# Analytical Investigations on Carrier Phase Recovery in Dispersion-Unmanaged n-PSK Coherent Optical Communication Systems

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## Abstract

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## 1. Introduction

## 2. Laser Phase Noise and Equalization Enhanced Phase Noise

_{Tx}and Δf

_{LO}are the 3-dB linewidths (assuming the Lorentzian distribution) of the transmitter laser and the LO laser, respectively, and T

_{S}is the symbol period of the coherent transmission system. It can be found that the variance of the laser phase noise decreases with the increment of the signal symbol rate R

_{S}= 1/T

_{S}.

_{LO}is the central frequency of the LO laser, which is equal to the central frequency of the transmitter laser f

_{Tx}in the homodyne optical communication systems, D is the CD coefficient of the transmission fiber, L is the length of the transmission fiber, R

_{S}is the signal symbol rate of the communication system, and λ = c/f

_{Tx}= c/f

_{LO}is the central wavelength of the optical carrier wave.

_{Eff}can be employed to describe the total phase noise variance in the EDC-based n-PSK coherent optical communication systems and it can be expressed as follows:

_{0}= 60.69 km. It means that at this transmission distance, the laser phase noise and the EEPN will have the same impact on the degradation of the performance of the 32-Gbaud n-PSK optical transmission systems.

## 3. Analysis of Carrier Phase Recovery Approaches

#### 3.1. One-Tap Normalized Least-Mean-Square (LMS) Carrier Phase Recovery

_{NLMS}(k) is the tap weight of the one-tap normalized LMS equalizer, d(k) is the desired output symbol after the carrier phase recovery, e(k) is the estimation error between the output symbol and the desired output symbol, and μ is the step size of the one-tap normalized LMS algorithm.

#### 3.2. Block-Wise Average Carrier Phase Recovery

_{BWA}is the block length in the block-wise average algorithm, and $\lceil x\rceil $ means the closest integer lager than x.

#### 3.3. Viterbi-Viterbi Carrier Phase Recovery

_{VV}is the block length of the Viterbi-Viterbi algorithm, and should be an odd value of e.g., 1, 3, 5, 7…

## 4. Results and Discussion

#### 4.1. Results

_{BWA}= 11 is employed in all subsequent analyses, if the value is not specified. Based on Equations (12) and (13), the performance of the block-wise average carrier phase recovery in the coherent optical communication systems using different modulation formats is shown in Figure 8b, where the block length is 11. It can be found in Figure 8b that the block-wise average carrier phase recovery algorithm is also very sensitive to the phase noise variance and the modulation formats, when phase noise variance is less than 0.1.

_{VV}= 11 is also selected as an example in the Viterbi-Viterbi carrier phase recovery to consider the mitigation of both the phase noise and the amplitude noise in practical applications. Based on Equation (15), the performance of the Viterbi-Viterbi carrier phase recovery in the coherent optical communication systems using different modulation formats is shown in Figure 11b, where the block length is 11. It is found in Figure 11b that the Viterbi-Viterbi carrier phase recovery algorithm is also very sensitive to the phase noise variance and the modulation formats, when phase noise variance is less than 0.15.

#### 4.2. Ideal Spectral Efficiency in Carrier Phase Recovery

_{C}in the n-PSK optical fiber communication systems (assuming an ideal hard-decision forward error correction coding) can be expressed as follows [53],

_{p}is the number of polarization states. The BER limits in Equation (16) can be obtained from the BER floors in the three carrier phase recovery approaches according to Equations (9), (12) and (15), respectively.

#### 4.3. Complexity of Carrier Phase Recovery Approaches

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Principle of equalization enhanced phase noise in electronic dispersion compensation based n-PSK coherent optical transmission system. PRBS: pseudo random bit sequence, N(t): additive white Gaussian noise (AWGN), e.g., amplified spontaneous emission (ASE) noise from optical amplifiers, ADC: analog-to-digital convertor.

**Figure 5.**Bit-error-rate (BER) floors versus phase noise variance in the one-tap normalized LMS carrier phase recovery in the coherent optical transmission systems using different modulation formats.

**Figure 6.**BER floors versus laser linewidths in the one-tap normalized LMS carrier phase recovery in the optical fiber transmission systems using different modulation formats. The indicated linewidth value is the 3-dB linewidth for both the Tx and the LO lasers.

