Array factor

The array factor A F {displaystyle AF} is the complex-valued far-field radiation pattern obtained for an array of N {displaystyle N} isotropic radiators located at coordinates r → n {displaystyle {vec {r}}_{n}} , as determined by: The array factor A F {displaystyle AF} is the complex-valued far-field radiation pattern obtained for an array of N {displaystyle N} isotropic radiators located at coordinates r → n {displaystyle {vec {r}}_{n}} , as determined by: A F ( r ^ ) = ∑ n = 1 N a n e j k r ^ ⋅ r → n , {displaystyle AF({hat {r}})=sum _{n=1}^{N}a_{n}e^{jk{hat {r}}cdot {vec {r}}_{n}},} where a n {displaystyle a_{n}} are the complex-valued excitation coefficients, and r ^ {displaystyle {hat {r}}} is the direction unit vector. The array factor is defined in the transmitting mode, with the time convention e j ω t {displaystyle e^{jomega t}} . A corresponding expression can be derived for the receiving mode, where a negative sign appears in the exponential factors.