signal from the time domain into a nonlinear frequency domain (spectrum), where the channel acts as a multiplicative filter on the signal. The nonlinear spectrum consists of a continuous and a discrete part. Both parts can be used to transmit information, either separately or jointly, and several schemes have been presented in theory and practice [@yousefi2014nftI-III; @dong2015; @aref2015ecoc; @aref2016ecoc; @geisler2016ecoc; @hari2016]. However, very little is known so far about the of the received signal in the nonlinear spectral domain when it is contaminated by channel noise.

In [@shevchenko2018tcom], a simplified communication system modulating only the imaginary part of the eigenvalues in the discrete nonlinear spectrum was presented. For this scheme, an approximation for the conditional of the channel can be obtained in closed form. In general, for a given channel, the capacity-achieving distribution is not known and is often different from the conventional distribution with equispaced signal points and uniform signaling. Hence, some form of shaping is required [@forney1984]. Two popular methods of shaping are probabilistic shaping and geometric shaping. In geometric shaping, the capacity-achieving distribution is mimicked by optimizing the position of the constellation points for equiprobable signaling [@sun1993_geometric_shaping] whereas probabilistic shaping uses uniformly spaced constellation points and approximates 