Does Damping Affect the Coherence Between Broadband Signals?

[jollage]

The coherence I'm asking about is also known as magnitude-squared coherence.

Let's say we have input A(t) and we look at the output B(t), the system L is linear but it has damping effect on the signals. In a long time, this damping will literally kill the high frequencies to zero.

My question is if A(t) contains a broadband spectrum, and let A(t) go through the system L, will the coherence between A(t) and B(t) be less than 1, or just equal to 1?

Jo

 

[Andy Resnick]

If I understand you correctly: if L is a deterministic linear system then the coherence properties of B(t) should be equal to the coherence properties of A(t). IIRC, the transfer function of L determines how the coherence propagates from A(t) to B(t), and while stochastic systems will degrade signal coherence, deterministic systems do not.

There is some discussion here: http://www.dtic.mil/dtic/tr/fulltext/u2/682486.pdf

 

[jollage]

Hi Andy,

Thanks for your reply.

OK, I see what you mean. By stochastic system, do you mean there is noise going into the linear system? Of course, in this case the coherence would decay.

Ley's consider the damping being very strong, literally killing all the high frequencies. Let's say we have A(t)=sin(2*pi*t)+sin(2*pi*9*t) as an input. Going through the linear system with damping, the high frequency 9 Hz is killed, so B(t)=c*sin(2*pi*t), c is some constant less than 1 because frequency 1 Hz is also damped. Do you mean the coherence between A(t) and B(t) is 1 for frequency 9Hz? I can't see that.

Jo

 

[Andy Resnick]

OK, I see what you mean. By stochastic system, do you mean there is noise going into the linear system? Of course, in this case the coherence would decay.

Not exactly- although noise is often modeled with stochastic equations. 'Stochastic' simply means that the time evolution of a system is not deterministic- to model a stochastic processes (e.g. diffusion- Brownian motion is the 'canonical problem') requires statistical analysis.

Ley's consider the damping being very strong, literally killing all the high frequencies. Let's say we have A(t)=sin(2*pi*t)+sin(2*pi*9*t) as an input. Going through the linear system with damping, the high frequency 9 Hz is killed, so B(t)=c*sin(2*pi*t), c is some constant less than 1 because frequency 1 Hz is also damped. Do you mean the coherence between A(t) and B(t) is 1 for frequency 9Hz? I can't see that.
Jo

I don't really understand how you are modeling the damping- but in any case, if you compute <A(t)A(t+τ)>/<A^2> and compare that to <B(t)B(t+τ)>/<B^2>, you will have your answer.

A possible reference for you is Marathay's "Elements of Optical Coherence Theory", which is more general than you need but is very complete.

 

[pmacias]

an,

The coherence between two signals, also known as magnitude-squared coherence, is a measure of the linear relationship between the two signals. It is a useful tool in understanding the behavior of a system, especially when there is damping involved. In your scenario, you have a linear system L with a damping effect on the input signal A(t) and output signal B(t).

The coherence between A(t) and B(t) will depend on the frequency content of A(t) and the damping effect of the system L. If A(t) has a broadband spectrum, it means that it contains a wide range of frequencies. As the signals pass through the system L, the high frequencies will be attenuated due to the damping effect. This means that the coherence between A(t) and B(t) will be less than 1, as the high frequencies will no longer have a strong linear relationship between the two signals.

In other words, the coherence between A(t) and B(t) will decrease as the high frequencies are damped out. However, if the system L has a minimal damping effect on the signals, the coherence may still remain close to 1. It is important to note that the coherence measure is not affected by the absolute magnitude of the signals, but rather the linear relationship between them.

I hope this helps clarify the concept of coherence between two signals. It is a valuable tool in understanding the behavior of a system and can provide insight into the effects of damping on signal processing.