A new digital Gaussian pulse shaping algorithm based on bilinear transformation

A new digital Gaussian pulse shaping algorithm based on bilinear transformation 1 first-author 1 corresp Corresponding author, glq@cdut.edu.cn Corresponding author, glq@cdut.edu.cn glq@cdut.edu.cn 2 2 The College of Nuclear Technology and Automation Engineering, Chendu University of Technology, Chengdu 610059, China The College of Nuclear Technology and Automation Engineering, Chendu University of Technology, Chengdu 610059, China Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031, China Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031, China Nuclear pulse signal needs to be transformed to a suitable pulse shape to remove noise and improve energy resolution of a nuclear spectrometry system. In this paper, a new digital Gaussian shaping method is proposed. According to Sallen-Key analog Gaussian shaping filter circuits, the system function of Sallen-Key analog Gaussian shaping filter is deduced on the basis of Kirchhoff laws. The system function of the digital Gaussian shaping filter based on bilinear transformation is deduced too. The expression of unit impulse response of the digital Gaussian shaping filter is obtained by inverse z-transform. The response of digital Gaussian shaping filter is deduced from convolution sum of the unit impulse response and the digital nuclear pulse signal. The simulation and experimental results show that the digital nuclear pulse has been transformed to a pulse with a pseudo-Gaussian, which confirms the feasibility of the new digital Gaussian pulse shaping algorithm based on bilinear transformation. Digital shaping Bilinear transformation Convolution sum I. INTRODUCTION 1 In a nuclear spectrometry system, the peak amplitude of nuclear pulse from the detector is proportional to energy of the incident particle, so the particle energy can be determined by measuring height of the pulse signal. A major cause of inaccuracy in the energy measurement is electronic noise, ballistic deficit, etc [ 1 1 ]. Shaping techniques are intended to improve the signal-to-noise ratio and extract height of the pulse signal accurately [ 2 2 - 12 12 2-12 ]. Gaussian pulse shaping, as a general shaping method in the nuclear spectrometry system, performs good in this regard [ 12 12 , 13 13 ]. With the rapid development of digital integrated circuits, and their advantages of flexibility and stability, studies on the digital Gaussian shaping algorithm for nuclear pulse signal has become popular during the past decades [ 12 12 - 21 21 12-21 ]. Valentin T. Jordanov studied digital Gaussian shaping algorithm of exponential signals [ 15 15 ]. Chen et al. made efforts in digital Gaussian shaping algorithm based on wavelet analysis [ 13 13 , 14 14 ]. Zhou et al. made advances in digital Gaussian shaping algorithm based on numerical differentiation [ 19 19 ]. The digital nuclear pulse signals are treated as quasi-Gaussian signals, and recursive algorithm is used to achieve the digital Gaussian shaping. In this paper, a digital Gaussian shaping algorithm based on bilinear transform is proposed. The unit impulse response of the digital Gaussian shaping system is deduced, and Gaussian shaping is realized by calculating the convolution sum of unit impulse response and digital nuclear pulse signal. II. SYSTEM FUNCTION OF THE ANALOG GAUSSIAN SHAPING SYSTEM 1 The analog Gaussian shaping in the nuclear spectrometry system, which is often implemented in Sallen-Key filter, as shown in Fig. 1 Fig. 1 , can shape a nuclear pulse signal as a pseudo-Gaussian signal. According to Kirchhoff’s law, the Eq. (1) can be obtained, <math display="block" id="M1"><mrow><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>/</mo><mi>R</mi><mo>=</mo><mi>C</mi><mtext>d</mtext><msub><mi>y</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>/</mo><mtext>d</mtext><mi>t</mi><mo>+</mo><mi>C</mi><mtext>d</mtext><mo