how determine correlation by looking at distributions?

[ggk]

Hi Experts,

I have an application related to DSP and probability/distributions I could *really* use some help with.

I have a real-time waveform captured using an oscilloscope. I take the FFT of this waveform and the magnitude spectrum shows a random-noise floor (looks like the profile of grass on your lawn) with dozens of spikes (sharp spurs) sticking up high above the noise floor. My goal is to determine which of these spurs are correlated to each other (if any).

Ideally, the problem is solved looking at the phase-angle of each spur: correlated spurs will always have exactly the same phase-angle. There are two problems with this... (1) real-life data is not "ideal" (it contains noise) and therefore the phases of KNOWN correlated spurs are actually several degrees different when measured, so this test is not conclusive, especially considering (2) even uncorrelated spurs' FFT coefficients have a specific phase and there is some chance these (uncorrelated) spurs can end up with the same phase as each other (misleading one to erronously conclude they *are* correlated).

Thus, I turn to probability to see if it can help... I can isolate each spur and do an inverse FFT (on that one spur only) to obtain a probability distribution function (PDF) (from the resulting real-time data after IFFT). If there are N spurs, I end up with N PDFs. In addition, I also pair-up all spurs in all possible combinations of two spurs, and for each combination take their inverse FFT to obtain their respective distributions (PDFs). That is, I would take two FFT coefficients (complex) and do one inverse FFT that includes both of these coefficients.

Then I analyze the data to determine whether spurs are correlated. I pick one pair of spurs to analyze -- let's say this pair contains spur1 and spur2. To determine correlation, I COMPARE (1) the PDF taken from real-time waveform resulting from the inverse FFT of the PAIR of spurs (IFFT of the spectrum containing both spur1 AND spur2, that is, ONLY spur1 and spur2 with everything else removed), with (2) the distribution resulting from spur1's PDF CONVOLVED with spur2's PDF... and if these two resulting distributions match (this is, they are the "same", or close enough) then the two spurs in question are NOT correlated (and likewise, if these two distributions do not match they ARE correlated).

I think this is mathematically sound, but would appreciate any comments, or if someone knows of a faster/better/more conclusive way to determine correlation I'd appreciate knowing.

Best regards, -GK

[fasteddie]

Would the cross correlation function be of any use ? ... I know that it can be implemented in Fourier domain ... but am not 100% sure its what you need here