Low complexity, modulation-transparent and joint polarization and phase tracking scheme based on the nonlinear principal component analysis

Qian Xiang; Yanfu Yang; Qun Zhang; Yong Yao

doi:10.1364/OE.27.017968

1. Introduction

The internet traffic demands driven by data centers, cloud services and high definition video streaming [1] require next generation optical networks to be more flexible and intelligent to meet the unpredictable and heterogeneous transmission demands. The promising solution for this transmission challenge is the bandwidth-variable transceivers (BVTs) which can adaptively adjust the modulation formats and bit-rate according to the channel condition and traffic demand to maximize network efficiency [2,3]. Therefore, the modulation transparent carrier recovery schemes are highly desired to implement intelligent optical receiver. Considering the polarization multiplexed multi-level modulated signals are susceptive to the random birefringence along fiber and time-varying laser phase noise, joint tracking scheme for polarization and phase is favorable compared with cascaded schemes due to the potentially reduced complexity. Furthermore, the polarization rotation rates between two orthogonal polarizations would be as fast as several Mrad/s under the scenarios of instantaneous external disturbance [4]. Thus, a modulation transparent and joint polarization/phase recovery scheme with fast polarization tracking capability is very highly demanded for next generation elastic optical networks (EONs).

Various methods are proposed to handle the impairment of polarization crosstalk and phase noise [5–11]. Those methods can be divided two categories: (1) modulation dependent scheme and (2) modulation-transparent scheme. The commonly used modulation dependent schemes such as constant modulus algorithm [5], blind phase search algorithm [6] and Kalman filtering [7] as well as Viterbi-Viterbi phase estimation (VVPE) [8] are usually used to jointly or separately track the state of polarization and phase noise. If these schemes are employed in EONs, the main challenge is that the modulation format, constellation information must be known in advance for parameter configuration in various DSP modules. Therefore, these modulation dependent schemes are not suitable to cope with the impairments in future dynamic optical network, especially for adaptive configuration modulation formats cases. Recently, Independent component analysis (ICA) and Kabsch algorithm are proposed to cope with polarization or phase noise in a modulation-transparent manner [9–11]. In Kabsch algorithm, the transmitted signal must be known in advance, which means the data-aided feature [9]. Regarding ICA, a nonlinear active function is employed to approximate the probability density function (PDF) of signals [10]. However, it is difficult to obtain the true PDF, especially when the scenarios of high polarization rotation rate or high order modulation format is considered. Thus, ICA would have relatively degraded performance, especially for high order modulation format, and it would not be considered as the optimal scheme for high order modulation formats, such as 16QAM signals. For a better performance, a modulation transparent and joint polarization and phase tracking scheme with fast polarization tracking capability is necessary for future BVTs.

Nonlinear principal component analysis (NPCA) based on the minimization of high-order statistics (HOS) is widely used for signal separation in blind source separation (BSS) [12–16]. To be specific, the involved cost function is constructed on the square error between the received signal and its reconstructed signal. Obviously, it does not require any information about the transmitted modulation format and channel condition. Actually, in coherent fiber communication, the impairments of polarization crosstalk and phase noise can be considered as the linear mixing problem if the narrow laser linewidth is considered. Thus, joint polarization and phase tracking can be considered as linear signal separation, which is equivalent to the problem of BSS. Therefore, NPCA can be theoretically used to track polarization and phase noise in modulation-format transparent manner. Moreover, compared with ICA, NPCA does not needs to construct the PDF of signal and outperforms ICA indeed [17]. Thus, NPCA is adopted in our scheme as a favorable technique to track polarization and phase noise in EONs.

In this paper, a low complexity, modulation-transparent and joint polarization and phase tracking scheme based on the NPCA is firstly proposed and demonstrated experimentally. HOS and the constraint of estimated matrix make NPCA fast convergence speed and singularity-free. Compared with the modulation dependent scheme such as CMA/MMA + VVPE, NPCA are investigated regarding polarization tracking capability, OSNR sensitivity and hardware complexity for 28GS/s polarization division multiplexing (PDM) QPSK/ 16QAM signals. Both simulation and experimental results confirm that NPCA shows comparable BER performance in back-to-back condition and excellent tracking performance over wide polarization rotation rate ranges, especially for 16QAM signals. Moreover, compared with CMA/MMA + VVPE, low computational complexity of NPCA is also confirmed, which makes NPCA more attractive in EONs.

