## 5 Results and Discussions
### Comparison of the Results  
Figure 7 compares the results obtained from the LDS and PA measurements. Each measurement was conducted five times under the same conditions, and the average velocity and SD were calculated. Notably, the velocity measured by the LDS was moving-averaged in advance within the instrument, necessitating the PA result to be moving-averaged. The green line PA(averaged) showcases the moving average velocities from the PA measurement, with the results indicating consistent velocity variations even during the folding back of the oscillation at 0.055 s. The acquired movements were consistent for both measurements. The maximum deviation between the LDS and PA was 26 mm/s at 0.031 s, primarily attributed to the temporal resolution of the respective measurements. The resolution of the LDS varied from 0.50 to 4.0 ms, while that of the PA was 2.8 ms. The resolution of the PA depends on the phase difference \(\varphi\) between LS and pulse signal and could potentially be improved up to 0.56 ms. By aligning the temporal resolution of both measurements, the deviation should be further reduced. The statistical uncertainties of the PA velocity measurement were calculated as 0.86 to 2.1 mm/s in the measurements. The average velocity \(v=-155\) mm/s was calculated at 0.10 s, the absolute uncertainty at 0.86 mm/s and the relative uncertainty at 0.56%. At an average velocity \(v=9.0\) mm/s at 0.055 s, the absolute uncertainty was 2.1 mm/s and the relative uncertainty was 24%. The large relative uncertainty at low velocities was attributed to the minimum measurable velocity determined by the sample rate of the digitizer of the measurement instrument. This limitation could be overcome by interpolating the CC coefficient for peak determination [27]. Figure 8 showcases the comparison of the results acquired from the LDS and PA measurements in the range of \(-50\) to \(50\) mm/s. The PA data were an ensemble average of five measurements. Comparing only the PA data with Figure 5, the velocity distribution in Figure 8 is similar to a cosine waveform. This is consistent with the fact that, theoretically, the output signal of the input sine signal is a cosine waveform. In addition, the positive maximum velocities were obtained around 0 and 0.20 s, and the negative maximum velocity was obtained around 0.10 s, which is consistent with the theoretical cosine waveform. We applied a time average to the PA results for comparing them with the LDS results internally averaged in the instrument. Figure 8 shows the comparison of the results with the moving average. While the LDS results were the average of every 64 samples with the sampling  
Figure 7: Comparative analysis of the measurement results based on the PA and the LDS. The gray line indicates the velocity measured by the LDS, while the green line represents the averaged velocities derived from the PA result. Notably, the LS oscillation exhibited a folding back at 0.055 s in all measurements. The PA measurement uncertainties ranged from 0.86 to 2.1 mm/s. The maximum deviation between the LDS and PA was 26 mm/s at 0.031 s.  
frequency at 0.5 ms, the PA results were averaged over 12 neighbor points, which were measured at the time resolution of 2.72 ms. As a result, PA (averaged) data are observed to be shifted 33 ms to the right side from the original PA data. This result indicates that the PA (averaged) data of velocity distribution with the SD are generally consistent with the LDS data. Relatively large deviation observed around the maximum velocities is expected due to the underestimatation occurring with the application of the central-difference scheme to the LDS data. Moreover, the consistency of the results was experimentally confirmed in velocity measurements with two additional cases described in Appendix A. The two cases include the object with a GC with a smaller diameter of 0.5 mm and with multiple cylinders with smaller diameters of 0.5 mm. These results indicate that the PA velocity measurement was able to monitor the movement of the object faithfully.