A review of acoustic imaging methods using phased microphone arrays
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Abstract
Phased microphone arrays have become a wellestablished tool for performing aeroacoustic measurements in wind tunnels (both openjet and closedsection), flying aircraft, and engine test beds. This paper provides a review of the most wellknown and stateoftheart acoustic imaging methods and recommendations on when to use them. Several exemplary results showing the performance of most methods in aeroacoustic applications are included. This manuscript provides a general introduction to aeroacoustic measurements for nonexperienced microphonearray users as well as a broad overview for general aeroacoustic experts.
Keywords
Phased microphone arrays Beamforming Acoustic imaging Deconvolution Aeroacoustic measurementsList of symbols
Latin symbols
 A
Source autopower (\(\text {Pa}^{2}\))
 \(\varvec{A}\)
Acoustic source map (\(\text {Pa}^{2}\))
 \(\tilde{\varvec{A}}\)
\(J \times J\) Propagation matrix whose columns contain the PSF of each grid point
 B
Array beamwidth of 3 dB down from beam peak maximum (\(\text {m}\))
 \(\varvec{B}\)
\(J \times J\) Global source crossspectral matrix for IBIA
 c
Speed of sound (m/s)
 \({\tilde{c}}\)
Propagation velocity (m/s)
 \(\varvec{c}\)
\(1 \times J\) propagation vector used in linear programming (\(\text {m}^{1}\))
 \(\varvec{C}\)
\(N \times N\) crossspectral matrix (CSM) (\(\text {Pa}^{2}\))
 \({\tilde{d}}\)
\(\root \of {{\tilde{D}}}\) (\(\text {Pa}^{\frac{1}{2}}\))
 D
Diameter of the microphone array (m)
 \({\tilde{D}}\)
Individual source amplitude in SODIX method (Pa)
 f
Sound frequency (Hz)
 F
Cost function in optimization problems
 \(\varvec{g}\)
\(N \times 1\) steering vector
 \(\varvec{G}\)
\(N \times K\) (unknown) matrix containing the steering vectors \(\varvec{g}\) to the unknown K source locations (\(\text {m}^{1}\))
 \(\tilde{\varvec{G}}\)
\(N \times J\) (known) matrix containing the steering vectors \(\varvec{g}\) to all J grid points (\(\text {m}^{1}\))
 \(\varvec{h}\)
Source component vector for CLEANSC [\(\text {m}^{1}\))
 \(i^{2}\)
Imaginary unit, \(1\)
 \(\varvec{I}\)
\(N \times N\) identity matrix
 J
Number of grid points
 k
Number of eigenvalues considered for Orthogonal Beamforming
 \(k_0\)
Acoustic wavenumber (\(\text {rad}\,\text {m}^{1}\))
 \(k_x, k_y\)
Wavenumbers in the x and y direction, respectively (\(\text {m}^{1}\))
 \(\varvec{k}\)
Wavevector \((k_x, k_y)\) (\(\text {m}^{1}\))
 K
Number of (incoherent) sound sources
 \(L_{p}\)
Sound pressure level (SPL) (dB)
 M
Mach number
 N
Number of microphones
 \(N_{\mathrm{speakers}}\)
Number of speakers for acoustic GPS
 p
Fourier transform of the recorded pressure at each microphone (Pa)
 \(\varvec{p}\)
\(N \times 1\) vector containing the Fourier transform recorded pressures at each microphone (Pa)
 \({\tilde{q}}\)
Sparsity parameter for IBIA
 r
Distance between the sound source and the observer (m)
 s
Sound source amplitude (Pa)
 \(\varvec{S}\)
Sound sourceamplitude vector (Pa)
 St
Strouhal number
 t
Time (s)
 \(\varvec{u}\)
Eigenvector of the crossspectral matrix
 \(\tilde{\varvec{u}}\)
Left singular vector of \(\varvec{G}\)
 \(\varvec{U}\)
Unitary matrix whose columns are eigenvectors of the crossspectral matrix
 \(\mathbf{v }\)
Parameter vector for the global optimization method
 \(\varvec{w}\)
Weighted steering vector
 \(\varvec{W}\)
\(J \times J\) diagonal matrix whose components are \(\left s_{j} \right\) for single microphone (Pa)
 \(\varvec{x}\)
Microphone position vector (x, y, z) (m)
 \({\tilde{x}}\)
Source autopower (\(\text {Pa}^{2}\)/Hz)
 \(\tilde{\varvec{x}}\)
\(J \times 1\) vector containing the unknown source autopowers (\(\text {Pa}^{2}\)/Hz)
 \(\varvec{y}\)
\(J \times 1\) vector containing the source autopowers obtained with CFDBF (\(\text {Pa}^{2}\)/Hz)
Greek symbols
 \(\delta\)
Noise parameter for compressivesensing beamforming (Pa)
 \(\varDelta t\)
Time delay (s)
 \(\varDelta x\)
Widthwise spacing of grid points (m)
 \(\epsilon\)
Artificial diagonal loading factor for RAB and GIBF
 \(\eta ^{2}\)
Regularization parameter used for IBIA
 \(\theta\)
Source emission angle (deg)
 \(\lambda\)
Acoustic wavelength (m)
 \(\mu _{0}\)
Artificial diagonal loading parameter for RAB
 \(\nu\)
Functional beamforming exponent
 \(\varvec{\xi }\)
Grid point position vector (m)
 \(\sigma\)
Eigenvalue of the crossspectral matrix (Pa)
 \({\tilde{\sigma }}\)
Singular value of \(\varvec{G}\)
 \(\varvec{\varSigma }\)
Diagonal matrix whose diagonal elements are eigenvalues of the crossspectral matrix
 \(\varvec{\varPsi }\)
\(J \times N\) regularized inverse of \(\varvec{G}\) (m)
Subscripts
 j
jth grid point
 k
kth sound source
 l
lth sound source
 m
mth microphone
 n
nth microphone
 \(\hbox {ref}\)
Reference microphone
 S
Referring to the signal subspace
Superscripts
 \(*\)
Complex conjugate transpose
 \(\dagger \eta\)
Pseudoinverse using Tikhonov regularization with regularization parameter \(\eta\)
 i
ith iteration
1 Introduction
Aircraft noise is an important social issue. To reduce the noise levels generated by flying aircraft, it is essential to accurately determine and analyze all the possible noise sources on board. Individual microphones only provide total noise levels, but do not give information about the locations and strengths of individual sound sources, such as engines, landing gears ,and highlift devices. The introduction of the phased microphone array solved this issue. A brief historical background and the main applications of phased microphone arrays for aeroacoustic measurements are summarized in the following subsections.
1.1 Historical background
The use of phased arrays dates back to World War II as radar antennas later developed for applications such as the sonar, radioastronomy, seismology, mobile communication, or ultrasound medical imaging [1]. The theory of electromagnetic antenna arrays was already applied in the field of acoustics for determining the direction of arrival of sound sources by Davids et al. [2]. The phased microphone array (also known as microphone antenna, acoustic telescope, and acoustic array or acoustic camera) was introduced by Billingsley [3, 4]. Using several synchronized microphones and a source localization algorithm [5, 6], the possibility to estimate the location and strength of sound sources was enabled. Since then, significant improvements have been made, for a large part by more powerful acquisition and computing systems [1], allowing higher sampling frequencies, longer acquisition times, larger numbers of microphones, and even realtime sound source localization. In the remaining of this paper, the term source localization is only related to the use of microphone arrays.
Acoustic imaging algorithms [6] are the essential link between the sound field measured at a number of microphone positions and the assessment of useful characteristics of noise sources, such as their locations and absolute levels. The main idea is to combine the data gathered by the microphone array with a sound propagation model to infer on the source parameters [7]. The conventional beamforming [5, 6] (see Sect. 3.1) is, perhaps, the most basic postprocessing approach for the signals recorded by the microphones, but it normally fails to provide the satisfactory results for practical applications. The localization and quantification of sound sources are limited by the geometry of the array. Most acoustic imaging methods are exhaustive search techniques where a selected grid containing the location of potential sound sources is scanned.

