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Abstract
A novel method assisted by feed-forward artificial neural network model for joint identification of in-phase/quadrature (IQ) time-skews and power-imbalances for coherent optical transmitters is proposed in this paper. This method not only reduces the complexity of hardware design but also significantly improves the accuracy of time-skew/power-imbalance estimation, therefore largely enhances the efficiency of coherent transmitter offline calibration. The proposed method works in the heterodyne detection manner to detect both the time- and power- imperfections. A modified 2 × 2 real-valued channel equalizer for QPSK and 16-QAM signals is applied to extract the channel coefficients, which can fully reconstruct the signals. A key tool using artificial neural networks is used to establish the explicit numerical relationships between the values of IQ time-skew/power-imbalance and the channel coefficients. Both simulation and experimental tests are carried out to verify the capability of the proposed method. Simulation results show that the mean square errors (MSE) of IQ time-skew and power-imbalance estimation can reach below 0.03% of the symbol period and 5.72 × 10−5 at the optical signal-to-noise ratio value of 20 dB. Experiment tests based on 100-Gb/s and 200-Gb/s coherent optical modules show that the mean absolute errors (MAEs) of estimated IQ time-skew and power-imbalance are 0.145 ps and 0.01 for 32-GBaud QPSK signal and 0.162ps and 0.006 for 32-GBaud 16-QAM signal. A demonstrative calibration process is applied to a coherent optical module with pre-set IQ time-skew and power-imbalance by improving the Q2 factor of 32-Gbaud 16-QAM signals from 12.9 dB to 20.3 dB in a coherent optical transmission link at back-to-back case.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the deployment of 100-Gb/s systems and the increasing demand for data transmission capacity, systems with bit rates of 400-Gb/s and 1-Tb/s per channel based on high-order quadrature amplitude modulation (QAM) running at high baud-rate combined with coherent optical detection technology has been widely investigated in long-haul transmission links [1–3]. Coherent in-phase/quadrature (IQ) transmitter is one of the key parts in the coherent optical module, which is used to convert complex electrical signals into in-phase and quadrature modulated optical signals. The imperfections at the coherent IQ transmitters such as IQ time-skew and power-imbalance may seriously degrade the system performance in such high order modulation format/baud rate transmission systems [4]. For example, the tolerance of transceiver IQ skew drastically drops from 8% of symbol period to around 4% at 0.5 dB OSNR penalty as modulation updated from 16-QAM to 64-QAM [5]. The optical internetworking forum (OIF) has specified the IQ time-skew should be smaller than 4 ps (12% of the symbol period) in 100 Gb/s coherent optical transmission modules [6]. Also, coherent optical transmitter calibration for IQ time-skew and power-imbalance is a key step in the coherent optical module manufacturing before commercialization.
Many works have been carried out to solve the imbalanced issues. One simple method is to measure the output optical power of modulator in I and Q branches independently [7]. However, this method is based on the assumption that the modulator is correctly biased when the swing voltage is zero. By interfering two tributary channels with specifically designed identical binary phase-shift keying (BPSK) signals, the value of IQ imbalance can be measured by the periodic power transfer function [8]. Another method relies on the adjustment of the finite impulse filter (FIR) in the digital domain at the transmitter side [9]. It is also shown that by imposing single sideband (SSB) comb signal at the transmitter side, the values of IQ time skew and power imbalance can be estimated based on the optical spectrum of the modulated optical signal [5]. Iterative K-means clustering procedure has also been proposed to estimate the IQ imbalances at the transmitter side considering the effects of IQ phase imbalance [10]. Several adaptive channel equalization methods have been also proposed at the receiver side to compensate the IQ time-skew and power-imbalance introduced by both the coherent transmitter and receiver to maintain transmission performance [11–16]. However, these methods can hardly distinguish the transmitter aspects from the overall impairments. Moreover, the use of adaptive equalization is unnecessary since the IQ time-skew and power-imbalance at the transmitter side are static or very slow drifting impairments. Therefore, it is preferable to estimate and compensate for the IQ time-skews and power-imbalance from the transmitter statically. It is noted that since the adaptive equalizer can compensate the IQ time-skew and power-imbalance at the receiver side, the channel equalization coefficients should be related to the true value of IQ time-skew and power-imbalance in the transmission module. Therefore, it is desired to figure out such relationships between the channel equalization coefficients and the value of IQ time-skew and power-imbalance.
