PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Rendering virtual source at various distances using binaural
Ambisonics scheme in dynamic virtual auditory display
Bosun XIE 1; Jianliang JIANG2
1

Acoustic lab, School of Physics and Optoelectronics, South China University of Technology, Guangzhou, China

2

Acoustic lab, School of Physics and Optoelectronics, South China University of Technology, Guangzhou, China

ABSTRACTIn dynamic virtual auditory display, free-field virtual source at various directions and distances isconventionally realized by filtering the input stimulus with near-field HRTFs. However, this conventional
method meets with some problems in practical use, due to the difficulty in acquiring near-field HRTFs andrequiring a large memory to storage them. In present work, a scheme for rendering virtual source at various
distances using binaural near-field compensated Ambisonics with appropriate order is proposed. Based on
the principle of spherical Bessel and harmonics decomposition of sound field, the directional and distanceinformation of sound field is first encoded into Ambisonics signals and then converted to headphone signals
by dynamic binaural synthesis using far-field HRTFs only. It is proved that the 5th order dynamic binaural
Ambisonics with 36-64 pairs of HRTF-based filters is able to recreate desired binaural pressures for various
source directions and distances up to 2 – 3 kHz, covering the frequency range for ITD (and its dynamic
variation) as well as low-frequency ILD as directional and distance localization cues, respectively. A
psychoacoustic experiment validates that the proposed scheme is able to recreate appropriate distances
localization perception outside the median plane.
Keywords: Binaural Ambisonics, HRTF, Virtual auditory display

1. INTRODUCTION
Dynamic virtual auditory display (VAD) is a spatial sound technique which aims to recreate theperception of virtual sources and other spatial auditory events in headphone rendering. With the recent
development of virtual reality and extending applications of dynamic VAD, rendering virtual sources
at various directions and distances is desired.
Actually, auditory directional localization is the comprehensive consequence of multiple cues (1,2).
Interaural time difference (ITD), interaural level difference (ILD), the spectral cue at high frequency,
the dynamic cue caused by head turning are responsible for directional localization in the free-field.
Auditory distance perception is also the comprehensive consequence of multiple cues, but is more
complex than directional localization(1-3). In the free-field, distance-dependent pressure (1/r law) or
loudness is a relative distance cue. Outside the median plane and at the near-field distance (less than
1.0 m), the distance-dependent ILD at low frequency and distance-dependent spectral cue caused by
the diffraction of head are absolute distance cues. Moreover, high-frequency attenuation caused by air
absorption is also a weak distance cue. In reflective room, the energy ratio of direct and reflectedsound is an effective distance cue.
Binaural pressures and their variation caused by head turning include the aforementione d
localization cues. In the free-field, the binaural pressures are determined by a pair of head -relatedtransfer functions (HRTFs)(1). Generally, HRTFs vary with frequency, source direction and distance.
They also depend on individual. At the far-field distance (large than 1.0 m), the HRTFs are
asymptotically independent of distance.
By duplicating the binaural pressures and their variation caused by head turning in a real soundfield, a dynamic VAD recreates localization information and then perception of virtual source
localization in headphone rendering. Of course, the aforementioned directional and distance1
2

