Diesel Engine Noise Source Visualization by Using Compressive Sens- ing Algorithms Tongyang Shi 1 Key Laboratory of Noise and Vibration Research, Institute of Acoustics, Chinese Academy of Sci- ence Beijing 100190, China J. Stuart Bolton Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University West Lafayette, IN 47906, USA Frank Eberhardt Cummins Inc. Columbus, IN 47201, USA

ABSTRACT To identify sound source locations by using Near-field Acoustical Holography (NAH), a large number of microphone measurements is generally required in order to cover the source region and ensure a sufficiently high spatial sampling rate: it may require hundreds of microphones. As a result, such measurements are costly, a fact which has limited the industrial application of NAH to identify sound source locations. However, recently, it has been shown possible to identify concentrated sound sources with a limited number of microphone measurements based on Compressive Sensing theory. In the present work, sound radiation from the front face of a diesel engine was measured by using one set of measurements from a thirty-five-channel combo-array placed in front of the engine. The locations of significant noise sources were then identified by using two algorithms: i.e., l 1 -norm min- imization and a hybrid approach which combined Wideband Acoustical Holography (WBH) and l 1 - norm minimization. It was found that both algorithms could successfully localize and visualize the major noise sources over a broad range of frequencies, even when using a relatively small number of microphones. Finally, comments are made on sound field reconstruction differences between the two algorithms. 1. INTRODUCTION

Diesel engine noise is a concern in both municipal environments and in an occupational health and safety context due to the use of diesel engines in heavy industry and transportation. In order to reduce engine noise levels, an accurate knowledge of the major noise source locations is required to guide the efficient application of noise control resources. Near-field Acoustical Holography (NAH) is a useful method for reconstructing sound fields based on microphone array measurements close to a sound source, which can be used to visualize and identify sound source locations 1,2 . However, when

1 charlesshi720@gmail.com

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applying conventional holography approaches, e.g., the Fourier transform method, or Statistically Optimized Near-field Acoustical Holography (SONAH) 3 , etc., a large number of measurements is usually required in order to avoid spatial aliasing and ensure good spatial resolution. Thus, visualizing diesel engine noise sources with conventional NAH methods can be time consuming, economically expensive and hard to perform in industrial environments.

Previously, Wideband Acoustical Holography (WBH) 4 with the monopole based Equivalent Source Method (ESM) was applied to visualize diesel engine noise based on a limited number of measurements 5 . In order to further increase sound field reconstruction accuracy, the compressive sensing concept was then combined with monopole based ESM 6 . Previous studies showed that with the compressive sensing concept, sound sources can be identified when the number of measurements is significantly smaller than the number of parameters that need to be determined 7,8 . In the present work, one set of thirty-five channel array measurements was taken in front of a diesel engine, and then the noise sources on the engine front face were identified by using two compressive sensing algorithms: l 1 -norm minimization and a hybrid approach 9 which combined Wideband Acoustical Ho- lography (WBH) and l 1 -norm minimization. Finally, the noise source reconstruction results of two algorithms were compared and commented upon. 2. THEORY

2.1. Introduction to Compressive Sensing Compressive sensing 10 , also known as compressive sampling, is a signal processing technique based on the principle that through optimization, a sparse signal can be recovered by finding solutions from an underdetermined linear system. With compressive sensing, the sparsity of a signal can be exploited to recover it from far fewer samples than would seem to be required by the Nyquist-Shan- non sampling theorem. However, two conditions are required under which the recovery is possible: sparsity and incoherence. Sparsity requires the signal to be sparse in the search domain, and incoher- ence is applied through the isometric property.

2.2. The l 1 -Norm Minimization To find the sparse solution from an underdetermined linear system, the first recovery proce- dure that comes to mind is to search for the sparsest vector which satisfies the measurement. This leads to solving the l 0 -norm minimization problem.

𝑚𝑖𝑛‖𝐳‖ ! 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐀𝐳= 𝐲 , (1)

where the signal that needs to be recovered is the vector 𝐳∈ℂ " , the m linear measurement vector resulting from signal 𝐳 is 𝐲∈ℂ # , and finally 𝐀∈ℂ #×" is the transfer matrix. Unfortunately, an al- gorithm that solves Eq. (1) for any matrix A and any right-hand side 𝐲 is generally computationally intractable. Therefore, l 1 -norm minimization was proposed in the literature as a tractable alternative, which approximates l 0 -norm minimization by minimizing the sum of the vector z . The l 1 -norm minimization approach considers the solution of

𝑚𝑖𝑛‖𝐳‖ % 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐀𝐳= 𝐲 , (2)

which is a convex optimization problem 11,12 and can be seen as a convex approximation of Eq. (1). If the problem is real-valued, Eq. (2) is equivalent to a linear program and in the complex-valued case it is equivalent to second order cone programming. In the present work, the l 1 -norm minimization is applied to an equivalent source model with a mesh of monopole point sources fixed at certain loca- tions in a plane.

