1. Introduction

Future coherent optical systems will be high-capacity, power-efficient and adaptive, which are expected to be realized by using varied modulation levels with probabilistic shaping. Carrier phase estimation is an essential digital signal processing (DSP) unit in high-capacity coherent optical systems and there are many phase estimation methods proposed in the literature. Conventional Viterbi-Viterbi (VV) algorithm is only applicable to the quadrature phase shift keying (QPSK) format [1]. Various improved versions can extend this method to high-order quadrature amplitude modulation (QAM), such as QPSK partition or proper selection of certain rings in the QAM constellation [2–5]. However, these methods exhibit either format dependency or a reduced linewidth tolerance due to the reduction of available symbols for estimation. This problem becomes more severe for probabilistically-shaped (PS) formats where the occurrence probabilities of different rings vary and can be very small. On the other hand, blind phase searching (BPS) is format transparent and can realize the optimal performance given a sufficient number of tested phases [6]. However, this method has high complexity. Consequently, methods such as 2-stage BPS [7] or the combination of BPS with the VV algorithm [8] were proposed to reduce the complexity. Decision aided maximum likelihood (DA-ML) can also achieve excellent performance but it also exhibits high complexity similar to the conventional BPS [9–10]. Recently, Kalman filtering [11–12] and principal component analysis (PCA) [13] were proposed to balance the performance and the complexity. Some other algorithms were also proposed for specific formats by exploiting the unique properties of these formats [14–15].

In this paper, we propose a novel phase estimation method for coherent optical systems, which is based on the fact that the Kullback-Leibler (KL) divergence between the signal constellation and the samples for estimation should be minimized when the phase is correctly compensated. This metric can be viewed as a generalized case of that in conventional BPS but avoids the pre-decisions so that phase estimation can be realized using a recursive algorithm without multiple phase tests. We compare the proposed KL method with previously reported 2-stage BPS, Kalman filtering, PCA and the PCA + BPS hybrid method. It is shown that the proposed method is format transparent and is particularly suitable for the PS formats. It exhibits better performance than Kalman filtering and the PCA + BPS hybrid method, and lower complexity than 2-stage BPS. This method also has a relaxed cycle-slip problem and is robust to the noise due to a smaller number of samples compared to 2-stage BPS and the PCA + BPS hybrid method. Therefore, the proposed method can be a promising solution for format-transparent phase estimation in coherent systems particularly using the PS formats.

2. Principle

2.1 Principle of the proposed method based on the KL divergence

Because phase estimation is performed after static dispersion compensation, polarization demultiplexing and equalization, the effect of the phase noise can be modelled as:

 

Fig. 1. Constellation of the PS 16QAM with an entropy of 3.6. The diamonds represent the samples r_n with the phase rotation of (a) π/12, (b) 0, and (c) -π/12, respectively.

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We assume the PDF of the samples without the phase noise, i.e. the constellation in Fig. 1, is f(x, y) and that of r_n·exp(-jφ_n) is ${f_{{r_n},{\varphi _n}}}(x,y)$, where x and y represent the real and imaginary variables in the constellation and φ_n is a phase variable. The metric to evaluate the distance between the two PDFs is the KL divergence or the relative entropy:

The above derivation shows that phase estimation can be realized by minimizing F(w_n), which can be viewed as a generalized case of that in conventional BPS, as proved in the Appendix. Figures 2(a) and (b) show F(w_n) versus phase error for PS 16QAM and PS 64QAM, respectively. E in the figure represents the entropy. The optical signal-to-noise ratios (OSNRs) are set such that the BERs in different cases are similar. These parameters match those in the simulations in the next section. It is seen that the value of F(w_n) is indeed minimal at the point of ϕ_n-φ_n,est=0 regardless of the OSNR values and the formats. A steeper slope around ϕ_n-φ_n,est=0 is observed for a larger E due to a higher OSNR. In general, a steeper slope implies that F(w_n) is more sensitive to the phase error and so it is easier to identify the point of ϕ_n-φ_n,est=0.

