Trends in Audio System Research based on
Physical Acoustics Models
Many studies have been conducted on audio systems
to achieve a greater sense of presence. These systems
can be classified into two basic groups: multi-channel
systems, which strive to achieve a greater sense of
presence by using more channels than the two used for
2-channel stereo, and sound field reproduction systems,
which attempt to reproduce a sound field accurately,
on the basis of physical acoustics theory. This article
introduces the theoretical background and examples
from recent research on the latter means, including Wave
Field Synthesis (WFS), which attempts to reproduce the
sound field in the entire listening area, and Ambisonics,
which attempts to reproduce the directionality of sound
at the listening position. It also explains methods for
converting between multi-channel audio signals that
have differing numbers of channels using a physical
acoustics approach.
1. Introduction
Sound field reproduction technology plays a very
important role in achieving a sense of presence with
audio. In particular, three-dimensional audio systems
have recently been shown to be effective in providing the
vertical movement of sound in addition to their front,
rear, left, and right directions. These systems include the
22.2 multi-channel sound format, which is an extension
of two-channel stereo. In what follows, however, we shall
discuss other methods that aim to reproduce a sound
field accurately based on the theory of physical acoustics.
Wave field synthesis (WFS), which has been under
research mainly in Europe, is a method that attempts to
reproduce the sound field over the entire listening area.
This type of sound field reproduction method is also being
researched in Japan. Research has also become quite
active on Ambisonics, which is a recording and playback
technology that attempts to reproduce the directionality
of the sound at the optimal listening position. In Section
2 of this article, we discuss methods that reproduce a
sound field using loudspeakers positioned around the
entire periphery of the reproduction area, and in Section
3, we describe methods such as WFS, which reproduce
the sound field by using loudspeakers positioned on one
side of the sound field. In Section 4, we discuss methods
such as Ambisonics, which attempt to reproduce the
directionality of sound. Finally, in Section 5, we apply the
idea of reproducing the directionality of sound to multichannel audio and outline a method for converting
between multichannel sound signals with differing
numbers of channels.
2. Sound Field Reproduction based on the Kirchhoff-Helmholtz Integral Theorem
The Huygens-Fresnel principle is a well-known theory
that explains wave phenomena such as propagation
and diffraction. The principle is illustrated in Figure
1, showing how the wavefront of a propagating wave
generates secondary wavefronts, and the envelope
of these forms the next wavefront. A more rigorous
description of this principle is the Kirchhoff-Helmholtz
Integral Theorem1).
The sound field of the arriving sound wave is given
by p(r, t), the sound pressure function. Here, r is the
position within the sound field, and t is time. The Fourier
transform*1 of sound pressure p(r, t) is written as p(r,
ω). Below, we describe the sound field in the frequency
domain by using the Fourier transform.
We shall consider a region V in the sound field, as
shown in Figure 2. In Figure 2, S is a closed curved surface
bounding V, rA is an arbitrary point inside V, and n is
the outward unit normal vector at the point r on S. The
Kirchhoff-Helmholtz Integral Theorem can be expressed
as:
.. (1)
Secondary wave source
Wave source
Figure 1: Huygens-Fresnel principle
n
V
S
Figure 2: Region V within a sound field
*1
10
A transform used to express time-domain functions in the
frequency domain.
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Here, j is the imaginary unit, k is the wave number,
and ∂/∂n is the partial differential with respect to n. In
Equation (1), the expression:
e − jk|r − r |
| r − rA |
Primary sound field
(hall, etc.)
Listening
area
........... (2)
A
is called a monopole function, and it closely approximates
the sound wave from an omni-directional sound source
when the sound source is sufficiently small relative to
the sound wavelength. Equation (2) expresses an omnidirectional sound source located at the point rA. The
function,
Secondary sound filed
(Reproduction space)
Recording and
playback system
Figure 3: Sound field reproduction using the Camras method
:Microphone
*2
An omni-directional microphone converts the sound pressure
on the front of a diaphragm to an electrical signal, while a
bi-directional microphone converts the gradient (difference)
of sound pressure between front and back surface of a diaphragm to an electrical signal.
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Sound
source
Listening
area
:Microphone
Reproduction field
is called a dipole function, and it closely approximates
the sound wave from a bi-directional sound source
when the sound source is sufficiently small relative
to the sound wavelength. Equation (3) expresses a bidirectional sound source with major axis in the direction
of n, located at point rA.
