# Clustering-Based Nonstationary Massive MIMO Channel Modeling at 1.4725 GHz
## 3 Channel Model
\(tr\{\cdot\}\) is the trace of matrix, and \(||\cdot||_{f}\) is the Frobenius norm. The search process of the stationary subinterval is actually the process of extracting the similarity of the correlation matrix. The algorithm assumes that there is no overlap between different stationary subintervals. We mainly focus on the influence of local scatterers on the link from the user to the base station. In this way, the signals from the user arrive at some or all elements of the antenna array after scattering. For the case where only some elements are under the coverage of the scatterer, the visible region of the scatterer only covers some elements, as shown in the above Figure 3.  
According to Cauchy Schwartz inequality, as to matrices \(A\) and \(B\), we have  
\[|\langle A,B\rangle|^{2}\leq||\,A\,||^{2}||\,B\,||^{2} \tag{15}\]  
Here \(A=cB\) is the necessary and sufficient condition for the equation to hold. \(A,B\) means the inner product, which can be decomposed by the singular value as below:  
\[\Theta_{1}\Theta_{2}=U_{1}\Lambda_{1}U_{1}^{H}U_{2}\Lambda_{2}U_{2}^{H} \tag{16}\]  
\(U_{i}\) and \(\Lambda_{i}\) represent the eigenvector matrix and the eigenvalue diagonal matrix of \(\Theta_{i}\), respectively. The value \(tr\{\Theta_{1}\Theta_{2}\}\) tends to zero, which means that there is only one module factor difference between matrices. Contrarily, if the value tends to 1, this means that the matrices are independent, that is, the matrices are extremely different. Equation (14) can be rewritten as the following deformation forms  
\[d_{corr}(\Theta_{1},\Theta_{2})=1-\frac{\langle vec\{\Theta_{1}\},vec\{ \{\Theta_{2}\}\}\rangle}{||vec\{\{\Theta_{1}\}}||_{2}||vec\{\{\Theta_{2}\}}|| _{2}} \tag{17}\]  
Therefore, it can be assumed that the fading of radio clusters from the same scatterer is stationary, so that the stationary region \(q\) can be defined, including the number of antenna elements \(r_{q}\). Here, the threshold of the correlation coefficient is 0.75. It is considered that the two correlation matrices are nonstationary if \(d_{corr}(\Theta_{1},\Theta_{2})>0.75\). Note that we suppose there is no overlap between regions, so there are  
\[\sum_{q=1}^{Q}r_{q}=M \tag{18}\]Then, the non-stationary Kronecker correlation matrix is  
\[R_{Tx}=\text{bdiag}\big{[}\Theta_{1},\Theta_{2},\cdots,\Theta_{Q} \big{]}\] \[=\left[ \begin{array}{cccc}\Theta_{1}&&0\\ &\Theta_{2}&&\\ &&\ddots&\\ 0&&\Theta_{Q}\end{array} \right]_{M\times M} \tag{19}\]  
Where \(\text{bdiag}[\cdot]\) is the operator of matrix diagonalization, which is realized by arranging the vector element to the diagonal position of the target matrix. \(\Theta_{q}\) is the \(r_{q}\times r_{q}\) square matrix. The cluster correlation distance can be calculated by a stepping rate of \(1\) on the array, and the M-element distance vector can be obtained. \[d_{EMD}=[d_{corr,1},d_{corr,2},\cdots,d_{corr,M}] \tag{20}\]  
If \(d_{corr,k}\) is larger than the preset threshold, and there are \((q-1)\) values within \(k=1,2,\ldots,M\), then the correlation matrix is divided into matrix blocks with number \(q\). In each matrix block with dimension \(r_{q}\), the cluster correlation distance is recalculated with stepping rate of \(2\), and a \(\big{(}r_{q}\text{--}3\big{)}\) element vector is obtained. The entry of matrix block larger than the threshold is searched, and the block is further split. This is repeated until the matrix dimension is smaller than the stepping rate. Finally, every splitting block is the stationary subinterval of the correlation matrix. In fact, the ECD algorithm is a linear projection algorithm, which aims to reduce dimensions in high-dimensional data space, such as a vector set of power, delay and angle in our application. By summarizing the linear weighted channel parameters, the parameters are converted to a one-dimensional object. Then, employing classification and clustering methods, to maximize the distance between classes, the correlation matrix is constructed.