question for theorists: zeros & phase

[Martin Czech]

:::Nothing really dramatic happends there... it just goes faster. You are
:::simply not view this in enougth academic way ;)

Yes, very fast indeed, too fast for the first simple approach,
wrong phase, thus ;->



:::> 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.
:::
:::This is true for zeros, it is equally true for poles, but then you are blasted
:::instead.

Poles are excluded here, stability.

:::But you would only experience an jump if you really pass throgth either a
:::zero or pole, this is actually _never_ done (think Heissenberg here), just
:::close enougth to count as it.

Yes , I think this is the practical way out, we can never put a zero
(or a pair) of zeros on the imag axis (exept for 0), so this situation
will never happen.  In the continous world (analog) this will solve the
problem by simply excluding it.

However, in the discrete world this is possible, a zero can be placed
on the unit circle.  But, it comes to my mind, the circle is a curve,
it is no longer straight, therefore the problem is not that troublesome
anymore, since the angle change near to the zero will be allways less then
180 DEG, so no ambiguity. If you get very close to the zero from both
sides (limes), the angle will expand to finally 180 DEG, but we know from which
side we are comming, so it is clear how to count the angle direction.

:::
:::We have a lot of signals around the imaginary axis, but we have a heck of a
:::problem getting real poles or zeros sit tigth at the imaginary axis.

See above.

Lots of true sentences excluded here...

:::No, staring at formulas will make you gray-haired, better to retreit to
:::a graphical view...
:::
:::           ^ imaginary axis
:::           |
:::         _-* 
:::      _-- /|\
:::     x   | | \
:::        /  |  \
:::-------|---+---o------> real axis
:::      /    |
:::     x     |
:::           |
:::           |
:::           |
:::
:::Here we have the s-plane view of a 2-pole (x) and 1-zero (o) filter, which
:::forms a simple bandpass filter which takes an op-amp and a few caps and
:::resistors to realize (just to show that we are talking real filters here).


Nice drawing. Isn't the zero located at 0 for a highpass/bandpass?
Or was it placed a bit off side to show low frequency speaktrough?

Ok, but for a theoretical system it is possible to have a zero at
least at 0.  The problem bites again, when analysing from -inf to +inf.
Do you mean that a theoretical system has no phase there?


Again some true sentenses and formulas snipped


:::and hopefully you figure the rest out.


Not really. We may exclude zeros on the imag axis for real continous systems,
but on the other hand, does this mean I can not analyse a theoretical system?
Strange.

:::The z-plane is pretty much the same.

I don't think so. See above.

m.c.