Machine learning (ML) technique, specifically the fully connected feed-forward neural network is chosen in this paper to establish the numerical relationships between the channel coefficients and the transmitter side IQ time-skews and power-imbalances, due to its capabilities and advantages in dealing with such “black-box” modeling problems. Similar applications of ML techniques have been widely used in the optical communication field for various optical applications such as impairment estimation and compensation [17–21]. Google has achieved the largest spectral efficiency-distance product of $66102$ b/s/Hz$\cdot$km over live-traffic carrying cable, using deep neural network based fiber nonlinearity compensation technique. Q$^2$ factor is improved by 0.4 dB compared to conventional channel equalization method [18]. A comprehensive study of the application of ML techniques to the optical networks has been illustrated in [20]. And, recently, we have demonstrated a multi-layer perceptron (MLP) based method to estimate and calibrate transmitter IQ skew and power-imbalance in 3 steps with 12.8 dB performance improvement in [22].
In this paper, we propose a novel method assisted by feed-forward artificial neural network model for joint identification of in-phase/quadrature (IQ) time-skews and power-imbalances for coherent optical transmitters. Such demands mainly derive from the coherent module manufacturing, where traditional testing requires very long time to tune the IQ time-skew and power-imbalance before product release. Different from using general coherent detection architecture (containing 4 balanced PDs), only a single photodetector (PD) is used to receive testing data, which significantly reduces the system costs. A modified real-valued adaptive equalizer is applied to obtain the channel coefficients with laser phase noise mitigation in the digital signal processing (DSP) scheme. Fully connected feed-forward neural network is used to model the projections between the channel coefficients and values of IQ time-skew and power-imbalance. By introducing machine learning techniques into the scheme, the proposed method exhibits high accuracy in time-skew/power-imbalance estimation, enlarges the range of estimation, and speeds up the calibration of the optical modules.
Validity of the proposed technique is demonstrated for both simulative and experimental cases. Simulation results show that the mean square errors (MSE) of IQ time-skew and power-imbalance estimation can reach below 0.03% of the symbol period and $5.72\times 10^{-5}$ at the optical signal-to-noise ratio value of 20 dB. In order to verify the feasibility of the proposed method in commercial coherent optical 100-Gb/s and 200-Gb/s transmission systems, we conduct an experimental demonstration to estimate the values of IQ time-skew and power-imbalance in coherent optical modules based on 32-GBaud QPSK and 16-QAM modulation. Experimental results shown that the mean absolute errors (MAEs) of estimated IQ time-skew and power-imbalance are 0.145 ps and 0.01 for 32-GBaud QPSK signals, and 0.162ps and 0.006 for 32-GBaud 16QAM signal. Skew and imbalance calibration is also conducted to show the effectiveness of the proposed method, by improving the Q$^2$ factor of 32-Gbaud 16-QAM signals from 12.9 dB to 20.3 dB, in a coherent optical transmission link at back-to-back case.
The paper is organized as the following. Sec. 2 explains principle and work flow of the proposed method, introduces a modified real-valued channel equalizer, and describes the structure of the neural network model build in this paper. Sec. 3 narrates the mechanism of feature engineering in the simulation scenario, and analyzes training and testing results of the ML based IQ time-skew and power-imbalance identification method. Sec. 4 extends the tests to the experimental scenario with 32-GBaud QPSK and 16-QAM signals and presents the results. And Sec. 5 concludes the paper. This paper is an extension work of [22], comprising a numerical simulation and all the detailed analyses.
2. Principle
To emphasize how severely system performance can be affected by imperfect IQ time-skew and power-imbalance, a simulation is first conducted to investigate the 1200-km fiber transmission performances of 32-GBaud 100-Gb/s coherent QPSK signal and 200-Gb/s coherent 16-QAM signal when suffering from IQ time-skews and power-imbalances from the transmitter side. The BER threshold is $2\times 10^{-2}$, corresponding to 20% feed forward error correction code. The effect of IQ time-skews is shown in Fig. 1(a). More than 1 dB and 2 dB OSNR penalties are observed for 100-Gb/s QPSK and 200-Gb/s 16-QAM signals, respectively. The 200-Gb/s 16-QAM signals are more sensitive to the imperfections of coherent optical transmitter. Performance degradations are much severer when the signal suffers IQ power-imbalances. As shown in Fig. 1(b), the OSNR penalties are more than 3 dB when the power ratio is larger than 1.25 for 200-Gb/s coherent optical 16-QAM signals.
