# Combined Channel Estimation with Interference Suppression in CPSS
## 3 Channel Estimation with Interference Suppression
### Noise Cancellation in CIR  
The noise term in Equation (11) can be cancelled in the CIR of time domain. To enhance resolution of the CIR of time domain, \(M=2N\) points Inverse Fast Fourier Transform (IFFT) is applied to \(\hat{H}_{LS}^{(2)}(k,l)\), so we have Equation (12):  
\[\hat{h}_{LS}^{(2)}(m,l)=h^{\prime}(m,l)+\frac{1}{M}\sum_{k=1}^{M-1}\frac{W(k,l )}{X(k,l)}e^{j2\pi\frac{km}{M}}. \tag{12}\]  
Because the IFFT of \(W(k,l)\) is also approximately Gaussian, the noise of the time domain can be denoted as \(w(m,l)\). Equation (12) can be rewritten as:  
\[\hat{h}_{LS}^{(2)}(m,l)=h^{\prime}(k,l)+w(m,l). \tag{13}\]  
Because the envelop feature of \(\hat{h}_{LS}^{(2)}(m,l)\) is known to the receiver, \(h^{\prime}(k,l)\) and \(w(m,l)\) can be classified by the machine learning mechanism. We proposed the classification of CIR and noise based on the \(K\)-nearest neighbor (KNN) algorithm [26]. The mean thought of KNN is to set up a training sample for each category, and then find out \(K\)-nearest neighbors which have the greatest similarity with the data to be measured from all the training sample sets. The data to be measured will be classified into the category that is the majority of its neighbors. According to the envelop feature of \(\hat{h}_{LS}^{(2)}(m,l)\), which can be decomposed into two sets theoretically, one of the sets is denoted as \(\widetilde{S}\), which represents the set of modulus of signal components. Because of the leakage of CIR, \(\widetilde{S}\) exists at the beginning and end of \(\hat{h}_{LS}^{(2)}(m,l)\):  
\[\widetilde{S}=\{|\hat{h}_{LS}^{(2)}(m,l)||0\leq m\leq 2M_{CP}\}\cup\{|\hat{h}_{ LS}^{(2)}(m,l)||M-M_{CP}\leq m\leq M\}, \tag{14}\]  
where the maximum multipath delay is denoted as \(M_{CP}\). The other set is denoted as \(\widetilde{W}\), which represents the set of modulus of noise components:  
\[\widetilde{W}=\{|\hat{h}_{LS}^{(2)}(m,l)||2M_{CP}<m<M-M_{CP}\}. \tag{15}\]  
For the noise components in \(\widetilde{S}\) realistically, we set up the training samples of signal component by the largest \(P\) samples in \(\widetilde{S}\). The initial set of training samples of signal component is denoted as \(G_{1}\):  
\[ \begin{split}& G_{1}=\{u_{i}|(u_{1}\geq u_{2}\geq...\geq u_{P-1} \geq u_{P})\wedge u_{i}\in\widetilde{S}\},\\ &\hat{\mu}_{1}=\frac{1}{P}\sum\limits_{m=1}^{P}u_{m},\\ &\hat{\sigma}_{1}=\frac{1}{P-1}\sum\limits_{m=1}^{P}|u_{m}-\hat{ \mu}_{1}|^{2},\end{split}  \tag{16}\]  
where \(\hat{\mu}_{1}\) and \(\hat{\sigma}_{1}\) are the mean and variance of the elements in \(G_{1}\). According to the principle of \(3\sigma\) in Gaussian distribution, the modulus of noise samples less than \(\sigma\) was a big probability. For there are signal components in \(\widetilde{W}\) realistically, we set up the noise training samples with modulus less than \(\sigma\) in \(\widetilde{W}\). The initial set of training samples of noise component is denoted as \(G_{2}\):  
\[ \begin{array}{l}G_{2}=\{v_{i}|(-\sigma<v_{i}<\sigma)\wedge v_{i}\in\widetilde{ W}\},\\ \hat{\mu}_{2}=\frac{1}{Q}\sum\limits_{m=1}^{Q}v_{m},\\ \hat{\sigma}_{2}=\frac{1}{q-1}\sum\limits_{m=1}^{Q}|v_{m}-\hat{\mu}_{2}|^{2}, \end{array}  \tag{17}\]  
where \(\sigma\) is the variance of the elements in \(\widetilde{W}\), \(\hat{\mu}_{2}\) and \(\hat{\sigma}_{2}\) are the mean and variance of the elements in \(G_{2}\). It can be seen from Equations (16) and (17) that the statistical properties of \(G_{1}\) and \(G_{2}\) are different. We should calculate the similarity by the standardized Euclidean distance, in order to eliminate the error by the distribution difference of samples. The data to be measured is denoted as \(q_{m}\in\{|\hat{h}_{LS}^{(2)}(m,I)|\},m=1,2,...,M.\) The steps of KNN classification are as follows. **Step 1. ** Calculate the standardized Euclidean distance between \(q_{m}\) and all the samples in \(G_{1}\) and \(G_{2}\). \(D_{m}\) is the set of all the calculation results:  
\[