Modified by special feature engineering, a powerful and low-order equalizer based on K-nearest neighbors (KNN) classifier is applied to improve performance of high-speed system with bandwidth-limited optics. The feature construction and feature weighting are specially designed to conduct an appropriate a feature engineering-based KNN (FE-KNN) scheme, which contains more data characteristics to enhance the equalization performance. Experimental comparisons of KNN classifier with/without feature engineering, decision feedback equalizer (DFE) and feed-forward equalizer (FFE) are implemented to prove the feasibility of our scheme in both 25-Gb/s NRZ and 50-Gb/s PAM-4 transmission experiments with 10-G optics system. The corresponding results show that, without the feature engineering, the performance achieved by the common KNN is not improved even in the case of hard decision (HD). In contrast, compared to the common 11-taps DFE, the performance achieved by FE-KNN with only 5 taps is improved by 1-dB at KP4-FEC threshold (BER=2.2E-4) for 25-Gb/s NRZ transmission. While, for 50-Gb/s PAM-4 case, 0.5-dB sensitivity improvement is achieved by our scheme compared to the common 11-taps DFE under the HD-FEC limit (BER=3.8E-3).
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Driven by current and forthcoming applications such as cloud computing, virtual/augmented reality and online games, higher capacity of the optical access network and short-reach communications is demanded . Different from the long-haul networks, these systems are cost-sensitive for their huge number of end-users [2,3]. Thus, how to ensure high capacity while maintaining low cost is paramount in these systems . A simple but effective method is utilizing low-bandwidth optical devices to transmit high rate data. Besides, owing to its low complexity and cost, the intensity modulation and direct detection (IM/DD) based systems have also been regarded as one of the mainstream technologies . Recently, many beyond 100-Gbps and even 200-Gbps IM/DD systems based on limited-bandwidth devices with advanced modulation formats, such as four/six/eight-level pulse amplitude modulation (PAM-4/6/8) were demonstrated [6–8]. While, non-return-to-zero (NRZ) [9,10] modulation is still highly attractive in real-time deployments due to its simplicity of modulation and the advantage of low-power and low-complexity transceiver integrated circuit (IC). However, for these narrow-band transceivers based high-speed systems, the inter-symbol interference (ISI) caused by the bandwidth limitation and the cumulative dispersion of the fiber would severely degrade system performance. Thus, some signal pre/post equalization schemes have been presented to resolve this problem . These demonstrated methods can be classified into optical equalization and electrical equalization depending on whether the signal is processed in optical or electrical domain. Compared to the electrical equalization based on mature digital signal processing (DSP), optical equalization technologies have many shortcomings, such as demanding additional optical devices, higher system cost and limited performance improvement [12,13]. Therefore, IM/DD system in combination with DSP technologies is considered one of the promising solutions for advanced access network transmissions .
For the electrical equalization cases, feed-forward equalizer (FFE) and decision-feedback equalizer (DFE) are the two most commonly used schemes [15,16]. And, high complexity Volterra equalizer (VE) are also widely proposed as an upgrade for DFE and FFE [17,18]. However, VE is mainly used to compensate for nonlinear distortions rather than ISI, which often requires numerous filter taps . Meanwhile, to further improve the performance of equalization, some advanced method based on machine learning (ML), such as the common support vector machines (SVM) [20,21] and neural network (NN) [22,23] have also been proposed. Whereas, NN has a complicated structure and calls for a huge amount of calculation, thus it is hard to be realized in real-time system, especially in low-cost optical access systems. In addition, SVM based on convex quadratic optimization problem is also complicated. And, for the multi-class tasks such as the equalization of PAM-4/8 case, multiple SVM are required to complete the equalization , which would increase the complexity of equalization. In contrast, K-nearest neighbor (KNN), as one of the top ten machine learning algorithms, is much easier to implement while maintaining satisfactory performance [24–26]. Compared to other algorithms, the main advantage of KNN is that it can better classify the data at the decision boundary . Furthermore, different from the common SVM, KNN is a multi-classifier that does not require multiple classifiers to equalize high-order modulated signals. However, to the best of our knowledge, for the reported KNN-based scheme in optical transmission system [24–26], only the common KNN was employed and the signals’ feature of system was not involved which only brought limited system performance improvement.
