## 5 Results and Discussion
### MIMO Diversity Characteristics  
The diversity properties of the proposed dual band-notch MIMO antenna are evaluated. The envelope correlation coefficient (ECC) [27] illustrates the degree of correlation or isolation among the MIMO elements. The intensity of the received pulse's signal-to-noise ratio defines the diversity gain (DG). The accepted ideal values for the ECC and DG are less than 0.5 and close to 10 dB. The antenna's S11 does not appropriately account for radiation efficiency and bandwidth. As a result, the total active reflection coefficient (TARC) for S11 can be employed [28]. TARC takes into account the coupling impact of the MIMO system and random signal combinations. The mean effective gain (MEG) measures the MIMO antenna's capacity to accept electromagnetic energy in fading environments [29]. The mean received power of a test antenna in relation to the ideal antenna is known as MEG. The channel capacity grows proportionally as the number of antenna components rises. However, placing many antennas close together creates a correlation and leads to channel capacity (CCL). This figure of merit quantifies the maximum rate of the channel's data transfer for reliable communication. The multiplexing efficiency statistic is a crucial diversity metric (ME). By calculating the efficiency (\(\eta\)) and degree of correlation (\(\rho_{ij}\)) among the system's components, the ME describes the MIMO system. The aforementioned diversity metrics are computed using Equations (15)-(21). The MIMO diversity features of the projected antenna are represented in Figure 22.  
\[ECC(\rho_{ij})=\frac{\left|\int_{0}^{2\pi}\int_{0}^{\pi}\left[XPR\cdot E_{ \theta i}E_{\theta j}^{*}P_{\theta}+E_{\phi i}E_{\phi j}^{*}P_{\phi}\right] \right|^{2}}{\int_{0}^{2\pi}\int_{0}^{\pi}\left[XPR\cdot E_{\theta i}E_{\theta i }^{*}P_{\theta}+E_{\phi i}E_{\phi i}^{*}P_{\phi}\right]d\Omega\times\int_{0}^ {2\pi}\int_{0}^{\pi}\left[XPR\cdot E_{\phi j}E_{\theta j}^{*}P_{\theta}+E_{ \phi j}E_{\phi j}^{*}P_{\phi}\right]d\Omega} \tag{15}\]  
where \(XPR\) is the cross-polarization ratio, \(P_{\theta}\) and \(P_{\phi}\) are the incoming wave's angular density functions at \(\theta\) and \(\phi\) fields, \(E_{\theta}\), \(E_{\phi}\) are the complex envelopes across \(\theta\) and \(\phi\) components.  
\[ECC(\rho_{ij})=\frac{-\sum_{n=1}^{N}S_{ni}^{*}\ S_{nj}}{\sqrt{(1-\sum_{n=1}^{ N}\left|S_{ni}\right|^{2})(1-\sum_{n=1}^{N}\left|S_{nj}\right|^{2})}} \tag{16}\]  
where \(S_{ni}\) and \(S_{nj}\) are the S-parameters of \(i\) and \(j\) ports, respectively, \(S_{ni}^{*}\) is conjugate of \(S_{ni}\).  
\[DG=10\sqrt{1-\left|\rho_{ij}\right|^{2}} \tag{17}\]  
\[\text{TARC}(\Gamma_{a}^{t})=\frac{\sqrt{\sum_{j=1}^{4}\left|S_{j1}+\sum_{m=2}^ {4}S_{jm}e^{i\theta_{m-1}}\right|^{2}}}{\sqrt{4}} \tag{18}\]  
\[MEG_{i}=0.5\ \eta_{i,rad}=0.5\left(1-\sum_{j=1}^{p}\left|S_{ij}\right|\right) \tag{19}\]  
\[CCL=-log_{2}\text{det}(\alpha)^{R} \tag{20}\]where \(a^{R}\) is the receiving antenna correlation matrix.  
\[ME(\eta_{max})=\sqrt{\eta_{1}\eta_{2}\big{(}1-\rho_{ij}\big{)}} \tag{21}\]  
Figure 22: Cont.  
Figure 22 shows the values of the diversity metrics as ECC < 0.001 (using S-parameter) and less than 0.3 (using a 3D pattern), DG close to 10 dB (using S-parameter) and >9.6 dB (using a 3D pattern), MEG < \(-\)3 dB, TARC < \(-\)10 dB, CCL < 0.3 bps/Hz, and ME <\(-\)2 dB across the operational frequency except for the notched bands.