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Audio Engineering Society

Convention Paper

Presented at the 125th Convention

2008 October 2–5 San Francisco, CA, USA

The papers at this Convention have been selected on the basis of a submitted abstract and extended precis that have

been peer reviewed by at least two qualiﬁed anonymous reviewers. This convention paper has been reproduced from

the author’s advance manuscript, without editing, corrections, or consideration by the Review Board. The AES takes

no responsibility for the contents. Additional papers may be obtained by sending request and remittance to Audio

Engineering Society, 60 East 42nd Street, New York, New York 10165-2520, USA; also see www.aes.org. All rights

reserved. Reproduction of this paper, or any portion thereof, is not permitted without direct permission from the

Journal of the Audio Engineering Society.

Reproduction of Virtual Sound Sources

Moving at Supersonic Speeds in Wave Field

Synthesis

Jens Ahrens and Sascha Spors

Deutsche Telekom Laboratories, Technische Universit¨at Berlin, Ernst-Reuter-Platz 7, 10587 Berlin, Germany

Correspondence should be addressed to Jens Ahrens (jens.ahrens@telekom.de)

ABSTRACT

In conventional implementations of wave ﬁeld synthesis, moving sources are reproduced as sequences of

stationary positions. As reported in the literature, this process introduces various artifacts. It has been

shown recently that these artifacts can be reduced when the physical properties of the wave ﬁeld of moving

virtual sources are explicitly considered. However, the ﬁndings were only applied to virtual sources moving

at subsonic speeds. In this paper we extend the published approach to the reproduction of virtual sound

sources moving at supersonics speeds. The properties of the actually reproduced sound ﬁeld are investigated

via numerical simulations.

1.INTRODUCTION

Since several decades, the problem of physically

recreating a given wave ﬁeld has been addressed

in the audio community. Independent of the cho-

sen approach, two rendering techniques exist: Data

based and model based reproduction [1]. The for-

mer case aims at perfectly reproducing a captured

sound ﬁeld. This situation will not be treated in

this paper. We concentrate on the latter case where

a sound scene is composed of a number of virtual

sound sources derived from analytical spatial source

models. For stationary virtual scenes accurate re-

production techniques exist. However, the reproduc-

tion of dynamic scenes implicates certain peculiari-

ties. This is mostly due to the fact that the speed

of sound in air is constant. When a source moves,

the propagation speed of the emitted wave ﬁeld is

not aﬀected. However, the emitted wave ﬁeld diﬀers

from that of a static source in various ways. For ex-

ample, in sources moving slower than the speed of

sound, the sound waves emitted in the direction of

motion experience an increase in frequency. Sound

Ahrens AND Spors Supersonic sources

waves emitted in opposite direction of motion expe-

rience a decrease in frequency. The whole of these

alterations is known as Doppler Eﬀect [2].

Typical implementations of sound ﬁeld reproduction

systems do not take the Doppler Eﬀect into account.

Dynamic virtual sound scenes are rather reproduced

as a sequence of stationary snapshots. Thus, not

only the virtual source but also its entire wave ﬁeld

is moved from one time instant to the next.

This concatenation leads to Doppler-like frequency

shifts. However, these frequency shifts occur due

to warping of the time axis rather than due to the

constant speed of sound, a circumstance which intro-

duces artifacts. Furthermore, this approach is lim-

ited to the reproduction of virtual sources moving

slower than the speed of sound. The artifacts have

been recently discussed in the literature in the con-

text of wave ﬁeld synthesis [3]. We are not aware

of an according publication focussing on alternative

sound ﬁeld reproduction methods. See [4, 5] for

a treatment of moving virtual sources in binaural

(HRTF-based) reproduction.

Various alternative implementations of the conven-

tional approach of concatenating stationary source

positions as outlined above are being applied both

frame-based as well as in a sample-by-sample fash-

ion. Most notably, in [3] it is proposed to incorporate

the retarded time of a moving source (see section 2)

into the driving function of a stationary source. Re-

sults presented ibidem show that this strategy still

leaves prominent artifacts.

