 Null-space feasibility test for baseline-estimator constraints

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.