# Detection of Background Water Leaks Using a High-Resolution Dyadic Transform

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## Abstract

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## 1. Introduction

- The first is the ${\mathcal{T}}_{e}$ cross-spectral density.
- The second is ${\mathcal{T}}_{e}$ coherence.

## 2. Theoretical Background

#### 2.1. Cross-Correlation Function

#### 2.2. ${\mathcal{T}}_{e}$ Transform

#### 2.3. Cross-Spectral Density Function

#### 2.4. Coherence Function

## 3. Method for the Detection of Water Background Leakage

#### 3.1. ${\mathcal{T}}_{e}$ Cross-Spectral Density Function

#### 3.2. ${\mathcal{T}}_{e}$ Coherence Function

## 4. Experimental Design

## 5. Results and Discussion

#### 5.1. Simulated Signal Scenario

#### 5.2. Real Experimentation Scenarios

#### 5.2.1. Results Obtained for Background Leakage of $1\text{}\mathrm{mm}$ in Diameter

#### 5.2.2. Results Obtained for Background Leakage of $4\text{}\mathrm{mm}$ in Diameter

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Cross-spectral density, with two synthetic signals: ${S}_{1}=\mathit{sin}\left(2\pi 10t\right)+\mathit{sin}\left(2\pi 100t\right)$ and ${S}_{1}=\mathit{sin}\left(2\pi 50t\right)+\mathit{sin}\left(2\pi 100t\right)$.

**Figure 3.**Coherence, with two synthetic signals: ${S}_{1}=\mathit{sin}\left(2\pi 10t\right)+\mathit{sin}\left(2\pi 100t\right)$ and ${S}_{1}=\mathit{sin}\left(2\pi 50t\right)+\mathit{sin}\left(2\pi 100t\right)$.

**Figure 4.**${\mathcal{T}}_{e}$ cross-spectral density, with two synthetic signals: ${S}_{1}=\mathit{sin}\left(2\pi 10t\right)+\mathit{sin}\left(2\pi 100t\right)$ and ${S}_{1}=\mathit{sin}\left(2\pi 50t\right)+\mathit{sin}\left(2\pi 100t\right)$.

**Figure 5.**${\mathcal{T}}_{e}$ cross-spectral density in Scale $2$, with two synthetic signals: ${S}_{1}=\mathit{sin}\left(2\pi 10t\right)+\mathit{sin}\left(2\pi 100t\right)$ and ${S}_{1}=\mathit{sin}\left(2\pi 50t\right)+\mathit{sin}\left(2\pi 100t\right)$. Absolute error $1\ast {10}^{-3}$.

**Figure 6.**${\mathcal{T}}_{e}$ coherence, with two synthetic signals: ${S}_{1}=\mathit{sin}\left(2\pi 10t\right)+\mathit{sin}\left(2\pi 100t\right)$ and ${S}_{1}=\mathit{sin}\left(2\pi 50t\right)+\mathit{sin}\left(2\pi 100t\right)$.

**Figure 7.**${\mathcal{T}}_{e}$ coherence in Scale $2$, with two synthetic signals: ${S}_{1}=\mathit{sin}\left(2\pi 10t\right)+\mathit{sin}\left(2\pi 100t\right)$ and ${S}_{1}=\mathit{sin}\left(2\pi 50t\right)+\mathit{sin}\left(2\pi 100t\right)$. Absolute error $1\ast {10}^{-3}$.

**Figure 8.**Signals were acquired from the first sensor with a background leak of $1\mathrm{mm}$ and $4\mathrm{mm}$ in diameter.

**Figure 9.**Signals were acquired from the second sensor with a background leak of $1\mathrm{mm}$ and $4\mathrm{mm}$ in diameter.

**Figure 10.**Cross-spectral density, with background leak ($1\mathrm{mm}$ in diameter) and flow rate $33.7\mathrm{mL}/\mathrm{s}$.

**Figure 11.**Coherence’s magnitude, with background leak ($1\mathrm{mm}$ in diameter) and flow rate $33.7\mathrm{mL}/\mathrm{s}$.

**Figure 12.**${\mathcal{T}}_{e}$ cross-spectral density, with background leak ($1\mathrm{mm}$ in diameter) and flow rate $33.7\mathrm{mL}/\mathrm{s}$.

**Figure 13.**${\mathcal{T}}_{e}$ cross-spectral density in Scale $4$, with background leak ($1\mathrm{mm}$ in diameter) and flow rate $33.7\mathrm{mL}/\mathrm{s}$.

**Figure 14.**${\mathcal{T}}_{e}$ cross-spectral density in Scale $3$, with background leak ($1\mathrm{mm}$ in diameter) and flow rate $33.7\mathrm{mL}/\mathrm{s}$.

**Figure 15.**${\mathcal{T}}_{e}$ coherence, with background leak ($1\mathrm{mm}$ in diameter) and flow rate $33.7\mathrm{mL}/\mathrm{s}$.

**Figure 16.**${\mathcal{T}}_{e}$ coherence in Scale $4$, with background leak ($1\mathrm{mm}$ in diameter) and flow rate $33.7\mathrm{mL}/\mathrm{s}$.

**Figure 17.**Cross-spectral density, with background leak ($4\text{}\mathrm{mm}$ in diameter) and flow rate $125.0\mathrm{mL}/\mathrm{s}$.

**Figure 18.**Coherence, with background leak ($4\mathrm{mm}$ in diameter) and flow rate $125.0\mathrm{mL}/\mathrm{s}$.

**Figure 19.**${\mathcal{T}}_{e}$ cross-spectral density, with background leak ($4\mathrm{mm}$ in diameter) and flow rate $125.0\mathrm{mL}/\mathrm{s}$.

**Figure 20.**${\mathcal{T}}_{e}$cross-spectral density in Scale $2$, with background leak ($4\mathrm{mm}$ in diameter) and flow rate $125.0\mathrm{mL}/\mathrm{s}$.

**Figure 21.**${\mathcal{T}}_{e}$ coherence, with background leak ($4\mathrm{mm}$ in diameter) and flow rate $125.0\mathrm{mL}/\mathrm{s}$.

**Figure 22.**${\mathcal{T}}_{e}$ coherence in Scale $2$, with background leak ($4\text{}\mathrm{mm}$ in diameter) and flow rate $125.0\mathrm{mL}/\mathrm{s}$.

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## Share and Cite

**MDPI and ACS Style**

Trutié-Carrero, E.; Seuret-Jiménez, D.; Nieto-Jalil, J.M.; Herrera-Díaz, J.C.; Cantó, J.; Escobedo-Alatorre, J.J.
Detection of Background Water Leaks Using a High-Resolution Dyadic Transform. *Water* **2023**, *15*, 736.
https://doi.org/10.3390/w15040736

**AMA Style**

Trutié-Carrero E, Seuret-Jiménez D, Nieto-Jalil JM, Herrera-Díaz JC, Cantó J, Escobedo-Alatorre JJ.
Detection of Background Water Leaks Using a High-Resolution Dyadic Transform. *Water*. 2023; 15(4):736.
https://doi.org/10.3390/w15040736

**Chicago/Turabian Style**

Trutié-Carrero, Eduardo, Diego Seuret-Jiménez, José M. Nieto-Jalil, Julio C. Herrera-Díaz, Jorge Cantó, and J. Jesús Escobedo-Alatorre.
2023. "Detection of Background Water Leaks Using a High-Resolution Dyadic Transform" *Water* 15, no. 4: 736.
https://doi.org/10.3390/w15040736