Hexagonal Array Factor in Closed Form

Abstract

A closed-form expression is produced for the array factor of the electronically scanned, uniform-amplitude, finite-size hexagonal array. The underlying arguments depend upon a rotational symmetry cleavage of the hexagon into three congruent parallelograms and, for each such parallelogram, the adoption of a reciprocal lattice in {u,v} observation space. In particular, the reciprocal lattice machinery succeeds in splitting the basic two-dimensional parallelogram array factor into a linear array sine function ratio (LASFR) product pair, which latter form makes contact with an earlier treatment of fairly general array configurations under the unifying viewpoint of affine transformations. Of course, as the individual parallelogram array factor contributions are summed, a need arises to acknowledge an element overcount at array center and along the three lines of parallelogram contiguity, and, when this is done, the composite array factor is found to reduce along its scanned boresight to the proper value 1 + 3N(N + 1) for its element population size, with N ≥ 0 being the number of hexagon rings. Two numerical implementations of the master array factor formula confirm a priori analytical anticipations as to the symmetric deployment in {u, v} space of satellite beam prominences, of both close-in and, when admitted into visible space u ² + v ² ≤ 1, genuine grating lobe character.