Performance of carrier phase recovery for electronically dispersion compensated coherent systems

doi:10.1364/OE.20.026568

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Performance of carrier phase recovery for electronically dispersion compensated coherent systems

Ramtin Farhoudi, Amirhossein Ghazisaeidi, and Leslie Ann Rusch »View Author Affiliations

Optics Express, Vol. 20, Issue 24, pp. 26568-26582 (2012)
http://dx.doi.org/10.1364/OE.20.026568

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Abstract

An analytical approach taking into account carrier phase estimation (CPE) is presented to predict performance of quadrature phase shift-keying (QPSK) systems using coherent detection. Using this approach, system performance is found as a function of symbol rate, local oscillator (LO) linewidth, chromatic dispersion (CD) and signal-to-noise ratio (SNR). A new expression is derived for the covariance matrix of the conditional probability density function (pdf) of the decision statistic. This pdf is used to find bit error rate (BER) semi-analytically. Our analytical derivation assumes perfect removal of data modulation which corresponds to an ideal decision feedback (DF) carrier recovery. The validity of the analytical pdf for predicting BER is verified for a wide range of system parameters of interest in long haul systems. In addition, our semi-analytical BER provides a lower bound for the Viterbi-Viterbi (VV) BER, while showing the analytical BER previously proposed in the literature shows an overly pessimistic prediction of VV BER performance. We show that inaccuracy in previous analysis stems from overly simple model for the CPE when compensating large accumulated dispersion electronically. Finally, we study extension of our results to quadrature amplitude modulation (QAM). Preliminary simulation results are promising but the accuracy of our semi-analytical approach for predicting BER should be investigated further.

1. Introduction

Coherent detection is a promising solution for next generation optical systems and networks due to several advantages over the already installed direct detection systems [1

E. Ip, A. Lau, D. Barros, and J. Kahn, “Coherent detection in optical fiber systems,” Opt. Express 16, 753–791 (2008). [CrossRef] [PubMed]

R. Saunders, M. Traverso, T. Schmidt, and C. Malouin, “Economics of 100 Gb/s transport,” in Proc. of OFC (2010), Paper. NMB.2.

–3

T. Hoshida, H. Nakashima, T. Tanimura, S. Oda, Z. Tao, L. Liu, W. Yan, L. Li, and J. Rasmussen, “Network innovations brought by digital coherent receivers,” in Proc. of OFC (2010), Paper. NMB.4.

]. Electronic equalization of fiber chromatic dispersion (CD) using digital signal processing (DSP) is one important advantage [4

E. Ip and J. Kahn, “Digital equalization of chromatic dispersion and polarization mode dispersion,” J. Lightwave Technol. 25, 2033–2043 (2007). [CrossRef]

]. The cost and complexity of optical dispersion compensating devices are avoided and, equally important, the system becomes more robust to nonlinearity [5

E. Ip and J. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave. Technol. 26, 3416–3425 (2008). [CrossRef]

V. Curri, P. Poggiolini, A. Carena, and F. Forghieri, “Dispersion compensation and mitigation of nonlinear effects in 111-Gb/s WDM coherent PM-QPSK systems,” IEEE Photon. Technol. Lett. 20, 1473–1475 (2008). [CrossRef]

X. Chen, C. Kim, G. Li, and B. Zhou, “Numerical study of lumped dispersion compensation for 40-Gb/s return-to-zero differential phase-shift keying transmission,” IEEE Photon. Technol. Lett. 19, 568–570 (2007). [CrossRef]

–8

C. Xie, “WDM coherent PDM-QPSK systems with and without inline optical dispersion compensation,” Opt. Express 17, 4815–4823 (2009). [CrossRef] [PubMed]

]. In addition, adaptive equalization can be fully exploited [9

D. Crivelli, H. Carter, and M. Hueda, “Adaptive digital equalization in the presence of chromatic dispersion, PMD, and phase noise in coherent fiber optic systems,” in Proc. of Globecom (2004), pp. 2545–2551.

Electronic CD compensation, in contrast to optical dispersion compensation, suffers from equalization enhanced phase noise (EEPN) which stems from the interaction of receive laser (or local oscillator, LO) phase noise with the taps of the dispersion equalizer. The penalty induced by EEPN increases with linewidth, symbol rate and dispersion, and limits the maximum allowable LO linewidth or system reach [10

C. Xie, “Local oscillator phase noise induced penalties in optical coherent detection systems using electronic chromatic dispersion compensation,” in Proc. of OFC (2009), Paper. OMT.4.

, 11

W. Shieh and K. Ho, “Equalization-enhanced phase noise for coherent-detection systems using electronic digital signal processing,” Opt. Express 16, 15718–15727 (2008). [CrossRef] [PubMed]

]. The EEPN impact is more serious for high spectral efficiency modulation formats like quadrature amplitude modulation (QAM) having a compact constellation [12

I. Fatadin and S. Savory, “Impact of phase to amplitude noise conversion in coherent optical systems with digital dispersion compensation,” Opt. Express 18, 16273–16278 (2010). [CrossRef] [PubMed]

The impact of EEPN on coherent systems has been studied in [10

C. Xie, “Local oscillator phase noise induced penalties in optical coherent detection systems using electronic chromatic dispersion compensation,” in Proc. of OFC (2009), Paper. OMT.4.

W. Shieh and K. Ho, “Equalization-enhanced phase noise for coherent-detection systems using electronic digital signal processing,” Opt. Express 16, 15718–15727 (2008). [CrossRef] [PubMed]

A. Lau, T. Shen, W. Shieh, and K. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express 18, 17239–17251 (2010). [CrossRef] [PubMed]

K. Ho, A. Pak, Tao Lau, and W. Shieh, “Equalization-enhanced phase noise induced timing jitter,” Optics Letters 36, 585–587 (2011). [CrossRef] [PubMed]

S. Oda, C. Ohshima, T. Tanaka, T. Tanimura, H. Nakashima, N. Koizumi, T. Hoshida, H. Zhang, Z. Tao, and J. Rasmussen, “Interplay between local oscillator phase noise and electrical chromatic dispersion compensation in digital coherent transmission system,” in Proc. of ECOC (2010), Paper. Mo.1.C.2.

