research papers
Highrate deadtime corrections in a general purpose digital pulse processing system
^{a}Dipartimento di Fisica e Chimica, University of Palermo, Viale delle Scienze, Edificio 18, Palermo 90128, Italy
^{*}Correspondence email: leonardo.abbene@unipa.it
Deadtime losses are well recognized and studied drawbacks in counting and spectroscopic systems. In this work the abilities on deadtime correction of a realtime digital pulse processing (DPP) system for highrate highresolution radiation measurements are presented. The DPP system, through a fast and slow analysis of the output waveform from radiation detectors, is able to perform multiparameter analysis (arrival time, pulse width, pulse height, pulse shape, etc.) at high input counting rates (ICRs), allowing accurate corrections even for variable or transient radiations. The fast analysis is used to obtain both the ICR and energy spectra with high throughput, while the slow analysis is used to obtain highresolution energy spectra. A complete characterization of the counting capabilities, through both theoretical and experimental approaches, was performed. The deadtime modeling, the throughput curves, the experimental timeinterval distributions (TIDs) and the counting uncertainty of the recorded events of both the fast and the slow channels, measured with a planar CdTe (cadmium telluride) detector, will be presented. The throughput formula of a series of two types of deadtimes is also derived. The results of deadtime corrections, performed through different methods, will be reported and discussed, pointing out the error on ICR estimation and the simplicity of the procedure. Accurate ICR estimations (nonlinearity < 0.5%) were performed by using the time widths and the TIDs (using 10 ns time bin width) of the detected pulses up to 2.2 Mcps. The digital system allows, after a simple parameter setting, different and sophisticated procedures for deadtime correction, traditionally implemented in complex/dedicated systems and timeconsuming setups.
1. Introduction
Quantitative analysis in Xray and γray experiments requires accurate and precise estimation of input rate (ICR or ρ) and photon energy, even at high conditions. High ICR environments are typical of synchrotron applications, medical Xray imaging, industrial imaging and security screening, and instrumentation with good counting and energyresolving capabilities (ERPC: energyresolved systems) is considered desirable (Fredenberg et al., 2010; Kraft et al., 2009; Iwanczyk et al., 2009; Szeles et al., 2008; Taguchi & Iwanczyk, 2013). At high ICRs, counting distortions, degradation of energy resolution and changes in energy calibration start to appear. Deadtime losses, pileup (tail and peak pileup) and baseline shifts (mainly due to thermal drifts, poor polezero cancellation and AC couplings) are the major drawbacks at high ICR environments and, therefore, highperformance spectrometers must be characterized by a well defined deadtime modeling, pileup rejection (PUR) and baseline restoration (BLR) (Gilmore, 2008; Knoll, 2000; ICRU, 1994; Laundy & Collins, 2003).
Concerning the counting process, the deadtime (DT or τ) of the systems is the major drawback, producing both count losses and distortions of the counting statistics. Generally, when the arrival of events is random in time (e.g. for Xrays from tubes, from radioactive decays and from synchrotron sources with a flat fill time structure) (Bateman, 2000), deadtimes are classified into two main categories: (i) nonparalyzable deadtime (also known as nonextendable, noncumulative or type I) (Yu & Fessler, 2000) and (ii) paralyzable deadtime (also known as extendable, cumulative or type II) (Yu & Fessler, 2000). Nonparalyzable deadtime is produced at each time an event is recorded and any arrival event from the recorded time to the τ period will not be recorded. In the paralyzable model, each arrival event, whether recorded or not, produces a deadtime τ and any new arrival event with a delay less than τ from the previous arrival event extends the deadtime and will not be recorded. This model results in paralysis, i.e. an increasing ICR will result in a lower measured output (OCR or R). A third model (also known as type III) (Yu & Fessler, 2000) can be defined when a PUR is used (this technique is generally used to mitigate pileup distortions in radiation measurements). The model of the deadtime of type III is similar to that of type II but the onset of paralysis is `twice as fast', since if two events arrive within τ of each other neither event will be recorded. The transmission/throughput functions (i.e. the relation among OCR, ICR and DT) of these deadtimes have been studied and widely presented in the literature (Arkani et al., 2013; DeLotto et al., 1964; Carloni et al., 1970; Pommé et al., 1999, Pommé, 1999; Yu & Fessler, 2000; Bateman, 2000).
