19th INTERNATIONAL CONGRESS ON ACOUSTICSMADRID, 2-7 SEPTEMBER 2007

NUMERICAL STUDY ON SOURCE DISTANCE DEPENDENCY
OF HEAD-RELATED TRANSFER FUNCTIONSPACS: 43.66.Qp
Otani, Makoto; Hirahara, Tatsuya
Faculty of Engineering, Toyama Prefectural University; Kurokawa 5180, Imizu, Toyama, 9390398, Japan; otani@pu-toyama.ac.jp

ABSTRACTThe source-distance dependency of head-related transfer functions (HRTFs) on a horizontal
plane is investigated numerically, using the boundary element method. HRTFs for planarsources, which are independent of distance, and spherical sources, which are located at
distances of up to 3.0 m at intervals of 1 cm, are simulated with a B&K 4128C scanned withlaser and CT scanner. The HRTF spectrum is compared between source distances anddirections using a head-centered coordinate system. The analysis reveals that the mainfeatures of the HRTF spectrum, such as spectral peaks and notches, show significant variations
with source distance. Furthermore, an ear-centered coordinate system is also incorporated inorder to clarify the effect of the angle of incident with respect to the outer ear. The comparison
between these two coordinate systems reveals that the source-distance dependency of theipsilateral HRTFs can be mainly attributed to the outer ear angle of incidence, whereas that of
the contralateral HRTFs is attributed to the presence of head. Results also show that, in an earcentered coordinate system, the ipsilateral HRTFs do not depend strongly on source distancelarger than 25 cm from the center of a head, whereas the contralateral HRTFs depend on
source distances less than 1.8 m.

INTRODUCTION
Sound localization is an important function of human auditory system. With interaural cues,
such as interaural level differences (ILD) and interaural time differences (ITD), as well as
spectral cues, we can perceive the direction and distance of a sound source. These cues are
involved in acoustic transfer functions from the sound source to each ear, i.e. the Head-Related
Transfer Functions (HRTFs). HRTFs are dependent on listener's head and pinnae shapes.
The binaural technique is a promising approach to reproduce or synthesize a desired auditory
space. This technique is achieved by controlling acoustic signals at a listener's ears, i.e.
binaural signals [1]. Binaural signals can be synthesized by HRTFs, enabling development of avirtual auditory display (VAD) system (e.g., [2]). Generally, HRTFs are measured in an anechoic
chamber for a finite number of source positions, yielding HRTF databases (e.g., [3]). Most
HRTF measurements are performed for a variety of source directions, but for only a single
distance to the head (e.g., 1 m). In a strict sense, VADs implementing such a limited set of
HRTFs can present virtual sound sources only at fixed distance from the head. Assuming that
HRTFs do not depend on source distance, one possible approach for presenting a virtual soundsource of various distances is to give distance-level decay to HRTFs, i.e. distance interpolation.
However, as Morimoto et al. reported, an HRTF spectrum for sources within 1 m to the headshows significant variation with source distance [4].
The characteristics of the HRTFs for sources within 1 m of the head, however, have not been
investigated thoroughly. With rigid sphere analysis, Brungart et al. showed that HRTFs within 1m distance vary significantly with source distances, whereas HRTFs over 1-m distance arealmost independent of source distance. ITDs are also independent of source distance, whereas
ILDs depend significantly on source distance [5]. Duda et al. confirmed these findings by
bowling ball measurements [6]. Brungart et al. reported that a nearer source yields a low-passfiltering effect; a nearer source increases an overall gain of ipsilateral HRTFs but decreases thatof contralateral HRTFs; a frequency and azimuthal angle of spectral notches at higher
frequencies vary with a source distance [7]. This phenomenon can be explained with “Auditory
parallax" [8], i.e., the angle of incidence to the outer ear varies with source distance in a head-centered coordinate system, even when the angle of incidence to the head center is held
constant.
As described above, the source-distance dependency of the HRTF for source distances
within 1 m is beginning to be revealed. The rigid sphere analysis, however, does not take outerears into account. Furthermore, Brungart et al.'s works were conducted in sparse sourcedistance resolution, 0.12, 0.25, 0.5 and 1.0 m. In this paper, the source-distance dependency of
the HRTFs is investigated in detail with numerically simulated HRTFs for spherical waves andplanar waves using the Boundary Element Method (BEM).

