1. Introduction

High order modulation signals with high spectral efficiency, such as M-ary phase-shift keying and M-ary quadrature-amplitude modulation (M-QAM), have been extensively applied to increase fiber transmission capacity and accommodate the upsurge of network traffic. These high order modulation signals are extremely susceptible to various noises in the fiber transmission systems and generally higher optical signal-to-noise ratio (OSNR) is required to obtain the desired bit error ratio (BER). Furthermore, the high-power signals are indispensable for sufficient OSNR and however, once these high-power signals are launched into optical fibers, the impairments from the fiber Kerr nonlinearity will be inevitable [1]. Thus, fiber Kerr nonlinearity degrades the high order modulation signal performance in long-haul coherent optical communication system.

To compensate fiber nonlinearity, many digital signal processing (DSP) algorithms have been proposed, such as digital back-propagation (DBP) [2], artificial neural networks (ANN) [3], support vector machines (SVM) [4–6] and k-means [7]. DBP utilizes the back-propagation algorithm in the digital domain to solve the inverse nonlinear Schrödinger equation of the fiber link based on the split-step Fourier method and calculate the transmitted signal from the received signal [2]. Although DBP-based fiber nonlinearity compensation techniques are effective, in a practical implementation the large number of iterations results in high complexity [8]. As machine learning equalization techniques, SVMs have shown prominent properties but the training complexity of SVMs is highly dependent on the size of a data set [9], and therefore the newton-based SVM was proposed to reduce classifier complexity [10]. In addition, ANNs have similar issues as that of SVMs, where learning time is too long and sometimes the purpose of learning cannot be achieved [3]. Hence, it is significant to investigate novel equalization techniques with reduced computational complexity and low data redundancy to mitigate fiber-nonlinearity-induced signal impairments.

In the abovementioned DSP algorithms to mitigate the fiber nonlinearity, the classical k-means algorithm has a simpler structure. Recently, k-means algorithm has been efficiently implemented on parallel and distributed computers for large-scale practical problems [11]. However, the classical k-means algorithm is sensitive to the initial centroids and different cluster centroids will lead to different clustering results. Actually, the probability of finding appropriate initial centroids is especially low for large data sets, which results in the local optimal solutions in the final results. In addition, with the increase of k value, it becomes challenging to select correctly the centroids of the clusters. There are some works about the k-means algorithm to improve the performance of the QPSK and 16-QAM systems [7,12]. In [12], the QPSK signal only needs 4 centroids and hence it is easy in the classical k-means algorithm to locate these centroids. In [7], although the k-means algorithm was used in 16-QAM simulation system, there was no any explanation as to anchoring 16 centroids’ position. Furthermore, the classical unsupervised k-means algorithm is an iterative algorithm in which the number of clusters must be determined before the execution. In each iteration, the k-means algorithm computes the distances between each data point and all centers, which requires large amount of computational resources especially for large-scale data sets [13]. In addition, in the classical k-means algorithm, the initial points are always randomly selected, which are either close each other or outliers of clusters and will result in unsatisfactory partitions. In principle, the initial points are required to be well separated to approximate each cluster in the sparse data space. Therefore, when the classical k-means algorithm is applied on the M-QAM coherent optical communication system, it is indispensable to quickly position the centroids of the clusters.

In this paper, we experimentally demonstrated novel k-means clustering-based fiber nonlinearity equalization techniques in 75-Gb/s 64-QAM coherent optical communication system. To overcome the problems of the classical k-means algorithm, we proposed the training-sequence assisted k-means algorithm and the blind k-means algorithm to significantly reduce calculation complexity and data redundancy and accurately track the centroids of the clusters. We explained in detail the proposed algorithms’ procedures and the vital method to rapidly search the optimum centroids of the clusters rather than iterations. By utilizing the proposed k-means algorithms, we can achieve improved BER performance. Both of the training-sequence assisted k-means algorithm and the blind k-means algorithm enable to achieve almost one order of magnitude improvement in the BER performance from 10⁻² to 10⁻³ in 75-Gb/s 64-QAM experiments.

