# PAM-\(N\)--Fundamental Limits in Chromatic Dispersive-Uncompensated Channels
## 2 General Theory  
In data center networks nonlinear effects can be neglected and therefore chromatic dispersion effects are the main cause of signal distortion. The amplitude of electromagnetic field \(A(z,t)\) obeys the dispersion equation (where nonlinear effects and higher-order dispersion effects are neglected) [12; 13]:  
\[\frac{\partial A}{\partial z}=i\frac{\beta_{2}}{2}\frac{\partial^{2}A}{ \partial t^{2}} \tag{1}\]  
where \(z\) is the fiber's length, \(\beta_{2}\) is the fiber's chromatic dispersion coefficient, and \(t=t^{\prime}-z/v\) is the difference between real time \(t^{\prime}\) and the time it takes for light to travel through the fiber \(z/v\). It is convenient to use dimensionless parameters: normalized distance \(\zeta\equiv B^{2}z\beta_{2}\) and normalized time \(\tau\equiv tB\), where \(B\) is the Baud Rate (BR). Using these parameters, the normalized dispersion equation can be written as follows:  
\[\frac{\partial A}{\partial\zeta}=i\frac{1}{2}\frac{\partial^{2}A}{\partial \tau^{2}} \tag{2}\]  
At the entrance to the fiber, i.e., at \(\zeta=0\), the data sequence can be written as follows:  
\[A(\tau,\zeta=0)=A_{0}\sum_{n}x_{n}g(\tau-n,0) \tag{3}\]  
where \(T=B^{-1}\) is the allocation time of a single pulse, which is equal to the reciprocal of the BR, and \(g(\tau,0)\) is the (amplitude) shape of a single pulse in the sequence. It has been shown that in practical cases, \(g(\tau,0)\) can be approximated with high accuracy using smooth rectangular pulses [14; 15]. The coefficients \(x_{n}\) carry the sequence's data. In the case of PAM-\(N\), their values are distributed pseudo-randomly. Thus, the probability, \(P\), that the \(n\)th pulse amplitude is equal to the \(q\)th symbol is:  
\[P\bigg{(}x_{n}=\sqrt{\frac{q}{N-1}}\bigg{)}=\frac{1}{N},\text{ for any }q=\ 0,1,2,\ldots N\ -\ 1. \tag{4}\]  
In particular, a PAM-4 sequence consists of the values \(x_{n}=\sqrt{0/3}\), \(\sqrt{1/3}\), \(\sqrt{2/3}\), and \(\sqrt{3/3}\) with equal frequency (0.25). At the end of the fiber (i.e., at \(z=L\)), the signal's amplitude is then,  
\[A(\tau,\zeta)=A_{0}\exp(-\alpha L)\sum_{n}x_{n}g(\tau-n,\zeta) \tag{5}\]  
where \(\alpha\) is the absorption coefficient, and [15; 16],  
\[g(\tau\zeta)=\frac{1}{\sqrt{-2\pi i\zeta}}\int\limits_{-\infty}^{\infty}g( \tau,0)\exp\bigg{(}-i\frac{(\tau-\tau t)^{2}}{\zeta}\bigg{)}d\tau \tag{6}\]  
is the shape of a single pulse at the end of the fiber. The \(m\)th symbol's intensity, which is measured at the center of the \(m\)th signals, which originally was:  
\[I(m,\zeta=0)=|A(\zeta_{L},m)|^{2}=|A_{0}|^{2}x_{m}^{2} \tag{7}\]\[I(m,\zeta)=\left|A_{0}\right|^{2}\exp(-2\kappa L)\left|\sum_{n}x_{n}g(m-n,\zeta) \right|^{2} \tag{8}\]  
In PAM-\(N\), there are \(N\) intensity levels. Therefore, there are \(N\)-1 different eyes, each of which has its own eye-opening. Each one of these levels introduces a thickening due to dispersion. In a noiseless channel, the eye-opening is the difference between the minimum of the upper levels and the maximum of the lower one. In other words, the eye-opening of the \(n\)th eye (i.e., the eye which is generated between the \(n\)th and the \((n+1)\)th levels), is defined as follows [16]:  
\[\Delta I_{n}(\zeta)=\min_{x_{m}=\sqrt{n+1/N-1}}\{I(\zeta,m)\}-\max_{x_{m}= \sqrt{n/N-1}}\{I(\zeta,m)\} \tag{9}\]  
Clearly, when \(\Delta I_{n}(\zeta_{\max})=0\) for any \(n\), the channel reaches its maximum capacity or maximum distance for a noiseless channel. Beyond this point (\(\zeta_{\max}\)), it is impossible to decode (error-free) the data without dispersion compensation modules that are passive (such as filters) or active (such as maximum likelihood sequence estimation (MLSE)). Clearly, the introduction of such a compensating module can increase the maximum distance substantially, but it is beyond the scope of this work. Moreover, since beyond this point bit error rate (BER) \(\sim\)0.5, the data is totally random and no forward error correction (FEC) can be applied.