NUMERICAL SIMULATION OF THE HRTFS
The numerical simulation method used in this work is based on the BEM, which enables afast HRTF computation for various positions [9, 10] of point and planar source [11]. HRTFs forplanar waves are approximately identical to HRTFs for spherical waves originating from infinite
distance, and are therefore independent of source distance. Hence, the HRTFs for planar
waves can be a benchmark for investigating the source-distance dependency of the HRTFs forspherical waves. HRTFs of Brűel & Kjær 4128C are simulated. A head model (Fig. 1a) wasconstructed from laser- and MicroCT-scanned surface geometrical data [10]. Both ear canals
are blocked at their entrances. The head model consists of 28,000 triangular elements whose
length is 5.64 mm in maximum. This element length corresponds to 1/4 wavelength of 15 kHz
and 1/3 wavelength of 20 kHz.
HRTFs for spherical and planar waves originating from sound sources located in a horizontal
plane are investigated. The head surface is assumed to be acoustically rigid. Transfer functions
are calculated up to 20 kHz at every 86 Hz. Figure 1(b) illustrates the position of the point
sources and planar source. The origin is set to the midpoint of both ears. θh represents an
azimuthal direction, in degrees. rh represents a distance from origin, in meters. The right ear
HRTFs are calculated for point sources located at θh between -175° and 180° at increments of 5
degrees, and rh between 0.1 m and 3.0 m at increments of 0.01 m. If θh = 60, rh = 0.1
corresponds to a point ~1 cm away from the surface of dummy head. Because the HRTFs for aplanar source are independent of source distance, only azimuthal direction varies. The receiver
is located adjacent to the entrance of the ear canal. All the calculated HRTFs are normalized bytransfer functions at the origin in the absence of the head.

(a) Computer model of the head (B&K 4128C)(b) Geometric configuration
Fig. 1 – Numerical condition

HRTF SPECTRUM VARIATION WITH SOURCE DISTANCE
Figure 2(a) shows HRTFs for point sources (spherical HRTF) located at θh = 60 and rh = 0.13,
0.25, 0.5, 1.0 and 3.0 and HRTFs for planar sources (planar HRTF) located at the sameazimuthal directions. Transfer functions up to 20 kHz are shown. As the source distanceincreases, the spectra of spherical HRTFs become closer to those of planar HRTFs. As thesource distance decreases, the spherical HRTFs gain below 6 kHz. Above 6 kHz, the amplitudeand frequency of their peaks and notches vary. Figure 2(b) shows the results for θh = -60. As asource distance decreases, the spherical HRTFs attenuate at full bandwidth. Also, thefrequency of notches around 2 kHz and 7–8 kHz vary with the source distance. Brungart et al.
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19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID

also observed these features in measured HRTFs [7].

