= Finite-input correction in pulse-detector calibration
{title2=$\sigma_h=\sqrt{\sigma_q^2-\sigma_p^2}$}
\b[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>,
$$
\sigma_h=\sqrt{\sigma_q^2-\sigma_p^2},
$$
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 gives
$$
\frac{\partial\sigma_h}{\partial\sigma_q}=\frac{\sigma_q}{\sigma_h},\qquad
\frac{\partial\sigma_h}{\partial\sigma_p}=-\frac{\sigma_p}{\sigma_h}.
$$
First-order propagation of <standard uncertainty> therefore yields
$$
u^2(\sigma_h)\simeq
\frac{\sigma_q^2u^2(\sigma_q)+\sigma_p^2u^2(\sigma_p)-2\sigma_q\sigma_p\operatorname{Cov}(\sigma_q,\sigma_p)}{\sigma_q^2-\sigma_p^2}.
$$
The <covariance> matters when both widths use the same timebase or fitted reference.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{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.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[Appendix C.1–C.2, printed pp. 36–37, Figure C1 and steps 1–7]{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.
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