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

Extensions of the orthogonal beamforming to identify multiple
multipole sources
Xingjian PAN 1; Weikang JIANG1;
1

Shanghai Jiao Tong University, China

ABSTRACT
Identification of sources with complex directivities is attracting great attentions in aeroacoustic
measurements these years. A multipole orthogonal beamforming algorithm was proposed to identify
multipoles with arbitrarily oriented axes with no prior knowledge about source types. Independent sourceswere separated by the eigenvalue decomposition of the cross spectral matrix of signals measured bymicrophone arrays. The influence of source directivities on source imaging was eliminated by normalizing
and squaring the eigenvectors. The delay-and-sum procedure was performed on each modified eigenvector tolocalize sound sources. The proposed algorithm was verified by simulations and experimental trials using a
combination of a loudspeaker and an Aeolian tone dipole source. The accuracy of estimated source positions
is improved and the resolution of the proposed algorithm is comparable to that of deconvolution approaches.
Keywords: Aeroacoustic measurements, Beamforming, Multipole sources

1. INTRODUCTION

Beamforming techniques have been widely applied to identify aeroacoustic noise sources such as
airframe noise, airfoil noise and broadband fan noise(1-3). Most delay-and-sum beamforming
algorithms and deconvolution approaches use source models of monopoles and produce source
imaging by compensating for the phase delay between assumed sources and microphones(4).
However, most aeroacoustic noise sources have non-uniform directivities and are often analogous tomultipole sources. For example, sources arising from flow-body interactions have a two-lobed
directivity pattern and are analogous to dipole sources. Beamforming methods based on themonopole model may lead to incorrect estimations of positions of aeroacoustic sources with complex
directivities.
Studies on beamforming methods for multipole sources have been carried out over recent years.
Schmidt et al. presented the multipole-detection algorithm based on the conventional
beamforming(5). It replaced the steering vector with a ‘steering matrix’ whose columns correspondto sources of different types. Pre-defined source types and orthogonal radiation patterns wererequired. A beamforming method for directional sources was proposed by Bouchard(6), in which thesource model based on the spherical harmonic expansion is used and a set of ‘sub-beamformers’ isdefined to response to different spatial modes of a source. The prior knowledge of source positions
was required. The L1 generalized inverse beamforming method was introduced by Suzuki(7). It
obtained eigenmodes through the eigen decomposition of the cross spectral matrix of array signals
and reconstructed the source distribution using generalized inverse techniques. The method wasrestricted to pre-defined monopoles and multipoles whose orientations were on the plane parallel tothe array.
The objective of this study is to develop a beamforming algorithm that is capable of estimating
the positions and strengths of multipole sources without knowing their types or orientations. The
basic idea of the orthogonal beamforming is used that individual eigenvalues and eigenvectors of thesignal subspace are linked to sources or source mechanisms(8). This idea is further extended thatthese eigenvectors also contain information of source directivities. Normalization and square
operations on the eigenvectors can remove the influence of source directivities on source imaging.
The outline of the paper is as follows. After the introduction, the multipole orthogonal
beamforming is presented in Section 2. Simulation results are discussed in Section 3. The proposed1

xj_pan@sjtu.edu.cn6253

method is verified by experimental trials using a loudspeaker and an Aeolian tone dipole setup inSection 4. Conclusions are drawn in Section 5.

2. THE MULTIPOLE ORTHOGONAL BEAMFORMING
2.1 Modeling the sound field
An array with N mic microphones is used for measurements. In the presence of a single multipole
source, the sound pressure p measured by the i-th microphone at x i is(1)p (xi ) = a (xi , x 0 , x s )q (x s ) ,where q is the sound pressure at a reference position x0 radiated from that source. Generally, thecenter of the microphone array is chosen as the reference position on the premise that q does not
approach zero. For a free sound field, the transfer function a is related to the source directivity andthe source positon and is given byr Fa (xi , x 0 , x s ) = 0 i e − jk ( ri − r0 ) ,ri F0(2)where ri = |x s - x i| indicates the distance between the source position and the microphone position. r0= |x s - x 0| represents the distance between the source position and the reference position. F i and F0
are the directivity functions at the i-th microphone and the reference position. The directivity
function is related to the source type and the space angle.
For N s multipole sources with various directivities, the vector of sound pressures measured by themicrophone array is given by=p Aq + n ,(3)where the Nmic-by-N s matrix A is the transfer matrix whose elements are transfer functions describedby Eq. (2). The noise signal n is assumed to be uncorrelated to source signals.
The cross spectral matrix of array signals can be written as
C = E{pp H } ,(4)where E{} represents the expectation operator and H donates the conjugate transpose. Since thecross spectral matrix is non-negative definite and Hermitian, it can be decomposed as(5)
C = UΛU H ,where U is a unitary matrix whose columns are eigenvectors of the cross spectral matrix C. Λ is adiagonal matrix whose diagonal elements are eigenvalues λ i of the cross spectral matrix.
2.2 Descriptions of the algorithm
It is assumed that the multiple sources to be identified are mutually incoherent and spatiallyorthogonal(9), and background noise is uncorrelated to source signals. Therefore, the eigenvectors of
the cross spectral matrix of array signals can be divided into two parts, including the signal subspacespanned by the first Ns eigenvectors and the noise subspace spanned by the rest. According to theorthogonal beamforming, eigenvectors of the signal subspace are linked to sound sources or source
mechanisms. Information of directivities and positions of sources is contained in the eigenvectors,
and the eigenvalues contain information of source strengths.
To eliminate the influence of source directivities on source imaging, elements of the unitarymatrix U are all normalized and squared asvij = (uij| uij |)2 ,(6)

