This paper is concerned with the mathematical analysis of experimental methods for the estimation of the power of an uncorrelated, extended aeroacoustic source from measurements of correlations of pressure fluctuations. We formulate a continuous, infinite dimensional model describing these experimental techniques based on the convected Helmholtz equation in R 3 or R 2 . As a main result we prove that an unknown, compactly supported source power function is uniquely determined by idealized, noise-free correlation measurements. Our framework further allows for a precise characterization of state-of-the-art source reconstruction methods and their interrelations.
Experimental aeroacoustics is concerned with the estimation of acoustic source power distributions, which are for instance caused by fluid structure interactions on scaled aircraft models inside a wind tunnel, from microphone array measurements of associated sound pressure fluctuations. In the frequency domain aeroacoustic sound propagation can be modelled as a random source problem for a convected Helmholtz equation. This article is concerned with the inverse random source problem to recover the support of an uncorrelated aeroacoustic source from correlations of observed pressure signals. We show that a variant of the factorization method from inverse scattering theory can be used for this purpose. We also discuss a surprising relation between the factorization method and a commonly used beamforming algorithm from experimental aeroacoustics, which is known as Capon’s method or as the minimum variance method. Numerical examples illustrate our theoretical findings.
Tag der mündlichen Prüfung: 20.07.2021 8. Conclusions and Outlook 115 A. Auxiliary Statements 117 B. Fundamentals from Calculus and Convex Analysis 127 Bibliography 131
This conference paper deals with computational methods to determine the wavenumber spectrum of acoustic data measured by a phased microphone array. Such problems occur e.g. within the analysis of pressure fluctuations due to a turbulent boundary layer on a surface such as a wind-tunnel
wall or the skin of an aircraft. The problem is closely related to the deconvolution of dirty beamforming maps in wavenumber domain, which seeks to determine the wavenumber spectrum by removing the influence of the shift-invariant point spread function from the beamforming result. Firstly,
we recall how this task can be formulated as a minimization problem and then discuss a specific solver for this problem, provided by the framework of the generalized FISTA algorithm. The resulting method takes advantage of the convolutional structure of the gradient of the data misfit functional
and allows further for a flexible regularization with L1 and L2 penalties as well as a nonnegativity constraint. Finally, the presented algorithmic framework is demonstrated with numerical examples.
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