PULSE SEPARATION CHARACTERISTICS BASED ON THEIR
APPROXIMATION
Jiří Jevický1
, Filip Mravec2
, Zdeněk Matěj2
Václav Přenosil2
1
Faculty of military Technology, University of Defence, Brno
jiri.jevicky@unob.cz
2
Faculty of Informatics, Masaryk University, Brno
Botanická 68a, 602 00 Brno, Czech Republic
xmravec@fi.muni.cz, matej.zdenek@gmail.com, prenosil@fi.muni.cz
Abstract: This article deals constructing of models to approximate the electrical impulses that
occur originate in the detector of a nuclear radiation. Radiation detector detects two kinds of
particles. Electrical impulses of both responses are different in the order of nanoseconds.
Mathematical models of the individual particles responses let easily distinguish what kind of
particles was detected. The article describes four separation methods and evaluated their
effectiveness.
Keywords: approximation; discrete orthogonal polynomials, 3D frequency histogram, scatter
plot
INTRODUCTION
Universal method of analyzing responses spectrometry detectors in mixed fields of fast neutrons
and gamma rays is an effective tool for many applications. One is used in radiation protection,
environmental protection, medicine, security, military, etc. Digital spectrometric detection can
reduce the dimensions of measuring equipment, reduce its power consumption, reduce time
measurements and measurements in the field. Separation of mixed energy field values by folders
belonging neutrons and gamma rays are usually made on the basis of analog signal processing
laboratory equipment and is therefore subject to laboratory conditions. Digital spectrometric
detection allows to measure routinely in sittu.
1 SEPARATION METHODS OF THE RADIATION DETECTOR RESPONSES
All of the separation characteristics of the considered C pulse f(t) are based on an approximation
of digitized pulse discrete orthogonal polynomials, ie the approximation:
(1)
where the basic function Pk(t) are orthogonal polynomials on the set of nodes, t = 0, 1, ... N, with
N ≥ n To generate a set of basic functions were used formula:
(2)
Separation by an initial pulse of edge demonstrates excellent separation properties themselves
coefficients c1, c3,, and c0, c5. Some functional expressions of these coefficients could then
separation capabilities even stronger.
)t(Pc)t(
n
0k
kk∑=
=ϕ
.n,....3,2k),t(p
)1Kn(k
)kN)(1k(
)t(P)t
2
N
)1kN(k
)1k2(2
)t(P
,
N
t
21)t(P
,1)t(P
2K1kk
1
0
=
+−
+−
−⎟
⎠
⎞
⎜
⎝
⎛
−
+−
−
=
−=
=
−−
For the separation of pulses by pulse decay is shown that the separation capacity of the
approximation coefficients is less significant. Good separation properties were detected only in
the coefficient c0, and somewhat less good factor at c2.
Therefore were examined the properties of other variously defined separation characteristics of
C, for example, assessing distance inflection points approximation Π(t), or expressions of
integrals approximation.
As promising appears the following separation characteristics and methods.
1.1 Koeficient c0 aproximace
First consider the response:
(3)
On Figure 1: is a scatter plot of the observed values of two-dimensional random variables (C, Q),
where Q is second characteristic are maxima of measured (digitized) pulses detected on a set of
measured impulse values {f0, f1, ... , fN}. Illustration images are created using the "Spectrum 2"
program (author Jevický) using data files "fotony.awd" and "neutrony.awd" (author Matěj,
Cvachovec).
Figure 1: Scatter plot of the random variables(C, Q) for characteristic c0
Given the purpose of the separation is possible to release from the transformation equations
transformation of coordinates Q and too the shift to a new center. New separation characteristic is
possible introduce simply by following formula:
(4)
where the angle α (and also the values of sine and cosine) is sufficient to calculate only once in
the process of calibration of the measuring device. For test data was used transformation:
CT = C ∗ 0,01341404257942208 + Q ∗ (-0,9999100276833298).
Scatter plot of the (CT, Q) is shown on Figure 2.
For ease of separation of photons (blue dots on the charts) and neutrons (red dots) is suitable
transformation of the coordinate system rotation by the angle α, which forms an axis regression
lines Q = a + b∗C for photons and neutrons with a vertical axis, with the center of rotation is the
intersection of the regression lines.
0cC =
α∗+α∗= sinQcosCCT
Figure 2: modified scatter plot of the variables(C, Q) for characteristic c0
According to formula (2) is P0(t) = 1 and hence the coefficient c0 of the approximation (1) is true:
(5)
which means that the characteristic C = c0 can be interpreted as the mean pulse. Since the time
step in the table of pulse measured values is 1 ns pulse (and because the values of f(t) at the
beginning and at the end of the table are virtually zero) can be the sum of the values in equation
(5) taken as an approximate value of the integral of the pulse f(t) at interval 0, N (more
precisely at the interval 0,5; N + 0,5 . Note, that the coefficient 1/(N + 1) in equation (5) is
constant for all pulses and therefore does not alter the properties of the actual values of the pulse
sum (or integral) as separation characteristics of pulse.
