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[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].

References

1. Balanis, C. A. Antenna Theory, Analysis and Design (3 ed.). p. 291.
2. "IEEE Standard for definitions of terms for antennas". IEEE Std. 2014.
3. Frid, Henrik (2020). Analysis and Optimization of Installed Antenna Performance. Stockholm, Sweden: KTH (PhD thesis). pp. 36–39. ISBN 978-91-7873-447-4.

See also

• Array antenna