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We study two discretisations of the nonlinear Fourier transform of AKNS-ZStype, ${\cal F}^E$ and ${\cal F}^D$. Transformation ${\cal F}^D$ is suitablefor studying the distributions of the form $u = \sum_{n = 1}^N u_n \,\delta_{x_n}$, where $\delta _{x_n}$ are delta functions. The poles $x_n$ arenot equidistant. The central result of the paper is the construction ofrecursive algorithms for inverses of these two transformations. The algorithmfor $({\cal F}^D)^{- 1}$ is numerically more demanding than that for $({\calF}^E)^{- 1}$. We describe an important symmetry property of ${\cal F}^D$. Itenables the reduction of the nonlinear Fourier analysis of the constant massdistributions $u = \sum_{n = 1}^N u_c \, \delta _{x_n}$ for the numericallymore efficient ${\cal F}^E$ and its inverse.

Discrete nonlinear Fourier transforms and their inverses