where u ij is the j-th element of the i-th eigenvector of the cross spectral matrix. A modified matrix
composed of elements v ij is generated as V = [v1 , v2 …v Nmic], where v ij is the (i, j)th element. The
normalization procedure aims to remove amplitudes of directivity functions contained in soundpressures. Because the directivity functions may be negative and increase the pressure phase by π,
the normalized elements are squared to compensate for the phase change. As a result, the signal
subspace is transformed into a subspace exclusive of source directivities. The first Ns modified
eigenvectors are then used to localize the multiple sources.
Using each eigenvalue λ i and the corresponding modified eigenvector v i , a matrix Di = λi v i vi H
can be calculated. The matrix is used to generate a beamforming map as6254

Bi (ξ j ) = h(ξ j ) H Di h(ξ j ) .

(7)

where ξj is the j-th grid point on the scanning plane. h is the steering vector whose i-th component
satisfies

1 −2 jk ( ri − r0 )
e.
Nm

(8)

uiH uih(ξ j ) H Di h(ξ j ) .
Nm(9)hi =Compared with the steering vector of the conventional beamforming, the phase of each element of
the modified steering vector is twice larger because the square procedure described by Eq. (6)enlarges the pressure phase. Source strengths estimated by Eq. (7) are much smaller than real ones
because the normalization procedure in Eq. (6) changes the amplitudes of sound pressures. This
underestimation can be corrected by compensating for the amplitudes as

Bi (ξ j ) =The source map generated by Eq. (9) corresponds to a single multipole source. The highest peakin the map is estimated as the source position. For a better resolution, the map is replaced by a ‘cleanbeam’ as−α ξ − ξ2Qi (ξ j ) = Bi _ max 10 j max .
The beamforming output of all the multipole sources is given byNs

Q(ξ j )= ∑ Qi (ξ j ) .

(10)
(11)

i =1

The number of sources Ns can be assessed by using the criterion of ‘entropic L-curve’ (10) or othercriteria based on the estimated eigenvalues.

3. NUMERICAL SIMULATIONSIn this section, two simulation cases were conducted to analyze the properties of the multipole
orthogonal beamforming. The first simulation case was to identify four quadrupoles with different
orientations. It was designed to evaluate the performance of the algorithm on localizing multipolesof the same type. The second one was to identify two quadrupoles and two dipoles with different
orientations. Its purpose was to investigate the capabilities of identifying multipoles of different
types. Beamforming results were compared with some existing algorithms including theconventional beamforming (CB)(4), CLEAN-SC(11) and the orthogonal beamforming (OB)(8).
3.1 Generation of simulation dataIn the first simulation case, four quadrupoles at the frequency of 4000 Hz were placed on the y-z
plane as shown in Figure 1(a). Generated sources in the second case were at the same frequency andset on the same positions as shown in Figure 1(b). An array of 56 microphones was installed in a
distance of 1m away from the source plane with the array center located on the x-axis. Relative
positions of the microphones and sources were shown in Figure 2. A 0.6m × 0.6m scanning plane
was set on the y-z plane with 0.02m spacing and 1681 grid points.

(a)

(b)

Figure 1 – The directivities of multipole sources in simulations. Case 1 (a) contained four quadrupoles and6255

Case 2 (b) contained two dipoles and two quadrupoles.

