= Threshold timing noise from amplitude noise
{title2=$\sigma_t\simeq\sigma_v/|s'(t_0)|$}
\b[Threshold timing noise from amplitude noise] occurs when voltage fluctuations move the apparent time at which a pulse crosses a fixed threshold. Let a clean electrical trace $s(t)$ cross at $t_0$ with nonzero slope $S=s'(t_0)$. For small additive noise $n$, linearizing $s(t_0+\delta t)+n=s(t_0)$ gives
$$
\delta t\simeq-\frac nS,\qquad \sigma_t\simeq\frac{\sigma_v}{|S|}.
$$
The result concerns a selected measurement feature. It is not automatically the source's physical arrival-time spread. The timing primer distinguishes reference-edge error from other jitter statistics.<Digital Design Debugging Guide by Tektronix>[Picking out Jitter, Diagrams5–6]{ref}
For an original readout example, use noise standard deviation $2\,\mathrm{mV}$ and crossing slope $0.20\,\mathrm{V/ns}$. The linear estimate is $10\,\mathrm{ps}$. At half that slope it becomes $20\,\mathrm{ps}$, even with the same noise amplitude. Both units and the local slope matter.
A pulse generator, receiver and <oscilloscope> can compare crossing times while the threshold position or signal slope changes. Small-noise linearization presumes a unique nearby crossing; ringing, large noise or multiple crossings require a fuller model. The relevant noise bandwidth must match the recorded trace.
A changing pulse scale can also shift a fixed threshold time even without additive noise. That amplitude-dependent time walk is a different effect from the random perturbation derived above. <Pulse timing jitter> describes the resulting timing observable, while the original optical arrival and detector readout contributions must be distinguished.
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