I'm working on some acceleration data, looking at vibrations and at what frequencies they're occuring. The recorded periods are fairly long (~1 minute) and I was wondering, what's the difference was between doing a FFT on the whole 1 minute data set and taking the magnitudes, vs doing fourier transforms for each second, taking the magnitudes, and then averaging them at the end?
This is the basics of spectral analysis using digital Fourier transform. Mountains of literature is written on the subject.
In short, when you do a FFT over the entire data set 1-minute long, you will have fairly high spectral resolution of about 1/60 Hz, or 16.6 mHz (milliHz). But your confidence interval of the amplitude at each frequency line will be very low, you will have a very high ~100% noise level.
When you break your data into smaller blocks, so for each second-long data your spectral resolution will be 1 Hz, low. But, if you take an average of all 60 periodograms, your noise level will be SQRT(60) =~8 times smaller, so you can be much more confident if you see some spectral peaks in your spectrum.
Note, that this works only on power spectrum estimates, Re^2 + Im^2.
So, with a limited data set, there is a trade-off between accuracy of estimates of spectral amplitudes, and frequency resolution of your power spectrum.