International Journal of Scientific and Research Publications, Volume 5, Issue 9, September 2015

ISSN 2250-3153

VHDL Simulation of Cusp-like Filter for High Resolution

Radiation Spectroscopy
Ms Kavita Pathak *, Dr. Sudhir Agrawal **
*
**

Electronics and Communication Department, Truba College of Engineering and Technology, Indore, M.P., India
Electronics and Communication Department, Truba College of Engineering and Technology, Indore, M.P., India

Abstract- One of the main objectives of a radiation spectroscopy is to have good energy resolution so that the results of the nuclear
physics experiments can be interpreted faithfully. Electronic noise introduced by the instrumentation systems put a major constraint on
the energy resolution. It has been proved theoretically by many researchers/authors that the signal to noise ratio can be maximized to
its best value if the electrical pulse obtained from a nuclear detector can be shaped into an infinite cusp form. Practically a cusp-like
pulse of finite length can be obtained by processing the detector signals in digital domain. This paper considers VHSIC Hardware
Description Language (VHDL) simulation aspect of cusp-like filtering algorithm. Such simulations provide the platform to verify the
algorithms for their implementation into a digital hardware such as Field Programmable Gate Array (FPGA) for real time applications.
The simulation considers exponentially decaying sequences as the input to the filter. In radiation spectroscopy applications, such
signals are obtained by digitizing the slow decaying exponential pulses derived from radiation detectors.
Index Terms- Cusp-like Filter, Energy Resolution, Signal to Noise Ratio, Radiation Spectroscopy, VHDL.

I. INTRODUCTION

n a typical radiation spectroscopy, the ionizing radiation quantum interacts with the sensitive volume of a nuclear detector and
releases its energy into the same. This process generates an equivalent charge which is further collected by the preamplifier to
form a voltage pulse. Peak of a voltage pulse obtained from preamplifier carries important information as the same is proportional to
energy of the radiation quantum. The capability of a spectrometer to distinguish radiation quantum closely separated in energy is
characterized by its energy resolution [1].The energy resolution is a crucial parameter to interpret the findings of a nuclear physics
experiment. A poor energy resolution gives rise to widening of spectral lines that may lead to wrong interpretation of the experiment.
Energy resolution is usually given by Full width at Half Maximum (FWHM) of the spectral peaks when the detector pulse is
approximated by the Gaussian shape [1].
= 2.35

(1)

Where is the standard deviation of the Gaussian pulse.

In case of spectroscopy the system FWHM is given by the following relation,

2
2
2
=
+

(2)

2
Where
mainly depends on statistical fluctuations in charge carrier generation in detector due to discrete nature of
2
radiation quantum.
depends on the noise contribution of preamplifier and other associated instrumentation. The noise
introduced by used instrumentation can be termed as electronic noise and it mainly depends on DC baseline fluctuations, pulse pileup
effects, electromagnetic and power supply disturbances [2]. Therefore to achieve the best energy resolution the noise must be
minimized. Various theories have been established to maximize the signal to noise ratio [1],[3],[4].The amount of electronic noise is
often expressed in terms of the equivalent noise charge (ENC). ENC is the amount of charge when applied to input of the
instrumentation system, will result in output voltage equal to RMS (i.e. Root Mean Square) level of output due to noise only[1]. ENC
can be decomposed into three terms: one proportional to current (or parallel) noise, another proportional to voltage (or series) noise
and the last proportional to 1/f noise. Efficiency of various noise shaping filters are characterized by their ENC value. If an impulse
signal is applied to a circuit having only current and voltage noise, the infinite cusp filter offers the best ENC value [5]. The modified
versions of infinite cusp filter have extensively been considered in spectroscopy applications to shape the detector pulse in presence of
all the three types of noise sources [6],[7]. Due to above mentioned reasons the infinite cusp has been considered as optimum detector
pulse shape to achieve the best possible signal to noise ratio and same has become the standard for comparing the performance of
other methods of pulse shaping. Figure 1 shows various pulse shapes and their signal to noise ratio relative to an infinite cusp.
Practically it is very difficult, rather impossible to obtain the infinite cusp shape using analog circuit techniques [8]. However the

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International Journal of Scientific and Research Publications, Volume 5, Issue 9, September 2015
ISSN 2250-3153

digital signal processing techniques can be exploited to obtain a finite length cusp-like shape. Such shaping will improve the
performance of a radiation spectrometer in terms of energy resolution by maximizing the signal to noise ratio. Additionally the same
techniques can also minimize the pile up losses. This paper is organized as follows. Section II briefly describes the cusp-like filtering
algorithm. It also briefly discusses the used methodology and heart of the developed simulation code. Section III discusses the
simulation results. Finally, the concluding remarks are included in Section IV.

Figure 1: Noise contribution of different pulse shapes relative to infinite cusp (top).

