# Formant bandwidths

I have some intuition about bandwidths from these videos (1 and 2) explaining how the frequency response corresponds to the Z-surface.

The only thing I could find on the internet about calculating the bandwidths is this matlab tutorial:

The bandwidths of the formants are represented by the distance of the prediction polynomial zeros from the unit circle.

[frqs,indices] = sort(angz.*(Fs/(2*pi)));
bw = -1/2*(Fs/(2*pi))*log(abs(rts(indices)));


but bw is not the distance to the unit circle but the log of the norm.

This is assuming the roots/poles are inside the unit circle or bw will be negative. But I'm unclear why the roots/poles shouldn't be outside the unit circle?

I can see that the relationship of the bandwidth is not linear with the root/pole position (relative to the unit circle), but why is it logarithmic?

Perhaps, if the root/pole is outside of the unit circle we should reflect log(x) around 1 i.e. log(2-x):

bw = -1/2*(Fs/(2*pi))*log(2-abs(rts(indices)));