Corresponding to: phbsxie@scut.edu.cn
403738218@qq.com

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information provided by binaural pressures (signals) are somewhat redundant. When some cue s areabsent, remaining cues may still enable localization to some extent. This feature is applicable tosimplify the scheme of VAD.
The scheme for rendering free-field virtual source at far-field distance and various directions indynamic VAD are relatively mature. It can be supplemented by far-field HRTF-based filtering andconstantly updating the HRTF data according to the temporary head position detected by head tracker(1). Many far-field HRTFs database are available and various high-efficient schemes for rendering
virtual sources at various directions have been developed.
In contrast, the schemes for rendering virtual sources at various distances in VAD are limited. The
distance perception in reflective field is often simulated by controlling the ratio of direct/reflectiveenergy in headphone rendering. But the accuracy of this method is limited. A conventional scheme forrecreating free-field virtual source is supplemented by filtering the input stimulus with a pair ofnear-field HRTFs and implementing the distance-related attenuation and linear delay, and constantly
updating the HRTF data according to the temporary head position detect by head tracker (4). However,this scheme suffers from following problems.
(1) Near-field HRTF database include HRTFs at various directions and distances. Due to thedifficulties in measurement, there are scarce near-field HRTF database available(5,6).
(2) Although a method for estimating near-field HRTFs on the basis of far-field measurement wassuggested (7), but it requires a far-field HRTF database with high directional resolution (1000 2000 measured directions). In addition, the scheme for estimation is too complicated to be
implemented in real time.
(3) The dimensionality of a full set of near-field HRTFs, whether they come from measurement or
estimation, is enormous. A large memory capacity is required to storage the data. This may bedifficulty in some practical uses, such as mobile devices.
(4) Dynamic VAD requires to update the HRTFs filters constantly according to the temporary positionof target virtual source with respect to head. Inappropriate updating scheme are liable to cause
audible artifact in reproduction (1).
Binaural Ambisonics is another scheme for synthesis binaural signals in dynamic VAD (8). By
combining the principle of spherical harmonics decomposition of target sound field and HRTFbased filtering, binaural Ambisonics has been used to render far-field virtual source at various
directions. One advantage of binaural Ambisonics is that it avoids the audible arti fact caused by
updating HRTFs.
In present work, a binaural Ambisonics-based scheme for rendering virtual source at various
directions distances is proposed. It overcomes the aforementioned problems in conventional scheme.

2. CONVENTIONAL BINAURAL RENDERING SCHEME
Spatial position is specified by distance 0 ≤ r < ∞, azimuth 0º ≤ θ < 360º and elevation -90º ≤I ≤90º.
Where I = -90°, 0° and 90° represents the bottom, horizontal and top direction, respectively; in thehorizontal plane, θ = 0°, 90° and 180° represents the front, right and back direction, respectively.
In conventional scheme, to rendering a free-field virtual source at position (rS, ΩS) = (rS, θS, IS), binauralsignals are recreated by filtering the input stimulus Ein(f) with a pair of near-field HRTFs, and thensupplementing the distance-related attenuation and linear delay:
1(1)
ED (rS , ΩS , f )HD (rS , ΩS , f )exp( j 2S f rS / c) Ein ( f )4S rS
Where α = L or R denotes the left or right ear, respectively; Hα (rS, ΩS, f) are HRTFs at the corresponding
position; f is frequency; and c = 343 m/s is the speed of sound.
In dynamic reproduction, the HRTFs in Eq.(1) are updated constantly.

3. BINAURAL AMBISONICS SCHEME FOR RENDERING VIRTUAL SOURCE AT
VARIOUS DISTANCES
Ambisonics is a series of flexible spatial sound reproduction system originally designed forloudspeaker reproduction. Based on spatial harmonics decomposition and each order approximationof sound field, near-field compensation high-order Ambisonics (NFC-HOA) is able to virtual sourceat various distances in loudspeakers reproduction (9).
Suppose that M loudspeakers are arranged in a spherical surface with radius r0. The position of the i th

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loudspeaker is (r0, Ωi). To create the virtual source at position (rS, ΩS), the loudspeaker signals for the (L-1)
order NFC-HOA are given by
E [ D]S NFC(2)Where E is an M × 1 column matrix or vector of loudspeaker signals:(3)
E [ E1 , E2 ,....EM ]T
2
SNFC is an L × 1 column matrix or vector of encoding signals:
S NFC
Ξl

T
Ein ( f )[Ξ0Y00(1) (ΩS ), Ξ1Y11(1) (ΩS ), Ξ1Y11(2) (ΩS ), Ξ1Y10(1) (ΩS ),.....Ξ L 1Y((1)
L 1)0 (ΩS )]Ξl (krS , kr0 ) hl (krS ) / hl (kr0 )l0,1....( L  1)

(4)

