PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

A field-matching method for sound field synthesis for large scale
sound reinforcement systems
Elena SHABALINA1
1d&b audiotechnik GmbH, Germany

ABSTRACT
There have been ongoing attempts to adapt existing Sound Field Synthesis methods for large scale soundreinforcement systems. In comparison to smaller indoor setups large scale systems have significant distances
between secondary sources and often irregular loudspeaker layouts due to rigging conditions. Theserestrictions make the use of WFS-related methods unfeasible. In addition, prominent andfrequency-dependent directivity patterns of concerts loudspeakers obstruct the use of usual level and delay
algorithm as it assumes omnidirectional secondary sources.
The paper proposes an alternative method based on a Field Matching technique that bypasses theabove-mentioned restrictions. Despite the typically low aliasing frequency of a concert sound system layout
the Field Matching approach was shown to provide an accurate localisation and a stable virtual source acrossthe audience area including front seats. The paper presents a theoretical background and a Matlab simulation
along with listening tests results. Limitation and challenges of the method for live sound applications arediscussed.
Keywords: Sound reinforcement, field-matching, sound field synthesis.

1. INTRODUCTION
There are several popular methods to reproduce a sound field of a combination of virtual sources through anarray of loudspeakers: WaveFieldSynthesis (1,2), Higher Order Ambisonics (HOA) (3), Vector Based
Amplitude Panning (3) to name just a few. While being well established, each of them poses restrictions onthe array geometry, loudspeakers characteristics or the audience area. Loudspeakers are often assumed to bemonopoles or dipoles and have to be located right next to each other. This works well for small rooms but not
so well for larger areas. Systems like HOA only work for a “sweet spot” and not for an audience area.
Modifications of WFS have been developed to account for loudspeaker directivity, however, loudspeakers
are required to be identical and located on a straight line or in a circle (4,5).
The main objective of this work is to find a method that doesn’t have any basic limitations on theloudspeakers layout and directivity.
The goal is to formulate a matrix equation for the driving functions and check if the driving functions provide
the desired result. The practicalities of numerically solving the equation are left for the future work.
The approach described below presents a version of the field matching method described (6).

2. PROBLEM LAYOUT
Ideally, a concert or theatre sound system designed for spatial sound reproduction should be able toprovide stable virtual source localization in the entire audience area. The loudspeakers or loudspeakerarrays are mainly located above the stage, along the sides and sometimes at the back. There can be a
separate row of smaller loudspeakers to cover the front row (so called “front fills”) and additional
loudspeakers to cover the back of the audience (so called “delays”). Due to rigging restrictions, costs
and logistics, especially in touring, the amount of loudspeakers in a setup is limited, and therefore thedistances between single loudspeakers can be up to several metres. The aliasing frequency for such a
setup is typically very low.
Concert loudspeakers have carefully designed directivity that tends towards cardioid at low
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Elena.Shabalina@dbaudio.com

frequency and becomes narrower at higher frequencies. The directivity patterns don’t introduce a large
error to a synthesized sound field if the loudspeakers are located close to each other and the audiencearea is small, so assuming monopoles or dipoles works fine for these applications. However, at large
distances to the listener and between the speakers the directivities don’t overlap enough and create
significant error.
The problem layout is shown in the fig. 1. For simplicity only one virtual source is shown, but their
number is not limited.

Figure 1 – Problem layout

3. ANALYTICAL SOLUTION
Our goal is to reconstruct the desired sound field at all the receivers. The desired field is the one thatthe virtual source/sources would create in the absence of loudspeakers. Knowing the positions ofvirtual sources and the receivers, the desired field can be calculated using eq. (1):ௌ‫݌‬ௗ ሺ‫ݎ‬௥ ǡ ݂ሻ ൌ ෍௦݁ ି௜௞௥ೝೞ‫ݎ‬௥௦(1)where ܵ is the number of virtual sources.
If needed, directivity and frequency response of the virtual source can be included:ௌ‫݌‬ௗ ሺ‫ݎ‬௥ ǡ ݂ሻ ൌ ෍ ‫ܣ‬௦ ሺ݂ሻ‫ܦ‬௦ ሺ݂ሻ௦݁ ି௜௞௥ೝೞ‫ݎ‬௥௦