2. Operating principle

If polarization mode dispersion and polarization dependent loss are neglected, after frequency offset estimation, the received signals are only impaired by the polarization crosstalk and phase noise. The received signals can be expressed mathematically as follows:

U (t) = [\begin{matrix} w_{x x} & w_{x y} \\ w_{y x} & w_{y y} \end{matrix}] Z (t) e^{j (θ (t))} + n (t) = J (t) Z (t) e^{j (θ (t))} + n (t)

in which

Z (t)

,

U (t)

,

J (t)

,

θ (t)

and

n (t)

denote the transmitted signal, the received signal, the fiber birefringence related Jones matrix, phase noise induced by laser, and additive Gaussian noise with zero mean, respectively. ‘t’ is the time index.

J (t)

is a unitary matrix.

The laser phase noises are modeled as a Wiener process and the different phase between two adjacent symbols is characterized by the normalized Gaussian distribution with zeros-mean and covariance of $2 π T_{s} Δ υ$ [6]. $Δ υ$ denotes the laser linewidth. At the case of small linewidth, the variation of phase noises is very slow over a short time. Thus, the impairments of polarization crosstalk and phase noises can be together expressed as the products of Jones matrix with the phase noise. Consequently, the received signals U can be considered as the linear mixing of the transmitted signal Z, given as follows:

U (t) = [\begin{matrix} e^{j (θ (t))} \cdot w_{x x} & e^{j (θ (t))} \cdot w_{x y} \\ e^{j (θ (t))} \cdot w_{y x} & e^{j (θ (t))} \cdot w_{y y} \end{matrix}] Z (t) + n (t) = M (t) Z (t) + n (t)

where M(t) denotes the linearly mixed matrix. Obviously, the impairments of polarization crosstalk and phase noise can be jointly compensated by estimating the inversely mixed matrix. Therefore, joint polarization and phase tracking scheme can be expressed mathematically as follows:

Z (t) = [\begin{matrix} w_{11} & w_{12} \\ w_{21} & w_{22} \end{matrix}] U (t) = W (t) U (t)

where

w_{11}

,

w_{12}

,

w_{21}

and

w_{22}

are the elements in the inversely mixed matrix W(t) which is also an unitary matrix.

As shown in Eqs. (2) and (3), the procedure of polarization and phase tracking scheme is similar with BSS. To be specific, U(t) and Z(t) are equivalent to the mixed signals and the source signals in BSS, respectively. NPCA based on HOS, as an important extension of PCA, is widely used for signal separation in BSS [13–17]. NPCA can be considered as the extended version of projection approximation subspace tracking in linear PCA [18]. Different from the singular value decomposition (SVD) in linear PCA, the eigenvectors of data in the projection approximation subspace tracking are estimated by minimizing the error between the data and the reconstructed data from its subspace [18]. However, it is noticed that linear PCA only involves second-order statistics, which makes linear PCA fail to achieve signal separation [16].

Compared with the linear PCA, HOS of the signal is introduced by a nonlinear active function, which makes NPCA separate the independent source signals from the mixed signals successfully. In NPCA, the source signals Z(t) in its subspace are obtained by the product of the inversely mixed matrix W and the mixed signals U(t). Next, the source signals Z(t) are activated by a nonlinear function g(.) for taking HOS into accounts. Then, the mixed signals are reconstructed by the product of the Hermitian transposition of W and the activated signals g(Z(t)). Finally, the inversely mixed matrix W can be updated by the minimization of mean-square representation error between the received and reconstructed signal. As described above, the cost function in NPCA is expressed mathematically as below:

