research papers
Countingloss correction method based on dualexponential impulse shaping
^{a}College of Nuclear Technology and Automation Engineering, Chengdu University of Technology, Chengdu 610059, People's Republic of China
^{*}Correspondence email: wangmingming13@qq.com
Under the condition of high
the phenomenon of nuclear pulse signal pileup using a single exponential impulse shaping method is still very serious, and leads to a severe loss in A real nuclear pulse signal can be expressed as a dualexponential decay function with a certain rising edge. This paper proposes a new dualexponential impulse shaping method and shows its deployment in hardware to test its performance. The signal of a highperformance silicon drift detector under high in an spectrometer is obtained. The result of the experiment shows that the new method can effectively shorten the deadtime caused by nuclear signal pileup and correct the counting rate.1. Introduction
Owing to the deadtime of a measurement system, the relationship between the incidence rate and et al., 2018) are used to correct the analysis system in energy The deadtime caused by nuclear signal pileup is the main reason affecting the accuracy. Pulse shaping methods (CanoOtt et al., 1999; Bolić & Drndarević, 2002; Ehrenberg et al., 1978; Imperiale & Imperiale, 2001; Jordanov, 2016; Jordanov & Knoll, 1994) can be used to correct the For the fast channel method, Huang et al. (2017) studied the singleexponential nuclear signal pileup discrimination system, and Hong et al. (2018) studied the deconvolution unit impulse shaping system – these two systems aimed at solving the problem of pileup deadtime. However, the deadtime correction of the single exponential fast shaping method is still barely satisfactory and therefore its practicality is limited.
is nonlinear under the condition of high radiation levels. The deadtime correction method or the fast and slow dualchannel measurement method (HongThis paper analyzes the dualexponential characteristic of the nuclear signal, and then the transfer function between dualexponential signal and impulse response is derived. The fastshaping method of the impulse response based on dualexponential signals can effectively reduce the deadtime and improve the
accuracy of the measurement system.2. Theory
2.1. Dualexponential impulse shaping theory
In the digital processing of pulses, pulse discrimination systems are used to discriminate the pulses' generation time and count the pulses. As shown in Fig. 1, the input of the transfer function model of the system is an ideal dualexponential pulse (Sun et al., 2017) and of which the output is an ideal impulse response function. The time when the impulse response emerges can be used to locate the generation time of the pulse, and the number of impulse responses can be used to correct the counting rate.
In the system, the input dualexponential pulse can be expressed as
where M and m represent the constants of the slow and fast components, respectively. The impulse response of the system is given as
By using a Z transformation for equations (1) and (2), the transform function of the shaping system is given by
where d_{1} = exp(−Ts/M), d_{2} = exp(−Ts/m) and Ts is the sampling rate period (corresponding to the ADC sampling rate). Then equation (3) can be rewritten as
After inverse Z transformation, the recursive model in the time domain can be obtained,
According to equation (1), pileup pulses are simulated with M = 50Ts and m = 2.5Ts. The simulated dualexponential pulses are illustrated in Fig. 2. Equation (5) is used to obtain the output signal of the discrimination system. It can be seen that the proposed method can identify the pileup pulse.
According to the decomposition principle of a cascade system, equation (3) can be written as a cascade equation, i.e.
where
Module H_{1} can shorten the slow component of the dualexponential and H_{2} can reduce the fast component of the dualexponential. H_{3} is a leading bit system, used to align the moment of pulses generation; H_{4} is an amplification system. The impulse pulse is finally obtained after one clock delay. Equations (7), (8), (9) and (10) can be obtained from the inverse transformation of equation (6),
2.2. Selection of the shaping parameters
2.2.1. Module H_{1}
In the system, module H_{1} shapes the slow component of the dualexponential pulse into an impulse signal to reduce the system deadtime. In Fig. 3(a), a dualexponential pulse is acquired and converted with a highperformance silicon drift detector (fast SDD) and ADC [20 Megasamples per second (MSPS)]. According to the fitting calculation, the decay constants of the slow and fast component are M ≃ 2.5 µs and m ≃ 83 ns, respectively. Therefore, the theoretical values of the parameters d_{1} and d_{2} in equation (5) are d_{1}^{ *} = exp(−1/50) and d_{2}^{ *} = exp(−3/5). Letting d_{1} = exp(−1/10), exp(−1/50) and exp(−1/200), respectively, according to equation (7), the output signals of the intermediate system v_{1} are displayed in Figs. 3(b), 3(c) and 3(d).