2. K-means algorithm

Classical k-means algorithm is widely used in pattern recognition and data mining to solve the data classification or clustering problems [13, 14]. The basic idea of the k-means clustering algorithm is partitioning n observations into k clusters. The concept diagram of k-means clustering is shown in Fig. 1. The objective of the algorithm is to classify the given data into k clusters by defining k centroids. The data classification is done by minimizing a chosen Euclidean distance measure between a data point and cluster center.

 

Fig. 1 Concept diagram of k-means clustering.

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Here, Euclidean distance between two points in space is as follows:

The partitioning process of the classical k-means algorithm is presented as follows. Firstly, we need to randomly select k objects from the original data set as the initial centroids. Secondly, for each object, we have to calculate the Euclidean distance with each centroid and assign it to the nearest cluster. Thirdly, we must update centroids by calculating the mean of cluster. Finally, we will iterate the abovementioned two procedures until the standard measure function begins to converge, otherwise return to the previous step.

In general, the quality of the clustering results given by the classical k-means algorithm can be measured by a standard measure function, such as, the summation of the square of the Euclidean distance between each cluster and its centroid.

The classical k-means algorithm has simple structure and when the k value is small, e.g. 2 or 3, its computation speed is usually fast [15]. However, the quality of the clustering results is sensitive to the initial centroids. Different initial cluster centers will lead to different clustering results, so it is very important to find k centroids accurately. Otherwise, the clustering results are unstable and inaccurate. Furthermore, with the increase of the input centroid k, the classical k-means clustering algorithm easily falls into local optima and it’s difficult to find the global optimal cluster centers [16]. Therefore, the classical k-means algorithm is very challenging to be used in 64-QAM systems. To solve these issues, we proposed two kinds of improved k-means clustering algorithms for fiber nonlinearity mitigation in 64-QAM coherent optical communication system, i.e., the training-sequence assisted k-means algorithm and the blind k-means algorithm. The key points of two algorithms are to determine the cluster centers with the training sequence and the density parameter of the data objects, which ensures the rationality of the cluster centers. Therefore, the proposed two kinds of algorithms can easily converge toward the global optima with improved stability and less calculating time.

2.1 Training-sequence assisted k-means algorithm

In the 64-QAM training-sequence assisted k-means algorithm, the cluster centroids distribution of 64-QAM constellation are learned during an initial training process and the cluster centroids are obtained by the mean function of the$l$training data set {(x¹, y¹), (x², y²), …,(x^l, y^l)}. Here x^l is a complex number and represents the received training data. y^l represents the label corresponding to x^l and l is the length of training data. These known cluster centers are used as the centroids and all data will be classified into different clusters.

The 64-QAM training-sequence assisted k-means algorithm includes the following steps:

2.2 Blind k-means algorithm

In order to reduce the data redundancy, we develop a novel blind k-means algorithm without any training sequence. Actually, the classical k-means algorithm belongs to the class of unsupervised algorithms, which greatly depends on the initial cluster centers and it is easy to converge toward the local optima. Since the clustering criterion function in the k-means algorithm is a non-convex squared error evaluation function, which drives the algorithm to deviate from the searching range of global optimal solution. Therefore, with the increase of the centroid k value, the classical k-means algorithm can hardly find the global optimal centroids. In the 64-QAM coherent optical communication system, 64 centroids are required. Thus, the most critical step in the blind k-means algorithm is to locate the 64 centroids accurately. In this work, we utilize the density parameter of the data objects to realize the blind 64 centroid tracking. The proposed blind centroid tracking method has the following advantages. (1) It can quickly locate the centroids’ position and significantly improve the quality of the cluster tracking results. (2) It can highly reduce computational complexity and does not need any iteration calculation. (3) It is especially applicable to large-scale data sets in the high-speed optical communication system.

