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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 $\mathbf{x}=\theta \mathbf{s}+H\mathbf{\beta }+\mathbf{ϵ}$, 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 $\hat{\theta }={\mathbf{a}}^T\mathbf{x}$, requireColumns $1,t,t^2,\dots$ 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 giveFor any feasible perturbation $\mathbf{v}$, $F^T\mathbf{v}=0$ and ${\mathbf{v}}^{T}C\mathbf{a}=0$, so the variance increases by ${\mathbf{v}}^{T}C\mathbf{v}$. 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.^[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.^[ref] The five-point design is an original idealization and assumes no pulse signal in its reference samples.

The null-space feasibility test for baseline-estimator constraints determines whether any measurement weights can preserve a specified pulse response while removing the requested backgrounds. Write the conditions as $F^T\mathbf{a}=\mathbf{b}$. The criterion in constraint redundancy in baseline-noise minimization is that $\mathbf{b}$ be orthogonal to every vector in the kernel of $F$.

A singular value decomposition makes this a finite test. Let $V_0$ contain an orthonormal basis for the kernel of $F$ and $V_r$ the complementary right singular vectors. ThenThe vector $\mathbf{e}$ is the component of the required responses outside the attainable response space. No covariance weighting can make an exactly infeasible set of responses feasible.

For an original three-reading example, suppose the sampled signal template is constant: $\mathbf{s}=(1,1,1{)}^T$. Requiring unit signal response while rejecting a constant baseline gives $F=(\mathbf{s},\mathbf{s})$ and $\mathbf{b}=(1,0{)}^T$. The kernel basis vector $(1,-1{)}^T/\sqrt{2}$ has inner product $1/\sqrt{2}$ with $\mathbf{b}$, so the conditions are incompatible. A constant signal cannot be distinguished from an unknown constant background by those readings alone.

Projecting $\mathbf{b}$ onto the attainable space gives $(1/2,1/2{)}^T$. The minimum-norm approximate weights are $(1/6,1/6,1/6{)}^T$, giving both signal and baseline response one half. Under independent unit-variance errors their variance is only $1/12$, but this low value belongs to a different, biased measurement response. It does not solve the requested cancellation problem.

A numerical feasibility residual for baseline constraints needs meaningful scaling of the conditions and a justified singular-value tolerance. Changing time units or allowing a small physical leakage changes the numerical interpretation; it must not silently turn an exact rejection requirement into an approximate one.

NIST's before/after pulse baselines motivate the physical need for a separable pulse and background.^[ref] Once feasibility is established, minimum-noise weights under baseline moment constraints uses the actual error covariance to choose among valid estimators. Extra timing information or a different signal template can restore identifiability when the original conditions fail.