## 4 Experiment Results and Discussion  
### Distributed Vibration Sensing  
The relationship between the differential voltage signal and strain was characterized through tests with different PZT strains, and a linear trend was observed, as shown in Figure 3.  
The sensitivity taken from the slope gradient of Figure 3 is 0.4141 \(\upmu\)V/\(\upmu\epsilon\). Due to the standard deviation of the intensity noise being 1.4124 \(\upmu\)V, the limit of detection (LoD) [21] can be calculated as follows:  
Figure 3: Relationship between applied strain and differential voltage. Figure 2: Schematic of the integrated system. PC: polarization controller; EOM: electro-optic modulator; OC: optical coupler; Cir: circulator; SMF: single-mode fiber; BPD: balanced photodetector. \[\text{LoD}=3.3*\frac{std(noise)}{sensitivity}=\frac{3.3*1.4124}{0.4141}\ \mu\varepsilon=11.26\ \mu\varepsilon, \tag{9}\]  
The normalized LoD, obtained by dividing the LoD by the square root of the detection bandwidth (400 MHz), is 563 \(\text{p}\varepsilon/\text{Hz}^{1/2}\). To verify vibration positioning accuracy, a commercial optical time-domain reflectometer (OTDR) was connected to a nominal end of the single-mode fiber (SMF1), and the center position of the optical fiber coiled around the PZT was measured to be at a distance of 100.51 km from the launch end. The total length of the fiber is 120.78 km. Figure 4a shows the differential voltage signals from the two detection ends, while Figure 4b presents the cross-correlation [22] results between the two detection ends. Figure 4c provides an enlarged view around the peak value. The maximum peak occurs at \(-406.94\ \upmu\text{s}\), and the vibration position is determined to be 101.04 km using Equation (9). Figure 4d displays the positioning results. When multiple vibration events occur along the sensing fiber, the traditional cross-correlation method treats the vibrations as a collective whole, and thus suffers from significant inaccuracies. Instead, a phase-spectrum time delay method based on frequency-domain analysis is used for resolving each vibration source of different frequencies [23]. Two PZTs were placed at distances of 50.32 km and 100.56 km, with frequencies of 160 Hz and 100 Hz, respectively. The resulting differential voltage signals are shown in Figure 5a. The time delays of 0.4205 ms and \(-0.0928\) ms shown in Figure 5b are calculated using the initial phase of each frequency component of the fast Fourier transform. They correspond to vibration positions of 51.11 km and 102.39 km. In the contour map shown in Figure 5c, the vibrations are resolved as a function of distance, frequency, and amplitude (intensity). To investigate the accuracy of vibration positioning within the context of integrated sensing applications, a specific vibration point was selected at a distance of 100.51 km, and 100 consecutive measurements were carried out. The piezoelectric transducer (PZT) was driven by a sinusoidal signal with an amplitude of 24 V and a frequency of 100 Hz. The measurement results are shown in Figure 6, which presents the histogram of the vibration position and its probability distribution. Figure 4: Single-point vibration positioning processing sequence. (**a**) Differential voltage signal; (**b**) cross-correlation results, red square: indicates area for magnification; (**c**) local magnification around the highest peak; (**d**) corresponding vibration position. The PZT was driven with a 200 Hz sinusoidal signal. To quantify the positioning accuracy, the root mean square error (RMSE) and standard deviation (STD) were used as metrics.