Prony's Method

[root <root@maori.colorado.edu>]

>From: gisinf@kitsap.engr.sgi.com (Franz Gisin)
Message-Id: <9812100756.ZM160641@kitsap.engr.sgi.com>
In-Reply-To: RUMSEY  IAN SCOTT <rumsey@ucsub.colorado.edu>
        "Re: Ringing in the Holidays (sort of)" (Dec  8,  3:09pm)
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Subject: Prony's Method
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Ian;

I also did some checking around, and it turns out that Prony (full name Gaspard
Clair Francois Marie Riche de Prony - whew!) was a French mathematician that
lives in the late 1700s/early 1800s. Apparently Prony's method fits a series of
damped exponential sine functions to waveforms that look like damped sinusoids.

The method is unstable if there is noise in the data (for example initial
transients that occur at the start of the FDTD analysis, etc.).  Some of the
papers I came across spend a great deal of time discussing how one goes about
"extracting" the right portion of the data before sending it to the Prony
algorithm.  This reminded me of the LC define Pulses scheme where we guess (??)
which portion of the waveform we should use for the Pulses analysis.

Osborne and Smyth, at the University of Queensland, Australia came up with a
modified (e.g. more stable) version in 1991, and I am looking into this.
 Several papers have been written on Prony and FDTD, and most of them seem to
get hung up on the instability aspects, so I am not sure that Prony's Method
(at least in the original form) is the way to go.  UMR has been doing some work
using the Pencil of Function (POF) method with FDTD, but a recent paper
published in the 1998 IEEE EMC Symposium Record says that POF also does not
work well for the class of problems we are working on here at SFSU (e.g. high Q
resonances inside electronic enclosures).

Gary Haussmann suggested we try loading (filling?) the inside of the enclosure
with a lightly resistive material.  The hope was that we could increase the
rate of dampening inside the enclosure without significantly impacting the
resonant frequencies.  Since this looks like an interesting approach, I thought
we would give it a try first. (Actually, one of the web sites we visited said
Prony's method was an "Estimation of sinusoid and exponential eigenalysis of
covariance matrices", and to be honest with you, it's way to close to Xmas to
get serious about this weighty kind of stuff ;-))

I'll keep you posted on the progress.

Regards,
Franz Gisin

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On Dec 8,  3:09pm, RUMSEY  IAN SCOTT wrote:
> Subject: Re: Ringing in the Holidays (sort of)
> Franz-
> 	I have an extremely rough understanding of Prony's method, so I
> may be able to point you in the right direction.  The way I see it,
> Prony's method approximates the signal you have with the impulse response
> of some digital filter over that time span.  To extrapolate the signal,
> the full impulse reponse of the filter is calculated.  The accuracy of
> the approximation depends on the number of poles and zeroes used in the
> digital filter representation (I think).  If you have MATLAB,
> I think there is a Prony's method function in the signal processing
> toolbox.  The book I looked in was Statistical Digital Signal Processing
> and Modeling by Monson H. Hayes, John Wiley & Sons, 1996.
> 	I hope this helps.  Maybe someone with a little more expertise
> will be able to supplement what I've told you.
>
> -Ian Rumsey
> University of Colorado
>
> On Sun, 29 Nov 1998, Franz Gisin wrote:
>
> > Does anybody have any experience using Prony's method (or any of it's
> > derivatives and/or equivalents) to extrapolate highly oscillatory LC FDTD
> > analysis results.
> >
> > We are modeling plane wave coupling into electronic enclosures through
> > apertures here at San Francisco State University, and have a number of
cases
> > were the plane wave, after it has passed through the aperture into the
> > enclosure itself, rings to no end.  In some cases, we are up to 30,000 time
> > steps, and a rough "visual extrapolation" indicates that we would have to
go
> > past 100,000 time steps before the ringing has decayed to a sufficiently
small
> > value.
> >
> > Thanks,
> >
> > Franz Gisin
> >
> > --
> >
>-- End of excerpt from RUMSEY  IAN SCOTT



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