1. Introduction

Optical transmission systems based on polarization division multiplexing (PDM), multilevel modulation formats, coherent detection, and digital signal processing (DSP) draw significant attention and are used in large capacity and long-haul optical transmission networks [1]. The PDM technology uses two orthogonal states of polarization (SOPs) at the same wavelength to double the transmission capacity [2]. However, PDM systems are more sensitive to polarization effects in optical fibers, such as polarization dependent loss (PDL), polarization mode dispersion (PMD), and rotation of SOP (RSOP) [3]. For a coherent detection PDM receiver, a multiple-input multiple-output (MIMO) algorithm, such as the constant modulus algorithm (CMA) or multiple modulus algorithm (MMA), is integrated in DSP to perform the equalization of the polarization effects [4]. On the other hand, chromatic dispersion (CD) is another static impairment, which results in the inter-symbol interference and needs to be compensated before the MIMO module. However, the CD compensation module cannot compensate CD exactly because of the mismatch between the evaluated CD and real value induced in the optical fiber channel, which results in the residual chromatic dispersion (RCD) (approximately several hundreds of picoseconds per nanometer) for the MIMO module to cope with [5,6]. Nevertheless, such MIMO algorithms as CMA/MMA cannot only handle the equalization of the polarization effects to some extent but also possess considerable tolerance to RCD.

It was found that in certain extreme environments, such as a lightning strike near the fiber cable, Faraday and Kerr effects can induce ultra-fast RSOP, as fast as several mega-radians per second, in fiber cables in the near-field covering several hundred kilometers [7]. Dr. Henry Yaffe, a renowned expert on the polarization effects in fibers and president of New Ridge Technologies, said, “I get a fair number of confidential phone calls and e-mails reporting coherent receivers unlocking due to very fast SOP transient events…” [8]. In addition, large PMD accumulated during long distance transmission in combination with ultra-fast RSOP impose a heavy burden on the MIMO algorithm (such as CMA/MMA), leading to the MIMO algorithm malfunction [9,10]. Therefore, in order to solve the extreme polarization impairments, it is necessary to find a feasible and stable polarization equalization scheme. Kalman filter was proposed as a fast convergent algorithm for equalizing polarization effects [11–13]. However, the Kalman filter structures were designed in time domain, while PMD and CD are induced in frequency domain, with the result that the PMD compensation using Kalman filters cannot achieve more than a symbol period [2,14,15]. Furthermore, the Kalman filter-based RSOP equalizations proposed in the literature are not immune to RCD, especially in the extreme polarization environments [11].

In this paper, we propose a new Kalman structure, which can jointly equalize large PMD (more than 200 ps) and ultra-fast RSOP (up to 3 Mrad/s) with large tolerance to RCD (over the range of ± 820 ps/nm in PDM-QPSK when the optical signal-to-noise rate (OSNR) is 15 dB or ± 500 ps/nm in PDM-16 QAM when the OSNR is 22 dB). Therefore, it functions properly in extreme polarization scenarios, such as a lightning strike, in which the transient change of the RSOP is combined with large PMD in the presence of RCD. In Section 2, we describe the principle of the proposed Kalman scheme in detail. First, we introduce a simplified fiber channel model to include the RSOP, RCD, and PMD. Then, according to the simplified fiber model, we design a Kalman structure based on extended Kalman filter (EKF), which enables the RSOP tracking in time domain and the compensation of RCD and PMD is carried out in frequency domain. Further, the appropriate state parameters are chosen to be monitored for the proposed Kalman scheme; the equalization operator for RCD and the equalization matrices for the PMD and RSOP are presented. In Section 3, first, the parameters are optimized and initialized. Next, the proposed Kalman scheme is validated on a 28 Gbaud PDM-QPSK/16 QAM coherent system platform. Simulation results for PDM-QPSK show that when the speed ranges of RSOP from 100 krad/s to 3 Mrad/s with differential group delay (DGD) as large as 200 ps and RCD of 300 ps/nm, the proposed Kalman scheme allows to achieve the OSNR requirements at a bit error rate (BER) of 3.8e-3 (7% forward error correction (FEC) threshold) from 13 dB to 13.7 dB. However, BERs of CMA in the OSNR range of 9-17 dB are all above the threshold when RSOP is above 500 krad/s. For PDM-16 QAM, the proposed Kalman scheme shows better performance for the mixing impairments under RCD from −500 ps/nm to 500 ps/nm with RSOP variation from 100 krad/s to 3 Mrad/s and PMD of 200 ps, BERs remain below 7% FEC threshold all the time. At the same time, when the RSOP speed is 100 krad/s with PMD of 200 ps, CMA-MMA is barely competent for RCD of ± 300 ps/nm, and with increasing of RCD or RSOP, the performance decreases sharply. Thus, the simulation result proves that the proposed Kalman scheme has better performance and stability than CMA-MMA. In Section 4, we made a computational complexity comparison of the proposed Kalman scheme and CMA/MMA. Finally, in Section 5, we made the conclusion.

