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.
|Title of host publication||Proceedings of the 11th International ITG Conference on Systems, Communications and Coding (SCC 2017)|
|Editors||G. Bauch, A. Klein|
|Place of Publication||Berlin-Offenbach, Germany|
|Number of pages||6|
|Publication status||Published - 2017|
|Event||SCC 2017 11th International ITG Conference on Systems, Communications and Coding - Hamburg, Germany|
Duration: 6 Feb 2017 → 9 Feb 2017
|Conference||SCC 2017 11th International ITG Conference on Systems, Communications and Coding|
|Abbreviated title||SCC 2017|
|Period||6/02/17 → 9/02/17|
Wahls, S. (2017). Second order statistics of the scattering vector defining the d-t nonlinear Fourier transform. In G. Bauch, & A. Klein (Eds.), Proceedings of the 11th International ITG Conference on Systems, Communications and Coding (SCC 2017) VDE Verlag.