## Abstract

We show that dispersion-enhanced phase noise (DEPN) induces performance degradations in both conventional CO-OFDM systems and reduced-guard-interval (RGI) CO-OFDM systems employing RF-pilot phase compensation. After analytically studying DEPN, we show that DEPN causes a 2 to 6 dB optical signal-to-noise ratio (OSNR) penalty at transmission distances of 3200 km and 1600 km for 28 and 56 Gbaud QPSK systems, respectively, using lasers with 2 MHz linewidths. At such distances, DEPN reduces the linewidth tolerance at 1 dB OSNR penalty to 250-500 kHz while in the back-to-back case the tolerance is 1-3 MHz for both systems. When fiber nonlinearity is included, we observe similar performance degradations.

© 2011 OSA

## 1. Introduction

Coherent optical orthogonal frequency-division multiplexing (CO-OFDM) has been extensively studied because of its advantages such as high spectral efficiencies, low required sampling rates, and flexible bandwidth scalability and allocation [1]. Recent experimental demonstrations of CO-OFDM have highlighted many successful attributes of this approach [2–5]. Conventional CO-OFDM is vulnerable to inter-carrier interference (ICI) caused by laser phase noise, since the symbol durations are typically very large in order to reduce the cyclic prefix overhead in dispersion-unmanaged transmission systems [6]. Therefore, high-cost external cavity lasers (ECL) with hundreds of kHz linewidth are required if pilot subcarrier based phase estimation is adopted [7–9]. However, using RF-pilot phase compensation, which was proposed in [7] to combat ICI, it has been demonstrated that the laser linewidth tolerances can be increased to several MHz [7–9]. This enables the use of low-cost distributed feedback (DFB) lasers without inducing large performance degradations. However, in previous works, the interaction between laser phase noise and the accumulated chromatic dispersion (CD), which is referred to as dispersion-enhanced phase noise (DEPN) [10, 11], was not studied for CO-OFDM systems employing RF-pilot phase compensation.

Reduced-guard-interval (RGI) CO-OFDM systems, in which CD is compensated prior to OFDM demodulation by means of a frequency domain equalizer, were recently proposed to realize low overhead transmission [4, 5]. RGI CO-OFDM systems suffer much less ICI compared to conventional CO-OFDM systems because much shorter symbol durations are used [4, 10]. However, it is still meaningful to evaluate the efficiency of the RF-pilot phase compensation schemes in RGI CO-OFDM systems because in addition to laser phase noise compensation, the RF-pilot is also able to simultaneously compensate laser frequency offset [12] and fiber nonlinearities [13–15].

DEPN plays an important role in both conventional and RGI CO-OFDM systems employing RF-pilot phase compensation since the dispersion-induced walk-off leads to phase misalignment between the RF-pilot tone and data subcarriers after transmission. Such phase misalignment reduces the capability of the RF-pilot tone to compensate laser phase noise. In this paper, we investigate the impact of DEPN on both 28 Gbaud dual polarization (DP) QPSK (112 Gb/s) and 56 Gbaud DP-QPSK (224 Gb/s) systems. We first analytically show that with RF-pilot phase compensation, DEPN will cause both residual phase shift (RPS) and ICI. Then, we find that ICI grows with increasing symbol duration, whereas RPS increases for decreasing symbol durations. Via simulation, we show that although using 2 MHz DFB lasers only induces approximately 1 dB optical signal-to-noise ratio (OSNR) penalties at bit error rates (BERs) = 10^{−3} for both 28 and 56 Gbaud DP-QPSK systems in the back-to-back case, the OSNR penalty is increased by 2-6 dB after 3200 km and 1600 km transmission for the 28 and 56 Gbaud systems respectively. Moreover, at such transmission distances, the laser linewidth tolerance of the RF-pilot compensation scheme at 1 dB OSNR penalty is reduced from 1-3 MHz to 250-500 kHz for both systems because of DEPN. Finally, when nonlinearities are considered, similar performance degradations are observed. Here we find that DEPN induced penalties in conventional CO-OFDM systems cannot be easily mitigated whereas DEPN-induced penalties in RGI CO-OFDM systems can be mitigated using maximum-likelihood algorithms [10, 16].

## 2. Analysis of dispersion-enhanced phase noise with RF-pilot phase compensation

The origin of DEPN is illustrated in Fig. 1(a) , which depicts the time domain positions of different subcarriers within one OFDM symbol relative to the RF-pilot tone at both the transmitter and receiver. The dispersion-induced walk-off between the RF-pilot tone and different subcarriers after transmission will limit its effectiveness for compensating phase noise, especially for the edge subcarriers far away from the RF-pilot tone. First, it will induce RPS since different subcarriers experience different phase shifts with respect to the RF-pilot tone. Second, the phase fluctuations experienced by the RF-pilot tone and data subcarriers are different due to the walk-off, which reduces the effectiveness of ICI compensation. These two effects are illustrated by the constellations in Fig. 1, as the edge subcarriers in Fig. 1(c) experience not only a larger phase noise, i.e. RPS, but also a larger amplitude noise, i.e. ICI, than the middle subcarrier in Fig. 1(b).