It can be seen that there still exists tailing in v_{1} when d_{1} < d_{1}^{ *}, and v_{1} becomes a bipolar signal when d_{1} > d_{1}^{ *}. The long tailing due to the slow component of the dualexponential pulse can be reduced as d_{1} = d_{1}^{ *}.
2.2.2. Module H_{2}
Module H_{2} shapes the fast component of the dualexponential pulse into an impulse signal to reduce the deadtime. H_{3} is only a delay to H_{2}. H_{4} is an amplification module; the performance of the module H_{2} can be obtained by analyzing the output v_{o}. In order to obtain an optimal d_{2}, the output of the impulse pulseshaping system v_{o} should be simulated and discussed. Under the condition d_{1} = d_{1}^{ *}, different parameters of d_{2} are simulated to test the performance of peak pileup identification. The simulation results are illustrated in Figs. 4(b), 4(c) and 4(d).
Fig. 4(b) indicates that there is a reverse impulse sequence in v_{o} when d_{2} > d_{2}^{ *}; and the reverse component can be reduced if d_{2} = d_{2}^{ *}.
Although the output is a synthetic polar impulse sequence when d_{2} < d_{2}^{ *}, the width of the shaped pulse is 250 ns.
The output in Fig. 4(c) is the response of the system with d_{2} = d_{2}^{ *} and the pulse width is 100 ns instead of the theoretical 50 ns. This is because the real nuclear signal is not a strictly dualexponential signal but only a dualexponentiallike pulse signal.
3. correction in the fast channel
The distribution of the intervals between random events is
where λ is the exponential representing the average pulse When the interval time of two consecutive pulses is less than the deadtime t_{pileup}, the fastshaping method cannot discriminate, and the two pulses will be misidentified as one pulse and result in a loss of When the interval time of two consecutive pulses is longer than the deadtime t_{pileup}, the pulse is recorded, and the probability of the measured pulse is given by
Therefore, the correction formula between the real pulse R_{real} and the measured pulse R_{measure} is
Equation (13) is an implicit equation and has two solutions. The slow channel cannot directly perform selfcorrection, and it needs to be corrected assisted by the fast channel. In the fast channel, when the is not severe, equation (13) can be used to perform the correction, and the smaller value of the two solutions should be taken as the result of the correction.
4. Experiment
4.1. Experimental conditions and comparative analysis
A data acquiring board embedded with a 20 MSPS 14bit ADC was developed by Sichuan XSTAR Technology (M&C Co. Ltd). An Amptek fastSDD detector was used to acquire the pulse signal; it has a 25 mm^{2} active area, a thickness of 500 µm and a 0.5 mil Be window. Dualexponentiallike pulse sequences with a less than 300 ns rising edge and of 3.2 µs were acquired. Some experimental pulses based on an XRF platform were also obtained from the Sichuan XSTAR.
An Xray tube with Ag target was used to irradiate an MnO_{2} sample; the tube voltage was set to 33.3 kV while the tube current varied from 3.9 µA to 160.8 µA. A vacuum pump was used to evacuate; the vacuum degree was about 0.093 MPa.
Fig. 5(a) shows a section of original pulses data acquired at 160.8 µA. The singleexponential impulse fastshaping method (Huang et al., 2017; Hong et al., 2018) was used to shape the acquired pulses, as shown in Fig. 5(b). The tailing of the fastshaping output narrowbeam signal has an exponential attenuation characteristic, so the singleexponential impulse shaping only eliminates the slow component of the dualexponential pulse while the fast component still exists. The time width of the fast component output signal in Fig. 5(b) is approximately 300 ns; it is close to the rise time of the acquired pulse. Fig. 5(c) shows the processing result of the collected pulses data using the dualexponential impulse shaping method proposed in this paper, and the output is a narrow impulse signal of which the width is about 100–150 ns.
4.2. Results analysis
The calculation results of the . The dualexponential impulse shaping method with parameters d_{1} = exp(−1/50) and d_{2} = exp(−3/5) is used for pulse shaping. R_{f1} and R_{f2} represent the obtained with the dualexponential impulse shaping method and the single exponential impulse shaping method, respectively.
under different currents is shown in Table 1

R_{f1c} and R_{f2c} are the corrected results of R_{f1} and R_{f2}, respectively, and the deadtime t_{pileup} of the dualexponential impulse shaping is 150 ns while that of the singleexponential impulse shaping is 300 ns.
