FFT Bin Width & Resolution
Get frequency resolution, capture time, and bin count from sample rate and FFT size before you set up a spectrum capture.
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The engineering
The FFT slices your sampled record into N evenly spaced frequency bins from 0 up to the sample rate; the spacing between them is simply the sample rate divided by the number of points. That spacing — the bin width Δf — is your frequency resolution, and it equals the reciprocal of how long you actually observed the signal. Want to resolve two tones 1 Hz apart? You need at least a one-second record, no matter how you window or zero-pad.
The one trap: resolution comes from capture time, not from throwing more points at a short record. Zero-padding to a bigger N interpolates the spectrum and makes it look finer, but it can't separate tones that a short window smeared together. Also remember only the first N/2+1 bins carry unique information for a real signal — the rest is the mirror image above Nyquist.
Where this math comes from
The discrete Fourier transform is old math, but it was computationally hopeless until James Cooley and John Tukey published the fast algorithm in 1965, dropping the cost from N² to N·log N and making real-time spectral analysis practical on the machines of the day. Suddenly the question wasn't whether you could compute a spectrum but how to set it up sensibly.
Fred Harris answered the practical half in his 1978 Proceedings of the IEEE paper on windows, which remains the reference every DAQ engineer cites when arguing about resolution, leakage, and scalloping loss. The bin-width relationships on this card are the arithmetic that precedes any of those window choices — decide Δf and capture time first, then pick the window.
- 1822Joseph FourierEstablishes that periodic signals decompose into harmonic components.
- 1928Harry NyquistFixes the sampling limit at half the sample rate — the top bin of any FFT.
- 1965Cooley & TukeyPublish the fast Fourier transform, making N-point spectra cheap.
- 1978Fred HarrisLandmark IEEE window paper ties bin width, leakage, and resolution together.
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