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

^{*}

## Abstract

**:**

## 1. Introduction

#### Notation

## 2. Review of NFDM

#### 2.1. Nonlinear Fourier Transform for Vanishing Signals

#### 2.2. NFDM Signal Generation

**Definition**

**1.**

**Definition**

**2.**

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

## 3. Upper Bounds on the Transmit Power of b-Modulators

#### 3.1. Power Bound for a Fixed Gap to Singularity

**Theorem**

**1.**

**Remark**

**4.**

**Remark**

**5.**

**Proof of Theorem**

**1.**

#### 3.2. Uniform Power Bound for Arbitrary Gaps to Singularity

**Theorem**

**2.**

**Remark**

**6.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2**

**.**We consider the b-modulator in Theorem 2. Let $0<W<\infty $. Then, there exists a finite constant ${\overline{E}}_{W}$ such that the energy of any generated $b\left(\xi \right)$ in $[-W,W]$ satisfies

**Proof of Lemma**

**2.**

**Proof of Theorem**

**2.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

NFT | Nonlinear Fourier Transform |

NFDM | Nonlinear Frequency Division Multiplexing |

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**Figure 1.**Nonlinear frequency domain multiplexing (NFDM) transmission of one block of ${N}_{b}$ bits.

**Figure 3.**In this example, $2W$ is exactly the 90% bandwidth: 90% ($\gamma =0.9$) of the energy corresponding to the left spectrum (E) are equal to the energy corresponding to right spectrum (${E}_{W}$). That is, $\gamma E={E}_{W}$.

**Figure 4.**The power bound from Theorem 1 for $W=6.0338$, $\gamma =0.9$ and $\epsilon \ge 0.1$. The transmit power of any b-modulator with these fundamental parameters must approach zero for long durations.

**Figure 5.**The left plot shows a $b\left(\xi \right)$ of the form Equation (20) for several values of the power control factor A, resulting in different gaps to singularity ${\epsilon}_{b}=1-{\parallel b\parallel}_{\infty}^{2}$. The right plot shows the corresponding integrand in Equation (50). The shaded areas thus represent the signal energy ${E}_{W}$ in the shown interval. Lemma 2 tells us that ${E}_{W}$ will stay below a finite bound no matter how small the gap to singularity becomes.

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Chimmalgi, S.; Wahls, S. Bounds on the Transmit Power of *b*-Modulated NFDM Systems in Anomalous Dispersion Fiber. *Entropy* **2020**, *22*, 639.
https://doi.org/10.3390/e22060639

**AMA Style**

Chimmalgi S, Wahls S. Bounds on the Transmit Power of *b*-Modulated NFDM Systems in Anomalous Dispersion Fiber. *Entropy*. 2020; 22(6):639.
https://doi.org/10.3390/e22060639

**Chicago/Turabian Style**

Chimmalgi, Shrinivas, and Sander Wahls. 2020. "Bounds on the Transmit Power of *b*-Modulated NFDM Systems in Anomalous Dispersion Fiber" *Entropy* 22, no. 6: 639.
https://doi.org/10.3390/e22060639