(1.1)

For simplicity, we assume an omnidirectional virtual source with flat frequency response.
The reconstructed field is a combination of the fields of all the loudspeakers with their respective
unknown driving functions:௅‫݌‬௥௘௖ ሺ‫ݎ‬௥ ǡ ݂ሻ ൌ ෍ ‫ݍ‬௟ ሺ݂ሻ‫ܦ‬௟ ሺ݂ሻ௟݁ ି௜௞௥ೝ೗‫ݎ‬௥௟(2)where ‫ݍ‬௟ ሺ݂ሻ is the unknown frequency dependent driving function for each loudspeaker, ‫ܦ‬௟ ሺ݂ሻ is its
directivity and ‫ ܮ‬is the number of loudspeakers.
Ideally, reconstructed field should be made equal to the desired field by finding appropriate driving
functions:(3)‫݌‬ௗ ሺ‫ݎ‬௥ ǡ ݂ሻ ൌ ‫݌‬௥௘௖ ሺ‫ݎ‬௥ ǡ ݂ሻௌ௅݁ ି௜௞௥ೝೞ݁ ି௜௞௥ೝ೗෍ൌ ෍ ‫ݍ‬௟ ሺ݂ሻ‫ܦ‬௟ ሺ݂ሻ‫ݎ‬௥௦‫ݎ‬௥௟
௦

(4)

௟

In order to solve Eq.4 for ‫ݍ‬௟ ሺ݂ሻ, we can write it in a matrix form for each frequency:

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෡ ൈ ‫ݍ‬Ԧ ൌ ܸǡሬሬሬԦ‫ܪ‬෡ ൌ ‫ܪ‬௥௟ ൌ ‫ܦ‬௟ ሺ݂ሻwhere ‫ܪ‬௘ ష೔ೖೝೝ೗௥ೝ

(5)is an ܴ ൈ ‫ ܮ‬matrix,(6)‫ݍ‬Ԧ is a vector of driving function values andௌሬԦ ൌ ܸ௥ ൌ ෍ܸ௦݁ ି௜௞௥ೝೞ‫ݎ‬௥௦The unknown vector of driving functions can be found by the matrix inversion:෡ ିଵ ൈ ܸǡሬሬሬԦ‫ݍ‬Ԧ ൌ ‫ܪ‬

(7)

(8)

Eq. 8 is a typical inverse problem.
Depending on the number of loudspeakers L and receiver points R, the system of equation can beunder defined, defined or over defined. [kim choi] Since receiver points are just virtual microphones
and their number can be easily adjusted, we don’t need to consider the first case. In case of R=L, matrix
inversion is straight forward. In case of R>L, the Eq. (8) is solved using a built in Least Mean Squares
method in Matlab.

4. SIMULATIONS
Eq. (8) was solved using Matlab. The setup consisted of 16 identical loudspeakers at the distance of
0.53 m between their centers. The simulation used measured loudspeaker’s directivity with 5°
precision and interpolation in between. The setup is too small for a live sound system, but isrepresentative for the first tests. The same setup was later used for an indoor listening test.
Figure 2 shows the normalized sound pressure level distribution of the desired and reproducted
fields. At low frequencies the reconstrusted field looks very similar to the desired field. However, at
higher frequencies interference patterns are clearly visible.

