Zero-overhead phase noise compensation via decision-directed phase equalizer for coherent optical OFDM

Mohammad E. Mousa-Pasandi; David V. Plant

doi:10.1364/OE.18.020651

1. Introduction

Following the recent surge of interest in digital signal processing (DSP) for optical fiber communications, the CO-OFDM has been intensively investigated as a powerful scenario for the future uncompensated transmission links [1,2]. One of the key features of DSP is the capability of sending pilot symbols (PSs) and pilot subcarriers (PSCs) which are known to the receiver to provide data-aided channel estimation [1–3]. To combat dynamic changes in channel characteristics, i.e. polarization mode dispersion (PMD), and to provide synchronization, the PSs are periodically inserted into the OFDM data symbol sequence. PSs have to be sent at a speed that is much faster than the speed of significant channel physical changes [3]. A PS overhead of 3% to 5% is often reported for CO-OFDM transmission systems [2–5]. However, the performance of CO-OFDM transmission links significantly suffers from the laser phase noise which requires not only tracking on a symbol-by-symbol basis but also extra equalization algorithms. By using the PSCs that are inserted in every symbol, such a fast time variation in the optical channel can be compensated [1,6]. An overhead of 5% to 10% is expected due to the PSC insertion. In [4,7], the authors proposed RF-pilot enabled PNC for CO-OFDM while ideally no extra optical bandwidth needs to be allocated. In this technique, PNC is realized by placing an RF-pilot tone in the middle of the OFDM signal band at the transmitter that is subsequently used at the receiver to revert the phase noise impairments. Inserting the RF-pilot typically results in 7% to 10% of power overhead [4,8].

We recently proposed a data-aided phase equalizer (DAPE) based on the combination of decision-directed and data-aided estimation schemes to increase the accuracy of RF-pilot enabled PNC or equivalently to reduce the required power overhead [3,8]. In this paper, motivated by recent progress in external-cavity laser (ECL) technology in manufacturing low linewidth telecom lasers, we investigate the feasibility of a pure decision-directed phase noise estimation and compensation for long-haul CO-OFDM transmission systems. This means that there will be no need for any extra overhead due to RF-pilot or PSC insertion. Decision-directed phase equalizer (DDPE) updates the equalization parameters, initially acquired by PSs, on a symbol-by-symbol basis after an initial decision making and then retrieves an estimation of the phase noise value for the time interval of one OFDM symbol by extracting and averaging the phase drift of all OFDM sub-channels. Subsequently, a second equalization is performed using this estimated phase noise value and afterwards, the equalized symbols are sent to the final decision making stage for detection. Considering the fact that decision-directed estimation algorithms are known to suffer from error propagation [9,10], we numerically study the safe range of laser linewidth that is required to guarantee the error-free transmission for a single-channel long-haul CO-OFDM system. For doing that, the bit-error-rate (BER) performances of DDPE and the CO-OFDM conventional equalizer (CE) after transmission over 2000 km of uncompensated SMF at 40 Gb/s are compared. We show that for the laser linewidth of 60 kHz and less, DDPE provides similar or even better performance than the CE with 5% PSC overhead. Moreover, for both DDPE and CE, the effect of fiber nonlinearity on the quality of the received signal is assessed and compared at two different received optical signal-to-noise ratio (OSNR) values. We also demonstrate that DDPE is capable of operating in conjunction with the digital back-propagation (BP) nonlinearity compensation scheme.

It is notable that since DDPE operates on a symbol-by-symbol basis and considering that OFDM symbol rate can be much lower than the actual transmitted bit rate, the DDPE implementation does not necessarily require very high-speed and high power consuming electronics. As the complexity analysis, we provide a brief analytical study to compare the number of required complex multiplications per bit in case of DDPE and CE and demonstrate that the extra complexity due to DDPE for the entire practical oversampling range of 1.2 to 2 and FFT size of 256 to 4096 is limited to 28%. For the particular parameters of our simulations in this paper (oversampling ratio of 2, FFT size of 2048 and PS overhead of 3%), DDPE is only about 15% more complex than the CE with 5% PSC overhead.

