= 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 is
$$
u_W^2=u_f^2+u_r^2-2\operatorname{Cov}(t_f,t_r)
=u_f^2+u_r^2-2\rho u_fu_r.
$$
This 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.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A.3, equation A-3]{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 $\sqrt{13}\approx3.606\,\mathrm{ns}$. Incorrectly treating the two full edge errors as independent would give $\sqrt{813}\approx28.51\,\mathrm{ns}$.
A common timing-scale error is different. If measured times obey $t_{\rm meas}=(1+g)t+c$, then $W_{\rm meas}=(1+g)W$: the offset cancels but the scale error remains. For a $100\,\mathrm{\mu s}$ <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 $\sqrt{14}\approx3.742\,\mathrm{ns}$.
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.<Pulse Width in Keysight BenchVue Function Generator documentation>[Pulse Width, threshold definition]{ref} These uncertainty calculations assume both crossings exist; missing or multiply selected edges need a separate observation model.
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