T. Xu, G. Jacobsen, S. Popov, J. Li, A. Friberg, and Y. Zhang, “Analytical estimation of phase noise influence in coherent transmission system with digital dispersion equalization,” Opt. Express 19, 7756–7768 (2011). [CrossRef] [PubMed]

–17

Q. Zhuge, C. Chen, and D. Plant, “Dispersion-enhanced phase noise effects on reduced-guard-interval CO-OFDM transmission,” Opt. Express 19, 4472–4484 (2011). [CrossRef] [PubMed]

]. Variance of EEPN was studied theoretically in [11

W. Shieh and K. Ho, “Equalization-enhanced phase noise for coherent-detection systems using electronic digital signal processing,” Opt. Express 16, 15718–15727 (2008). [CrossRef] [PubMed]

] and a power penalty expression was suggested by adding the EEPN variance to the amplified spontaneous emission (ASE) noise variance. It was found in [11

W. Shieh and K. Ho, “Equalization-enhanced phase noise for coherent-detection systems using electronic digital signal processing,” Opt. Express 16, 15718–15727 (2008). [CrossRef] [PubMed]

] that this penalty increases linearly with linewidth, fiber length and symbol rate. Simulation results reported in [10

C. Xie, “Local oscillator phase noise induced penalties in optical coherent detection systems using electronic chromatic dispersion compensation,” in Proc. of OFC (2009), Paper. OMT.4.

, 12

] suggest that this penalty should instead increase exponentially with fiber length.

A two dimensional probability density function (pdf) of received symbols before decision was derived analytically in [13

A. Lau, T. Shen, W. Shieh, and K. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express 18, 17239–17251 (2010). [CrossRef] [PubMed]

] and an elliptically shaped pdf was predicted in presence of EEPN. However, the impact of the carrier phase estimator (CPE) was not considered in the analysis. The sum of transmitter and receiver phase noises was used for phase tracking, essentially ignoring EEPN. The impact of EEPN on Viterbi-Viterbi (VV) algorithm was also studied briefly by simulation in [13

A. Lau, T. Shen, W. Shieh, and K. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express 18, 17239–17251 (2010). [CrossRef] [PubMed]

] which indicated that time-variation of VV phase estimate does not follow sum of the true transmitter and receiver phase noises closely.

The analytical expression for the BER floor due to EEPN was found in [11

W. Shieh and K. Ho, “Equalization-enhanced phase noise for coherent-detection systems using electronic digital signal processing,” Opt. Express 16, 15718–15727 (2008). [CrossRef] [PubMed]

], assigning an effective laser linewidth in the presence of EEPN. A CPE based on one-tap normalized least mean square filter was used to validate the analysis. The impact of EEPN on reduced-guard-interval coherent optical orthogonal frequency-division multiplexing (CO-OFDM) systems was investigated in [17

Q. Zhuge, C. Chen, and D. Plant, “Dispersion-enhanced phase noise effects on reduced-guard-interval CO-OFDM transmission,” Opt. Express 19, 4472–4484 (2011). [CrossRef] [PubMed]

As described above, almost all the analytical results concerning EEPN are based on the assumption that the estimated phase by the CPE is approximately the sum of transmitter and receiver phase noises. This is certainly the case for an ideal CPE in optically dispersion-compensated systems. However, this model is no longer valid when the EEPN contribution becomes significant.

An analysis of the EEPN was also presented in [18

G. Colavolpe, T. Foggi, E. Forestieri, and M. Secondini, “Impact of phase noise and compensation techniques in coherent optical systems,” J. Lightwave Technol. 29, 2790–2800 (2011). [CrossRef]

] with and without a digital coherence enhancement (DCE) technique developed in [18

G. Colavolpe, T. Foggi, E. Forestieri, and M. Secondini, “Impact of phase noise and compensation techniques in coherent optical systems,” J. Lightwave Technol. 29, 2790–2800 (2011). [CrossRef]

M. Secondini, G. Meloni, T. Foggi, G. Colavolpe, L. Poti, and E. Forestieri, “Phase noise cancellation in coherent optical receivers by digital coherence enhancement,” in Proc. of ECOC (2010), Paper. P4.17.

–20

G. Colavolpe, T. Foggi, E. Forestieri, and M. Secondini, “Phase noise sensitivity and compensation techniques in long-haul coherent optical links,” in Proc. of Globcom (2010), pp. 1–6.

] to reduce EEPN. It was shown that the EEPN contribution can be considered as an additive zero-mean Gaussian noise whose variance can be approximated with the variance derived in [11

W. Shieh and K. Ho, “Equalization-enhanced phase noise for coherent-detection systems using electronic digital signal processing,” Opt. Express 16, 15718–15727 (2008). [CrossRef] [PubMed]

]. However, the assumption on the CPE is the same as [13

A. Lau, T. Shen, W. Shieh, and K. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express 18, 17239–17251 (2010). [CrossRef] [PubMed]

An experimental verification of EEPN simulation results is reported in [21

Q. Zhuge, X. Xu, Z. El-Sahn, M. Mousa-Pasandi, M. Morsy-Osman, M. Chagnon, M. Qiu, and D. Plant, “Experimental investigation of the equalization-enhanced phase noise in long haul 56 Gbaud DP-QPSK systems,” Opt. Express 20, 13841–13846 (2012). [CrossRef] [PubMed]

]. A reduction in the phase error variance of the received symbols in the presence of uncompensated dispersion is reported and it is shown that this reduction is proportional to the fiber dispersion.

In this paper, we examine estimated phase noise in the presence of EEPN with a more realistic model for the CPE. The CPE in our derivation has perfect removal of data modulation, but retains additive noise that can be reduced by means of a moving average filter. An expression for the estimated phase noise as a function of transmitter and receiver phase noises is provided. The derivation corresponds to the performance of a CPE with ideal decision feedback (DF). The expression that we provide for CPE phase shows the same behavior reported experimentally in [21

]; the phase variance is reduced by increasing dispersion.

With the expression for the DF CPE phase, we derive the semi-analytical pdf of the decision statistic in this paper after the CPE. We show that a simple modification of the covariance matrix in the conditional pdf reported in [13

A. Lau, T. Shen, W. Shieh, and K. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express 18, 17239–17251 (2010). [CrossRef] [PubMed]

] provides a much better approximation of the pdf of a practical CPE such as VV. For example our analysis correctly predicts the circularly shaped pdf following CPE rather than the elliptically shaped pdf predicted in [13

A. Lau, T. Shen, W. Shieh, and K. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express 18, 17239–17251 (2010). [CrossRef] [PubMed]

]. Using our semi-analytical pdf, we calculate BER for a differential quadrature phase shift keying (QPSK) system and compare it with BER from Monte-Carlo (MC) simulation of the system employing either a VV or ideal DF algorithm. We show that our semi-analytical pdf provides an accurate estimate of the system BER and power penalty for the DF CPE method. In addition, simulation results suggest that this semi-analytical BER can be considered as a lower bound for the BER of VV algorithm.