This paper is structured as follows. We explain the DDPE principles in section 2. In section 3, we review the CO-OFDM transmission link and numerically study the performance of DDPE. In section 4, the complexity of DDPE is studied and section 5 concludes the paper.

2. DDPE description

Figure 1 depicts the detailed diagram of DDPE, consisting of two equalization stages and Fig. 2 shows the corresponding constellation points after each equalization stage, for the case of quadrature phase shift keying (QPSK) modulation format. The first equalization stage is in charge of equalizing the effect of the dispersion of the optical channel, similar to the CE as reported in [1,2]. The equalization parameters of the first equalization stage are initially acquired by using PSs and then get updated on a symbol-by-symbol basis using the initial decision making stage based on the decision-directed estimation scheme. After the initial decision making stage, the phase rotation angle of the constellation points, due to the laser phase noise, is extracted for each received symbol. By using this angle, the second equalization stage compensates the rotation of constellation points due to the phase noise. Afterward, the equalized symbol is sent to the final decision making stage for detection. In this technique, as long as the amount of rotation does not result in incorrect initial decision making for the majority of the constellation points in each received symbol, the phase noise can be fully retrieved and compensated. As one can expect, DDPE performance for denser constellation formats is more sensitive to the optical channel impairments.

Fig. 1 DDPE diagram.

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Fig. 2 Graphical illustration of the equalized received constellation points after (a) the first and (b) the second equalization stage.

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Assume n and k denote the indexes for the received symbol (time index) and the OFDM subcarrier (frequency index), respectively. The subcarrier-specific received complex value symbol, $R_{n, k}$ , is equalized, in the first equalization stage, by applying the 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:

{\hat{S}}_{n, k} = R_{n, k} / {\tilde{H}}_{n - 1, k}

where

{\hat{S}}_{n, k}

is the subcarrier-specific equalized complex value symbol.

{\tilde{H}}_{1, k}

is normally derived by using the PSs that are inserted at the beginning of each block of OFDM data symbols.

{\hat{S}}_{n, k}

is then detected by the demodulator as the first decision making stage. Presuming that the decision was correct, the received symbol,

R_{n, k}

, can be further divided by the detected symbol,

{\bar{S}}_{n, k}

, in order to calculate a new channel transfer factor,

{\hat{H}}_{n, k}

:

{\hat{H}}_{n, k} = R_{n, k} / {\bar{S}}_{n, k}

We call this new channel transfer factor the ideal channel transfer factor since if we knew it before demodulation and could apply it as the denominator in Eq. (1), then perfect equalization and decision making would be achieved.

{\hat{H}}_{n, k}

is basically the updated version of

{\tilde{H}}_{n - 1, k}

and includes the information of the optical channel drifts i.e. laser phase noise in the time interval of the symbol number n. At this point, an estimation of laser phase noise can be provided. For that, we average the difference between the phase term of the ideal channel transfer factor and the phase term of the previously estimated transfer factor over all OFDM sub-channels:

Δ ϕ_{D D P E, n} = (\sum_{i = 1}^{N} (\arg {{\hat{H}}_{n, i}} - \arg {{\tilde{H}}_{n - 1, i}})) / N

where N is the total number of subcarriers. Equation (3) tries to extract the phase drift of the optical channel in the time interval of the symbol number n, assuming that the drift due to other impairments such as 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, as the second equalization stage,

{\hat{S}}_{n, k}

can be further divided by

e^{j Δ ϕ_{D D P E, n}}

to provide PNC and the new resulting equalized symbol is again sent to the demodulator for the final decision making. Since the calculation of Eq. (3) is done after the demodulation and is dependent on Eq. (1), a fairly reliable initial equalization is necessary to prevent error propagation. Therefore, for relatively high laser phase noise scenarios this technique is not capable of proper PNC and requires the assistance of the PSCs or the RF-pilot, equivalently overhead, to avoid error propagation, as been presented in [3,8]. However, if the laser linewidth is less than a specific threshold, DDPE can independently estimate and compensate the phase noise without requiring any overhead.

