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Abstract
When we implement the equalizations of polarization effects using a Kalman filter (KF) in a coherent optical fiber communication system, we will require to multiply many matrices. If the state vector describing the system has a dimension of n, the state error covariance matrix P will have the dimension of n × n, and other matrices used in the Kalman filter will also have the dimension of n × l (l is the dimension of the measurement vector). If n is very large, the KF-based algorithm will suffer from significant complexity, which results in an impractical KF-based polarization demultiplexing algorithm. In this paper, we propose a new structured KF-based polarization demultiplexing algorithm in which the state error covariance matrix P is diagonalized, which we call the diagonalized Kalman filter (DKF). We theoretically analyze the rationality of the DKF, and the validity of the DKF was verified in both 64 Gbaud polarization-division multiplexed (PDM) QPSK and 16QAM Nyquist coherent optical simulation systems. Compared with the conventional KF, simulation results proved that under a rotation of state of polarization from 1 to 10 Mrad/s for QPSK and 1 to 5 Mrad/s for 16QAM, a differential group delay from 15 to 75 ps, and a residual chromatic dispersion of 100 ps/nm, the OSNR penalties for the DKF are only within 0.5 dB for QPSK at the threshold BER = 3.8 × 10−3, and within 2 dB for 16-QAM at the threshold BER = 2 × 10−2, respectively, compare to the case of no impairment. In the meantime, for the proposed DKF, a computational complexity reduction of over 30% is achieved, compared with conventional KF, at the expense of about no more than 50 symbols convergence delay.
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1. Introduction
The Kalman filter (KF) is an efficient recursive filter and has many applications in the technical field, including the guidance, navigation, and control of aircraft [1]. In 2010, the KF was introduced for optical communications [2]. Since then, almost all linear impairments such as chromatic dispersion (CD), rotation of state of polarization (RSOP) [3–6], polarization mode dispersion (PMD) [7,8], carrier frequency offset, and carrier phase noise [9,10] can be compensated using KF-based algorithms. In particular, under the polarization state transient in optical fiber caused by the Faraday effect under a lightning scenario, the traditionally used multi-input–multi-output digital signal processing (DSP) algorithms, such as the constant modulus algorithm (CMA), will fail [11–13]. Our group proposed a time–frequency domain KF structure, which can jointly equalize an ultra-fast RSOP (up to 2 Mrad/s) and large PMD (with a differential group delay (DGD) exceeding six symbol periods) [14]. In addition, under the inevitable residual chromatic dispersion (RCD), which is induced by the mismatch between the true CD and equalized CD [15–17], we offered a joint compensation scheme for extreme polarization impairments and RCD [18]. Therefore, a well-designed Kalman filter would be a promising polarization demultiplexing algorithm in future optical fiber communication systems.
However, in practical communication systems, the complexity of an algorithm is an important consideration. The complexity of the KF is a critical factor limiting its application in practice and primarily results from matrix multiplications. According to the principle of the KF, if the state vector x has a dimension of n, that is, the state vector has n parameters to describe the system, the state error covariance square matrix P will have the dimension of n × n, and other matrices used in the KF will have dimensions of n × l (e.g., the observation matrix, ${\mathbf H} = \frac{{\partial h({\mathbf x})}}{{\partial {\mathbf x}}}$ is the Jacobian matrix of h(x), which connects the measurement vector z and the state vector in the form of ${{\mathbf z}_k} = h({{\mathbf x}_k}) + {{\mathbf v}_k}(noise)$, and used for calculation of Kalman gain). If n is very large, the KF-based algorithm will suffer from significant complexity, which results in an impractical KF-based polarization demultiplexing algorithm. For example, in short-reach optical communication, where only the RSOP was equalized, Ref. [4] selected the three angles of RSOP impairment as state vectors to be monitored by a KF, which made P a 3 × 3 square matrix. In contrast, for a long-haul optical communication system, the equalization of an ultra-fast RSOP and a large PMD in the presence of RCD should be jointly equalized. Here, the selected state vector is of seven dimensions, three for the RSOP, three for the PMD, and one for the RCD; hence, P is a 7 × 7 square matrix. Therefore, a large number of multi-dimensional matrix multiplication operations are required to update x and P during each iteration, which means a large complexity of the KF-based polarization demultiplexing. Therefore, reducing the complexity of the conventional KF-based polarization multiplexing algorithm is a method of overcoming this problem.
