# Selecting Target Range with Accurate Vital Sign Using Spatial Phase Coherency of FMCW Radar
## 2 Methods
### Spatial Phase Coherency  
To select the range-bins that contain displacement of VSs, we introduce a new method that selects range-bins by considering the coherency between \(P(t,r)\) over multiple range-bins. As shown in Figure 2, displacement by VSs is spatially distributed in multiple range-bins, and the displacement is simultaneously observed in the \(P(t,r)\) of the range-bins. Thus, the coherency between \(P(t,r)\) in range-bins that contain VS displacement should be high. Coherency is our criterion for selecting a range-bin that contain vital information, and an SPC index to quantify coherency was devised, as in (5).  
\[SPC(t,r,r^{\prime})=\frac{\sum_{s=t-t_{0}}^{t}P(s,r)\cdot P(s,r^{\prime})}{ \sigma_{P}(r)\cdot\sigma_{P}(r^{\prime})}\ \ \ \text{for}\ \ \ r\
eq r^{\prime}, \tag{5}\]  
where the mean of \(P(t,r)\) over \(t\) is eliminated beforehand, and \(\sigma_{P}(r)\) is the standard deviation of \(P(t,r)\).  
Then, the pair of range-bins that have the highest SPC value are selected as the VS target range \(r_{T}(t)\), as in (6):  
\[r_{T}(t)=\arg\max_{\{r,r^{\prime}\}}SPC(t,r,r^{\prime}). \tag{6}\]  
Figure 2: Measured vital-sign signals. (**a,b**) \(M(t,r)\) and \(P(t,r)\) of respiration and heartbeat, (**c,d**) temporal correlation coefficient between \(P(t,r)\) and referenced vital-sign, and (**e,f**) respiration and heartbeat in several range-bins.  
To test whether SPC could measure coherency, signals were generated and SPC was applied. Figure 3a shows generated \(P(t,r)\). To generate \(P(t,r)\), peak points of sinusoids were detected, the location of peak points were disordered by uniform noise, and the points were interpolated. From \(r_{1}\) to \(r_{9}\), the boundary of uniform distribution was expanded to generate further disorder. Figure 3b shows the SPC between \(P(t,r_{1})\) and \(P(t,r)\) in other range-bins. As \(P(t,r)\) was disordered from \(r_{2}\) to \(r_{9}\), the SPC decreased.  
As shown in Figure 4, SPC was applied to a real FMCW radar signal shown in Figure 4a,b and the capability of SPC in selecting the range-bins that contain vital information was tested. Figure 4c,d shows the calculated SPC. The SPCs for respiration and heartbeat were high for range-bins where \(P(t,r)\) was spatially coherent such as \(0.50\leq r\leq 0.60\) m in Figure 4a, and \(r=0.75\leq r\leq 0.85\) m in Figure 4b. In addition, maximum SPC at each range-bin was compared to the correlation coefficient between \(P(t,r)\) and referenced VS. The SPC and correlation coefficient represented similar trends over the range, and both indices had maximum values at the same range-bin. Figure 4e,f shows \(M(t,r)\) and \(P(t,r)\) in the pair of range-bins that had the highest SPC. For respiration, \(P(t,r_{R1})\) and \(P(t,r_{R2})\) in Figure 4e are coherent, and are in proportion with the referenced respiration. For heartbeat, \(P(t,r_{H1})\) and \(P(t,r_{H2})\) in Figure 4f are coherent, and are proportional to the referenced heartbeat. However, the \(M(t,r)\) in the same range-bins does not show VS displacement.  
To summarize, displacement by VSs was represented in several range-bins of \(P(t,r)\); accurate respiration and heartbeat signals were extracted by selecting a pair of range-bins that had the highest SPC.  
Figure 3: Generated signal and SPC. (**a**) Generated \(P(t,r)\), and (**b**) SPC with respect to \(P(t,r_{1})\).