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

Shengjiao Cao; Changyuan Yu; Pooi-Yuen Kam

doi:10.1364/OE.21.022847

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]. 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 [2–4] and optical domain [5–8]. The frequency offset can be divided into a fraction and an integer part of the carrier spacing (f₀) 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] (which is the method employed in [8]) and Moose [3], or cyclic prefic (CP) [4]. 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] 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 X_ki, the frequency-domain complex modulation symbol associated to the k-th subcarrier and i-th OFDM symbol:

x_{N * i + n} = (1 / N) \sum_{k = 0}^{N - 1} X_{k i} e^{j 2 π k n / N},

where N is the DFT size. Assuming perfect time synchronization, the received signal sampled at {0, 1/(Nf₀), …, n/(N f₀), …} would be:

y_{N * i + n} = e^{j 2 π (N * i + n) ε / N + j ϕ_{N * i + n}} \sum_{l = 0}^{L - 1} h_{l} x_{N * i + n - l} + w_{N * i + n},

where the signal is distorted by channel distortion h_l, frequency offset Δf = εf₀, laser phase noise ϕ_n and additive white Gaussian noise w_n. 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:

Y_{k, i} = \sum_{l = 0}^{N - 1} Ψ_{k - l, i} H_{l} X_{l, i} + W_{k, i}, Ψ_{m, i} = \frac{1}{N} \sum_{n = 0}^{N - 1} e^{j 2 π (N * i + n) ε / N + j ϕ_{N * i + n}} e^{- j 2 π m n / N},

where X_k,i, Y_k,i, H_k and W_k,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], Moose [3] and CP [4]) are included, and the derivation can be easily extended to other estimators. For fair comparison, we modify the Schmidl [2] 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:

{\hat{ε}}_{S c h m i d l} = ∠ {\sum_{n = 0}^{N - 1} y_{N + n} y_{n}^{*}} / 2 π, {\hat{ε}}_{M o o s e} = ∠ {\sum_{n = 0}^{N - 1} Y_{k 1} Y_{k 0}^{*}} / 2 π,

where the two equations are the definition of Schmidl [2] and Moose [3] estimators, respectively. Taking the effect of CP into consideration, the actual frequency offset is Δf = εf₀N/(N + CP). Similarly, the CP estimator [4] 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:

{\hat{ε}}_{C P} = ∠ {\sum_{d = 0}^{D - 1} \sum_{n = 0}^{C P - 1} y_{(N + C P) * d + n + N} y_{(N + C P) * d + n}^{*}} / 2 π .

We rewrite the relationship between the two training symbols of Schmidl estimator as follows:

y_{n} = r_{n} + w_{n}, y_{N + n} = r_{n} e^{j 2 π ε + j (ϕ_{N + n} - ϕ_{n})} + w_{N + n},

where

r_{n} = e^{j 2 π n ε / N + j ϕ_{n}} \sum_{l = 0}^{L - 1} h_{l} x_{n - l}

according to Eq. (2). Following a similar approximation in [2], we can derive the estimation variance for Schmidl estimator from the tangent of the phase error:

\tan [2 π (\hat{ε} {}_{S c h m i d l}- ε)] = (\sum_{n = 0}^{N - 1} Im [y_{N + n} y_{n}^{*} e^{- 2 π j ε}]) / (\sum_{n = 0}^{N - 1} Re [y_{N + n} y_{n}^{*} e^{- 2 π j ε}]),

For $| \hat{ε} {}_{S c h m i d l}- ε | ≪ 1 / 2 π$ , Eq. (7) can be approximated as:

\hat{ε} {}_{S c h m i d l}- ε \approx \frac{1}{2 π} (\sum_{n = 0}^{N - 1} Im [X]) / (\sum_{n = 0}^{N - 1} Re [X]), X = (r_{n} e^{j (ϕ_{N + n} - ϕ_{n})} + w_{N + n} e^{- 2 π j ε}) (r_{n}^{*} + w_{n}^{*})

With high signal-to-noise ratio, Eq. (8) may be further approximated by:

\hat{ε} {}_{S c h m i d l}- ε \approx \frac{Δ ϕ_{S c h m i d l}}{2 π} + \frac{\sum_{n = 0}^{N - 1} Im (w_{N + n} r_{n}^{*} e^{- j 2 π ε} + r_{n} w_{n}^{*})}{2 π \sum_{n = 0}^{N - 1} {| r_{n} |}^{2}}

