Robert Adams <robert.adams@analog.com> wrote in news:0c83dc02-4ddd-48f9-
8826-3d9340c8176e@p25g2000hsf.googlegroups.com:

> Suppose I take a signal X(n) and apply it to an unknown system that
> consists of a linear filter in parallel with some sort of non-linear
> processing, to create an output signal Y(n).
> 
> 
> If the non-linear portion of the unknown system were set to zero, then
> given X and Y, I could design an adaptive filter that, when fed with
> X, would eventually converge and produce Y. However, if the non-linear
> part is significant, I cannot produce Y from X using a linear filter.
> 
> 
> My question is this; given 2 signals X and Y, can I estimate how much
> they are linearly related without actually running an adaptive filter
> and looking at the convergence error?
> 
> I recently read about the "coherence" function, which seemed promising
> but I am not that familiar with it.
> 
> 
> Bob
> 

Coherence will be unity for a perfectly linear system with no noise, and 
nonlinearity and/or noise will pull it down.

You're near the right place.  You can use a Welch Periodogram to find the 
gain and phase of the linear portion, and you can use coherence to put a 
confidence interval around the whole shebang.  Don't forget that 
coherence is a function of frequency.


-- 
Scott
Reverse name to reply

On 2 Mar, 02:50, Robert Adams <robert.ad...@analog.com> wrote:
> Suppose I take a signal X(n) and apply it to an unknown system that
> consists of a linear filter in parallel with some sort of non-linear
> processing, to create an output signal Y(n).
>
> If the non-linear portion of the unknown system were set to zero, then
> given X and Y, I could design an adaptive filter that, when fed with
> X, would eventually converge and produce Y. However, if the non-linear
> part is significant, I cannot produce Y from X using a linear filter.
>
> My question is this; given 2 signals X and Y, can I estimate how much
> they are linearly related without actually running an adaptive filter
> and looking at the convergence error?
>
> I recently read about the "coherence" function, which seemed promising
> but I am not that familiar with it.

You are on the right track. Check out

Bendat and Piersol: "Random data"

and the two books by Bendat alone. Bendat wrote one book
on the analysis of nonlinear systems.

Rune

>Suppose I take a signal X(n) and apply it to an unknown system that
>consists of a linear filter in parallel with some sort of non-linear
>processing, to create an output signal Y(n).
>
>
>If the non-linear portion of the unknown system were set to zero, then
>given X and Y, I could design an adaptive filter that, when fed with
>X, would eventually converge and produce Y. However, if the non-linear
>part is significant, I cannot produce Y from X using a linear filter.
>
>
>My question is this; given 2 signals X and Y, can I estimate how much
>they are linearly related without actually running an adaptive filter
>and looking at the convergence error?
>
>I recently read about the "coherence" function, which seemed promising
>but I am not that familiar with it.
>
>
>Bob
>
>********************************************
Hello Bob,

the answer at exactly what you are asking, namely "how much" linearly
dependent is x (the input of the system) to y (the output of the system),
is given by the so called correlation coefficient. The correlation
coefficient is given by the formula

r = [E(x-mean(x))*(y-mean(y))] / [std(x)*std(y)]

When r = + 1 or r = - 1 then x and y are related via a 100% linear
relationship, namely y = a * x + b. As r -> 0 then the relationship
between x and y tends to be less linear. When r = 0 then x and y are said
to be uncorrelated. That means that there is nothing linear between x and
y, however x and y maybe related via a nonlinear model.

Manolis C. Tsakiris

Suppose I take a signal X(n) and apply it to an unknown system that
consists of a linear filter in parallel with some sort of non-linear
processing, to create an output signal Y(n).


If the non-linear portion of the unknown system were set to zero, then
given X and Y, I could design an adaptive filter that, when fed with
X, would eventually converge and produce Y. However, if the non-linear
part is significant, I cannot produce Y from X using a linear filter.


My question is this; given 2 signals X and Y, can I estimate how much
they are linearly related without actually running an adaptive filter
and looking at the convergence error?

I recently read about the "coherence" function, which seemed promising
but I am not that familiar with it.


Bob