Abstract

Optical spectrum (OS) is a vital characteristic of optical signals. Ultra-high resolution (UHR) OS can provide more detailed and accurate information for optical performance monitoring and optical link quality diagnosis. By comparing actual signal UHR-OS observed at in-line monitoring points with the theoretical ideal ones, various signal distortions can be readily identified and more accurately estimated. But in the future flexible heterogeneous optical networks optical signals with different symbol rates, modulation formats and pulse shaping schemes may coexist in the same system. Hence the ideal reference OS of the channel to be monitored can’t be assumed to be fixed or known in advance. It may also be impossible to undertake a reference OS measurement at or near the transmitter as the route path may be dynamically generated. To solve this problem we proposed an automatic ideal reference optical spectrum retrieval (OSR) method according to the actually observed ones. The OSR method can tolerate large OS distortions due to non-ideal optical links or transmitters by the integration of two machine learning techniques, namely unsupervised principle component analysis (PCA) and supervised multiclass support vector machines (SVMs) for feature extraction and UHR-OS classification, respectively. Extensive simulations conducted for nine types of optical signals commonly used show that this method performs very well in the presence of various significant distortions caused by non-ideal optical links or transmitters.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In optical fiber communication systems, optical spectrum (OS) is a very important characteristic for optical performance monitoring and optical link quality diagnosis. For common bulk-grating based optical spectrum analyzers (BG-OSAs) the available resolution is about 0.02nm [1], while for ultra-high resolution optical spectrum analyzers (UHR-OSAs) based on stimulated Brillouin scattering (SBS) [2–5] or coherent detection [6] the resolution can be better than 0.1pm. Such ultra-high resolution OS (UHR-OS) provides more detailed and accurate spectrum information which is desirous for the aforementioned monitoring and diagnostic purpose.

The common OS distortions suffered by the optical signals generally come from non-ideal optical link or transmitters. These distortions are diagrammatically illustrated in Fig. 1. OSNR degradation caused by accumulated amplified spontaneous emission (ASE) noise makes the lower part of the OS submerge below the noise pedestal, and meanwhile reduces the slope of the OS edge [7]. Filtering narrowing effect (FNE) and off-center filtering effect (OFE) caused by cascaded optical nodes containing non-ideal optical filtering elements makes OS outermost part severely attenuated, especially when the signal bandwidth is close to the ITU-defined channel grid(CG) [8, 9]. Fiber nonlinear effect (NLE) which is not neglectable in large capacity long-haul systems generates new frequency components thus broadening the OS [10]. In practical optical transmitters optical modulator bias voltage drift (BVD), drive voltage swing variation (DVSV), extinction ratio degeneration (ERD) and laser frequency drift (LFD), linewidth variation and insufficient bandwidth (IB) commonly occurs due to non-ideal components used, environment changes and components aging [11]. Non-ideal modulation leads to additional spectral lines in m-ary phase shift keying (mPSK) and quadrature amplitude modulation (mQAM) signal OS, while reduces the height of the spectral lines in OOK signal OS, as will be explained later in this paper. Transmitter IB attenuates the OS outermost part similar to FNE, while LFD shifts signal OS toward the edge of the grid slot which may result in larger OFE.

 figure: Fig. 1

Fig. 1 OS distortions induced by non-ideal optical link or transmitters. (solid line: the ideal OS, dashed line: the distorted OS).

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The above distortions may lead to severe system bit error rate (BER) performance degeneration [8, 9, 12]. These distortions can be readily and more accurately diagnosed and estimated by comparing the actual signal UHR-OS observed at the in-line monitoring points with the ideal reference ones. For example non-intrusive in-band OSNR estimation can be realized by comparing the actual OS with the ideal data-carrying OS without noise or distortions [13–15]. The in-band OSNR estimation accuracy is higher when narrow bandwidth optical or electrical band-pass filters (BPF) are utilized. In [15] the experiments showed that the highest OSNR estimation accuracy can be achieved with the lowest BPF bandwidth when it is varied from 500MHz to 8GHz. Thus if measured and reference ideal UHR-OS can be acquired, the OSNR estimation accuracy of the above techniques can be significantly improved as UHR-OS resolution can easily satisfy the narrow bandwidth requirement. But in the future flexible heterogeneous optical networks which can support various kinds of data traffic, different types of optical signals in terms of modulation format (MF), symbol rate (SR) and pulse shaping scheme (PSS) may co-propagate in the same optical link. The signal type of certain channels may also be changed over time because of add-drop operation or adaptive transmitter adjustment [16]. Hence the reference ideal OS of the channel to be monitored can’t be assumed to be fixed or known in advance. In [13–15], it is suggested that one can undertake a reference OS measurement at or near the transmitter. But it may be impossible in the flexible network as the optical signal route path may be dynamically generated. Therefore it is desirous to retrieve the ideal reference UHR-OS according to the actually observed ones. By now many machine learning based optical signal classification methods have been proposed. With these methods the signal type hence the ideal reference OS can be obtained. But these methods utilize time-domain intensity information obtained with high speed optical detectors [17–19], Stokes space representation [20, 21] or signal field samples recovered by digital coherent receivers [22, 23], thus are not suitable for the UHR-OS based signal type classification and distortion analysis.

In this paper, we propose a novel automatic reference optical spectrum retrieval (OSR) method to retrieve the reference ideal UHR-OS from the distorted ones for various types of optical signals based on the integration of two machine learning techniques, namely unsupervised principle component analysis (PCA) for UHR-OS feature extracting and supervised multiclass support vector machines (SVMs) for UHR-OS classification. Extensive numerical simulations conducted for nine different types of optical signals show that an average retrieval accuracy (RA) of 96.97% can be achieved in the presence of significant OSNR degeneration, FNE, BVD, DVSV and ERD. The robustness of the proposed method to OFE, fiber nonlinearities, laser effects is also investigated

