research papers
Counting-loss correction for
using unit impulse pulse shapingaCollege of Nuclear Technology and Automation Engineering, Chengdu University of Technology, No.1 the Third Section East, Er Xianqiao Road, Chengdu, Sichuan 610059, People's Republic of China
*Correspondence e-mail: zjb@cdut.edu.cn
High-precision measurement of X-ray spectra is affected by the statistical fluctuation of the X-ray beam under low-counting-rate conditions. It is also limited by
resulting from the dead-time of the system and pile-up pulse effects, especially in a high-counting-rate environment. In this paper a detection system based on a FAST-SDD detector and a new kind of unit impulse pulse-shaping method is presented, for counting-loss correction in The unit impulse pulse-shaping method is evolved by inverse deviation of the pulse from a reset-type preamplifier and a C-R shaper. It is applied to obtain the true incoming rate of the system based on a general fast–slow channel processing model. The pulses in the fast channel are shaped to unit impulse pulse shape which possesses small width and no undershoot. The in the fast channel is corrected by evaluating the dead-time of the fast channel before it is used to correct the in the slow channel.Keywords: counting loss correction; X-ray spectroscopy; unit impulse pulse shaping; FAST-SDD detector.
1. Introduction
There are many sources of dead-time in radiation measurement systems, including amplifier overload, pile-up rejection, analog-to-digital converter (ADC) and multi-channel analyzer (MCA) store-to-memory (Twomey et al., 1991). Dead-time correction methods are extensively used to correct the due to dead-time to obtain the true incoming rate. In analog spectroscopy systems, the conversion time of an ADC is regarded as a major source of dead-time and it is often compensated in the counting-loss correction (Roscoe & Furr, 1977; Lindstrom & Fleming, 1995). due to pile-up pulse effects is also analyzed and corrected (Bolotin et al., 1970).
With the development of digital radiation spectroscopy, the high-speed ADC has reduced the DSP code, and 137Cs spectra were obtained at rates of 5.9 × 104 counts s−1 (CPS) without dead-time (Odell et al., 1999). Moreover, Cardoso et al. (2004) analyzed the dead-time in a digital spectrometer through a computational simulation tool, considering the input pulse rate, the existence of pile-up pulse effects and the complexity of algorithms for pulse processing as the primary dead-time sources. In addition, high-speed ADCs and trapezoidal pulse-shaping algorithms are used by digital pulse processors, such as PX4 and DP5, to eliminate the dead-time associated with peak acquisition and digitization. It is believed that the dead-time in digital spectroscopy is related to pulse shaping (Amptek; https://amptek.com/).
due to long conversion time. Also, the application of field programmable gate arrays (FPGAs) and digital signal processing (DSP) improves the digital pulse processing speed, reducing the coming from analog circuits. Some experiments regarding dead-time in digital spectroscopy systems have already been discussed. For instance, algorithms for pulse shaping and processing were implemented in theIn order to create high-precision and high-rate spectra, a general fast–slow channel processing model is established. The slow channel has long shaping time and is used for high-resolution spectra while the fast channel has a short shaping time used for counting-loss correction in the slow channel. Abbene & Gerardi evaluated the dead-time in the fast and slow channel according to the classical dead-time models, paralyzable model and non-paralyzable model (Abbene & Gerardi, 2015; Knoll, 2000). For the low-counting-rate condition, Boromiza et al. (2017) made use of the principle followed by the true and used the detected instead of the true to calculate the correction factor.
In this paper, depending on the fast–slow processing model, the pulses in the fast channel are handled by the unit impulse pulse-shaping method. The true incoming rate is obtained by the corrected
in the fast channel. Also, the pulses in the slow channel are shaped to a trapezoidal pulse with long shaping time.2. Detection system
A structure chart of the detection system is shown in Fig. 1, including a FAST-SDD detector, digital pulse processor (DPP) and PC software. The FAST-SDD detector (XR-100SDD) is made by Amptek and equipped with a reset-type preamplifier. The energy resolution (full width at half-maximum) of the detector is 125 eV at 5.89 keV peak. The DPP is designed to complete four major tasks: (i) extract the detector output and amplify its amplitude; (ii) convert the analog pulse to a digital pulse for pulse shaping; (iii) shape the sampled pulse with a fast–slow channel processing model; (iv) complete amplitude analysis creating energy spectra and send the spectra data to PC software. The front-end circuit, ADC, FPGA and micro-controller unit (MCU) are integrated to complete the four tasks, respectively.
