Modulation Format Identification Utilizing Polar-Coordinate-System-Based Features for Digital Coherent Receivers

Abstract

Modulation format identification (MFI) is one of the most critical functions embedded in digital coherent receivers in elastic optical networks (EONs). In view of inherent amplitude and phase characteristics of received signals, different modulation formats exhibit a set of notable features in the polar coordinate system, based on which an MFI scheme incorporating the Gaussian weighted k-nearest neighbors (KNN) algorithm was proposed to identify polarization division multiplexed (PDM)-QPSK/-16QAM/-32QAM/-64QAM/-128QAM signals. The performance of the proposed scheme was numerically verified in 28GBaud coherent optical communication systems. The numerical simulation results show that, to achieve 100% correct identification rates for all of the five modulation formats, the required minimum optical signal-to-noise ratios (OSNRs) were less than their relevant thresholds corresponding to the 20% forward error correction (FEC). The tolerable ranges of the residual chromatic dispersion (CD) for QPSK, 16QAM, 32QAM, 64QAM, and 128QAM were −1920 ps/nm~1920 ps/nm, −720 ps/nm~360 ps/nm, −1200 ps/nm~1680 ps/nm, −600 ps/nm~360 ps/nm, and −600 ps/nm~480 ps/nm, respectively. Meanwhile, the results demonstrate the maximum tolerable differential-group delay (DGD) for the QPSK, 16QAM, 32QAM, 64QAM, and 128QAM signals were 34 ps, 16 ps, 20 ps, 6 ps, and 1.2 ps, respectively. In addition, the simulated results also show that the proposed MFI scheme is robust against the fiber nonlinearities, even if the launch power is increased to 4 dBm.

1. Introduction

To meet the dynamic and heterogeneous demands of data traffic associated with mobile internet, cloud computing, big data, and internet of things, optical networks are evolving from static network architectures to flexible and elastic ones [1,2]. For such a flexible and elastic optical network, an increase in the number of variable network operation parameters (e.g., modulation format, baud rate, spectrum allocation, etc. [3]) often gives rise to the improved network flexibility and elasticity [4]. To dynamically vary the network operation parameters, the receivers must require modulation format information to ensure the operation of its embedded digital signal processing (DSP) modules and demodulation of received signals, as such without the assistance from a corresponding supervisory control layer.

Previously reported modulation format identification (MFI) schemes for the digital coherent receivers can be generally classified into the following two categories: (1) Schemes based on Stokes space [5,6,7,8,9,10,11,12,13]. These schemes are not sensitive to carrier phase noise, frequency offset, and polarization mixing. However, it is difficult for these schemes to identify high-order modulation formats that have a large number of clusters. (2) Schemes based on signal characteristics obtained after the constant modulus algorithm (CMA) equalization [14,15,16,17,18,19,20,21,22]. The CMA is capable of compensating for the residual chromatic dispersion (CD) and polarization mode dispersion (PMD) effects, thus resulting in the enhanced MFI performance tolerance against residual CD and PMD. However, as the CMA equalization is placed in the middle of DSP chain where phase noise is not eliminated yet by the carrier phase recovery algorithm, only amplitude information can therefore be identified in these schemes [14,15,16,17,18,19,20,21,22]. To minimize the influence of phase noise, some approaches including differential-phase [23], fractal dimension (FD) and gray-level co-occurrence matrix (GLCM) [24], principal component analysis (PCA), and singular value decomposition (SVD) [25] can be employed in the MFI schemes. Meanwhile, in order to utilize detailed phase information, a blind phase search strategy of 4QAM (4QAM-BPS) can also be introduced to roughly recover the phase of the CMA-equalized signals before the MFI operation [26]. Nevertheless, in addition to complicated feature-extraction [24], these schemes [23,25,26] can just identify the highest order modulation format of up to 64QAM.