When a KF converges, the trace of P remains stable for every iteration [19]. This property demonstrates the concept that we can only update the diagonal elements of P and set the nondiagonal elements of P to zero for every iteration. This treatment changes P from a square matrix to a diagonal matrix, meaning that the multiplications of nonzero square matrices become multiplications between diagonal matrices, which is the resource of complexity reduction. Specifically, as we discuss in Section 3.3, when two n-dimensional real square matrices are multiplied, n 3 real multiplications and n 3 real additions are required. However, when two diagonal real square matrices are multiplied, the complexity is only n real multiplications and n real additions. With diagonalized matrix treatment instead of nonzero square treatment, we must check if the trace of P remains stable. Fortunately, based on theoretical and simulation analyses, this diagonalized treatment only slightly changes the stability of the trace of P and hence the polarization demultiplexing ability when using this diagonalized Kalman filter (DKF).
In this paper, we propose an effective polarization demultiplexing algorithm based on such a DKF, in which we maintain P as a diagonal matrix during the iterations of the DKF, which significantly reduces the complexity of the KF. Through theoretical analysis and simulation verification, we proved that, compared with the conventional KF, for QPSK the proposed DKF suffers from an OSNR penalty of only 0.5 dB at the threshold BER = 3.8 × 10−3, when we jointly equalize the RSOP (up to 10 Mrad/s) and PMD (DGD = 45 ps) in the presence of an RCD (100 ps/nm) for QPSK, with a complexity reduction rate over 30%, compared to the case of no impairment. For 16QAM, the DKF experiences an OSNR penalty of 2 dB at the threshold BER = 2 × 10−2, under the impairments of up to 5 Mrad/s RSOP, combined with 45 ps DGD and 100 ps/nm RCD, while the complexity reduction rate over 30% is achieved. Therefore, the proposed DKF can be a good substitute for the conventional Kalman scheme used in optical fiber communications.
2. Principle
2.1 Role of the covariance matrix of the state vector in a Kalman Filter
The KF is an optimal estimation algorithm based on the least mean square error deviation. It defines a state vector x to be estimated step by step, and it defines a measurement z vector to aid in updating the state vector. The stochastic process to be estimated and the measurement relation can be described as [20]
The KF operation process is as follows: The first step is to initialize the state vector x and the covariance matrix P of the state vector using Eq. (2). After the initialization, the iterations of Eqs. (3 )–(5) are repeated. For the k-th iteration, $\mathrm{\hat{x}}_\textrm{k}^\textrm{ - }$ and $\textrm{P}_\textrm{k}^\textrm{ - }$ are first obtained from the k−1-th step posteriori estimation $\mathrm{\hat{x}}_{\textrm{k - 1}}^\textrm{ + }$ and $\textrm{P}_{\textrm{k - 1}}^\textrm{ + }$ using Eq. (3). Subsequently, the Kalman gain is calculated using Eq. (4). In the update process, the posteriori state $\mathrm{\hat{x}}_\textrm{k}^\textrm{ + }\; $is updated with the linear combination of the a priori estimate $\mathrm{\hat{x}}_\textrm{k}^\textrm{ - }$ and the weighted difference between the actual measurement z k and the measurement prediction $h({\mathrm{\hat{x}}_\textrm{k}^\textrm{ - }} )$ using Eq. (5). $\textrm{P}_\textrm{k}^\textrm{ + }$ is also updated.
The covariance matrix P has an important role in the calculations in Eqs. (3 )–(5). If the state vector has n dimensions, P will have n × n dimensions, ${\mathbf P} \in \mathrm{\mathbb{R}}^{n \times n}$. The higher the dimension, the greater the computational complexity of the KF, because there are many matrix multiplications in which P is the multiplier.
We know that if the KF converges, we obtain
Based on the analyses above, the covariance matrix P has the role of not only the source of computational complexity, but also the convergence performance of the KF. In this study, we aimed to reduce the computational complexity by diagonalizing P, in which only the diagonal elements of P are considered and the nondiagonal elements are set as zero. A diagonalized P matrix will reduce the complexity of the matrix multiplications. However, how will the calculation accuracy and convergence performance be? A KF based on a diagonalized P matrix is called a DKF. In Section 2.2, we discuss the feasibility of the DKF.2.2 Constitution of the diagonalized Kalman filter
This section describes the constitution of the DKF. Using a three-dimensional state vector such as ${{\mathbf x}_k} = {[{x_{k1}},{x_{k2}},{x_{k3}}]^T}$ an example, the posteriori state vector is $\hat{{\mathbf x}}_k^{+}{=} {[\hat{x}_{k1}^{+},\hat{x}_{k2}^{+},\hat{x}_{k3}^{+}]^{T}}$. Thus the corresponding covariance matrix is ${\mathbf P}_k^{+}{=} E[{\mathbf e}_k^{+} {\mathbf e}_k^{+T}] = E[({{{\mathbf x}_k} - \hat{{\mathbf x}}_k^{+}} ){({{{\mathbf x}_k} - \hat{{\mathbf x}}_k^{+}} )^{T}}]$, which can be expressed as follows [19]:
Based on the above assumption that $\hat{x}_{k\textrm{1}}^ +{=} E({x_{k\textrm{1}}}),\begin{array}{c} {} \end{array}\hat{x}_{k2}^ +{=} E({x_{k2}}),\begin{array}{c} {} \end{array}\hat{x}_{k3}^ +{=} E({x_{k3}})$, Eq. (7) can be written as a diagonal matrix:
2.3 Polarization equalization using the DKF
In this paper, we provide an example to evaluate the performance of the DKF in which we will complete depolarization in a coherent polarization-division multiplexed quadrature phase shift keying (PDM-QPSK) communication system with impairments including ultra-fast RSOP, large PMD, and residual CD (RCD).
