# PAM-\(N\)--Fundamental Limits in Chromatic Dispersive-Uncompensated Channels
## 3 Ideal Numerical Analysis  
To confront these differences, we first choose a re-shapeable smooth rectangular spectral filter, which is characterized by two parameters: the normalized spectral width \(\Delta/T\), which is the filter's amplitude (not intensity) full width at half maximum (FWHM), and the normalized transition width \(\delta/T\)[14; 15], i.e.,  
\[G(\omega)=\frac{1}{2}\bigg{[}\mathrm{erfc}\bigg{(}\frac{\omega-\pi\Delta}{ \delta}\bigg{)}-erfc\bigg{(}\frac{\omega+\pi\Delta}{\delta}\bigg{)}\bigg{]} \tag{11}\]  
Figure 2 presents the shape of a signal's spectrum after smooth rectangular filtering (Equation (11)) as a function of the normalized frequency\(f/\Delta=\omega/(2\pi\Delta)\). The lower panel shows an enlarged portion of the upper graph. The dashed line is the slope of the spectrum. As can be seen in this figure, \(\delta/2\pi\Delta\) is a measure of the spectral transition.  
In this case, \(g(\tau,0)\) in Equations (3) and (5) should be replaced with the (inverse) Fourier transform of Equation (11), as follows:  
\[g(\tau,0)=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}G(\omega)\exp(i\omega \tau)d\omega=\frac{\sin(\pi\tau\Delta)}{\pi\tau\Delta}\exp\bigg{[}-\bigg{(} \frac{\delta\tau}{2}\bigg{)}^{2}\bigg{]} \tag{12}\]  
Figure 2: (**a**) The spectrum of the signal after smooth rectangular filtering and (**b**) the enlarged portion.  
It should be stressed that because \(\tau\) is dimensionless and \(\omega\) is its Fourier counterpart, then \(\omega\) is also dimensionless, and the physical angular frequency is \(\omega/T=\omega B\).  
In addition, \(g(\tau,0)\) experiences the following distortion after a distance \(\zeta\):  
\[g(\tau,\zeta)=\frac{1}{2\Delta}\exp\left(-i\frac{t^{2}}{2\beta z 2}\right)\sqrt{\frac{i}{2\pi\zeta}}\times\left[\text{erf}\left(-\frac{t-|\pi \zeta\Delta|}{\sqrt{i2\zeta(1-i\zeta\delta^{2}/2)}}\right)-\text{erf}\left(- \frac{t+|\pi\zeta\Delta|}{\sqrt{i2\zeta(1-i\zeta\delta^{2}/2)}}\right)\right] \tag{13}\]  
Figure 3 presents results of a simulation where the maximum distance \(\zeta_{\text{max}}\), which is the distance beyond which the first eye (from the N-1 ones) is shut, is presented as a function of \(\delta\) for different values of \(\Delta\). It should be stressed that each one of the different eyes presents different results for \(\zeta_{\text{max}}(\Delta,\delta)\); however, Figure 3 presents the shortest one. That is, if \(\zeta^{\text{ (4)}}_{\text{max}}(\Delta,\delta)\) is the distance for which eye \(q\) is shut, then \(\zeta_{\text{max}}(\Delta,\delta)=\min\limits_{q}\left\{\zeta_{\text{max}}^{ \text{(4)}}(\Delta,\delta)\right\}\).  
As can be seen, the scenario where \(\Delta=1\) and \(\delta=0\) is not the best scenario (as might be expected from [16]), and in fact it is possible to reach longer distances by increasing both \(\Delta\) and \(\delta\) by a few percent.  
To derive a formula for \(\zeta_{\text{max}}\), we will focus, for simplicity, on the lower eye-opening (despite the fact that each of the eyes is shut at different distances and is affected differently by different spectral filters: these differences are relatively small, especially in relation to their \(\zeta_{\text{max}}\)).