The expression 𝑃 ! for the sound pressure at position x generated by a monopole with unit source strength located at 𝑥 ! and at frequency w and wave number k is:

& !"#‖%!%&‖

𝑃 ! (𝑥 |𝑥 ! ) =

' p ‖)*) & ‖ . (3)

And ideally, the reconstructed sound pressure at the m- th microphone position should be equal to the measured sound pressure, 𝑝 # , on the hologram surface: i.e.,

𝑝 # = ∑ 𝐴 #, 𝑞 , - ,.% , (4)

where I is the number of monopoles, the 𝑞 , , i = 1, 2…, I are the complex amplitudes of those mono- poles which are the unknowns, and 𝐴 #, = exp (−𝑗𝑘‖𝑥 # −𝑥 , ‖/4𝜋‖𝑥 # −𝑥 , ‖) , represents the sound pressure at microphone m due to a unit excitation of the i -th monopole. Equation (4) can be written in matrix-vector form as:

𝐩= 𝐀𝐪 , (5)

where A is an M ´ I matrix with members 𝐴 #, , and where M is the total number of measurements, q is the source strength vector containing all the 𝑞 , ’s and p is the sound pressure measurement vector. In order to cover the potential sound source region with sufficient spatial resolution to pin- point the noise source of diesel engine, the number of monopoles is generally much greater than the number of measurements. From a previous study it was found that the major noise sources at the front of a diesel engine (described in Section 3) are usually different pulleys which occupy a relative small area compared with the whole diesel engine front face. Thus, the solution of Eq. (5) can be assumed sparse, so the compressive sensing concept can be applied in this problem to identify noise sources. Since in an experiment some measurement error is inevitable, Eq. (2) was modified to allow a 5% relative error between the reconstructed sound field and the measurement: i.e.,

𝑚𝑖𝑛‖𝐪‖ % 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 ‖𝐀𝐪−𝐩‖/‖𝐩‖ < 5% . (6)

The open-source convex optimization algorithm package CVX 13 was used to solve this l 1 -norm min- imization problem.

2.3. The Hybrid Method The l 1 -norm minimization can recover major noise sources by solving an underdetermined linear system. But in a previous study it was also found that when the second norm relative error is set to a value smaller than the real noise level, in order to satisfy the relative error constraint condition, the optimized solution from Eq. (6) tends to create weak, false noise source, i.e., ghost sources. Those ghost sources may mislead the observer as to the true noise source locations. But, if the relative error was set to a value larger than the real noise level, then it is also possible that the solution could miss true noise sources. To overcome the relative error selection difficulty, a hybrid method was proposed by Shi et al 9 . The hybrid method consists of two steps: first the solution is obtained through l 1 -norm minimization with a small relative error between reconstruction and measurement. Then secondly, the obtained solution (most likely containing ghost sources) is used as the initial solution for the WBH process. In this way, the solution from l 1 -norm minimization should contain all the major noise sources plus some ghost sources. Then in the iterative WBH process, the ghost sources created in the

l 1 -norm minimization solution are removed by enforcing the cardinality threshold in the WBH algo- rithm, with the result that the solution is concentrated at the correct source locations. More details on the WBH algorithm and the hybrid process can be found in previous studies 9,14 .

In the hybrid method used in the present work, the l 1 -norm minimization was formulated as:

𝑚𝑖𝑛‖𝐪‖ % 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 ‖𝐀𝐪−𝐩‖/‖𝐩‖ < 1% . (7)

The parameters used in WBH were:

𝛼= 0.5; 𝐷 ! = 5.1 dB; ∆𝐷= 1.0 𝑑B; 𝐷 /01 = 60 𝑑B; 𝑒 /01 = 0.05 . (8)

