  In the presence of additive Gaussian noise, the statistics of the nonlinear
Fourier transform (NFT) of a pulse are not yet completely known in closed form.
In this paper, we propose a novel approach to study this problem. Our
contributions are twofold: first, we extend the existing Fourier Collocation
(FC) method to compute the whole discrete spectrum (eigenvalues and spectral
amplitudes). We show numerically that the accuracy of FC is comparable to the
state-of-the-art NFT algorithms. Second, we apply perturbation theory of linear
operators to derive analytic expressions for the joint statistics of the
eigenvalues and the spectral amplitudes when a pulse is contaminated by
additive Gaussian noise. Our analytic expressions closely match the empirical
statistics obtained through simulations.
