We present a method for computing a spherical harmonic representation of a sound field based on observations of the sound pressure along the equator of a rigid spherical scatterer. Our proposed solution assumes that the captured sound field is height invariant so that it can be represented by a two-dimensional (2D) plane wave decomposition (PWD). The 2D PWD is embedded in a three-dimensional representation of the sound field, which allows for perfectly undoing the effect of the spherical scattering object. If the assumption of height invariance is fulfilled, then the proposed solution is at least as accurate as a conventional spherical microphone array of the same spherical harmonic order, which requires a multiple of the number of sensors. Our targeted application is binaural rendering of the captured sound field. We demonstrate by analyzing the binaural output signals that violations of the assumptions that the solution is based on—particularly height invariance and consequently also horizontal propagation—lead to errors of moderate magnitude.
References
1.
Abhayapala
, T. D.
, and Ward
, D. B.
(2002
). “Theory and design of high order sound field microphones using spherical microphone array
,” in Proceedings of the 2002 IEEE International Conference on Acoustics, Speech, and Signal Processing
, May 13–17, Orlando, FL, Vol. 2
, pp. 1949
–1952
.2.
Ahrens
, J.
(2012
). Analytic Methods of Sound Field Synthesis
(Springer
, Heidelberg, Germany
).3.
Ahrens
, J.
, and Andersson
, C.
(2019
). “Perceptual evaluation of headphone auralization of rooms captured with spherical microphone arrays with respect to spaciousness and timbre
,” J. Acoust. Soc. Am.
145
(4
), 2783
–2794
.4.
Ahrens
, J.
, Helmholz
, H.
, Alon
, D.
, and Amengual Gari
, S.
(2021a
). “Audio examples accompanying the article ‘Spherical harmonic decomposition of a sound field based on observations along the equator of a rigid spherical scatterer’, JASA (2021)
,” https://zenodo.org/record/4805266/export/hx#.YPWDuehKhPY (Last viewed May 31, 2021).5.
Ahrens
, J.
, Helmholz
, H.
, Alon
, D.
, and Amengual Gari
, S.
(2021b
). “Audio examples accompanying the paper ‘The far-field equatorial array for binaural rendering’, IEEE ICASSP (2021)
,” https://zenodo.org/record/4807499#.YPWEOOhKhPY (Last viewed May 31, 2021).6.
Ahrens
, J.
, Helmholz
, H.
, Alon
, D.
, and Amengual Gari
, S.
(2021c
). “The far-field equatorial array for binaural rendering
,” in Proceedings of ICASSP 2021—2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
, June 6–11, Toronto, Canada.7.
Ahrens
, J.
, and Spors
, S.
(2008
). “An analytical approach to sound field reproduction using circular and spherical loudspeaker distributions
,” Acta Acust. united Acust.
94
(6
), 988
–999
.8.
Ahrens
, J.
, and Spors
, S.
(2012
). “Wave field synthesis of a sound field described by spherical harmonics expansion coefficients
,” JASA
131
(3
), 2190
–2199
.9.
Algazi
, V. R.
, Duda
, R. O.
, and Thompson
, D. M.
(2004
). “Motion-tracked binaural sound
,” J. Audio Eng. Soc.
52
(11
), 1142
–1156
.10.
Bernschütz
, B.
(2013
). “A spherical far field HRIR/HRTF compilation of the Neumann KU100
,” in Proceedings of the 40th Italian (AIA) Annual Conference on Acoustics and the 39th German Annual Conference on Acoustics (DAGA)
, March 18–21, Meran, Italy, pp. 592
–595
.11.
Bernschütz
, B.
(2016
). “Microphone arrays and sound field decomposition for dynamic binaural recording
,” Ph.D. thesis, Technische Universität Berlin
, Berlin, Germany
.12.
Bernschütz
, B.
(2020
). “Spherical far-field HRIR compilation of the Neumann KU100
,” https://zenodo.org/record/3928297##.X-8-zKpKhBw (Last viewed May 31, 2021).13.
