Beamforming matrix regularization and inverse problem for sound

AIAA 2013-2212 Aeroacoustics Conferences May 27-29, 2013, Berlin, Germany 19th AIAA/CEAS Aeroacoustics Conference

Beamforming matrix regularization and inverse problem for sound source localization: Application to aero-engine noise T. Padois,∗ A. Berry,∗ P.-A. Gauthier,∗ GAUS, Sherbrooke, Quebec, J1K2R1, Canada

and N. Joshi† Downloaded by Thomas Padois on September 19, 2014 | http://arc.aiaa.org | DOI: 10.2514/6.2013-2212

Acoustics, Pratt and Whitney Canada, Longueuil, Quebec, J4G1A1 Canada.

Phased-microphone arrays associated with beamforming have become a standard technique to localize aeroacoustic sources. The limitations of beamforming have been overcome thanks to deconvolution technique (DAMAS or CLEAN-SC) or iterative process (L1-GIB). However, the computational cost of these methods can be large or assumptions on the source coherence have to be done. In this paper we present a technique based on inverse methods initially developed for sound field extrapolation. The aim is to use a beamforming regularization matrix to penalize the non-signal region in the inverse problem. First, this Hybrid Method is applied to laboratory experiments to demonstrate its effectiveness. Then noise data of an aero-engine measured over a half circular, far-field microphone array are used. The source maps obtained show that the Hybrid Method provides better spatial resolution than beamforming, similar to Clean-SC and results in less iterations of DAMAS.

Nomenclature Matrix or Vector Q Source power W Weight vector C Cross Spectral Matrix (CSM) G Green function q Source strength L Regularization matrix p Microphone sound pressure Parameter λ Penalization parameter M Number of microphones S Number of virtual sources Upperscript H Hermitian transpose Subscript s Virtual source positions m, n Microphone BF Beamforming ∗ GAUS,

University of Sherbrooke, Sherbrooke Quebec J1K2R1 Canada. Pratt and Whitney Canada, Longueuil, Quebec, J4G1A1 Canada.

† Acoustics,

1 of 12 American Institute of Aeronautics and Astronautics Copyright © 2013 by T. Padois,[|#3#|] A. Berry,[|#3#|] P.-A. Gauthier and N. Joshi. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Downloaded by Thomas Padois on September 19, 2014 | http://arc.aiaa.org | DOI: 10.2514/6.2013-2212

I.

Introduction

Over the last two decades, phased-microphone arrays have become a standard technique to localize aeroacoustic sources.1, 2 The basic approach in beamforming consists in delaying and summing microphone signals. The advantage of beamforming is its simplicity and robustness. One of the main problem in beamforming is the poor spatial resolution at low frequency which somehow limits the interpretation of source power maps. To overcome this issue, techniques have been developed in the past years. We can cite the deconvolution technique called DAMAS3 (that has been extended to DAMAS24 and DAMAS-C5 ). The aim of DAMAS is to extract a source distribution from a beamforming source map by iteratively deconvolving the map. The width of the main lobe is clearly reduced with this technique and side lobes are suppressed. However the computational cost is very high and an uncorrelated monopole has to be used as the reference solution. An alternative approach, called CLEAN-SC,6 has been developed. The aim is to suppress side lobes and noise correlated with the main source. However the CLEAN-SC method cannot discriminate coherent sources. Recently, a new algorithm has been developed by Suzuki.7 After decomposing the microphone cross spectral matrix into eigenmodes, an inverse problem is defined and solved iteratively. The spatial resolution of this approach is clearly improved but the computation time is large due to iteration steps. In this paper we propose an algorithm, called Hybrid Method, initially developed for sound field extrapolation,8 based on an inverse problem with beamforming matrix regularization. The advantage of this regularization is to penalize the non-signal region in the inverse problem. Moreover, there is no assumptions on the nature of the sources (correlated or uncorrelated) and the computation time is low if few scan points are required. In Section II of this paper, the Hybrid Method, CLEAN-SC and DAMAS are introduced and compared with simulated data. Section III is devoted to the experimental tests. First a laboratory experiment is set to evaluate the effectiveness of the proposed method. Then the method is applied to aeroengine data to illustrate its capabilities.

