#
Back-to-Back Performance of the Full Spectrum Nonlinear Fourier Transform and Its Inverse^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Nonlinear Fourier Transform

## 3. Algorithms

#### 3.1. The Full Spectrum Inverse Nonlinear Fourier Transform

#### 3.2. The Search-Based Nonlinear Fourier Transform

#### 3.3. The Eigenvalue Removal Nonlinear Fourier Transform

#### 3.4. Complexity of the NFT Algorithms

## 4. Simulation Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AIR | Achievable Information Rates | NFT | Nonlinear Fourier Transform |

AWGN | Additive White Gaussian Noise | NLSE | Nonlinear Schrödinger Equation |

DBP | Digital Back-Propagation | NMSE | Normalized Mean Square Error |

DRA | Distributed Raman Amplification | PCTW | Phase Conjugated Twin Waves |

DT | Darboux Transform | PMD | Polarization Mode Dispersion |

EDFA | Erbium-Doped Fiber Amplifier | PSK | Phase Shift Keying |

ER | Eigenvalue Removal | RRC | Root Raised Cosine |

FT | Fourier Transform | SDM | Space Division Multiplexing |

INFT | Inverse Nonlinear Fourier Transform | SE | Spectral Efficiency |

MI | Mutual Information | WDM | Wavelength-Division Multiplexing |

MSSI | Mid-Span Spectral Inversion | ZS | Zakharov-Shabat (System) |

NFDM | Nonlinear Frequency Division Multiplexing |

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**Figure 1.**Block diagram of joint spectrum modulation (based on [60]).

**Figure 5.**Altered pulses in the eigenvalue removal NFT. (

**a**) Initial pulse q(t) with continuous and discrete spectrum (

**b**) After removal of λ

_{1}= j0.25 (

**c**) After also removing λ

_{2}= j (only continuous spectrum remains).

**Table 1.**Minimum and maximum NMSEs of three spectral parameters for two tested NFTs in two scenarios ($K=2,\phantom{\rule{4pt}{0ex}}K=3$).

Continuous Spectrum | Discrete Eigenvalues | b-Values | ||||
---|---|---|---|---|---|---|

min | max | min | max | min | max | |

${\mathrm{NMSE}}_{\mathrm{s}}$$(K=2)$ | $5.3\times {10}^{-5}$ | $1\times {10}^{-2}$ | $5\times {10}^{-7}$ | $4\times {10}^{-4}$ | $6.2\times {10}^{-7}$ | $7.9\times {10}^{-5}$ |

${\mathrm{NMSE}}_{\mathrm{r}}$$(K=2)$ | $5.3\times {10}^{-5}$ | $1\times {10}^{-2}$ | $3.8\times {10}^{-7}$ | $2\times {10}^{-4}$ | $5.2\times {10}^{-7}$ | $7.6\times {10}^{-5}$ |

${\mathrm{NMSE}}_{\mathrm{s}}$$(K=3)$ | $5.3\times {10}^{-5}$ | $1\times {10}^{-2}$ | $3.5\times {10}^{-6}$ | $2.9\times {10}^{-3}$ | $5.2\times {10}^{-6}$ | $1\times {10}^{-3}$ |

${\mathrm{NMSE}}_{\mathrm{r}}$$(K=3)$ | $5.3\times {10}^{-5}$ | $1\times {10}^{-2}$ | $3\times {10}^{-6}$ | $2.7\times {10}^{-3}$ | $4.2\times {10}^{-6}$ | $1\times {10}^{-3}$ |

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**MDPI and ACS Style**

Leible, B.; Plabst, D.; Hanik, N.
Back-to-Back Performance of the Full Spectrum Nonlinear Fourier Transform and Its Inverse. *Entropy* **2020**, *22*, 1131.
https://doi.org/10.3390/e22101131

**AMA Style**

Leible B, Plabst D, Hanik N.
Back-to-Back Performance of the Full Spectrum Nonlinear Fourier Transform and Its Inverse. *Entropy*. 2020; 22(10):1131.
https://doi.org/10.3390/e22101131

**Chicago/Turabian Style**

Leible, Benedikt, Daniel Plabst, and Norbert Hanik.
2020. "Back-to-Back Performance of the Full Spectrum Nonlinear Fourier Transform and Its Inverse" *Entropy* 22, no. 10: 1131.
https://doi.org/10.3390/e22101131