Deconvolution techniques such as DAMAS [8, 9, 10] (developed in 2004, see Sect. 3.5) or CLEANSC [11] (developed in 2007, see Sect. 3.4) can be seen as postprocessing methods of the results obtained using the conventional beamforming, assuming hypotheses such as source coherence or positive source powers.

Several inverse methods have been proposed, which, in contrast to beamforming algorithms, aim at solving an inverse problem accounting for the presence of all sound sources at once. This way, interferences between potentially coherent sources can be taken into account [7]. This inverse problem is typically underdetermined and inverse methods are often sensitive to measurement noise. Hence, regularization procedures are required. Different regularization techniques are available, depending on the source sparsity constraint set by the user [7, 12]. Acoustic imaging methods such as generalized inverse beamforming [13] (developed in 2011, see Sect. 3.12) or the Iterative Bayesian inverse approach [14, 15] (developed in 2012, see Sect. 3.13) are included in this group.
In general, measurements with phased microphone arrays provide certain advantages with respect to measurements with individual microphones when performing acoustic measurements. For experiments in wind tunnels with open and closedtest sections, as well as in engine test cells, the background noise suppression capability of the source localization algorithms is very useful [1, 16, 17, 18], as well as the removal of reflections from the walls [19, 20]. Beamforming can also be applied to moving objects, such as flying aircraft or rotating blades, provided that the motion of the source is tracked accurately [1]. Moreover, microphone arrays are useful tools for studying the variability of the noise levels generated by different aircraft components, within the same aircraft type [21, 22, 23, 24] to improve the noise prediction models in the vicinity of airports. Nowadays, the microphone array has become the standard tool for analyzing noise sources on flying aircraft [25, 26, 27, 28, 29, 30], trains [31, 32, 33], cars [34, 35, 36], snowmobiles [37], and other machineries, such as wind turbines [38, 39].
1.2 Windtunnel measurements
As in the field of aerodynamics, windtunnel measurements offer a controlled environment to perform acoustic measurements on scaled models of aircraft or aircraft components. It is, however, difficult to replicate the exact conditions present at an aircraft in flight. As shown by Stoker et al. [40], differences occur when results from a standard windtunnel measurement with a closedtest section are compared to the results obtained from flight tests. The differences can be explained by lack of model fidelity, installation effects, a discrepancy in the Reynolds number (see Sect. 1.2.4), and the applicability of the assumptions made in phasedarray processing. Depending on the size of the model, scale effects need to be taken into account for the soundgeneration mechanisms [41]. Windtunnel acoustic measurements feature convection of sound waves, which can be corrected for [41, 42]. A major issue is the high background noise level, but mitigation techniques are available [43, 44, 45, 46].
Windtunnel measurements can be performed in open jets or in closedtest sections, each of these options having different challenges:
1.2.1 Closedtest sections
Closedtest sections offer wellcontrolled aerodynamic properties. Acoustic measurements can be performed nonintrusively by mounting microphones flush in the floor, ceiling, or walls of the wind tunnel. However, the amplitudes of the nearfield pressure fluctuations inside the turbulentboundary layer (TBL) are generally much larger than the acoustic signal from the model. Suppression of these nearfield pressure fluctuations can be realized by mounting the microphones in cavities covered by a perforated plate or wire mesh at some distance from the TBL [47, 48, 49]. This solution takes advantage of the fact that TBL pressure fluctuations feature short wavelengths, which decay exponentially with distance. Microphones recessed in a cavity are offered commercially too [50]. A more radical solution for the TBL issues is to replace the windtunnel walls by Kevlar sheets, as in the stability tunnel of the Virginia Polytechnic Institute and State University [51]. In addition, acoustic measurements are hampered by reflections by the test section walls [19, 20]. In general, acoustic measurements in closedtest sections are dominated by high noise levels, either due to the TBL pressure fluctuations or due to the noise from the windtunnel circuit. The noise influence can be reduced substantially by subtracting the noise influence on the crossspectral matrix (CSM) before the finalsource location analysis [45, 52] (see Sect. 3.1).
1.2.2 Open jets
The test chamber surrounding the jet is usually acoustically treated, so that most reflections are suppressed. Moreover, the background noise levels are lower than in a closedtest section and the microphones can be placed outside the flow, not being subject to turbulence. However, the aerodynamic conditions are less well controlled and corrections are needed to account for refraction through the shear layer, which produces some disturbances (distortion in phase) that need to be taken into account [42, 53, 54, 55]. Furthermore, the turbulence in the shear layer causes spectral broadening [56] and decorrelation [57], see Sect. 2.4.
1.2.3 Comparability of windtunnel measurements
Therefore, openjet wind tunnels are recommended for measuring models emitting lowfrequency noise with a lowsource strength, whereas closedsection wind tunnels are preferred for measuring highfrequency noise sources [60]. For middle frequencies, both test sections provide comparable acoustic performance. In practice, farfield noise measurements can almost only be performed in openjet wind tunnels, since it is typically possible to place the microphone array further away from the source than in closedsection wind tunnels.
1.2.4 Reynolds number dependence
In standard windtunnel measurements, a sole discrepancy in Reynolds number at otherwise similar conditions can lead to a difference in results. The effect of a varying Reynolds number on the noise generated was investigated by Ahlefeldt [62]. Here, acoustic measurements were performed on a smallscale aircraft model in highlift configuration at both a realflight Reynolds number and a lower Reynolds number corresponding to the standard windtunnel conditions. Measurements were performed in the European Transonic Wind tunnel (ETW) which, due to its pressurized and cryogenic environment, enabled a variation of Reynolds number up to realflight Reynolds numbers. Other parameters were kept unchanged. Thus, Reynolds number effects on aeroacoustic behavior were separated from the effects of model fidelity and Mach number M.
Several sources with significant Reynolds number dependence were found and exemplary differences at selected Strouhal numbers are shown in Fig. 2. Several dominant sources can be found at the realflight Reynolds number, but are not present at standard conditions. Contrary to that, sources are present in the standard measurement but not at the realflight Reynolds number, as can be seen for example in the slat region. Locally integrated sources from the slat and the flap are shown in Fig. 3. The strong tonal components in the spectrum in the slat region measured at the standard windtunnel conditions (lower Reynolds number) disappeared at realflight Reynolds numbers.
1.3 Aircraft flyover measurements
Measurements on flying aircraft provide the most reliable results of engine and airframe noise emissions of a certain aircraft type [1], especially if the measurements are taken under operational conditions [64]. However, lesscontrolled experimental conditions, like propagation effects [65], moving sources [25, 66], and localization of noise emitter, need to be considered. Therefore, the microphone signals have to be deDopplerized by resampling the original time series by linear interpolation [25]. Interpolation errors were shown to be small if the maximum frequency of analysis is restricted to one tenth of the sampling frequency and for flight Mach numbers up to 0.81 and flyover altitudes as low as 30 m [26]. Upsampling can be performed numerically before interpolation to alleviate this requirement [67]. Additional considerations need to be taken when applying deconvolution algorithms to moving sources [68].
1.4 Engine noise tests
Sound source localization techniques can also be applied to open air engine test beds and indoor engine test cells [1, 69, 70, 71, 72] and to ducted rotating machinery [73, 74, 75].
1.5 Outline of the manuscript
A brief explanation of the hardware requirements and considerations can be found in Sect. 2, as well as some guidelines for distributing microphones in an array. A selection of several widely used acoustic imaging methods are presented in Sect. 3, including some highresolution techniques and inversion and deconvolution methods. A list discussing the performance of each method considered for the most common aeroacoustic applications (flyover and windtunnel measurements) is also included. Some results selected from the previous publications are shown in Sect. 4. Finally, Sect. 5 contains the conclusions.
2 Experimental and hardware considerations