Deadtime also affects the counting statistics, even if the original process can be described by a simple ; Muller, 1967, 1971, 1972; Pommé, 1999, 2008) and experimental (Arkani & Raisali, 2015; Denecke & de Jonge, 1998; Hashimoto et al., 1996; Pommé et al., 1999) works, the recorded counts of a counting system with deadtime can be characterized by a nonPoisson counting uncertainty and by timeinterval distributions (TIDs) different from the typical exponential shape.
As shown well in both theoretical (Choi, 2009Of course, deadtime distortions strongly depend on the ρτ product, and small values are considered desirable (ρτ << 1). Therefore, at high ICRs, the counting systems should be characterized by small deadtime values to minimize the distortions and simplify the corrections.
conditions, generally related to theGenerally, to control the length and type of deadtime, a well defined deadtime is imposed on every event counted. Deadtime of type I or type II, greater than the deadtime of the counting chain, is typically imposed on the recorded counts. However, this approach fails at high ICRs, first, because long deadtimes strongly reduce the throughput of the system and, second, it does not take into account additional counting losses due to pileup. The presence of pileup requires a more complex analysis of the deadtime losses, often modeled as the series arrangement of two deadtimes (Choi, 2009; DeLotto et al., 1964; Muller, 1972; Pommé, 2008).
Deadtime corrections can be divided into two main categories (Pommé, 2008; Michotte & Nonis, 2009; Redus et al., 2008): (i) the livetime mode and (ii) the realtime mode. Livetime correction is hardware implemented. Livetime is incremented by counting a timed pulse train of known frequency only in the time intervals when the system is free to record the events. The realtime mode, offline software implemented, is based on the knowledge of the throughput formula and the deadtime value (i.e. by applying the inversion of the throughput formula).
Differential methods for spectral counting correction (classified as livetime modes) have been proposed and used to also investigate variable and transient radiations (lossfree counting and zero deadtime methods) (Westphal, 2008; Upp et al., 2001). These methods are based on the concept of adding N counts, rather than simply a single count, to a pulse height channel whenever an event was stored (N should equal 1 plus a weighting factor representing the estimated number of events that were lost since the last event was stored).
In this work, we will present the abilities on deadtime correction, investigated through both theoretical and experimental approaches, of a realtime digital pulse processing (DPP) system, recently developed by our group, for highrate highresolution radiation measurements. Currently, several spectroscopic systems are developed by using DPP techniques (Arkani et al., 2014; Arnold et al., 2006; Bolić & Drndarević, 2002; Cardoso et al., 2004; Gerardi et al., 2007; Dambacher et al., 2011; Meyer et al., 2001; Nakhostin & Veeramani, 2012; Papp & Maxwell, 2010), where the detector output signals, i.e. the output signals from chargesensitive preamplifiers (CSPs), are directly fed into fast digitizers and then processed by using digital algorithms. As widely recognized, the digital approach gives many benefits against the analog one, among which: (i) the possibility to implement custom filters and procedures, which are challenging to realise in the analog approach, (ii) stability and reproducibility (insensitivity to pickup noise as soon as the signals are digitized) and (iii) the possibility to perform multiparameter analysis for detector performance enhancements and new applications. Concerning the deadtime, DPP systems are free from the deadtime due to the A/D conversion and data storage time of the traditional multichannel analyzers (MCAs). Moreover, by employing parallel or pipelined procedures, treatment deadtime (the deadtime that can arise when the online algorithms are applied to treat the incoming data) can be eliminated. Generally, deadtime in DPP systems is mainly due to the digital pulse shaping, allowing simple deadtime modeling and the possibility to obtain low ρτ values even at high ICRs.
Our system, based on an innovative processing architecture, is able to perform an accurate estimation of the true ICR, a fine pulse height (energy) and shape (peaking time) analysis even at high ICRs. Through two pipelined shaping branches (fast and slow channels), the system is able to minimize and correct the typical high rate distortions (deadtime distortions, pileup and baseline shifts) in radiation measurements and, due to the pipelined analysis, no treatment deadtime is introduced. Generally, the fast channel is used to obtain the ICR and energy spectra with high throughput, while the slow channel is used to obtain energy spectra with high energy resolution. The event/pulse data from both channels (arrival time, pulse height, pulse width, peaking time), provided in listing mode, together with some housekeeping data (the starting time of the packed data acquisition, the sum of the time widths of the fast shaped pulses, the number of both fast and slow detected pulses, etc.), allow the correction of transmission deadtime and corrections even for variable or transient radiations.