As shown by the authors in [6], the mentioned arti-

facts occurring in conventional implementations can

be avoided when the physical properties of the wave

ﬁeld of moving sound sources are a priori taken into

account. However, the approach in [6] was exclu-

sively applied to virtual sources moving slower than

the speed of sound. In this paper, we extend this

approach to the reproduction of virtual sources mov-

ing at supersonic speeds. Our work can also be re-

garded as an extension of the approach presented

in [7] which focuses on the reproduction of the fre-

quency content present in supersonic booms of air-

crafts but does not physically reproduce the actual

wave front.

Note that the considerations presented in this paper

are of relevance only for sound ﬁeld reproduction ap-

proaches which employ time delays in the procedure

of yielding the loudspeaker driving signals.

α∆x

x

y

y=y0

r

n

x

x0

x0

Fig. 1: The coordinate system and geometry used

in this paper. The dots •denote the positions of the

secondary sources used for wave ﬁeld synthesis. The

grey-shaded area denotes the listening area.

2.THE WAVE FIELD OF A MOVING SOURCE

The fundamental prerequisite for model-based sound

ﬁeld reproduction is the knowledge of the sound ﬁeld

that is to be recreated. In this section, we derive

analytical expressions of the sound ﬁeld of a moving

sound source. For simplicity, we assume a monopole

source. However, the presented approach also allows

for the treatment of arbitrary source types. The

derivation below follows [8, 9].

The time-domain free-ﬁeld Green’s function of a

stationary sound source at position xs, i.e. its

spatio-temporal impulse response, is denoted by

g(x−xs, t). See ﬁgure 1 for a sketch of the

coordinate system. The time-domain Green’s

function of a moving sound source is then

gx−xs(˜

t(x, t)), t −˜

t(x, t), whereby ˜

t(x, t) de-

notes the time instant when the impulse was emit-

ted. Confer to ﬁgure 2. gx−xs(˜

t(x, t)), t −˜

t(x, t)

is referred to as retarded Green’s function [8]. ˜

t(x, t)

is dependent on the location of the receiver xand

the time tthat the receiver experiences.

Assume a monochromatic harmonic source oscillat-

ing at angular frequency ωs. Its source function s0(˜

t)

reads in complex notation

s0(˜

t) = a0·ejωs˜

t.(1)

In order to yield the wave ﬁeld produced by a mov-

ing source with spatio-temporal impulse response

gx−xs(˜

t(x, t)), t −˜

t(x, t)driven by the signal

s0(˜

t), we model s0(˜

t) as a dense sequence of weighted

Dirac pulses. Each Dirac pulse of the sequence

multiplied by gx−xs(˜

t(x, t)), t −˜

t(x, t)yields the

wave ﬁeld created by the respective Dirac pulse. To

AES 125th Convention, San Francisco, CA, USA, 2008 October 2–5

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Ahrens AND Spors Supersonic sources

x

y

xs(˜

t(x, t)) xs(t)

x

|x|

|x−xs(˜

t(x, t))|

v

Fig. 2: Derivation of the Green’s function of a moving sound source.

yield the wave ﬁeld emitted due to the entire se-

quence of Dirac pulses, we integrate over ˜

tas

s(x, t) =

∞

Z

−∞

s0(˜

t)·gx−xs(˜

t), t −˜

td˜

t , (2)

whereby we temporarily altered the nomenclature

for convenience (˜

t=˜

t(x, t)).

Assuming a moving monopole sound source, its

Green’s function explicitly reads

gx−xs(˜

t(x, t)), t −˜

t(x, t)=

=1

4π

δt−˜

t(x, t)−|x−xs(˜

t(x,t))|

c

|x−xs(˜

t(x, t))|.(3)

Note that

τ(x, t) = |x−xs(˜

t(x, t))|

c(4)

is referred to as retarded time [8]. It denotes the du-

ration of sound propagation from the source to the

receiver. In the remainder of this paper, M=v

cde-

notes the Mach number, with vbeing the speed of

the sound source.