To illustrate DEPN mathematically, we assume a system model, in which only laser phase noise, CD and amplified spontaneous emission (ASE) noise are considered. The received OFDM signal *r*(*t*) (RF-pilot tone is not included) can then be expressed similar to [10] as

*N*is the number of subcarriers,

_{c}*c*denotes the symbol transmitted on the

_{k}*k*th subcarrier, Δ

*f*is the subcarrier frequency spacing, and

*z*(

*t*) is the ASE noise.

*T*is the dispersion-induced walk-off between the

_{k}*k*th subcarrier and the RF-pilot tone which is calculated as in [10].

*ϕ*(

_{t}*t*) and

*ϕ*(

_{r}*t*) denote the laser phase noise caused by the transmitter and receiver lasers, respectively. They are both modeled as Wiener processes with a variance of

*2πβt*and

*β*denotes the linewidth for each individual laser.

The RF-pilot tone is then filtered out and its phase is

where*ϕ*(t) is the phase of the filtered ASE noise, the power of which is determined by the pilot-to-signal ratio and the bandwidth of the filter used to extract the RF-pilot [7]. Next, the phase is inverted and applied to the OFDM signal for compensating the phase noise as follows

_{n}After CD equalization and FFT, the received data subcarriers can be expressed by [10,17]

where*Z*(k) is the DFT of ${z}^{\prime}(t)$.

*ICI*(

*k*) is obtained by $ICI(k)={\displaystyle {\sum}_{l=0,l\ne k}^{N-1}{c}_{l}\cdot {I}_{k}(l-k)}$, and

*I*(

_{k}*p*) including

*I*(0) is given by

_{k}*T*is the sample duration.

_{s}RPS results from the dependence of the *I _{k}*(0) term in Eq. (4) on the subcarrier index

*k*, whereas ICI results from the

*ICI*(

*k*) term in Eq. (4) which arises from other subcarriers interfering with the

*k*th subcarrier [10, 17]. However, both terms depend on the [

*ϕ*(

_{t}*n*) -

*ϕ*(

_{t}*n + D*)] term in Eq. (5), which represents the dispersion-induced difference between the transmit laser phase applied to the RF-pilot tone and that applied to the data subcarrier. Therefore, both RPS and ICI will be enhanced with the increase of the walk-off

_{k}*D*, which is proportional to the transmission distance

_{k}*L*and the baud rate.

The relationship between the impairments (RPS and ICI) and the number of subcarriers is not straight forward based on the above analytical results. Therefore, we use simulations to study it as shown in Fig. 2
. We use normalized variance as a metric, which can be mapped to corresponding BER values as shown in [10]. For RPS, the normalized variance is defined as the power of phase noise (rad^{2}), whereas for ICI it is defined as the amplitude noise normalized to the average power of the signal. First, in the back-to-back case, the noise variance is larger with shorter symbol duration (top left constellation for each figure). This has been explained in [9] by the larger spectral leakage into the RF-pilot for shorter symbol duration. Next, by comparing Fig. 2(a), 2(b) and 2(c), it is seen that as the number of subcarriers increases, the variance of RPS is reduced due to the enhanced correlation between the phases acquired by data subcarriers and the RF-pilot tone, which results from the longer time duration in which their samples overlap. In contrast to RPS, the variance of ICI increases for longer symbol durations because each symbol becomes more vulnerable to ICI for a given phase difference. These two key observations can be also verified by the top right constellations of each figure as the phase noise dominates for systems with 80 subcarriers, whereas amplitude noise dominates for systems with 1280 subcarriers.

## 3. Simulation results and discussions

#### 3.1 Channel including only CD and ASE noise

Extensive simulations for both conventional and RGI CO-OFDM systems employing RF-pilot phase compensation have been conducted to evaluate the impact of DEPN on system performance. 28 Gbaud and 56 Gbaud QPSK systems were investigated. The fiber channel included CD with dispersion parameter *D* = 17 ps/(nm·km) and ASE noise. The linewidths of the transmit and receive lasers were both 2 MHz for all simulations, if not otherwise specified. For conventional CO-OFDM, the number of subcarriers was set to 1280 and 2560 for the 28 and 56 Gbaud systems respectively in order to make the cyclic prefix overhead identical at *L* = 3200 km (28 Gbaud) and 1600 km (56 Gbaud). For RGI CO-OFDM systems, a frequency domain equalizer compensates CD prior to OFDM demodulation [4]. CD was completely compensated without introducing any extra OSNR penalty for all systems. The bandwidth overhead for the RF-pilot tone was approximately 1.3%. The pilot-to-signal ratio was fixed at −12 dB and a 5th order Butterworth low pass filter with a bandwidth of 100 MHz was used to filter out the RF-pilot tone.