stretchy="false">(</mo><msub><mi>y</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>/</mo><mtext>d</mtext><mi>t</mi><mn>,</mn></mrow></math> <math display="block" id="M2"><mrow><msub><mi>y</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>R</mi><mfrac><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mi>R</mi></mrow></mfrac><mn>,</mn></mrow></math> <math display="block" id="M3"><mrow><msub><mi>y</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>y</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>R</mi><mi>C</mi><mtext>d</mtext><msub><mi>y</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>/</mo><mtext>d</mtext><mi>t</mi><mn>.</mn></mrow></math> Thus, the relationship between input signal f ( t ) and output signal y ( t ) can be expressed as <math display="block" id="M4"><mrow><msup><mi>R</mi><mn>2</mn></msup><msup><mi>C</mi><mn>2</mn></msup><msup><msup><mi>y</mi><mo>′</mo></msup><mo>′</mo></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>R</mi><mi>C</mi><msup><mi>y</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Let the Laplace transform of f ( t ) and y ( t ) be F ( s ) and Y ( s ), respectively. Using differential properties of Laplace transform, we have the Laplace transform of y’ ( t ) and y" ( t ) be sY ( s ) and s 2 Y ( s ), respectively, and we have <math display="block" id="M5"><mrow><msup><mi>R</mi><mn>2</mn></msup><msup><mi>C</mi><mn>2</mn></msup><msup><mi>s</mi><mn>2</mn></msup><mi>Y</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mi>R</mi><mi>C</mi><mi>s</mi><mi>Y</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Thus, the system function of the analog Gaussian shaping system is <math display="block" id="M6"><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>/</mo><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mo>/</mo><mo stretchy="false">(</mo><msup><mi>R</mi><mn>2</mn></msup><msup><mi>C</mi><mn>2</mn></msup><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mi>R</mi><mi>C</mi><mi>s</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mn>.</mn></mrow></math> III. SYSTEM FUNCTION OF THE DIGITAL GAUSSIAN SHAPING SYSTEM 1 In the process of transforming Gaussian shaping system from analog domain to digital domain, bilinear transformation is used for conversion in order to maintain the basic properties of the system function unchanged and overcome the spectrum aliasing phenomenon. The principle of bilinear transformation is secondary mapping method, as shown in Fig. 2 Fig. 2 . The mapping process includes three steps, (1) The s -plane is mapped onto the strip-like region from – π / T to π/T of the s 1 -plane. A conversion can be done as <math display="block" id="M7"><mrow><mi>s</mi><mo>=</mo><mn>2</mn><mtext>tan</mtext><mo stretchy="false">(</mo><msub><mi>s</mi><mn>1</mn></msub><mi>T</mi><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo><mo>/</mo><mi>T</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>s</mi><mn>1</mn></msub><mi>T</mi></mrow></msup><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>s</mi><mn>1</mn></msub><mi>T</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mn>2</mn><mo>/</mo><mi>T</mi><mo stretchy="false">]</mo><mn>.</mn></mrow></math> (2) Mapping the s 1 -plane onto the z -plane; the conversion relationship can be expressed as <math display="block" id="M8"><mrow><mi>z</mi><mo>=</mo><msup><mi>e</mi><mrow><msub><mi>s</mi><mn>1</mn></msub><mi>T</mi></mrow></msup><mn>.</mn></mrow></math> (3) Mapping the s -plane onto the z -plane; the following transform equation between s and z domain based on the Eqs. (7) and (8) can be obtained, <math display="block" id="M9"><mrow><mi>s</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mn>2</mn><mo>/</mo><mi>T</mi><mo stretchy="false">]</mo><mn>.