We also investigate the extension of our results to quadrature amplitude modulation (QAM). Our simulation results show that the same analytical expression for the CPE phase estimate accurately predicts performance for ideal DF CPE. The semi-analytical pdf of the decision statistic is also applicable to QAM, as no assumption on the modulation format is made in deriving this pdf. The accuracy of this BER needs to be investigated further in the case of other CPEs.

Using our analytical pdfs for evaluating BER via numerical integration is faster than MC simulation for BERs by several orders of magnitude, even at low BER, e.g. 10⁻⁴. In our semi-analytical method, a number of covariance matrices must be evaluated for a few two dimensional Gaussian pdfs, which can be performed at low complexity. Although we present the analysis for an ideal DF carrier recovery, we can approximate the BER for another carrier recovery if we know the SNR penalty of that specific carrier recovery compared to the ideal DF. In addition to finding BER, having an accurate approximation of the pdfs is useful to implement soft forward error correction (FEC) algorithms, as log-likelihood ratios must normally be provided to the decoder.

This paper is organized as follows. We first consider the system model in section II and introduce the notation to be used. The expression for the CPE phase estimate and the analytical pdf of the decision statistic are presented in section III. Simulation results are provided in section IV to validate the analytical expressions. The extension to QAM is presented in section V. Finally, we draw the conclusions in section VI.

2. System model

In this section, we introduce the coherent system model to be used in our analysis. A simplified block diagram of a coherent system is shown in Fig. 1 where the input symbols x_k are transmitted using pulse shape p(t). Single polarization transmission is considered here, but it is possible to extend the analysis to the dual-polarization case. The fiber is assumed to show only chromatic dispersion (CD) with transfer function H_CD(jω) given by

H_{C D} (j ω) = exp (- j \frac{1}{2} β_{2} L ω^{2})

(1)

where β₂ is the dispersion coefficient and L is the fiber length. In addition to the CD effects introduced by impulse response h_CD(t), the transmitted signal is corrupted by noise sources ϕ_T(t), ϕ_R(t) and n(t), respectively the transmitter and receiver laser phase noises and additive noise due to optical amplifiers. We have not considered the nonlinear effects in our model. Nonlinear effects can be taken into account by modifying the noise sources. For example in [22

M. Magarini, A. Spalvieri, F. Vacondio, M. Bertolini, M. Pepe, and G. Gavioli, “Empirical modeling and simulation of phase noise in long-haul coherent optical transmission systems,” Opt. Express 19, 22455–22461 (2011). [CrossRef] [PubMed]

], it is shown that the phase noise can still be modeled as a Wiener process whose parameters are calculated based on a empirical model.

Fig. 1 Block diagram of a coherent system.

The received signal is mixed with the local oscillator and photo-detected. Mixing is modeled by multiplication by a phase noise process (i.e., zero frequency offset is assumed), and the photodetection and RF front end are modeled by a single impulse response h_oe(t). The signal is sampled at arbitrary rate 1/T (typically T is half the symbol duration).

Electronic dispersion equalization is performed using a finite-impulse response (FIR) filter corresponding to the inverse transfer function of the fiber. The taps of the equalizer w_n are given by [23

S. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16, 804–817 (2008). [CrossRef] [PubMed]

]

w_{n} = \sqrt{\frac{j T^{2}}{2 π β_{2} L}} exp (- j \frac{T^{2}}{2 β_{2} L} n^{2})

(2)

where

- N \leq n \leq N, N = ⌊ \frac{π β_{2} L}{T^{2}} ⌋

(3)

After CD compensation, carrier phase recovery is performed by derotating the signal using the estimated phase ϕ̂_k. This estimated phase can be obtained by several different methods, as we discuss in the following sections.

3. Analysis in the case of perfect data remodulation

In this section we first provide an expression for the estimated phase ϕ̂_k using remodulation with perfect knowledge of the transmitted symbols (i.e., ideal decision feedback). We use this expression to find the pdf of the decision statistic y_k in Fig. 1. The details of the analysis can be found in the appendix.

3.1. Analytical expression for CPE phase estimate

An essential part of a coherent receiver is the CPE where the phase offset is estimated and the equalized signal is derotated with the estimated phase. Algorithms used in the CPE typically consist of two stages; in the first stage, data modulation is eliminated and in the second stage the obtained raw phase estimate is filtered to reduce noise [24

E. Ip and J. Kahn, “Feedforward carrier recovery for coherent optical communications,” J. Lightwave Technol. 25, 2675–2692 (2007). [CrossRef]

M. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Lightwave Technol. 27, 901–914 (2009). [CrossRef]

–26

D. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” J. Lightwave Technol. 24, 12–21 (2006). [CrossRef]

]. For example, data modulation can be eliminated by using decision feedback or raising to the power of M (for M-PSK signal) in the VV algorithm.

In this paper, we estimate the best possible performance of the CPE assuming ideal removal of modulation. When modulation is completely removed, the residual signal consists of equalization enhanced phase noise (EEPN) and noise stemming from ASE. To remove the additive noise, a moving average filter is used. While the Wiener filter would give lowest mean square error for the estimate, the suboptimal moving average filter is analytically tractable and gives a reasonable approximation of the performance of the phase estimate with perfect data remodulation.

The number of taps in the CD compensating filter, 2N + 1, can be considerable for long fiber runs. However, the block length 2N_B + 1 of the moving average filter is chosen large enough to average additive noise, yet short enough to assure that the tracked laser phase noise remains relatively constant. Let r_m be the output of the CD compensating filter for the m^th symbol (see appendix for details). The phase estimate is

{\hat{ϕ}}_{k} = Arg {\sum_{m = k - N_{B}}^{k + N_{B}} r_{m} x_{m}^{*}}

(4)

We define

Δ_{n, k} = ϕ_{R} (k T - n T) - ϕ_{R} (k T)

(5)

to be the phase increment in the receiver LO phase noise for the k^th sample at the n^th filter tap in the CD filter. In section 7 (Appendix 1), we show that the output of the moving average filter for ideal data remodulation is

{\hat{ϕ}}_{k} \approx ϕ_{R} (k T) + ϕ_{T} (k T) + \sum_{n} Re {w_{n} p_{d} (- n T)} Δ_{n, k} + n_{k}^{'} + n_{k}^{″}

(6)

where Re denotes real part, p_d(t) = p(t) ⊗ h_CD(t) ⊗ h_oe(t) (⊗ being the convolution) is the overall system impulse response. n′_k and n″_k are the noise contribution from intersymbol interference (ISI) attributable to EEPN and the residual ASE noise respectively. The former noise source n′_k contains zero-mean independent symbols that add incoherently and is thus averaged out in the moving average filter; see appendix for more details. This term is an unpredictable part of the EEPN as it contains symbols unknown at the time of applying the CPE, despite ideal decision feedback.