Figure 2 – The schematic representation of the simulation setup: the × symbols denote the positions ofmicrophones and the red circles denote the positions of sources. The microphone array was set parallelly
1m away from the source plane.
3.2 Results and discussions
Source maps of the first case produced by using four beamforming algorithms are shown in
Figure 3. Beamforming patterns of quadrupoles generated by CB are all separated into two side lobes
with small outputs in real source positions. Positions of sources estimated by CLEAN-SC and OB
deviate from the correct ones considerably although their beamforming patterns are focused. The
inaccurate estimation of source positions is influenced by source directivities. The directivity
functions of quadrupoles act as weighting parameters in the beamforming procedure. The steeringvectors of beamforming algorithms based on the monopole model only consider the influence ofmeasurement distances and cannot compensate for the directivity functions. Consequently,
beamforming outputs are not only determined by the phase delay between assumed sources andmicrophones but also influenced by the amplitude variance caused by source directivities. It leads toa distorted source map. The multipole orthogonal beamforming eliminates the amplitude difference
and recovers the source map by normalizing and squaring elements of eigenvectors. A small
deviation may be caused by the decomposition procedure.
Source strengths estimated by the four beamforming algorithms are nearly the same. However,different from a monopole source, sound pressures radiated from a multipole source are not only
related to the source strength and the measurement distance but also the directivity function and thespace angle. Microphone arrays placed at different space angles may output different source levels.
It impedes the further investigation of the nature of sources. Further research on the estimation ofabsolute source strengths is needed.

(a)

6256

(b)

(c)
(d)
Figure 3 – Source maps of four quadrupoles with different orientations produced by four beamformingalgorithms, including (a) CB, (b) CLEAN-SC, (c) OB and (d) the multipole orthogonal beamforming.
Source maps of two dipoles and two quadrupoles in the second case are shown in Figure 4.
Similarly, beamforming algorithms based on the monopole model lead to an inaccurate estimation ofsource positions. The deviation is related to the types and orientations of sources. For dipole sources
with their axes closer to the normal of the microphone array, the source positions are estimated moreaccurately. The proposed multipole orthogonal beamforming can still localize the sources accurately
regardless of their types and orientations.

(a)(b)(c)(d)
Figure 4 – Source maps of two dipoles and two quadrupoles with different orientations produced by four
beamforming algorithms, including (a) CB, (b) CLEAN-SC, (c) OB and (d) the multipole orthogonal
beamforming.

4. EXPERIMENTAL VERIFICATION

Experimental trials were conducted to investigate the performance of the multipole orthogonal
beamforming in a more practical environment. The experimental setup was shown in Figure 5(a),including a microphone array, a loudspeaker, a 4mm circular cylinder and an air duct. The open jet
flow produced by the air duct was around 35 m/s. An Aeolian tone dipole source could be induced by
the flow over the circular cylinder. The loudspeaker was set above the cylinder and had a uniform
directivity as a monopole source. The microphone array was installed parallel to the flow direction ina distance of 0.5 m.
Two cases were tested when the circular cylinder was parallel and vertical to the ground, as
shown in Figure 7 (b) (c). In theory, the oriented axis of the Aeolian tone dipole source is6257

perpendicular to the plane composed of the flow direction and the cylinder. Therefore, the dipole
axis was respectively parallel and vertical to the array in Case 1 and 2. It is well established that theStrouhal number St = fD/U is approximately equal to 0.21 for the flow around a circular cylinder, andthe dipole frequency equals the vortex shedding frequency when the Reynold number Re is more than
5000. For this experimental setup, the flow velocity was around 35 m/s, and the corresponding source
frequency was 1835 Hz. Given that the flow velocity was unsteady, a 1/3 octave band centered at 2000
Hz was analyzed. Four beamforming algorithms were applied and compared, including CB,CLEAN-SC, OB and the proposed multipole orthogonal beamforming.

(a)

(b)

(c)