For convenience, we assume the virtual source to

move uniformly along the x-axis in positive x-

direction (cf. to ﬁgure 2). As outlined in [6], ar-

bitrary trajectories can be approximated by assum-

ing a piece-wise uniform motion and an appropriate

translation and rotation of the coordinate system.

At time t= 0 the source is located at position xs(0).

For this particular source trajectory, the integral in

equation (2) can be solved via the substitution

u=˜

t(x, t) + τ(x, t) (5)

and the exploitation of the sifting property of the

delta function [10]. It turns out that the integral has

diﬀerent solutions for M < 1, M= 1, and M > 1.

In the following sections, we present solutions to the

integral in (2) for subsonic (M < 1) as well as super-

sonic (M > 1) sound sources and brieﬂy comment

on the case of sources moving at the speed of sound

(M= 1).

2.1.Sound sources moving at subsonic speeds

For M < 1, the integral boundaries in (2) can be

kept and the solution, i.e. the sound ﬁeld sM<1(x, t)

of a source moving at a speed v < c reads then

sM<1(x, t) = 1

4π·s0(˜

t(x, t))

Ψ(x, t),(6)

whereby

˜

t(x, t) = t−MΦ(x, t) + Ψ(x, t)

c(1 −M2),

Ψ(x, t) = pΦ2(x, t) + y2(1 −M2),

Φ(x, t) = x−vt −xs(0) .

AES 125th Convention, San Francisco, CA, USA, 2008 October 2–5

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Ahrens AND Spors Supersonic sources

−2 0 2

−1

0

1

2

3

4

5

−1

−0.5

0

0.5

1

x→[m]

y→[m]

(a) ℜ{s(x, t0)}

−2 0 2

−1

0

1

2

3

4

5

−1

−0.5

0

0.5

1

x→[m]

y→[m]

(b) ℜ{pWFS(x, t0)}. The loudspeaker array indicated by

the dotted line is situated symmetrically around the y-

axis at y0= 1 m and its overall length is 8 m. The

loudspeakers are positioned at intervals of ∆x= 0.1 m.

Tapering is applied.

Fig. 3: Simulated wave ﬁelds of a source oscillating monochromatically at fs= 500 Hz and moving along

the x-axis in positive x-direction at v= 120 m

s. Due to the employment of the complex notation for time

domain signals (see equation (1)), only the real part ℜ{·} of the considered wave ﬁeld is depicted. The wave

ﬁelds have been scaled to have comparable levels. The values of the sound pressure are clipped as indicated

by the colorbars.

A snapshot of the wave ﬁeld of a moving sound

source described by equation (6) is depicted in ﬁgure

3(a).

For M= 0, i.e. a static source, equation (6) reads

sM=0(x, t) = 1

4π·s0(t−τ)

|x−xs|(7)

which corresponds to the familiar expression for the

sound ﬁeld of a static harmonic monopole sound

source [6].

2.2.Sound sources moving at supersonic speeds

For sound sources moving at supersonic speeds, the

integral in (2) has to be split into a sum of two in-

tegrals after the substitution (5) reading

sM>1(x, t) =

∞

Z

u1

(·)du +

∞

Z

u2

(·)du, (8)

whereby

u1,2=1

v±(xs(0) −x) + ypM2−1.

(·) denotes the argument of the integral in (2).

The solution yields the wave ﬁeld sM>1(x, t) of a

monopole sound source moving at a supersonic speed

vreading

sM>1(x, t) =

=

s1(x, t) + s2(x, t) for Φ(x, t)2+y2(1 −M2)

≥0

and xs(0) + vt ≥x

0 elsewhere ,

(9)

with

s1,2(x, t) = 1

4π

s0(˜

t1,2(x, t))

Ψ(x, t),

˜

t1,2(x, t) = t−MΦ(x, t)±Ψ(x, t)

c(1 −M2),

The most prominent property of the wave ﬁeld of a

supersonic source is the formation of the so-called

Mach cone, a conical sound pressure front following

the moving source. See ﬁgure 4(a). Note that the

AES 125th Convention, San Francisco, CA, USA, 2008 October 2–5

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Ahrens AND Spors Supersonic sources

−4 −2 0 2

−3

−2

−1

0

1

2

3

−1

−0.5

0

0.5

1

x→[m]

y→[m]

(a) Wave ﬁeld sM>1(x, t) of a supersonic source.