2. Methodology

2.1 Simplified fiber channel and equalization operators

For the combined effect of RSOP and PMD in a fiber channel, the generally recognized model is RSOP1 + PMD + RSOP2, which is presented in Fig. 1(a) and Eq. (1):

 

Fig. 1 (a) Simplification of a generalized combined model of PMD and RSOP; (b) optical transmission model.

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It was proven in our previous work that this model can be simplified as a new PMD with the new principal state of polarization (PSP) and the same DGD combined with a new RSOP (new-RSOP + new-PSP PMD) [11], which is illustrated as in Fig. 1(a). Hence, for the effects of CD, RSOP and PMD we consider, an optical linear transmission impairments model is established as shown in Fig. 1(b).

In this paper, we consider only the linear signal impairments in the fiber channel, including frequency offset, PMD, RSOP, CD, carrier phase noise and amplified spontaneous emission (ASE) noise. Hence, in the receiver module, the received signal $q\left(t\right)$ can be expressed as [12]:

Equation (2) also indicates that CD and PMD are induced in frequency domain and must be equalized in frequency domain.

In addition, according to the PMD theory [19], the first-order PMD can be represented as a vector $\stackrel{\rightharpoonup }{\tau }=\Delta \tau \widehat{p}$in the Stokes space, it consists of two parts, the first part is represented by its direction $\widehat{p}$ which corresponds to the direction of the slow PSP, and the second part is represented by its magnitude DGD $(\Delta \tau ).$ Hence, we choose a Jones matrix $U_{eq}\left(\omega \right)$ as the first- order PMD compensation operator as shown in Eq. (4) [14].

In addition, according to the relation ${\beta }_{\text{2}}\text{=}-\left({\lambda }^{2}D\right)/2\pi c$ (D is the CD coefficient), the channel frequency response transfer function caused by CD can be described as [21]:

In section 2.2, we will describe the application of the proposed new Kalman filter architecture to achieve a joint equalization of the RCD, PMD, and RSOP using the above three equalization operators. Further, the RSOP tracking will be conducted in time domain, while the RCD and PMD compensations will be carried out in frequency domain. In addition, we will carry out the equalizations of the phase noise and frequency offset after the Kalman filter. Hence, we can schematically draw the DSP equalization module frame including our proposed joint equalization module as shown in Fig. 2.

 

Fig. 2 The proposed polarization effect treatment module.

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2.2 The principle and implementation of the proposed Kalman scheme

As mentioned above, we use a specially designed window-split Kalman scheme for the compensation of the joint effects of the RCD, PMD, and RSOP. To describe the all procedures of the proposed Kalman scheme in detail is a trivial thing. However, the three following issues are important in designing our proposed Kalman scheme, and should be explained: 1) correct state vector selection; 2) appropriate measurement and innovation vectors; 3) exact compensation operators. Next, we will elaborate on above three issues in turn.

Issue 1): The appropriate state vector for Kalman filter can be selected using the following procedure. As the PMD vector $\stackrel{\rightharpoonup }{\tau }$ can be represented as $\stackrel{\rightharpoonup }{\tau }={\left({\tau }_1,{\tau }_2,{\tau }_3\right)}^T,$ the values ${\tau }_1,{\tau }_2,{\tau }_3$can be selected as a part of the Kalman state parameters to monitor PMD variation. According to the Jones matrix of RSOP equalization in Eq. (3), we can find that RSOP in optical fibers generally has three parameters $\kappa ,\xi ,\eta ,$ so we can choose them as the state parameter to track the RSOP. In addition, for the reason that ρ represents the RCD value accumulated in the fiber links, we select ρ as the parameter to monitor RCD. Thus, we choose the state vector of Kalman filter as:

