Array measurements can be contaminated by strong noise, especially when dealing with microphones located near or in a flow. The denoising of these measurements is crucial to allow efficient data analysis or source imaging. In this paper, a denoising approach based on a Probabilistic Factor Analysis is proposed. It relies on a decomposition of the measured cross-spectral matrix (CSM) using the inherent correlation structure of the acoustical field and of the flow-induced noise. This method is compared with three existing approaches, aiming at denoising the CSM, without any reference or background noise measurements and without any information about the sources of interest. All these methods make the assumption that the noise is statistically uncorrelated over the microphones, and only one of them significantly impairs the off-diagonal terms of the CSM. The main features of each method are first reviewed, and the performances of the methods are then evaluated by way of numerical simulations along with measurements in a closed-section wind tunnel.
References
1.
R. O.
Schmidt
, “Multiple emitter location and signal parameter estimation
,” in Proceedings of the RADC Spectrum Estimation Workshop
, Rome, Italy (October 3–5, 1979
), pp. 243
–258
.2.
J. Y.
Chung
, “Rejection of flow noise using a coherence function method
,” J. Acoust. Soc. Am.
62
(2
), 388
–395
(1977
).3.
W.
Burdic
, Underwater Acoustic System Analysis
(Prentice-Hall
, Englewood Cliffs, NJ
, 1991
).4.
J. A.
Cadzow
, “Signal enhancement-a composite property mapping algorithm
,” IEEE Trans. Acoust. Speech Signal Process.
36
(1
), 49
–62
(1988
).5.
P.
Forster
, “Generalized rectification of cross spectral matrices for arrays of arbitrary geometry
,” IEEE Trans. Signal Process.
49
(5
), 972
–978
(2001
).6.
P.
Forster
and T.
Asté
, “Rectification of cross spectral matrices for arrays of arbitrary geometry
,” in Proceedings of the 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing
, Phoenix, AZ (March 15–19, 1999
), pp. 2829
–2832
.7.
B. A.
Fenech
, “Accurate aeroacoustic measurements in closed-section hard-walled wind tunnels
,” Ph.D. thesis, University of Southampton
, Southampton, UK
, 2009
.8.
J.
Bulté
, “Acoustic array measurements in aerodynamic wind tunnels: A subspace approach for noise suppression
,” in Proceedings of the 13th AIAA/CEAS Aeroacoustics Conference
, Rome, Italy (May 21–23, 2007
).9.
B.
Arguillat
, D.
Ricot
, C.
Bailly
, and G.
Robert
, “Measured wavenumber: Frequency spectrum associated with acoustic and aerodynamic wall pressure fluctuations
,” J. Acoust. Soc. Am.
128
(4
), 1647
–1655
(2010
).10.
P.
Druault
, A.
Hekmati
, and D.
Ricot
, “Discrimination of acoustic and turbulent components from aeroacoustic wall pressure field
,” J. Sound Vib.
332
(26
), 7257
–7278
(2013
).11.
E.
Salze
, C.
Bailly
, O.
Marsden
, E.
Jondeau
, and D.
Juvé
, “An experimental characterization of wall pressure wavevector-frequency spectra in the presence of pressure gradients
,” in Proceedings of the 20th AIAA/CEAS Aeroacoustics Conference
, Atlanta, GA (June 16–20, 2014
), p. 2909
.12.
K.
Ehrenfried
, L.
Koop
, A.
Henning
, and K.
Kaepernick
, “Effects of wind-tunnel noise on array measurements in closed test sections
,” in Proceedings of BeBeC-2006
, Berlin, Germany (November 22–23, 2006
).13.
L.
Koop
and K.
Ehrenfried
, “Microphone-array processing for wind-tunnel measurements with strong background noise
,” in Proceedings of the 14th AAIA/CEAS Aeroacoustics Conference
, Vancouver, British Columbia, Canada (May 5–7, 2008
), p. 2907
.14.
D.
Blacodon
, “Spectral estimation method for noisy data using a noise reference
,” Appl. Acoust.
72
(1
), 11
–21
(2011
).15.
D.
Blacodon
, “Array processing for noisy data: Application for open and closed wind tunnels
,” AIAA J.
49
(1
), 55
–66
(2011
).16.
C. J.
Bahr
and W. C.
