Physics Wave Prediction Methods

Two methods exist to predict physics signal: Time Convolution Method (TCM) and Response Transformation Method (RTM). Both rely on knowledge of 6 parameters (see Fig. 3):

τcali: Calibration pulse decay constant. (~340 ns). In TCM method fixed at 340 ns.
fstep = offset in the injected calibration signal (~0.065). In TCM method fixed at 0.065 ns.
τ0 = sqrt(LC) (~5.4ns)
τr = rC (~ 1ns)
Td: drift time of electrons in liquid argon (~ 400 ns). In TCM method fixed at 450ns.
Tdiff: delay time between the starting times of the calibration and physics pulses (~20 ns).

Note: based on the ATLAS notes by D. Prieur, ATL-LARG-PUB-2005-001, and M. Aleksa et al., ATL-COM-LARG-2006-003.

This technique relies on Calibration (Delay) and Physics Waveforms obtained during the test beam period.
Based on the model that the physics pulse Pp(t) can be expressed as a convolution of the calibration pulse Pc(t) and the convolution function G(t-t'), which takes into account the differences between the injection points for the physics and calibration pulses.
The last 3 parameters, τ0, τr, Tdiff are free and are found by fitting the predicted physics signal to the real physics pulse. Factor Mphys/Mcali is taken into account in the fit. τcali and fstep are fixed (measured directly).
Since at the CTB only the central part of the barrel module (sampling region 2) was exposed to high energy electron beam, it was possible to obtain an estimate of the ionization pulses from data only for a small fraction of the readout cells.

II. Response Transformation Method (RTM), a.k.a. "Milano Method"

Note: LArCalorimeter/LArCalibUtils/LArRTMPhysWavePredictor is the corresponding offline simulation package.

RTM: Finding τcali and fstep.

where Iinj^cali is the injected step current, Hdet(s) describes the effect of the calorimeter cell properties on the injected step current current and Hreadout(s) the effect of the calibration line, preamplifier and shaper (see Fig. 3, right).

τcali and fstep may be found by minimimzing the Uout^cali - Vout^cali difference:

where ΔU^step is the minimizing function and whre Vout^cali is obtained by sampling the output pulse every 1 ns. τ'cali and f'step are parameter estimators which should converge to true values of τcali and fstep when the Uout^cali - Vout^cali difference is minimized.

Eq. 3 holds true only if Hdet(s) and Hreadout(s) contain small time constants (compared to Vout^cali), thus only the tail portion of Vout^cali can be minimized. It has been found that for a calibration wave extending to 800 ns, the "safe" fitting range starts at t > tmin = 550 ns.

Predicting Physics Waveforms from Calibration (Delay) Pulses