## 2 Phase Estimation Algorithm for Displacement Monitoring  
FMCW radar essentially transmits frequency-modulated continuous wave signals toward a target at a distance d\({}_{0}\), and the returning echo is a delayed version of the transmitted signal, with the delay denoted by \(\tau_{0}\). In an FMCW system designed with T\({}_{\rm c}\) and B as parameters--where T\({}_{\rm c}\) is the chirp duration, B is the chirp sweep bandwidth, and c is the speed of light--the range is determined by the beat frequency \(f_{\rm b}\), which represents the difference between the transmitted waveform and the echo as described in Figure 2. The range d\({}_{0}\) between the radar and the target can be expressed by Equation (2). In this study, we focus on a specific \(f_{\rm b}\), meaning that a specific d\({}_{0}\) is fixed.  
\[d_{0}=\frac{cT_{c}f_{b}}{2B} \tag{2}\]  
If the target undergoes a very small displacement while \(f_{b}\) remains the same between two chirps, the echo delay \(\Delta\tau\) of the second echo is minimal, and the change in \(f_{b}\) is less than the system's resolution. This scenario is illustrated in Figure 3.  
The phase of the beat signal from the first chirp and its echo can be described by Equation (3), while the phase of the beat signal for the second chirp is described by Equation (4). The difference between these two phases can be calculated using Equation (5).  
\[\varphi_{1}=2\pi f_{c}\tau_{0} \tag{3}\]  
\[\varphi_{2}=2\pi f_{c}(\tau_{0}+\Delta\tau) \tag{4}\]  
\[\Delta\varphi=\varphi_{2}-\varphi_{1}=2\pi f_{c}\Delta\tau \tag{5}\]  
The small displacement \(d\) that occurs between the first and second chirp results in a delay \(\Delta\tau\), as expressed in Equation (6). The carrier frequency \(f_{c}\) is defined in Equation (7).  
Figure 3: Small displacement by two chirps.  
Figure 2: FMCW waveform.  
Consequently, the phase difference \(\Delta\phi\) due to the small displacement is given by Equation (8), and d can be derived from Equation (9).  
\[2d=c\Delta\tau \tag{6}\]  
\[f_{c}=\frac{c}{\lambda} \tag{7}\]  
\[\Delta\varphi=\frac{4\pi d}{\lambda} \tag{8}\]  
\[d=\frac{\lambda}{4\pi}\Delta\varphi \tag{9}\]  
According to Equation (9), a displacement causes a phase shift in the beat signal, with the wavelength \(\lambda\) being a fixed system parameter. For instance, if \(d\) is very small, on the order of 1 \(\upmu\)m, and \(\lambda\) is 3.8 mm (i.e., 79 GHz), then \(\Delta\phi\) would be 0.19 degrees. This implies that the sensor system must have the accuracy to detect such a small phase shift (\(\Delta\phi\)). Therefore, this study will examine the accuracy and precision of phase measurements through experimental approaches.  
The phase accuracy is dependent on the radar MMIC, and it is not dependent on the supported RF frequency. If the same phase shift (e.g., 1 degree) is captured by both a 24 GHz (\(\lambda\) = 12.5 mm) and a 79 GHz (\(\lambda\) = 3.8 mm) radar system, the corresponding displacement in the 24 GHz system would be 3.3 times larger than that in the 79 GHz system, according to Equation (9). Therefore, a higher-frequency system enables more precise measurements under the same phase shift conditions. Based on this principle, this study adopts a millimeter-wave FMCW radar MMIC.