Ylm(1) (ΩS ) and Ylm(2) (ΩS ) with l = 0,1,…(L-1) and m = 0,1…l are the real-valued spherical harmonics
functions; hl(krS) and hl(kr0) are spherical Hankel function of the second kind.
[D] is an M × L2 decoding matrix. When the number of loudspeakers satisfies:
M t L2(5)Matrix [D] can be found from the pseudoinverse solution of L2 × M matrix [Y] with its entries being thespherical harmonics components of loudspeaker directions:(6)[ D] pinv{[Y ]} [Y ]T {[Y ][Y ]T }1ª Y00(1) (Ω1 )
Y00(1) (Ω2 ) .... Y00(1) (ΩM ) º« (1)»Y (Ω )
Y11(1) (Ω2 ) ..... Y11(1) (ΩM ) »[Y ] « 11 1
«
»
:
« (1)
»
(1)
(1)
¬«Y( L 1),0 (Ω1 ) Y( L 1),0 (Ω2 ) ..... Y( L 1),0 (ΩM ) ¼»

(7)

[D] is only relevant to the directions of M loudspeakers and independent from the target source position.
And Eq.(5) is the minimal number of loudspeakers required for (L-1) order NFC-HOA.
In addition, given the radius rH of a spherical region, an (L-1) order NFC-HOA is able to reconstructtarget sound field within some frequency range. The upper frequency limit can be estimated by (9)( L  1)cf d f max
(8)
exp(1)S rH
The upper frequency limit increases with the order.
In binaural NFC-HOA scheme (BNFC-HOA), the signals in Eq.(2) are converted into binaural signals
for headphone rendering by filtering the each loudspeaker signal with a pair of HRTFs of corresponding
loudspeaker position, supplementing the distance-related attenuation and linear delay of each loudspeakers
and the summing up them:
1 M(9)E 'D (rS , ΩS , f )¦ HD (r0 , Ωi , f )exp( j2S f r0 / c) Ei
4S r0 i 1
In other words, the NFC-HOA signals are reproduced by M virtual loudspeakers created by HRTF-based
binaural synthesis. The upper frequency limit of accurately reconstruct binaural pressures in BNFC-HOA
rendering can be evaluated by letting the rH in Eq.(8) be the mean radius 0.0875 m of human head.
To summarize, the BNFC-HOA scheme includes following steps:(1) For a target virtual source at position (rS, ΩS), create the NFC-HOA encoding signals SNFC
according to Eq.(4).
(2) Given the arrangement of M virtual loudspeakers, the encoding signals SNFC are decoded into M
loudspeakers signals according to Eq.(2) and Eq.(6).
(3) The M loudspeaker signals are converted into binaural signals according to Eq.(9) and the renderedby headphone.
(4) When the listener’s head turns, or the position of target source changes, the direction of target
source with respect to head changes. Update the encoding signals SNFC according to the temporary
direction of target source with respect to head.
Compared with the conventional scheme outlined in the Sec.2, the BNFC-HOA scheme has following
advantages:(1) The HRTF-based filters in Eq.(9) are independent of target source position with respect to head.
They only require the far-field HRTFs at a constant loudspeaker distance r0 rather than thenear-field HRTFs at various distances.
(2) The position information of target source with respect to head is included in the encoding signals

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SNFC. For dynamic binaural synthesis, when the head turns or the position of target source changes,
it is only required to update the encoding signals SNFC, avoiding the audible artifacts caused by
updating the HRTF filters.
On the other hand, it can be estimated from Eq.(8) that an 44 order BNFC-HOA and 2025 virtualloudspeakers at least are needed to reconstruct binaural signals accurately up to 20 kHz. Accordingly, 2025
pair of HRTF-based filters are needed to implement the signal processing in Eq.(9). Such a large number offilters make the signal processing difficult. However, the localization information included in the binauralsignals is somewhat redundant. If the BNFC-HOA can be simplified to an appropriate order but it is still
able to provide enough information for directional and distance localization, the problem is solved. The
order will be selected by psychoacoustic experiment and analysis.

