= Noise cost of polynomial baseline cancellation
{title2=$u^2=\mathbf a^TC\mathbf a$}
The \b[noise cost of <polynomial> baseline cancellation] is the change in random measurement uncertainty when extra background trends are removed. For a linear pulse estimate $\widehat U=\mathbf a^T\mathbf x$ and reading-error <covariance matrix> $C$,
$$
u^2(\widehat U)=\mathbf a^TC\mathbf a.
$$
The <moment conditions for baseline-drift cancellation> constrain $\mathbf a$ to reject specified <polynomial> backgrounds. They do not make the remaining random error vanish.<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}
For an original comparison, assume a pulse occupies only the central sample and the other samples contain background alone. Three samples at $-h,0,h$ with weights $\Delta t(-1/2,1,-1/2)$ cancel constant and linear backgrounds. Five samples at $-2h,-h,0,h,2h$ can instead use
$$
\mathbf a=\Delta t(1/6,-2/3,1,-2/3,1/6)^T.
$$
These weights preserve the central signal and have zero moments of orders $0,1,2,3$. For independent reading errors of equal <variance> $\sigma^2$, the three-sample <variance> is $3\Delta t^2\sigma^2/2$, while the five-sample <variance> is
$$
u^2(\widehat U)=\frac{35}{18}\Delta t^2\sigma^2.
$$
With $\sigma=1\,\mathrm{mW}$ and $\Delta t=1\,\mathrm{\mu s}$, the standard uncertainties are approximately $1.225$ and $1.394\,\mathrm{nJ}$. The second design removes quadratic and cubic drift as well, but its larger reference weights increase noise. This compares two specified estimators; it is not a claim that collecting more background data must worsen precision.
NIST's pulse-energy apparatus used before/after baseline measurements for background and thermal-drift correction.<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} The <polynomial> comparison above is an original model for choosing such measurements. Choosing <minimum-noise weights under baseline moment constraints> can exploit additional reference samples and their measured correlations. Pulse leakage into a reference or drift outside the cancelled <polynomial> family still creates bias.
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