## Experimental analysis of singularity-avoidance techniques for CMA equalization in DP-QPSK 112-Gb/s optical systems |

Optics Express, Vol. 19, Issue 19, pp. 18655-18664 (2011)

http://dx.doi.org/10.1364/OE.19.018655

Acrobat PDF (1169 KB)

### Abstract

We experimentally investigate the singularity problem in DP-QPSK 112-Gb/s receivers using the CMA. Three algorithms are compared: Constrained, Two-Stage, and Multi-User. Although these algorithms have been individually evaluated, they have not been compared by extensive experiments. The transmission setup emulates amplifier noise; first-order PMD; and chromatic dispersion. It is shown that all algorithms effectively mitigate singularities. However, under certain conditions, the Multi-User and the Constrained algorithms – both used for system startup – outperformed the Two-Stage, which does not distinguish between system operation and startup. In light of its effectiveness and low computational complexity, we recommend the Constrained algorithm.

© 2011 OSA

## 1. Introduction

*x*

_{1}and

*x*

_{2}to produce outputs

*y*

_{1}and

*y*

_{2}. The equalizer is usually represented by matrix

**F**= [

**f**

*],*

_{ij}*i*,

*j*= 1, 2, where

**f**

*arranged in the so-called “butterfly structure”. The “Standard CMA” is implemented in a way that filters*

_{ij}**f**

_{11}and

**f**

_{12}are driven by output

*y*

_{1}, while

**f**

_{21}and

**f**

_{22}by output

*y*

_{2}. Since the statistical properties on which CMA is based are equal for input signals

*x*

_{1}and

*x*

_{2}, and since both sets of filters are adjusted independently, the recovered signals are subject to delay and permutation ambiguity [1]. Therefore, the equalizer can recover correctly both signals, or it can recover the same signal for both outputs, or else, a time shifted version of the same signal. The last two cases are known in the literature as the “singularity problem” [1–4]. It is well-known that the choice of the filter initial tap setup affects the filter convergence properties [1, 4] and, consequently, the occurrence of singularities. Indeed, it has been a common practice to use a “single spike” initialization [5

5. J. Johnson, R., P. Schniter, T. Endres, J. Behm, D. Brown, and R. Casas, “Blind equalization using the constant modulus criterion: a review,” Proc. IEEE **86**, 1927–1950 (1998). [CrossRef]

**f**

_{11}and

**f**

_{22}are set to “1” and the remaining coefficients to “0”. The single spike initialization itself already helps mitigating the singularity problem.

6. K. Kikuchi, “Polarization-demultiplexing algorithm in the digital coherent receiver,” IEEE/LEOS Summer Topical Meetings pp. 101–102 (2008). [CrossRef]

7. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. **31**, 488–493 (1941). [CrossRef]

6. K. Kikuchi, “Polarization-demultiplexing algorithm in the digital coherent receiver,” IEEE/LEOS Summer Topical Meetings pp. 101–102 (2008). [CrossRef]

**f**

_{11}and

**f**

_{12}using the CMA, while inferring

**f**

_{21}and

**f**

_{22}from

**f**

_{11}and

**f**

_{12}. In a further development, [1] suggests using the same approach to initialize the filter coefficients, but removing the constraint after convergence is achieved. We denote this approach as being the “Constrained CMA”. The “Two-Stage CMA” proposed in [3], in turn, uses a two-stage butterfly filter with a constrained first stage, and an unconstrained second stage. While the Constrained CMA is less complex, the Two-Stage CMA allows advanced performance monitoring, and does not need to distinguish between startup and tracking phases. In a different approach, [8] proposes to extend the Standard CMA cost function by adding the cross-correlation between outputs, thus avoiding singularities. Known as “Multi-User CMA” (MU-CMA), the algorithm was also investigated in optical communications [9

9. A. Vgenis, C. S. Petrou, C. B. Papadias, I. Roudas, and L. Raptis, “Nonsingular constant modulus equalizer for PDM-QPSK coherent optical receivers,” IEEE Photon. Technol. Lett. **22**, 45–47 (2010). [CrossRef]

## 2. Singularity avoidance techniques

### 2.1. Constrained CMA

7. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. **31**, 488–493 (1941). [CrossRef]

