# Clustering-Based Nonstationary Massive MIMO Channel Modeling at 1.4725 GHz
## 2 Channel Measurement
### Signal Model  
Describing a wireless communication system, the received signal can be decomposed into a large number of multipath signals. Furthermore, multipath can be classified and merged to form clusters according to the correlation of parameters. For a given multipath, the propagation can be characterized by power, delay, polarization, Doppler velocity, departure angle, arrival angle and so on. Considering a 3D (3 dimension) model, the angle of elevation should also be included. Therefore, we can treat the cluster as a multipath set that the multipath parameters in the set are similar. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Quantity** \\ \hline Central frequency & 1.4725 GHz \\ Bandwidth & 100 MHz \\ Transmit power & 25 dBm \\ Excitation signal & Zad-Off Chu sequence \\ Code length & 2047 \\ Tx antenna number & 128 \\ Rx antenna number & 2 \\ Tx antenna height & 30 m \\ Rx antenna height & 2 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Measurement system specifications. Figure 2: The measurement system setup. The different locations of the antenna element determine the inconsistency of the scattering environment. As a result, the observed incident waves were different from the antennas, whether in quantity, in energy, or in angle of radio waves, as shown in Figure 3. In this case, the correlation coefficient between different antenna elements in the correlation matrix will be inconsistent [17]. Although the Massive MIMO system's complexity and difficulty will be increased because of the non-stationarity in the array domain, some methods still can be employed to reduce the complexity. In the far-field conditions, we can divide the antenna elements into several adjacent subsets according to their spatial positions, so that the antenna arrays in each subset are compact enough; that is, we can assume that the large-scale fading of all elements in the subset is the same as that of the target user [18]. With this assumption, the channel in each subset satisfies the generalized stationary non-correlation. In this way, we can use the traditional modeling theory to carry out channel modeling based on stationary characteristics in a subset window, which satisfies the stationary modeling conditions. Then the subset combines to the whole channel model, which can accurately reflect the characteristics of Massive MIMO channel. Considering a system with transmit antennas of M and single antenna at the receiver, for the traditional MIMO channel, the small-scale propagation matrix can be expressed as  
\[H=\left[ \begin{array}{cccc}h_{11}&h_{12}&\cdots&h_{1M}\\ h_{21}&h_{22}&\cdots&h_{2M}\\ \vdots&\vdots&&\vdots\\ h_{K1}&h_{K2}&\cdots&h_{KM}\end{array} \right] \tag{1}\]  
where \(h_{km}\) presents the sub-channel impulse response between the \(m\)-th antenna element and the \(k\)-th user. Given the transmit signal \(x_{m}(t)=a_{0}\ e^{iwt}\delta(t)\), then we have  
\[h_{km}(t)=\sum_{l=1}^{L_{km}}a_{l}\ e^{i[\omega(t-\tau_{l})]} \tag{2}\]  
where \(a_{l}\) is the amplitude of the \(l\)-th incident signals, and \(L_{km}\) is the total number of multipath. \(\tau_{l}\) is the time-delay. Here, we assume that the antenna has a uniform radiation pattern and ignore the influence of Doppler shift. Figure 3: Nonstationary of Multipath over the Antenna Array. Usually, we introduce correlation matrix \(R\) to research the correlation, which can be obtained by the long-term observations of channel matrix \(H\). \[R=\text{ivec}\Big{(}E\Big{(}\text{vec}(H)\cdot\text{vec}(H)^{H}\Big{)}\Big{)} \tag{3}\]  
where \(\text{vec}(\cdot)\) presents the operator of converting Matrix to Vector, and \(\text{ivec}(\cdot)\) presents the inverse operation. According to [PERSON]'s channel model, we have  
\[H=R_{Rx}^{1/2}\cdot W\cdot R_{Tx}^{1/2} \tag{4}\]  
where \(W\) presents the matrix that the entry obeys independent identical distribution complex Gaussian distribution with zero-mean. \(R_{Rx}\) and \(R_{Tx}\) are the correlation matrixes on BS and MS, respectively.