, 1/(Nf0), …, n/(N f0), …} would be:
yN*i+n=ej2π(N*i+n)ε/N+jϕN*i+nl=0L1hlxN*i+nl+wN*i+n,
(2)
where the signal is distorted by channel distortion hl, frequency offset Δf = εf0, laser phase noise ϕn and additive white Gaussian noise wn. We assume a finite impulse response of length L samples for the fiber, which is constant within a certain OFDM frame. The frequency domain samples can be written as:
Yk,i=l=0N1Ψkl,iHlXl,i+Wk,i,Ψm,i=1Nn=0N1ej2π(N*i+n)ε/N+jϕN*i+nej2πmn/N,
(3)
where Xk,i, Yk,i, Hk and Wk,i are the frequency domain transmitted symbol, received symbol, channel transfer function and AWGN noise, respectively.

3. Correlation-based fraction part estimation

In this section, we investigate the performance of correlation-based fraction part estimators in the presence of chromatic dispersion and laser phase noise. Three methods (Schmidl [2

2. T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997). [CrossRef]

], Moose [3

3. P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun. 42(10), 2908–2914 (1994). [CrossRef]

] and CP [4

4. J. van de Beek, M. Sandell, and P. O. Borjesson, “ML estimation of time and frequency offset in OFDM systems,” IEEE Trans. Signal Process. 45(7), 1800–1805 (1997). [CrossRef]

]) are included, and the derivation can be easily extended to other estimators. For fair comparison, we modify the Schmidl [2

2. T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997). [CrossRef]

] method to include two identical training symbols instead of one. Ignoring phase noise and additive noise, there is 2πε phase shift between the first and second training symbol, both in time and frequency domain. Thus, the maximum likelihood estimation is obtained from cross-correlation between two received symbols:
ε^Schmidl={n=0N-1yN+nyn*}/2π,ε^Moose={n=0N-1Yk1Yk0*}/2π,
(4)
where the two equations are the definition of Schmidl [2

2. T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997). [CrossRef]

] and Moose [3

3. P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun. 42(10), 2908–2914 (1994). [CrossRef]

] estimators, respectively. Taking the effect of CP into consideration, the actual frequency offset is Δf = εf0N/(N + CP). Similarly, the CP estimator [4

4. J. van de Beek, M. Sandell, and P. O. Borjesson, “ML estimation of time and frequency offset in OFDM systems,” IEEE Trans. Signal Process. 45(7), 1800–1805 (1997). [CrossRef]

] takes cross-correlation between the cyclic prefix and the data from which cyclic prefix is generated. A better estimation can be achieved by taking average over D consecutive symbols:

ε^CP={d=0D1n=0CP1y(N+CP)*d+n+Ny(N+CP)*d+n*}/2π.
(5)

We rewrite the relationship between the two training symbols of Schmidl estimator as follows:
yn=rn+wn,yN+n=rnej2πε+j(ϕN+n-ϕn)+wN+n,
(6)
where rn=ej2πnε/N+jϕnl=0L1hlxnl according to Eq. (2). Following a similar approximation in [2

2. T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997). [CrossRef]

], we can derive the estimation variance for Schmidl estimator from the tangent of the phase error:

tan[2π(ε^-Schmidlε)]=(n=0N1Im[yN+nyn*e2πjε])/(n=0N1Re[yN+nyn*e2πjε]),
(7)

For |ε^-Schmidlε|1/2π, Eq. (7) can be approximated as:

ε^-Schmidlε12π(n=0N1Im[X])/(n=0N1Re[X]),X=(rnej(ϕN+n-ϕn)+wN+ne2πjε)(rn*+wn*)
(8)

With high signal-to-noise ratio, Eq. (8) may be further approximated by:
ε^-SchmidlεΔϕSchmidl2π+n=0N1Im(wN+nrn*ej2πε+rnwn*)2πn=0N1|rn|2
(9)
Var[ε^Schmidl]=(1/2π)2[σSchmidl2+N0/(NEs)]
(10)
where ΔϕSchmidl=1/Nn=0N1(ϕN+nϕn) and Es = |rn|2 is the symbol energy. As {ϕn} is a wiener process, the variance of Δϕ is calculated as σSchmidl2=2π[(2N2+1)/3N+CP]vTs, with v as the combined laser linewidth and Ts as the sample interval. We can easily prove that ΔΦMoose=ΔϕSchmidl and Moose estimator shares the same variance as Schmidl. In Schmidl (or Moose) estimator, the channel distortion affects the two signals yN+n and yn in the same way as long as the guard interval is longer than channel memory L. The estimation result of Schmidl (or Moose) estimator will be robust to linear channel distortion. However, the linear channel distortion for the cyclic prefix and data from which the cyclic prefix is generated are different. Thus, the CP estimator works accurately only for zero channel distortion case. Assuming distortion has been removed prior to estimation, CP estimator follows a similar derivation as Schmidl (Eq. (8-10) but comes with a different variance for phase difference ΔϕCP:

