Skip to main content
Book cover

Analytic Methods of Sound Field Synthesis pp 57–113Cite as

Continuous Secondary Source Distributions

  • 1508 Accesses

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

Abstract

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.

Keywords

  • Plane Wave
  • Sound Source
  • Secondary Source
  • Sound Field
  • Convolution Theorem

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-642-25743-8_3
  • Chapter length: 57 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   119.00
Price excludes VAT (USA)
  • ISBN: 978-3-642-25743-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   159.99
Price excludes VAT (USA)
Hardcover Book
USD   179.99
Price excludes VAT (USA)
Learn about institutional subscriptions
Fig. 3.1
Fig. 3.2
Fig. 3.3
Fig. 3.4
Fig. 3.5
Fig. 3.6
Fig. 3.7
Fig. 3.8
Fig. 3.9
Fig. 3.10
Fig. 3.11
Fig. 3.12
Fig. 3.13
Fig. 3.14
Fig. 3.15
Fig. 3.16
Fig. 3.17
Fig. 3.18
Fig. 3.19
Fig. 3.20
Fig. 3.21
Fig. 3.22
Fig. 3.23
Fig. 3.24
Fig. 3.25
Fig. 3.26
Fig. 3.27
Fig. 3.28
Fig. 3.29
Fig. 3.30
Fig. 3.31
Fig. 3.32

Notes

  1. 1.

    Note that the term “two-dimensional” does not represent the fact that observations are carried out in a plane. A two-dimensional problem in acoustics is independent of one of the spatial dimensions. An example are height-invariant sound fields, i.e., sound fields that do not exhibit any variation along the z-axis. Because of this height invariance, two-dimensional sound field synthesis requires line-like secondary sources (Williams 1999, Sect 8.6.1 and 8.6.2).

References

  • Abramowitz, M., & Stegun, I. A. (Eds.). (1968). Handbook of mathematical functions. New York: Dover Publications Inc.

    Google Scholar 

  • 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.

    CrossRef  Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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 

  • Ahrens, J., & Spors, S. (2010d, May). On the scattering of synthetic sound fields. In 130th Convention of the AES (p. 8121).

    Google Scholar 

  • 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.

    CrossRef  Google Scholar 

  • Arfken, G., & Weber, H. (2005). Mathematical methods for physicists (6th ed.). San Diego: Elsevier Academic Press.

    Google Scholar 

  • 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 

  • Berkhout, A. J. (1987). Applied seismic wave theory. Amsterdam: Elsevier Publishing Company.

    Google Scholar 

  • Berkhout, A. J., de Vries, D., & Vogel, P. (1993). Acoustic control by wave field synthesis. JASA, 93(5), 2764–2778.

    Google Scholar 

  • Betlehem, T., & Abhayapala, T. D. (2005). Theory and design of sound field reproduction in reverberant rooms. JASA, 117(4), 2100–2111.

    Google Scholar 

  • 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 

  • Colton, D., & Kress, R. (1998). Inverse acoustic and electromagnetic scattering theory (2nd ed.). Berlin: Springer.

    Google Scholar 

  • Copley, L. G. (1968). Fundamental results concerning integral representations in acoustic radiation. JASA, 44, 28–32.

    MATH  Google Scholar 

  • Corteel, E. (2006). Equalization in an extended area using multichannel inversion and wave field synthesis. JAES, 54(12), 1140–1161.

    Google Scholar 

  • D. de Vries, (2009). Wave field synthesis. AES Monograph. New York: AES.

    Google Scholar 

  • 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 

  • 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 

  • Driscoll, J. R., & Healy, D. M. (1994). Computing fourier transforms and convolutions on the 2-sphere. Advances in Applied Mathematics, 15(2), 202–250.

    MathSciNet  MATH  CrossRef  Google Scholar 

  • Fazi, F. (2010). Sound Field Reproduction. Ph.D. thesis, University of Southampton.

    Google Scholar 

  • 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 

  • 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 

  • Fazi, F., Nelson, P., & Potthast, R. (2009, June). Analogies and differences between 3 methods for sound field reproduction. In Ambisonics Symposium.

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • Girod, B., Rabenstein, R., & Stenger, A. (2001). Signals and systems. New York: Wiley.

    Google Scholar 

  • Giroire, J. (1982). Integral equation methods for the Helmholtz equation. Integral Equations and Operator Theory, 5(1), 506–517.