In this paper, an MFI scheme using features in the polar coordinate system is proposed for digital coherent receivers. Firstly, the phase of a CMA-equalized signal was roughly recovered by using 4QAM-BPS, as the phase pre-compensation does not require any modulation format information. The signal was then converted into the polar coordinate system to better characterize the notable features of a modulation format [27]. After performing normalization and quantization, a three-dimensional polar coordinate graph was generated based on the quantized amplitude, phase, and number of symbols in each grid. Unlike most of the previously reported MFI schemes that tend to learn overall global features, the proposed scheme makes full use of effective local features of the polar coordinate graph. A total of 20 specific local zones were selected, and the number of symbols in these zones were grouped into six-dimensional (6 × 1) features. This not only utilized the local features precisely but also reduced the computational complexity of the subsequent identification algorithm correspondingly. After that, the Gaussian weighted k-nearest neighbors (KNN) algorithm was applied to classify the features extracted from the polar coordinate graph. In order to verify the feasibility of the proposed scheme, numerical simulations of 28 GBaud polarization division multiplexed (PDM)-QPSK/-16QAM/-32QAM/-64QAM/-128QAM signals were conducted. The results show that, to achieve a 100% correct identification rate for all of the five modulation formats, the required minimum optical signal-to-noise ratio (OSNR) values were less than the thresholds corresponding to the 20% forward error correction (FEC) correcting bit error rate (BER) of 2.4 × 10⁻². Furthermore, the simulated results also showed that the proposed MFI scheme was robust against residual CD, PMD, and fiber nonlinearities.

2. Operating Principle

As shown in Figure 1, the proposed MFI scheme was placed after the modulation format-independent algorithms. E_x and E_y denote the received X-polarization and Y-polarization digital signals after coherent detection and analog to digital convention (ADC). The CD compensation algorithm, timing recovery algorithm, and CMA are applied to compensate for the CD impairments, timing-jitters, and polarization mixing. It should be noted that the CMA can only achieve complete polarization demultiplexing for constant-modulus signal. By using the proposed MFI scheme, the identified modulation format information is provided to the subsequent modulation format-dependent algorithms. For mQAM (m > 4) signals, further polarization demultiplexing is achieved by multi-modulus algorithm (MMA). After that, the frequency offset is estimated and compensated. Finally, the carrier phase noise of incoming PDM signals is compensated by the carrier phase recovery algorithm.

As shown in Figure 1, there are three steps in the proposed MFI scheme. Firstly, in order to utilize more detailed phase information, 4QAM-BPS is introduced to roughly recover the phase of the CMA-equalized signal, regardless of the modulation format. The constellation diagrams in the rectangular coordinate system before or after 4QAM-BPS are depicted in Figure 2, where the linewidth (LW) of the laser is assumed to be 100 kHz. After that, the signal is then converted into the polar coordinate system for further improving feature extraction. Secondly, after normalization and quantization, a three-dimensional polar coordinate graph is generated based on the quantized amplitude, phase, and number of symbols in each grid. According to the notable amplitude and phase features of different modulation formats present in the polar coordinate system, 20 specific zones are selected, and the number of symbols in these zones are grouped into six-dimensional features. Thirdly, the six-dimensional features are identified by the Gaussian weighted KNN algorithm to obtain the modulation format information.

2.1. Polar Coordinate Transformation and Feature Extraction

The signals are transformed into the polar coordinate system by a process expressed below:

(1)$r=\sqrt{I^2+Q^2},$

(2)$\theta =\arctan {\displaystyle \frac{Q}{I}},$

where I and Q represent the real and imaginary parts of the signal, while r and $\theta$ indicate the amplitude and phase after in the polar coordinate system. The constellation diagrams in the rectangular coordinate system and the polar coordinate system are shown in Figure 3.

After normalization, the range of amplitude r is from 0 to 2.5, while the range of phase $\theta$ is from −3.2 to 3.2. Then, the amplitude r is equally divided into 50 intervals, while the phase $\theta$ is equally divided into 80 intervals. Based on amplitude, phase, and number of symbols in each grid, the three-dimensional polar coordinate graphs of these different modulation formats are generated as illustrated in Figure 4.

To clearly illustrate the feature extraction procedure, the vertical view of three-dimensional polar coordinate graphs is shown in Figure 5, where the amplitude levels for QPSK, 16QAM, 32QAM, 64QAM, and 128QAM are 1, 3, 5, 9, and 13, respectively. Meanwhile, the phase distributions of these five modulation formats are also different. In comparison with global feature extraction, extracting effective local features can reduce the computational complexity for some features with high similarity, thus improving the identification performance.

In order to extract effective local features, 20 specific zones are selected, as illustrated in Figure 5. The number of symbols within these zones can be expressed as

(3)$T_k={\displaystyle \sum _{i=r_{start,k}}^{r_{end,k}}{\displaystyle \sum _{j=c_{start,k}}^{c_{end,k}}{M}_{i,j},k=1,2,3,\dots 20}},$

where $T_k$ represents the number of symbols in the k-th local zone; $M_{i,j}$ indicates the number of symbols in the grid of the i-th row and the j-th column; $r_{s\mathrm{tart}}$ and $r_{\mathrm{end}}$ denote the starting index and ending index of rows in the k-th local zone, respectively; and $c_{s\mathrm{tart}}$ and $c_{\mathrm{end}}$ are the starting index and ending index of the column in the k-th local zone, respectively.