Figure 5 – The experimental setup to produce an Aeolian tone dipole source and a monopole source. (a)shows the trial facilities. (b) shows the setup for Case 1, which produces a dipole whose axis parallel to thearray. (c) shows the setup for Case 2, which produces a dipole whose axis vertical to the array.
Sound maps of the first case are shown in Figure 6. For the dipole source whose axis parallel tothe array, CB, CLEAN-SC and OB all produce distorted beamforming patterns. The beamforming
pattern of the dipole source is separated into two side lobes with a small output in the real source
position. It is easy to misinterpret this beamforming pattern as two individual sources. The multipole
orthogonal beamforming localizes the dipole source accurately and the resolution is comparable tothat of the CLEAN-SC algorithm. Source levels estimated by OB and multipole OB are 2 dB lowerthan the outputs of CB and CLEAN-SC. It is because side lobes contributed from the backgroundnoise have been removed by the eigenvalue decomposition. Source maps of the second experimental
case are presented in Figure 7. All four beamforming algorithms estimate the source positions
accurately. In this case, the dipole axis is vertical to the microphone array and the dipole source
looks like a monopole source from the array direction. Directivity functions of the dipole source overthe microphone array vary a little and have a negligible influence on source imaging.
In general, the impact of source directivities on source imaging can be eliminated by thenormalization and square operation described by Eq. (6). The multipole orthogonal beamforming canlocalize multiple independent multipole sources regardless of their types and orientations. Thisalgorithm can be applied to identify airframe noise sources or airfoil noise sources in the wind tunnel,
which have complex directivities.

6258

(a)

(b)

(c)
(d)
Figure 6 – Source maps of a monopole source and an Aeolian-tone dipole whose axis parallel to the array.
Maps are produced by four beamforming algorithms, including (a) CB, (b) CLEAN-SC, (c) OB and (d) the
multipole orthogonal beamforming.

(a)

(b)

(c)
(d)
Figure 7 – Source maps of a monopole source and an Aeolian-tone dipole whose axis vertical to the array.
Maps are produced by four beamforming algorithms, including (a) CB, (b) CLEAN-SC, (c) OB and (d) the
multipole orthogonal beamforming.

5. CONCLUSIONS

A new beamforming method was developed to identify multiple independent multipole sources
regardless of their types or orientations. Compared with beamforming algorithms based on the monopolemodel, the multipole orthogonal beamforming estimates positions of multipole sources more accurately,
especially when the source axis deviates from the normal of the microphone array. Compared with othermultipole beamforming algorithms, it needs no prior knowledge of source types, orientations or positions.
The resolution of this algorithm is comparable to that of deconvolution approaches with moderatecomputational effort. Further research can be conducted on the estimation of source types and absolute

6259

source strengths.

ACKNOWLEDGEMENTS
This work was supported by National Natural Science Funds of China (Grant No. 11574212).

REFERENCES

1. Stoker R, Guo Y, Streett C, Burnside N, editors. Airframe Noise Source Locations of a 777 Aircraft inFlight and Comparisons with Past Model-Scale Tests. 9th AIAA/CEAS Aeroacoustics Conference andExhibit; 2003 2003-05-12. Hilton Head, South Carolina: American Institute of Aeronautics andAstronautics.
2. Porteous R, Prime Z, Doolan CJ, Moreau DJ, Valeau V. Three-dimensional beamforming of dipolar
aeroacoustic sources. J Sound Vib. 2015;355:117-34.
3. Dougherty RP, Walker BE, Sutliff DL. Locating and Quantifying Broadband Fan Sources using In Duct
Microphones. 16th AIAA/CEAS Aeroacoustics Conference 2010 31st AIAA Aeroacoustics Conference.
2010:531-49.
4. Sarradj E. Three-dimensional acoustic source mapping with different beamforming steering vector
formulations. Adv Acoust Vib (USA). 2012:12.
5. Schmidt R, Franks R. Multiple source DF signal processing: An experimental system. IEEE
Transactions on Antennas and Propagation. 1986;34(3):281-90.
6. Bouchard C, Havelock DI, Bouchard M. Beamforming with microphone arrays for directional sources.
J Acoust Soc Am. 2009;125(4):2098-104.
7. Suzuki T. L1 generalized inverse beam-forming algorithm resolving coherent/incoherent, distributedand multipole sources. J Sound Vib. 2011;330(24):5835-51.
8. Sarradj E. A fast signal subspace approach for the determination of absolute levels from phased
microphone array measurements. J Sound Vib. 2010;329(9):1553-69.
9. Dong B, Antoni J, Pereira A, Kellermann W. Blind separation of incoherent and spatially disjoint soundsources. J Sound Vib. 2016;383:414-45.
10.Dong B, Antoni J, Zhang E. Blind separation of sound sources from the principle of least spatial entropy.
J Sound Vib. 2014;333(9):2643-68.
11. Sijtsma P, editor CLEAN Based on Spatial Source Coherence. 13th AIAA/CEAS Aeroacoustics
Conference (28th AIAA Aeroacoustics Conference); 2007 2007-05-21. Rome, Italy: American Instituteof Aeronautics and Astronautics.

6260