El citoesqueleto de microtúbulos y las proteínas marcadoras de

The conversion of signal sound files into comprehensible and measurable parameters is a significant task in this research since there is a need for values that can represent the cases under study. Figure IV-1 highlights the two forms of analyses conducted in this research: single signal analysis and dual signal analysis.

Figure IV-2: Frequency Distribution Analysis

The single signal analysis represents the full analysis of one single file where the required parameters are derived and developed. The first step for single signal analysis is Fast Fourier

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Transform (FFT). FFT is conducted via a written Matlab code using the recorded sound file and converts the file into viewable and readable data represented in decibels, as displayed in the right image in Figure IV-2. Additionally, the FFT can provide the frequency distribution of the sound file, highlighting the repetitions and amplitude at each given frequency. The frequency distribution of a given sound file after analysis can be viewed in Figure IV-2, where the signal is converted to a percentage of the dominance of each frequency band starting with 0 Hz and ending with 500 Hz. The development of the acoustic signal analysis model is summarized in Figure IV-3. The first step is to determine whether the analysis is for a single file for the purpose of detection or more than one file for the purpose of pinpointing. If the analysis is for more than one file, then correlation analysis is performed to determine the relationships between the two sound files collected from the sensors surrounding the leak. However, a single signal analysis begins with conducting fast Fourier transform to determine the amplitude and the frequency curves of the signal.

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The next step is to calculate the average amplitude as displayed in Equation IV-1. The value of the amplitude in decibels is calculated at each data point in the signal. The sum of those values is then divided by the total number of data points in the signal sequence.

𝜇𝜇𝑆𝑆𝑖𝑖𝑆𝑆𝑛𝑛𝑆𝑆𝑆𝑆 = ∑ 𝐴𝐴𝑐𝑐𝐴𝐴𝑆𝑆𝑖𝑖𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑖𝑖

𝑁𝑁 𝑖𝑖=1

𝑁𝑁 (IV-1)

Where:

µSignal (dB) = the average sound amplitude of sound file under study.

Amplitudei (dB) = sound amplitude at instant i in the sound file under study. N = the total number of data points within the sound file under study.

Following the calculation of the average sound amplitude, the next step will be to determine the standard deviation of the signal. Equation IV-2 shows the calculation approach for σSignal by using the previously developed µSignal in Equation IV-1. The equation is the same as a basic standard deviation equation but rewritten to fit the particular problem at hand.

𝜎𝜎𝑆𝑆𝑖𝑖𝑆𝑆𝑛𝑛𝑆𝑆𝑆𝑆 = �1_𝑛𝑛 ∑ (𝐴𝐴𝑚𝑚𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑚𝑚𝑁𝑁𝑖𝑖=1 𝑖𝑖 − 𝜇𝜇𝑆𝑆𝑖𝑖𝑆𝑆𝑛𝑛𝑆𝑆𝑆𝑆)2 (IV-2)

Where:

σSignal (dB) = the standard deviation of the amplitudes within the sound file under study.

Following the development of the average signal amplitude and the standard deviation of the signal amplitude, the level of the signal and the spread are developed. The level of a signal represents the dominant or governing amplitude of the signal, i.e. the highest amplitude of the signal. Equation IV-3 displays the calculation of a sound file’s level through computing the sound amplitude in decibels at each instant i. The level of a signal is the peak of the signal. Thus, the equation determines the highest value in the sequence.

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𝐿𝐿𝑚𝑚𝐿𝐿𝑚𝑚𝐴𝐴 (𝐴𝐴𝑑𝑑) = max ({𝑆𝑆𝐴𝐴𝑆𝑆𝑚𝑚𝑚𝑚𝐴𝐴(𝐴𝐴) ∶ 𝐴𝐴 = 1, … , 𝑁𝑁}) (IV-3)

Following the calculation of the level of the signal, another representation of the variation of the recorded sound is calculated. The parameter, in this case, would be the spread of the recording in the sound files. The spread represents the distance between the highest and lowest points in the signal sequence. Thus, the spread is determined by calculating the level or the maximum amplitude found in the recording as in equation IV-3. The level value determined has the minimum sound value subtracted from it. The spread represents the span of amplitudes within a signal.

𝑆𝑆𝐴𝐴𝑆𝑆𝑚𝑚𝑚𝑚𝐴𝐴 (𝐴𝐴𝑑𝑑) = max({𝑆𝑆𝐴𝐴𝑆𝑆𝑚𝑚𝑚𝑚𝐴𝐴(𝐴𝐴) ∶ 𝐴𝐴 = 1, … , 𝑁𝑁}) − min({𝑆𝑆𝐴𝐴𝑆𝑆𝑚𝑚𝑚𝑚𝐴𝐴(𝐴𝐴) ∶ 𝐴𝐴 = 1, … , 𝑁𝑁}) (IV-4)

After calculating the spread, the analysis moves from single sound file analysis to the analysis of two sound files representing the same phenomenon. In this case, the study aims at determining the level of similarity between the two signals by determining the correlation lag and also the time signal delay between the two signals. Regarding the correlation lag, the equations under study are discrete because the used loggers provide 16 seconds of recordings each time they record a sound and the values are consistent, so they do not increase or decrease. As a result, the discrete equation of correlation, Equation IV-5, has been used to determine the correlation lag value. The equation shows similarities with the convolution equation and convolution concepts. The aspired output of this equation is the displacement or lag (n) between the two signals. The

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approach calculates the complex conjugate of function (f ), (f*), then moves ahead to determine the displacement (n) of function (f ) against function (g).

(𝑓𝑓 ∗ 𝑆𝑆) [𝑚𝑚] ≝ ∑∞ 𝑓𝑓∗_{[𝑚𝑚] 𝑆𝑆[𝑚𝑚 + 𝑚𝑚]}

𝑐𝑐= −∞ (IV-5)

Where:

f = the initial sound function under study for the leak event.

g = the second correlation sound function under study for the leak event. n = the lag between the two leak event functions.

Accordingly, after determining the lag values, another critical relational parameter is the time delay values. The standard cross-correlation equation for time delay, Equation IV-6, has been used to identify the discrepancy in time between the two leak event signals. The equation aims at determining the time difference between two signals at their point of similarity. The main unit for assessment is seconds. This value allows the user to realize which signal arrived before the other and with what time difference.

𝜏𝜏𝐴𝐴𝐴𝐴𝑆𝑆𝑆𝑆𝑑𝑑 = 𝑚𝑚𝑆𝑆𝑆𝑆𝐴𝐴max ((𝑓𝑓 ∗ 𝑆𝑆)(𝐴𝐴)) (IV-6)