🚀 go-pugleaf

rob@kaa.eng.ohio-state.edu (Rob Carriere) writes:
<In article <2074@cs-spool.calgary.UUCP> smit@enel.ucalgary.ca (Theo Smit)
<writes: 
<>In article <1389@mrsvr.UUCP> kohli@gemed.ge.com (Mr. Bad Judgment) writes:
<>>1. Are the AR "parameters" the poles of the waveform?
<>>	If so, does this mean that an AR parameter of magnitude
<>>	> 1 indicates an unstable system (assuming no zero to
<>>	counteract it)?
<
<[AR parameters are the coefficients of a polynomial whose _roots_
<are the poles]
<
<[stability force deleted]
<>Mostly this is used to counter arithmetic errors; if your system is unstable
<>the poles will likely be well outside the unit circle and you have no hope in
<>heck of doing much about it.
<
<If your system is unstable and you force the estimate to be stable you have a
<meaningless estimate.
<
	That is true.  On the other hand, the system I am
modeling is stable, consisting exclusively of damped sinusoids
of various frequencies and additive noise (so why not use
Prony's method?  I'm doing that, too).  Marple's book suggests
(and rightfully so) that if the results of doing AR analysis
generates poles which have magnitude greater than one, attempts
to model the system will be disappointing because of the
instability in the model.  This may be easily confirmed upon
inspection of an AR(1) system, for example, where

	X(t) = A(t)X(t-1) + n(t).

	X(t) may be expected to grow without bound if |A(t)| >
1.0.  If the AR coefficients are the coefficients of
polynomials whose roots are the poles are the system (we know
they are), what are the consequences of scaling the
coefficients?  It seems to me that the frequency components
should be preserved, and if |variance of n(t)| were also scaled
down, the modeled X(t) sequence would be stable, but scaled
incorrectly.  Is this wrong?  I would like to have stable
estimates for this known-stable system, but most of the
recommended AR techniques in Marple generate unstable
parameters (Marple makes note of this possibility in a couple
of places, but I didn't see any recommendations about how to
fix the problem.  I *think* there's an implication that the
"next better" algorithm should be used).

Opinions?  Facts?
Either are welcome, just please be kind.

Thanks again,
Jim Kohli
GE Medical Systems

P.S.  Thank you Theo, Rob, and Stephen, the discussion has been
interesting and enlightening (so far!).