2. OSR using PCA and SVM

2.1 UHR-OS of different types of optical signals

As we know every type of optical signals has their unique OS [24, 25]. The OS difference comes from the different combinations of MF, SR and PSS. For example Fig. 2 shows the theoretical ideal UHR-OS of nine types of optical signals commonly used. They are also listed in Table 1. The MF considered includes on-off keying (OOK), binary phase shift keying (BPSK), quadrature phase-shift keying (QPSK) and 16-ary quadrature amplitude modulation (16QAM). The bit rates ranges from 10Gb/s to 400 Gb/s. The PSS includes not-return-to-zero (NRZ), return-to-zero (RZ) and Nyquist pulse shaping [25]. First, as we can see the UHR-OS of OOK signals are unique in having spectral lines (impulses) with interval equal to SR. It is noteworthy that such spectral lines are only visible in UHR-OS. On the contrary, the ones for BPSK, QPSK and 16QAM signals don’t have such spectral lines. This is because their constellations are symmetric relative to the origin of the complex plane, while OOK constellations are not [24]. But as shown in Fig. 1, the aforementioned non-ideal modulation will destroy the symmetry, thus leading to prominent spectral lines as those appears in OOK signal UHR-OS. Secondly, as we can see the OS main lobe width (MLW) increases with increasing SR when MF and PSS are fixed. Thirdly, the OS is also dependent on the PSS. The MLW decreases with increasing pulse duty cycle (DC). The Nyquist pulse-shaped signals have much narrower MLW compared to the other signals with NRZ and RZ pulse shapes. As can be observed in Fig. 2(a) and Fig. 2(b), the MLW of NRZ-OOK is about half of that of the 33%RZ-OOK with the same SR. When the signal MLW is close to CG, the MLW may be significant reduced because of FNE. This will make the signals with close MLW harder to distinguish from each other.

 figure: Fig. 2

Fig. 2 UHR-OS of nine commonly used optical signals without distortions.

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Tables Icon

Table 1. Nine of the commonly used optical signals and their SVM classification labels

It is noteworthy that some optical signals with specific combinations of SR, MF and PSS will have the same OS. For example mPSK and mQAM signals with the same SR and PSS have the same theoretical OS. Thus one ideal reference UHR-OS can be used for distortion analysis of several different types of optical signals. This leads to one especial aspect of the OS based distortions analysis techniques namely that they are modulation format independent to some extent. For example format independent in-band OSNR estimation has been realized for 25GBaud dual polarization QPSK, 16QAM and 64QAM signals, utilizing the fact that they have the same OS (the calibration parameters need not be modified) [15].

2.2 Principles of the OSR method

The OSR method proposed is based on the integration of PCA and SVM [26–28]. The flow chart of the OSR method is shown in Fig. 3. Machine learning techniques are used as a powerful statistical signal processing tools for UHR-OS characterization and to learn the distortions from the observed UHR-OS, so that an optimal UHR-OS classification model can be build.

 figure: Fig. 3

Fig. 3 The Schematic diagram of the OSR process.

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First, in order to train the classification model, one need to collect a large data set, called training data set, containing a large number of UHR-OS samples each of which represents one observed UHR-OS with a specific combination of different distortions. The UHR-OS sample is normalized and represented by a vector X=[x1,x2,xp] which consists of p spectral power values corresponding to the frequency sampling grid of the UHR-OS. We note that, throughout this paper, scalars, vectors and matrices are presented by lowercase, capital and bold-face capital letters, respectively.

As the OS resolution is very high, p can be very large which means SVM classification has to be performed in a very high dimensionality space. It is difficult and computation expensive. Thus in the next step, PCA (also known as the Karhuner-Loève transform) is used for data feature extracting and dimensionality reduction. In this paper the training data set is represented by one big matrix Xn,p=[X1,X2,Xn]T . PCA performs the transformation

Yn,k=Xn,p×Wp,k,
which maps each sample vector consisting of p elements to a new sample vector consisting of only k elements (k<<p). Here Wp,k is constituted by the first k principal components (PCs) which are the eigenvectors of the covariance matrix of Xn,p defined byCp,p=Xn,pTXn,p. Cp,phas up to p eigenvectors (PCs) and eigenvalues. Let vi and λi be the i-th eigenvector and eigenvalue ofCp,p, respectively. The computed eigenvectors are ranked according to their eigenvalues and amongst them the first k eigenvectors are selected such that the following criterion is satisfied,
r=i=1kλi/i=1pλi>thr,
where the value of thr is typically chosen to be above 0.9. PCA leads to a much more compact representation of the UHR-OS samples.

Based on the extracted feature vectors multi-class SVM is utilized for automatic classification [28]. Given a set of training samples, each marked for belonging to one of the two categories, SVM attempts to find an optimal hyperplane to partition the training samples into two groups, and makes the margin between the hyperplane and the samples as wide as possible. To identify the aforementioned nine optical signals, a multiclass SVM classifier is constructed by the one-versus-one approach. As the input vectors are often not linearly separable, SVM maps the input vectors into a higher-dimensional feature space through the kernel function, making the separation easier in that space. In this work we adopt the commonly used Gaussian radial basis function (RBF). In practice, outlier vectors due to noise or some other reasons may lie in the margin between the two classes of samples. In this case the exact separation of all of the training samples including the outliers leads to poor generalization. Thus to strike a tradeoff between minimizing the classification error and maximizing the margin, we adopt the slack variable c in the multi-class SVM model. The effectiveness of multi-class C-SVM model depends on the selection of the kernel’s parameter γ and the slack parameter c. In this work we find the best combination of γ and c in the training process by a grid search with exponentially growing sequence of γ and c.

After the training process the classification model is obtained and used to identify the observed UHR-OS X with unknown signal type. In the pre-processing step, X is also normalized and centered on the ITU-T grid to mitigate LFD effect. Then it is multiplied by Wp,k and transformed into feature vector Y=X×Wp,kwith much fewer elements. Y is then input into the SVM classifier. The classification label output by the SVM classifier is used to retrieve the reference UHR-OS from the ideal UHR-OS database.

3. Simulation results

3.1 System configuration and the training process

To verify the OSR method proposed above, the nine types of optical signals shown in Fig. 2 are selected in the test. The simulation system setup is shown in Fig. 4. In the numerical simulation commercial software VPI TransmissionMaker V.9.1 is used. The nine different types of optical signals are generated using the most commonly used modulation schemes [25]. The signals are launched into the fiber via the wavelength division multiplexer. The CG is assumed to be 100GHz to accommodate all of the nine signals. In the optical link multichannel optical band pass filters (OBPFs) with 100GHz bandwidth are used to emulate FNE and OFE imposed by the optical nodes containing non-ideal filtering elements. At the monitoring point, a small portion of the optical signal is tapped off the link using an optical coupler. The UHR-OS of the channel to be monitored is measured and sent to the OSR module.

 figure: Fig. 4

Fig. 4 The Schematic diagram of the setup of the simulation system. SSMF: standard single mode fiber, EDFA: erbium doped fiber amplifier, OBPF: multichannel optical band pass filter, UHR-OSM: UHR-OS measurement.

