Notes · Speech from first principles
Reflection coefficient
also: PARCOR · k
At a junction between two tube sections of area \(A_k\) and \(A_{k+1}\), continuity of pressure and volume velocity forces a single number to govern how much of a wave passes and how much reflects:
\[r_k = \frac{A_k - A_{k+1}}{A_k + A_{k+1}}.\]A blocked tube gives \(r = +1\) (reflect in phase — a rigid wall); an opening into infinite space gives \(r = -1\) (inverted — an open end). The two boundary conditions of the single tube are just the endpoints of this one formula. Chaining junctions is the Kelly–Lochbaum model of the vocal tract.
The remarkable part: the Levinson–Durbin recursion that fits an all-pole filter to a signal produces exactly these coefficients at each stage. In speech coding they’re called PARCOR coefficients — a direct description of the shape of your throat, recovered from nothing but the sound.