To reduce the dimension of features and decrease the entire complexity of this scheme, the number of symbols in the 20 specific zones are grouped into six dimensional features, and can be defined as

(4)$Z_1=T_1+T_2,$

(5)$Z_2=T_3,$

(6)$Z_3=T_4+T_5+T_6,$

(7)$Z_4=T_7+T_8+T_9+T_{10},$

(8)$Z_5=T_{11}+T_{12}+T_{13}+T_{14},$

(9)$Z_6=T_{15}+T_{16}+T_{17}+T_{18}+T_{19}+T_{20},$

where $Z_i$ is the i-th dimensional feature. As illustrated in Figure 5, $Z_1$ is the sum of the symbol number in the two local zones, $T_1$ (27:36, 46:54) and $T_2$ (27:34, 66:74). The first two numbers in the parentheses indicate the starting and ending index of the rows, and the last two numbers in the parentheses denote the starting and ending index of the columns. It should be noted that both the rows and columns are counted from the top left corner. These two zones correspond to two clusters of the QPSK signal. $Z_2$ denotes the number of symbols in the local zone $T_3$ (31:35, 14:28), which corresponds to two clusters in the second amplitude of the 16QAM signal. As illustrated in Figure 5, the polar coordinate graph shows symmetry properties. Due to less clusters, for the low order modulation format (16QAM), the distribution of clusters in the polar coordinate graph is easily recognizable. Therefore, even if the number of symbols is only extracted in the local zone $T_3$ rather than more local zones corresponding to the second amplitude of the 16QAM signal, the feature $Z_2$ is still able to ensure the identification performance based on the symmetry properties. $Z_3$ is the sum of symbol number in the three local zones, $T_4$ (30:31, 15:24), $T_5$ (30:32, 36:44), and $T_6$ (30:31, 55:64). There are six clusters of the 32QAM signal in these zones. $Z_4$ represents the sum of symbol number in the four local zones, $T_7$ (22:25, 9:14), $T_8$ (22:25, 27:34), $T_9$ (22:25, 47:54), and $T_{10}$ (22:25, 66:72), which correspond to four clusters in the ninth amplitude of the 64QAM signal. Meanwhile, $Z_5$ indicates the sum of symbol number in the four local zones, $T_{11}$ (42:43, 15:18), $T_{12}$ (42:43, 22:27), $T_{13}$ (42:43, 34:38), and $T_{14}$ (42:43, 44:47), which correspond to four clusters in the second amplitude of the 64QAM signal. As a high order of a modulation format requires more complex features to be extracted, $Z_4$ and $Z_5$ are mainly applied to identify the features for 64QAM signals. Finally, $Z_6$ denotes the number of symbols in the local zones, $T_{15}$ (27:28, 4:6), $T_{16}$ (27:28, 23:26), $T_{17}$ (27:28, 43:46), $T_{18}$ (27:28, 54:58), $T_{19}$ (27:28, 63:66), and $T_{20}$ (27:28, 74:77). Extracting the number of symbols in more zones is conducive to the accurate identification of the 128QAM signal since the cluster distribution of 128QAM is more dispersed than any other modulation format. Based on the approach mentioned above, the extracted six-dimensional features $Z_1$, $Z_2$, $Z_3$, $Z_4$, $Z_5$, and $Z_6$ are identified by the Gaussian weighted KNN algorithm.

2.2. Gaussian Weighted KNN Algorithm

KNN is a classification algorithm that does not require a complicated training process compared to other machine learning algorithms. For a testing sample, the classification is achieved by using the Euclidean distances between the testing sample and several training samples in the feature space. Nevertheless, one of the disadvantages of the KNN algorithm is that when the data in the training set are skewed, i.e., one of the classes is much more frequent, the prediction may be biased [28]. To address such a problem, the Gaussian weighted KNN algorithm is applied. The Euclidean distance between the testing sample and training sample in feature space is provided into a Gaussian function for optimizing the sample weight at different distances. The Euclidean distance is closer, and the weight of the distance is higher. In the Gaussian weighted KNN algorithm, k samples, which are closest to the testing sample, are found in the training set. Then, among these k training samples, if the sum of weights of one class is largest, the testing sample is identified as this class. Assuming the feature space $\chi$ is an six-dimensional real vector space $R^6$, $x_i,x_j\in \chi$, $x_i={({x_i}^{\left(1\right)},{x_i}^{\left(2\right)},\dots {x_i}^{\left(6\right)})}^T$, $x_j={({x_j}^{\left(1\right)},{x_j}^{\left(2\right)},\dots {x_j}^{\left(6\right)})}^T$, the Euclidean distance $L_p$ between the training sample $x_j$ and testing sample $x_i$ is defined as

