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The array factor <math> AF </math> is the complex-valued far-field radiation pattern obtained for an array of <math> N </math> isotropic radiators located at coordinates <math> \vec{r}_n </math>, as determined by
:<math> AF(\hat{r}) = \sum_{n=1}^N a_n e^{jk\hat{r}\cdot\vec{r}_n},</math>
where <math> a_n </math> are the complex-valued excitation coefficients, and <math> \hat{r} </math> is the unit direction vector.
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