4. BINAURAL PRESSURE ERROR IN BNFC-HOA
The upper frequency limit of BNFC-HOA can be approximately estimated from Eq.(8). A more strict
method for evaluating BNFC-HOA is analyzing the error of complex-valued binaural pressures (signals)| E 'D (rS , ΩS , f )  ED (rS , ΩS , f ) |2
H˄
r
,
Ω
,
f
)
10log
(dB)
(10)
10
D S
S
| PD (rS , ΩS , f ) |2
Where Eα(rS, ΩS, f)and E’α(rS, ΩS, f) are target binaural pressures (signals) and binaural pressures in
BNFC-HOA rendering, and given by Eq.(1) and Eq.(9), respectively.
To evaluate the influence of BNFC-HOA order on the binaural pressures error, Figure.1 plots thepressure error of the left and right ear for the (L-1) = 3 and 5 order BNFC-HOA. The target source locatesat distance of rS = 0.25 m and lateral direction of ΩS = (θS, IS) = (90º, 0º) in the horizontal plane. For the(L-1) order rendering, M = (L+1)2 virtual loudspeakers are arranged uniformly on a spherical surface withradius r0 = 1.0 m. The HRTFs used are obtained by 3D-laser-scanned model of KEMAR artificial head
and BEM-based calculation (10). The sample frequency and length of HRTFs are 44.1 kHz and 512
points, respectively. The upper frequencies evaluated from Eq.(8) are also plotted in the figure.

(a) The 3 order(b) The 5 orderFigure 1 – Error of complex-valued binaural pressures for target source
distance rS = 0.25 m and direction ΩS = (θS, IS) = (90º, 0º)
It is observed that
(1) Below the upper frequency limit, the error for the pressure at right ear (ipsilateral to target source)
is generally small (less than -10 dB). The error for the left ear (contralateral to the target source) islarger than that of right ear but is still small. This is due to the fact that the diffraction of soundwave by head makes the pressure spectra at the contralateral ear
complex and difficult to bereconstructed.
(2) Above the upper frequency limit, the errors increases obviously, with maximum reaching 8 dB ~12 dB for the right ear and 35 dB for the left ear.
(3) The upper frequency limit increases with order. However, even for the 5 BNFC-HOA, it is only
able to reconstruct binaural pressures accurately below the upper frequency limit of 2.3 kHz.
The aforementioned case of rS = 0.25 m, ΩS = (θS, IS) = (90º, 0º) is the worst case. It can be proved thatthe error reduces when the target source distance increases, or target source deviates from the horizontal
lateral direction to the front (back) or high (low) elevation.

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5.

