= Shot noise spectral convention
{title2=$S_I^{(1)}(f)=2qI,\quad S_I^{(2)}(f)=qI$}
Write an ideal point-transfer model using the <Dirac delta function>:
$$
i(t)=q\sum_k\delta(t-t_k),\qquad \delta i(t)=i(t)-I.
$$
For this arrival stream with <independent increments>, distinct arrivals contribute the product of the means, which cancels when centering. The same-arrival term leaves the <autocovariance>
$$
\mathbb E[\delta i(t)\delta i(t+s)]=q^2\lambda\delta(s)=qI\delta(s).
$$
Its <Fourier transform> gives the two-sided <power spectral density>, while folding negative <frequencies> onto positive ones gives the one-sided convention:
$$
S_I^{(2)}(f)=qI\quad(-\infty<f<\infty),\qquad
S_I^{(1)}(f)=2qI\quad(f>0),\qquad
[S_I]=\mathrm{A^2/Hz}.
$$
These spectra describe $\delta i$, excluding the spectral line from the mean <direct current>. The <amplitude spectral density> is $\sqrt{2qI}$ in $\mathrm A/\sqrt{\mathrm{Hz}}$; it is not a <root mean square> <electric current> until a measurement <signal bandwidth> is specified. Approximate <white noise> requires <frequencies> low enough that the single-transfer pulse shape and any correlations can be neglected. Neither the ideal impulses nor a flat spectrum extending to infinity describe a real detector at every timescale.
All spectra above are per <hertz>, with $f$ in cycles per second. One possible <angular frequency> convention defines a density $\mathcal S_I^{(2)}(\omega)$ integrated using $d\omega$ alone. Since $\omega=2\pi f$, it has
$$
\mathcal S_I^{(2)}(\omega)=\frac{S_I^{(2)}(\omega/(2\pi))}{2\pi},\qquad
\operatorname{Var}(i_{\rm filtered})=\int_{-\infty}^{\infty}\mathcal S_I^{(2)}(\omega)|H(\omega/(2\pi))|^2\,d\omega.
$$
Another common convention keeps the numerical density $qI$ but integrates it with $d\omega/(2\pi)$. State the integration measure as well as whether a spectrum is one-sided: a change of variable and a folding of signs are separate operations.
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