(10)$L_p(x_i,x_j)={({\displaystyle \sum _{l=1}^6|{x_i}^{\left(l\right)}-{x_j}^{\left(l\right)}{|}^2})}^{{\displaystyle \frac{1}{2}}},$

Meanwhile, the Gaussian function can be expressed as

(11)$f\left(L_p\right)=a{e}^{{\displaystyle \frac{-{\left(L_p-b\right)}^2}{2c^2}}},$

where a, b, c $\in$ R are the coefficients. By weighting the Euclidean distance, the Gaussian weighted KNN algorithm can overcome the problem caused by skewed training sets, as shown in Figure 6.

The green square shown in Figure 6 represents a testing sample, and the Gaussian weighted KNN algorithm calculates the Euclidean distance between the testing sample and training samples (blue triangles denote the training samples of category A, while red circles indicate the training samples of category B) to find k training samples, which are closest to the testing sample. For instance, assuming k = 3, then three training samples that are closest to the testing sample are found. These three training samples consist of two training samples of category A and one training sample of category B. The Euclidean distances between the testing sample and these three training samples are then weighted by using the Gaussian function in Equation (11), and the Gaussian weighted values are denoted as $w_1$, $w_2$, and $w_3$, respectively. Since the sum of the weighted values ($w_1$ + $w_2$) of category A is greater than that ($w_3$) of category B, the testing sample is thus identified as category A.

3. Results and Analysis

To verify the validity of the proposed MFI scheme, a series of simulations of 28 GBaud PDM-QPSK/-16QAM/-32QAM/-64QAM/-128QAM signals were conducted by using VPI Transmission Maker 9.8. As illustrated in Figure 7, the optical carrier with a wavelength of 1550 nm and a linewidth of 100 kHz was generated by an external cavity laser (ECL). Then, the optical carrier was split by a polarization beam splitter (PBS) and modulated by two I/Q modulators. After a polarization beam combiner (PBC), the PDM signals were launched into the back-to-back (BTB) or long-distance transmission link. The Set OSNR module was applied to adjust the value of OSNR in BTB case, and the OSNR ranges of these five modulation formats were 7~26 dB, 14~33 dB, 14~37 dB, 14~38 dB, and 19–42 dB, respectively. The long-distance transmission link was composed of M × 80 km (M = 25 for QPSK, M = 13 for 16QAM, M = 5 for 32QAM, M = 3 for 64QAM, M = 1 for 128QAM) spans of single-mode fibers (SMFs). The dispersion parameter, PMD parameter, attenuation, and nonlinear coefficient of the SMF were D = 16 ps/nm/km, D_PMD = 0.1 ps/km^1/2, α = 0.2 dB/km, and γ = 1.267 km⁻¹W⁻¹, respectively. The noise figure of the erbium doped fiber amplifier (EDFA) was 5 dB. At the receiving end, the received signals were detected with the assistance from a local oscillator (LO). Then, the digital signals were obtained after ADC and provided to an off-line DSP module. To verify the accuracy of the proposed MFI scheme, in the BTB case, the training set and the testing set were obtained by extracting 100 independent samples for each OSNR value of each modulation format according to the ratio of 8:2. In the long-distance transmission case, the same established training set was also applied, while 20 independent samples for each launch power of each modulation format were included in the testing set. It should be noted that the samples extracted in an excessive low OSNR case cannot be employed as training samples since deteriorated training samples can actually lead to decreased performance. Therefore, the samples from the 64QAM signals with an OSNR of 14 dB and the 128QAM signals with OSNRs of 19 dB and 20 dB were only applied for testing rather than training.

The required minimum number of symbols is regarded as one of the most important factors since it determines the response speed and complexity of the MFI scheme. To explore the required minimum number of symbols, in Figure 8, the required minimum OSNR as a function of symbol number was plotted, where the range of the symbol number was from 5000 to 9000, and the step size was 1000. In theory, if the number of symbols is more, the features are more obvious and easy to be identified. However, since the proposed MFI scheme only depends on the features in local zones, for high-order modulation formats, excessive symbols not only enhance the feature in specific local zones but also result in confused features in other local zones, especially in the low OSNR case. This causes slight performance degradation in the 9000 symbols case. Therefore, considering the trade-off between the MFI performance and complexity, the optimal number of symbols is 8000, and thus the number of symbols is fixed at 8000 in the following numerical simulations.

