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Nonlinear Fourier division Multiplexing (NFDM) can be realized frommodulating the discrete nonlinear spectrum of an $N$-solitary waveform. Togenerate an $N$-solitary waveform from desired discrete spectrum (eigenvalueand discrete spectral amplitudes), we use the Darboux Transform. We explain howto the norming factors must be set in order to have the desired discretespectrum. To derive these norming factors, we study the evolution of nonlinearspectrum by adding a new eigenvalue and its spectral amplitude. We furthersimplify the Darboux transform algorithm.  We propose a novel algorithm (to the best of our knowledge) to numericallycompute the nonlinear Fourier Transform (NFT) of a given pulse. The NFTalgorithm, called forward-backward method, is based on splitting the signalinto two parts and computing the nonlinear spectrum of each part from boundary($\pm\infty$) inward. The nonlinear spectrum (discrete and continuous) derivedfrom efficiently combining both parts has a promising numerical precision. Thismethod can use any of one-step discretization NFT methods, e.g. Crank-Nicolson,as an NFT kernel for the forward or backward part. Using trapezoid rule ofintegral, we use an NFT kernel (we called here Trapezoid discretization NFT) inforward-backward method which results discrete spectral amplitudes with a verygood numerical precision.  These algorithms, forward-backward method and Darboux transform, are used in[1],[2] for design and detection of phase-modulated 2-soliton pulses, and morerecently, in [3] for design and detection of more complex pulses with 7eigenvalues and modulation of spectral phase. For those soliton pulses, thediscrete spectral amplitudes (in particular, phase) of both eigenvalues arequite precisely estimated using the forward-backward method.

Control and Detection of Discrete Spectral Amplitudes in Nonlinear Fourier Spectrum