Figure 3
shows the OSNR penalty as a function of the transmission distance *L* for both conventional CO-OFDM and RGI CO-OFDM systems. In the back-to-back case, OSNR penalties for all systems are around 1 dB, which is consistent with previous works [7–9]. After transmission, systems with fewer subcarriers are more vulnerable to DEPN in terms of OSNR penalty compared to systems with larger number of subcarriers. This is because systems with fewer subcarriers suffer more from RPS, which leads to an increased BER than the ICI experienced by systems with a large numbers of subcarriers even when RPS and ICI have equal variances. For the 28 Gbaud system, the increase in the penalty is insignificant for *L* ≤ 2000 km, however at *L* = 3200 km penalties increase to 3.3-5.5 dB depending on the number of subcarriers. When the baud rate is doubled to 56 Gbaud, the penalties reach 3.8-6.8 dB at *L* = 1600 km, although they are still only 1 dB in the back-to-back case. Therefore, the impact of DEPN should be considered when designing long-haul transmission systems, especially for high baud rate systems.

Figure 4
shows the ONSR penalty versus the laser linewidth *β* for various scenarios. For the back-to-back case, linewidth tolerances at 1 dB penalty are larger than 1 MHz for all systems. It should be pointed out that the performance for larger linewidths can be further improved if the spectral overhead and filter bandwidth are optimized as shown in [9]. After transmission over 3200 km for the 28 Gbaud systems and 1600 km for the 56 Gbaud systems, the linewidth tolerance is reduced to 250-500 kHz, which means ECLs might be required even with RF-pilot phase compensation.

#### 3.2 Optical channel

For a real optical channel, fiber nonlinearities and ASE noise will dominate after thousands of km of transmission, and nonlinearities will also interact with CD. Therefore, it is important to investigate the impact of DEPN when nonlinearities exist. For the following simulations the systems were simulated in Optisystem 9.0. The length of each span is 80 km and the noise figure of inline EDFAs is 5 dB. Table 1 summarizes the additional simulation conditions, and other system parameters are the same as in the previous subsections.

The obtained BER versus OSNR for various scenarios is shown in Fig. 5
. Comparing the performance of systems with *β* = 2 MHz without the grouped maximum-likelihood (GML) algorithm and *β* = 0 MHz (without any phase compensation), we notice that OSNR penalties are larger than 4 dB and might reach 8 dB for RGI systems (*N _{c}* = 80). The inherent penalty of the RF-pilot phase noise compensation is around 1 dB in the back-to-back case as previously observed from Fig. 3. Therefore, we claim that most of these large penalties arise from DEPN, which is seen to be also significant in a real optical transmission system when nonlinearities are included. The increased OSNR penalties compared to those in Fig. 3 can be explained by the raised noise floor due to the addition of nonlinear impairments. Next, since RPS is a continuous phase noise [10], it can be mitigated by estimating the phases for different groups of subcarriers using a maximum-likelihood algorithm we refer to as a GML algorithm [10, 16]. After using GML, the required OSNR is significantly reduced for RGI CO-OFDM systems in which RPS dominates over ICI. On the other hand, the improvement for conventional CO-OFDM systems is negligible because ICI is the dominant impairment. Because GML can only partially compensate for RPS, the OSNR difference is a lower bound of the DEPN-induced penalty for RGI CO-OFDM systems, which is approximately 5.5 dB. Based on the results presented in Fig. 5, we conclude that although RGI CO-OFDM systems with shorter symbol duration are more vulnerable to DEPN than conventional CO-OFDM systems in terms of BER performance, RGI CO-OFDM systems are more tolerant to laser phase noise when a GML algorithm is employed.

## 4. Conclusions

In this paper, we evaluate the impact of DEPN on CO-OFDM systems using RF-pilot phase compensation. The origin of DEPN is analytically studied and the dependence of DEPN on OFDM symbol duration is illustrated. In addition, for reasonable transmission distances, (e.g. 3200 km for 28 Gbaud QPSK signals and 1600 km for 56 Gbaud QPSK signal) the presence of DEPN limits laser linewidths to 250-500 kHz. When fiber nonlinearities are included, we observe similar performance degradations.

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