PROCEEDINGS of the
23rd International Congress on Acoustics

9 to 13 September 2019 in Aachen, Germany

Attempt to Classify the Microphone Array Deconvolution Methods
in Aeroacoustics
Christophe PICARD1; Quentin LECLERE2
1
2

MicrodB, France

Laboratoire Vibrations Acoustique (LVA), University of Lyon, INSA-Lyon, France

ABSTRACTIn the field of experimental aeroacoustics in wind tunnel, the number of existing methods is important and isincreasing continuously so that it becomes difficult for Industrials to have a clear view of the respective
advantages of each of them and make appropriate choice. Benchmark works are also plentiful and often focus
on the performance of these methods in terms of spatial localization and computing time, more rarely in termsof quantitative estimation. We are interested here by these deconvolution methods that we aim to classify,
and even to show in what extent all of these methods are equivalent. The approach adopted is to define acommon protocol for controlling and monitoring the different algorithms: define same initial condition, same
convergence parameters and same exit conditions. All of the considered deconvolution algorithms are
implemented with respect to the same protocol. Convergence and calculation time are monitored at eachfrequency. Finally acoustic power of source area are extracted from the resulting maps and compared together.
Indeed, it is observed that within the range of few dBs the results are quite similar and we propose to classify
the algorithms with their equivalent Lp-norm minimization function.
Keywords: Microphone array technique, Deconvolution, Aeroacoustics, Wind tunnel measurements,
Airframe noise

1. INTRODUCTION
Aeroacoustic measurements based on acoustic arrays in wind tunnels can provide a detailed
understanding of complex noise sources. In particular for airframe noise in aeronautics, wind tunnel
tests help to investigate new aircraft concepts, verify performance of innovative designs and validate
prediction models. Indeed, thanks to increasing computational power, more and more CFD models areused in design phases, and wind tunnel testing is the ultimate way to validate these models long before
the aircraft can actually fly. However, wind tunnel testing is often very expensive and must be
performed efficiently to get the most out of the limited testing time.
Microphone array techniques provide a considerable amount of data from which it is possible toextract valuable information on the origin of sound such as location or power of acoustic sources. The
most common microphone array method is certainly the conventional beamforming algorithm (CBF[1], also known as “delay-and-sum” [2]) because of its simplicity, robustness and computational
efficiency. However it suffers from a lack of resolution in the low frequency range and from theinability to properly quantify the source power, especially in the case of complex source distributions
like those generally encountered in aeroacoustics. These drawbacks have been tackled in the 2000’swith advanced beamforming-based algorithms such as deconvolution techniques. In the field ofexperimental aeroacoustics in wind tunnel, the number of existing deconvolution methods is important(Clean-PSF [3], DAMAS [4], SEM [5], Clean-SC [6] and NNLS [7], and the many derived versions).
All of these methods in fact share the same theoretical basis. Indeed the physical model is the sameand the techniques only differ from the final inversion algorithm, while, in aeroacoustics, the modelin general assumes an uncorrelated monopole distribution in free field [8].
We are interested here by these deconvolution methods that we aim to classify, and even to show
1
2

christophe.picard@microdb.fr
quentin.leclere@insa-lyon.fr

6245

in what extent all of these methods are equivalent. The approach adopted is to define a common
protocol for controlling and monitoring the different algorithms: define same initial condition, same
convergence parameters and same exit conditions. The DLR1 case (Dornier-728 half model) [9] takenfrom the Array Analysis Benchmark [10] is used. All of the already mentioned deconvolution
algorithms are implemented with respect to the same protocol. Convergence and calculation time aremonitored at each frequency. Finally acoustic power of source area are extracted from the obtained
maps and compared together. Indeed, it is observed that within the range of 1-2dB the results are quite
similar and we propose to classify the algorithms with their equivalent Lp-norm minimization function.

