Distance-varying filters to synthesize head-related transfer functions in the horizontal plane from circular boundary values

This paper derives a continuous-space model to describe variations in magnitude of complex head-related transfer functions (HRTFs) along angles and radial distances throughout the horizontal plane. The radial part of this model defines a set of horizontal-plane distance-varying filters (HP-DVFs) that are used to synthesize the HRTFs for arbitrary sound source positions on the horizontal plane from initial HRTFs obtained for positions on a circular boundary at a single distance from the head of a listener. The HP-DVFs are formulated in terms of horizontal-plane solutions to the three-dimensional acoustic wave equation, which are derived by assuming invariance along elevation angles in spherical coordinates. This prevents the free-field inaccurate distance decay observed when assuming invariance along height in cylindrical coordinates. Furthermore, discontinuities along the axis connecting the ears are also overcome, which appear when assuming invariance along the polar angle in interaural coordinates. This paper also presents a magnitude-dependent band-limiting threshold (MBT) for restricting the action of filters to a limited angular bandwidth, which is necessary in practice to enable discrete-space models that consider a finite number of sources distributed on the initial circle. Numerical experiments using a model of a human head show that the overall synthesis accuracy achieved with the proposed MBT outperforms the one achieved with the existing frequencydependent threshold, especially at low frequencies and close distances to the head.