On the condition number of a normal matrix in near-field to far-field transformations

In near-field scanning, the model parameters for the antenna under test can be determined from measured probe outputs by solving a linear system of equations in the least-squares sense. The model parameters are far-field pattern values, cylindrical or spherical-wave expansion coefficients, or equivalent surface-source values. The normal matrix of this linear system of equations is sometimes extremely ill-conditioned. This occurs when certain sets of model parameters lie outside the spatial bandwidth of the operator that computes the probe output. One remedy is to restrict the sets of model parameters allowed and perform upsampling if needed to achieve the desired accuracy. These ideas are illustrated through analysis and examples that involve both 2-D and 3-D scanning geometries.