For an application, I need to obtain the Fourier transform of an observed period of time from an electroencephalogram (actually a STFT, but that isn't too relevant). However, I worry that the observed periods might sometimes be too short with respect to the wave lengths that I expect to observe. My intuition tells me that the observed period should be well above a particular wave length in order for the Fourier transform to accurately capture its presence.
Is my intuition correct? How long should the observed period be with respect to a wave length to estimate its presence?
We should use appropriately matching units; the observed period needs to be significantly longer than the reciprocal of the frequency - equivalently, wavelength divided by speed.
For more precision, there's the uncertainty principle for the Fourier transform; the product of the time uncertainty of a signal and the frequency uncertainty is at least a constant. If you cut things off with a too-short signal, you'll spread out the observed frequencies to a broader range than they should be.
Checking a reference - EEG data is mostly concentrated in the $10$ Hz range, and variations of less than $1$ Hz in frequency are a big deal. To capture that, you'll need significantly more than a second of data.