2. MICROPHONE ARRAY TECHNIQUES
2.1 General principles
Many acoustic imaging techniques have been developed over the past 30 years. These techniques
have in common the formalization of the theoretical problem which is to estimate the position andpower level of acoustic sources from pressure measurements on a microphone array.
The principle of identifying sources consists in positioning a microphone array facing the object
to study and try to locate the main noise areas through specific processing of microphone signals,
based on a source model chosen a priori. For aeroacoustic applications, the usual hypothesis is to
assume that the phenomena are well represented by a distribution of uncorrelated monopoles [8]. Thiscan be justified by the fact that the pressure correlation scales are lower than the spatial resolution of
the method in general. One then seeks the model adequacy to the measurement data.
One distinguishes far field and near field zones of the array (respectively Fraunhofer and Fresnel
zones). It is admitted [8] that it is advantageous to measure a source in the near field of the array, inparticular to overcome the background noise. The source emission is then spherical and one performs
a focused array processing where the position of the source is sought, that is to say the emission centerof the spherical wave. This model is the most commonly used (in aeroacoustics anyway) and usuallyprovides interesting and useful results. But it must be noticed that this is an approximation of the
propagation model and that the differences between the actual acoustic propagation and the
propagation model can result in errors, especially on the source amplitudes.
In practice, one chooses a region of the scan area, where the presence of sources is sought for. For
practical reasons linked to mechanical design, one generally arrange the microphones on a plane. Fora typical 2D configuration, an area of candidate source points is then defined in a plane parallel to themicrophone array.
2.2 Expression of the direct problem
By solving the wave equation in the frequency domain for a source in an open field and discretizing
the solution, the complex pressure vector ࡼ ሺ‫ ܯ‬ൈ ͳሻ received at the ‫ ܯ‬microphones of the array atfrequency ݂ can be expressed as:ࡼ ൌ G௝ ‫ݍ‬௝ ,(1)where ‫ݍ‬௝ is the complex amplitude of a source at point ࣈ௝ and G௝ is the Green function vector ሺ‫ ܯ‬ൈͳሻ in free space between the source and the microphones ‫ ܯ‬at points ࢞௜ , ݅ ‫ א‬ሾͳ ǥ ‫ܯ‬ሿ. The frequency
dependence of the equations is omitted in the notation of this paper.
In the presence of strong background noise perturbation, as often it is the case in aeroacoustics, it
is wise to proceed with ensemble averages in the frequency domain, assuming stationarity property ofsignals. For this, one uses the cross spectral matrix (CSM) of the measured signals on the array, matrix
of complex values, expressed as:(2)࡯ ൌ ‫ۄ כࡼࡼۃ‬,with ‫ ۄ ڮۃ‬designating the ensemble average and the superscript ‫ כז‬being the Hermitian complex
conjugate operator.
Applying this to equation (1), in the presence of ܵ sources and assuming uncorrelated source
model, i.e. incoherent sources, leads to the following expression of the modelled CSM:
࡯௠௢ௗ ൌ σௌ௝ୀଵ G௝ G௝‫ܳ כ‬௝ ,(3)with ܳ௝ being the power spectral density of the sources at each considered point ࣈ௝ . Methods
considering other source models also exist, see [11] and [12] for a review.

6246

2.3 Expression of the inverse problem
Starting from equation (3), the sound source identification consists in the estimation of the power
spectral density of the sources knowing the measured CSM ࡯௠௘௦ (measured pressure signals on thearray). This is an inverse problem that can be solved using a classical optimization procedure. For
that, an objective function - error function or cost function - is defined, traducing the adequacy of themodel to the measurement data. A L2-norm is chosen here to quantify the difference between themeasured CSM and the modelled CSM, i.e.:ߝ ൌ ԡ࡯௠௘௦ െ ࡯௠௢ௗ ԡଶ .
(4)
The objective is then to minimize this cost function by considering its gradient regarding the
unknown, i.e. searching an estimation of unknown ܳ௝ satisfying:డఢడொೕൌ Ͳ or ܳ௝ ൌ Ͳ, subject to ܳ௝ ൒ Ͳ for all considered candidate source point ݆.

