 Numerical feasibility residual for baseline constraints

A numerical feasibility residual for baseline constraints measures how closely computed pulse-estimator weights satisfy their requested responses. For $F^T\mathbf{a}=\mathbf{b}$, evaluateThis is not the same diagnostic as the projected unattainable response in null-space feasibility test for baseline-estimator constraints. That test asks whether a solution exists at the chosen matrix rank; the residual checks the weights actually returned.

Small numerical components need physical scaling. If the measurement model is $\mathbf{x}=\theta \mathbf{s}+{\sum }_j{\beta }_j{\mathbf{h}}_j+\mathbf{ϵ}$, an estimator intended to preserve the signal and reject the backgrounds instead has biasThus an unweighted Euclidean residual norm is not a universal error bound. Coefficient magnitudes and the units of each constraint matter.

For an original example, suppose the constant-background residual is $r_0=10^{-6}$ and the linear-time residual is $r_1=10^{-12}\mathrm{s}$. A background offset $1\mathrm{mV}$ contributes $1\mathrm{nV}$ through the first residual. A drift rate $1mV/\mu \mathrm{s}=10^{3}V/s$ also contributes $1\mathrm{nV}$ through the second. The much smaller raw numerical value is not a much smaller physical leakage. Expressing time in microseconds changes $r_1$ to $10^{-6}\mu \mathrm{s}$ without changing that leakage.

A response-normalized tolerance for baseline-feasibility residuals can specify allowed response errors ${\delta }_j>0$ in matching units and check $\eta ={\max }_j|r_j|/{\delta }_j\le 1$. When coefficient bounds $|\theta |\le B_s$, $|{\beta }_j|\le B_j$ are justified, the corresponding bias bound is $B_s|r_s|+{\sum }_j{B}_j|r_j|$. A noise-minimizing estimator still needs a separate variance calculation from the actual covariance.^[ref]

NIST's before/after pulse baselines provide the physical setting for rejecting background and drift.^[ref] Report the chosen constraint scaling and tolerated leakage. A small solver residual alone establishes neither low variance nor negligible bias for every possible background.