The front-end circuit, as shown in Fig. 2, is composed of a one-stage C-R shaper and three-stage linear amplifier circuit. The C-R shaper can shape the detector output to an pulse shape and the shaping time constant is 3.2 µs. The amplitude and offset of the pulse are adjustable in the three-stage linear amplifier circuit. Then, the amplified pulse is sampled by the following ADC which is operated at 20 Msps (megasamples per second) with 12-bit resolution. The fast–slow channel processing model is realized in the FPGA (Xilinx, XC3S400). The digitized pulse is processed in the fast and slow channel in parallel. Peak detection and counting-loss correction are also accomplished in the FPGA. An STM32F103VET6 chip, which has two serial peripheral interfaces (SPIs) and one CAN interface, is selected as the MCU. The output of the FPGA is transmitted to the MCU by one of the SPI buses, and communication between the MCU and PC software is established by the CAN interface. In addition, the other SPI bus of the MCU is connected to a digital-to-analog convertor (DAC) for amplitude and offset adjustment in the front-end circuit.
In the DPP, the pulse in the slow channel is shaped to a trapezoidal pulse shape, and the pile-up pulse identification technique is also applied to improve the et al., 2015). In order to obtain the true incoming rate, the pulse in the fast channel is usually shaped to a small width by triangle pulse shaping or single delay line (SDL) shaping (Abbene & Gerardi, 2015; Abbene et al., 2010). Fig. 3(a) shows the acquired pulses by ADC. Triangle pulse shaping with 800 ns width and SDL shaping, whose delay time is one clock cycle, are applied to shape the acquired pulses. It can be seen that the shaped pulse after triangle pulse shaping has a relative large width. There are pile-up pulses in the shaped pulses. Although the smaller shaping time can reduce the pulse width, the reduced rising time of the triangle pulse could deteriorate the signal-to-noise ratio. SDL shaping can separate the pile-up pulses in the triangle pulse shaping; however, the shaped pulses possess undershoot.
in the slow channel (Zhou3. Unit impulse pulse shaping
SDL shaping is accomplished by subtracting from the original pulse its delayed and attenuated fraction, and it is easily realized in FPGAs. The width of the shaped pulse is equal to the sum of the delay time and peaking time of the input pulse. In our detection system, the input pulse of SDL shaping is an
shape, thus there exists undershoot in the shaped pulse. The depth of the undershoot is related to the rising time and falling time of the input pulse.The
pulse can be described bywhere A represents the amplitude of the pulse, and τ and θ are the falling part and rising part, respectively.
Pulses with different values of τ/θ are simulated and the shaped pulses with SDL shaping are illustrated in Fig. 4. It can be seen that the undershoot depth over pulse amplitude ratio (U/A) decreases with increasing τ/θ. The undershoot could be eliminated if τ/θ approaches infinity. It is clear that a step pulse can be taken as an pulse with infinite falling time. To be precise, the FAST-SDD detector output is step-shaped.
The processing flow of the detector output is shown in Fig. 5. The unit impulse pulse is generated in the detector after the incident ray is deposited. It is converted to a step pulse after the reset-type preamplifier, and the step pulse is shaped to an pulse by the followed C-R shaper. Therefore, only by the reverse process with the C-R shaper and reset-type preamplifier can the unit impulse pulse be obtained. Signal recovery from a C-R shaper and amplifier by deconvolution has been discussed elsewhere (Jordanov, 1994, 2016). In this paper, a realizable algorithm in the FPGA is used to obtain the unit impulse pulse.
3.1. Output of the reset-type preamplifier
The C-R shaper is shown in Fig. 6. According to Kirchhoff's current law, which states that the sum of the current into a junction is equal to the sum of the current out of the junction, the current transmission equation of the circuit can be written as
Numerical differentiation can stand for differential operation in equation (2) as the time increment dt does not approach zero. The differential operation in equation (2) can be written as
where Vout is the ADC output, Vin is the preamplifier output and Ts is the ADC sampling time.
