= Numerical feasibility residual for baseline constraints
{title2=$\mathbf r=F^T\widehat{\mathbf a}-\mathbf b$}
A \b[numerical feasibility residual for baseline constraints] measures how closely computed pulse-estimator weights satisfy their requested responses. For $F^T\mathbf a=\mathbf b$, evaluate
$$
\mathbf r=F^T\widehat{\mathbf a}-\mathbf b.
$$
This 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 $\mathbf x=\theta\mathbf s+\sum_j\beta_j\mathbf h_j+\boldsymbol\epsilon$, an estimator intended to preserve the signal and reject the backgrounds instead has bias
$$
\operatorname{Bias}(\widehat\theta)=\theta r_s+\sum_j\beta_jr_j.
$$
Thus 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 $1\,\mathrm{mV/\mu s}=10^3\,\mathrm{V/s}$ 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 $10^{-6}\,\mathrm{\mu s}$ 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 $\eta=\max_j|r_j|/\delta_j\leq1$. When coefficient bounds $|\theta|\leq B_s$, $|\beta_j|\leq B_j$ are justified, the corresponding bias bound is $B_s|r_s|+\sum_jB_j|r_j|$. A noise-minimizing estimator still needs a separate <variance> calculation from the actual <covariance>.<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 provide the physical setting for rejecting background and drift.<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} Report the chosen constraint scaling and tolerated leakage. A small solver residual alone establishes neither low <variance> nor negligible bias for every possible background.
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