In order to numerically calculate individual head-related transfer functions (HRTFs), a boundary element model (BEM) was developed. This model makes it possible to calculate the sound pressure at the head that is caused by different external sound sources with frequencies up to 20,000 Hz.


In engineering, the traditional BEM is widely used for solving problems. However, the computational effort of the BEM grows quadratically with the number of unknowns. This is one reason why the traditional BEM cannot be used for large models, even on highly advanced computers. In order to calculate the sound pressure at the head at high frequencies, very fine meshes need to be used. These meshes result in large systems of equations. Nevertheless, to be able to use the BEM, the equations must be combined with the Fast Multipole Method (FMM). With the FMM, the resulting matrices can be kept smaller, thus allowing the numeric solving of the Helmholtz equation with feasible effort and almost no accuracy loss as compared to the traditional BEM.


The geometry of the head (especially the form of the outer ear or pinna) acts as a kind of filter. This geometry is very important in localizing sound in the vertical direction and distinguishing between sounds coming from the front or the back. The BEM model can be used to numerically calculate these filter functions, which are dependent on the position and the frequency of the sound source.


FWF (Austrian Science Fund): Project #P18401-B15


  • Kreuzer, W., Majdak, P., Chen, Z. (2009): Fast multipole boundary element method to calculate head-related transfer functions for a wide frequency range, in: J. Acoust .Soc. Am. 126, 1280-1290.
  • Kreuzer, W.  and Chen, Z. S. (2008). "A Fast Multipole Boundary Element Method for calculating HRTFs," AES preprint  7020, AES Convention, Vienna.