Broadcast Technology No.46, Autumn 2011 ● C NHK STRL10

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. IntroductionSound 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 multi-channel audio and outline a method for converting between multichannel sound signals with differing numbers of channels.

2. Sound Field Reproduction based on the Kirch-hoff-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 A transform used to express time-domain functions in the frequency domain.

Secondary wave source

Wave source

n

SV

Figure 1: Huygens-Fresnel principle

Figure 2: Region V within a sound field

.. (1)

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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:

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 omni-directional sound source located at the point rA. The function,

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 bi-directional 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

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

Prim

ary

field

:Microphone

Listeningarea

Sound source

:Microphone

Reproduction field

Repr

oduc

tion

field

Figure 4: Sound field boundary control method

Primary sound field (hall, etc.)

Listening area

Secondary sound filed (Reproduction space)

Recording and playback system

Secondary sound filed (Reproduction space)

Figure 3: Sound field reproduction using the Camras method

*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 dia-phragm to an electrical signal.

e

r r

jk r rA

A

− −

−

| |

| |........... (2)

........... (3)

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radius of S2 to infinity2). The Rayleigh I integral is given by:

and the Rayleigh II integral by:

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 multi-channel 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

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

Plane

nV

Rr

(x )y z, ,

S1

S2

z=z1

x, y

z

Secondary wave source

Wave source

Recording system Reproduction system

Wav

e so

urce

W

z=z1

z=z0

l

n

Figure 5: Boundary surfaces S1, S2

Figure 6: Wave-front synthesis using the Raleigh integral

Figure 7: Sound field synthesis using WFS

.......... (4)

.......... (5)

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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 omni-directional 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:

Here, T indicates the transpose of a matrix. The sound pressure at P can be expanded with spherical harmonic functions as14):

Here, Q is the output of the sound source, and is 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,

represents spherical harmonic functions15)*5 expressing

W

+

+

+

－

－

－

Forward X

Leftward Y

Upward Z

x

r

y

z

θ

ϕ

Listening point P

Figure 9: Ambisonics spatial recording method 12)

Figure 10: Polar coordinate system

*3 These methods are sometimes called Higher-Order Ambisonics (HOA).

*4 Bessel functions are particular solutions of the Bessel differen-tial 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 compo-nents of wave motion in the angular direction. The coeffi-cients shown in the square root in Eqn. (8) are normalizing coefficients which make the spherical harmonic functions orthonormal.

Omni-directional

Bi-directional

Constant component

1st-order cosine component

1st-order sine component

S

=

+

+

+

2π

2π

2π

2π

π/2C

0

0

0

0

・・Figure 8: Fourier expansion of the directional patterns of sound injected at the point of sound collection

r

r r r r= ( )cos cos sin cos sinθ ϕ θ ϕ ϕ T ........... (6)

... (7)

........... (8)

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the change in sound pressure in the angular directions. ( )P xn

m in the spherical harmonic functions are associated-Legendre functions of the first kind*6, given by: 15)

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:

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:

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.

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:

The basic equations for first-order Ambisonics are:

From the difference of the second and fourth expressions in Equation (13), we get:

From Equations (13) and (14), we get:

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

*6 Legendre functions are solutions to the Legendre differential equations, and for integer n, are parameters appearing in Leg-endre’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.

*7 If the distance between the origin and listening point is not small, the sound wave from the loudspeaker can be approxi-mated by a planar wave.

r

θ

ϕ

Observation Reproduction

Listening point

Listening point P

Input

Figure 11: Synthesis of sound pressure from a flat surface wave using speakers

P

r

P x00 1( ) ,=

P xx

11

2 1 212

−−

=−( ) ( ) ,

/

P x x10 ( ) ,=

P x x11 2 1 21( ) ( ) ,/= −

a k Q n mll

N

y( ) ( , )=

= = =1

0 0

e a k Q e n mj ll l

l

N

yjθ ψϕ φcos ( ) cos ( , )

=

= = = −1

1 1

sin ( ) sin ( , )ϕ φl ll

N

a k n m=

= = =1

1 0Q y

e a k e n mj ll l

l

Nj−

=

−= = =θ ψϕ φcos ( ) cos ( , )1

1 1Q y

a kll

N

( )=

=1

Q y

cos cos ( ) cos cosθ ϕ ψ φl l ll

N

a k=

=1

Q y

sin cos ( ) sin cosθ ϕ ψ φl l ll

N

a k=

=1

Q y

sin ( ) sinϕ φl ll

N

a k=

=1

Q y

cos cos ( ) cos cosθ ϕ ψ φl l ll

N

a k=

=1

Q y

sin cos ( ) sin cosθ ϕ ψ φl l ll

N

a k=

=1

Q y

.. (9)

.. (10)

PP

......... (11)

......... (12)

..... (13)

......... (14)

......... (15)

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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

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:

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 triple-stimulus 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

channel audio signaln channel audio signalm

n m

H̃ Wd

H̃dHd

H

W

W

d

=t qq× matrix

Coincidence of physical properties of sound

Sound propagation in original space

Sound propagation in original space

InverseFourier

transform

Fourier transform

F F -1 t

5.0

4.0

3.0

2.0

1.0

Imperceptible

Perceptible, but not annoying

Slightly annoying

Annoying

Very annoying

Figure 12: Converting signals between multi-channel audio systems

Table 1: Impairment scale for double-blind triple-stimulus with hidden-reference method

H̃ W Hd d= ......... (16)

*8 The sum of sound propagation characteristics from loudspeak-ers to the listening point.

*9 For linear simultaneous equations, this situation is when the number of variables is greater than the number of equations.

*10 A solution that can be expressed as an equation rather than one obtained by numerical computation.

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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 six-channel 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. ConclusionWe described the theory and current research activity on

sound field reproduction methods that strive to reproduce the physical properties of sound field accurately. Two-channel 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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