# Acoustic Emission Monitoring in Prestressed Concrete:
## 3 Materials and Methods
### Data Processing  
The analysis of the measurement data was conducted using two distinct processes, each comprised of multiple steps. First, the time-domain features peak amplitude \(A_{\text{peak}}\) and energy \(E\) were derived. To achieve this, signals were standardized at a preamplification level of 0 dB, and a consistent threshold of 90 dB was established. Following this, both features were calculated directly from the raw data of the time-amplitude signal. The peak amplitude is typically measured in decibels (\(dB_{AE}\)), whereas energy is calculated over the signal's duration and is expressed in \(V^{2}\cdot s\). For enhanced comparability, the energy was also converted to decibels in this study using the formula  
\[E_{\text{(dB)}}=10\cdot\log_{10}(E_{(V^{2}\cdot s)}). \tag{4}\]  
The features were then represented in decibels against the distance between the source and sensor, as illustrated in Section 4. To account for the variance in individual measurements, a 90 % confidence interval was depicted as an envelope curve for each feature. In processing the data, the two types of attenuation--geometric and dispersive--could not be distinguished in detail due to insufficient knowledge about their specific contributions to the attenuation observed. Nevertheless, converting the data to a logarithmic scale linearized the exponential function's appearance, facilitating the determination of the attenuation coefficient through linear regression across the entire range of values, i.e., all distances on the \(x\)-axis [21]. Additionally, to enhance the empirical distribution statistically, a bootstrapping method with 1000 iterations per attenuation profile was employed, justified by the adequate size of the original sample set. The regression's coefficient and the residual standard deviation (RSD) were determined, with the latter calculated as follows:  
\[\text{RSD}=\sqrt{\frac{\sum_{i=1}^{n}(y_{i}-\hat{y_{i}})^{2}}{n-2}} \tag{5}\]  
where \(n\) is the number of observations, \(y_{i}\) is the observed, and \(\hat{y_{i}}\) is the predicted value of the dependent variable. The RSD quantifies the spread of the differences between the observed values and those predicted by the model, thus serving as an indicator for evaluating the model's fit to the data. Both coefficients and RSDs are documented in Section 4. Utilizing these two statistical parameters, the characteristics of the phenomena under investigation were assessed. In the second process, the focus was on the frequency dependence of attenuation. To this end, signals were normalized to a preamplification level of zero dB. A threshold is generally not required for frequency analysis, but was used with 90 dB to determine the signal onset. The signals were then trimmed to a unified length of 1200 us with pre- and post-triggers of 200 and 1000 us, respectively. This segmented signal was then processed, resampling at 10 MHz and applying a Hanning window to mitigate leakage effects, before being transformed into the frequency domain with Fourier transformation. To investigate the frequency dependence of attenuation, the resulting frequency spectrum was segmented into partial frequency bands, each 10 kHz wide, spanning from 0 to 100 kHz, as illustrated in Figure 5a. The investigation's upper frequency limit was set at 100 kHz. This was based on the observation that the frequency spectra of the measured signals from both signal sources show a significant decrease within the 70 to 100 kHz range, already evident near the source location (approximately 0.3 m), as depicted in Figure 5. The subsequent increase between 115 and 120 kHz remains below the amplitudes noted in the range of 10 to 60 kHz and is, therefore, excluded from the evaluation, presumably resulting from a harmonic of the sensor's resonance frequency, around 60 kHz. For each of the delineated partial frequency bands, the mean spectral amplitude was calculated. These calculations served as the basis for creating the attenuation profiles shown in Section 4.