Betlehem
, T.
, and Poletti
, M.
(2019
). “Measuring the spherical-harmonic representation of a sound field using a cylindrical array
,” in Proceedings of ICASSP 2019—2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
, pp. 955
–959
.14.
Blauert
, J.
(1996
). Spatial Hearing: The Psychophysics of Human Sound Localization
(MIT
, Cambridge, MA
).15.
Chen
, H.
, Abhayapala
, T.
, and Zhang
, W.
(2015
). “Theory and design of compact hybrid microphone arrays on two-dimensional planes for three-dimensional soundfield analysis
,” J. Acoust. Soc. Am.
138
(5
), 3081
–3092
.16.
Galindo
, M. B.
, Coleman
, P.
, and Jackson
, P. J.
(2020
). “Microphone array geometries for horizontal spatial audio object capture with beamforming
,” J. Audio Eng. Soc.
68
(5
), 324
–337
.17.
Gumerov
, N.
, and Duraiswami
, R.
(2005
). Fast Multipole Methods for the Helmholtz Equation in Three Dimensions
(Elsevier
, Amsterdam
).18.
Helmholz
, H.
, Alon
, D. L.
, Gari
, S. A.
, and Ahrens
, J.
(2020
). “Instrumental evaluation of sensor self-noise in binaural rendering of spherical microphone array signals
,” in Proceedings of e-Forum Acusticum
, December 7–11, Lyon, France.19.
Kaiser
, F.
, Pomberger
, H.
, and Zotter
, F.
(2012
). “Investigations on cylindrical microphone arrays
,” in Proceedings of Spatial Audio in Today's 3D World: 25th AES UK Conference
, March 25–27, York, UK.20.
Koyama
, S.
, Furuya
, K.
, Wakayama
, K.
, Shimauchi
, S.
, and Saruwatari
, H.
(2016
). “Analytical approach to transforming filter design for sound field recording and reproduction using circular arrays with a spherical baffle
,” J. Acoust. Soc. Am.
139
(3
), 1024
–1036
.21.
Lebedev
, V. I.
(1977
). “Spherical quadrature formulas exact to orders 25–29
,” Sib. Math. J.
18
(1
), 99
–107
.22.
Li
, Z.
, and Duraiswami
, R.
(2006
). “Headphone-based reproduction of 3D auditory scenes captured by spherical/hemispherical microphone arrays
,” in Proceedings of the 2006 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP)
, May 14–19, Toulouse, France.23.
Lübeck
, T.
, Helmholz
, H.
, Arend
, J. M.
, Pörschmann
, C.
, and Ahrens
, J.
(2020
). “Perceptual evaluation of mitigation approaches of impairments due to spatial undersampling in binaural rendering of spherical microphone array data
,” J. Audio Eng. Soc.
68
(6
), 428
–440
.24.
Meyer
, J.
(2001
). “Beamforming for a circular microphone array mounted on spherically shaped objects
,” J. Acoust. Soc. Am.
109
, 185
–193
.25.
Meyer
, J.
, and Elko
, G.
(2002
). “A highly scalable spherical microphone array based on an orthonormal decomposition of the soundfield
,” in Proceedings of the 2002 IEEE International Conference on Acoustics, Speech, and Signal Processing
, May 13–17, Orlando, FL, pp. 1781
–1784
.26.
Parthy
, A.
, Epain
, N.
, van Schaik
, A.
, and Jin
, C. T.
(2011
). “Comparison of the measured and theoretical performance of a broadband circular microphone array
,” J. Acoust. Soc. Am.
130
(6
), 3827
–3837
.27.
Rabenstein
, R.
, Spors
, S.
, and Ahrens
, J.
(2014
). “Sound field synthesis
,” edited by in Academic Press Library in Signal Processing
, Vol. 4
, edited by S.
Theodoridis
and R.
Chellappa
(Academic
, Chennai, India
), Chap. 32, pp. 915
–979
.28.
Rafaely
, B.