In the proposed DKF, the RCD equalization operator is expressed as follows [16]:
where $\mathrm{\rho \;\ }$denotes the value of the accumulated RCD in the fiber links with units of ps/nm. And $\mathrm{\lambda }$ represents the wavelength of light, which is chosen as 1550 nm. $\mathrm{\omega }$ stands for the angular frequency and c is the speed of light, which is approximated to 3 × 10−8 m/s.The first-order PMD equalization matrix in the Jones space can be expressed as [21]
The RSOP equalization matrix can be expressed as [4]:
According to Eqs. (11 )–(13), we extract the state vector as
in which all the parameters describe the RCD, PMD, and RSOP.For the innovation, in the constellation space, the equalized QPSK signals should converge into a ring with a normalized radius r = 1. Similarly, the 16QAM signals should converge into three rings with normalized radii r = 1, $\sqrt {\textrm{5}} $, 3. Therefore, we select innovation as
As we know, chromatic dispersion (CD) and PMD are the impairments due to the physics mechanism in frequency domain, while RSOP induced in time domain. Therefore, the RCD and PMD equalization (using the equalization operators in Eq. (11) and (12)) should be implemented in frequency domain, and RSOP (using Eq. (13)) in time domain. In Ref. [14] we designed a window-split structure for the Kalman filter in which we collected a symbol sequence in a symbol window with a window length Lw, so that we could make Fourier transformation of the time-domain symbol sequence in the window into frequency domain, in order for us to make RCD and PMD equalizations, and then we made the inverse Fourier transformation of the symbol sequence into time domain, in order for us make RSOP equalization. After one iteration of equalization, the window will slide forward a step of symbols with the slide step length Δs for the next iteration. The readers can get more information of the window-split structure in Ref. [14].
Figure 1 shows a flow chart of the KF and DKF. The part of the left half in Fig. 1 is the signal path through which the compensation processes of the signals are conducted. Through the window-split structure, RCD and PMD are compensated in the frequency domain, and the RSOP is equalized in the time domain [14]. The right half shows the recursive processes for the KF or DKF. The parameters (ρ), (τ 1, τ 2, τ 3), and (γ, α, β) used for the equalization matrices are updated and provided by the iterations of the KF/DKF. The operations of KF and DKF both follow Eqs. (2 )–(4). For KF, we use Eq. (5) to update the posteriori error covariance matrix P, and for DKF, we use Eq. (10).
3. Simulation
3.1 Simulation platform
To verify the performance of the proposed DKF, a simulation platform of both 64 Gbaud PDM-QPSK and 64 Gbaud PDM-16QAM Nyquist coherent optical communication system was constructed in this study, as shown in Fig. 2: Two orthogonal polarized optical signals were generated at the transmitter, and the root-raised-cosine shaping with the roll-off factor was set as 0.1 in the arbitrary wave generator (AWG). Subsequently, the optical signals passed through the fiber channel with impairments such as amplifier spontaneous emission noise, CD, combined RSOP, and PMD. To emulate the polarization effect including RSOP and the first order PMD, we take the model of RSOP1 + PMD + RSOP2, as defined in [14], in which RSOP1 and RSOP2 are independent time-varying RSOP matrices, and PMD matrix takes the form as Eq. (1) in [14]. The PMD matrix includes the DGD Δτ, the fast and slow principal states of polarization (PSP). If the angels in matrices RSOP1 and RSOP2 varies, we have proved that the ultimate PSPs of fiber system will also vary with the speed almost as the same as the RSOP speed, with DGD fixed. The readers can get more information in [14] and [4]. At the receiver end, the distorted signals passed through the optical Gaussian optical bandpass (OBPF) with a bandwidth of 1.25${\times} $64 GHz. In addition, 300 kHz linewidth carrier phase noise and 300 MHz carrier frequency offset were induced by the continuous wave (CW) laser at the transmitter and local oscillator (LO) laser at the receiver. At the receiver, the optical signals were transformed into electrical signals and sent to the DSP module for equalization and evaluation. In the DSP module, the signals were resampled as two samples per symbol. A fixed CD compensation was implemented. Subsequently, we applied the proposed DKF to jointly equalize the RCD, PMD, and RSOP and used the KF as a comparison. Subsequently, the improved Mth-power algorithm and blind phase search were performed for carrier frequency estimation and carrier phase recovery [22,23]. Finally, the bit error rate (BER) was calculated to evaluate the performance of the DKF/KF.