In this paper, we propose a specially-designed feature engineering–based KNN (FE-KNN) classifier scheme to improve the performance of bandwidth limited IM/DD system. Here, as a pre-processor, the feature engineering is used to generate the feature vector for achieving better data characteristics, thereby further improving algorithm performance. To verify the feasibility of our proposed scheme, a 25-Gb/s NRZ and 50-Gb/s PAM-4 per wavelength transmission IMDD-based system with 10-GHz class optics is constructed. And, the common FFE/DFE and KNN classifier are also measured for further comparison analysis. Experimental results show that, for the case of 25-Gb/s NRZ, even with only 5 taps, compared to the 11-taps DFE and FFE schemes, our method can achieve 1-dB sensitivity improvement in both 20 km standard single mode fiber (SSMF) and back-to-back (B2B) transmission under the KP4-FEC threshold (BER = 2.2E-4). While, for 50-Gb/s PAM-4 transmission, 0.5-dB sensitivity improvement is achieved by our scheme under the HD-FEC threshold (BER = 3.8E-3).
2.1 Channel Model of band-limited system
The system model of band-limited PON or short reach system is given to describe the feature of transmission signals. Generally, in these systems, due to the limited bandwidth, the signals would be severely distorted, and the corresponding schematic diagrams are presented in Fig. 1. Here, T0 represents the current symbol sampling period, and T1 and T−1 represent the current adjacent symbol samplings’ period respectively. It is easily found that, the waveform of current symbol will be superimposed on the adjacent symbols’ waveform, which would generate severe ISI. In this way, the system performance would be degraded. As depicted in Fig. 1, the red line represents the superimposed signal, which can be described as
2.2 Principle of proposed FE-KNN equalizer
In general, for the KNN classifier, the received data can be divided into training data and testing data. For the training data, the classification is performed and certain at the receiver end. And, the testing data is the effective data transmitted through system, which are not classified before equalization operation. These data can form corresponding training and testing set, respectively. The training set is composed by the feature vector and the corresponding label, which can be denoted as
To solve this problem, a modified algorithm, KNN based on feature engineering (FE-KNN) is proposed, and the corresponding processing steps is presented in Fig. 2. Firstly, for the training set, a feature construction engineering scheme suitable for the band-limited system is designed, which is constituted by feature construction and feature weighting. Next, the training sequence generator is used to produce the label of training set. Then, the feature vectors for testing set are constructed by the same feature engineering as in the training set. At last, the training set and the testing set are injected into KNN classifier to realize the equalization. The detailed principle and effectiveness of feature engineering are described as below.
As for a feature engineering, how to obtain the suitable features is a crucial step. Based on the character of received signal as described in the Eq. (1), the previous, follow-up and current period samples are selected as the features after downsampling processing, and the sampled value is considered as the feature value. Figure 3 is presented to illustrate the effectiveness of feature construction. As shown in Fig. 3(a), the down-sampling data (one-dimensional) of NRZ transmission in band-limited system is employed. Here, different colors represent different categories of data. It can be noted that, a large number of different types of data are overlapping, thus only using the common KNN classifier is not effective in dealing with this case. As opposed to this, the data scatter plot after feature construction is shown in the Fig. 3(b). For simplicity of presentation, only two features are utilized to construct the feature vector, namely the data only have two-dimensional coordinates. And, blue and black points, circled by two black circles, represent two categories of training data. As we can see, after feature constructing, the problem of data overlapping is avoided and the KNN-based equalizer can correctly classify the testing data.
In addition, for the common KNN, since the configuration of weight for feature was not involved as described in Eq. (3), each feature has the same effect for the Euclidian distance calculation. In contrast, the weight of the band-limited channel is not consistent in Eq. (1), namely the center weight is obviously greater than its adjacent ones. Consequently, with this character, the traditional KNN cannot be directly used as equalizer to deal with the ISI induced by the band-limited system. For this purpose, we give the weighting concept for the feature of KNN classifier. In this way, the feature vector of the n-th symbol based on our proposed feature engineering can be described as
Figure 4 is utilized to further demonstrate the importance of feature weighting. To explain the effect more intuitively, the data in the Fig. 4 is not the real experimental data. From the Fig. 4(a), it can be observed that the data has a skewed boundary, which can be denoted as y = 2x. This phenomenon will result in inconsistent importance of the distances in the two dimensions and points near the boundary are easily misclassified. Zoom in a part of Fig. 4(a), as shown in Fig. 4(b), we can see that, without the feature weighting, the point closest to testing point A is point B with a different category, rather than point C with the same category. In contrast, the effect of feature weighting are shown in the Fig. 4(c). After feature weighting, KNN classifier can find the closest point with the same category to point A. Therefore, feature weighting can further improve the accuracy of data classification at the decision boundary. According to our tests, the improvement is more pronounced in PAM-4 and not that obvious in NRZ, so this processing was only implemented for PAM-4.