The summation over n in Eq. (6) has non-zero mean and can be tracked by the CPE, along with the laser phase drifts ϕ_R(kT) and ϕ_T(kT). We call this summation the predictable part of the EEPN as it varies slowly as compared with n′_k. Note that when there is no electronic dispersion equalization in the system, the equalizer can be imagined as a one tap filter of w₀ = 1. As Δ_0,k = 0, the EEPN contribution is zero and the estimated phase reduces to ϕ̂_k = ϕ_T +ϕ_R.

3.2. Modifying the symbol error pdf

In this section, we find the pdf of the decision statistic y_k = r_ke^−jϕ̂_k (shown in Fig. 1). We neglect transmitter phase noise, i.e., ϕ_T(t) = 0, as the contribution of the transmitter phase noise in the presence of EEPN is negligible. In section 8 (Appendix 2) we use Eq. (6) to show the decision statistic y_k, in the case of perfect remodulation and a moving average filter to suppress noise, can be written as

y_{k} \approx x_{[k]} + \sum_{n = - N}^{N} s_{n, k} Δ_{n, k} + {\tilde{n}}_{k}

(7)

where ñ_k with variance σ̃² is a combination of ASE noise and the noise in phase estimation process and

s_{n, k} = j \sum_{i} x_{i} w_{n} p_{d} [(k - n) T - i T_{s}] - j x_{[k]} Re {w_{n} p_{d} (- n T)}

(8)

The index of the symbol at the k^th sample time is given by [k]

In [11

W. Shieh and K. Ho, “Equalization-enhanced phase noise for coherent-detection systems using electronic digital signal processing,” Opt. Express 16, 15718–15727 (2008). [CrossRef] [PubMed]

, 13

A. Lau, T. Shen, W. Shieh, and K. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express 18, 17239–17251 (2010). [CrossRef] [PubMed]

, 18

G. Colavolpe, T. Foggi, E. Forestieri, and M. Secondini, “Impact of phase noise and compensation techniques in coherent optical systems,” J. Lightwave Technol. 29, 2790–2800 (2011). [CrossRef]

] they assume perfect knowledge of the true phase noise ϕ_R(kT) and use this as the phase estimate ϕ̂_k although it is highly suboptimal and essentially ignores EEPN. While the expression for y_k in Eq. (7) remains unchanged, s_n,k is the case of ϕ̂_k = ϕ_R(kT)

s_{n, k} = j \sum_{i} x_{i} w_{n} p_{d} [(k - n) T - i T_{s}]

(9)

which matches the results in [13

A. Lau, T. Shen, W. Shieh, and K. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express 18, 17239–17251 (2010). [CrossRef] [PubMed]

Using Eq. (7), we can find the pdf of y_k following the same procedure in [13

A. Lau, T. Shen, W. Shieh, and K. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express 18, 17239–17251 (2010). [CrossRef] [PubMed]

], but with the more realistic and higher performance estimate ϕ̂_k in Eq. (6). Hence our calculations are based on the analysis leading to Eq. (8) that includes the EEPN term depending on the desired bit x_[k], and not exclusively the ISI term as in previous analysis. We write Δ_n,k as a sum of independent and identically distributed (iid) Gaussian random variables Δ_n,k − Δ_n_±1,_k each having zero mean and variance σ² = 2πΔν_LOT (Δν_LO being the LO linewidth)

\begin{array}{r} y_{k} = x_{[k]} + \sum_{n = - N}^{- 1} [(Δ_{n, k} - Δ_{n + 1, k}) \sum_{m = - N}^{n} s_{m, k}] + \\ \sum_{n = 1}^{N} [(Δ_{n, k} - Δ_{n - 1, k}) \sum_{m = n}^{N} s_{m, k}] + n_{k} \end{array}

(10)

Suppose 2M is the system memory multiplied by the oversampling rate (2 in this paper). Given symbol pattern X = [x_[M−k],..., x_[k−1], x_[k+1],..., x_[M+k]] and desired symbol x_[k], we can calculate s_n,k in Eq. (8). The conditional pdf f(y_k|X) is a complex Gaussian pdf with mean vector [Re{x_[k]}, Im{x_[k]}]^tr and covariance matrix C which is given in section 8 (Appendix 2). This covariance matrix is a function of s_n,k, σ̃² and σ². The conditional pdf f(y_k|x_[k]) is a Gaussian mixture, that is, the normalized sum over pdfs f(y_k|X) for different symbol patterns X

f (y_{k} | x_{[k]}) = \frac{1}{| P |} \sum_{X \in P} f (y_{k} | X)

(11)

where P is the set of all possible symbol patterns X and |P| is its cardinality. The total pdf f(y_k) is an average of pdfs f(y_k|x_[k]) over all possible symbols x_[k] (or constellation points). Although the number of symbol patterns could be very large, we show by simulation that a summation over a limited number of patterns chosen randomly is sufficient to find a relatively accurate estimate of the pdf. For calculating s_n,k, we need to calculate an infinite sum in Eq. (8) which can be truncated in practice to the finite system memory 2M is finite to a sum from −M to M. It can be shown that M ≈ N. Thus, for L = 3000 km, R_s = 28 Gbaud and two samples per symbol, M and N are around 640.

