I'm trying to understand a few concepts about Fourier Transforms (mainly in the context of signal processing).
Let's suppose a signal is sampled at 10kHz and that the FFT size is 1000. If 1000 samples are processed through this FFT (real only, assuming rectangular window), and if we take the amplitude of the result (first half only and discarding the phase), this gives 500 bins of width 10Hz (except the first one), since the maximum/Nyquist frequency is 5000Hz.
- bin 0, centered at 0Hz goes from 0Hz to 5Hz
- bin 1, centered at 10Hz goes from 5Hz to 15Hz
- bin 2, centered at 20Hz goes from 15Hz to 25Hz
- bin 3, centered at 30Hz goes from 25Hz to 35Hz
- (and so on...)
Although such a spectrum can be plotted as a line, I tend to think of this as a histogram, some sort of "integral" (I'm using this term loosely, feel free to correct me), representing the energy falling within the range of each bin.
Following this, if the goal was to find the energy corresponding to a wider bin (say 15Hz to 35Hz), I would "add" the values for bins 2 and 3 (or take the square root of the sum of the square values). Firstly, does it make sense to group bins this way, to get a single value for a contiguous group of bins (or is it complete non-sense, and nothing can be said of range 15Hz->35Hz from those two bins)?
Then, let's say someone wants to know the "energy" between two frequencies that don't fall on bin delimiters, say 5Hz->6Hz and 151Hz->3002Hz. I've heard it could be possible to use linear interpolation:
- For 5->6Hz, take the 1/10th of the value for bin 5->15Hz,
- For 151->3002Hz, take the 4/10th of the value for 145->155Hz (4/10 for going from 151 to 155), all the bins from 155Hz to 2995Hz and 7/10th of the bin 2995->3005Hz (7/10 for going from 2995 to 3002).
I'm not sure whether this is correct. While the error margin may be low on the 151->3002Hz band, because the two incomplete bins on the edges might be negligible compared to the number of full bins in the middle, it sounds like whatever value obtained for 5->6Hz using this method could be quite far from what the actual value for such a bin would have been if a larger FFT size had been used (and thus had narrower bins allowing for 5->6Hz to fall on actual bins).
Does this kind of interpolation make any sense at all mathematically, especially for lower frequencies? Would it work for even narrower bands, for example 5.1Hz to 5.3Hz?