# Design and Analysis on Decoupling Techniques for MIMO Wireless Systems in 5G Applications
## 3 Evaluation Parameters of Mutual Coupling
### Envelope Correlation Coefficient (ECC)  
A good diversity effect is a key to ensuring the communication quality of MIMO systems. The effect of diversity depends on the correlation of antenna signals. The envelope correlation coefficient, as a measurement parameter, is used to measure the degree of similarity between different signals and can also be used as a basis for judging the mutual coupling of different ports. The most accurate \(ECC\) calculation method is based on the far-field pattern function. it can be given by Equation (8):  
\[ECC(i,j)=\frac{\left|\int_{\mathcal{A}\pi}\left[\overset{\rightarrow}{F_{1}}( \theta,\varphi)\cdot\overset{\rightarrow}{F_{2}}(\theta,\varphi)\right]d \Omega\right|^{2}}{\int_{\mathcal{A}\pi}\left|\overset{\rightarrow}{F_{1}}( \theta,\varphi)\right|^{2}d\Omega\iint_{\mathcal{A}\pi}\left|\overset{ \rightarrow}{F_{2}}(\theta,\varphi)\right|^{2}d\Omega} \tag{8}\]  
\(\overset{\rightarrow}{F_{i}}(\theta,\varphi)\) is a function of the radiation pattern of the field when port i is excited, and the other ports are matched under the condition of \(Z_{0}=50\Omega\), \".\" represents the Hermite product. However, due to the complex measurement process and the limitation of available pattern characterization tools, this method is time-consuming. [PERSON] et al. [27] showed how to compute the \(ECC\) from the \(S\)-parameter description of the two-antenna system. The equation has been derived using the law of energy conservation, which has the advantage that it does not require the computation or the measurement of the radiation pattern of the antenna system. It is given by Equation (9):  
\[ECC(1,2)=\frac{\left|S_{11}^{*}S_{12}+S_{21}^{*}S_{22}\right|^{2}}{\left\{1- \left(\left|S_{11}\right|^{2}+\left|S_{21}\right|^{2}\right)\right\}\left\{1- \left(\left|S_{22}\right|^{2}+\left|S_{12}\right|^{2}\right)\right\}} \tag{9}\]  
Further, [PERSON] and [PERSON] [28] deduced that in a MIMO system consisting of \(N\times N\) antenna elements, the \(ECC\) between antenna elements and antenna elements can be expressed by Equation (10):  
\[ECC(i,j,N)=\frac{\left|\sum_{n=1}^{N}S_{in}^{*}S_{n}\right|^{2}}{\left|\prod_{k =i,j}\left[1-\sum_{n=1}^{N}S_{k,n}^{*}S_{n,k}\right]\right|} \tag{10}\]  
However, Equation (10) is only applicable to the ideal case where the antenna is loss-free. In practical applications, the realistic power losses of the antenna structures should be taken into account. [PERSON] et al. [29] proposed a modification of the \(S\)-parameter method which included the antenna losses. It is expressed in the two-antenna system as Equation (11):  
\[ECC(1,2)=\frac{\left|-\left(S_{11}^{*}S_{12}+S_{21}^{*}S_{22}\right)-\left(L_{ 12}^{\prime}+L_{12}^{\prime\prime}\right)\right|^{2}}{\left\{1-\left(\left|S_{ 11}\right|^{2}+\left|S_{21}\right|^{2}\right)-\left(L_{11}^{\prime}+L_{11}^{ \prime\prime}\right)\right\}\left\{1-\left(\left|S_{22}\right|^{2}+\left|S_{12 }\right|^{2}\right)-\left(L_{22}^{\prime}+L_{22}^{\prime}\right)\right\}} \tag{11}\]  
The elements \(L_{ij}^{\prime}\) and \(L_{ij}^{\prime\prime}\) are obtained from the power loss calculation. In addition, a modification of the \(S\)-parameter method that took into account the effects of the radiation efficiency was proposed in [30]. However, instead of giving the exact value, it gave an upper bound for \(ECC\). It is expressed by Equation (12):  
\[ECC(i,j,N)=\frac{-\sum_{n=1}^{N}S_{in}^{*}S_{nj}}{\sqrt{\left(1-\sum_{n=1}^{N }\left|S_{ni}\right|^{2}\right)\left(1-\sum_{n=1}^{N}\left|S_{nj}\right|^{2} \right)\eta_{rad,i}\eta_{rad,j}}}+\sqrt{\left(\frac{1}{\eta_{rad,i}}-1\right) \left(\frac{1}{\eta_{rad,j}}-1\right)} \tag{12}\]  
where \(\eta_{rad,i}\) and \(\eta_{rad,\ j}\) are the radiation efficiency of antenna elements \(i\) and \(j\), respectively. Additionally, once the efficiency goes below 60%, the errors start increasing dramatically. The smaller the value of \(ECC\), the lower the mutual coupling effect. When \(ECC=0\), it means that the relationship between the two ports is completely independent.