As mentioned in Section 2.2, k is a key parameter for the Gaussian weighted KNN algorithm. As illustrated in Figure 9, the required minimum OSNR as a function of k was shown to evaluate the optimal value of k. The range of the k value was from 1 to 9. For QPSK, 16QAM, 64QAM, and 128QAM, the required minimum OSNR remained unchanged over the entire k range. On the other hand, when the value of k was decreased to 1, the required minimum OSNR for 32QAM was significantly increased. Under the condition of low OSNR, the extracted feature of the testing sample was seriously interfered by noise. If the k was set to be 1, the identification of KNN completely relied on the one training sample closest to the testing sample. In such a case, the chance of misjudgment is increased by the single training sample. Therefore, the optimal value of k was set to be 3, which was employed in the following numerical simulations.

The correct identification rate versus OSNR is illustrated in Figure 10, and the vertical dashed lines are the OSNR thresholds corresponding to the 20% FEC for the five modulation formats. Except for 16QAM, the required minimum OSNR for the other four modulation formats were significantly lower than the corresponding thresholds. Even for 16QAM, the required minimum OSNR (15 dB) was less than the theoretical 20% FEC limit (15.85 dB). The results indicate the outstanding performance of the proposed MFI scheme.

Meanwhile, the tolerance against residual CD and PMD for the proposed MFI scheme is also plotted in Figure 11 and Figure 12, respectively. The testing set contained 20 independent samples for each value of CD (five modulation formats, 2020 testing samples in total). Considering the 100% correct identification rate as a metric to evaluate the MFI performance, the tolerable ranges of the residual CD for QPSK( 12 dB), 16QAM (19 dB), 32QAM (22 dB), 64QAM (24 dB), and 128QAM (25 dB) were −1920 ps/nm~1920 ps/nm, −720 ps/nm~360 ps/nm, −1200 ps/nm~1680 ps/nm, −600 ps/nm~360 ps/nm, and −600 ps/nm~480 ps/nm, respectively. The results indicate that the proposed scheme has good tolerance against residual CD.

The results illustrated in Figure 12 also show the tolerance to PMD. The range of differential-group delay (DGD) was from 0 ps to 34 ps with a step size of 2 ps. A total of 20 independent samples for each value of DGD (five modulation formats, 1160 testing samples in total) were employed for testing. The results demonstrate the maximum tolerable DGD for the QPSK (12 dB), 16QAM (19 dB), 32QAM (22 dB), and 64QAM (24 dB) signals were 34 ps, 16 ps, 20 ps, and 6 ps, respectively. From Figure 12b, it can also be seen that the maximum tolerable DGD for 128QAM was much less than that of other lower-order modulation formats when the value of OSNR was 25 dB. For 128QAM, when the value of OSNR was low, features that had already been blurred by noise were very sensitive to the influence of DGD. On the other hand, it was noteworthy that once the OSNR value increased, the tolerable DGD for the 128QAM was able to be significantly raised.

To evaluate the impact of fiber nonlinearities, a series of long-distance transmission simulations were conducted for the QPSK (2000 km), 16QAM (1040 km), 32QAM (400 km), 64 QAM (240 km), and 128QAM (80 km) signals. The correct identification rates of these five modulation formats in the long-distance transmission links are illustrated in Figure 13. For QPSK, 16QAM, 32QAM, and 64QAM, the decrease in identification accuracy caused by the nonlinear effects occurred only when their corresponding launch powers were increased to 8 dBm, 8 dBm, 6 dBm, and 10 dBm, respectively. In particular, the correct identification rate of 128QAM still remained at 100% even when its launch power was increased to 10 dBm. These results indicate that the proposed MFI scheme is robust against the fiber nonlinearities.

Similar to the MFI scheme using the Calinski–Harabasz index [26], the proposed MFI scheme also makes use of 4QAM-BPS to roughly recover the phase of the CMA-equalized signal, only the proposed MFI scheme can still achieve 100% correct identification rates for QPSK, 16QAM, 32QAM, and 64QAM when their corresponding OSNRs are no less than the thresholds corresponding to the 20% FEC. Furthermore, in contrast to the MFI scheme using the Calinski–Harabasz index [26], the proposed MFI scheme is able to identify an additional higher-order modulation format (128QAM).

