FFTs: Are low frequency peaks believable for short signals?

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Hello All,
I have a question regarding FFTs. Let's say we have a signal that is sampled at 20kHz. I know that the frequency resolution is driven by the length of my data array - resolution increases with a longer data array.
Now, does the length of that data array also play a role in the minimum resolvable frequency? Say I conduct two ffts on a 20kHz signal. In case a, the signal is 20k entries long, while in case b it is only 2k entries long.
Can I trust low-frequency peaks (say ~1Hz) from either of these cases? In case a, my data array encompasses one period (20k samples at 20kHz ~ 100% of the period), and it therefore seems reasonable that I can trust a 1Hz peak. However, in case b the data array only encompasses a small portion of it (2k samples at 20kHz ~ 10% of the period). Can I trust a low-frequency peak in this case?
This seems equivalent to a very long extrapolation and just doesn't feel right. Can anyone shed some light on this? Am I missing something about the method? If there is a low-frequency trustworthiness limit? How do you define it if it does exist?
Thanks!
  1 Comment
Adam
Adam on 15 Apr 2015
One thing you must remember about FFT analysis is that there is an assumption that your signal is infinitely repeating, therefore the FFT result is as if your 2k samples were repeated an infinite number of times in succession. So in that sense the signal for the FFT is as long as it needs to be for as small a frequency as is required. But obviously that FFT assumption of a repeating signal becomes less and less valid in such cases usually.
For a signal of 20kHz though I'm not sure what useful information you would be able to interpret at 1Hz anyway.
I am familiar with using seismic signals so the sampling frequencies I work with are more of the order of 250-1000 Hz. Even then the 1Hz component of the signal is a little on the low side to be considered useful. So from that perspective I can't really help with regard to whether the 1Hz component for your signal has any significance.

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