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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 $\hat{U}={\mathbf{a}}^T\mathbf{x}$, with weights expressed in the required time and calibration units. Adding an unknown common power offset $c$ givesThus exact cancellation requires ${\mathbf{a}}^{T}1=0$. 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 $T=8.0\mu \mathrm{s}$ but mistakenly scales the baseline with $\hat{T}=7.8\mu \mathrm{s}$. With identical calibration, the residual offset is $(T-\hat{T})c$. 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 $c_0+c_1{t}_i$, both components cancel whenThese 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.^[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.^[ref]

The uncertainty of a three-point baseline integral depends on both the sample uncertainties and their correlations. With a known fixed gate duration $T$, Simpson baseline integration for pulse-energy correction uses the linear combinationIf $C_b$ is the covariance matrix of the baseline estimates, thenFor fixed weights this variance identity is exact; it does not require Gaussian errors. Its use as a complete measurement uncertainty still requires an adequate baseline model. The general covariance propagation rule retains these cross terms.^[ref]

An original equal-precision model assigns each sample standard uncertainty ${\sigma }_b$ and common pairwise correlation coefficient $\rho$, with $-1/2\le \rho \le 1$. Substitution givesFor $T=10\mu \mathrm{s}$ and ${\sigma }_b=0.10\mathrm{mW}$, independent samples give $u(B)=0.707\mathrm{nJ}$. At $\rho =0.60$ this rises to $0.894\mathrm{nJ}$; a perfectly common additive error gives $1.00\mathrm{nJ}$. Three readings under one uncertain offset need not behave like independent measurements.

The final corrected pulse integral has another covariance. If $Y$ is the raw gate integral and $\hat{U}=Y-B$, thenThe covariance between a raw pulse integral and its baseline correction matters when both share a readout offset. If that offset is $c$, it contributes $Tc$ to both $Y$ and $B$ and cancels exactly in their difference. Treating these appearances as independent would invent uncertainty.

Use repeated matched blank acquisitions to estimate the relevant covariance structure, keeping the gate and readout sequence comparable. A contaminated midpoint, an inadequate baseline shape or uncertain timing adds effects outside the fixed-weight sample-noise calculation. Numerical quadrature exactness alone does not establish a sound pulse-energy uncertainty budget.