## 2 System Model
### Time Jitter Effect  
The TJ effect is a stochastic process, which can be described from a statistical distribution of the random value (RV) \(T\), with mean value \(\mu_{T}\) and variance \(\sigma_{T}^{2}\), where the latter parameter corresponds to the TJ effect strength and the former corresponds to a systematic time shift either leading or lagging. As declared in the introduction, in this work, the TJ is modeled by the truncated normal distribution [37], which is expressed by the following PDF:  
\[f_{T}=\frac{1}{\sqrt{2\pi}\sigma_{T}\big{(}\Phi_{T_{2}}-\Phi_{T_{1}}\big{)}}\exp \left[-\frac{(T-\mu_{T})^{2}}{2\sigma_{T}^{2}}\right],\ T_{1}\leq T\leq T_{2}, \tag{4}\]  
where \(\Phi_{T_{i}}=0.5\Big{\{}1+erf\Big{[}(T_{i}-\mu_{T})/\Big{(}\sqrt{2}\sigma_{T} \Big{)}\Big{]}\Big{\}}\) for \(i=1\), 2 indicates the truncated edges of the current timeslot. Assuming that TJ is symmetrically distributed around the timeslot center, without loss of generality, hereinafter the mean value of the TJ's PDF is considered as zero (\(\mu_{T}=0\)). Figure 3a depicts the corresponding PDF curve as a function of \(T\), for various TJ strength parameters. Figure 3b is a replica of Figure 1 for \(z=0.75z_{c2}\), duplicated for two random jitters, \(\Delta T\)= \(-8\) and \(+5\) ps. In both cases, the pulse amplitude received at the timeslot center (\(T=0\)) is much lower than the maximum one at the pulse peak. In the latter case, the red curve is superior to the black one, whereas, in the former case, the reverse situation is valid, implying that the down-chirped pulses outmatch the up-chirped pulses only when the instantaneous TJ is relatively small, depending on the distance propagated.  
The value of \(T\) is extracted from Equation (2), as a function of \(I_{j}\), denoting the jitter influence on received irradiance.  
\[T=\pm T_{pw}\sqrt{\ln\big{(}B/I_{j}\big{)}}. \tag{5}\]  
Considering the timeslot duration as \(T_{sl}\), its edges lie at instants \(T=\pm T_{sl}/2\); hence, the irradiance value is bound between \(I_{j,\min}=B\exp\left[-\big{(}T_{sl}/T_{pw}\big{)}^{2}/4\right]\) and \(B\), which is the maximum irradiance at \(T=0\). Supposing the simplistic OOK modulation scheme, the bit  
Figure 2: (**a**) The amplitude of the propagating pulse along \(z\)-axis, supposing the original amplitude equal to unity; (**b**) the corresponding pulse width evolution along z-axis.  
rate, \(R\), is related to the timeslot, as \(T_{sl,OOK}=R^{-1}\). Next, in order to estimate the joint PDF, an RV transformation has to be performed as follows:  
\[f_{I_{j}}\big{(}I_{j}\big{)}=\frac{f_{T}\big{[}\sqrt{\ln\big{(}B\big{/}I_{j} \big{)}}\big{]}}{\left|dI_{j}/dT\right|_{T=+T_{pw}\sqrt{\ln\big{(}B\big{/}I_{j} \big{)}}}}+\frac{f_{T}\big{[}-\sqrt{\ln\big{(}B\big{/}I_{j}\big{)}}\big{]}}{ \left|dI_{j}/dT\right|_{T=-T_{pw}\sqrt{\ln\big{(}B\big{/}I_{j}\big{)}}}}. \tag{6}\]  
Lastly, the PDF as a function of \(I_{j}\), caused by the GVD and the TJ effect, is derived as follows:  
\[f_{I_{j}}\big{(}I_{j}\big{)}=\frac{T_{pw}}{\sigma_{T}Z\sqrt{2\pi\ln\big{(}B \big{/}I_{j}\big{)}}\,I_{j}}\exp\left[-\frac{T_{pw}^{2}\ln\big{(}B\big{/}I_{j }\big{)}}{2\sigma_{T}^{2}}\right], \tag{7}\]  
where \(Z=\Phi_{T_{2}}-\Phi_{T_{1}}\).