4. Simulation results

We consider the performance of three phase estimation methods using both MC and semi-analytical techniques. The methods are

ideal decision feedback for removal of modulation and a moving average filter to remove noise (ideal DF)
Viterbi-Viterbi for removal of modulation and a moving average filter to remove noise (VV)
Using an ideal pilot tone (PT) to extract perfect knowledge of ϕ_R(t) +ϕ_T (t)

In the case of ideal DF we use our previous analysis to generate a semi-analytical prediction of the BER based on Eq. (11) using Eq. (8), as well as an MC simulation where ideal removal of modulation is followed by a moving average filter. In the case of VV, we only generate MC BER curves using VV algorithm. In the PT case, a continuous wave PT provides a separable signal at the receiver input whose phase is the sum of transmitter and receiver phase noise, ϕ_R(t)+ϕ_T(t). We compare BER from MC for this PT method with an analytical formula where ϕ_R(t) + ϕ_T(t) is assumed for phase noise cancelation of received symbols [11

W. Shieh and K. Ho, “Equalization-enhanced phase noise for coherent-detection systems using electronic digital signal processing,” Opt. Express 16, 15718–15727 (2008). [CrossRef] [PubMed]

, 18

G. Colavolpe, T. Foggi, E. Forestieri, and M. Secondini, “Impact of phase noise and compensation techniques in coherent optical systems,” J. Lightwave Technol. 29, 2790–2800 (2011). [CrossRef]

]. Note that the PT is clearly suboptimal, but was examined in [11

W. Shieh and K. Ho, “Equalization-enhanced phase noise for coherent-detection systems using electronic digital signal processing,” Opt. Express 16, 15718–15727 (2008). [CrossRef] [PubMed]

, 13

A. Lau, T. Shen, W. Shieh, and K. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express 18, 17239–17251 (2010). [CrossRef] [PubMed]

] due to tractability of the analysis. Unfortunately the known total phase noise at the input provided by PT is before CD compensation, while the phase noise derotation is after CD compensation when the filtered phase noise no longer resembles the input phase noise.

4.1. System parameters

We perform numerical simulations of a coherent system and compare the analytical prediction of BER with the MC simulation results. We consider simulation of quadrature PSK (QPSK) in this section and study the quadrature-amplitude (QAM) modulation in the next section. In the case of QPSK transmitter differential and Gray coding are used. A root-raised cosine pulse having roll-off factor 1 is used as the transmitted pulse p(t).

The fiber span of length L is assumed to have only second-order CD with dispersion coefficient β₂ = 21.6 ps²/km. It is assumed that fiber loss is perfectly compensated with amplifier gains and the additive Gaussian noise due to amplifiers is added at the end of the fiber span. The transmitter and receiver laser phase noises (each having linewidth Δν/2) are modeled as a Wiener process.

Temporal resolution of the simulation is T′ = T_s/8 which is also used in generating the Wiener process. In the coherent receiver, the received signal is first filtered by a filter matched to the transmitted pulse shape and then sampled at 2 (T = T_s/2). This is followed by the CD compensation which is performed using the filter in Eq. (2); filter length varies as a function of fiber length examined.

After CD compensation, carrier phase recovery is performed per one of the three methods discussed previously. In the DF case, data modulation is first removed using perfect knowledge of the transmitted symbols. Next the signal is filtered to remove noise using an 11-tap moving average filter. The ideal DF provides an upper-bound of performance that can be achieved by any CPE.

The VV case has data modulation removed by raising to the appropriate power (2 for BPSK, 4 for QPSK, etc.). The phase is then unwrapped (exploiting the differential encoding) and filtered for noise using the same 11-tap moving average filter for the DF case. Finally, in the PT case no CPE is used, the phase estimate is simply ϕ_R(t)+ϕ_T(t) using perfect knowledge of the laser phase noise at transmitter and receiver.

4.2. Accuracy of semi-analytical pdfs

Two dimensional pdfs of the decision statistic f(y_k) are estimated using MC simulation (assuming ideal DF as the CPE) and calculated semi-analytically for the DF and PT cases. MC estimates of the pdf are obtained by transmitting 5 × 10⁶ symbols. In order to calculate the semi-analytical pdfs, 50 symbol patterns X chosen randomly are used in Eq. (11) for each of the four possible symbols x_k to calculate f(y_k|x_k).

The estimated pdf and the semi-analytical pdf using DF are both circularly-shaped while the pdf in [13

A. Lau, T. Shen, W. Shieh, and K. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express 18, 17239–17251 (2010). [CrossRef] [PubMed]

] is ellipticity-shaped; the tails of each elliptic constellation points extends toward the decision boundary which explains why we will observe worse BER performance for the PT case. The one-dimensional cross section of these pdfs at Im{ŷ_k} = 0 is shown in Fig. 2(a).

Fig. 2 (a) Cross section at Im{ŷ_k} = 0 of two-dimensional pdfs i) obtained by MC simulation (solid), ii) as developed in Eq. (11) (dashed) and iii) as reported in [13

A. Lau, T. Shen, W. Shieh, and K. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express 18, 17239–17251 (2010). [CrossRef] [PubMed]

] (dot-dashed);L = 3000 km and Δν = 10 MHz, (b) error in pdf as a function of number of patterns for averaging in Eq. (11); pdf plotted in (a) uses 50 patterns.

The semi-analytical pdf in Eq. (11) can be calculated very fast as the number of patterns required to get an accurate approximate for the true pdf is small. This is investigated in Fig. 2 where the error in pdf is defined as

ε = \int_{y_{k} \in S} | f_{M C} (y_{k}) - f (y_{k}) | d y_{k}

(12)

where f_MC(y_k) stands for the pdfs obtained by MC. The integration is over entire complex plane S. It can be observed from Fig. 2(b) that the error ε converges to an error floor very fast by increasing the number of patterns.

4.3. BER for three phase estimates

In this section, we compare BER estimates found with MC simulations and analytical expressions. Semi-analytical BER is calculated for DF using the conditional pdfs f(y_k|x_k) by Eq. (11). Each of the four conditional pdfs (two dimensional Gaussian pdfs) is calculated by averaging over 50 random pattern vectors X, where the length of the vector corresponds to system memory 2M. We use 50 patterns to find the sum in Eq. (11) as the total number of patterns is excessively large − 4²^M⁻¹ for the case of QPSK. We have confirmed via simulation that this number is sufficient to obtain accurate results. The conditional pdfs are integrated over QPSK decision regions (quarters of the complex plane) to find the symbol error rate (SER). As the SER of a Gray-coded QPSK system is twice its BER, and this BER is half of the BER of differential QPSK system, the calculated SER is equal to the BER of differential QPSK. We have generated MC estimates of the BER of the differential QPSK system with ideal DF.

