= Moment conditions for baseline-drift cancellation
{title2=$\sum_i a_i t_i^q=0$}
\b[Moment conditions for baseline-drift cancellation] specify which polynomial backgrounds a linear measurement removes. Let readings at known times $t_i$ enter the corrected pulse estimate as $\widehat U=\sum_i a_ix_i$, with the integration/calibration units included in $a_i$. A background $b(t)=\sum_{q=0}^p c_qt^q$ contributes nothing whenever
$$
\sum_i a_it_i^q=0\qquad(q=0,1,\ldots,p).
$$
Substitution gives this result directly. The zeroth condition removes a common offset; the first additionally removes linear drift. These conditions concern the actual weights and times, not only the number of background samples.
An original one-bin signal model measures a pulse at $t=0$, with uncontaminated reference samples at $-h$ and $h$. Use
$$
\widehat U=\Delta t\left[x_0-\frac{x_-+x_+}{2}\right].
$$
The weights sum to zero and have zero first time moment. Constant and linear backgrounds cancel exactly, but a quadratic background $c_2t^2$ leaves bias $-\Delta t\,c_2h^2$. With chosen $h=2\,\mathrm{\mu s}$, $\Delta t=1\,\mathrm{\mu s}$ and $c_2=0.05\,\mathrm{mW/\mu s^2}$, the bias is $-0.20\,\mathrm{nJ}$. Here the pulse is stipulated to occupy the central bin; the formula is not a universal integral estimator for an arbitrary waveform.
Cancellation has a <noise cost of polynomial baseline cancellation>. For independent equal-variance reading errors $\sigma^2$, the chosen estimate has variance $(3/2)\Delta t^2\sigma^2$, even though its constant and linear bias terms vanish. General correlations use $\mathbf a^TC\mathbf a$ instead.<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 historical pulse-energy apparatus used before/after baselines for background and thermal-drift compensation.<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 test above is an original general analysis. Pulse contamination in a reference, uncertain timing or drift outside the assumed polynomial family can leave a residual even when the nominal algebraic conditions hold.
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