    MathSciNet  MATH  CrossRef  Google Scholar 

  • Gumerov, N. A., & Duraiswami, R. (2004). Fast multipole methods for the Helmholtz equation in three dimensions. Amsterdam: Elsevier.

    Google Scholar 

  • 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.

    CrossRef  Google Scholar 

  • Lindner, F., Völk, F., & Fastl, H. (2011, March). Simulation und psychoakustische Bewertung von Übertragungsfehlern bei der Wellenfeldsynthese. In DAGA.

    Google Scholar 

  • 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 

  • Morse, P. M., & Feshbach, H. (1953). Methods of theoretical physics. Minneapolis: Feshbach Publishing, LLC.

    MATH  Google Scholar 

  • Morse, P. M., & Ingard, K. U. (1968). Theoretical acoustics. New York: McGraw-Hill Book Company.

    Google Scholar 

  • Neukom, M. (2007, October). Ambisonic panning. In 123th Convention of the AES.

    Google Scholar 

  • Nieto-Vesperinas, M. (2006). Scattering and diffraction in physical optics. Singapore: World Scientific Publishing.

    MATH  Google Scholar 

  • 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 

  • Poletti, M. A. (2000). A unified theory of horizontal holographic sound systems. JAES, 48(12), 1155–1182.

    Google Scholar 

  • Poletti, M. A. (2005). Three-dimensional surround sound systems based on spherical harmonics. JAES, 53(11), 1004–1025.

    Google Scholar 

  • Poletti, M., Fazi, F., & Nelson, P. (2010). Sound-field reproduction systems using fixed-directivity loudspeakers. JASA, 127(6), 3590–3601.

    Google Scholar 

  • Rabenstein, R., Steffen, P., & Spors, S. (2006). Representation of twodimensional wave fields by multidimensional signals. EURASIP Signal Processing Magazine, 86(6), 1341–1351.

    MATH  Google Scholar 

  • 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 

  • Spors, S. (2005). Active listening room compensation for spatial sound reproduction systems. PhD thesis, University of Erlangen-Nuremberg.

    Google Scholar 

  • 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 

  • Spors, S., Rabenstein, R., & Ahrens, J. (2008, May). The theory of wave field synthesis revisited. In 124th Convention of the AES.

    Google Scholar 

  • Spors, S., Ahrens, J. (2008b). Towards a theory for arbitrarily shaped sound field reproduction systems. In Acoustics 08.

    Google Scholar 

  • 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 

  • 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 

  • Start, E. W. (1996, May). Application of curved arrays in wave field synthesis. In 100th Convention of the AES, (p. 4143).

    Google Scholar 

  • Start, E. W. (1997). Direct sound enhancement by wave field synthesis. PhD thesis, Delft University of Technology.

    Google Scholar 

  • The SoundScape Renderer Team. (2011). The SoundScape Renderer. http://www.tu-berlin.de/?id=ssr.

  • Toole, F. E. (2008). Sound reproduction: The acoustics and psychoacoustics of loudspeakers and rooms. Oxford: Focal Press.

    Google Scholar 

  • Travis, C. (2009, June). New mixed-order scheme for ambisonic signals. In Ambisonics Symposium.

    Google Scholar 

  • Verheijen, E. N. G. (1997). Sound reproduction by wave field synthesis. PhD thesis, Delft University of Technology.

    Google Scholar 

  • 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 

  • 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.

    CrossRef  Google Scholar 

  • Weisstein, E. W. (2002). CRC Concise encyclopedia of mathematics. London: Chapman and Hall/CRC.

    CrossRef  Google Scholar 

  • Williams, E. G. (1999). Fourier acoustics: Sound radiation and nearfield acoustic holography. London: Academic.

    Google Scholar 

  • Wittek, H. (2007). Perceptual differences between wavefield synthesis and stereophony. PhD thesis, University of Surrey.

    Google Scholar 

  • 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.

    CrossRef  Google Scholar 

  • 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 

Download references

Author information

Authors and Affiliations

  1. Deutsche Telekom Laboratories, Technische Universität Berlin, Ernst-Reuter-Platz 7, 10587, Berlin, Germany

    Jens Ahrens

Authors
  1. Jens Ahrens
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Ahrens, J. (2012). Continuous Secondary Source Distributions. In: Analytic Methods of Sound Field Synthesis. T-Labs Series in Telecommunication Services. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25743-8_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-25743-8_3

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-25742-1

  • Online ISBN: 978-3-642-25743-8

  • eBook Packages: EngineeringEngineering (R0)