## 3 Overview of PhyC-SN  
### Data Transmission and Collection in PhyC-SN  
Figure 1 shows the system model of the considered WSN. Since multiple sensors access to a FC via wireless communication, a star type network topology is assumed. Sensing results are modeled as multi-level discrete information. The transmitter of PhyC-SN transmits the continuous carrier whose frequency is a discrete value decided by inverse discrete Fourier transform (IDFT) and is selected in accordance with the conversion table between the discrete frequency and the sensing result, where the carrier with discrete frequency is referred to as subcarrier similar to an orthogonal frequency division multiplexing (OFDM). For example, if the sensing results are at fifth level, the sensor sends the fifth subcarrier to the FC. The modulation scheme of PhyC-SN is the same as a frequency shift keying. The maximal levels of sensing results are equal to the number of subcarriers. All the sensors send subcarrier to FC, simultaneously. Therefore, FC receives the signal composed of the mixed subcarriers.  
When the \(k\)th sensor (\(k=1,2,\ldots,K\) and \(K\) is the total number of sensors) sends \(n_{k}\)th subcarrier (\(n_{k}\in{1,2,\ldots,N}\) and \(N\) is the total number of subcarriers) to the FC, the transmit signal in the \(m\)th time interval is decided using IDFT as follows.  
\[x_{k}(m)=\exp\left(j2\pi\frac{n_{k}m}{N}\right) \tag{1}\]  
Wireless communication is modeled as the multipath fading with \(L\) paths and independent identical distribution for each path and each user. The impulse response and the delay time in the \(l\)th path of the \(k\)th sensor are \(h_{k,l}\) and \(\tau_{k,l}\), respectively, where \(\tau_{k,l}\) includes the time difference caused by detection timing mismatch.  
In \(m\)th time interval, the signal received by the FC is given as follows.  
\[y(m)=\sum_{k=1}^{K}\sum_{l=1}^{L}h_{k,l}\exp\left(j2\pi\frac{n_{k}m -n_{k}\tau_{k,l}+m\epsilon_{k}}{N}\right)\] \[+n_{0}(m), \tag{2}\]  
where \(n_{0}(m)\) is a noise component and \(\epsilon_{k}\) (\(-1.0<\epsilon<1.0\)) is the CFO derived by the frequency mismatch of local oscillator between \(k\)th sensor and FC. The value of CFO is normalized by the frequency interval of subcarriers.  
FC analyzes a frequency spectrum of received signal by DFT. In PhyC-SN, all the sensors send the continuous wave for certain time duration. We assume the access timing difference among the sensors to be smaller than a time duration of DFT detection. Since DFT can detect symbol duration of all the subcarriers, the distortion of subcarrier orthogonality is avoided.  
The \(v\)th subcarrier component is given as follows.  
\[Y(v)=\sum_{m=1}^{N}y(m)\exp\left(-j2\pi\frac{mv}{N}\right),\] \[=\sum_{k=1}^{K}\gamma_{k}\sum_{m=1}^{N}\exp\left(j2\pi\frac{n_{k} m+m\epsilon_{k}-mv}{N}\right)+\sum_{m=1}^{N}n_{0}(m)\exp\left(-j2\pi\frac{mv}{N}\right) \tag{3}\]  
where \(\gamma_{k}\) is a CSI of the wireless channel between \(k\)th user and FC and it is defined as follows.  
\[\gamma_{k}=\sum_{l=1}^{L}h_{k,l}\exp\left(j2\pi\frac{-n_{k}\tau_{k,l}}{N}\right) \tag{4}\]  
From the conversion table between the subcarrier number and the sensing results, FC can recognize all the sensing results and then can also evaluate the median and the outlier of all the sensing results at once.  
If the second term of Equation (3), which is the noise component, is ignored, the subcarrier component in \(v\
eq n_{k}\) appears and it is an ICI. For simple explanation, the spectrum of subcarrier in \(v=n_{k}\) and \(v\
eq n_{k}\) are the main lobe and side-lobe, respectively. The ICI components mask the subcarrier components sent by the other sensor and generate the image subcarrier that causes the false subcarrier detection by FC [6].