Issue 2): The measurement vector choice is based on what happens to the signal after equalization. As known, after equalization of polarization effects, the constellation of QPSK signal should converge to a circle with the radius r₁, which is the similar situation when we use the CMA as the polarization demultiplexing method. However, for a 16 QAM signal, the constellations of the signal should converge to three circles with the radii r₁, r₂, r₃. Hence, for a dual-polarization signal, we constructed a measurement matrix with the components having the following form:

Issue 3): As mentioned above, the RCD and PMD are induced in frequency domain; hence, their equalization should be also implemented in the frequency domain, while RSOP should be tacked in the time domain. To conduct RCD and PMD compensations in frequency domain, we use a window-split structure as in [11]. The window with length Lw covers several symbols from the data stream. The signal queue covered in the window is converted into the frequency domain by FFT. Afterward, RCD is compensated using Eq. (6), and then PMD is compensated using Eq. (4). Next, the signal queue is inversely transformed into the time domain by IFFT to equalize RSOP by Eq. (3). For the next Kalman iteration, the window slides several symbols forward by a step Δs to get a new sequence of symbols into Kalman filter for the next iterative compensation; then, the iteration repeats again. The window-split structure is shown in Fig. 3.

 

Fig. 3 Window-split structure.

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After discussion of the three important issues, we can outline the EKF used in this paper [22,23]. The following equations describe the main mathematic structure of the EKF:

The flow chart of the proposed Kalman scheme is illustrated in Fig. 4. The procedures in the left frame are responsible for the treatment of the received signal, while the right frame covers the prediction and correction procedures which described by Eqs. (12)–(15).

 

Fig. 4 The flow chart of the proposed Kalman scheme.

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To begin with Kalman process, the initialization is important, in which the initial estimation ${\widehat{x}}_0$ and covariance $P_0$ need to be assigned. At the same time, the noise covariance matrices $Q_{k-1},R_k$ are also required to be optimized, the appropriate initial values of $Q_{k-1},R_k$ provide better performance compared with improper choice. After initialization, the prediction process of Eq. (13) and the correction process of Eqs. (14) and (15) start repeating until the Kalman filter reaches convergence. As mentioned above $F$ is taken as the unit matrix. Through the prediction process, we can get the prior estimation of ${\widehat{x}}_{k|k-1}$at moment k from the previous posterior ${\widehat{x}}_{k-1}.$ In the correction process, as described in Eq. (15), updating of the state vector ${\widehat{x}}_k$ depends on two terms. One comes from the prior estimation ${\widehat{x}}_{k-1},$ the other corresponds to the innovation $d_k$ scaled by Kalman gain $K_k$ [11].

The window covered symbols from the received signal are transformed into frequency domain by FFT for the RCD and PMD compensations, then the equalized signal is converted into the time domain by IFFT, and the RSOP recovery is implemented. Next, the equalized signal will be sent to the right frame for the correction and prediction of the Kalman filter for the next iteration until the output signal makes the innovation in Eqs. (9) and (14) reach minimum, which means that the Kalman filter has come to a convergence and the impairments induced by the RCD, PMD and RSOP are equalized, and the signal is recovered [11].

3. Simulation Results

3.1 Simulation platform setup

To verify the feasibility of the proposed Kalman scheme for the joint equalization of ultra-fast RSOP, RCD and large PMD in coherent optical communication system, we established a 28 Gbaud PDM-QPSK/16 QAM coherent system as a simulation platform as shown in Fig. 5. At the transmitter, two branches of 28 Gbaud PDM-QPSK/16 QAM signal streams generated by two I/Q modulators construct two mutually orthogonal polarization signals. The roll-off factor is set as 0.1. In fiber channel, the CD, RSOP and PMD impairments have been considered, and we also add ASE noise into the signal, similar to the model in Fig. 1. Besides, we assume that 300kHz carrier phase noise is introduced due to laser linewidth. In addition, 300 MHz frequency offset is introduced between the lasers at transmitter and receiver. At the receiver, the received signal first passes through the coherent receiver, and then, enter the DSP module. The fixed CD compensation is carried out in the CD compensation module. the proposed Kalman scheme is used separately for the x and y branch data streams to recover the received signal from the RCD, PMD, and RSOP impairments (CMA and CMA-MMA algorithms were chosen for comparison). Besides, we employed a fourth-power scheme to estimate the frequency offset of either PDM-QPSK or PDM-16 QAM, and then we used blind phase search (BPS) to implement carrier phase recovery for PDM-QPSK and PDM-16 QAM. Finally, the BER was calculated.

