question for theorists: zeros & phase

[Martin Czech]

Hello theorists, I'm just putting together a little program to
compute phase and magnitude for continuous and discrete filters. This
is not difficult, all is written in the books.  We can read there
phase=arctan{Im/Re}. Oh, really? This is lovely academia. What happens if
the pole/zero is very very close to the imaginary axis (unit circle)? Ther
phase vector suddenly turns allmost 180 DEG. Due to the ambiguity of the
arctan you get artefacts, phase jumps, oh well, one can eliminate that,
but it causes some headache.

But one real phase jump remains: a zero on the imaginary axis, or on
the unit circle.  In both cases the phase vector comes closer and closer
to the zero, and then it jumps 180 DEG to the other side (especially in
the continuous case).

It is clear to me that just at the zero the vector vanishes, so the
phase does have no meaning there. Especially in the case of continuous
filters the question remains, if the phase jumps by +180 or -180 DEG at
the zero. Both would lead to the same phase vector, but different bode
plots. To make things more confusing, zeros on the left side (inside
unit circle) have a positive rotation whereas zeros on the right side
(outside circle) have a negative rotation. So one can not argument by
slowly approaching the imaginary axis (circle).

Since I expect a nonambigous bode plot for a stable pole/zero
configuration there must be another restriction, I guess impulse
response ?

??

m.c.