 Moment conditions for baseline-drift cancellation

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 $\hat{U}={\sum }_i{a}_i{x}_i$, with the integration/calibration units included in $a_i$. A background $b(t)={\sum }_{q=0}^p{c}_q{t}^q$ contributes nothing wheneverSubstitution 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 $h$. UseThe weights sum to zero and have zero first time moment. Constant and linear backgrounds cancel exactly, but a quadratic background $c_2{t}^2$ leaves bias $-\Delta t{c}_2{h}^2$. With chosen $h=2\mu \mathrm{s}$, $\Delta t=1\mu \mathrm{s}$ and $c_2=0.05mW/\mu {\mathrm{s}}^2$, 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 ${\mathbf{a}}^{T}C\mathbf{a}$ instead.^[ref]

NIST's historical pulse-energy apparatus used before/after baselines for background and thermal-drift compensation.^[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.