II. A.

Theory

Direct problem

The direct acoustic problem is first introduced. The figure 1 shows the microphone array geometry typically used by Pratt and Whitney for exterior, static aero-engine tests.9 The array geometry is semi-circular and is composed of M microphones located in the horizontal plane in the far-field of the engine (the engine axis corresponding to the x-axis in figure 1). We consider an acoustic source at location xs .

Figure 1. Illustration of microphone array geometry and coordinate system.

ˆ (xm ) and can be written in matrix form The acoustic pressure recorded by the microphones is denoted p ˆ (xm ) = G(xm , xs )q(xs ), p

(1)

where G(xm , xs ) is the free-field Green function representing the acoustic radiation between the source and the microphones and q(xs ) the strength of the source. The bold character are vectors or matrices and the

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dimensions are p(xm ) = [M, 1], G(xm , xs ) = [M, S] and q(xs ) = [S, 1] (with S the number of point sources in the scan region). B.

Inverse problem: Tikhonov regularization

Basically, the aim of inverse methods is to estimate the strength of the source q that creates the acoustic ˆ onto the microphone array, knowing the free-field Green function G. One way to define inverse pressure p problems is to solve the following minimization problem qλ = argmin{kˆ p − Gqk22 + λ2 Ω(q)2 },

(2)

10

with Ω(.) a discrete smoothing norm and λ a penalization parameter. The vector 2-norm is denoted by k.k2 . It is well know that the minimization problem is ill-conditioned,11–13 meaning that the solution can be very sensitive to measurement noise or model uncertainties, it is why a discrete smoothing norm is used to regularize the solution qλ . We can define the discrete smoothing norm as

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Ω(q) = kLqk2 ,

(3)

where L is a square weighting matrix. The optimal solution of this minimization problem is14, 15 ˆ, qλ = (GH G + λ2 LH L)−1 GH p

(4)

where (.)H is the Hermitian transpose. Basically, in the classical Tikhonov regularization the discrete smoothing norm is the identity matrix L = I. Finally, the solution of the inverse problem with the Tikhonov regularization can be written as ˆ. qλ = (GH G + λ2 I)−1 GH p (5) C.

Inverse problem: Beamforming regularization matrix

The Tikhonov regularization adds signals to the product (GH G) to impove the conditionning of the inverse problem, but no information about the acoustic problem is added. One way to perform a better regularization is to take into account the result obtained by focused beamforming. The main idea is to use a discrete smoothing norm which depends on the beamforming results, given a priori information about the acoustic problem, instead of the identity matrix. The aim of beamforming is to delay and sum all microphone signals in relation to a virtual source position. When the source position is equal to the real source position, the sum is maximum. The beamforming response denoted qBF can be written basically ˆ. qBF = GH p

(6)

The new discrete smoothing norm in relation to beamforming, called beamforming regularization matrix, can be introduced by    ˆ| |GH p −1 , (7) L = diag ˆ k∞ kGH p where |.| denotes elementwise absolute value of the argument and k.k∞ is the infinite norm. The term diag(A) means that the [1, S] vector A is mapped on the main diagonal of a [S, S] matrix. The discrete smoothing norm is normalized by the infinity norm of the beamforming to ensure that the regularization is normalized in terms of beamforming output. Before introducing the beamforming regularization matrix, the solution of the minimization problem Eq. 4 is written in terms of general-form inverse problem qλ and standard-form inverse problem qλ ˆ, qλ = L−1 ([L−1 ]T GH GL−1 + λ2 I)−1 [L−1 ]T GH p

(8)

T

where (.) denotes the matrix transposition. Therefore the general-form inverse problem can be related to the standard-form inverse problem thanks to the beamforming regularization matrix ˆ, qλ = L−1 qλ = L−1 (GH G + λ2 I)−1 GH p