The deadtime modeling, the throughput curves, the experimental TIDs and the counting uncertainty of the recorded events of both the fast and the slow channels, measured with a planar CdTe (cadmium telluride) detector, will be presented. The results of deadtime corrections, performed by different methods, will be also reported and discussed, pointing out the error on ICR estimation, the simplicity of the procedure and the easy implementation in a realtime mode.
The counting capabilities together with the pulse shape and height abilities, presented in our previous works (Abbene et al., 2013a,b; Gerardi & Abbene, 2014), will give a complete overview of our digital strategy on the development of highrate highresolution radiation systems.
2. DPP system
In this section, we will present a brief description of our DPP system. A detailed description of the system is reported in our previous work (Gerardi et al., 2014). The DPP system consists of a digitizer and a PC, where the user can control all digitizing functions, the acquisition and the analysis. The pulse processing analysis is performed by using a custom DPP firmware, developed by our group and uploaded to the digitizer. We used a commercial digitizer (DT5724, CAEN SpA, Italy) (https://www.caentechnologies.com), housing four highspeed ADCs (16bit, 100 MS/s), four buffers of external memory (8 MByte wide each) and four channel FPGAs (ALTERA Cyclone EP1C20). Each ADC channel, AC coupled, is characterized by three fullscale ranges (±1.125 V, ±0.5625 V and ±0.2813 V). The digital pulse processing is carried out by the channel FPGAs, in which our DPP method is implemented (DPP firmware). Each channel FPGA packs output data and sends them to another FPGA (ROC FPGA) that collects asynchronously the packets from all four channels and transmits them, via USB channel (or via optical link), to the PC. The PC runs a C++ program able to control all digitizer functions, to acquire packed data, to produce online histograms, display and to store all received information in dedicated binary files.
By using a common external clock, N digitizers can be assembled and synchronized to realise a digitizing system with 4×N channels. The acquisition start of each unit can be synchronized using a daisychain cascade, with the starting pulse coming from the master unit. In this way, the timing of each unit can use the same time base and starts from zero synchronously.
The DPP firmware was developed by our group and successfully used for both offline and online analysis (Abbene et al., 2010, 2012, 2013a,b, 2015; Abbene & Gerardi, 2011; Gerardi & Abbene, 2014). The DPP method is able to perform multiparameter analysis (event arrival time, pulse shape, pulse height, pulse time width, etc.) even at high ICRs. A general overview of the method is presented below (see also Fig. 1). The DPP method is based on two pipelined shaping steps: a fast and a slow shaping. The preamplifier output waveform (CSP output waveform) is shaped by using the classical single delay line (SDL) shaping technique (Knoll, 2000). SDL shaping is obtained by subtracting from the original pulse its delayed (by using a programmable delay time) and attenuated fraction. SDL shaping gives short rectangular output pulses with fast rise and fall times. In fact, the falling edge of the pulse is a delayed mirror image of the leading edge. These features make SDL shaping very appealing for timing and pulse shape and height analysis (PSHA) at both low and high counting rates. Through the fast SDL shaping the following operations are performed: (i) pulse detection and timetag triggering, (ii) time width measurement of the SDLshaped pulses, (iii) fast pulse height analysis (PHA), that provides energy spectra with high throughput, and (iv) pileup rejection for the slow branch. Concerning the pulse detection, the trigger is generated and timestamped through the ARC (amplitude and rise time compensation) timing marker (at the leading edge of the SDL pulses) and its amplitude (25% of the peak value) defines the new amplitude threshold (ARC threshold) for SDL pulse width estimation. The estimation of the ARC cross timing is improved by using a linear interpolation (time resolution < 1 ns). The width of each SDL pulse is calculated from the difference between the times when the leading and the falling edges cross the ARC threshold. Through the fast branch, the system is able to provide, for each detected event, the following results: (i) trigger time stamp, (ii) pulse width, (iii) fast pulse height. Fig. 2 shows the CSP output waveform and the fast shaped pulses, related to Xrays from an Agtarget Xray tube impinging on a CdTe detector with an ICR of 2.2 Mcps (cps = counts s^{−1}) (the experimental setup is described in §4).