Figure 2 shows the schematic diagram of the blind centroid tracking method, where we take a simple QPSK constellation with phase offset as an example to explain the principle of the proposed method. Firstly, we use the QPSK signal with the widely scattered constellation points and the rotated phase to emulate the distorted signal by the amplified spontaneous emission noise (ASE) noise and the fiber nonlinearity as the original signal, as shown in Fig. 2(a). Secondly, we define the density parameter of the data set and extract the density-based spatial constellation cluster as the input data set to estimate the initial centroids’ position, as shown in Fig. 2(b), where the black snowflakes denote the obtained first-generation centroids. Thirdly, we calculate the distance between the first-generation centroids and each data and reorganize the clusters according to the minimum distance, as shown in Fig. 2(c). Fourthly, we calculate the mean of each cluster to achieve the updated centroids, as shown in Fig. 2(d). The extracted density-based spatial constellation clusters and their optimal centroids are shown in Fig. 2(e). Finally, we apply the obtained optimal centroids, marked by the black snowflakes, to classify the original signal, as shown in Fig. 2(f).

 

Fig. 2 Blind k-means schematic diagram in QPSK system.

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Then, we apply the abovementioned blind centroid tracking method on the 64-QAM signal, the procedure is described as follows:

To sum up, when the k value is large, the classical k-means algorithm suffers from a O(Nkt) computational effort dependence on the data size N and the number of iterations t. Moreover, the classic k-means randomly selected k points as the initial centroids of clusters, the probability that each initial centroid of clusters falls on a “real” cluster is small. Therefore, for 64-QAM signals, the classical k-means algorithm has very high computational complexity and always cannot achieve good classification results. On the other hand, the proposed blind k-means method is scalable, which is promising for large-scale data sets to achieve high-quality clustering results. Firstly, the density function provides high-quality initial clusters and mitigate the noise effects. Secondly, the proposed method in conjunction with the demodulation function can approximately distinguish these high-quality 64 clusters. Finally, we use the two-stage centroids tracking techniques to accurately anchor the optimum centroids and reduce the computational complexity.

3. Experimental setup

Figure 3 shows our experimental setup of 75-Gb/s 64-QAM coherent optical communication system. Firstly, an arbitrary waveform generator (AWG) with an 8-bit digital-to-analog converter (DAC) running at 50-GSamples/s emits two 12.5-Gbaud electrical signals with the de-correlated 2¹⁵–1 pseudo-random binary sequence (PRBS). Then these electrical signals are modulated to the continuous optical wave from an external cavity laser (ECL) with line-width of 100 kHz by an in-phase/quadrature (I/Q) modulator. The modulated optical 64-QAM signal is amplified by an erbium-doped fiber amplifier (EDFA) with 5-dB noise figure. The following variable optical attenuator (VOA) is used to adjust the launched signal power into the 50-km single mode fiber (SMF) with 0.2-ps/√km polarization mode dispersion. Before detected by the coherent receiver, the output signal from the 50-km SMF is amplified by another EDFA and another VOA attached at the output is used to adjust the power of the received signal to 5.5 dBm in the coherent receiver. The parameters of 50-km SMF are as follows: attenuation α = 0.2 dB/km; dispersion D = 18 ps/nm·km; and non-linear coefficient of γ = 1.2 W⁻¹km⁻¹. A 100-kHz local oscillator (LO) was used and superimposed with the polarization aligned signal in a 90° optical hybrid. After that, two balanced photo detectors were connected to the hybrid outputs and a 100 Gs/s real-time oscilloscope as analog-to-digital (A/D) converter to acquire data. Offline processing includes chromatic dispersion (CD) compensation, FIR adaptive equalization, clock recovery, carrier-phase estimation with blind phase search, k-means nonlinearity mitigation algorithm, decoding and finally bit error rate (BER) evaluation.