In order to comprehensively compare these two schemes, the comparison of their average execution times is shown in Figure 14. These two schemes are implemented by MATLAB R2022b, which run on a graphics workstation equipped with a Core i9-13900K CPU at 3 GHz and 128 GB RAM. The graphics card is RTX A6000 with 48 GB of memory. It should also be noted that, unlike clustering algorithms, the Gaussian weighted KNN is a supervised algorithm. However, without enormous computations in training, the training phase of the Gaussian weighted KNN algorithm consists of storing the training samples only [28]. For identifying a testing sample, the average execution time required for the proposed scheme is only 59.2% of the MFI scheme using the Calinski–Harabasz index.

4. Discussion

In order to utilize detailed phase information, the 4QAM-BPS is applied in the proposed MFI scheme to roughly recover the phase of the CMA-equalized signals before the MFI operation. The capability of the phase pre-compensation algorithm is one of the important factors to determine MFI performance. In our future work, there is still more room for improvement in phase pre-compensation.

5. Conclusions

In this paper, a novel MFI scheme has been proposed for digital coherent receivers. According to the notable features in the polar coordinate system, the incoming PDM signals of the digital coherent receivers can be identified by the Gaussian weighted KNN algorithm. The proposed MFI scheme is able to achieve 100% correct identification rates for five modulation formats when the values of OSNR are no less than the corresponding theoretical 20% FEC limit. Meanwhile, the proposed MFI scheme is also robust against the effects of residual CD, PMD, and fiber nonlinearities. Furthermore, compared with other relevant MFI schemes, the proposed MFI scheme significantly reduces the average execution time. Therefore, the proposed MFI scheme may be regarded as a good candidate for implementing next-generation EONs.

Author Contributions

Conceptualization, S.L. (Shuai Liang) and M.H.; methodology, S.L. (Shuai Liang) and M.H.; software, S.L. (Shuai Liang), R.X. and S.L. (Shuang Liang); validation, S.L. (Shuai Liang), R.X. and S.L. (Shuang Liang); formal analysis, S.L. (Shuai Liang) and M.H.; writing—original draft preparation, M.H. and S.L. (Shuai Liang); writing—review and editing, J.T., W.J. and L.C.; supervision, M.H.; project administration, M.H. and J.T.; funding acquisition, M.H. and J.T. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

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.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures

Figure 1. The proposed MFI scheme embedded in the DSP chain of the digital coherent receiver.

Figure 2. The constellation diagrams in the rectangular coordinate system before or after 4QAM-BPS.

Figure 3. The constellation diagrams in (a) the rectangular coordinate system and (b) the polar coordinate system.

Figure 4. The three-dimensional polar coordinate graph of five modulation formats: (a) QPSK, (b) 16QAM, (c) 32QAM, (d) 64QAM, and (e) 128QAM.

Figure 5. The vertical view of three-dimensional polar coordinate graphs for different modulation formats at different OSNR values, specifically: (a) QPSK at 26 dB, (b) 16QAM at 33 dB, (c) 32QAM at 37 dB, (d) 64QAM at 38 dB, and (e) 128QAM at 42 dB.

Figure 6. Schematic diagram of identification by the Gaussian weighted KNN algorithm.

Figure 7. The simulation setup of the PDM communication system. OBPF: optical band-pass filter, LPF: low-pass filter, PRBS: pseudo-random bit sequence.

Figure 8. The required minimum OSNR versus different number of symbols for five modulation formats.

Figure 9. The required minimum OSNR versus different values of k for five modulation formats.

Figure 10. The correct identification rate versus OSNR for five modulation formats.

Figure 11. The tolerance with respect to the residual CD for the five modulation formats.

Figure 12. The tolerance with respect to the DGD for (a) QPSK, 16QAM, 32QAM, 64QAM, and (b) 128QAM.

Figure 13. The correct identification rate in long-distance transmission for five modulation formats.

Figure 14. The comparison of the average runtime for identifying a testing sample with two MFI schemes.

