Continuous Secondary Source Distributions

  • Jens Ahrens
Part of the T-Labs Series in Telecommunication Services book series (TLABS)


This chapter presents a number of analytic solutions to the problem of sound field synthesis in three and 2.5 dimensions, whereby continuous distributions of secondary sources are assumed. A focus lies on the explicit solution of the synthesis equation, which provides a perfect solution for enclosing secondary source distributions. The explicit solution is derived for spherical, circular, planar, and linear geometries. It is then shown that the well-known Near-field Compensated Higher Order Ambisonics approach is equivalent to the explicit solution for spherical secondary source distributions. The recently proposed Spectral Division Methods is identified as the extension of Near-field Compensated Higher Order Ambisonics to planar and linear secondary source distributions. Apart from the explicit solution, an implicit solution exists, which has become known as Wave Field Synthesis. The latter is derived from the Rayleigh Integral and its modern formulation for arbitrary complex secondary source distributions is outlined.


Plane Wave Sound Source Secondary Source Sound Field Convolution Theorem 
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  1. Abramowitz, M., & Stegun, I. A. (Eds.). (1968). Handbook of mathematical functions. New York: Dover Publications Inc.Google Scholar
  2. Ahrens, J., & Spors, S. (2008). An analytical approach to sound field reproduction using circular and spherical loudspeaker distributions. Acta Acustica utd. with Acustica, 94(6), 988–999.CrossRefGoogle Scholar
  3. Ahrens, J., & Spors, S. (2009a, August). An analytical approach to 2.5D sound field reproduction employing circular distributions of non-omnidirectional loudspeakers. In 17th European Signal Processing Conference (EUSIPCO) (pp. 814–818).Google Scholar
  4. Ahrens, J., & Spors, S. (2009b, October). On the secondary source type mismatch in wave field synthesis employing circular distributions of loudspeakers. In 127th Convention of the AES.Google Scholar
  5. Ahrens, J., & Spors, S. (2010a, March). An analytical approach to 2.5D sound field reproduction employing linear distributions of non-omnidirectional loudspeakers. In IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (pp. 105–108).Google Scholar
  6. Ahrens, J., & Spors, S. (2010b, March). An analytical approach to 3D sound field reproduction employing spherical distributions of non-omnidirectional loudspeakers. In IEEE International Symposium on Communication, Control and Signal Processing, (ISCCSP).Google Scholar
  7. Ahrens, J.,& Spors, S. (2010c, May). Applying the ambisonics approach on planar and linear arrays of loudspeakers. In 2nd International Symposium on Ambisonics and Spherical Acoustics.Google Scholar
  8. Ahrens, J., & Spors, S. (2010d, May). On the scattering of synthetic sound fields. In 130th Convention of the AES (p. 8121).Google Scholar
  9. Ahrens, J., & Spors, S. (2010e). Sound field reproduction using planar and linear arrays of loudspeakers. IEEE Transactions on Speech and Audio Processing, 18(8), 2038–2050.CrossRefGoogle Scholar
  10. Arfken, G., & Weber, H. (2005). Mathematical methods for physicists (6th ed.). San Diego: Elsevier Academic Press.Google Scholar
  11. Bamford, J. S. (1995). An analysis of ambisonics sound systems of first and second order. M.Sc. thesis, University of Waterloo, Ont. Canada.Google Scholar
  12. Berkhout, A. J. (1987). Applied seismic wave theory. Amsterdam: Elsevier Publishing Company.Google Scholar
  13. Berkhout, A. J., de Vries, D., & Vogel, P. (1993). Acoustic control by wave field synthesis. JASA, 93(5), 2764–2778.Google Scholar
  14. Betlehem, T., & Abhayapala, T. D. (2005). Theory and design of sound field reproduction in reverberant rooms. JASA, 117(4), 2100–2111.Google Scholar
  15. Caulkins, T., Warusfel, O. (2006, May). Characterization of the reverberant sound field emitted by a wave field synthesis driven loudspeaker array. In 120th Convention of the AES (p. 6712).Google Scholar
  16. Colton, D., & Kress, R. (1998). Inverse acoustic and electromagnetic scattering theory (2nd ed.). Berlin: Springer.Google Scholar
