Σ-separability of the far-field beam-pattern due to an arbitrary excitation-function over an elliptical-rim aperture

This study is the first in the phase-mode analysis literature to analytically characterise the far-field azimuth-elevation (φ-θ) beam-pattern due to any arbitrary excitation-function on an elliptical-rim aperture. The key findings include: such an arbitrary excitation-function's far-field beam-pattern is not always φ-θ separable. Nonetheless, this beam-pattern is Σ-separable in the sense that it is a sum of φ-θ separable terms. Moreover, this sum may be approximated (with arbitrarily small truncation-error) as a finite-term summation. A ceiling of this truncation-error is derived.