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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 [141]. 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 $ A_i$ were determined based on X-ray images of the vocal tract, and the corresponding characteristic impedances were proportional to $ 1/A_i$ . 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 [132].

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


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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