## Analytical modeling of a single channel nonlinear fiber optic system based on QPSK |

Optics Express, Vol. 20, Issue 25, pp. 27740-27755 (2012)

http://dx.doi.org/10.1364/OE.20.027740

Acrobat PDF (1185 KB)

### Abstract

A first order perturbation theory is used to develop analytical expressions for the power spectral density (PSD) of the nonlinear distortions due to intra-channel four-wave mixing (IFWM). For non-Gaussian pulses, the PSD can not be calculated analytically. However, using the stationary phase approximations, we found that convolutions become simple multiplications and a simple analytical expression for the PSD of the nonlinear distortion is found. The PSD of the nonlinear distortion is combined with the amplified spontaneous emission (ASE) PSD to obtain the total variance and bit error ratio (BER). The analytically estimated BER is found to be in good agreement with numerical simulations.

© 2012 OSA

## 1. Introduction

1. I. Shake, H. Takara, K. Mori, S. Kawanishi, and Y. Yamabayashi, “Influence of inter-bit four-wave mixing in optical TDM transmission,” Electron. Lett. **34**, 1600–1601 (1998). [CrossRef]

3. P. V. Mamyshev and N. A. Mamysheva, “Pulse-overlapped dispersion managed data transmission and intrachan-nel four-wave mixing,” Opt. Lett. **24**, 1454–1456 (1999). [CrossRef]

5. S. Kumar, “Intrachannel four-wave mixing in dispersion managed RZ systems,” IEEE Photon. Technol. Lett. **13**, 800–802 (2001). [CrossRef]

6. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express **16**, 15777–15810 (2008). [CrossRef] [PubMed]

12. S. Turitsyn, M. Sorokina, and S. Derevyanko, “Dispersion-dominated nonlinear fiber-optic channel,” Opt. Lett. **37**, 2931–2933 (2012). [CrossRef] [PubMed]

6. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express **16**, 15777–15810 (2008). [CrossRef] [PubMed]

7. D. Yang and S. Kumar, “Intra-channel four-wave mixing impairments in dispersion-managed coherent fiber-optic systems based on binary phase-shift keying,” J. Lightw. Technol. **27**, 2916–2923 (2009). [CrossRef]

8. A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling monlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express **20**, 7777–7791 (2012). [CrossRef] [PubMed]

9. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol. **30**, 1524–1539 (2012). [CrossRef]

*M*

^{3}where

*M*is the number of samples in frequency domain. In this paper we have developed an analytical expression for the PSD of intrachannel nonlinear distortion. With analytic simplifications, we found that the computational cost scales as ∼

*N*

^{2}

*M*/8 where

*N*is the total number of significant neighboring signal pulses. Typically,

*N*is smaller than

*M*leading to significant reduction in computational time. However, the direct comparison between these two approaches is not appropriate since Ref. [9

9. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol. **30**, 1524–1539 (2012). [CrossRef]

11. A. Mecozzi and Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol. **30**, 2011–2024 (2012). [CrossRef]

12. S. Turitsyn, M. Sorokina, and S. Derevyanko, “Dispersion-dominated nonlinear fiber-optic channel,” Opt. Lett. **37**, 2931–2933 (2012). [CrossRef] [PubMed]

## 2. Mathematical derivation of power spectral density

*T*is the symbol interval,

_{s}*p*(

*t*, 0) is the pulse shape function at

*z*= 0, and

*x*and

_{n}*y*are real random variables that take values ±1 with equal probability, respectively. The evolution of the optical field envelope is governed by the NLSE in the lossless form [7

_{n}7. D. Yang and S. Kumar, “Intra-channel four-wave mixing impairments in dispersion-managed coherent fiber-optic systems based on binary phase-shift keying,” J. Lightw. Technol. **27**, 2916–2923 (2009). [CrossRef]

*β*

_{2}is the dispersion coefficient,

*γ*is the nonlinear coefficient,

*a*

^{2}(

*z*) = exp(−

*αz*) between amplifiers, and

*α*is fiber loss coefficient.