J_{1} (W) = E {| e (t) |^{2}} = E {| U (t) - W^{H} g (Z (t)) |^{2}}

in which e(t) and E(.) denote the reconstructed error and the expectation operation, and superscript ‘H’ denotes the Hermitian transposition. The nonlinear active function in NPCA is always chosen to be g(y) = tanh (y) for sub-Gaussian source signal which is commonly used in communication system. Obviously, as shown in Eq. (4), NPCA can achieve signal separation successfully without any prior information of the source signals. Meanwhile, with the proper nonlinear active function g, the cost function in Eq. (4) is equivalent to the kurtosis of the recovered signal [19]. Kurtosis K, given as K(y) = E{|y|⁴}-2(E{|y|})²-|E{y²}|², is widely used for signal separation, because it is susceptive to the type of signals.

Owing to the advantages of modulation transparency, NPCA based polarization and phase tracking scheme is proposed to make BVTs more flexible and intelligent. Furthermore, to reduce the complexity of calculating the exponential term, nonlinear active function is simplified as its first and third terms Taylor expression, given as g(y) = y-y³/3, because Kurtosis is only involved with the fourth-order moments.

The schematic diagram of NPCA based polarization and phase tracking scheme can be found in Fig. 1. It consists of three stages: 1) joint polarization and phase recovery, 2) signal reconstruction and 3) parameter updating. First, the received signals U(t) are processed by Eq. (3) to achieve joint polarization and phase recovery. Next, HOS of the recovered signals Z(t) is introduced by the nonlinear active function g(.). Then, the received signals are reconstructed by the product of the activated signal g(Z(t)) and the Hermitian transposition of W. The reconstructed error e(t) is obtained by calculating the difference between the received signals and the reconstructed signals. Finally, the parameter W is updated according to the reconstructed error and activated signals, given as follows [14]:

\begin{array}{l} W (t) = W (t - 1) + μ \cdot \frac{\partial J_{1} (W)}{\partial W} \\ \approx W (t - 1) + μ \cdot [^{e^{*} (t) \cdot g^{T} (W (t - 1) \cdot U (t - 1))]^{T}} \end{array}

where the superscript ‘*’ and ‘T’ denotes the complex conjugate and transpose operation, respectively.

μ

is the step size. Nonlinear active function for the complex value is always expressed as the combination of real part

g (ℜ (y))

and imaginary part

g (ℑ (y))

, given as

g (y) = g (ℜ (y)) + j g (ℑ (y))

. Meanwhile, the proposed scheme is also immune to the singularity problem in CMA/MMA because NPCA is implemented under the constraint of

W^{H} W = I

[13].

Fig. 1 The schematic diagram of the proposed NPCA.

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Additionally, the proposed NPCA can also be implemented in parallel manner, as shown in Fig. 2. To be precise, the input signals are separated into the blocks with size of N, and it is processed in block by block. The procedure of signal recovery for the signals in k-th block, such as U (1), U (2), …, U(N), is achieved by Eq. (3). Then, the recovered signals are sent to the signal reconstruction and error computing module to calculate the activated signal and reconstructed error for each symbol. Different from the parameter updating in serial implementation, the estimated matrix W in parallel implementation is adjusted according to the average value of the reconstructed errors and activated signals over blocks [20,21], given as follows:

W (k) = W (k - 1) + \frac{μ}{N} \cdot \sum_{i = 1}^{N} {(e^{*} (i) \cdot g^{T} (W (k - 1) \cdot U (i)))}^{^{T}}

where k is the index of block, and ‘i’ denotes the index of symbol in block.

Fig. 2 The schematic diagram of NPCA in parallel implementation.

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For emulating the polarization tracking effect, the endless polarization rotation is added in digital domain, given as below:

R (t) = [\begin{matrix} \cos (f T_{s} t) & \sin (f T_{s} t) \\ - \sin (f T_{s} t) & \cos (f T_{s} t) \end{matrix}]

where f and Ts denote polarization rotation frequency and the symbol duration.