 

Fig. 5 Simulation platform

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3.2 The parameters settings for the proposed Kalman scheme and CMA-MMA

First, in order to ensure the objectivity of the simulation results and obtain the optimal parameters, we performed 50 simulations per tested parameters and the data length of each simulation is setting 2¹⁶ symbols. For the proposed Kalman scheme, as described in section 2.2, we have to make the appropriate initialization.

For the real calculation of the proposed Kalman scheme, we modify the state vector as the normalized state vector, which is dimensionless one

The initialization of the noise covariance Q and R have a great influence on the performance of the Kalman filter. Therefore, the initialization is important. According to Ref [11], the Q and R are approximately diagonal. Following the optimization in Ref [11], we set the optimal initialization of Q and R as: for QPSK Q₁ = diag ([2e⁻², 2e⁻², 2e⁻², 1e⁻⁵, 1e⁻⁵, 1e⁻⁵, 8e⁻⁶]), R₁ = diag ([4,4]), and for 16QAM Q₂ = diag ([2e⁻³, 2e⁻³, 2e⁻³, 1e⁻⁶, 1e⁻⁶, 1e⁻⁶, 8e⁻⁶]), R₂ = diag ([5,5]).

The slide step Δs is given by 4-symbol. For the CMA/CMA-MMA, no matter whether the CMA or MMA is used, the iteration number is set to 5. For PDM-16 QAM, first, CMA with fewer symbols is used for pre-equalization, then the coefficients in CMA are used as the initial parameter in MMA for processing the left symbols. The window length L_w of the window-split structure will determine the performance of the proposed Kalman scheme, and the number of taps and the step size of CMA/MMA will affect the effectiveness of CMA/MMA. Generally, the larger window length is, the higher the complexity of the algorithm becomes. However, if the window length is small, the performance of the proposed Kalman scheme drops dramatically because of the severe impairments. Figures 6 and 7 depict the BER performance corresponding to different window lengths for the Kalman filter and different taps for the CMA/MMA in the process of RCD compensation under the same situation of impairment. Figures 8 and 9 depict the BER performance corresponding to different step sizes of the CMA/MMA in the process of RCD compensation under the same situation of impairment.

 

Fig. 6 BER vs. RCD with different window lengths of Kalman filter and different filter lengths of CMA in PDM-QPSK signals. (a) DGD = 50 ps RSOP = 100 krad/s (b) DGD = 100 ps RSOP = 500 krad/s.

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Fig. 7 BER vs. RCD with different window lengths of Kalman filter and different filter lengths of CMA-MMA in PDM-16 QAM signals. (a) DGD = 50 ps RSOP = 100 krad/s (b) DGD = 100 ps RSOP = 500 krad/s.

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Fig. 8 BER vs. RCD with different step sizes of CMA in PDM-QPSK signals. (a) DGD = 50 ps RSOP = 100 krad/s (b) DGD = 100 ps RSOP = 500 krad/s.

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Fig. 9 BER vs. RCD with different step sizes of the CMA-MMA in PDM-16 QAM signals. (a) DGD = 50 ps RSOP = 100 krad/s (b) DGD = 100 ps RSOP = 500 krad/s.

