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Correlated threshold and amplitude errors in RC timing must be propagated together. An input level and a comparator threshold can share a reference, so treating their measurement uncertainties as independent can overestimate or underestimate the uncertainty of the RC threshold-crossing time. The correct cancellation depends on which levels share an error, including the initial baseline.

First hold the time constant $\tau$ fixed and take a known zero initial output. Write $H>0$ for the step's high level and $0<\theta <H$ for the threshold. The RC threshold-crossing time and its two sensitivity coefficients areThe first-order law of propagation of uncertainty retains the covariance term. If the two input standard uncertainties are $u_{\theta },u_H$, and their correlation coefficient is $\rho$, thenThe negative cross term follows from the opposite signs of the two sensitivities: raising the threshold delays a crossing, whereas raising the input makes it earlier. Setting covariance to zero is an assumption about the measurement process.^[ref]

For an original example, let $H=1\mathrm{V}$, $\theta =0.50\mathrm{V}$, $\tau =1\mu \mathrm{s}$, $u_H=20\mathrm{mV}$ and $u_{\theta }=10\mathrm{mV}$. The nominal crossing is $0.69315\mu \mathrm{s}$, andIf the two errors are independent random variables, the result is $28.28\mathrm{ns}$; $\rho =0.9$ gives $8.94\mathrm{ns}$; perfect anticorrelation gives $40.00\mathrm{ns}$. These are standard uncertainties from a local approximation, not automatically coverage intervals.

There is an exact cancellation under a more specific model. With zero baseline, if $H=G{H}_0$ and $\theta =G{\theta }_0$ with the same positive gain $G$, the ratio $a$ is fixed andThis ratiometric cancellation of RC timing gain errors is a ratiometric measurement principle. With $u_{\theta }=a{u}_H$, the zero at $\rho =1$ reflects exact common scaling, rather than a claim that every source of timing uncertainty has vanished. Uncertainty in $\tau$, for example, still gives $u_t/t\approx u_{\tau }/\tau$.

A common offset in RC timing needs the baseline to be specified. If the initial output remains fixed at zero while only $H$ and $\theta$ receive the same offset $B$, thenIn the numerical example, a small shared offset with standard uncertainty $10\mathrm{mV}$ gives about $10\mathrm{ns}$ of timing uncertainty. Perfect positive correlation between these two levels alone does not ensure cancellation.

If the baseline shares the offset too, it must be included. The nonzero-baseline model from residual baseline in RC pulse-width discrimination isAdding $B$ to all three levels leaves both differences, and hence the time, unchanged. This is a change of common zero reference. Holding the baseline fixed while shifting only the two other levels is a different experiment. The extra sensitivity is $c_b=-\tau /(H-V_b)$; at $V_b=0$, the three voltage sensitivities sum to zero, as required by the exact common-offset invariance.

An exact numerical check can use bounded errors with prescribed correlation. Take two independent random variables $Z,W$ with the Rademacher distribution and setThe four equally likely states have the specified variances and covariance. At the chosen error scales, all states retain $0<\theta <H$, so the exact logarithmic crossing exists in every state. Evaluating those four times checks the local approximation without silently assigning a crossing to an invalid input pair. Near a threshold or pulse-duration boundary, missing crossings and nonlinear effects require separate treatment.

For a full measurement uncertainty calculation, include the baseline and time constant when their uncertainties matter. With input vector $(\theta ,H,V_b,\tau )$, Jacobian matrix $J$ and covariance matrix $\Sigma$, the first-order result is $u_t^2\approx J\Sigma J^{\mathsf{T}}$. The measurement model must represent significant corrections and uncertainty sources, rather than only a physical law.^[ref] A bench check can vary a common amplitude reference and then an independent threshold offset while recording actual levels and crossings on an oscilloscope; the baseline should be recorded in both tests.

Nonuniqueness of bounded uncertainty laws with a fixed covariance means that specifying finite bounds, expected values and a covariance matrix can still leave several different joint probability distributions. They agree on the variance of every linear combination of the errors, but can disagree on threshold probabilities and nonlinear outputs. A particular construction in bounded errors with prescribed correlation therefore requires a distributional assumption beyond its second moments.

Here is an original example with a common allowed box $[-\sqrt{2},\sqrt{2}{]}^2$. In model A, $X$ and $Y$ are independent fair signs: the four points $(1,1),(1,-1),(-1,1),(-1,-1)$ each have probability $1/4$. In model B, the four points $(\sqrt{2},0),(-\sqrt{2},0),(0,\sqrt{2}),(0,-\sqrt{2})$ each have probability $1/4$. Both are bounded, have zero expected value, and satisfyFor model B, each squared coordinate is two at two of the four points and zero otherwise, proving its unit variance directly. Its zero covariance does not make the coordinates independent random variables: when $X$ is nonzero, $Y$ must vanish. The common box is an allowed deterministic bound; its boundary need not be attained by every model.

The linear output $U=X+Y$ has expected value zero and variance two under either model. Yet under model A it takes $-2,0,2$ with probabilities $1/4,1/2,1/4$, while model B gives only $-\sqrt{2},+\sqrt{2}$ with equal probabilities. Consequently,Even a linear output's threshold risk is not fixed by its variance. For the nonlinear output $F=X^2{Y}^2$, model A gives $F=1$ always and model B gives $F=0$ always. Thus the complete output law, and even a nonlinear output's expected value, can differ.

The law of propagation of uncertainty computes first-order output variance from sensitivity coefficients and input covariance; it does not identify all probabilities. NIST correspondingly requires distributional justification to assign a confidence level to an uncertainty interval.^[ref] The finite counterexample above is an original exact calculation. For guaranteed linear limits, robust linear constraints for bounded covariance models must use the chosen support of a probability distribution, rather than treating a covariance matrix as a support description.