By substituting equation (3) into equation (2), the recursive relation between Vin and Vout is obtained,
where K = Ts/(RC). RC is called the shaping time constant and equals 3.2 µs. Assuming that the three-stage linear amplifier circuit only amplifies the pulse amplitude and does not change the shape of the input pulse, the preamplifier output can be recovered by equation (4). Fig. 7(a) shows the digital realization of equation (4) in the FPGA. The sampled pulse is delayed by one clock cycle in the register and is then subtracted from the prompt signal which is amplified (1 + K) times. The preamplifier output is the addition of the obtained signal and the one-clock-cycle delay.
Fig. 8(a) shows the result of the recovered preamplifier output from a sampled pulse with K = 0.015625. It indicates that the preamplifier output, which is a step shape, can be recovered by using equation (4).
3.2. Unit impulse pulse shaping
The response of the incident ray in the detector can be expressed by the unit impulse function δ(t), and the step function u(t) is used to represent the preamplifier output. The unit impulse function has a differential relation with the step function, as δ(t) = du(t)/dt, and the relationship can be described by the following equation in a discrete-time domain,
Equation (5) indicates that the unit impulse pulse can be recovered by subtracting the preamplifier output from its one-clock cycle delay. The digital realization of equation (5) is shown in Fig. 7(b). The recovered unit impulse pulse from the step pulse in Fig. 8(a) is pictured in Fig. 8(b).
A conclusion can be drawn that the unit impulse signal in the detector can be obtained from the ADC output in two steps, as equations (4) and (5) stated. Fig. 9 shows the unit impulse pulse-shaping results for the acquired pulses in Fig. 3(a). It can be seen that the undershoot due to SDL shaping is eliminated. Not only is the shaped pulse amplitude unattenuated by the unit impulse pulse shaping but also the pulse width remains unchanged.
4. Experimental tests
4.1. Dead-time correction in the fast channel
Although the unit impulse pulse has a short width, which is also equal to the sum of the original pulse delay time and its peaking time, the shaped pulse can also be lost due to the dead-time in the fast channel. The ).
in the fast channel should be corrected before application. The paralyzable dead-time model and non-paralyzable dead-time models are widely used for dead-time correction (Knoll, 2000The time duration T from starting to process one pulse to being capable of processing another pulse is defined as the dead-time of the radiation system. In the paralyzable model, each arrival event produces a time duration T and any new arrival event during T will extend T. In contrast, the arrival event does not extent T in the non-paralyzable model. The dead-time models are written as
where m represents the detected and n is the true incoming rate. These two models describe the behavior of an idealized system; a real system often displays a behavior that is intermediate. The dead-time model in the designed system is paralyzable.
It can be seen from equation (6) that n cannot be solved explicitly in the paralyzable model. The parameters m and T must be achieved primarily. Abbene & Gerardi (2015 established the relation, as shown in equation (7), between the detected and the X-ray tube current to estimate the dead-time of the fast channel,
where A is a constant, Rfast, Tfast represent the and dead-time in the fast channel, respectively, and I is the X-ray tube current. Equation (7) is similar to the paralyzable model in equation (6). The product of A and I stands for the true incoming rate as the number of the generated pulses is proportional to the X-ray tube current with the X-ray tube voltage being constant.
The experiments were carried out by adjusting the X-ray tube current from 3.9 µA to 70.6 µA while the voltage remained constant. The OriginPro2015 tool was used to construct user-defined relations which conformed to the paralyzable model. The in the fast channel and its corresponding current is plotted in Fig. 10. Rfast accords with a positive exponent and the fitting parameter R2 = 0.9999. The dead-time of the fast channel can be calculated as Tfast = 482 ns. The availability of the estimated dead-time can also be confirmed in Fig. 8(b) where the shaped pulse width is about 500 ns.
