Array factor

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

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

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