The PUR performs a selection of time windows of the CSP waveform for the slow shaping (Fig. 3). Each selected time window of the CSP waveform is termed `Snapshot', while the width of this window, userchosen, is termed `Snapshot Time' (ST). The selection is related to the reference time of each fast SDL pulse (it occurs near the maximum amplitude of the related CSP pulse), i.e. to the time when the falling edge of the SDL pulse crosses the ARC threshold. If two detected fast SDL pulses are within ST/2 of each other, then neither pulse will be selected; i.e. a pulse is accepted if it is not preceded and not followed by another pulse in the ST/2 time window periods. We stress that the PUR only works on the temporal positions of the CSP pulse peaks, i.e. it selects the snapshots before any useful operation for slow shaping. The slow shaping is characterized by two main features: (i) it performs the PSHA on each selected snapshot, and (ii) due to an automatic baseline restoration (based on the analysis on single pulses), it allows high rate measurements. The pulse height analysis (that provides energy spectra with high energy resolution of each PUR selected event) is performed by applying an optimized lowpass filter (e.g. trapezoidal filter) to all the samples of each slow SDLshaped pulse. The energy resolution strongly depends on the ST values; as the shaping time of classic analog systems, long ST values give better energy resolution. Through the slow branch, the system is able to provide, for each selected pulse, the following results: (i) trigger time stamp, (ii) pulse height and (iii) the peaking time. The shape (peaking time) of the pulses and its correlation with the pulse height is very helpful for improving the detector performance. Pulse shape discrimination (PSD) techniques were successfully used, in our previous works (Abbene & Gerardi, 2011; Abbene et al., 2012, 2013a,b, 2015; Gerardi & Abbene, 2014), to minimize incomplete charge collection effects, pileup and charge sharing.
We stress that this PSHA, performed on isolated time windows containing a single CSP pulse, allows a strong reduction of the corruptions that the traditional analysis produces to adjacent pulses (residual tails at the end of shaped pulses), thus minimizing baseline shifts at high ICRs.
The output results from both channels are provided in listing mode, where each list is characterized by a userchosen number of eventsequences (typical fast channel sequence: arrival time, fast energy and pulse width; typical slow channel sequence: arrival time, slow energy and peaking time). Moreover, to perform investigations, with high time resolution, on variable and transient radiations (multiscaling and spectral modes), each data list is tied to some housekeeping data, such as: the starting time of the packed data acquisition, the sum of the time widths of the detected pulses (total detection deadtime), total number of fast shaped pulses, total number of analysed events (after PUR), total number of pileup events, etc. These data, continuously updated, allow the analysis of the time evolution of the total rate to be performed and allow the detection and the measurement of any transmission deadtime. Of course, the time resolution of this analysis depends on the and the chosen number of packed eventsequences. Moreover, the data within each list (i.e. the sequences: arrival time, energy, etc.) allow a finer analysis of the time evolution of the energy spectra (e.g. changes of the rate of some energy lines in the spectrum) and losscounting corrections can be easily performed.
3. Deadtime modeling
In this section, the deadtime modeling, throughput functions, timeinterval distributions and counting uncertainties of the two shaping channels will be presented and discussed.
3.1. Deadtime of the fast channel
As will be shown in the experimental results (§5), the deadtime of the fast channel can be modeled as a single paralyzable deadtime (type II). The pulse detection is performed in the fast channel by looking for the fast SDL output pulses exceeding an amplitude threshold (leading edge detection). Pulses that are large enough to cross this threshold are counted. The width of the fast SDL output pulses at the threshold causes an extending deadtime (type II). If a second pulse arrives while the first pulse is still above the threshold, the second pulse overlays the first, and extends the deadtime by its width from its arrival time. Because the system counts threshold crossings, it will count only the first pulse. If τ_{F} is the fast deadtime, R_{F} the output and ρ the input the throughput function is given by the following relation (Gilmore, 2008; Knoll, 2000),
As is widely reported in classic textbooks on radiation detection (Gilmore, 2008; Knoll, 2000), equation (1) is obtained by calculating the probability of timeintervals, between consecutive events, longer than τ_{F}, i.e. by integrating the exponential timeinterval distribution, typical of a Poisson process, between τ_{F} and ∞. As discussed in the Introduction, deadtime also affects the shape of the TID. The TID of the recorded events after a single deadtime of type II can be described by the following function (Pommé et al., 1999; Pommé, 2008; Muller, 1971),
where is the Heaviside step function. Due to the effect of the deadtime, the TID, described by (2), is represented by a piecewise polynomial function, i.e. characterized by a different shape from the exponential function, typical of a Poisson process.
