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Coherence

Started by Robert Adams March 1, 2008
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

>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
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
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