Abstract-Electromagnetic field transformations are important for electromagnetic simulations and for measurements. Especially for field measurements, the influence of the measurement probe must be considered, and this can be achieved by working with weighted field transformations. This paper is a review paper on weighted field transformations, where new information on algorithmic properties and new results are also included. Starting from the spatial domain weighted radiation integral involving free space Green's functions, properties such as uniqueness and the meaning of the weighting function are discussed. Several spectral domain formulations of the weighted field transformation integrals are reviewed. The focus of the paper is on hierarchical multilevel representations of irregular field transformations with propagating plane waves on the Ewald sphere. The resulting Fast Irregular Antenna Field Transformation Algorithm (FIAFTA) is a versatile and efficient transformation technique for arbitrary antenna and scattering fields. The fields can be sampled at arbitrary irregular locations and with arbitrary measurement probes without compromising the accuracy and the efficiency of the algorithm. FIAFTA supports different equivalent sources representations of the radiation or scattering object: 1) equivalent surface current densities discretized on triangular meshes, 2) plane wave representations, 3) spherical harmonics representations. The current densities provide for excellent spatial localization and deliver most diagnostics information about the test object. A priori information about the test object can easily be incorporated, too. Using plane wave and spherical harmonics representations, the spatial localization is not as good as with spatial current densities, but still much better than in the case of conventional modal expansions. Both far-field based expansions lead to faster transformations than the equivalent currents and in particular the orthogonal spherical harmonics expansion is a very attractive and robust choice. All three expansions are well-suited for efficient echo suppression by spatial filtering. Various new field transformation and new computational performance results are shown in order to illustrate some capabilities of the algorithm.

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Various formulations of the inverse equivalent surface-source problem and corresponding solution approaches are discussed and investigated. Starting from the radiation integrals of electric and magnetic surface current densities, the probe-corrected inverse equivalent source formulation is set up together with different forms of side constraints such as the zero-field or Love condition. The linear systems of equations resulting from the discretized forms of these equations are solved by the normal residual (NR) and normal error (NE) systems of equations. As expected and as demonstrated by the solution of a variety of inverse equivalent surface-source problems, related to synthetic as well as realistic antenna near-field measurement data, it is found that the iterative solution of the NE equations allows for a better control of the solution error and leads in general to a slightly faster convergence. Moreover, the results show that the incorporation of the zero-field condition into the solution process is in general not beneficial, which is also supported by the structure of the NE systems of equations. If desired, Love surface current densities, or just fields in general, can more easily be computed in a post-processing step. The accuracy of the obtained near-fields and far-fields depends more on the stopping criterion of the inverse source solver than on the particular choice of the equivalent surface-source representation, where the zero-field condition may influence the stopping criterion in a rather unpredictable way.

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