# Metamaterial Based Design of Compact UWB/MIMO Monopoles Antenna with Characteristic Mode Analysis
## 2 UWB-MIMO with Characteristic Modes Theory and Metamaterial Structure  
### Characteristic Mode Theory  
Modal analysis saw its initiation in the mid-1970s when [PERSON] built up a generalized approach to enlarge the field transmitted from a conducting structure [28]. Nonetheless, the reformulation refined by [PERSON] and [PERSON] [29] considered the mode's functional calculation utilizing the method of moments. This formulation is the reason for a large majority of the work, which uses mode analysis. As per [PERSON], the part of an incident electric field tangential to the surface of at least one impeccably conducting body characterized by surface S is related to the current through a linear operator is explained in the following equation:  
\[Z(J)=E_{itan}, \tag{1}\]  
where \(Z(J)\) represents the symmetric operator of the induced current on the conductor surface S due to the incident E-field, and \(E_{itan}\) is the incident E-field in the scatter problems or input power.  
This linear operator can be gotten from [PERSON]'s equations utilizing the magnetic vector potential A and the electric scalar potential \(\phi\) of the current flow from each point on a superficial level S and incorporating the appropriate free-space Green's function the whole surface. The characterizing of two vector functions' inner product on the surface S is given in the following equation:  
\[(BC)=\iint_{t}B.Cds, \tag{2}\]  
where \(BC\) is the inner product over complex space, and \(ds\) is the surface integral function.  
If the Z operator is consider to be symmetric, then we have following equation:  
\[(Z(B),C)=(B,Z(d)), \tag{3}\]  
where \(Z(d)\) is the Z operator over complex space This property is a result of the reciprocity of [PERSON]'s equations in free space. By setting \(Z=R+J_{x}\), the following weighted eigenvalue problem is defined as in equation:  
\[Z(I_{n})=V_{n}R(I_{n}). \tag{4}\]  
\(V_{n}\) is the eigenvalues, the \(I_{n}\) is the modal current, and \(R\) is a weight operator. By using \(V_{n}=1+j\) and canceling the common terms \(R(I_{n})\) in (4), then the Equation (4) will be in the following form:  
\[X(I_{n})=\lambda_{n}R(I_{n}). \tag{5}\]  
Since X and R are a real symmetric operator, \(\lambda_{n}\) is the eigenvalue, and the resultant eigenvalues and eigenvectors will be purely real. Therefore, we consider the eigenvectors \([I_{n}]\) to be the modal surface current for the perfectly conducting surface \(S\) defined by the operator \(Z\). For an electrically small radiator, a couple of modes are needed to describe the antenna's behavior. Modes with small \(\lambda_{n}\) are good radiators. Those with large \(\lambda_{n}\) are poor radiators, while mode with \(\lambda_{n}\) = 0, it is resonance [30; 31]. The extend of eigenvalues, thus, tells how well a specific mode transmits. The modal significance is defined as in the following equation:  
\[MS=|\frac{1}{1+j\lambda_{n}}|, \tag{6}\]  
where \(MS\) is the modal significance which is related to the conducting body's shape, dimension, and frequency. But, it is port independent. The characteristic angle is defined in Reference [32; 33] as:  
\[\theta=180-tan^{-1}(\lambda_{n}). \tag{7}\]  
Equation (7) above gives a clear explanation of how energy is stored in the \(2\times 2\) MIMO antenna. When the angle is at 180 degrees, the mode resonates when it is lessthan 180 degrees; the mode stores magnetic energy, and, when it is more significant than 180 degrees, the mode stores electric energy.