= Effective noise bandwidth
{title2=$B_{\rm n}=\int_0^\infty|H(f)|^2df/|H_0|^2$}
For a <linear filter> with chosen nonzero reference voltage gain $H_0$, define its effective noise bandwidth by
$$
B_{\rm n}=\frac{1}{|H_0|^2}\int_0^\infty |H(f)|^2\,df.
$$
This is the width of an ideal rectangular <linear filter> with gain magnitude $|H_0|$ passing the same mean square of input <white noise>. Consequently
$$
v_{o,\rm rms}=|H_0|\sqrt{4k_{\rm B}TRB_{\rm n}}.
$$
Use the full voltage transfer from the <open-circuit voltage> source to the readout, including any <electrical load>, when applying this formula. A nominal <signal bandwidth> stated between two <cutoff frequencies> does not specify the integral: gradual tails and a nonuniform passband matter. For an ideal rectangular unit-gain passband $[f_1,f_2]$, $B_{\rm n}=f_2-f_1$. For an unloaded <RC low-pass filter> with $H_0=1$ and <cutoff frequency> $f_c=1/(2\pi RC)$,
$$
B_{\rm n}=\int_0^\infty\frac{df}{1+(f/f_c)^2}
=\frac\pi2 f_c=\frac1{4RC}.
$$
The last equality follows by substituting $x=f/f_c$ and integrating $1/(1+x^2)$ as $\arctan x$. Using the nominal $f_c$ as the noise bandwidth underestimates the <variance> by a factor $\pi/2$, or the <root mean square> voltage by $\sqrt{\pi/2}$.
\Image[/thermal-electrical-noise-spectrum-and-bandwidth.png]
{title=Equal-area bandwidth for filtered <thermal electrical noise>}
{description=Original analytical illustration for $T=300\,\mathrm K$, $R=10\,\mathrm{k}\Omega$, and $C=1\,\mathrm{nF}$. The left panel shows the one-sided <open-circuit voltage> <power spectral density> and the output spectrum of an unloaded <RC low-pass filter>. The right panel compares the squared <frequency response> with an equal-area ideal rectangular <linear filter>; the shaded RC area continues beyond the plotted range. The nominal corner is $15.915\,\mathrm{kHz}$, whereas the <effective noise bandwidth> is $25\,\mathrm{kHz}$. Calculated curves, not measured data.}
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