Fig. 2. Simulation platform diagram. CW: continuous wave; PBS: polarization beam splitter; IQ Mod.: in-phase and quadrature-phase modulator; AWG: arbitrary waveform generator; OBPF: optical Gaussian optical bandpass; LO: local oscillator
3.2 Simulation verification
3.2.1 Parameter initialization
To implement KF/DKF in the aforementioned simulation platform, we initialized the state vector and error covariance matrix and obtained the optimal noise matrices Q and R. For the window-split structure, the appropriate window length Lw and slide step $\mathrm{\Delta s}$ were also considered.
It is clear that a smaller window length Lw and a larger step $\mathrm{\Delta s}$ lowers the complexity per symbol; however, the performance of the algorithm cannot be guaranteed. Based on the trade-off between complexity and performance, we selected Lw = 16 and $\mathrm{\Delta s}$ = 4, and the reason why we chose the parameter will be discussed in Fig. 4 from section 3.2.2. For QPSK, we defined the initial ${\mathrm{\hat{x}}_\textrm{0}}\,\textrm{ = }\,{\textrm{(1,1,1,0,0,0,0,)}^\textrm{T}}$ and P0 was set as a unit matrix with dimensions of 7, which was the number of elements of the selected state vector. For the KF, Q was set as a diagonal matrix with diagonal elements (10−5, 10−5, 10−5, 10−9, 10−9, 10−9, 10−7), and R was set as a diagonal matrix with diagonal elements (10−2, 10−2). For DKF, the values of Q and R were set as (10−5, 10−5, 10−5, 10−8, 10−8, 10−8, 10−6) and (10−2, 10−2), respectively. For 16QAM, Q and R were set as (10−3, 10−3, 10−3, 10−6, 10−6, 10−6, 10−5) and (103, 103) for KF, (10−3, 10−3, 10−3, 10−7, 10−7, 10−7, 10−6) and (102, 102) for DKF. These parameters were set based on experiences for the best performance and can be adjusted slightly under different conditions of impairments.
3.2.2 BER performance
In this section, we will exhibit the BER performance of the proposed DKF. At first, we should make optimization of the parameters Lw and $\mathrm{\Delta s}$. After the optimization as shown in following discussion, we chose Lw = 16 and $\mathrm{\Delta s}$ = 4 based on the trade-off between the performance and complexity of algorithms.
Figure 3 depicts the BER performance vs. RSOP under the condition of different window length Lw and slide step $\mathrm{\Delta s}$. Evidently smaller slide step $\mathrm{\Delta s}$ causes better performance, while indicates more iteration of the algorithm (see the red lines, either solid lines for KF or dashed lines for DKF, have the better BER performance than the blue lines). Larger Lw is generally beneficial to the PMD compensation, but harmful for fast RSOP equalization. We compare Fig. 3 (a) and (c) for PDM-QPSK, Fig. 3 (b) and (d) for PDM-16QAM, and conclude that $\mathrm{\Delta s}$ is the key factor affecting the performance of the algorithm, and Lw seems not make apparent difference under the given impairment conditions. But we also simulated the cases of Lw = 8 and Lw = 64 (not show here), which both caused the significant BER performance degradations.
Fig. 3. Performance evaluation: (a) and (b) BER vs. RSOP in PDM-QPSK and PDM-16QAM. (c) and (d) BER vs. RSOP in PDM-QPSK and PDM-16QAM.
Fig. 4. Performance evaluation: (a) and (b) BER vs. OSNR in PDM-QPSK and PDM-16QAM. (c) and (d) BER vs. RSOP in PDM-QPSK and PDM-16QAM. (e) and (f) BER vs. RCD in PDM-QPSK and PDM-16QAM
Based on the above analysis considering complexity and performance, we chose Lw = 16 and $\mathrm{\Delta s}$ = 4 to balance the performance and complexity of KF/DKF. Detailed complexity analysis can be found in Table 1, Table 2 and Table 3 in the section 3.3.