</mn></mrow></math> This allows the one-to-one mapping relationship from the s -plane to the z -plane. Thereby spectrum aliasing phenomenon is eliminated. The system function of the digital Gaussian shaping system can be obtained by Eqs. (6) and (9) when letting M = RC/T <math display="block" id="M10"><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mo>+</mo><mn>4</mn><msup><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>2</mn><msup><mi>z</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">[</mo><mn>4</mn><msup><mi>M</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>M</mi><mo>+</mo><mn>1</mn><mo>+</mo><mo stretchy="false">(</mo><mn>2</mn><mo>−</mo><mn>8</mn><msup><mi>M</mi><mn>2</mn></msup><mo stretchy="false">)</mo><msup><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><mn>4</mn><msup><mi>M</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>M</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>z</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo stretchy="false">]</mo><mn>.</mn></mrow></math> Let a =4 M 2 +2 M +1, b =2-8 M 2 , and c =4 M 2 -2 M +1, the inverse z transform is carried out to Eq. (10). Then the unit impulse response of the digital Gaussian shaping system is <math display="block" id="M11"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>2</mn><msub><mi>k</mi><mn>1</mn></msub><mo>/</mo><mi>a</mi><mo stretchy="false">)</mo><mo>{</mo><mo stretchy="false">[</mo><mo>−</mo><mi>b</mi><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo stretchy="false">]</mo><mo>/</mo><mo stretchy="false">(</mo><mn>2</mn><mi>a</mi><msup><mrow><mo stretchy="false">)</mo><mo>}</mo></mrow><mi>n</mi></msup><mi>u</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>k</mi><mn>2</mn></msub><mo>/</mo><mi>a</mi><mo stretchy="false">)</mo><mo>{</mo><mo stretchy="false">[</mo><mo>−</mo><mi>b</mi><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo stretchy="false">]</mo><mo>/</mo><mo stretchy="false">(</mo><mn>2</mn><mi>a</mi><msup><mrow><mo stretchy="false">)</mo><mo>}</mo></mrow><mi>n</mi></msup><mi>u</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">(</mo><msub><mi>k</mi><mn>3</mn></msub><mo>/</mo><mi>a</mi><mo stretchy="false">)</mo><mo>{</mo><mo stretchy="false">[</mo><mo>−</mo><mi>b</mi><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo stretchy="false">]</mo><mo>/</mo><mo stretchy="false">(</mo><mn>2</mn><mi>a</mi><msup><mrow><mo stretchy="false">)</mo><mo>}</mo></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>u</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mn>2</mn><msub><mi>k</mi><mn>4</mn></msub><mo>/</mo><mi>a</mi><mo stretchy="false">)</mo><mo>{</mo><mo stretchy="false">[</mo><mo>−</mo><mi>b</mi><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo stretchy="false">]</mo><mo>/</mo><mo stretchy="false">(</mo><mn>2</mn><mi>a</mi><msup><mrow><mo stretchy="false">)</mo><mo>}</mo></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>u</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where, k 1 =( b -4 a )/[2( b 2 -4 ac ) 1/2 ] +1/2, k 2 =(- b +4 a )/[2( b 2 -4 ac ) 1/2 ] +1/2, k 3 = - a /( b 2 -4 ac ) 1/2 , and k 4 = a /( b 2 -4 ac ) 1/2 . Let z = e jω , the discrete-time Fourier transform of h ( n ) can be obtained by Eq. (10), <math display="block" id="M12"><mrow><mi>H</mi><mo stretchy="false">(</mo><msup><mi>e</mi><mrow><mi>j</mi><mi>ω</mi></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mo>+</mo><mn>4</mn><msup><mi>e</mi><mrow><mo>−</mo><mi>j</mi><mi>ω</mi></mrow></msup><mo>+</mo><mn>2</mn><msup><mi>e</mi><mrow><mo>−</mo><mn>2</mn><mi>j</mi><mi>ω</mi></mrow></msup><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>j</mi><mi>ω</mi></mrow></msup><mo>+</mo><mi>c</mi><msup><mi>e</mi><mrow><mo>−</mo><mn>2</mn><mi>j</mi><mi>ω</mi></mrow></msup><mo stretchy="false">)</mo><mn>.