At this point, to update the equalization parameters for the next received symbol, we apply a simple recursive filtering procedure using both the previously estimated channel transfer factor, ${\tilde{H}}_{n - 1, k}$ , and the ideal channel transfer factor, ${\hat{H}}_{n, k}$ . 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 subcarrier-specific channel transfer factor for the n ^th received symbol can then be updated as:

{\tilde{H}}_{n, k} = (1 - γ) {\hat{H}}_{n, k} + γ {\tilde{H}}_{n - 1, k} e^{Δ ϕ_{D D P E, n}}

where γ is the weighting parameter and can take any value between 0 and 1. A large value of γ boosts the role of previously estimated channel transfer factor,

{\tilde{H}}_{n - 1, k}

, while conversely, a smaller value of γ increases the effect of ideal channel transfer factor,

{\hat{H}}_{n, k}

. γ controls the recursion and can either be a fixed or an adaptive value [3]. In this study, we chose a fix γ value of 0.15.

3. Simulation of DDPE performance in CO-OFDM transmission system

Figure 3 depicts the simulated transmission link setup. Simulations are performed in MATLAB. The principle of operation of CO- OFDM is well-known and the specific usage of each block diagram can be found elsewhere [1–3]. The original data at 40 Gb/s were first divided and mapped onto 1024 frequency subcarriers with QPSK modulation format, 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. The resulting electrical OFDM data signal is then electro-optic converted using an IQ Mach-Zehnder modulator (IQ-MZM). The optical transmission link consists of 25 uncompensated SMF spans with dispersion parameter of 17 ps/nm.km, nonlinear coefficient of 1.5 W⁻¹.km⁻¹, PMD coefficient of 0.5 ps/√km 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. Split step Fourier method is used to simulate the optical fiber medium. The laser phase noise is modeled using the well-established model, described in [11]. 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. For each simulation point, 100 different random sets of time-domain realizations of laser phase noise have been simulated to mimic the continuous time characteristics of the optical channel. At the optical receiver, an optical filter with the bandwidth of 0.4 nm is applied to reject the out-of-band ASE noise. The receiver is based on intradyne CO-OFDM scenario in which the local oscillator (LO) wavelength is close to the transmitter wavelength. The OFDM signal then beats with the LO signal in an optical 90° hybrid to obtain the I and Q components of the signal. In this paper, each OFDM block consists of 2 pilot and 62 data symbols resulting in 3% of PS overhead.

Fig. 3 Simulation setup.

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In Fig. 4 , an example of the received constellation points for the cases of (a) no PNC and (b) PNC using DDPE are shown. Lasers with the linewidth of 30 kHz are employed at both transmitter and receiver sides and the launch power to each fiber span is set to −4 dBm. As one can see, when no PNC is applied, the rotation of constellation points due to the phase noise results in a poor separation of constellation points however, by using DDPE, all constellation points can be perfectly separated. This illustrates that DDPE is capable of compensating the effect of laser phase noise. To characterize the DDPE performance, we investigate the BER of the received signal versus the received OSNR for different laser linewidth values and compare it to the performance of CE at the same raw bit rate including a PSC overhead of 5%. For this study, we set the fiber launch power to −4 dBm and consider an identical linewidth for both lasers at transmitter and receiver sides. As seen in Fig. 5 , DDPE provides a better performance than CE for the laser linewidth of 20 kHz, 40 kHz and 60 kHz when the received OSNR is higher than 12 dB. This slightly better signal quality is coming from the fact that the recursive filtering of Eq. (4) suppresses the effect of ASE noise on the estimated channel transfer factor resulting in more accurate equalization. For the case of laser linewidth of 80 kHz, as seen in Fig. 5(d), the DDPE performance is severely compromised as an OSNR penalty of 2 dB is observed to achieve the BER of 10⁻³. This is due to the strong effect of error propagation that a pure decision-directed equalizer, i.e. DDPE, does not perform as reliable as a data-aided one, i.e. CE. Figure 6 compares the BER performance of DDPE for different laser linewidth values in one figure. As we expected, DDPE cannot provide a good PNC due to the error propagation for relatively higher phase noise scenarios, i.e. laser linewidths of 80 kHz and 100 kHz. However, for relatively lower phase noise scenarios, it provides a good equalization and the forward-error-correction (FEC) threshold, the commonly-reported BER value of 10⁻³, is achieved at the OSNR values of 11.6 dB, 11.9 dB and 12.7 dB for the laser linewidth values of 20 kHz, 40 kHz and 60 kHz, respectively. The slight OSNR penalty between the performance of 20 kHz, 40 kHz and 60 kHz scenarios is attributed to the inter-carrier interference (ICI) originated from the cross-leakage between subcarriers due to the phase noise, as elaborated in [2,6]. Similarly, due to the cross-leakage between subcarriers, an OFDM symbol with relatively shorter duration shows better performance against phase noise. Therefore, by increasing the number of filled subcarriers or equivalently increasing the oversampling ratio, slightly better performance is expected.

