MIMO antenna parameters such as Envelope Correlation Coefficient (ECC) and Diversity Gain (DG) can be found from far-field radiation patterns by following these steps:

Here are the mathematical formulas for calculating ECC and DG from far-field radiation patterns:

ECC:

ECC = \int_0^{2\pi} \int_0^{\pi} F_1(\theta, \phi) F_2(\theta, \phi) \sin(\theta) d\theta d\phi

DG:

DG = \frac{\left( \sum_{i=1}^N |F_i(\theta, \phi)|^2 \right)^2}{N \sum_{i=1}^N |F_i(\theta, \phi)|^4}

where:

• Fi​(θ,ϕ) is the far-field radiation pattern of the $i$th antenna element in the MIMO array
• θ and ϕ are the spherical coordinates of the direction of observation

Once the ECC and DG have been calculated, they can be used to assess the performance of the MIMO array. A lower ECC indicates that the antenna elements are less correlated, which is desirable for MIMO systems. A higher DG indicates that the antenna array is better able to resolve multiple signals arriving from different directions, which is also desirable for MIMO systems.

Here are some additional tips for calculating ECC and DG from far-field radiation patterns:

• It is important to use a sufficiently dense grid of measurement points to accurately capture the shape of the radiation patterns.
• When calculating the ECC, it is important to normalize the radiation patterns so that they have the same integral power.
• When calculating the DG, it is important to consider the polarization of the antenna elements.

ECC and DG are important parameters for evaluating the performance of MIMO antenna arrays. By calculating these parameters from far-field radiation patterns, engineers can design MIMO antennas that meet the specific requirements of their applications.