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Chapter2. Theory 21 Contrary to deconvolution methods, SODIX does not assume a uniform source strength, but instead models the CSM Cmod using the steering vectors g and the mn jm point source strength Djm of the sound pressures of each source j towards each microphone m, Eq. 2.12. mod ∗ Cmn (2.12) J = ∑gjmgjnDjmDjn j=1 ⃗ ⃗ The complex steering vectors are defined as in Eq. 2.13, with rjm = ξj − ξm being the distance between microphone and source position and k = 2π f c−1 the angular wavenumber [11]. gjm = 1 eikrjm (2.13) rjm The goal is to reconstruct the measured CSM Cmn such that the mean square error F(D) between the measured and modelled matrix is minimized, Eq. 2.14 [11, 22]. M mod2 F(D) = ∑ Cmn − Cmn (2.14) m,n=1 This problem is not mathematically well-posed, as there can potentially be more unknown source strengths Djm than there are microphones M and point sources J, for solving for Eq. 2.14 results in a set of JM non-linear equations [10]. For this reason, smoothing functions G1 and G2 were introduced, assuming an even dis- tribution of source directivity (G1) and source distribution (G2) [22]. Further iter- ations on the SODIX methods introduced a cost function F to restrain the solver from negative solutions for D, as it now solves for djm = Djm [11]. The cost function then is 2 M J ∗22 F(d) = ∑ Cmn − ∑ gjmgjndjmdjn . (2.15) m,n=1 j=1 Using the smoothing functions, the equation that has to be minimized instead of Eq. 2.15 is defined as G(d) = F(d) + σ1G1(d) + σ2G2(d). (2.16) The slack variables σ1 and σ2 in Eq. 2.16 are used to control the intensity of the smoothing. σ1 controls the source directivity smoothing; with increasing val- ues greater than zero, the directivity of each source becomes smoother. Values σ1 = 0.0001 . . . 1 are typically used. σ2 controls the source distribution smoothing;PDF Image | Investigation of wing installation effects on the sound field of a model jet engine
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