(5)All deconvolution methods intend to solve the optimization problem (5), or close forms of it. Whatdifferentiates them lies in the algorithms that are implemented to find a solution. The conventional
beamforming method and general principles of deconvolution methods are presented hereafter.
2.4 Conventional Beamforming method
The most commonly used method in the industry for the identification of acoustic sources is theconventional beamforming (CBF) for which an estimate of the power spectral density is search ed foreach source point scanned one after the other among the chosen set of candidate points. This means
considering the following modelled CSM :࡯௠௢ௗ ൌ G௝ G௝‫ܳכ‬௝ .
(6)Deriving the optimization procedure leads to the following estimate of the source power spectral
density ‫ܣ‬௝ at the considered scan point ࣈ௝ :‫ܣ‬௝ ൌ w௝‫࡯ כ‬௠௘௦ w௝ ,(7)
where w࢐ ൌ G௝ ΤG௝‫כ‬G௝ is viewed as a spatial filter.
In the case of a unique real source at point ࣈ௝ , the linear regression determines ‫ܣ‬௝ is the best
estimation of the power spectral density of the real source [8].
2.5 Deconvolution methods
The conventional beamforming method suffers from severe limitations. The beamforming process
in fact performs a spatial convolution between the true source distribution and the response of thearray to a unique point source, called Point Spread Function, PSF, whose shape depends on the numberof microphones, their spatial arrangement, the frequency and the relative position of the center of thearray to the scan point. This generates artefacts as main lobes, side lobes and grating lobes that po llute
the resulting source map. These non-physical artefacts may render physical interpretation difficult.
The PSF at point ࣈ௝ due to a source at point ࣈ௞  can be expressed as:ଶR௝௞ ൌ ൫w௝‫ כ‬G௞ G‫כ‬௞ w௝ ൯ ൌ ฮw௝‫ כ‬G௞ ฮ .
(8)
It is then possible to express the CBF result ‫ܣ‬௝ as a product of the array directivity pattern R௝௞
and the true power spectral density ܳ௞ :‫ܣ‬௝ ൌ R௝௞ ܳ௞ .
(9)Considering a set of ܰ candidate points the CBF map result, ሺܰ ൈ ͳሻ vector ‫ۯ‬, is expressed as
the product of the complete PSF, i.e. ሺܰ ൈ ܰሻ matrix R, and the ሺܰ ൈ ͳሻ vector of source candidates‫ۿ‬, thus forming a second linear system assuming incoherent sources (also called in the field as the
DAMAS problem [4]):(10)‫ ۯ‬ൌ R‫ۿ‬,which requires a second inversion method of a linear system. The special feature of equation (10) thus
obtained is that all terms of the system are real and positive. A positivity constraint thus has to beconsidered on the unknowns ‫ۿ‬.
It is necessary to introduce here a quantity used in the optimization algorithm, called the residue anddefined by :‫ ܚ‬ൌ ‫ ۯ‬െ R‫ۿ‬.

(11)