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(9)

with G = GL−1 and qλ = L−1 qλ . The standard-form inverse problem qλ can be seen as an inverse problem regularized by the beamforming whereas the general-form inverse problem qλ is regularized and scaled by the beamforming. The aim of this study is to compare the source power maps provided by the various microphone array techniques described previously. Therefore the Cross Spectral Matrix (CSM) is introduced in the following. The beamforming output denoted QBF defined in terms of source power can be expressed QBF = qBF qBF H = WH CW,

(10)

where W stands for the weight vector6 and is expressed as

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W=

G , |G|2

(11)

ˆ .ˆ The matrix C is basically the microphone CSM defined by C = p pH . The discrete smoothing norm depends on the microphone sound pressures, the next step is to express it in terms of the CSM of microphone sound pressures    ˆ| |WH p −1 L = diag . (12) ˆ k∞ kWH p ˆ | is a column vector formed by the absolute values of WH p ˆ components. The sth component The term |WH p Hˆ of W p is s sX X X p H H p H p Wms ˆm p ˆH Wms ˆm |2 = Diag(QBF ). (13) | Wms p ˆm | = | n Wns = m

m

m,n

ˆ | is formed by the square root of the components of Diag(QBF ) where Diag is the vector Therefore |WH p ˆ k∞ is the maximum value of |WH p ˆ | over all formed by the main diagonal of the matrix. Similarly, kWH p possible values of s and can also be expressed as p ˆ k∞ = kDiag(QBF )k∞ . kWH p (14) Finally the beamforming regularization matrix can be expressed in terms of the CSM C as " !# p Diag(QBF ) −1 L = diag p . kDiag(QBF )k∞

(15)

Finally, the normalized power of the general-form Qλ = qλ qH λ and the standard-form inverse problems are straightforward H −1 H Qλ = L−1 Qλ (L−1 )H = L−1 kJλ k∞ (Jλ GH )C(GJH ) , (16) λ )kJλ k∞ (L with Jλ = (GH G + λ2 I)−1 .

(17)

As the beamforming regularization matrix is expressed in terms of the weight vector, the regularized steering vector (free-field Green function) has to be equal to G = WL−1 . In the following, Eq. 16 is called Hybrid Method. One can notice that the expression of the Hybrid Method is similar to the beamforming. Therefore, it is possible to iterate the Hybrid Method with the result of the Hybrid Method at the previous step. Moreover, deconvolution algorithms based on beamforming such as DAMAS or Clean-SC, can be implemented with the Hybrid Method. D.

Clean-SC

Clean-SC is one the most commonly used deconvolution technique in aeroacoustics. Clean-SC removes side lobes and spots spatially coherent with the main lobe.6 Moreover the size of the main lobe is reduced. The iterative process of Clean-SC is briefly introduced here, for more information reader is referred to.6 First of all, the beamforming source power map is computed, then the peak value is searched. A new CSM is constructed due to a single coherent source at this location. The new CSM is subtracted to the initial and the process is repeated iteratively. At the end, the peak values are replaced by point sources and added to the residual CSM. In this paper the source power maps given by Clean-SC are compared to beamforming and the Hybrid Method. 4 of 12 American Institute of Aeronautics and Astronautics

E.

DAMAS

Several deconvolution algorithms have been described in the literature. The Deconvolution Approach for the Mapping of Acoustic Sources (DAMAS) is one of the more efficient, but the computation time is very large. We choose to investigate the influence of the Hybrid Method on the DAMAS results. For sake of conciseness, DAMAS algorithm is briefly presented. For more information, reader should refer to the literature.3 The ˆX ˆ =Y ˆ where A ˆ and X ˆ stand for aim of the DAMAS algorithm is to solve the following inverse problem A ˆ the DAMAS matrix and noise source matrix at each grid point respectively. The matrix Y is equal to the beamforming map QBF . Then, this inverse problem is solved iteratively. The source power maps obtained are clearly improved with better resolution and no side lobes.

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F.

