## 3 Method to Control the BS Correlation
Therefore, the combined signal varies depending on the initial phases of the probes, even though the set of received signals from the probes are the same since the terminal antenna in a fading emulator does not move, which results in the generation of an incoming signal to any BS antenna. Based on the above-mentioned principle, the setting of the initial phases of the probes is very important to embody a realistic BS correlation for the uplink channel. Figure 3 shows the implementation model used to achieve the BS correlation. In Figure 3, the virtual point sources illustrated by the green circles are positioned to generate a geometrical phase difference that realizes the desired BS correlation. Therefore, it is different from the actual BS antenna arrangement and the channel model used to evaluate a MIMO antenna using the fading emulator or Monte Carlo simulation. The received signals of each BS antenna can be observed using different initial phase matrices depending on the desired BS correlation \(\mathbf{\Phi}_{m}\). When the reference source is the virtual point source of BS #1 placed at the center of the implementation model, the virtual point source of BS #_m_ is arranged at a distance \(d_{1m}\) from that of BS #1. Then, the distance \(d_{1m}\) is decided depending on the desired BS correlation using [PERSON] model. Therefore, the geometrical phase difference between the virtual point source of BS #_m_ and probe #_i_\(\alpha_{mi}\) is given by  
\[\alpha_{mi}=\frac{2\pi d_{1m}}{\lambda}\cos\phi_{i}. \tag{3}\]  
The initial phase matrix corresponding to BS #_m_\(\mathbf{\Phi}_{m}\) is the sum of the initial phase matrix for BS #1, which is generated by a random number, and the geometrical phase difference matrix using Equation (3), and it is defined as  
\[ \begin{array}{rcl}\mathbf{\Phi}_{m}&=&\mathbf{\Phi}_{1}+\mathbf{A}_{m}\\ &=&\left[\begin{array}{cc}\varphi_{11}+\alpha_{m1}&\varphi_{21}+\alpha_{m2} &\cdots&\varphi_{K1}+\alpha_{mK}\end{array}\right]\end{array}  \tag{4}\]  
In [PERSON] model, spatial correlation is calculated as a function of the antenna separation. However, in this paper, it was necessary to determine the distance between the virtual point sources \(d\) according to the desired BS correlation. In [21], a third-order approximate formula for the relationship between the spatial correlation and antenna separation was derived via the least-squares method. However, the accuracy of the estimated formula is poor when the spatial correlation is close to 1. The spatial correlation is expected to be close to 1 in a four-element BS antenna depending on the direction of the incident wave; therefore, a high-accuracy 4 \(\times\) 4 correlation matrix cannot be realized using the conventional method. In this paper, the geometrical phase difference required to complete the desired BS correlation is determined via the bisection method using [PERSON] model, as depicted in Figure 4. First, the distance between the virtual point sources is set to the range from 0 to 0.4 \(\lambda\), as denoted by the blue arrows in Figure 4, so that the spatial correlation satisfies 0-1. Then, the spatial correlation at \(d\) = 0.2 \(\lambda\), which is the midpoint of the blue arrow, is calculated using a zeroth-order Bessel function of the first kind, as indicated in Equation (2). If the absolute value of the difference between the calculated spatial correlation and the desired BS correlation, \(|\Delta\rho|\), is less than the threshold value, which is a sufficiently small  
Figure 3: Implementation model for BS correlation. value, the distance between the virtual point sources is set. When it is greater than the threshold value and the calculated spatial correlation is greater than the desired BS correlation, the range of the distance between the virtual point sources is modified to the upper half of the original range. In the other case, the range is changed to the lower half.