## Abstract

We report an adaptive weighted channel equalizer (AWCE) for orthogonal frequency division multiplexing (OFDM) and study its performance for long-haul coherent optical OFDM (CO-OFDM) transmission systems. This equalizer updates the equalization parameters on a symbol-by-symbol basis thus can track slight drifts of the optical channel. This is suitable to combat polarization mode dispersion (PMD) degradation while increasing the periodicity of pilot symbols which can be translated into a significant overhead reduction. Furthermore, AWCE can increase the precision of RF-pilot enabled phase noise estimation in the presence of noise, using data-aided phase noise estimation. Simulation results corroborate the capability of AWCE in both overhead reduction and improving the quality of the phase noise compensation (PNC).

©2010 Optical Society of America

## 1. Introduction

OFDM was originally designed for wireless transmission, however, recently it has gained a great deal of attention in optical communications considering its ease of equalization and therefore, robustness with respect to the fiber transmission impairments such as chromatic dispersion (CD) and polarization mode dispersion (PMD) [1,2]. OFDM transmits high-speed serial information through multiple lower-speed sub-channels. This reduction in the baud-rate leads to a reduction in inter-symbol interference (ISI) and therefore a simplification of the equalization process at the receiver.

One of the key features of digital signal processing (DSP) in optical fiber transmission systems is the capability of sending pilot symbols (PSs) which are known to the receiver to provide channel estimation. In optical OFDM, zero forcing estimation (ZFE) method has been often used to extract the channel information and to calculate the equalization parameters [2–5]. For that, by comparing the received PS with the transmitted PS whose subcarriers are known, the channel transfer function for each subcarrier can be extracted by a single complex division. The accuracy of this estimation is often limited by the presence of noise and pattern dependent nonlinearities. To increase the accuracy of channel estimation, usually a time domain averaging method is used in which multiple pilot symbols are used to extract the channel transfer function matrices [2,4,6]. The improvement can also be obtained by averaging in the frequency domain over multiple subcarriers as presented in [6]. To combat dynamic changes in channel characteristics, the PSs are periodically inserted into the OFDM data symbol sequence so that the channel estimation can be performed periodically in order to track the dynamic behavior of the channel. This basically presumes a stationary (non-varying) channel for the block of data symbols between each two consecutive sets of PSs; therefore, PSs should be sent at a speed that is much faster than the speed of significant channel physical changes. In fiber-optic communications, these physical changes are mainly originating from the PMD and the laser phase noise. The PMD varies due to the mechanical and temperature fluctuation on the time scale of a millisecond [6,7] however, there are reports on microsecond PMD changes under severe mechanical stress [8]. In [4], Jansen reported 25.8 Gb/s CO-OFDM by employing 2 pilot symbols for each 25 data symbols resulting in a 4% overhead only due to the PS insertion. Buchali, in [9], reported a PS overhead of 5%. The PS overhead can be alternatively expressed as an increase in the required optical signal-to-noise ratio (OSNR), defined as ΔOSNR_{PS}

The output of a single-frequency laser is not monochromatic and exhibits some phase noise which results in a finite linewidth of laser output, normally ranging from 100 kHz to several megahertz [2]. Therefore, laser phase noise needs to be tracked on a symbol-by-symbol basis. By using the pilot subcarriers that are inserted in every OFDM symbol, such a fast time variation of the channel can be compensated. The pilot subcarriers are equally distributed over the OFDM spectrum and their state of modulation is known at the receiver. Shieh, in [10], studied the number of required pilot subcarriers for CO-OFDM systems. In [4,11], the authors proposed RF-pilot enabled PNC for CO-OFDM systems while ideally no extra optical bandwidth needs to be allocated. With this technique, PNC is realized by placing an RF-pilot tone in the middle of the OFDM band at the transmitter that is subsequently used at the receiver to revert phase noise impairments. However, there is a trade-off on the RF-pilot power: for low-power RF-pilot, the accumulated spontaneous emission (ASE) noise reduces the degree with which the phase noise can be compensated for, whereas for high-power RF-pilot the OSNR of the OFDM signal becomes too low and the received signal quality degrades [4,11].