**T**(

*jω*) of an optical system composed of retarders and rotators is unitary, i.e.:

**f**

_{11}and

**f**

_{12}be adapted by the CMA applied to output

*y*

_{1}. After

*y*

_{1}converges to a constant modulus, the algorithm sets

**f**

_{22}= –

*TR*[

**f**

_{11}]

^{*}and

**f**

_{21}= –

*TR*[

**f**

_{12}]

^{*}, where

*TR*[

**f**

*] means time-reversed*

_{ij}**f**

*, i.e.,*

_{ij}**f**

_{11}and

**f**

_{12}are driven by

*y*

_{1}, and

**f**

_{21}and

**f**

_{22}by

*y*

_{2}. Note however that, although Eq. (1) applies to optical fiber channels with certain distortions (e.g. PMD), it may not hold for the complete channel model, including pulse shaping and fractionally-spaced sampling. In addition, PDL and CD also do not fit in the hypotheses behind Eq. (1). In practice, however, even if the symmetry of Eq. (1) for the complete channel is not fulfilled, previous works [1] have observed that the filter initialization by Constrained CMA drives the equalizer output to a point that, although sub-optimal, lies in the vicinity of the singularity-free optimal solution. Thus, after initialization, the independent adaptation of the four filter coefficients leads the equalizer, finally, to the singularity-free optimal solution. These observations are corroborated by the experimental results in this paper.

### 2.2. Two-stage CMA

**F1**and

**F2**, where constraints

**f1**

_{22}= –

*TR*[

**f1**

_{11}]

^{*}and

**f1**

_{21}= –

*TR*[

**f1**

_{12}]

^{*}are imposed to the first stage. The second stage updates filters

**f2**

_{11}and

**f2**

_{12}; and

**f2**

_{21}and

**f2**

_{22}independently, as in the Standard CMA. Although the first stage produces T/2-spaced samples, the filter coefficients are adjusted every two samples. The second stage, in turn, processes T/2-spaced samples to produce T-spaced samples, as in the Standard CMA. This special structure also allows for additional performance monitoring and parameter estimation [3]. Even though the Two-Stage CMA and the Constrained CMA rely on the same property, they have fundamental differences: the Two-Stage CMA does not distinguish between initialization and operation, and its second stage always run freely. Thus, there are two possible outcomes. First, the first stage drives the equalizer to a sub-optimal solution, and the second stage produces an optimal, singularity-free solution. Second, the first stage drives the equalizer to a sub-optimal solution, and the second stage leads the solution back to a singularity. The final outcome depends on the interaction of noise and other fiber effects during the equalizer convergence phase.

### 2.3. MU-CMA

*J*(

*F*) for a two-user case was originally defined as:

*r*(

_{ij}*δ*) is the cross-correlation function between outputs

*y*

_{1}and

*y*

_{2}, defined as

*δ*∈ ℕ. In practice, values

*δ*

_{1}and

*δ*

_{2}are bounded by the equalizer length. It can be shown that

*J*is minimized using following the stochastic gradient algorithm: where

_{MU–CMA}*μ*is the step size, and Δ

*[*

_{l}*k*] is given by:

**X**[

*k*] = [

**x**

_{1}[

*k*]

**x**

_{2}[

*k*]]

*, where*

^{T}**x**

_{1}[

*k*] and

**x**

_{2}[

*k*] are line vectors with

*M*fractionally spaced input samples. In practice, the exact cross-correlation

*r*(

_{ij}*δ*) is not available, and must be estimated from the data. In this paper we implemented the MU-CMA algorithm as in Fig. 1(a). The filter coefficients are updated periodically, but not in every symbol, to allow a sufficiently accurate estimation of

*r*(

_{ij}*δ*) from a limited size set of symbols, called correlation estimation vector (CEV). The calculation applies for 0 ≤

*δ*≤ ⌊

*M*/2⌋, which, using a T/2 spaced equalizer, corresponds to the possible delays between

*y*

_{1}and

*y*

_{2}. The cross-correlation estimation schemes for the minimum (

*δ*= 0) and maximum (

*δ*= ⌊

*M*/2⌋) delays are depicted in Figs. 1(b) and 1(c), where the arrows indicate multiplications involved in the estimation process. Note that the case where

*δ*= ⌊

*M*/2⌋ requires the interval between consecutive filter iterations to be at least CEV length + ⌊

*M*/2⌋ symbols long.

