PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Reducing Number of Transfer Function Measurement in Local
Sound Field Reproduction using Acoustic Modeling
Qiaoxi ZHU1; Xiaojun QIU1; Philip COLEMAN2; Ian BURNETT1
1

Centre for Audio, Acoustics and Vibration, Faculty of Engineering and IT, University of Technology Sydney,Sydney, Australia
2Institute of Sound and Recording, University of Surrey, Guildford, UKABSTRACT
Broadband local sound field reproduction over an extended spatial region is a challenging problem when
only limited transfer function measurements are available. An approach based on acoustics modeling isproposed in this paper to reduce the required number of transfer function measurements in local sound fieldreproduction. The proposed method only requires measuring the transfer functions from each source to a few
samples over the boundary of the controlled region, and the transfer functions to the samples inside thecontrolled region are then estimated through efficient acoustic modelling. Simulations demonstrate that theproposed method requires fewer transfer function measurements than existing methods such as the leastsquares and the spatial harmonic decomposition methods.
Keywords: Sound field control, Spatial audio, Acoustic modelling

1. INTRODUCTION
Local sound field reproduction delivers a sound field within a given region of space (1-3). A simple
approach uses multiple microphones as matching points and derives the loudspeaker weights basedon the least square (LS) match (4, 5). It allows arbitrary loudspeaker placement but requires theknowledge of the transfer functions between the loudspeakers and the matching points. One remaining
challenge is to design effective systems with a practical number of microphones in the transferfunction acquisition.
According to the Kirchhoff-Helmholtz integral equation, an interior sound field can be expressedby the integral of the sound pressure and its gradient over the boundary of the region (6). Therefore,
the transfer functions from each source to the region samples can be estimated from those to thesamples over the boundary, as long as there are sufficient measurements on the boundary.
Unfortunately, limited measurements with a small number of microphones are often used in practicedue to the cost and efficiency.
The spatial harmonic decomposition (SHD) method (7, 8) provides a practical way to determine
the transfer function between each loudspeaker and every point in the reproduction region. In thismethod, the transfer functions over a region are effectively parameterized by the superposition ofspatial harmonics. For two dimensional sound field reproduction, the spatial harmonics are thecylindrical wave functions, and the required number of boundary samples is proportional to the squareof the product of the frequency and the size of the region. However, this method becomes impracticalwhen a large number of measurements are required to reproduce a sound field over a broad region ora broad frequency band.
In this paper, an approach based on acoustic modeling (AM) is proposed to facilitate broadband
local sound field reproduction with fewer transfer function measurements. In the proposed method,an acoustic model is constructed based on the prior information of the loudspeaker location. Then, the122684

Qiaoxi.Zhu@uts.edu.au; Xiaojun.Qiu@uts.edu.au; Ian.Burnett@uts.edu.au
p.d.coleman@surrey.ac.uk

acoustic model is used to estimate the transfer functions over the reproduction region frommeasurement at a few samples over the boundary. Compared to the spatial harmonic method, theacoustic model can better match the actual transfer functions by using information on the control
region and the loudspeaker location.

2. LOCAL SOUND FIELD REPRODUCTION
2.1 Least Squares Method
The least squares method minimizes the reproduction error over the desired region at eachfrequency,
2min Hw  p des 2 ,w

(1)

where pdes represents the desired sound pressure values at V sound field samples produced by a virtualsource, and H represents the transfer function matrix between L loudspeakers and the sound fieldsamples. The loudspeaker weights are derived asw  H  p des ,(2)

where the upper script “+” denotes the Moore–Penrose inverse, that H+=(HHH+δI)1 HH and δ > 0 is
the regularization parameter to avoid ill-conditioning. The spacing between the samples should beless than half a wavelength (λ/2), to ensure accurate reproduction over the region (4). Considering acircular region with a radius of r, the required number of samples isV 2 kr  ,2πr 22π(3)where the wavenumber k = 2π/λ. The required number V can be very large for an enlarged reproduction
region or a broad frequency band, resulting in impractical numbers of the transfer function
measurement.
2.2 Spatial Harmonic Decomposition Method
In the spatial harmonic decomposition method (7), the transfer function matrix H is estimated fromHM. HM denotes the transfer functions measured at M samples over the boundary of the reproduction
region. The parameterization of the acoustic transfer functions over the circular region is obtained
through spatial Fourier transformation fSHD over the measured samples,aSHD  fSHD  H M  .

(4)Then, the transfer function matrix H is estimated with the inverse transformation for sound fieldsamples, that1
HSHD  fSHD  aSHD  .

(5)Based on the transfer function estimation, the loudspeaker weights are derived as

+ p .
w SHD  H
SHD des(6)In this method, the required number of measured boundary samples is(7)M  2kre  1 .
Compared to (3), the spatial harmonic decomposition method requires much smaller number ofsamples when kr >> 1.