The terms of the integral in Equation (1) give the sum
of sound pressures at point rA due to an omni-directional
sound source driven by the sound pressure gradient,
∂p(r, ω)/∂n, and a bi-directional sound source driven by
the sound pressure, p(r, ω), both placed at point r on
the boundary. Thus, the sound pressure at the point
rA is expressed by the integral over all points r on the
boundary surface S2).
The Kirchhoff-Helmholtz Integral Theorem shows that
the source sound field can be reproduced by measuring
the sound pressure and sound pressure gradients at the
boundaries of a listening region in the primary sound
field such as a concert hall and then reproducing them at
the boundary of the secondary sound field (reproduction
space). This idea was first proposed in the 1960’s by
Camras3). In the Camras method shown in Figure 3,
omni-directional and bi-directional microphones*2
are placed at the boundaries of the source listening
region to record the sound pressure and sound-pressure
gradient, and these values are used to drive monopole
and dipole sound sources placed at the boundaries of the
reproduction space to reproduce the primary sound field.
Besides the fact that the reproduction space requires
infinite sound sources, another difficulty with the Camras
method is that there are no loudspeakers with monopole
and dipole characteristics that cover the whole frequency
bandwidth of sound.
Ise et al. have proposed a method of “boundary
surface control principle” to resolve these difficulties4).
The boundary surface control method is shown in
Figure 4. The circles arranged along the boundary in the
Primary field
........... (3)
Reproduction
field
Figure 4: Sound field boundary control method
figure represent microphones. The main feature of the
method is that instead of placing loudspeakers around
the boundary of the reproduction region, microphones
are placed there, and the signals input to loudspeakers
placed around the reproduction space are controlled
such that the signals picked up by these microphones
are the same as those recorded by microphones on the
boundary of the primary listening space. Thus, with
boundary surface control principle, it is necessary
to solve the equations for the inverse transmission
characteristic from the loudspeakers to the microphones
in the reproduction space. In other words, it is a inverse
multiple-input, multiple output (MIMO) transmission
problem. Solving for the inverse characteristic is not easy,
but once a solution has been found for the reproduction
space, it can be applied to any primary sound fields.
3. Sound Field Reproduction Based on the Rayleigh
Integral
As shown in Figure 5, the Kirchhoff-Helmholtz
Integral Theorem can be transformed into a Rayleigh I
or II integral by dividing the boundary surface into a flat
surface, S1, and a partial sphere, S2, and increasing the
11
radius of S2 to infinity2). The Rayleigh I integral is given
by:
n
V
........... (4)
r
R
(x , y , z )
and the Rayleigh II integral by:
S1
S2
........... (5)
x, y Plane
z
z=z1
12
Figure 5: Boundary surfaces S1, S2
Secondary wave source
Wave source
Figure 6: Wave-front synthesis using the Raleigh integral
Recording system
Reproduction system
l
Wave source
Here, the normal vector n and the z-axis are in the
same direction, so ∂n can be replaced by ∂z. The only
differences between the Rayleigh integrals and the
Kirchhoff-Helmholtz Integral Theorem are that the terms
of the integrand are just the sound-pressure gradient, ∂p(r,
ω)/∂z, or sound pressure, p(r, ω), and that the integration
is only over a planar surface S1 between the primary and
secondary fields instead of a closed surface surrounding
the secondary field (listening region). Figure 6 shows
an example of wavefront synthesis using the Rayleigh
integrals.
For the Rayleigh I integral, the gradient of sound
pressure, ∂p(r, ω)/∂z, is measured using bi-directional
microphones arranged on a planar surface, and this
signal is used to drive monopole sound sources to recreate
the primary sound field. On the other hand, for the
Rayleigh II integral, sound pressure is measured using
omni-directional microphones positioned on the plane,
and these signals are used to drive dipole sound sources
with their major axis aligned in the z-direction to recreate
the primary sound field. Reproducing a sound field based
on the Rayleigh integrals is the same as the method using
the Kirchhoff-Helmholtz Integral Theorem described in
Section 2 in that a signal measured in the primary sound
field is used to recreate the sound field in a different space.
Using this concept, Berkhout et al. have proposed
Wave Field Synthesis (WFS) based on Rayleigh
integrals5)6). A feature of WFS is that the recording system
used in the primary field and the playback system in the
reproduction field can be treated separately. For WFS,
the reproduction is generally approximated using a
one-dimensional linear loudspeaker array rather than
a planar loudspeaker array on an infinite plane. Figure
7 shows recording and reproduction systems for WFS.
