# Nonlinear Frequency-Modulated Waveforms Modeling and Optimization for Radar Applications

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## Abstract

**:**

## 1. Introduction

## 2. Modeling and Optimization of NLFM Waveform

#### 2.1. Modeling of NLFM Waveforms

- Step 1:
- Input the coefficient vector of the Legendre polynomial $\mathbf{a}=[{a}_{1},{a}_{2},{a}_{3},{a}_{4},{a}_{5}]$;
- Step 2:
- Define a frequency function of time based on the coefficient vector of the Legendre polynomial using Equation (1);
- Step 3:
- Integrate the frequency function to obtain the phase function using Equation (2);
- Step 4:
- Output the NLFM signal waveform using Equation (3);
- Step 5:
- Assess the signal performance.

**a**are [1 1 1 1 1], [1 1 2 1 −1], [1 1 3 1 1], and [0.2 1 3 2 1]. As can be seen from the numerical results, small differences in the coefficients produce widely varying time-frequency functions as well as different performance IRFs.

#### 2.2. Optimization of NLFM Waveform

- Step 1:
- Initialize each firefly $\mathbf{a}$ as $a\left(i\right)=1+\mathrm{rand},1\le i\le 5.$
- Step 2:
- Step 3:
- Rank the fireflies by light intensity and find the current global best solution (corresponding to the brightest one).
- Step 4:
- Move all fireflies toward the brighter ones by Equation (8).
- Step 5:
- Repeat Steps 2–5 until the iteration times reach the threshold of the objective function.

## 3. Numerical Results

#### 3.1. Numerical Results of the NLFM Signal

#### 3.2. Application of the Optimized NLFM Signal for Radar-Target Range and Velocity Measurements

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Modeling the NLFM signal with four coefficient vectors of LPs. The plots in the left column show the time-frequency relationship of the signal, and the right column shows the corresponding IRFs.

**Figure 5.**Comparisons of the optimized NLFM signal and the LFM signal: (

**a**) time-frequency function, (

**b**) waveforms, (

**c**) IRF, and (

**d**) zoom in the IRF.

SNR (dB) | LFM | WLFM | NLFM |
---|---|---|---|

0 | 7.302 | 1.441 | 0.665 |

15 | 7.254 | 1.378 | 0.641 |

30 | 7.249 | 1.365 | 0.630 |

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**MDPI and ACS Style**

Xu, Z.; Wang, X.; Wang, Y.
Nonlinear Frequency-Modulated Waveforms Modeling and Optimization for Radar Applications. *Mathematics* **2022**, *10*, 3939.
https://doi.org/10.3390/math10213939

**AMA Style**

Xu Z, Wang X, Wang Y.
Nonlinear Frequency-Modulated Waveforms Modeling and Optimization for Radar Applications. *Mathematics*. 2022; 10(21):3939.
https://doi.org/10.3390/math10213939

**Chicago/Turabian Style**

Xu, Zhihuo, Xiaoyue Wang, and Yuexia Wang.
2022. "Nonlinear Frequency-Modulated Waveforms Modeling and Optimization for Radar Applications" *Mathematics* 10, no. 21: 3939.
https://doi.org/10.3390/math10213939