The anti-aliasing requirements for area and timing mesurements with the Fluorescence Detector read-out channel

This note calculates the requirements of 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 errors in area and timing 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 criterion, that is band limitedness within ωN = π 2T to allow shape reconstruction [1]. The input signal of the anti-aliasing filter i(t) is supposed to be a rectangle with duration ∆ > T , (approximating the expected trapezoidal shape): i(t) = Q ∆ [1(t)− 1(t−∆)]←→I(s) = Q s∆ [1− exp(−s∆)] (1) Three different n-th order anti-aliasing filters with the same cut-off frequency ωc are analyzed: multiple RC, Bessel and Butterworth. Their transfer functions are, respectively, [3]: H RC(s) = 1 Dn RC (s) = 1 (1 + s ωRC ) n H Be(s) = 1 Dn Be(s) = 1 ∑n k=0 bk( s ωD )k bk = n!/(n − k)! 2n!/(2n − k)! 2k k! H Bu(s) = 1 Dn Bu(s) = 1 ∏n k=1( s ωc − sk) sk = exp(i 2k + n− 1 2n π)(2)