How to determine if distributions are correlated?

[ggk]

Hi Everyone,

I am analyzing real data using fast-fourier transforms (FFT) in Matlab. The FFT magnitude spectrum show some background noise floor with several sharp spurs popping up high out of the background noise. I need to figure out conclusively which of these spurs are correlated with which other spurs (if any).

To simplify this problem let me just analyze two spurs, to see if they are correlated or not. Let me call them spur1 and spur2. I process the data to obtain three probability distribution functions (PDFs):

1) I isolate spur1 and do an inverse FFT only on spur1 to obtain its' respective real-time waveform (a sinusoid of a certain frequency). I take the PDF of this waveform (PDFspur1).

2) I isolate spur2 and do an inverse FFT only on spur2 to obtain its' respective real-time waveform (a sinusoid of a different frequency). I take the PDF of this waveform (PDFspur2).

3) I isolate both spur1 and spur2 from the rest of the spectrum and take an inverse FFT on the spectrum containing both spur1 and spur2. This results in a real-time waveform whose PDF I'll call PDFspur12.

I want to conclusively determine if spur1 and spur2 are correlated with each other. How do I do it?

One thought I have is, if the distibutions (PFDs) are statistically independent (that is, uncorrelated) then PDFspur12 should EQUAL PDFspur1 CONVOLVED with PDFspur2. If they are NOT equal, then they are not correlated.

I think this is mathematically sound, but I'd appreciate any comments/feedback, especially if you know a better/faster/more conclusive way to determine correlation. -GK

 

[rbeale98]

what do you mean by two signals of different frequencies being correlated?

 

[ggk]

Correlation meaning the spurs share the same origin (source). For example, a square-wave has odd harmonics. If you see these harmonics in the FFT magnitude spectrum, you can tell they are correlated because each spur has the same phase-angle and they are integer multiples of the fundamental frequency. In my case, however, there are so many spurs that phase-angle and multipler (if it's known, which isn't always known in my case) aren't sufficient to determine correlation.