Regression Technique for Electromagnetic Field Sampling and Reconstruction

Abstract

The presented work deals with an alternative technique for sampling and reconstructing the electromagnetic field radiated by any kind of antenna or equivalent currents, and measured in its far field region. Based on the electrical size of the radiating object, it truncates the vector spherical modal field expansion series. Then, each component of the field is projected on the classical Fourier space for the polar dependency. This is carried out by minimizing the variance of the residual noise, or, in other words, by applying the Tikhonov-Phillips regularization scheme. This results is not only a numerically well-posed problem, but also in the statistical independence of the resulting coefficients as their co-variance matrix is diagonal. Afterwards, the azimuth dependency is projected on the real valued Gegenbauer, also known as ultra-spherical, polynomial family, once again following the Tikhonov-Phillips regularization scheme. Once again, it does not only result in a numerically well-posed problem, leading to a statistical independence of the obtained spherical modal coefficients. Moreover this double regression technique leads to the smallest two-dimensional Cartesian grid of angular sampling positions, a very useful result for the far field antenna characterization industry where measurement time has to be reduced as much as possible. Additionally to this, both optimum estimators and stable regularizer are also extracted. Then, a statistical analysis of the residual error is performed by extracting and analyzing the noise properties and also creating a statistical filter that rejects any mode that is not statistically significant through the definition of a modal signal-to-noise ratio. This result turns out to be very useful when this technique is applied in a compressive-sensing-like radiated far field antenna analysis. At last, these estimators are modified to attend cases where the measured data do not form a full column ranked matrix. This corresponds to the case where measurements data are lacking in the previously defined smallest two-dimensional Cartesian grid of angular sampling positions. This technique is then applied to several different antenna measurements, as shown on fig. 1, where can be noticed a very good matching between in the far $E_{\theta}$ and $E_{\varphi}$ field 2D map in amplitude comparison of both the measured electric field and the reconstructed one for a flat metallic Luneburg lens antenna designed at the frequency 12 GHz in the frame of MERLIN, which is a joint laboratory of Thalès Alenia Space and IETR. More details about this radiating structure are available in the reference [1].