## 2 System Model  
A UWB impulse radar detects changes in the propagation delay of signals reflected by targets such as respiration and heart rate signals. The range between the human chest and radar is [33; 37]  
\[d(t)=d_{0}+r(t)=d_  
where \(s(t)\) is the transmitted impulse signal. In discrete form, this can be expressed as [33]  
\[\begin{split}\mathbf{R}[m,n]&=a_{v}s(m\delta_{T}-\tau_{ v}(nT_{s}))+\sum_{i}a_{i}s(m\delta_{T}-\tau_{i})\\ &\quad=a_{v}s(m\delta_{R}-v\tau_{v}(nT_{s}))+\sum_{i}a_{i}s(m \delta_{R}-v\tau_{i}/2)\\ &\quad=\mathbf{h}[m,n]+\mathbf{c}[m]\,,\end{split} \tag{5}\]  
where \(T_{s}\) is the pulse repetition time, \(m=0,\dots,\,M-1\) in fast time and \(n=1,\dots,\,N-1\) in slow time. The sampling interval in fast time is \(\delta_{T}\) and \(\delta_{R}=v\delta_{T}/2\). \(\mathbf{h}[m,n]\) is the desired human motion signal and \(\mathbf{c}[m]\) corresponds to the static signals which are slow time-invariant.  
Figure 2 shows the received signal due to human respiration where the dashed line denotes a fast time bin. In a real environment, the received signal can contain static and non-static clutter, linear trend, human motion, and AWGN. Therefore, the received signal can be expressed as  
\[\mathbf{R}[m,n]=\mathbf{h}[m,n]+\mathbf{c}[m]\,+\mathbf{a}[m,n]+\mathbf{w}[m,n ]+\mathbf{q}[m,n]+\mathbf{g}[m,n]\,+\mathbf{z}[m,n], \tag{6}\]  
where \(\mathbf{a}[m,n]\) is the linear trend, \(\mathbf{w}[m,n]\) is AWGN, \(\mathbf{q}[m,n]\) is the non-static clutter, \(\mathbf{g}[m,n]\) is other clutter (e.g., due to system electronics), and \(\mathbf{z}[m,n]\) is the clutter caused by moving objects other than the target (e.g., persons in the environment). Figure 1 presents the ideal received signal matrix and the corresponding matrix with AWGN at an SNR of 0 dB is given in Figure 3. This shows that it is difficult to extract vital sign signals when the SNR is low. To avoid range ambiguities and frequency aliasing requires that  
\[1/T_{s}\geq 2(\max(f_{r},f_{h})) \tag{7}\]  
Further, the signals should be received within one pulse interval so that  
\[T_{w}+\max\{\tau_{v}(t)\}-\min\{\tau_{v}(t)\}<T_{s} \tag{8}\]  
where \(T_{w}\) is the \(-6\) dB UWB signal pulse width.  
Figure 3: The resulting matrix with AWGN at an SNR of 0 dB.  
Figure 2: The ideal pulses acquired from one human subject.  
_UWB Impulse Radar_  
The UWB impulse radar developed by the Key Laboratory of Electromagnetic Radiation and Sensing Technology, Chinese Academy of Sciences, is employed here. The radar parameters are given in Table 1. The center frequency is 400 MHz and the pulse repetition frequency (PRF) is 600 kHz. The data were obtained using six segments with a segment time window of 124 ns and \(M_{s}=682\) samples per segment. The number of samples in fast time is \(M=4092\) and \(N_{A}\) samples were averaged during data acquisition. Thus, a signal is obtained every \(M_{s}N_{A}/\text{PRF}=0.0341\) s. The slow time sampling frequency is 29 Hz, so \(N=512\) pulses are received every 17.6 s. Both equivalent-time and real-time sampling [39] are employed as the performance is better than with an analog receiver employing only equivalent-time sampling [40]. Figure 1 gives the two-dimensional (slow time and range) matrix \(R\) obtained using the UWB radar with a subject at a distance of 6 m from the antenna in an outdoor environment (this will be described in Section 4). The vital sign signals are not apparent due to the significant signal attenuation which is typical of real environments. Therefore, a new method for vital sign detection is proposed in this paper.