Kelly-Lochbaum Vocal Tract Model

• Vocal tract $\approx$ piecewise cylindrical acoustic tube
• Ladder filter with physical interpretation
• Possibly the earliest digital waveguide acoustic model
• Wave impedance in section $i$:
  $$R_i = \frac{\rho c}{A_i}$$
  where $A_i =$ cross-sectional area of tube
• Reflection coefficient at $i$th cylinder-cylinder junction:
  $$k_i \mathrel{\stackrel{\mathrm{\Delta}}{=}}\frac{R_i - R_{i-1}}{R_i + R_{i-1}}$$

Subsequent work

• Articulatory speech synthesis
• Linear Predictive Coding (LPC) of speech

Recent Extensions

• Kelly-Lochbaum + nasal tract + neck radiation + damping for female singing voice (Perry Cook)
• Sparse acoustic tubes (Lacroix)
• Piecewise conical tubes (Välimäki and Karjalainen)

Recent Music Applications (This Course)

• Stringed instruments
• Wind instruments
• Brass instruments
• Singing Voice
• Membrane simulation (drums, gongs, ...)
• Reverberation (complex networks, plates)