Singing Kelly-Lochbaum Vocal Tract

In 1962, John L. Kelly and Carol C. Lochbaum published a software version of a digitized vocal-tract analog model [251,252]. This may be the first instance of a sampled traveling-wave model of the vocal tract, as opposed to a lumped-parameter transmission-line model. In other words, Kelly and Lochbaum apparently returned to the original acoustic tube model (a sequence of cylinders), obtained d'Alembert's traveling-wave solution in each section, and applied Nyquist's sampling theorem to digitize the system. This sampled, bandlimited approach to digitization contrasts with the use of bilinear transforms as in wave digital filters; an advantage is that the frequency axis is not warped, but it is prone to aliasing when the parameters vary over time (or if nonlinearities are present).

At the junction of two cylindrical tube sections, i.e., at area discontinuities, lossless scattering occurs.^E.14As mentioned in §E.5.4, reflection/transmission at impedance discontinuities was well formulated in classical network theory [33,34], and in transmission-line theory.

The Kelly-Lochbaum model can be regarded as a kind of ladder filter [301] or, more precisely, using later terminology, a digital waveguide filter [442]. Ladder and lattice digital filters can be used to realize arbitrary transfer functions [301], and they enjoy low sensitivity to round-off error, guaranteed stability under coefficient interpolation, and freedom from overflow oscillations and limit cycles under general conditions. Ladder/lattice filters remain important options when designing fixed-point implementations of digital filters, e