In this paper we discuss a novel technique to control the spatial distribution of sound level within a synthesized sound field. The problem is formulated by separating the sound field into regions with high acoustic level, so-called bright regions, and zones with low acoustic level (zones of quiet) by time independent virtual boundaries. This way, the propagating sound field obtains a static spatial shape, which we call sound figure. This problem is treated with a generic approach for creating sound figures. We give an analytic solution to the problem and highlight, how our findings can be applied using established sound field synthesis techniques. We furthermore show the limitations of our approach, provide simulation results to prove the concept and discuss some application areas.
This is a preview of subscription content,to check access.
Access this article
A manifold of dimension \(n\) is a topological space that resembles an \(n-\)dimensional Euclidean space in a neighborhood of each point (Boothby 1975).
A non-overlapping manifold \(\Omega \) does not exhibit any nodes. This property is necessary for defining differential operators on \(\Omega \).
In differential geometry, the Laplace operator can be generalized to operate on functions defined on surfaces in Euclidean space and, more generally, on Riemannian and pseudo-Riemannian manifolds. This more general operator goes by the name Laplace–Beltrami operator. The Laplace–Beltrami operator, like the Laplacian, is the divergence of the gradient.
Note that the triangle does not represent the dependency between \(k_x\) and \(k_y\) but should exemplary reference to the general complex amplitude of the spectrum.
Abhayapala, T., & Wu, Y. (2009). Spatial soundfield reproduction with zones of quiet. In 127th audio engineering society convention. New York, USA.
Ahrens, J., & Spors, S. (2010). Sound field reproduction using planar and linear arrays of loudspeakers. IEEE Transactions on Audio, Speech, and Language Processing, 18(8), 2038–2050.
Arfken, G. (2005). Mathematical methods for physicists (6th ed.). Boston: Elsevier.
Berkhout, A. (1987). Applied seismic wave theory. Amsterdam: Elsevier Science.
Berkhout, A., & Wapenaar, C. (1989). One-way versions of the Kirchhoff integral. Geophysics, 54(4), 460–467.
Berkhout, A., De Vries, D., & Vogel, P. (1993). Acoustic control by wave field synthesis. The Journal of the Acoustical Society of America (JASA), 93, 2764–2778.
Boothby, W. M. (1975). An introduction to differentiable manifolds and Riemannian geometry. New York: Academic Press.
Choi, J., & Kim, Y. (2002). Generation of an acoustically bright zone with an illuminated region using multiple sources. Journal of the Acoustical Society of America (JASA), 111, 1695–1700.
Chung, F. (1997). Spectral graph theory. In Conference board of the mathematical sciences. American Mathematical Society.
Courant, D., & Hilbert, R. (1953). Methods of mathematical physics. New York: Interscience.
Daniel, J. (2000). Représentation de champs acoustiques, application à la transmission et à la reproduction de scènes sonores complexes dans un contexte multimédia. Ph.D. thesis, Université Paris 6.
Fazi, F. (2010). Sound field reproduction. Ph.D. thesis. University of Southampton, UK. http://eprints.soton.ac.uk/158639/
Fazi, F., & Nelson, P. (2011). Sound field reproduction with an array of loudspeakers. Rivista Italiana di Acustica, 35(1), 1–11.
Fazi, F., Nelson, P. A., & Potthast, R. (2009). Analogies and differences between three methods for sound field reproduction. In 1st ambisonic symposium. Graz, Austria.
Giroire, J. (1982). Integral equation methods for the helmholtz equation. Integral Equations and Operator Theory, 5(1), 506–517.
Golub, G., & Van Loan, C. (1996). Matrix computations (3rd ed.). Baltimore: Johns Hopkins University Press.
Gumerov, N. A., & Duraiswami, R. (2004). Fast multipole methods for the Helmholtz equation in three dimensions. Amsterdam: Elsevier.
Helwani, K., Spors, S., & Buchner, H. (2011). Spatio-temporal signal preprocessing for multichannel acoustic echo cancellation. In Proceedings of the IEEE international conference on acoustics, speech, and, signal processing (ICASSP).
Kleinberg, J., & Tardos, E. (2005). Algorithm design. Reading, MA: Addison-Wesley.
Lanczos, C. (1997). Linear differential operators. New York: Courier Dover.
Menzies, D. (2012). Sound field synthesis with distributed modal constraints. Acta Acustica united with Acustica, 98(1), 15–27.
Oppenheim, A. (1999). Discrete-time signal processing (2nd ed.). Upper Saddle River, NJ: Prentice-Hall.
Poletti, M. A. (2000). A unified theory of horizontal holographic sound. Journal of the Audio Engineering Society, 48(12), 1155–1182.
Poletti, M., & Abhayapala, T. (2011). Interior and exterior sound field control using general two-dimensional first-order sources. The Journal of the Acoustical Society of America (JASA), 129(1), 234–244.
Rabenstein, R., Steffen, P., & Spors, S. (2006). Representation of two-dimensional wave fields by multidimensional signals. Signal Processing, 86(6), 1341–1351.
Rosenberg, S. (1997). The Laplacian on a Riemannian manifold: An introduction to analysis on manifolds. Cambridge, MA: Cambridge University Press.
Rossing, T. D. (2007). Springer handbook of acoustics. Berlin: Springer.
Shin, M., Lee, S., Fazi, F., Nelson, P., Kim, D., Wang, S., et al. (2010). Maximization of acoustic energy difference between two spaces. The Journal of the Acoustical Society of America (JASA), 128, 121–131.
Spors, S. (2007). Extension of an analytic secondary source selection criterion for wave field synthesis. In 123th audio engineering society convention. New York, USA.
Spors, S., Helwani, K., & Ahrens, J. (2011). Local sound field synthesis by virtual acoustic scattering and Time-Reversal. In 131st audio engineering society convention. New York, USA.
Spors, S., Rabenstein, R., & Ahrens, J. (2008). The theory of wave field synthesis revisited. In 124th audio engineering society convention (Vol. 24). Amsterdam, Netherlands
Teutsch, H. (2007). Modal array signal processing: principles and applications of acoustic wavefield decomposition. Berlin: Springer.
Tikhonov, A., & Samarskii, A. (1963). Equations of mathematical physics. New York: Dover.
Williams, E. (1999). Fourier acoustics: Sound radiation and nearfield acoustical holography. New York: Academic Press.
Zotter, F., & Spors, S. (2013). Is sound field control determined at all frequencies? How is it related to numerical acoustics? In Audio engineering society conference: 52nd international conference (pp. 1–9). Guildford, UK.
We thank the reviewers for their thorough reading of the manuscript and highly appreciate the comments and suggestions.
About this article
Cite this article
Helwani, K., Spors, S. & Buchner, H. The synthesis of sound figures. Multidim Syst Sign Process 25, 379–403 (2014). https://doi.org/10.1007/s11045-013-0261-4