# Numerical Approximation of Dirichlet-to-Neumann Mapping and its Application to Voice Generation Problem

• Takashi Kako
• Kentarou Touda
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 40)

## Summary

In this paper, we treat the numerical method for the Helmholtz equation in unbounded region with simple cylindrical or spherical shape outside some bounded region and apply the method to voice generation problem. The numerical method for the Helmholtz equation in unbounded region is based on the domain decomposition technique to divide the region into a bounded region and the rest unbounded one. We then treat the approximation of the artificial boundary condition given through the DtN mapping on the artificial boundary. We apply the finite element approximation to discretize the problem. In applying the method to the voice generation problem, it is essential to compute the frequency response function or the formant curve. We give variational formulas for the resolvent poles with respect to the variation of vocal tract boundary which determine the peaks of frequency response function known as formants, and we propose the use of variational formulas to design the location of formants.

## Keywords

Helmholtz Equation Frequency Response Function Vocal Tract Variational Formula Formant Curve

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## References

1. A. Bayliss and E. Turkel. Radiation boundary conditions for wave-like equations. Comm. Pure and Appl. Math., 33(6):707–725, 1980.
2. B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of waves. Math. Comp., 31(139):629–651, 1977.
3. B. Engquist and A. Majda. Radiation boundary conditions for acoustic and elastic wave calculations. Comm. Pure Appl. Math., 32(3):313–357, 1979.
4. O. G. Ernst. A finite-element capacitance matrix method for exterior helmholtz problems. Numer. Math., 75(2):175–204, 1996.
5. K. Feng. Finite element method and natural boundary reduction. In Proceeding of the International Congress of Mathematicians, Warsaw, Poland, 1983.Google Scholar
6. J. Flanagan. Speech analysis, synthesis, and perception. Springer, Berlin-New York, 1972.Google Scholar
7. S. Furui. Digital speech processing, synthesis, and recognition. Marcel Dekker, 1989.Google Scholar
8. E. Heikkola, Y. A. Kuznetsov, P. Neittaanmaki, and J. Toivanen. Fictitious domain methods for the numerical solution of two-dimesional scattering problems. J. Comput. Phys., 145:89–109, 1998.
9. T. Kako and T. Kano. Finite element method for the helmholtz equation and numerical simulation of the wave propagation in vocal tract. In GAKUTO Int. Series Math. Sci. and Appli., volume 12, pages 55–63, 1999.
10. R. Kent and C. Read. The acoustic analysis of speech. Singular Publ. Group, San Diego, 1992.Google Scholar
11. M. Lenoir, M. Vullierme-Ledard, and C. Hazard. Variational formulations for the determination of resonant states in scattering problems. SIAM J. Math. Anal., 23(3):579–608, 1992.
12. H. Nasir. A mixed type finite element method for radiation and scattering problems with applications to structural-acoustic coupling problem in unbounded region. PhD thesis, The University of Electro-Communications, Japan, 2003.Google Scholar
13. H. Nasir, T. Kako, and D. Koyama. A mixed-type finite element approximation for radiation problems using fictitious domain method. J. Comput. Appl. Math., 52:377–392, 2003.
14. O. Poisson. Étude numérique des pôles de résonance associés à la diffraction d'ondes acoustiques et élastiques par un obstacle en dimension 2. M2AN, 29(2):819–855, 1995.

## Copyright information

© Springer-Verlag Berlin Heidelberg 2005

## Authors and Affiliations

• Takashi Kako
• 1
• Kentarou Touda
• 1
1. 1.Department of Computer ScienceThe University of Electro-CommunicationsJapan