Fig. 1. OSNR penalty for 32-GBaud coherent optical 100-Gb/s QPSK and 200-Gb/s 16-QAM signals with different values of (a) IQ time-skew and (b) IQ power-imbalance.
In order to separate the IQ time-skews and power-imbalances brought in by the transmitters from those resulted from fiber channel and receivers, heterodyne coherent detection is applied to distinguish the imperfections from the receiver side. The heterodyne coherent detection scheme for single polarization case based on one single PD is shown in Fig. 2. In this case, digital signal processing (DSP) technique based on Kramers-Kronig (KK) detection algorithm is used to cancel the signal-to-signal beating noise (SSBN) after analog-to-digital converter (ADC) and resampling as shown in Fig. 2. In order to obtain the effective channel coefficients containing the effects of IQ time-skews and power-imbalances, channel equalization process based on conventional real-value constant modulus algorithm (R-CMA) is then applied followed by blind phase search method for carrier phase estimation. It is noted that carrier phase can be tracked since conventional R-CMA has certain ability to combat with the IQ time-skews and power-imbalances [11]. Then the estimated carrier phase is applied to eliminate the effects of carrier phase for the received samples after KK detection. Finally, modified real-value based CMA (MR-CMA) is used to obtain the channel coefficients as signal features to be fed into neural networks. It should be pointed out that such scheme can also be used for polarization multiplexing system, in which the two polarization parameters can be split and analyzed separately.
Fig. 2. Principle of the proposed method for detection and calibration of IQ time-skew and power-imbalance.
2.1 Modified R-CMA for QPSK format
In this work, we select QPSK as the main modulation format, which is the most widely used format in the 100G transponder. In order to improve the channel equalization effects, we also propose an R-CMA to compensate the IQ time-skews and power-imbalances at the transmitter side. In the single-polarization and QPSK modulation format case, the channel adaptive filters are updated using stochastic gradient descent optimization to achieve minimization of the constant modulus cost function. The process of filter coefficient $H_{(i(q)i(q)}$ updates can be expressed as,
where $\mu$ is the step size of stochastic gradient descent, $X_{in}$ is the received signal after re-sampling at time interval of $T/2$, and $T$ is the period of the signal. $X_{out}$ is the recovered signal after the channel adaptive filter, which can be calculated by In Eqs.(1)–(4), error function of R-CMA is $E_I=E_Q=1-|Re(X_{out})|^2-|Im(X_{out})|^2$. In R-CMA, signal in I and Q branch are recovered independently, which can mitigate the effects of IQ time-skew and power-imbalance. In this paper, we modify the error function of conventional R-CMA by setting $E_I=1-|Re(X_{out})|^2$ and $E_Q=1-|Im(X_{out})|^2$, respectively. The main advantage of the MR-CMA is that the error function considers I and Q branches independently, which is more effective when signal suffering from IQ power-imbalance at the transmitter side. For higher modulation format, such as 16-QAM, the channel coefficients can be extracted by cooperating the MR-CMA and constellation partition [23]. In our simulation and experimental demonstration, 1000 iterations are required to ensure that the performance is converged to a stable condition.We also conduct a simulation to verify the effectiveness of the MR-CMA. The QPSK symbols are duplicated by 50 times to emulate the up-sampling operation followed by low-pass filter for pulse shaping. Then IQ time-skews and power-imbalances are added to the generated QPSK symbols. The number of channel filters is set to 9. Figures 3(a) and 3(b) shows the constellation points of the recovered signals based on the modified and the conventional R-CMAs, when the value of IQ power-imbalance is set to 1.1 dB. Apparently, the MR-CMA, Q$^2$ factor of which achieves 17.8 dB, outperforms 3.7 dB compared to the conventional R-CMA, Q$^2$ factor of which is 14.1 dB. Therefore, it is convincing that the coefficients generated by the MR-CMA contain more accurate information about the imperfections.
