## 3 Verification Method
### Data Acquisition and Transformation Methodology  
In this study, instead of relying solely on the magnitude of the acceleration vector during the preprocessing stage, we employed a method that calculates the tilt angles of each axis relative to gravitational acceleration, enabling a more precise analysis of the sensor data. This approach involves computing the resultant vector of acceleration along the X, Y, and Z axes based on gravitational acceleration (R), as defined by the mechanical principle in Equation (1). \[\mathrm{R}=\ 9.8\ \mathrm{m}/\mathrm{s}^{2}=\sqrt{\mathrm{x}^{2}+\mathrm{y}^{2 }+\mathrm{z}^{2}} \tag{1}\]  
Such an analytical strategy enhances the tracking performance of angular velocity, allowing for a more refined understanding of the device's orientation in three-dimensional space. Furthermore, to maximize the utility of the data, we integrated trigonometric functions into the analysis to convert the measured acceleration data into angular displacement. By applying this method, the device's motion can be measured more reliably in reference to gravity, enabling the quantification of displacement while minimizing the influence of external forces. This technique serves as a critical component in applications requiring high-precision orientation and motion tracking, such as navigation assistance systems and human motion analysis. In this analysis, the direction of the gravitational acceleration vector is defined as angles A\({}_{\mathrm{xr}}\), A\({}_{\mathrm{yr}}\), and A\({}_{\mathrm{zr}}\), with respect to each axis. These directional components are visually illustrated in Figure 2. To quantify rotational motion, accelerometer data was transformed using trigonometric functions to calculate angular velocity, as described in Equation (2). Furthermore, as shown in Equation (3), a complementary filter approach was used to estimate tilt angles: the angle estimated from the gyroscope was processed with a High Pass Filter to eliminate low-frequency noise, while the angle estimated from the accelerometer was processed with a Low Pass Filter to reduce high-frequency noise. The two filtered outputs were then combined to estimate the final tilt angle. \[\mathrm{cos}(\mathrm{A}_{\mathrm{xr}})=\mathrm{Rx}/\mathrm{R},\ \mathrm{ cos}\big{(}\mathrm{A}_{\mathrm{yr}}\big{)}=\mathrm{R}\mathrm{y}/\mathrm{R}, \mathrm{cos}(\mathrm{A}_{\mathrm{zr}})=\mathrm{R}\mathrm{z}/\mathrm{R} \tag{2}\]\[\mathrm{A_{ij}}=\mathrm{R}\times\left(\mathrm{A_{ij-1}}+\mathrm{GyroscopicData}+ \mathrm{\Delta t}\right)+\left(1-\mathrm{R}\right)\times\left(\mathrm{AccelerometerAngle}\right) \tag{3}\]  
In this study, data was collected using a six-axis sensor at a high-resolution sampling rate of 255 Hz. This enabled the precise detection of subtle vibrational movements induced by heartbeats within the cardiovascular system. In the proposed method, as shown in Equation (4), the Root Mean Square (RMS) differences in the continuously measured data along the X, Y, and Z axes were calculated to quantitatively assess these fine movements. \[\mathrm{d}=\sqrt{\left(\mathrm{A_{xx}}-\mathrm{A_{x-1}}\right)^{2}+\left( \mathrm{A_{yr}}-\mathrm{A_{yr-1}}\right)^{2}+\left(\mathrm{A_{zr}}-\mathrm{A_{ zr-1}}\right)^{2}} \tag{4}\]  
To suppress low-frequency drift and enhance the visibility of periodic cardiac components, a high-pass filter with a cutoff frequency of 0.5 Hz was applied. This drift correction step effectively preserves essential heart rate signals without introducing distortion in the target frequency band (0.8-2.0 Hz). To illustrate the effects of the preprocessing steps, Figure 3 presents the time-domain signals before and after filtering, while Figure 4 shows the corresponding power spectral densities (PSDs). As shown, the band-pass filter effectively removes baseline drift and high-frequency noise, while the Welch-based PSD analysis improves spectral stability and enhances the visibility of the dominant frequency component within the expected heart rate range.