After the feature engineering, the training set needs to get its corresponding label. As the traditional adaptive algorithms, the label of the proposed equalizer is provided by the training sequence generator. At last, the complete training and testing set are injected into the KNN classifier. Note that, the KNN algorithm can remain unchanged, all the modifications have been completed in the feature engineering.
3. Experimental setup
As shown in Fig. 5, an experiment is constructed to verify the feasibility of our scheme. Here, both transmitter and receiver are employed with 10-Gb/s application scenario. At the transmitter side, 25-Gb/s NRZ and 50-Gb/s PAM-4 signals are chosen as transmitted data, which are both pseudo random binary sequence with a length of 215-1 (PRBS15) generated offline. The data including 250000 symbols and the specific training sequence are loaded into a pulse pattern generator (PPG) to obtain electrical signals. As for PAM-4 signal, one output signal of PPG is delayed by 37-bits, and another output is attenuated by 6 decibels, and then the two outputs are combined into the four-level signal by a wideband power combiner. Next, these signals are used to drive 8-GHz direct modulation laser (DML) with 1311-nm center wavelength, whose launch power is measured to be 10 dBm. After 20-km SSMF transmission, a variable optical attenuator (VOA) is employed to adjust the received optical power for further performance test. At the receiver end, the optical signals are detected by an avalanche diode (APD, 3-dB bandwidth ∼7 GHz) and then captured by a digital oscilloscope (DSO, LeCroy SDA845Zi-A) with 80-GSa/s sampling rate. The sampled data are then sent to offline DSP module deployed by Matlab, including the upsampling, timing recovery, downsampling, equalization (including FFE/DFE, normal KNN and proposed FE-KNN) and BER calculations. Besides, the eye diagrams of NRZ and PAM-4 signals received by 10-GHz APD are shown in the Fig. 5(i) and 5(ii), respectively. It is easy to observe that unbalanced eye diagram is tilted and crossed, which will cause big BER for signal decision. And, Fig. 5(iii) is the system frequency response with 7-GHz of 3-dB bandwidth.
It is noted that, the detailed offline processing procedures are presented in Fig. 5. Here, to realize the data clock recovery (DCR), the up-sampling operation for data sampled by the DSO should be first implemented to achieve an integer multiple of the symbol rate. After that, by utilizing the corresponding timing recovery algorithm , the data are then injected into the down-sampling and the equalization operation models. The employed equalizers are the normal FFE, DFE and our proposed FE-KNN with various tap numbers. The equalizer coefficients of DFE and FFE are determined by the RLS algorithm, which functions by circularly adjusting the tap coefficients until the error converges to some certain degree. After equalization, the distorted signals are recovered for BER measurements to further evaluate the system performances.
4. Experimental results and further discussion
4.1 25-Gb/s NRZ experimental results
To prove the superiority of our scheme, various characteristics of the 25-Gb/s NRZ transmission are demonstrated. Firstly, according to our test, the numbers of feature is the most important parameter for the performance of FE-KNN equalizer. Here, the numbers of the previous and the follow-up samples are set from 0 to 5. Note that the feature number of our scheme is equal to the number of filter taps. And the BER result is depicted as a contour diagram, shown in Fig. 6(a). It is easily got that, when the number of previous samples and follow-up samples are set to 2 or 3, the satisfactory performance can be obtained for our scheme. In other word, for the case of 5/7 features (namely 5/7-taps), our scheme can achieve better BER performance. Here, the 5 features are composed of 2 previous samples, 1 current sample and 2 follow-up samples. While, in the case of too many features, the excess features are actually independent of the current data. This will cause the FE-KNN to calculate some unrelated distances, resulting in the accuracy of classification degradation. Therefore, the deteriorative BER performance is achieved.