Benchmark problem

In this section, the source power maps obtained with the microphone array technique previously presented are compared via numerical simulations. We consider a two-dimensional domain as shown on figure 1. A 188 circular microphone array with a radius of 1.86 m is used, the center of the circle is the origin (this geometry corresponds to the experimental geometry). To generate acoustic waves, we consider a monopolar source located at (xs = 0 and ys = 0). The monopole radiation is expressed in the frequency domain and given by.7 The source frequency is 1 kHz and the source strength is set to 1. The scan zone where the sources are searched is a square with 0.8 m side. The scan zone is sampled with (21 × 21) points. Beamforming is performed without diagonal removing of the CSM C. The loop factor of Clean-SC6 is set to 1. DAMAS is performed over 100 iterations and either the beamforming source power map QBF or the Hybrid Method source power map Qλ is used to initiate the iterations. The source power maps generated by the different microphone array techniques are presented on figure 2. In each case, the peak value is located at the source position and the source power level is correctly estimated. Beamforming exhibit a source power map with a large main lobe and strong side lobes. Clean-SC and the Hybrid Method clearly improve the spatial resolution by narrowing the main lobe and removing the side lobes. The best resolution is provided by DAMAS but at the expense of a much larger computational cost. To compare the DAMAS performance when using beamforming or the Hybrid Method at the initial step, the Root Mean Square (RMS) of the source power map is shown on figure 2.e over the iterations. DAMAS with beamforming needs 56 iterations to converge to a steady value whereas with the Hybrid Method only 4 iterations are necessary. The smaller number of iterations can be explained by the initial source power map given by the Hybrid Method which is cleaner than the beamforming source power map. To conclude, the Hybrid Method provides a resolution similar to Clean-SC, decreases the number of iterations of DAMAS and correctly estimates the source power.

III. A.

Application of the sound source localization methods

Laboratory experiments: Set-up

To demonstrate the performances of the microphone array techniques, experimental tests have been carried out in a hemi-anechoic room. In the next section the different methods are applied on a Pratt and Whitney aero-engine, for which the acoustic pressure was recorded by an irregular circular array. Therefore, the laboratory set up is a small replica of the real tests. We used a 94 custom-made circular microphone array described in.16 The radius of the microphone array is 1.86 m. The microphones are 6 mm electret capsules. The sensitivity of each microphone is calibrated at 1 kHz. Custom-made preamplifiers are used and the acoustic signal is digitized using sound cards connected to a computer. The acoustic signal was sampled at 48 kHz during 10 seconds. Two kinds of sources were used. First, we used two loudspeakers placed at 0.6 m from the center of the circular array along the x-axis. Then we used an open cylindrical waveguides fitted with two back to back loudspeakers, the aim being to create a highly directional source. The length of each waveguide (between speaker membrane and duct termination) is 0.45 m and the total length of the system is 1.65 m. The duct was maintained above the floor using a wood structure (0.3 m above the ground). The input signal is a white noise. A picture of the experimental set-up, in the waveguide system configuration, is shown on figure 3.

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a)

b)

c)

d)

e)

f)

Figure 2. Comparison of the source power maps for a monopolar sound source at 1 kHz. a) Scan zone, b) beamforming, c) Hybrid Method, d) Clean-SC, e) DAMAS, f ) Root Mean Square of the DAMAS source power maps. The black dots are the microphone positions. The black cross is the source position. The source power level is in dB and one color level represents 1 dB.

B.

