Proper determination of baseline for WB-based detectors

Description

In 2025-e run, data from all PMT-based detectors (S,V, ECAL, HCAL, VHCAL, VETO, LYSO) were acquired with the Waveboard digitizers, operating at 233.32 MHz, 14b, 1 bit = 0.06105 mV

The front-end of waveboard digitizers was recently modified removing all AC-decoupling capacitors. The reason for this was to avoid the dependency of the baseline value on the instantaneous beam intensity, specifically for WB channels reading signals from centermost ECAL cells - after a capacitor, the average DC signal is zero. For negative pulses, the larger the frequency of the pulses, the larger the positive baseline to have zero DC value.

This can be easily shown: assume that the shape of a single pulse is parameterized by a function P(t), so that \int_{-\inf}^{+\inf}P(t)=-A_{P} - with A_{P}>0. Important, the function P(t) is non-zero only for a narrow interval [0,\tau], so that the integral can be actually be restricted only to this range. The full signal shape in a large interval [0,T] thus reads f(t)=\sum_{i=1}^N P(t-t_i)+B, where N is the number of pulses in the interval. The interval should be considered large enough to have a large number of pulses. Then, \int_{0}^T f(t)=-N\cdot <P> +B\times T. After the capacitor, <f(t)>=0, so B=<P> \times T/N = <P> \times f, where f is the typical frequency of pulses.

Having removed all DC-decoupling capacitors, the system is no longer subject to this effect, however is more subject to low-frequency noise. Indeed, in the WB we observed low-frequency noise induced by DC-DC converters. The frequency of this is about kHz, so that for a single waveform (approximately 300 ns) the baseline is constant, but it varies at maximum 50 ADC counts for different events - 3 mV.

In summary, this calls for an event-by-event, cell-by-cell determination of the baseline