# Clustering-Based Nonstationary Massive MIMO Channel Modeling at 1.4725 GHz
## 3 Channel Model
The final MCD distance is the RMS model of the weighted sum of time-delay and angle domain distance as follows:  
\[d_{MCD}(i,j)=P_{j}\sqrt{||MCD_{AOA,ij}||^{2}+||MCD_{AOD,jj}||^{2}+||MCD_{\tau, ij}||^{2}} \tag{11}\]  
At the same time, we can calculate the matrix of distance as  
\[D_{k\times L}=\left[\begin{array}{cccc}d(1,1)&d(1,2)&\ldots&d(1,L)\\ d(2,1)&d(2,2)&\ldots&d(2,L)\\ \ldots&\ldots  
where \(\left\{p_{ij}\right\}_{k\times L}\) is a two-dimensional power matrix.  
According to the programming results, the multipath is assigned to the nearest cluster, and the cluster cores are recalculated again. An iteration procedure is used to make sure that all the multipath components are divided into corresponding clusters.  
Taking the outdoor LOS scenario as an example, the measured PDP of the antenna sub-channel is plotted as shown in the curve in Figure 7a. The average noise power and signal threshold are counted according to the estimation algorithm of bottom noise, which is identified by the horizontal real line segment in the figure. The signal implicates fluctuating nature. Peaks and valleys can be seen on the curve. According to the theory of radio wave propagation, the top points represent the arrival of multipath components with strong energy. Among all the multipath components, due to spatial scattering, we assume that the LOS path exists. It is obvious that the time-delay is the smallest and the power is the strongest as to the LOS component. Therefore, based on the traditional multi-path peak search algorithm, we can reasonably set the number of clusters as the number of peak points on the PDP, and we define the cluster core as the corresponding multipath component of the peak points. Please refer the marked points in Figure 7a for the core location.  
The inherent flaw of KPowerMeans algorithm is that the random hypothesis of the initial cluster core might cause the classification result to converge to the local optima rather than the global optimal solution. This problem can be overcome by using the method of peak searching on PDP. Determining the number of clusters and the initial cluster core by measurement data can greatly improve the effectiveness and reliability of the results. However, the peak search algorithm only considers the distribution of multipath in the time-delay domain, so the classification result is necessary rather than sufficient. On the other hand, the best advantage of the KpowerMeans algorithm is that the contributions of the spatial parameters are fully considered. By analyzing the results of multipath in the spatial domain, the AOA and AOD parameter pairs are obtained, and the MCD distance in the delay domain and angle domain are calculated synthetically to cluster the multipath.  
Taking the outdoor LOS scenario as example, the results of the final multipath clustering are shown in Figure 7b. In the figure, the blue solid dot represents the spatial distribution of multipath in the sub-channel of array element, and the red solid dot represents the clustering result, namely the statistical distribution of cluster core. The implementation of the MCD algorithm is simple, and the clustering performance is optimized compared with the traditional algorithm.  
Figure 7: Cluster distribution of antenna element channel. (**a**) The PDP of the channel between No. 1 transmit and No. 1 receive element; (**b**) multipath spatial distribution.  
In the second step, the ECD algorithm is used to program the stationary subinterval of the antenna array based on the cluster visible region theory. Because of obstacles, the radio reaches only several elements of the antenna array; that is, only some elements of the array are covered by some scatterer. It can be assumed that the radio wave coming from the same scatterer is coincident. Therefore, the stationary region can be defined. In the process of programming the stationary subinterval in the array dimension, it is key to determine the similarity. The distance criterion is still used here, and the distance of the correlation matrix is introduced to the work. The accuracy of measurements directly impacts the model. The matrix of distance is the metric to the similarity between two correlation matrices. The expression of the distance formula of the correlation matrix is [18]  
\[d_{corr}(\Theta_{1},\Theta_{2})=1-\frac{tr\{\Theta_{1}\Theta_{2}\}}{\Big{|} \Big{|}\Theta_{1}\Big{|}\Big{|}\rho_{2}\Big{|}\Big{|}_{f}} \tag{14}\]  
Here \(\{\Theta_{i}|i=1,2,\ldots\}\) is the autocorrelation expression of sub-channel matrix.