= Shared-offset cancellation in a baseline-corrected pulse integral
\b[Shared-offset cancellation in a baseline-corrected pulse integral] occurs when the raw integral and its background estimate assign exactly the same coefficient to a common additive error. For sampled readings $\mathbf x$, let the corrected energy be $\widehat U=\mathbf a^T\mathbf x$, with weights expressed in the required time and calibration units. Adding an unknown common power offset $c$ gives
$$
\widehat U(\mathbf x+c\mathbf1)-\widehat U(\mathbf x)
=c\,\mathbf a^T\mathbf1.
$$
Thus exact cancellation requires $\mathbf a^T\mathbf1=0$. It is a property of the measurement weights, not an assumption that the offset is negligible. The parent <covariance between a raw pulse integral and its baseline correction> gives the corresponding variance identity.
An original mismatch example integrates the raw trace over $T=8.0\,\mathrm{\mu s}$ but mistakenly scales the baseline with $\widehat T=7.8\,\mathrm{\mu s}$. With identical calibration, the residual offset is $(T-\widehat T)c$. If $c$ has standard uncertainty $0.30\,\mathrm{mW}$, its contribution is $0.060\,\mathrm{nJ}$. It would have vanished for matched durations. A calibration mismatch between raw and blank paths similarly changes their effective coefficients even when their nominal durations agree.
The same linear-algebra test extends to a shared linear drift. If each sample at time $t_i$ gains an error $c_0+c_1t_i$, both components cancel when
$$
\sum_i a_i=0,\qquad \sum_i a_it_i=0.
$$
These <moment conditions for baseline-drift cancellation> require the actual sample times and weights. Removing a constant offset does not automatically remove drift, and timing uncertainty can spoil the second condition.
NIST's thermocouple pulse-energy apparatus used before/after baseline averages to compensate background and thermal drift.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[§3.5, printed p. 8; §7.2, Leakage Effect]{ref} The weight test above is an original general formulation. For uncertain residual errors, propagate the complete joint covariance rather than treating repeated appearances of the same offset as independent.<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}
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