We assume symbol rate of 28 Gbaud, L = 3000 km and total linewidth of Δν = 10 MHz (5 MHz for LO and transmitter laser). In Fig. 3(a) we present BER versus signal-to-noise ratio (SNR), including the AWGN case (an optically compensated with no EEPN) given by the solid line. For the assumed parameters, the main source of SNR penalty (as compared with AWGN case) is EEPN. The BER obtained from the semi-analytical pdf Eq. (11), where Eq. (8) is used in the covariance matrix, is given in the dashed curve in red. The accuracy of the semi-analytic result is confirmed by the MC BER given by the blue curve with square markers. Having confirmed our semi-analytic BER result for ideal DF, we next examine the utility of this result as a lower bound for the VV carrier recovery. An MC simulation of BER for VV carrier recovery is shown in Fig. 3(a) with diamonds markers in green. The non-ideal modulation removal of the VV technique leads to a clearly visible penalty. Finally, we reproduce the BER found in [18

G. Colavolpe, T. Foggi, E. Forestieri, and M. Secondini, “Impact of phase noise and compensation techniques in coherent optical systems,” J. Lightwave Technol. 29, 2790–2800 (2011). [CrossRef]

] by integrating the two dimensional pdf in [13

A. Lau, T. Shen, W. Shieh, and K. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express 18, 17239–17251 (2010). [CrossRef] [PubMed]

] over the decision boundaries. This semi-analytical BER, shown in black triangles, assumed a pilot tone (PT) provided the true total phase noise at the receiver input that was then used for derotating the symbol after CD compensation. Note that the PT results allowed an analytical attack leading to the following simple expression (plotted in the black dashed curve) that closely matched the semi-analytical PT BER results

BER = 2 Q (1 / \sqrt{σ_{n}^{2} + σ_{e e p n}^{2}})

(13)

where

σ_{e e p n}^{2} = π^{2} L Δ ν_{L O} R_{s}

, Q(·) is the Q-function and signal power is normalized to one. The factor of two accounts for differential detection. Despite the perfect side information, the BER found (either analytical or semi-analytical) is not a bound, and is in fact a pessimistic predictor of VV performance. This is due to the highly non-optimal strategy of derotating the received symbols with this side information.

Fig. 3 BER versus (a) SNR (b) fiber length L (c) linewidth-symbol time product ΔνT_s (d) SNR for different baud rates from MC simulation (markers) and semi-analytical pdfs (dashed). Baud rate in (a)–(c) is 28 Gbaud.

We next fix the SNR to 15 dB and examine the impact of other system parameters on BER: BER versus fiber length, linewidth and baud rate are shown in Figs. 3(b)–(d), respectively. In Figs. 3(b)–(c), where a fixed baud rate of 28 Gbaud is assumed, a good match between our semi-analytical predication and MC validation is evident. In addition, changing the baud rate in Fig. 3(d) does not affect accuracy of the results. The prediction provides a lower bound for VV performance over the range of values considered.

We use the semi-analytical BER to calculate the penalty in SNR at BER of 3.8 × 10⁻³ and compare it with the penalty obtained by MC simulation. The penalty is measured with respect to the system having just additive Gaussian noise whose theoretical BER is known. The result is shown in Fig. 4 for different fiber lengths and laser linewidths.

Fig. 4 Penalty in dB at the BER of 3.8×10⁻³ from MC simulation (markers) and analytical pdf (dashed). DF is assumed for carrier recovery and baud rate is 28 Gbaud.

It can be observed from Fig. 3 that our semi-analytical BER is a lower bound for the BER of the VV carrier recovery so it can be useful in practice to estimate a lower bound for the required SNR, minimum LO laser linewidth and system reach in presence of EEPN. In addition, having a closed-form expression for conditional pdfs, gives more insight into the stochastic properties of the decision statistic that can be exploited in taking measures to reduce EEPN impact, for example, via employing soft decision FEC.

5. Extension to QAM

The derivation of the analytical expression for ϕ̂_k in Eq. (6) is presented in section 7 (Appendix 1) for the case of M-PSK signaling. The constant amplitude of M-PSK signaling was exploited in the derivation. While 16-QAM and 64-QAM are not constant amplitude, we nonetheless examined via simulation the accuracy of Eq. (6) for these cases over the parameter ranges on interest. We ran two separate MC simulations. In one case the CPE estimate ϕ̂_k was generated with ideal DF remodulation followed by a moving average filter of the unwrapped phase. In the second case the MC values for phase noise and AWGN were used in Eq. (6) to produce the CPE estimate ϕ̂_k. While not presented here, the BER predictions of these two simulations were very close for a wide range of system parameters in the case of both 16 and 64-QAM.

Having verified the accuracy of Eq. (6) by simulation for QAM modulation, we adopt this expression and follow the same derivation for the conditional pdfs. Note that the constant amplitude of QPSK was not needed in deriving Eq. (10). The mean vectors [Re{x_[k]}, Im{x_[k]}]^tr and the covariance matrix depending on Eq. (8) are calculated for 16-QAM symbols to produce the Gaussian pdfs f(y_k|X). The analytical pdfs are integrated to find the BER.

We again assume symbol rate of 28 Gbaud, L = 3000 km and total linewidth of total linewidth of 2 MHz (1 MHz for LO or transmitter laser). In Fig. 5(a) we present BER versus SNR, including the AWGN case. The dashed curve is the BER from semi-analytical pdfs, which shows a good prediction of the BER by MC simulation (square markers). Again the SNR penalty compared with the AWGN curve is mainly due to EEPN here. Accuracy of the semi-analytical BER is verified versus fiber length and linewidth in Figs. 5(b)–(c). An SNR of 20 dB is used for the 16-QAM system. A good match between theory and simulation can be observed at large distances or linewidth, however, there is deviation from theory in small distances or narrow linewidth at this SNR. Future investigations should examine the applicability of these equations for a lower bound for realistic carrier recovery techniques for QAM.

Fig. 5 BER for 16QAM at 28 Gbaud for ideal DF and for our prediction of ideal DF performance from (10) and (11). BER versus (a) SNR (b) fiber length L (c) linewidth-symbol time product ΔνT_s.

6. Conclusion

Interaction of dispersion and phase noise in coherent systems is studied analytically in presence of CPE. Based on this analysis, a more realistic model for the CPE in terms of transmitter and receiver phase noise is proposed. Based on this CPE model, an analytical pdf of decision statistic previously studied in the literature is improved. It is shown than the BER prediction of previously presented pdf is pessimistic. The analysis is validated by MC simulations for a QPSK system using ideal DF CPE and it is shown that our model predicts well the system BER employing this ideal CPE. It is shown by simulation that the proposed semi-analytical BER can be served as a lower bound for the BER by using VV carrier recovery. It is also shown that the proposed model can be extended to the QAM and it provides fairly accurate prediction of the BER in the case of 16-QAM employing ideal DF. However, accuracy of our semi-analytical BER to estimate BER for other carrier recovery techniques should be studied further.