9. A. Vgenis, C. S. Petrou, C. B. Papadias, I. Roudas, and L. Raptis, “Nonsingular constant modulus equalizer for PDM-QPSK coherent optical receivers,” IEEE Photon. Technol. Lett. **22**, 45–47 (2010). [CrossRef]

## 3. Experimental setup

^{11}mutually delayed pseudo-random bit sequences (PRBS) at 7 Gb/s, multiplexed later into one sequence at 28 Gb/s. This sequence and its inverted version are split and delayed by 52 bits to form the in-phase (I) and quadrature (Q) components that drive two nested MZM based modulators, whose optical input is a tunable DBF laser. The resulting QPSK signal is then submitted to an RZ pulse carver. A block that consists of a polarization beam splitter, a 300 symbols delay line polarization maintaining fiber in one of the branches, and a polarization beam combiner, produces the polarization multiplexed signal. A polarization scrambler, a first order PMD emulator and a tunable CD compensator generate the desired transmission conditions. On the receiver side, optical noise is loaded to fine-tune the OSNR. The polarization diversity coherent receiver contains four pairs of balanced photodetectors. The four eletrical outputs are sampled at 50 GSamples/s by a 4 channel oscilloscope (8 bits nominal resolution) and then stored for offline post-processing.

## 4. Experimental results and analysis

^{6}samples. During offline processing we submitted the data sequences to normalization and orthogonalization procedures, followed by resampling by a factor of 28/25, to obtain the rate of 2 Samples/symbol. After equalization and source separation by the tested filter, we performed carrier recovery, decision and decoding. We then computed the cross-correlation between two decoded bit sequences for delays up to half filter length, setting the threshold for singularity occurrence to 0.5, computed over 50,000 bits. We repeated this procedure 1, 000 times with different data strings for each tested transmission condition. The filter length was set to 15 taps for all tested algorithms (the Two-Stage CMA used 15 taps in each stage). The Constrained and MU-CMA equalizers used a convergence phase of 5, 000 iterations before switching to the Standard CMA (see Fig. 2). In order to assess the convergence properties of the investigated algorithms, the signal to noise ratio (SNR) of the equalizer constellation was estimated from the data [10

10. T. Benedict and T. Soong, “The joint estimation of signal and noise from the sum envelope,” IEEE Trans. Inform. Theory **13**, 447–454 (1967). [CrossRef]

*M*/2⌋)]=135,000 symbols. Additional 10,000 symbols were included for the Standard CMA convergence, giving a total of 145,000 symbols used for convergence and initialization.

^{–4}.

*y*

_{1}and

*y*

_{2}outputs in presence of amplifier noise at OSNR = 17 dB. Figs. 4(c) and 4(d) show the box plots for the same data, indicating the smallest observation, lower quartile, median, upper quartile, and largest observation. The Two-Stage algorithm exhibits the highest median for the estimated SNR in

*y*

_{1}. This is an artifact from using two 15-taps filters, while the others use a single 15-taps filter. However, one striking remark is that the estimated SNR in

*y*

_{2}is in average lower than in

*y*

_{1}. This indicates that the algorithm privileges the orientation that is used for adapting the first stage. We can also conclude that the symmetry relationship employed in the first stage is not optimal, and some mismatch has to be compensated in the second stage.

*y*

_{1}and

*y*

_{2}for the system under residual chromatic dispersion, at CD = 50 ps/nm, OSNR = 18 dB. The results confirm the conclusions obtained when amplifier noise was the only source of distortion. The convergence properties for the system impaired by first-order PMD is shown in Figs. 6(a) and 6(b); and Figs. 6(c) and 6(d) (DGD = 20 ps, OSNR = 18 dB). One can observe a single case where the Standard CMA and the Constrained CMA equalizers did not attain convergence for

*y*

_{1}. Such

*outliers*were not included in the corresponding box plot to maintain the figure scaling. Once again, the MU-CMA outperformed the remaining algorithms for output

*y*

_{1}, however, its behavior for

*y*

_{2}was slightly inferior.