6247

It is interesting to note at this point that starting from equation (4) but considering all ܰ scanned
points at once, i.e. considering equation (3) for the modelled CSM instead of equation (6) (as forconventional beamforming), Blacodon and Elias [5] derived the optimization process (as shortly
described in section 2.3), and resulted in the same linear system to be solve as equation (10).
A fundamental question arises about this problem. Indeed some consider a linear system and others aquadratic system (nonlinear), both under constraint, depending on considering the unknown vector ‫ۿ‬as linear or as a quadratic quantity, what it is physically. This question has no answer within thecommunity to our knowledge, and should have a consequence on the choice of solving algorithms .
Whatever, it is expected that the positivity constraint has a strong regularizing effect and tends tofoster sparsity of the solution. Thus any algorithm used in applied mathematics should findequivalently a local minimum even without any penalty (regularization).
To overcome artefacts due to the directivity of the array, the deconvolution process consists in two
steps: 1/ estimate the CBF result at each point of the chosen scan grid with equation (7) and 2/ estimate
the source power density taking into account all the scanning points at the same time by solving
equation (10), iteratively in general.
2.5.1 Approach adopted
For this study, we consider 5 deconvolution algorithms: Clean-PSF [3], DAMAS [4], SEM [5],NNLS [7] and CIRA [13]. Even if Clean-SC [6] is one of the most popular deconvolution methods inthe field of aeroacoustics, it isn’t considered here because it differs from the others in the sense thatit modifies steering vector during the iterative process. Note that for SEM, the selected algorithm is a
simple Conjugate Gradient (easier to modify) as chosen initially (CONMIN) and instead of LBFGS
as finally proposed by the authors.
The implementation in an industrial context of this type of algorithm requires a lot of attention inparticular with regard to the initial condition, the management of convergence and positivity
constraint, and the exit conditions of the algorithm. Our experience is that the initial condition haveno important impact on the convergence speed and the result, the final map. On the other handmanagement of convergence and exit conditions of the algorithm have a more crucial impact.
There are also other implementation tricks useful for aeroacoustics (CSM diagonal removal (DR),consideration of convection effects on acoustic propagation (Amiet)) that are actually applied in thiswork. Note that the DR option requires consistent operations to be applied in the algorithm,
particularly with regard to the positivity constraint to be applied to the initial CBF map or to thepositive matrix R. All of these implementation details are not detailed in this publication.
After a literature revue on the methods and algorithms, the approach adopted, based on making
compromise and synthesis, is to define a common protocol for controlling and monitoring the chosenalgorithms during the different parts of a deconvolution algorithm:
x initial condition
There’s two options for the initial map (also called clean map in Clean-PSF): either set to zero or
initialized by the CBF map, possibly renormalized. In the literature, the initial map is set to zeroor to the CBF map. For all algorithms we choose here the first option for the sake of consistency
between algorithms, because Clean-PSF algorithm can’t support the other choice.
x management of convergence
We are talking about the convergence parameter which can be called sometimes relaxation
parameter, step factor, loop factor or energy injection factor, and about the positivity constraint.
Convergence parameter is certainly the most delicate part to settle in this work because the chosenalgorithms consider this kind of parameter in very different ways. SEM (CG) and NNLS containa convergence parameter that is automatically calculated at each iteration. The convergenceparameter for CIRA is defined once at each frequency but several options exist. And, finally,
Dougherty proposed to integrate a convergence parameter in Clean-PSF to counteract somehow
the spatial extent of aeroacoustic sources, parameter settled empirically (typically between 0.1
and 0.25). In the end, harmonization between algorithms is not feasible and we choose to apply
the best/optimized know-how concerning the convergence parameter, for the algorithms that
require a user defined choice.
Imposition of positivity is an absolute necessity for our deconvolution methods. The mostconsistent option is to zero all negative solutions founded in the map at each iteration. It waspossible to impose this constraint on all the algorithms chosen in this work.
x exit conditions of the algorithm

6248

This is the most confusing part encountered in the bibliography or even the less well documented.
There is no consensus on this issue. In summary we usually find two associated criteria for theexit of the algorithm: an energy criterion and a maximum number of iterations, a safety condition
that sets the limit of the exit condition. In terms of energy criterion we can list non-exhaustively
the following conditions. We then propose to evaluate in this work the following energy criteria
at each iteration ࢏: the L0-norm of the resulting map (clean map), called here ‫݊Ͳܮ‬ሺ௜ሻ , the L1-norm of the clean map, called here ‫݊ͳܮ‬ሺ௜ሻ . Note that because of the quadratic natureof the physical quantity to identify, this represents in fact the squared L2-norm of thesource level in the map, the L2-norm of the clean map, called here ‫݊ʹܮ‬ሺ௜ሻ , the value of the cost function (equation (4) at iteration ሺ݅ሻ) ߝ ሺ௜ሻ , eventually normalized by
the energy in the CSM ԡ࡯௠௘௦ ԡଶ , i.e. ܰߝ ሺ௜ሻ ൌ ߝ ሺ௜ሻ Τԡ࡯௠௘௦ ԡଶ. A special care must be taken
in case of DR option is selected, for the calculation of ࡯௠௘௦ , going back to the optimization procedure (§2.3), a local minimum can be founded whenequation (5) is satisfied. In Fact equation (5) is equal to the residue (equation (11)),expressed in the iterative process as ࢘ሺ௜ሻ ൌ ‫ ۯ‬െ R‫ۿ‬ሺ௜ሻ . Then the L2-norm of the residue,
defined by ‫ ݎ݊ʹܮ‬ሺ௜ሻ ൌ ฮ࢘ሺ௜ሻ ฮ, can be used.
The question of defining a relevant threshold for each criteria then arise. Threshold values ofenergy criteria are often chosen empirically. To try to clarify this question, the solution adopted
here is to observe also the evolution of the criteria between two successive iterations, i.e. forexample หߝ ሺ௜ሻ െ ߝ ሺ௜ିଵሻ ห.
Reminding that it is expected to compare and classify the 5 algorithms, following a (as much as)common and global protocol, it seems necessary to consider also the industrial process of analyzing
acoustic imaging results. Indeed, in general, engineers are interested in the integrated source power
on areas related to components to optimize. That’s why it is decided to monitor also the source power
integrate in the following areas [9]: FLAP INT, SLAP INT and FLAP TIP. It defined finally all thecriteria we have monitored in this study (next section).
All of the considered deconvolution algorithms are implemented with respect to the same pseudocode.
2.5.2 Modus operandi adopted
The study is composed of two parts. Firstly the chosen algorithms will be monitored for several
discrete frequencies: 2021Hz, 3018Hz, 4394Hz, 5010Hz, 6006Hz, 7002Hz and 8496Hz. A maximum
number of iterations is fixed arbitrarily from 200 in the lowest frequency band to 100 in the highest)
and the energy convergence criteria (previous section) are monitored at each iteration. The objective
is to observe the behavior of each algorithm and try to identify the most relevant exit criteria for eachalgorithm, finding supposed local minima. Then, secondly, more complete calculation will be done inthird octave bands and integrated energy on defined areas will be compared.

