When you calculate an FFT (or more strictly speaking a DFT, FFT is just a fast algorithm for calculating it), the algorithm treats the signal as if it was repeated indefinitely, aka a periodic or cyclic extension.
1 ) That means if you have the samples for one whole cycle of sinewave, and I mean samples 0 to N-1, where N samples make one exact cycle, this periodic extension to the signal will behave as you might expect it to. You will see one sinewave frequency.
2 ) If you have collected 1.5 cycles of sinewave, then the FFT will again do its periodic extension, and the large step between the last and first sample will get repeated as well, as if it's part of the signal. The result of that will be a splat of power in every output bin. Not what you want, but easy to spot.
Perhaps a worse case than that is where you collect 1.001 cycles of sinewave. It's difficult to see the error in the time domain, and the power in all the other frequencies might be low enough for you to miss what's gone wrong, and blame something else.
3 ) This periodic extension works fine for any number of whole cycles. You can collect one cycle, or two cycles, or 10, as long as they are whole cycles, you'll get the answer you expect. You will have more samples, for the same frequency range, so your output frequency resolution is higher for a large number of cycles than for one. We normally express this as the frequency resolution (in Hz) being the inverse of the length of the time record (in seconds).
4 ) But the real world isn't that tidy, and we are always having to cope with taking the FFT of an incomplete number of cycles. This is where windowing comes in. A window tapers the record off smoothly to near zero at both ends, so that there's a much reduced step when the periodic extension happens.
As a window reduces the effective length of the time domain data, by de-emphasising the data at the ends, the frequency resolution is worse than without a window. Generally, we need to capture several cycles to see a reasonable spectrum. There are different window shapes that are optimised for better frequency resolution, or better suppression of the power splat from the repeated step. There's a whole section of mathematics devoted to the construction and performance of these windows.
If the signal frequency is varying in time, whether linearly or non-linearly, then we need what is called a Short-Time FFT, STFT. In this, you split the signal up into short sections, and perform an FFT on each section. There is a tradeoff here between time and frequency resolution. For a short record, resolution is limited by the FFT. For a long record, resolution is limited by how much the signal frequency varies in that time. For any given rate of variation, there's an optimum window length. As before, you need at least 5 or so cycles in each record to have reasonable resolution.
The window choice for an STFT can make a small difference. With a varying signal, it tends to be better to use a wider window like a Hamming to minimise loss of frequency resolution, at the expense of poor suppression of the non-signal bins, compared to for instance a Blackman Harris or Gaussian 4. As the window loses data at the edge of the bins, successive records should be overlapped, typically by 50%, more overlap would result in little more benefit.
The STFT is a non-parametric method, which can be used on any signal, or multiple signals, any rate or linearity of variation, but only averagely well. If you have a good model of your process, you can get better resolution by model fitting, for instance with MUSIC or ESPRIT algorithms. These come at a cost of complexity, and performing very poorly on signals outside their model.
If you know you have a single frequency only, then a better method might be to estimate the frequency directly. One route is via a PLL, ideally digitally at this low rate. The other route is via FM demodulation, for which we first pass the signal through a Hilbert filter to make it into an analytic signal. Either technique has to be designed with an assumption of what the minimum frequency is. Either technique should be better than STFT for resolution, and cheaper in terms of calculations, because it's only dealing with a single frequency.