Optical OFDM and its impressive channel estimation capacity provide the opportunity to exploit more complex channel estimation techniques to reach higher throughput and performance. In this paper, we introduce a data-aided equalizer, AWCE, for optical OFDM systems and study its performance for the CO-OFDM transmission system via numerical simulations. This equalizer updates the equalization parameters on a symbol-by-symbol basis and operates based on decision-directed channel estimation which is an alternative to the pilot-assisted channel estimation: it uses previous demodulator decisions to help estimate channel fading factors, happened during the current symbol period. In decision-directed channel estimation, since the equalization parameters are not from pilot symbols, they are not as reliable, so the estimator’s performance may suffer from error propagation. However, it does not require the overhead of the pilot-assisted estimators. To overcome the shortcomings of decision-directed and pilot-assisted channel estimators, we combine the two methods as suggested in [12,13], therefore, AWCE utilizes both receiver decisions and pilot symbols so can produce better results when the receiver decisions are correct. In this way, by updating the equalization parameters on a symbol-by-symbol basis, receiver can track the slight drifts in the optical channel. Consequently, this allows the system to increase the periodicity of sending the PSs which leads into the overhead reduction. AWCE can also improve the performance of the RF-pilot enabled PNC by providing a more accurate estimation using both data-aided and RF-pilot enabled phase noise estimation techniques. It is notable that since AWCE operates on a symbol-by-symbol basis and regarding the fact that OFDM symbol rate can be much lower than the actual bit rate, the implementation of AWCE does not necessarily require very high speed electronics. Numerical simulations show superior performance of AWCE for a single-channel 40 Gb/s CO-OFDM transmission over 2000 km uncompensated single-mode fiber (SMF) in the presence of noise, nonlinearities, high-speed PMD changes and laser phase noise.

This paper is structured as follows. In section 2, we explain the AWCE principles, adaptive weighting parameter and how it optimizes the performance of the equalizer. In section 3, we review the CO-OFDM transmission link and numerically verify the performance of AWCE in this system. Section 4 concludes this paper.

## 2. Theory of operation

Assume *n* denotes the index for the received symbol (time index) and *k* is the index for the OFDM subcarrier (frequency index). The subcarrier-specific received complex value symbol, ${R}_{n,k}$, is equalized by applying a zero-forcing technique based on the previously estimated transfer factor, ${\tilde{H}}_{n-1,k}$, that is taken as a prediction of the current channel transfer factor:

We call this new channel transfer factor as the ideal channel transfer factor since if we knew it before demodulation and could apply it as the denominator in Eq. (1), then a perfect equalization and decision making would be achieved. ${\widehat{H}}_{n,k}$ is basically the updated version of ${\tilde{H}}_{n-1,k}$ and includes the information of the optical channel drifts in the time interval of the symbol number *n*. A low-pass filter (LPF) can be applied to ${\widehat{H}}_{n,k}$ to suppress the high- frequency detected noise without applying time averaging over several channel transfer functions in each block as presented in [14]. At this point, a more accurate estimation of laser phase noise can be provided that can subsequently lead into better PNC. For that, one extra step is required in which we average the difference between the phase term of the ideal channel transfer function and the phase term of the previously estimated transfer function over all subcarriers:

*N*is the total number of OFDM subcarriers including all data subcarriers. Equation (4) tries to extract the phase drift of the optical channel in the time interval equal to the duration of one OFDM symbol and assumes that the drift due to the PMD is negligible. This is a good assumption since PMD variations are low-speed (in the range of kHz) in comparison to the typical CO-OFDM symbol rate. Now, $\Delta {\varphi}_{pilot,n}$ in Eq. (2) can be replaced by $\Delta {\varphi}_{AWCE,n}$ to provide a more accurate PNC and the new resulting ${\widehat{S}}_{n,k}$ is again sent to the demodulator for better decision making. As we see in Eq. (4), since $\Delta {\varphi}_{AWCE,n}$ is calculated by averaging over all OFDM sub-channels, it is capable of suppressing the effect of ASE noise and therefore, providing a more accurate phase noise estimation. However, because the calculation of Eq. (4) is done after the demodulation and is dependent on Eq. (2), a fairly reliable RF-pilot enabled phase noise estimation is necessary to prevent error propagation.

To update the equalization parameters for the next received symbol, we apply a simple recursive filtering procedure using both the previously estimated channel transfer function, ${\tilde{H}}_{n-1,k}$, and the ideal channel transfer function, ${\widehat{H}}_{n,k}$. The recursion is performed independently for each subcarrier and a time-domain correlation is implicitly utilized. No channel statistics such as correlation function or signal-to-noise ratio (SNR) are needed. The estimated channel response for *n*
^{th} received symbol can be updated as:

*γ*is the adaptive weighting parameter and can take any value between 0 and 1. A large value of

*γ*boosts the role of previously estimated channel transfer function, ${\tilde{H}}_{n-1,k}$, in Eq. (5) and conversely, a smaller value of

*γ*increases the effect of ideal channel transfer function, ${\widehat{H}}_{n,k}$.

