## 2 Psychoacoustic Principles of Auditory Distance Perception
### Cues for Auditory Distance Perception
Moreover, prior knowledge on the sound source may influence the distance estimation performance when a high-frequency spectral cue is used. In general, spectral cues caused by high-frequency attenuation provide weak information for relative distance perception. (3) Curve wavefront and near-field HRTF cues  
In a free field, the sound field radiated by a point source is a spherical wave with a curved wavefront at proximal distance to the source. The scattering and diffraction of the anatomical structures, such as the head and pinnae, to the spherical wave result in distance-dependent binaural pressure even if the effect of the \(1/r_{5}\) law has been compensated. This physical course can be described by near-field head-related transfer functions (HRTFs), which are defined as the normalized pressure at two ears caused by a point source [7]:  
\[H_{\alpha}(r_{5},\Omega_{s},f)=\frac{P_{\alpha}(r_{5},\theta_{S},\phi_{S},f)}{ P_{0}(f)}, \tag{3}\]  
where \(P_{\alpha}\) denotes the pressure at either ear; \(\alpha=\) L or R denotes left or right ear, respectively; and \(P_{0}\) is the pressure at the position of the head center in the absence of the head. Generally, HRTFs vary with frequency \(f\), source distance \(r_{S}\), and direction (\(\theta_{s}\), \(\phi_{S}\)) with respect to the head center. For a source distance farther than 1.0 m, HRTFs are asymptotically independent of the source distance and called far-field HRTFs. HRTFs also vary from individual to individual due to the individual differences in the anatomical structures and the dimensions of the head and pinnae. Figure 1 illustrates the left and right HRTF magnitudes of a human subject for two horizontal azimuths and three source distances [8]. Figure 1a,b indicate two horizontal source azimuths of \(\theta_{s}=0^{\circ}\) and \(90^{\circ}\), which denote the front and left directions, respectively. The variations in HRTF magnitudes with source distance are obvious at a lateral direction of \(\theta_{s}=90^{\circ}\). Figure 1: The left and right HRTFs magnitudes of a human subject for two horizontal azimuths and three source distances. (**a**) \(\theta_{s}=0^{\circ}\); (**b**) \(\theta_{s}=90^{\circ}\). The auditory distance perception cues encoded in near-field HRTFs are termed near-field HRTF cues, which are valid at proximal source distances of less than 1.0 m (to head center), especially less than 0.5 m.  
Near-field HRTFs involve two distance-related physical effects. The first effect is the distance-related interaural level difference (ILD). Outside the median plane, when a source approaches the head, the scattering and shadow effects of the head cause an increase in the pressure at the ipsilateral ear and reduction in the pressure at the contralateral ear. Such a variation in binaural pressures results in a slight variation in interaural time difference (ITD), but a large variation in ILD that increases with the reduction in source distance. As an example, Figure 2 illustrates the ILD versus frequency for a horizontal frontal source at \(\theta_{s}=0^{\circ}\), horizontal lateral source at \(\theta_{s}=90^{\circ}\), and three different source distances \(r_{S}\) of 0.25, 0.5, and 1.0 m. The ILDs are evaluated from the HRTFs of the same individual as in Figure 1. For the lateral source, the magnitude of ILD increases obviously with the reduction in source distance. For a source distance \(r_{S}\) of 0.25 m and at low frequency of 250 Hz, the ILD is about 10 dB. The distance-dependent ILD, especially at mid and low frequencies below 3 kHz, provides absolute and strong distance perception information for lateral source at proximal distances of less than 1.0 m [9; 10; 11]. In addition to ILD, the distance-dependent head scattering and shadow also change the spectra of binaural pressure and change the perceived timbre, which may also provide information for distance perception. Another physical effect involved in near-field HRTFs is the parallax effect [12; 13].