## 2 The Principles of Polybinary Transformation
When the time-packing factor equals 0.45, which means severe ISI distortion, we observe in Figure 7d that all the transformed signal curves are largely deformed. We see that the number of curve peaks has changed for the 1-level polybinary transformed and duobinary transformed signals, compared with the case of a factor of 1.00. Moreover, the curve corresponding to the tribinary transformed signal also has four unchanged peaks, although the curve is apparently deformed. Therefore, we speculate that we can achieve good decoding performance when using the combination algorithm of tribinary transformation + MLSE, and it is not a wise choice to use 1-level polybinary transformation + MLSE or duobinary transformation + MLSE. These statements are also verified by the results in Figure 6a. Therefore, the tribinary transformation + MLSE is more suitable for the case of a 0.45 time-packing factor. For other time-packing factors, we find similar relations to those mentioned above. Based on these relations, we can find the appropriate combinations of n-level polybinary transformation and MLSE in order to achieve the best decoding performance with certain time-packing factors. In this paper, we hope to enhance the SE further by adopting as small a time-packing factor as possible in the FTN system. Therefore, in the next section, we will focus on the case of 0.55-0.45 time-packing factors. The SE can be enhanced up to approximately 3.27 bit/s/Hz-4.89 bit/s/Hz under this time-packing factor range. The Performance of Phase Noise Equalizing Algorithm in FTN System under Extremely Small Time-Packing Factor  
All the discussions in the previous section are made under the assumption of no FO and no PN, which is not the case in a real optical fiber communication system. In this section, the received signal will pass through module 2 in \[p_{i}=E\sum_{k=0}^{n-2}d_{m-k} \tag{2}\]3 The PN induced by the linewidth of the laser both at the transmitter side and receiver side will influence the final quality of the decoding signal, which needs to be equalized by related algorithms. Generally, the most effective algorithm is the BPS algorithm, which can equalize PN for QPSK signals and other QAM signals, such as 16 QAM, 32 QAM, etc. BPS can achieve optimal performance based on the exact knowledge of the constellation point locations in the QAM signal. Because a smaller time-packing factor induces severe ISI and causes the serious overlap of constellation points, BPS will face the challenge of performance degradation in the FTN system. The lesser the overlap, the better performance of BPS can be achieved in the FTN system. Moreover, the BPS is placed in the front of the decoding algorithm. In other words, the effectiveness of BPS will affect the performance of subsequent decoding algorithms. We need to explore an effective method to make the BPS algorithm workable again in the FTN system. We take the 0.50 time-packing factor as an example. We find that the constellation points are not distinguishable in Figure 4a. Therefore, when we choose the original signal as the input signal with the 0.50 time-packing factor, the BPS will work ineffectively. According to the aforementioned analysis, we know that the tribinary transformation can greatly reduce the influence of ISI on the original QPSK signal when the time-packing factor equals 0.50. As shown in Figure 4c, based on the tribinary transformation, the constellation points become much more distinguishable, with the result that the BPS works again in the FTN system. To allow the BPS algorithm to work effectively in the FTN system, we need to make some modifications of the BPS algorithm according to the following two aspects: 1 the number of constellation points should be changed from 4 to 16 (4-QAM to 16-QAM); 2 the ideal locations of the constellation points should also be modified, and the new, ideal locations of constellation points should be expressed as in Equation (3), from that of 4-QAM to 16-QAM. For the sake of clarity, we call this type of BPS algorithm the modified-BPS algorithm.