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2.3 Acoustic Modelling Based MethodIn the proposed method, an acoustic model is constructed to calculate the transfer function between
the ith loudspeaker (i=1, 2, …, L) and the reproduction region from the measured transfer functions
at M boundary samples. The acoustic model includes a set of randomly distributed monopole point
sources around the known location of the ith loudspeaker. The ‘cloud’ of point sources around theloudspeaker location is applied because the exact loudspeaker location and radiation feature are notalways available in practice. It is used to model the unknown spatial sound radiation features of theloudspeaker to the reproduction region. The relationship between the transfer functions to the M
measured boundary samples and those to the V sound field samples is1 H
Ti  h A,i  h HA-M,i h A-M,i   I  hA-M,
i,


(8)

where hA,i and hA-M,i are respectively the transfer function matrix between the acoustic model of theith loudspeaker and the sound field samples or the boundary samples, and λ>0 is the regularization
parameter to avoid ill-conditioning. The transfer functions hA,i and hA-M,i are theoretically calculatedby the acoustic model, rather than obtained from measurements. Based on this relationship, theestimation of H isHh AM,2 ... h AM, L AM  h AM,1(9)h AM,1  Th
i M,i(10)whereand hM,i is the ith column of the measured transfer function matrix HM over the boundary samples.
Then, the loudspeaker weights are derived as

+ p .
w AM  H
AM des(11)Compared to modelling each loudspeaker as a monopole point source in the authors’ previous work(9), the new model can be adaptive to the unknown loudspeaker’s radiation feature and robust to theloudspeaker’s location mismatch.

3. SIMULATIONS
3.1 Comparison among Different Methods
The proposed acoustic modelling based method is compared with the least squares method and thespatial harmonic decomposition method. As shown in Fig. 1(a), the desired sound field is produced
by a point monopole source located at (2.0, 0.0) m, and is labeled as virtual source in the figure. The
reproduction region is centered at (0.0, 0.0) m, with a radius of 0.2 m. A uniform line array of 5
loudspeakers is used to reproduce the local sound field. The loudspeakers are located at x = 0.4 m
and y = 0.4 m ~ 0.4 m. Each loudspeaker is set as a point monopole source. Since the spacing between
the loudspeakers is 0.2 m, the aliasing frequency of the array is 850 Hz.
The least squares, spatial harmonic decomposition and the proposed acoustic modelling based
methods are compared, when the transfer functions are measured at 16 boundary samples. The
reproduction error is present in Fig. 1(b), which is defined asV

 p ( f )  p ( f )MSE( f )  10 log10 v 1v

V p (f)v 12v

2.

(12)

v

where p v is the desired sound pressure and p v is the reproduced sound pressure at the vth sound fieldsample. The metric is evaluated by V = 5025 grid samples, with 0.005 m spacing over the reproduction
region.

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Reproduction Error (dB)

y (m)

(a)
(b)
Figure 1 – (a) The system geometry of the local sound field reproduction employing transfer function
measurement at 16 boundary samples. (b) The reproduction error of the least squares, spatial harmonicdecomposition and the proposed acoustic modelling based methods, employing transfer function
measurement at 16 boundary samples, and the least squares method employing full transfer function
measurement over the reproduction region as reference.
As shown in Fig. 1(b), the SHD method performs the worst and suffers the inaccuracy caused by
the zeros of the Bessel functions impacting the measurement. The proposed method employing 16samples has better accuracy than the least squares method with the same samples and achieves thesame performance with the least squares method employing 5025 samples (Full Meas.) over the region.
The ‘Full Meas.’ indicates the ‘ideal’ reproduction error achieved by the same loudspeakers withsufficient microphones, which is free of the degradation caused by reduced sound field sampling.
3.2 Reproduction Error with Reduced Number of Samples
Figure 2 presents the performance of different methods employing fewer transfer function
measurements. With decreasing number of transfer function measurements, all the three methods have
performance degradation. However, the proposed acoustic modelling based method performs better
than the least squares and spatial harmonic methods with the same transfer function measurements
employed. Furthermore, the proposed acoustic modelling based method achieves reproduction error
less than 10 dB over 100 Hz to 2500 Hz, even with 2 boundary samples. Table 1 presents theperformance averaged over 40 uniform samples between 100 Hz and 4000 Hz. It shows the proposed
method using 4 microphones achieves the same performance with the least squares method using 5025
microphones throughout the region. Besides, the proposed method using 2 microphones has only 1.4
dB broadband degradation compared to the least squares method using 5025 microphones throughout
the region.
Table 1 – The mean square error (dB) of the local sound field reproduction averaged over 100 ~ 4000 Hz,
using the LS, SHD and AM methods, employing different number of sound field samples in the transfer
function measurement, respectively.
MethodNumber of measured sound field samples
124165025

LS

1.5

2.1

7.1

11.5

14.6

SHD

6.4

4.3

3.7

6.6

AM

6.4

13.2

14.6

14.62687

Reproduction Error (dB)

y (m)

(b)

y (m)

Reproduction Error (dB)

(a)

(d)

y (m)

Reproduction Error (dB)

(c)

(e)
(f)
Figure 2 – The system geometry and the reproduction error of the least squares, spatial harmonic
decomposition and the proposed acoustic modelling based methods, employing transfer function
measurement at 4 boundary samples in (a) and (b), 2 boundary samples in (c) and (d), 1 boundary sample in(e) and (f), and the least squares method employing full transfer function measurement over thereproduction region as reference.

4. CONCLUSION
An approach based on acoustic modeling is proposed to reduce the number of the transfer function
measurement for implementing local sound field reproduction. In the proposed method, the acoustic
model is constructed for each loudspeaker by the information of its location. The acoustic model is2688

applied to estimate the transfer function between each loudspeaker and each sample of thereproduction region. Simulation shows that the proposed method performs better than the spatial
harmonic decomposition method and the least squares method when employing the same number oftransfer function measurements. Furthermore, the proposed method requires much fewer transfer
function measurements to achieve the same the broadband local sound field reproduction with theconventional least squares method. However, the simulation only considers sound sources with simple
radiation features. Future work includes experimental validation on a real-world local sound field
reproduction system.

ACKNOWLEDGEMENTS
This research is supported under the Australian Research Council’s Linkage Project funding
scheme (LP160100616).

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