= Noise-limited dynamic range
{title2=$D=x_{\max}/x_{\min}$}
A \b[noise-limited dynamic range] is the ratio between an upper usable signal level and a lower level defined by a noise criterion, with the observation conditions specified. The lower bound might require a chosen <signal-to-noise ratio>; the upper bound can be set by <amplifier clipping> or <detector saturation>. The ratio depends on <bandwidth>, integration time, signal convention and acceptance threshold, not only on displayed digits.
For a linear response $V=Kx$ after offset correction, let the output-noise <standard deviation> be $\sigma_V$, the minimum accepted signal be $k\sigma_V$, and the output ceiling be $V_{\max}$. Then
$$
x_{\min}=\frac{k\sigma_V}{K},\qquad
x_{\max}=\frac{V_{\max}}K,\qquad
D=\frac{V_{\max}}{k\sigma_V}.
$$
These equations assume the stated noise and ceiling apply throughout the interval. Changing $K$ alone shifts the input interval without changing $D$ under those assumptions. A real amplifier can change noise and bandwidth with gain; the primary specification reports these dependencies separately.<Datasheet DLPCA-200 by FEMTO>[p. 2, integrated RMS input noise and gain-dependent bandwidth; footnote, shielded open-input measurement condition]{ref}
For an original voltage-amplitude example, take a usable RMS signal ceiling of $1\,\mathrm V$, RMS zero-mean noise $100\,\mathrm{\mu V}$, and $k=5$. The lower accepted signal is $0.5\,\mathrm{mV}$ and $D=2000$. As a voltage-amplitude ratio at fixed impedance, this is $20\log_{10}(2000)\approx66.02\,\mathrm{dB}$. If the input quantity instead denotes <optical power>, its power ratio uses $10\log_{10}D$; the physical quantity must accompany a <decibel> statement.
Noise density and total noise are different. For a flat one-sided output <power spectral density> $S_V$ over a specified <effective noise bandwidth> $B$, the model gives $\sigma_V^2=S_VB$. Increasing $B$ fourfold doubles $\sigma_V$ and halves $D$ if the ceiling and threshold factor remain fixed. A spectrum with drift or nonwhite components need not obey that scaling.
A <noise-limited detection threshold> is consequently an operational choice. Five times RMS noise is not a universal minimum detectable signal, nor by itself a statement of false-alarm probability without a noise distribution and observation rule. To characterize a particular instrument, measure its zero-input fluctuations under the actual filtering and loading conditions, then test a stable source near the proposed lower and upper bounds.
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