A probabilistic approach to large circular and spherical arrays

Antenna arrays with elements distributed at random in a three-dimensional space are studied. Arbitrary excitation and nonisotropic elements are considered. The distribution of the sidelobe level below a certain value for all directions of observation, as well as other probabilistic properties are determined approximately. These general results are then applied to circular and spherical arrays. In particular, for the latter case, conical log-spiral antennas are considered in some detail. It is found that for large arrays with high resolution, the required number of elements can be several orders of magnitude smaller than what is commonly required on the basis of one element per
(\lambda/2)^{2}
Finally, a few experiments simulated by the Monte Carlo method were conducted and excellent agreements with theory have been found in all cases.