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In the training process a training data set containing 32400 UHR-OS samples with different combinations of distortions are generated by numerical simulation. The variation ranges of the distortions are listed in Table 2. OSNR is changed from 6dB to 30dB in step of 3dB by loading different amount of ASE noise to the signal. For FNE, the number of the in-line OBPFs is varied from 1 to 30 with step of 4. As regard to the OBPF type, a realistic 100GHz bandwidth fourth order Gaussian OBPF model has been implemented which is characterized by 1dB and 3dB bandwidths of 0.4nm and 0.6nm, respectively, according to the commercial AWG chips used in 100GHz grid DWDM system [29]. The UHR-OS is interpolated to consist of 161 elements with 5pm wavelength interval. The wavelength interval is selected to balance the competing requirements of low computation load and more details.

Tables Icon

Table 2. The variation ranges of distortions

In the training process, PCA is first applied to extract the feature vectors. Figure 5(a) shows the eigenvalues for a few PCs in descending order. It is obvious that the eigenvalues rapidly converge to zero. Figure 5(b) shows the values of the parameter r as a function of the number of PCs selected k. It is found that the value of r is above 0.99 for just 10 PCs. This implies that the extracted feature vector consisting of 10 elements is sufficient for the subsequent classification. Then the feature vectors extracted by PCA are used to train the SVM models. The LibSVM which is a widely used SVM library [30] is adopted to construct and train the SVM models. To find out the best combination of γ and c, lnγand lnc are swept in the range of [-8, 8] in step of 0.5. Figure 6 shows the variations of the SVM classification accuracy versus different combinations of γ and c. The optimal lnγand lnc are found to be 1.5 and 6, respectively.

 figure: Fig. 5

Fig. 5 (a) Eigenvalues for a few PCs in descending order, (b) Parameter r as a function of the number of PCs selected.

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 figure: Fig. 6

Fig. 6 The variations of the classification accuracy of all training samples versus γ and c in the SVMs training process.

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3.2 Testing results and analysis

Once the training phase is over, the performance of OSR model is evaluated by using an independent set of data namely testing data set consisting of 32400 samples with different combinations of the distortions. The overall OSR results are summarized in Table 3. An overall average RA of 96.97% can be achieved despite significant distortions caused by OSNR degeneration, BVD, DVSV, ERD and FNE. Here RA for each type of optical signals is defined by the ratio of the number of the correctly classified samples to that of the testing samples of each signal type (32400/9). The overall average RA is defined by the ratio of the number of the correctly classified UHR-OS samples to that of the total testing samples (32400).

Tables Icon

Table 3. The OSR results using PCA and SVM. The average OSR accuracy is 96.97%.

To Investigate RA of different signals, RAs for each of the nine types of optical signals are shown in Fig. 7(a) for OSNR in the range of 6-30dB and 10-30dB, respectively. In the former case the RA of 85.42% for No.6 signal is relatively lower. As shown in Fig. 7(b) No.6 signal contribute a share of 53.50% to the total errors. We found that the errors come from misidentifying No.6 as No.5 or No.7 signal as Table 3 shows. From Fig. 7(a) we can observe that the misclassification rate is only 2.04% (RA=97.96%) when 30dB≥OSNR≥10dB, whilst for 30dB≥OSNR≥6dB it increases to 14.58% (RA=85.42%). Hence, the classification of No.6 is more challenging when OSNR is below 10dB. This is expected since No.6 exhibits a high degree of similarity to both No.5 and No.7 UHR-OS when only the very top of the UHR-OS is above the ASE noise pedestal. It is noteworthy that No.6 and No.5 become similar because when non-ideal modulation effects are present the two have similar spectral lines as shown in Fig. 8.

 figure: Fig. 7

Fig. 7 (a) The RAs for each of the nine types of optical signals (solid line: OSNR varied in the range of 6-30dB, dashed line: OSNR varied in the range of 10-30dB). (b) The contribution of each type of optical signals to the total retrieval errors.

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 figure: Fig. 8

Fig. 8 UHR-OS of No.5 (a) and No.6 (b) signals with non-ideal modulation. (a) BVD = −0.16Vπ, DVSV = 16%, ER = 18dB. (b) BVD = −0.16Vπ, DVSV = −16%, ER = 24dB.

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On the other hand Fig. 7(a) shows that when 30dB≥OSNR≥10dB the RA of 91.32% for No.7 signal become relatively lower. We found that most of the errors come from misidentifying it as No.6 signal. This is because No.7 signal suffers most from the FNE because its MLW is larger than the CG as shown in Fig. 2(g). After traversing 30 OBPFs, the FNE induced MLW reduction makes No.7 UHR-OS exhibit a high similarity as No.6 UHR-OS even when OSNR is high, as shown in Fig. 9. So for OSNR above 10dB FNE becomes the main limit to the overall average RA because of No.7 signal. As shown in Fig. 7(b) No.7 signal contribute a share of 61.67% to the total errors when OSNR≥10dB.

 figure: Fig. 9

Fig. 9 The UHR-OS of No.6 (blue) and No.7 (red) UHR-OS filtered by 1 (a) and 30 OBPFs (b), respectively, OSNR is set at 24dB.

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To further investigate the impact of FNE on the RA, sample sets with different FNE are generated to test the model. Here RA is defined by the ratio of the number of the correctly classified samples to that of the testing samples with the same FNE. Figure 10(a) shows the RA variations against the FNE. As expected, the RA decreases with increasing number of OBPFs, but a RA of 95.60% can still be achieved after traversing 30 OBPFs which are more than sufficient for a large regional network containing many non-ideal filtering elements [9]. When RA = 95.60%, we observed 35.39% of the errors come from misidentifying No.7 as No. 6 signal for the aforementioned reason. If No.7 signal is not included (only 8 types of signals are considered), a RA of 96.81% can still be achieved after traversing 30 OBPFs as shown in Fig. 10(a).

 figure: Fig. 10

Fig. 10 (a) The variations of the overall average RA against the number of OBPFs crossed. (b) The RA against the different drift ranges of the OBPFs when the number of OBPFs is 17 and 30, respectively. For the 8 types of signals case No.7 signal is not included.