3. Numerical simulation and discussion

In this section, compared with the modulation dependent scheme CMA/MMA + VVPE, the proposed NPCA are investigated detailly regarding polarization tracking capability and OSNR sensitivity penalty as well as hardware complexity. 28GS/s PDM QPSK/16QAM signals are chosen as an example to verify the polarization and phase tracking ability of NPCA. The data length is set to 2¹⁶. The laser linewidth is set to 100 KHz. The filter length of CMA/MMA is set to 1 with only polarization crosstalk considered here. The block length of NPCA and CMA/MMA + VVPE is set to 40 for the parallel implementation. Besides, QPSK partitioning based VVPE is used to estimate phase noise for 16QAM signals. For emulating the fast time-varying endless polarization rotation, the polarization rotation is added in digital domain by adjusting f as shown in Eq. (7).

Firstly, the polarization tracking capability of CMA/MMA + VVPE and NPCA is investigated under different step size. With 7% FEC threshold, OSNR is set to 13 dB and 20 dB for QPSK and 16QAM signals, respectively. As shown in Fig. 3, the polarization tracking capability of two schemes is plotted as a function of polarization rotation rate, and varies under different step size. It is easy to find that both schemes with larger step size show better polarization tracking performance under fast polarization rotation condition, compared with the case of small step size. However, it is opposite under the scenarios of low polarization rotation rate. This phenomenon is attributed to the fact that the estimation accuracy and tracking speed in both schemes are strictly dependent on its step size. Generally, large step size means fast convergence speed and poor estimation accuracy, and small step size always leads to high estimation accuracy and slow convergence speed. As shown in Fig. 3(a), with 1 dB Q factor degradation, the largest polarization tracking capability of CMA with the step size of 3e-1 is around 5 Mrad/s, which is 7 times than that of CMA with the step size of 5e-2. As a comparison, with the same threshold, NPCA can tolerate the polarization rotation rate as fast as 40 Mrad/s, which is around 8 times than that of CMA. Meanwhile, for 16QAM signals, NPCA also shows faster polarization tracking capability than that of MMA, as shown in Fig. 3(b). Compared with CMA/MMA, the smaller decision error and the constraint of the estimated matrix make NPCA fast polarization tracking capability and avoid the singularity problem in CMA/MMA. However, it is worthy to notice that MMA shows better polarization tracking performance than NPCA under the scenarios of slow polarization rotation rate. The main reason behinds this phenomenon is that large step size makes NPCA track the phase noise and polarization state successfully, but the large estimation error would degrade the performance of NPCA, especially for 16QAM signals. Compared with CMA/MMA + VVPE, the simulation results confirm the advantages of fast polarization tracking capability and modulation transparency in NPCA.

Fig. 3 The polarization tracking capability of NPCA and CMA/MMA + VVPE, (a) QPSK and (b) 16QAM signals.

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Meanwhile, the Q-factor also varies with the linewidth, as shown in Fig. 4. OSNR is set to 13 dB and 20 dB for QPSK and 16QAM signals, respectively. The step size of NPCA and CMA as well as MMA are set to 1, 1e-1 and 3e-2, respectively. With 1 dB Q-factor degradation, Fig. 4(a) indicates that the proposed NPCA can tolerate the linewidth up to 1.3 MHz for QPSK. With the same threshold, the linewidth tolerance of NPCA is around 300 KHz for 16QAM signals, as depicted in Fig. 4(b). As a comparison, VVPE shows better linewidth tolerance for both QPSK and 16QAM signals. The main reason for this phenomenon is that the large step size in NCPA must be used for tracking the phase and polarization successfully, but it also induces in residual phase. The residual phase will degrade the recovered signals, especially for high order modulation formats. Furthermore, for 16QAM signals, there is only 0.3 dB Q-factor penalty between NPCA and VVPE under 100 KHz linewidth. Thus, although the linewidth tolerance is limited, NPCA can be used to deal with the phase noise in the practical system.

Fig. 4 The linewidth tolerance of NPCA and CMA/MMA + VVPE, (a) QPSK and (b) 16QAM signals.

