## Modified constant modulus algorithm for polarization-switched QPSK |

Optics Express, Vol. 19, Issue 8, pp. 7734-7741 (2011)

http://dx.doi.org/10.1364/OE.19.007734

Acrobat PDF (741 KB)

### Abstract

By using a generalized cost function, a modified constant modulus algorithm (CMA) that allows polarization demultiplexing and equalization of polarization-switched QPSK is found. An implementation that allows easy switching between the conventional and the modified CMA is described. Using numerical simulations, the suggested algorithm is shown to have similar performance for polarization-switched QPSK as CMA has for polarization-multiplexed QPSK.

© 2011 OSA

## 1. Introduction

2. M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Express **17**, 10814–10819 (2009). [CrossRef] [PubMed]

3. E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightwave Technol. **27**, 5115–5126 (2009). [CrossRef]

^{−3}.

4. P. Poggiolini, G. Bosco, A. Carena, V. Curri, and F. Forghieri, “Performance evaluation of coherent WDM PS-QPSK (HEXA) accounting for non-linear fiber propagation effects,” Opt. Express **18**, 11360–11371 (2010). [CrossRef] [PubMed]

5. P. Serena, A. Vannucci, and A. Bononi, “The performance of polarization switched-QPSK (PS-QPSK) in dispersion managed WDM transmissions,” in European Conference on Optical Communication (ECOC) (2010), paper Th.10.E.2. [CrossRef]

6. M. Sjödin, P. Johannisson, H. Wymeersch, P. Andrekson, and M. Karlsson, “Experimental comparison of polarization-switched QPSK and polarization-multiplexed QPSK at 30 Gbit/s,” Opt. Express (submitted). [PubMed]

*PM-CMA*and to the new algorithm as

*PS-CMA*.

*Notation:*Vectors are denoted in bold letters (e.g.,

**a**), and matrices in capital bold letters (e.g.,

**A**). Transposition is written as

**a**

^{T}, conjugation as

**a**, and conjugate transpose is denoted by

^{*}**a**

^{H}. The identity matrix is written as

**I**and the expectation operator is denoted by 𝔼[·].

## 2. The problem with PM-CMA when using PS-QPSK

*E*+

_{x,r}*jE*,

_{x,i}*E*+

_{y,r}*jE*)

_{y,i}^{T}[2

2. M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Express **17**, 10814–10819 (2009). [CrossRef] [PubMed]

3. E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightwave Technol. **27**, 5115–5126 (2009). [CrossRef]

_{PM}= {(±1 ±

*j*, ±1 ±

*j*)

^{T}/2} for any choice of signs. It may seem obvious that PM-CMA cannot be used for PS-QPSK since the two polarizations do not have a constant modulus, but PS-QPSK can alternatively be described as a subset of PM-QPSK [2

2. M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Express **17**, 10814–10819 (2009). [CrossRef] [PubMed]

_{PS}= {(± 1 ±

*j*, ±1 ±

*j*)

^{T}/2} for any sign combination with even parity, i.e.,

*E*> 0. The mapping from

_{x,r}E_{x,i}E_{y,r}E_{y,i}_{PS}to ℳ

_{PS}can be done with a unitary Jones matrix and can therefore be performed both by the fiber and by the demultiplexing algorithm. To show this, we introduce the two unitary matrices The

**T**

_{1}matrix is a

*π*/4 rotation of linearly polarized light and

**T**

_{2}is a phase retardation of the y polarization. Applying the

**T**

_{1}rotation to

_{PS}and ℳ

_{PM}, respectively, the constellations in Figs. 1a and 1b are obtained. (Only the x polarization is plotted since the y polarization is identical.) The constellation in Fig. 1a corresponds to ℳ

_{PS}and in Fig. 1b we see four additional points since all points no longer overlap after a

**T**

_{1}rotation.

**T**

_{1}

**T**

_{2}

**T**

_{1}and for this example we choose

*φ*=

*π*/6. The resulting constellations for PS-QPSK and PM-QPSK are plotted in Figs. 1c and 1d, respectively. For PM-QPSK, the constellation no longer has a constant modulus and this will be corrected by PM-CMA. However, for PS-QPSK the constellation conserves its constant modulus property since all points lie on the circle. Thus, the problem with PM-CMA for PS-QPSK is

*not*that there is no demultiplexing matrix that gives a constant modulus constellation. Instead, the problem is that obtaining a constant modulus constellation after equalization is not sufficient to have proper demultiplexing for PS-QPSK.

