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

Optics Express, Vol. 21, Issue 19, pp. 22847-22853 (2013)

http://dx.doi.org/10.1364/OE.21.022847

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### 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

*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]

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. 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]

*f*

_{0}) and estimated separately. In [8], 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]

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]

## 2. System Model

*X*, the frequency-domain complex modulation symbol associated to the

_{ki}*k*-th subcarrier and

*i*-th OFDM symbol:where

*N*is the DFT size. Assuming perfect time synchronization, the received signal sampled at {0, 1/(

*Nf*

_{0}), …,

*n*/(

*N f*

_{0}), …} would be:where the signal is distorted by channel distortion

*h*, frequency offset Δ

_{l}*f*=

*εf*

_{0}, laser phase noise

*ϕ*and additive white Gaussian noise

_{n}*w*. We assume a finite impulse response of length

_{n}*L*samples for the fiber, which is constant within a certain OFDM frame. The frequency domain samples can be written as:where

*X*,

_{k,i}*Y*,

_{k,i}*H*and

_{k}*W*are the frequency domain transmitted symbol, received symbol, channel transfer function and AWGN noise, respectively.

_{k,i}## 3. Correlation-based fraction part estimation

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]

**45**(12), 1613–1621 (1997). [CrossRef]

**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]

*f*=

*εf*

_{0}

*N*/(

*N*+

*CP*). Similarly, the CP estimator [4

**45**(7), 1800–1805 (1997). [CrossRef]

*D*consecutive symbols:

**45**(12), 1613–1621 (1997). [CrossRef]

*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.

*D*=

*N*/

*CP*= 8. Figure 2 and 3 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,

*E*/

_{s}*N*

_{0}) 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

*σ*

^{2}>>

*N*

_{0}/(

*NE*). 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

_{s}*N*,

*CP*and

*D*as long as we hold

*D*=

*N*/

*CP*and

*CP*≤

*N*. 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

*h*for

_{l}*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

*ε*has been compensated for,

_{f}*m*+

*ε*) being an integer, we can conclude that

_{i}*m*+

*ε*= 0 and

_{i}*ε*will shift the pilot-tone (peak in the received spectrum)

_{i}*ε*positions away. We can thus calculate

_{i}*ε*by:where

_{i}*I*= argmax

*|*

_{k}*Y*| (

_{k}*k*= 0,…,

*N*-1). Note that fiber chromatic dispersion will not affect the energy of the received signal, which is formulated as a constant envelope function

*L*being the entire length of the transmission link and

*β*

_{2}the GVD coefficient. Thus, the probability of correct detection (

*P*) is calculated as:where

_{c}*X*is a real number representing the DC and

_{p}*X*can be any point from the signal constellation. From Eq. (17), we can conclude that the error probability is only dependent on the pilot to average signal power ratio (

_{k}*E*/

_{p}*E*,

_{s}*E*= |

_{p}*X*|

_{p}^{2},

*E*= E[|

_{s}*X*|

_{s}^{2}]), SNR (

*E*/

_{s}*N*

_{0}) and DFT size (

*N*). In Fig. 5 (a) we plot

*P*versus pilot to average signal power ratio at different SNR in a back to back transmission with QPSK format. As predicted, the probability curve depends on SNR value and DFT size, but it is unaffected by

_{c}*f*

_{0},

*ε*or dispersion. For constant modulus format, we can further reduce Eq. (17) to:

_{i}*P*is ploted versus

_{c}*E*/

_{p}*E*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.

_{s}## 5. Conclusion

*f*

_{0}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

## 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. |

2. | T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. |

3. | P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun. |

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. |

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. |

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. |

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

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### References

- 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]
- T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun.45(12), 1613–1621 (1997). [CrossRef]
- P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun.42(10), 2908–2914 (1994). [CrossRef]
- 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]
- 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.
- 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]
- 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]
- 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.

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