# Multiport Single Element Mimo Antenna Systems: A Review
## 2 Mathematical Concepts of MIMO  
In this section, the mathematical concepts for MIMO antenna design are discussed [10]. Since the MIMO antenna system has multiple antennas at the transmit side (\(\text{M}_{\text{t}}\)) and receive end (M\({}_{\text{r}}\)) as shown in Figure 1, the MIMO system capacity (C) can be expressed by Equation (1). \[\text{C}=\text{E}\bigg{[}\log_{2}\det\bigg{(}\text{I}_{\text{Mr}}+\frac{\text{P} _{\text{T}}}{\text{M}_{\text{t}}+\sigma_{\text{n}}^{2}}\bigg{)}\text{H}\cdot \text{H}^{\text{H}}\bigg{]} \tag{1}\]  
where P\({}_{\text{T}}\)--total input power, \(\sigma_{\text{n}}^{2}\)--noise power, H--channel matrix, H\({}^{\text{H}}\)--Hermitian transpose of channel matrix and I\({}_{\text{Mr}}\)--identity matrix. Mutual coupling occurs when antenna elements are placed in closed proximity, which results in isolation reduction thus affecting performance. The envelope correlation coefficient (ECC) is required because the S-parameters such as S\({}_{12}\) or S\({}_{21}\) measured in between the ports are inadequate to encompass the effect of all S-parameters. The mutual coupling and return loss at the ports can be used to determine ECC, which helps to find the diversity performance of the MIMO antennas [11]. The acceptable and standard value of ECC should be less than 0.5 for portable devices. However, the ECC calculation through S-parameters is effective only in case of lossless antenna substrates but in case of lossy antenna substrate, the ECC must be calculated/measured from the antennas' far-field radiation patterns. In Table 1, the acronyms and their meanings are included. The mean effective gain (MEG) is an important performance parameter and can be defined as the ratio of the mean power received to the mean incident power of the antenna. It is used to calculate the average received signal strength of each antenna [12]. Another performance parameter, which is the total active reflection coefficient (TARC) is the ratio of square root of total reflected powers to the total incident powers [13; 14]. The TARC is a function of frequency and is expressed in Equations (2) and (3). \begin{table}
\begin{tabular}{l l} \hline \hline \multicolumn{2}{c}{**Acronyms**} & \multicolumn{1}{c}{**Meanings**} \\ \hline SISO & Single Input Single Output \\ \hline MIMO & Multiple Input Multiple Output \\ \hline TARC & Total Active Reflection Coefficient \\ \hline ECC & Envelop Correlation Coefficient \\ \hline DG & Diversity Gain \\ \hline DCS & Defected Ground Structure \\ \hline DMN & Decoupling and matching network \\ \hline CDRN & Coupled Resonator Decoupling Network \\ \hline CSRR & Complementary Split Ring Resonators \\ \hline ACS & Asymmetric Coplanar Strip \\ \hline CPW & Co Planar Waveguide \\ \hline CE & Common Element \\ \hline PIFA & Planar Inverted-F antenna \\ \hline iMAT & Isolated Mode Antenna Technology \\ \hline EBG & Electromagnetic Bandgap \\ \hline MEG & Mean Effective Gain \\ \hline WiFi & Wireless Fidelity \\ \hline UWB & Ultra-Wide Band \\ \hline LTE & Long-Term Evolution \\ \hline WilMax & Worldwide Interoperability for Microwave Access \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Acronyms. \[\Gamma=\sqrt{\frac{\text{available power -- radiated power}}{\text{available power}}} \tag{2}\]  
\[\Gamma_{\text{a}}^{\text{t}}=\frac{\sqrt{\sum_{\text{n=1}}^{\text{N}}\left| \text{b}_{\text{n}}\right|^{2}}}{\sqrt{\sum_{\text{n=1}}^{\text{N}}\left| \text{a}_{\text{n}}\right|^{2}}} \tag{3}\]  
where, [b] = [S][a] and [S] is the antenna's scattering matrix, vector [a] is the sum of the available incident power at all the ports, vector [b] is the power reflected to the source and \(\text{a}_{\text{n}}\), \(\text{b}_{\text{n}}\) are the nth element of the vector [a] and vector [b] respectively. Similarly, the antenna Diversity Gain (DG) is a well-known performance parameter used to verify the efficacy of the diversity. It can be defined as the ratio of rise in SNR of mixed signal from a multiple antenna to the SNR from a single antenna in the system. The DG can be calculated using Equation (4).