−4 −2 0 2

−3

−2

−1

0

1

2

3

−1

−0.5

0

0.5

1

x→[m]

y→[m]

(b) Backward travelling component s1(x, t).

−4 −2 0 2

−3

−2

−1

0

1

2

3

−1

−0.5

0

0.5

1

x→[m]

y→[m]

(c) Forward travelling component s2(x, t).

Fig. 4: Wave ﬁeld of a source traveling at 600 m

/s

(M≈1.7).

Mach cone is a direct consequence of causality.

For the receiver this has two implications: (1)

He/She does not receive any sound wave before the

arrival of the Mach cone, (2) after the arrival of the

Mach cone the receiver is exposed to a superposition

of the wave ﬁeld which the source radiates into back-

ward direction s1(x, t) and the wave ﬁeld s2(x, t)

which the source had radiated into forward direc-

tion before the arrival of the Mach cone. s1(x, t)

carries a frequency shifted version of the emitted sig-

nal propagating in opposite direction to the source

motion (ﬁgure 4(b)), s2(x, t) carries a time-reversed

version of the emitted signal following the source

(ﬁgure 4(c)). The latter is generally also shifted in

frequency.

2.3.Sound sources moving at the speed of sound

The integral in (2) can also be solved for M= 1. In

that case, the lower integral boundary is ﬁnite, the

upper boundary is inﬁnite. The result then resem-

bles the circumstances for M > 1, i.e the receiver is

not exposed to the source’s wave ﬁeld at all times.

It is rather such that the source moves at the lead-

ing edge of the sound waves it emits. The wave ﬁeld

can not surpass the source. The leading edge of the

wave ﬁeld is termed sound barrier.

Unlike for M > 1, the resulting wave ﬁeld is not

composed of two diﬀerent components. It contains

only one single component carrying the frequency

shifted input signal.

Informal listening suggests that it can not be as-

sumed that the human ear is aware of the details of

the properties of the wave ﬁeld of a transonic source

(a source moving exactly at the speed of sound).

We therefore do not present an explicit treatment

here. For convenience, we propose to assume that

the wave ﬁeld of a transonic source is perceptually

indistinguishable from the wave ﬁeld s1(x, t) of a

source moving at a speed slightly faster than the

speed of sound c.

3.WAVE FIELD SYNTHESIS

In this section, we demonstrate how a moving vir-

tual sound source can be reproduced using the ﬁnd-

ings derived in section 2. Exemplarily, we use wave-

ﬁeld synthesis (WFS) employing a linear array of

secondary sources (loudspeakers).

The theoretical basis of WFS employing linear sec-

ondary source arrays is given by the two-dimensional

Rayleigh I integral [11, 12]. It states that a linear

AES 125th Convention, San Francisco, CA, USA, 2008 October 2–5

Page 5 of 11

Ahrens AND Spors Supersonic sources

distribution of monopole line sources is capable of

reproducing a desired wave ﬁeld (a virtual source)

in one of the half planes deﬁned by the secondary

source distribution. The wave ﬁeld in the other half

(where the virtual source is situated) is a mirrored

copy of the desired wave ﬁeld. For convenience, the

secondary source array is assumed to be parallel to

the x-axis at y=y0as depicted in ﬁgures 1 and

3(b). The listening area is chosen to be at y > y0.

The two-dimensional Rayleigh I integral determines

the sound pressure pWFS (x, t) created by such a

setup reading

pWFS(x, t) =

∞

Z

−∞ −∂

∂ns(x, t)|x=x0

| {z }

d(x0,t)

∗tg(x, t)dx0.

(10)

s(x, t) denotes the sound ﬁeld of the virtual source

and ∂

∂nthe gradient in the direction normal to the

secondary source distribution (confer also to ﬁgure

1). The asterisk ∗tdenotes convolution with respect

to time.

