Array factor

The array factor $AF$ is the complex-valued far-field radiation pattern obtained for an array of $N$ isotropic radiators located at coordinates ${\vec {r}}$, as determined by

$AF({\hat {r}})=\sum _{n=1}^{N}a_{n}e^{jk{\hat {r}}\cdot {\vec {r}}_{n}},$

where $a_{n}$ are the complex-valued excitation coefficients, and ${\hat {r}}$ is the unit direction vector.