Second Order Statistics of the Scattering Vector Defining the D-T Nonlinear Fourier Transform

The impact of time-domain noise on the nonlinear Fourier transform is currently not well understood. Most, if not all, available results are based on perturbation theory and become exact only in the low-noise regime. In this paper, it is pointed out that the mean and the (conventional and complementary) covariance of the scattering vector [a(z) b(z)](exp T) that is used to define the discrete-time nonlinear Fourier transform can be computed exactly if a known deterministic signal is contaminated with circular symmetric white noise. Since the scattering vector is a polynomial in z(exp -1), also the second-order statistics of its coefficient vector are derived. This result is finally used to determine the second-order statistics of an arbitrary multipoint scattering vector, in which the values of the scattering vector for several arguments are stacked. The results are illustrated in a numerical example, and potential extensions are discussed.