The document discusses spatial audio and sound localization techniques. It describes how head-related transfer functions (HRTFs) capture the spatial cues the human auditory system uses to localize sound sources. HRTFs are measured impulse responses between sound sources and the ears. Convolving a sound source with HRTFs for that source's location produces binaural audio. Ambisonics uses spherical harmonics to represent soundfields with a small number of channels and can render spatial audio in virtual reality.

1. Build Your Own VR Display
Spatial Sound
Nitish Padmanaban
Stanford University
stanford.edu/class/ee267/

2. Overview
• What is sound?
• The human auditory system
• Stereophonic sound
• Spatial audio of point sound sources
• Recorded spatial audio
Zhong and Xie, “Head-Related Transfer Functions
and Virtual Auditory Display”

3. What is Sound?
• “Sound” is a pressure wave propagating in a medium
• Speed of sound is where c is velocity, is density of
medium and K is elastic bulk modulus
• In air, speed of sound is 340 m/s
• In water, speed of sound is 1,483 m/s
c = K
r r

4. Producing Sound
• Sound is longitudinal vibration
of air particles
• Speakers create wavefronts by
physically compressing the air,
much like one could a slinky

5. The Human Auditory System
pinna
Wikipedia

6. The Human Auditory System
pinna
cochlea
Wikipedia
• Hair receptor cells pick up
vibrations

7. The Human Auditory System
• Human hearing range:
~20–20,000 Hz
• Variation between
individuals
• Degrades with age
D. W. Robinson and R. S. Dadson, 1957
Hearing Threshold in Quiet

8. Stereophonic Sound
• Mainly captures differences between the ears:
• Inter-aural time difference
• Amplitude differences from path length
and scatter
Wikipedia
time
L
R
t + Dt
t
L R
Hello,
SIGGRAPH!

9. Stereo Panning
• Only uses the amplitude differences
• Relatively common in stereo audio tracks
• Works with any source of audio
Line of
sound
0
0.5
1
L R
0
0.5
1
L R
0
0.5
1
L R

10. Stereophonic Sound Recording
• Use two microphones
• A-B techniques captures
differences in time-of-arrival
• Other configurations work too,
capture differences in amplitude
Rode
Olympus
Wikipedia
X-Y technique

11. Stereophonic Sound Synthesis
L R
R
time
amplitude
L
time
amplitude• Ideal case: scaled & shifted Dirac peaks
• Shortcoming: many positions are identical
Input
time
amplitude
Input

12. Stereophonic Sound Synthesis
• In practice: the path lengths and scattering are more
complicated, includes scattering in the ear, shoulders etc.
R
time
amplitude
L
time
amplitude
R
time
amplitude
L
time
amplitude

13. Head-Related Impulse Response (HRIR)
• Captures temporal responses at all possible sound directions,
parameterized by azimuth and elevation
• Could also have a distance parameter
• Can be measured with two microphones in ears of mannequin &
speakers all around
Zhong and Xie, “Head-Related Transfer Functions and Virtual Auditory Display”
q
q f
L R

14. Head-Related Impulse Response (HRIR)
• CIPIC HRTF database: http://interface.cipic.ucdavis.edu/sound/hrtf.html
• Elevation: –45° to 230.625°, azimuth: –80° to 80°
• Need to interpolate between discretely sampled directions
V. R. Algazi, R. O. Duda, D. M. Thompson and C. Avendano, "The CIPIC HRTF Database,” 2001

15. Head-Related Impulse Response (HRIR)
• Storing the HRIR
• Need one timeseries for each location
• Total of samples, where is the number of
samples for azimuth, elevation, and time, respectively
hrirL t;q,f( )
hrirR t;q,f( )
2×Nq ×Nf ×Nt Nq,f,t

16. Head-Related Impulse Response (HRIR)
Applying the HRIR:
• Given a mono sound source and its 3D position
L R
s t( )
s t( )

17. Head-Related Impulse Response (HRIR)
Applying the HRIR:
• Given a mono sound source and its 3D position
1. Compute relative to center of listener’s head
L R
q,f( )
s t( )
s t( )
q,f( )

18. Head-Related Impulse Response (HRIR)
Applying the HRIR:
• Given a mono sound source and its 3D position
1. Compute relative to center of listener’s head
2. Look up interpolated HRIR for left and right ear at these
angles
time
time
amplitude
hrirL t;q,f( )
hrirR t;q,f( )
amplitude
q,f( )
s t( )

19. Head-Related Impulse Response (HRIR)
Applying the HRIR:
• Given a mono sound source and its 3D position
1. Compute relative to center of listener’s head
2. Look up interpolated HRIR for left and right ear at these
angles
3. Convolve signal with HRIRs to get the sound
at each ear
time
time
amplitudeamplitude
sL t( )= hrirL t;q,f( )*s t( )
sR t( )= hrirR t;q,f( )*s t( )
hrirL t;q,f( )
hrirR t;q,f( )
q,f( )
s t( )

20. Head-Related Transfer Function (HRTF)
frequency
amplitude
hrtfL wt;q,f( )
hrtfR wt;q,f( )
amplitude
sL t( )= hrirL t;q,f( )*s t( )
sR t( )= hrirR t;q,f( )*s t( )
• HRTF is Fourier transform of HRIR! (you’ll find the term HRTF
more often that HRIR)
sL t( ) = F-1
hrtfL wt;q,f( )×F s t( ){ }{ }
sR t( ) = F-1
hrtfR wt;q,f( )× F s t( ){ }{ }
time
time
amplitude
hrirL t;q,f( )
hrirR t;q,f( )
frequency

