PIXIE-4 User’s Manual
V2.69
©
XIA
2015. All rights reserved.
xxxvi
+
−
=
−
−
+
−
−
=
+
−
=
k
L
k
i
i
G
L
k
G
L
k
i
i
k
x
V
V
LV
1
1
2
,
(6.2)
where the filter length is
L
and the gap is
G
. The factor
L
multiplying
k
x
V
,
arises because the
sum of the weights here is not normalized. Accommodating this factor is trivial.
While this relationship is very simple, it is still very effective. In the first place, this is the
digital equivalent of triangular (or trapezoidal if
G
≠ 0) filtering which is the analog industry’s
standard for high rate processing. In the second place, one can show theoretically that if the
noise in the signal is white (i.e. Gaussian distributed) above and below the step, which is
typically the case for the short shaping times used for high signal rate processing, then the
average in Eqn. 6.2 actually gives the best estimate of V
x
in the least squares sense. This, of
course, is why triangular filtering has been preferred at high rates. Triangular filtering with
time variant filter lengths can, in principle, achieve both somewhat superior resolution and
higher throughputs but comes at the cost of a significantly more complex circuit and a rate
dependent resolution, which is unacceptable for many types of precise analysis. In practice,
XIA’s design has been found to duplicate the energy resolution of the best analog shapers while
approximately doubling their throughput, providing experimental confirmation of the validity
of the approach.
6.2 Trapezoidal Filtering in the Pixie-4
From this point onward, we will only consider trapezoidal filtering as it is implemented in the
Pixie-4 according to Eqn. 6.2. The result of applying such a filter with Length L=1
s and Gap
G=0.4
s to a
-ray event is shown in Figure 6.3. The filter output is clearly trapezoidal in shape
and has a rise time equal to L, a flattop equal to G, and a symmetrical fall time equal to L. The
basewidth, which is a first-order measure of the filter’s noise reduction properties, is thus
2L+G.
This raises several important points in comparing the noise performance of the Pixie-4 to
analog filtering amplifiers. First, semi-Gaussian filters are usually specified by a
shaping time
.
Their rise time is typically twice this and their pulses are not symmetric so that the basewidth
is about 5.6 times the shaping time or 2.8 times their rise time. Thus a semi-Gaussian filter
typically has a slightly better energy resolution than a triangular filter of the same rise time
because it has a longer filtering time. This is typically accommodated in amplifiers offering
both triangular and semi-Gaussian filtering by stretching the triangular rise time a bit, so that
the
true
triangular rise time is typically 1.2 times the selected semi-Gaussian rise time. This
also leads to an apparent advantage for the analog system when its energy resolution is
compared to a digital system with the same nominal rise time.
One important characteristic of a digitally shaped trapezoidal pulse is its extremely sharp
termination on completion of the basewidth 2L+G. This may be compared to analog filtered
pulses whose tails may persist up to 40% of the rise time, a phenomenon due to the finite
bandwidth of the analog filter. As we shall see below, this sharp termination gives the digital
filter a definite rate advantage in pileup free throughput.