## 2 Principle of the Proposed Scheme
### K-means Cluster Algorithm at the Receiving End of Data Center  
IPM signals are impacted by ASE, fiber dispersion, and other phase noise distortions while they are transmitted from one DC to another. The nonlinear phase noise caused by the Kerr effect will dominate the modulated signal when OSNR is high sufficient. Kerr effect is an electro-optic effect, which indicated that the refractive index is proportional to the square of the applied electric field. Since the refractive index is nonlinear, as the electric field intensity varies in the optical fiber, the refractive index fluctuates, and the signal phase shifts as well, resulting in nonlinear phase noise [23; 24; 25]. To simulate the nonlinear phase noise generated by the Kerr effect during optical pulse propagation, the Generalized Nonlinear Schrodinger Equation (GNLSE) can be used as the mathematical modeling to describe the process of optical pulse propagation equation [26]. GNLSE, on the other hand, cannot provide an analytical solution for any input light pulse. As a consequence, a numerical method called Split-Step Fourier Transform (SSFT) should be employed to model the propagation of light pulses in a single-mode fiber (SMF) [27]. Accordingly, the essential DSP module is required to recover optical signals at the receiving end of the data center. In response to the frequency offset generated by laser and phase rotation caused by nonlinear phase noise (NLPN), the most common use recovery algorithm is BPS. However,  
Figure 1: Different orders of constellation based on IPM (**a**) M = 16; (**b**) M = 64; (**c**) M = 256.  
the complexity of BPS grew dramatically as the modulation order increased. Additionally, the BPS algorithm could only compensate \((-\pi/4,\pi/4)\) phase shift of the range, which is clearly insufficient when there is a factor of frequency offset.  
K-means cluster algorithm is an unsupervised learning, which will classify comparable objects into the same cluster. It has the advantages of low complexity, easy implementation, and fast convergence. Its good classification characteristics could be employed to demodulate optical signals that have been greatly affected by NLPN and avoid using BPS algorithm with high complexity. The principle of K-means is mainly to figure out the associated centroid according to the sum of minimum Euclidean distance. The specific procedure of K-means is as follows. Firstly, randomly choose K points as the initial centroids; we set the initial centroids here as the coordinates of the transmission constellation to make the algorithm rapidly converge to the optimal. Then calculate the Euclidean distance between each point and K initial centroids, and assign which clusters the point belongs to. After resetting the locations of the centroid, repeat the above steps until the centroid will not change.  
As an example, the transmitting and receiving procedure for 64-QAM is as follows. The bit stream to be transmitted is modulated into optical 64-QAM signal at the transmitter, and then the numerical solution of GNLSE using SFFT algorithm to simulate the optical pulse propagation in SMF is calculated. N symbols affected by dispersion and NLPN are received at the receiving end and classified using the principle of the above-mentioned K-means. The acquired centroid is one-to-one corresponding to the points of the ideal 64-QAM constellation to complete the demodulation. Figure 2 shows the cluster classification result after deploying the K-means algorithm to 64-QAM; it is evident that even if the optical signal phase rotated seriously, the algorithm could still classify and demodulate constellation points more accurately.