# Estimation and Classification of NLFM Signals Based on the Time–Chirp Representation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Nlfm Signals and Their Polynomial Approximations

## 3. The CPF-Based Estimator of the IFR

## 4. Frequency Estimation of NLFM Signals Based on CPF

## 5. Classification of Signals Based on Phase Polynomial Coefficiencies Obtained from CPF

- Binary classification—in this case, there are only two classes;
- Multiclass classification—in this case, there are more than two classes, and the classifier can only report one of them as output;
- Multilabel classification—in this case, the classifier is allowed to choose many answers. This type of classification can be simply considered as a combination of multiple independent binary classifiers.

- true positives (TP): the actual value is positive and the prediction is also positive;
- true negatives (TN): the actual value is negative and the prediction is also negative;
- false positives (FP): the actual value is negative, but the prediction is positive (type I error);
- false negatives (FN): the actual value is positive, but the prediction is negative (type II error).

- Accuracy (ACC)$$ACC=\frac{TP+TN}{TP+TN+FP+FN}$$
- Precision also known as positive predictive value (PPV)$$PPV=\frac{TP}{TP+FP}$$
- Sensitivity also known as recall, hit rate or true positive rate (TPR)$$TPR=\frac{TP}{TP+FN}$$

_{0}for noiseless NLFM is shifted to a new random location:

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The accuracyof nonlinear frequency approximations, (

**a**) the function ${f}_{1}(t,\alpha ,\gamma )$ and its Taylor polynomial approximations of the order $M\in \{5,9,13\}$, (

**b**) the error of approximations.

**Figure 2.**The accuracyof phase function approximations, (

**a**) the function ${\varphi}_{1}(t,\alpha ,\gamma )$ and its Taylor polynomial approximations of order $M\in \{6,10,14\}$; (

**b**) the error of approximations.

**Figure 4.**The RMSEof estimation of the instantaneous value of the frequency: (

**a**) ${f}_{1}(t,\alpha ,\gamma )$, (

**b**) ${f}_{2}(t,{B}_{l},{B}_{c})$.

**Figure 5.**The RMSEof estimation of the instantaneous value of the frequency: (

**a**) ${f}_{3}(t,{k}_{1},{k}_{2})$; (

**b**) ${f}_{LFM}\left(t\right)$.

**Figure 9.**Example ofthe realizations of the coefficients ${a}_{\varphi 2}$, ${a}_{\varphi 4}$ and ${a}_{\varphi 6}$ obtained by CPF for four signal classes in the case of (

**a**) $SNR=5$ dB; (

**b**) $SNR=1$ dB; (

**c**) $SNR=-1$ dB; and (

**d**) $SNR=-5$ dB.

**Figure 12.**Accuracy (42) metrics of MLP and LVQ classifiers.

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**MDPI and ACS Style**

Swiercz, E.; Janczak, D.; Konopko, K.
Estimation and Classification of NLFM Signals Based on the Time–Chirp Representation. *Sensors* **2022**, *22*, 8104.
https://doi.org/10.3390/s22218104

**AMA Style**

Swiercz E, Janczak D, Konopko K.
Estimation and Classification of NLFM Signals Based on the Time–Chirp Representation. *Sensors*. 2022; 22(21):8104.
https://doi.org/10.3390/s22218104

**Chicago/Turabian Style**

Swiercz, Ewa, Dariusz Janczak, and Krzysztof Konopko.
2022. "Estimation and Classification of NLFM Signals Based on the Time–Chirp Representation" *Sensors* 22, no. 21: 8104.
https://doi.org/10.3390/s22218104