Fig. 3. Recovered constellation points (a) based on modified real-valued channel equalization and (b) based on conventional real-valued channel equalization.
2.2 Artificial neural network based power-imbalance/time-skew identification
Finally, we use ANN to find the relationship between the extracted channel coefficients and the value of IQ time-skews/power-imbalances as shown in Fig. 4. After feature engineering, the channel coefficients are fed into multi-layer perceptrons as inputs to establish the projection between channel coefficients and time-skew and power-imbalances. The normalized corresponding time-skew and power-imbalance values are set as training targets in the neural network training stage and as ground truth values in the algorithm testing stage. Number of layers and detailed training parameters vary depending on the quality of training data. Due to the difference between simulation and experimental data, details of network structure and analysis of feature engineering are discussed separately in the following two Secs. 3 and 4.
Fig. 4. Artificial neural network for modeling the projections between channel coefficients and IQ power-imbalance and time-skew.
3. Simulation result
Simulation is first conducted to prove the concept of the proposed method. Specifically, in this section, we use 1 unit = 2% symbol period as measurement unit for IQ time-skew. And power-imbalance is measured in the unit of unity through out the paper as it is the ratio of amplitude between the I and Q branches. In order to emulate the effect of carrier phase noise in Fig. 2, the linewidth of laser is set at 100 kHz. The OSNR is set at 20 dB in the simulation.
3.1 Channel coefficients analyses and feature engineering for simulation data
Several sample channel coefficients sets $[H_{ii},H_{iq},H_{qi},H_{qq}]$ calculated by Eqs. (1)–(4) corresponding to different time-skew and power-imbalance values are presented in Figs. 5(a) –5(c). as shown in Fig. 5, the curves associate the coefficients to the variations of time-skews and power-imbalances. It can be seen that changes in $H_{iq}$ and $H_{qi}$ are not as explicit as changes in $H_{ii}$ and $H_{qq}$, therefore these two sets of coefficients are neglected, leaving only $H_{ii}$ and $H_{qq}$ as inputs to the neural network training. In Fig. 5(a), time-skew is constantly set to 0, while power-imbalance is gradually increase from $1.00$ to $1.50$. It can be found that when time-skew is perfectly compensated/calibrated, both $H_{ii}$ and $H_{qq}$ reveal symmetric shapes, and the differences or ratios between the values with same indices in $H_{ii}$ and $H_{qq}$ exhibit the trend of power-imbalance changes. In Fig. 5(b), power-imbalance is constantly set to $1.00$ , while time-skew is increased from 0% to 36%. In this case, one of the $H_{ii}$ and $H_{qq}$ coefficients still maintains a symmetric shape, but the other one deforms when time-skew increases. The difference between the two sides separated by the peak in either $H_{ii}$ and $H_{qq}$, where deformation appears, can indicate the level of time-skew. In Fig. 5(c), when time-skew and power-imbalance are added to the system at the same time, phenomena described in Figs. 5(a) and 5(b) superpose, making the deformation even more complicated, which further makes it hard to separate the contributions of the two impacting factors.
Fig. 5. Featured coefficients for simulation data with (a) varying power-imbalance and perfect time-skew; (b) varying time-skew and perfect power-imbalance; and (c) increasing power-imbalance and time-skew. Time-skew is in the unit of percentage, and power-imbalance is in the unit of unity.
Combining the features and trends observed above, the feature vector $\mathbf {h}=[h_1,h_2, \ldots ,h_N]$ is formed by omitting $H_{iq}$ and $H_{qi}$ coefficients and rearranging the order of $H_{ii}$ and $H_{qq}$ coefficients by placing the set with higher peak values as the first coefficient set. Length of feature vector N here for simulation data is 18. The rearranged feature vectors $\mathbf {h}$, which will be fed directly to the artificial neural network after normalization, corresponding to Figs. 5(a)–5(c) are shown in Figs. 6(a)–6(c), respectively.
Fig. 6. Feature vectors as inputs to MLP for simulation data with (a) varying power-imbalance and perfect time-skew; (b) varying time-skew and perfect power-imbalance; and (c) increasing power-imbalance and time-skew.