 

Fig. 2. (a) and (b) F(w_n) as a function of the phase error. F(w_n) is normalized such that its minimal value is set as zero for all cases. (c) and (d) the standard deviation of estimation errors versus the number of samples for BPS and the proposed KL method. (a) and (c) PS 16QAM with different entropies (represented by E); (b) and (d) PS 64QAM with different entropies.

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In practice, a limited number of samples should be used for phase estimation. In this case, the minimal value may not always occur at ϕ_n-φ_n,est=0, resulting in estimation errors. Figures 2(c) and (d) show the standard deviation (STD) of estimation errors versus the number of samples. The results of the conventional BPS with a sufficient number of tested phases are also given for comparison. In both methods, we use 200 sets of samples with the number of samples in each set varying from 6 to 300. In the proposed KL method, we calculate the F(w_n) versus ϕ_n-φ_n,est curve similar to Figs. 2(a) and (b) and find the ϕ_n-φ_n,est that gives the minimal F(w_n). Then we calculate the STD of ϕ_n-φ_n,est based on the 200 sets of samples. In the conventional BPS, the samples are pre-decided and the metric is the sum of the distances instead of F(w_n). From the figure, it is seen that the metric F(w_n) exhibits the same STD of estimation errors as that in the conventional BPS for both PS 16QAM and PS 64QAM. It is also seen that a steeper slope around the point of ϕ_n-φ_n,est=0 in Figs. 2(a) and (b) indeed results in a small STD. Note that the STD does not fully reflect the influence of the estimation error on the bit error rate (BER) performance. For example, the BER of PS 64QAM is degraded more significantly than that of PS 16QAM even under the same STD of estimation errors.

The STD in Figs. 2(c) and (d) represents the static performance of F(w_n), given an arbitrary but unknown phase error. It does not consider the speed of phase change that is reflected by linewidth. In practice, we may utilize F(w_n) in the same way as conventional BPS, i.e. test multiple phases, calculate F(w_n) using r_n and f(x,y) for each phase, and the optimal phase is that gives the minimal F(w_n). Additional results show that the use of F(w_n) in this way results in the same performance as conventional BPS. However, this implementation results in high complexity by testing multiple phases. Instead, because pre-decisions are not required in Eq. (4), the optimal w_n can be obtained by a recursive algorithm:

The derivations above assume that the elements in r_n have the same phase. When the linewidth is large, the phases within r_n cannot be viewed as a constant and in this case the samples with the phase deviating more from the common phase error of r_n may result in a larger Q(R{r_i·w_n}, I{r_i·w_n})·r_i* and thus introduce additional noise to the algorithm. Therefore, we improve Eq. (8) by normalizing Q(R{r_i·w_n}, I{r_i·w_n})·r_i* as:

Instead of complex multiplications in Eq. (8-1), we extract the phases of Q(x, y) and r_i to normalize the impact of all samples in Eq. (9-1). In practice, Angle(r_i) and Angle(Q(x, y)) can be readily realized by look-up tables. The complexity of the proposed method will be analyzed and compared with 2-stage BPS, PCA, and Kalman filtering in the next subsection.

After phase compensation, the PDF f(x, y) is used in the MLSD or BMD using SD-FEC for symbol/bit decision. In MLSD, the decision is made by:

2.2 Complexity analysis and comparison

We will compare the complexity of the proposed KL method with those of 2-stage BPS, PCA, PCA + BPS hybrid and Kalman filtering. In the square QAM, there is π/2 phase ambiguity. In this paper, this problem is solved by comparing the estimated phases of adjacent blocks [6], together with pilot symbols inserted every 120 symbols [18]. Note that as shown later, in contrast to BPS and PCA, the proposed KL method does not have the cycle-slip problem for small and moderate linewidths and only requires to correct the phase ambiguity for large linewidths. For simplicity, the complexity here excludes the correction of the phase ambiguity. The calculation is based on a block with 2 m samples.

