= Null-space feasibility test for baseline-estimator constraints
{title2=$V_0^T\mathbf b=0$}
The \b[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. Then
$$
V_0^T\mathbf b=0,
\qquad\text{equivalently}\qquad
\mathbf e=(I-V_rV_r^T)\mathbf b=0.
$$
The 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 $(1,-1)^T/\sqrt2$ has inner product $1/\sqrt2$ 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.<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} 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.
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