## 2 The Principles of Polybinary Transformation
In order to conduct an in-depth analysis, we attempt to convert the two-dimensional constellation PDF along with the real and imagined axes in Figures 2 and 4 into a one-dimensional constellation projection PDF only along with the real axis, as shown in Figure 7. We take four cases in which the time-packing factor equals 1.00, 0.95, 0.65 and 0.45 as examples, respectively. Figure 7a-d show the abovementioned constellation projection PDF graphs corresponding to the time-packing factors of 1.00, 0.95, 0.65 and 0.45, respectively. Note that, in Figure 7a-d, the received signals are both exercised with 1-level polybinary transformation (equivalent to the original signal), duobinary transformation and tribinary transformation. \[p_{i}=E\sum_{k=0}^{n-2}d_{m-k} \tag{2}\]0 _Cont._  
\[p_{i}=E\sum_{k=0}^{n-2}d_{m-k} \tag{2}\]1 The performance of OSNR versus BER for MLSE, duobinary MLSE and tribinary MLSE under different time-packing factors: (**a**) the time-packing factor changes from 0.45 to 0.60; (**b**) the time-packing factor changes from 0.65 to 0.80; (**c**) the time-packing factor changes from 0.85 to 1.00. The signal has no distortion effect induced by ISI with the time-packing factor of 1.00. Therefore, to analyze the system conveniently, we set the case with the time-packing factor of 1.00 as the standard benchmark for the following comparison, as shown in Figure 7a. The original signal has a 4 QAM constellation projection PDF curve, the duobinary transformed signal has a 9 QAM constellation projection PDF curve, and the tribinary transformed signal has a 16 QAM constellation projection PDF curve, as shown in Figure 7a. In Figure 7a, the three curves are the standard benchmark. For the smaller time-packing factors, the comparison results and analysis are shown as follows. \[p_{i}=E\sum_{k=0}^{n-2}d_{m-k} \tag{2}\]2 The normalized frequent and continuous versus real values of constellation points for original signal, after duobinary transformation and after tribinary transformation (**a**) when the time-packing factor equals 1.00; (**b**) when the time-packing factor equals 0.95; (**c**) when the time-packing factor equals 0.60; (**d**) when the time-packing factor equals 0.45. When the time-packing factor equals 0.95, in which little distortion is induced by ISI, we observe that all the curves in Figure 7a,b are similar. This means that with the time-packing factor of 0.95, we achieve almost the same performance with the combinations of 1-level polybinary transformation + MLSE (or MLSE), duobinary transformation + MLSE and tribinary transformation + MLSE, indicating that they can all eliminate ISI in the FTN system with the 0.95 time-packing factor. However, Figure 6c shows that the MLSE has better decoding performance with the 0.95 time-packing factor. Therefore, the MLSE is more suitable for the 0.95 time-packing factor. 2. When the time-packing factor equals 0.65, which is the case with medium ISI distortion, we observe that the original signal curve has a central peak (which is approximately 0.87), as shown in Figure 7c, which does not appear when the time-packing factor equals 1.00, indicating apparent distortion induced by ISI. Moreover, the tribinary transformed signal curve generates two maximum peaks at \(\pm\)2, while the peaks at \(\pm\)1 are severely decreased. Thus, we speculate that using the combinations of 1-level polybinary transformation + MLSE (or MLSE) and tribinary transformation + MLSE cannot achieve good decoding performance. However, we see that the duobinary transformed signal curve has the same locations of all peaks, as is the case of factor 1.00, as shown in Figure 7a,c. Hence, we suggest that it is appropriate to adopt the combination algorithm of duobinary transformation + MLSE, in order to achieve good decoding performance. All the abovementioned speculations are verified by the results in Figure 6b. Therefore, the duobinary transformation + MLSE is more suitable for the case of a 0.65 time-packing factor. 3.