The driving function d(x0, t) for a loudspeaker at po-

sition x0is thus yielded by evaluating the gradient

of the desired virtual sound ﬁeld in direction normal

to the loudspeaker distribution at the position of the

respective loudspeaker.

Due to the fact that the physical requirements can

not be perfectly fulﬁlled in practical implementa-

tions, the virtual source’s wave ﬁeld is not perfectly

reproduced in the receiver’s half-space. Equation

(10) requires an inﬁnitely long continuous distri-

bution of secondary sources, practical implementa-

tions can only employ a ﬁnite number of discrete

loudspeakers. The array has thus a ﬁnite length.

Furthermore, equation (10) requires secondary line

sources which are positioned perpendicular to the

receiver plane [12]. Practical implementations typ-

ically employ loudspeakers with closed cabinets as

secondary sources. These are more accurately de-

scribed by point sources rather than line sources.

This fact is known as secondary source mismatch

and has to be compensated for as

dcorr(x, t) = f(t)∗td(x, t).(11)

f(t) is a ﬁlter with frequency response F(ω) =

2√2πjkdref , the asterisk ∗tdenotes convolution with

respect to time, and dref denotes the reference dis-

tance from the secondary source array, to which the

amplitude of the reproduced wave ﬁeld is referenced.

See [12] for a thorough treatment of the properties

of WFS.

For convenience, we do not explicitly compensate for

the secondary source mismatch in the analytical ex-

pressions for the driving functions. However, in the

simulations this compensation is performed.

3.1.Driving function for subsonic sources

For a virtual harmonic monopole sound source of

angular frequency ωsmoving uniformly along the x-

axis as described in section 2, the driving function

d(x, t) derived from (6) and (10) reads [6]

dsub(x, t) = y(1 −M2)

Ψ(x, t)1

Ψ(x, t)+jωs

c(1 −M2)×

×s(x, t).(12)

Note that dsub(x, t) in equation (12) implicitly in-

cludes static virtual sources.

The wave ﬁeld reproduced by a linear WFS array

driven by equation (12) is depicted in ﬁgure 3(b).

The overall length of the loudspeaker array is 8 m.

The virtual source moves at a speed v= 120 m

salong

the x-axis in positive x-direction (M≈1

3).

3.2.Driving function for supersonic sources

The driving function for supersonic sources derived

from (9) and (10) reads

dsup(x, t) = d1(x, t) + d2(x, t) =

=y(1 −M2)

Ψ(x, t)1

Ψ(x, t)+jωs

c(1 −M2)×

×s1(x, t) +

+y(1 −M2)

Ψ(x, t)1

Ψ(x, t)−jωs

c(1 −M2)×

×s2(x, t).(13)

3.3.Driving function for transonic sources

As outlined in section 2.3, we propose to reproduce

s1(x, t) of a virtual source moving slightly faster

than the speed of sound in order to approximate a

transonic source. The appropriate driving function

is then d1(x, t).

4.RESULTS

In this section, we present a number of simulations

in order to analyze the properties of the proposed

AES 125th Convention, San Francisco, CA, USA, 2008 October 2–5

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Ahrens AND Spors Supersonic sources

0 0.5 1 1.5 2

350

400

450

500

550

600

650 [dB]

−80

−70

−60

−50

−40

−30

−20

t→[s]

f→[Hz]

(a) Real source. The emitted frequency is

500 Hz.

0 0.5 1 1.5 2

350

400

450

500

550

600

650 [dB]

−80

−70

−60

−50

−40

−30

−20

t→[s]

f→[Hz]

(b) Truncation artifacts. The length of the

array is 40 m. The emitted frequency is

500 Hz.

0 0.5 1 1.5 2

3000

3500

4000

4500

5000 [dB]

−80

−70

−60

−50

−40

−30

−20

t→[s]

f→[Hz]

(c) Spatial aliasing. The desired signal is

the S-shaped one in the middle. The emit-

ted frequency is 4000 Hz.