</mn></mrow></math> The amplitude spectrum of H (e jω ) is shown in Fig. 3 Fig. 3 . The spectral width of the digital Gaussian shaping system increases with M decreasing. This makes the high-frequency noise filtration reduce. IV. GAUSSIAN SHAPING OF THE NUCLEAR PULSE SIGNAL 1 A. Mathematical description of the nuclear pulse signal 2 The structure of a nuclear spectrometer system with digital shaping is shown in Fig. 4 Fig. 4 . If the charge generated by an impinging particle is instantaneously collected by the detector, the output signal from the preamplifier can be approximately expressed by a continuous-time exponential signal <math display="block" id="M13"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>B</mi><mo stretchy="false">(</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mo>/</mo><msub><mi>τ</mi><mn>1</mn></msub></mrow></msup><mo>−</mo><msub><mi>B</mi><mn>1</mn></msub><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mo>/</mo><msub><mi>τ</mi><mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo><mi>u</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where, B is magnitude of the signal; u ( t ) is a unit step signal; τ 1 and τ 2 are time constants of the exponential signal; and B 1 is proportionality coefficient. To simplify the calculation, let B 1 =1. If the ADC sampling period is T , the sampled discrete-time signal f ( n ) is <math display="block" id="M14"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>B</mi><mo stretchy="false">(</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>n</mi><mi>T</mi><mo>/</mo><msub><mi>τ</mi><mn>1</mn></msub></mrow></msup><mo>−</mo><msub><mi>B</mi><mn>1</mn></msub><msup><mi>e</mi><mrow><mo>−</mo><mi>n</mi><mi>T</mi><mo>/</mo><msub><mi>τ</mi><mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo><mi>u</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> B. Digital Gaussian shaping of the simulated double exponential signal 2 The response of the input signal f ( n ) after the digital Gaussian system is described as <math display="block" id="M15"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>*</mo><mi>h</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mo>−</mo><mi>∞</mi></mrow><mi>∞</mi></munderover></mstyle><mi>f</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>m</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where, h ( n ) is the unit impulse response of the system; and y ( n ) is the zero state response of the system. Take double exponential Gaussian signal as input of the digital Gaussian shaping system, and the sampling period T =0.01 μs, B =1.6, τ 1 =0.23 μs. Fig. 5 Fig. 5 shows the response y ( n ) of digital Gaussian shaping system, calculated and plotted by Matlab at τ 2 = 0.4 and 1.8 μs for different of M values. Width of the shaping waveform increases with M , and the symmetry improves with larger M . An advantages of the digital Gaussian shaping system is that one needs just changing the M value of the parameter, rather than adjusting the hardware. C. Digital Gaussian shaping of the real sampled nuclear pulse signal 2 Signals from a 60 Co source were measured by a NaI(Tl) detector and an Agilent DSO-X 2012A oscilloscope. The response y ( n ) of the digital Gaussian shaping system is calculated by Eq. (15) in MATLAB, after the digital nuclear pulse signals are stored, with sampling frequency of 200 MHz. Fig. 6 Fig. 6 shows the measured pulse signal and its corresponding waveform of the digital shaping signal at M = 80 and 160. The shaping waveform at M =160 is wider, and the symmetry is better, than those at M =80. The shaping waveform is preferably Gaussian waveform. Decreasing the M value, the shaping waveform will be narrowed in time domain. Also, a low shaping time reduces the noise filtration and resulted in a poor energy resolution. At M =0.5 ( Fig. 7 Fig. 7 ), significant noises in shaping waveform can be seen. V. DISCUSSION AND CONCLUSION 1 Pulse shaping digitization is the key to digitization of nuclear instruments. After deducing unit impulse response of the digital Gaussian shaping system by bilinear transformation, the convolution sum of the digital nuclear pulse signal and unit impulse response of the derived digital Gaussian system can be carried out. The digital nuclear pulse signal is shaped as pseudo-Gaussian signal and the noises are removed. A higher M leads to better symmetry of