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As regard to OFE, according to the performance of commercial WDM products the typical center frequency drift is typically smaller than 4GHz [31]. In the simulation each of the OBPFs has center frequency randomly drifted to the right or left of the nominal value within the drift range. Figure 10(b) shows the variation of the RA against the drift ranges. Here RA is defined by the ratio of the number of the correctly classified samples to that of the testing samples filtered by the OBPFs with the same center frequency drift range. The number of OBPFs is set at 17 and 30, respectively to test the model in a smaller and larger size networks. As expected, the RA decreases with larger OFE. For the 9 types of optical signals the lowest RA is 94.42% and 91.79% after traversing 17 and 30 OBPFs, respectively. When RA = 91.79% we observed 33.98% of the errors come from misidentifying No.7 signals as it suffers the largest OFE distortions. If No.7 signal is not included (only 8 types of signals are considered), the lowest RA increases to 95.62% and 93.34% for the two cases, respectively.

To investigate the NLE impact the optical signals are transmitted over 640km EDFA amplified fiber link. The system has five channels with the channel to be monitored in the middle. The launch power is set at 0 and 4dBm, respectively. A total of 25920 UHR-OS samples with nonlinear distortions are generated. The obtained RAs are 96.33% (0dBm) and 94.95% (4dBm), respectively. As expected NLE leads to RA degeneration but it is not significant (RA is decreased by about 0.64% and 2.02% for the two cases respectively). When RA = 94.95%, we observed 52.67% of the errors come from misidentifying No.6 as No.7 signal because NLE broadens No.6 UHR-OS making it harder to be distinguished from No.7 UHR-OS with very close MLW.

We also investigate the laser effect such as linewidth and LFD on the RA. Here RA is defined by the ratio of the number of the correctly classified samples to that of the testing samples with the same linewidth or LFD. Figure 11(a) shows the variation of the RA against the laser linewidth. As we can see the linewidth variation has little impact on the RA. Figure 11(b) shows the variation of the RA against LFD. In the simulation the laser frequency is shifted while the center frequency of the OBPFs is assumed to be fixed on the nominal value. So when the laser center frequency is shifted off its nominal value all of the OBPFs will have center frequency above or below the signal center frequency. So the OFE experienced by the signal is larger compared to the case investigated in Fig. 10(b) where the OBPFs’ center frequency are randomly shifted within the shift range. The number of OBPFs is also set at 17 and 30. As we can see after traversing 17 OBPFs the RA is 95.27%, while after traversing 30 OBPFs the RA is 94.65%. When RA = 94.65%, we observed 33.07% of the errors come from misidentifying No.7 signals. This is because No.7 signals suffers the largest LFD induced OFE distortions. If No.7 signal is not included (only 8 types of signals are considered), the lowest RA increases to 95.97% and 95.93% for the two cases, respectively.

 figure: Fig. 11

Fig. 11 The influence of laser linewidth (a) and wavelength drift (b) on the RA when the number of OBPFs is 17 and 30, respectively. For the 8 types of signals case No.7 signal is not included.

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4. Conclusions and discussions

In this paper we propose an OSR method based on the integration of two machine learning techniques. With this method the ideal reference UHR-OS can be directly obtained only from the ones observed at arbitrary intermediate monitoring point. Thus the OS based optical performance monitoring and optical link quality diagnosis become more convenient to realize, especially in the homogenously flexible networks. Extensive numerical simulation shows that for the nine commonly used optical signals selected, an overall average RA of 96.97% can be achieved for OSNR in the range of 6-30dB and in the presence of significant FNE, BVD, DVSV and ERD. This method is also robust to OFE, NLE and laser effects. The testing results show that NLE and laser linewidth variations have relatively smaller impact, and the RA degeneration is mainly due to the signals with MLW close to the CG because they suffer the largest filtering distortions induced by FNE, OFE and LFD. We note that when the network has too many filtering elements or non-ideal filtering elements with bandwidth much narrower than the CG, filtering distortions will become very significant. Significant filtering distortions will put a mask on the signal spectra and erase most of the OS characteristics, thereby degrading the classification ability of the proposed method. If possible, the above condition should be avoided in network design because significant filtering distortions will entail a large optical signal-to-noise penalty [8, 9]. If narrow bandwidth filtering elements must be used, filtering distortions can also be mitigated by using optical wave-shaping devices as equalizing filters [9]. For a well-designed network where filtering distortions are not too significant, a good performance can still be achieved as has been demonstrated.

Recently flexible WDM grid has been standardized by ITU-T [9]. According to the standard, WDM grid slots of different sizes can be allocated in steps of 12.5 GHz. In this paper the WDM grid is assumed to be 100GHz. For different grid, such as 50GHz, or flexible grid system a preliminary identification of the WDM grid is required. Identification of the grid can be easily realized as WDM signal spectra consisting of multiple channels to be monitored can be readily acquired with UHR-OSAs. Specific classification models for different grids can be trained by the same method using the UHR-OS samples of the optical signals that can be accommodated by the grid slots. Then the suitable classification model can then be selected according to the grid slot obtained.

Lately artificial neural networks (ANNs) have been used for amplitude histograms (AHs) based signal type identification and distortion analysis [17, 23]. AHs, like UHR-OS, also have unique and distinctive patterns for different type of optical signals. So ANNs may also be applicable to UHR-OS based signal type identification and distortions analysis. We select SVMs instead of ANNs because SVMs are based on sound theory and have a simple geometric interpretation [30]. Another advantage of SVMs is that whilst ANNs can suffer from multiple local minima, the solution to an SVM is often global and unique. However in recent years, deep ANNs have been developing very rapidly and have become more powerful [23, 32]. As the comparison of the UHR-OS classification methods based on SVMs and ANNs is complicated this issue will be investigated in our future work.

The method proposed opens new possibilities to extract more information from the UHR-OS by machine learning techniques such as signal type, in-band OSNR, magnitude of FNE, etc. It also provides a new approach to enhance the function of OSAs so that they can have the ability of intelligent signal identification and OS distortion diagnosis for various optical signals with only software upgrades.

Funding

Major Equipment Development Project (Grant No.2013YQ160487); Natural Science Foundation of China (Grant No. 61475053); Fundamental Research Funds for the Central Universities (Grant No. 2015TS045).

