# Wavelet-Based Estimation of Damping from Multi-Sensor, Multi-Impact Data
## 4 Proposed Algorithms  
In this section, as shown in Algorithm 1, we present two wavelet-based algorithms designed to extract the damping ratio and natural frequency from \(K\) impulse response recordings, denoted as \(\mathrm{Signal}^{k}[n]\). These methods involve three main steps to determine the natural frequencies (\(\omega_{n_{j}}\)) and damping ratios (\(\zeta_{j}\)). In this context, each \"sample\" represents a repeated measurement of the same impulse response, taken under varying impact amplitudes (\(B^{k}\)) and times of impact (\(\tau_{0}^{k}\)) (refer to Equation (4)). The repetition of measurements is essential as it helps average out the effects of noise and variability, leading to more accurate and robust estimates of damping and frequency. A flowchart illustrating this algorithm can be found in Appendix A. ```
Input: \(\mathrm{Signal}^{k}[n]\) where \(k=1,2,\ldots,K\) represents the sample and \(K\) is the number of samples Output: \(\omega_{n_{j}}\) and \(\zeta_{j}\) for \(j=1,2,\ldots,N\) where \(N\) is the number of modes for\(k=1,2,\ldots,K\)do \(Y^{k}\leftarrow\) Compute the magnitude of \(\mathrm{FFT}(\mathrm{Signal}^{k}(n))\) endfor \(AveY\leftarrow\) Average of \(Y^{k}\) \(N\leftarrow\) Number of significant peaks in \(AveY\) \(\omega_{d_{j}}\leftarrow\) Frequency corresponding to each significant peak in \(AveY\) for each \(\omega_{d_{j}}\)do for\(k=1,2,\ldots,K\)do \(W^{k}\leftarrow\) Compute \(\mathcal{W}_{\Psi,\mathrm{Signal}^{k}[n]}(s_{0},\tau)\) at scale \(s_{0}\) corresponding to \(\omega_{d_{j}}\) [Equation (21)] \(\tau_{0}^{k}\leftarrow\) get_max_energy_time(\(W^{k}\)) [Maximum energy to estimate time of impact ((see Equation (4)))] ifCWT-1 then \(Z^{k}\gets W^{k}(\tau_{0}^{k}:end)\) [Alignment in CTWT-1] else ifCTWT-2 then \(Z^{k}\leftarrow\) [\(W^{k}(\tau_{0}^{k}:end)\) ] [Alignment in CTWT-2] endif \(Z^{k}\gets Z^{k}/|Z^{k}(\tau_{0}^{k})|\) [Normalizing to compensate the effect of \(B^{k}\) (see Equation (4))] endfor \(AveZ\leftarrow\) Average of \(Z^{k}\) ifCTWT-1 then \(\lambda_{j}\leftarrow\) Slope of \(\ln(|AveZ|)\) {Equation (31)} elseifCTWT-2 then \(\lambda_{j}\leftarrow\) Slope of \(\ln(AveZ)\) endif \(\omega_{n_{j}}\leftarrow\sqrt{\lambda_{j}^{2}+\omega_{d_{j}}^{2}}\) [Equation (3)] \(\zeta_{j}\leftarrow\lambda_{j}/\omega_{n_{j}}\) endfor
```  
**Algorithm 1** CTWT-1 and CTWT-2 Methods  
**Estimation of Damping Frequencies**: The initial phase of the algorithm focuses on the analysis of the FFT for each sample. By evaluating the FFT's magnitude, represented by \(Y^{k}\), we identify significant peaks. These peaks are indicative of the system's modes, with the damping frequencies, \(\omega_{d_{j}}\), corresponding to these peaks. Since individual samples might provide varying results, we average out the damping frequencies using the mean of all \(Y^{k}\)s.  
2. **CTWT and Sample Alignment**: Assuming that the system behaves similarly to an SDOF dynamic around each damped frequency, we determine the corresponding scale. At this designated scale, the CTWT is computed for every sample, denoted by \(W^{k}\). A fundamental step in this phase involves the alignment of samples. Due to the variability in impact times across repeated measurements, we use the maximum energy point as an alignment reference. This reference point, which is an estimate of the time of impact, is obtained from the get_max_energy_time function. This function finds the maximum point after applying a Savitzky-Golay finite impulse response (FIR) smoothing filter to the input data. Data recorded prior to this point are disregarded. To account for the varying magnitudes of each impact, a normalization factor is incorporated before the averaging step. 3. **Averaging and Extraction of Damping Ratio**: As the wavelet transform is linear, it retains the additive characteristics of Gaussian noise. The CTWT-1 approach involves averaging the **complex-valued**\(Z^{k}\) values directly in the wavelet domain. This method aims to reduce noise while preserving the full complex signal information before further processing.