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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.

Range-switching hysteresis uses different thresholds for changing from a more sensitive instrument range to a less sensitive one and for changing back. Within the intervening interval, the current range is retained. The selected range therefore depends on the previous state as well as the present measurement indication. This prevents small fluctuations near a single boundary from repeatedly triggering gain ranging.

A concrete electronic analogue is an electronic comparator with two switching thresholds. The primary circuit example specifies upper and lower thresholds of $2.5\mathrm{V}$ and $2.2\mathrm{V}$ and explains why one threshold can produce unwanted transitions with a noisy or slowly changing input. Those are comparator circuit values, not specifications for an autoranging measurement instrument.^[ref]

For an original two-range current measurement, let $H$ denote high sensitivity and $L$ low sensitivity. Use an input-current estimate $x$, expressed on the same calibrated scale in either range. Switch $H\to L$ when $x\ge 8\mathrm{nA}$, switch $L\to H$ when $x\le 6\mathrm{nA}$, and otherwise keep the previous state. The hysteresis interval has width $2\mathrm{nA}$.

Starting in $H$, the indication sequence $(7.9,8.1,7.9,8.1,5.9)\mathrm{nA}$ selects $(H,L,L,L,H)$. A single threshold at $8\mathrm{nA}$ would instead select $(H,L,H,L,H)$ and switch at every step after the first. At a steady underlying $7\mathrm{nA}$ with bounded readout fluctuations of $\pm 0.2\mathrm{nA}$, neither boundary is crossed; either previous range remains selected.

The thresholds should lie within the usable overlap established by range-overlap calibration. If the two ranges give inconsistent current estimates, a switch can itself move the indication across the opposite threshold and create a cycle. The comparison must therefore use the corrected measurand estimate, not identical raw output voltages from different gains.

Hysteresis is not temporal averaging and does not ensure that the new reading has settled. Reserve headroom for a rapidly rising input, reject readings during the required settling time, and verify that the less sensitive range still meets its noise-limited detection threshold. An input that jumps beyond the current range can cause amplifier clipping before the controller reacts, even when its two-threshold rule is correct.