In the proposed method, we calculate the complexity based on Eq. (9), which is used in the simulations in the next section. The calculation of 2m r_i·w_n in Eq. (9-1) requires 4 × 2m real multiplications and 2 × 2m real additions. For Angle(Q(R{r_i·w_n}, I{r_i·w_n})), we build up a two-dimensional look-up table with each dimension having a q-bit resolution. During the operation, we quantize the real and imaginary parts of r_i·w_n and read the corresponding Angle(Q(R{r_i·w_n}, I{r_i·w_n})) from the look-up table. The complexity of the quantization is q×2 × 2m comparisons. As shown later, q of 5∼6 can achieve the optimal performance for both PS 16QAM and PS 64QAM. Angle(r_i) can be obtained in a similar way with q×2 × 2m comparisons for the quantization. Therefore, the total complexity of Eq. (9-1) is 4 × 2m+2 real multiplications, 2 × 2m+2m+2 × 2m real additions and q×4 × 2m comparisons. Finally, the complexity of the normalization in Eq. (9-2) is 5 real multiplications and 1 real addition. The above process is iterated for U times. Note that r_i only needs to be quantized once regardless of U. Therefore, the total complexity is (8m+7)U real multiplications, (10m+1)U real additions, and 4mq(U+1) comparisons. As shown later, U of 1∼2 is sufficient for small and moderate linewidths but more iterations are required for large linewidths. In this paper, U is 4 unless otherwise stated.

In 2-stage BPS [7], the compensation of r_i using each tested phase requires 4 × 2m real multiplications and 2 × 2m real additions. The obtained signals are then compared with the constellation of PS 16QAM or PS 64QAM to make pre-decisions. This requires 2m × log₂(K) comparisons. The distances between the compensated r_i and their pre-decisions are then calculated and summed, which requires 2m×2 real multiplications and 2m×4-1 real additions. The above process is iterated with the tested phases at the first and second stages as B₁ and B₂, respectively. B₁-1 and B₂-1 comparisons are also needed at each stage to find the optimal phase. Therefore, the total complexity is 12m(B₁+B₂) real multiplications, (12m-1)·(B₁+B₂) real additions and (2mlog₂(K) + 1)·(B₁+B₂)-2 comparisons. It is shown later that B₁ and B₂ should be larger than 8 and 4 respectively to avoid the performance penalty.

In the PCA method [13], the square operation for the 2m samples requires 4 × 2m real multiplications and 2 × 2m real additions. The calculation of the covariance matrix in Eq. (2) of [13] requires 4 × 2m real multiplications and 4×(2m-1) real additions. Updating the principal component in Eq. (3) of [13] needs 8 real multiplications and 4 real additions, while the normalization in Eq. (4) of [13] requires 5 real multiplications and 1 real addition. Finally, the calculation of the phase requires 1 real multiplication and a look-up table. The total complexity is 16m+14 real multiplications and 12m+1 real additions.

As shown later, the PCA method exhibits significant performance degradation for the PS formats. This is because the low occurrence probability of high-power constellation points in the PS formats weakens the principal component and consequently degrades the performance. Therefore, we also compare the PCA + BPS hybrid method in this paper. In this case, the total complexity is 16m+14 + 12mB₂ real multiplications, 12m+1+(12m-1)B₂ real additions and (2mlog₂(K) + 1)B₂-1 comparisons, where the number of tested phases at the second stage, B₂, should be larger than 4 to avoid the performance penalty.

Finally, we calculate the complexity of Kalman filtering, with the block diagram depicted in Fig. 2 of [11]. Note that different from the sample-by-sample estimation in [11], the calculation here is based on a block with 2 m samples. The influence of the number of samples will be investigated in the next section. The complexity of the pre-decision and the phase rotations before and after the pre-decision is 8 × 2 m real multiplications, 4 × 2 m real additions, and 2m × log₂(K) comparators. The complexity to calculate the Kalman gain is 6 × 2 m real multiplications and 2 m real additions. The complexity to update the phase is 4 × 2 m real multiplications and 6 × 2 m real additions. Finally, updating the covariance matrix requires 4 × 2m+2 real multiplications and 4 × 2m-1 real additions. Therefore, the total complexity is 44m+2 real multiplications, 30m-1 real additions and 2m × log₂(K) comparators.

