Optics Communications 290 (2013) 49–54

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Optics Communications

0030-40

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/optcom

Polarization de-multiplexing based on T-CMN

Lu Jinhua, Hu Guijun n, Sun Yunbo, Shi Jian

Department of Optics Communication, College of Communication Engineering, Jilin University, 5372 Nanhu Road, Changchun 130012, PR China

a r t i c l e i n f o

Article history:

Received 15 March 2012

Received in revised form

17 September 2012

Accepted 20 October 2012Available online 7 November 2012

Keywords:

Optical fiber communication

Polarization de-multiplexing

T-CMN algorithm

18/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.optcom.2012.10.032

esponding author. Tel.: þ86 431 85171693.

ail address: hugj@jlu.edu.cn (H. Guijun).

a b s t r a c t

In order to mitigate the crosstalk in polarization multiplexing (PM) systems and achieve more effective

performance of de-multiplexing, the fixed-point algorithm based on complex maximization of

negentropy (T-CMN) is presented for the first time to implement polarization de-multiplexing in a

2�2 PM system. The algorithm successfully implements de-multiplexing, furthermore some very

properties of the T-CMN, such as strong robustness, inverse-free of the matrix and transparent to the

modulation format, make it more suitable for polarization de-multiplexing.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

The quantity demand for information is constantly booming inrecent years. As a transmission medium with enormous band-width and tremendous capacity, optical fiber is playing a vital rolein communication systems. However, it is far away from satisfy-ing the requirement of users for information by merely improvingtransmission rate in a single channel. Consequently, all kinds offiber communication techniques which include polarization mul-tiplexing (PM) have been brought up to solve the problem [1].

The polarization property of single mode fiber (SMF) is utilizedin polarization multiplexing technique by regarding two ortho-gonal polarization states as two independent channels. Thistechnique can double transmission capacity and spectral effi-ciency of the existing systems by carrying two independentlymodulated data at the same wavelength, therefore, it hasattracted researchers’ interests significantly [2–4]. However, thecrosstalk between the polarizations occurs in PM systems due torandom polarization rotation and linear distortions such aschromatic dispersion (CD), polarization mode dispersion (PMD)and polarization dependent loss (PDL). In consequence, what wedetect at the receiver end is the weighted stack of two sourcesignals [5–7]. So polarization de-multiplexing is an integral partof PM systems.

Usually, polarization de-multiplexing can be implemented inthe optical domain and the electrical domain [8,9]. But classicalschemes are difficult to realize, which mainly limit the develop-ment of PM technique. Recently, digital signal processing (DSP)

ll rights reserved.

technique, with features of low hardware requirement and easilyrealization, provides more room for the development of polariza-tion de-multiplexing schemes in the digital domain.

The coherent detection technique cooperating with the DSPalgorithm is utilized at the receiver end to implement de-multiplexing in this scheme [10–12]. The algorithms commonly usedin the DSP include the constant modulus algorithm (CMA) [13] andthe independent component analysis (ICA) [14,15]. The StochasticGradient Descent Constant Modulus Algorithm (SGD-CMA) and theRecursive Least Squares constant modulus algorithm (RLS-CMA) arethe most popular algorithms of the CMA. Although these algorithmshave been used for many years, there are still some shortages. Forexample, the convergence of the SGD-CMA is sensitive to the iterationstep-size, and the performance of de-multiplexing relies on theselected step-size [16]. In spite of avoiding the sensitivity of iterationstep-size, calculating the inverse of correlation matrix of input datawill affects the robustness of the RLS-CMA [17]. Meanwhile, the CMAis only effective for the constant modulus modulation formats, suchas quadrature phase shift keying (QPSK). The de-multiplexing per-formance of the tensor-based ICA algorithm used earlier seems to bevery effective, but kurtosis is applied as the criteria for non-Gaussianin this algorithm, which is sensitive to the singular value of mixedsignals. Furthermore, the tensor-based ICA algorithm does need tosolve the inverse of channel matrix, which can significantly increasethe complexity of hardware [14]. In order to overcome thesesituations, the T-CMN algorithm is introduced to implement polar-ization de-multiplexing in PM systems in this paper. De-multiplexingperformance of this algorithm has been analyzed by constellation, eyediagram, and symbol error rate (SER). The results indicate that the T-CMN can separate mixed signals perfectly. Moreover, it avoidscalculating the inverse of channel matrix, and has a great advantagein hardware implementation.