Fig. 5: Spectrograms illustrating artifacts apparent

in the reproduced wave ﬁeld of a subsonic source

(v= 40 m

/s). The virtual source passes the receiver

at t≈1 s.

approach with focus on the case of M > 1. The case

of M < 1 is thoroughly treated in [6].

We assume a linear array of secondary monopole

sources. The secondary sources are placed at an in-

terval of ∆x= 0.1 m throughout the simulations.

The loudspeaker array is situated parallel to the x-

axis and symmetrically around the y-axis at y0= 1

m. Its overall length is 14 m except where stated

explicitly.

As inherent to WFS, the reproduced wave ﬁeld only

approximates the desired one for y > y0. Due to the

fact that we assume secondary monopole sources,

the reproduced wave ﬁeld on the other side of the

loudspeaker array (where y < y0) is a mirrored ver-

sion.

4.1.Artifacts apparent in the reproduced wave

ﬁeld

As outlined in [6], the reproduced wave ﬁeld suf-

fers from two major artifacts: (1) echo-like artifacts

due to spatial truncation of the secondary source ar-

ray, and (2) spatial aliasing when the frequency con-

tent of the reproduced wave ﬁeld is above the spatial

aliasing frequency. Figure 5 shows spectrograms of

the reproduced wave ﬁeld observed at xR= [0 −4]T.

The loudspeaker array similar to the one used in the

simulations in ﬁgure 3, i.e. the loudspeaker array is

situated symmetrically around the y-axis at y0= 1

m and its overall length is 8 m. The loudspeakers

are positioned at intervals of ∆x= 0.1 m

In ﬁgure 5(b) a pre- and a post-echo additional to the

desired signal are apparent. The shorter the array

the closer in time to the desired signal the echoes oc-

cur. These truncation artifacts can be signiﬁcantly

reduced by the application of tapering (i.e. an at-

tenuation of the secondary sources towards the very

ends of the array) [11, 6].

Figure 5(c) depicts the spectrogram of a virtual

source reproduced above the spatial aliasing fre-

quency. For the given array with a loudspeaker spac-

ing of ∆ = 0.1 m the spatial aliasing frequency is

approximately 1700 Hz [13].

Finally, another artifact resulting from spatial trun-

cation of the secondary source distribution is an in-

correct amplitude envelope of the receiver signal.

This circumstance can be observed when comparing

e.g. ﬁgures 5(a) and 5(b). At the very ends of the

depicted time window, the receiver signal due to the

real source is signiﬁcantly higher in amplitude than

AES 125th Convention, San Francisco, CA, USA, 2008 October 2–5

Page 7 of 11

Ahrens AND Spors Supersonic sources

−4 −2 0

−1

0

1

2

3

4

5

−1

−0.5

0

0.5

1

x→[m]

y→[m]

(a) ℜ{s(x, t0)}

−4 −2 0

−1

0

1

2

3

4

5

−1

−0.5

0

0.5

1

x→[m]

y→[m]

(b) ℜ{pWFS(x, t0)}. No limitation of the temporal band-

width. Strong aliasing artifacts are apparent (see text).

−4 −2 0

−1

0

1

2

3

4

5

−1

−0.5

0

0.5

1

x→[m]

y→[m]

(c) ℜ{pWFS(x, t0)},fmax = 3000 Hz.

−4 −2 0

−1

0

1

2

3

4

5

−1

−0.5

0

0.5

1

x→[m]

y→[m]

(d) ℜ{pWFS(x, t0)},fmax = 2000 Hz.

Fig. 6: Simulated wave ﬁelds of a source oscillating monochromatically at fs= 500 Hz and moving along

the x-axis in positive x-direction at v= 600 m

/s(M≈1.7). Due to the employment of the complex notation

for time domain signals (see equation (1)), only the real part ℜ{·} of the considered wave ﬁeld is depicted.