2.4. Principal Component Analysis As discussed in Section 2.1., one condition required to implement the compressive sensing concept is that the solution needs to be sparse. Despite the fact that major noise sources on the diesel engine front face are usually at concentrated locations, it is still possible that all the major sound sources may occupy a large region and jeopardize the sparsity condition. To avoid this, and to separate uncorrelated noise sources, e.g., combustion noise sources and mechanical noise sources 15 , principal component analysis was applied to the measured signal 16 . After the uncorrelated noise sources were decomposed, each noise source was reconstructed separately and added together to visualize the total sound field. In this section, the basis of the principal component analysis and the reconstruction of each partial field will be reviewed briefly. The sound field measured by the field microphones on the measurement surface can be ex- pressed as a product of the data measured by a set of reference signals multiplied by appropriate transfer functions 17 : that is, one can express this relation in matrix form,

𝐓 𝐫 , (9)

𝐩= 𝐇 𝒓𝒑

where 𝐇 56 is the transfer matrix that relates the measurement surface microphone signals, 𝐩= [𝑝 % , . . 𝑝 # ] 7 , and the reference signals 𝐫= [𝑟 % , . . 𝑟 8 ] 7 , in the frequency domain. Here n is the total number of references, m is the total number of measurement microphones, and the superscript T denotes the transpose operator. When considering statistically random sources, Eq. (7) can be ex- pressed in terms of cross-spectral matrices: i.e.,

9 𝐂 55

9 𝐂 55 𝐇 56 , (10)

*% 𝐂 56 = 𝐇 56

𝐂 66 = 𝐂 56

*% 𝐂 56 , the superscript H denotes the Hermitian operator, and the cross-spectral ma- trices 𝐂 55 , 𝐂 56 , and 𝐂 66 , are defined as,

where 𝐇 56 = 𝐂 55

𝐂 55 = E[𝑟 ∗ 𝑟 7 ] , (11)

𝐂 56 = E[𝑟 ∗ 𝑝 7 ] , (12)

𝐂 66 = E[𝑝 ∗ 𝑝 7 ] , (13)

where the superscript * denotes the complex conjugate and E[…] represents the expectation operator. In order to decompose the total sound field into a set of uncorrelated sound fields, Singular Value Decomposition (SVD) was performed on the composite sound field: i.e.,

𝐂 55 = 𝐔Σ𝐕 9 = 𝐔Σ𝐔 9 , (14)

where å is a diagonal matrix composed of singular values, and 𝜆 , , U , V are unitary matrices that comprise the left and right singular vectors. Since 𝐂 55 is a positive, semi-definite Hermitian matrix, U and V are identical matrices in this case. The decomposed partial fields are than expressed as:

𝐩a = 𝐂 56 7 𝐔 ∗ Σ * '

( = 𝐇 56 7 𝐔Σ * '

( , (15)

where the i -th column of vector 𝐩a represents the i -th uncorrelated noise source. 3. DIESEL ENGINE EXPERIMENT

The experiment was conducted in the Cummins Walesboro Noise and Vibration Lab. The test engine was an ISF-3.8, which is a four-cylinder 3.8-liter diesel engine. A Brüel & Kjær 36-channel microphone array was placed parallel to the diesel engine front face, 0.58 m from the diesel engine. The measurement position was relatively distant from the engine because there was an intake pipe placed in front of the engine, so that a closer measurement was not possible. The setup of the exper- iment is shown in Figure 1. Two diesel engine operation conditions were considered: 750 rpm, idle and 1000 rpm, idle. During the test, it was noted that one microphone was not working properly, so measurements were made with the remaining 35 microphones. The measurements in the time domain were acquired using a Brüel & Kjær LAN-XI system, the sampling frequency was chosen as 25.6 kHz, and the measurement duration was 10 seconds for each test. Then the acquired signal was transformed from the time domain to the frequency domain with 3200 Fourier transform points cor- responds to a segment length that was 1/50 of the total time length: i.e., 0.2 s. A Hann window was applied to the data in the time domain, and the overlaps between each segment were half a segment length, so the result in the frequency domain was averaged 100 times. Then Welch’s averaged peri- odogram method 18 was used to calculate the cross-power spectral density matrix, after which the principal component analysis was performed to separate the uncorrelated noise sources. In this ex- periment, the measurement channels were also used as reference channels.

Figure 1: Measurement setup in Cummins Walesboro Noise Lab in Columbus, IN.

The equivalent source plane of monopoles was placed 0.6 m from the measurement micro- phone array, that is 0.02 m behind the diesel engine front face. The reconstruction plane was 0.58 m from the measurement array: i.e., at the position of the diesel engine front face. In order to cover the whole engine front face, the plane of monopole was 0.68 m wide by 0.48 m tall, and the monopole spacing was 0.01 m in both directions, so there were 3381 monopoles in the equivalent source model.