  17. Copley, L. G. (1968). Fundamental results concerning integral representations in acoustic radiation. JASA, 44, 28–32.zbMATHGoogle Scholar
  18. Corteel, E. (2006). Equalization in an extended area using multichannel inversion and wave field synthesis. JAES, 54(12), 1140–1161.Google Scholar
  19. D. de Vries, (2009). Wave field synthesis. AES Monograph. New York: AES.Google Scholar
  20. Daniel, J. (2001). Représentation de champs acoustiques, application à à la transmission et à à la reproduction de scènes sonores complexes dans un contexte multimédia [Representations of Sound Fields, Application to the Transmission and Reproduction of Complex Sound Scenes in a Multimedia Context]. PhD thesis, Université Paris 6. text in French.Google Scholar
  21. Daniel, J. (2003, May). Spatial sound encoding including near field effect: Introducing distance coding filters and a viable, new ambisonic format. In 23rd International Conference of the AES.Google Scholar
  22. Driscoll, J. R., & Healy, D. M. (1994). Computing fourier transforms and convolutions on the 2-sphere. Advances in Applied Mathematics, 15(2), 202–250.MathSciNetzbMATHCrossRefGoogle Scholar
  23. Fazi, F. (2010). Sound Field Reproduction. Ph.D. thesis, University of Southampton.Google Scholar
  24. Fazi, F., Brunel, V., Nelson, P., Hörchens, L., & Seo, J. (2008a, May). Measurement and Fourier-Bessel analysis of loudspeaker radiation patterns using a spherical array of microphones. In 124th Convention of the AES 2008.Google Scholar
  25. Fazi, F. M., Nelson, P. A., Christensen, J. E. N., Seo, J. (2008b, October). Surround system based on three dimensional sound field reconstruction. In 125th Convention of the AES.Google Scholar
  26. Fazi, F., Nelson, P., & Potthast, R. (2009, June). Analogies and differences between 3 methods for sound field reproduction. In Ambisonics Symposium.Google Scholar
  27. Fazi, F.,& Nelson, P. (2010a, May). Nonuniqueness of the solution of the sound field reproduction problem. In 2nd International Symposium. On Ambisonics and Spherical Acoustics.Google Scholar
  28. Fazi, F., & Nelson, P. (2010b, August). Sound field reproduction using directional loudspeakers and the equivalent acoustic scattering problem. In 20th International Congress on Acoustics.Google Scholar
  29. Gauthier, P. -A., & Berry, A. (2006). Adaptive wave field synthesis with independent radiation mode control for active sound field reproduction: Theory. JASA, 119(5), 2721–2737.Google Scholar
  30. Girod, B., Rabenstein, R., & Stenger, A. (2001). Signals and systems. New York: Wiley.Google Scholar
  31. Giroire, J. (1982). Integral equation methods for the Helmholtz equation. Integral Equations and Operator Theory, 5(1), 506–517.MathSciNetzbMATHCrossRefGoogle Scholar
  32. Gumerov, N. A., & Duraiswami, R. (2004). Fast multipole methods for the Helmholtz equation in three dimensions. Amsterdam: Elsevier.Google Scholar
  33. Kirkeby, O., Nelson, P. A., Hamada, H., & Orduna-Bustamante, F. (1998). Fast deconvolution of multichannel systems using regularization. IEEE Transactions on Speech and Audio Processing, 6(2), 189–195.CrossRefGoogle Scholar
  34. Lindner, F., Völk, F., & Fastl, H. (2011, March). Simulation und psychoakustische Bewertung von Übertragungsfehlern bei der Wellenfeldsynthese. In DAGA.Google Scholar
  35. Lopez, J. J., Gonzalez, A., Fuster, L. (2005, October). Room compensation in wave field synthesis by means of multichannel inversion. In IEEE Workshop on Applied of Signal Processing to Audio and Acoustics (WASPAA) (pp. 146–149).Google Scholar
  36. Morse, P. M., & Feshbach, H. (1953). Methods of theoretical physics. Minneapolis: Feshbach Publishing, LLC.zbMATHGoogle Scholar
  37. Morse, P. M., & Ingard, K. U. (1968). Theoretical acoustics. New York: McGraw-Hill Book Company.Google Scholar
  38. Neukom, M. (2007, October). Ambisonic panning. In 123th Convention of the AES.Google Scholar
  39. Nieto-Vesperinas, M. (2006). Scattering and diffraction in physical optics. Singapore: World Scientific Publishing.zbMATHGoogle Scholar
