## 4 Proposed Sensing Results Separation
### Sensing Results Separation with Duplication of Sensing Results
#### 4.3.2 Data Tracking Based on Vector of Detected Subcarrier  
As FC can detect the duplication of the sensing result by counting the type of subcarriers, the estimated CSI and CFO are not available for data tracking because the estimated CSI and CFO are different from those of each sensor. In the proposed separation, the data tracking based on the vector of subcarrier is proposed. We assume high time correlation of CSI and CFO during two time slots at least. We consider the duplication of sensing results does not occur in \(t\)th time slot and occurs in \((t+1)\) time slot. From Equation (13), the spectrum of subcarrier, which certain sensors commonly select in \((t+1)\)th time slot is the linear combination among the spectrum of the subcarriers, which these select in \(t\)th time slot. Therefore, we consider the connection of the data set between the two time slots. In \(t\)th time slot, the \(i\in\{1,2,\ldots K\}\)th label is assigned to the data set including sensing result, CSI, and CFO. In \((t+1)\)th time slot, \(j\in\{1,2,\ldots,K^{\prime}\}\), \(K^{\prime}<K\) is assigned to the data set, where \(K^{\prime}\) is the types of detected subcarriers in \((t+1)\)th time slot. The problem of connecting data set is similar to that without duplication of sensing results but the data set in \((t+1)\) time slot can be connected to multiple data sets in \(t\)th time slot. As a result, the total number of connection patterns, \(S^{\prime}\), is given as follows. \[S^{\prime}={K^{\prime}}^{K}-\sum_{k=1}^{K^{\prime}-1}\ {K^{\prime}}^{C} \mathcal{C}_{k}k^{K}, \tag{16}\]  
where \(\ a\mathcal{C}_{b}\) is the number of combinations for \(b\) samples selected from \(a\) samples. \(\boldsymbol{u^{\prime}}_{s}\) is defined as the \(s(\in 1,2,\ldots,S^{\prime})\)th class of connection combination. In \((t+1)\)th time slot, the group of sensors, which select the \(j\)th subcarrier is defined as \(\boldsymbol{p_{j}}\). The \(j\)th subcarrier is reconstructed using subcarrier components in \(t\)th time slots and it is given as follows. \[\Lambda_{j}=\sum_{i\in\boldsymbol{p_{j}}}\gamma_{i}\sum_{m=1}^{N}\exp\left(j2 \pi\frac{me_{i}}{N}\right). \tag{17}\]  
\(\Lambda_{j}\) is referred to as the prediction of \(j\)th subcarrier. The spectrum detected in \(t\)th time slot is switched into \(\Lambda_{j}\) and then the new spectrum is given as follows. \[Y(n_{i})=\Lambda_{j}(i\in\boldsymbol{p_{i}}). \tag{18}\]  
As a result, the following cost function for the transition from \(j\)th data set to \(i\)th data set is given as  
\[g(i,j)=\left(\frac{n_{i}-n_{j}}{D}\right)^{2}+\left|\frac{Y(n_{i})-Y(n_{j})}{E }\right|^{2}. \tag{19}\]  
Since the subcarrier component is composed of the CSI and CFO, which are particular amount for each sensor, it is also particular amount. If CSI and CFO exhibit time continuity, we can consider subcarrier component also has time continuity. Therefore, the transition with minimum cost of Equation (19) is a most powerful. Therefore, selecting the suitable transition pattern is considered to be the following optimization problem. \[s^{*}=\arg\min_{s}\sum_{\{i,j\}\in\boldsymbol{u}_{s}^{\prime}}g(i,j). \tag{20}\]  
After recovering the above optimization problem, the data separation for each sensor is achieved. After deciding the connection between the data sets, the data set in \((t+1)\)th time slot connected to multiple data set in \(t\)th time slot requires the update of CSI and CFO because the individual CSI and CFO are not obtained due to large interference. In the proposed separation, the data set connected to multiple data sets is divided into the individual data set using that in \(t\)th time slot. Figure 5 shows the image of dividing the data set in \((t+1)\)th time slots. Although the sensing results are common for the divided data set, the CFO and the CSI of the data set in \((t+1)\)th time slot are overwritten by the CFO and the CSI of data in \(t\)th time slot. Therefore, the number of data sets in \((t+1)\) slots is increased, and it is equal to \(K\).