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Estimating signals with irrational spectra

Started by Tim Wescott March 14, 2012
OK.  This is sort of off the wall, and I know that the best practical 
answer is probably "approximate it with a lumped-paremeter system and use 
a Kalman filter".  But:

Assume that you have a signal that has an irrational spectra (lots of 1/f 
effects, specifically).  Is there a way to estimate the signal content 
_directly_, without approximating it as a signal with a rational spectrum 
and implementing a Kalman filter?

If not, is there a way to take a signal with an arbitrary spectrum and 
get a measure of how well future values of that signal might be 
estimated, _without_ approximation with a rational spectrum?

I know I'm being moderately incoherent here -- feel free to ask me 
leading questions to clarify what I'm trying to ask.

-- 
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.
Why am I not happy that they have found common ground?

Tim Wescott, Communications, Control, Circuits & Software
http://www.wescottdesign.com
On &#2013266102;ro, 14 mar 2012 22:43:02 in article news:<CMSdnesRabV7jfzSnZ2dnUVZ_hudnZ2d@web-ster.com>
Tim Wescott wrote:
> OK. This is sort of off the wall, and I know that the best practical > answer is probably "approximate it with a lumped-paremeter system and use > a Kalman filter". But: > > Assume that you have a signal that has an irrational spectra (lots of 1/f > effects, specifically). Is there a way to estimate the signal content > _directly_, without approximating it as a signal with a rational spectrum > and implementing a Kalman filter? > > If not, is there a way to take a signal with an arbitrary spectrum and > get a measure of how well future values of that signal might be > estimated, _without_ approximation with a rational spectrum? > > I know I'm being moderately incoherent here -- feel free to ask me > leading questions to clarify what I'm trying to ask.
For arbitrary spectra I don't have any knowledge. In case of 1/f^alpha effects you can look for keywords: fractional order systems, fractional calculus, FARMA (fractional ARMA). In practical case even if you get consistent estimates numerical approximation goes thru (rational) Taylor expansion of gamma function. My experience in this subject is nearly null ;) Some www references: http://mechatronics.ece.usu.edu/foc/cdc10tw/slides/05_Dominik_cdc10afc_tutorial.pdf http://mechatronics.ece.usu.edu/foc/ http://www.ims-bordeaux.fr/CRONE/toolbox/pages/pageDynamiqueSITEExt.php?guidPage=faq http://fomcon.net/introduction/fractional-order-calculus/
On 3/14/2012 2:43 PM, Tim Wescott wrote:
> OK. This is sort of off the wall, and I know that the best practical > answer is probably "approximate it with a lumped-paremeter system and use > a Kalman filter". But: > > Assume that you have a signal that has an irrational spectra (lots of 1/f > effects, specifically). Is there a way to estimate the signal content > _directly_, without approximating it as a signal with a rational spectrum > and implementing a Kalman filter? > > If not, is there a way to take a signal with an arbitrary spectrum and > get a measure of how well future values of that signal might be > estimated, _without_ approximation with a rational spectrum? > > I know I'm being moderately incoherent here -- feel free to ask me > leading questions to clarify what I'm trying to ask. >
Tim, I'm a little confused: - We are used to linear system analysis where a lumped parameter system function is often a rational function. And, evaluating it as a function of real frequency will yield something close enough to a "spectrum". - We are also used to doing Fourier Transsforms of rather arbitrary waveforms (or "functions"?). So, my first WAG would be to compute the Fourier Transform, assert that it is "good forever" and then call it good. No rational / irrational stuff to worry about. But, I wasn't entirely clear if you wanted spectrum values or time domain values at the end. If the starting Fourier Transform is in a finite window .. which it surely must be .. then you essentially want to expand the window somehow. In a DFT context that means increasing the frequency resolution. So, I guess that works in a continuous FT context just as well.... A narrower sinc kernel. I hope this gets you started because I don't have time right now to figure out what this implies.... Fred
On 3/16/2012 2:33 PM, Fred Marshall wrote:
> I hope this gets you started because I don't have time right now to > figure out what this implies....
Tim, I thought about it some more. I'm not sure this will help but maybe it will suggest something. It seems you could look at the Fourier Transform dual problem: - Take a complex spectrum. I will assume that it is not intentionally bandlimited but that there is a practical limit beyond which there is no useful information. - Compute the Inverse Fourier Transform to get a continuous, infinite time function. Now, ask the question: How can I understand the spectral content beyond the limits I chose from the IFT numbers? The answer is "you can't because there was no useful information there in the first place". OK. Now, lowpass the same spectrum .. frequency limit it. This is the dual of a time window. And, having done this we will assert that the IFT can be a sum of sincs. (Well, so could the first one but no matter). Now take the Inverse FT of this lowpassed version. Now, ask the question: From the IFT numbers that result, how can I understand the spectral content beyond the limits of the frequency limit that I chose? I think the answer would be "you can't because you took it all away to begin with". But, to get a corresponding broader spectrum would only mean that you need to assume a narrower sinc in the IFT. The classical way of doing that (in the dual sense) is to append zeros to the spectrum. So, that's a useful case to ponder. Just as we say: "By appending zeros to a time sequence to "increase" frequency resolution you get no new frequency information" We can also say: "By appending zeros to a frequency sequence to "increase" temporal resolution, you get no new time information" All you get is an interpolation. So, if you start with the interpolation, how can you expect to get back anything but the zero-extended spectrum? Leaving the dual and going to the initial question context: "If you start with an interpolation in frequency, how can you expect to get back anything in time but a zero-extended sequence?" So now, it seems to me that your question about the Kalman Filter makes a lot of sense. BUT a Kalman Filter works in the time domain in its fundamental form anyway, yes? Once you have computed a spectrum from a finite time sequence then all the nice intra-window temporal qualities are lost. It's a mathematical transformation or even perhaps we should call it an abstraction. The Kalman Filter needs *time* to figure things out and starting with a spectrum gives no such time. It's always in a sort of "transient" state. Maybe someone will suggest a counter argument that's more hopeful. Maybe a more hopeful answer would be limited to the type of cases, etc. But, I think this fairly well focuses on the issues you face. I hope this doesn't lead to computing an FFT at every sample and then fiddling with a matrix of FFT results, etc. But that actually may fit your situation than the single spectrum I've been assuming. I suppose thought that it doesn't? Fred