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From: rob@kaa.eng.ohio-state.edu (Rob Carriere)
Newsgroups: comp.dsp
Subject: Re: AR and MA questions
Message-ID: <3455@quanta.eng.ohio-state.edu>
Date: 9 Nov 89 08:52:10 GMT
References: <1389@mrsvr.UUCP>
Sender: news@quanta.eng.ohio-state.edu
Reply-To: rob@kaa.eng.ohio-state.edu (Rob Carriere)
Organization: Ohio State Univ, College of Engineering
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In article <1389@mrsvr.UUCP> kohli@gemed.ge.com (Mr. Bad Judgment) writes:
>
>I've been working a little bit with some ARMA stuff
>and have run into some questions which I can't
>generalize an answer to from my experience:
>
>1. Are the AR "parameters" the poles of the waveform?

Not quite.  The AR parameters are the coefficients of a polynomial whose roots
are the poles.  Note that for an AR(1) process it doesn't matter (root = coeff)

>	If so, does this mean that an AR parameter of magnitude
>	> 1 indicates an unstable system (assuming no zero to
>	counteract it)?

No, an _AR_coefficient_ >1 is quite innocent.  It is the poles you have to
worry about. 

>2. (And the MA parameters *are* the zeroes?)

Same thing here: the zeros are the roots of the polynomial formed by the MA
coefficients.  Again, for MA(1), it doesn't matter.

>3. Is the noise process usually neglected when
>	using the AR and MA parameters to model the
>	time series?  If not, is a Gaussian model

The underlying model is:

                  ______________
                 |              |
white noise ---> |  ARMA Filter | ---> observed signal
                 |              |
                  --------------

Here ``ARMA Filter'' means the filter with the transfer function given by the
estimated ARMA parameters.  The white noise is usually assumed to be Gaussian.

What you seem to be looking for is:              Noise
                                 _______________   |
                                |               |  |
Deterministic white signal ---> |  Prony Filter | -+-> observed signal
                                |               |
                                 ---------------

This is handled in the literature as ``Prony Estimation''  (and is very
similar to ARMA stuff).  The noise model is indeed usually assumed to be iid
Gaussian.

>	used?  If the noise process is calculated
>	using the calculated ARMA parameters and
>	the original sampled data, doesn't that mean
>	that your model will be identical to the
>	sampled data?

Yes.  Your model will (ideally) match the spectral shape of the process you
observed.  That is is the meaning and purpose of the whole exercise, to
estimate the spectral content of the signal.

>
>Thanks for your thoughts!

Such as they are at 3:41 am :-)
If you want some background literature, I'm personally rather fond of 

@book{Mar87,
      author = "Marple, S.~Lawrence,~Jr.",
      title  = "Digital Spectral Analysis with applications",
      publisher = "Prentice-Hall",
      year   = 1987
}

It discusses everything mentioned here (and quite a bit more) and comes with a
disk full of FORTRAN routines that implement all the common algorithms.

SR