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I have an arbitrary recorded digial signal, on which I have run a Fourier transform. I'm not sure what conventions are on a case like this, but I have 1024 frequency bins. Second bin is the highest magnitude, at about 9500. From there they approximately exponentially decay. The first 13 bins are all above 1000. The last 500 bins are under 150, averaging around 60.

It would be useful for me to define some of these frequencies as "signal" and others as "noise." Is there a canonical way to do that? Perhaps by saying "all frequencies past X are less than 5% of peak" or "this gives us an SNR of 10:1" or "these frequencies have a value less than one standard deviation above mean" or something like that?

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  • $\begingroup$ There are many cases where all the FFT peaks are noise, and the signal is buried down in the noise floor. So the answer depends on your data, your system model (clean linear 2nd degree diff.eq?), and whether they are related. $\endgroup$ – hotpaw2 Sep 4 at 2:22
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    $\begingroup$ I think signal is defined as being useful information, and noise is any disturbance relative to the signal. In a philosophical sense, I suppose if you don’t know what you’re looking for then everything is noise. $\endgroup$ – Dan Szabo Sep 4 at 2:45
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There can't be. One man's signal is another man's noise. In fact, a communication system making the absolute most of a bandwidth would be spectrally white, just like white noise, and hence be indistinguishable from noise to anyone but the receiver for that specific system.

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    $\begingroup$ What if the largest bins come from a strong interferer? $\endgroup$ – MBaz Sep 3 at 14:53
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    $\begingroup$ @StephenCollings that's Really not the case. I swear. OFDM looks white within its bandwidth. Spread-spectrum systems can bring communication signals below noise power level. $\endgroup$ – Marcus Müller Sep 3 at 15:38
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    $\begingroup$ I don't know – to me it really seems you're just wrong, I'm afraid. $\endgroup$ – Marcus Müller Sep 3 at 17:00
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    $\begingroup$ Point is, you really can't tell signal from noise, a lot of the time. You can look at the bands in which GPS operates: The noise floor is solidly above the signal power density, and you hence can't see GPS signals in the spectrum. And GPS isn't by far the most frequency-spread system I've encountered. What about UWB? What about the original Hertz' spark gap transmitter? So, really, the presumption that a signal must be somehow detectable at any case through considering the spectrum is wrong, sadly. $\endgroup$ – Marcus Müller Sep 3 at 17:07
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    $\begingroup$ "I don't KNOW what the signal is; it's literally unknown to anyone. But there's obviously a signal of some kind there" -- It looks like you're defining "signal" as a narrow band that looks different from the rest of the spectrum in a wider band. If that's the case, then that's your answer! The main thing is, you're in uncharted territory and you must come up with your own, hopefully precise definitions for what you want. $\endgroup$ – MBaz Sep 3 at 17:37

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