Since the noise sources that radiate energy to the far field are of interest, the holography results will be shown in terms of acoustic intensity. The sound pressure, 𝑃 ! , at x , generated by a monopole at 𝑥 ! is given by Eq. (3). Then, by using the Euler equation, the particle velocity reconstruction result at 𝑥 ! , which is directed along the line joining the monopole and the field position, is:

;

%

>?‖)*) & ‖ ) ! (16)

𝑉=

<= (1 +

where r is the density of air, c is the speed of sound in the air, and k is the wave number. Next, the velocity component normal to the reconstruction surface, 𝑉 8 , was calculated. Finally, the intensity reconstruction result can be calculated by multiplying the pressure and normal velocity to give the intensity: i.e.,

%

@ Re(𝑃𝑉 ∗ ) . (17)

𝐼=

Figure 2 shows the principal component analysis result at 750 rpm, idle operation condition. The first and second sources are contributing most of the energy to the total sound field, so the first two uncorrelated sound sources were reconstructed and added together to visualize the sound field. Based on the spectrum, the sound field was reconstructed to identify the sound source locations at several peak frequencies: i.e., 160 Hz, 744 Hz, and 1040 Hz.

Figure 2: Principal component analysis result for diesel engine operating at 750 rpm, idle.

The reconstructed intensity distribution results computed using both the l 1 -norm minimization and the hybrid methods are shown in Figures 3, 4 and 5. At 160 Hz, both algorithms identified similar noise source locations: i.e., the crank pulley and the pipe in front of the engine. When comparing the hybrid process reconstructed result with l 1 -norm minimization result, the hybrid process eliminated small sources, e.g., the hot spots at the corners, and assigned more weighting to the pipe position. It is noted that 160 Hz is a very low frequency which generally makes it difficult to identify accurate sound source location, since the wavelength is 2.14 m, which is larger than the engine dimension, so the reconstruction result can only be used as a reference even though the reconstruction result indi- cates reasonable source location. At 744 Hz, both algorithms also identified similar sound source locations, which were the front pulleys, pipe and a sound source on the left-hand side which was half

out of the picture. The hot spot on the left-hand side may indicate sound coming from the left side of the engine. The reconstruction result obtained using the hybrid method gives a clearer indication of the sound source location compared with the l 1 -norm minimization. Finally, at 1040 Hz, both algo- rithms found the noise sources that were close to the hub pulley.

(a) (b) Figure 3: Diesel engine front face sound intensity field reconstruction results at 160 Hz: (a) l 1 -norm minimization, (b) the hybrid method.

(a) (b) Figure 4: Diesel engine front face sound intensity field reconstruction results at 744 Hz: (a) l 1 -norm minimization, (b) the hybrid method.

(a) (b) Figure 5: Diesel engine front face sound intensity field reconstruction results at 1040 Hz: (a) l 1 - norm minimization, (b) the hybrid method.

The second test condition was the engine running at 1000 rpm, idle. The principal component analysis result is shown in Figure 6, and based on these spectra, the sound field was reconstructed at peak frequencies 104 Hz, 728 Hz, and 1720 Hz. The intensity reconstruction results are shown in Figures 7, 8 and 9.

Figure 6: Principal component analysis result for diesel engine operating at 1000 rpm, idle.

At 104 Hz, the l 1 -norm minimization and the hybrid process identified the crank pulley, front pipe, and the pulley at the upper right corner as noise sources, and it seems that the hybrid method eliminated the ghost source found by the l 1 -norm minimization at the top of the engine. Again, at very low frequency the holography result can only be used as a general guide. At 728 Hz, both algorithms found the front crank pulley, front pipe, and a noise source at the lower left corner, which could be sound from left side of the engine. At 1720 Hz, the algorithms identified the major sound source to be the crank pulley at the center of the engine, and the upper left pulley as a secondary noise source. Then the hybrid method also identified the pulley on the right-hand side as a noise source which was not found with l 1 -norm minimization.

It is noted that since the diesel engine is a complex noise source and the number of measure- ments was relatively small, the reconstruction results can only be used as a reference, and there is still a need for further near-field measurements to confirm the results.

ae ae ee ee ee

(a) (b) Figure 7: Diesel engine front face sound intensity field reconstruction results at 104 Hz: (a) l 1 -norm minimization, (b) the hybrid method.

(a) (b) Figure 8: Diesel engine front face sound intensity field reconstruction results at 738 Hz: (a) l 1 -norm minimization, (b) the hybrid method.