*γ*can control the recursion and prevents error propagation. If the previous equalization is fairly accurate and a good decision is made then obviously ${\tilde{H}}_{n-1,k}$was precise enough and we would like to boost its role in Eq. (5) and a large value of

*γ*is desired. However, if the previous equalization is not that accurate then we are interested in increasing the role of ${\widehat{H}}_{n,k}$to update the equalization parameters therefore, a smaller value for

*γ*is preferred. As we see,

*γ*should be defined in a way that can assess the quality of the previous equalization. Due to the noise, nonlinearities, inaccurate PNC and small PMD drifts in the optical channel, the equalized received constellation points do not exactly lay on the ideal constellation points, defining error vectors. We use these error vectors to assess the precision of the previous equalization. Keeping the above-mentioned requirements in mind, we define

*γ*as:where $\overrightarrow{r}$ and

*d*respectively denote the error vector of one received constellation point and the distance between ideal constellation point and the closest decision line, as it is shown in Fig. 1(a) for the case of quadrature phase-shift keying (QPSK) constellation. The $avg$ implies an averaging over all the equalized constellation points in each received symbol. Figure 1(b) and Fig. 1(c) show two different scenarios. When previously estimated channel transfer function is accurate, we expect a random distribution of the equalized received constellation points around the ideal constellation points due to the noise as depicted in Fig. 1(b). In this case, the term $\left|avg\left\{\overrightarrow{r}/d\right\}\right|$ in Eq. (6) limits to zero and a large value of

*γ*is obtained, increasing the role of ${\tilde{H}}_{n-1,k}$. In Fig. 1(c), the scenario in which the previous estimation is not accurate enough is shown. In this case, the term $\left|avg\left\{\overrightarrow{r}/d\right\}\right|$ in Eq. (6) increases which makes the value of

*γ*decrease. This consequently boosts the role of ${\widehat{H}}_{n,k}$ in Eq. (5) to track the drifts.

The adaptive weighting parameter, *γ*, can also be used in a different way: it can be first transformed by a function *f* and then, applied to the Eq. (5). *f* can be any non-decreasing function such as a sigmoid nonlinear function as been used for reliability factors of weighted decision feedback equalizers in [15]. Regarding the fact that our main focus in this paper is the concept of the data-aided equalizer and for the sake of simplicity, we use the identity function as *f*.

## 3. Simulation of AWCE performance in CO-OFDM transmission system

Figure 2
depicts the simulated transmission link setup. Simulations are performed in MATLAB. The principle of operation of optical OFDM is well-known and the specific usage of each block diagram can be found elsewhere [1–4]. The original data at 40 Gb/s were first divided and mapped onto 1024 frequency subcarriers with QPSK modulation, and subsequently transferred to the time domain by an IFFT of size 2048 while zeros occupy the remainder. A cyclic prefix of length 350 is used to accommodate dispersion. Following this, an up-conversion stage shifts the OFDM signal to 10-30 GHz band. The OFDM signal is electrically up-converted using an electrical intermediate frequency (IF) carrier to modulate the signal upon applying a complex electrical I/Q mixer. To insert the RF-pilot in the middle of the OFDM signal for the phase noise estimation, we set the first OFDM channel to 0 and apply a small DC offset in I and Q tributaries of the IQ-mixer [4]. The DC offset will be up-converted with the OFDM signal and as a result a small RF-pilot will be present at the IF frequency. In Fig. 3
, the electrical spectrum after up-conversion is shown in which the RF-pilot can be seen. The ratio between RF-pilot power and the power of all subcarriers is called as the pilot to signal power ratio (PSR). The resulting up-converted electrical OFDM signal is then electro-optic converted using a chirp-less Mach-Zehnder modulator (MZM). Then, the transmitted signal is optically filtered to suppress the sideband. Transmission link consists of 25 uncompensated single mode fiber (SMF) spans with dispersion parameter of 17 ps/nm.km, nonlinear coefficient of 1.5 W^{−1}.km^{−1} and loss parameter of 0.2 dB/km. Spans are 80 km long and separated by erbium doped fiber amplifiers (EDFAs) with the noise figure of 6 dB. Injected power to each fiber span is set to −4 dBm. Split step Fourier method is used to simulate the optical fiber medium. At the optical receiver, an optical filter with the bandwidth of 0.4 nm is applied to reject out-of-band ASE noise. The receiver is based on heterodyne CO-OFDM scenario in which the optical OFDM signal is converted into a real valued electrical OFDM signal at an IF. Using a subsequent electrical I/Q demodulator, the real and imaginary components are available in the baseband.