L. Jinhua et al. / Optics Communications 290 (2013) 49–5450

2. System model

The block diagram of the 2�2 PM system is shown in Fig. 1.The beam emitted by laser passes through the polarization beamsplitter (PBS) and then is split into two orthogonal polarizations.Then this couple of polarizations are respectively fed into theMach–Zehnder modulators (MZM) that each driven by 56 Gb/sNRZ code which is properly pre-coded to generate QPSK or16QAM optical signal. Finally, the two polarizations are combinedby an ideal polarization beam combiner (PBC) and launched intothe optical fiber link. At the receiver end, the detected opticalsignal passes through the PBS and then is divided into x and y,while so does the beam emitted by local oscillator (LO). The abovefour polarization components are detected by four balanced-detectors (BD) after pass two 901 optical hybrids and the detectedsignals are recovered by digital demodulation (DD) followingdigital signal processing (DSP) module.

The DSP module mainly consists of analog digital conversion(ADC), chromatic dispersion compensation (CDC), polarization de-multiplexing (PD), carrier phase estimation (CPE) and digitalanalog conversion (DAC). Moreover the CD is compensated byhorizontal digital filters; the T-CMN is applied in polarization de-multiplexing, and the Viterbi–viterbi algorithm is employed torecover carrier phase.

3. T-CMN algorithm

As a method of signal processing, ICA extracts independentcomponents only from mixed data, so it can be applied as a meansof de-multiplexing and recovers the source signals from thedetected signals.

In complex ICA, the detected signals are usually the linearmixtures of the statistically independent components:

x¼ As ð1Þ

where s¼[s1,y,sM]T is the source signals vector, x¼[x1,y,xN]T isthe observed signals vector, and the source signals and theobserved signals are all complex valued, and A is an unknownN�M matrix of full column rank, referred to as the mixing matrixrepresenting the instantaneous linear mixing channel. There aretwo principles that the complex ICA must follow: the sourcesignals have to be statistically independent and at most one of thesource signals may be Gaussian [18]. First, the two respectivelymodulated pseudo random binary sequence (PRBS) signals

PBS PBC

I

Q

I

Q

SMF

PB

P

Signal 1

MZM

MZM 2�

� 2

MZM

MZM

LO

Laser

Signal 2

Fig. 1. The block diagram o

transmitted in the simulation accord with the requirement ofindependence. Second, the statistical distribution of the QPSK or16QAM modulation format belongs to sub-Gaussian distribution,so the requirement of non-Gaussian distribution is fulfilled.Finally, in polarization de-multiplexing, the influence to polariza-tion states caused by all kinds of linear distortions in the opticalfiber can be regarded as a 2�2 unitary matrix. From all above, thecomplex ICA can be utilized to implement polarization de-multiplexing.

T-CMN was proposed by Novey and Adalı [19], and negentropyis used to measure the non-Gaussian of signal in this algorithm.Different from Ref. [14]in which kurtosis is used to measure non-Gaussianity, negentropy which has stronger numerical stabilityand is insensitive to outliers is used as non-Gaussianity criterionin this algorithm. The negentropy in complex ICA is defined as:

Jneg yRk ,yI

k

� �¼H yR

gauss,yIgauss

� ��H yR

k ,yIk

� �ð2Þ

where HðyRgauss,y

IgaussÞ is the entropy of a complex Gaussian ran-

dom variable with the same covariance as y. Therefore, maximiz-ing negentropy can be implemented by minimizing H yR

k ,yIk

� �. If

we make use of differential entropy as the cost function, it isimportant to get the prior knowledge of probability densityfunction (PDF), which is difficult for actual systems. In order toovercome this problem, we can substitute a nonlinear function forthe negentropy approximately. The T-CMN cost function isdefined as:

JðwÞ ¼ Ef9G wHx� �

92g ð3Þ

In which superscript H denotes the conjugate transpose ofcomplex number, G is an arbitrary nonlinear function, w is thecolumn vector of WH, and x¼[x1,y,xN]T is the column vector ofobserved signals.

