Frontiers in Computing and Intelligent Systems 
ISSN: 2832-6024 | Vol. 3, No. 1, 2023 

 

182 

Research on Pulse Shaper Based on Trapezoidal 
Function 

Zhenwei Han 1, 2, *, Kefei Song1, Yang Liu 1 

1 Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China 
2 University of Chinese Academy of Sciences, Beijing 100049, China 

* Corresponding author: Zhenwei Han (Email: hanzhenwei1234@163.com) 

 

Abstract: In order to improve the time resolution and spatial resolution of photon-counting imaging system, the pulse shaper 

based on trapezoidal function is studied. The signal with an exponential decay shape from charge sensitive amplifier is converted 

to digital discrete signal by ADC and then the discrete number sequence is filtered and shaped by digital signal processing 

technology. The amplitude linearity and time width of the trapezoidal shaper are simulated and verified using software-generated 

single exponential signals and double exponential signals. While the continuous pulses are applied to the trapezoidal shaper, the 

separation characteristics of the trapezoidal algorithm for pulses pile-up is verified, and the setting method of the rise edge and 

fall edge parameters of the trapezoidal shaper is proposed, which can realize the high counting capability of the system.  

Keywords: Trapezoidal function; Pulse shaper; Pulses pile-up; Photon counting. 

 

1. Introduction 

Photon counting imaging detection plays an important role 

in solar activity monitoring, atmospheric and marine 

environment monitoring. The far ultraviolet radiation 

imaging measurement technology can realize the effective 

and accurate monitoring of the global ionosphere distribution, 

which is of great significance to the global navigation satellite 

system. Based on the X-ray, extreme ultraviolet and 

ultraviolet band photon count imaging detection technology, 

researcher can establish the solar, space environment and 

atmospheric environment monitoring system, monitoring the 

distribution and diffusion of atmospheric pollution, 

monitoring the thickness, distribution and composition of 

seawater oil pollution. 

Since the photon to be detected and the charge pulse output 

by the position-sensitive anode of the detector are 

independent events, the pulse amplitude and pulse interval 

follow Poisson distribution. Photoelectric pulse signal can be 

regarded as high-speed random signal, which generally 

presents the form of peak pulse and contains noise signal. 

Therefore, amplitude extraction is not easy, and the signal 

attenuation width of charge sensitive amplifier is wide, which 

is easy to generate signal accumulation, which adds difficulty 

to amplitude extraction. The amplitude information of the 

photoelectric pulse is closely related to the position of the 

incident photon, so the accurate extraction of the amplitude of 

the photoelectric pulse is very important. 

When processing the photoelectric pulse signal, special 

filtering processing is needed to make the amplitude 

extraction easier. Filtering process is to extract the signal to 

be analyzed from the input signal and ignore the interference 

signal. Here, the filtering processing of photoelectric pulse 

signal means that the pulse amplitude information of 

photoelectric pulse signal can be accurately presented, the 

accumulation of nuclear signal can be reduced, and noise 

interference can be eliminated as much as possible. The 

method of pulse shaper can greatly improve the extraction of 

pulse amplitude. Good pulse shaper can effectively separate 

stacking signals, deduct the amplitude of tail stacking, and 

provide a relatively flat amplitude information platform for 

subsequent processing to extract more accurate amplitude 

information. In order to improve the practicability of the 

photoelectric pulse shaper algorithm, the trapezoidal shaper 

algorithm is studied and improved in this paper. 

2. The algorithm of Trapezoidal shaper  

In photon counting imaging system, the detector usually 

uses low noise charge sensitive preamplifier to realize the 

initial conversion of charge pulse signal. The feedback 

resistor of the charge sensitive preamplifier is typically large 

to improve charge collection efficiency. The feedback resistor 

with shunt capacitance results in a long fall time at the charge 

sensitive preamplifier output. The output signal from the 

charge sensitive amplifier can be approximated by double 

exponential model: 

1 2/ /( )= ( )t t

iV t A e e − −-k            (1) 

whereτ1 is the fast time constant of double exponential 

signal andτ2 be the slow time constant of double exponential 

signal. The rise time and decay time of pulse signal are 

determined byτ1 andτ2. The A is the input pulse amplitude, 

and k is the proportionality constant. 

Let the sampling period be TS. After digital discretization 

of Formula 1, Z transformation of Vi(t) can be obtained by (2). 

1 2/ /
( )=

S S
i T T

z z
V z A

z e z e
 − −

 
− 

− − 
     (2) 

Trapezoidal shaper is to convert the input exponential 

attenuation signal into trapezoidal signal of equal proportion, 

and the time domain signal waveform of trapezoidal pulse is 

shown in Fig.1. 