## 3. Description of the PS-CMA

### 3.1. Obtaining the cost function for PS-CMA

*P*

_{0}is the average power level for each polarization [7

7. D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. **28**, 1867–1875 (1980). [CrossRef]

*P*

_{0}and

*s*

_{1}= 0 for all symbols in ℳ

_{PM}. However, for PS-QPSK the alphabet ℳ

_{PS}is mapped to only two points on the Poincaré sphere since (

*s*

_{0},

*s*

_{1},

*s*

_{2},

*s*

_{3}) = (1,±1,0,0). Thus, minimizing

*P*and ii) that

*Q*must be positive for PS-QPSK. This more general cost function can be used to find update expressions for both PS-CMA and PM-CMA and switching between these two algorithms is done simply by changing the numerical parameters. This is convenient if PS-QPSK serves as a fall-back for PM-QPSK [4

4. P. Poggiolini, G. Bosco, A. Carena, V. Curri, and F. Forghieri, “Performance evaluation of coherent WDM PS-QPSK (HEXA) accounting for non-linear fiber propagation effects,” Opt. Express **18**, 11360–11371 (2010). [CrossRef] [PubMed]

*Q*= 1/2 for PS-CMA and should take

*Q*= −1/2 for PM-CMA. The parameter

*P*is equal to the total average power for PS-QPSK and must be set to the average power of each polarization

*P*

_{0}to reproduce the PM-CMA. We may also need to adjust the step size.

### 3.2. Equalizer update expressions

### 3.3. Singularities in the PS-CMA and the PM-CMA

*singularity problem*. This behavior is not penalized by the cost function and various ways to work around the problem have been suggested. For example, the problem is avoided in the single-tap case by enforcing the condition that the demultiplexing matrix should be unitary [10, 11

11. I. Roudas, A. Vgenis, C. S. Petrou, D. Toumpakaris, J. Hurley, M. Sauer, J. Downie, Y. Mauro, and S. Raghavan, “Optimal polarization demultiplexing for coherent optical communications systems,” J. Lightwave Technol. **28**, 1121–1134 (2010). [CrossRef]

**A**, a demultiplexing matrix

**B**, and two singular constant matrices according to The total transfer matrix

**BA**=

**I**when

**B**=

**A**

^{−1}and this corresponds to zero cost in the absence of noise. However, the PM-CMA cost function does not define

**B**uniquely since also

**B**=

**M**

_{1}

**A**

^{−1}leads to zero cost for any symbol from ℳ

_{PM}. This is the singularity problem for PM-CMA and there is a similar problem for PS-CMA, which we see by studying

**B**=

**M**

_{2}

**A**

^{−1}. Choosing any symbol from ℳ

_{PS}, the cost is zero and the second channel will carry zero power. The third bit, which can be viewed as the selection of the launch polarization, has therefore been lost. It is fair to say that PS-CMA is neither better nor worse than PM-CMA in this respect but it is easy to detect the failure for PS-CMA by checking the output channel power.

## 4. Numerical evaluation of PS-CMA and PM-CMA

_{PS}or ℳ

_{PM}and complex white Gaussian noise (AWGN) is added. For the convergence study, the fiber Jones matrix,

**A**, is drawn uniformly from the set of 2 × 2 unitary matrices and is then held constant. The demultiplexing matrix

**B**is initially set to

**I**. Running many simulations using different

**A**matrices we compute the probability of being above a given SNR penalty threshold at every iteration. We have set the convergence threshold value to be 1 dB SNR penalty and used the found probabilities as a measure of the convergence rate. The step size has been selected as to maximize the final probability of being below 1 dB penalty in all cases. The polarization tracking study is made by choosing the fiber matrix time evolution to be where

*ϕ*(

*t*) is a linearly evolving phase. Running the algorithms on long sequences of symbols, we find the averaged value of the SNR penalty. Plotting this penalty as a function of the angular frequency of

*ϕ*provides a quantitative measurement of the algorithm tracking capability.

