to $\SI{1.45}{\dBm}$ for which the model [()]{} is applicable. While this approach is not a rigorous proof, the results indicate that memorylessness is a valid assumption. Although the transmission scheme is different in [@shevchenko2018tcom], the underlying condition that any two pulses need to be sufficiently separated is the same. Hence, we can treat the -based transmission system in this work as a memoryless channel.

In a practical system, we assume distributed Raman amplification and noise with received power spectral density $\sigma^2$ to compensate for the lossy fiber and be able to use the to relate the input and the output. The conditional of such a system has been derived via a perturbative approach and the Fokker-Planck equation method [@derevyanko2005] and is used to design a communication system in [@shevchenko2018tcom]. It is given by

$$\begin{aligned}
    p_{Y|X}(\psi|\lambda) &= \frac{2}{\sigma^2}\sqrt{\frac{{\mathfrak{I}\{\psi\}}}{{\mathfrak{I}\{\lambda\}}}}{\operatorname{e}^{-2\frac{{\mathfrak{I}\{\lambda\}}+{\mathfrak{I}\{\psi\}}}{\sigma^2}}}{\operatorname{I}_{1}{\mathopen{}\mathclose\bgroup\originalleft}(\frac{4\sqrt{{\mathfrak{I}\{\lambda\}}{\mathfrak{I}\{\psi\}}}}{\sigma^2}{\aftergroup\egroup\originalright})},
\label{eqn:pdf_yx_sac}\end{aligned}$$

where $\psi$ is the received symbol as in [Fig.]{}, and ${\operatorname{I}_{1}{\mathopen{}\mathclose\bgroup\originalleft}(\cdot{\aftergroup\egroup\originalright})}$ is the modified Bessel function of the first kind of order one. The power spectral density of the received noise $\sigma^2$ is normalized and relates to real world units as $\sigma^2 = \gamma\sqrt{L^3}\sigma_0^2 / {\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{2|\beta_2|}{\aftergroup\egroup\originalright})$. The is defined as $\mathrm{SNR}\triangleq 4{\mathbb{E}_{X}\!{\mathopen{}\mathclose\bgroup\originalleft}\{{{\mathfrak{I}\{\lambda\}}}{\aftergroup\egroup\originalright}\}
}/\sigma^2$. It is important to note that the model [()]{} 