Table 1. Complexity comparison per symbol for the KF and DKF
Table 2. The detailed Numerical comparison per symbol of KF/DKF for QPSK
Table 3. The detailed Numerical comparison per symbol of KF/DKF for 16QAM
Next, the performance of algorithms for jointly compensating RSOP and PMD in presence of RCD was studied as in Fig. 4.
Figure 4 shows the performance of KF and DKF. The left figures were obtained on the 64Gbuad-PDM-QPSK system and the right ones were obtained on the 64Gbuad-PDM-16QAM system.
Figure 4 (a) and (b) depicts the BER performance vs. optical signal-to noise ratio (OSNR) with a DGD of 45ps and RCD of 100 ps/nm. The RSOP speed was set from 1 to 10 Mrad/s and 1 to 5 Mrad/s respectively. Under aforementioned impairments, DKF did not show apparent performance degeneration compared with KF. To be specific, with the threshold BER = 3.8 × 10−3 (corresponding to a hardware decision 7% forward error correction threshold), compared to KF (corresponds to solid lines), DKF shows almost same performance (corresponds to dashed lines) for PDM-QPSK with the small OSNR penalty of less than 0.05 dB as shown in Fig. 4(a), and for PDM-16QAM, DKF exhibits little larger performance degradation with OSNR penalty of 0.1 dB.
Figure 4 (c) and (d) depict the BER performances as the function of RSOP speed, with a DGD ranging from 15 to 75 ps in the presence of 100ps/nm RCD. The OSNRs are set as 18 dB in (c) and 26 dB in (d), respectively. We see again that KF and DKF exhibit almost same performances for PDM-QPSK, and for PDM-16QAM, there are only a little difference between them (the solid lines and dashed lines are also corresponded to the cases of KF and DKF respectively).
Figure 4 (e) and (f) depict the BER performance vs. RCD with DGD 45 ps and RSOP speeds ranging from 1 to 10 Mrad/s in Fig. 4(e) and 1 to 5 Mrad/s in Fig. 4(f), respectively. The ONSRs are set as 18 dB for PDM-QPSK in (e) and 26 dB for PDM-16QAM in (f), respectively. As shown in Fig. 4(e), with the threshold BER = 3.8×10−3, for PDM-QPSK, at the RSOP speed 5 Mrad/s, KF has the RCD tolerance of 170 ps/nm, while DKF has the RCD tolerance of 150 ps/nm. In Fig. 4(f), for PDM-16QAM system, with the threshold BER = 2.0×10−2 and also at the RSOP speed of 5 Mrad/s, KF and DKF have a little larger RCD tolerance performance difference,120 ps/nm for KF and 110 ps/nm for DKF.
3.2.3 Impairments tracking ability
One of the good features of Kalman filter is rapid convergence in equalizing the impairments. In this section, we investigate the convergence performance of the proposed DKF. As an example, we take Lw = 16 and $\mathrm{\Delta s}$ = 4, OSNR = 18 dB, RSOP speed = 10 Mrad/s, DGD = 45 ps, and RCD = 100 ps/nm. As mentioned in section 2.1, the moment when the covariance matrix P becomes a steady state $\textrm{Tr }({\mathbf P}_k^ + ) = \textrm{Tr }({\mathbf P}_{k - 1}^ + )$ means the Kalman filter gets to convergence. In Fig. 5, we check $\textrm{Tr }({\mathbf P}_k^ + )$ of the symbols in the 64 GBuad-PDM-QPSK system either using KF (corresponds to black line) or DKF (corresponds to red line). We observed in Fig. 5, that the two curves representing KF and DKF tends to be stationary after approximately 400-th symbol, and the convergence speed of KF is faster than DKF.
To be specific, we investigate the convergence features of the 3 components ${\tau _1},\textrm{ }{\tau _2},\textrm{ }{\tau _3}$ of PMD, and the parameter ρ of RCD, as shown in Fig. 6.
Fig. 6. Performance evaluation of PMD vector tracing and RCD value tracing; (a) and (b) for the KF, (c) and (d) for the DKF
Figure 6(a) and (c) depict the PMD tracking curves for KF and DKF, respectively. 6(b) and (d) depict the RCD tracking for KF and DKF, respectively. To more accurately describe the difference between KF and DKF in tracking performance, we calculated the tracking error based on the tracking curves and the root mean square errors (RMSE). RMSE follows the form of $\sqrt {\sum\limits_{i = 1}^N {(predicte{d_i} - actua{l_i})} /N} $, where the predictedi represents the predicted value of the i-th observation, the actuali represents the true value of the i-th observation, N is the total number of observations. The total number of symbols sent into the KF/DKF is 220 in this manuscript. We find in Fig. 6, that the RMSE values of DKF are a little larger than KF on average, but this tracking errors are still small and acceptable.
