Re: System Identification

Rune Allnor wrote:

> On 8 Jul, 16:01, Vladimir Vassilevsky <antispam_bo...@[EMAIL PROTECTED]
>
> wrote:
> 
>>While trying to find the parameters of the pole only system of the 2nd
>>order from the impulse response, I am running into the following
problem:
>>
>>If the coefficients of the system are calculated using the AR method,
>>the solution appears to be wrong.
> 
> 
> The spectrum of the data you posted last week doesn't comply to the
> 2nd order model. There are at least one set of spectrum lines (f=0.2)
> and a 'hill' (f=0.4) in the spectrum. That's three poles right there.
> Since the data are real, you need 3 more poles for the complex
> conjugates. Try an AR model of order 6 or higher, I'd say 8-10 or so.

NOW I understand what you are saying! Thank you for pointing me.

The LPC model can be surprisingly wrong even if the data mismatch is 
small. This happens due to the nature of the LPC method. The way to deal 
with the problem is increasing the order of the model. It took me a 
whole day to realize that.

>>But if the coefficients of the system are found by search for the best
>>matching impulse response of the model to the data, the solution is
correct.
> 
> Sure. One pair of poles completely domminates the impulse response.

This is what puzzled me. If one pair of poles is clearly dominating, 
then why AR(2) couldn't find it in one shot.

>>The AR minimizes the sum of errors at every step, whereas the impulse
>>response matching optimizes the whole thing. They are not equvalent.
>>
>>Didn't figured out what to do yet.
> 
> You need to approach the data on their own terms. The fact that
> the end target is a 2-pole model means nothing if the data don't
> comply to the AR(2) model.

Yes, yes, and yes.

> The best you can hope for is that the
> data respond well to an AR(P) order of 'sufficiently high' order.
> If that is the case, you can inspect the pairs of roots and see
> if any of them match the criteria you look for.

This implies the numeric solving of the polynomial, and estimation of 
the amplitudes. I.e. the "full" Prony method. I wonder if Greg Berchin 
method of FDLS can be a quicker way to solution; it seems like the 
selection of the reference points is clear in this case.

> 
> Rune

Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com