Fig. 4 An example of received constellation points at 40 Gb/s after 2000 km transmission using laser linewidth of 30 kHz (a) without any PNC (b) with zero-overhead PNC based on DDPE.

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Fig. 5 The BER performance of DDPE, blue solid curves, and CE with 5% PSC overhead, red dashed curves, for laser linewidth of (a) 20 kHz, (b) 40 kHz, (c) 60 kHz and (d) 80 kHz.

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Fig. 6 The BER Performance of DDPE for different laser linewidth values.

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Regarding the fact that fiber nonlinearity is one of the main impairments in CO-OFDM transmission systems [1–3], we investigate the DDPE performance versus fiber launch power to assess the behavior of our proposed PNC technique in the presence of strong nonlinearity. Figure 7 compares the BER performance of DDPE and CE versus fiber launch power at two received OSNR values of 13 dB and 15.3 dB. In this study, the laser linewidth is set to 60 kHz for all scenarios. As seen in Fig. 7, although both equalizers suffer from strong nonlinearity and the signal quality significantly degrades as the launch power increases, DDPE shows slightly higher sensitivity to nonlinearity, as can be observed for launch power range of higher than −4 dBm. However, its performance is similar or even better than CE for the launch power range of less than −4 dBm. To characterize the behavior of DDPE and CE in conjunction with the nonlinearity compensation schemes, we simulate the same scenario in the presence of digital back-propagation algorithm, BP [12]. In our study, the BP employs two steps per fiber span. As we see, in the presence of the BP compensation scheme, DDPE performs slightly better than CE and can support the error-free threshold of 10⁻³ even for a high launch power of up to 0 dBm for both received OSNR values of 13 dB and 15.3 dB.

Fig. 7 The BER performance of DDPE and CE versus launch power with and without BP nonlinearity compensation scheme at two different received OSNR values of 13 dB and 15.3 dB. The linewidth of the lasers at both transmitter and receiver sides is set to 60 kHz.

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4. System complexity

The complexity of an equalization technique directly affects the implementation cost of the transmission link regarding the required electronic hardware and the power consumption [13]. In this section, a brief analysis of the complexity of DDPE and CE is provided. The complexity of each equalizer is evaluated in terms of the number of required complex multiplications per bit taking into account the fast Fourier transform (FFT) operation, the channel estimation, the equalization and the updating process of the equalization parameters. In this study, the same complexity for multiplication and division is considered. The oversampling of OFDM signal is denoted by $N / U$ where N and U are the FFT size and the number of used subcarriers, respectively. M represents the number points in the signal constellation hence, every symbol contains $U \log_{2} M$ useful bits by assuming that all data subcarriers are using the same modulation format.

