Array factor

In order to simplify the mathematics, a number of assumptions are typically made:

1. All antennas are identical in every respect.
2. The antennas are uniformly spaced.
3. The signal phase shift between radiators is constant.

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,^[2] with the time convention ${\displaystyle e^{j\omega t}}$. A corresponding expression can be derived for the receiving mode, where a negative sign appears in the exponential factors, as derived in reference.^[3]

• Array antenna