Path: utzoo!utgpu!jarvis.csri.toronto.edu!cs.utexas.edu!usc!samsung!think!mintaka!mit-eddie!uw-beaver!ubc-cs!alberta!calgary!!smit
From: smit@.ucalgary.ca (Theo Smit)
Newsgroups: comp.dsp
Subject: Re: AR and MA questions
Message-ID: <2173@cs-spool.calgary.UUCP>
Date: 27 Nov 89 16:58:47 GMT
References: <1456@mrsvr.UUCP>
Sender: news@calgary.UUCP
Reply-To: smit@enel.ucalgary.ca (Theo Smit)
Organization: U. of Calgary, Calgary, Alberta, Canada
Lines: 59

In article <1456@mrsvr.UUCP> kohli@gemed.ge.com (Mr. Bad Judgment) writes:
[...previous discussion deleted]
>	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
[]
>	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
>Opinions?  Facts?
>Thanks again,
>Jim Kohli
>GE Medical Systems
>P.S.  Thank you Theo, Rob, and Stephen, the discussion has been
>interesting and enlightening (so far!).

I assume by damped sinusoids you mean sinusoids with exponentially decaying
amplitude? (Are you by any chance doing MRI data? We've done a lot of
experimentation on modeling MRI)
Let's assume that we have an exponentially decaying unit step. Then the time
series of the output is:
			        n
			x(n) = a  , a < 1.0, 0 <= n < inf.

The AR(1) model is:	       -1
			1 + a z

and if a < 1.0, the thing is stable. However, if we look at the thing 
backwards, we see an exponentially increasing series. If we model this
sequence, we get an AR coefficient of 1/a*, that is, outside the unit circle.
The coefficient is also conjugated from the forward model coefficient (I
think. I'm doing this from memory), so the overall angle of the pole in the
Z-plane stays the same, it's just been reflected outside the unit circle.
Obviously, this situation is unstable.
If your algorithm applies forward _and_ backward prediction to decaying data,
your results will not be correct, since the model cannot provide a stable
solution in both directions. The coefficients will be the result of a 
best-fit in both directions, and will not mean much of anything. This
is a problem I found in using the Burg algorithm; anything that's not
stationary doesn't get modeled worth &*@Q$.

About the pole radius scaling - if you don't care what the absolute amplitude
is (ie in spectral analysis, you're usually only interested in relative
magnitude), there is no problem. The pole scaling results in reduced
resolution in the frequency domain, similar to what windowing the data
does. If you're trying to model the data exactly enough to be able to
recreate it, a different approach will be necessary.

Hope this helps,

Theo Smit (smit@enel.ucalgary.ca)


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