WFS assumes a virtual field, rather than the primary
field, and it inputs a signal obtained by a simulation
of sound propagation in this virtual field to the sound
reproduction system. For example, by positioning
loudspeakers for the 11 forward channels of 22.2
multi-channel sound system7) in the virtual space and
synthesizing the wavefront using the loudspeaker array,
WFS is able to reproduce the frontal sound of 22.2 multichannel audio content.
Research and development on WFS is very active in
Europe, and a WFS system using a multi-actuator panel
is being developed at the Institute for Research and
Coordination Acoustic/Music (IRCAM) in France. Other
projects include, the WFS circle, which uses a circular
speaker array, at the Deutsch Telecom Laboratories of
W
n
z=z0
z=z1
Figure 7: Sound field synthesis using WFS
the Berlin Institute of Technology and the WFS theater
at the Fraunhofer Institute for Digital Media Technology
(IDMT)8). Even vehicle-mounted WFS systems have been
developed9).
4. Sound Field Reproduction Based on Spherical
Harmonic Expansion
Cooper and Shiga have proposed a Fourier
representation for the directionality of sound observed at
the listening point10). The directional pattern of sound at a
particular listening point can be expressed as a function,
S, of angle, θ, and S(θ) can be decomposed into a constant
component, first-order sine and cosine components,
second-order sine and cosine components, and so on, as
shown in Figure 8. Omni-directional microphones have
the same directionality as the constant component, and
bi-directional microphones have the same directionality
as the first-order components, so these microphones
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Constant component
0
1st-order cosine component
Bi-directional
C
2π
+
0
Omni-directional
2π
+
0
1st-order sine component
2π
=
0
π/2
2π
・
・
+
Figure 8: Fourier expansion of the directional patterns of sound injected at the point of sound collection
can be used in a system to make observations of the
constant and first-order components. Gerzon has devised
the Ambisonics method11) based on such directionality
patterns. The method offers a hierarchical approach
to sound recording, transmission and reproduction,
which corresponds to monophonic, two channel
stereo, horizontal surround, or 3D audio. Ambisonics
approximates the directional pattern of sound by
expanding the sound pressure function at the listening
point in terms of the spherical harmonic functions up to
a certain order. Figure 9 shows an Ambisonic recording
and reproduction system using only zero and first-order
terms. W in Figure 9 is the directionality of the omnidirectional microphone, which is used to measure the
zeroth order coefficient of the expansion. X, Y, and
Z are the directional characteristics of bi-directional
microphones with their major axes aligned with the x,
y, and z-axes, respectively. These are used to measure
the first-order coefficients. The Ambisonics concept has
recently been extended by expanding the incoming
sound using higher-order spherical-harmonic functions*3.
We explain this extended Ambisonics concept below13).
Consider a planar sound wave arriving from an
arbitrary direction, (ψ, φ). Here, ψ is the azimuth and φ is
the elevation of the arrival angle. Also, as shown in Figure
10, the listening point P is indicated by a vector r whose
direction is specified by (θ, φ) and whose magnitude is
r. P is expressed in rectangular coordinates by:
r = (r cos θ cos ϕ
r sin θ cos ϕ
r sin ϕ )
+
ϕ
S
y
θ
x
Figure 10: Polar coordinate system
Here, T indicates the transpose of a matrix. The sound
pressure at P can be expanded with spherical harmonic
functions as14):
r
.... (7)
is
Here, Q is the output of the sound source, and
the complex conjugate of Y. Moreover, jn(z) is a spherical
Bessel function of the first kind*4 expressing the change
in sound pressure in the radial (r) direction. Accordingly,
........... (8)
These methods are sometimes called Higher-Order Ambisonics
(HOA).
*4
Bessel functions are particular solutions of the Bessel differential equation. Spherical Bessel functions are a type of function
defined using Bessel functions.
*5
Spherical harmonic functions appear when the wave equation
is expressed in polar coordinates, and they are the components of wave motion in the angular direction. The coefficients shown in the square root in Eqn. (8) are normalizing
coefficients which make the spherical harmonic functions
orthonormal.
*3
−
W
+
−
−
Figure 9: Ambisonics spatial recording method 12)
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represents spherical harmonic functions15)*5 expressing
+
Leftward Y
Listening point P
........... (6)
T
Upward Z
Forward X
z
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the change in sound pressure in the angular directions.
P mn (x) in the spherical harmonic functions are associatedLegendre functions of the first kind*6, given by: 15)
P
... (9)
Note that the expression within the braces in Equation
(9) terminates when the power of the x coefficient reaches
zero.
