Active reflection coefficient

The ARC for antenna element ${\displaystyle m}$ in an array of ${\displaystyle N}$ elements is calculated by:[3]

${\displaystyle \Gamma _{S}=\sum _{n=1}^{N}S_{mn}{\frac {a_{n}}{a_{m}}},}$

where ${\displaystyle a_{n}}$ are the excitation coefficients and ${\displaystyle S_{mn}}$ are the coupling coefficients.

In a linear array with inter element spacing ${\displaystyle a}$, uniform amplitude tapering and scan angle ${\displaystyle \theta _{0}}$, the following excitation coefficients are used: ${\displaystyle a_{n}=e^{-jkna\sin \theta _{0}}}$. By inserting this expression into the general equation above, we obtain:[4]

${\displaystyle \Gamma _{S}(\theta _{0})=e^{jkma\sin \theta _{0}}\sum _{n=1}^{N}S_{mn}e^{-jkna\sin \theta _{0}}.}$