= Baseline-drift bias in pulse-energy integration
{title2=$B_U=\int_{\rm gate}(b-\widehat b)\,dt$}
\b[Baseline-drift bias in pulse-energy integration] is the systematic energy error caused by using an inaccurate baseline over the integration gate. Relative to the true signal energy in the same gate,
$$
B_U=\widehat U-U_g=\int_{\rm gate}[b(t)-\widehat b(t)]\,dt.
$$
This error is separate from energy excluded by the gate. Historical NIST measurements bracketed each run with baseline observations and subtracted their average to compensate background and thermal drift.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[§3.5; §7.2, Type B component4, Leakage Effect]{ref}
An original curvature example shows a limit of endpoint bracketing. On $-T/2\leq t\leq T/2$, suppose $b(t)=b_0+ct^2$. Both endpoint baselines equal $b_0+cT^2/4$, so their average and the straight interpolation between them coincide. However, the true integrated baseline is $b_0T+cT^3/12$. Subtracting the endpoint level over the whole gate therefore gives
$$
B_U=-\frac{cT^3}{6}.
$$
For a $20\,\mathrm{\mu s}$ gate whose endpoints are $3\,\mathrm{mW}$ above its midpoint baseline, this bias is $-(2/3)(3\,\mathrm{mW})(20\,\mathrm{\mu s})=-0.040\,\mathrm{\mu J}$. A true gated signal energy of $1.00\,\mathrm{\mu J}$ would be reported as $0.960\,\mathrm{\mu J}$ under these assumptions.
Thus <quadratic baseline curvature in gated energy integration> survives a correction that is exact for a linear baseline. For fixed curvature, the error grows with the cube of gate duration. Additional pulse-free samples can test the baseline model, but pulse tails in those samples can themselves corrupt the estimate. <Pulse energy baseline correction> supplies the broader measurement context.
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