A Mathematical Perspective on Acoustic Array Imaging Methods

Abstract

Microphone arrays are a widely used measurement device for the localization and quantification of (aero-)acoustic sources. Using the tools of mathematics, the common formulations of array imaging methods can be further formalized. In this conference paper we present two specific mathematical frameworks namely a representation of Beamforming functionals by characteristic scalar products and an infinite dimensional inverse problem on function spaces. For each aspect we illustrate how a theoretical analysis can be carried out and mathematical statements can be derived. Major focus is put on the analysis of the mean squared error of Beamforming functionals and the study of uniqueness of the inverse source problem.