However, at low ρτ_{F} product values and at long time intervals (if compared with the involved deadtime), the TID tends towards the Poisson exponential shape. In the following, we will present some calculated TIDs (Fig. 4) of a single deadtime of type II [by using equation (2)] and we will discuss the limits of the exponential approximation of the TID. By using equation (2), we calculated the TIDs at three different ρτ_{F} product values of 0.03, 0.3 and 3. The ρτ_{F} values are related to the experimental conditions presented in this work: ρ values from 220 kcps to 2.2 Mcps and a deadtime τ_{F} equal to 138 ns (the estimation of the fast deadtime τ_{F} will be presented in §5). Fig. 4(a) shows the calculated TID by using ρ = 220 kcps (ρτ_{F} = 0.03). The TID is zero between 0 and τ_{F}, constant between τ_{F} and 2τ_{F}, and at time intervals longer than 2τ_{F} can be modeled with an exponential function. Indeed, by performing an exponential fitting, at times >2τ_{F}, we obtained ρ_{FITTING} = 220 kcps, that is equal to the ρ used for the calculus of the TID. Fig. 4(b) shows the TID at 2.2 Mcps (ρτ_{F} = 0.3). Here, the TID follows an exponential shape at times longer than 5τ_{F}. Fig. 4(c) shows two TIDs at 2.2 Mcps but characterized by different ρτ_{F} product values: ρτ_{F} = 0.3 (dashed gray line, with τ_{F} equal to 138 ns) and ρτ_{F} = 3 (solid gray line, with τ_{F} equal to 1.38 µs). The different slope of the two TIDs is clearly evident. In particular, the exponential fitting of the TID with ρτ_{F} = 3 gives ρ_{FITTING} = 1.3 Mcps, even at time intervals > 20τ_{F}. Therefore, at high ρτ_{F} product values, the exponential fitting, despite the good agreement with the calculated TID data, gives an error on the ρ estimation.
Since the deadtime models of type I and II lead to identical results in the limit of small deadtime losses (i.e. small ρτ values), the exponential modeling of the TID of the fast shaped pulses can be compared with the similar behaviour of the nonparalyzable (type I) deadtime, which is characterized by zero value for t ≤ τ_{F} and by an exponential TID for t > τ_{F} (Muller, 1967; Pommé, 1999).
These results justify the following exponential modeling of the TID of the fast shaped pulses for small ρτ_{F} values. At small ρτ_{F} values (ρτ_{F} ≤ 0.03), we will use the following piecewise model for the TID of a single deadtime τ_{F} of type II (fast channel),
The counting uncertainty of the recorded counts are also affected by deadtime. In particular, the relative uncertainty on the recorded counts N_{F} of the fast channel (type II) is given by the following relation (Pommé et al., 1999; Yu & Fessler, 2000),
3.2. Deadtime of the slow channel
The slow channel performs a multiparameter analysis (arrival time, energy and peaking time) on each pulse selected by the PUR, within a time window of the CSP waveform equal to ±ST/2, centered at the peak position. The value of this window, chosen by the user, generally represents the best compromise between the energy resolution and the throughput in the slow energy spectra (the ST acts as the shaping time constant of an analog shaping amplifier). The deadtime of the slow channel should be modeled as a single deadtime of type III, i.e. characterized by the following throughput function (Yu & Fessler, 2000),
where R_{S} is the output and τ_{S} is the slow deadtime equal to ST/2. However, due to the finite width of the pulses of the fast channel, higher throughputs than the values expected from equation (5) would be observed (i.e. a lower deadtime than ST/2) (Pommé et al., 1999, 2008; Michotte & Nonis, 2009; Yu & Fessler, 2000). Indeed, the slow channel should be modeled through the cascade of two paralyzable deadtimes: the first of type II (fast deadtime equal to τ_{F}) and the second of type III (slow deadtime τ_{S} equal to ST/2).