To mimic the continuous time characteristics of the optical channel, 50 different random sets of time-domain realizations of laser phase noise and PMD have been simulated. The laser phase noise is modeled using the well-established model, described in [16]. This model assumes that the laser phase undergoes a random walk where the steps are individual spontaneous emission events which instantaneously change the phase by a small amount in a random way. We considered a laser linewidth of 100 kHz for both transmitter and receiver sides. The dynamic response of the PMD is simulated using dynamic wave plate model as proposed in [17]. This model generates a continuous PMD variation and correlation between adjacent time samples. The PMD coefficient of the fiber medium was set to 0.5 ps/√km and 1600 wave plates were taken into account. Other simulation parameters were properly adjusted to emulate a fast PMD speed in the range of microsecond-scale to millisecond-scale.

As discussed in section 2, AWCE is capable of improving the performance of PNC. Figure 4
compares the BER performance of AWCE versus PSR, with and without data-aided PNC for two different received OSNR values of 13 dB and 16 dB. In this simulation, each OFDM block consists of 2 pilot symbols and 62 data symbols (an overhead of 3% due to PS insertion). In both received OSNR cases, for lower values of PSR, data-aided PNC slightly improves the performance however, as PSR increases it significantly increases the precision of phase noise estimation and therefore, better BER results are obtained. This is due to the fact that data-aided PNC relies on the correct decision making and a fairly good RF-pilot enabled phase noise estimation is required to achieve a pronounced improvement. As one can see, to achieve the forward-error-correction (FEC) threshold, the commonly-reported BER value of 10^{−3}, AWCE with data-aided PNC requires 1 and 2.2 dB less RF-pilot power for the received OSNR values of 16 and 13 dB, respectively, showing higher improvement for the noisier scenario.

In Fig. 5
, we investigate the capability of AWCE on PS overhead reduction. For that, we fixed the PSR at −10 dB for all simulations. As one can see for the case of ZFE, when we reduce the PS overhead from 3% (blue curve) to 0.3% (red curve), the signal quality degrades. This degradation introduces an OSNR penalty of 1.2 dB at the FEC threshold. However, when we apply AWCE, without data-aided PNC, to the signal with PS overhead of 0.3%, almost the same performance as ZFE with PS overhead of 3% can be achieved (see the black curve in Fig. 5). This demonstrates that AWCE can significantly reduce the PS overhead, here by a factor of 10, while providing the same signal quality. This improvement is due to the fact that AWCE updates the equalization parameters on a symbol-by-symbol basis and can track slight drifts in the optical channel. Moreover, AWCE with data-aided PNC can further improve the signal quality. As seen by the green curve in Fig. 5, to achieve the BER of 10^{−3}, AWCE with data-aided PNC and PS overhead of 0.3% requires 2.9 and 1.8 dB less OSNR than ZFE with PS overhead of 0.3% and 3%, respectively. This is due to the fact that AWCE not only tracks the PMD drifts but also provides enhanced phase noise estimation and consequently more accurate compensation.

Figure 6 compares the received constellation of QPSK signal after equalization by ZFE and AWCE with data-aided PNC. The overhead of PS insertion, the received OSNR and the PSR for both cases are set to 0.3%, 16 dB and −10 dB, respectively. It can be clearly observed that AWCE provides better equalization and subsequently separated constellation points.

## 4. Conclusion

We reported a novel data-aided equalizer, AWCE, for long-haul CO-OFDM transmission systems and numerically studied its transmission performance over a 2000 km uncompensated link at 40 Gb/s. This equalizer improves the quality of PNC by combing the data-aided and RF-pilot enabled phase noise estimation techniques. The BER of 10^{−3} for the received OSNR values of 16 and 13 dB was achieved by 1 and 2.2 dB less RF-pilot power, respectively. Furthermore, AWCE combines the pilot-assisted and decision-directed estimation techniques and by doing that, is capable of reducing the PS overhead. A PS overhead reduction by a factor of 10 was demonstrated in the presence of high-speed PMD changes. Simulations confirmed that to achieve the BER of 10^{−3}, by using a PS overhead of 0.3% and PSR of −10 dB, AWCE with data-aided PNC requires 2.9 dB less OSNR than ZFE.