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17. F. N. Khan, Y. Zhou, A. P. T. Lau, and C. Lu, “Modulation format identification in heterogeneous fiber-optic networks using artificial neural networks,” Opt. Express 20(11), 12422–12431 (2012). [CrossRef]   [PubMed]  

18. M. C. Tan, F. N. Khan, W. H. Al-Arashi, Y. Zhou, and A. P. Tao Lau, “Simultaneous optical performance monitoring and modulation format/bit-rate identification using principal component analysis,” J. Opt. Commun. Netw. 6(5), 441–448 (2014). [CrossRef]  

19. J. Thrane, J. Wass, J. Diniz, M. Piels, J. C. M. Diniz, R. Jones, and D. Zibar, “Machine Learning Techniques for Optical Performance Monitoring from Directly Detected PDM-QAM Signals,” J. Lightwave Technol. 35(4), 868–875 (2017). [CrossRef]  

20. R. Borkowski, D. Zibar, A. Caballero, V. Arlunno, and I. T. Monroy, “Stokes Space-Based Optical Modulation Format Recognition for Digital Coherent Receivers,” IEEE Photonics Technol. Lett. 25(21), 2129–2132 (2013). [CrossRef]  

21. P. Isautier, J. Pan, R. Desalvo, and S. E. Ralph, “Stokes Space-Based Modulation Format Recognition for Autonomous Optical Receivers,” J. Lightwave Technol. 33(24), 5157–5163 (2015). [CrossRef]  

22. S. M. Bilal, G. Bosco, Z. Dong, A. P. T. Lau, and C. Lu, “Blind modulation format identification for digital coherent receivers,” Opt. Express 23(20), 26769–26778 (2015). [CrossRef]   [PubMed]  

23. F. N. Khan, K. Zhong, X. Zhou, W. H. Al-Arashi, C. Yu, C. Lu, and A. P. T. Lau, “Joint OSNR monitoring and modulation format identification in digital coherent receivers using deep neural networks,” Opt. Express 25(15), 17767–17776 (2017). [CrossRef]   [PubMed]  

24. E. Ip and J. M. Kahn, “Power Spectra of Return-to-Zero Optical Signals,” J. Lightwave Technol. 24(3), 1610–1618 (2006). [CrossRef]  

25. I. P. Kaminow, T. Li, and A. E. Willner, Optical Fiber Telecommunications V Volume B Systems and Networks (Elsevier Science Limited, 2008), chap. 2.

26. I. Jolliffe, Principal Component Analysis (John Wiley & Sons Ltd, 2002).

27. V. Vapnik, The Nature of Statistical Learning Theory (Springer Science & Business Media, 2013).

28. B. Schölkopf and A. J. Smola, Learning with kernels: support vector machines, regularization, optimization, and beyond (Massachusetts Institute of Technology, 2002).

29. Finisar, “DWDM AWG Chips” http://pdf.directindustry.com/pdf/finisar/dwdm-awg-chips/35039-216627.html

30. C. C. Chang and C. J. Lin, “LIBSVM: a library for support vector machines,” ACM Trans. Intell. Syst. Technol. 2(3), 27 (2011). [CrossRef]  

31. T. Saito, T. Ota, T. Toratani, and Y. Ono, “16-ch arrayed waveguide grating module with 100-GHz spacing,” Furukawa Review 19, 47–52 (2000).

32. Y. Bengio, “Learning deep architectures for AI,” Found. Trends Mach. Learn. 2(1), 1–127 (2009). [CrossRef]  

References

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  17. F. N. Khan, Y. Zhou, A. P. T. Lau, and C. Lu, “Modulation format identification in heterogeneous fiber-optic networks using artificial neural networks,” Opt. Express 20(11), 12422–12431 (2012).
    [Crossref] [PubMed]
  18. M. C. Tan, F. N. Khan, W. H. Al-Arashi, Y. Zhou, and A. P. Tao Lau, “Simultaneous optical performance monitoring and modulation format/bit-rate identification using principal component analysis,” J. Opt. Commun. Netw. 6(5), 441–448 (2014).
    [Crossref]
  19. J. Thrane, J. Wass, J. Diniz, M. Piels, J. C. M. Diniz, R. Jones, and D. Zibar, “Machine Learning Techniques for Optical Performance Monitoring from Directly Detected PDM-QAM Signals,” J. Lightwave Technol. 35(4), 868–875 (2017).
    [Crossref]
  20. R. Borkowski, D. Zibar, A. Caballero, V. Arlunno, and I. T. Monroy, “Stokes Space-Based Optical Modulation Format Recognition for Digital Coherent Receivers,” IEEE Photonics Technol. Lett. 25(21), 2129–2132 (2013).
    [Crossref]
  21. P. Isautier, J. Pan, R. Desalvo, and S. E. Ralph, “Stokes Space-Based Modulation Format Recognition for Autonomous Optical Receivers,” J. Lightwave Technol. 33(24), 5157–5163 (2015).
    [Crossref]
  22. S. M. Bilal, G. Bosco, Z. Dong, A. P. T. Lau, and C. Lu, “Blind modulation format identification for digital coherent receivers,” Opt. Express 23(20), 26769–26778 (2015).
    [Crossref] [PubMed]
  23. F. N. Khan, K. Zhong, X. Zhou, W. H. Al-Arashi, C. Yu, C. Lu, and A. P. T. Lau, “Joint OSNR monitoring and modulation format identification in digital coherent receivers using deep neural networks,” Opt. Express 25(15), 17767–17776 (2017).
    [Crossref] [PubMed]
  24. E. Ip and J. M. Kahn, “Power Spectra of Return-to-Zero Optical Signals,” J. Lightwave Technol. 24(3), 1610–1618 (2006).
    [Crossref]
  25. I. P. Kaminow, T. Li, and A. E. Willner, Optical Fiber Telecommunications V Volume B Systems and Networks (Elsevier Science Limited, 2008), chap. 2.
  26. I. Jolliffe, Principal Component Analysis (John Wiley & Sons Ltd, 2002).
  27. V. Vapnik, The Nature of Statistical Learning Theory (Springer Science & Business Media, 2013).
  28. B. Schölkopf and A. J. Smola, Learning with kernels: support vector machines, regularization, optimization, and beyond (Massachusetts Institute of Technology, 2002).
  29. Finisar, “DWDM AWG Chips” http://pdf.directindustry.com/pdf/finisar/dwdm-awg-chips/35039-216627.html
  30. C. C. Chang and C. J. Lin, “LIBSVM: a library for support vector machines,” ACM Trans. Intell. Syst. Technol. 2(3), 27 (2011).
    [Crossref]
  31. T. Saito, T. Ota, T. Toratani, and Y. Ono, “16-ch arrayed waveguide grating module with 100-GHz spacing,” Furukawa Review 19, 47–52 (2000).
  32. Y. Bengio, “Learning deep architectures for AI,” Found. Trends Mach. Learn. 2(1), 1–127 (2009).
    [Crossref]

