8.3. Pulse Shaping 483
The above equation implies that the output pulse decays exponentially with time
with a width that can be controlled by the time constantτd. In radiation measure-
ment systems the main objective is to measure the height of the pulse, which is
proportional to the energy deposited by the radiation in the active volume of the
detector. However, as can be seen in Fig.8.3.5, the output pulse from a CR dif-
ferentiator has a sharp peak, which makes it difficult to measure the pulse height.
Another disadvantage of the sharp peak is that it is most affected by the high fre-
quency noise in the system. Obviously the solution to the problem is to somehow
make this peak rounded. To solve this problem the output of the CR differentiator
can be passed through an RC integrator.
An RC integrator with a time constantτi=RiCiis shown in Fig.8.3.4. The
figure also depicts the behavior of the circuit in response to a step input pulse. In
such a case the time profile of the output pulse can be approximated by
Vout=1−e−t/τi. (8.3.4)If now the fast rising output of CR differentiator is passed through this RC
integrator circuit, the result will be a well shaped pulse with rounded maximum
(see Fig.8.3.6 and Fig.8.3.5).
If we assume that the preamplifier output is a step function then the pulse profile
after passing through subsequent CR and RC stages can be approximated by
Vout=
τd(τde−t/τd+τie−t/τi)
τdτi(τd−τi). (8.3.5)
As we saw above the differentiator and integrator act like high-pass and low-pass
filters respectively and can be combined together to form a band pass filter or pulse
shaper where the shaping takes place in two steps. At first step the preamplifier
pulse goes through the high pass filter, which attenuates the noisy low frequency
components from the signal. The resulting signal is then fed into the low pass
filter, which allows only the low frequency clean signals to pass through. The high
frequency components consisting mainly of noise are attenuated at this stage.
According to equation 8.3.5 the rise and decay times of the final pulse depend
on the time constants of the CR and RC shapers. Generally the time constants are
chosen such that a pulse with rounded peak, short rise time, and long decay time is
obtained. This can, for example be ensured by setting the two decay constants to
be equal (τd=τi=τ). In this case equation 8.3.5 becomes
Vout=t
τe−t/τ. (8.3.6)However the practical shapers used in radiation detection systems do not produce
output pulses that can be approximated by a step function. The reason is that the
shaping time of the preamplifier and the shaper do not differ by several orders of
magnitude as required by the step function approximation. Realistic preamplifiers
produce exponentially decaying pulses, which resemble the output of a CR filter
in response to a step input pulse (cf. Fig. 8.3.5). If this pulse is then input into
a CR-RC shaper, it inhibits a significant undershoot in the decaying part of the
output pulse (see Fig. 8.3.6). This undershoot can be a serious problem for high
rate systems where a new pulse can arrive before the previous one has fully decayed.