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Figures 6(a) and 7(a) depict the BER performance depending on the RCD with DGD of 50 ps and RSOP speed of 100 krad/s, at OSNR of 15 dB for PDM-QPSK, and OSNR of 22 dB for PDM-16 QAM, respectively. We can see that using 7% FEC threshold (BER = 3.8e-3) as the criterion, with above polarization impairment, CMA and CMA-MMA can work normally, with the RCD tolerance around ± 300 ps/nm for PDM-QPSK, 250 ps/nm for PDM-16QAM. However, for more severe polarization impairment, say DGD = 100 ps, and RSOP = 500 krad/s, CMA and CMA-MMA cannot achieve 7% FEC threshold even RCD is absent. As any rate, the tap number 15 behaves better, and is chosen as the tap for both CMA and MMA for the following discussions. On the contrary, Kalman based scheme performs well for the cases of weak and more severe polarization impairments. The window length plays a key role for the equalization performance. We can see that too short window length cannot provide enough window for equalization of PMD and RCD in frequency domain, although it means, at the same time, the low complexity and fast response for RSOP in time domain. We can also see that too long window length means high complexity and slow response for RSOP. The optimum length is 16 for PDM-QPSK and 32 for PDM-16QAM (Figs. 6 and 7), which can enhance the RCD tolerance to around ± 800 ps/nm for PDM-QPSK, and around ± 750 ps/nm for PDM-16QAM. Therefore, in the following discussions for proposed Kalman scheme, the window length is set to 16 for PDM-QPSK, and 32 for PDM-16QAM. Figures 8(a) and 9(a) depict the BER performance depending on the RCD with DGD of 50 ps and RSOP of 100 krad/s, at OSNR of 15 dB for PDM-QPSK, and OSNR of 22 dB for PDM-16 QAM, respectively. The impairment environment in Figs. 8(b) and 9(b) is DGD of 100 ps, RSOP of 500 krad/s, also at OSNR of 15 dB for PDM-QPSK, and OSNR of 22 dB for PDM-16 QAM, respectively. For PDM-QPSK, we can see from Figs. 8(a) and 8(b) that the optimal step size of CMA is 1e-3 with tap 15. For PDM-16 QAM (Figs. 9(a) and 9(b)), first, the CMA with tap of 15 and step size of 1e-3 is carried out as the pre-equalization with fewer symbols, then the MMA with filter length of 15 is implemented for processing the left symbols. From Figs. 9(a) and 9(b), we can find that when the step size of MMA is 1e-3, the equalization performance is best. In summary, for the PDM-QPSK and PDM-16 QAM, in the impairment circumstance of PMD from 50 to 100 ps and RSOP speed from 100 krad/s to 500 krad/s, the optimal step size for either CMA or MMA is 1e-3. We find this optimal value can be extended to the RSOP speed of 1 Mrad/s, or even 3 Mrad/s. Therefore, in the following discussions, the step sizes of CMA and MMA are taken as the optimal value 1e-3.

3.3 Performance comparison

The equalization performance of the proposed Kalman scheme was verified on the 28 GBaud PDM-QPSK/16 QAM simulation platform as shown in Fig. 5. For the comparison, the CMA or CMA-MMA was used as the reference. In addition, the polarization demultiplexing module (either using the proposed Kalman scheme or CMA/MMA) was followed by the frequency offset estimation (fourth-power method) and carrier phase recovery (BPS). Thus, the compensation performance of CMA/MMA and the proposed Kalman scheme could be compared under the same situation of impairment. Figures 10–12 show the overall comparison of the performance of the proposed Kalman scheme with CMA/CMA-MMA in different impairment environments.

 

Fig. 10 Performance evaluation: BER vs. OSNR (a) and (c) in PDM-QPSK, (b) and (d) in PDM-16 QAM.

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Fig. 11 Performance evaluation: (a) and (b) BER vs. RSOP in PDM-QPSK and PDM-16 QAM. (c) and (d) BER vs. RCD in PDM-QPSK and PDM-16 QAM.

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Fig. 12 Performance evaluation of RCD tracking for the proposed Kalman scheme: (a) and (b) RCD tracking error curve in PDM-QPSK and PDM-16 QAM.