When the Xray tube current is 3.9 µA, the R_{f} = R_{real}. When the other experiment variables are the same, the amount of emitted beam from the Xray tube is proportional to the tube current; then the true R_{cal} is derived from the R_{f} measured at the current of 3.9 µA,
is relatively low, thenwhere k is a constant; when the current is weak, k = R_{f}/I and I represents the Xray tube current.
Fig. 6 shows the relationship between and current. As the current increases, the increases. Also, as the Xray tube current increases, the pileup pulse increases.
It can be seen from Fig. 6 that the of dualexponential impulse shaping is higher than that of single exponential impulse shaping. After the correction, the of dualexponential impulse shaping is closer to the real value. Table 1 shows R_{f1}, R_{f2} and their errors relative to R_{cal}, and Table 2 shows R_{f1c}, R_{f2c} and their relative errors with R_{cal} as the tube current increases.

The relative errors of R_{f1} and R_{f2} gradually increase as the tube current increases. When the current is maximum at 160.8 µA, the pulse incidence rate is 746 kcps theoretically. The pulses identified with the dualexponential impulse shaping method is 622 kcps, and the relative error is −16.67%. The pulse identified with single exponential impulse fast shaping is 541 kcps, and the relative error is −27.49%. The results using the deadtime correction are shown in Table 2.
After deadtime correction, the
acquired using dualexponential impulse shaping is 690 kcps with a relative error of −7.9%, while that acquired using the single exponential impulse fastshaping method is 660 kcps with a relative error of −11.62%. Therefore, the dualexponential impulse shaping algorithm has a better pulse recognition ability.5. Conclusion
This paper proposed a new dualexponential impulse shaping method for pulse shaping in the fast channel. The output signal of the fast SDD is a dualexponentiallike pulse signal with a fast and slow component. The application of the new method can not only eliminate the long tailing of the slow component but also weaken the tailing of the fast component. A comparison between the new method and the single exponential impulse fastshaping method is made. The output pulse width of the fast channel with the new method is about 100–150 ns, while for the single exponential impulse fastshaping method it is approximately 300 ns; the deadtime of the pulse identification can be reduced significantly with the new method. Different Xray tube currents are set: Mn in the high purity sample MnO_{2} is measured for 100 s with the fast SDD, as the current is maximum at 160.8 µA and the loss is most severe. Before the correction, the relative error with regard to the real is −16.67% with the new method, while for the single exponential impulse fastshaping method it is −27.49%; after the loss correction, the relative error with regard to the real is −7.9% with the new method, while for the single exponential impulse fastshaping method it is −11.62%. Thus, the obtained by the proposed dualexponential impulse pulseshaping method is closer to the real As mentioned in the discussion above, the new method proposed in this paper can not only shorten the deadtime but also can be used to effectively correct the under the condition of a high counting rate.
Funding information
The following funding is acknowledged: National Natural Science Foundation of China (grant No. 11975060); Major science and technology projects of Sichuan Province (grant No. 19ZDZX).
References
Bolić, M. & Drndarević, V. (2002). Nucl. Instrum. Methods Phys. Res. A, 482, 761–766. Google Scholar
CanoOtt, D., Tain, J. L., Gadea, A., Rubio, B., Batist, L., Karny, M. & Roeckl, E. (1999). Nucl. Instrum. Methods Phys. Res. A, 430, 488–497. CAS Google Scholar
Ehrenberg, J. E., Ewart, T. E. & Morris, R. D. (1978). J. Acoust. Soc. Am. 63, 1861–1865. CrossRef Web of Science Google Scholar
Hong, X., Zhou, J., Ni, S., Ma, Y., Yao, J., Zhou, W., Liu, Y. & Wang, M. (2018). J. Synchrotron Rad. 25, 505–513. Web of Science CrossRef IUCr Journals Google Scholar
Huang, Y., Gong, H. & Li, J. (2017). J. Tsinghua Univ. (Sci. Technol.), 57, 512–524. Google Scholar
Imperiale, C. & Imperiale, A. (2001). Measurement, 30, 49–73. Web of Science CrossRef Google Scholar
Jordanov, V. T. (2016). Nucl. Instrum. Methods Phys. Res. A, 805, 63–71. Web of Science CrossRef CAS Google Scholar
Jordanov, V. T. & Knoll, G. F. (1994). Nucl. Instrum. Methods Phys. Res. A, 345, 337–345. CrossRef CAS Web of Science Google Scholar
Sun, C., Rao, K.Y., Guo, J.F., Zhang, H.B., Dong, Y.J. & Wu, J.P. (2017). Nucl. Electron. Detect. Technol. 37, 752–756. Google Scholar
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