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In the following, BER of two schemes with optimal step size are investigated under different OSNR condition. The azimuth angle along fiber links is set to pi/6. The step size of NPCA and CMA as well as MMA are set to 1, 1e-1 and 3e-2 for the trade-off under different polarization rotation condition. With the target BER of 7% FEC threshold, the OSNR sensitivity in back-to-back (BTB) scenarios for QPSK and 16QAM signals are 12.94 dB and 19.24 dB and used as the reference to calculate the OSNR sensitivity penalty. As shown in Fig. 5, CMA/MMA + VVPE requires 13.12 dB and 19.72 dB to achieve the 7% FEC threshold for QPSK and 16QAM signals, respectively. The corresponding OSNR sensitivity penalties is around 0.18 dB and 0.48 dB. As a comparison, NPCA shows comparable BER performance with CMA + VVPE for QPSK signals. However, more OSNR is required for NPCA to achieve the same BER threshold for 16QAM signals, which is mainly introduced by the residual phase in NPCA. It is noticed that NPCA can also be used for polarization and phase tracking because the OSNR sensitivity penalty of NPCA is still smaller than 1 dB, compared with CMA/MMA + VVPE.

Fig. 5 BER as a function of OSNR for NPCA and CMA/MMA + BPS, (a) QPSK signals and (b) 16QAM signals.

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The hardware complexity of CMA/MMA + VVPE and NPCA for polarization and phase tracking are concluded as shown in Table 1. The complexity is analyzed based on the principle that the product of two complex value needs 4 real multipliers and 2 real adders [22]. NPCA consists of three stages: (1) signals recovery, (2) signals active and reconstruction and (3) parameter updating. At the first stage, 16N real multipliers and 10N real adders are used to achieve signal recovery. Then, 28N real multipliers and 14N real adders are needed to activate the recovered signal and reconstruct the received signal. The procedure of parameter updating needs 16N + 12 real multipliers and 15N + 5 real adders. Thus, the overall hardware complexity of NPCA is 60N + 12 real multipliers and 39N + 5 real adders. As a comparison, to achieve polarization recovery, CMA needs 44N + 12 real multipliers and 28N + 4 real adders as well as 2N look-up tables (LUTs). Moreover, 28N + 6 real multipliers and 14 + 4 real adders are need to achieve phase recovery by VVPE for QPSK. Similarly, 44N + 12 real multipliers and 32N + 4 real adders as well as 6N LUTs are needed in MMA. 42N + 6 real multipliers and 22N + 4 real adders are needed in QPSK partitioning based VVPE for 16QAM signals. As shows in Table 1, with the block length N of 40, NPCA just needs 2414 real multipliers and 1525 real adders, which is around 70% of MMA + VVPE for 16QAM signals. Obviously, compared with CMA/MMA + VVPE, NPCA must be a favor technique for polarization and phase tracking in BVTs, because power consumption in DSP will be decreased.

Table 1. Hardware complexity for CMA/MMA + VVPE and NPCA

View Table

4. Experimental result and discussion

To verify the polarization and phase tracking capability of NPCA, the experimental system is carried out under the scenarios of back-to-back (BTB) propagation, as shown in Fig. 6. A programmable arbitrary waveform generator (Keysight: M9502A) with the 3dB bandwidth of 32 GHz is used to generate 4 channels 28 GS/s electrical driver signals. An external cavity laser is used as the optical source. After the electrical amplifier, the electrical driver signals are added to dual polarization IQ modulator for generating signal lights. Besides, an ASE noise source with tunable power is used to adjust OSNR in BTB. In the receiver, another external cavity laser is used as the local oscillator lights which beats with signal in the optical hybrid. Moreover, the detected signal is sampled by 80 GS/s real-time oscilloscope (Keysight: DSA-X 96204Q). In the following, DSP algorithms are employed to compensate the related impairments, such as timing error, and IQ imbalance as well as frequency offset. Timing recovery is achieved by the cubic spline interpolation. After timing recovery, the endless polarization rotation is added in the electrical domain. Next, IQ imbalance is compensated by Gram-Schmidt orthogonalization produce (GSOP). Then, frequency offset is estimated using maximum fast Fourier transform. Finally, NPCA and CMA/MMA + VVPE are employed to track the polarization and phase noise simultaneously for comparison.

