Re: estimation problem? How to seperate two random process?

From: John Bailey (john_bailey_at_rochester.rr.com)
Date: 10/28/04 

Date: Thu, 28 Oct 2004 13:31:37 GMT

On Wed, 27 Oct 2004 12:09:20 -0700, "lucy" <losemind@yahoo.com> wrote:

>Hi all,
>
>I guess this is an estimation problem?
>
>Suppose my observations random process C is in fact a mixture of random
>processes A and B, ie. C_t=A_t+B_t...
>
>What can I do to separate the mixture C_t into A_t and B_t(reconstruct A_t
>and B_t)? Under what condition can I reconstruct A_t and B_t perfectly?
>
>Any thoughts?

Quoting C Algorithms for Real Time DSP, a text by Embree, "One of the
most basic theorems of adaptive and optimal filter theory is the Wold
Decomposition. This theorem states that any real-world process can
be decomposed into a deterministic component and a noise process.
The noise process can be modeled as the output of a linear filter,
excited at its input by a white noise signal." That's all it says.

Would someone please share references or at least a formal statement
of the theorem and its basis assumptions?

With no limits on the number of observations and absent Embree's
statement, I would have used a simple Bayesian hypothesis testing
approach--determine which values of x, where A and B are mixed in
proportions x and 1-x; best explains the given data. That should at
least yield a probabilistic distribution of likely values of x.

John Bailey
http://home.rochester.rr.com/jbxroads/mailto.html