The vertical olive-green lines drawn in Fig. 6 indicate the convergence beginning of KF and DKF algorithms. As shown in Fig. 6 (a) and (b), KF converges approximately at the 400-th symbol for PMD tracking and 150-th symbol for RCD tracking, respectively. In Fig. 6 (c) and (d), DKF converges approximately at the 450-th symbol for PMD tracking, 320-th symbol for RCD tracking. DKF converges 12.5% slower than KF under such extreme conditions, which is an expense of the diagonalization operation. Note that this 12.5% convergence delay corresponds to the relatively severe impairments of RSOP speed of 10 Mrad/s and DGD 45 ps (approximate 3.5 × symbol period). When the impairments are not so severe, both KF and DKF will converge faster and the time difference of convergence between them will be smaller. Considering the benefit of reduction in complexity of over 30% (see section 3.3), it is worth to sacrifice this amount of less than 12.5% convergence speed.
Furthermore, above convergence speed is evaluated by the symbols passed before convergence. Now we evaluate the convergence features of KF and DKF by the time elapsed. Based on ordinary personal computer (CPU: Intel Core i7-10700 CPU @ 2.90 GHz. RAM: 32.0 GB. Graphics card: NVIDIA GeForce GTX 1660 Ti.), a simulation was conducted on a 64GBaud-PDM-QPSK optical communication system platform we built, and 220 symbols with two samples per symbol were sent into the KF/DKF algorithms. The elapsed time were 33.938744 seconds for KF and 32.999200 seconds for DKF, respectively. Since KF converges at the 400-th symbol, and DKF converges at the 450-th symbol, KF takes 400 / 220 ${\times} $33.938744 s = 0.0129 s, and DKF takes 450 / 220 ${\times} $32.999200 s = 0.0142 s, respectively. DKF converges 10.08% slower than KF. Note that the above evaluation is based on the offline real-time processing, the practical feedback delay is not considered. Since the complexity of DKF is 30% less than KF (see section 3.3), the feedback delay will decrease accordingly, which is beneficial to the practical application.
In short, DKF converges around 10% slower than KF due to the diagonalization operation. However, considering the complexity reduction, DKF wins KF in the benefit competition in the polarization demultiplexing in optical fiber communication system.
3.3 Complexity comparison
In this section, we compare the complexities of the KF and DKF. Because multipliers consume more resources than adders and occupy a large proportion in the calculation, we consider the multipliers as the indicator of complexity of the algorithm.
Different impairments needed to be equalized means different sizes of the state vectors (with n denoting the dimension). For example, in order to equalize the combined effects of RCD, PMD, and RSOP, n = 7 (${\mathbf x} = ({{\tau_1},{\tau_2},{\tau_3},\alpha ,\beta ,\kappa ,\rho } )$) and the window-split structure must be applied. When we want to equalize PMD and RSOP without considering RCD, the dimension of state vector is 6 (${\mathbf x} = ({{\tau_1},{\tau_2},{\tau_3},\alpha ,\beta ,\kappa } )$). For the short-reach transmission system, only the RSOP with dimension of 3 will be considered (${\mathbf x} = ({\alpha ,\beta ,\kappa } )$), and the window-split structure will not be used.
We made the complexity comparison of KF/DKF per symbol in Table 1, and detailed numerical comparison for QPSK in Table 2, for 16-QAM in Table 3.
Kalman filter is applied to compensate impairments by updating the state vector x and the posteriori error covariance matrix P based on matrix multiplications, as shown in the right half part of flow chart in Fig. 1. DKF follows the same procedures, except that the matrix P in DKF takes a diagonal matrix. Therefore, the complexity of KF has the order of O(n 3), and that of DKF is O(n). Specifically, when only the matrix multiplication part of KF/DKF is considered as in Eqs. (1 )–(5), the KF includes ${C_{KF\_i}} = {n^3} + 6{n^2} + 10n$ multiplications, meanwhile, DKF has ${C_{DKF\_i}} = 17n$ multiplications. The complexity of this part is independent of compensation. Moreover, this is the only part that reduces the complexity of the Kalman scheme.