Using the CPE model, the link between the LO phase noise and the estimated phase by the CPE can be mathematically described. We are currently working to exploit this link to devise algorithms that compensate EEPN in the system. On the other hand, the analytical pdf of decision statistic is useful in practice when maximum-likelihood sequence estimation (MLSE) algorithms or soft decision FEC are employed to reduce EEPN impact. In addition, prediction of the system penalty without resorting to MC simulations could be of interest in the system design where connection between this penalty and dispersion, linewidth and electrical filter bandwidth needs to be determined.

Appendices

7. Appendix 1

We consider an M-PSK signal for which |x_i| = 1 and assume that the transmitter phase noise is zero. The output of the CD filter is given by

r_{k} = \sum_{n = - N}^{N} w_{n} \sum_{i = - \infty}^{\infty} x_{i} p_{d} [(k - n) T - i T_{s}] e^{j ϕ_{R} [(k - n) T]} + n_{k}

(14)

where T_s and T are symbol and sampling time respectively and p_d(t) = p(t) ⊗ h_CD(t) ⊗ h_oe(t) (⊗ being convolution). We assume the CD compensating filter {w_n} perfectly compensates the chromatic dispersion, i.e.,

\sum_{n = - N}^{N} w_{n} \sum_{i = - \infty}^{\infty} x_{i} p_{d} [(k - n) T - i T_{s}] = x_{[k]}

(15)

where [k] gives the index of the desired symbol at the k^th sample. Under this assumption, the output of the CD compensating filter is

r_{k} = x_{[k]} (1 + q_{k}) e^{j ϕ_{R} (k T)} + n_{k}

(16)

where

q_{k} = \sum_{n = - N}^{N} w_{n} (e^{j Δ_{n, k}} - 1) \sum_{i = - \infty}^{\infty} \frac{x_{i}}{x_{[k]}} p_{d} [(k - n) T - i T_{s}]

(17)

and where we define

Δ_{n, k} = ϕ_{R} (k T - n T) - ϕ_{R} (k T)

(18)

to be the phase increment in the receiver LO phase noise for the k^th sample at the n^th filter tap in the CD compensating filter. Consider the product w_n (e^jΔ_n,k − 1). For small n the phase increment is small and we use the small angle approximation w_n (e^jΔ_n,k − 1) ≈ jΔ_n,kw_n; for larger n the coefficient w_n is decaying exponentially and the product will be close to zero. Hence

q_{k} \approx \sum_{n = - N}^{N} j Δ_{n, k} w_{n} \sum_{i = - \infty}^{\infty} \frac{x_{i}}{x_{[k]}} p_{d} [(k - n) T - i T_{s}]

(19)

We assume the filter output r_k is ideally remodulated, removing x_[k]. We further assume that the phase noise ϕ_R(kT) is nearly constant within the block length 2N_B +1 of the moving average filter, to obtain

\begin{array}{r} \frac{1}{2 N_{B} + 1} \sum_{m = k - N_{B}}^{k + N_{B}} r_{m} x_{[m]}^{*} \approx \frac{1}{2 N_{B} + 1} e^{j ϕ_{R} (k T)} \sum_{m} q_{m} \\ + e^{j ϕ_{R} (k T)} + {\tilde{n}}_{k}^{″} \end{array}

(20)

where ñ″_k is proportional to n″_k in Eq. (6) due to ASE noise. The sum over q_m can be written as

\begin{array}{l} \sum_{m} q_{m} = j \sum_{m} \sum_{n} Δ_{n, k} w_{n} p_{d} (- n T) + \\ \sum_{m, n, x_{i} \neq x_{[m]}} \frac{x_{i}}{x_{[m]}} j Δ_{n, k} w_{n} p_{d} [(m - n) T - i T_{s}] \end{array}

(21)

The first term over m becomes multiplication by the number of taps of the moving average filter 2N_B + 1. Drawing on our assumption that the block length is chosen such that phase noise ϕ_R(kT) is nearly constant over the moving average, we approximate the phase increments (relative to the n^th CD filter tap timing) to also remain unchanged, Δ_n,m ≈ Δ_n,k, hence we use Δ_n,k in the summations.

The second term in Eq. (21) isolates the ISI contribution. We call this term ñ′_k as it is proportional to the noise n′_k in Eq. (6). This summation is characterized by terms with random phase changes due to ISI that add incoherently. This zero mean term adds to the noise, which will be approximated as Gaussian and lumped with other noise at the output of the moving average filter. We now have

\sum_{m} q_{m} \approx (2 N_{B} + 1) \sum_{n} j Δ_{n, k} w_{n} p_{d} (- n T)

(22)

and the output of the moving average filter is

e^{j ϕ_{R} (k T)} {1 + 1 + \sum_{n} j Δ_{n, k} w_{n} p_{d} (- n T)} + {\tilde{n}}_{k}^{'} + {\tilde{n}}_{k}^{″}

(23)

Taking the argument of the moving average filter output to get our phase estimate ϕ̂_k, we have

{\hat{ϕ}}_{k} \approx ϕ_{R} (k T) + Arg {1 + \sum_{n} j Δ_{n, k} w_{n} p_{d} (- n T)} + n_{k}^{'} + n_{k}^{″}

(24)

Consider the summation. The phase increment Δ_n,k is small for small index n since the phase holds constant over 2N_B samples. As the tap index n grows, so does the phase increment, however, the filter coefficients {w_n} decay exponentially. Therefore the sum is small compared to one and we can use the approximation Arg{1 + h} ≈ Im{h}.

{\hat{ϕ}}_{k} \approx ϕ_{R} (k T) + \sum_{n} Δ_{n, k} Re {w_{n} p_{d} (- n T)} + n_{k}^{'} + n_{k}^{″}

(25)

θ_{k} = \sum_{m, n, i \neq [m]} Δ_{n, k} Re \frac{x_{i}}{x_{[m]}} w_{n} p_{d} [(m - n) T - i T_{s}]

(26)

using arguments similar to those just presented.

