## 2 The Principles of Polybinary Transformation
With the above modifications, the modified-BPS algorithm can compensate for PN in the FTN system with a smaller time-packing factor, as small as 0.50. Then, the MLSE algorithm can effectively achieve sufficient decoding performance to eliminate ISI. Therefore, when we make good use of the combination of the appropriate n-level polybinary transformation, the modified-BPS algorithm and MLSE, the received FTN QPSK signal can be recovered with different time-packing factors. Moreover, for other FTN QAM signals, such as 16 QAM, 32 QAM and 64 QAM, similar methods can be adopted. \[\left\{ \begin{array}{c}1+1j\\ -1+1j\\ -1-1j\\ 1-1j\end{array} \right.\Rightarrow\left\{ \begin{array}{c}2.1+2.1j,\quad 2.1+5.4j,\quad 5.4+5.4j,\quad 5.4+2.1j\\ -2.1+2.1j,\quad-2.1+5.4j,\quad-5.4+5.4j,\quad-5.4+2.1j\\ -2.1-2.1j,\quad-2.1-5.4j,\quad-5.4-5.4j,\quad-5.4-2.1j\\ 2.1-2.1j,\quad\quad 2.1-5.4j,\quad 5.4-5.4j,\quad 5.4-2.1j\end{array} \right. \tag{3}\]  
To verify the performance of the modified-BPS algorithm for compensating for PN in the FTN system with a smaller time-packing factor, we perform simulations on our platform, as shown in Figure 5. Note that the received signal will pass through module 2 in Figure 5, which means that we take the linewidth of the lasers into account. The linewidth of the laser at the transmitter is set to be 300 kHz, 500 kHz, 800 kHz and 1000 kHz, respectively. We also assume that other channel impairments besides ISI have been equalized by corresponding DSP algorithms. The time-packing factor changes from 0.45 to 0.55. In module 2, the sequence that the received signal passes through the algorithm is as follows: (1) firstly, we utilize the duobinary or tribinary transformation for the received signal to eliminate the overlap of the constellation points; (2) secondly, the signal passes through the modified-BPS to compensate for PN after duobinary or tribinary transformation; (3) thirdly, the signal passes through the MLSE algorithm to eliminate ISI after compensating for PN. Figure 8a-c show the BER as a function of OSNR, which demonstrates the effectiveness of the modified-BPS algorithm to equalize PN in relation to the linewidths of 300 kHz, 500 kHz, 800 kHz and 1000 kHz in the FTN system with the different time-packing factors. In this paper, the linewidth of laser \(\Delta\)f is defined as the full width at half maximum (FHWM). Figure 9a,b show the required OSNR and OSNR penalty versus the product of the linewidth and symbol period (\(\Delta\)f-Ts) for the modified-BPS algorithm in the FTN system with the different time-packing factors, respectively. We call the product of the linewidth and symbol period (\(\Delta\)f-Ts) the linewidth symbol rate. The FEC threshold is 3.0 \(\times\) 10\({}^{-2}\), corresponding to soft-decision 20% FEC. Figure 8: Under the different time-packing factors, the performance of OSNR versus BER is shown for the modified-BPS algorithm to equalize PN when the linewidth equals 300 kHz, 500 kHz, 800 kHz and 1000 kHz, respectively, and (**a**) the time-packing factor equals 0.45; (**b**) the time-packing factor equals 0.50; (**c**) the time-packing factor equals 0.55. Figure 9: The performance of required OSNR and OSNR penalty versus linewidth symbol rate (\(\Delta\)f-Ts) under the time-packing factor changes of 0.45–0.55: (**a**) required ONSR versus \(\Delta\)f-Ts; (**b**) OSNR penalty versus \(\Delta\)f-Ts. We can see in Figure 8a-c that the modified-BPS algorithm can effectively compensate for 300 kHz, 500 kHz, 800 kHz and 1000 kHz linewidths with the time-packing factor of 0.45-0.55. Figure 9a shows the required OSNRs for the modified-BPS algorithm with the \(\Delta\)f-Ts parameter values of 0, 1.07 \(\times\) 10\({}^{-5}\), 1.79 \(\times\) 10\({}^{-5}\), 2.86 \(\times\) 10\({}^{-5}\) and 3.57 \(\times\) 10\({}^{-5}\) when the time-packing factor takes values within 0.45-0.55.