## 5. Numerical results for the single-tap equalizer

**B**

*in every iteration. The blue and the red curves show the result for PS-QPSK and PM-QPSK, respectively, when noise has been added to make the BER = 10*

_{k}^{−3}. The black curve shows the case for PS-QPSK with equal SNR as for PM-QPSK.

^{−3}. Comparing the results at equal amount of added noise, PS-CMA performs somewhat better than PM-CMA. It is seen that without orthogonalization, both algorithms have a probability floor due to the singularity. At the SNR values used, the probability of failure is 2–4%. The step size used for each case is indicated close to each curve.

^{−3}rad/symbol corresponds to 10 Mrad/s at 10 Gbaud, which is a very high polarization rotation rate. For the longest step size at BER = 10

^{−3}, PS-CMA shows an increased steady-state SNR penalty. However, as seen in Fig. 2b, both better steady-state and tracking performance than PM-CMA can be obtained by reducing the step size.

*μ*is not dimensionless but depends on the signal power. Since PS-QPSK is a 4D modulation format, we have used unit energy 4D symbols. Thus, the energy per polarization is 1/2, which leads to an increase of

*μ*by a factor of four. For the convergence phase, it is beneficial to use a long step size in order to rapidly converge close to the demultiplexing matrix but for the following tracking phase, the step size can be reduced to improve the steady state performance. As seen from the tracking study, both algorithms are capable of tracking polarization changes that are much faster than what is typically seen in practice. Therefore, the step size should be decreased in order to reduce the steady-state penalty to a negligible value.

## 6. The impact from PMD and PDL

**A**matrix. This modified matrix,

**Ã**, was set up as the product of a random unitary matrix and a PDL matrix that had one transparent axis and one orthogonal lossy axis. The angle for the lossy axis was randomly selected from a uniform distribution. The obtained average SNR penalty has been plotted as a function of the amount of PDL in Fig. 3. In the worst possible alignment of the PDL [14], only one of the channels is attenuated and the following noise loading will lead to an SNR degradation equal to the amount of PDL. In the best case, the PDL is equally divided between the two PM channels. The worst and best cases are indicated by the dashed black lines. The averaged penalty is almost identical for all three simulated cases. As an example, a PDL of 3 dB causes an average SNR penalty of 2.4 dB. In these simulations, the convergence has been guaranteed by using an initial matrix that is sufficiently close to the system transfer matrix.

15. M. Kuschnerov, F. N. Hauske, K. Piyawanno, B. Spinnler, M. S. Alfiad, A. Napoli, and B. Lankl, “DSP for coherent single-carrier receivers,” J. Lightwave Technol. **27**, 3614–3622 (2009). [CrossRef]

## 7. Conclusion

## Acknowledgments

## References and links

1. | H. Bülow, “Polarization QAM modulation (POL-QAM) for coherent detection schemes,” in Optical Fiber Communication Conference (OFC) (2009), paper OWG2. |

2. | M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Express |

3. | E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightwave Technol. |

4. | P. Poggiolini, G. Bosco, A. Carena, V. Curri, and F. Forghieri, “Performance evaluation of coherent WDM PS-QPSK (HEXA) accounting for non-linear fiber propagation effects,” Opt. Express |

5. | P. Serena, A. Vannucci, and A. Bononi, “The performance of polarization switched-QPSK (PS-QPSK) in dispersion managed WDM transmissions,” in European Conference on Optical Communication (ECOC) (2010), paper Th.10.E.2. [CrossRef] |

6. | M. Sjödin, P. Johannisson, H. Wymeersch, P. Andrekson, and M. Karlsson, “Experimental comparison of polarization-switched QPSK and polarization-multiplexed QPSK at 30 Gbit/s,” Opt. Express (submitted). [PubMed] |

7. | D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. |

8. | A. Hjørungnes and D. Gesbert, “Complex-valued matrix differentiation: Techniques and key results,” IEEE Trans. Signal Process. |

9. | S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express |

10. | K. Kikuchi, “Polarization-demultiplexing algorithm in the digital coherent receiver,” in IEEE/LEOS Summer Topical Meetings (LEOSST) (2008), paper MC2.2. |

