The anti-aliasing requirements for amplitude mesurements with the Fluorescence Detector read-out channel

This note calculates the requirements on the anti-aliasing filter for the read-out of the fluorescence detector of the AUGER experiment. The filter order and its cut-off frequency are related to the error in amplitude reconstruction. 1 System transfert functions The Fluorescence Detector read-out channel consist of a current preamplifier followed by a low pass anti-aliasing filter and a sampling ADC with period T = 100ns. The filter output signal should ideally satisfy the Nyquist criteria, that is band limitedness within fN = 1 2T = ωN 2π to allow shape reconstruction [1]. The input signal of the anti-aliasing filter i(t) is assumed to be a rectangle with duration ∆ > T , (approximating the expected trapezoidal shape): i(t) = I0[1(t) − 1(t−∆)]←→I(s) = I0 s [1− exp(−s∆)]. (1) Three different n-th order anti-aliasing filters normalized at the same cutoff frequency ωc are considered: a multiple RC, a Bessel and a Butterworth. Their transfer functions are, respectively, [4]: H RC(s) = 1 (1+ s ωRC ) H Be(s) = 1 ∑ n i=0 bi( s ωD ) bi = n!/(n− i)! 2n!/(2n− i)! 2i i! H Bu(s) = 1 ∏ n i=1 ( s ωc −si) si = exp(j 2i + n− 1 2n π) (2)