For every received symbol, CE applies one FFT which needs $N / 2 \times \log_{2} N$ complex multiplications [13]. As the channel estimation, we assume that CE periodically estimates the channel transfer factors every $1 / R_{P S}$ symbols using one known OFDM symbol. Since the channel transfer factors must be evaluated for every used subcarrier, its calculation requires $U \times R_{P S}$ complex multiplications per symbol. CE also estimates the phase noise drift only for data symbols by using $U \times R_{P S C}$ known subcarriers where $R_{P S C}$ is the PSC overhead ratio. As a result, it requires $U \times R_{P S C} \times (1 - R_{P S})$ complex multiplications per symbol to estimate the phase noise. As the equalization operation, one complex multiplication is required for every used subcarrier only for every data symbol resulting in $U \times (1 - R_{P S})$ complex multiplications. Moreover, for every data symbol, the equalization parameters need to be updated using the estimated phase noise so $U \times (1 - R_{P S})$ more complex multiplications are required. Table 1 summarizes the number of required complex multiplications per symbol for the subsystems using CE. Therefore, the total number of complex multiplications per bit for CE can be expressed as

Table 1. Number of required complex multiplications per symbol for CE.

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\begin{array}{l} N_{C E} = (\frac{N}{2} \log_{2} N + U \times (1 - R_{P S}) + U \times R_{P S} + U \times R_{P S C} \times (1 - R_{P S}) + U \times (1 - R_{P S})) / U \log_{2} M = \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} (\frac{N}{2 U} \log_{2} N + R_{P S} + (2 + R_{P S C}) \times (1 - R_{P S})) / \log_{2} M \end{array}

For the case of DDPE, similar to the CE, the receiver needs one FFT operation for every received symbol. As the channel estimation, the channel transfer factors are evaluated and updated for every received symbol. Considering that the channel transfer factors are evaluated for every used subcarrier, the channel estimation requires U complex multiplications every symbol. As the equalization operation, in each equalization stage, one complex multiplication is applied for every used subcarrier for every data symbol, resulting in $2 \times U \times (1 - R_{P S})$ complex multiplication per symbol. As the updating process of the equalization parameters, one more complex multiplication is needed for every used subcarrier in every data symbol. This is due to the recursive filtering procedure, as seen in the second term of Eq. (4), and results in $U \times (1 - R_{P S})$ complex multiplications per symbol. Table 2 summarizes the number of required complex multiplications per symbol for the subsystems of DDPE. Consequently, the total number of complex multiplications per bit for DDPE can be expressed as

Table 2. Number of required complex multiplications per symbol for DDPE.

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\begin{array}{l} N_{D D P E} = (\frac{N}{2} \log_{2} N + U + 2 \times U \times (1 - R_{P S}) + U \times (1 - R_{P S})) / U \log_{2} M = \\ \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} (\frac{N}{2 U} \log_{2} N + 1 + 3 \times (1 - R_{P S})) / \log_{2} M \end{array}

By comparing Eq. (5) and Eq. (6) and considering the fact that $R_{P S}$ and $R_{P S C}$ are in the range of 2% to 4% and 5% to 10%, respectively for a typical CO-OFDM system [1–3], one can mathematically see the extra complexity of DDPE. Figure 8 shows the percentage of DDPE extra complexity in terms of number of required complex multiplications versus FFT size for different oversampling ratios. The PS overhead of 3%, PSC overhead of 5% and the modulation format of QPSK are considered. As seen in Fig. 8, when FFT size and oversampling ratio increase, the extra complexity of DDPE versus CE decreases. Moreover, although the DDPE is naturally a more complex technique, the extra complexity for the practical oversampling range of 1.2 to 2 and FFT size of 256 to 4096 is limited to 28%. For the particular set of parameters in our simulations in section 3, the oversampling ratio of 2 and the FFT size of 2048, an extra complexity of only 15% is experienced.

Fig. 8 The percentage of DDPE extra complexity to CE (in terms of the number of required complex multiplications per bit) versus the FFT size for different oversampling ratios. The PS overhead is 3%. The PSC overhead of CE is 5%.