3.2 Result
Datasets with 231 combinations of time-skew and power-imbalance are collected, in which time-skews increases from 0% to 40% stepping up by 2% symbol period, and power-imbalance increases from 1.00 to 1.50 stepping up by 0.05 in absolute value (absolute values of power-imbalance are converted to values in dB for presentation in some figures and texts in this paper). Among the 231 combinations, 20% of the cases are randomly selected for model testing and the rest 80% cases are used as training sets. 100 samples are collected for each skew and imbalance combination.
A four-layer perceptron with 60 and 30 neurons in the two successive hidden layers is built for time-skew and power-imbalance estimation for the simulation data. Dropout rate of 0.3 is set for all layers except for the output layers, and Parametrized Rectified Linear Unit (PReLU) is set as activation functions for all layers. PReLU is an extension of the ReLU activation function. Mathematic expression of PReLU is $f(x) = (x, \alpha x)$, where $\alpha$ is the coefficient of leakage and is learned along with other parameters of the MLP. 1% of the training set are used as validation set during model training, which enables the early stopping mechanism to avoid overfitting. Both networks are trained with Adam optimizer and use MSE as loss function.
Training and testing results for power-imbalance are presented in Fig. 7(a), and the two purple curves to the right axis are the MSE error for each power-imbalance case. The averaged MSE for training set is $1.63 \times 10^{-4}$. Testing results show good consistency with training results, for which averaged MSE is $5.72\times 10^{-5}$. Training and testing results for time-skew are presented in Fig. 7(b), and the two purple curves to the right axis are the MSE error for each time-skew case. Averaged MSE is 0.049% symbol period for training set and 0.030% for testing set, respectively. Training and testing results for both power-imbalance and time-skew reveal good consistency.
Fig. 7. (a) Training and testing results of time-skew for simulation data; (b) Training and testing results of power-imbalance for simulation data.
In general, simulation results prove that the modified real-valued CMA channel coefficients for QPSK signals indeed contain information for calculating IQ power-imbalance and time-skew, and simple structured multi-layer neural network models are capable of establishing the projections between the coefficients and the two factors of interest with high accuracy. Therefore, we promote to further experimental tests for algorithm verification.
4. Experimental setup and results
The experimental setup for IQ time-skew and power-imbalance estimation is shown in Fig. 8. The four-channel electrical QPSK and 16-QAM signals are generated by an arbitrary waveform generator (AWG, Keysight M8195A) operating at 32 GSa/s, which are then launched into dual-polarization coherent optical transmitter integrated with four electrical amplifiers. The bias of the dual-polarization IQ modulator is adjusted by our designed automatic bias control board [24]. For simplification, the values of IQ time-skew and power-imbalance in one polarization are adjusted by the AWG. Time-skew is set to increase from 0 ps to 10 ps with 1 ps step, and every 1 ps equals to 3% of the symbol period. And power-imbalance is obtained by keeping the output voltage of I branch at 200 mV for QPSK signal and 100 mV for 16-QAM signal. The output voltage of Q branch is tuned from 200 mV to 160 mV with 10 mV step for QPSK signal and from 100 mV to 80 mV with 5 mV step for 16-QAM signal. The real values of IQ time-skew and power-imbalance can be determined by directly connecting the two output ports of the AWG to the two input ports of an oscilloscope. For simplicity, we only detect the signal in one polarization. The single-polarization optical signal is obtained by one polarization controller (PC) and one polarizer, which is then combined with another laser source by an optical coupler. The optical spectrum at the input port of PD is also shown in the inset of Fig. 8. A PD with bandwidth of 70 GHz is utilized to convert the optical signal into electrical signal, which is further sampled by a Lecory oscillator scope operating at 160 GSa/s, and processed off-line. It is noted that the two laser sources used are external-cavity laser (ECL) with line-width of 100 kHz. The frequency difference between the two laser sources is 25 GHz.
Fig. 8. Experimental setup for the identification of IQ time-skew and power-imbalance at the transmitter side based on heterodyne coherent optical detection with only one single PD.