11. | I. Roudas, A. Vgenis, C. S. Petrou, D. Toumpakaris, J. Hurley, M. Sauer, J. Downie, Y. Mauro, and S. Raghavan, “Optimal polarization demultiplexing for coherent optical communications systems,” J. Lightwave Technol. |

12. | L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in Optical Fiber Communication Conference (OFC) (2009), paper OMT2. |

13. | U. Madhow, |

14. | T. Duthel, C. R. S. Fludger, J. Geyer, and C. Schulien, “Impact of polarisation dependent loss on coherent POLMUX-NRZ-DQPSK,” in Optical Fiber Communication Conference (OFC) (2008), paper OThU5. |

15. | M. Kuschnerov, F. N. Hauske, K. Piyawanno, B. Spinnler, M. S. Alfiad, A. Napoli, and B. Lankl, “DSP for coherent single-carrier receivers,” J. Lightwave Technol. |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

(060.2330) Fiber optics and optical communications : Fiber optics communications

(060.4080) Fiber optics and optical communications : Modulation

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: February 16, 2011

Revised Manuscript: March 27, 2011

Manuscript Accepted: March 27, 2011

Published: April 6, 2011

**Citation**

Pontus Johannisson, Martin Sjödin, Magnus Karlsson, Henk Wymeersch, Erik Agrell, and Peter A. Andrekson, "Modified constant modulus algorithm for polarization-switched QPSK," Opt. Express **19**, 7734-7741 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-8-7734

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### References

- H. Bülow, “Polarization QAM modulation (POL-QAM) for coherent detection schemes,” in Optical Fiber Communication Conference (OFC) (2009), paper OWG2.
- M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Express 17, 10814–10819 (2009). [CrossRef] [PubMed]
- E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightwave Technol. 27, 5115–5126 (2009). [CrossRef]
- P. Poggiolini, G. Bosco, A. Carena, V. Curri, and F. Forghieri, “Performance evaluation of coherent WDM PS-QPSK (HEXA) accounting for non-linear fiber propagation effects,” Opt. Express 18, 11360–11371 (2010). [CrossRef] [PubMed]
- P. Serena, A. Vannucci, and A. Bononi, “The performance of polarization switched-QPSK (PS-QPSK) in dispersion managed WDM transmissions,” in European Conference on Optical Communication (ECOC) (2010), paper Th.10.E.2. [CrossRef]
- M. Sjödin, P. Johannisson, H. Wymeersch, P. Andrekson, and M. Karlsson, “Experimental comparison of polarization-switched QPSK and polarization-multiplexed QPSK at 30 Gbit/s,” Opt. Express (submitted). [PubMed]
- D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. 28, 1867–1875 (1980). [CrossRef]
- A. Hjørungnes and D. Gesbert, “Complex-valued matrix differentiation: Techniques and key results,” IEEE Trans. Signal Process. 55, 2740–2746 (2007). [CrossRef]
- S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16, 804–817 (2008). [CrossRef] [PubMed]
- K. Kikuchi, “Polarization-demultiplexing algorithm in the digital coherent receiver,” in IEEE/LEOS Summer Topical Meetings (LEOSST) (2008), paper MC2.2.
- I. Roudas, A. Vgenis, C. S. Petrou, D. Toumpakaris, J. Hurley, M. Sauer, J. Downie, Y. Mauro, and S. Raghavan, “Optimal polarization demultiplexing for coherent optical communications systems,” J. Lightwave Technol. 28, 1121–1134 (2010). [CrossRef]
- L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in Optical Fiber Communication Conference (OFC) (2009), paper OMT2.
- U. Madhow, Fundamentals of Digital Communication (Cambridge Univ. Press, 2008).
- T. Duthel, C. R. S. Fludger, J. Geyer, and C. Schulien, “Impact of polarisation dependent loss on coherent POLMUX-NRZ-DQPSK,” in Optical Fiber Communication Conference (OFC) (2008), paper OThU5.
- M. Kuschnerov, F. N. Hauske, K. Piyawanno, B. Spinnler, M. S. Alfiad, A. Napoli, and B. Lankl, “DSP for coherent single-carrier receivers,” J. Lightwave Technol. 27, 3614–3622 (2009). [CrossRef]

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