2017 (4)

2016 (2)

2015 (3)

2014 (3)

2013 (1)

R. Borkowski, D. Zibar, A. Caballero, V. Arlunno, and I. T. Monroy, “Stokes Space-Based Optical Modulation Format Recognition for Digital Coherent Receivers,” IEEE Photonics Technol. Lett. 25(21), 2129–2132 (2013).
[Crossref]

2012 (1)

2011 (2)

G. Gariépy, G. He, and G. W. Schinn, “Non-intrusive measurement of In-Band OSNR of high bitrate polarization-multiplexed signals,” Opt. Fiber Technol. 17(5), 518–522 (2011).
[Crossref]

C. C. Chang and C. J. Lin, “LIBSVM: a library for support vector machines,” ACM Trans. Intell. Syst. Technol. 2(3), 27 (2011).
[Crossref]

2009 (1)

Y. Bengio, “Learning deep architectures for AI,” Found. Trends Mach. Learn. 2(1), 1–127 (2009).
[Crossref]

2006 (1)

2005 (1)

J. M. S. Domingo, J. Pelayo, F. Villuendas, C. D. Heras, and E. Pellejer, “Very high resolution optical spectrometry by stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 17(4), 855–857 (2005).
[Crossref]

2002 (1)

D. M. Baney, B. Szafraniec, and A. Motamedi, “Coherent optical spectrum analyzer,” IEEE Photonics Technol. Lett. 14(3), 355–357 (2002).
[Crossref]

2000 (1)

T. Saito, T. Ota, T. Toratani, and Y. Ono, “16-ch arrayed waveguide grating module with 100-GHz spacing,” Furukawa Review 19, 47–52 (2000).

Al-Arashi, W. H.

Arlunno, V.

R. Borkowski, D. Zibar, A. Caballero, V. Arlunno, and I. T. Monroy, “Stokes Space-Based Optical Modulation Format Recognition for Digital Coherent Receivers,” IEEE Photonics Technol. Lett. 25(21), 2129–2132 (2013).
[Crossref]

Baney, D. M.

D. M. Baney, B. Szafraniec, and A. Motamedi, “Coherent optical spectrum analyzer,” IEEE Photonics Technol. Lett. 14(3), 355–357 (2002).
[Crossref]

Bengio, Y.

Y. Bengio, “Learning deep architectures for AI,” Found. Trends Mach. Learn. 2(1), 1–127 (2009).
[Crossref]

Bilal, S. M.

Bohn, M.

T. Rahman, A. Napoli, D. Rafique, B. Spinnler, M. Kuschnerov, I. Lobato, B. Clouet, M. Bohn, C. Okonkwo, and H. de Waardt, “On the mitigation of optical filtering penalties originating from ROADM cascade,” IEEE Photonics Technol. Lett. 26(2), 154–157 (2014).
[Crossref]

Borkowski, R.

R. Borkowski, D. Zibar, A. Caballero, V. Arlunno, and I. T. Monroy, “Stokes Space-Based Optical Modulation Format Recognition for Digital Coherent Receivers,” IEEE Photonics Technol. Lett. 25(21), 2129–2132 (2013).
[Crossref]

Bosco, G.

Caballero, A.

R. Borkowski, D. Zibar, A. Caballero, V. Arlunno, and I. T. Monroy, “Stokes Space-Based Optical Modulation Format Recognition for Digital Coherent Receivers,” IEEE Photonics Technol. Lett. 25(21), 2129–2132 (2013).
[Crossref]

Chang, C. C.

C. C. Chang and C. J. Lin, “LIBSVM: a library for support vector machines,” ACM Trans. Intell. Syst. Technol. 2(3), 27 (2011).
[Crossref]

Chen, Z.

Chiadò Piat, A.

Clouet, B.

T. Rahman, A. Napoli, D. Rafique, B. Spinnler, M. Kuschnerov, I. Lobato, B. Clouet, M. Bohn, C. Okonkwo, and H. de Waardt, “On the mitigation of optical filtering penalties originating from ROADM cascade,” IEEE Photonics Technol. Lett. 26(2), 154–157 (2014).
[Crossref]

Cugini, F.

de Waardt, H.

J. M. Fabrega, M. Svaluto Moreolo, L. Martín, A. Chiadò Piat, E. Riccardi, D. Roccato, N. Sambo, F. Cugini, L. Potì, S. Yan, E. Hugues-Salas, D. Simeonidou, M. Gunkel, R. Palmer, S. Fedderwitz, D. Rafique, T. Rahman, H. de Waardt, and A. Napoli, “On the Filter Narrowing Issues in Elastic Optical Networks,” J. Opt. Commun. Netw. 8(7), A23–A33 (2016).
[Crossref]

T. Rahman, A. Napoli, D. Rafique, B. Spinnler, M. Kuschnerov, I. Lobato, B. Clouet, M. Bohn, C. Okonkwo, and H. de Waardt, “On the mitigation of optical filtering penalties originating from ROADM cascade,” IEEE Photonics Technol. Lett. 26(2), 154–157 (2014).
[Crossref]

Desalvo, R.

Diniz, J.

Diniz, J. C. M.

Domingo, J. M. S.

J. M. S. Domingo, J. Pelayo, F. Villuendas, C. D. Heras, and E. Pellejer, “Very high resolution optical spectrometry by stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 17(4), 855–857 (2005).
[Crossref]

Dong, Z.

Fabrega, J. M.

Fedderwitz, S.

Gariépy, D.

Gariépy, G.

G. Gariépy, G. He, and G. W. Schinn, “Non-intrusive measurement of In-Band OSNR of high bitrate polarization-multiplexed signals,” Opt. Fiber Technol. 17(5), 518–522 (2011).
[Crossref]

Gunkel, M.

