= Minimum-noise weights under baseline moment constraints
{title2=$\mathbf a=C^{-1}F(F^TC^{-1}F)^{-1}\mathbf b$}
\b[Minimum-noise weights under baseline moment constraints] choose a linear pulse estimator with the smallest <variance> among those that preserve a specified signal and remove specified backgrounds. Write the reading vector as $\mathbf x=\theta\mathbf s+H\boldsymbol\beta+\boldsymbol\epsilon$, where $\mathbf s$ is a known signal template, the columns of $H$ contain baseline shapes, and the error <covariance matrix> $C$ is positive definite. For $\widehat\theta=\mathbf a^T\mathbf x$, require
$$
F^T\mathbf a=\mathbf b,\qquad F=(\mathbf s\ H),\quad\mathbf b=(1,0,\ldots,0)^T.
$$
Columns $1,t,t^2,\ldots$ in $H$ implement the <moment conditions for baseline-drift cancellation>. The <noise cost of polynomial baseline cancellation> depends on which feasible weights are chosen.
If the constraint columns are independent, <Lagrange multipliers> give
$$
\mathbf a=C^{-1}F(F^TC^{-1}F)^{-1}\mathbf b,\qquad
\operatorname{Var}(\widehat\theta)=\mathbf b^T(F^TC^{-1}F)^{-1}\mathbf b.
$$
For any feasible perturbation $\mathbf v$, $F^T\mathbf v=0$ and $\mathbf v^TC\mathbf a=0$, so the <variance> increases by $\mathbf v^TC\mathbf v$. This proves the minimum and connects the estimator to <generalized least squares>. The <covariance> propagation is the full quadratic form, not a sum that discards correlations.<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}
For an original one-bin pulse example, use five equally spaced times $-2h,-h,0,h,2h$, a signal present only at the centre, and independent errors of <variance> $\sigma^2$. To remove just constant and linear backgrounds, the optimal dimensionless weights are $(-1/4,-1/4,1,-1/4,-1/4)$. Their <variance> is $5\sigma^2/4$, lower than $3\sigma^2/2$ for the centre and its two nearest references alone. Multiplying the amplitude estimate by a bin duration $\Delta t$ gives energy <variance> $5\Delta t^2\sigma^2/4$. Extra uncontaminated references can reduce noise when the cancelled background family stays the same.
The inverse requires genuinely independent constraints. <Constraint redundancy in baseline-noise minimization> calls for removing redundant columns and checking feasibility; a signal template inside the nuisance span cannot be preserved while that span is rejected. NIST's before/after pulse baselines provide the physical measurement context.<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 five-point design is an original idealization and assumes no pulse signal in its reference samples.
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