Huge differences emerge between the simulation data and the experimentally collected data. Major differences are induced by the unknown frequency responses of the opto-electronic devices, which makes accurate simulation of these devices impossible. Besides, the coherent heterodyne detection scheme brings to the experimental data time-variant frequency offset, while in simulation only time-invariant frequency offset is added. What’s more, time-variant bias variations and nonlinearity of IQ modulation cannot be perfectly simulated, which also contributes to the differences between experimental and simulation data.
These differences in data will be reflected also in the extracted channel coefficients. Therefore, although the NN model trained by simulation data in Sec. 3.2 can be applied to the experimental data, to pursue higher accuracy and better practical application, we choose to update the NN model using the experimental data.
4.1 Channel coefficients analyses and feature engineering for QPSK experimental data
Same as for the simulation tests, several sample feature coefficients for the QPSK experimental data are calculated by Eqs. (1)–(4), processed by the procedures described in Sec. 3.1, and shown in Fig. 9. Length of feature vector N = 22 here for the experimental data. Difference and similarity of patterns are explained as the following. First, fluctuations of the $H_{iq}$ and $H_{qi}$ coefficients become much severer, but still few explicit patterns can be observed, thus these two sets of coefficients are excluded as well in all the experimental tests. Secondly, in Fig. 9(a), when time-skew maintains at 0 ps, no perfect symmetric shapes appear as in Fig. 6. However, the $H_{ii}$ and $H_{qq}$ sets exhibit nearly identical shapes, and the ratios between the two peak values partially reveal the change of power-imbalances. Thirdly, in Fig. 9(b), the effects of time-skew on the deformation of the $H_{ii}$ and $H_{qq}$ shapes are similar to what is shown in Fig. 6, in which increase of time-skew will enlarge the deformation of the coefficients’ shape, and difference or ratio between the values at the two sides of the peak values can reveal this change. Therefore, despite the difference in channel coefficients extracted from experimental data, we still believe the rearranged feature vectors of $H_{ii}$ and $H_{qq}$, as shown in Fig. 9, are appropriate inputs for training a neural network model.
Fig. 9. Feature vectors as inputs to MLP for QPSK signal with (a) varying power-imbalance and perfect time-skew; (b) varying time-skew and perfect power-imbalance; and (c) increasing power-imbalance and time-skew.
4.2 Experimental results for QPSK signal
In the experiment test, 55 combinations of time-skew and power-imbalance are set for data generation, with 11 time-skew values and 5 power-imbalance values. And for each combination, 100 samples are collected. Same as the simulation tests, 80% of the combinations are randomly selected and fed as training sets, and the rest 20% cases are used as ground truth in model testing. All the skew-imbalance combinations collected in this experiment are shown in the heatmap-like figure below in Fig. 10. Each square cell in the figure represents a combination of skew-imbalance pair, and the x-coordinate indicates its time-skew in ps and the y-coordinate indicates its power-imbalance in unity. The black cells represent the test sets, and white cells represent the train sets.
A three-layer perceptron with 30 neurons in the hidden layer is built for time-skew and power-imbalance identification. Dropout rate of 0.3 is set for all layers except for the output layers, and PReLU is set as activation functions for all layers. 1% of the training set are used as validation set during model training, which enables the early stopping mechanism to avoid overfitting. Both networks are trained with Adam optimizer, but use MAE as loss function. Change of loss function from MSE to MAE considers the emergence of highly noisy samples, and MAE is stricter dealing with outliers.
Training and testing results for time-skew are presented in Fig. 11(a), and the two purple curves to the right axis are the MAE error for each time-skew case. Averaged MAE is 0.120 ps for training data and 0.145 ps for testing data, respectively. Training and testing results for power-imbalance for experimental data are presented in Fig. 11(b), and the two purple curves to the right axis are the MAE error for each power-imbalance case. Averaged MAE is 0.009 for training data and 0.010 for testing data, respectively. Training and testing results for both time-skew and power-imbalance reveal high consistency and low error in evaluating the parameters of interest.
Fig. 11. (a) Training and testing results of time-skew for QPSK signal; (b) training and testing results of power-imbalance for QPSK signal.
