File:Hall2006 Freq sampling.png

Summary
Sampling and localization. (a) A 128 ms, 32 Hz signal, padded to 256 ms. (b) Its spectrum, computed by Fast Fourier Transform (FFT); the short window means that the spectrum is sparsely sampled, since ∆f = 1/0.256 = 3.9 Hz. (c) The same 128 ms, 32 Hz signal, padded to 1024 ms. (d) Its spectrum; the longer window has improved the sampling in frequency (∆f = 1/1.024 = 0.977 Hz), but the uncertainty in the signal’s frequency is the same, indicated by the relatively wide central peak in the spectrum. (e) A 768 ms, 32 Hz signal, padded to 1024 ms. (f) Its spectrum, which has the same sampling as (d), but much better frequency localization. Adapted from Stockwell (1996).