(a) 60 deg.
(b) -60 deg.
Fig. 2 – HRTFs for spherical and planar waves
Variation in spectral peaks and notches
Figure 3 shows the spherical HRTFs at every 30-degree azimuthal direction. The abscissa
represents frequency, and the ordinate represents point source distance. The angles written
around the head denote azimuthal direction θh. The gray shaded area indicates HRTFs that
could not be calculated because the point source would have been located inside the head (θh =0, rh < 0.12).
Results of θh = 30–150 show that the HRTF spectrum gains at lower frequencies withdecreasing source distance. The depth and frequency of the notches above 5 kHz changes witha source distance. Generally, except for a spectral notch at reference point b (θh = 90, 10–18
kHz) and notches at θh = 150, notches become deeper and peaks become higher withdecreasing source distance, yielding HRTF spectra with significantly emphasized peaks andnotches. The ipsilateral HRTFs of a rigid sphere do not have such notches at higher frequencies[7], indicating that these notches are produced by a pinna.
The spectral notch at reference point a (θh = 30, 9 kHz) moves to lower frequency withbecoming deeper, as the source distance decreases. Spectral notches at reference points a' (θh= 0, 9 kHz) and a"(θh = -30, 9 kHz), as
well as a notch at 20 kHz in the same
figure, all have the same characteristics.
The notches at reference point a andat 20 kHz also appear at θh = 60 for verysmall source distances (rh<0.02).
A spectral notch at reference point b(θh = 90, 10–18 kHz) is disappearing asa source distance decreases, its
frequency, however, is almost constantunlike the notch at reference point a.
This is because the incident angle to theouter ear (θe) is constant regardless of asource distance as sources are locatedat just right (θh = 90).
A spectral notch at reference point cdemonstrates a typical example of anotch frequency that varies with asource distance. The notch appears at 5
kHz for rh = 3.0, but moves to 6-7 kHz
for rh = 0.1. This phenomenon is due toa source-distance dependency of θe. For
rh = 3.0, θh = 150 corresponds to ~151
degrees of θe, but for rh = 0.1, the sameθh corresponds to ~193 degrees of θe,generating a 42-degree difference of θe.
In contrast, a spectral notch at around19 kHz for the same θh does not vary its
Fig. 3 – Spherical HRTFs at every 30 deg. azimuth
frequency for rh > 0.2.
in a head-centered coordinate system.
3
19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRIDSpectral notches in the contralateral HRTFs, such as those at reference points d, e and f,vary their depths and frequencies significantly with source distance. These notches are caused
by wave diffraction and interference due to a head. Furthermore, a source-distance dependencyof such notches is attributed to the presence of the head, as Brungart showed by a rigid sphere
analysis [7]. Due to the presence of the head, a sound wave originating from a contralateralsource does not reach directly to the receiver ear. Instead, it diffracts on the surface of the head
and travels along the head surface to receiving the outer ear. Hence, the contralateral HRTFs
spectrum does not depend on θe, but does depend significantly on θh.
Effect of an incident angle to the outer ear
The source-distance dependency of the HRTFs, especially in the ipsilateral HRTFs, is thought
to be due to a variation of θe. Figure 4 shows the spherical HRTFs calculated in the earcentered coordinate system. The origin is located at the receiver point located at the entrance tothe ear canal; re is the distance between the point source and the receiver point. The grayshaded area indicates HRTFs that could not be calculated; this area is much larger than thecorresponding area in Fig. 3. Note that, unlike Fig. 3, the ordinate (re) ranges from 0.01 m to 3.0m.
When re = 3.0, the spherical HRTFs for both coordinate systems (Figs. 3 and 4) show almost
the same characteristics of the planar HRTFs, indicating that these two coordinate systems do
not yield any significant difference in this case. Unlike Fig. 3, spectral notches at θe = 0(reference point a) and 30 (reference point a') do not depend significantly on source distance forre > 0.1; this is because θe does not vary with source distance. In contrast, the HRTFs gain
prominently below 3 kHz for re < 0.1. Additionally, a spectral notch at reference point a moves
slightly for re < 0.1. The former phenomenon is due to the presence of the head, as also shown
for all ipsilateral HRTFs (θe = 0–180). The latter phenomenon can be explained by a sourcedistance dependency for the outer ear. Brungart et al. reported that the outer ear's frequency
response varies with source distances within 4 cm to the outer ear [13]. Considering thedimensional ratio of the outer ear andthe head, the outer ear would have onlya slight influence on the source-distancedependency of the HRTFs, unless thesource is located adjacent to the outerear, in which case its influence would besignificant.
The HRTF spectra for θe = 90 are
almost identical to those shown in Fig. 3.
Note that the ordinate values aredifferent between the two figures. A
small spectral notch is seen at 10 kHz
for a source located at 2 cm to theentrance to the ear canal (re = 0.02).
The spectral notch at reference point
c for θe = 150 case does not varysignificantly in either depth or location
for re > 0.25. Below 0.25 m, however,
the
notch
becomesshallower,
disappearing at re = 0.05. A notch at 19
kHz shows the same feature. Spectral
notches appear at 4, 8, and 11–20 kHz
for θe = 150. A source location of θe =150 and re = 0.03 corresponds to thebackside of the pinna. The HRTF for θe= 180 has a similar spectral notch.
Hence, these notches are produced if asource is located near the backside of
the pinna.
Unlike in Fig. 3, deep spectral notches,
Fig. 4 – Spherical HRTFs at every 30 deg.
shown for the cases of θe = -60
azimuth in an ear-centered coordinate system.
(reference point e) and -120 (reference4
19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID

point f), move to lower frequency as source distance decreases, because these notches are
due to the presence of the head. Not only source distance from the head but also the angle ofincidence to the head center varies, as the origin is set to the outer ear. On the other hand, aresult for θe = -90 is similar to that shown in Fig. 3, as sources are located on the interaural axis.
As shown above, spectral notches at higher frequencies of the ipsilateral HRTFs spectra (θe =0–180), which are produced due to the pinna, do not depend significantly on a source distancelarger than tens of centimeters from the outer ear. This indicates that a source-distancedependency of higher-frequency spectral notches of the ipsilateral HRTFs is due to a changing
incident angle to the outer ear (θe) in a head-centered coordinate system. In contrast, the pinna
increases its impact on the source-distance dependency of the HRTFs as the source becomescloser to it, generating spectral notches that are not seen for further sources.
Objective evaluation
Spectral distances (SDs) to the planar HRTFs are calculated for the spherical HRTFs in bothcoordinate systems. SDs are calculated as follows:

H spherical (r , θ , ω ) ⎞1 K ⎛⎜⎟ ,20 log10∑
K h =1 ⎜⎝H planar (θ , ω ) ⎟⎠2

SD =where K is calculated frequency; ω, angular frequency; Hspherical(r, θ, ω), an HRTF for a point
source located at a distance of r and an azimuthal angle of θ; Hplanar(r, θ), an HRTF for a planarsource located at an azimuthal angle of θ. θ = θh and r = rh for a head-centered coordinate
system, whereas θ = θe and r = re for an ear-centered one.