Horne
, “Advanced background subtraction applied to aeroacoustic wind tunnel testing
,” in Proceedings of the 21st AIAA/CEAS Aeroacoustics Conference
, Dallas, TX (June 22–26, 2015
).17.
S.
Jaeger
, W.
Horne
, and C.
Allen
, “Effect of surface treatment on array microphone self-noise
,” in Proceedings of the 6th AIAA/CEAS Aeroacoustics Conference
, Lahaina, HI (June 12–14, 2000
), p. 1937.18.
R. P.
Dougherty
, “Directional acoustic attenuation of planar foam rubber windscreens for phased arrays
,” in Proceedings of BeBeC-2012
, Berlin, Germany (February 22–23, 2012
).19.
Q.
Leclère
and C.
Picard
, “Acoustic beamforming through a thin plate using vibration measurements
,” J. Acoust. Soc. Am.
137
(6
), 3385
–3392
(2015
).20.
D.
Lecoq
, C.
Pézerat
, J.-H.
Thomas
, and W.
Bi
, “Extraction of the acoustic component of a turbulent flow exciting a plate by inverting the vibration problem
,” J. Sound Vib.
333
(12
), 2505
–2519
(2014
), Vol. 1.21.
J.
Benesty
, J.
Chen
, and Y.
Huang
, Microphone Array Signal Processing
(Springer Science & Business Media
, New York
, 2008
).22.
S.
Boll
, “Suppression of acoustic noise in speech using spectral subtraction
,” IEEE Trans. Acoust. Speech Signal Process.
27
(2
), 113
–120
(1979
).23.
S. H.
Jensen
, P. C.
Hansen
, S. D.
Hansen
, and J. A.
Sorensen
, “Reduction of broad-band noise in speech by truncated QSVD
,” IEEE Trans. Speech Audio Process.
3
(6
), 439
–448
(1995
).24.
C. R.
Lowis
, “In-duct measurement techniques for the characterisation of broadband aeroengine noise
,” Ph.D. thesis, University of Southampton
, Southampton, UK
, 2007
.25.
P.
Sijtsma
, A.
Dinsenmeyer
, J.
Antoni
, and Q.
Leclere
, “Beamforming and other methods for denoising microphone array data
,” in Proceedings of the 25th AIAA/CEAS Aeroacoustics Conference
, Delft, the Netherlands
(May 20–23, 2019
), p. 2653
.26.
J.
Hald
, “Removal of incoherent noise from an averaged cross-spectral matrix
,” J. Acoust. Soc. Am.
142
(2
), 846
–854
(2017
).27.
M.
Grant
and S.
Boyd
, “CVX: Matlab software for disciplined convex programming, version 2.1 [computer program]
,” http://cvxr.com/cvx (Last viewed 3/31/2020).28.
M.
Grant
and S.
Boyd
, “Graph implementations for nonsmooth convex programs
,” in Recent Advances in Learning and Control
, edited by V.
Blondel
, S.
Boyd
, and H.
Kimura
(Springer-Verlag Limited
, New York
, 2008
), pp. 95
–110
.29.
R. H.
Tütüncü
, K.-C.
Toh
, and M. J.
Todd
, “Solving semidefinite-quadratic-linear programs using sdpt3
,” Math. Program.
95
(2
), 189
–217
(2003
).30.
R. P.
Dougherty
, “Cross spectral matrix diagonal optimization
,” in Proceedings of the 6th Berlin Beamforming Conference
, Berlin, Germany (February 29–March 1, 2016
).31.
Q.
Leclere
, N.
Totaro
, C.
Pézerat
, F.
Chevillotte
, and P.
Souchotte
, “Extraction of the acoustic part of a turbulent boundary layer from wall pressure and vibration measurements
,” in Proceedings of INTER-NOISE and NOISE-CON Congress and Conference
, San Francisco, CA (August 9–12, 2015
), pp. 816
–824
.32.
S. H.
Yoon
and P.
Nelson
, “A method for the efficient construction of acoustic pressure cross-spectral matrices
,” J. Sound Vib.
233
, 897
–920
(2000
).33.
L.
Yu
, J.
Antoni
, and Q.
Leclère
, “Spectral matrix completion by cyclic projection and application to sound source reconstruction from non-synchronous measurements
,” J. Sound Vib.
