### Vocal Tract Analog Models

There is one speech-synthesis thread that clearly classifies under computational physical modeling, and that is the topic of*vocal tract analog models*. In these models, the vocal tract is regarded as a

*piecewise cylindrical acoustic tube*. The first mechanical analogue of an acoustic-tube model appears to be a hand-manipulated leather tube built by Wolfgang von Kempelen in 1791, reproduced with improvements by Sir Charles Wheatstone [140]. In

*electrical*vocal-tract analog models, the piecewise cylindrical acoustic tube is modeled as a cascade of electrical

*transmission line*segments, with each cylindrical segment being modeled as a transmission line at some fixed characteristic impedance. An early model employing four cylindrical sections was developed by Hugh K. Dunn in the late 1940s [120]. An even earlier model based on two cylinders joined by a conical section was published by T. Chiba and M. Kajiyama in 1941 [120]. Cylinder cross-sectional areas were determined based on X-ray images of the vocal tract, and the corresponding characteristic impedances were proportional to . An impedance-based, lumped-parameter approximation to the transmission-line sections was used in order that analog LC ladders could be used to implement the model electronically. By the 1950s, LC vocal-tract analog models included a side-branch for nasal simulation [131].

The theory of transmission lines is credited to applied mathematician Oliver Heaviside (1850-1925), who worked out the

*telegrapher's equations*(sometime after 1874) as an application of Maxwell's equations, which he simplified (sometime after 1880) from the original 20 equations of Maxwell to the modern vector formulation.

^{A.12}Additionally, Heaviside is credited with introducing complex numbers into circuit analysis, inventing essentially Laplace-transform methods for solving circuits (sometime between 1880 and 1887), and coining the terms `impedance' (1886), `admittance' (1887), `electret', `conductance' (1885), and `permeability' (1885). A little later, Lord Rayleigh worked out the theory of waveguides (1897), including multiple propagating modes and the cut-off phenomenon.

^{A.13}

**Next Section:**

Singing Kelly-Lochbaum Vocal Tract

**Previous Section:**

Dudley's Vocoder