## 3 Experiment and Results
Nevertheless, the presence of random noises in the raw backscattering traces of the phase-sensitive OTDR made it difficult to truly detect the vibration event (location information and vibration  
Figure 1: (**a**) Experimental setup of the phase-sensitive optical time-domain reflectometer (OTDR)-based vibration sensor. Above are the images showing the vibration setup in order to induce: (**b**) physical perturbation using a mechanical actuator, and (**c**) acoustic perturbation using a sound-speaker on the 9 m fiber loop placed at specific position of the fiber-under-test (FUT). frequency) and could lead to false alarms in real-life applications. Therefore, we present the WSVD method to denoise the backscattering traces to extract the accurate information of the vibration event. We performed the WSVD on the raw backscattering traces assembled in the form of image matrix \(A\), according to the theory described in Equations (1)-(5). In the decomposition of the image matrix as given in Equation (2), we obtained the \(m\) number of singular values and the associated left and right singular vectors from the \(m\) number of RBS traces. Figure 3 describes the normalized magnitudes of singular values as a function of the number of RBS traces. In the figure, the magnitudes are normalized, and the first normalized singular value (=1) is omitted. Our analyses showed that in the original signal, the first set of largest 30 singular values having the magnitudes in the range of 1-0.015 represent the slowly varying components having energies greater than those of the vibration event. The next set of 145 singular values having the magnitudes in the range of 0.015-0.0065 represents the energies of the vibration event. The rest of the singular values having magnitudes lower than the 0.0065 represents the energies of the noise components. Importantly, even though the number (\(m\)) of RBS traces in the dataset changed from 610 to 310 in a step of 100, the magnitudes of their \(m\) number of singular values varied slightly, and followed nearly similar decreasing trends. The serials of the set containing the singular values of interest, due to slight change in the associated magnitudes with the change in the \(m\), could be slightly shifted. Thus, it is expected that there will be no considerable effect of changing the number of RBS traces on the amplitudes of the associated singular vectors, and hence the SNR of the location information will not be significantly changed. According to the Nyquist theorem, the number of backscattering traces in the original signal must be at least twice the frequency of the applied vibration to be measured accurately. For \(m=610\) traces (and so 610 singular values), we carefully chose the energy components of interest corresponding to the set of singular values in the magnitude range of 0.015-0.0065, and obtained the reconstructed image matrix according to Equation (3). In the said range, there are about 145 singular values from the serial number 30 to 175, which makes about 4-25% of the total 610 values. This means that by selecting the 4-25% of the associated energy components to make matrix \(\tilde{A}\), we were able to filter out the slowly varying amplitude fluctuation and substantial residual noise having energies higher or lower than those of the applied vibration from the original signal. Thereafter, to increase the SNR of the vibration event and to preserve the frequency of the event, the weight factorsincluded in the reconstructed image matrix \(\tilde{A}\), following Equation (5). This completed the process of denoising the original signal using WSVD, and produced \(a\) denoised signal in the form of weighted image matrix \(\tilde{A}_{\text{w}}\). Figure 4a,b presents the location information and the periodic change in amplitude caused by the weak vibration event of 102 Hz, respectively, using the WSVD-based denoising method.