5. LISTENING TESTS
5.1 Setup
Listening tests were conducted to evaluate the performance of the method. The setup shown in
Figure 3 consisted of 16 loudspeakers located with spacing of 53cm, same as the simulation in 4. The
loudspeakers were hidden behind an acoustically transparent curtain to exclude visual cues. Fourlisteners positions were defined to represent the entire listening area. Virtual sources are located in the“stage” area behind the loudspeakers; one virtual source is placed far off the stage on the side. The test
signals were three pulses of broadband pink noise. Each listener position was provided with a disk
marked in degrees on the floor. Participants were asked to write down the perceived direction of thesource which was later compared with the geometrical direction.
On day one, 32 respondents took part in the test. Every participant was tested at two listeners
positions, so that every position has 16 independent results. On the first day only locations of virtual
sources were tested (Table 1).
On day two, 14 respondents were available and provided 7 independent results for each listener
position. Three more virtual sources were added as well as two real sources (loudspeakers) that werelocatedbehindan
acousticallytransparent
curtain(Table2).

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Figure 2 – Simulated normalized sound pressure level distribution of desired and reconstructed fields1159

Figure 3 – Listening test setup
5.2

Results
Table 1 – Listening test results, day1
Listener positions

Virtual sources1

coordinates

2

3

4average(1, 1)

1.67

1.6

5.40

4.00

3.17

(5, 8)

2.33

2.20

4.67

4.00

3.30

(12, 3)
34.33
13.33
9.47
4.00
15.28
Table 1 presents the average deviation of the perceived direction of the virtual source from thegeometrical direction in degrees.
Table 2 – Listening test results, day2
Listener positions
Virtual sources1

coordinates

2

3

4average(1, 1)

2.39

4.34

10.73

3.57

5.26

(5, 8)

2.39

1.71

4.34

1.09

2.38

(12, 3)

34.24

16.17

13.34

14.38

19.53

(3, 4)

2.41

3.71

7.75

9.73

5.90

(7, 6)

3.76

3.22

6.92

0.26

3.54

(8,10)

2.03

0.87

3.67

6.26

3.21

7.87

5.00

7.79

5.88

6.14

Real sources
coordinates(1, 1)(3, 3.3)
1.91
3.52
6.67
6.58
6.25
Table 2 presents the average deviation of the perceived direction of the virtual source from the1160

geometrical direction in degrees.
5.3 Discussion
The virtual source with coordinates (12, 3), located far outside the “stage” area was localized by far
the worst, with average deviation of 17.4°. Virtual sources within the stage were localized with theaverage deviation of 3.8°. The overall average deviation is 6.8°.
For real sources, the average was about 6.2° in average for both sources and all listener positions.
With all virtual source positions taken into account, localization of real sources was slightly better
than virtual sources. However, if the virtual source outside the stage area is discarded as an unrealistic
scenario, the localization of remaining virtual sources was (surprisingly) better than the localization of
the real sources.
All in all, listening tests show that the field-matching method provides sufficiently accurate
localization despite aliasing.

6. CONCLUSIONSThe paper shows how to derive an explicit expression for driving functions for a sound field
synthesis setup without posing initial restrictions on loudspeakers’ layout or directivity. Preliminary
tests show that the method provides reliable localization with precision of about 7°, despite visible
aliasing effects at higher frequencies.
Future work may include analyzing numerical methods to solve Eq. (8), possibly introducing
regularization to obtain smooth filters as driving functions.

ACKNOWLEDGEMENTS
The author would like to thank the team of d&b audiotechnik for participating the listening tests,
and specially Julian Ebert for his help with the setup.

REFERENCES
1. Berkhout A.J., de Vries D., Vogel P., Acoustic Control by WaveFieldSynthesis, The Journal of the
Acoustical Society of America 93, 2764 (1993)
2. Ahrens J., Analytical Methods of Sound Field Synthesis, Springer, 2012
3. Roginska A., Geluso P., Immersive Sound, Routledge, 2018
4. de Vries D., Sound Reinforcement by WFS: Adaptation of the Synthesis Operator to Loudspeaker
Directivity Characteristic, J. Audio Eng. Soc., Vol.44, No. 12, 1996.
5. Ahrens J. and Spors S., Sound field reproduction employing non-omnidirectional loudspeakers, 126th
Conv. Audio Eng. Soc. (2009), preprint 7741.
6. Kim Y.-H., Choi J.-W., Sound Visualization and Manipulation, Wiley, 20131161