Fig. 6 28GS/s PDM QPSK/16QAM experimental systems and the procedure of DSP

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Similar with the simulation results, the polarization tracking capability of two schemes is also as a function of polarization rotation rate, as shown in Fig. 7. With 1 dB Q factor degradation. CMA/MMA can tolerate 4 Mrad/s and 15 Mrad/s polarization rotation rate for QPSK and 16QAM signals, respectively. Meanwhile, for QPSK signals, the proposed NPCA with the step size of 1e-1 can keep its performance under the scenarios of polarization rotation as fast as 40 Mrad/s. Moreover, the largest polarization tracking capability of NPCA is around 20 Mrad/s for 16QAM signals. As a comparison, the polarization tracking capability of NCPA is around 10 times and 1.3 times than that of CMA/MMA for QPSK and 16QAM signals, respectively. Moreover, NPCA also shows comparable polarization tracking performance with MMA under slow polarization rotation condition, as shown in Fig. 7(b). The reason for this phenomenon is that the laser linewidth in the experimental system is less than 100KHz and consequently negligible penalty is introduced due to the residual phase noise. The experimental results, which is consistent with simulation results and theoretical analysis, further confirm the advantage of the proposed scheme in polarization tracking ability.

Fig. 7 Polarization tracking capability of NPCA and CMA/MMA + VVPE, (a) QPSK and (b) 16QAM signals

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As depicted in Fig. 8, BER is plotted as a function of OSNR. The step size of NPCA and CMA/MMA is set to 1 and 5e-2, respectively. NPCA shows comparable performance with the conventional schemes for both QPSK and 16QAM signals. Moreover, the experimental result confirm NPCA can achieve the same performance with MMA + VVPE. Consequently, the excellent capability of NPCA for tracking the state of polarization and phase noise simultaneously is verified experimentally. Furthermore, the parallel implementation of NPCA also confirm the practical application in the real-time receiver.

Fig. 8 BER as a function of OSNR for 28GS/s PDM QPSK and 16QAM signals, (a) QPSK and (b) 16QAM signals

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5. Conclusion

In this paper, a low complexity, modulation-transparent and joint polarization and phase tracking scheme is proposed and demonstrated via both simulation and experiment. With HOS and the constraint of estimated matrix, NPCA has the advantage of fast convergence speed and singularity-free. Both simulation and experimental results confirm that NPCA can joint track polarization and phase noise simultaneously and it is also suitable for dealing with arbitrary modulation formats in EONs. Compared with the format dependent scheme of CMA/MMA + VVPE, NPCA can achieve fast polarization tracking performance over wide polarization rotation rates ranges. Moreover, around 30% computational resource can be reduced with both polarization and phase recovery considered together. Therefore, owing to its robust and flexible performance as well as simple implementation, NPCA can be considered as a candidate for joint polarization and phase tracking in future dynamic and reconfigurable optical networks.

Funding

Shenzhen Municipal Science and Technology Plan Project (JCYJ 20150529114045265) and National Natural Science Foundation of China (61671053 and 61871030)

Acknowledgements

Prof. Lu Chao is thanked for his support on experimental platform.

References

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Operations	CMA + VVPE(QPSK)	MMA + VVPE(16QAM)	NPCA(QPSK)	NPCA (16QAM)
Real multiplier	72N + 18	86N + 18	60N + 12	60N + 12
Real adder	42N + 8	54N + 8	39N + 5	39N + 5
LUTs	6N	12N	0	0

Low complexity, modulation-transparent and joint polarization and phase tracking scheme based on the nonlinear principal component analysis

Abstract

1. Introduction

2. Operating principle

3. Numerical simulation and discussion

4. Experimental result and discussion

5. Conclusion

Funding

Acknowledgements

References

Cited By

Figures (8)

Tables (1)

Equations (7)

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