In the following, a simple modeling of the cascade of type II (with ρτ_{F} << 1) and type III will be presented. To our knowledge, the modeling of the cascade of deadtimes of type II with type III has not been presented in the literature. For an exponential TID, the probability that an event can be preceded or followed by another event, within the time interval τ_{S}, is the same. We define P_{LOSS} as the probability that one pulse is rejected by the arriving of a new event, within the interval (0, τ_{S}). The probability that one event is accepted by our PUR, i.e. no event is present in the two time intervals (−τ_{S}, 0) and (0, τ_{S}), is equal to (1 − P_{LOSS})(1 − P_{LOSS}). At low ρτ_{F} product values (ρτ_{F} << 1), P_{LOSS}, by using the TID of equation (3), is given by
Therefore, the output R_{S}^{ *} of the slow channel can be given by
Notice that the same result is obtained, without approximation, if the fast deadtime is of type I.
The relative uncertainty on the recorded counts N_{S} of the slow channel can be written as (Pommé et al., 1999; Yu & Fessler, 2000)
where f is a fraction of the counts in the slow energy spectrum. Equation (8) is derived by the theoretical relation for a single deadtime of type III; we take into account the cascade of deadtime of type II and deadtime of type III by using the cascade corrected total deadtime (i.e. 2τ_{S} − τ_{F}).
4. Experimental procedures
To investigate the counting capabilities of the DPP system, a planar CdTe detector was used (XR100TCdTe, S/N 6012, Amptek, USA) (https://www.amptek.com), with a thickness of 1 mm (absolute efficiency of 64% at 100 keV) and equipped with a resistivefeedback CSP (decay time constant of the resistivefeedback circuit is around 100 µs). The gain of the CSP is 0.82 mV keV^{−1} and the rise time of the CSP output pulses is around 60 ns (59.5 keV Xrays). As is well known, CdTe/CdZnTe detectors (1–2 mm thick) are very appealing for in the energy range 1–100 keV (Auricchio et al., 2011; Del Sordo et al., 2009; Owens, 2006; Takahashi & Watanabe, 2001; Turturici et al., 2014, 2015).
The highrate spectroscopic abilities of the DPP system, connected to the CdTe detector, were investigated in our previous works (Abbene et al., 2013a,b; Gerardi & Abbene, 2014). Table 1 shows the spectroscopic response of the system at low and high rates, in terms of energy resolution (FWHM), at 59.5 keV (^{241}Am source). The electronic noise of the CdTe detector coupled to the DPP system (by using ST = 30 µs) is 0.4 keV (FWHM). The results highlight, beside the excellent highrate spectroscopic abilities, the flexibility of the system to perform measurements for both optimum energy resolution or high throughput.

In this work, we measured the response of the system to an Agtarget Xray tube (Amptek, Inc. USA) with Al (1 mm thick) and Ag (25.4 µm thick) filters. Xray spectra were measured by using a tube voltage of 30 kV and tube current values between 5 µA and 60 µA (ICR up to 2.2 Mcps).
5. Measurements and results
In this section, experimental results on the
capabilities of the system, through the fast and slow channels, are shown. As will be presented in the following subsections, the DPP system is characterized by two main features: (i) the deadtime modeling of both the fast and the slow channel is well defined and (ii) thanks to the low deadtime values of the fast channel, accurate estimation of the true ICR can be performed.5.1. Deadtime correction and counting rates in the fast channel
Fig. 5 shows the measured throughput curve (i.e. the R_{F} versus tube current) of the fast channel. Each experimental point was obtained by evaluating the mean value of R_{F} values of 400 acquisitions (each acquisition consists of 20000 events). The experimental curve is in good agreement with the typical throughput function of the single paralyzable deadtime model [the deadtime model described by equation (1)].