References

1. Gerstel, O.; Jinno, M.; Lord, A.; Yoo, S.J.B. Elastic optical networking: A new dawn for the optical layer?. IEEE Commun. Mag.; 2012; 50, pp. s12-s20. [DOI: https://dx.doi.org/10.1109/MCOM.2012.6146481]

2. Hao, M.; He, W.; Jiang, X.; Liang, S.; Jin, W.; Chen, L.; Tang, J. Modulation Format Identification Based on Multi-Dimensional Amplitude Features for Elastic Optical Networks. Photonics; 2024; 11, 390. [DOI: https://dx.doi.org/10.3390/photonics11050390]

3. Chatterjee, B.C.; Sarma, N.; Oki, E. Routing and Spectrum Allocation in Elastic Optical Networks: A Tutorial. IEEE Commun. Surv. Tutor.; 2015; 17, pp. 1776-1800. [DOI: https://dx.doi.org/10.1109/COMST.2015.2431731]

4. Rottondi, C.; Barletta, L.; Giusti, A.; Tornatore, M. Machine-Learning Method for Quality of Transmission Prediction of Unestablished Lightpaths. J. Opt. Commun. Netw.; 2018; 10, pp. A286-A297. [DOI: https://dx.doi.org/10.1364/JOCN.10.00A286]

5. Yang, L.; Xu, H.; Bai, C.; Yu, X.; You, K.; Sun, W.; Guo, J.; Zhang, X.; Liu, C. Modulation Format Identification Using Graph-Based 2D Stokes Plane Analysis for Elastic Optical Network. IEEE Photonics J.; 2021; 13, 7901315. [DOI: https://dx.doi.org/10.1109/JPHOT.2021.3056138]

6. Mai, X.; Liu, J.; Wu, X.; Zhang, Q.; Guo, C.; Yang, Y.; Li, Z. Stokes space modulation format classification based on non-iterative clustering algorithm for coherent optical receivers. Opt. Express; 2017; 25, pp. 2038-2050. [DOI: https://dx.doi.org/10.1364/OE.25.002038] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/29519052]

7. Hao, M.; Yan, L.; Yi, A.; Jiang, L.; Pan, Y.; Pan, W.; Luo, B. Stokes Space Modulation Format Identification for Optical Signals Using Probabilistic Neural Network. IEEE Photonics J.; 2018; 10, 7202213. [DOI: https://dx.doi.org/10.1109/JPHOT.2018.2836151]

8. Jiang, L.; Yan, L.; Yi, A.; Pan, Y.; Bo, T.; Hao, M.; Pan, W.; Luo, B. Blind Density-Peak-Based Modulation Format Identification for Elastic Optical Networks. J. Light. Technol.; 2018; 36, pp. 2850-2858. [DOI: https://dx.doi.org/10.1109/JLT.2018.2827118]

9. Zhao, R.; Sun, W.; Xu, H.; Bai, C.; Tang, X.; Wang, Z.; Yang, L.; Cao, L.; Bi, Y.; Yu, X. et al. Blind modulation format identification based on improved PSO clustering in a 2D Stokes plane. Appl. Opt.; 2021; 60, pp. 9933-9942. [DOI: https://dx.doi.org/10.1364/AO.439749]

10. Yi, A.; Yan, L.; Liu, H.; Jiang, L.; Pan, Y.; Luo, B.; Pan, W. Modulation format identification and OSNR monitoring using density distributions in Stokes axes for digital coherent receivers. Opt. Express; 2019; 27, pp. 4471-4479. [DOI: https://dx.doi.org/10.1364/OE.27.004471]

11. Cho, H.J.; Varughese, S.; Lippiatt, D.; Desalvo, R.; Tibuleac, S.; Ralph, S.E. Optical performance monitoring using digital coherent receivers and convolutional neural networks. Opt. Express; 2020; 28, pp. 32087-32104. [DOI: https://dx.doi.org/10.1364/OE.406294] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/33115171]

12. Yu, X.; Bai, C.; Xu, H.; Sun, W.; Yang, L.; Zheng, H.; Hu, W. A modified PSO assisted blind modulation format identification scheme for elastic optical networks. Opt. Commun.; 2020; 476, 126280. [DOI: https://dx.doi.org/10.1016/j.optcom.2020.126280]

13. Wang, M.; Liu, J.; Zhang, J.; Zhang, D.; Guo, C. Modulation format identification based on phase statistics in Stokes space. Opt. Commun.; 2021; 480, 126481. [DOI: https://dx.doi.org/10.1016/j.optcom.2020.126481]

14. Liu, G.; Proietti, R.; Zhang, K.; Lu, H.; Yoo, S.J.B. Blind modulation format identification using nonlinear power transformation. Opt. Express; 2017; 25, pp. 30895-30904. [DOI: https://dx.doi.org/10.1364/OE.25.030895]

15. Yi, A.; Liu, H.; Yan, L.; Jiang, L.; Pan, Y.; Luo, B. Amplitude variance and 4th power transformation based modulation format identification for digital coherent receiver. Opt. Commun.; 2019; 452, pp. 109-115. [DOI: https://dx.doi.org/10.1016/j.optcom.2019.07.016]