(2004
). “Plane-wave decomposition of the sound field on a sphere by spherical convolution
,” J. Acoust. Soc. Am.
116
, 2149
–2157
.29.
Rafaely
, B.
(2005
). “Analysis and design of spherical microphone arrays
,” IEEE Trans. Speech Audio Proc.
13
(1
), 135
–143
.30.
Rafaely
, B.
, and Avni
, A.
(2010
). “Interaural cross correlation in a sound field represented by spherical harmonics
,” J. Acoust. Soc. Am.
127
(2
), 823
–828
.31.
Teutsch
, H.
, and Kellermann
, W.
(2006
). “Acoustic source detection and localization based on wavefield decomposition using circular microphone arrays
,” J. Acoust. Soc. Am.
120
(5
), 2724
–2736
.32.
Thomas
, M. R. P.
, Ahrens
, J.
, and Tashev
, I.
(2014
). “A method for converting between cylindrical and spherical harmonic representations of sound fields
,” in Proceedings of the 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
, May 4–9, Florence, Italy, pp. 4723
–4727
.33.
Thresh
, L.
, Armstrong
, C.
, and Kearney
, G.
(2019
). “A direct comparison of localisation performance when using first, third and fifth order ambisonics for real loudspeaker and virtual loudspeaker rendering
,” in Proceedings of the 143rd Convention of the AES
, October 18–21, New York, Paper 9864.34.
Tiana-Roig
, E.
, Jacobsen
, F.
, and Fernandez-Grande
, E.
(2011
). “Beamforming with a circular array of microphones mounted on a rigid sphere (L)
,” J. Acoust. Soc. Am.
130
(3
), 1095
–1098
.35.
Trevino
, J.
, Koyama
, S.
, Sakamoto
, S.
, and Suzuki
, Y.
(2014
). “Mixed-order ambisonics encoding of cylindrical microphone array signals
,” Acoust. Sci. Technol.
35
(3
), 174
–177
.36.
Ward
, D. B.
, and Abhayapala
, T. D.
(2001
). “Reproduction of a plane-wave sound field using an array of loudspeakers
,” IEEE Trans. Speech Audio Process.
9
(6
), 697
–707
.37.
Weller
, T.
, Favrot
, S.
, and Buchholz
, J. M.
(2011
). “Application of a circular 2D hard-sphere microphone array for higher-order ambisonics auralization
,” in Proceedings of Forum Acusticum
, June 26–July 1, Aalborg, Denmark, pp. 2269
–2274
.38.
Wierstorf
, H.
, Geier
, M.
, and Spors
, S.
(2011
). “A free database of head related impulse response measurements in the horizontal plane with multiple distances
,” in Proceedings of the 130th Convention of the AES
, May 13–16, London, UK, e-Brief 6.39.
Williams
, E.
(1999
). Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography
(Academic
, New York
).40.
Zaunschirm
, M.
, Schörkhuber
, C.
, and Höldrich
, R.
(2018
). “Binaural rendering of ambisonic signals by head-related impulse response time alignment and a diffuseness constraint
,” J. Acoust. Soc. Am.
143
(6
), 3616
–3627
.41.
Zaunschirm
, M.
, and Zotter
, F.
(2014
). “Measurement-based modal beamforming using planar circular microphone arrays
,” in Proceedings of the EAA Joint Symposium on Auralization and Ambisonics
, April 3–5, Berlin, Germany, pp. 75
–80
.42.
Zotkin
, D. N.
, Duraiswami
, R.
, and Gumerov
, N. A.
(2010
). “Plane-wave decomposition of acoustical scenes via spherical and cylindrical microphone arrays
,” IEEE Trans. Audio Speech Lang. Process.
18
(1
), 2
–16
.43.
Zotter
, F.
(2009
). “Sampling strategies for acoustic holography/holophony on the sphere
,” Proceedings of NAG/DAGA
, March 23–26, Rotterdam, Netherlands, pp. 1107
–1110
.© 2021 Acoustical Society of America.
2021
Acoustical Society of America
You do not currently have access to this content.