**Figure 7.**BER floors versus transmission distances in the one-tap normalized LMS carrier phase recovery in the coherent optical transmission systems using different modulation formats, considering the equalization enhanced phase noise. Both the Tx and LO lasers linewidths are 1 MHz.

**Figure 8.**BER floors versus phase noise variances in the block-wise average carrier phase recovery in the coherent optical transmission systems. (

**a**) Different block lengths in the 8-PSK transmission system; (

**b**) different modulation formats with the block length of 11.

**Figure 9.**BER floors versus laser linewidths in the block-wise average carrier phase recovery in the coherent optical transmission systems using different modulation formats. The block length is 11, and the indicated linewidth value is the 3-dB linewidth for both the Tx and the LO lasers.

**Figure 10.**BER floors versus transmission distances in the block-wise average carrier phase recovery in the coherent optical transmission systems using different modulation formats. The block length is 11, and the linewidth of both the Tx and the LO lasers are 1 MHz.

**Figure 11.**BER floors versus phase noise variances in the Viterbi-Viterbi carrier phase recovery in the coherent optical transmission systems. (

**a**) Different block lengths in the 8-PSK transmission system; (

**b**) different modulation formats with the block length of 11.

**Figure 12.**BER floors versus laser linewidths in the Viterbi-Viterbi carrier phase recovery in the coherent optical transmission systems using different modulation formats. The block length is 11, and the indicated linewidth value is the 3-dB linewidth for both the Tx and the LO lasers.

**Figure 13.**BER floors versus transmission distances in the Viterbi-Viterbi carrier phase recovery in the coherent optical transmission systems using different modulation formats. The block length is 11, and the linewidth of both the Tx and the LO lasers are 1 MHz.

**Figure 14.**BER floors versus different phase noise variances in the three carrier phase recovery algorithms in the 8-PSK optical fiber communication systems. (

**a**) Block length of the BWA and VV algorithms is 5; (

**b**) block length of the BWA and VV algorithms is 11; (

**c**) block length of the BWA and VV algorithms is 17.

**Figure 15.**BER floors versus different phase noise variances in the three carrier phase recovery algorithms in the optical fiber communication systems using different modulation formats. Block lengths of the BWA and VV algorithms are both 11. (

**a**) QPSK system; (

**b**) 8-PSK system; (

**c**) 16-PSK system; (

**d**) 32-PSK system.

**Figure 16.**BER floors versus different transmission distances in the three carrier phase recovery algorithms in the optical fiber communication systems using different modulation formats. Linewidths of the transmitter and LO lasers are both 1 MHz, and block lengths of the BWA and VV CPR algorithms are both 11. (

**a**) QPSK system; (

**b**) 8-PSK system; (

**c**) 16-PSK system; (

**d**) 32-PSK system.

One-Tap Normalized LMS | Block-Wise Average | Viterbi-Viterbi |
---|---|---|

5 | n | n |

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## Share and Cite

**MDPI and ACS Style**

Xu, T.; Jacobsen, G.; Popov, S.; Li, J.; Liu, T.; Zhang, Y.; Bayvel, P. Analytical Investigations on Carrier Phase Recovery in Dispersion-Unmanaged *n*-PSK Coherent Optical Communication Systems. *Photonics* **2016**, *3*, 51.
https://doi.org/10.3390/photonics3040051

**AMA Style**

Xu T, Jacobsen G, Popov S, Li J, Liu T, Zhang Y, Bayvel P. Analytical Investigations on Carrier Phase Recovery in Dispersion-Unmanaged *n*-PSK Coherent Optical Communication Systems. *Photonics*. 2016; 3(4):51.
https://doi.org/10.3390/photonics3040051

**Chicago/Turabian Style**

Xu, Tianhua, Gunnar Jacobsen, Sergei Popov, Jie Li, Tiegen Liu, Yimo Zhang, and Polina Bayvel. 2016. "Analytical Investigations on Carrier Phase Recovery in Dispersion-Unmanaged *n*-PSK Coherent Optical Communication Systems" *Photonics* 3, no. 4: 51.
https://doi.org/10.3390/photonics3040051