## Acknowledgements

The authors would like to thank Dr. Javad Haghighat for his collaboration and Dr. Pegah Seddighian and Prof. Lawrence R. Chen for the fruitful discussions. The authors gratefully acknowledge the financial support from the NSERC/Bell Canada Industrial Research Chair.

## References and links

**1. **W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express **16**(2), 841–859 (2008). [CrossRef] [PubMed]

**2. **W. Shieh, X. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come?” J. Opt. Netw. **7**(3), 234–255 (2008). [CrossRef]

**3. **I. B. Djordjevic and B. Vasic, “Orthogonal frequency division multiplexing for high-speed optical transmission,” Opt. Express **14**(9), 3767–3775 (2006). [CrossRef] [PubMed]

**4. **S. L. Jansen, I. Morita, T. Schenk, N. Takeda, and H. Tankada, “Coherent Optical 25.8-Gb/s OFDM Transmission Over 4160-km SSMF,” J. Lightwave Technol. **26**(1), 6–15 (2008). [CrossRef]

**5. **S. L. Jansen, I. Morita, T. C. W. Schenk, N. Takeda, and H. Tanaka, “Theoretical and experimental investigations of direct-detected RF-tone assisted optical OFDM systems,” J. Lightwave Technol. **27**(10), 1332–1339 (2009). [CrossRef]

**6. **X. Liu and F. Buchali, “Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM,” Opt. Express **16**(26), 21944–21957 (2008). [CrossRef] [PubMed]

**7. **M. Karlsson, J. Brentel, and P. A. Andrekson, “Long-term Measurement of PMD and Polarization Drift in Installed Fibers,” J. Lightwave Technol. **18**(7), 941–951 (2000). [CrossRef]

**8. **P. Krummrich, and K. Kotten, “Extremely fast (microsecond timescale) polarization changes in high speed long haul WDM transmission systems,” in *Optical Fiber Communication Conference*, OSA Technical Digest Series (CD) (Optical Society of America, 2004), paper FI3.

**9. **F. Buchali, R. Dischler, and X. Liu, “Optical OFDM: A Promising High-Speed Optical Transport Technology,” Bell Labs Tech. J. **14**(1), 125–148 (2009). [CrossRef]

**10. **X. Yi, W. Shieh, and Y. Tang, “Phase Estimation for Coherent Optical OFDM,” IEEE Photon. Technol. Lett. **19**(12), 919–921 (2007). [CrossRef]

**11. **S. L. Jansen, I. Morita, N. Takeda, and H. Tanaka, “20-Gb/s OFDM transmission over 4,160-km SSMF enabled by RF-Pilot tone phase noise compensation,” in *Optical Fiber Communication Conference*, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper PDP15.

**12. **M. Rim, “Optimally combining decision-directed and pilot-symbol-aided channel estimation,” Electron. Lett. **39**(6), 558–560 (2003). [CrossRef]

**13. **J. Ran, R. Grunheid, H. Rohling, E. Bolinth, and R. Kern, “Decision-directed channel estimation method for OFDM systems with high velocities,” in *Proceedings of IEEE Vehicular Technology Conference*, (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 2358–2361.

**14. **M. E. Mousa Pasandi, J. Haghighat, and D. V. Plant, “Adaptive weighted channel equalizer for direct-detection optical OFDM transmission systems,” in *Proceedings of IEEE Photonics Society Summer Topicals’09*, (Institute of Electrical and Electronics Engineers, New York, 2009), pp. 85–86.

**15. **J. Palicot and A. Goupil, “Performance analysis of the weighted decision feedback equalizer,” Signal Process. **88**(2), 284–295 (2008). [CrossRef]

**16. **R. W. Tkach and A. R. Chraplyvy, “Phase Noise and Linewidth in an InGaAsP DFB Laser,” J. Lightwave Technol. **4**(11), 1711–1716 (1986). [CrossRef]

**17. **C. Xie, D. Werner, and H. F. Haunstein, “Dynamic Performance and Speed Requirement of Polarization Mode Dispersion Compensators,” J. Lightwave Technol. **24**(11), 3968–3975 (2006). [CrossRef]