Third International Conference on Spoken Language Processing (ICSLP 94)
The Kelly-Lochbaum (KL)  and Distinctive Region (DRM)  vocal tract models are concatenations of uniform loss-less tubelets of equal and unequal lengths respectively. The transfer matrixes of these models lead to resonance conditions that are polynomials F(x,Sj) in x = cos(2pirfl/c)f with / as the tubelet length, f the resonance frequency, c the sound speed and Sj the vocal tract cross sections. Usually, finding the resonance frequencies consists of numerically searching for the roots of F(x, Sj) and taking their arccosine to isolate the frequencies. In this article, we explore two methods for arriving instead at an explicit link fi = fi(Sj) between the cross sections Sj and resonance frequencies fi in the framework of KL and DRM vocal tract models. We show the following. Firstly, equation F(fi,Sj) = 0 can be solved analyti- cally provided that the number of tubelets is less than, or equal to, nine and provided that they are of equal length. Secondly, approximate analytical solutions of equation F(fi, Sj) = 0 can be obtained by representing the relations between log(fi) and log(Sj) by means of a Taylor expansion of the implicitly defined function F(fi(Sj), Sj) = 0. The results show that these approximations are valid for cross sections that typically vary between 1 or 2 cm2 and 10 cm2. Approximate analytical solutions of F(fi(Sj),Sj) = 0 can be arrived at for models with an arbitrary number of tubelets of unequal length.
Bibliographic reference. Schoentgen, Jean / Ciocea, S. (1994): "Explicit relations between resonance frequencies and vocal tract cross sections in loss-less kelly-lochbaum and distinctive region vocal tract models", In ICSLP-1994, 611-614.