The wave ﬁelds have been scaled to have comparable levels. The values of the sound pressure are clipped as

indicated by the colorbars. The loudspeaker array in ﬁgures 6(b)-6(d) is indicated by the dotted line. It is

situated symmetrically around the y-axis at y0= 1 m and its overall length is 14 m. The loudspeakers are

positioned at intervals of ∆x= 0.1 m.

the receiver signal due to the virtual source. In the

center of the plot, i.e. when the source is behind the

secondary sources from the receivers point of view,

the amplitude due to the virtual source is similar to

that due to the real source.

4.2.Direct application of the driving function for

M > 1

Figure 6(b) shows a simulation of a WFS system

reproducing the wave ﬁeld depicted in ﬁgure 6(a).

The virtual source moves at v= 600 m

/s, i.e. M≈

1.7. Due to the omnidirectionality of the secondary

sources, the reproduced wave ﬁeld in ﬁgure 6(b) is

AES 125th Convention, San Francisco, CA, USA, 2008 October 2–5

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Ahrens AND Spors Supersonic sources

3456789

x 10−3

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 104

t→[s]

f1,2(x1, t)→[Hz]

f1

f2

(a) f1,2(x1, t). Negative frequencies indicate time

reversal of the input signal.

3456789

x 10−3

0

5

10

15

20

t→[s]

1

Ψ(x1,t)

(b) 1

Ψ(x1,t).

Fig. 7: Details of the wave ﬁeld of a source of v=

600 m

/s(M≈1.7) oscillating at fs= 500 Hz ob-

served at x1= [1 1]T. The Mach cone arrives at

t≈4·10−3s.

symmetric with respect to the secondary source con-

tour. Note that strong artifacts are apparent. It can

be shown that these artifacts occur due to temporal

as well as spatial aliasing.

This can by veriﬁed by analyzing the instantaneous

frequencies f1(t) and f2(t) of the reproduced wave

ﬁeld components s1(x, t) and s2(x, t). Confer to ﬁg-

ure 7(a). It can be seen that f1(t) and f2(t) are

inﬁnite at the singularity of the Mach cone, i.e. at

the moment of the arrival of the Mach cone. After

the arrival they decrease quickly to moderate values.

The former means that f1(t) and f2(t) will exceed

any limit imposed on a reproduction system due to

discrete treatment of time and discretization of the

secondary source distribution.

4.3.Modiﬁed driving function

In order to prevent temporal aliasing in digital sys-

tems due to discretization of the time, it is desirable

to limit the bandwidth of the temporal spectrum of

the driving function. Typical bandwidths in digital

systems are 22050 Hz for systems using a tempo-

ral sampling frequency of 44100 Hz and 24000 Hz

for systems using a temporal sampling frequency of

48000 Hz.

In order to prevent respectively reduce spatial alias-

ing of the WFS system under consideration, it is

desirable to further limit the bandwidth of the tem-

poral spectrum of the driving function to values in

the order of the spatial aliasing frequency which is

typically a few thousand Hertz. Recall that the crit-

ical frequency above which spatial aliasing occurs in

the given secondary source array is approximately

1700 Hz (confer to section 4.1).

A simple means to limit the bandwidth is to sim-

ply fade-in the driving signal from a moment on

when its temporal frequency has dropped below a

given threshold. This strategy also avoids the cir-

cumstance that the amplitude of the driving sig-

nal is inﬁnite at the moment of arrival of the Mach

cone. Real-world implementations of WFS systems

can not reproduce arbitrarily high amplitudes.

Confer to ﬁgure 7(b). It depicts the factor Ψ(x, t)−1

which determines the amplitude of the wave ﬁeld

around the Mach cone.

The simulations in ﬁgures 6(c) and 6(d) show the

reproduced wave ﬁeld when the driving function is

faded-in after the instantaneous frequency of the

driving function has dropped below 3000 Hz (ﬁgure

6(c)) respectively 2000 Hz (ﬁgure 6(d)). The alias-

ing artifacts are signiﬁcantly reduced.

Note that the shorter the fade-in of the driving func-

tion is the better the impulsive property of the Mach

cone is preserved. However, shorter fade-ins result

in stronger spatial aliasing since they impose more

high frequency content onto a signal.

