## 3 Signal Model  
UWB radar detects the range of a human subject via transmitting pulses and observing the time delay of the echo signal. The time delay depends on the variable distance from radar to subject. The changing distance caused by respiration and heartbeat can be expressed by [17]  
\[d(t)=d_{0}+A_{r}sin(2\pi f_{r}t)+A_{h}sin(2\pi f_{h}t) \tag{1}\]  
where \(d_{0}\) is the distance from the antenna to the chest, \(A_{r}\) and \(f_{r}\) are the amplitude and frequency of respiration and \(A_{h}\) and \(f_{h}\) are the amplitude and frequency of heartbeat. For vital signal detection in UAV-mounted UWB radar, heartbeat is much smaller than respiration and interference. We only take the respiratory signal as the basis for distinguishing the life state of the wounded subject. Additionally, the UAV platform random motion leads to an additional interference component. Then, the changing distance can be expressed by  
\[d(t)=d_{0}+X(t)+d_{UAV}(t)=d_{0}+A_{r}sin(2\pi f_{r}t)+d_{UAV}(t) \tag{2}\]  
where \(X(t)\) is the respiratory signal and \(d_{UAV}(t)\) is the platform movement. With a single human subject and stationary background environment, the radar system response is [18]  
\[h(\tau,t)=a_{r}\delta(\tau-\tau_{r}(t))+\sum_{i}a_{i}\delta(\tau-\tau_{i}) \tag{3}\]  
where \(a_{r}\delta(\tau-\tau_{r}(t))\) is the response of the human target and \(\sum\limits_{i}a_{i}\delta(\tau-\tau_{i})\) is the response of other static clutter. In the scenario of outdoor injured people respiration detection using UAV-mounted radar, the static object refers to the ground. Due to platform movement, the static object no longer behaves as static, thus the response of radar can be further expressed as  
\[h(\tau,t)=a_{r}\delta(\tau-\tau_{r}(t))+a_{g}\delta(\tau-\tau_{g}(t)) \tag{4}\]  
Time delay \(\tau_{r}(t)\) is given by  
\[\tau_{r}(t)=\frac{2d(t)}{c} \tag{5}\]  
and time delay \(\tau_{g}(t)\) is given by  
\[\tau_{g}(t)=\frac{2d_{1}(t)}{c} \tag{6}\]  
where \(c=3\times 10^{8}\) m/s is the speed of light, \(d_{1}(t)=d_{0}+d_{UAV}(t)\) is the changing distance from the antenna to the ground. \(a_{r}\delta(\tau-\tau_{r}(t))\) is the response of the human target and \(a_{g}\delta\left(\tau-\tau_{g(t)}\right)\) is the response of the static ground that appears as the dynamic clutter changing with time in the radar echo signal and reflecting the platform motion signal. The received signal can be denoted by [19]  
\[R(\tau,t)=s(\tau)*h(\tau,t)=a_{r}s(\tau-\tau_{r}(t))+a_{g}s\big{(}\tau-\tau_{g }(t)\big{)} \tag{7}\]  
where \(s(\tau)\) is the transmitted pulse. The two-dimension raw echo data in the form of discrete are denoted by  
\[R[m,n]=a_{r}s(m\delta_{T}-\tau_{r}(nT_{s}))+a_{g}s\big{(}m\delta_{T}-\tau_{g }(nT_{s})\big{)}=r[m,n]+g[m,n] \tag{8}\]  
where \(m=\{1,2,\cdots,M\}\) is the number of fast time samples representing the range points, and \(n=\{1,2,\cdots,N\}\) is the number of slow time samples representing the time points. \(r[m,n]\) represents the interferenced respiratory signal and \(g[m,n]\) represents the echo signal of the ground. In real scenario detection, the received raw data can be expressed as  
\[R[m,n]=r[m,n]+g[m,n]+z[m,n] \tag{9}\]  
where \(z[m,n]\) is unavoidable noise interference. Adopting a complex exponent to represent the transmitted signal, the echo signal that carries the respiration component in Equation (7) can then be expressed as [20]  
\[r(t)=A_{r}e^{i(2\pi f_{c}t+\theta(t)+\varphi_{0})} \tag{10}\]  
where \(f_{c}\) is the central operating frequency, \(\varphi_{0}\) is the initial phase and \(\theta(t)\) is the phase shift caused by human respiration and platform motion.