## 5. Conclusion

## Acknowledgment

## References and links

1. | 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,” OSA/OFC/NFOEC(2009). |

2. | S. Faruk, Y. Mori, C. Zhang, and K. Kikuchi, “Proper polarization demultiplexing in coherent optical receiver using constant modulus algorithm with training mode,” OptoElectron. and Commun. Conf. Tech. Digest(2010). |

3. | C. Xie and S. Chandrasekhar, “Two-stage constant modulus algorithm equalizer for singularity free operation and optical performance monitoring in optical coherent receiver,” OSA/OFC/NFOEC pp. 1–3 (2010). |

4. | S. J. Savory, “Digital coherent optical receivers: algorithms and subsystems,” IEEE J. Quantum Electron. |

5. | J. Johnson, R., P. Schniter, T. Endres, J. Behm, D. Brown, and R. Casas, “Blind equalization using the constant modulus criterion: a review,” Proc. IEEE |

6. | K. Kikuchi, “Polarization-demultiplexing algorithm in the digital coherent receiver,” IEEE/LEOS Summer Topical Meetings pp. 101–102 (2008). [CrossRef] |

7. | R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. |

8. | C. B. Papadias and A. Paulraj, “A space-time constant modulus algorithm for SDMA systems,” Vehicular Tech. Conf. |

9. | A. Vgenis, C. S. Petrou, C. B. Papadias, I. Roudas, and L. Raptis, “Nonsingular constant modulus equalizer for PDM-QPSK coherent optical receivers,” IEEE Photon. Technol. Lett. |

10. | T. Benedict and T. Soong, “The joint estimation of signal and noise from the sum envelope,” IEEE Trans. Inform. Theory |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: June 17, 2011

Revised Manuscript: July 28, 2011

Manuscript Accepted: August 14, 2011

Published: September 9, 2011

**Citation**

Valery N. Rozental, Thiago F. Portela, Diego V. Souto, Hugo B. Ferreira, and Darli A. A. Mello, "Experimental analysis of singularity-avoidance techniques for CMA equalization in DP-QPSK 112-Gb/s optical systems," Opt. Express **19**, 18655-18664 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-19-18655

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### References

- 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,” OSA/OFC/NFOEC(2009).
- S. Faruk, Y. Mori, C. Zhang, and K. Kikuchi, “Proper polarization demultiplexing in coherent optical receiver using constant modulus algorithm with training mode,” OptoElectron. and Commun. Conf. Tech. Digest(2010).
- C. Xie and S. Chandrasekhar, “Two-stage constant modulus algorithm equalizer for singularity free operation and optical performance monitoring in optical coherent receiver,” OSA/OFC/NFOEC pp. 1–3 (2010).
- S. J. Savory, “Digital coherent optical receivers: algorithms and subsystems,” IEEE J. Quantum Electron.16, 2120–2126 (2010).
- J. Johnson, R., P. Schniter, T. Endres, J. Behm, D. Brown, and R. Casas, “Blind equalization using the constant modulus criterion: a review,” Proc. IEEE86, 1927–1950 (1998). [CrossRef]
- K. Kikuchi, “Polarization-demultiplexing algorithm in the digital coherent receiver,” IEEE/LEOS Summer Topical Meetings pp. 101–102 (2008). [CrossRef]
- R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am.31, 488–493 (1941). [CrossRef]
- C. B. Papadias and A. Paulraj, “A space-time constant modulus algorithm for SDMA systems,” Vehicular Tech. Conf.1, 86–90 (1996).
- A. Vgenis, C. S. Petrou, C. B. Papadias, I. Roudas, and L. Raptis, “Nonsingular constant modulus equalizer for PDM-QPSK coherent optical receivers,” IEEE Photon. Technol. Lett.22, 45–47 (2010). [CrossRef]
- T. Benedict and T. Soong, “The joint estimation of signal and noise from the sum envelope,” IEEE Trans. Inform. Theory13, 447–454 (1967). [CrossRef]

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