In addition, we counted the amount of calculation in the left half of the algorithm flow chart in Fig. 1. DKF has the same complexity as KF in this part. At first, we analyzed the impairments compensation for PMD, RCD and RSOP based on the window split structure. Note that these mathematical operator like exp $({\cdot} )$, cos$({\cdot} )$ , sin $({\cdot} )$ , $\sqrt \cdot $ are considered as lookup table calculations. The symbols are sent into the KF/DKF by two samples per symbol, so when the window length is Lw, the FFT/IFFT complexity is $2{L_w}\log _2^{2{L_w}}$. For Lw symbols, there are 2Lw multiplications needed for RCD compensation, 8Lw multiplications needed for PMD compensation, and 8Lw multiplications needed for RSOP compensation. Therefore, the complexity of this part is ${C_{EM\_w}} = 2 \times 2{L_w}\log _2^{2{L_w}} + 2{L_w} + 8{L_w} + 8{L_w} = (4\log _2^{2{L_w}} + 18){L_w}$ for KF/DKF. Next, we focus on the calculation of the Jacobian matrix, which is ${C_{JM\_w}} = (4{L_w} + (2m - 1) \times m + 2) \times n + {m^2}$, where m is the same factor in Eq. (15). m changes with the modulation format, m = 1 for QPSK and m = 3 for 16QAM (see Ref. [14]). It is worth to note that for window-split structure, among the symbols in Lw only $\mathrm{\Delta s}$ symbols are compensated, so the complexity per each symbol should be divided by $\mathrm{\Delta s}$.
Furthermore, if only the time domain impairment RSOP is considered, the window-split structure will not be used, and the complexity will be calculated differently for compensation part, which is ${C_{EM}} = 24n$. The complexity of calculating Jacobian matrix is ${C_{JM}} = (4 \times 2 + (2m - 1) \times m + 2) \times n + {m^2}$.
With the Table 2 and Table 3, we can draw a conclusion that when jointly compensating RSOP and PMD in presence of RCD (n = 7), DKF reduces over 30% complexity compared with KF. When RSOP and PMD are considered (n = 6), the reduction rate is around 25%. The larger $\mathrm{\Delta s}$, the lower the complexity for both KF and DKF, but the reduction rate remains the same. So, Lw = 16 and $\mathrm{\Delta s}$ = 4 is chosen to balance the performance and complexity of KF/DKF in this paper. We also see that the reduction rate increases as the dimension of the state vector (n) increases.
4. Conclusion
A new structured Kalman filter polarization demultiplexing algorithm based on diagonalized matrix treatment is proposed in this study called the diagonalized Kalman filter, in which the error covariance matrix P is used as a diagonal matrix to reduce the matrix multiplication complexity during the iteration process. The premise of the diagonalization simplification operation is that the elements of the chosen state vector are independent of each other according to the theoretical analyses in the constitution of the DKF. Numerical simulation results obtained on the 64Gbaud-PDM-QPSK/16QAM systems platform verified that the DKF provides no apparent BER performance degradation when jointly compensating RSOP (up to 10 Mrad/s for QPSK and up to 5 Mrad/s for 16QAM) and DGD (up to 75ps) in presence of RCD of 100 ps/nm. At the expense of about 10% convergence delay of DKF compared to KF, more than 30% reduction in complexity when using DKF. Based on the trade-off, DKF wins KF in the benefit competition in the polarization demultiplexing in optical fiber communication system.
Funding
National Natural Science Foundation of China (62071065, 62101062); China Postdoctoral Science Foundation (2020M680463).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.