VIRTUAL SOURCE LOCALIZATION EXPERIMENT

5.1 Method
A virtual source localization experiment was conducted to examine the effect of BNFC-HOA order todirectional and distance localization. According to the analysis in Sec.3, the (L-1) = 3 and 5 order
BNFC-HOA with (L+1)2 = 25 and 49 virtual loudspeakers were examined. For comparison, the binaural
rendering with conventional method described in Sec. 2 was also examined. Therefore, there were threeschemes in total to be examined.
All three schemes were implemented in a dynamic VAD. The dynamic VAD was based on a PC with aWindows platform and software written in C++ language. An electromagnetic head tracker (Polhemus
FASTRAK) detected the orientation of the subject’s head. It was able to detect the head turning in threedegrees of freedom, including turning around the left-right axes (pitch), around the front-back axes (tilting
or rolling), and around the up-down axes (rotation or yaw). According to the position of the target source
relative to the temporary orientation of the subject’s head, the PC synthesized binaural signals according to
the schemes described in Sec.1 and 2. The HRTFs used in binaural synthesis were identical to those used in
Sec.4. The resultant binaural signals were rendered by an in-ear headphone (Etymotic Research ER-2).
Because the ER-2 headphone exhibits a flat magnitude response measured at the end of an occluded-ear
simulator, the equalization of the headphone to eardrum transmission was omitted. The details of the
dynamic VAD used in the experiment are referred to (11). The update rate and system latency time of the
VAD were 60 Hz and 25.4 ms, respectively.
Two stimuli with different bandwidths were used in the experiment, including pink noise with fullaudible bandwidth and low-pass filtered pink noise with cut off frequency fmax.H given by Eq.(8). Due to theleft-right symmetry, five horizontal target source directions withθS = 0, 45°, 90°, 135° ,and 180° were tested.
For each target direction, there were six target source distances at rS = 0.25, 0.3, 0.4, 0.5, 0.75 and 1.0 m. To
test the ability of the schemes in the reconstruction of binaural distance cues, the binaural signals given by
Eq.(1) and Eq.(8) are equalized (divided) by a factor 1/ rS to exclude the distance-related loudness cue.
The experiment was conducted in a sound-proof listening room where the level of background noise
was less than 30 dBA. Binaural signals were presented at a sound pressure level equivalent to a free-fieldpresentation of approximately 75 dB. Eight subjects participated in the experiment. The subjects were from23 to 30 years old and had normal hearing.
During the experiment, subjects judged the perceived virtual source direction and distance and reported
using an electromagnetic tracker (Polhemus FASTRAK). The tracker included two receivers. One receiver
was fixed on the subject’s head surface to monitor the position and orientation of the head. Another receiver
was fixed at one end of a wooden rod. The subject pointed the rod at the position of the perceived virtual
source and a computer recorded the result. The binaural signals for each case were played twice. Therefore,
there were 8 subjects × 2 repetitions = 16 judgments in each case. And the binaural signals for all cases andrepetitions were played in a random order. During the experiment, subjects were encouraged to turn their
head.
Statistical analysis is applied to the raw experimental results. In each case (each scheme, stimulus,
target position), mean angular error between the perceived and target angles across judgments is used toevaluate the perceived directional error in rendering:
1 N
(11)
':
¦| arccos[rI (n) x rS ]Nn1
Where rS is the unit vector from the origin to the target direction. rI (n) is the unit vector from the origin to
the n th judgment. The dot denotes the scalar multiplication of two vectors. N is the total number of
judgments. If reversal errors (front-back or up-down confusion) occur in the raw localization results,
reversal is resolved through spatial reflection and the percentage of confusion is calculated.
The mean perceived distance across judgments is used to evaluate performance of distance localization:
1 N
(12)
rI
¦ rI (n)Nn1Where rI (n) is the perceived distance of the nth judgment.
In addition, ANOVA is also applied to the raw data.
5.2 Experiment results
No reversal errors occur in all raw localization results. This is due to the fact that dynamic cue

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caused by head turning was included in the rendering. Dynamic cues deleted the reversal errors inlocalization.
For all three schemes with 1/ r S equalization, and two signal bandwidths, all target distances anddirections, the mean angular errors are less than 18.8°. Larger mean angular errors usually occur for
proximal target distance. As target distance increases, the mean angular errors generally reduce. For
example, in the case of conventional scheme with full bandwidth pink noise and target azimuthθ S =90º, the mean angular errors are 17.7º for rS = 0.25 m and 12.2º for r S = 1.00 m. This is due to theerror in reporting the perceived directions using FASTRAK. At proximal distance, a small error in
the position of the receiver of FASTRAK causes a larger error in reported angle. A multi -way
ANOVA on the mean angular errors indicates that, at a significance level of 0.05, the effects ofscheme, stimuli bandwidth and target direction are insignificant to the mean angular errors.
Therefore, the 3 or 5 order BNFC-HOA yields directional localization performance similar toconventional scheme. This is consistent with previous experiment (12). For conciseness, the detailresults and analysis of directional localization are omitted here.
A multi-way ANOVA on the mean perceived distance indicates that, at a significance level of 0.05,
the effect of stimuli bandwidth is insignificant; but the effect of scheme and target direction issignificant.
Figure.2 plots the mean perceived distance and corresponding standard deviation for conventional
scheme with full audible bandwidth pink noise. For target direction at the front θ S = 0° and back θ S =180°, conventional scheme is unable to create the perceived virtual source at various distance.
However, for target direction at θ S = 45°, 90°, 135° , especially at 90º, conventional scheme is able tocreate the perceived virtual source at various distances, although deviation between the mean
perceived and target distances exists. This result is consistent with some previous experiments on
distance perception for the real and virtual source under the free field condition (3). Actually,
distance perception is biased. Under the free-field condition, if the distance-dependent loudness cue
is excluded, distance perception in the median plane is difficult due to the constant ILD. While target
source departs from the median plane, especially at the lateral direction, distance-dependent ILD
provides information for distance perception.
Figure.3 and Figure.4 plot the mean perceived distance and corresponding standard deviation forthe 3 and 5 order BNFC-HOA, respectively, with full audible bandwidth pink noise. For the 3 order
BNFC-HOA, the performance of perceived distance is unsatisfied. However, the 5 order BNFC -HOA
yields distance perceived results similar to those of conventional scheme. The results of multi -way
ANOVA indicate that, at a significance level of 0.05, the difference between the mean perceived
distance of the 5 order BNFC-HOA and conventional scheme is insignificant.

