## 3 Results and Discussion
### MIMO Characteristics  
One of the main parameters determining how well the antennas are isolated from one another is the envelope correlation coefficient (ECC) [21]. Lower ECC values indicate that the antenna elements are less correlated and more isolated [22]. An ECC value of 0.5 or lower is preferable for better performance [23]. Equations (3) and (4) can be used to calculate the ECC in terms of S-parameters and far-field radiation patterns, respectively. \[ECC_{\text{s-parameters}}=\frac{\left|S_{ii}*S_{ij}+S_{ji}*S_{ji}\right|^{2}}{ \left(1-\left|S_{ii}\right|^{2}-\left|S_{ij}\right|^{2}\right)\left(1-\left|S_ {ji}\right|^{2}-\left|S_{ji}\right|^{2}\right)} \tag{3}\]  
\[ECC_{far-field}=\frac{\left|\iint_{4\pi}\left[\overset{\rightarrow}{R_{i}} \left(\theta,\varphi\right)\times\overset{\rightarrow}{R_{j}}\left(\theta, \varphi\right)\right]\,d\Omega\right|^{2}}{\iint_{4\pi}\left|\overset{ \rightarrow}{R_{i}}\left(\theta,\varphi\right)\right|^{2}d\Omega\,\iint_{4 \pi}\left|\overset{\rightarrow}{R_{j}}\left(\theta,\varphi\right)\right|^{2} d\Omega} \tag{4}\]  
The reflection coefficient is represented by \(S_{ii}\), while the transmission coefficient is represented by \(S_{ij}\). \(\overset{\rightarrow}{R_{i}}\left(\theta,\varphi\right)\) and \(\overset{\rightarrow}{R_{j}}\left(\theta,\varphi\right)\) denote the three-dimensional radiation patterns of the ith and jth antennas, respectively, and \(\Omega\) represents the solid angle. As depicted in Figure 14, the recommended MIMO antenna's simulated and measured ECC is extremely low: less than 0.0002 in the desired bands. A low ECC indicates minimal correlation among the signals received by different antenna elements, resulting in higher data rates and improved spectral efficiency. The diversity gain (DG) parameter measures how effectively the introduction of additional antennas (MIMO antennas) reduces the transmission loss or degradation that can occur in wireless communication due to factors like fading, interference, and attenuation [24; 25]. The DG can be calculated by using Equation (5):  
\[DG=10\sqrt{1-\left|ECC_{s-parameters}\right|^{2}} \tag{5}\]  
A diversity gain of higher than 9.991 dB is achieved, nearly reaching the optimal value of 10 dB, as shown in Figure 15.  
Figure 13: (**a**) The 3D radiation pattern at 2.5 GHz; (**b**) 3D radiation pattern at 6.2 GHz; (**c**) simulated and measured radiation patterns for XZ-plane (H-plane) at 2.5 GHz; (**d**) simulated and measured radiation patterns for YZ-plane (E-plane) at 2.5 GHz; (**e**) simulated and measured radiation patterns for XZ-plane (H-plane) at 6.2 GHz; (**f**) simulated and measured radiation patterns for YZ-plane (E-plane) at 6.2 GHz. A channel capacity loss (CCL) between antenna elements may arise as a result of correlation effects [26]. The formula provided in Equation (6) can be utilized for calculating the CCL:  
\[CCL=-log_{2}\Big{(}1-|S_{ii}|^{2}-\big{|}S_{ij}|^{2}\Big{)} \tag{6}\]  
Figure 16 shows that in the desired bands, the simulated and measured values of CCL are less than 0.1 bps/Hz, which is below the acceptable limit of 0.5 bps/Hz [27]. The only parameter employed to determine the unpredictable phase shifts between incoming signals is the total active reflection coefficient (TARC). The TARC is defined as  
Figure 14: Simulated and measured ECC values of the proposed MIMO antenna. Figure 15: Simulated and measured diversity gain of the proposed MIMO antenna. the square root of the ratio of the total reflected power to the total incident power [28; 29]. Equation (7) can be used to determine the TARC based on S-parameters:  
\[TARC=\frac{\sqrt{\left|S_{ii}+S_{ij}e^{\left|\theta\right|}\right|^{2}+\left|S_{ ji}+S_{ii}e^{\left|\theta\right|}\right|^{2}}}{\sqrt{2}} \tag{7}\]  
TARCs were obtained for different excitation phases (\(\theta\)) which vary from \(0^{\circ}\) to \(180^{\circ}\) in steps of \(30^{\circ}\).