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) 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, 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. NOTATION Bold face Matrix or vector Underbar Random quantity Id Identity matrix of size dx d 0d Zero matrix of size dx d I Identity matrix of appropriate dimensions 0 Zero matrix of appropriate dimensions d[n] Discrete Dirac delta E{ Expectation Complex conjugate Transpose Conjugate transpose R*,I* Real and imaginary part j Imaginary number