# Combined Channel Estimation with Interference Suppression in CPSS
## 3 Channel Estimation with Interference Suppression  
### Interference Suppression in CIR  
We perform LS estimation to Equation (3), and the CIR of frequency domain is denoted as \(\hat{H}_{LS}^{(1)}(k,l)\)  
\[\hat{H}_{LS}^{(1)}(k,l)=\frac{Y(k,l)}{X(k,l)}=H(k,l)+\frac{J(k_{l},l)}{X(k,l)} +\frac{W(k,l)}{X(k,l)}. \tag{4}\]  
It can be seen from Equation (4) that there are three categories of signals in the \(l\)th symbol in \(k\)th carrier: CIR, interference and noise. The existence of interference is uncertain, and the signal feature is unknown. System performance will be deteriorated with the interference incorrectly cancelled as AWGN [23]. It is necessary to detect and suppress the interference before noise cancellation. Usually, NBI is added to partial subcarriers by high-powered signals. Thus, the receive signal power of the subcarriers that are interfered with will be higher than the others. As in CIR, the modulus of the interfered subcarriers will be higher than the others. On the basis of the feature, we introduced machine learning techniques into interference cluster. Because the interference does not necessarily exist in CIR, we proposed using the clustering of interfered subcarriers based on the global \(K\)-means algorithm [24]. The main thought of the \(K\)-means algorithm is to start from the data to be measured, first randomly generate \(K\) clusters, and then calculate the similarity between the data to be measured and the cluster center, and continuously update both the cluster center and the cluster in the iterative calculation until the convergence of the cluster center point is completed. For interference clustering, the set of subcarriers with no interference is denoted as cluster \(C_{1}\), and the set of interfered subcarriers is denoted as cluster \(C_{2}\). Therefore, interference detection is equal to the cluster of samples in CIR. The steps of \(K\)-means interference clustering are as follows:  
**Step 1. ** The data to be measured is denoted as \(m_{i}\) and \(m_{i}\in\{|\hat{H}_{LS}^{(1)}(k,l)|\},i=1,2,...,N\). The cluster center of \(C_{1}\) is defined as the mean of \(m_{i}\), which is denoted as \(m_{t}\):  
\[m_{t}=\frac{1}{N}\sum_{i=1}^{N}m_{i}. \tag{5}\]  
Figure 1: Power spectrum of NBI. (**a**) single tone jamming; (**b**) multitone jamming. **Step 2. ** Euclidean distance is used to measure the similarity between \(m_{t}\) and \(m_{i}\). The calculation formula is given as Equation (6):  
\[d_{i,t}=m_{i}-m_{t}. \tag{6}\]  
**Step 3. ** The cluster center of \(C_{2}\) is denoted as \(m_{s}\), which represents the data of the longest distance from \(m_{t}\) in Equation (6). The distance between \(m_{s}\) and \(m_{i}\) is calculated as Equation (7):  
\[d_{i,s}=m_{i}-m_{s}. \tag{7}\]  
**Step 4. **\(m_{i}\) will be classified by the distance calculated in Equations (6) and (7):  
\[\left\{ \begin{array}{ll}m_{i}\in C_{1}&\text{if }d_{i,t}<d_{i,s},\\ m_{i}\in C_{2}&\text{if }d_{i,t}\geq d_{i,s}.\end{array} \right. \tag{8}\]  
**Step 5. ** The cluster center \(m_{s}\) will be updated as the cluster \(C_{2}\) was updated. Step 3 and Step 4 will be repeated until all data to be measured are classified. The calculation formula of \(m_{s}\) updating is given as Equation (9):  
\[m_{s}=\frac{1}{Q}\sum_{i\in C_{2}}m_{i}, \tag{9}\]  
where \(Q\) is denoted as the number of samples in \(C_{2}\). According to the interference cluster scheme, the modulus of samples in \(C_{2}\) are bigger than those in \(C_{1}\). The power of a few subcarriers is singularly high for \(Q\ll N\). It is fair to hypothesize that the corresponding subcarriers have been interfered with. When \(Q\) is not much less than \(N\), the corresponding subcarriers are not interfered with or the interference is not significant to the subcarriers. Nevertheless, the subcarriers may be subject to non-narrowband interference and it is not the scope of this study. It is necessary to suppress interference for \(Q\ll N\).