# Detection of Quasi-Static Trapped Human Being Using Mono-Static UWB Life-Detection Radar
## 2 Material and Methods
### Vital Sign Detection Using a Mono-Static UWB Life-Detection Radar
#### 2.2.3 Algorithm for Stationary Operating Mode  
To remove respiration response from static objects around the human test subject, the time-invariant background must be estimated and subtracted from the measured data. Different approaches for estimating the static background have been proposed in Reference [37]. However, in cases where people are detected according to their micro-movement, respiration or cardiac activity, the reflected signals contained in measured radar echoes have the same time-delay and are always misinterpreted as static clutter using the background subtraction methods. Thus, people are almost invisible under these circumstances. To overcome this drawback, the adaptive background subtraction (ABS) method based on the exponential averaging and a vector of weighting coefficient \(\alpha\) can be used [38]. \[p_{n}(m)=\alpha_{n}(m)\times p_{n-1}(m)+(1-\alpha_{n}(m))\times q_{n}(m)[M \times 1] \tag{3}\]  
Here, \(p_{n}(m)\) and \(q_{n}(m)\) are one-dimensional (1D) vectors with the size and contain'sampled background estimation' and'measured impulse response' in the raw echo matrix \(S(m,\,n)\), respectively. \(M\) is the number of fast-time sampling points and \(m=0,\,1,\,\ldots,\,\)M - 1, \(N\) is the number of slow-time sampling points and \(n=0,\,1,\,\ldots,\,N-1\). The new background estimation takes a fraction of the previous estimation and a fraction of the measured radar echo. The time-variant weighted coefficient \(\alpha\), which has the size \([M\times 1]\), can be adjusted between 0 and 1 adaptively. According to Reference [38], \(\alpha=0.95\) has a reasonable balance between background attenuation and micro-movement reservation. According to (2), denote \(S_{\text{raw}}(k,\,m)\) as the slow-time range matrix obtained at a single observation point, where \(k=0,\,1,\ldots,\,K-1\) is the range cell index, and \(m=0,\,1,\ldots,\,M-1\) is the slow-time index. To eliminate the responses from static objects around the trapped human being, the time-invariant clutter/interference has to be estimated and subtracted from the slow-time range matrix, \(S_{\text{raw}}(k,\,m)\), using the pre-processing method, including the adaptive background subtraction, the linear trend suppression and the advance normalization (AN) [39]. The corresponding output slow-time range matrix is  
Figure 4: The slow-time range matrix including two types of vital signs. (**a**) Non-periodic non-weak respiration. (**b**) Quasi-periodic weak respiration. \(S(k,m)\), and it can be used to estimate the almost unchanged distance from the UWB life-detection radar for the quasi-static trapped human beings with non-periodic but non-weak respiration by the proposed detection method based on the ordered statistics constant false alarm rate (OS-CFAR) method. In Figure 5, we define \(\mathbf{x}_{m}\), \(\mathbf{q}_{m}\) and \(\mathbf{z}_{m}\) as the \(m\)th column vector, adjacent cancellation vector and background subtraction vector of \(S(k,m)\), respectively. Define \(\mathbf{b}_{m}\) as the \(m\)th background, which can be estimated by \(\mathbf{b}_{m-1}\), \(\mathbf{z}_{m}\) and the scale weighted factor \(\mathbf{a}m\) as follows. \[\mathbf{q}_{m}\ =\ \mathbf{x}_{m}-\mathbf{x}_{m-1} \tag{4}\]  
\[\mathbf{z}_{m}\ =\ \mathbf{x}_{m}-\mathbf{b}_{m-1} \tag{5}\]  
\[\mathbf{b}_{m}\ =\ \mathbf{b}_{m-1}\ +\ (1-\mathbf{a}_{m-1})\cdot\mathbf{z}_{m} \tag{6}\]  
\[\mathbf{a}_{m}\ (k)=\ 1\,\ k\ \in\ \mathbf{s}_{m}\ \text{or}\ k\ \in\ \mathbf{p}_{m}\mathbf{a}_{m}\ (k)=\ a\,\ \text{otherwise} \tag{7}\]  
where \(k=0\), \(1,\dots,K\)-\(1\) is the range cell index. According to Reference [38], \(\alpha=0.95\) is the empirical constant scale weighted factor [39]. \(\mathbf{s}_{m}\) is the correlation output between the last output \(\mathbf{s}_{m-1}\) and the OS-CFAR output \(\mathbf{s}_{0m}\). Generally, with \(\mathbf{z}_{m}\) being the input of the OS-CFAR method, \(\mathbf{s}_{0m}\) might contain all the possible distances of the quasi-static trapped human beings away from the UWB life-detection radar.