cross-spectrum question 2

[mathlete7]

Hi,
Does anyone have a good definition of what a "cross-spectrum" is or know of any resources that give a good deformation of the physical meaning is?
I'm using it in the context of fluctuating pressure levels due to a turbulent boundary layer. It seems like most information I can find on cross-spectra only show some equations and don't explain what the physical meaning is. It seems that the cross-spectra essentially scales the pressure spectral density levels across the surface of the OML, however I'm looking for a more formalized definition.
Thanks much...

 

[BobM3]

It gives an indication of the relationship of two signals as a function of frequency. Specifically, do a fourier transform of each signal, form the complex conjugate of one of them and then multiply the two transforms frequency by frequency.

It can be useful in characterizing a system's response as a function of frequency. If a system has a 100 hertz natural frequency and you excite it with a 100 hertz input then you would see a high number for the CSD at 100 hertz. You'd also get the phase relationship between the two signals. If you excited that same system with a 10 hertz input (and there were no 10 hertz natural frequencies in it) then you would see a low number for the cross spectral density at 10 hertz.

 

[Vibac]

When used to describe TBLs the cross-spectrum contains information about both the frequency content and the lenght-scale content. BobM3 just explained the frequency part. If eg. the TBL is convected above a plane surface you typically use two length-scale parameters: One (x1) in the direction of the flow and one (x2) perpendicular to the flow but in the same plane as the surface.

Evaluating the cross-spectrum density function of the pressure at a certain frequency f and certain lenght scales x1 and x2 will tell you how much power is located here. If you examine the dependence on x1 and x2 you'll get an idea of the typical "size" of the TBL "structure" in the two directions. You'll typically find that the TBL has longer correlation lenght in the direction of the flow than perpendicular to it.

Hope this helps a little, havn't worked with this for several years...

 

[SomptingGuy]

I can't really comment in the right context, but...

A cross spectrum relates two signals that may or may not be correlated.

If they are correlated, it will give information that relates the amplitude and phase between them as a function of frequency.

A single cross spectrum is rare. It's more usual to average them and also average the autopowers of the two contributing signals. That way upper & lower FRF estimates can be made, along with coherence.

- Steve

 

[mathlete7]

Thanks for your responses everyone! They were very helpful