3. IMPLEMENTATION
3.1 Test case description
The test case selected here is the AIAA array analysis benchmark case DLR 1, Aircraft Half Modelin closed wind tunnel. This benchmark problem consists of a test configuration with a Dornier -728
half model of scale 1:9.24 in high-lift configuration (landing) placed in the center of the cryogenic
wind tunnel at the DLR Kryo-Kanal Koeln (DNW-KKK), refer to [9] and [10] for a more complete
description of the test case.
The microphone array has a diameter of 1 m and is composed of 135 microphones arranged inspiral arms mounted onto the sidewall. The sampling frequency of microphone signals is 120 kHz.
The benchmark data is acquired for 9 couples of parameters: 3 model angles of attack (3°, 5° and9°) and 3 free stream velocities (Mach numbers M=0.15, 0.2 and 0.25) during 30 seconds each. The
region of interest of the half model, defined in the frame of the benchmark, is a 1.05 m x 1.45 m
observation plane at a distance of y = 1.045 m (position of the wing root) from the microphone array.
Three grid resolutions of the source maps have been proposed in the frame of the benchmark: 1
cm, 2 cm and 5 cm leading to different numbers of equidistant scanning points respectively of 15476,

6249

3869 and 660.
In this paper, results for the case of M=0.25 (88m/s) and 3° angle are presented, for a grid of 2 cm
resolution (3869 points).
3.2 Algorithms monitoring
The adopted mode of analysis of all the results begins with the observation of the source power byarea. Indeed, in general, we observe a relatively rapid convergence on these quantities [4]. We thendeduce a number of iterations that could be considered as sufficient or satisfying a convergencecriterion (to be determined). We note that the source powers by area are quite coh erent between themfrom one algorithm to another. Then we observe the criteria L0, L1 and L2 (calculated on the source
map at each iteration) and finally the relevant criteria on the value of the functional and on the residue.
Not surprisingly, two groups of algorithms are identified. The first (view as a L2-norm class ofalgorithm) composed of CG (SEM), CIRA and NNLS (very close but with NNLS a little more sparse)
and DAMAS (even more sparse) and the other group (view as a L1-norm class of algorithm) composedof Clean-PSF which provides the sparsest maps (often impossible to visualize). The degree of sparsity
is observed with criterion L0 which always decreases for the first group and always increases for thesecond, as expected. Thus, CG(SEM), CIRA and NNLS find a variable number of sources dependingon the frequency. This is not the case for DAMAS and Clean-PSF (see Figure 1).