*a*

^{2}(

*z*) is the loss/gain profile that includes the step amplification at the beginning of each span. Using the perturbation technique [13

13. S. Kumar and D. Yang, “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightw. Technol. **23**, 2073–2080 (2005). [CrossRef]

*u*

_{0}represents the 0

*order solution which satisfies The first order correction*

^{th}*u*

_{1}is Taking the Fourier transform of Eq. (6), we find where ℱ denotes the Fourier transform. Assuming the perfect dispersion compensation at the receiver and with

*ũ*

_{1}(

*f*, 0) = 0, Eq. (7) is solved to yield, where

*L*is the total transmission distance. The solution of Eq. (5) with the initial condition given by Eq. (1) is where From Eq. (8), we have where “*” denotes convolution. Using Eq. (11) in Eq. (14), we find

_{tot}*N*/2 to

*N*/2. Substituting Eq. (15) into Eq. (10), we find The distortion due to fiber nonlinearity is

*δũ*=

_{NL}*γu*

_{1}. The PSD of the nonlinear distortion is defined as where

*E*{} denotes the ensemble average. Using Eq. (17), Eq. (19) can be written as

### 2.1. ND-IFWM

*l*≠

*m*≠

*n*and

*l*′ ≠

*m*′ ≠

*n*′. For QPSK signals, we have where

*δ*is Kronecker delta function. Using Eqs. (21)–(23), Eq. (20) becomes, Equation (24) can be rewritten as

*lT*,

_{s}*mT*, and

_{s}*nT*generate a echo pulse at

_{s}*qT*= (

_{s}*l*+

*m*−

*n*)

*T*[1

_{s}1. I. Shake, H. Takara, K. Mori, S. Kawanishi, and Y. Yamabayashi, “Influence of inter-bit four-wave mixing in optical TDM transmission,” Electron. Lett. **34**, 1600–1601 (1998). [CrossRef]

5. S. Kumar, “Intrachannel four-wave mixing in dispersion managed RZ systems,” IEEE Photon. Technol. Lett. **13**, 800–802 (2001). [CrossRef]

*q*term on the right-hand side (

^{th}*RHS*) of Eq. (25) represents the nonlinear distortion on the

*q*symbol interval. Due to symmetry, the ensemble average of the nonlinear distortion should be the same on each symbol interval. In the other words, each term on the

^{th}*RHS*of Eq. (25) should be equal, which yields where

### 2.2. D-IFWM

### 2.3. Correlation between D-IFWM and ND-IFWM

*l*=

*m*≠

*n*and

*l*′ =

*m*′ ≠

*n*′. Now since Therefore, from Eq. (20), we have

*ρ*(

_{NL}*f*) = 0. In other words, there is no correlation between D-IFWM and ND-IFWM.

### 2.4. Total PSD

*ρ*

_{ND}_{−IFWM}(

*f*), and

*ρ*

_{D}_{−IFWM}(

*f*) are given by Eqs. (26) and (31), respectively. For Gaussian pulses,

*X*̃

*(*

_{l,m,n}*f*) can be calculated as (see Appendix A) and

*T*

_{0}is 1/

*e*pulse width.

### 2.5. Stationary phase approximation

*p*̃(

*f*,

*z*) in Eq. (13), stationary phase approximation can be employed to approximate

*X*̃

*(*

_{l,m,n}*f*). Stationary phase method is a standard technique for evaluating the integrals of the form [14] where

*y*(

*x*) is a fast-varying function of

*x*over most of the range of integration and

*G*(

*x*) is a slowly varying function. At the rapidly varying regions of

*y*(

*x*), the contribution to the integral is approximately zero as the area under the high frequency sinusoids with its slowly varying envelope

*G*(

*x*) is close to zero. The only significant contributions to the integral occurs in the regions where

*dy*/

*dx*= 0, i.e. at the points where the phase is stationary. At the vicinity of stationary phase point,

*x*

_{0},

*y*(

*x*) may be written as Using Eq. (42), Eq. (41) may be approximated as Now returning to Eq. (16), it has double convolutions which can be analytically integrated using the stationary phase approximation when the dispersion is sufficiently large. Now Eq. (16) becomes (see Appendix B) Note that the convolution in Eq. (16) is hard to evaluate numerically unless the pulse shape is Gaussian. But the stationary phase approximation translates the convolutions into simple multiplications as shown in Eq. (45), which can be easily computed. When the Nyquist pulse such as sinc pulse is used, Eq. (45) can be further simplified. A sinc pulse has a rectangular spectrum, where

*B*= 1/

_{s}*T*. Using Eq. (46), Eq. (45) can be approximated as where

_{s}**Property 1:**

*X*

_{l,m,l+m}is invariant under the exchange of

*l*and

*m*. This property holds true in general (see Eq. (16)) even without the stationary phase approximation.

