# Respiration Rate Extraction of Moving Subject Using Velocity Change in FMCW Radar
## 2 Methods  
### FMCW Radar Signal Processing  
The FMCW radar transmits a linearly modulated radio frequency signal and receives the frequency signal reflected from the target. The combination of the transmitted and received frequency signals is passed through a low-pass filter to extract the signal \(x(t,\ n,\ m)\) of a target moving at a speed \(v_{0}\). \(f_{r}\) is the frequency corresponding to distance \(r\), \(M_{0}\) is the reflected power from \(r\), and \(P\) is the time delay of the reflected propagation from \(r\)[5; 6; 7]. \[x(t,n,m)=M_{0}\cdot\cos\left(2\pi f_{r}n+\frac{4\pi\cdot f_{c}\cdot T_{c}\cdot v _{0}}{c}\cdot m+\varphi\right),\]  
\[f_{r}=\frac{2\cdot BW\cdot\gamma}{c\cdot T_{c}\cdot F_{s}},\]  
\[P=\frac{4\pi\cdot f_{c}\cdot T_{c}\cdot v_{0}}{c}\cdot m+\varphi, \tag{1}\]  
In Equation (2.1), \(m\) denotes the number of chirps, and \(n\) is the sample index of the chirps. When calculating \(f_{r}\), \(BW\) is the frequency bandwidth, \(c\) is the speed of light, \(T_{c}\) is the chirp duration, \(\varphi\) is the constant of time delay, \(F_{s}\) is the sampling frequency, and \(fc\) is the center frequency. \(P(t,r)\) and \(M(t,r)\) were extracted using \(X(t,\ r_{k},m)\) via the first-order discrete Fourier transform (DFT) of \(x(t,\ n,\ m)\). \[X(t,r_{k},m)=\sum_{n=0}^{T_{c}}x(t,n,m).e^{-\frac{(2\pi+k\pi}{t_{c}}},\]  
\[M(t,r_{k})=2|X(t,r_{k})|  
Note that the magnitude component \(M(t,\ r_{k})\) in \(x(t,n,m)\) is inversely proportional to the chest displacement \(\Delta(t)\). As \(\Delta(t)\) changes, the magnitude component continuously changes.  
Most of the current research detects the respiration rate by exploring \(M(t,\ r_{k})\) or \(P\). In the figure below, Figure 2a shows the magnitude \(M(t,\ r_{k})\) at various distances \(\ r_{k}\) and Figure 2b shows the magnitude change in \(\ r_{k}=0.525\) m according to the chest's up and down movement. As shown in Figure 2, the magnitude \(M(t,\ r_{k})\) and \(P\) in Figure 2b continuously change, since the reflected power or the time delay of the received radar signal changes. The respiration detection can be obtained by exploring the magnitude or phase change in the target distance.  
An additional experiment was conducted to demonstrate that a respiration measurement is possible with the radar used in this paper. The experiment shown in Figure 3 was conducted using a dummy (\"Sakamoto Baby Touch\"-Vital Sings Simulator, M179) whose chest displacement, due to respiration, was less than 0.05 m (5 cm). The respiratory rate per minute was set at 20 and the heart rate per minute at 120. The distance between the radar and the dummy was about 0.6 m.  
Figure 3: Experiment to prove that respiration can be measured using radar.  
Figure 2: Radar signal extracted for respiration measurement. (**a**) The magnitude \(M(t,\ r_{k})\) at various distances \(r_{k}\), (**b**) the magnitude change in \(r_{k}=0.525\) m according to the chest’s up and down movement.  
If \(r_{k}+\Delta(t)\) in Equation (3) includes the displacement pertaining to the movement of the human body and micromovements caused by respiration, distorted \(M(t,r_{k})\) and \(P(t,\ r_{k})\) appear. Therefore, a distorted IF signal is extracted when the subject is in motion.  
To extract the velocity to be used in the proposed signal processing method, the second-order DFT is obtained for \(X(t,\ r_{k},m)\).  
\[Z(t,r_{k},v_{0})=\sum_{m=0}^{N_{chirp}}X(t,r_{k},m)\cdot e^{-\frac{|zn\pm m}{N _{chirp}}},\]  
\[Z(t,r_{k},v_{0})=\left\{ \begin{array}{c}0,\ v\
eq v_{0}\\ \frac{M_{0}}{2}\cdot e^{i\varphi},\ v=v_{0}\end{array} \right., \tag{4}\]  
\(Z(t,r_{k},v_{0})\) extracted from Equation (4) only appears in proportion to the reflected power at \(r_{k}\) when the object has a velocity change of \(v_{0}\) at the position \(r_{k}\), as depicted in Figure 1b.