Figure 2 – Mean perceived distance and corresponding standard deviation
for conventional scheme with full audible bandwidth pink noise

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Figure 3 – Mean perceived distance and corresponding standard deviation
for the 3 order BNFC-HOA with full audible bandwidth pink noise

Figure 4 – Mean perceived distance and corresponding standard deviation
for the 5 order BNFC-HOA with full audible bandwidth pink noise

6. DISCUSSION
The results in Sec.5 indicate that both the 3 and 5 order BNFC-HOA yield appropriate directional
localization performance in the horizontal plane. Previous experiment also indicated that the 3 order
dynamic binaural Ambisonics yields appropriate directional localization performance in other target
direction (such as in the median plane)(12). Actually, it can be estimated from Eq.(17) that the 3 or 5-order
BNFC-HOA are able to accurately reconstruct binaural pressures up to 1.4 kHz and 2.3 kHz, respectively,
which basically or completely cover the frequency range (below 1.5 kHz) of ITD as a lateral localization
cue as well as the dynamic variation of ITD as a vertical and front-back localization cue. Although the 3 or
5-order BNFC-HOA are unable to provide accurate high-frequency spectral cue, the information provided
by low-frequency ITD and its dynamic variation are basically enough for directional localization due to theredundancy in various directional localization cues.
The results in Sec.5 also indicate that the 5-order BNFC-HOA yields appropriate distance perception
performance. Actually, the 5-order BNFC-HOA are able to accurately reconstruct binaural pressures up to
2.3 kHz, which approaches the frequency range (below 3.0 kHz) of ILD as distance perception cue.
Therefore, according to the analysis and experimental results, a (L-1) = 5 order BNFC-HOA is5231

suggested for rendering virtual source at various directions and distances in the proposed scheme.
Accordingly, M = L2 = 36, or a little bit more, M = (L+1)2 = 48 virtual loudspeakers (or pairs of
HRTF-based filters) are needed in the scheme of Eq.(9). When one single virtual source is rendered, thescheme of Eq.(9) is more complicated than the conventional one in Eq.(1). However, when multiple virtual
sources are rendered at the same time, such as in the case of virtual auditory environment where multiple
sources for direct sound and image sources for reflections are rendered, the scheme of Eq.(9) is more
effective than the conventional one. This is due to the fact that for the scheme of Eq.(9), multiple virtual
sources share the same set of HRTF-based filters. The number of HRTF-based filters does not increase with
the number of virtual sources.

7. CONCLUSIONS
A 5-order BNFC-HOA scheme is able to provide appropriate information for directional anddistance localization, and therefore to recreate the directional and distance perception of free -fieldvirtual source in dynamic virtual auditory display. The proposed scheme only requires the far-field
HRTFs at 36 ~ 48 directions, rather than the near-field HRTFs or far-field HRTFs with high directional
resolution. It also avoids the audible artifact caused by updating the HRTFs in dyna mic VAD withconventional scheme. The present work only deals with the localization performance of the proposed
scheme. The future work will address the timbre problem.

ACKNOWLEDGEMENTS
This work was supported by the National Natural Science Foundation of China (11674105) andState Key Lab of Subtropical Building Science, South China University of Technology.

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