Finally, this analysis assumed zero transmitter phase to simplify our derivation. As we have a linear system, and we have assumed the CD compensating filter exactly compensated the chromatic dispersion, we can use the superposition principle to include the transmitter phase noise and the ISI phase contribution to conclude

{\hat{ϕ}}_{k} \approx ϕ_{R} (k T) + ϕ_{T} (k T) + θ_{k} + \sum_{n} Δ_{n, k} Re {w_{n} p_{d} (- n T)} + n_{k}^{'} + n_{k}^{″}

(27)

8. Appendix 2

The decision statistic y_k is the result of derotating the output of the CD compensating filter by ϕ̂_k. From Eq. (14), y_k = r_ke^−jϕ̂_k is

y_{k} = \sum_{n} w_{n} \sum_{i} x_{i} p_{d} [(k - n) T - i T_{s}] e^{j ϕ_{R} ((k - n) T)} e^{- j {\hat{ϕ}}_{k}} + n_{k}

(28)

Using Eq. (27) with zero transmitter phase noise, ϕ_T = 0,

\begin{array}{l} y_{k} & = & \sum_{n} w_{n} \sum_{i} x_{i} p_{d} [(k - n) T - i T_{s}] \\ e^{j Δ_{n, k}} \cdot e^{- j Re \sum_{l} w_{l} p_{d} (- l T) Δ_{l, k}} + {\tilde{n}}_{k} \end{array}

(29)

where we lump noise from ϕ̂_k into a new additive noise term. We can write the exponentials in Eq. (29) as

\begin{array}{l} e^{j Δ_{n, k}} \prod_{l} e^{- j Δ_{l, k} Re {w_{l} p_{d} (- l T)}} \\ \approx (1 + j Δ_{n, k}) \prod_{l} (1 - j Δ_{l, k} Re {w_{l} p_{d} (- l T)}) \\ \approx 1 + j Δ_{n, k} - \sum_{l} j Δ_{l, k} Re {w_{l} p_{d} (- l T)} \end{array}

(30)

where we used the small angle approximation and only kept terms of first order in the product. Thus we have

\begin{array}{l} y_{k} & = & (1 - \sum_{l} j Δ_{l, k} Re {w_{l} p_{d} (- l T)}) \\ \cdot \sum_{n} w_{n} \sum_{i} x_{i} p_{d} [(k - n) T - i T_{s}] \\ + \sum_{n} j Δ_{n, k} w_{n} \sum_{i} x_{i} p_{d} [(k - n) T - i T_{s}] + {\tilde{n}}_{k} \\ \approx & x_{[k]} - x_{[k]} \sum_{l} j Δ_{l, k} Re {w_{l} p_{d} (- l T)} \\ + \sum_{n} j Δ_{n, k} w_{n} \sum_{i} x_{i} p_{d} [(k - n) T - i T_{s}] + {\tilde{n}}_{k} \end{array}

(31)

using Eq. (15). Defining

s_{n, k} = j \sum_{i} x_{i} w_{n} p_{d} [(k - n) T - i T_{s}] - j x_{[k]} Re {w_{n} p_{d} (- n T)}

(32)

we can write

y_{k} = x_{[k]} + \sum_{n} Δ_{n, k} s_{n, k} + {\tilde{n}}_{k}

(33)

where ñ_k is the cumulative noise in the estimation process with variance σ̃².

The conditional pdf f(y_k|X) is a complex Gaussian pdf with mean vector [Re{x_[_k_]}, Im{x_[_k_]}]^tr and a 2 × 2 covariance matrix C depending on s_n,k and σ̃² and σ² = 2πΔν_LO. To find elements of this matrix, we consider real and imaginary parts of the right hand side of Eq. (10). The diagonal elements of C correspond to the variance of real and imaginary parts (each being a sum of iid Gaussian random variables, the variance is the sum of variances). The off-diagonal elements of C correspond to the covariances of the real and imaginary parts, which can be easily simplified using the fact that many terms in the covariance are products of independent random variables with zero mean. This matrix is the same as that in [13

A. Lau, T. Shen, W. Shieh, and K. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express 18, 17239–17251 (2010). [CrossRef] [PubMed]

], however using Eq. (8) for s_n,k. We repeat the equations for the covariance matrix here for completeness.

C = σ^{2} [\begin{matrix} c_{11} + {\tilde{σ}}^{2} / σ^{2} & c_{12} \\ c_{21} & c_{22} + {\tilde{σ}}^{2} / σ^{2} \end{matrix}]

(34)

where

c_{11} = \sum_{i = - N}^{- 1} {(\sum_{n = - N}^{i} Re {s_{n, k}})}^{2} + \sum_{i = 1}^{N} {(\sum_{n = i}^{N} Re {s_{n, k}})}^{2},

(35)

c_{22} = \sum_{i = - N}^{- 1} {(\sum_{n = - N}^{i} Im {s_{n, k}})}^{2} + \sum_{i = 1}^{N} {(\sum_{n = i}^{N} Im {s_{n, k}})}^{2},

(36)

\begin{array}{r} c_{12} = c_{21} = \sum_{i = - N}^{- 1} (\sum_{n = - N}^{i} Re {s_{n, k}} \sum_{n = - N}^{i} Im {s_{n, k}}) + \\ \sum_{i = 1}^{N} (\sum_{n = i}^{N} Re {s_{n, k}} \sum_{n = i}^{N} Im {s_{n, k}}) \end{array}

(37)

References and links

1.	E. Ip, A. Lau, D. Barros, and J. Kahn, “Coherent detection in optical fiber systems,” Opt. Express 16, 753–791 (2008). [CrossRef] [PubMed]
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18.	G. Colavolpe, T. Foggi, E. Forestieri, and M. Secondini, “Impact of phase noise and compensation techniques in coherent optical systems,” J. Lightwave Technol. 29, 2790–2800 (2011). [CrossRef]
19.	M. Secondini, G. Meloni, T. Foggi, G. Colavolpe, L. Poti, and E. Forestieri, “Phase noise cancellation in coherent optical receivers by digital coherence enhancement,” in Proc. of ECOC (2010), Paper. P4.17.
20.	G. Colavolpe, T. Foggi, E. Forestieri, and M. Secondini, “Phase noise sensitivity and compensation techniques in long-haul coherent optical links,” in Proc. of Globcom (2010), pp. 1–6.
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22.	M. Magarini, A. Spalvieri, F. Vacondio, M. Bertolini, M. Pepe, and G. Gavioli, “Empirical modeling and simulation of phase noise in long-haul coherent optical transmission systems,” Opt. Express 19, 22455–22461 (2011). [CrossRef] [PubMed]
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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: August 27, 2012
Revised Manuscript: November 5, 2012
Manuscript Accepted: November 6, 2012
Published: November 12, 2012

Citation
Ramtin Farhoudi, Amirhossein Ghazisaeidi, and Leslie Ann Rusch, "Performance of carrier phase recovery for electronically dispersion compensated coherent systems," Opt. Express 20, 26568-26582 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-26568

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References

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