FFT Resolution & Windows
Frequency resolution, record length, ENBW and spectral leakage for an FFT analyser set-up.
Analyser set-up
The time–frequency trade-off
An FFT of N samples taken at rate f_s produces spectral lines spaced by the inverse of the record length:
Finer frequency resolution always costs a longer acquisition. To separate two tones 0.5 Hz apart you need at least 2 seconds of data (more in practice), because the window widens each peak. Analysers traditionally display N/2.56 lines: the anti-aliasing filter needs a guard band between the useful range (f_s/2.56) and Nyquist.
Windows and leakage
The FFT assumes the record repeats periodically. If a tone does not complete an integer number of cycles, the discontinuity spreads energy into neighbouring lines: leakage. Window functions taper the record to zero at both ends, trading main-lobe width for lower side lobes:
| Window | ENBW (bins) | Scallop loss | Use for |
|---|---|---|---|
| Rectangular | 1.00 | 3.92 dB | Transients, impact tests, order-tracked signals |
| Hann | 1.50 | 1.42 dB | General-purpose random and periodic signals |
| Hamming | 1.36 | 1.75 dB | Close tones of similar level |
| Blackman-Harris | 2.00 | 0.83 dB | High dynamic range, weak tones near strong ones |
| Flat top | 3.77 | ~0.01 dB | Accurate amplitude of tones (calibration) |
ENBW (equivalent noise bandwidth) = N\cdot\sum w^2/(\sum w)^2 is used to convert power spectra to power spectral density:\mathrm{PSD} = \mathrm{power}\,/\,(\mathrm{ENBW}\cdot\Delta f). Scallop loss is the worst-case amplitude error for a tone falling halfway between two lines.
Averaging and overlap
Averaging n spectra reduces the random error of a noise estimate by √n. Overlapping records (typically 50 % for Hann, 67 % for flat top) recovers data down-weighted by the window and shortens the measurement time.