Statistics of the Nonlinear Discrete Spectrum of a Noisy Pulse
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In the presence of additive Gaussian noise, the statistics of the nonlinear Fourier transform (NFT) of a pulse is still an open problem. 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 FC is more accurate than some other known NFT algorithms. Second, we apply perturbation theory of linear operators to derive analytic expressions for the statistics of both 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.
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