Notes · Speech from first principles

All-pole filter

also: LPC · linear predictive coding

A lossless chain of tubes, closed at one end and open at the other, has a transfer function with poles and no zeros:

\[V(z) = \frac{G}{1 - \sum_{k=1}^{N} a_k z^{-k}}.\]

So the vocal tract is an all-pole filter — exactly, not by approximation. Linear predictive coding exploits this: fitting an all-pole model to a slice of speech is not generic curve-fitting, it is inverting the physics — given the sound, recover the tube. The fit produces, at each order, the reflection coefficients of the tube ladder.

The all-pole assumption is also where the model breaks. Nasals and laterals hang a side branch off the tract that cuts notches — zeros — into the spectrum, and no all-pole filter can represent a zero, only approximate it with extra poles. The cepstrum is the other classical route to the same formant envelope, one that doesn’t assume the filter’s shape in advance.

← All notes