 Constraint redundancy in baseline-noise minimization

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.