Guo, Z.

He, G.

Heras, C. D.

J. M. S. Domingo, J. Pelayo, F. Villuendas, C. D. Heras, and E. Pellejer, “Very high resolution optical spectrometry by stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 17(4), 855–857 (2005).
[Crossref]

Hugues-Salas, E.

Ip, E.

Isautier, P.

Jones, R.

Kahn, J. M.

Ke, C.

Khan, F. N.

Kuschnerov, M.

T. Rahman, A. Napoli, D. Rafique, B. Spinnler, M. Kuschnerov, I. Lobato, B. Clouet, M. Bohn, C. Okonkwo, and H. de Waardt, “On the mitigation of optical filtering penalties originating from ROADM cascade,” IEEE Photonics Technol. Lett. 26(2), 154–157 (2014).
[Crossref]

Lau, A. P. T.

Li, L.

Lin, C. J.

C. C. Chang and C. J. Lin, “LIBSVM: a library for support vector machines,” ACM Trans. Intell. Syst. Technol. 2(3), 27 (2011).
[Crossref]

Liu, D.

Lobato, I.

T. Rahman, A. Napoli, D. Rafique, B. Spinnler, M. Kuschnerov, I. Lobato, B. Clouet, M. Bohn, C. Okonkwo, and H. de Waardt, “On the mitigation of optical filtering penalties originating from ROADM cascade,” IEEE Photonics Technol. Lett. 26(2), 154–157 (2014).
[Crossref]

Lu, C.

Lu, Y.

Martín, L.

Monroy, I. T.

R. Borkowski, D. Zibar, A. Caballero, V. Arlunno, and I. T. Monroy, “Stokes Space-Based Optical Modulation Format Recognition for Digital Coherent Receivers,” IEEE Photonics Technol. Lett. 25(21), 2129–2132 (2013).
[Crossref]

Motamedi, A.

D. M. Baney, B. Szafraniec, and A. Motamedi, “Coherent optical spectrum analyzer,” IEEE Photonics Technol. Lett. 14(3), 355–357 (2002).
[Crossref]

Napoli, A.

J. M. Fabrega, M. Svaluto Moreolo, L. Martín, A. Chiadò Piat, E. Riccardi, D. Roccato, N. Sambo, F. Cugini, L. Potì, S. Yan, E. Hugues-Salas, D. Simeonidou, M. Gunkel, R. Palmer, S. Fedderwitz, D. Rafique, T. Rahman, H. de Waardt, and A. Napoli, “On the Filter Narrowing Issues in Elastic Optical Networks,” J. Opt. Commun. Netw. 8(7), A23–A33 (2016).
[Crossref]

T. Rahman, A. Napoli, D. Rafique, B. Spinnler, M. Kuschnerov, I. Lobato, B. Clouet, M. Bohn, C. Okonkwo, and H. de Waardt, “On the mitigation of optical filtering penalties originating from ROADM cascade,” IEEE Photonics Technol. Lett. 26(2), 154–157 (2014).
[Crossref]

Okonkwo, C.

T. Rahman, A. Napoli, D. Rafique, B. Spinnler, M. Kuschnerov, I. Lobato, B. Clouet, M. Bohn, C. Okonkwo, and H. de Waardt, “On the mitigation of optical filtering penalties originating from ROADM cascade,” IEEE Photonics Technol. Lett. 26(2), 154–157 (2014).
[Crossref]

Ono, Y.

T. Saito, T. Ota, T. Toratani, and Y. Ono, “16-ch arrayed waveguide grating module with 100-GHz spacing,” Furukawa Review 19, 47–52 (2000).

Ota, T.

T. Saito, T. Ota, T. Toratani, and Y. Ono, “16-ch arrayed waveguide grating module with 100-GHz spacing,” Furukawa Review 19, 47–52 (2000).

Palmer, R.

Pan, J.

Pelayo, J.

J. M. S. Domingo, J. Pelayo, F. Villuendas, C. D. Heras, and E. Pellejer, “Very high resolution optical spectrometry by stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 17(4), 855–857 (2005).
[Crossref]

Pellejer, E.

J. M. S. Domingo, J. Pelayo, F. Villuendas, C. D. Heras, and E. Pellejer, “Very high resolution optical spectrometry by stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 17(4), 855–857 (2005).
[Crossref]

Piels, M.

Potì, L.

Rafique, D.

J. M. Fabrega, M. Svaluto Moreolo, L. Martín, A. Chiadò Piat, E. Riccardi, D. Roccato, N. Sambo, F. Cugini, L. Potì, S. Yan, E. Hugues-Salas, D. Simeonidou, M. Gunkel, R. Palmer, S. Fedderwitz, D. Rafique, T. Rahman, H. de Waardt, and A. Napoli, “On the Filter Narrowing Issues in Elastic Optical Networks,” J. Opt. Commun. Netw. 8(7), A23–A33 (2016).
[Crossref]

T. Rahman, A. Napoli, D. Rafique, B. Spinnler, M. Kuschnerov, I. Lobato, B. Clouet, M. Bohn, C. Okonkwo, and H. de Waardt, “On the mitigation of optical filtering penalties originating from ROADM cascade,” IEEE Photonics Technol. Lett. 26(2), 154–157 (2014).
[Crossref]

Rahman, T.

J. M. Fabrega, M. Svaluto Moreolo, L. Martín, A. Chiadò Piat, E. Riccardi, D. Roccato, N. Sambo, F. Cugini, L. Potì, S. Yan, E. Hugues-Salas, D. Simeonidou, M. Gunkel, R. Palmer, S. Fedderwitz, D. Rafique, T. Rahman, H. de Waardt, and A. Napoli, “On the Filter Narrowing Issues in Elastic Optical Networks,” J. Opt. Commun. Netw. 8(7), A23–A33 (2016).
[Crossref]

T. Rahman, A. Napoli, D. Rafique, B. Spinnler, M. Kuschnerov, I. Lobato, B. Clouet, M. Bohn, C. Okonkwo, and H. de Waardt, “On the mitigation of optical filtering penalties originating from ROADM cascade,” IEEE Photonics Technol. Lett. 26(2), 154–157 (2014).
[Crossref]

Ralph, S. E.

Riccardi, E.

Roccato, D.

Saito, T.

