Is that another way of saying that even if the original complex signal is aperiodic, the resultant sine waves will be periodic?
An elegant summary, sir!
Understood. But I'm not talking about using an infinitely long signal - just a short one, e.g. somebody saying "hello".
Of course. However, the reason I mentioned that was to tie in with things like live stream encoding, which are essentially indefinately long. You have no choice but to encode such things in discrete chunks.
Ahhh I think I understand what you're saying. A FT on a whole 10 second signal may create 10 million pure tones, whereas a FT on each of the 10 one-second chunks would create 10 much smaller spectra, maybe of 1000 pure tones each, because each chunk is less complex (by virtue of the fact that it is shorter). Is that kind of what you mean?
Exactly that. To give you an idea, MP3 typically encodes chunks of source audio that are 576 samples long. If there is a transient (a sudden, sharp signal) in the source data, it uses a chunk of 192 samples. These slices are but only a few milliseconds long at 44kHz.
In contrast, the machines that I used to use to get NMR spectra for the compounds I was preparing would FFT a signal that was several seconds long, repeating the process several times and averaging the results until a sharp, well defined spectrum was obtained. These machines had horsepower could encode normal mp3 many times realtime, but would take several minutes to produce the FFT for the NMR spectrum.