the shaping waveform, wider pulse, and better effect of noise filtering, with flat top pulse and small ballistic deficit. But a wider pulse will increase the pulse pile-up. Pile-up distortion is a major drawback in nuclear spectrometry system at high count rate. Narrowing the width of the pulse can lead to the reduction of the pulse pile-up distortion, but a low shaping time reduces the noise filtration and results in a poor energy resolution. In practical applications, the shaping parameter values can be changed neatly to meet different shaping needs without adjustment of the hardware. The study provides a new method for achieving the digital Gaussian shaping of the nuclear pulse signal. References E Gatti , A Geraci , G Ripamonti . Automatic synthesis of optimum filters with arbitrary constraints and noises: a new method . Nucl Instrum Methods Phys Res Sec A , 1996 , 381 : 117 - 127 . DOI: 10.1016/0168-9002(96)00653-5 http://doi.org/10.1016/0168-9002(96)00653-5 Automatic synthesis of optimum filters with arbitrary constraints and noises: a new method Emilio Gatti , Giancarlo Ripamonti , Sergio Rescia . A new class of weighting functions: energy resolution improvement from an input shunt inductor . Nucl Instrum Methods Phys Res Sec A , 1987 , 260 : 171 - 186 . DOI: 10.1016/0168-9002(87)90401-3 http://doi.org/10.1016/0168-9002(87)90401-3 A new class of weighting functions: energy resolution improvement from an input shunt inductor E Gatti , M Sampietro , P F Manfredi . Optimum filters for detector charge measurements in presence of 1/f noise . Nucl Instrum Methods Phys Res Sec A , 1990 , 287 : 513 - 520 . DOI: 10.1016/0168-9002(90)91571-R http://doi.org/10.1016/0168-9002(90)91571-R Optimum filters for detector charge measurements in presence of 1/f noise Giancarlo Ripamonti , Andrea Castoldi , Emilio Gatti . Multiple delay line shaping: a new class of weighting functions suitable for digital signal processing . Nucl Instrum Methods Phys Res Sec A , 1994 , 340 : 584 - 593 . DOI: 10.1016/0168-9002(94)90142-2 http://doi.org/10.1016/0168-9002(94)90142-2 Multiple delay line shaping: a new class of weighting functions suitable for digital signal processing Valentin T Jordanov , F Knoll Glenn . Digital synthesis of pulse shapes in real time for high resolution radiation spectroscopy . Nucl Instrum Methods Phys Res Sec A , 1994 , 345 : 337 - 345 . DOI: 10.1016/0168-9002(94)91011-1 http://doi.org/10.1016/0168-9002(94)91011-1 Digital synthesis of pulse shapes in real time for high resolution radiation spectroscopy Valentin T Jordanov , F Knoll Glenn , Alan C Huber , et al . Digital techniques for real-time pulse shaping in radiation measurements . Nucl Instrum Methods Phys Res Sec A , 1994 , 353 : 261 - 264 . DOI: 10.1016/0168-9002(94)91652-7 http://doi.org/10.1016/0168-9002(94)91652-7 Digital techniques for real-time pulse shaping in radiation measurements Angelo Geraci , Emilio Gatti . Optimum filters for charge measurements in the presence of 1/f current noise . Nucl Instrum Methods Phys Res Sec A , 1995 , 361 : 277 - 289 . DOI: 10.1016/0168-9002(95)00214-6 http://doi.org/10.1016/0168-9002(95)00214-6 Optimum filters for charge measurements in the presence of 1/f current noise J Wang , T M Fan , Y G Qian . Nuclear electronics . Beijing (China) : Atomic Energy Press , 1983 , 135 - 165 . (in Chinese) Nuclear electronics E Gatti , A Geraci , G Ripamonti . Optimum filter for l/f current noise smoothed-to-white at low frequency . Nucl Instrum Methods Phys Res Sec A , 1997 , 394 : 268 - 270 . DOI: 10.1016/S0168-9002(97)00593-7 http://doi.org/10.1016/S0168-9002(97)00593-7 Optimum filter for l/f current noise smoothed-to-white at low frequency P O’Connor , G Gramegna , P Rehak , et al . Ultra Low Noise CMOS preamplifier-shaper for X-ray spectroscopy . Nucl Instrum Methods Phys Res Sec A , 1998 , 409 : 315 - 321 . DOI: 10.1016/S0168-9002(97)01289-8 http://doi.org/10.1016/S0168-9002(97)01289-8 Ultra Low Noise CMOS preamplifier-shaper