## 4 Proposed Sensing Results Separation
### Sensing Results Separation with Duplication of Sensing Results  
#### 4.3.1 Detecting Duplication of Sensing Results  
We assume certain sensors, or all the sensors take common sensing result. Figure 4 shows the image of the data set in the \(t\)th and \((t+1)\)th time slots. We define the sensors taking the sensing result in the \(j\)th data set as the sensor group of \(\mathbf{p}_{j}\in\mathcal{I}^{D_{j}}\), where \(\mathcal{I}\) is the integer field and \(D_{j}\) is the number of sensors taking the sensing data in \(j\)th data set.  
In the \((t+1)\)th time slot, the \(n_{j}\)th spectrum detected via DFT is given as follows  
\[Y(n_{j})=\sum_{k\in\mathbf{p}_{j}}\gamma_{k}\sum_{m=1}^{N}\exp\left(j2\pi\frac{m \epsilon_{k}}{N}\right). \tag{13}\]  
The noise components are also ignored. Since the subcarrier sent by each sensor is interfered with each other, the CSI and the CFO in each sensor node is hardly estimated. In the proposed separation,  
Figure 4: Example of labeling problem for data separation with the duplication of sensing results.  
regardless of the duplication of sensing results, FC selects the spectrum with the maximal power and estimates the CSI and the CFO from the selected spectrum. Even under the duplication of the sensing results, it hardly occurs that the power of side-lobe of subcarrier is larger than that of the main lobe of subcarrier as large as the absolute value of CFO is smaller than 0.5. Therefore, it is reasonable assumption that the selected spectrum with maximal power is equal to the main lobe of the subcarrier even under the duplication of sensing results. From the \(j\)th subcarrier, the CFO, \(\hat{\epsilon_{j}}\), is estimated in accordance with Equation (9). Note that the estimated CFO, \(\hat{\epsilon_{j}}\), is different from the individual CFO of each sensor because the mutual interference among subcarriers is exceedingly large. The CSI of the \(j\)th subcarrier is estimated as  
\[\hat{\gamma_{j}}=\frac{Y(n_{j})}{\sum_{m=1}^{N}\exp(j2\pi\frac{m \hat{\epsilon_{j}}}{N})}. \tag{14}\]  
As a result, the constructed ICI replica is given as  
\[\hat{Y}(n_{j})=\frac{Y(n_{j})}{\sum_{m=1}^{N}\exp(j2\pi\frac{m \hat{\epsilon}}{N})}\sum_{m=1}^{N}\exp\left(j2\pi\frac{n_{k}m+m\hat{\epsilon} -mv}{N}\right). \tag{15}\]  
It is different from the original ICI given by Equation (3). Therefore, the effect of suppressing ICI is limited. However, the main lobe of the constructed replica is matched to the original one because the detected spectrum, \(\hat{Y}(n_{j})\) is the same as the original one \(Y(n_{j})\). Therefore, the main lobe of the subcarrier can be removed. In the proposed separation, the scheme for counting the types of subcarriers is proposed, where a type of subcarrier is the subcarrier selected by one or more sensor nodes.  
We define the protocol, which is the selection of subcarrier in terms of maximal power criterion, the estimation of CSI and CFO, and the cancellation by constructed ICI replica as the iterative protocol. After the iterative protocol, FC can detect the subcarrier with power exceeding a certain threshold. If it is true, the iterative protocol is performed again and then comparison between the power of subcarrier and the certain threshold is conducted. Otherwise, it is stopped. If the number of constructed ICI replicas during the iterative protocols is equal to the number of sensors, \(K\), the duplication of sensing results does not occur. Otherwise, it occurs.  
If the CFO is so large that the power of ICI is large, the residual ICI is still large even after ICI cancellation. Especially, if the duplication of sensing results occurs, it is large because the mitigation of ICI is limited. Therefore, the CFO should be so small that ICI is not so large.