Window Amplitude & Scallop Loss
Coherent gain, scallop loss, ENBW, and processing loss for common FFT windows — read off the amplitude penalty before you trust an FFT peak.
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The engineering
Every window trades amplitude accuracy against leakage. Coherent gain is how much a window shrinks an on-bin tone — you divide the FFT magnitude by it to recover true amplitude. Scallop loss is the extra dip a tone suffers when it lands halfway between bins, and worst-case processing loss (scallop + noise-bandwidth penalty) bounds how bad a single-bin amplitude reading can get.
Rule of thumb: use flat-top when you care about amplitude accuracy (scallop ≈ 0.01 dB, so a peak reads within ~0.1% regardless of frequency), and Hann/Blackman when you care about resolving nearby tones. Rectangular has the best resolution but a brutal 3.92 dB scallop — a real tone can read nearly 40% low if it falls between bins.
ENBW (in bins) is the factor that turns a windowed power-spectral-density bin into a proper noise measurement: multiply your bin width by ENBW to get the effective noise bandwidth per bin.
Where this math comes from
Windowing predates the FFT — spectral analysts in the 1940s and 50s tapered data records to fight leakage, and names like Tukey, Hann (von Hann's cosine bell), and Hamming came out of that era of hand-tuned smoothing. But the definitions were scattered and inconsistent, and every lab reported window merits differently.
Fred Harris, an engineer at San Diego State and the Naval Ocean Systems Center, cleaned it all up in his 1978 Proceedings of the IEEE paper 'On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform.' He tabulated coherent gain, scallop loss, ENBW, and worst-case processing loss for dozens of windows in one consistent framework — the reference every DAQ and spectrum-analyzer designer still reaches for. This card evaluates his tabulated figures.
- 1898Julius von HannCosine-bell smoothing later formalized as the Hann window.
- 1949Richard Hamming & John TukeyOptimized raised-cosine window minimizing the largest side lobe.
- 1958Ralph Blackman & John TukeyThree-term cosine window with deep side-lobe suppression.
- 1978Fred J. HarrisUnifies window figures of merit in the definitive IEEE survey — the source for this card.
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