= Input-referred noise
\b[Input-referred noise] expresses observed or modeled output noise as an equivalent disturbance at a system's input. For a locally linear response $y=Gx$ with constant nonzero gain over the relevant measurement band, an output-noise <root mean square> $n_{\rm out}$ corresponds to
$$
n_{\rm in}=\frac{n_{\rm out}}{|G|}.
$$
This is a way to compare noise with the input signal; it does not locate every physical noise source at the input. Walt Kester's ADC tutorial explicitly models converter noise as an equivalent series input source and describes its measurement through repeated codes under a fixed input.<MT-004 by Walt Kester>[pp1–2, input-referred noise model and grounded-input histogram]{ref}
For an original amplifier with voltage gain $100$, output noise $1\,\mathrm{mV}$ rms is equivalent to $10\,\mathrm{\mu V}$ rms at the input. Changing gain can change this quotient without removing the device's own physical noise. State the measurement bandwidth and processing, since a longer averaging interval or different filter changes the observed rms value.
An original two-stage example shows why the position of noise matters. Let gains be $G_1,G_2$ and independent input-equivalent noise contributions be $e_1,e_2$ for the respective stages. The output-noise variance is
$$
n_{\rm out}^2=(G_1G_2)^2e_1^2+G_2^2e_2^2,
$$
so referring it to the original input gives
$$
e_{\rm total}^2=e_1^2+\frac{e_2^2}{G_1^2}.
$$
With $e_1=2\,\mathrm{\mu V}$, $e_2=30\,\mathrm{\mu V}$ and $G_1=10$, this is $\sqrt{13}\simeq3.61\,\mathrm{\mu V}$. Increasing $G_1$ to $100$ reduces the equivalent total to about $2.02\,\mathrm{\mu V}$, approaching the first-stage contribution. The independent-noise assumption matters; correlated contributions require <covariance> terms.
A quiet output code does not by itself establish a small total error, and rms noise is not the same criterion as <input-referred resolution>. Kester distinguishes noise-based resolution from dynamic converter measures and notes the conditions needed for averaging to help.<MT-004 by Walt Kester>[pp3–5, noise-free resolution, averaging and dynamic measures]{ref} Retain offsets, gain uncertainty and <instrumental drift> separately in the <measurement model>.
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