Now, let us consider the problem of synthesizing the
sound pressure at the listening point P, for a planar
sound wave arriving from the direction (ψ, φ), using N
loudspeakers arranged on a sphere of diameter σ centered
on the origin. In this case, the listening point is within
the sphere (r<σ). This situation is shown in Figure 11. We
define the direction of the first speaker as (θl,φl) and the
input to that speaker as al(k). We can approximate the
sound wave from the loudspeaker with a planar wave*7.
The synthesized acoustic pressure from all N loudspeakers
can be expanded in terms of the spherical harmonic
functions of Equation (7) as follows:
r
... (10)
When Equations (7) and (10) are equal, a planar wave
arriving from the (ψ, φ) direction is reproduced by the
loudspeakers. These equations are solved by using the
orthogonality of the spherical harmonic functions and
limiting the degree of the expansion to a constant, M.
This yields the basic Ambisonics equation, given by:
P
*7
Legendre functions are solutions to the Legendre differential
equations, and for integer n, are parameters appearing in Legendre’s differential equation. Associated-Legendre functions
are solutions to m-th derivatives of the Legendre differential
equation. In Eqn. (9), n and m are parameters for these.
If the distance between the origin and listening point is not
small, the sound wave from the loudspeaker can be approximated by a planar wave.
Observation
Reproduction
Input
Listening point P
rϕ
θ
Listening point
Figure 11: Synthesis of sound pressure from a flat surface
wave using speakers
14
We give an example computation limiting the degree
of the expansion in Equation (11) to one (M=1). The
associated-Legendre functions of the first kind are:
P 00 ( x ) = 1,
P 1−1 ( x ) =
(1 − x 2 ) −1/ 2
,
2
......... (12)
P ( x ) = x,
0
1
P 11 ( x ) = (1 − x 2 ) 1/ 2 ,
The basic equations for first-order Ambisonics are:
N
l =1
N
l =1
N
l =1
N
l =1
al ( k ) = Q y
e jθl cos ϕ l al ( k ) = Qy e jψ cos φ
sin ϕ l al ( k ) = Q y sin φ
e − jθl cos ϕ l al ( k ) = Q y e − jψ cos φ
(n = 0, m = 0)
(n = 1, m = −1)
(n = 1, m = 0)
...... (13)
(n = 1, m = 1)
From the difference of the second and fourth
expressions in Equation (13), we get:
N
l =1
N
P
......... (11)
*6
The right side of Equation 11 gives the orthogonal
expansion coefficients of an acoustic wave arriving from
the (ψ, φ) direction, and the left side gives the orthogonal
expansion coefficients of the sound wave reproduced
using N loudspeakers.
l =1
cos θ l cos ϕ l al ( k ) = Q y cos ψ cos φ
......... (14)
sin θ l cos ϕ l al ( k ) = Q y sin ψ cos φ
From Equations (13) and (14), we get:
N
l =1
N
l =1
N
l =1
N
l =1
al ( k ) = Q y
cos θ l cos ϕ l al ( k ) = Q y cos ψ cos φ
......... (15)
sin θ l cos ϕ l al ( k ) = Q y sin ψ cos φ
sin ϕ l al ( k ) = Q y sin φ
The right sides in Equation (15) are the signals observed
by microphones W, X, Y, and Z, in Figure 9, respectively.
The angles on the left side of Equation (15), θl, and φl
(l=1, ..., N), are defined by the loudspeaker positions.
Thus, by inputting signals, al(k), satisfying Equation
(15) to the loudspeakers, the arriving wavefront can be
reproduced.
Spherical microphones have been extensively
researched16) for Ambisonic recording, and a microphone
array with 32 compact microphone capsules that is able
to observe Ambisonics coefficients up to the 4th order
has been commercialized17). In addition, sound field
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reproduction methods using spherical harmonic function
expansions have become the main-stream in sound-field
reproduction theory18)19). There have even been attempts
to handle WFS and Ambisonics in a unified way20).
5. Conversion between Multi-channel Audio Signals
with Differing Numbers of Channels
The 22.2 multi-channel audio system normally
requires 24 speakers to be installed in the reproduction
space. However, such an installation would be difficult
in typical home listening environments. Accordingly, we
are advancing research on signal conversion methods
that enable reproduction of the same physical sound
properties at the listening point with different numbers
of channels and speaker placements21). Rather than
recreating the primary sound field, as described in
Sections 2 and 3, we attempt to recreate a reproduced
sound field such as a mixing studio (the original sound
field) equipped with a multichannel sound system in a
different space. The method described below reproduces
the sound direction at the listening point as well as the
method discussed in Section 4.