(a) (b) Figure 9: Diesel engine front face sound intensity field reconstruction results at 1720 Hz: (a) l 1 -norm minimization, (b) the hybrid method. 4. CONCLUSIONS

In the present article, an equivalent source model composed of a monopole distribution at fixed locations in combination with two compressive sensing concept algorithms was discussed. This acoustical holography process was applied to a diesel engine test with one set of 35 channel meas- urements, and the major noise sources at different frequencies could be successfully visualized and localized even though there were a relatively small number of measurements at a relatively large measurement distance from the engine. 5. ACKNOWLEDGEMENT

The authors are grateful for the financial support of this research provided by Cummins, Inc. 6. REFERENCE

1. P.A. Nelson and S.H. Yoon, “Estimation of acoustic source strength by inverse methods: Part I,

conditioning of the inverse problem”, J. Sound Vib . 233(4) , 639-664 (2000). 2. S.H. Yoon, and P.A. Nelson, “Estimation of acoustic source strength by inverse methods: Part II,

experimental investigation of methods for choosing regularization parameters”, J. Sound Vib . 233(4) , 665-701 (2000). 3. J. Hald, “Basic theory and properties of statistically optimized near-field acoustical holog-

raphy”, J. Acoust. Soc. Am. 125(4) , 2105-2120 (2009). 4. J. Hald, “Fast wideband acoustical holography”, J. Acoust. Soc. Am. 139(4) , 1508-1517 (2016).

5. T. Shi, Y. Liu, J.S. Bolton, F. Eberhardt, and W. Frazer, “Diesel engine noise source visualization

with wideband acoustical holography” SAE Technical Paper, No. 2017-01-1874. 6. G. Chardon, L. Daudet, A. Peillot, F. Ollivier, N. Bertin and R. Gribonval, “Near-field acoustic

holography using sparse regularization and compressive sampling principles”, J. Acoust. Soc. Am. 132(3) , 1521-1534 (2012). 7. T. Shi, Y. Liu, and J.S. Bolton, “Noise source identification in an under-determined system by

convex optimization”, Proc. INTER-NOISE and NOISE-CON 2018, Chicago, Illinois, USA, Vol. 258 , No. 6, pp. 1308-1318. Institute of Noise Control Engineering. 8. T. Shi, W. Thor, J.S. Bolton, “Near-field acoustical holography incorporating compressive sens-

ing”, Proc. INTER-NOISE and NOISE-CON 2019, San Diego, California, USA, Vol. 260 , No.1, pp. 683-698 . Institute of Noise Control Engineering. 9. T. Shi, Y. Liu, and J.S. Bolton, “Spatially sparse sound source localization in an under-determined

system by using a hybrid compressive sensing method”, J. Acoust. Soc. Am. 146(2) , 1219-1229 (2019). 10. M. Fornasier, and R. Holger, "Compressive Sensing." Handbook of Mathematical Methods in

Imaging 1, 187-229 (2015). 11. J. Friedman, T. Hastie, and R. Tibshirani, “Sparse inverse covariance estimation with the graph-

ical lasso”, Biostatistics , 9(3) , 432-441. 12. S. Chen, and D. Donoho, “Basis pursuit”, Proc. 1994 28th Asilomar Conference on Signals, Sys-

tems and Computers, Vol. 1 , pp. 41-44. IEEE. 13. M. Grant, and S. Boyd, “CVX: Matlab software for disciplined convex programming”, version

2.1, 2014. 14. T. Shi, Y. Liu, and J.S. Bolton, “The use of wideband acoustical holography for noise source

visualization”, Proc. INTER-NOISE and NOISE-CON 2016, Providence, Rhode Island, USA, Vol. 252, No. 2, pp. 479-490. Institute of Noise Control Engineering. 15. F.R. Eberhardt, “Study of the Feasibility of Estimation Combustion Noise Radiation in Reverber-

ant Environment”, MSME Thesis , Purdue University , 2011 16. M. Lee, and J.S. Bolton, “Scan-based near-field acoustical holography and partial field decom-

position in the presence of noise and source level variation”, J. Acoust. Soc. Am. , 119(1) , 382-393 (2006). 17. H.S. Kwon and J.S. Bolton, “Partial Field Decomposition in Near-field Acoustical Holography

by the Use of Singular Value Decomposition and Partial Coherence Procedures”, Proc. NOISE- CON 98 , PP. 649-654 (1998). 18. P. Welch, “The use of fast Fourier transform for the estimation of power spectra: a method based

on time averaging over short, modified periodograms”, IEEE Transactions on Audio and Elec- troacoustics , 15(2) , pp.70-73 (1967).