phase response & zeros

[Martin Czech]

Hello, I'm just finishing my little phase-plot program.  As usuall
things turn out to take 10x longer then estimated.  But I think I have
something usefull now, better than the Anacad we use in the company. No
GUI of course, all file driven.

I did a few test cases again and they looked quite strange.  It was
a single zero/pole near the imag axis/unit circle.  Strange enough the
phase response looks then same for zeros close to the axis, but being
on different sides.

The program works correctly, the point is that both  zeros really start
with different angles, but as the analysis point approaches they rush
to allmost the  same angle (local symmetry), and from then on they walk in
the same way.  So if the zeros come closer and closer to the axis,
the zone of different results becomes closer and closer, until we are
actually ON the axis, where we have a singularity problem, as it was
pointed out in the previous mails. Since we get the same value when
approaching from the two sides, we could DEFINE the value in the point
of singularity as steady extension, however.

Example: zero @ 0.00001 and -0.00001.  Of course, in the begining you
get +180 and -180 deg, but when passing the zeros the result will be
quickly 90 deg, no difference.

The example works even better at the unit circle with 0.9999 and 1.0001.

Note that bilinear transformation enforces zeros at -1.


So, for meaningfull computation, zeros have to stay away from the axis,
and the point of analysis should not start from 0 (we could DEFINE all these
singular points, but this would only mean more programming). Since
the phase response is computed step by step, the zone of ambiguity is
not visible.

The question where to set a zero (left or right) does simply not matter
for such a program.

The problem was, that my pencil drawings put the zero not close enough
to the axis, so the rush of phase pointers could not be visualised.

When I deald with this stuff @ university 10 years ago, such a
visualisation would have been extremely helpfull. From today I'd say that
I had no clue at all at that time, it was all academic and pro forma,
but nobody knew what it was all about in terms of engineering.

Math professors tend to stick inside the math world, but the
interpretation of math symbols, equations etc. is of equal importance.

Without interpretion, or with a strange interpretation 1+1 = 3 could
be true.  (see D. Hofstaedter, EGB on this topic).

The time schedule in university was so , that one could hardly deal with
the formal operations, no time for interpretation of the results. Just
trying to survive the next exam. 

This makes me angry, wasted years.

OTOH the education was for free...

m.c.