# 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 [1]
${\displaystyle AF({\hat {r}})=\sum _{n=1}^{N}a_{n}e^{jk{\hat {r}}\cdot {\vec {r}}_{n}},}$
where ${\displaystyle a_{n}}$ are the complex-valued excitation coefficients, and ${\displaystyle {\hat {r}}}$ is the direction unit vector. The array factor is defined in the transmitting mode, with the time convention ${\displaystyle e^{j\omega t}}$.