# Metamaterial Based Design of Compact UWB/MIMO Monopoles Antenna with Characteristic Mode Analysis
## 5 Diversity Analysis of the MIMO Antenna  
In this section, some merits are used to evaluate the performance of the proposed MIMO antenna. These are the envelope correlation coefficient, total active reflection coefficient, channel capacity loss, mean effective gain, and diversity gain. They are evaluated using the S-parameters extracted from either the electromagnetic solver or the vector network analyzer. The envelope correlation coefficient (ECC) and total active reflection coefficient (TARC) are essential parameters to quantify signal interference between MIMO channels [38,39] to secure the MIMO antenna's capability. TARC can be computed for a 2 \(\times\) 2 MIMO antenna using the following equation:  
\[\Gamma_{a}^{t}=\sqrt{\frac{\sum\limits_{i=1}^{N}|b_{i}|^{2}}{\sum\limits_{i=1}^ {N}|a_{i}|^{2}}}, \tag{20}\]  
where \(\Gamma\) is the symbol of reflection coefficient, \(t\) is the total, \(a\) is the active, \(a_{i}\) is the incident signal, and \(b_{i}\) is the reflected signal. \[TARC=\sqrt{\frac{|S_{11}+S_{12}e^{i}\theta|^{2}+|S_{22}+S_{21}e^{i}\theta|^{2}} {2}}, \tag{21}\]  
where \(\theta\) is the phase difference between the excitation difference of our MIMO antenna. The MIMO antenna's TARC plot is given in Figure 28, with a difference of \(\theta\) from 0\({}^{\circ}\) to 180\({}^{\circ}\) with an interval of 30\({}^{\circ}\). The plot reveals that \(0\leq\Gamma_{a}^{t}\leq 1\). The ECC (\(\rho e\)) is computed from the far-field radiation pattern as in Equation (22) [40]. \[\rho e=\frac{|\iint_{4\pi}[\vec{F}_{1}(\theta,\phi)\times\vec{F}_{2}(\theta, \pi)d\Omega]|^{2}}{|\iint_{4\pi}[\vec{F}_{1}(\theta,\phi)]|^{2}d\Omega|\iint 4\pi[\vec{F}_{2}(\theta,\pi)]|^{2}d\Omega}, \tag{22}\]  
where \(\vec{F}_{1}(\theta,\phi)\) is the farfield property of the MIMO array when port i is excited. The proposed MIMO antenna ECC was calculated from 1 to 12 GHz, as shown in Figure 29; it can be observed that ECC's value is less than 0.002 through the UWB bandwidth and fulfills good diversity standard for a MIMO system [41]. Figure 27: The measured and simulated far-field pattern for E and H plane at 10.6 GHz. Channel capacity can be defined as a data rate supported in a particular channel, and that channel is a fading environment [42; 43]. Considering a high signal to noise ratio value, CCL can be evaluated from either measured or simulated S-parameters [44; 45]. The CCL graph is shown in Figure 30; the plot uncovers that the channel capacity loss is less than 0.35 over the whole UWB span because, in practical terms, \(CCL<0.4\) bps/Hz. The capacity loss is given in the following equation:  
\[C_{Loss=-}\log_{2}det(\alpha^{R}), \tag{23}\]  
\[\alpha^{R}= \begin{bmatrix}\alpha_{11}&\alpha_{12}\\ \alpha_{21}&\alpha_{22}\end{bmatrix} , \tag{24}\]  
\[\alpha_{ii}=1-(\sum_{j=1}^{N}|S_{ij}|^{2}), \tag{25}\]  
\[\alpha_{ij}=-(S_{ii}^{*}S_{ij}+S_{ji}^{*}S_{ij}), \tag{26}\]  
where \(\alpha^{R}\) shows the S-Parameter matrix, and \(S_{ii}\)\(S_{ij}\) represent \(S_{11}\)S11, \(S_{12}\), \(S_{21}\), and \(S_{22}\), respectively. Figure 28: Total active reflection coefficient (TARC) of MIMO antenna system when \(\theta\) changed from 0 to 180 degrees. Figure 29: Envelope correlation coefficient (ECC) of MIMO antenna system. The diversity gain of MIMO antennas can be computed from ECC using the relation [46]:  
\[DG=10\sqrt{1-ECC^{2}}, \tag{27}\]  
where \(DG\) is the diversity gain. In Figure 31, the diversity gain of nearly 10 dB is depicted; this signifies that the antenna has low polarization diversity that can give low mutual coupling. The antenna mean effective gain (MEG) is defined as the ratio of the average power received at the microwave circuit to the sum of the average power of the vertically and horizontally polarized waves received by an isotropic antenna [47]. Figure 32 shows the MEG, which is less than \(-3\) dB. It can be calculated as in following equation:  
\[MEG=0.5[1-\sum_{j=1}^{N}|S_{ij}|]. \tag{28}\]  
The N is the number of antenna elements.