What does the word 'bin' mean in the context of FFT's?

Question

Everywhere I read about the FFT, I read about frequency bins. I can't find anything about what is precisely meant with 'bin'. What I wonder about is if this is actually meant to imply that the FFT returns binned data in the sense of the word as used in statistics.

I think this question can be rephrased into this example: if I have bins centered at 1Hz, 2Hz, 3Hz, etcetera, what would happen to a frequency component at 2.4Hz? If the frequency bins are actually bins, I think this would mean that the amplitude is simply added to the 2Hz bin.

If the bins are indeed bins as I interpret the word, I would really like to know why the FFT results are binned. If not, then I am curious why they are called bins.

Answer 1

A finite length DFT transform has a finite number of basis vectors, which are all single frequency complex sinusoids. If the results weren't binned then you would need an infinitely long DFT or FFT result to represent any continuous spectrum. However, Parseval's theorem says that all the energy in a finite input vector ends up in the finite FFT result vector. So any single between-bin-center-in-frequency pure sinusoid (or between DFT basis vector frequency, windowed by any finite length FFT) ends up binned, spread mostly into FFT result bins very near in frequency.

"very near" because:

Each FFT result bin is not rectangular, but a filter in the shape of Sinc function (more precisely, a periodic Sinc function or Dirichlet kernel), since that is how a pure sinusoid correlates with any one DFT complex basis vector when swept across the frequency range from 0 to Fs/2. The result is also conjugate mirrored in the full length FFT result for strictly real input.

Another interpretation is that there exist no in-between bin frequencies, that any cut-off sinusoid is really a repetitively discontinuous periodic waveform. But that mythical input is not representative of how FFTs are actually used in many forms of DSP signal analysis (windowing, peak estimation, low offset overlapping windows, phase vocoders, etc.). Although knowing about this circularity is important when using an FFT for fast convolution.