= Constraint redundancy in baseline-noise minimization
{title2=$\ker F\perp\mathbf b$}
\b[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 is
$$
\mathbf z\in\ker F\quad\Longrightarrow\quad\mathbf z^T\mathbf b=0.
$$
Indeed, 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 $G=F^TC^{-1}F$. A redundant-column version of the optimum is
$$
\mathbf a=C^{-1}FG^+\mathbf b,
$$
where $G^+$ is the <Moore-Penrose inverse>. Feasibility ensures $GG^+\mathbf b=\mathbf b$. 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.<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}
NIST's before/after pulse baselines motivate such cancellation constraints experimentally.<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} 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.
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