 Noise-limited dynamic range

A 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 $V_{\max }$. ThenThese 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.^[ref]

For an original voltage-amplitude example, take a usable RMS signal ceiling of $1\mathrm{V}$, RMS zero-mean noise $100\mu \mathrm{V}$, 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 $20{\log }_{10}(2000)\approx 66.02\mathrm{dB}$. 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 ${\sigma }_V^2=S_{V}B$. 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.