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Finite-input correction in pulse-detector calibration removes the known input duration from an inferred receiver width, while propagating uncertainty in both widths. Under the nonnegative, normalized, finite-variance convolution model of known-input calibration of a pulse detector,where $p$ is the known input profile, $q$ the measured output and $h$ the receiver's impulse response. The subtraction concerns temporal variances, not arbitrary reported full widths.

Differentiating on the positive-width branch givesFirst-order propagation of standard uncertainty therefore yieldsThe covariance matters when both widths use the same timebase or fitted reference.^[ref]

An original near-resolution example has ${\sigma }_p=0.95\mathrm{ns}$ and ${\sigma }_q=1.00\mathrm{ns}$, each with independent standard uncertainty $0.01\mathrm{ns}$. The corrected receiver width is $0.31225\mathrm{ns}$ and its propagated standard uncertainty is about $0.04417\mathrm{ns}$, or $14.1\%$. Width measurements individually precise to about one percent can thus give a much less precise difference-derived width.

NIST's primary bandwidth-correction procedure used roughly 120 ps optical inputs and recorded the detector, cable and oscilloscope as one response chain at different gains and signal levels.^[ref] A sufficiently short characterized input reduces this resolution-limit conditioning in finite-input detector calibration. Near a zero variance difference, linear uncertainty propagation may fail; a nonpositive measured difference does not represent a negative physical detector width. Report an interval or limit under the stated model rather than silently replacing the difference by zero. Negative ringing also violates the nonnegative-profile assumption behind the variance subtraction.

Optical tail and detector-response ambiguity arises because a measured temporal trace combines the optical waveform with the receiver's dynamics. In a calibrated linear time-invariant system,where $P$ is incident radiant power and $h$ the unit-area detector impulse response. One trace does not generally identify both unknown functions. The parent bounding unobserved optical energy with an exponential tail model gives an impulsive-input example; two finite-width profiles provide another exact ambiguity.

Define $h_{\tau }(t)={\tau }^{-1}{e}^{-t/\tau }$ for $t\ge 0$, and zero otherwise. Consider model A with optical power $P=U{h}_{2\mathrm{ns}}$ and detector response $h_{10\mathrm{ns}}$, and model B with these two time constants exchanged. By commutativity of convolution, both give exactlyTheir optical energies after $20\mathrm{ns}$ are nevertheless $U{e}^{-10}$ and $U{e}^{-2}$. For original chosen $U=100\mathrm{nJ}$, those are about $0.00454$ and $13.53\mathrm{nJ}$. Identical recorded tails therefore coexist with very different true late optical energy.

Known-input calibration of a pulse detector supplies information missing from this inverse problem. NIST measured the detector/cable/oscilloscope response with optical excitation much shorter than the response before calculating waveform distortions.^[ref] The example above is an independent mathematical construction. A fitted electrical decay constant alone cannot establish a global optical-tail envelope.