Through a curve fitting with the following function,
where I is the tube current and A is a constant, we estimated, with a confidence level (CL) of about 99.7%, τ_{F} = (138.0 ± 0.6) ns. Therefore, it is possible to estimate the input ρ by applying the realtime deadtime correction, i.e. by solving the throughput equation (1) iteratively. Of course, this method requires the experimental measurement of the throughput curve and therefore the measurement of multiple Xray spectra at different ICRs (multiple measurements). In the following, a different method, able to perform accurate estimation of ρ with a single measurement (i.e. by performing a measurement at a single ICR condition), will be presented.
As discussed in §3 and reported in the literature (Arkani & Raisali, 2015; Denecke & de Jonge, 1998; Pommé et al., 1999), due to the small deadtime τ_{F} of the fast channel (138 ns), the simple exponential fitting of the experimental TID, at time intervals >5τ_{F}, can give an accurate estimation of the input up to 2.2 Mcps (ρτ_{F} = 0.3). Fig. 6 shows the experimental TID, through the fast channel at 2.2 Mcps; the trigger times of the eventdata, histogrammed with a time bin width of 10 ns, were used. The exponential fitting, performed at time intervals >5τ_{F}, gives ρ_{TID} = (2232000 ± 6000) cps (CL = 99.7%). The estimated ρ_{TID} from the measured TIDs versus the tube current is characterized by a very good linear behavior (nonlinearity < 0.5%), as shown in Fig. 7. Moreover, to check the ρ_{TID} values, we also estimated τ_{F} by fitting the throughput curve (R_{F} versus ρ_{TID}; for simplicity, this curve was not reported in the paper) with the single paralyzable function [equation (1)], obtaining a deadtime τ_{F,TID} = (137 ± 0.9) ns (CL = 99.7%), in good agreement with the deadtime (138 ± 0.6 ns) estimated from the experimental throughput curve (i.e. R_{F} versus tube current).
The digital system, through the fast channel, is able to perform the estimation of the true ρ by using several deadtime correction methods. In the following, we summarize all techniques used to estimate the true ρ, pointing out if each method needs a single measurement of multiple measurements:
(i) ρ_{REAL}, obtained through the realtime correction [i.e. by using equation (1)] from the measured throughput curves (multiple measurements);
(ii) ρ_{TID}, estimated from the exponential best fit of the measured TIDs (single measurement);
(iii) ρ_{LIVE}, obtained through the relation N_{F}/(T_{acq} − T_{width}), where N_{F} is the total number of the detected pulses by the fast channel, T_{acq} is the total real acquisition time, while T_{width} is the total detection deadtime, calculated as the sum of the time widths of the fast shaped pulses (single measurement);
(iv) ρ_{TW}, obtained by using a different realtime correction, based on the paralyzable throughput function [equation (1)] and the deadtime τ_{FAST,TW} estimated through the mean value of the time widths of the detected pulses (single measurement). Fig. 8 shows the time width distribution of the fast pulses at ρ = 752 kcps. From the time width data, we obtained a constant τ_{FAST,TW} = (129 ± 10) ns (CL = 99.7%) for all counting rates (from 200 kcps to 2.2 Mcps).
The ρ values (related to ρ_{TID}), estimated through the various correction methods, are shown in Fig. 9. We used ρ_{TID} as the reference input due to the good linearity with the tube current. To better point out the counting corrections, the R_{F} values are also reported in Fig. 9. At 200 kcps, all correction methods are characterized by low errors, <0.8%. By applying the realtime correction, up to counting rates of about 2.2 Mcps, the uncertainty of ρ_{REAL} is <0.6%, while the error on ρ_{TW} is <1.6%. For comparison purposes, the live correction was also reported. As clearly shown in Fig. 9, the error on ρ_{LIVE} (<7.8% at 2.2 Mcps) is greater than for the other correction methods, mainly due to its major sensibility to the pulse pileup (Pommé, 2008; Michotte & Nonis, 2009); moreover, ρ_{LIVE} values are always lower than the expected values as happens in the live time correction.
