Pulse-width uncertainty from correlated edge timing

Pulse-width uncertainty from correlated edge timing depends on the joint errors of the two crossing times, not just their separate uncertainties. For specified leading and trailing crossings $t_r,t_f$, let $W=t_f-t_r$. Their variance identity isThis is exact for the stated difference of random times. Approximating the edge errors from waveform measurements may require a local model. NIST's primary uncertainty law explicitly retains covariance terms.^[ref]

A common time translation illustrates cancellation. Suppose measured edges have errors $c+e_r$ and $c+e_f$, with independent zero-mean components. The pulse width error is $e_f-e_r$; the common $c$ disappears. For an original example, take standard deviations $20\mathrm{ns}$ for $c$, $2\mathrm{ns}$ for $e_r$, and $3\mathrm{ns}$ for $e_f$. Individual edge uncertainties are about $20.10$ and $20.22\mathrm{ns}$, but pulse width uncertainty is only $\sqrt{13}\approx 3.606\mathrm{ns}$. Incorrectly treating the two full edge errors as independent would give $\sqrt{813}\approx 28.51\mathrm{ns}$.

A common timing-scale error is different. If measured times obey $t_{\mathrm{meas}}=(1+g)t+c$, then $W_{\mathrm{meas}}=(1+g)W$: the offset cancels but the scale error remains. For a $100\mu \mathrm{s}$ pulse width, a $10$ parts-per-million standard scale uncertainty contributes $1\mathrm{ns}$. If independent of the residual edge errors above, it raises the combined standard uncertainty to $\sqrt{14}\approx 3.742\mathrm{ns}$.

Nor does every shared voltage error make the edges positively correlated. A fixed threshold shift can move a rising crossing later and a falling crossing earlier, increasing pulse width uncertainty. The waveform slopes and shared references determine the covariance sign. A measurement should therefore record its level convention and model the two edges together. Keysight's documented pulse width uses the rising and next falling 50-percent crossings.^[ref] These uncertainty calculations assume both crossings exist; missing or multiply selected edges need a separate observation model.