**Property 2:**

*p*(

*t*) is real and symmetric, it follows that

*p*̃(

*f*) is symmetric and from Eq. (45), it is easy to see that there is a mirror symmetry,

### 2.6. Variance

### 2.7. Computational cost

*N*= 6 and computational cost associated with SPM, IXPM and IFWM. When Property 1 and Property 2 are not used, the computational cost per frequency calculations are as shown in Table 1.

*N*(

*N*−1)/2. If both Property 1 and Property 2 are used, the cost per frequency for ND-IFWM and D-IFWM are

*N*

^{2}/4 and

*N*/2, respectively, and total computational cost per frequency (ND-IFWM + D-IFWM) is

*N*

^{2}/4+

*N*/2. If there are

*M*samples in the frequency domain, total computational cost is (

*N*

^{2}/4+

*N*/2)

*M*. In addition, if |

*l*+

*m*| >

*N*/2, |

*n*| >

*N*/2, and from Eq. (26), it follows that the signal pulse centered at

*nT*with |

_{s}*n*| >

*N*/2 does not contribute significantly for the formation of the echo pulse at

*t*= 0 and hence, such a triplet may be ignored. With this approximation, total computational cost scales as ∼

*N*

^{2}

*M*/8 for large

*N*. Validation of the stationary phase approximation is carried out in section 3.1.

## 3. Results and discussions

*α*= 0.2 dB/km, fiber nonlinear coefficient

*γ*= 1.1 (W.km)

^{−1}, symbol rate = 25 Gbaud, and modulation = QPSK. Gaussian pulses with full width at half maximum (FWHM) of

*T*= 20 psec are launched to the fiber to obtain Figs. 2, 3 and 6. Amplifiers spacing is 80 km. Multi-span fiber-optic system is simulated here. The dispersion is uncompensated in each span. At the receiver, the transmission fiber dispersion is fully compensated either optically or electrically. Laser phase noise, polarization effects, and the coherent receiver imperfections are ignored since the primary focus of this paper is to validate our analytical model for the fiber nonlinear impairments. For numerical simulation, a pseudo-random bit sequence (PRBS) of length 2

_{FWHM}^{15}− 1 is used for the calculation of the PSD as well as BER. A Gaussian filter with a full bandwidth of 100 GHz is used as the receiver filter. The significant number of neighbors,

*N*= 20. Equation (45) provides a guideline for choosing the frequency resolution. The minimum frequency shift of the pulse spectrum is

*πT*/

_{s}*δ*. If the frequency resolution Δ

*f*is larger than

*πT*/

_{s}*δ*, errors occur in the computation of

*X*̃

_{l,m,l+m}(

*f*,

*z*). For a 20-span system and for |

*β*

_{2}| = 21 ps

^{2}/km, Δ

*f*=

*πT*/

_{s}*δ*= 0.189 GHz. With the 100 GHz bandwidth of the receiver filter, number of frequency samples,

*M*= 527. Since

*N*is much smaller than

*M*, in this example the computational cost savings would be ∼

*O*((527/20)

^{2}).

*iθ*) where

*θ*is found adaptively. We used adaptive least mean square (LMS) equalizer to compensate the phase shift. We assumed the following parameters for the LMS algorithm: Number of filter taps = 10, number of training sequence = 2

^{10}, number of samples/symbol = 2, and step size = 0.1. The numerical PSD due to the nonlinear distortion is computed by subtracting the optical field envelope at the transmitter from that at the receiver (after dispersion compensation and the phase shift removal) and then taking the Fourier transform of the difference. To account for the bit-pattern variations, numerical simulations were performed 20 times with different bit patterns and the average PSD is computed. Over a range of powers that is of practical interest for QPSK-based system (−6 dBm to 0 dBm), the discrepancy between the analytical model and the numerical model is less than 4% and 12% in 5-span (Fig. 2(a)) and 20-span (Fig. 2(b)) systems, respectively. Figures 3(a) and 3(b) show the analytical and numerical variances versus the accumulated dispersion for 5-spans and 20-spans systems, respectively. As can be seen, there is a good agreement between numerical simulations and analytical results for a 5-span system. For a 20-span system, there is a small discrepancy at large launch powers which is probably due to the truncation of the field up to the first order (see Eq. (4)). For the 5-span system, when the dispersion is small, the variance of the nonlinear distortion is quite small. However, it grows quickly and beyond 7 ps/nm.km, it decays slowly. For the 20-span system, the variance decreases slowly with the transmission fiber dispersion.