II. CUSP-LIKE FILTERING ALGORITHM

The finite length cusp-like filtering algorithm can be described in its recursive form [9]. The recursive forms are often faster in
execution and same can easily be implemented into digital hardware. The following recursive equations were considered for the
VHDL simulation.
d_K[i] = x[i] x[i K]
d_1[i] = x[i] x[i 1]
b[i] = d_K[i] K d_1[i L], i 0
c[i] = b[i] + c[i 1], i 0
s[i] = M2 c[i] + s[i 1], i 0
t[i] = M1 c[i] + s[i], i 0
y[i] = t[i] + y[i 1], i 0

(3)
(4)
(5)
(6)
(7)
(8)
(9)

Where X[i] is digitized input exponential signal and Y[i] is output of the filter respectively. The value of M is given by the equation
(10). The value of M1 and M2 can be selected to satisfy the equation (11). The symbol (*) in the above equations is a multiplication
operator.
=

( )
1

= 2

(10)
(11)

Where is digitizers clock frequency and is time constant of exponential input signal. The duration of the rising (falling) edge of
the cusp-like shape is decided by the value of L, and the value of K is equal to 2L+1. For high count applications a detector pulse is
required to be processed to the short duration [10][11]. The short duration pulse of cusp-like shape can be obtained by appropriately
choosing the value of L to minimize the pile up losses. Figure 2 is block diagram representation of the recursive equations given
above. The nature of input signal considered for the simulation was similar as considered in our previous paper [12] and the same has
been shown in figure 3. Heart of the simulation code is a 'generate statement' which instantiates the various components required for
the filtering operation and the same is given below.

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International Journal of Scientific and Research Publications, Volume 5, Issue 9, September 2015
ISSN 2250-3153

Figure 2: Block diagram of cusp-like filter

Figure 3: Input sequence to cusp-like filter

begin
Cusp-like filter : for i in m to n generate
SUB-1 : subtractor port map (x(i), x(i-K), d_K(i));
SUB-2 : subtractor port map (x(i), x(i-1), d_1(i));
MUL1 : multiplier port map (K, d_1(i-L), a(i));
SUB-3 : subtractor port map (d_K(i), a(i), b(i));
ACC-1 : adder port map (c(i-1), b(i), c(i));
MUL-2 : multiplier port map (M1, c(i), p(i));
MUL-3 : multiplier port map (M2, c(i), r(i));
ACC-2 : adder port map (s(i-1), r(i), s(i));
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International Journal of Scientific and Research Publications, Volume 5, Issue 9, September 2015
ISSN 2250-3153

ADD : adder port map (s(i), p(i), t(i));

ACC-3 : adder port map (y(i-1), t(i), y(i));
end generate Cusp-like filter;
The filtering scheme comprises of three subtractors, three multipliers, three accumulators and an adder respectively. The subtractors
(SUB-1, SUB-2 and SUB-3) and multipliers (MUL-1, MUL-2 and MUL-3) are generated by instantiating the components 'subtractor'
and 'multiplier' respectively. Whereas the adder and accumulators are generated by instantiating the component 'adder'. The first two
signals in parentheses (after the keyword 'port map') for each component are mapped to input ports while the third signal is mapped
to output port respectively. All the signals are one dimensional array of size 'n'.

III. SIMULATION RESULTS

The simulations were performed for non-piled up input sequences as well as for piled up input sequences by considering different
values of shaping parameters (i.e. K and L). Figure 4 shows the simulation result for single input sequence while the figure 5 shows
the simulation result for piled up input sequence respectively. Both the figures show plots for x[i], c[i], t[i] and y[i] respectively. The
value of L, K, M1, M2 and M was selected/calculated as 7, 15, 200, 4 and 50 respectively in both the cases.

(A)

(C)

(B)

(D)

Figure 4: Cusp-like filtering simulation result for non piled up detector signal (A) input sequence-x[i],
(B) sequence-c[i], (C) sequence-t[i] and (D) output sequence-y[i]

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International Journal of Scientific and Research Publications, Volume 5, Issue 9, September 2015
ISSN 2250-3153

(A)

(B)

(C)

(D)

Figure 5: Cusp-like filtering simulation result for piled up detector signal (A) input sequence-x[i],
(B) sequence-c[i], (C) sequence-t[i] and (D) output sequence-y[i]

IV. CONCLUSION
The VHDL simulation code for cusp-like filter was developed. The same was verified by observing the desired simulation results. It
was also observed that the value of 'L' decides the rise/fall time. This parameter should be chosen carefully to minimize the pile up
losses. The algorithm can be implemented into a digital hardware such as Field Programmable Gate Array (FPGA) to carry out the
cusp-like filtering in real time applications.

ACKNOWLEDGMENT
The authors would like to thank Ms. Pallavi Pahadiya, Assistant Professor at Truba College of Engineering and Technology for her active
participation in technical discussions and providing resources for the research work.
References
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V. Radeka and N. Karlovac, "Least-square-error amplitude measurement of pulse signals in presence of noise", Nucl . Instr. and Meth., Vol. 52, 1967, pp 86-92.

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International Journal of Scientific and Research Publications, Volume 5, Issue 9, September 2015
ISSN 2250-3153

[4]
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E. Gatti and P.M . Manfredi, "Processing the signals from solid-state detectors in elementary-particle physics ", Nuovo Cimento, Vol. 9, 1986, p38.
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AUTHORS
First Author Ms Kavita Pathak, B.E. (Electronics and communication), Final year, M.Tech (Digital Communication) Student, Truba
College of Engineering and Technology, Indore, M.P., India. Email id: - kavitapathak.2012@gmail.com
Second Author Dr. Sudhir Agrawal, B.E, M.Tech, Ph.D, Professor and Director, Truba College of Engineering and Technology, Indore,
M.P., India. Email id: - sudhiragrawal@trubainstitute.ac.in

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