Meanwhile, Fig. 6(b) shows the comparison of FFE, DFE and our scheme with difference taps at -23-dBm received power as an example. Obviously, within the appropriate taps, our scheme can achieve the best performance, which can be increased by about an order of magnitude compared to FFE and DFE. Besides, the complex DFE did not achieve a significant performance improvement over FFE and the same is true in subsequent tests. For more than 7 taps cases, the BER performance is greatly worsen, which has the same phenomenon as in Fig. 6(a). This also can be attributed to the accuracy of classification degradation induced by too many features. And, even with more than 11-taps, the DFE and FFE cannot achieve the performance as the 5-taps FE-KNN. Therefore, it is verified that our scheme demands less number of taps, which would effectively reduce DSP complexity in real applications.
Then, the convergence rates versus the length of training samples based on DFE, FFE and FE-KNN at -23-dBm received power is illustrated, as shown in Fig. 6(c). Here, the tap number for our proposed FE-KNN equalizer is 5, and for DFE/FFE is 11. In this way, for our scheme, the latency caused by tap delayer is shorter than DFE and FFE. It can be noted that, although our scheme does require more training data to achieve its optimal performance, but it can achieve better performance than DFE and FFE as long as the training length is greater than 1000. Besides, different from DFE and FFE, FE-KNN does not require the iterative algorithm to calculate the tap coefficients. This feature makes FE-KNN less complex than DFE and FFE during the training processing. And, due to the over-fitting problem caused by iterative calculations for getting the tap coefficient, BER performance of DFE/FFE has larger jitter, which is avoided for our FE-KNN.
Next, the effect of the timing error induced by the clock recovery block or timing jitter of the ADC in terms of the receive sensitivity in practical system is also investigated. To emulate this timing error, the sampled data output by the DSO is first upsampled from 80-GSa/s to 800-GSa/s. In this case, for the 25G-Baud data, 32 sampling points can be achieved for each symbol, namely 32 samples/symbol. With this operation, the resolution of evaluating timing offset is 1.25 ps (1s/25G/32 = 1.25 ps). Figure 7(a) depicts the calculated BER versus timing offset for traditional FFE, DFE and our scheme with different timing offsets. As can be seen, our scheme can achieve the best performance among the investigated equalization methods, especially near the best sampling time. As for the timing jitter, a Gaussian-distributed random index with zero mean and different variance is added to the number of columns during the down-sampling process for simulating the timing jitter of ADC from ± 1.25 ps to ± 10 ps. And, the calculated BER result is shown in the Fig. 7(b). It can be found that DFE has only a slight improvement compared to FFE. On the contrary, the FE-KNN obtains a prominent improvement, and the BER achieved by FE-KNN always maintains below 1E-3. Moreover, the ADC resolution evaluation for different equalizations is also given, as shown in the Fig. 7(c). Here, by utilizing the method in Ref. , to emulate the ADC resolution in offline, the sampled data considered as occupying all the bits are re-quantified into 1-, 2-,3-,……,8-bit (according with the ADC bit resolution), and then are used to calculate the BER for each bit. From this figure, it is easily got that, for the investigated equalizers, almost the same trend can be achieved for the ADC resolution from 1-bit to 8-bit. And, with the ADC resolution varying from 5-bit to 8-bit, the BER curves shows no fluctuations, thus it poses no extra requirement for ADC in real deployments, and 5-bit resolution is enough. Figure 7(d) depicts the influence of the value of the K for the proposed scheme. It is obviously observed, the impact of K is limited. Considering the trade-off between performance and calculation complex, the value of K is fixed at 5 for other tests (including both NRZ and PAM-4 system) in this paper.
Finally, Fig. 8 presents the BER versus received power for different equalizer in both B2B and 20 km transmission case. And, the normal KNN without feature construction and the scheme without any equalization are also measured for contrastive analysis. By preliminary test, we can get that almost the same results can be achieved in the case of B2B and 20 km. So, for clarity sake only the B2B results are also plotted in this figure. Here, for a fair comparison, the parameters of the investigated equalizers have been adjusted to their respective optimal values (11-taps and 2000-training length for DFE/FFE, 5-taps and 4000-training length for KNN). Besides, as we can see, due to the fact that normal KNN can't effectively classify serious overlapping data caused by limited bandwidth, its performance is almost the same as the case without any equalization. And for these two cases, the BER is so high that the signals even cannot be recovered. In contrast, the proposed FE-KNN scheme always obtains the optimal BER performance whether in B2B or 20 km transmission case. Furthermore, compared to common DFE and FFE, 1-dB sensitivity improvement can be achieved for our scheme at KP4-FEC threshold (BER = 2.2E-4).