T. Saito, T. Ota, T. Toratani, and Y. Ono, “16-ch arrayed waveguide grating module with 100-GHz spacing,” Furukawa Review 19, 47–52 (2000).

Sambo, N.

Schinn, G. W.

G. Gariépy, G. He, and G. W. Schinn, “Non-intrusive measurement of In-Band OSNR of high bitrate polarization-multiplexed signals,” Opt. Fiber Technol. 17(5), 518–522 (2011).
[Crossref]

Searcy, S.

Simeonidou, D.

Spinnler, B.

T. Rahman, A. Napoli, D. Rafique, B. Spinnler, M. Kuschnerov, I. Lobato, B. Clouet, M. Bohn, C. Okonkwo, and H. de Waardt, “On the mitigation of optical filtering penalties originating from ROADM cascade,” IEEE Photonics Technol. Lett. 26(2), 154–157 (2014).
[Crossref]

Svaluto Moreolo, M.

Szafraniec, B.

D. M. Baney, B. Szafraniec, and A. Motamedi, “Coherent optical spectrum analyzer,” IEEE Photonics Technol. Lett. 14(3), 355–357 (2002).
[Crossref]

Tan, M. C.

Tao, Z.

Tao Lau, A. P.

Thrane, J.

Tibuleac, S.

Toratani, T.

T. Saito, T. Ota, T. Toratani, and Y. Ono, “16-ch arrayed waveguide grating module with 100-GHz spacing,” Furukawa Review 19, 47–52 (2000).

Villuendas, F.

J. M. S. Domingo, J. Pelayo, F. Villuendas, C. D. Heras, and E. Pellejer, “Very high resolution optical spectrometry by stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 17(4), 855–857 (2005).
[Crossref]

Wass, J.

Xing, C.

Yan, S.

Yu, C.

Zhang, C.

Zhang, K.

Zheng, Z.

Zhong, K.

Zhong, Y.

Zhou, X.

Zhou, Y.

Zhu, L.

Zhu, X.

Zibar, D.

J. Thrane, J. Wass, J. Diniz, M. Piels, J. C. M. Diniz, R. Jones, and D. Zibar, “Machine Learning Techniques for Optical Performance Monitoring from Directly Detected PDM-QAM Signals,” J. Lightwave Technol. 35(4), 868–875 (2017).
[Crossref]

R. Borkowski, D. Zibar, A. Caballero, V. Arlunno, and I. T. Monroy, “Stokes Space-Based Optical Modulation Format Recognition for Digital Coherent Receivers,” IEEE Photonics Technol. Lett. 25(21), 2129–2132 (2013).
[Crossref]

ACM Trans. Intell. Syst. Technol. (1)

C. C. Chang and C. J. Lin, “LIBSVM: a library for support vector machines,” ACM Trans. Intell. Syst. Technol. 2(3), 27 (2011).
[Crossref]

Found. Trends Mach. Learn. (1)

Y. Bengio, “Learning deep architectures for AI,” Found. Trends Mach. Learn. 2(1), 1–127 (2009).
[Crossref]

Furukawa Review (1)

T. Saito, T. Ota, T. Toratani, and Y. Ono, “16-ch arrayed waveguide grating module with 100-GHz spacing,” Furukawa Review 19, 47–52 (2000).

IEEE Photonics Technol. Lett. (4)

R. Borkowski, D. Zibar, A. Caballero, V. Arlunno, and I. T. Monroy, “Stokes Space-Based Optical Modulation Format Recognition for Digital Coherent Receivers,” IEEE Photonics Technol. Lett. 25(21), 2129–2132 (2013).
[Crossref]

J. M. S. Domingo, J. Pelayo, F. Villuendas, C. D. Heras, and E. Pellejer, “Very high resolution optical spectrometry by stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 17(4), 855–857 (2005).
[Crossref]

D. M. Baney, B. Szafraniec, and A. Motamedi, “Coherent optical spectrum analyzer,” IEEE Photonics Technol. Lett. 14(3), 355–357 (2002).
[Crossref]

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Figures (11)

Fig. 1
Fig. 1 OS distortions induced by non-ideal optical link or transmitters. (solid line: the ideal OS, dashed line: the distorted OS).
Fig. 2
Fig. 2 UHR-OS of nine commonly used optical signals without distortions.
Fig. 3
Fig. 3 The Schematic diagram of the OSR process.
Fig. 4
Fig. 4 The Schematic diagram of the setup of the simulation system. SSMF: standard single mode fiber, EDFA: erbium doped fiber amplifier, OBPF: multichannel optical band pass filter, UHR-OSM: UHR-OS measurement.
Fig. 5
Fig. 5 (a) Eigenvalues for a few PCs in descending order, (b) Parameter r as a function of the number of PCs selected.
Fig. 6
Fig. 6 The variations of the classification accuracy of all training samples versus γ and c in the SVMs training process.
Fig. 7
Fig. 7 (a) The RAs for each of the nine types of optical signals (solid line: OSNR varied in the range of 6-30dB, dashed line: OSNR varied in the range of 10-30dB). (b) The contribution of each type of optical signals to the total retrieval errors.
Fig. 8
Fig. 8 UHR-OS of No.5 (a) and No.6 (b) signals with non-ideal modulation. (a) BVD = −0.16 V π , DVSV = 16%, ER = 18dB. (b) BVD = −0.16 V π , DVSV = −16%, ER = 24dB.
Fig. 9
Fig. 9 The UHR-OS of No.6 (blue) and No.7 (red) UHR-OS filtered by 1 (a) and 30 OBPFs (b), respectively, OSNR is set at 24dB.
Fig. 10
Fig. 10 (a) The variations of the overall average RA against the number of OBPFs crossed. (b) The RA against the different drift ranges of the OBPFs when the number of OBPFs is 17 and 30, respectively. For the 8 types of signals case No.7 signal is not included.
Fig. 11
Fig. 11 The influence of laser linewidth (a) and wavelength drift (b) on the RA when the number of OBPFs is 17 and 30, respectively. For the 8 types of signals case No.7 signal is not included.

Tables (3)

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Table 1 Nine of the commonly used optical signals and their SVM classification labels

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Table 2 The variation ranges of distortions

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Table 3 The OSR results using PCA and SVM. The average OSR accuracy is 96.97%.

Equations (2)

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Y n,k = X n,p × W p,k ,
r= i=1 k λ i / i=1 p λ i >thr,

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