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Constraint redundancy in baseline-noise minimization occurs when some signal-preservation or baseline-rejection conditions are linear combinations of others. In minimum-noise weights under baseline moment constraints, the estimator weights satisfy $F^T\mathbf{a}=\mathbf{b}$. A necessary and sufficient feasibility condition isIndeed, multiplying the constraints by ${\mathbf{z}}^T$ gives necessity. Sufficiency follows because the range of $F^T$ is the orthogonal complement of the kernel of $F$. Redundant conditions are harmless only when their required responses obey the same dependencies.

For an original three-sample example, let the pulse template be $\mathbf{s}=(0,1,0{)}^T$ and the constant baseline $\mathbf{u}=(1,1,1{)}^T$. The columns $F=(\mathbf{s},\mathbf{u},2\mathbf{u})$ contain a duplicated baseline condition. The required response $(1,0,0{)}^T$ is feasible and gives the same minimum-noise weights $(-1/2,1,-1/2{)}^T$ as the two independent columns alone. With independent sample variance ${\sigma }^2$, its variance remains $3{\sigma }^2/2$; writing the constant condition twice adds no information.

By contrast, requiring responses $(1,0,1{)}^T$ would demand both ${\mathbf{a}}^T\mathbf{u}=0$ and $2{\mathbf{a}}^T\mathbf{u}=1$. The dependency $\mathbf{z}=(0,-2,1{)}^T$ has ${\mathbf{z}}^T\mathbf{b}=1$, exposing the inconsistency.

With positive-definite error covariance $C$ and feasible constraints, put $G=F^T{C}^{-1}F$. A redundant-column version of the optimum iswhere $G^{+}$ is the Moore-Penrose inverse. Feasibility ensures $G{G}^{+}\mathbf{b}=\mathbf{b}$. Without the null-space feasibility test for baseline-estimator constraints, this formula can silently return approximate responses instead of the requested exact cancellations. Variance still uses the full covariance quadratic form.^[ref]

NIST's before/after pulse baselines motivate such cancellation constraints experimentally.^[ref] A singular value decomposition can diagnose numerical dependence, but a small singular value caused by time units or poor scaling is not automatically permission to discard a physical drift-rejection requirement.

The noise cost of polynomial baseline cancellation is the change in random measurement uncertainty when extra background trends are removed. For a linear pulse estimate $\hat{U}={\mathbf{a}}^T\mathbf{x}$ and reading-error covariance matrix $C$,The moment conditions for baseline-drift cancellation constrain $\mathbf{a}$ to reject specified polynomial backgrounds. They do not make the remaining random error vanish.^[ref]

For an original comparison, assume a pulse occupies only the central sample and the other samples contain background alone. Three samples at $-h,0,h$ with weights $\Delta t(-1/2,1,-1/2)$ cancel constant and linear backgrounds. Five samples at $-2h,-h,0,h,2h$ can instead useThese weights preserve the central signal and have zero moments of orders $0,1,2,3$. For independent reading errors of equal variance ${\sigma }^2$, the three-sample variance is $3\Delta t^2{\sigma }^2/2$, while the five-sample variance isWith $\sigma =1\mathrm{mW}$ and $\Delta t=1\mu \mathrm{s}$, the standard uncertainties are approximately $1.225$ and $1.394\mathrm{nJ}$. The second design removes quadratic and cubic drift as well, but its larger reference weights increase noise. This compares two specified estimators; it is not a claim that collecting more background data must worsen precision.

NIST's pulse-energy apparatus used before/after baseline measurements for background and thermal-drift correction.^[ref] The polynomial comparison above is an original model for choosing such measurements. Choosing minimum-noise weights under baseline moment constraints can exploit additional reference samples and their measured correlations. Pulse leakage into a reference or drift outside the cancelled polynomial family still creates bias.

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.