A block diagram of the signal conversion method is
shown in Figure 12. First, the sound from a loudspeaker
is expressed using the monopole function (2). Thus, if
n loudspeakers are used, the original sound field is
expressed as the sum of n monopole functions. When
considering the sound direction, the physical properties of
sound in the original field are three-dimensional vectors.
Accordingly, the sound propagation characteristic*8, Hd,
in the source space as shown in Figure 12, is a 3 × n
matrix, which converts the loudspeaker signal into
*8
The sum of sound propagation characteristics from loudspeakers to the listening point.
3D vector sound physical values. Similarly, the sound
propagation characteristic,
in the reproduction field
is a 3 × m matrix. The required conversion matrix, W,
satisfies the equation:
......... (16)
H̃d W = Hd
Actually, Equation (16) is a system of three simultaneous
equations for the x, y, and z components. Thus, if m>3,
there are more variables than equations, the system is
underdetermined*9, and there are an infinite number
of solutions. To obtain an analytical solution*10 to the
equation, we can divide the reproduction space into
spherical triangles with three neighboring loudspeakers
as the vertices. In other words, the method uses three
neighboring loudspeakers to form a virtual sound image
at the positions of the loudspeakers in the original space.
Subjective evaluation experiments were carried out
in order to evaluate the conversion method. A 22.2
multichannel sound signal without the two low frequency
effect channels was converted into a fewer number of
channels for various loudspeaker arrangements. The
impairment with the conversion was assessed from
the two spatial sound impressions, sound localization
and sound envelopment, with the “double-blind triplestimulus with hidden reference” method22). The subject
was asked to assess the impairment on two sound stimuli
compared with the reference 22-channel sound, according
to a continuous five-grade impairment scale shown in
Table 1. One of the two stimuli was the same as the
reference (referred to as the hidden reference). The other
*9
*10
For linear simultaneous equations, this situation is when the
number of variables is greater than the number of equations.
A solution that can be expressed as an equation rather than
one obtained by numerical computation.
n channel audio signal
t
m channel audio signal
q
n × m matrix
W
F
=W
F
Fourier
transform
-1
q t
Inverse
Fourier
transform
Sound propagation
in original space
Hd
Hd
Coincidence of physical
properties of sound
H̃d W
Sound propagation
in original space
H̃d
Figure 12: Converting signals between multi-channel audio systems
Table 1: Impairment scale for double-blind triple-stimulus with hidden-reference method
5.0
Imperceptible
4.0
Perceptible, but not annoying
3.0
Slightly annoying
2.0
Annoying
1.0
Very annoying
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stumulus (referred to as the object) was the converted
sound with a reduced number of channels. After the
experiment, the difference grade was calculated for each
object by subtracting the grade to the hidden reference
from that to the object. The first experiments reduced the
22-channel signal, composed of 9 top, 10 middle, and 3
bottom loudspeakers, to 10 channels in three different
loudspeaker arrangements. The loudspeakers in these
three arrangements were allocated to the top, middle,
and bottom layers in proportions of 4/5/1, 3/6/1, and
3/5/2, respectively. In the experiment, subjects were 38
peoples. The conversion method obtained a difference
grade of more than -0.8 for both spatial impressions in
all loudspeaker arrangements. Further experiments were
then conducted on the converted sound signals with
eight and six channels. We prepared three loudspeaker
arrangements for each number of channels. Top, middle,
and bottom loudspeaker allocations for eight channels
were 3/4/1, 2/5/1, and 2/4/2, respectively, and for six
channels they were 2/4/0, 1/5/0, and 1/4/1. The subjective
evaluations with 38 subjects yielded difference scores of
about -1.0 for both types of spatial impression with the
eight-channel conversion and less than -1.0 for the sixchannel conversion. In other words, the 8 channel sound
converted by the method gave almost the same spatial
impressions (difference grades of more than -1.0) as the
original 22 channel sound.
6. Conclusion
We described the theory and current research activity on
sound field reproduction methods that strive to reproduce
the physical properties of sound field accurately. Twochannel and multi-channel audio systems do not attempt
to reproduce physical properties accurately; instead, they
attempt to create a sound field in the studio that gives
a strong sense of presence. However, the conversion
between different multi-channel audio signals depends
on matching the physical properties of sound including
a sound pressure. From the practical viewpoint, WFS,
which attempts to reproduce physical properties of
sound, tries to reduce the number of loudspeakers within
the range in which the degradation in the reproduced
sound is imperceptible. The future immersive audio
systems will adjust the balance between the accurate
reproduction of the physical properties of sound and
the approximate reproduction whose degradation is
imperceptible by listeners.
(Akio Ando)
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