4.3 Channel coefficients analyses and feature engineering for 16-QAM experimental data
Rearranged sample feature vectors for experimental data in 16-QAM format are calculated by Eqs. (1)–(4), processed by the procedures described in Sec. 3.1, and shown in Figs. 12(a)–12(c). Length of feature vector N = 22 here for 16-QAM signal, which is the same as for the QPSK experimental data. Changing pattern and trend of feature vectors shown in Fig. 12 are consistent with the features of simulation data and QPSK data.
Fig. 12. Feature vectors as inputs to MLP for 16-QAM signal with (a) varying power-imbalance and perfect time-skew; (b) varying time-skew and perfect power-imbalance; and (c) increasing power-imbalance and time-skew.
4.4 Experimental results for 16-QAM signal
For the 16-QAM data, as well, 55 combinations of time-skew and power-imbalance are set for data generation, with 11 time-skew values and 5 power-imbalance values, with 100 samples per skew-imbalance combination. All the skew-imbalance combinations collected in this experiment are shown in the heatmap-like figure below in Fig. 13. Each square cell in the figure represents a combination of skew-imbalance pair, and the x-coordinate indicates its time-skew in ps and the y-coordinate indicates its power-imbalance in unity. The black cells represent the test sets, and white cells represent the train sets.
Configurations and training parameters of the neural network model build for 16-QAM data are the same as for the QPSK data. Training and testing results for time-skew are presented in Fig. 14(a), and averaged MAE is 0.157 ps for training data and 0.162 ps for testing data, respectively. Training and testing results for power-imbalance for experimental data are presented in Fig. 14(b), and averaged MAE is 0.007 for training data and 0.006 for testing data, respectively.
Fig. 14. (a) Training and testing results of time-skew for 16-QAM signal; (b) E Training and testing results of power-imbalance for 16-QAM signal.
Experimental results for both QPSK and 16-QAM data show high accuracy in power-imbalance and time-skew estimation, and therefore prove the robustness and capability of the proposed method for fast and accurate IQ power-imbalance and time-skew adjustment for coherent optical transmitters.
Finally, we provide an example to show the effectiveness of the proposed method for dual-polarization 32-GBaud 16-QAM signal in the back-to-back case. The output voltage of I and Q branches of the AWG are set to 100 mv and 95 mv, respectively. The IQ time-skew is set to 5 ps. After DSP offline processing, which includes re-sampling to 2 samples per symbol, decision-directed adaptive channel equalization, and carrier phase recovery using V&V algorithm, the obtained constellations are displayed in Fig. 15(a). As shown in Fig. 15(a), since conventional channel equalizer cannot deal with the IQ time-skew and power-imbalance issues, the recovered signal exhibits large distortions. Then we apply the proposed scheme to estimate the IQ time-skew and power-imbalance in this experimental configuration. The estimated IQ time-skew is 4.97 ps and IQ power-imbalance is 1.051 times. After calibrating the corresponding parameters at the AWG, the recovered constellations are shown in Fig. 15(b). The Q$^2$ factor is improved from 12.9 dB to 20.3 dB, and very clear constellation points can be observed after calibration.
Fig. 15. Recovered constellations before (a) and after calibration (b) from condition of IQ time-skew of 5ps and power-imbalance of 1.05 times.
5. Conclusions
A novel machine learning assisted method for joint identification of IQ time-skews and power-imbalances for coherent optical transmitters is proposed in this paper. At the OSNR value of 20 dB, the MSE of the estimated IQ time-skew and power-imbalance can reach below 0.03% of the symbol period and $5.72\times 10^{-5}$ in the simulation. The MAEs of estimated IQ time-skew and power-imbalance are 0.145 ps and 0.01 for 32-GBaud QPSK signal, and 0.162 ps and 0.006 for 32-GBaud 16-QAM signal in the experimental demonstration. The calibration process is also conducted to verify the effectiveness of the proposed method in coherent optical transmission link at back-to-back case. This method not only reduces the complexity of hardware design, but also dramatically improves the accuracy of time-skew/power-imbalance estimation, therefore largely enhances the efficiency of coherent transmitter calibration. The results show the potential of the proposed method as a fast calibration method to fine tune and mitigate the slow time-varying transmitter impairments, and reveal the possibility for future automatic calibration.
Funding
National Natural Science Foundation of China (61705171); Natural Science Foundation of Hubei Province (2019CFB356).
Disclosures
The authors declare no conflicts of interest.
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