(a) Head-centered coordinate system (b) Ear-centered coordinate system
Fig. 5 – SDs for various source distances and azimuthal angles in (a) a head- and (b) an earcentered coordinate systems.
Results for each coordinate system are shown in Figs. 5(a) and (b), respectively. The
abscissa represents the azimuthal angle, and the ordinate represents the distance from eachorigin point, i.e. the head center or the entrance to the ear canal. Colors closer to blue represent
smaller SD values; dark blue part represents a SD of less than 1 dB. The gray shaded area
indicates that a SD could not be calculated because the source is located inside the head. As
shown in Fig. 5(a), SDs are larger for the contralateral HRTFs than for the ipsilateral HRTFs,
revealing that the contralateral HRTFs depend more on source distance. This is because thesource-distance dependency of the head has a relatively large effect, as the head is present
between the source and the receiver ear. In contrast, SDs are generally smaller for theipsilateral HRTFs as the presence of the head has a smaller effect. For the ipsilateral HRTFs,
the largest distance when SDs are less than 1 dB is rh = 0.38 at θh = 170. For the contralateral
HRTFs, it is rh = 1.69 at θh = -90.
In an ear-centered coordinate system (Fig. 5b), this distance is re = 1.79 at θe = -110 for thecontralateral HRTFs, indicating that the contralateral HRTFs have a significant dependency on
source distance even in an ear-centered coordinate system. In contrast, it is re = 0.18 at θe = 75
for the ipsilateral HRTFs. re = 0.18 at θe = 75 corresponds to a source distance of about 0.25 m
in a head-centered coordinate system. Hence, these results demonstrate that the ipsilateral
HRTFs depend less on source distance, compared with a head-centered coordinate system.
Assuming that a source-distance dependency of the HRTFs is sufficiently small when SDs to
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19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID

the planar HRTFs are less than 1 dB, source distance has negligible effects on the HRTFs forsources located further than 1.7 m in a head-centered coordinate system and 1.8 m in an earcentered one. For the ipsilateral HRTFs, the criterion decreases to ~0.4 m in a head-centeredcoordinate system and ~0.2 m in an ear-centered coordinate system.

CONCLUSIONSIn this paper, the source-distance dependency of HRTFs is investigated in detail, withnumerically simulated HRTFs both for a spherical and planar wave. In a head-centeredcoordinate system, HRTF spectra vary significantly with decreasing source distance. As thesource distance decreases, the ipsilateral HRTFs gain at lower frequencies, and thecontralateral HRTFs attenuate at full bandwidth. Spectral notches vary their depths, widths, andfrequencies significantly. In an ear-centered coordinate system, higher-frequency spectral
notches of the ipsilateral HRTFs, which are produced due to a pinna, do not vary significantly
for sources located further than several tens of centimeters from the outer ear, and show aweak source-distance dependency. The comparison between both coordinate systems reveals
that the source-distance dependency of higher-frequency spectral notches in the frontal andipsilateral HRTFs can be attributed to changing angle of incidence with respect to the outer ear
in a head-centered coordinate system. Furthermore, the source-distance dependency of a pinna
is significant for a source adjacent to the pinna, producing spectral notches that are not seen forfurther sources.
SDs to the planar HRTFs are calculated in order to objectively evaluate the source-distancedependency of the HRTFs. Results show that SDs are less than 1 dB for a point source locatedfurther than 1.7 m and 1.8 m in a head-centered and ear-centered coordinate system,
respectively. For the ipsilateral HRTFs, they are 0.4 m and 0.2 m, respectively.
From these results, we can conclude the following: The HRTFs for sources further than 2 m
can be distance-interpolated from HRTFs in the same range based on a head-centeredcoordinate system. It is also reasonable that the ipsilateral HRTFs for sources located from tensof centimeters to 2 m can be distance-interpolated from those in the same range. For thecontralateral HRTFs in the same range, however, careful handling is required, because theirdependency on source distance is larger. When interpolating the HRTFs for nearby sources(within tens of centimeters), the source-distance dependency of the ipsilateral HRTFs can be
reduced by using an ear-centered coordinate system. However for the contralateral HRTFs, thesource-distance dependency is much larger due to the presence of the head, requiring more
careful handling. Furthermore, the accuracy of distance interpolation decreases significantly
because the HRTFs for sources adjacent to the pinna differ considerably from those for moredistant sources.
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