 

Fig. 3 Experimental setup of 64-QAM coherent optical communication system with k-means nonlinearity mitigation algorithm. (AWG: arbitrary waveform generator, EDFA: erbium-doped fiber amplifier, VOA: variable optical attenuator, SMF: single mode fiber, LO: local oscillator.)

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4. Results and discussions

In order to verify the performance of our proposed k-means-clustering-based fiber nonlinearity equalization algorithms, we apply them to process the data with carrier-phase estimation and evaluate the BER of the received signal. Figure 4 shows the constellation diagrams of 64-QAM signal with back-to-back (BTB) and launched optical power (LOP) to 50-km SMF at −9.98 dBm, −5.45 dBm and 5.07 dBm, respectively. The constellation diagrams are different for various LOPs and the details will be discussed later. Next, we utilize the proposed two algorithms to process the data at the 5.07-dBm LOP and explain how to achieve the optimum 64 centroids, where the algorithms are used to optimize symbol decision and improve BER performance but have no influence on the constellation diagram of 64-QAM signal.

 

Fig. 4 Constellation diagrams of 64-QAM signal with BTB and the different launched optical power.

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Firstly, we apply the training-sequence assisted k-means algorithm in the 64-QAM transmission experiment. Figure 5 shows the constellation diagrams of 64-QAM training data (left) and signal data (right), where the launched power into the 50-km SMF is set to 5.07 dBm and the severely ASE noise and fiber nonlinearity made the symbol constellation points widely scattered and the constellation diagram rotated. The optimum 64 centroids can be easily found according to the mean value of the training data, as shown in the left part of Fig. 5. Here, the red points represent the centroids and the blue points represent the data. The structure of the training-sequence assisted k-means algorithm is simpler. After that, the signal data can be classified, as shown in the right part of Fig. 5. The length of the training sequences accounts for 10% of the total data length.

 

Fig. 5 Constellation diagrams of 64-QAM signal with the launched power at 5.07dBm training data (left) and signal data (right)with the training-sequence assisted k-means algorithm.

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Secondly, we apply the 64-QAM blind k-means algorithm to extract the 64 centroids from the abovementioned severely distorted signal. Figure 6 shows the constellation diagrams of 64-QAM signal in the different steps of the blind k-means algorithm, where the key step is to search the optimum 64 centroids. As shown in Fig. 6(i), firstly, the 64 centroids are found by step (e) and however, the obtained centroids are not accurate. Then we use these inaccurate centroids to find the adjacent clusters and use those clusters to obtain the optimal 64 centroids, as shown in Fig. 6(ii). Finally, all data are assigned into the 64 clusters, as shown in Fig. 6(iii). Here, the red points represent the centroids and the blue points represent the data. Moreover, according to the cluster labels, we can evaluate the BER of the received signal.

 

Fig. 6 Constellation diagrams of 64-QAM signal with the launched power at 5.07dBm in the different steps of the blind k-means algorithm.

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Next, we vary the launched 64-QAM signal power into the 50-km SMF to measure BER curves and observe the signal impairments caused by the ASE noise and the fiber nonlinearity. Limited by the available equipment, the generated 64-QAM signal is transmitted on the 50-km SMF to verify the feasibility of the proposed algorithms. We will apply the proposed k-means algorithms in the abovementioned experiment to compensate the signal impairments and improve the BER performance. Figure 7 shows the measured BER curves versus launched signal power P_in into the 50-km SMF, marked by diamonds. As the launched signal power increases, the OSNR of the received signal becomes higher, which contributes to a better BER performance, as shown by the left part of the diamond-marked curve in Fig. 7. In this launched signal power range, the signals with lower launched signal power always suffer from the severe ASE noise and the constellation points are widely scattered as shown by the constellation diagram of −9.98 dBm launched signal power in Fig. 4. On the other hand, once the launched signal power exceeds a value, the BER performance of the received signal is deteriorated by the fiber Kerr nonlinearity, as shown by the right part of the diamond-marked curve in Fig. 7. The constellation diagram rotates because of the phase rotation caused by the fiber Kerr nonlinearity and the outer symbol constellation points with larger power are severely distorted, as shown by the constellation diagram of the 5.07-dBm launched signal power in Fig. 4. The optimal launched signal power is around −5.45 dBm and the balance between the ASE noise and the fiber Kerr nonlinearity is achieved and the corresponding constellation diagram has distinct constellation points.