372
, 31
–49
(2016
).34.
J.
Wright
, A.
Ganesh
, S.
Rao
, Y.
Peng
, and Y.
Ma
, “Robust principal component analysis: Exact recovery of corrupted low-rank matrices via convex optimization
,” in Proceedings of Advances in Neural Information Processing Systems
, Vancouver, BC, Canada (December 7–10, 2009
), pp. 2080
–2088
.35.
A.
Finez
, A.
Pereira
, and Q.
Leclère
, “Broadband mode decomposition of ducted fan noise using cross-spectral matrix denoising
,” in Proceedings of Fan Noise 2015
, Lyon, France (April 15–17, 2015
).36.
S.
Amailland
, J.-H.
Thomas
, C.
Pézerat
, and R.
Boucheron
, “Boundary layer noise subtraction in hydrodynamic tunnel using robust principal component analysis
,” J. Acoust. Soc. Am.
143
(4
), 2152
–2163
(2018
).37.
A.
Sobral
, T.
Bouwmans
, and E.-H.
Zahzah
, “Lrslibrary: Low-rank and sparse tools for background modeling and subtraction in videos
,” in Robust Low-Rank and Sparse Matrix Decomposition: Applications in Image and Video Processing
(CRC Press
, Boca Raton, FL
, 2015
).38.
See https://github.com/andrewssobral/lrslibrary/ (Last viewed March 31,
2020
).39.
J.
Hald
, “Denoising of cross-spectral matrices using canonical coherence
,” J. Acoust. Soc. Am.
146
(1
), 399
–408
(2019
).40.
41.
D.
Gamerman
and H. F.
Lopes
, Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference
(Chapman and Hall/CRC
, London
, 2006
).42.
A.
Gelman
, J. B.
Carlin
, H. S.
Stern
, D. B.
Dunson
, A.
Vehtari
, and D. B.
Rubin
, Bayesian Data Analysis
(Chapman and Hall/CRC
, London
, 2014
).43.
J.
Antoni
, C.
Vanwynsberghe
, T.
Le Magueresse
, S.
Bouley
, and L.
Gilquin
, “Mapping uncertainties involved in sound source reconstruction with a cross-spectral-matrix-based Gibbs sampler
,” J. Acoust. Soc. Am.
146
(6
), 4947
–4961
(2019
).44.
T.
Eltoft
, T.
Kim
, and T.-W.
Lee
, “On the multivariate laplace distribution
,” IEEE Signal Process. Lett.
13
(5
), 300
–303
(2006
).45.
E.
Sarradj
, G.
Herold
, P.
Sijtsma
, R.
Merino-Martinez
, A.
Malgoezar
, M.
Snellen
, T.
Geyer
, C.
Bahr
, R.
Porteous
, D.
Moreau
, and C.
Doolan
, “A microphone array method benchmarking exercise using synthesized input data
,” in Proceedings of the 23rd AIAA/CEAS Aeroacoustics Conference
, Denver, CO (June 5–9, 2017
).46.
A.
Dinsenmeyer
, J.
Antoni
, Q.
Leclere
, and A.
Pereira
, “On the denoising of cross-spectral matrices for (aero) acoustic applications
,” in Proceedings of the 7th Berlin Beamforming Conference
, Berlin, Germany (March 5–6, 2018
).47.
G. M.
Corcos
, “Resolution of pressure in turbulence
,” J. Acoust. Soc. Am.
35
(2
), 192
–199
(1963
).48.
A.
Dinsenmeyer
, Q.
Leclere
, J.
Antoni
, and E.
Julliard
, “Comparison of microphone array denoising techniques and application to flight test measurements
,” in Proceedings of the 25th AIAA/CEAS Aeroacoustics Conference
, Delft, the Netherlands (May 20–23, 2019
), p. 2744.49.
P.
Ahrendt
, “The multivariate Gaussian probability distribution
,” in Technical Report
(Technical University of Denmark
, Lyngby, Denmark
, 2005
).50.
K. B.
Petersen
and M. S.
Pedersen
, “The matrix cookbook (version: November 15, 2012
),” http://www.math.uwaterloo.ca/∼hwolkowi/matrixcookbook.pdf (Last viewed 3/31/2020)© 2020 Acoustical Society of America.
2020
Acoustical Society of America
You do not currently have access to this content.