## 2 Proposed PS-NPS-8-PAM-Based DWDM System  
Figure 1 shows the proposed LDPC-coded PS-NPS-8-PAM signal generator. The input was a pseudorandom binary sequence (PRBS), and after the distribution matcher (DM), there was an array of different amplitudes that satisfy a certain probability distribution. After binary labelling and LDPC encoding [36], we obtained a block of LDPC-coded binary bits. After mapping to the PAM constellation points, we performed the Nyquist pulse shaping, followed by the DWDM multiplexing.  
Figure 1: Low-density parity-check (LDPC)-coded, probabilistic shaping, Nyquist pulse shaping, 8-level pulse-amplitude modulation (PS-NPS-8-PAM) signal generator. DM: Distribution matcher.  
The PS distribution performances are highly dependent on a selected distribution function. In this paper, we used an exponential distribution [33], in which different constellation points \(a_{i}\) were transmitted with probabilities determined using:  
\[\mathrm{P}(a_{\mathrm{i}})=\exp(-\lambda\|a_{\mathrm{i}}\|)/Z(\lambda),\ \lambda\geq 0, \tag{1}\]  
where \(Z(\lambda)\) is the normalization function used to ensure that the probabilities of occurrence of symbols sum up to one, which means \(Z(\lambda)\) is defined as:  
\[Z(\lambda)=\sum_{\mathrm{i}}\exp(-\lambda\|a_{\mathrm{i}}\|). \tag{2}\]  
Nyquist pulse shaping is widely used in DWDM systems to improve the spectral efficiency [40]. It often uses a low-pass raised-cosine (RC) filter with the frequency response being:  
\[\mathrm{H}_{\mathrm{RC}}(\omega)=\left\{\begin{array}{cc}\mathrm{T}_{\mathrm{ S}}&0\leq|\omega|<\pi(1-\mathrm{ROF})/\mathrm{T}_{\mathrm{S}}\\ \frac{\mathrm{T}_{\mathrm{S}}}{2}\Big{(}1-\sin\Big{[}\frac{\mathrm{T}_{ \mathrm{S}}}{2\times\mathrm{ROF}}\Big{(}|\omega|-\frac{\pi}{\mathrm{T}_{ \mathrm{S}}}\Big{)}\Big{]}\Big{)},&\pi(1-\mathrm{ROF})/\mathrm{T}_{\mathrm{S} }\leq|\omega|<\pi(1+\mathrm{ROF})/\mathrm{T}_{\mathrm{S}}\\ 0&|\omega|>\pi(1+\mathrm{ROF})/\mathrm{T}_{\mathrm{S}}\end{array}\right., \tag{3}\]  
where \(\omega\) is the angular frequency, \(T_{\mathrm{s}}\) is the symbol duration, and ROF is the roll-off factor mentioned before. A small ROF value can make the frequency response an almost rectangular shape and be able to reduce the channel spacing but comes with a longer memory length and a higher generation complexity. In this study, we used the square-root raised cosine (SRRC) filter, whose transfer function is given as:  
\[\mathrm{H}_{\mathrm{SRRC}}(\omega)=\sqrt{\mathrm{H}_{\mathrm{RC}}(\omega)}. \tag{4}\]  
Figure 2 shows the frequency response for two different ROF values of neighboring DWDM channels, with the channel spacing set to 50 GHz. It is clear that a smaller ROF had a better spectral efficiency with a lower inter-channel crosstalk.  
Figure 3 shows the whole DWDM system. On the transmitter side, we generated the PS-NPS-8-PAM signals using the generator in Figure 1 and sent them to the modulator for each channel with a different frequency. Then, after transmission, at the receiver, we used a super Gaussian filter to select every target frequency (wavelength).  
Figure 2: Frequency response with different roll-off factors (ROFs): (**a**) \(\mathrm{ROF}_{1}=\mathrm{ROF}_{2}=0.1\); (**b**) \(\mathrm{ROF}_{1}=1\), \(\mathrm{ROF}_{2}=0.1\); and (**c**) \(\mathrm{ROF}_{1}=\mathrm{ROF}_{2}=1\).