1.2 The difference of the coefficients c0 a c2 of the approximation
In this case, considering the following characteristics of the pulse separation:
(6)
where the factor 10 is only for the purpose of scaling the axis C "pleasant" greater value. Scatter
plot (C, Q) for this characteristic is on Figure 3.
Using nonlinear regression hyperbolic transformation was found next pulse characteristic:
(7)
Scatter plot (CT, Q) for the transformed separation characteristic CT is on Figure 4.
1.3 Quotient of active edge approximations integral of the pulse and the maxima
approximations of the pulse
In this method as the separation characteristics of the pulse is considered:
(8)
∑
∑
∑
=
=
=
+
===
N
0i
N
0i
2
0
N
0i
0
00
0
0 )i(f
1N
1
))i(P(
)i(P)i(f
)P,P(
)P,f(
c
〈 〉
〈 〉
)cc(10C 20 −=
00676,0Q1,0
C
CT
−∗
=
∫
ϕ
ϕ
ϕ
=
kon
max
t
tmax
I dt)t(
1
C
Figure 3: Scatter plot of variables (C, Q) for characteristic 10(c0-c2)
Figure 4: modified scatter plot of the variables(C, Q) for characteristic
10(c0-c2)
Figure 5: Scatter plot of variables (C, Q) for characteristic CI
where φmax is the global maximum of the approximation φ(t) pulse f(t), tφmax ∈ 0, N s the time
at which the functions φ(t) takes its maximum and tkon = tφmax + d. The length of the integration
interval was chosen d = 300. The scatter chart (C, Q) for this characteristic is shown on Figure 5.
1.4 Integral from approximation of the active edge pulse
In this method the separation characteristic of the pulse is:
(9)
where tφmax ∈ 0, N is the time at which the approximation φ(t) pulse attains its maximum and
tkon = tφmax + d. The length of the integration interval was chosen d = 300. The scatter plot of
variables (C, Q) for this characteristic is shown on Figure 6.
Figure 6: Scatter plot of variables (C, Q) for characteristic CII
Approximately linear dependence of random variables C and Q allows analogously as in the case
of separation characteristic C = c0 (see 1.1 ) to calculate the appropriate transformation by
rotating of the coordinate system (C, Q).
Figure 7: modified scatter plot of the variables(C, Q) for characteristic CII
∫
ϕ
ϕ=
kon
max
t
t
II dt)t(C
〈 〉
〈 〉
The coefficients for the characteristic CII were calculated from the following transformation
formula:
CII = C*0,02012942734552012 + Q*(-0,9997973825504552).
Scatter plot (CII, Q) for the transformed separation characteristic CII is shown on Figure 7.
2 OVERALL VIEW OF THE SEPARATION
The four types of separation characteristics C was prepared from data file "_data_10000.awd"
containing records of one million pulses. After removal of deformed pulses was pair separation
characteristics (C, Q) calculated from 985.313 pulses.
Thus obtained four statistical samples (by methods discussed in the previous chapters) were
divided into a total of 1,024,512 classes.
All four dimensional frequency tables were stored in files in CSV format. The following figures
are shown both frequency histograms 3D dimensional random variables (C, Q) and 2D scatter
plots frequency tables.
Figure 8: C characteristics according to the method from chapter 1.1 see file
_data_10000_Aupr.csv
Figure 9: C characteristics according to the method from chapter 1.2 for file
_data_10000_Bupr.csv
Figure 10: C characteristics according to the method from chapter 1.3 for
file _data_10000_Cupr.csv
Figure 11: C characteristics according to the method from chapter 1.4 for
file _data_10000_Dupr.csv
CONCLUSION
The described models necessary separation proved as easy usable. To implement ones into the
digital spectrometer, all models appear to be readily applicable to technical equipment. Digital
spectrometer of the project SPECTRUM is equipped with FPGA. Described models can be
implemented into the FPGA and thus significantly reducing the width of stream spectrometric
system data channel. This ability will be a major contribution to the ON-LINE application.
LITERATURE
[1] Glenn F. Knoll. Radiation Detection and Measurement, third edition, 2000 John Wiley &
Sons, Inc. ISBN-0-47-0738-5.
[2] EJ-444 Dual Phosphor Thin Scintillator, www.eljentechnology.com
[3] T.K. Alexander, F.S. Goulding. An amplitude-insensitive system that distinguishes pulses of
different shapes, Nuclear Instruments and Methods, Volume 13, August–October 1961,
Pages 244-246, ISSN 0029-554X, 10.1016/0029-554X(61)90198-7.
[4] M.L. Roush, M.A. Wilson, W.F. Hornyak. Pulse shape discrimination, Nuclear Instruments
and Methods, Volume 31, Issue 1, 1 December 1964, Pages 112-124, ISSN 0029-554X,
10.1016/0029-554X(64)90333-7.
[6] J.M. Adams, G. White. A versatile pulse shape discriminator for charged particle separation
and its application to fast neutron time-of-flight spectroscopy. Nuclear Instruments and
Methods, 156(3):459-476, 1978. Sons, Inc. ISBN-0-47-0738-5.
Acknowledgement
The work presented in this paper has been supported under the research project SPECTRUM, No.
TA01011383 by Technology Agency of the Czech Republic.