Nonlinear Fourier Spectrum Characterization of Time-Limited Signals

Abstract

Addressing the optical communication systems employing the nonlinear Fourier transform (NFT) for the data modulation/demodulation, we provide an explicit proof for the properties of the signals emerging in the so-called b-modulation method, the nonlinear signal modulation technique that provides explicit control over the signal extent. We present details of the procedure and related rigorous mathematical proofs addressing the case where the time-domain profile corresponding to the b-modulated data has a limited duration, and when the bound states corresponding to specifically chosen discrete solitonic eigenvalues and norming constants, are also present. We also prove that the number of solitary modes that we can embed without violating the exact localisation of the time-domain profile, is actually infinite. Our theoretical findings are illustrated with numerical examples, where simple example waveforms are used for the b-coefficient, demonstrating the validity of the developed approach. We also demonstrate the influence of the bound states on the noise tolerance of the b-modulated system.

Shortening Solitons for Fiber-Optic Transmission

“…These conditions are also sufficient [18]. To give a precise statement, we require the Paley-Wiener space PW 1 σ .…”

Neural networks for computing the continuous nonlinear Fourier spectrum in focusing nonlinear Schrodinger equation: NFT-Net

“…In practical applications, we do not typically deal with the signals defined on the whole infinite t-axis, but rather operate with the truncated wave-forms, meaning that q(t) is non-zero only inside the finite interval of t. In this case, the NF spectrum of the signal is completely characterised by the coefficient b(ξ ) from ( 7), which becomes band-limited, appended with the finite discrete set of solitonic parameters {ξ n , c n } 25,26 . When, in addition, the discrete NF spectrum is absent, as it is in the case considered, the whole NF spectrum can be defined using just b(ξ ) profile 24 , while the coefficient a(ξ ) can be expressed through b(ξ ) in the following way:…”

“…If needed, we then can use both computed quantities to find the reflection coefficient (10). We note that within the b-modulation concept, which has turned out to be the most efficacious NFDM method developed so far, we utilise the b(ξ ) functions as information carriers [24][25][26][27] .…”

“…In our current study we specifically address the continuous NF spectrum: our goal is to compute the profile r(ξ ) given the localised q(t) shape. Then, we mention that the continuous NF spectrum modulation using the special technique coined b-modulation [24][25][26][27] has provided the highest NFDM data rates so far 12,28 . Thus, in this paper we also address the recovery of the b-coefficient, b(ξ ) ∈ C, ξ ∈ R, given q(t).…”

Bounds on the Transmit Power of b-Modulated NFDM Systems in Anomalous Dispersion Fiber

“…Theorem 2 is formulated such that it is applicable to the carriers typically used in NFDM systems. One of the commonly used carriers is the sinc function [ 10 , 11 , 12 , 32 ] The function is real-analytic [ 33 ], square-integrable, and decays to zero as . To apply Theorem 2, we need to show that .…”