Limited spatial resolution, especially at low frequencies, i.e., the capability to separate two different sound sources placed at a small distance from each other. This is related to the beamwidth of the main lobe in the source map.

The presence of sidelobes or “spurious sources”, due to the array response function, which can be misidentified as real sources. This phenomenon, as well as the spatial resolution, is characterized by the array point spread function (PSF), which is the array response (beam pattern) to a unitarystrength point source.

Background noise suppression. This is especially interesting for noisy environments, such as wind tunnels.

Reliability of both the estimated location and amplitude of the sound sources.
2.1 Hardware requirements
Processing multiple microphone signals increases the signaltonoise ratio (SNR) and the spatial resolution compared to a measurement with only one microphone [41]. The choice of the microphones highly depends on the particular experiment to be performed [76]. Characteristics such as the dynamic range, the frequency range, and the sensitivity have to be selected with care. In general, smaller microphones can measure up to higher frequencies and larger microphones have higher sensitivities. The directivity of the microphones has to be taken into account, as well, especially for higher frequencies. Most of these specifications are provided by the manufacturer.
All microphone signals have to be simultaneously sampled by the data acquisition system. The sampling frequency should be at least twice the maximum frequency of interest, according to the sampling theorem. As mentioned in Sect. 1.3, it is recommended to have a sampling frequency ten times the maximum frequency of interest in the case of flyover measurements or to perform upsampling [67].
Monitoring the data acquisition during the experiment is recommended, to check that the frequency spectra obtained are valid.
Moreover, for microphone arrays installed in small plates, the sound waves scatter at the edges of the plate. Reflections from walls in wind tunnels and from the ground in flyover measurements should also be taken into account. These phenomena produce phase and amplitude errors on the measured signal [77, 78]. Thus, it is recommended to either use hard plates (for complete reflection) or on an acoustically transparent structure, using sound absorbing materials, for example. The setup choice depends on the experiment to be performed and, of course, on the budget available.
2.2 Microphonearray calibration
2.2.1 Amplitude and phase calibration of individual microphones
All the microphones in a phased array should be individually calibrated in both amplitude and phase. Normally, the microphone manufacturer provides some initial calibration data sheets per frequency. Additional calibrations can be performed employing a calibration pistonphone which generates a sinusoidal signal of known amplitude at a certain frequency, typically 250 Hz or 1 kHz.
2.2.2 Metrological determination of the microphone positions
A precise calibration of the microphone positions is crucial for accurate aeroacoustic measurements [41]. Small sound sources with omnidirectional radiation patterns and with known sound pressure levels are recommended for the calibration of the microphone array in both source location and quantification. Broadband white noise signals are preferred, instead of tonal sound at single frequencies to avoid coherence problems [79].
2.3 Microphone distribution guidelines
A detailed study of the optimization of array microphone distribution is out of the scope of this manuscript, but arrays consisting of spirals or of several circles with an odd number of regularly spaced microphones seem to perform best [1, 41]. Interesting studies using the optimization methods [84] and thorough parametric approaches leading to parettooptimal arrangements [85] can be found in the literature.
The microphone positions in the first large test campaign of aircraft flyovers [86] were optimized with a genetic algorithm. To study moving sources, such as aircraft flyovers, the array shape can be elongated in the flight direction [87] to compensate for the loss of resolution due to emission angles other than 90\(^\circ\). The Boeing Company also used an elliptical array shape for their flyover tests in the QTD2 program [88] consisting of 614 microphones in five subarrays with an overall size of approximately 91 m by 76 m, likely the largest array ever used in flyover tests.
2.4 Microphone weighting and coherence loss
Different weighting or “shading” functions can be applied to the signals of each microphone to obtain better acoustic imaging results [61, 89, 90]. Moreover, the beamwidth can be kept roughly constant by selecting smaller subarrays for higher frequencies [35] to minimize coherence loss. This requires clustering the microphones in the center of the array. Shading can be applied per onethirdoctave band to reduce the coherence loss, to compensate for the nonuniform microphone density, and to reduce the sidelobe levels.
3 Acoustic imaging methods
A vast list of acoustic imaging algorithms exists in the literature [7]. Some of them are based on the deconvolution of the sound sources, i.e., the removal of the effect of the PSF [41] of the sound sources, such as CLEANSC, DAMAS, etc. These methods aim at enhancing the results of the conventional beamforming [6], but their computational time is considerably larger. Most of the listed methods require a scan grid, where all the grid points are considered as potential sound sources. This section aims to summarize widely used acoustic imaging methods for aeroacoustic experiments.
3.1 Conventional beamforming (CFDBF)
The conventional beamforming [5, 6] is a very popular method, since it is robust, fast, and intuitive. However, it suffers from the Sparrow resolution limit^{1} [96] and presents a high sidelobe level, especially at high frequencies.
It can be applied both in the time domain [44, 97, 98, 99] or in the frequency domain [61]. The former is normally applied to moving sources [66] and the latter is more commonly used for stationary sources due to the lower computational time required.
The main diagonal of \(\varvec{C}\) can be removed to neglect the contribution of noise which is incoherent for all the array microphones. This can be especially useful for cases with wind noise or TBL noise, such as in closedsection wind tunnels [41, 61]. However, precaution has to be taken when removing the main diagonal of \(\varvec{C}\), because, then, \(\varvec{C}\) is no longer positivedefinite and the PSF can get negative values (which are not physical) and the negative sidelobes of strong sources can eliminate weaker sources in some cases. This fact is explained by the appearance of negative eigenvalues in \(\varvec{C}\) when the main diagonal is removed, since the sum of the eigenvalues of a matrix is always equal to the sum of its diagonal elements (zero, in this case). Hence, all the methods based on the CFDBF algorithm will suffer from this issue with the diagonal removal process.
When directly applied to distributed sound sources, CFDBF (and other methods assuming the presence of point sources) can lead to erroneous source levels. Therefore, integration methods, such as the source power integration (SPI) [29, 61, 94] technique, have been proposed to deal with this issue and limit the influence of the array PSF. This method can be extended to consider line sources [101] when distributed sound sources, such as trailingedge noise [102, 103, 104, 105, 106, 107], are expected. This integration technique has showed the most accurate results in a simulated benchmark case [101, 108] representing a trailingedge noise measurement in a closedsection wind tunnel, with respect to the other wellknown acoustic imaging methods.
3.2 Functional beamforming
For single sound sources, the PSF, which has a value of one at the correct source locations and alias points and a value less than one elsewhere, is powered to the exponent \(\nu\). Therefore, powering the PSF at a sidelobe will lower its level, leaving the true source value virtually identical [109] if an adequate grid is used [30]. For ideal conditions, the dynamic range of functional beamforming should increase linearly with the exponent value, \(\nu\). Thus, for an appropriate exponent value, the dynamic range is significantly increased.
The computational time for the functional beamforming is basically identical to the CFDBF one, since the only relevant operation added is the eigenvalue decomposition of \(\varvec{C}\).
The application of the diagonal removal method aforementioned to functional beamforming is even more prone to errors, since this algorithm is based on the eigenvalue decomposition of the CSM. Mitigation of the diagonal removal issue is possible with CSM diagonal reconstruction methods [112, 113].
In the previous work, functional beamforming has been applied to numerical simulations [30, 109, 110], controlled experiments with components in a laboratory [109, 110], and to fullscale aircraft flyover measurements under operational conditions [30, 114, 115].
A similar integration method as the SPI technique was used for quantifying noise sources in flyover measurements [107]. A somewhat similar timedomain technique based on the generalized mean of the generalized cross correlation has been developed recently [116, 117].
3.3 Orthogonal beamforming
Let the \(N \times K\) matrix \(\varvec{G}\) contain the transfer functions (i.e., the steering vectors) between the K sources and N microphones \(\left[ \varvec{g}_{1} \ \ldots \ \varvec{g}_{K} \right]\) (see Eq. 3). As shown in [120], the matrix \(\varvec{\varSigma }_S\) is mathematically similar to \((\varvec{G}^*\varvec{G}) \varvec{C}_{S}\) and, therefore, has the same eigenvalues. Here, \(\varvec{C}_{S}\) is the CSM of the source signals. The main idea behind orthogonal beamforming is that each eigenvalue of \(\varvec{\sigma }_S\) can be used to estimate the absolute source level of one source, from the strongest sound source within the map to the weakest, assuming orthogonality between steering vectors.
In a second step, these sources are mapped to specific locations. This is done by assigning the eigenvalues to the location of the highest peak in a special beamforming sound map, which is purposely constructed from a rankone CSM that is synthesized only from the corresponding eigenvector. Hence, the map is the output of the spatial beamforming filter for only one single source and the highest peak in this map is an estimate of the source location. The main diagonal of the reduced CSM for each eigenvalue may be removed to reduce uncorrelated noise. The beamforming map can be constructed on the basis of vector–vector products and is, therefore, computationally very fast.
An important parameter in the eigenvalue decomposition, which has to be adjusted by the user, is the number of eigenvalues k that span the signal subspace \(\varvec{U}_S\). In a practical measurement, a reliable approach is to estimate the number of sources K and choose a value \(k> K\). If the last eigenvalues represent sources that only marginally contribute to the overall sound level, the most important sound sources will be correctly estimated using a value of k considerably smaller than N. The influence of the choice of k is illustrated in Fig. 16 for a trailingedge noise experiment in an openjet wind tunnel.
Since the eigenvalue decomposition of the CSM results in a reduced number of point sources in the map, which is always less than or equal to the number of microphones, the sum of all source strengths within the map is never greater than the sum of the microphone autospectral densities. Hence, the sum of the acoustic source strengths is never overestimated.
3.4 CLEANSC

The CSM is calculated from a large number of time blocks, so that the ensemble averages of the crossproducts \(\varvec{p}_{k}\varvec{p}^{*}_{l}, k\ne l\), can be neglected.

There is no decorrelation of signals from the same source between different microphones (e.g., due to sound propagation through turbulence).

All sound sources present are incoherent.

There is no additional incoherent noise.
This method works well for the case of a welllocated sound source and it is especially suitable for closedsection windtunnel measurements.
Another algorithm called TIDY [123] is similar to CLEANSC but works in the time domain, using the crosscorrelation matrix instead of the CSM. TIDY has been used for imaging jet noise [70, 123] and motor vehicle passby tests [36].
Recently, a higher resolution version of CLEANSC (HRCLEANSC) has been developed and applied successfully to simulated data [124] and to experimental data using two speakers in an anechoic chamber [125, 126].
3.5 DAMAS
DAMAS considers incoherent sound source distributions and attempts to determine each source power by solving the inverse problem of Eq. (15), subject to the constraint that source powers are nonnegative. The problem is commonly solved using a Gauss–Seidel iterative method, which typically requires thousands of iterations to provide a “clean” source map. Moreover, for practical grids, the large size of \(\tilde{\varvec{A}}\) can become an issue. The computation time of DAMAS employed this way is proportional to the third power of the number of grid points, \(J^{3}\). In most applications, \(\tilde{\varvec{A}}\) is singular and not diagonal dominant [127] and the convergence towards the exact solution is not guaranteed. DAMAS has no mechanism to let the iteration converge towards a welldefined result and the solution may depend on how the grid points and the initial values are ordered [127].
The inverse problem in Eq. (15) can also be evaluated using efficient nonnegative leastsquares (NNLS) solvers [127, 128, 129]. In case sparsity of the vector \(\tilde{\varvec{x}}\) is considered, the inverse problem can also be solved with greedy algorithms such as the Orthogonal Matching Pursuit (OMP), which approximates the solution in a considerably lower computation time, proportional to \(J^{2}\) instead of \(J^{3}\). The different steps of OMP are summarized in [130]. The Least Angle Regression Lasso algorithm (LarsLasso) also benefits of the sparsity of \(\tilde{\varvec{x}}\), but requires the choice of a regularization factor by the user, which can be quite complicated for nonexperienced users [129]. Herold et al. [129] compared the NNLS, OMP, and LarsLasso approaches on an aeroacoustic experiment, and found that only NNLS and LarsLasso (using an appropriate regularization factor) surpass the classic DAMAS algorithm in terms of overall performance.
DAMAS was later extended to allow for source coherence (DAMASC) [131]. Whereas the computational challenges in the use of DAMASC have limited its widespread application, the conventional DAMAS has shown its potential with coherent source distributions in jetnoise analyses [132]. This jetnoise study also demonstrated the use of in situ pointsource measurements for the calibration of array results. A similar method to DAMASC called noise source localization and optimization of phasedarray results (LORE) was proposed by Ravetta et al. [133, 134]. LORE first solves an equivalent linear problem as DAMAS using an NNLS solver, but considering the complex point spread function [133, 134], which contains information about the relative source phase. The output obtained is optimized solving a nonlinear problem. Satisfactory results were obtained in the simulated and experimental cases in a laboratory featuring incoherent and coherent sound sources [133, 134]. Whereas this method is faster than DAMASC, it does not provide accurate results when using diagonal removal and when calibration errors are present in the microphone array [133, 134].
3.5.1 DAMAS2
Further versions of DAMAS have been proposed in the literature [135, 136, 137], especially for reducing the high computational cost that it implies. For example, DAMAS2 assumes that the array’s PSF is shift invariant. The term shift invariant describes the property that the characteristics of the PSF do not vary relative to the source position even if the absolute position of that source in the steering grid is changed. Thus, if a source is translated by a certain offset, the entire PSF will also translate with it. The error involved in this assumption is small in astronomy applications [122] where the distance between the source and the observer is huge compared to the size of the array or of the source itself, but, in aeroacoustic measurements, the PSF can vary significantly within the source region [127]. The distortion of the PSF away from the center of the scanning domain can be alleviated by including spatialdifferentiation terms [138]. When applied, the size of the PSF has to be chosen large enough to prohibit wrapping, which has been implemented in DAMAS2.1 [139]. In addition, a fast implementation of DAMASC based on similar techniques to reduce computational costs as DAMAS2 has been introduced [140], exploiting the benefits of convolution using Fourier transforms. Embedded versions of DAMAS2 and a Fourierbased nonnegative leastsquares (NNLS) approach of DAMAS were proposed by Ehrenfried and Koop [127] which do account for the shift variation of the PSF and are potential faster alternatives compared to the original DAMAS algorithm.
3.6 Wavenumber beamforming
Exemplary plots of the wavenumber domain for \(f=1480\,\text {Hz}\) are shown in Fig. 7. Due to the relation \({\tilde{c}} = 2\pi f/\left\ \varvec{k}\right\\) with \({\tilde{c}}\) being the propagation velocity, and \(\varvec{k}=(k_x, k_y)\) the wavevector of a source, each position in the map represents a different propagation velocity. Sources with a propagation speed equal to or higher than the speed of sound are located in the ellipticshaped acoustic domain shown in both plots in Fig. 7 with a solid black line. In the closedsection windtunnel test of Fig. 7 (left), acoustic sources are present, while the flight test data of Fig. 7 (right) appear to be free of dominant acoustic content at the frequency shown. The elongated spot on the righthand side outside the acoustic domain is a representation of the pressure fluctuations caused by the subsonic TBL flow. In the windtunnel data, this elongated spot is seen to be parallel to the \(k_y\)axis, indicating a flow component only in the xdirection. In the flight test plot, the elongated shape is rotated slightly about the origin, which indicates a flow direction that is not aligned with the array’s x and yaxes. The wavenumber domain can be used to easily separate between different propagation mechanisms.
3.7 Linear programming deconvolution (LPD)
Linear programming deconvolution (LPD) [145] is basically a faster alternative than DAMAS to solve the inverse problem introduced in Eq. (15). It considers an additional constraint that no correct model of the beamform map \(\tilde{\varvec{A}}\tilde{\varvec{x}}\) would exceed the beamform source map obtained by CFDBF \(\varvec{y}\) anywhere. This difference (\(\varvec{y}\tilde{\varvec{A}}\tilde{\varvec{x}}\)) represents the effect of uncorrelated sound sources that were present in the measurement but not in the model, such as background noise, microphone self noise, and longrange reflections [145].
However, a disadvantage of LPD is that it does not work well with diagonal removal [145]. An alternative approach is to add an extra element to \(\tilde{\varvec{x}}\) which represents the incoherent noise level. The matrix \(\tilde{\varvec{A}}\) would now be \(J \times (J+1)\) and the extra column is filled with ones.
This method has been applied to a distribution of aeroacoustic point sources in a laboratory [145] and was shown to provide a better resolution than the Sparrow resolution limit.
A combination of LPD and functional beamforming has been reported by Dougherty [110] showing better results due to the higher dynamic range offered by functional beamforming.
The application of LPD, however, breaks up continuous source distributions into spots. This method is thus appropriate for discrete sources and for situations where spatial resolution is more important than dynamic range.
3.8 Robust adaptive beamforming (RAB)
A different approach to calculate \(\epsilon\), based on the white noise gain constraint idea from Cox et al. [146], can be found in [150]. In this approach, however, a different value of \(\epsilon\) is calculated for each grid point, consequently, increasing the computational cost considerably.
An application of this method for aircraft flyover measurements [30] can be found in Figs. 20 and 21.
Capon beamforming can be extended to treat potentially coherent sources [151].
3.9 Spectral estimation method (SEM)
The spectral estimation method (SEM) [152] is intended for the location of distributed sound sources. It is based on the idea of describing sound sources by the mathematical models that depend on several unknown parameters. It is assumed that the power spectral density (PSD) of the sources can then be expressed in terms of these parameters. This method is also known in the literature as covariance matrix fitting (CMF) [153, 154, 155].
The choice of the source model is based on the fact that an extended sound source may only be viewed as an equivalence class between source functions radiating the same pressure field on a phased microphone array. In many applications, a majority of sound sources have smooth directivity patterns. This means that if the aperture angle of a microphone array seen from the overall source region is not too large, the directivity pattern of each region may be considered as isotropic within this aperture, neglecting its directivity.
SEM has shown its efficiency in noiseless environments, on numerical simulations [152] and on data measured during experiments performed with an aircraft model in the openjet anechoic wind tunnel CEPRA 19 (see Fig. 13) [16, 17, 152].
One big advantage of SEM over deconvolution methods using beamforming is that the main diagonal of the CSM can be excluded from the optimization without violating any assumptions. SEM can even be used to reconstruct the diagonal without the influence of the spurious contributions. The resulting source distribution is relatively independent on the array pattern and the assumed source positions can be restricted to the known regions on an aircraft.
To take into account the inevitable background noise in practical applications, an extension of SEM has been proposed: the spectral estimation method with additive noise (SEMWAN) [18]. This method is based on a prior knowledge of the noise signal and it has the advantage of being able to reduce the smearing effect due to the array response and, at the same time, the inaccuracy of the results caused by noise sources, which can be coherent as well as incoherent, with high or low levels. This technique is well suited for applications in wind tunnels, since, for example, a noise reference or record of the environmental noise can be obtained prior to the installation of the model in the test section [18, 45]. Figures 10 and 11 present the results of SEM and SEMWAN applied in a closedsection windtunnel experiment.
Another important issue arises when the acoustic measurements are performed in a nonanechoic closedsection wind tunnel. In this situation, the pressures collected by the microphones are not only due to the direct paths of the acoustic sources, but are also due to their unwanted reflections on the unlined walls, thereby losing accuracy when calculating the power spectra. To remove this drawback, a multimicrophone cepstrum method, aiming at removing spurious echoes in the power spectra, has been developed and tested successfully with the numerical and experimental data [46].
3.10 SODIX
SODIX (Source Directivity Modeling in the CrossSpectral Matrix) is an extension of SEM that can model sound sources with arbitrary directivities. The method was initially developed for noise tests with engines in open static test beds to separate the various broadband sound sources of turbofan engines, which are known to have sizable directivities. However, the method can be applied to any problem where the directivities of the sound sources are of interest.
To reduce the number of possible solutions and to support physical results, smoothing terms can be added to the cost function [69, 71, 156]. The smoothing terms prevent spurious peaks in the source amplitudes and can, therefore, also improve the dynamic range of the results.
 1.
The amplitudes \({\tilde{d}}^2_{j,m}\) of each point source may have different values for every microphone.
 2.
The phase of a sound wave radiated by a point source spreads spherically, i.e., according to the complex argument in the steering vector \(g_{j,m}\).
The analysis of broadband noise during static engine tests with large linear microphone arrays demonstrated that SODIX can calculate reliable results over a wide range of angles around the engine [71, 72]. Phase jumps between different radiation lobes become a problem mainly for tonal noise. They violate the second assumption in the source model and, therefore, lead to source distributions and directivities that are not representative. However, the outstanding capability of SODIX is the free modeling of the directivities of the sound sources: SODIX can not only account for the multipole characteristics of the sound sources, but also for directivities due to source interference within coherent sources, such as jet mixing noise.
As in the case of SEM (see Sect. 3.9), the main diagonal of the CSM can be removed from the calculations without violating any assumptions. This makes SODIX, like SEMWAN (see Sect. 3.9), suitable for closedsection windtunnel applications.
The spatial resolution of SODIX was compared to that of CFDBF and various deconvolution methods in [78]. The results showed that SODIX overcomes the Sparrow resolution limit and, therefore, provides superresolution. However, the computational effort for SODIX is rather high, because the number of unknowns to be determined, i.e., \(J \times N\), can be very large.
SODIX was also successfully applied to engine indoor tests [72], ground tests with an Airbus A320 aircraft [157, 158, 159], measurements of a counter rotating open rotor (CROR) model in an openjet wind tunnel [160, 161], and measurements of modelscale turbofannozzles in an openjet wind tunnel [162].
3.11 Compressivesensing beamforming
This method has been adopted in the identification of spinning modes for turbofan noise using microphonearray measurements. The required number of sensors can be much less than the number required by the sampling theorem as long as the incident fan noise is sparse in spinning modes [167, 168, 169].
It should be noted that compressive sensing, unlike the beamforming methods discussed, is an inverse method that aims to determine the phase and amplitude of the source distribution. The linear algebra problems at the core of inverse methods are often severely underdetermined. Selecting a particular solution to display requires that the undetermined components of the solution be established somehow. The conventional approach of minimizing the \(L_2\) norm of the solution can give solutions that look incorrect, because they have many small nonzero elements. Compressive sensing can be considered as a cosmetic improvement that tends to cluster the nonzero sources. If there is reason to believe that the true sources have this kind of distribution, then the compressivesensing result may be more accurate than the conventional inverse solution. Several of the techniques discussed below are also inverse methods that handle the illconditioned problem in slightly different ways.
3.12 Generalized inverse beamforming (GIBF)
Benefits of this algorithm are not only the treatment of the source coherence, but also to include any types of prescribed sources in \(\tilde{\varvec{G}}\). Hence, in principle, multipoles in an arbitrary orientation can be detected and different types of sources can be collocated at the same grid point. An application to duct acoustics, in which overdetermined problems are typically solved, was also studied in [173]. Several regularization methods for GIBF have been recently applied to airfoilnoise measurements in openjet wind tunnels [12, 174].
3.13 Iterative Bayesian inverse approach (IBIA)
The parameter \({\tilde{q}}\), defining the shape of the a priori probability density function of sources, determines the power of the solution norm used in the regularization process. A value of \({\tilde{q}}=2\) keeps the initial value \(\varvec{\varPsi }^0\) (Eq. (28)), which means no sparsity. Thus, the amount of sparsity increases as \({\tilde{q}}\) decreases, down to 0 for which a strong sparsity is requested.
This method has been tested with the experimental data from a halfaircraft model in a closedsection wind tunnel, as shown in Fig. 12.
3.14 Global optimization methods
In Ref. [175] a method is presented where the search for the locations and amplitudes of sound sources is treated as a global optimization problem. The search can be easily extended to more unknowns, such as additional geometrical parameters, and more complex situations with, for example, multiple sound sources or reflections being present. The method is essentially gridfree and can overcome the Sparrow resolution limit when sources are positioned close together.
The presence of sidelobes will, however, hamper the optimization as they act as local optima against which the global optimum needs to be found. In the literature, a number of mathematical methods are presented which allow for optimization problems with many unknowns and with the capability to escape from local optima, in contrast to local search techniques, e.g., gradient methods. These methods are generally denoted as global optimization methods. Wellknown examples are genetic algorithms [176], simulated annealing [177], and ant colony optimization [178].
In [175], a variation of the genetic algorithm, called differential evolution [179], is proposed as a global optimization method. This type of optimization method mimics natural evolution. They use populations of solutions, where promising solutions are given a high probability to reproduce and worse solutions have a lower probability to reproduce.
This work can be seen as an alternative approach to DAMAS and SEM. Whereas DAMAS assesses the performance of a solution based on the agreement between modeled and measured beamformed outputs, SEM is based on the comparison between the modeled and measured pressure fields. In contrast, in this technique, the locations of the sources are sought using a global optimization method, instead of considering a predefined grid of potential source locations. This way, estimates for source positions and source strengths are obtained as a solution of the optimization and do not need to be obtained from a source plot.
Some first experimental results of this technique with the simulated data and a single speaker in an anechoic room are presented in reference [175].
3.15 Applications

Conventional beamforming (CFDBF): [5, 6] should be a standard procedure in all cases, since it provides a fast overview of the sound sources characteristics. However, its spatial resolution and dynamic range are usually not suitable for several applications. Integration methods can be applied for distributed sources [101].

Functional beamforming: [109, 110] greatly increases the dynamic range compared with CFDBF in a comparable computational time. It works well in windtunnel (both in openjet and closedsection) and in aircraft flyover measurements [30, 115, 180, 181]. However, diagonal removal produces considerable errors and diagonal denoising methods are recommended [112].

Orthogonal beamforming: [118, 119, 120, 121] is based on the eigenvalue decomposition of the CSM, has a low computational cost, and never overestimates the strength of the acoustic sources. This method has been applied in trailingedge noise measurements in an openjet windtunnel [155, 182, 183, 184].

CLEANSC: [11] is a widely used deconvolution technique that cleans the source map obtained with CFDBF iteratively, removing the parts of it that are coherent with the real sources. Thus, the dynamic range is greatly improved. The spatial resolution can be increased beyond the Rayleigh resolution limit using the new highresolution version HRCLEANSC [125]. This technique has been applied to windtunnel [11] and aircraft flyover experiments [30].

DAMAS: [8, 9, 10] is a deconvolution method that solves an inverse problem iteratively to remove the influence of the array geometry from the obtained results. Enhancements in dynamic range and spatial resolution are obtained, but it has a high computational cost. The extension DAMASC [131] is suitable for analyzing source coherence, but it requires even higher computational resources. Further and similar versions of DAMAS have been proposed [127, 133, 134, 135, 136, 137, 138, 139, 140] to reduce the computational time. This technique is normally used in jetnoise analyses [132] and in windtunnel tests [8, 9, 10].

Wavenumber beamforming: [52, 141, 142, 143, 144] is a useful technique when mechanisms resulting in different propagation speeds and directions are present, provided that farfield conditions apply. This method has been used in windtunnel experiments [52, 141, 142, 143] and to characterize an aircraft boundarylayer flow [144].

Linear programming deconvolution: [145] is a faster alternative to DAMAS. It has been applied to aeroacoustic point sources in a laboratory [145], showing a better resolution than the Sparrow resolution limit. It can be combined with the functional beamforming providing even better results [110].

Robust adaptive beamforming: [149] attempts to maximize the SNR. It works well for clean data, but it is quite sensitive to noise and errors in the data. It has been used for aeroacoustic sources and flyover measurements [30]. This method can be extended to treat potentially coherent sources [151].

Spectral estimation method (SEM): [152] is intended for the location of distributed sound sources. It offers a better dynamic range and spatial resolution than CFDBF. Removing the main diagonal of the CSM does not violate any assumption for this method. The effect of background noise and reflections can also be taken into account using SEMWAN [18] or cepstrum [46], respectively. This method has been applied in openjet and closedsection windtunnel experiments [16, 17, 152].

SODIX: [69, 71, 156] is an extension of SEM for experiments where the directivities of the sound sources are of interest. It provides a better resolution than the Sparrow resolution limit. SODIX has been mostly applied to static engine tests on freefield and indoor test beds, [69, 71, 72] and measurements in openjet wind tunnels [160, 161, 162].

Compressivesensing beamforming: [163, 164, 165] assumes spatially sparse distributions of sound sources and requires a lower number of microphones. This inverse technique has been used to identify spinning modes of turbofan engines [167, 168].

Generalized inverse beamforming: [13] considers partially coherent sound sources using inversion techniques. Multipoles in an arbitrary direction can be considered as well. An application to duct acoustics was studied in [173] and to airfoil noise in [12, 174].

Iterative Bayesian inverse approach (IBIA): [7, 14, 15] is based on a userdefined level of sparsity of the sound sources. The results are somewhat between those of CFDBF (no sparsity) and CLEANSC (strong sparsity). It has been applied to measurements in closedsection wind tunnels [185].

Global optimization methods: [175] can be used for searching the locations and amplitudes of sound sources without using a scan grid. Other parameters, such as the sound speed, can be added to the optimization problem to obtain more information about the sound field. Differential evolution was applied to a experiment with a speaker in an anechoic room [175].
Summary of the main characteristics of the acoustic imaging methods introduced
Method  Parameters to be set  Typical use  Other characteristics 

Conventional beamforming  None  General purpose  Integration techniques for extended sources Timedomain version 
Functional beamforming  Power parameter v  Aircraft flyover measurements  Sensitive to diagonal removal 
Orthogonal beamforming  Number of eigenvalues k  Limited use, see text  The sum of the acoustic source strengths is not overestimated 
CLEANSC  Damping parameter Number of iterations  Airframe noise measurements  Allows diagonal removal Superresolution version 
DAMAS  Number of iterations  Airframe noise measurements  Superresolution Allows diagonal removal Faster versions 
Wavenumber beamforming  None  Measurements featuring different wave propagation speeds  Farfield formulation 
Linear programming deconvolution  Same as DAMAS  Same as DAMAS  Superresolution Does not work with diagonal removal Can be combined with functional beamforming 
RAB  Diagonal loading parameter \(\mu _{0}\)  Limited use, see text  Very sensitive to the uncorrelation assumption 
SEM  Number of iterations or the maximum error in solutions  Airframe noise measurements  High resolution with positivity constrained Allows diagonal removal Distributed sound sources 
SODIX  Optional regularization function (smoothness constraint of directivity) with 2 parameters  Engine noise measurements (directional sources)  Superresolution Robust also for illposed problems Allows diagonal removal 
Compressivesensing beamforming  Regularization parameter \(\delta\)  Duct acoustics  Imposed sparsity degree (\(L_{1}\)norm) 
GIBF  Requires SNR to set up regularization  Windtunnel experiments and duct acoustics  Imposed sparsity degree (\(L_{1}\)norm) 
IBIA  Degree of sparsity \({{\tilde{q}}}\)  General purpose  Fully automatic regularization Possibility to tune the degree of sparsity (\(L_{q}\)norm) 
Global optimization methods  Number of unknowns to search Settings for the optimization algorithm  General purpose  This method does not require a predefined scan grid 
The beamforming methods that determine only the incoherent source strengths are conventional, functional, orthogonal, wavenumber, and robust adaptive beamforming, as well as CLEANSC, DAMAS, linear programming deconvolution, and SEM. SODIX adds source directivity to the results, and DAMASC adds source coherence as an output (CLEANSC considers it implicitly). The remaining methods attempt to find the amplitude, phase, and, in some cases, the partial coherence of sources using different regularization schemes. A summary of all these characteristics is presented in Table 1.
In general, more complex methods require considerably more computational time than CFDBF. Hence, depending on the experiment requirements, this can pose some constraints when selecting the most suitable method.
Other benchmark cases analyzing the performance of some of these methods for specific acoustic applications can be found in the literature [30, 154, 186, 187]. Moreover, a broad effort in the aeroacoustics community was started by NASA Langley researchers to establish a common set of benchmark problems for the purpose of testing and validation of new analysis techniques [188, 189]. This ongoing working group has met at several forums, shared the initial results for the chosen test cases, and has recently presented the results from the first release of benchmark problems [108, 185, 190, 191].
4 Results
This section aims to illustrate some representative acoustic imaging results for each of the main aeroacoustic applications introduced in Sect. 1, according to the acoustic imaging methods discussed in Sect. 3.15. The purpose of these results is to represent the typical examples that are normally found in practical aeroacoustic experiments, but not to provide an exhaustive list of all the possible applications of each single method.
4.1 Windtunnel measurements
4.1.1 Closed wind tunnels
Airfoil noise can also be studied at closedsection wind tunnels [107, 193].
4.1.2 Openjet wind tunnels
Additional studies about the noisegeneration mechanisms for trailingedge noise, flaps, slats, and landing gears involving microphone arrays were performed at the Quiet Flow Facility in NASA Langley [94, 195, 196, 197, 198, 199].
4.2 Aircraft flyover measurements
Figures 20 and 21 present the results of different acoustic imaging methods applied to two aircraft flyovers which belong to a measurement campaign in Amsterdam Airport Schiphol by Delft University of Technology, where 115 landing aircraft were recorded with a considerable smaller array [23, 24, 30, 114, 115], featuring 32 microphones and a diameter of 1.7 m (i.e., about seven times fewer microphones and seven times smaller than for the array used in Fig. 17). The average aircraft height overhead was approximately 67 m. These flyovers were selected, because they presented a strong tonal component at the presented frequencies: 1630 and 7140 Hz, respectively [30].
Figure 20 shows an Airbus A321 emitting sound at 1630 Hz, where the main noise source seems to be the noselanding gear, probably due to a cavity [115]. Similar results were observed by Michel and Qiao [86]. Functional beamforming (with \(\nu =100\)) and CLEANSC present the highest dynamic ranges.
Figure 21 shows a Fokker 70 emitting sound at 7140 Hz, where the main noise sources appear to be located at the main landing gear wheels. Once again, functional beamforming (with \(\nu =100\)) and CLEANSC present the highest dynamic ranges, but CLEANSC only shows one of the two sources (left). Functional beamforming does not seem to have this problem.
Figures 20 and 21 show that, even with a relatively cheap and small experimental setup, satisfactory results can be obtained. Additional measurements on aircraft flyovers can be found in the literature [26, 27, 29, 61, 86, 201, 203, 204]. Researchers from NASA Langley have developed fielddeployable microphone phased arrays for flight tests [205], including acoustic measurements of small Unmanned Aerial System (sUAS) vehicles [83]. Investigations of the use of array processing techniques to crossvalidate the computational and experimental results of airframe noise analysis have also been performed [206, 207].
4.3 Static engine noise tests
Exemplary SODIX results from static tests with a longcowl turbofan engine at low engine speed using a linear array with 248 microphones [72] are shown in Fig. 22. The microphones were aligned parallel to the engine axis on the testbed floor at a distance of approximately 11.2 m to the engine axis. A linear grid of point sources with a spacing of \(\varDelta x\approx 0.25 \lambda\) between grid points is placed on the engine axis, where \(\lambda\) represents the acoustic wavelength.
The source map on the left of Fig. 22 shows the source directivities as a function of the source emission angle \(\theta _{jm}\) that is defined between the engine axis and the connection from a source j to a microphone m with \(\theta _{jm} = 0^\circ\) in flight direction. SODIX models strong sources at the axial positions of the intake and the nozzle. Jet sources appear close to the nozzle exit, which is reasonable for a Strouhal number of \(St = 4.6\). All sources present strong directivities with peak radiation angles in the forward arc for the intake and in the rear arc for the nozzle and the jet. The dynamic range of these results is greater than 20 dB.
To evaluate the contributions of the single sources in the far field, the source amplitudes were extrapolated to the positions of microphones at a distance of 150 ft = 45.72 m which are commonly used for noise certification purposes: with the assumption of an axisymmetric sound radiation of the engine, the source amplitudes, \({\tilde{d}}^2\), at emission angles corresponding to the farfield microphones [gray lines in Fig. 22 (left)], are scaled according to the 1/r distance law, where r is the distance between the source and the observer.
Dotted white lines around the positions of the intake and the nozzle indicate the parts of the source grid that are used for the calculation of the farfield contributions from the intake (\(x=5.4\) m) and the nozzle (\(x=0\) m) in Fig. 22 (left). The farfield contribution of the jet is calculated from all source downstream of the nozzle region.
The farfield results are shown in the right plot in Fig. 22. The intake dominates in the forward arc up to \(\theta \approx 75^\circ\). The sound field radiated from the nozzle shows a maximum at \(\theta \approx 115^\circ\) and the jetnoise peaks at \(\theta \approx 125^\circ\). The sum of all sources (SODIX total) agrees very well with the measured data of the farfield microphones.
The capability to model the directive sound sources of a turbofan engine makes SODIX a useful tool for the development and the validation of new engine technologies. Other methods, which often use monopole sources, would average the sound field over the aperture of the array and, therefore, lead to the inaccurate results.
5 Conclusions
Phased microphone arrays are useful tools for estimating the location and strength of sound sources. Aeroacoustic experiments present important challenges, such as noisy environments like wind tunnels or moving sources like flying aircraft. A wide variety of 14 acoustic imaging methods is presented in this paper and the performance of each method for aeroacoustic applications is assessed. This selection spans from the simple conventional beamforming algorithm to deconvolution and inversion methods, which normally imply higher computational cost. Although there is no such thing as a perfect method, recommendations are given for nonexperienced users to obtain the best results, depending on the desired application.
Footnotes
 1.
Many authors normally refer to the Rayleigh resolution limit [95], i.e., the first zero of the firstorder Bessel function (similar to the Airy disk in optics), whereas the Sparrow resolution limit is defined as the angular distance where the sum of the PSF of the two sources produces a flat profile. The Rayleigh resolution limit, on the other hand, shows a distinct dip between both sources. The Rayleigh resolution limit is defined as \(1.22\lambda /D\) and the Sparrow resolution limit as \(0.95\lambda /D\), where \(\lambda\) is the acoustic wavelength and D is the array diameter. Thus, the Sparrow resolution limit is about 22% lower than the Rayleigh resolution limit. However, both criteria are based on the assumption of a continuous disk as a receiver, rather than an array with a finite number sensors. Hence, both criteria represent an approximation.
Notes
Acknowledgements
The authors would like to kindly thank Dr. Takao Suzuki from the Boeing Company for his suggestions about generalized inverse beamforming.
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