What's the minimal time to distinguish 1 Hz difference in frequency?

Question

Given a fragment of a single-frequency wave, what's the minimal length of it to calculate its frequency with error less than 1 Hz? Or if there are two fragments with frequency difference of 1 Hz, how long does it take to distinguish between them?

Answer 1

This is an excellent question, one that has kept mathematicians, engineers and physicists busy for almost a couple of centuries. However, as posed, your question may be subject to multiple different interpretations. More context would be needed for a more precise answer.

As was noted in the comments, if you assume that you get access to a small $contiguous$ fragment of the signal, you'll be able to recover the full signal, regardless of the frequency of the wave.

But in practice, there is no such thing as contiguous capture of the signal. Things get discretized. And that's where the frequency of your wave matters. When you want to reconstruct the original signal from what you captured, noise is introduced via 2 sources of discretization:

Time discretization: Most devices $sample$ the signal, which means that the signal's values get recorded at specific moments in time. For instance, if you assume you sample at regular intervals, then Shannon's sampling theorem tells you that you need to sample at least at twice the frequency of the signal (and in practice, much more than 2x if you want to be robust). If you don't respect that principle, then you may be unable to distinguish two signals. The most intuitive example of this is the Wagon-wheel effect in movies.

Value discretization: In addition to time discretization, the magnitude of the signals being captured is usually discretized. This process if called quantization. In essence, instead of representing each value with infinite precision as a real number, it is encoded via a scheme that can only accommodate a finite number of possible values. One usually speaks of $n$-bit quantization if $n$ bits are used to quantize the values (so, $2^n$ possible values). Quantization also adds noise to the signal when you want to reconstruct it.