### 3.1. Stationary phase approximation with raised-cosine pulse

*p*̃(

*f*) is of the form [15

15. U. Madhow, “Chapter 3: Demodulation,” in *Fundamentals of Digital Communication* (Cambridge University Press, 2008). [CrossRef]

*a*is the roll-off factor, and

*T*is the symbol time interval.

_{s}*X*̃

*can not be calculated exactly for this pulse shape. Using the stationary phase approximation (Eq. (45)), total PSD and the variance is calculated. The following parameters are used for raised-cosine pulse. Roll-off factor*

_{l,m,n}*a*= 1 and symbol time interval

*T*= 40 psec are assumed. Figures 4(a) and 4(b) show the variance as a function of the launch peak power for a 5-span and 20-span systems, respectively. In the range of −6 dBm to 0 dBm, the discrepancy between the analytical model and the numerical model is less than 7% in Figs. 4(a) and 4(b). Figures 5(a) and 5(b) show the dependence of the variance on the fiber dispersion for a 5-span and 20-span systems, respectively. For a 5-span system, when the dispersion is very low, we see that the stationary phase approximation becomes inaccurate. This inaccuracy is due to the fact that the phase (∝

_{s}*β*

_{2}

*f*

^{2}) does not vary rapidly at low dispersions. However, practical fiber-optic systems use fibers with moderate to large dispersions to suppress nonlinear effects and therefore, stationary phase approximation leads to reasonably accurate results for dispersion parameter range that is of practical interest.

### 3.2. BER calculation

*ρ*(

_{ASE}*f*) is ASE noise PSD due to amplifiers, where

*N*is number of amplifiers,

_{A}*L*is amplifiers’ spacing,

_{a,j}*α*is fiber loss,

*h*is Planck constant,

*f̄*is the mean frequency of the channel, and

*n*is spontaneous noise factor. Using Eq. (59), total noise variance

_{sp}*σ*is calculated and the probability of error

_{tot}*P*for QPSK is given by [15

_{e}15. U. Madhow, “Chapter 3: Demodulation,” in *Fundamentals of Digital Communication* (Cambridge University Press, 2008). [CrossRef]

*Q*is Q-function [15

15. U. Madhow, “Chapter 3: Demodulation,” in *Fundamentals of Digital Communication* (Cambridge University Press, 2008). [CrossRef]

*SNR*is the signal to noise ratio, for Gaussian pulses with 50% duty cycle, and The Q-factor is defined as

*n*= 10 dB, and number of spans = 20. We chose a relatively large noise figure intentionally so as to reduce the computational time of Monte Carlo simulations at large launch power. We found that the maximum discrepancy between the analytical and numerical Q-factor is less than 0.6 dBQ

_{sp}_{20}which is attributed to non-Gaussian distribution of the IFWM pdf [7

7. D. Yang and S. Kumar, “Intra-channel four-wave mixing impairments in dispersion-managed coherent fiber-optic systems based on binary phase-shift keying,” J. Lightw. Technol. **27**, 2916–2923 (2009). [CrossRef]

**27**, 2916–2923 (2009). [CrossRef]

17. M. Nazarathy, “Accurate evaluation of bit-error rates of optical communication systems using the Gram-Charlier series,” IEEE Trans. Commun. **54**, 295–301 (2006). [CrossRef]

## 4. Conclusion

## 5. Appendix A: Gaussian pulse case

*π*

^{2}

*β*

_{2}

*z*=

*δ*. Using Eq. (68) in Eq. (76), we find

## 6. Appendix B: Stationary phase approximation

*p*(

*t*) is real and symmetric with respect to origin so that

*p*̃(

*f*) is real. Since

*θ*(

*f*

_{1}) is varying rapidly, the dominant contribution to the integral in Eq. (93) comes when or Substituting Eq. (95) in Eq. (92), we find Under the stationary phase approximation, Eq. (93) becomes where ″ denotes differentiating twice. Using Eqs. (95) and (98) in Eq. (97), we find Substituting Eq. (99) in Eq. (90), we find where Substituting Eq. (100) in Eq. (75) and simplifying, we obtain where Since

*θ*

_{2}is varying rapidly, we use the stationary phase approximation again to evaluate the integral in Eq. (102). Differentiating Eq. (104) with respect to