Table 1 summarizes the complexity of different methods. Figure 3 shows the required number of multiplications and additions versus 2 m. In the figure, U in the proposed method is 4 or 2. In 2-stage BPS, B₁=8 and B₂=4. B₂ in the PCA + BPS hybrid method is 4. These parameters match the simulations in the next section. It is seen that PCA exhibits the least complexity. However, this method shows significant performance degradation for the PS formats. Combination of PCA and BPS reduces the performance penalty but at the expense of higher complexity. Kalman filtering requires a smaller number of additions but more multiplications than the proposed KL method with U=4. As shown later, this method also has a less linewidth tolerance than the KL method with U=2. Finally, 2-stage BPS has the highest complexity.

Table 1. Complexity of different phase estimation methods

View Table

 

Fig. 3. Required number of (a) real multiplications and (b) real additions versus the number of samples. In (a), the curves for PCA and the proposed KL method with U=2 overlap.

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3. Simulation setup and results

3.1 Simulation setup

Simulations were performed to verify the proposed theory. Five methods were compared: 2-stage BPS, PCA, PCA + BPS hybrid, Kalman filtering and the proposed KL method. All methods were based on the same system parameters. Figure 4 depicts the simulation setup. A bit sequence was sent to a CCDM encoder to generate PS 16QAM or PS 64QAM with different entropies. Generally, the CCDM encoder should be followed by an FEC encoder, which was not considered in this paper for simplicity. The generated symbols were up-sampled to two samples per symbol and passed a finite impulse response (FIR) filter to pre-compensate the responses of the drivers and digital-to-analogue converters (DACs) so that the signal after the modulator exhibited a raised-cosine spectral profile with a roll-off factor of 0.5. The pre-filtered signals were sent to 100-GS/s DACs with an 8-bit resolution, amplified by drivers, and modulated a CW light with a dual-polarization (DP) I/Q modulator. In the optical link, the OSNR was controlled by a variable optical attenuator (VOA) followed by an Erbium-doped fiber amplifier (EDFA). The out-of-band optical noise was filtered by an optical filter with 60-GHz 3-dB bandwidth. A DP coherent receiver with a 5-order Bessel response was used for detection. The 3-dB bandwidth of the receiver was 40 GHz. The powers of the local oscillator and the signal were 10 dBm and 0 dBm, respectively. The electronic signals were sampled by 100-GS/s analogue-to-digital converters (ADCs). In DSP, the signals were polarization demultiplexed and filtered by FIR filters. The signals of two polarizations were combined for phase estimation since the phase offset between them was generally static and could be readily corrected [15]. The number of samples per polarization in each block was 30 unless otherwise stated. The performance was evaluated in terms of either BER using MLSD before the FEC or NGMI. The number of simulated symbols was 300,000.

 

Fig. 4. Simulation setup

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3.2 Simulation results

Figure 5 shows the performance of different methods for (a) PS 16QAM with an entropy of 3.6 and (b) PS 64QAM with an entropy of 5. The linewidth is used for both the transmitter laser and the local oscillator. In all methods, the parameters are optimized to balance the performance and complexity. Specifically, in the proposed method, U is 4 and α in Eq. (9) is optimized. Angle(Q(x, y)) is realized by a look-up table with the resolution q of 6 bits and 8 bits for PS 16QAM and PS 64QAM, respectively. In 2-stage BPS, B₁ and B₂ are 8 and 4, respectively. In Kalman filtering, the noise parameters R_n and Q_n as defined in [11] are 10⁻⁵ and 10⁻⁴, respectively. In the PCA + BPS hybrid method, B₂ is 4 and the BPS searching range is [-π/16+φ_{n, stage1}, π/16+φ_{n, stage1}], where φ_{n, stage1} is the estimated phase in the first PCA stage. From Fig. 5, it is seen that PCA exhibits the worst performance in both PS 16QAM and PS 64QAM. As shown later, this method works well for the conventional QAM. However, in the PS formats, the principal component is weakened due to lower occurrence probabilities of high-power constellation points, resulting in performance degradation especially for PS 64QAM. Combination of PCA with BPS can improve the performance. However, even for B₂=4 under which the complexity of the hybrid method is greatly increased, its performance is still poorer than that of the proposed KL method. On the other hand, Kalman filtering has slightly higher complexity than the KL method but also shows a less tolerance to the linewidth. Finally, 2-stage BPS has the best performance but requires the highest complexity as shown in Fig. 3.

