Physics and Engineering of Radiation Detection

486 Chapter 8. Signal Processing

Detailed discussion about transfer functions and their derivations is beyond the
scope of this book and the reader is referred to standard texts in electronics (11; 6).
The Laplace transform of the transfer function of the CR-CR-RC circuit with a step
input is given by

H(s)=

τp

(τps+1)

τds

(τds+1)(τis+1)

, (8.3.7)

wheresis a complex variable of the Laplace transform. The subscriptsd,i,andpin
τrefer to CR differentiator, RC integrator, and preamplifier (which we have taken
to be a CR differentiator with step input) respectively.
It is apparent that the above function has singularities (points at which the
function becomes infinite) ats=− 1 /τp,− 1 /τd,− 1 /τi. These are calledpolesof the
function. One of the effects of these poles is the appearance of significant undershoot
in the decaying part of the pulse (see Fig.8.3.7).

t/τp

0 12345 6

out

V

0

0.2

0.4

0.6

0.8

(^1) Preamplifier Output
t/τ
0123456
out
V
-0.2
0
0.2
0.4
0.6
0.8
(^1) Shaper Output
Figure 8.3.7: Typical output of a preamplifier having shaping timeτp
(upper plot). The response of a CR-RC shaper to this preamplifier output
is shown in the lower plot. The graph has been generated usingτd=τi=
τ. The significant undershoot in the decaying part of the pulse reduces
system resolution and must therefore be either reduced or accounted for
in the data.