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Ardo Illaste edited res_fitfunction.md
almost 10 years ago
Commit id: 6b036ec8eebc759ea1697b21bdf61d891ecfe226
deletions | additions
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index 7cb290e..0abb32f 100644
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...
\[
f(A, d, \tau_{d}, \tau_{r}, \mu, t, \sigma) = g \ast G
\]
For notational purposes we shall represent the fit function parameters by
\(\mathbf{p}= \(\mathbf{r}=
\left[
\begin{array}{c c c c c}
A&d&\tau{d}&\tau_{r}&\mu
\end{array}
\right]
\). The smoothing parameter \(\sigma\) will be fixed for all pixels. Therefore, for the *i*-th pixel the *k*-th event is represented by
\(f(\mathbf{p}_{i,k},t)\). \(f(\mathbf{r}_{i,k},t)\).
The entire raw signal for the *i*-th pixel can be represented as:
\[
\label{eq_ithpix}
F_i(t) = b(\mathbf{q}_i, t) + \sum_{k=0}^{m}
f(\mathbf{p}_{i,k},t) f(\mathbf{r}_{i,k},t) + W + R
\]
, where \(b\) is a n-th order polynomial with \(\mathbf{q}_i\) being the polynomial coefficients for the *i*-th pixel, summation is performed over all *m* events in the pixel, \(W\) represents noise and \(R\) is the remaining residual not captured in the baseline nor events. Ideally, \(R=0\), but achieving this is limited by the accuracy of the event region detection (we cannot fit what we do not detect) and whether or not our fitting function is general enough to be able to approximate various types of events.