 

Fig. 5. BER versus linewidth for different methods in (a) PS 16QAM with an entropy of 3.6 at 23-dB OSNR and (b) PS 64QAM with an entropy of 5 at 27-dB OSNR.

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In order to understand the results in Fig. 5 more clearly, we investigate each of previously reported methods in more details and compare them with the proposed KL method. Figure 6 shows the performance of the PCA + BPS hybrid method for different searching ranges at the BPS stage. B₂ in this method is 4. The results for the KL method with U of 4 are also given. Because the performance of the PCA method is poor as shown in Fig. 5, the searching range of the BPS stage, I, should be carefully optimized. When I is small, once ϕ_n-φ_{n, stage1} is beyond this range, the performance is degraded regardless of the resolution of the BPS stage. When I is large, the resolution is reduced under the fixed B₂. Consequently, in Fig. 6(a), it is seen that a smaller I of π/16 results in a less tolerance to the linewidth, while by using a larger I of π/4, penalties are observed for small and moderate linewidths. In Fig. 6(b), there are penalties for small and moderate linewidths even by using I of π/16. This is because the performance of PCA is so poor in this case that ϕ_n-φ_{n, stage1} is beyond the BPS searching range even for small linewidths. From both figures, it is confirmed that the proposed KL method shows better performance than the PCA + BPS hybrid method under optimized searching range.

 

Fig. 6. BER versus linewidth for the PCA + BPS hybrid method and the proposed method in (a) PS 16QAM and (b) PS 64QAM. The entropies and OSNRs are the same as those in Fig. 5. I is the searching interval of the BPS stage.

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Figure 7 shows the performance of Kalman filtering with different noise parameters R_n. The performances of the proposed KL method with U of 4 and 2 are also given. Under a fixed Q_n, the value of R_n balances the estimation accuracy and the tracking speed. When R_n is larger, Kalman filtering can compensate the phase noise more accurately for small and moderate linewidths but has a less tolerance to the linewidth. In contrast, when R_n is smaller, it can track the phase change more quickly but at the expense of the penalty for small and moderate linewidths. Note that Kalman filtering is also capable to compensate amplitude noise, which is not observed in Fig. 7 but becomes more visible as the number of samples reduces as shown later. The results also show that the performance of Kalman filtering with the optimized R_n is even not as good as that of the proposed KL method with U of 2.

 

Fig. 7. BER versus linewidth for the Kalman filtering method and the proposed KL method in (a) PS 16QAM and (b) PS 64QAM. The entropies and OSNRs are the same as those in Fig. 5.

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Figure 8 shows the performance of 2-stage BPS under different values of B₁ and B₂. The BER of the proposed method with U of 4 and 8 are also given. In Fig. 8(a), there is a slight penalty for small and moderate linewidths when B₁ and B₂ in the BPS method are 4 and 4, respectively. This penalty becomes significant in Fig. 8(b). This is because the performance of PS 64QAM is more sensitive to the phase noise. In addition, if the searching range is wrongly identified at the first stage, the performance is degraded regardless of the resolution at the second stage. Therefore, B₁ should be larger than B₂ to avoid an inaccurate searching range at the second stage. It is also seen that although the metric F(w_n) has the same STD of estimation errors as that in BPS as shown in Fig. 2, the proposed method still has a less linewidth tolerance due to the convergence of the recursive algorithm. A larger U can enhance the tolerance but increase the complexity. Thus, U of 4 is used to balance the performance and complexity.

 

Fig. 8. BER versus linewidth for the 2-stage BPS method and the proposed KL method in (a) PS 16QAM and (b) PS 64QAM. The entropies and OSNRs are the same as those in Fig. 5.