16. Jiang, L.; Yan, L.; Yi, A.; Pan, Y.; Hao, M.; Pan, W.; Luo, B. Blind optical modulation format identification assisted by signal intensity fluctuation for autonomous digital coherent receivers. Opt. Express; 2020; 28, pp. 302-313. [DOI: https://dx.doi.org/10.1364/OE.372406] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/32118959]

17. Khan, F.N.; Zhong, K.; Al-Arashi, W.H.; Yu, C.; Lu, C.; Lau, A.P.T. Modulation Format Identification in Coherent Receivers Using Deep Machine Learning. IEEE Photonics Technol. Lett.; 2016; 28, pp. 1886-1889. [DOI: https://dx.doi.org/10.1109/LPT.2016.2574800]

18. Khan, F.N.; Zhong, K.; Zhou, X.; Al-Arashi, W.H.; Yu, C.; Lu, C.; Lau, A.P.T. Joint OSNR monitoring and modulation format identification in digital coherent receivers using deep neural networks. Opt. Express; 2017; 25, pp. 17767-17776. [DOI: https://dx.doi.org/10.1364/OE.25.017767]

19. Yu, Z.; Wan, Z.; Shu, L.; Hu, S.; Zhao, Y.; Zhang, J.; Xu, K. Loss weight adaptive multi-task learning based optical performance monitor for multiple parameters estimation. Opt. Express; 2019; 27, pp. 37041-37055. [DOI: https://dx.doi.org/10.1364/OE.27.037041]

20. Zhao, Z.; Yang, A.; Guo, P.; Tang, W. A Modulation Format Identification Method Based on Amplitude Deviation Analysis of Received Optical Communication Signal. IEEE Photonics J.; 2019; 11, 7201807. [DOI: https://dx.doi.org/10.1109/JPHOT.2019.2893658]

21. Xiang, Q.; Yang, Y.; Zhang, Q.; Yao, Y. Joint and accurate OSNR estimation and modulation format identification scheme using the feature-based ANN. IEEE Photonics J.; 2019; 11, 7204211. [DOI: https://dx.doi.org/10.1109/JPHOT.2019.2929913]

22. Zhao, Z.; Yang, A.; Guo, P.; Tan, Q.; Xin, X. A modulation format identification method based signal amplitude sorting and ratio calculation. Opt. Commun.; 2020; 470, 125819. [DOI: https://dx.doi.org/10.1016/j.optcom.2020.125819]

23. Feng, J.; Jiang, L.; Yan, L.; Yi, A.; Pan, W.; Luo, B. Intelligent Optical Performance Monitoring Based on Intensity and Differential-Phase Features for Digital Coherent Receivers. J. Light. Technol.; 2022; 40, pp. 3592-3601. [DOI: https://dx.doi.org/10.1109/JLT.2022.3149412]

24. Huang, Z.; Zhang, Q.; Xin, X.; Yao, H.; Gao, R.; Jiang, J.; Tian, F.; Liu, B.; Wang, F.; Tian, Q. et al. Modulation Format Identification Based on Signal Constellation Diagrams and Support Vector Machine. Photonics; 2022; 9, 927. [DOI: https://dx.doi.org/10.3390/photonics9120927]

25. Jiang, J.; Zhang, Q.; Xin, X.; Gao, R.; Wang, X.; Tian, F.; Tian, Q.; Liu, B.; Wang, Y. Blind Modulation Format Identification Based on Principal Component Analysis and Singular Value Decomposition. Electronics; 2022; 11, 612. [DOI: https://dx.doi.org/10.3390/electronics11040612]

26. Zhang, W.; Yue, Z.; Ye, J.; Xu, H.; Wang, Y.; Zhang, X.; Xi, L. Modulation format identification using the Calinski–Harabasz index. Appl. Opt.; 2022; 61, pp. 851-857. [DOI: https://dx.doi.org/10.1364/AO.448043]

27. Li, F.; Lin, X.; Shi, J.; Li, Z.; Chi, N. Modulation format recognition in a UVLC system based on reservoir computing with coordinate transformation and folding algorithm. Opt. Express; 2023; 31, pp. 17331-17344. [DOI: https://dx.doi.org/10.1364/OE.491377]

28. Morais, R.M.; Pedro, J. Machine Learning Models for Estimating Quality of Transmission in DWDM Networks. J. Opt. Commun. Netw.; 2018; 10, pp. D84-D99. [DOI: https://dx.doi.org/10.1364/JOCN.10.000D84]