References
1. Z. Paul and M. Howard, Fundamentals of Kalman Filtering: A Practical Approach, 3rd Edition, (AIAA, 2009).
2. T. Marshall, B. Szafraniec, and B. Nebendahl, “Kalman filter carrier and polarization-state tracking,” Opt. Lett. 35(13), 2203–2205 (2010). [CrossRef]
3. Y. Yang, G. Cao, K. Zhong, X. Zhou, Y. Yao, A. P. T. Lau, and C. Lu, “Fast polarization-state tracking scheme based on radius-directed linear Kalman filter,” Opt. Express 23(15), 19673–19680 (2015). [CrossRef]
4. N. Cui, X. Zhang, Z. Zheng, H. Xu, W. Zhang, X. Tang, L. Xi, Y. Fang, and L. Li, “Two-parameter-SOP and three-parameter-RSOP fiber channels: problem and solution for polarization demultiplexing using Stokes space,” Opt. Express 26(16), 21170–21183 (2018). [CrossRef]
5. S. Ziaie, F. P. Guiomar, N. J. Muga, A. Nespola, G. B. Bosco, A. Carena Carena, and A. N. Pinto, “Adaptive Stokes-Based Polarization Demultiplexing for Long-Haul Multi-Subcarrier Systems,” IEEE Photonics Technol. Lett. 31(10), 759–762 (2019). [CrossRef]
6. N. J. Muga and A. N. Pinto, “Extended Kalman Filter vs. Geometrical Approach for Stokes Space-Based Polarization Demultiplexing,” J. Lightwave Technol. 33(23), 4826–4833 (2015). [CrossRef]
7. Y. Feng, L. Li, J. Lin, H. Xu, W. Zhang, X. Tang, L. Xi, and X. Zhang, “Joint tracking and equalization scheme for multi-polarization effects in coherent optical communication systems,” Opt. Express 24(22), 25491–25501 (2016). [CrossRef]
8. N. J. Muga and A. N. Pinto, “PMD tolerance in Stokes space based polarization de-multiplexing algorithms,” Opt. Quantum Electron. 49(6), 1–10 (2017). [CrossRef]
9. L. Li, Y. Feng, W. Zhang, N. Cui, H. Xu, X. Tang, L. Xi, and X. Zhang, “Extended Kalman filter for carrier frequency offset and carrier phase noise,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2017), paper STu3M.7.
10. J. Jignesh, B. Corcoran, and A. Lowery, “Parallelized unscented Kalman filters for carrier recovery in coherent optical communication,” Opt. Lett. 41(14), 3253–3256 (2016). [CrossRef]
11. M. Kuscherov and M. Herrmann, 2016, “Lightning affects coherent optical transmission in aerial fiber,” lightwave, http://www.lightwaveonline.com/articles/2016/03/lightning-affects-coherent-optical-transmission-in-aerial-fiber.html.persona
12. P. M. Krummrich, D. Ronnenberg, W. Schairer, D. Wienold, F. Jenau, and M. Herrmann, “Demanding response time requirements on coherent receivers due to fast polarization rotations caused by lightning events,” Opt. Express 24(11), 12442–12457 (2016). [CrossRef]
13. F. Pittalà, C. Stone, D. Clark, M. Kuschnerov, C. Xie, and A. Haddad, “Laboratory measurements of SOP transients due to lightning strikes on OPGW cables,” in Opt. Fiber Commun. Conference, OSA Technical Digest (online) (Optical Society of America, 2018), paper M4B.5.
14. Z. Zheng, N. Cui, H. Xu, X. Zhang, W. Zhang, L. Xi, Y. Fang, and L. Li, “Window-split structured frequency domain Kalman equalization scheme for large PMD and ultra-fast RSOP in an optical coherent PDM-QPSK system,” Opt. Express 26(6), 7211–7226 (2018). [CrossRef]
15. G. Goldfarb and G. Li, “Chromatic dispersion compensation using digital IIR filtering with coherent detection,” IEEE Photonics Technol. Lett. 19(13), 969–971 (2007). [CrossRef]
16. M. Chen, Y. Shi, C. Qiu, H. Chen, and S. Xie, “Residual chromatic-dispersion monitoring and dynamic compensation in 40-Gb/s systems,” IEEE Photonics Technol. Lett. 19(15), 1142–1144 (2007). [CrossRef]
17. R. Corsini, A. Peracchi, E. Matarazzo, T. Foggi, J. Nijhof, G. Meloni, L. Poti, R. Magri, and E. Ciaramella, “Blind adaptive chromatic dispersion compensation and estimation for DSP-based coherent optical systems,” J. Lightwave Technol. 31(13), 2131–2139 (2013). [CrossRef]
18. W. Yi, Z. Zheng, N. Cui, L. Qiu, X. Zhang, N. Zhang, W. Zhang, and L. Xi, “Joint equalization scheme of ultra-fast RSOP and large PMD in presence of residual chromatic dispersion,” in Opt. Fiber Commun. Conference (OFC), paper Th2A.51 (2019), OSA Technical Digest (Optical Society of America, 2019).
19. D. Simon, Optimal State Estimation Kalman, H∞, and Nonlinear Approaches (Wiley, 2006).
20. J. F. Bellantoni and K. W. Dodge, “A square root formulation of the Kalman-Schmidt filter,” AIAA J. 5(7), 1309–1314 (1967). [CrossRef]
21. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U. S. A. 97(9), 4541–4550 (2000). [CrossRef]
22. S. J. Savory, “Digital coherent optical receivers: algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1164–1179 (2010). [CrossRef]
23. C. Xie, “Polarization and nonlinear impairments in fiber communication systems,” in Enabling Technologies for High Spectral-Efficiency Coherent Optical Communication Networks, X. Zhou and C. Xie, eds. (Wiley, 2016).