*f*

_{2}and setting the result to zero yields Proceeding as before, Eq. (102) can be simplified as

## References and links

1. | I. Shake, H. Takara, K. Mori, S. Kawanishi, and Y. Yamabayashi, “Influence of inter-bit four-wave mixing in optical TDM transmission,” Electron. Lett. |

2. | R. J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel crossphase modulation and four-wave mixing in high-speed TDM systems,” Electron. Lett. |

3. | P. V. Mamyshev and N. A. Mamysheva, “Pulse-overlapped dispersion managed data transmission and intrachan-nel four-wave mixing,” Opt. Lett. |

4. | A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. |

5. | S. Kumar, “Intrachannel four-wave mixing in dispersion managed RZ systems,” IEEE Photon. Technol. Lett. |

6. | M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express |

7. | D. Yang and S. Kumar, “Intra-channel four-wave mixing impairments in dispersion-managed coherent fiber-optic systems based on binary phase-shift keying,” J. Lightw. Technol. |

8. | A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling monlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express |

9. | A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol. |

10. | P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-Linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. |

11. | A. Mecozzi and Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol. |

12. | S. Turitsyn, M. Sorokina, and S. Derevyanko, “Dispersion-dominated nonlinear fiber-optic channel,” Opt. Lett. |

13. | S. Kumar and D. Yang, “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightw. Technol. |

14. | J. D. Jackson, |

15. | U. Madhow, “Chapter 3: Demodulation,” in |

16. | A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proceedings ECOC, (2010), pp. 1–3. |

17. | M. Nazarathy, “Accurate evaluation of bit-error rates of optical communication systems using the Gram-Charlier series,” IEEE Trans. Commun. |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: August 21, 2012

Revised Manuscript: November 12, 2012

Manuscript Accepted: November 15, 2012

Published: November 29, 2012

**Citation**

Shiva Kumar, Sina Naderi Shahi, and Dong Yang, "Analytical modeling of a single channel nonlinear fiber optic system based on QPSK," Opt. Express **20**, 27740-27755 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-25-27740

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### References

- I. Shake, H. Takara, K. Mori, S. Kawanishi, and Y. Yamabayashi, “Influence of inter-bit four-wave mixing in optical TDM transmission,” Electron. Lett.34, 1600–1601 (1998). [CrossRef]
- R. J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel crossphase modulation and four-wave mixing in high-speed TDM systems,” Electron. Lett.35, 1576–1578 (1999). [CrossRef]
- P. V. Mamyshev and N. A. Mamysheva, “Pulse-overlapped dispersion managed data transmission and intrachan-nel four-wave mixing,” Opt. Lett.24, 1454–1456 (1999). [CrossRef]
- A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett.12, 292–294 (2000).
- S. Kumar, “Intrachannel four-wave mixing in dispersion managed RZ systems,” IEEE Photon. Technol. Lett.13, 800–802 (2001). [CrossRef]
- M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express16, 15777–15810 (2008). [CrossRef] [PubMed]
- D. Yang and S. Kumar, “Intra-channel four-wave mixing impairments in dispersion-managed coherent fiber-optic systems based on binary phase-shift keying,” J. Lightw. Technol.27, 2916–2923 (2009). [CrossRef]
- A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling monlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express20, 7777–7791 (2012). [CrossRef] [PubMed]
- A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol.30, 1524–1539 (2012). [CrossRef]
- P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-Linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett.23, 742–744 (2011). [CrossRef]
- A. Mecozzi and Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol.30, 2011–2024 (2012). [CrossRef]
- S. Turitsyn, M. Sorokina, and S. Derevyanko, “Dispersion-dominated nonlinear fiber-optic channel,” Opt. Lett.37, 2931–2933 (2012). [CrossRef] [PubMed]
- S. Kumar and D. Yang, “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightw. Technol.23, 2073–2080 (2005). [CrossRef]
- J. D. Jackson, Classical Electrodynamics (Wiley, 1998).
- U. Madhow, “Chapter 3: Demodulation,” in Fundamentals of Digital Communication (Cambridge University Press, 2008). [CrossRef]
- A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proceedings ECOC, (2010), pp. 1–3.
- M. Nazarathy, “Accurate evaluation of bit-error rates of optical communication systems using the Gram-Charlier series,” IEEE Trans. Commun.54, 295–301 (2006). [CrossRef]

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