AES 125th Convention, San Francisco, CA, USA, 2008 October 2–5

Page 9 of 11

Ahrens AND Spors Supersonic sources

Finally, it has to be considered that spatial aliasing

is not necessarily audible under all circumstances.

5.PERCEPTUAL ASPECTS

Informal listening suggests that the human audi-

tory system is not aware of all the properties of the

wave ﬁeld of supersonic sources. Especially the fact

that the wave ﬁeld contains a component carrying

a time-reversed version of the source’s input signal

is confusing. Depending on the speciﬁc situation, it

might be preferable to exclusively reproduce s1(x, t),

i.e. the component of the wave ﬁeld carrying the non-

reversed input signal.

Furthermore, only the localization when exposed to

s1(x, t) is plausible since s1(x, t) assures localization

of the source in its appropriate location (however

with some bias due to the retarded time τ). Expo-

sure of the receiver to s2(x, t) suggests localization

of the source in the direction where it “comes from”.

This also seems unnatural. Finally, the exposure of

the receiver to a superposition of s1(x, t) and s2(x, t)

suggests the localization of two individual sources.

6.CONCLUSIONS

An approach to the reproduction of the wave ﬁeld

of virtual sound sources moving at supersonic speeds

was presented. The approach constitutes an exten-

sion to a treatment of the reproduction of the wave

ﬁeld of virtual sound sources moving at subsonic

speeds previously published by the authors. It was

shown that the reproduced wave ﬁeld suﬀers from

spatial aliasing artifacts due to the fact that the in-

stantaneous frequency of the virtual sound ﬁeld is

inﬁnite at the moment of arrival of the Mach cone.

As workaround, it was proposed to fade-in the driv-

ing signal for a given secondary source right after

the instantaneous frequency of the driving signal has

dropped below a desired threshold. A short fade-in

preserves the impulsive quality of the Mach cone.

In order to optimize the reproduction of the sound

ﬁeld of supersonic virtual sources, it is necessary to

perform preceptive experiments investigating which

properties of the virtual wave ﬁeld have to be repro-

duced in order to evoke a plausible perception both

in terms of frequency content and localization.

ACKNOWLEDGEMENTS

We thank Holger Waubke of Austrian Academy of

Sciences for providing us with the notes of his lecture

on theoretical acoustics [9].

7.REFERENCES

[1] R. Rabenstein and S. Spors. Multichannel

sound ﬁeld reproduction. In Benesty, J.,

Sondhi, M., Huang, Y, (Eds.), Springer Hand-

book on Speech Processing and Speech Com-

munication, Springer Verlag, 2007.

[2] C. Doppler. ¨

Uber das farbige Licht der Dop-

pelsterne und einiger anderer Gestirne des

Himmels. In Abhandlungen der k¨oniglichen

b¨ohmischen Gesellschaft der Wissenschaften, 2,

pp. 465–482, 1842.

[3] A. Franck, A. Gr¨afe, T. Korn, and M. Strauß.

Reproduction of moving virtual sound sources

by wave ﬁeld synthesis: An analysis of artifacts.

32nd Int. Conference of the AES, Hillerød,

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[4] H. Strauss. Simulation instation¨arer

Schallfelder f¨ur virtuelle auditive Umge-

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Verlag, D¨usseldorf, 2000.

[5] Y. Iwaya and Y. Suzuki. Rendering moving

sound with the doppler eﬀect in sound space.

Applied Acoustics, Technical note, 68:916–922,

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[6] J. Ahrens and S. Spors. Reproduction of mov-

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to the doppler eﬀect. In 124th Convention of

the AES, Amsterdam, The Netherlands, May

17–20 2008.

[7] N. Epain and E. Friot. Indoor sonic boom repro-

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[8] J.D. Jackson. Classical Electrodynamics. Wiley,

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[9] H. Waubke. Aufgabenstellung zur Seminarar-

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[11] E.W. Start. Direct sound enhancement by wave

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Ahrens AND Spors Supersonic sources

[12] S. Spors, R. Rabenstein, and J. Ahrens. The

theory of wave ﬁeld synthesis revisited. In

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