# Through-Floor Vital Sign Searching for Trapped Person Using Wireless-Netted UWB Radars
## 3 Through-Floor Vital Sign Search
### Searching Path Design
(2) Supposing \(K^{2}\) observation points are required for 2D observation, in this manuscript, there are \(K^{2}=36\) mono-static UWB life-detection radars to be arranged on the desired 2D observation aperture. Here, we set \(K=6\). (3) According to the concept of one-way 2D effective aperture corresponding to the 2D MIMO array, \(N_{T}\times N_{R}=36\) virtual elements are taken as the observation points where the mono-static UWB life-detection radars are placed. Here, we set \(N_{T}=4\) and \(N_{R}=9\). (4) Convert to the problem of \(N_{T}=4\) and \(N_{R}=9\) 2D MIMO array topology design. * Step 3: Determine the transmitters' positions: The \(N_{T}=4\) transmitters are placed symmetrically at the four vertices of a square with a width of \(\Delta_{A}\). The Cartesian coordinate is established and the location of four transmitters are as follows: (3) \[\mathbf{r}_{T}=\left\{ \begin{array}{cc}(x_{T1},y_{T1})&(x_{T2},y_{T2})\\ (x_{T3},y_{T3})&(x_{T4},y_{T4})\end{array} \right\}=\left\{ \begin{array}{cc} (\Delta_{A}/2,\Delta_{A}/2)&(\Delta_{A}/2,-\Delta_{A}/2)\\ (-\Delta_{A}/2,-\Delta_{A}/2)&(-\Delta_{A}/2,\Delta_{A}/2)\end{array} \right\}\] (4)
* Step 4: Determine the receivers' positions: (1) The selection range of \(N_{R}=9\) receivers are expressed as (4) where \(\mathbf{s}=\left[-4\frac{\Delta_{A}}{N_{R}}:\frac{\Delta_{A}}{N_{R}}:4\frac{ \Delta_{A}}{N_{R}}\right]\), and when \(i\
eq j\), there will be \(x_{R_{i}}\
eq x_{R_{j}}\) and \(y_{R_{i}}\
eq y_{R_{j}}\). (2) The effective virtual elements can be expressed as \(\mathbf{r}_{\text{eff}}\), including 36 positions. \[\mathbf{r}_{\text{eff}} =\mathbf{r}_{\text{T}}+\mathbf{r}_{\text{R}}\] (5) \[=\left\{ \begin{array}{ll}(x_{T1}+x_{R1},y_{T1}+y_{R1})&(x_{T2}+x_{R1},y_{ T2}+y_{R1})&(x_{T3}+x_{R1},y_{T3}+y_{R1})&(x_{T4}+x_{R1},y_{T4}+y_{R1})\\ (x_{T1}+x_{R2},y_{T1}+y_{R2})&(x_{T2}+x_{R2},y_{T2}+y_{R2})&(x_{T3}+x_{R2},y_{ T3}+y_{R2})&(x_{T4}+x_{R2},y_{T4}+y_{R2})\\ (x_{T1}+x_{R3},y_{T1}+y_{R3})&(x_{T2}+x_{R3},y_{T2}+y_{R3})&(x_{T3}+x_{R3},y_{ T3}+y_{R3})&(x_{T4}+x_{R3},y_{T4}+y_{R3})\\ (x_{T1}+x_{R4},y_{T1}+y_{R4})&(x_{T2}+x_{R4},y_{T2}+y_{R4})&(x_{T3}+x_{R4},y_{ T3}+y_{R4})&(x_{T4}+x_{R4},y_{T4}+y_{R4})\\ (x_{T1}+x_{R5},y_{T1}+y_{R5})&(x_{T2}+x_{R5},y_{T2}+y_{R5})&(x_{T3}+x_{R5},y_{ T3}+y_{R5})&(x_{T4}+x_{R5},y_{T4}+y_{R5})\\ (x_{T1}+x_{R6},y_{T1}+y_{R6})&(x_{T2}+x_{R6},y_{T2}+y_{R6})&(x_{T3}+x_{R6},y_{ T3}+y_{R6})&(x_{T4}+x_{R6},y_{T4}+y_{R6})\\ (x_{T1}+x_{R7},y_{T1}+y_{R7})&(x_{T2}+x_{R7},y_{T2}+y_{R7})&(x_{T3}+x_{R7},y_{ T3}+y_{R7})&(x_{T4}+x_{R7},y_{T4}+y_{R7})\\ (x_{T1}+x_{R8},y_{T1}+y_{R8})&(x_{T2}+x_{R8},y_{T2}+y_{R8})&(x_{T3}+x_{R8},y_{ T3}+y_{R8})&(x_{T4}+x_{R8},y_{T4}+y_{R8})\\ (x_{T1}+x_{R9},y_{T1}+y_{R9})&(x_{T2}+x_{R9},y_{T2}+y_{R9})&(x_{T3}+x_{R9},y_{ T3}+y_{R9})&(x_{T4}+x_{R9},y_{T4}+y_{R9})\end{array} \right\}\] (6)  
(3) Divide the 2D observation aperture (\(2\Delta_{A}\times 2\Delta_{A}\)) into \(K\times K=6\times 6\) grids, as shown by the red grids in Figure 2. Meanwhile, each grid must contain only one effective virtual element from \(\mathbf{r}_{\text{eff}}\), which is determined by the \"two basic principles\" in Step 1. (4) Evenly divide each grid into \(\sqrt{N_{R}}\times\sqrt{N_{R}}=3\times 3\) positions, as shown by the black circles in Figure 3. Thus, there are \((3\times 6)\times(3\times 6)=324\) optional positions to place effective virtual elements. (5) According to Step 4.1, by traversing all possible \(\mathbf{r_{R}}\), the 36 effective virtual elements from the resultant \(\mathbf{r_{eff}}\) would distribute randomly into 324 optional positions. We define a scoring rule: (i) Give the grid zero score when there are no virtual elements in it; (ii) otherwise, give the grid one score; (iii) the highest total score of all grids equal to 36. The 36 score indicates that the corresponding traversing result of \(\mathbf{r_{R}}\) is satisfied with the \"two basic principles\" in Step 1. For example, as shown in Figure 4, by optimizing the MIMO array topology, the effective aperture with different scores can be obtained.