The advantage of the T-CMN is matching cost functions withsource signals, which improves the stability of algorithm. Thebasic iteration of this algorithm is executed to each row vector wof demixing matrix W, respectively:

w’�EfGnðyÞgðyÞxgþEfgðyÞgnðyÞgwþEfxxTgEfGn

ðyÞg’ðyÞgwn ð4Þ

where GðdÞ is a nonlinear function, gðdÞ is the derivative of GðdÞ,and g0ðdÞ is the derivative of gðdÞ, and the constraint :w:2

¼ 1. Themajor steps of the symmetric regularized T-CMN algorithmemployed in this paper are presented in Table 1.

DD

DD

S

BS

90° Optical H

ybrid

BD

DSP

f the 2�2 PM system.

L. Jinhua et al. / Optics Communications 290 (2013) 49–54 51

4. Result analysis

For the simulations, the wavelength of laser is 1550 nm, theaverage power is 1 mW, the linewidth is 0.1 MHz. And the lengthof SMF is 20 km, the attenuation is 0.2 dB/km, the dispersion is16.75 ps/nm/km, the dispersion slope is 0.075 ps/nm2/km, thepolarization mode dispersion coefficient is 0.02 ps/sqrt(km). Inorder to evaluate the effect of the T-CMN, we compare the T-CMNwith the SGD-CMA which is most popular. Meanwhile, we per-form several simulations, and get the best performance of de-multiplexing used SGD-CMA with the step-size m¼0.005.

The detected signals are de-multiplexed by the SGD-CMA andT-CMN algorithms, respectively, and eye diagrams (figures aboveare the eye diagrams for In-phase component, and the figuresbelow are the eye diagrams for Quadrature component) areshown in Fig. 2. All these eye diagrams are got after phaseestimation. It is obvious that the crosstalk between two polariza-tion components is very serious without de-multiplexing. The de-multiplexing performance of the SGD-CMA is better, and

Table 1The major steps of the symmetric regularized T-CMN algorithm.

Whiten the observed data x to give z¼Vx.

Initialize wi, i¼1,y,n to make :wi:¼ 1.

Update wi,

wi’�EfGnðyÞgðyÞxgþEfgðyÞgnðyÞgwiþEfxxT gEfGn

ðyÞg’ðyÞgwin .

Regularize W¼[w1,y,wn]T using W’(WWH)�1/2W.

If W is non-convergent, return to step 3.

Estimate source signals using y¼WHx.

-0.6 -0.4 -0.2 0 0.2 0.4-1

-0.5

0

0.5

1

Time

Am

plitu

de

Eye Diagram for In-PhaseSignal

-0.6 -0.4 -0.2 0 0.2 0.4-1

-0.5

0

0.5

1

Time

Am

plitu

de

-1

-0.5

0

0.5

1

Am

plitu

de

-1

-0.5

0

0.5

1

Am

plitu

de

Eye Diagram forQuadrature Signal

-0.6 -0.4 -0.2

Eye DiagramSignal (S

-0.6 -0.4 -0.2

Eye DiaQuadrature Sig

Fig. 2. Eye diagrams of x polarization without polarization de-multiplexing(a), w

successful polarization de-multiplexing is implemented by thisalgorithm. The eye opening of the eye diagram is significantlyimproved with the T-CMN polarization de-multiplexer, whichshows the performance of the T-CMN polarization de-multiplexeris perfect. The eye diagram of y polarization is similar, and so isthe situation of the other polarization state.

Then we study the de-multiplexing performance of the T-CMNby constellation. Fig. 3(a) shows the constellation of x polarizationwithout polarization de-multiplexing, and Fig. 3(b) and (c) showthe constellations of x polarization with the SGD-CMA and T-CMNpolarization de-multiplexers, respectively.

Due to dispersion, crosstalk and phase excursion duringtransmission, original constellation at the receiver end is formedof multiple circles. After dispersion compensation, polarizationde-multiplexing and phase estimation, the constellation is trans-formed into four standard spots. Comparing these two constella-tions, we can get the conclusion that both the SGD-CMA andT-CMN algorithms can implement polarization de-multiplexingperfectly. But the convergence of SGD-CMA is dependent closelyon iteration step-size, and the effect of de-multiplexing is influ-enced by the selected step-size, so the robustness of this algo-rithm is not strong. Avoiding this problem completely, the T-CMNhas nothing to do with step-size, so this algorithm has a goodrobustness.

