Solving inverse equivalent surface source problems by the normal error system of normal equations

For near-field antenna measurements, the most versatile post-processing techniques are inverse equivalent surface source solvers, since they are able to provide suppression of parasitic echo influences by source localization and offer diagnostic information about the behavior of the antenna under test (AUT) [1]-[3]. We focus on the fast irregular antenna field transformation algorithm (FIAFTA), which offers multi-level fast multipole method acceleration and full probe correction [4], [5]. For diagnostics, the equivalent sources are modeled as electric and magnetic surface current densities, which are in turn discretized by the low-order divergence-conforming Rao-Wilton-Glisson functions. Therefore, the number of current unknowns can be considerably larger than the degrees of freedom in the radiated AUT fields (determined purely by the AUT size), rendering the inverse problem underdetermined in most scenarios with a nullspace in the forward operator. Furthermore, a left nullspace is observed when the AUT near field is oversampled for noise and measurement error suppression.