Localization of two correlated loudspeakers

In this section we compare the source power maps obtained by the microphone array techniques for the case of two loudspeakers driven by two correlated white noises. Considering that the center of the circular array is the coordinate system origin, the x-axis is parallel to the waveguide system and the y-axis is perpendicular (see figure 3). The loudspeakers are located at (xs = [−0.6; 0.6] m and ys = 0 m). The dimension of the scan zone (where the sources are searched) is 1 m in the x and y direction with a scan grid of 41 points in each directions. The time signals are divided into 16384 points. Each block is filtered by a Hanning window and 50% overlap is used to compute the microphone CSM. Finally, a symetric virtual array is used to compute the source maps in order to enforce an axial symmetry of all computed source maps. The loop factor of Clean-SC is set to 1 and DAMAS is performed over 100 iterations. All microphone array techniques are

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Figure 3. Experimental set-up with the waveguide system configuration. The 94 custom-made circular microphone array is set on the ground.

performed at 1 kHz. The source power maps are presented on figure 4. Beamforming is able to detect the two source positions but with strong side lobes. As the two source signals are correlated, Clean-SC detects only one source and in this case the strongest source is detected. DAMAS with beamforming provides a better resolution of the source power map. The two source positions are well detected but small side lobes are still present. The Hybrid Method provides a clean source power map with only two main lobes at source positions, all the sides lobes are removed. Finally, DAMAS with Hybrid Method provides the best results with narrow main lobes. Moreover only 4 iterations are needed to converge the RMS of the source map whereas at least 50 iterations are necessary for DAMAS with beamforming. C.

Localization of two uncorrelated loudspeakers in a duct

In this section the two loudspeakers are replaced by the waveguide system described in section III.A. The goal of this experiment is to create highly directional sources at the waveguide terminations, unlike the two loudspeakers. The duct terminations are located at (xs = [−0.82; 0.82] m and ys = 0 m). Moreover, the amplitude of the right source is decreased to evaluate the ability of the methods to distinguish two sources with different levels. The same parameters are used for the scan zone and the CSM computation (see section III.B), the loop factor of Clean-SC is set to 1 and DAMAS is performed over 100 iterations. All microphone array techniques are performed at 1 kHz. The source power maps are presented on figure 5. The best result is provided by DAMAS with the Hybrid Method. Only two narrow spots appear at source positions. DAMAS with beamforming exhibits side lobes close to the main source. Moreover, the RMS of the source power maps provided by DAMAS and the Hybrid Method is converged after 15 iterations whereas the beamforming is not entirely converged after 60 iterations. Clean-SC and the Hybrid Method give the same source power maps in this case. It has to be noticed that in the case of two sources with different levels, here more than 7 dB, it is better to use the Hybrid Method without the beamforming scaling, i.e. the Qλ source power maps. In summary, the semi-circular array is able to localize two sources with different levels and directional radiation pattern. D.

localization of aero-engine sources

To validate the Hybrid Method for aeroacoustic sources, it is applied to the noise radiated by an aero-engine. The aim is to separate the noise coming from the inlet or exhaust of the aero-engine. The measurement was performed in free-field condition on a static engine with a circular microphone array located above a hard ground at 45 m from the engine. A total of 17 microphones were distributed at polar angles ranging between 20˚ and 160˚ from the engine axis. The sound pressure signals were sampled at 25 kHz during 30 seconds. The time signals are divided into 8192 points. Each block is filtered by a Hanning window and 50% overlap is used to compute the microphone CSM. Diagonal removing of the CSM is also used. Finally, a

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a)

b)

c)

d)

e)

f)

Figure 4. Comparison of the source power maps for two correlated loudspeakers driven by a white noise. a) beamforming, b) Hybrid Method, c) DAMAS with beamforming, d) DAMAS with Hybrid Method, e) CleanSC, f ) Root Mean Square of the DAMAS source power maps. The black crosses are the source positions. The source power level is in dB and one color level represents 1 dB.

symetric virtual array is used to compute the source power maps, thus providing a total of 34 measurement points. Two specific frequencies are selected to examine inlet/exhaust noise separation, 250 Hz and 631 Hz. The scan zone is adapted for each frequency and the number of scan points is close to 3000 (depending on the scan zone size). The source power maps at 250 Hz given by beamforming, Clean-SC and the Hybrid Method are compared on figure 6. Considering the relatively small number of microphones and the large distance between the source and the array, the conditions are not well-inclined to efficiently localize the real souce positions. As expected at low frequency, jet noise is predominating and only a large spot appears at the exhaust position. The Hybrid Method and Clean-SC remove side lobes and at this frequency only noise coming from the exhaust is detected. The source power maps are also computed at a higher frequency of 631 Hz (see figure 7). In this case, the Clean-SC loop factor is set to 0.1 to get a clean source power map and therefore the number of iterations increase whereas the Hybrid Method does not require iteration process.