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For QPSK, Figs. 10(a) and 10(c) depict the BER performance depending on the OSNR with DGD of 200 ps and RCD of 300 ps/nm. The speed of RSOP changes from 100 krad/s to 3 Mrad/s. We can see that when the RSOP speed is 100krad/s, which is a low speed, the CMA can work normally (under 7% FEC threshold), at the OSNR greater than 14 dB. But when the RSOP speed is greater than 500krad/s, the BER corresponding to CMA are all above the 7% FEC threshold. On the contrary, the proposed Kalman scheme exhibits excellent and stable performance in the wide range of RSOP from 100 krad/s to 3 Mrad/s with the DGD of 200 ps and the RCD of 300 ps/nm. It is worth noting that for the proposed Kalman scheme the required OSNR only ranges from 13.1 dB to 13.7 dB corresponding to a wide range of RSOP speed (100 krad/s ~3 Mrad/s) when using 7% FEC criterion. Compare with the case of no impairment, the OSNR penalty is only 0.7 dB. For 16 QAM, Figs. 10(b) and 10(d) present the BER performance corresponding to the proposed Kalman scheme and the CMA-MMA as the function of OSNR ranging from 17 dB to 25 dB under the same impairment conditions as for QPSK. With the same impairment cases as for QPSK, the required OSNR for 16QAM at the level of 7% FEC threshold are only from 21.2 dB to 22 dB. The OSNR penalty here is only 1 dB compare with no impairment. Figure 11(a) presents the BER vs. RSOP speed for the QPSK with DGD ranging from 30 ps to 200 ps when OSNR is set to 15 dB and RCD is set to 300 ps/nm. We can read out from Fig. 11(a) that, for QPSK signal, the CMA is competent for compensating large DGD with slow RSOP; with the fact that, for the DGD ranging from 30 ps to 200 ps, the CMA provides the BER values nearly around 7% FEC threshold when RSOP speed is 300 krad/s. However, with the larger RSOP speed (more than 800 krad/s), the BER performance of CMA degrades seriously, which proves that the CMA cannot response fast RSOP. On the other hand, the proposed Kalman scheme exhibits a better and stable BER performance, with the fact that all the BER values are below the 7% FEC threshold, even with the large DGD of 200 ps. For 16QAM signal, Fig. 11(b) illustrates the BER performance vs. RSOP speeds when using either CMA-MMA or the proposed Kalman scheme, with the DGD ranging from 30 ps to 200 ps, with OSNR of 22 dB and the RCD is 300 ps/nm. It indicates that CMA-MMA can provide a satisfied performance only when RSOP speed is lower than 300 krad/s, and the DGD is less than 50 ps. Figure 11(c) depicts the BER performance vs. RCD for the QPSK with the RSOP speed ranging from 100 krad/s to 3 Mrad/s when OSNR is set to 15 dB and DGD is set to 200 ps. With a slow RSOP speed, say 100 krad/s, the CMA can provide an RCD tolerance of approximately ± 400 ps/nm. However, as the RSOP speed is more than 500 kad/s, the CMA nearly cannot provide any RCD tolerance. On the contrary, the proposed Kalman scheme shows a good performance for either slow or ultra-fast RSOP with the almost the same RCD tolerance up to ± 820 ps/nm. Figure 11(d) exhibits the BER performance vs. RCD for PDM-16 QAM with a similar impairment environment as for QPSK when OSNR is set to 22 dB. Similarly, with a slow RSOP speed, say 100 krad/s, CMA-MMA has an approximate RCD tolerance of ± 300 ps/nm. As the RSOP speed increases, the performance of the CMA-MMA degrades dramatically. On the other hand, the proposed Kalman scheme has an RCD tolerance of approximately ± 700 ps/nm when the RSOP speed is 100 krad/s. Larger RSOP speed can only induce a little RCD tolerance degradation; for example, when the RSOP speed increases to 3 Mrad/s, the RCD tolerance reduces to approximately ± 500 ps/nm. In addition, we evaluate the RCD tracking capacity when employing the proposed Kalman filter as shown in Fig. 12(a) for QPSK and Fig. 12(b) for 16 QAM. The impairment environment is assumed to be 200 ps DGD, 3 Mrad/s RSOP, 820 ps/nm RCD under 15 dB OSNR for the PDM-QPSK and 500 ps/nm RCD under 22 dB OSNR for the PDM-16 QAM. We can see that the RCD tracking curve can rapidly reach the true value of the RCD at about 500 symbols for the QPSK, and the error value is around ± 5 ps/nm. For the PDM-16 QAM, the error value is around ± 8 ps/nm after about 400 symbols. This suggests that we can make a small modification for the proposed Kalman scheme, that after about 500 symbols we can stop the RCD tracking, only RCD compensation is implemented using the RCD value which has been evaluated by the previous tracking. In this way, we can reduce the complexity for the proposed Kalman scheme.

4. Computational complexity

In this section, we will evaluate complexity of the proposed Kalman scheme. The CMA (for QPSK) and CMA-MMA (for 16QAM) are chosen as the comparison counterparts. All the comparisons are based on their optimum parameters as mentioned in above sections.

As shown in Fig. 4, for the proposed Kalman filter, the implementation calculation includes three parts. 1) the updating calculations of the state vector using the equations of Eqs. (13)–(15) and the detail calculations of the innovation $d_k$(Eqs. (8) and (9)) and Kalman gain $K_k$through the Jacobi matrix $H=\partial h\left(x\right)/\partial x$ (Eqs. (A4)–(A6) in section 6 Appendix); 2) the signal recovery by equalization operators, Eqs. (3), (4), and (6); 3) the calculations for the window-split structure processing, along with the FFT/IFFT transforms. The comparison results are shown in Table 1. Among the aforementioned calculations, we record a complex multiplication as 4 real multiplications and 2 real additions, a complex additions as two real additions, and a complex exponential operations exp(j •) as 2 looking up table (LUT) operations. For the Kalman algorithm, we define L_w as the length of the window and Δs as the window slide step. For CMA-MMA, N₁ is defined as the taps number of CMA, N₂ is the taps number of MMA, S₁ represents the number of symbols of the pre-equalization for CMA, S₂ represents the number of the left symbols for MMA, and CNT is the repetition iteration number.