In order to evaluate the robustness of the T-CMN in depth, weget the SER under various optical signal to noise ratio (OSNR) ofthe two algorithms, as shown in Fig. 4. From this figure, it can beseen that the de-multiplexing performance of the T-CMN is stillgood even under low OSNR. Comparing these two algorithms, theT-CMN achieves better performance all the time, where theT-CMN outperforms the SGD-CMA by 1.62 dB at SER¼10�3.

-1

-0.5

0

0.5

1

Am

plitu

de

-1

-0.5

0

0.5

1

Am

plitu

de

0 0.2 0.4Time

for In-PhaseGD-CMA)

0 0.2 0.4Time

gram fornal (SGD-CMA)

-0.6 -0.4 -0.2 0 0.2 0.4Time

Eye Diagram for In-PhaseSignal (T-CMN)

-0.6 -0.4 -0.2 0 0.2 0.4Time

Eye Diagram forQuadrature Signal (T-CMN)

ith the SGD-CMA(b) and T-CMN(c) polarization de-multiplexer, respectively.

-1 -0.5 0 0.5 1

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

1

In-Phase

Qua

drat

ure

After DispersionCompensation - X

-1 -0.8

-0.6

-0.4

-0.2 0 0.2 0.4 0.6 0.8 1

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

1

In-Phase

Qua

drat

ure

After Carrier PhaseEstimation - X (SGD-CMA)

-1 -0.8

-0.6

-0.4

-0.2 0 0.2 0.4 0.6 0.8 1

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

1

In-Phase

Qua

drat

ure

After Carrier PhaseEstimation - X (T-CMN)

Fig. 3. Constellations of x polarization without polarization de-multiplexing (a), with the SGD-CMA (b) and T-CMN (c) polarization de-multiplexers, respectively.

Fig. 4. QPSK SER vs. OSNR performance of the SGD-CMA and T-CMN polarization

de-multiplexers.

Fig. 5. 16QAM SER vs. OSNR performance of the T-CMN polarization de-

multiplexer.

L. Jinhua et al. / Optics Communications 290 (2013) 49–5452

In addition, for purpose of substantiating that the T-CMN is notmerely effective for the constant modulus modulation formats,we bring in 16QAM modulation format. Fig. 5 shows the SERperformance of the T-CMN versus the OSNR, with 16QAMmodulation format is adopted. Quantitatively, for example, at aSER of 10�3, the requirement of the OSNR is 22.61 dB, higher thanthe QPSK by 6.34 dB, and it is reasonable.

Fig. 6(a–c) illustrate the constellations of x polarization of the16QAM modulation format, with the OSNR¼20 dB (lower thanthe critical value 22.61 dB), 25 dB and 30 dB. From these figures,we can observe that when the OSNR is higher than the criticalvalue, the de-multiplexing performance of T-CMN is ideal. Andthe de-multiplexing performance is improved dramatically withthe OSNR increases.

-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

In-Phase

Qua

drat

ure

After Carrier Phase Estimation - X (T-CMN)

-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

In-Phase

Qua

drat

ure

After Carrier Phase Estimation - X (T-CMN)

-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

In-Phase

Qua

drat

ure

After Carrier Phase Estimation - X (T-CMN)

Fig. 6. Constellations of x polarization of the 16QAM modulation format under the OSNR of 20 dB (a), 25 dB (b) and 30 dB (c).

L. Jinhua et al. / Optics Communications 290 (2013) 49–54 53

5. Conclusion

The T-CMN algorithm is presented in PM systems for the firsttime to solve the problem of crosstalk between two polarizations inthis paper. We perform the simulations in a 2�2 PM system, andde-multiplex the mixed signals successfully even under low OSNR.Compared with the most popular SGD-CMA, the T-CMN not only hasa good robustness, but also can be applied in many modulatedformats. Furthermore, this algorithm avoids the problem of thetensor-based ICA, which has to solve the inverse of matrix, and has agreat advantage in hardware implementation. All above indicatethat the T-CMN is an ideal de-multiplexing algorithm.

Acknowledgements

This work was supported partly by National Natural ScienceFoundation of China (NSFC) under grant 61177066 and 61077054,and also was supported by Science and Technology Developmentof Jilin Province under grant 20120761 and International CooperationFoundation of Changchun Bureau of Science and Technology undergrant 2011105.

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