Closed-Form Directivity Expression for Arbitrary Volumetric Antenna Arrays

It is proposed a closed-form expression of directivity for an arbitrary volumetric antenna arrays using a general element pattern expression of type $\sin^u{(\theta)}\cos^v{(\theta)}$, with $v > -\frac{1}{2}$ and $u > -1$, and $u, v \in \mathbb{Z}$. Variations of this expression for different values of $v$ and $u$ are analyzed from the analytical and numerical perspectives. The parameters found in the closed-form expression are related to the order $v$ and $u$ of the element patterns, the rectangular spatial coordinate of each antenna element, the magnitude and phase excitation coefficients (complex excitation) of all elements, and the desired angle in spherical coordinates $(\theta_0, \phi_0)$. The expression found in this work has been validated by numerical results, considering distinct configuration scenarios.