Determination of Smoothed Cross-Sectional-Area Functions of the Vocal Tract from Formant Frequencies
Paul G. Mermelstein, Manfred R. Schroeder · The Journal of the Acoustical Society of America · 1965
Approximate formant frequencies for a given vocal-tract configuration can be calculated by integrating Webster's horn equation. No general methods are available for the solution of the inverse problem, that of determining the cross-sectional area from the formant frequencies. Previous attempts involved approximations of the tract shape in terms of a small number of parameters of a given model based on the physical constraints acting on the vocal tract [see, for example, K. N. Stevens and A. S. House, J. Acoust. Soc. Am. 27, 484–493 (1955)]. The procedure presented here elimininates the necessity for such a priori assumptions. The logarithm of the area function is represented by a spatial Fourier cosine series of a symmetrical tract of length twice that of the original tract. Starting with a uniform tract closed at one end, a steepest-descent search in the multidimensional space of Fourier coefficient is found to converge rapidly to a configuration yielding low-order formant frequencies nearly identical to those specified. Results are presented for articulations of sustained vowels calculated from the first three formant frequencies and represented by 3 Fourier coefficients. The area functions so obtained are found to be good smoothed approximations to those derived from x-ray data.