Ultraspherical/Gegenbauer polynomials to unify 2D/3D Ambisonic directivity designs
Abstract: This report on axisymmetric ultraspherical/Gegenbauer polynomials and their use in Ambisonic directivity design in 2D and 3D presents an alternative mathematical formalism to what can be read in, e.g., my and Matthias Frank's book on Ambisonics or J\'er^ome Daniel's thesis, Gary Elko's differential array book chapters, or Boaz Rafaely's spherical microphone array book. Ultraspherical/Gegenbauer polynomials are highly valuable when designing axisymmetric beams and understanding spherical t designs, and this report will shed some light on what circular, spherical, and ultraspherical axisymmetric polynomials are. While mathematically interesting by themselves already, they can be useful in spherical beamforming as described in the literature on spherical and differential microphone arrays. In this report, these ultraspherical/Gegenbauer polynomials will be used to uniformly derive for arbitrary dimensions D the various directivity designs or Ambisonic order weightings known from literature: max-DI/basic, max-rE , supercardioid, cardioid/inphase. Is there a way to relate higher-order cardioids and supercardioids? How could one define directivity patterns with an on-axis flatness constraint?
- P. Craven and M. A. Gerzon, “Coincident microphone simulation covering three dimensional space and yielding various directional outputs,” U.S. Patent, no. 4,042,779, 1977.
- D. H. Cooper and T. Shiga, “Discrete-matrix multichannel stereo,” J. Audio Eng. Soc., vol. 20, no. 5, pp. 346–360, 1972.
- J. Daniel, “Représentation des champs acoustiques, application à la transmission et à la reproduction de scènes sonores complexes dans un contexte multimédia,” Ph.D. dissertation, Université Paris 6, 2001.
- B. Devaraju, “Understanding filtering on the sphere,” Ph.D. dissertation, Universität Stuttgart, 2015. [Online]. Available: https://elib.uni-stuttgart.de/bitstream/11682/4002/1/BDevarajuPhDThesis.pdf
- P. Delsarte, J.-M. Goethals, and J. J. Seidel, “Spherical codes and designs,” Geometriae Dedicata, vol. 6, no. 3, 1977.
- J. Daniel, J.-B. Rault, and J.-D. Polack, “Acoustic properties and perceptive implications of stereophonic phenomena,” in AES 6th Int. Conf.: Spatial Sound Reproduction, 1999.
- G. W. Elko, “Superdirectional microphone arrays,” in Acoustic Signal Processing for Telecommunication, J. Benesty and S. L. Gay, Eds. Kluwer Academic Publishers, 2000.
- ——, “Differential microphone arrays,” in Audio Signal Processing for Next-Generation Multimedia Communication Systems, Y. Huang and J. Benesty, Eds. Springer, 2004.
- P. Felgett, “Ambisonic reproduction of directionality in surround-sound systems,” Nature, vol. 252, pp. 534–538, 1974.
- L. Gegenbauer, “Über die Functionen Cnν(x)superscriptsubscript𝐶𝑛𝜈𝑥C_{n}^{\nu}(x)italic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT ( italic_x ),” Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Classe, vol. 57, no. 8, 1877. [Online]. Available: https://viewer.acdh.oeaw.ac.at/viewer/image/MN_2Abt_75_1877/904/LOG_0074/
- M. A. Gerzon, “The design of precisely coincident microphone arrays for stereo and surround sound,” in prepr. L-20 of 50th Audio Eng. Soc. Conv., 1975.
- M. Gräf and D. Potts, “On the computation of spherical designs by a new optimization approach based on fast spherical fourier transforms,” Numer. Math., vol. 119, 2011. [Online]. Available: http://homepage.univie.ac.at/manuel.graef/quadrature.php
- R. H. Hardin and N. J. A. Sloane, “Mclaren’s improved snub cube and other new spherical designs in three dimensions,” Discrete and Computational Geometry, vol. 15, pp. 429–441, 1996. [Online]. Available: http://neilsloane.com/sphdesigns/dim3/
- A. G. Law and M. B. Sledd, “Normalizing orthogonal polynomials by using their recurrence coefficients,” Proc. American Mathematical Society, vol. 48, no. 2, pp. 505–507, 1975.
- F. Zotter and M. Frank, “All-round ambisonic panning and decoding,” Journal of the Audio Eng. Soc., 2012.
- F. Zotter, S. Riedel, and M. Frank, “All-round ambisonic decoding: Spread and correlation,” in Fortschritte der Akusti, DAGA, Stuttgart, 2022. [Online]. Available: https://pub.dega-akustik.de/DAGA_2022/data/articles/000344.pdf
- F. Zotter, M. Zaunschirm, M. Frank, and M. Kronlachner, “A beamformer to play with wall reflections: The icosahedral loudspeaker,” Computer Music Journal, vol. 41, no. 3, 2017.
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