Threshold timing noise from amplitude noise (${\sigma }_t\simeq {\sigma }_v/|s^{\prime }(t_0)|$)

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 $S=s^{\prime }(t_0)$. For small additive noise $n$, linearizing $s(t_0+\delta t)+n=s(t_0)$ givesThe 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.^[ref]

For an original readout example, use noise standard deviation $2\mathrm{mV}$ and crossing slope $0.20V/ns$. 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.