System Identification

Rune Allnor wrote:

>>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 data you posted were 100 samples long (or theerabout). The order
> estimator I used was of order 10. Computing the normal equations
> means that it uses the average of 100 independent spectra.
> 
> The impulse transient dominates the first ~200 samples, wheras the
> spectrum lines (power line noise?) remains for the duration of the
> recording. So all in all, the energy seems to be 'similar' even
> if the amplitudes are vastly different.
> If you want the f=0.2 lines, let the recording go on for 10 times
> as long and the AR(2) model is more likely to home in on them.
> If you want the start transient, truncate the recording earlier
> and the AR(2) model might home in on the dominant poles.


This is not it. I tried to use only the meaningful data at the 
beginning; I also tried to make the unbiased WSS data by combining the 
the shifted versions of the input with random amplitude and phase. The 
AR(2) result is no different. However if the data is decimated by the 
factor of 5..15, then AR(2) works quite well.

Here is why:

AR tries to minimize the sum of errors at each step:

E[i] = x[i] - A1*x[i-1] - A2*x[i-2] - ....

This is the different problem from finding the best approximation of the 
impulse response! Those problems are identical only in the case of the 
ideal data.

Besides, the mean difference between x[i] and x[i-1] is at the order of 
Amplitude * Fc/Fsa. This makes the model very sensitive to imperfections.

> In between, well, use a 'reasonably' high AR(P) model and
> see what happens.

Higher order models appear to be not too diferent from AR(2):
The dominating poles are still real, however the next pair of poles is 
close to what is expected.


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