21. • HRTF is Fourier transform of HRIR! (you’ll find the term HRTF
more often that HRIR)
• HRTF is complex-conjugate
symmetric (since the HRIR must
be real-valued)
Head-Related Transfer Function (HRTF)
frequency
amplitudeamplitude
frequency
hrtfL wt;q,f( )
hrtfR wt;q,f( )
sL t( )= hrirL t;q,f( )*s t( )
sR t( )= hrirR t;q,f( )*s t( )
sL t( ) = F-1
hrtfL wt;q,f( )×F s t( ){ }{ }
sR t( ) = F-1
hrtfR wt;q,f( )× F s t( ){ }{ }

22. Spatial Sound of N Point Sound Sources
L R
s2 t( )
• Superposition principle holds, so just sum the contributions of
each s1 t( )
q2
,f2
( )
q1
,f1
( )
sL t( ) = F-1
hrtfL wt;qi
,fi
( )× F si t( ){ }{ }
i=1
N
å
sR t( ) = F-1
hrtfR wt;qi
,fi
( )× F si t( ){ }{ }
i=1
N
å

23. Spatial Audio for VR
• VR/AR requires us to re-think audio, especially spatial audio!
• User’s head rotates freely  traditional surround sound
systems like 5.1 or even 9.2 surround isn’t sufficient

24. Spatial Audio for VR
Two primary approaches:
1. Real-time sound engine
• Render 3D sound sources via HRTF in real time, just
as discussed in the previous slides
• Used for games and synthetic virtual environments
• A lot of libraries available: FMOD, OpenAL, etc.

25. Spatial Audio for VR
Two primary approaches:
2. Spatial sound recorded from real environments
• Most widely used format now: Ambisonics
• Simple microphones exist
• Relatively easy mathematical model
• Only need 4 channels for starters
• Used in YouTube VR and many other platforms

26. Ambisonics
• Idea: represent sound incident at a point (i.e. the listener)
with some directional information
• Using all angles is impractical – need too many sound
channels (one for each direction)
• Some lower-order (in direction) components may be
sufficient  directional basis representation to the rescue!
q,f

27. Ambisonics – Spherical Harmonics
• Use spherical harmonics!
• Orthogonal basis functions on the surface of a sphere, i.e.
full-sphere surround sound
• Think Fourier transform equivalent on a sphere

28. Ambisonics – Spherical Harmonics
0th order
1st order
2nd order
3rd order
Wikipedia
Remember, these
representing functions on
a sphere’s surface

29. Ambisonics – Spherical Harmonics
1st order approximation
 4 channels: W, X, Y, Z
W
X Y Z
Wikipedia

30. Ambisonics – Recording
• Can record 4-channel Ambisonics via special microphone
• Same format supported by YouTube VR and other
platforms
http://www.oktava-shop.com/

31. Ambisonics – Rendered Sources
W = S ×
1
2
X = S ×cosq cosf
Y = S ×sinq cosf
Z = S ×sinf
• Can easily convert a point sound source, S, to the 4-
channel Ambisonics representation
• Given azimuth and elevation , compute W, X, Y, Z asq,f
omnidirectional component (angle-independent)
“stereo in x”
“stereo in y”
“stereo in z”

32. Ambisonics – Playing it Back
LF = 2W + X +Y( ) 8
LB = 2W - X +Y( ) 8
RF = 2W + X -Y( ) 8
RB = 2W - X -Y( ) 8
• Easiest way to render Ambisonics: convert W, X, Y, Z
channels into 4 virtual speaker positions
• For a regularly-spaced square setup, this results in
LF
LB
RF
R
L R

33. Ambisonics – Omnitone
• Javascript-based first-order Ambisonic decoder
Google, https://github.com/GoogleChrome/omnitone

34. References and Further Reading
• Google’s take on spatial audio: https://developers.google.com/vr/concepts/spatial-audio
HRTF:
• Algazi, Duda, Thompson, Avendado “The CIPIC HRTF Database”, Proc. 2001 IEEE Workshop on
Applications of Signal Processing to Audio and Electroacoustics
• download CIPIC HRTF database here: http://interface.cipic.ucdavis.edu/sound/hrtf.html
Resources by Google:
• https://github.com/GoogleChrome/omnitone
• https://developers.google.com/vr/concepts/spatial-audio
• https://opensource.googleblog.com/2016/07/omnitone-spatial-audio-on-web.html
• http://googlechrome.github.io/omnitone/#home
• https://github.com/google/spatial-media/

35. References and Further Reading
• Google’s take on spatial audio: https://developers.google.com/vr/concepts/spatial-audio
HRTF:
• Algazi, Duda, Thompson, Avendado “The CIPIC HRTF Database”, Proc. 2001 IEEE Workshop on
Applications of Signal Processing to Audio and Electroacoustics
• download CIPIC HRTF database here: http://interface.cipic.ucdavis.edu/sound/hrtf.html
Resources by Google:
• https://github.com/GoogleChrome/omnitone
• https://developers.google.com/vr/concepts/spatial-audio
• https://opensource.googleblog.com/2016/07/omnitone-spatial-audio-on-web.html
• http://googlechrome.github.io/omnitone/#home
• https://github.com/google/spatial-media/
Demo