| Summary: | This article solves the problem of detecting water leaks with a minimum size of down to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mrow><mtext> </mtext><mi>mm</mi></mrow></mrow></semantics></math></inline-formula> in diameter. Two new mathematical tools are used to solve this problem: the first one is the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">T</mi><mi>e</mi></msub></mrow></semantics></math></inline-formula> cross-spectral density and the second is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">T</mi><mi>e</mi></msub></mrow></semantics></math></inline-formula> coherence. These mathematical tools provide the possibility of discriminating spurious frequency components, making use of the property of multi-sensitivity. This advantage makes it possible to maximize the sensitivity of the frequency spectrum. The wavelet function used was Daubechies <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>45</mn></mrow></semantics></math></inline-formula>, because it provides an attenuation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>150</mn><mrow><mtext> </mtext><mi>dB</mi></mrow></mrow></semantics></math></inline-formula> in the rejection band. The tools were validated with two scenarios. For the first scenario, a synthetic signal was analyzed. In the second scenario, two types of background leakage were analyzed: the first one has a diameter of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mrow><mtext> </mtext><mi>mm</mi></mrow></mrow></semantics></math></inline-formula> with a signal-to-noise ratio of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2.82</mn><mrow><mtext> </mtext><mi>dB</mi></mrow></mrow></semantics></math></inline-formula> and flow rate of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>33.7</mn><mrow><mtext> </mtext><mi>mL</mi></mrow><mo>/</mo><mi mathvariant="normal">s</mi></mrow></semantics></math></inline-formula>, and the second one has a diameter of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>4</mn><mrow><mtext> </mtext><mi>mm</mi></mrow></mrow></semantics></math></inline-formula> with a signal-to-noise ratio of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>9.73</mn><mrow><mtext> </mtext><mi>dB</mi></mrow></mrow></semantics></math></inline-formula> with a flow rate of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>125.0</mn><mrow><mtext> </mtext><mi>mL</mi></mrow><mo>/</mo><mi mathvariant="normal">s</mi></mrow></semantics></math></inline-formula>. The results reported in this paper show that both the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">T</mi><mi>e</mi></msub></mrow></semantics></math></inline-formula> cross-spectral density and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">T</mi><mi>e</mi></msub></mrow></semantics></math></inline-formula> coherence are higher than those reported in scientific literature.
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