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5. Conclusion

We reported the feasibility of zero-overhead PNC based on decision-directed phase equalization (DDPE) for CO-OFDM transmission systems and numerically investigated its performance at 40 Gb/s after 2000 km transmission over uncompensated SMF. We compared the BER performance of DDPE and the CO-OFDM conventional equalizer (CE) for different laser linewidth values. By comparing the DDPE and CE, we demonstrated that DDPE can perform as reliable as CE for the laser linewidth range of less than 60 kHz at a launch power of −4 dBm. We also compared their performances against fiber launch power and showed that although DDPE is more vulnerable than CE to higher launch power settings, it can be adopted in the transmission systems employing the digital back-propagation (BP) nonlinearity compensation scheme. Moreover, the complexity of DDPE and CE in terms of the number of required complex multiplications per bit was analytically studied showing that the extra complexity due to DDPE is limited to 28% for the entire practical range of oversampling and FFT size of typical CO-OFDM systems. Considering the recent advances in telecom laser technology and by using this novel decision-directed PNC technique, CO-OFDM systems can remove the required overhead for PNC to achieve higher throughput.

Acknowledgements

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. M. E. Mousa-Pasandi and D. V. Plant, “Data-aided adaptive weighted channel equalizer for long-haul optical OFDM transmission systems,” Opt. Express 18(4), 3919–3927 (2010). [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. 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]

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

7. 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.

8. M. E. Mousa-Pasandi, and D. V. Plant, “Improvement of Phase Noise Compensation for Coherent Optical OFDM via Data-Aided Phase Equalizer,” in Optical Fiber Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2010), paper JThA10.

9. 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.

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

11. 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]

12. E. Ip and J. M. Kahn, “Compensation of Dispersion and Nonlinear Impairments Using Digital Backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008). [CrossRef]

13. S. L. Jansen, B. Spinnlera, I. Moritab, S. Randelc, and H. Tanakab, “100GbE: QPSK versus OFDM,” Opt. Fiber Technol. 15(5), 407–413 (2009). [CrossRef]

CE subsystem	Number of complex multiplications per symbol
FFT	$\frac{N}{2} \log_{2} N$
Channel Estimation (Pilot Symbols)	$U \times R_{P S}$
Channel Estimation (Pilot Subcarriers)	$U \times R_{P S C} \times (1 - R_{P S})$
Equalization	$U \times (1 - R_{P S})$
Updating the Equalization Parameters	$U \times (1 - R_{P S})$

DDPE subsystem	Number of complex multiplications per symbol
FFT	$\frac{N}{2} \log_{2} N$
Channel Estimation	U
Equalization (First Stage)	$U \times (1 - R_{P S})$
Equalization (Second Stage)	$U \times (1 - R_{P S})$
Updating the Equalization Parameters	$U \times (1 - R_{P S})$

CE subsystem	Number of complex multiplications per symbol
FFT	$\frac{N}{2} \log_{2} N$
Channel Estimation (Pilot Symbols)	$U \times R_{P S}$
Channel Estimation (Pilot Subcarriers)	$U \times R_{P S C} \times (1 - R_{P S})$
Equalization	$U \times (1 - R_{P S})$
Updating the Equalization Parameters	$U \times (1 - R_{P S})$

DDPE subsystem	Number of complex multiplications per symbol
FFT	$\frac{N}{2} \log_{2} N$
Channel Estimation	U
Equalization (First Stage)	$U \times (1 - R_{P S})$
Equalization (Second Stage)	$U \times (1 - R_{P S})$
Updating the Equalization Parameters	$U \times (1 - R_{P S})$

Zero-overhead phase noise compensation via decision-directed phase equalizer for coherent optical OFDM

Abstract

1. Introduction

2. DDPE description

3. Simulation of DDPE performance in CO-OFDM transmission system

4. System complexity

5. Conclusion

Acknowledgements

References and links

Cited By

Figures (8)

Tables (2)

Equations (6)

Optics Express