 

Fig. 7 Measured BER curves vs. launched signal power into 50-km SMF P_in (dBm) in the 75-Gb/s 64-QAM coherent optical communication system: without k-mean algorithm (diamond-marked); with blind k-means algorithm (triangle-marked); with training-sequence assisted k-means algorithm (circle-marked).

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The improved BER curves by the training-sequence assisted k-means algorithm and the blind k-means algorithm are shown by the circle-marked and the triangle-marked curves in Fig. 7. From Fig. 7, it can be observed that the training-sequence assisted k-means algorithm can obtain a little bit better BER performance than that of the blind k-means algorithm, as shown by the circle-marked and the triangle-marked curves. With the increase of the fiber nonlinearity, the performance of the k-means algorithms begin to decline dramatically with the launched signal power beyond −5.45 dBm, as shown in Fig. 7 and the BER improvement is smaller. To sum up, when the launched signal power varies from −9.98 dBm to −5.45 dBm, the k-means algorithms can well compensate the signal impairments dominated by the ASE noise. On the other hand, when the launched signal power varies from −5.45 dBm to 5.07 dBm and the signal impairments is dominated by the fiber nonlinearity, where the outside clusters experience serious ASE noise and nonlinear phase shifts and the distorted constellation points are overlapped, the k-means algorithms have less compensation performance. It is also challenging for any soft-decision algorithms to effectively distinguish these overlapped points. Therefore, from Fig. 7, it can be seen that BER performance of 75-Gb/s 64-QAM signal is improved by the training-sequence assisted k-means algorithm and the blind k-means algorithm, the improved BER is below forward error correction (FEC) threshold floor at 1.0 × 10⁻³ BER.

In Fig. 5, we use 10% training sequences of the total data length in the 64-QAM training-sequence assisted k-means algorithm. It should be noted that if we change the training overhead, the raw bit-rate was adjusted accordingly. In order to figure out the influence of the training sequences length on the performance, we choose the different training sequences with various sequence length of 2%, 5%, 10%, 15% and 20% of the total data length. Figure 8 shows the influence of the training sequences length on the improved BER performance. From Fig. 8, it is evident that the curve is smooth and fast convergent and the length from 10% to 20% of the training sequences has no significant influence on the BER improvement performance. The 10% training overhead are enough to be utilized to find optimum 64 centroids. Once the optimum 64 centroids are found, the signal data can be classified and BER of the received signal can be evaluated. Therefore, higher training overhead cannot improve BER performance more. In addition, the training sequence cannot be too much small. When the length of the training sequence with the totally distributed 64 points is reduced, the selected centroids are sensitive to the signal noise, which will degrade the system performance.

 

Fig. 8 The improved BER vs. training overhead of the training-sequence assisted k-means algorithm for 75-Gb/s 64-QAM signal at the 50-km of transmission for a launched optical power (LOP) of 5.07 dBm.

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5. Summary

In conclusion, we proposed two kinds of novel k-means-based fiber nonlinearity equalization algorithms and experimentally demonstrated them in 75-Gb/s 64-QAM coherent optical communication system. The proposed k-means-based techniques can reduce clustering complexity and data redundancy and we achieved better BER improvement performance in experiments. Moreover, both of the the training-sequence assisted k-means algorithm and the blind k-means algorithm have simpler structure and can be used to handle the large-scale data and increase processing efficiency. In pratical 64-QAM long-hual coherent optical communication system, the proposed low data-redundancy and computational complexity k-means algorithm is a suitable candidate to mitigate the fiber nonlinearity.