Table 1. Computational complexity comparison

View Table

As can be seen from Table 1, for the PDM-QPSK signal, if we take the length of the window L_w = 16 and Δs = 4, the total computational complexity of the proposed Kalman scheme is 424.25/382.75 (real multiplication/real addition). For CMA, we take N₁ = 15 and CNT = 1, and we can get the total computational complexity of 548/360 (real multiplication/real addition). We can see that the proposed Kalman scheme has a little less computational complexity compared to CMA. For the PDM-16QAM signal, if we take the length of the window L_w = 32 and Δs = 4, the total computational complexity of the proposed Kalman scheme is 773.75/728.75 (real multiplication/real addition), while for CMA-MMA, we take N₁ = N₂ = 15, CNT = 1, S₁ = 5000, S₂ = 2¹⁶−S₁, the total computational complexity is 506.6/369.2 (real multiplication/real addition). Thus, for PDM-16QAM, the proposed Kalman scheme has a little higher computational complexity than CMA-MMA. However, with this little sacrifice, we obtain much better performance by using the proposed Kalman scheme.

5. Conclusions

In this paper, we propose a special Kalman structure based equalization scheme which can jointly cope with the compensations of RCD, ultra-fast RSOP, and large PMD, in which the compensations for RCD and PMD are implemented in frequency domain, while the tracking of RSOP is carried out in time domain. The performance of the proposed scheme was verified in a 28 Gbaud PDM-QPSK/16 QAM system platform, and the performance comparison with CMA/CMA-MMA was performed. The proposed Kalman scheme can be competent for relatively large PMD (more than 200 ps), ultra-fast RSOP (up to 3 Mrad/s), and RCD (up to ± 820 ps/nm for PDM-QPSK at OSNR of 15 dB and ± 500 ps/nm for PDM-16 QAM at OSNR of 22 dB). The overall performance of the proposed Kalman scheme exhibits better compared with the CMA/CMA-MMA, including larger impairments tolerance for RCD, PMD, and RSOP, and lower computational complexity. In addition, the proposed Kalman scheme also shows a small OSNR penalty, stable BER performance under either weak or severe impairments of RCD, PMD, and RSOP. To summarize, the proposed Kalman scheme can be a good candidate for polarization demultiplexing method in some extreme polarization scenarios.

6 Appendix: The details of the updating of the state vector and the recovery of the signal

This appendix provides some details of the calculations for the proposed window-split Kalman scheme.

According to Fig. 4, the flow chart of the proposed Kalman scheme, the main calculations include the updating of the state vector ${\widehat{x}}_k$ and the updating of the signal from the received $q_t$ to recovered $u_{rec}$. The recovery from $q_t$ to $u_{rec}$ is carried out through being multiplied by the equalization operators $g_{eq}\left(\omega \right),$ $U_{eq}\left(\omega \right)$ in frequency domain, and $M_{eq}$ in time domain. The required parameters $\left({\tau }_1,{\tau }_2,{\tau }_3,\kappa ,\xi ,\eta ,\rho \right)$ in these equalization operators are actually the components of the state vector x, which come from the updating of x. The updating from $q_t$ to $u_{rec}$ is implemented in the window-split structure, and the received $q_t$ sequence in the window and the window sliding forward are illustrated in Fig. 13.

 

Fig. 13 The window-split structure and the window sliding forward.

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The signal in the window can be expressed as:

(A1)$$q_t=[q_i{q}_{i+1}\cdot \cdot \cdot q_{i+L_w-1}].$$

So the whole the signal recovery procedure from the received $q_t$ to the recovered $u_{rec}$ can be illustrated in Fig. 14 (the superscript x, y means the signal includes its x and y components):

 

Fig. 14 The signal recovery from the received one to the recovered one.

