[sdiy] SVF & phase

[Magnus Danielson]

From: Tom May <tom at tommay.net>
Subject: Re: [sdiy] SVF & phase
Date: 28 Nov 2002 11:54:06 -0800

Dear Tom,

> Magnus Danielson <cfmd at swipnet.se> writes:
> 
> >    There is exactly one solution to placement of equalently many poles and
> >    zeros which produces no phase shift - such that the position of one pole is
> >    matched by the position of a zero, and that all poles and zeros is matched
> >    accordingly. Now, this effectively makes z1 = p1, z2 = p2 etc. and those
> >    makes perfert cancelation. There is no way for any amplitude response to
> >    change (which is BTW the absolute response of H(s) when s = jomega), since
> >    H(s) is now exactly 1!
> 
> Is it not also the case that there will be zero phase shift if all the
> poles and zeros are symmetric about the jw axis?

No, you think the right way, but phase of poles and zeros work with opposite
sign, due to their side of the division. So, instead of canceling, they will
instead add in phase, however, the amplitude responce will experience the
cancelation. What you then just have created is the perfect condition for the
generic allpass filter!

Good try! ;O)

> In practice this implies right half-plane or jw axis poles which will be
> unstable.

Well, you can place your zeros wherever you like, but your poles better stay on
the left half-plane (called so since half of the s-plane, to the left of the
jw axis, is safe, where the other half is unsafe), as tradition and common
sense dictates. So then the zeros would turn up on the right halfplane, as you
say, symmetric about the jw-axis.

> But if you don't have the number of poles == number of zeros
> constraint, you can get rid of the poles altogether and have a filter
> with zeros only (which can be implemented as a FIR filter).

In theory it seems like a good plan, but not until you look more carefully at
it. A true zero's only filter would have infinit gain at infinit frequencies.
That will never happend in the real world. Also, you actually *never* have a
filter with zero zeros. Any real implementation would actually have EXACTLY as
many poles as zeros. The only thing you will do is to toss them far out of the
way, like out towards infinity or down into DC. They *NEVER* reach those
targets, but they come darn close for much of our daily work. Realizing that
you allways have an equal amount of poles and zeros in any real filter is maybe
one of the things which initially seems to contradict whatever you've learned
even in the more advanced courses at the uni, but once you've over that fact
you realize you finally can understand things and it's easy to prove why real
filters have this property. Also, when you know it, you start to learn that
since I now have n zeros, maybe I could put them into better use then tossing
them out to infinity all the time...

For those interested in classical responses, Cauer filters actually use zeros
more intelligent than that of Butterworth, Bessel-Thomson, Chebychev etc.

Cheers,
Magnus