= Uncertainty of a three-point baseline integral
The \b[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 combination
$$
B=\mathbf w^T\mathbf b,\qquad
\mathbf b=(b_0,b_m,b_1)^T,\qquad
\mathbf w=\frac T6(1,4,1)^T.
$$
If $C_b$ is the <covariance matrix> of the baseline estimates, then
$$
u^2(B)=\mathbf w^TC_b\mathbf w.
$$
For 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.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2.2, equation13]{ref}
An original equal-precision model assigns each sample <standard uncertainty> $\sigma_b$ and common pairwise <correlation coefficient> $\rho$, with $-1/2\leq\rho\leq1$. Substitution gives
$$
u^2(B)=T^2\sigma_b^2\frac{1+\rho}{2}.
$$
For $T=10\,\mathrm{\mu s}$ 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 $\widehat U=Y-B$, then
$$
\operatorname{Var}(\widehat U)=\operatorname{Var}(Y)+\operatorname{Var}(B)-2\operatorname{Cov}(Y,B).
$$
The <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.
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