### Abstract

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.

Original language | English |
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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 |

Publisher | VDE Verlag |

Number of pages | 6 |

ISBN (Print) | 978-3-8007-4362-9 |

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

Conference | SCC 2017 11th International ITG Conference on Systems, Communications and Coding |
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Abbreviated title | SCC 2017 |

Country | Germany |

City | Hamburg |

Period | 6/02/17 → 9/02/17 |

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## Cite this

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.