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Due to the Nyquist rate, you need to sample twice as fast as the highest frequency you want to measure. Then when you do a FFT the number of samples determines the number of bins and thus the frequency resolution.

But what about the lowest frequency you want to measure? I would think your sampling duration needs to be at least as long as one cycle of your lowest frequency.

Is this true, or can you register $20$ $Hz$ from a dozen samples at $44$ $kHz$?

If this is true, how long do you need to sample, and how do you limit the number of bins?

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  • $\begingroup$ Rule of thumb - one period of your wave. $\endgroup$
    – jojeck ♦
    Commented Mar 6, 2015 at 12:36

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Using the DFT, the lowest bin is indeed corresponding to one period (i.e., the frequency which have exactly one period within the length of the DFT).

Conceptually, the DFT is correlating the input signal with different sinusoids. Now, to capture the phase information, these are "complex sinusoids", but if you knew the phase beforehand, you could just correlate with a real-valued sinusoid with the same phase to obtain the amplitude of the sinusoid.

Hence, it would conceptually be possible to register the existence of 20 Hz using a dozen sample at 44 kHz if you knew the phase of the signal (or by making multiple computations with different phase shifts). The drawback is that there are (probably/most likely) other signals which will be falsely detected as 20 Hz, because they correlate as well as the 20 Hz signal (and/or cancel the 20 Hz signal).

This is in no way a comprehensive answer, but more an insight that what the DFT and most other transforms do is to correlate the input signal to a set of basis functions, to see how they compare. With enough knowledge of your input signal you can find simpler basis functions for your particular case, but there is a trade-off between flexibility (ability to handle different cases) and simplicity (how little information/computation is needed).

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  • $\begingroup$ So is there a way to reduce the resolution of the transform without reducing the number of samples? $\endgroup$
    – Pepijn
    Commented Mar 6, 2015 at 14:29
  • $\begingroup$ @Pepijn: yes, do not compute those bins of the DFT. Or do you mean something else? From an FFT perspective, the benefit is not that great, though. Called "Pruned FFT". $\endgroup$
    – Oscar
    Commented Mar 6, 2015 at 16:09
  • $\begingroup$ I also believe that DFT assumes that the measured signal is periodic, thus repeating that signal will not contain signal with a longer period as the measured signal. This is also why windowing is often useful, because repeating the signal will often be discontinues. $\endgroup$
    – fibonatic
    Commented Mar 7, 2015 at 2:29

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