for X-ray spectroscopy Siro Buzzetti , Chiara Guazzoni , Antonio Longoni . EROIC: a BiCMOS pseudo-gaussian shaping amplifier for high-resolution X-ray spectroscopy . Nucl Instrum Methods Phys Res Sec A , 2003 , 512 : 150 - 156 . DOI: 10.1016/S0168-9002(03)01889-8 http://doi.org/10.1016/S0168-9002(03)01889-8 EROIC: a BiCMOS pseudo-gaussian shaping amplifier for high-resolution X-ray spectroscopy N Menaa , P D’Agostino , B Zakrzewski , et al . Evaluation of real-time digital pulse shapers with various HPGe and silicon radiation detectors . Nucl Instrum Methods Phys Res Sec A , 2011 , 652 : 512 - 515 . DOI: 10.1016/j.nima.2010.08.095 http://doi.org/10.1016/j.nima.2010.08.095 Evaluation of real-time digital pulse shapers with various HPGe and silicon radiation detectors Shiguo Chen , Shiyin Ji , Wansong Liu , et al . Gaussian pulse shaping of exponential decay signal based on wavelet analysis . Acta Phys Sinica , 2008 , 57 : 2882 - 2887 . (in Chinese) DOI: 10.3321/j.issn:1000-3290.2008.05.041 http://doi.org/10.3321/j.issn:1000-3290.2008.05.041 Gaussian pulse shaping of exponential decay signal based on wavelet analysis Shiguo Chen , Shiyin Ji , Wansong Liu , et al . Recursive implementation of Gaussian pulse shaping based on wavelet analysis . Acta Physica Sinica , 2009 , 58 : 3041 - 3046 . (in Chinese) DOI: 10.3321/j.issn:1000-3290.2009.05.027 http://doi.org/10.3321/j.issn:1000-3290.2009.05.027 Recursive implementation of Gaussian pulse shaping based on wavelet analysis Valentin T Jordanov . Real time digital pulse shaper with variable weighting function . Nucl Instrum Methods Phys Res Sec A , 2003 , 505 : 347 - 351 . DOI: 10.1016/S0168-9002(03)01094-5 http://doi.org/10.1016/S0168-9002(03)01094-5 Real time digital pulse shaper with variable weighting function João M Cardoso , Simões J Basilio , M B A Correia Carlos . Dead-time analysis of digital spectrometers . Nucl Instrum Methods Phys Res Sec A , 2004 , 522 : 487 - 494 . DOI: 10.1016/j.nima.2003.11.389 http://doi.org/10.1016/j.nima.2003.11.389 Dead-time analysis of digital spectrometers Gaetano Gerardi , Leonardo Abbene , Angelo La Manna , et al . Digital filtering and analysis for a semiconductor X-ray detector data acquisition . Nucl Instrum Methods Phys Res Sec A , 2007 , 571 : 378 - 380 . DOI: 10.1016/j.nima.2006.10.113 http://doi.org/10.1016/j.nima.2006.10.113 Digital filtering and analysis for a semiconductor X-ray detector data acquisition Valentin T Jordanov . Exponential signal synthesis in digital pulse processing . Nucl Instrum Methods Phys Res Sec A , 2012 , 670 : 18 - 24 . DOI: 10.1016/j.nima.2011.12.042 http://doi.org/10.1016/j.nima.2011.12.042 Exponential signal synthesis in digital pulse processing Jianbin Zhou , Wei Zhou , Jiarong Lei , et al . Study of time-domain digital pulse shaping algorithms for nuclear signals . Nucl Sci Tech , 2012 , 23 : 150 - 155 . Study of time-domain digital pulse shaping algorithms for nuclear signals Juan Lanchares , Garnica Oscar , JoséL Risco-Martín , et al . Real-time evolvable pulse shaper for radiation measurements. Real time digital pulse processing for X-ray and gammaray semiconductor detectors . Nucl Instrum Methods Phys Res Sec A , 2013 , 727 : 73 - 83 . DOI: 10.1016/j.nima.2013.05.164 http://doi.org/10.1016/j.nima.2013.05.164 Real-time evolvable pulse shaper for radiation measurements. Real time digital pulse processing for X-ray and gammaray semiconductor detectors L Abbene , G Gerardi , F Principato . Nucl Instrum Methods Phys Res Sec A , 2013 , 730 : 124 - 128 . DOI: 10.1016/j.nima.2013.04.053 http://doi.org/10.1016/j.nima.2013.04.053 Nucl Instrum Methods Phys Res Sec A 10.13538/j.1001-8042/nst.26.010402 Supported by National High Technology Research and Development Program of China (No. 2012AA061803) and Higher Education and Teaching Reform Project of Chendu University of Technology (No. 13JGY25) 2014-07-20 2014-08-09 2015-02-09 2015-02-20 2021-04-11 2015 Nuclear Science and Techniques 1 26