For a low nT << 1), the paralyzable model can be written approximately as the non-paralyzable model, as
and small dead-time (Defining the correction factor D = n/m, equation (8) can be written as
Then, the true incoming rate Rtrue can be obtained from
4.2. Counting-loss correction in the slow channel
The ratio of the true incoming rate Rtrue and the from the slow channel Rslow is used to correct the in the slow channel,
where Rmeai is the measured in the i channel and Ri is the corrected counting rate.
The X-ray tube (the rated voltage and current are 50 kV and 1 mA) is operated at 49 kV and the current is adjusted, varying from 39.2 µA to 784.3 µA. The emitting beams are filtered by a combined filter (0.5 mm Ag + 0.5 mm Ag + 0.2 mm Cu) before irradiating a manganese (Mn) sample. The measured counting rates from the fast channel and slow channel are plotted in Fig. 11. Each of the data points is the mean of five measurements. Fig. 11 demonstrates that both the corrected Rfast * and Rfast have good linear relation with the current. However, the measured from the slow channel Rslow tends to decrease because of the increase in the true incoming rate.
Table 1 lists the values of Rfast and Rfast * and their relative error with the calculated Rcal. Rcal is obtained according to the proportional relation between the current and the true incoming rate. When the current is 39.2 µA, there exists few pile-up pulses and Rfast can be considered as the true incoming rate. Table 1 shows that both Rfast and Rfast * become closed to Rcal when the is less than 36.8 kCPS. Therefore, both Rfast and Rfast * can be applied for the counting-loss correction in the slow channel when the true incoming rate is low (less than 36.8 kCPS).
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The filter is removed to study the relation between Rfast, Rfast * and the current under high-rate conditions. The recorded values are shown in Fig. 12. In this experiment, with current increases at the initial stage, the from both the fast channel and slow channel increases. The rate of change of the is decreased because more and more pile-up pulses are rejected, especially in the slow channel. The from the slow channel is significantly less than the from the fast channel as the current remains increasing. When the current is in excess of 27.5 µA, where Rfast * = 259.8 kCPS, Rslow is in decline. In this situation, a large amount of pile-up pulses are abandoned in the slow channel.
The values of Rfast and Rfast * and their relative error with Rcal are listed in Table 2. It can be seen that the relative error between Rslow and Rcal increases with the current. It is more than 70% when the current is 27.5 µA. In this case, the calculated counting-rate relative errors with Rfast * and Rfast are −2.25% and −13.13%, respectively. Rfast * should be applied for the counting-loss correction in the slow channel instead of Rfast.
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The corrected spectra corresponding to the currents 27.5 µA, 39.2 µA, 51.0 µA and 62.7 µA are displayed in Fig. 13. It can be seen that the in the high-energy region increases with the current. However, the of the 5.89 keV peak decreases with the current. The spectra in one figure are depicted in Fig. 14. It is shown that the peak position shifts to high energy and the energy resolution is reduced when the current increases. All of these changes in the spectra are caused by the pile-up pulse effects which could enlarge the amplitude of an individual pulse. In conclusion, Rfast * can be used to correct the in the slow channel under the condition that the is within a 265.8 kCPS limit where nT = 0.13.
5. Conclusions
A new kind of unit impulse pulse-shaping method is proposed for pulse shaping in the fast channel. The method is derived based on the reversed analysis of the detector output processing flow, which consists of a reset-type preamplifier and a C-R shaper. The amplitude and width of the shaped pulse are no less than that of the pulse processed with SDL shaping. The presented method can also eliminate undershoot which exists in SDL output.
The dead-time in the fast channel is estimated by measuring the Rfast is corrected by the estimated dead-time before being used for correction in the slow channel. For the condition where the incoming rate is less than 36.8 kCPS, both Rfast and Rfast * have small relative error with Rcal. The quantity of pile-up pulses in the fast channel increases with the X-ray tube current, inducing a increase in the high-energy region, peak shift and energy resolution deterioration. Rfast * can be used to correct the in the slow channel when the incoming rate reaches up to 265.8 kCPS.
with different X-ray tube currents.Funding information
The following funding is acknowledged: General Program of National Natural Science Foundation of China (grant No. 11475036 to Yingjie Ma); National Key Research and Development Program of China (grant No. 2016YFC1402505 to Jianbin Zhou).
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