Figure 1 – Evolution of criteria L0-norm (left) and L1-norm (right) over frequencies at a converge
state (judged as such).
L0-norm criterion is never used as an exit conditions of a deconvolution algorithm as far as weknow. Although imposing a number of lighted source points is difficult to conceive for engineers, thenumber of sources being unknown a priori and can be one of the unknowns of the problem, we observea great similarity of the visual representation by doing so. Figure 2 shows maps calculated at 8496Hz
imposing a number of visible source around 360 (determined according to the power level dynamics)
for CG(SEM), CIRA and NNLS algorithms. It is interesting to note then that the number of iterations
is quite different, respectively 100, 26 and 10. This highlights the intrinsic tendency to sparsity ofeach algorithm.
Criterion L1 is interesting to observe: there is generally fast convergence towards a stable value
but whose value depends on the frequency. Similarly, criterion L2 is always increasing withasymptotic behavior. It's interesting regarding the number of sources: for the first group the moreiterates, the less we have enlightened source points, the more we tend towards sparsity but the energyof the map increases slightly. We therefore operate a concentration of energy on identified source
points. For the other criteria, we observe a relatively good consistency of the ߝ criterion for DAMAS
and ‫ ݎ݊ʹܮ‬for CG(SEM), CIRA, NNLS and even Clean-PSF. But the evidence is less clear than for
L1-norm criteria.
Generally, we therefore observe the regularizing effect and the tendency to sparsity of the positivity
constraint with a gradation toward sparsity from CG(SEM), CIRA, NNLS, DAMAS to Clean-PSF. In
our view, the effect is essentially of the order of the visualization of the maps because the powers per
areas are consistently estimated between the 5 algorithms.
The analysis of those criteria showed that it is difficult to define common thresholds for all
algorithms and each energy criterion defined in §2.5.1. However the evolution of the criteria overiterations seemed more suited to that. Thus we finally selected commonly to the 5 algorithms threecriteria and defined three thresholds constant on the frequency band. Indeed, the intention was to drive
the algorithms in the most unified way and the thresholds have been chosen the least restrict ive so as
to be suitable for all algorithms over a wide frequency band. The choice of the three criteria was basedon the greatest representativeness of the quantities of interest of the problem (refer to §2.5.1):
adequacy of the model to the measurement on the array (หߝ ሺ௜ሻ െ ߝ ሺ௜ିଵሻ ห), source map energy considering6250

the physical quantity of interest (ห‫݊ͳܮ‬ሺ௜ሻ െ ‫݊ͳܮ‬ሺ௜ିଵሻ ห) and minimization of the residue error in the
convolution problem (ห‫ ݎ݊ʹܮ‬ሺ௜ሻ െ ‫ ݎ݊ʹܮ‬ሺ௜ିଵሻ ห).
Then it has been possible to roll out the calculations throughout the complete frequency band.

Figure 2 – Comparison of maps obtained using CG(SEM) (left), CIRA (middle) and NNLS (right)
algorithms by imposing a fixed number of lighted source points (~360) at 8496Hz .
3.3 1/3 octave calculation analysis
Calculations were done on the third octave bands from 2kHz to 8kHz, monitoring the source power
levels on the three areas defined on the map (FLAP INT, SLAP INT and FLAP TIP), the average
iteration number and the average CPU time each per third octave band.

Figure 3 – Source power level par area (FLAP INT left, SLAP INT in the middle and FLAP TIP in
the right).
The behavior already observed during fine-band calculations for monitoring (§3.2) is confirmedwith third octave calculations: the source power level per area is quite well uniformly founded byeach algorithms (figure 3), within the range of few dBs, while the visual representation of the source
map is very different, highlighting the nature of each algorithm in terms of sparsity representation of
the sources. Only the CLEAN-PSF algorithm deviates from the general trend. We believe that this is
not due to the intrinsic nature of the algorithm, its natural tendency to sparsity (L1-norm class ofalgorithm). The experience gained during this study indeed encourages us to think that the iteration
number is not enough high. In other words, the exit conditions defined in a common way to all thealgorithms are not restrictive enough for this algorithm. It may well be that the differentiation of thealgorithms adopted in this work (L2-norm class vs. L1-norm class) is not valid in the end.
Concerning the average iteration number and the average relative CPU time per third octave band
observed during calculations, we see that they are relatively independent of frequency excepting for
DAMAS. Note that CPU time is calculated relatively to the CBF CPU time calculation. The
monitoring also indicates that sparsity of the source model has a computational cost .