Var [{\hat{ε}}_{S c h m i d l}] = {(1 / 2 π)}^{2} [σ_{S c h m i d l}^{2} + N_{0} / (N E_{s})]

where

Δ ϕ_{S c h m i d l} = 1 / N \sum_{n = 0}^{N - 1} (ϕ_{N + n} - ϕ_{n})

and E_s = |r_n|² is the symbol energy. As {

ϕ_{n}

} is a wiener process, the variance of

Δ ϕ

is calculated as

σ_{S c h m i d l}^{2} = 2 π [(2 N^{2} + 1) / 3 N + C P] v T_{s}

, with v as the combined laser linewidth and T_s as the sample interval. We can easily prove that

Δ Φ_{M o o s e} = Δ ϕ_{S c h m i d l}

and Moose estimator shares the same variance as Schmidl. In Schmidl (or Moose) estimator, the channel distortion affects the two signals y_N₊_n and y_n 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

Δ ϕ_{C P}

:

Var [{\hat{ε}}_{C P}] = {(1 / 2 π)}^{2} [σ_{C P}^{2} + N_{0} / (N E_{s})]

Δ ϕ_{C P} = \sum_{d = 0}^{D - 1} \sum_{n = 0}^{C P - 1} [ϕ_{(N + C P) * d + n + N} - ϕ_{(N + C P) * d + n}] / (D * C P)

σ_{C P}^{2} = [(1 - C P^{2}) / (3 C P) + N + 2] / D

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. 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 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/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/(NE_s). 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 $σ_{S c h m i d l}^{2} > σ_{C P}^{2}$ for any values of 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 $| ε | \leq 0.5$ and CP method has the most accurate estimation at nonzero laser linewidth.

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)

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Fig. 2 Analytical and simulation curves of estimation variance versus SNR for v = 0,1,100 kHz, using Schmidl, Moose and CP estimator.

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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.

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In Fig. 4, 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 h_l 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.

Fig. 4 Simulation curves for estimation variance versus SNR under various dispersion values (0, 1700ps/nm, 17000ps/nm) using: (a)Schmidl/Moose; (b) CP.

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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 ( $ϕ_{n} = 0$ ) and that ε_f has been compensated for, $Ψ_{m}$ can be calculated as:

Ψ_{m} = \frac{1}{N} \frac{\sin (π (m + ε_{i}))}{\sin (π (m + ε_{i}) / N)} e^{(j π (m + ε) (1 - \frac{1}{N}))},

With (m + ε_i) being an integer, we can conclude that $Ψ_{m} = 1$ when m + ε_i = 0 and $Ψ_{m} = 0$ otherwise. The resulted received symbol would be:

Y_{k} = H_{k + ε_{i}} X_{_{k + ε_{i}}} + W_{k}

From Eq. (14), we observe that ε_i will shift the pilot-tone (peak in the received spectrum) ε_i positions away. We can thus calculate ε_i by:

{\hat{ε}}_{i} = {\begin{matrix} I, (0 \leq I \leq N / 2 - 1) \\ I - N, (N / 2 \leq I \leq N - 1) \end{matrix},

where 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

H (ω) = \exp (- j ω^{2} β_{2} L / 2)

, with L being the entire length of the transmission link and β₂ the GVD coefficient. Thus, the probability of correct detection (P_c) is calculated as:

P_{c} = \prod_{k = 1}^{N - 1} P ({| X_{p} + W_{p} |}^{2} > {| X_{k} + W_{k} |}^{2})

where X_p is a real number representing the DC and X_k 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 (E_p/E_s, E_p = |X_p|², E_s = E[|X_s|²]), SNR (E_s/N₀) and DFT size (N). In Fig. 5 (a) we plot P_c 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 f₀, ε_i or dispersion. For constant modulus format, we can further reduce Eq. (17) to:

Fig. 5 Probability of correct detection versus pilot to average signal power ratio: (a) for different DFT size, SNR, f₀, ε_i and dispersion; (b) for different SNR and laser phase noise.

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P_{c} = P^{N - 1} ({| X_{p} + W_{p} |}^{2} > {| X_{s} + W_{s} |}^{2})

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), P_c is ploted versus E_p/E_s 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.

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

Abstract

1. Introduction

2. System Model

3. Correlation-based fraction part estimation

4. Pilot-tone-assisted integer part estimation

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (18)

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