## 4 Method for Implementing a 4 \(\times\) 4 Correlation Matrix
### Design of Implementation Model for the Conventional BS Antenna Arrangement  
Figure 9 shows the relationships between conventional BS antenna arrangements and the incoming wave angle \(\phi\). In this paper, two types of four-element BS antennas are considered: a circular and a linear array installed at equal intervals, as illustrated in Figure 9a,b, respectively [33]. The distance between adjacent antennas was set to 5 \(\lambda\). Figure 8: Arrangement of the virtual point sources to achieve the BS correlation at \(\phi=0^{\circ}\) in Figure 5: (**a**) before translating; (**b**) after translating. Figure 9: Relationship between the BS antenna arrangement and the incident wave: (**a**) circular array; (**b**) linear array. First, the 4 \(\times\) 4 correlation matrix is derived using Equation (5), considering the angle of the incident wave and the distances between the individual BS antennas. Figure 10 shows the BS correlation as a function of the incoming wave angle. As shown in Figure 10, the 4 \(\times\) 4 correlation matrix varies significantly depending on the BS antenna arrangement and the incoming wave angle \(\phi\). In the case of the circular array, illustrated in Figure 9a, the angle of the incident wave for the BS array constructed by BS #1 and BS #2 at the incoming wave angle \(\phi\) is the same as that constructed by BS #3 and BS #4, whereas the angle of the incident wave for the BS array constructed by BS #1 and BS #4 at \(\phi\) is the same as that constructed by BS #2 and BS #3. Consequently, \(\rho_{12}\) and \(\rho_{14}\) coincide with \(\rho_{34}\) and \(\rho_{23}\), respectively, regardless of the incoming wave angle \(\phi\), and there are four types of BS correlation characteristics, as shown in Figure 10a. However, in the case of a linear array, illustrated in Figure 9b, the angle of the incident wave for any BS array is the same; however, the distance between the BS antennas is different. The distance between BS #1 and #2 is the same as that between BS #2 and #3 or between BS #3 and #4, whereas the distance between BS #1 and #3 is the same as that between BS #2 and #4. Consequently, there are three types of BS correlation characteristics, as exhibited in Figure 10b. Second, based on the BS correlation characteristics shown in Figure 10, the distances between the virtual point sources are estimated via the bisection method using [PERSON]  
Figure 10: BS Correlation characteristics vs. incoming wave angle: (**a**) circular array; (**b**) linear array. model. Figure 11 shows the virtual point source separation as a function of the incoming wave angle, which ranges from 0\({}^{\circ}\) to 90\({}^{\circ}\) considering the symmetry of the BS arrangement. As described in Figure 11, the virtual point source separations according to the designed implementation model, indicated by dots in the figure, are in good agreement with the ideal distance that is required to achieve the desired BS correlation shown in Figure 10. Therefore, the 4 \(\times\) 4 correlation matrix to be mounted on the MIMO-OTA apparatus to embody the desired BS correlation can be generated by the proposed method, which determines the implementation model depending on the arrangement of the BS antenna and the incoming wave angle. As shown in the circular array, illustrated in Figure 9a, when the incoming wave angle is 45\({}^{\circ}\) the spatial correlation \(\rho_{13}\) is 1 because the array formed by BS #1 and #3 is parallel to the incident wave. Therefore, the virtual point sources for BS #1 and BS #3 have the same coordinates; therefore, there are only three virtual point sources in the implementation model. Furthermore, when the incoming wave angle is 0\({}^{\circ}\), the spatial correlations \(\rho_{12}\) and \(\rho_{34}\) are 1 because two arrays (BS #1 and #2, BS #3 and #4) are parallel to the incident wave. Hence, there are only two virtual point sources in the implementation model. Even in the case of few virtual point sources, their arrangement can be set using the proposed method, as shown in Figure 12a,b.