vectors of are capitalized bold, e.g., $\bm{X}$. The of an $X$ is written as $p_X(x)$ and its expectation as ${\mathbb{E}_{X}\!{\mathopen{}\mathclose\bgroup\originalleft}\{{x}{\aftergroup\egroup\originalright}\}
}$. The conditional of $Y$ given $X$ is denoted as $p_{Y|X}(y|x)$. The of an $X$ is denoted by $P_X(x)$. The transpose of a vector or matrix is given as $(\cdot )^\mathsf{T}$. A set is denoted by a capitalized Greek letter, e.g., $\Lambda$, and its cardinality by $|\Lambda|$. We write $\log_a(\cdot)$ for the logarithm of base $a$ and $\ln(\cdot)$ for the natural logarithm.

Nonlinear Fourier Transform-based Transmission System {#sec:system}
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Pulse Propagation and the Nonlinear Fourier Transform {#subsec:NFT}
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Pulse propagation in optical fibers is governed by a partial differential equation, the stochastic , $$\begin{gathered}
        \jmath\frac{\partial u(\tau,\ell)}{\partial \ell} + \jmath\frac{\alpha}{2}u(\tau,\ell) - \frac{\beta_2}{2}\frac{\partial^2 u(\tau,\ell)}{\partial \tau^2} + \gamma u(\tau,\ell) |u(\tau,\ell)|^2  \\= n(\tau,\ell ),\nonumber$$ where $u(\tau,\ell)$ denotes the envelope of the electrical field as a function of the position $\ell$ along the fiber and time $\tau$, $\alpha$ the attenuation, $\beta_2$ the second order dispersion, $\gamma$ the nonlinearity parameter, and $n(\tau, \ell)$ is a white Gaussian process in time and in space with spectral density $\sigma_0^2$. The spectral density depends on the system and for distributed Raman amplification is given as $\sigma_0^2 = \alpha K_T 