Control and Detection of Discrete Spectral Amplitudes in Nonlinear Fourier Spectrum

Abstract

Nonlinear Fourier division Multiplexing (NFDM) can be realized from modulating the discrete nonlinear spectrum of an $N$-solitary waveform. To generate an $N$-solitary waveform from desired discrete spectrum (eigenvalue and discrete spectral amplitudes), we use the Darboux Transform. We explain how to the norming factors must be set in order to have the desired discrete spectrum. To derive these norming factors, we study the evolution of nonlinear spectrum by adding a new eigenvalue and its spectral amplitude. We further simplify the Darboux transform algorithm. We propose a novel algorithm (to the best of our knowledge) to numerically compute the nonlinear Fourier Transform (NFT) of a given pulse. The NFT algorithm, called forward-backward method, is based on splitting the signal into two parts and computing the nonlinear spectrum of each part from boundary ($\pm \infty$) inward. The nonlinear spectrum (discrete and continuous) derived from efficiently combining both parts has a promising numerical precision. This method 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 of integral, we use an NFT kernel (we called here Trapezoid discretization NFT) in forward-backward method which results discrete spectral amplitudes with a very good 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 more recently, in [3] for design and detection of more complex pulses with 7 eigenvalues and modulation of spectral phase. For those soliton pulses, the discrete spectral amplitudes (in particular, phase) of both eigenvalues are quite precisely estimated using the forward-backward method.

14 pages

References in corpus (1)

• Transmission of Waveforms Determined by 7 Eigenvalues with PSK-Modulated Spectral Amplitudes

Cited by in corpus (10)

• Fast Inverse Nonlinear Fourier Transformation using Exponential One-Step Methods, Part I: Darboux Transformation
• Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schrödinger equation
• Introducing phase jump tracking -- a fast method for eigenvalue evaluation of the direct Zakharov-Shabat problem
• Time-Bandwidth Product Perspective for Multi-Soliton Phase Modulation
• Bound-state soliton gas as a limit of adiabatically growing integrable turbulence
• Silicon Photonics DWDM NLFT Soliton Transmitter
• Nonlinear Fourier Transform of Truncated Multi-Soliton Pulses
• Successive Eigenvalue Removal for Multi-Soliton Spectral Amplitude Estimation
• An Efficient Nonlinear Fourier Transform Algorithm for Detection of Eigenvalues from Continuous Spectrum
• Processing of optical signals by "surgical" methods for the Gelfand-Levitan-Marchenko equation