The time-domain expression of trapezoidal pulse can be 

expressed as a piecewise function: 
4

0

0

( )= ( )i

i

V t y t
=

                   (3) 

1

( )
y ( )=

a

Atu t
t

t
                   (4) 



 

183 

2

( ) ( )
y ( )= a a

a

A t t u t t
t

t

− −
−               (5) 

 

Figure 1. Trapezoid shape in time domain 

3

( ) ( )
y ( )= b b

a

A t t u t t
t

t

− −
−               (6) 

4

( ) ( )
y ( )= c c

a

A t t u t t
t

t

− −
           (7) 

where ta represents the rise time of the trapezoid pulse, tb–

ta is the flat top width of the trapezoid pulse, and tc is the 

bottom width of the trapezoid pulse. And A is the trapezoid 

pulse amplitude. 

Let the sampling period be TS. After digital discretization 

of Equation (3), Z transformation of Vo(t) can be obtained by 

(8). 
1

0 1 2

(1 )
( )=

(1 )

a b cn n n

a

Az z z z
V z

n z

− − −−

−

− − +

−
         (8) 

where na=ta/TS, nb=tb/TS and nc=tc/TS. 

The transfer function of trapezoidal shaper can be given by 

(9). 

1 2

1

0

1 2

/ /

(1 )
( )= =

(1 )

a b c

S S

n n n

i
a T T

V z z z z z
H z

z zV z
n z

z e z e
 

− − −−

−

− −

− − +

 
− − 

− − 

（ ）

（ ）   (9)

 

3. Simulation and results of trapezoidal 
pulse shaper 

As illustrated in Fig. 2, the ideal exponential decay pulse is 

first used as the input pulse and corresponds to two trapezoid 

pulses. The rise edge time of the exponential pulse is 0, and 

the tail attenuation duration is 100us. The ordinate is the pulse 

amplitude, the abscissa is the sampling point, and the interval 

of each point is 20ns. The shape of trapezoid is symmetrical 

in rise time and fall time, so it was an isosceles trapezoid for 

optimum signal to noise ratio. The trapezoidal flat top is 

beneficial to extract the input pulse amplitude. 

However, the actual input charge pulse rise edge is not ideal. 

Due to the charge sensitive preamplifier response time, the 

rise time of input pulse is about tens of ns. In view of this 

input pulse characteristic, we optimized the trapezoidal 

shaping method to make the flat top time of the trapezoid as 

short as possible to ensure the stability of the algorithm. The 

rapidity of the algorithm and the ease of fpa implementation 

are also important concerns. 

In the high-counting rate application pulse pile-up would 

certainly appear on the falling edge of the charge sensitive 

preamplifier output when the interval time between the two 

adjacent events is shorter than the fall time. In order to 

mitigate pulse pile-up, the charge sensitive preamplifier is 

usually followed by a trapezoidal pulse shaper, which can 

regulate the shape of the output pulse to reduce the fall time 

of the charge sensitive preamplifier output. Meanwhile, the 

smooth pulse generated by the trapezoidal pulse shaper 

improves the signal-to-noise ratio (SNR) of the analog front-

end chains, which facilitates the measurement of the pulse 

amplitude due to a flap maximum instead of a steeple shape. 

When the photon count rate is higher, the two input pulses 

are closer together, as shown in Fig.4 and Fig.5. If the 

trapezoidal shaper width is larger than the interval of two 

input pulses, the overlapped input pulses cannot be separated 

effectively. Measures to optimize trapezoidal shaping 

parameters were taken by reducing the flat top width of the 

trapezoid pulse tb– ta and increasing the rise time of the 

trapezoid pulse ta. However, the optimized tb’ must be greater 

than tb otherwise the SNR will decrease rapidly. The falling 

edge of the trapezoid pulse tc – tb is reduced to less than the 

rise time of the trapezoid pulse ta, which is beneficial to 

reduce the pulse accumulation. 

 

Figure 2. Simulation of trapezoidal shape for ideal exponential 
decay pulse 

 

Figure 3. Simulation of trapezoidal shape for double exponential 
pulse 

In order to verify the processing capability of the above 

method on the noisy pulse, the pulse output of CSA is used as 

the input. The tail decay time of the output signal from the 

charge sensitive amplifier is 200us. Discrete digital 

waveforms by the sampling period 50ns are reconstructed in 

Fig.6. The curve after trapezoid shaper reduces the tail width 

of the input pulse significantly and extracts the amplitude of 

the input pulse linearly. 

 



 

184 

 

Figure 4. Simulation of trapezoidal shape for pulse pile-up 

 

Figure 5. Simulation of optimized trapezoidal shape 

 

Figure 6. Test result of trapezoidal shape for the output signal from 
charge sensitive amplifier 

4. Conclusions 

It can be concluded from the above discussion that the 

trapezoidal shaper parameters should be selected according to 

the actual demand. In high resolution applications, the rise 

edge of the trapezoidal shaper can be as large as possible and 

the duration of the flat top can be extended. In the application 

of high counting rate, the rising edge of trapezoidal shaper can 

be shortened appropriately, the duration of flat top can be 

greatly reduced, and the falling edge of tail can be greatly 

reduced, but the undershoot compensation should be careful. 

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