4.2 50-Gb/s PAM-4 experimental results
The effectiveness of our proposal is also experimentally verified in the 50-Gb/s PAM-4 per wavelength system. For the case of PAM-4 signal, the most influential factor is still the number of features. We specifically tested the impact of feature numbers for FE-KNN, and the BER result is depicted as a contour diagram, shown in Fig. 9(a). As the mentioned results of NRZ, the best BER performance can be achieved with the feature configuration of 2 previous samples, 2 follow-up samples and 1 current samples, namely 5-taps case. It can be seen that, for a specific baud rate and transceiver bandwidth, the length of the ISI is fixed, so the optimal tap number is also fixed. In an actual system, the number of taps needed to be adjusted according to the baud rate and the bandwidth of the transceiver.
And, the comparison of FFE, DFE and our scheme with difference taps are shown in Fig. 9(b). Since the transceiver bandwidth and baud rate of the 25-Gb/s NRZ and 50-Gb/s PAM-4 signals is identical, the same tap number are configured for equalization. Obviously, with only 5-taps, our scheme can achieve the best performance among these equalizers. Performance of the FFE/DFE equalizers has a positive correlation with the employed tap numbers, and at least 11-taps are needed. As mentioned above, due to the influence of unrelated features on calculating the distance calculation, BER performance for FE-KNN is seriously degraded in the case of more than 5 taps. This can further verify that our scheme demands less number of taps. For time jitter of the oscilloscope and ADC resolution, the conclusions are also similar, so these results will not be given here.
As we have proved in section 2.2, since the signal is mainly affected by ISI, weighting of the central feature can further improve the performance of the KNN classifier. The BER curves with different weights of the center feature is shown in the Fig. 9(c). It can be noted that, in different cases of received optical power, feature weighting can achieve a certain degree of improvement. And, it is obviously observed that, for the case of coefficient c = 1.3, the best BER performance can be achieved, which can be regarded as an empirical value. Since the method mainly optimizes the judgment of the data located at the decision boundary, the promotion is not as great as the feature construction. But the approach is very simple, namely multiplying a tap by a fixed coefficient, so it is still highly applicable in real deployed systems.
Finally, the BER performances of FE-KNN, KNN with only feature construction (FC-KNN) and normal DFE/FFE are compared in Fig. 10. Due to the negligible nonlinear effect of the O-band DML transmission, the performance of the B2B and 20km transmissions is almost the same, so only the results of the 20km transmission are given. The parameters of all researched equalizers have been adjusted to their respective optimal values (11-taps for DFE/FFE, 5-taps for FE-KNN and FC-KNN). It is noted that, since the signal aliasing of PAM4 is more severe than that of NRZ, the FC-KNN achieves better performance than the FFE and is close to the DFE. And, FE-KNN can achieve the best performance. With 5-taps, FE-KNN can improve the receiver sensitivity by 0.5 dB and 1.5 dB at 7% HD-FEC limit (BER = 3.8E-3) compared to the DFE and FFE with 11-taps, respectively. According to the above results, the proposed FE-KNN scheme can prominently enhance the transmission performance and power budget for the high-speed band-limited IM/DD-based system.
In this paper, we demonstrated a new equalizer based on machine learning classifier for IMDD-based band-limited optical system. A special feature engineering is proposed to improve the KNN classifier for NRZ and PAM-4 modulated signals. The feature construction transforms original data into the data that is more suitable for the KNN classifier. And, the feature weighting can further improve the classification of data, especially data that are close to the decision boundary which are easily misjudged. A comparison between the proposed FE-KNN scheme, the common KNN and FFE/DFE is performed to prove its feasibility in 25-Gb/s NRZ and 50-Gb/s PAM-4 signal transmission experiment with 10-G class optics system. Experimental results show that, compared to the common DFE and FFE, our proposed scheme can achieve 1-dB sensitivity improvement in 25-Gb/s NRZ transmission case at KP4 FEC limit (BER = 2.2E-4). And, for the 50-Gb/s PAM-4 signal, 0.5-dB sensitivity improvement is achieved by our scheme compared to the common 11-taps DFE at HD-FEC limit (BER = 3.8E-3). We believe the proposed FE-KNN equalizer algorithm is promising for high capacity transmission within band-limited systems. In the future work, we will focus on shortening the length of the training set required for convergence to achieve a better performance.
National Natural Science Foundation of China (61431009, 61501157); Natural Science Foundation of Zhejiang Province (Y201533646).
We acknowledge support of laboratory equipment provided by Shanghai Jiao Tong University.
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