4. CONCLUSIONSWith the aim of classifying the microphone array deconvolution methods in aeroacoustics, acommon protocol for controlling and monitoring the different algorithms selected has been definedand applied to a use case taken from the Array Analysis Benchmark (DLR1).
We actually observe that the necessary positivity constraint of the deconvolution problem
considered here has a strong regularizing effect and tends to foster sparsity of the solution. Thus the
5 selected algorithms coming from applied mathematics find equivalently sound power levels on areas
defined in the source map for industrial analysis process, both in fine band as in third octave band.
This is a comforting result that must convince engineers that when we want to solve an inverse
problem, the most important is not the choice of the algorithm as such but rather the way we define
the inverse problem and the way the algorithm is controlled. Thus the classification of the algorithms

6251

deduced from this work lies in the sparsity of the obtained source model (and of its consequence interms of visual representation) and in the computation time. The intuition of the classification of thealgorithms on the Lp-norm nature is not demonstrated, for the selected algorithms and with the applied
protocol.
The importance of the algorithm exit conditions is highlighted through this work. The intention todrive the algorithms in the most unified way (on the threshold choice at least) probably does not makemuch sense except for the demonstration proposed here. A more rigorous study of the convergence ofthe deconvolution algorithms should be done now.
Finally, we must also keep in mind that the observation conducted in this work may probably bestrongly linked to the test case itself. Thus for a generalization of the conclusions, it is necessary toapply this protocol to others aeroacoustic use cases.

ACKNOWLEDGEMENTS
The authors wish to acknowledge the array methods community for their input with regard to thevalue of this work, selecting datasets of interest, defining metrics, and providing feedback at various
meetings and additionally, they wish to thank the team at DLR DNW-KKK facilities for their efforts
in acquiring data for analysis.
This work was conducted in the frame of the Labcom P3A (ANR-13-LAB2-0011-01).

REFERENCES
1
2
3
4

5
6
7
8
9
10

11

12

13

6252

J. Billingsley et R. Kinns, «The Acoustic Telescope,» Journal of Sound and Vibration, vol. 48, 1976.
D. H. Johnson et D. E. Dudgeon, Array Signal Processing: Concepts and Techniques, Prentice Hall,
1993. 0-13-048513-6, 1993.
R. Dougherty et W. Stoker, «Sidelobes supression for phased array aeroacoustic measurement,» chez
AIAA/CEAS 1998-2242, 1998.
T. F. Brooks et W. M. Humphreys, «A deconvolution approach for the mapping of acoustic sources(DAMAS) determined from phased microphone arrays,» Journal of Sound and Vibration n°294, p.
856–879, 2004.
D. Blacodon et G. Elias, «Level estimation of extended acoustic sources using a parametric method,»
Journal of Aircraft, vol. 41, n° %16, pp. 1360-1369, 2004.
P. Sijtsma, «CLEAN Based on Spatial Source Coherence,» International Journal of Aeroacoustics,
n° %16, pp. 357-374, 2007.
K. Ehrenfried et L. Koop, «Comparison of Iterative Deconvolution Algorithms for the Mapping of
Acoustic sources,» AIAA Journal, vol. 45, n° %17, p. 1584, 2007.
G. Elias, «Experimental Techniques for Sources Location,» Von Karman Institute for Fluid Dynamics,
VKI LS 1997-07, Aeroacoustic and active noise control., 1997.
T. Ahlefeldt, «Aeroacoustic measurements of a scaled half-model at high Reynolds numbers,» AIAA
Journal, vol. 51, n° %112, pp. 2783-2791, 2013.
C. J. Bahr, W. M. Humphreys, D. Ernst, T. Ahlefeldt, C. Spehr, A. Pereira, Q. Leclère, C. Picard, R.
Porteus, D. J. Moreau, J. Fischer et C. J. Doolan, «A comparison of microphone phased array
methods applied to the study of airframe noise in wind tunnel testing,» chez 23rd AIAA/CEAS
Aeroacoustics Conference, 2017.
Q. Leclère, A. Pereira, C. Bailly, J. Antoni et C. Picard, «A unified formalism for acoustic imaging
based on microphone array measurements,» International Journal of Aeroacoustics, vol. 16,
n° %14-5, pp. 431-456, 2017.
A. Finez, C. Picard, T. Le Magueresse, Q. Leclère et A. Pereira, «Microphone array techniques basedon matrix inversion,» Von Karman Institute for Fluid Dynamics, VKI Lecture Series STO-EN-AVT287-05, 2017.
C. Picard, F. Lepercque, T. La Magueresse, O. Minck, R. Hallez et J. Lanslots, «Microphone Array
Deconvolution Methods for Efficient Aeroacoustic Testing in Wind Tunnels,» chez Inter.Noise
2019, Madrid, 2019.