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Then we denote components for $g_{eq}\left(\omega \right),$ $U_{eq}\left(\omega \right),$ $M_{eq}$ as:

(A2)$$g_{eq}\left(\omega \right)=A_0,U_{eq}\left(\omega \right)=\left(\begin{array}{cc}{A}_1& A_2\\ A_3& A_4\end{array}\right),M_{eq}=\left(\begin{array}{cc}{A}_5& A_6\\ A_7& A_8\end{array}\right).$$

where A₀ represents the RCD compensation expression as in the right side of Eq. (6). coefficients A₁, A₂, A₃, A₄ represent the elements in the PMD compensation matrix as in Eq. (4), and coefficients A₅, A₆, A₇, A₈ represent the elements in the RSOP compensation matrix as in Eq. (3).

In order to complete the signal recovery in Fig. 4, we need the updated parameters $\left({\tau }_1,{\tau }_2,{\tau }_3,\kappa ,\xi ,\eta ,\rho \right)$ coming from the state vector updating. From Fig. 4, we see that the updating of ${\widehat{x}}_k$depends on the innovation $d_k=z_k-h_k({\widehat{x}}_{\text{k}|k-1})$ scaled by the Kalman gain $K_k,$ through the relation ${\widehat{x}}_k={\widehat{x}}_{k|k-1}+K_k{d}_k.$ In order to obtain $d_k$ and $K_k,$ we must calculate the Jacobi matrix $H=\partial h\left(x\right)/\partial x.$ Taking the QPSK signal as an example, the innovation vector can be expressed as:

(A3)$$d_k=\left[\begin{array}{c}0\\ 0\end{array}\right]-h_k\left({\widehat{x}}_{k|k-1}\right),h_k\left({\widehat{x}}_k\right)=\left[\begin{array}{c}\left(u_{x,k}{u}_{x,k}^{\ast }-1\right)\\ \left(u_{y,k}{u}_{y,k}^{\ast }-1\right)\end{array}\right].$$

where u and u* represent the recovered signal after equalization by the Kalman filter and its complex conjugate, respectively. The innovation expression in (A3) means that, after equalization of the polarization effects, the constellation of QPSK signal should converge to a circle with the radius of 1.

Based on the innovation expressed in (A3), we can calculate the Jacobi matrix $H=\partial h\left(x\right)/\partial x.$ Without loss of generality, we take the first element ${\tau }_1$ in the state vector as an example to get $H_{{\tau }_1,k}=\partial h\left({\widehat{x}}_k\right)/\partial {\tau }_1.$

(A4)$$H_{{\tau }_1,k}={\left[\begin{array}{c}{u}_x\cdot {\left(\frac{\partial u_x}{\partial {\tau }_1}\right)}^{\ast }+u_x^{*}\cdot \left(\frac{\partial u_x}{\partial {\tau }_1}\right)\\ u_y\cdot {\left(\frac{\partial u_y}{\partial {\tau }_1}\right)}^{\ast }+u_y^{*}\cdot \left(\frac{\partial u_y}{\partial {\tau }_1}\right)\end{array}\right]}_k.$$

where

(A5)$$\begin{array}{l}\frac{\partial u_x}{\partial {\tau }_1}=A_5\cdot \frac{\partial q_{Ugt}^x}{\partial {\tau }_1}+A_6\cdot \frac{\partial q_{Ugt}^y}{\partial {\tau }_1},\\ \frac{\partial u_y}{\partial {\tau }_1}=A_7\cdot \frac{\partial q_{Ugt}^x}{\partial {\tau }_1}+A_8\cdot \frac{\partial q_{Ugt}^y}{\partial {\tau }_1}.\end{array}$$

and

(A6)$$\begin{array}{l}\frac{\partial q_{Ugt}^x}{\partial {\tau }_1}=ifft\left(\frac{\partial A_1}{\partial {\tau }_1}\cdot A_0\cdot q_f^x+\frac{\partial A_2}{\partial {\tau }_1}\cdot A_0\cdot q_f^y\right),\\ \frac{\partial q_{Ugt}^y}{\partial {\tau }_1}=ifft\left(\frac{\partial A_3}{\partial {\tau }_1}\cdot A_0\cdot q_f^x+\frac{\partial A_4}{\partial {\tau }_1}\cdot A_0\cdot q_f^y\right).\end{array}$$

where ifft denotes the inverse fast Fourier transformation. Therefore, we can finally obtain H matrix according to above calculation rules, and then the Kalman gain $K_k$. In addition, we can realize the updating of the state vector, and finally the recovery of the signal.

Funding

National Natural Science Foundation of China (61571057, 61527820, 61575082).

Acknowledgments

The authors also thank the anonymous reviewers for their substantial and inspiring comments.

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