Var[ε^CP]=(1/2π)2[σCP2+N0/(NEs)]
(11)
ΔϕCP=d=0D1n=0CP1[ϕ(N+CP)*d+n+Nϕ(N+CP)*d+n]/(D*CP)
(12)
σCP2=[(1CP2)/(3CP)+N+2]/D
(13)

Assuming the performance of CP and Schmidl (or Moose) estimators are identical for zero dispersion and zero phase noise case (D = N/CP), we expect CP to be performing better than Schmidl (or Moose) under nonzero laser phase noise case. This is because CP estimator has a smaller variance of phase noise difference between the two signals taken for cross-correlation.

To verify the derived variance expressions, we built a CO-OFDM system using MATLAB. The transmitter and receiver block diagram of our CO-OFDM is shown in Fig. 1
Fig. 1 The schematic of CO-OFDM system (Mod: modulation, Demod: demodulation, S/P: serial to parallel, P/S: parallel to serial, DAC: digital to analog converter, ADC: analog to digital converter)
. The system employs QPSK modulation with a DFT/IDFT size of 256 and a cyclic prefix of 32 samples. The signal is sampled at 10 Gsample/s. To match the performance of CP with Schmidl/Moose for the zero phase noise case, we set D = N/CP = 8. Figure 2
Fig. 2 Analytical and simulation curves of estimation variance versus SNR for v = 0,1,100 kHz, using Schmidl, Moose and CP estimator.
and 3
Fig. 3 (a) Analytical and simulation curves for estimation variance versus laser linewidth (v) at SNR = 15 dB, using Schmidl, Moose and CP estimator; (b) Estimation variance versus relative frequency offset for v = 0, 1, 10, 100 kHz, using Schmidl, Moose and CP estimator.
shows the simulation result in a back-to-back transmission. Figure 2 shows the estimation accuracy in terms of variance (Var[ε]) versus signal to noise ratio (SNR, Es/N0) with different laser linewidths. All the simulation results (black symbol) match perfectly with the analytical curves (red line). At higher laser linewidth, say 100 kHz, the variance curves are no longer sensitive to SNR for all three methods, as σ2>>N0/(NEs). CP estimator is more tolerant to laser phase noise than Schmidl (or Moose) estimator, e.g., it has nearly 10 times smaller variance than the other two methods at 100 kHz. Figure 3(a) compares the laser linewidth tolerance of the three estimators at 15-dB SNR. In addition to the fact that no training symbol is required for CP estimator, it performs the best in the presence of laser phase noise. We can easily prove that σSchmidl2>σCP2 for any values of N, CP and D as long as we hold D = N/CP and CPN. Figure 3(b) depicts the variance versus relative FO ε for different methods and different laser linewidths . All the methods have the same estimation range |ε|0.5and CP method has the most accurate estimation at nonzero laser linewidth.

In Fig. 4
Fig. 4 Simulation curves for estimation variance versus SNR under various dispersion values (0, 1700ps/nm, 17000ps/nm) using: (a)Schmidl/Moose; (b) CP.
, we incorporate linear channel distortion (chromatic dispersion) with different dispersion values (0, 1700, 17000 ps/nm) into the system. As predicted, Schmidl (or Moose) estimator performs almost the same under different amount of dispersion. The small deviation from the ideal curve is due to the small nonzero components hl for l≥CP generated by fiber chromatic dispersion. However, the degradation is almost negligible for as large as 17000-ps/nm dispersion (1000 km of standard single mode fiber with 17 ps/nm/km dispersion parameter). To the contrary, the accuracy of the CP estimator is severely degraded by dispersion as small as 1700 ps/nm, especially for smaller laser phase noise case. The conclusions are expected to be the same for polarization mode dispersion (PMD). We can transmit identical training symbols in different polarizations and thus PMD will affect the received signals in a similar way as CD in the single polarization case.

4. Pilot-tone-assisted integer part estimation

In this section, we will fully derive the pilot-tone-assisted integer part estimation method from its OFDM model in frequency domain (Eq. (3). A pilot-tone with larger energy at DC is inserted at the center of the spectrum. Assuming zero laser phase noise (ϕn=0) and that εf has been compensated for, Ψm can be calculated as:

Ψm=1Nsin(π(m+εi))sin(π(m+εi)/N)e(jπ(m+ε)(11N)),
(14)

With (m + εi) being an integer, we can conclude that Ψm=1when m + εi = 0 and Ψm=0 otherwise. The resulted received symbol would be:

Yk=Hk+εiXk+εi+Wk
(15)

Pc=PN1(|Xp+Wp|2>|Xs+Ws|2)
(18)

As indicated by Eq. (18), a smaller SNR or larger DFT size requires higher pilot to average signal power ratio to achieve error free detection, which is verified by simulation. In Fig. 5(b), Pc is ploted versus Ep/Es for different laser linewidths, where degradation is hardly noticeable under 100 kHz. Larger laser linewidth (500 kHz, 1 MHz) affects the curves to a small extend but different curves still converge to 0 at almost the same speed. Laser phase noise affects the received signal through ICI, which will corrupt the peak in a similar way as AWGN noise.