  40. Petrausch, S., Spors, & S., Rabenstein, R. (2005). Simulation and visualization of room compensation for wave field synthesis with the functional transformation method. In 119th Convention of the AES (p. 6547).Google Scholar
  41. Poletti, M. A. (2000). A unified theory of horizontal holographic sound systems. JAES, 48(12), 1155–1182.Google Scholar
  42. Poletti, M. A. (2005). Three-dimensional surround sound systems based on spherical harmonics. JAES, 53(11), 1004–1025.Google Scholar
  43. Poletti, M., Fazi, F., & Nelson, P. (2010). Sound-field reproduction systems using fixed-directivity loudspeakers. JASA, 127(6), 3590–3601.Google Scholar
  44. Rabenstein, R., Steffen, P., & Spors, S. (2006). Representation of twodimensional wave fields by multidimensional signals. EURASIP Signal Processing Magazine, 86(6), 1341–1351.zbMATHGoogle Scholar
  45. Sonke, J. -J., Labeeuw, J., & de Vries, D. (1998, May). Variable acoustics by wavefield synthesis: A closer look at amplitude effects. In 104th Convention of the AES (p. 4712).Google Scholar
  46. Spors, S. (2005). Active listening room compensation for spatial sound reproduction systems. PhD thesis, University of Erlangen-Nuremberg.Google Scholar
  47. Spors, S., Buchner, H., Rabenstein, R., & Herbordt, W. (2007). Active listening room compensation for massive multichannel sound reproduction systems using wave-domain adaptive filtering. JASA, 122(1), 354–369.Google Scholar
  48. Spors, S., Rabenstein, R., & Ahrens, J. (2008, May). The theory of wave field synthesis revisited. In 124th Convention of the AES.Google Scholar
  49. Spors, S., Ahrens, J. (2008b). Towards a theory for arbitrarily shaped sound field reproduction systems. In Acoustics 08.Google Scholar
  50. Spors, S., & Ahrens, J. (2010a, May). Analysis and improvement of preequalization in 2.5-dimensional wave field synthesis. In 128th Convention of the AES.Google Scholar
  51. Spors, S., & Ahrens, J. (2010c, March). Reproduction of focused sources by the spectral division method. In IEEE International Symposium on Communication Control and Signal Processing(ISCCSP).Google Scholar
  52. Start, E. W. (1996, May). Application of curved arrays in wave field synthesis. In 100th Convention of the AES, (p. 4143).Google Scholar
  53. Start, E. W. (1997). Direct sound enhancement by wave field synthesis. PhD thesis, Delft University of Technology.Google Scholar
  54. The SoundScape Renderer Team. (2011). The SoundScape Renderer.
  55. Toole, F. E. (2008). Sound reproduction: The acoustics and psychoacoustics of loudspeakers and rooms. Oxford: Focal Press.Google Scholar
  56. Travis, C. (2009, June). New mixed-order scheme for ambisonic signals. In Ambisonics Symposium.Google Scholar
  57. Verheijen, E. N. G. (1997). Sound reproduction by wave field synthesis. PhD thesis, Delft University of Technology.Google Scholar
  58. de Vries, D. (1996). Sound reinforcement by wavefield synthesis: Adaptation of the synthesis operator to the loudspeaker directivity characteristics. JAES, 44(12), 1120–1131.Google Scholar
  59. Ward, D. B., & Abhayapala, T. D. (2001). Reproduction of a plane-wave sound field using an array of loudspeakers. IEEE Transactions on Speech and Audio Processing, 9(6), 697–707.CrossRefGoogle Scholar
  60. Weisstein, E. W. (2002). CRC Concise encyclopedia of mathematics. London: Chapman and Hall/CRC.CrossRefGoogle Scholar
  61. Williams, E. G. (1999). Fourier acoustics: Sound radiation and nearfield acoustic holography. London: Academic.Google Scholar
  62. Wittek, H. (2007). Perceptual differences between wavefield synthesis and stereophony. PhD thesis, University of Surrey.Google Scholar
  63. Wu, Y. J., & Abhayapala, T. D. (2009). Theory and design of soundfield reproduction using continuous loudspeaker concept. IEEE Transactions on Audio, Speech and Language Processing, 17(1), 107–116.CrossRefGoogle Scholar
  64. Zotter, F., Pomberger, H., & Frank, M. (2009, May). An alternative ambisonics formulation: Modal source strength matching and the effect of spatial aliasing. In 126th Convention of the AES.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jens Ahrens
    • 1
  1. 1.Deutsche Telekom LaboratoriesTechnische Universität BerlinBerlinGermany

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