Minimum Energy Method (MEM) microphone array back-propagation for measuring musical wind instruments sound hole radiation

Abstract

Using a 128 microphone array, the sound source distribution of musical wind instruments from their blowing and finger holes are measured. The Japanese \emph{shakuhachi} flute, the Chinese \emph{dizi} transverse flute, the Balinese \emph{suling} bamboo flute and flue organ pipes are investigated. The sound radiation is measured by a rectangular microphone array in the near field, and back-propagated onto the flute radiation plane using the Minimum Energy Method (MEM). Here, the radiation is assumed to be composed of as many sound sources as there are microphones, where the source positions can be chosen arbitrarily in space. Using a regularization parameter $\alpha$ the virtual sound sources, which are monopoles for $\alpha = 0$ are narrowed for higher $\alpha$. Calculating the reconstruction energy on the radiation plane while using different $\alpha$, this energy becomes minimal for the correct $\alpha$. The ill-posedness of the solved Fredholm integral at the presence of measurement noise is met by easing $\alpha$ slightly, therefore stabilizing the solution in a robust way. This is especially necessary for problems like musical wind instruments, where the sound sources are the air itself and often a precise sound source position cannot be known in advance. The flutes show complex radiation behavior and sometimes even a coupling of the sound field between finger holes outside the flute. Coupling between organ pipes can also lead to synchronization between the pipes.