= Filtered shot noise
{title2=$\sigma_v^2=\int_0^\infty S_I^{(1)}(f)|K(f)|^2\,df$}
Let $K(f)$ be the complete current-to-voltage <frequency response> from detector input to readout, in $\mathrm{V/A}$. For a real stable <linear time-invariant system>, the shot contribution to the output voltage <variance> is
$$
\sigma_{v,\rm shot}^2=\int_0^\infty 2qI|K(f)|^2\,df
=\int_{-\infty}^{\infty}qI|K(f)|^2\,df.
$$
If $K(f)=R_FG H(f)$ with feedback <electrical resistance> $R_F$, a reference <voltage gain> $G$, and dimensionless normalized <frequency response> $H$, then
$$
B_{\rm n}=\int_0^\infty |H(f)|^2\,df,\qquad
\sigma_{v,\rm shot}^2=2qI R_F^2G^2B_{\rm n}.
$$
This is the <effective noise bandwidth> for unit reference magnitude. The <transimpedance amplifier> factor $R_F$ converts <electric current> to voltage; $G$ multiplies voltage. Neither is an additional <noise> source merely because it amplifies an existing fluctuation. In contrast, a real <amplifier> adds input voltage <noise> and input current <noise>, and its feedback <resistor> contributes <thermal electrical noise>. Those terms need their own transfers to the output. In particular, input voltage <noise> generally has a different transfer from an injected detector <electric current>. Simply multiplying every input specification by $R_FG$ is incorrect.
For a unit-gain exponential <low-pass filter>, with <time constant> $\tau>0$,
$$
h(t)=\frac1\tau e^{-t/\tau}\quad(t\geq0),\qquad
H(f)=\frac1{1+2\pi\mathrm i f\tau},\qquad
B_{\rm n}=\frac1{4\tau}=\frac\pi2 f_c,\qquad f_c=\frac1{2\pi\tau}.
$$
The <cutoff frequency> $f_c$ is not $B_{\rm n}$. An independent time-domain derivation sums independent event contributions to give
$$
\operatorname{Var}(i_{\rm filtered})
=\lambda q^2\int_0^\infty h(t)^2\,dt
=\frac{qI}{2\tau}.
$$
The same reasoning for a rectangular averaging gate has $h=1/T$ on $[0,T]$ and zero elsewhere, hence <effective noise bandwidth> $1/(2T)$ and <variance> $qI/T$, agreeing with the <shot noise counting model>. Equal numerical durations $T=\tau$ do not make these two different <linear filters> equivalent.
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