A Weighted Fresnel Approximation for the Delays Used in Focused Beamforming

Abstract

Beamforming is a linear technique aimed at processing array signals in order to enhance incoming signals from a selected steering direction and to abate incoming signals from any other direction1. Thanks to its flexibility, beamforming can be successfully employed in many application fields with different objectives. In any case, if the array works under far-field conditions, the delays required by the beamforming operation can be easily computed in an exact way, whereas, if it works under near-field conditions, the focalization of beamforming is required to take into account the curvature of waves. In the latter case, a fast computation of the exact delays is often prohibitive, then an approximate version is preferred: generally, the Fresnel approximation (obtained by the expansion of the time-independent free-space Green’s function^2,3) is adopted1. Moreover, the Fresnel approximation makes it possible to apply the Fast Fourier Transform (FFT) in the implementation of beamforming even when focalization is necessary⁴, thus resulting in a great computational profit. Despite its simplicity and advantages, the Fresnel approximation has a well defined region of validity² that forces potential steering directions to be contained inside a narrow scanning region. This constraint is heavy in applications (e.g., acoustic imaging) that require a wide region of view, for both medical and underwater investigations. To avoid this drawback, some imaging techniques that do not need the Fresnel approximation have been devised^5,6 but, unfortunately, they increase the computational load and/or the system complexity.