Analytical calculation of effective common volumes in bistatic scatter problems

The need to calculate effective common volumes (ECVs) formed by the intersection of antenna beams arises in several communications (and radar) engineering problems. Order of magnitude calculations are easily made using assumptions such as top-hat antenna radiation patterns, disparate beam dimensions at the common volume, etc. Accurate calculations can be made in a straight forward way using numerical methods. It is thought, however, that an analytical formula allowing accurate calculations to be made without the need for a computer program would be useful. The paper derives such a formula and compares it for accuracy with results from a numerical program. The only assumptions used in the derivation are that the antennas have Gaussian radiation patterns and the antenna beams are approximately cylindrical in the region of the common volume. Within the constraints of these assumptions the formula is exact