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After comparisons with previously reported methods, we will characterize the proposed KL method in more details. Figure 9 shows the performance of the KL method using Eq. (6) or Eq. (7) as the PDF in Q(x, y) and Eq. (9) or Eq. (8) for the recursive algorithm. It is observed that different PDFs do not influence the performance. However, Eq. (9) results in better performance than Eq. (8). As discussed previously, although Eq. (8) can minimize the KL divergence, it assumes that the phase within r_n is a constant. This assumption is not valid for large linewidths, in which the samples with the phases deviating more from the common phase error of r_n may have a larger impact and thus reduce the stability. In contrast, Eq. (9) normalizes the impact of all samples and thus improves the tolerance to the linewidth.

 

Fig. 9. BER versus linewidth for the proposed KL method with different PDF representations and metrics in the recursive algorithm. (a) PS 16QAM; (b) PS 64QAM. The entropies and OSNRs are the same as those in Fig. 5. U is 4.

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Figure 10 depicts the performance versus (a) the number of iterations and (b) the resolution of the lookup table. For moderate linewidths, 1∼2 iterations are sufficient to achieve the optimal performance. Four iterations can realize the near-optimal performance for 5-MHz and 2-MHz linewidths in PS 16QAM and PS 64QAM, respectively. From Fig. 10(b), it is seen that the performance becomes stable when the resolution increases to 6 bits in both PS 16QAM and PS 64QAM. This confirms that the proposed KL method can indeed achieve the performance advantage with the complexity depicted in Fig. 3.

 

Fig. 10. BER versus (a) the number of iterations and (b) resolution of the lookup table for Q(x, y) in the proposed KL method. The entropies and OSNRs for PS 16QAM and PS 64QAM are the same as those in Fig. 5.

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In the above investigations, the problem of the π/2 phase ambiguity is solved by comparing the estimated phases of adjacent blocks, together with pilot symbols inserted every 120 symbols. However, these processes are not necessary for all methods. Figure 11 shows the performance when the cycle slips are not compensated (a) by comparing the phases of adjacent blocks and (b) by the pilot symbols. It is found that the cycle-slip problem always exists in 2-stage BPS, PCA and the PCA + BPS hybrid method such that their performances are very poor if the cycle slips are not compensated by comparing the estimated phases of adjacent blocks. In contrast, the cycle-slip problem does not exist for small and moderate linewidths in the proposed KL method and Kalman filtering, whose performances in Fig. 11(a) are not degraded compared to those in Fig. 5(a). On the other hand, when the linewidth is large, there is a probability that the phase difference between adjacent blocks is too large to be recovered, resulting in cycle slips in all methods. Therefore, as shown in Fig. 11(b), if the pilot symbols are not used, the linewidth tolerances of all methods are reduced compared to those in Fig. 5(a). The results verify that the proposed KL method and Kalman filtering outperform other methods on the cycle-slip problem and only require the correction of the phase ambiguity for large linewidths.

 

Fig. 11. BER versus linewidth when the cycle slips are not compensated (a) by comparing the phases of adjacent blocks and (b) by pilot symbols inserted every 120 symbols. The format is PS 16QAM with an entropy of 3.6. The OSNR is 23 dB. In (a), the BERs of 2-stage BPS, PCA and the PCA + BPS hybrid method are higher than 5 × 10⁻² and so are not plot in the figure.

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We also investigate the performance of different methods under other entropies, as shown in Fig. 12. In the figure, U in the proposed method is 4. B₁ and B₂ in 2-stage BPS are 8 and 4, respectively. B₂ in the PCA + BPS hybrid method is 4. Other parameters are optimized in a similar way as in Fig. 5. Comparison with Fig. 5(a) confirms that PCA does not work properly in the PS formats. However, this method shows good performance in the conventional QAM as shown in Fig. 12(b), due to a strong principal component. Kalman filtering always shows a smaller linewidth tolerance than the proposed KL method. On the other hand, the proposed KL method exhibits degraded performance compared to 2-stage BPS. This is because searching the minimal F(w_n) is based on the recursive algorithm whose convergence speed may not be sufficient for large linewidths. Note that by using the algorithm, the complexity is much less than the one using multiple phase tests to find the minimal F(w_n).

 

Fig. 12. BER versus linewidth using different methods for (a) PS 16QAM with an entropy of 3.2 at 21-dB OSNR and (b) conventional 16QAM at 25-dB OSNR.

