# PAM-\(N\)--Fundamental Limits in Chromatic Dispersive-Uncompensated Channels
## 4 Analytical Derivation of Maximum Distance  
As was shown in [16], the alternating sequences are the most dispersion sensitive and most destructive to signal encoding. Therefore, the lower eye is mainly affected by two alternating sequences (see Figure 4). The sequence that is responsible for shutting the eye from below is \(x_{n}=\ldots 0,1,0,1,0,1,0,1\ldots\), i.e.,  
\[A_{01}(\tau)=A_{0}\sum_{n=-\infty}^{\infty}rect_{1}(\tau-2n) \tag{14}\]  
\[\text{where }rect_{1}(x)\equiv \begin{cases}1&|x|\leq 1/2\\ 0&|x|>1/2\end{cases} .\]  
Figure 3: Distance at which first eye-opening vanishes, with \(\zeta_{\text{max}}\) as function of low pass filter (LPF) parameters \(\Delta\) and \(\delta\). Because Equation (14) can be written as a Fourier series:  
\[A_{01}(\tau)=A_{0}\Bigg{\{}\frac{1}{2}-\sum_{n=1}^{\infty}\frac{\sin(\tau n/2)}{( \tau n/2)}\cos(\tau n\tau)\Bigg{\}} \tag{15}\]  
after applying the spectral filter \(G(\omega)\), the filtered sequence can be written as follows:  
\[\frac{A_{Below}(\zeta=0,\tau)}{A_{0}}=\frac{1}{2}-\sum_{n=1}^{\infty}\frac{ \sin(\tau n/2)}{(\tau n/2)}G(\tau n)\cos(\tau n\tau) \tag{16}\]  
Similarly, the sequence that shuts the eye from above is \(x_{n}=\ldots 0,a,0,a,0,a,0,a\ldots\), where \(a\equiv(N-1)^{-1/2}\), i.e.,  
\[\frac{A_{Above}(\zeta=0,\tau)}{A_{0}}=\sqrt{\frac{1}{Q-1}}\Bigg{[}\frac{1}{2} +\sum_{n=1}^{\infty}\frac{\sin(\tau n/2)}{(\tau n/2)}G(\tau n)\cos(\tau n\tau) \Bigg{]} \tag{17}\]  
The amplitude is presented in the upper panel of Figure 4, and the intensity is presented in the lower one. The four levels are represented by horizontal dotted lines. The small arrows indicate the closing direction. After a distance \(\zeta\), these signals experience the following distortion:  
\[\frac{A_{Below}(\tau,\zeta)}{A_{0}}=\frac{1}{2}-\sum_{n=1}^{\infty}\frac{\sin( \tau n/2)}{(\tau n/2)}G(\tau n)\exp\biggl{(}\frac{in^{2}\pi^{2}\zeta}{2} \biggr{)}\cos(\tau n\tau) \tag{18}\]  
and  
\[\frac{A_{Above}(\tau,\zeta)}{A_{0}}=\sqrt{\frac{1}{N-1}}\Bigg{[}\frac{1}{2}+ \sum_{n=1}^{\infty}\frac{\sin(\tau n/2)}{(\tau n/2)}G(\tau n)\exp\biggl{(} \frac{in^{2}\pi^{2}\zeta}{2}\biggr{)}\cos(\tau n\tau)\Bigg{]} \tag{19}\]  
Figure 4: Two most detrimental sequences, which are responsible for closing the lower eye in 4-level pulse amplitude modulation (PAM-4). However, because \(\delta<<1\) in these cases, the higher harmonics (\(n>1\)) have negligible contributions (because \(G(\pi n)<<G(\pi)\) for \(n>1\)), and the eye-opening, which is measured at the center of the symbol, is approximately:  
\[\frac{\Delta I(\zeta)}{I_{0}}\equiv\left|\frac{A_{Above}(\tau,\zeta)}{A_{0}} \right|^{2}-\left|\frac{A_{Below}(\tau,\zeta)}{A_{0}}\right|^{2}=\frac{1}{N-1} \left|\frac{1}{2}+\frac{2}{\pi}G(\pi)\exp\left(\frac{i\pi^{2}\zeta}{2}\right) \right|^{2}-\left|\frac{1}{2}-\frac{2}{\pi}G(\pi)\exp\left(\frac{i\pi^{2}\zeta }{2}\right)\right|^{2} \tag{20}\]  
Therefore, the distance for which the eye-opening is closed, \(\zeta_{\max}\) (i.e., \(\Delta I(\zeta_{\max})=0\)) is:  
\[\zeta_{\max}=\frac{2}{\pi^{2}}\mathrm{acos}\left\{\frac{\left|1+\left(\frac{4 }{\pi}\right)^{2}G^{2}(\pi)\right|\left(\frac{N-2}{N}\right)}{\frac{8}{\pi}G( \pi)}\right\} \tag{21}\]  
In Figure 5, this function is plotted as a function of the filter's amplitude at the first harmonics, i.e., as a function of \(G(\pi)\), for various \(N\).