Non-uniform FFTs in Near-field/Far-field Transformations

Near-field antenna characterization consists of measuring, on proper scanning surfaces, the near-field radiated by an Antenna Under Test (AUT) and of determining, from those measurements, the far-field pattern (Near-Field/Far-Field – NFFF – transformation) [1]. Whenever possible, both the amplitude and the phase of the near-field are acquired on a sole scanning surface, so that the NFFF transformation involves only linear operations on the data [1]. When, on the contrary, measuring the phase becomes expensive or difficult, the only amplitude of the near-field is usually collected on two measurement surfaces and the NFFF transformation involves non-linear operations on the data, typically consisting of the iterative optimization of a cost functional [2]. In recent years, significant advancements have been achieved in the framework of the near-field antenna characterization which have concerned new scanning geometries (plane-polar [3], spiral and helicoidal [4]), advanced non-uniform sampling techniques [5] capable to significantly reducing the acquisition time and the ill-conditioning as compared to uniform samplings, and new processing schemes [2,6]. Whenever an estimate or a measurement of the “complex” (i.e., amplitude and phase) near-field is available, transforming a near-field into the far-field conceptually requires a simple Fourier transform operation. However, when non-uniform grids are involved, such Fourier transformation cannot be performed by a standard Fast Fourier Transform (FFT) algorithm, having a convenient asymptotic computational complexity growing as NlogN. Then, in the case of complex measurements, following the acquisition of non-uniform samples, the most simple way to transform the data to the far-field is to perform an interpolation stage prior to the FFT. This establishes a trade-off between accuracy and computational complexity [7]. Such a trade-off becomes even more relevant in the case of algorithms relying on phaseless data which, at each iteration step, require evaluating the spectrum from estimates of the near-field on non-uniform lattices or calculating the near-field on non-uniform grids from estimates of the spectrum. In the last years, Non-Uniform FFT (NUFFT) algorithms [8] have been developed enabling to evaluate Fourier transforms from non-uniform grids to uniform ones (Non-Equispaced Data – NED - NUFFT), from uniform grids to non-uniform ones (Non-Equispaced Results – NER - NUFFT) and from non-uniform grids to non-uniform ones (“type-3” NUFFT). Such algorithms enable to accurately performing the interpolation stage in a computationally convenient way, so that their computational burden is proportional to that of a standard FFT. Furthermore, NUFFTs can be made available as library routines to strongly simplify their usage and, accordingly, their exploitation in NFFF transformations. Purpose of this paper is to show how NUFFT algorithms can be conveniently employed in near-field characterization algorithms, operating both with complex or phaseless data.