5. Conclusion

In this paper, we carried out a comprehensive analysis to examine the performance of our recently proposed correlation-based and pilot-tone-assisted frequency offset compensation method in coherent optical OFDM system. We have analytically derived the fraction part estimation accuracy in the presence of laser phase noise for various correlation-based methods. Furthermore, we re-propose our pilot-tone-assisted integer part estimation method with a full derivation based on the OFDM model. Its estimation accuracy is proved to be independent of f0 and dispersion, dependent of DFT size, pilot to average signal power ratio, SNR and laser phase noise. In the future we will further investigate the performance in the presence of nonlinear phase noise.

Acknowledgments

The authors would like to thank the supports of AcRF Tier 1 Grant R-263-000-631-112 and NUSRI Grant R-2012-N-009.

References and links

1.

T. Pollet, M. Van Bladel, and M. Moeneclaey, “BER sensitivity of OFDM systems to carrier frequency offset and wiener phase noise,” IEEE Trans. Commun. 43(2), 191–193 (1995). [CrossRef]

2.

T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997). [CrossRef]

3.

P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun. 42(10), 2908–2914 (1994). [CrossRef]

4.

J. van de Beek, M. Sandell, and P. O. Borjesson, “ML estimation of time and frequency offset in OFDM systems,” IEEE Trans. Signal Process. 45(7), 1800–1805 (1997). [CrossRef]

5.

F. Buchali, R. Dischler, M. Mayrock, X. Xiao, and Y. Tang, “Improved frequency offset correction in coherent optical OFDM systems,” in IEEE Proc. ECOC 2008, paper Mo.4.D.4.

6.

S. Fan, J. Yu, D. Qian, and G.-K. Chang, “A fast and efficient frequency offset correction technique for coherent optical orthogonal frequency division multiplexing,” J. Lightwave Technol. 29(13), 1997–2004 (2011). [CrossRef]

7.

S. L. Jansen, I. Morita, T. C. W. Schenk, N. Takeda, and H. Tanaka, “Coherent Optical 25.8-Gb/s OFDM Transmission over 4,160-km SSMF,” J. Lightwave Technol. 26(1), 6–15 (2008). [CrossRef]

8.

S. Cao, S. Zhang, C. Yu, and P.-Y. Kam, “Full-range pilot-assisted frequency offset estimation for OFDM systems,” in Proceedings of OFC/NFOEC 2013, paper JW2A.53.

OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(060.2330) Fiber optics and optical communications : Fiber optics communications

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: June 5, 2013
Revised Manuscript: July 18, 2013
Manuscript Accepted: August 10, 2013
Published: September 20, 2013

Citation
Shengjiao Cao, Changyuan Yu, and Pooi-Yuen Kam, "A performance investigation of correlation-based and pilot-tone-assisted frequency offset compensation method for CO-OFDM," Opt. Express 21, 22847-22853 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-22847


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. T. Pollet, M. Van Bladel, and M. Moeneclaey, “BER sensitivity of OFDM systems to carrier frequency offset and wiener phase noise,” IEEE Trans. Commun.43(2), 191–193 (1995). [CrossRef]
  2. T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun.45(12), 1613–1621 (1997). [CrossRef]
  3. P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun.42(10), 2908–2914 (1994). [CrossRef]
  4. J. van de Beek, M. Sandell, and P. O. Borjesson, “ML estimation of time and frequency offset in OFDM systems,” IEEE Trans. Signal Process.45(7), 1800–1805 (1997). [CrossRef]
  5. F. Buchali, R. Dischler, M. Mayrock, X. Xiao, and Y. Tang, “Improved frequency offset correction in coherent optical OFDM systems,” in IEEE Proc. ECOC 2008, paper Mo.4.D.4.
  6. S. Fan, J. Yu, D. Qian, and G.-K. Chang, “A fast and efficient frequency offset correction technique for coherent optical orthogonal frequency division multiplexing,” J. Lightwave Technol.29(13), 1997–2004 (2011). [CrossRef]
  7. S. L. Jansen, I. Morita, T. C. W. Schenk, N. Takeda, and H. Tanaka, “Coherent Optical 25.8-Gb/s OFDM Transmission over 4,160-km SSMF,” J. Lightwave Technol.26(1), 6–15 (2008). [CrossRef]
  8. S. Cao, S. Zhang, C. Yu, and P.-Y. Kam, “Full-range pilot-assisted frequency offset estimation for OFDM systems,” in Proceedings of OFC/NFOEC 2013, paper JW2A.53.

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4 Fig. 5
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited