An improved eigenvalue background noise reduction method for acoustic beamforming

Mechanical Systems and Signal Processing 140 (2020) 106702

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An improved eigenvalue background noise reduction method for acoustic beamforming J. Fischer ⇑, C. Doolan School of Mechanical and Manufacturing Engineering, UNSW Australia, Sydney, NSW 2052, Australia

a r t i c l e

i n f o

Article history: Received 7 August 2019 Received in revised form 28 January 2020 Accepted 31 January 2020

Keywords: Aeroacoustics Beamforming Background noise reduction Eigenvalue decomposition

a b s t r a c t This paper presents a modification to an existing eigenvalue identification and subtraction method [Bahr. & Horne, Advanced background subtraction applied to aeroacoustic wind tunnel testing, 21st AIAA/CEAS Aeroacoustics Conference 2015] to locate sound sources in noisy environments. That novel method is compared with two others: classical background noise removal and the original eigenvalue identification and subtraction method. In addition, an improved version of the proposed method is proposed to obtain better level estimate. Four test cases are proposed, involving numerical and experimental studies. Overall, it is found that the new method provides a very accurate location when compared with other techniques, and levels are comparable with those obtained with classical methods. This was especially found in cases where the background noise is very high compared with the source level, with signal-to-noise ratio down to
60 dB. Ó 2020 Elsevier Ltd. All rights reserved.

1. Introduction The reduction of noise arising from aircraft, wind turbines and most recently drones is a challenging topic. In order to understand the nature of these flow induced noise sources, acoustic experiments are carried out. These are usually performed in wind tunnels as the environment is more controlled than in an in situ environment. One of the most popular tools to identify and analyse acoustic sources is known as beamforming [1–6]. It requires a set of numerous synchronized microphones, usually disposed in a plane with an appropriate design, and is principally used to find the location and strength of acoustic sources. For that, the microphone signals are delayed and summed with respect to several positions where the source, assumed to be a monopole, is sought. In most of experimental tunnel testings, the acoustic signal can be considered stochastic, and thus the CBF algorithm is used in the frequency-domain (by Fourier transform) and is then known as Conventional Beamforming (CBF). However, in some cases the measured signals are intermittent [7–9], so the time-domain formulation of the algorithm is then prefered [10–12]. Even though CBF methods have improved over the past years, acoustic measurements in wind tunnel still suffer from some drawbacks. First, reverberant walls create undesired noise sources close to the actual source, but can be lowered by either working in a anechoic tunnel or using some dereverberation algorithms. In the case of a rectangular test section, Guidati et al. [13] have proposed a method where the image sources due to the wall were estimated theoretically and their effect was incorporated into the beamforming process. The beamforming maps obtained with this so-called reflection canceller on experimental trailing edge noise were better resolved. This lead to the development of a similar method called the Image Source Model

⇑ Corresponding author. E-mail address: [email protected] (J. Fischer). https://doi.org/10.1016/j.ymssp.2020.106702 0888-3270/Ó 2020 Elsevier Ltd. All rights reserved.

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J. Fischer, C. Doolan / Mechanical Systems and Signal Processing 140 (2020) 106702

(ISM) that uses a modeled Green’s function in the beamforming steering vector formulation in order to remove the effect of reflections on experimental acoustic data [14–16]. A different approach was imagined by Sijtsma & Holthusen [17] where the beamforming algorithm was changed to account for the influence of a mirror source coherent with the main peak. It was found that the focused beamformer spectra obtained with this method were similar to the ones with no reflections. More recently, Fischer & Doolan [18] have showed that by filtering the cross-correlation functions from the microphones, the reflections can be removed and the acoustic maps were greatly improved when compared with conventional methods. Another very common source of background noise is caused by the wind tunnel drive fan. In some cases the signal from the test model is strong enough to not be affected by it. However, it is quite frequent that the background noise obscures the source signal, which makes the acoustic analysis difficult. A very popular tool to lower background noise in CBF maps is to cancel the self noise of the microphones [1]. This is done by setting the diagonal elements of the cross-spectra matrix (CSM) to zero. However, it leads to non-positive-definite matrices, which is in many cases unwanted. Another widely used method is known as background noise removal (BNR) [19]. It consists in measuring the CSM of the flow noise without the model, and subtracting it from the CSM with the model in flow. That method assumes that the source and facility noise are uncorrelated and that the facility noise measured without and with the model are exactly the same. However, a model placed in the flow will induce some blockage which will lead the wind tunnel fan to work at a higher power setting to reach the same speed. To overcome that issue, some methods have been developed to numerically lower the effect of background noise. Adaptive filtering [20] uses a reference sensor near the background noise source and filters noise from the acoustic array measurements in the time domain. Koop & Ehrenfried [21] have worked in the wavenumber domain to remove the background noise by assuming the latter is caused by plane waves. Finally, the background noise can be separated from the source signal via an eigenvalue decomposition (EVD) of the microphone signal CSM [22]. More recently, Bahr & Horne [23,24] have proposed a denoising algorithm based on and Eigenvalue Identification and Subtraction (EIS). It works under the assumption that the background noise eigenvectors are orthogonal to the source eigenvectors. In conventional situations, the method does not show much improvement when compared with BNR. However, EIS is a robust method that provides good results in cases where the background noise data are slightly corrupted [23]. In addition, some recent work [25] has demonstrated the efficiency of EIS on an experimentally controlled source (speaker) with flow. In this paper, a modification to Bahr & Horne’s method [23] is proposed. It is found that the new method provides a much better location accuracy with a relatively accurate level estimation. Thus, a second method is developed which corrects the level. The paper will first describe this new technique before comparing its output with existing methods. Four different test cases are considered: a numerical source with numerical (1) and experimental (2) background noise, and experimental flow noise cases with known (3) and unknown (4) sources.

2. Processing methods 2.1. Conventional Beamforming An acoustic array composed of M microphones where the mth microphone is located at xm is used to measure the sound of a given source. Each microphone provides a time signal which is projected in the frequency domain using a Fourier transform. The CSM of the frequency vectors Pð f Þ ¼ ½p1 ; . . . ; pM T is constructed using the following convention: H

Cð f Þ ¼ Pð f ÞPð f Þ

ð1Þ

where superscript H represents the hermitian transpose and the superscript X denotes the averaging of the quantity X in a number of discrete time blocks using Welch’s periodogram [26]. The algorithm then steers the array signals to several positions in a so-called focusing plane where the source is sought. The result is called beamformer output and takes the following expression: H

Z ðyn ; f Þ ¼

hðyn ; f Þ Cð f Þhðyn ; f Þ ; M ðM
1Þ

ð2Þ

where hðyn ; f Þ ¼ hðxm ; xs ; f Þ is called the steering vector and stands for the normalized Green’s function between the microphone located at xm and the focusing point at yn . In order to reduce the self-noise of the microphones, the diagonal elements of the CSM are set to 0. This process removes the microphone self-noise and thus provides a cleaner acoustic map [27]; the reason for the M ðM
1Þ denominator in Eq. (2) is to account for this adjustment in the beamformer output level. The expression of the steering vector is the following:

hðxm ; yn ; f Þ ¼

g ðxm ; yn ; f Þ ; jg ðxm ; yn ; f Þj

ð3Þ

where the operator j:j produces the absolute value. In CBF, it is assumed that the acoustic sources are monopolar, so the steering vector uses the free-field Green’s function for a monopole:

J. Fischer, C. Doolan / Mechanical Systems and Signal Processing 140 (2020) 106702

g ðxm ; yn ; f Þ ¼

expð
jkjjxm
yn jj2 Þ ; 4pjjxm
yn jj2

3

ð4Þ

where k ¼ 2pf =c0 denotes the wavenumber at frequency f in a medium which speed of sound is denoted by c0 , and jj:jj2 2

denotes the Euclidean l -norm. Note that with the definition of the steering vector from Eq. (3), the denominator in Eq. (4) is unnecessary. The steering vector expression in Eq. (4) does not take into account the refraction of waves through the shear layer. This effect is approximated by shifting the grid by M 0 h where M 0 is the Mach number of the test jet and h is the perpendicular distance from the body surface to the flow shear layer (see Padois et al. [28]). This approximation is considered sufficient due to the low Mach number and the small height of the wind tunnel opening. 2.2. Beamforming parameters The main objective of beamforming is to find the exact source location with the correct level. Optimising both of these parameters usually leads to a compromise; however, no steering vector formulation produces both correct source location and level estimation [29]. If the actual source position and level are known, the error in location and level can be estimated. Considering a source located at rs , if the CBF map provides a maximum at a location rmax , then the error in location is defined as:

DR ¼

jjrs
rmax jj2 d

ð5Þ

where d denotes the distance between the array and focusing plane. Similarly, if Ls designates the exact source level and Lmax stands for the level of the maximum on the CBF map, the error in level is defined as:

DL ¼ Ls
Lmax :

ð6Þ

2.3. Background Noise Removal (BNR) The most popular denoising method in CBF is known as Background Noise Removal (BNR). This method relies on the assumption that the measured signal can be decomposed into two uncorrelated components which are the source and the background noise (fan, boundary layer noise, electronic noise, etc.). In terms of Cross-Spectral Matrix (CSM), this can be written as:

G ¼ Gs þ Gd

ð7Þ

Practically, the measured CSM G is obtained experimentally. The background CSM Gd is that which is measured with the flow but without the model. The procedure consists in calculating the denoised CSM by using Gs ¼ G
Gd . The latter is then used as the input CSM in the CBF algorithm. 2.4. Eigenvalue Identification and Subtraction (EIS) An advanced background noise subtraction based on eigenvalue decomposition has been introduced by Bahr & Horne [23]. First, the eigenvalues and eigenvectors of the background noise CSM Gd are calculated, such that

Gd ¼ Xd Kd XHd

ð8Þ

where Kd is an M  M diagonal matrix containing the eigenvalues ki;d and Xd is an M  M matrix of the eigenvectors that satisfies the property XHd Xd ¼ I (the identity matrix).
1=2

The matrix Kd is then defined; it is a diagonal matrix containing the square root of the inverse of the eigenvalues from pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi Kd , namely 1= ki;d . Note that if some eigenvalues are exactly equal to zero, the corresponding value 1= ki;d is set to zero by default. This matrix is then used to build a ‘‘prewhitening” operator:
1=2

Bd ¼ Xd Kd

ð9Þ

The latter operator is applied to the measured CSM G in order to get a prewhitened CSM:

b ¼ BH GBd G d

ð10Þ

which yields (using Eq. (7)):

b¼c G Gs þ I

ð11Þ

In order to obtain a proper estimate of Gs , some more eigenvalue analysis needs to be achieved. An eigendecomposition of

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b is performed, such that: G

  b¼X bH þ I ¼ X bH þ X bsK bsK b sIX bH ¼ X bs K bH bK bH ¼ X bsX cs X bs þ I X bX G s s s s

ð12Þ

b are the same as G b s , such that The relationship established in the previous equation shows that the eigenvectors of G b b b s should be b b b b X ¼ X s . It is also seen that the eigenvalues of G are related to those of G s by k i ¼ k i;s þ 1. The eigenvalues of G b greater than or equal to 0 (Hermitian positive semidefinite matrix), so the eigenvalues of G must be greater than or equal b d. b s while those equal to 1 are related to the noise G to 1. Eigenvalues greater than 1 correspond to the source of interest G b and only keeping the eigenvalues b s can then be obtained by performing an eigenvalue decomposition on G An estimate of G b s , in that are greater than 1. This is equivalent to only retaining the positive eigenvalues, and corresponding eigenvectors X b s are obtained, the original source CSM can be reconstructed cs ¼ K b
I. Once the reduced-size matrices K cs and X the matrix K using the prewhitened operator from Eq. (10) by:

 H b s B
1 Gs ¼ B
1 G d d

ð13Þ

1=2 H where B
1 d ¼ Kd Xd . The CSM Gs is then implemented in the CBF algorithm in order to produce the beamforming maps. A summary of the EIS methodology is proposed below:

(1) Enforce Gd to be positive semi-definite. (2) Compute Kd and Xd from the EVD of Gd ¼ Xd Kd XHd . (3) Compute Bd ¼ Xd K
1=2 . d H b (4) Compute G ¼ B GBd . d

b¼X b from the EVD of G bK b H. b and X bX (5) Compute K cs ¼ K b
I, where I denotes the identity matrix. Retain only the positive values in K cs . This leads to a dimen(6) Compute K cs from K. b sion reduction of K cs ¼ X b while retaining only the eigenvectors columns associated with the retained eigenvalues of K cs . (7) Define X H b b b b (8) Compute G s ¼ X s K s X . s

H (9) Compute B
1 d ¼ Kd Xd .  H b s B
1 . (10) Compute Gs ¼ B
1 G d d 1=2

2.5. Eigenvalue Identification, Organization and Subtraction (EIOS) b s is already An improvement of the EIS method is given in this section. The idea is the following: in step (10), the matrix G theoretically ‘‘denoised” from the background noise (due to the subtraction process at step (6)), so a better choice of the operator B
1 d could produce better results for the final CSM Gs . The change that is proposed appears in step (9): originally with EIS, B
1 d was built by associating each eigenvalue in Kd with their corresponding eigenvector in XHd . In the proposed method, all the eigenvectors are equally weighted, which means   that the eigenvalues in Kd are all the same. To be clearer, step (9) is replaced with 90 :  0 P 1=2 H 9 Compute K1=2 ¼ N1 Ni¼1 k1=2 I. Then compute B
1 d ¼ Kd Xd . i d b s contains exclusively information from the signal: the noise information was removed in step This step is legitimate as G   cs (the negative values are related to the noise). Thus, step 90 is just a dif(6) when retaining only the positive values in K ferent way of organizing the weighting of each eigenvector by changing the eigenvalues. That new method is designated by Eigenvalue Identification, Organization and Subtraction (EIOS). In order to have a better representation of the difference between the two methods, their matrix representation is detailed. With EIS, each eigenvalue ki is different, resulting in different weighting for the corresponding eigenvectors, as seen in Eq. (14):

0

K

1=2 H d Xd

k1

B0 B ¼B B .. @. 0

0

 0

11=2 0

k2 ...

 0 C C . . .. C C . . A

0

   kM

x1;1

B x2;1 B B B .. @. xM;1

x1;2

   x1;M

x2;2 ... xM;2

 .. . 

1H

0

1=2

k1 x1;1

B B k1=2 x x2;M C C B C ¼ B 2 1;2 C B. ... A B .. @  xM;M k1=2 M x1;M

where superscript * denotes the complex conjugate operator.

1=2

k1 x2;1  k1=2 2 x2;2

.. .  k1=2 M x2;M

1=2

   k1 xM;1

1

C C     k1=2 2 xM;2 C C C . . .. C . . A 1=2     kM xM;M

ð14Þ

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On the other hand, EIOS uses one identical value k for all the elements in the eigenvalue matrix, which results in a similar weighting for all the eigenvectors, as shown in Eq. (15):

0

k

B B0 Kd1=2 XHd ¼ B B .. @. 0

0  0

10

x1;1

CB 0 CB x2;1 B .. C CB . . A@ .. 0  k xM;1 k .. .

 .. .

1H

0

kx1;1

x1;2

   x1;M

x2;2 .. .

 .. .

x2;M .. .

B  C B kx1;2 C C ¼B B. C B. A @.

xM;2

   xM;M

kx1;N

kx2;1

   kxM;1

1

.. .

C    kxM;2 C C C . . .. C A . .

kx2;M

   kxM;M

kx2;2

ð15Þ

1=2

where k denotes the average value of all the ki from Eq. (14). By doing this, all eigenvectors are boosted equally: initially with EIS, some of the eigenvectors had more influence on the output CSM Gs than others due to the diversity of the eigenvalues; e.g. the eigenvectors corresponding to the lowest eigenvalues would not have as much influence on the output as the eigenvectors related to higher eigenvalues. This is especially true when some of the eigenvalues are close to 0, thus almost cancelling the corresponding eigenvectors. With EIOS, all of the eigenvectors are weighted with the same value, which results in enhancing the influence of the eigenvectors that were initially related to the lowest eigenvalues. This increases the amount of information that is extracted from the ‘‘denoised” b s at step (10) in the methodology, thus improving the accuracy of the beamforming algorithm for source location. matrix G b s is different A result of equal-weighting of the eigenvectors is that the Euclidian norm of the newly obtained matrix G from that obtained with EIS. In the latter, each eigenvalue is multiplied by its corresponding set of eigenvectors, as shown in Eq. (14). With EIOS however, even though an average eigenvalue k is used, the energy (or Euclidian norm) of the matrix H K1=2 d Xd is different, which alters the energy content of Gs and eventually results in a poor estimation of the source level. This is due to the fact that the eigenvectors are not weighted with their respective eigenvalue (as was the case for EIS); in other words, the Euclidian norm of Eqs. (14) and (15) are different. In order to solve this problem, a level correction method is proposed.

2.6. Eigenvalue Identification, Organization and Subtraction with Level Correction (EIOS-LC) In order to obtain a better estimate of the source level, the following method is used. In essence, it uses EIOS to find the source location, and then BNR to estimate the source level at that location. The so-called EIOS with Level Correction (EIOS-LC) methodology is summarized below. (1) (2) (3) (4) (5)

Plot the beamforming map using the CSM Gs obtained with EIOS. Save the location of the maximum peak rmax from the previous map. Plot the beamforming map using the CSM G
Gd obtained with BNR. Save the level Lref on the previous map at the location rmax . Correct the level of the map obtained with EIOS at step (1) with the level value Lref obtained at step (4).

All the methods described above (BNR, EIS, EIOS and EIOS-LC) will now be compared on several numerical and experimental tests cases. 3. Test case 1: evaluation of a simulated source with a simulated background noise 3.1. Numerical parameters The first test case consists of an exclusively numerical simulation. The monopole source and the background noise are generated in the time domain using two uncorrelated white noise signals with a Gaussian distribution. The data were sampled at f s ¼ 65; 536 Hz during T ¼ 32 s. The source is set to 40 dB and the background noise level is chosen in order to obtain a Signal-to-Noise Ratio (SNR) of
60,
50,
40,
30,
20,
10, 0 and +10 dB. Fig. 1 shows the case where the SNR is set to +10 dB. In that case, the level of the source with noise is virtually the same as the one of the source alone. The acoustic emission is recorded using a numerical set of microphones, also known as an acoustic array. The array geometry is based on a multi-arm logarithmic spiral [1] which has an optimized design for investigating broadband signals. The parameters that are used are 9 equally spaced arms of 7 microphones each, leading to a total of 64 microphones (including the centre one). The minimum and maximum diameters of the spiral are set to 1 m and 1.5 m respectively and the spiral angle of curvature is h ¼ 0:7p. These parameters were found to optimize both the beamforming map resolution and the side lobe levels. The array, located in the (x; y) plane with the centre microphone as the coordinate system reference, is placed at a distance z ¼ 1 m from the source. The numerical source is located at (x; y; z) = (0;0;1) m. The block-averaging procedure to create the CSM was performed using Welch’s periodogram [26]: the Fourier transform was performed over time blocks in the signals. Each time block consisted of N FFT ¼ 8192 discrete points, resulting in a frequency resolution of 6 Hz, and was multiplied by a Hann window function. In addition, an overlapping of 50% was used in order to increase the total number of blocks to 512.

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J. Fischer, C. Doolan / Mechanical Systems and Signal Processing 140 (2020) 106702

Fig. 1. Spectra of the source, background and total signals with an SNR of +10 dB. The source and background are white noise signals.

3.2. Results For each SNR, the error in location DR and level DL have been calculated. Fig. 2 shows the evolution of DR (a) and DL (b) according to several SNR and using each of the denoising methods: BNR, EIS, EIOS and EIOS-LC. For each SNR, the CBF maps were calculated in 1/3rd octave bands between 500 Hz and 10 kHz. The results are shown with the symbols (BNR),  (EIS), D (EIOS) and  (EIOS-LC). In addition, the mean and standard deviation of DR and DL were calculated over the 1/3rd octave frequency bands for each SNR; these are shown by the bold line and coloured area, respectively. The error in location in Fig. 2(a) shows that the source position is accurately found with each method for SNR ranging from +10 dB to
30 dB. Below that threshold, the BNR method is not efficient, as the mean value deviates from 0. However, the other methods display accurate location estimation even at an SNR of
60 dB. In terms of level, Fig. 2(b) shows that EIOS never recovers the level, even with positive SNR. The other methods however are accurate for SNR values from +10 dB down to
20 dB. Below that, the mean DL increases and reaches a peak at the lowest SNR of 2 dB with EIOS-LC, 5 dB with BNR and 15 dB with EIS. When looking at the EIOS method, the trend is exactly the same as for EIS except that the level is 2 dB higher, even for a positive SNR. It must also be noted that the standard deviation over the frequency range is quite important with

Fig. 2. Location error DR (a) and level error DL (b) on Test Case 1: a numerical source with numerical background noise, using BNR, EIS, EIOS and EIOS-LC. The individual results in each 1/3rd frequency band are denoted by (BNR),  (EIS), D (EIOS) and  (EIOS). The mean and standard deviation are respectively shown in bold line and coloured area.

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BNR and EIOS-LC, but it is almost 0 with EIS and EIOS. The conclusion of this first test is that the location is accurately found with EIS, EIOS and EIOS-LC, and the mean level is best estimated with EIOS-LC. 4. Test case 2: evaluation of a simulated source with experimental background noise 4.1. Numerical parameters The situation is very similar with the previous case. The only difference is that the background noise is now obtained from an experimental wind-tunnel measurement. In order to investigate a wide range of SNRs, the background noise amplitude was varied. That process enabled to investigate several SNR values. The frequency f ¼ 5 kHz was used as a reference; the background noise spectra was then moved from
10 dB to +60 dB in 10 dB increments relative to the source spectra at that particular frequency, which corresponds to SNR ranging from 10 dB to
60 dB. It must be noted that unlike Test Case 1, now the SNR is not constant over the frequency range due to the spectral shape of the background noise. Fig. 3 shows an example of the background and source spectra obtained with an SNR of +10 dB. 4.2. Results As the SNR is not constant with the frequency, it is not possible to calculate the mean and standard deviation for each SNR. Thus, the mean and standard deviation of DR and DL from Fig. 2 will be used as a reference. The lines in Fig. 4 thus represent these reference plots and the markers designate the results from the current set up. In terms of location (Fig. 4(a)), each method provides good accuracy from SNR = +10 dB down to SNR =
30 dB. For lower SNRs, the BNR results deviate slightly from 0 while the EIS shows a large difference (the sources are found approximately 0.8 m from the actual location which is the domain boundary). The EIOS and EIOS-LC methods locate the source accurately in all cases. For the level however, the EIOS results are contained in the range [0;20] dB over the whole range of SNRs. The level is better recovered with the other methods down to SNR=-30 dB. Below that, BNR and EIOS-LC deviate slightly while EIS is quite inaccurate, with errors reaching 30 dB at the lowest SNR. These results are consistent with the ones obtained in Test Case 1 (Section 3). The EIOS-LC method shows very accurate results in terms of location and a good estimate of the source level, comparable with BNR. 4.3. Two sources The case of two source is now investigated. The situation is the same as above, except two sources are present, located at (x; y) = (
0.2,0) m and (0.2,0) m. The two sources are at the same level (40 dB) and the background noise spectra is the one that corresponds to an SNR of
40 dB (at 5 kHz, see Section 4.1). As the number of sources has increased, the results will be presented as beamforming maps instead. Fig. 5 shows the maps obtained with the four methods (BNR, EIS, EIOS and EIOS-LC) for three 1/3rd octave bands f ¼ 1 kHz, 2 kHz and 4 kHz. The circles denote the exact source locations and the cross correspond to the maximum on each side of the domain, respectively x < 0 and x > 0. The reference map, obtained with CBF with no background noise, is presented in the first line (Clean map). At the lowest frequency f ¼ 1 kHz, EIS does not locate any source, BNR locates one (but is contaminated with numerous sidelobes), and EIOS and EIOS-LC display two lobes of equal energy around each source location. The latter also contains sidelobes but they are found at the edge of the domain, so relatively far from the sources. The maps at f ¼ 2 kHz draw the same conclusions, with a note on the EIOS and EIOS-LC locations which are now perfectly found at the actual source locations. At

Fig. 3. Spectra of the source, background and total signals with an SNR of +10 dB at f ¼ 5 kHz. The source is a white noise signal and the background is an experimental wind tunnel fan noise.

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J. Fischer, C. Doolan / Mechanical Systems and Signal Processing 140 (2020) 106702

Fig. 4. Location error DR (a) and level error DL (b) on Test Case 2: a numerical source with experimental background noise, using BNR, EIS, EIOS and EIOS-LC. The individual results in each 1/3rd frequency band are denoted by (BNR),  (EIS), D (EIOS) and  (EIOS). The mean and standard deviation are the same as in Fig. 2.

f ¼ 4 kHz, all methods are able to locate the two sources, but EIOS and EIOS-LC show the lowest amount of sidelobes. It can also be noted that the level is the closest to the reference case with EIOS-LC for all the frequency bands: from 1 kHz to 4 kHz, the difference is respectively 6 dB, 5 dB and 0 dB. 4.4. Multiple sources In this section, 9 sources of equal level, placed in a square, are now considered located at x ¼
0:2, 0 and 0.2 and y ¼
0:2, 0 and 0.2. The background noise is identical to that used in the previous section. The results for each method are presented in Fig. 6. Contrary to the case with 2 sources, here none of the methods are able to locate the sources at f ¼ 1 kHz; only EIOS and EIOS-LC seem to depict energy in the source region but none of them are clearly found. At f ¼ 2 kHz, EIS is still inefficient, BNR finds 2 of the sources, and EIOS and EIOS-LC show a lobe at each source location. However, the levels of each lobe are different. At f ¼ 4 kHz, EIS is able to find 2 sources but high sidelobes are present all over the map; BNR finds a maximum at each source location but only 6 of them have a level above the sidelobes; EIOS and EIOs-LC perfectly depict all of the 9 sources with equal levels and lower sidelobes, with a difference of 2 dB from the reference case with EIOS-LC. This shows that the effectiveness of EIOS-LC over the other methods is still observed even if numerous sound sources are present. Also, when 2 sources are considered (Section 4.3), EIOS-LC works accurately for all the frequency of interest from 1 kHz to 4 kHz. However, when the number of sources is increased to 9, the sources are properly located only from 2 kHz, and show similar levels only at 4 kHz. This demonstrates that increasing the number of sources tends to reduce the frequency range in which EIOS-LC works accurately.

5. Test case 3: evaluation of an experimental controlled source in flow 5.1. Experimental setup In this experiment, a speaker is placed behind a low Mach number flow in an anechoic wind tunnel. The wind tunnel is the Anechoic Wind Tunnel (AWT) at UNSW, Sydney. It is an open circuit tunnel with an open jet located inside an anechoic room (size 3 m  3 m  2 m) and with a cut-off frequency of 300 Hz. The inlet has a square section of 0.46 m  0.46 m. The flow speed can go up to U 1 ¼ 20 m/s which is the speed used for this experiment. The speaker generates a white noise signal between 0 Hz and 20 kHz at different levels ranging from 1% to 100% of its capacity. The detailed values are 1, 2, 3, 4, 5, 10, 20, 40, 60, 80 and 100%. The 1/3rd octave band spectra for each speaker level are displayed in Fig. 7. The speaker spectra

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J. Fischer, C. Doolan / Mechanical Systems and Signal Processing 140 (2020) 106702

1 kHz

2 kHz

4 kHz

Clean map

BNR

EIS

EIOS

EIOS-LC

Fig. 5. CBF maps with 2 sources using BNR, EIS, EIOS and EIOS-LC (lines). The Clean map (first line) is the reference case (no background noise). Results are shown at 1/3rd octave band frequencies f ¼ 1 kHz, 2 kHz and 4 kHz (columns). The source exact locations are denoted by  and the maximum on the map by .

(coloured lines) are obtained without the flow while the background spectra (bold line) is measured with the flow at 20 m/s and with no speaker. An acoustic array with the same geometry as the one described in Section 3.1 is used. It is located at a distance z ¼ 1:5 m away from the speaker. Fig. 8 shows a sketch of the experiment, with the acoustic array on the left, the inlet in the middle and the speaker on the right. Note that the array and the speaker are located 1 m and 0.5 m away from the center of the flow, respectively.

5.2. Results As for Section 4.2, the error in location and level are displayed on top of the reference mean and standard variation from Test Case 1. This is shown in Fig. 9.

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1 kHz

2 kHz

4 kHz

CBF

BNR

EIS

EIOS

EIOS-LC

Fig. 6. CBF maps with 9 sources using BNR, EIS, EIOS and EIOS-LC (lines). The Clean map (first line) is the reference case (no background noise). Results are shown at 1/3rd octave band frequencies f ¼ 1 kHz, 2 kHz and 4 kHz (columns). The source location is denoted by .

First, the location is investigated in Fig. 9(a). The four methods provide good results for SNR between 10 dB and
15 dB. From
15 dB to
40 dB, the BNR and EIS sources are found far away from the actual source location (0.5 m away on average). On the other side, the EIOS and EIOS-LC usually find the source at an accurate location, except for a few cases where the error is about 0.1 m. Finally, for SNRs below
40 dB, all the methods do not recover the source location. It must be noted that the EIOS and EIOS-LC still provide some good results, but only for a few cases. As for the level, Fig. 9(b) shows that BNR, EIS and EIOS-LC are accurate until an SNR of
20 dB. Then both methods decay with the same trend, reaching an error or 45 dB for an SNR of
50 dB. The EIOS results however are comprised between 0 and 20 dB with very little cases of exact level recovery. Again, similar observations as Test Case 1 and 2 are found: the EIOS-LC method is very accurate to recover the source location even at very low SNR, and the level estimation is similar to the BNR results. The other methods are efficient in terms of location and level only for SNR down to
20 dB.

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Fig. 7. Background noise level (black) and speaker levels (color) that were used in the experiment.

460 Flow inlet Acoustic array

1500

Flow direction

Speaker

x z 500

1500 Fig. 8. Sketch of the experimental set up with the speaker behind the flow. Dimensions in mm, drawing is not to scale.

6. Test case 4: evaluation of an experimental uncontrolled source in flow 6.1. Experimental setup The experiments are conducted in the same wind tunnel as in Section 5.1. Here, the speaker is removed and a 3D NACA0012 airfoil with flat tip is installed in the flow, as shown in Fig. 10. The airfoil has a chord c ¼ 200 mm and a span s ¼ 200 mm and is set at 0 angle of attack. A plate is attached at the lower section of the inlet and the airfoil is mounted on the centreline, 62 mm downstream of the inlet. The flow speed is set to U 1 ¼ 20 m/s and the array plane is located at 0.97 m from the airfoil trailing edge. 6.2. Results The acoustic spectra measured with the centre array microphone is presented in Fig. 11 between 300 Hz and 10 kHz (the lowest value corresponds to the cut-off frequency of the anechoic chamber). The background noise, which is obtained without the model in the flow, is also shown in dashed line. It appears that no difference is found between the two spectra, which implies that the noise source is of lower amplitude than the background noise (negative SNR). Thus, the acoustic maps will be calculated over several 1/3rd octave bands between 300 Hz and 10 kHz. A comparison of the different denoising methods is shown in Fig. 12 at three 1/3rd octave bands f ¼ 500 Hz, 2.5 kHz and 4 kHz. The results of BNR, EIS, EIOS and EIOS-LC are displayed in each line. The green square denotes the NACA airfoil location, the black circles stand for the microphones position and the flow goes from left to right.

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Fig. 9. Location error DR (a) and level error DL (b) on Test Case 3: a speaker behind a flow, using BNR, EIS, EIOS and EIOS-LC. The individual results in each 1/ 3rd frequency band are denoted by (BNR),  (EIS), D (EIOS) and  (EIOS). The mean and standard deviation are the same as in Fig. 2.

Acoustic array 460

1500

Flow inlet y

260 30

z

170 NACA 0012

1500 Fig. 10. Sketch of the experimental set up with the 3D NACA0012 airfoil in the (y; z) plane. Dimensions in mm, drawing is not to scale.

At f ¼ 500 Hz, the main noise source with BNR and EIS is coming from the outlet on the right which is on the fan side, with a level around 58 dB. The results also show that the airfoil region (green square) is at a level lower than 48 dB. With EIOS and EIOS-LC, the source location is found near the airfoil trailing edge. Also, the level found with the latter method is 40 dB, which is consistent with the comment at the beginning of this paragraph: as the background noise due to the fan is too high, the source level has to be lower than the BNR maps values. At 2.5 kHz, the maximum on the CBF map using BNR is located near the airfoil trailing edge [30], but a few side lobes are surrounding this region which leads to some doubts concerning the existence of that source. EIS is still displaying a source near the outlet. The EIOS and EIOS-LC maps now display a source located exactly at the airfoil trailing edge, and the level of the source with the latter is of 29 dB. At 4 kHz, the maps obtained with BNR and EIS are too noisy to draw any conclusion. The only useful information is that the level around the trailing edge region with BNR is around 20 dB. On the other side, the EIOS and EIOS-LC results still show a source at the airfoil trailing edge, with a level of 16 dB, which is consistent with the BNR map.

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Fig. 11. Spectra of the array centre microphone with and without the 3D NACA0012 airfoil in flow, U 1 ¼ 20 m/s.

500 Hz

2.5 kHz

4 kHz

BNR

EIS

EIOS

EIOS-LC

Fig. 12. CBF maps of Test Case 4: a 3D NACA0012 airfoil using BNR, EIS, EIOS and EIOS-LC (lines). Results are shown at 1/3rd octave band frequencies f ¼ 500 Hz, 2.5 kHz and 4 kHz (columns). The airfoil is represented by the green square. The maximum on the map is denoted by .

These results show the efficiency of the EIOS method on an experimental flow induced noise source. They confirm that EIOS-LC is a useful method that can drastically improve the resolution of acoustic maps in noisy environments and also provide a good estimate of the source level. This is particularly the case at low frequencies, where conventional beamforming methods usually fail to provide an accurate location.

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7. Conclusion This paper has presented a novel background noise removal method based on an existing eigenvalue decomposition method. The new method (EIOS) was found to improve the source location accuracy in low SNR cases, but the level was not exactly recovered. The latter issue was fixed by proposing an extension to this method, known as EIOS-LC. It is a combination of EIOS and BNR, which provide a good estimation of respectively the source location and level. The method was compared with others on 4 different test cases involving numerical simulation and experimental data. First, the BNR, EIS, EIOS and EIOS-LC methods were used on a numerical source with a numerical background noise with varying SNR (Test Case 1). This reference case has shown that all the eigenvalue methods could locate the source at lower SNR than the traditional BNR. However, the level was found to be more accurate with BNR and EIOS-LC than with the two other methods. Then, the numerical background noise was replaced with an experimental wind tunnel dataset (Test Case 2). The SNR was varied again by shifting the background spectra at several levels from the source spectra. In that case, only the EIOS and EIOSLC could accurately recover the exact source location over the whole SNR range, and EIOS-LC showed results that were comparable with the BNR method. The number of sources was then increased to 2 and 9, and it was found that EIOS-LC was still showing better results than the two other methods, in terms of location and level. It was also found that increasing the number of sources tends to reduce the frequency range in which EIOS is accurate. Test Case 3 involved an experimental set up with a speaker located behind a flow at 20 m/s. The speaker level was changed to eleven different levels in order to provide a variety of SNRs. All the methods were unable to accurately locate the source at some point, but the EIOS and EIOS-LC worked well down to an SNR of
40 dB while the other methods started to deviate from an SNR of
15 dB. The trends for the level error were similar between BNR, EIS and EIOS-LC while for EIOS the actual level source was rarely found. Finally, the methods were used on a finite NACA0012 airfoil in the same wind tunnel and at the same flow speed (Test Case 4). It was found that from 500 Hz to 4 kHz, the EIOS-LC was always showing a source located either close or exactly at the airfoil trailing edge and with a proper level estimation. The other methods however did not show convincing results. Overall, the new EIOS-LC method has demonstrated great improvement in terms of sound source localization when compared with other methods. The results on experimental data are quite impressive as the CBF maps were clean and the source exactly found. This method is very promising when investigating noise sources in environments with a high background noise. CRediT authorship contribution statement J. Fischer: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing original draft, Writing - review & editing, Visualization. C. Doolan: Conceptualization, Methodology, Resources, Writing review & editing, Supervision, Project administration, Funding acquisition. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement The authors wish to thank Ryan Cooper for providing the airfoil data for Test Case 4. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, athttps://doi.org/10.1016/j.ymssp. 2020.106702. References [1] T.J. Mueller, Aeroacoustic Measurements, Springer, 2002. [2] P. Chiariotti, M. Martarelli, P. Castellini, Acoustic beamforming for noise source localization – reviews, methodology and applications, Mech. Syst. Signal Process. 120 (2019) 422–448. [3] A. Cigada, F. Ripamonti, M. Vanali, The delay & sum algorithm applied to microphone array measurements: numerical analysis and experimental validation, Mech. Syst. Signal Process. 21 (2007) 2645–2664. [4] Z. Chu, Y. Yang, Comparison of deconvolution methods for the visualization of acoustic sources based on cross-spectral imaging function beamforming, Mech. Syst. Signal Process. 48 (2014) 404–422. [5] C.J. Bahr, W.M. Humphreys, D. Ernst, T. Ahlefelds, C. Spehr, A. Pereira, Q. Leclère, C. Picard, R. Porteous, D.J. Moreau, J. Fischer, C.J. Doolan, A comparison of microphone phased array methods applied to the study of airframe noise in wind tunnel testing, in: 23rd AIAA/CEAS Aeroacoustics Conference, Denver, Colorado, 2017.. [6] G. Raman, C. Ramachandran, K. Srinivasan, R. Dougherty, Advances in experimental aeroacoustics, Int. J. Aeroacoust. 12 (2013) 579–637.

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