FW: AW: Re: digital contents

[Don Tillman]

   From: Paolo Predonzani <predo at dist.dist.unige.it>
   Date: Mon, 23 Sep 1996 16:36:26 +0200 (MET DST)

   > How *far* would these poles move? Usually, the signal amplitude
   > thru the ladder is small in comparison with the bias current. (?)

   It's not so small. The open loop poles almost touch (0,0) in the s-plane.
   As to the closed-loop poles I never examined them in detail because I
   would have to find the roots of a 4-th order polinomial. Currently I don't
   have an algorithm for that. 

I'm not sure that I understand you here.  If you mean the positions of
the (linear) poles with feedback, I can tell you that the poles start
at -w (real, negative), and, as feedback is applied, they split off in
an X pattern at 45/135/-45/-135 degrees.  Eventually the two rightmost
poles hit the axis and it oscillates.

I've had a lot of success using an iterative method to solve n-th
order polynomials, up to about 12-th order or so.  Newton's method,
while fine for real numbers, fails completely for complex numbers
because the second dimension makes it just too easy to overshoot.  So
instead you have to make a 2-dimensional parabolic approximation to
the curve at the point in question, and that's easily solvable
generating a fine point for the next iteration.

I have some software to do this; it was part of a speaker design
program I wrote once -- I wanted to derrive pole positions for
arbitrary filters whose characteristics were defined by their
polynomials (ie., Bessel).  

Code fragment available to synth-diy'ers on request, but be warned,
it's written in Lisp.

Or you can use Mathematica.

  -- Don