Determination of the Vocal-Tract Shape from Measured Formant Frequencies

We model the vocal tract as a lossless acoustic tube and consider the relationship between the resonant frequencies and the cross-sectional area function. Empirical results show that if the logarithm of the area function is band limited preserving only 2n Fourier components, the lowest n pole and n zero frequencies of the admittance function measured at the lips uniquely determine the area coefficients. The formant frequencies determined from the speech signal, the normal resonant frequencies of the vocal tract, correspond to the admittance poles. They alone do not suffice for unique area determination unless the even area coefficients are constrained to specified values or some other physical contraints are imposed. The output-admittance zeros are not obtainable from the speech signal, as they correspond to resonant modes for closed-lip boundary conditions. The above results allow the synthesis of band-limited vocal-tract shapes from admittance functions that have only their low-frequency singularities specified. They can be generalized to any system exhibiting wave propagation primarily in one dimension only, for example, nonuniform transmission lines, and allow a low spatial-frequency synthesis of the system from two sets of low-order eigenvalues.