 Equivalent noise bandwidth

The equivalent noise bandwidth of a finite window is the width of an ideal rectangular frequency response that passes the same white-noise power when both responses have the same peak gain. It quantifies noise collected by a spectral amplitude estimate, rather than the separation needed to distinguish two tones. For real coefficients $w_n$ with nonzero sum and sampling frequency $f_s$,$B_e$ is in hertz; $b_e$ is in bins of an $N$-sample record. The primary software documentation gives $B_e$ as the exact conversion factor between its squared spectrum and power-spectral-density scalings.^[ref]

An original variance derivation explains the formula. Let independent zero-mean sample noise ${ϵ}_n$ have variance ${\sigma }^2$, and form the complex amplitude estimate at frequency ${\nu }_0$,Cross terms vanish by the independence and zero means, soThe flat two-sided input noise spectral density is ${\sigma }^2/f_s$. Integrating it over the peak-normalized window response gives the same result by Parseval identity. A one-sided real-signal convention folds negative-frequency power into positive frequencies; factors of two must be handled consistently, rather than changing the window’s bandwidth ratio.

For a rectangular window, $\sum w_n=N$ and $\sum w_n^2=N$, giving $b_e=1$. For the periodic Hann window with $N>2$, the cosine sums give $\sum w_n=N/2$ and $\sum w_n^2=3N/8$, hence $b_e=1.5$. These agree with the instrument manual’s tabulated values.^[ref]

For an original numerical comparison, take $f_s=4096\mathrm{Hz}$ and $N=256$, so the original frequency grid spacing is $16\mathrm{Hz}$. Rectangular and periodic Hann bandwidths are respectively $16$ and $24\mathrm{Hz}$. With independent sample noise of standard deviation $0.20\mathrm{V}$, the RMS magnitude of the zero-mean complex noise estimate is $0.0125\mathrm{V}$ for rectangular weighting and about $0.01531\mathrm{V}$ for Hann weighting. The latter variance is 1.5 times greater, despite Hann’s lower distant sidelobes. It is the RMS complex-error magnitude, not automatically the standard deviation of a positive spectral-magnitude display.

Zero padding leaves $f_s$, the actual window coefficients and $B_e$ unchanged. If a 256-sample Hann record is padded to 1024 points, the display spacing becomes $4\mathrm{Hz}$ and the same $24\mathrm{Hz}$ bandwidth spans six displayed bins. Replacing $N$ by the padded FFT length in a formula while leaving the coefficient sum unchanged confuses these two bin units.

The noise covariance between windowed Fourier bins also matters when combining several spectral values: a taper can correlate neighboring estimates. Bandwidth alone does not specify their joint statistics, and colored noise requires weighting its actual spectral density. A proposed check repeats the same estimator on independent synthetic noise records with known variance, then on measured noise with a characterized acquisition response, comparing empirical mean squared complex error with the formula. Report window convention, detrending, one-sided/two-sided scaling and record duration so the test can be reproduced.