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a)

b)

c)

d)

e)

f)

Figure 5. Comparison of the source power maps for two uncorrelated loudspeakers set in a duct and driven by a white noise. a) beamforming, b) Hybrid Method, c) DAMAS with beamforming, d) DAMAS with Hybrid Method, e) Clean-SC, f ) Root Mean Square of the DAMAS source power maps. The black crosses are the source positions. The source power level is in dB and one color level represents 1 dB.

Clean-SC and the Hybrid Method clearly exhibit the inlet source which can be interpreted as a fan tone emerging from the broadband jet noise.

IV.

Conclusion

In this paper, several source localization methods, designed for aeroacoustic problems, have been presented. The Hybrid Method which is based on an inverse problem with beamforming regularization has been introduced. Then Clean-SC and DAMAS have been briefly presented. To validate the microphone array techniques, the source power maps for a simulated monopole have been compared. The Hybrid Method gives a source power map similar to Clean-SC in terms of resolution and provides correct source power estimates. Using DAMAS with the Hybrid Method clearly improves the results by decreasing the number of iterations. Experiments have been carried out in the laboratory with a semi-circular microphone array similar to the full-scale aero-engine test. The cases of two correlated loudspeakers and two directional sources with different levels have been studied. The Hybrid Method provides source power maps similar to Clean-SC and DAMAS with the Hybrid Method leads to fewer iterations. Finally, the Hybrid Method has been applied to 9 of 12 American Institute of Aeronautics and Astronautics

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a)

b)

c) Figure 6. Comparison of the source power maps of the aero-engine at 250 Hz. a) beamforming, b) Hybrid Method and c) Clean-SC. The engine is sketched at the origin. The source power level is in dB and one color level represents 1 dB.

real aeroacoustic data (aero-engine noise). Despite the small number of microphones and the large distance between source and array, the Hybrid Method provides correct results as compared to Clean-SC in terms of resolution and side lobes. Overall, the results demonstrate the good potential of the Hybrid Method to localize aeroacoustic sources.

Acknowledgments The authors wish to thank NSERC (Natural Sciences and Engineering Research council of Canada) and Pratt and Whitney Canada for their financial support.

References 1 Michel,

U., “History of acoustic beamforming,” 1st Berlin Beamforming Conference, BEBEC, Berlin, 2006. T. J., Aeroacoustic measurements, Springer, 2002. 3 Brooks, T. F. and Humphreys, W. M., “A deconvolution approach for the mapping of acoustic sources (DAMAS) determined from phased microphone arrays,” Journal of Sound and Vibration, Vol. 294, No. 4, 2006. 4 Dougherty, R. P., “Extension of DAMAS and benefits and limitations of deconvolution in beamforming,” 11th AIAA/CEAS Aeroacoustics Conference (26th AIAA Aeroacoustics Conference), AIAA 2005-2961, 23-25 May Monterey, California, 2005. 5 Brooks, T. F. and Humphreys, W. M., “Extension of DAMAS Phased Array Processing for Spatial Coherence Determination (DAMAS-C),” 12th AIAA/CEAS Aeroacoustics Conference (27th AIAA Aeroacoustics Conference), AIAA 2006-2654, 8-10 May Cambridge, Massachusetts, 2006. 6 Sijtsma, P., “CLEAN based on spatial source coherence,” International Journal of Aeroacoustic, Vol. 6, No. 4, 2007, 2 Mueller,

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a)

b)

c) Figure 7. Comparison of the source power maps of the aero-engine at 631 Hz. a) beamforming, b) Hybrid Method and c) Clean-SC. The engine is sketched at the origin. The source power level is in dB and one color level represents 1 dB.

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