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The above results are obtained when the number of samples per polarization is 30. In practice, the block length is chosen to balance the linewidth tolerance and the noise. Figure 13 shows the performance of different methods when the number of samples per polarization is (a) 8 and (b) 4. In both figures, the performance of Kalman filtering with a larger R_n is also shown in addition to that with the optimized R_n of 10⁻⁴. Note that this optimal R_n is different from that in Fig. 5 because of a different number of samples. U in the proposed method is 4. B₁ and B₂ in 2-stage BPS are 8 and 4, respectively. B₂ in the PCA + BPS hybrid method is 4. Other parameters are optimized in a similar way as in Fig. 5. It is seen that PCA is more sensitive to the number of samples. This is because a smaller number of samples cannot provide sufficient statistics for PCA. In the PCA + BPS hybrid method, inaccurate estimation at the PCA stage may result in a wrong searching range at the second stage. Consequently, this method exhibits a penalty for small and moderate linewidths in Fig. 13(a) while this penalty becomes more significant in Fig. 13(b). Kalman filtering with the optimized R_n has a similar sensitivity to the number of samples compared to the proposed KL method, but always shows a smaller linewidth tolerance. Note that Kalman filtering with a larger R_n may mitigate the noise effect particularly in Fig. 13(b). However, the tolerance to the linewidth is also reduced significantly compared to that with the optimized R_n. Finally, 2-stage BPS works well in Fig. 13(a). However, it shows a penalty for small and moderate linewidths in Fig. 13(b), implying that this method is more sensitive to the reduction of the number of samples compared to the proposed method.

 

Fig. 13. BER versus linewidth using different methods for PS 16QAM with an entropy of 3.6 at 23-dB OSNR. The number of samples per polarization is (a) 8 and (b) 4. In (b), the BER of the PCA method is higher than 5 × 10⁻² and so is not plot in the figure.

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Finally, we investigate the performance at lower OSNRs, which correspond to the cases using SD-FEC and so the metric of NGMI is employed. Figure 14(a) shows NGMI versus linewidth at 19-dB OSNR. It is seen that conclusions similar to Fig. 5 can be drawn. Compared to other methods, the PCA method shows a NGMI penalty even at small linewidths. Kalman filtering exhibits a smaller linewidth tolerance than 2-stage BPS, the proposed KL method, and the PCA + PBS hybrid method. The performance of 2-stage BPS is the best but this method has the highest complexity. Figure 14(b) depicts NGMI versus OSNR at 1-MHz (solid) and 5-MHz (dashed) linewidths. For all methods, NGMI reduces as the OSNR decreases. At 1-MHz linewidth, the performance of the PCA method is poorer than other methods. On the other hand, when the linewidth increases to 5 MHz, the performance of Kalman filtering is degraded more significantly due to a limited linewidth tolerance. The proposed KL method makes a balance between the performance and the complexity.

 

Fig. 14. (a) NGMI versus linewidth at 19-dB OSNR. (b) NGMI versus OSNR at 1-MHz (solid) and 5-MHz (dashed) linewidths. In (a) and (b), the format is PS 16QAM with an entropy of 3.6.

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4. Summary

We have proposed a novel format-transparent phase estimation method in coherent optical systems by minimizing the KL divergence between the signal constellation and the samples for estimation. The proposed metric is a generalized case of that in conventional BPS but eliminates the requirement of pre-decisions so that phase estimation can be realized by a recursive algorithm. The complexity and the performance of the proposed method are compared with previously reported 2-stage BPS, Kalman filtering, PCA, and the PCA + BPS hybrid method. It is shown that the proposed method is particularly suitable for the PS formats in which the PCA method does not work properly. It exhibits better performance than Kalman filtering and the PCA + BPS hybrid method and lower complexity than the 2-stage BPS method. It is also shown that the proposed KL method has a relaxed cycle-slip problem and is robust to the noise due to a smaller number of samples compared to 2-stage BPS and the PCA + BPS hybrid method. The study confirms that the proposed method can be a promising solution for format-transparent phase estimation in coherent optical systems particularly using the PS formats.