We present a low-cost and light-weight measurement equipment based on a software-defined radio (SDR) and a radio-frequency over fiber (RFoF) connection. The main purpose of the hardware are near-field antenna measurements, but possible use-cases also include imaging and other inverse-source scenarios. The two features “low-cost” and “light-weight” of the hardware are important for the use on an unmanned aerial vehicle: Crashes should not financially ruin the operator and payload is of course a central issue for any flying object. Regarding the RF properties of the hardware, the RFoF connection has the great benefit of extremely low loss as compared to coaxial cables, while the SDR offers a great deal of flexibility concerning measurement frequency, bandwidth, and signal filtering. One transmit channel of the SDR is employed to provide a coherent signal to an antenna and two receive channels can capture two components of the field, e.g., two linearly independent polarizations. As an important part of the RF circuitry, a drift compensation is implemented as a countermeasure against changing temperature conditions.

Phase retrieval, in particular for the operators arising from near-field measurements, is a non-convex task suffering from a severe lack of reliability due to local minima and false solutions. Approaches to tackle this issue, in turn, suffer from strict and often unrealistic oversampling requirements and possibly unfeasible computational complexities---in particular when larger scenarios are considered. In this paper, we analyze the sampling requirements for convex phase retrieval based on partially coherent observations which are captured with multi-probe arrays. We discuss requirements for the orientation and positioning of the probe antennas in the probe arrays. Following these conditions, a recently introduced linearized method is able to reconstruct a unique global solution reliably. The theoretical deliberations are corroborated with simulated near-field data.

According to the electromagnetic uniqueness theorem, the radiation behavior of an antenna under test (AUT) can be recovered from measurements of two tangential components of its radiated fields on an enclosing surface. In practice, measurements conducted in the radiating near-field (NF) of an AUT utilize probe antennas of finite size. Thus, instead of discrete field values, spatially blurred probe signals are acquired. The probe influence can be compensated, however, this commonly requires precise knowledge about the probe antenna. In this work, the concept of NF far-field (FF) transformations (NFFFTs) with full correction of the influence of unknown probe antennas is introduced. Two nonconvex and two convex formulations are presented and their relation to the similar task of phase retrieval is highlighted. Simulation and measurement results illustrate the validity of the concept, shed light on the required complexity of measurement setups, and illustrate limitations of the approach. Special attention is paid to the case of high practical relevance when AUT and probe are identical.

The accuracy of near-field antenna measurements combined with near-field far-field transformations (NFFFT) is deteriorated if objects are located in close vicinity to the antenna under test (AUT). We compare two equivalent sources based modeling techniques to suppress the influence of structures in the vicinity of the AUT. Simulation results for perfect electrically conducting objects showcase the capabilities and limitations of both approaches. The presented methods can easily extend existing NFFFTs in order to remove the influence of mounting structures or parts of the positioning system in anechoic near-field measurement facilities.

The electric field integral equation (EFIE) for perfect electrically conducting objects is augmented by a weak form combined source side condition. Thereby, the solution of the resulting combined source integral equation (CSIE) is unique and the interior resonance problem is circumvented. The discretization of electric and magnetic surface current densities and the testing of the electric field is solely performed with Rao-Wilton-Glission functions. Simulation results of scattering by a sphere demonstrate the good iterative solver convergence and excellent accuracy, as compared to EFIE as well as magnetic and combined field integral equations.

The accuracy problem of the classically Rao-Wilton-Glisson (RWG) discretized magnetic field and combined field integral equations originates from the strong singularity of the identity operator. With a novel basis transformation scheme, i.e. two subsequent weak-form rotations of the RWG basis functions, the accuracy of the identity operator discretization is improved. The presented discretization scheme is immediately compatible with fast algorithms and is combined with the multilevel fast multipole algorithm. The good accuracy and the good conditioning behavior of the formulation are verified in several numerical simulations, also including electrically large problems.

Starting from the monopolar representation of the Rao-Wilton-Glisson (RWG) functions, an approximate inverse (AI) of the RWG Gram matrix is constructed, where only 3x3 matrices for all individual triangles have to be inverted. The inversion of the Gram matrix is, however, only approximate, since it results as the inverse of a product of singular matrices, which is not easily obtained. The AI is shown to be an effective preconditioner for the RWG Gram matrix and is in particular useful for ill-conditioned Gram matrics on distorted meshes.

Reducing near-field measurement times is an important challenge for future antenna measurement systems. We propose to incorporate knowledge about material parameters of the antenna measurement environment within the simulation model. To do so, a method-of-moments code with surface discretization is implemented as a side constraint to the near-field far-field transformation problem performed with the fast irregular antenna field transformation algorithm. Transformation and source reconstruction results of synthetic measurement data demonstrate the effectiveness of the proposed method.

Distributed spherical harmonics expansions are a powerful approach for the efficient solution of inverse source problems. They allow for a relatively accurate representation of the geometric support of the source distribution without the overhead of handling a mesh and mesh-specific basis functions representing the commonly utilized surface current densities. Standard spherical harmonics expansions distributed over a Huygens surface around the source domain are, however, redundant. Therefore, we discuss ways towards the reduction of this redundancy, where we are in particular interested in the construction of spherical harmonics with predominant radiation into the actual solution domain. Several sets of such spherical harmonics are presented and their utilization with measured and synthetic near-field data is demonstrated.

The characterization of unknown antennas under test (AUTs) from measurements of the radiated near field (NF) is commonly performed with known probe antennas. By accounting for the behavior of a given probe in the processing of the NF data, the obtained quantities of interest, e.g., the far-field (FF) behavior of the radiator, are free of biases and distortions caused by the probe sensor. We discuss an NF FF transformation (NFFFT) which can fully compensate the effect of the probe, while only requiring knowledge about its electrical size, position, and orientation. An iterative nonconvex technique based on alternating projections is discussed, as well as a convex approach utilizing bilinear forms is provided. Connections to the related problem of phase retrieval are highlighted. Simulation results showcase the validity of NFFFTs with full probe correction of unknown probes and shed some light on potential limitations.

Inverse source solutions for field transformations and diagnostics are mostly working with surface current densities on appropriately defined Huygens surfaces around the test object. This gives excellent modeling flexibility, but the handling of the discretized surface current representation requires also substantial computational effort. Distributed spherical harmonics expansions of low order, for example based on a solution space partitioning as used in the multilevel fast multipole method, have somewhat reduced modeling flexibility, but they can save considerable computational effort, and they are, thus, an excellent choice of expansion functions for many practical inverse source problems. We discuss different forms of distributed spherical harmonics expansions comprising purely scalar spherical modes and vector spherical modes. Moreover, we discuss how the different surface source expansions consisting of electric and magnetic surface currents with Love condition or without, or consisting of directive Huygens radiators can be related to corresponding distributed spherical harmonics expansions.

The magnetic field surface integral equation for perfect electrically conducting scatterers suffers from accuracy problems when discretized with lowest-order Rao-Wilton-Glisson (RWG) functions. For high-frequency scattering scenarios, one of the various reported countermeasures are hierarchical higher-order (HO) functions. We demonstrate that the accuracy of these HO methods of up to 1.5th order may be further improved by employing a weak-form discretization scheme for the identity operator inside the magnetic field integral equation (MFIE), in particular for scatterers with sharp edges. As expected, the presented numerical results indicate that this approach becomes less effective for increasing order. Moreover, since the weak-form discretization overcomes only the anisotropy problems of the standard discretizations, parts of the accuracy problems of the MFIE persist for HO discretizations if the testing is performed with non dual-space conforming functions.