We calculated the standard deviation of the recorded counts of 400 measurements. Fig. 10 shows the ratio between the measured standard deviation of N_{F} and (N_{F})^{1/2} (i.e. the expected standard deviation in a Poisson process) versus the ρτ_{F} product values. At 2.2 Mcps (i.e. ρτ_{F} = 0.3), the counting uncertainty is clearly less than the value expected from Poisson statistics, with a percentage deviation of about 30%. The experimental points are in agreement with equation (4) and with the values obtained in the literature, in both simulations and experiments (Pommé et al., 1999; Yu & Fessler, 2000) with counting systems characterized by a single paralyzable deadtime. This result points out that the system is able to associate the proper standard deviation on the fast recorded counts.
5.2. Deadtime correction and counting rates in the slow channel
Fig. 11 shows the measured throughput curve (i.e. R_{S} versus tube current) of the slow channel by using ST = 3 µs. Each experimental point was obtained by evaluating the mean value of R_{S} of 400 acquisitions (each acquisition consists of the selected events by the PUR from the 20000 events from the fast channel; the number of the selected events changes with the rate). The experimental curve was fitted with the following equation,
where I is the tube current and B is a constant, giving a total deadtime τ equal to (2.87 ± 0.04) µs (confidence level CL = 99.7%). This value is equal to (2τ_{S} − τ_{F}), clearly pointing out the good agreement between the experimental curve and the throughput function of the cascade of type II and type III deadtimes up to ρτ_{F} = 0.3 [equation (7)].
Fig. 12 shows the measured TIDs, through the slow channel, at 200 kcps, 752 kcps and at 2.2 Mcps, with a time bin width of 100 ns. The shapes of the distributions show an agreement with both simulated and experimental TIDs (single deadtime of type III) in the literature (Pommé et al., 1999). These results show that the low deadtime of the fast channel produces negligible effects in the TIDs of the slow channel. However, due to the high distortions of the slow deadtime, the measured TIDs from the slow channel do not allow accurate ρ estimations through a simple exponential fitting. Therefore, each slow channel highresolution spectrum should be tied to the ρ information provided by the fast channel, characterized by very low deadtime distortions.
In the following, we present some appealing strategies, in terms of both simplicity and accuracy, that can be used to provide the scaling ratio for the spectral counts and its error with a single measurement:
(i) Estimation of ρ through the exponential fitting of the TID from the fast channel and calculation of the scaling ratio K = ρ_{TID}/R_{S}; this is the best strategy in terms of accuracy, but requiring the implementation of a bestfit procedure;
(ii) Estimation of τ_{F} through the mean value of the time width of the fast pulses and calculation of by inversion of the formula (1); the scaling ratio is given by
(iii) Estimation of τ_{F} through the mean value of the time width of the fast pulses; since R_{S} follows the relation
it is possible to estimate the scaling ratio K^{**} through
At 2.2 Mcps, by using the estimated τ_{F,TW} = (129 ± 10) ns, the measured R_{S}, R_{F} and 2τ_{S} = ST = 3 µs, a ρ value (2200000 ± 50000 cps) was obtained, through equation (13), with a percentage deviation of 1.4% from the ρ_{TID} (2232000 ± 6000 cps); therefore, taking into account this maximum error, it is possible to correct the counts in the slow spectra though this simple procedure up to 2.2 Mcps.
Fig. 13 shows the ratio between the measured standard deviation of N_{S} and (N_{S})^{1/2} versus the ρ(2τ_{S} − τ_{F}) product values. We calculated the standard deviation of the recorded counts of 400 measurements. As in the fast channel, the counting uncertainty is different from the value expected from Poisson statistics (maximum percentage deviation of about 15%), but this difference is much smaller than the measured fast channel one. Moreover, the experimental points are in agreement with equation (8) and with the values obtained in the literature, in both simulations and experiments (Pommé et al., 1999) with counting systems characterized by a single pileup rejection (type III).
Of course, a smaller difference with the Poisson counting uncertainty can be gradually obtained when small fractions of the spectral counts (ROI) are considered (Pommé et al., 1999).
6. Conclusions
The high rate abilities of a realtime DPP system on deadtime correction are presented. The system through a fast and a slow channel is able to provide counting and energy spectra at different resolution and throughput conditions. The results of Xray spectra measurements (up to 2.2 Mcps) highlight two main features of the DPP system: (i) the deadtime modeling of both the fast and the slow branch is well defined: a single deadtime of type II for the fast channel and a cascade of deadtime of type II and type III for the slow channel; and (ii) thanks to the low deadtime values of the fast channel, low deadtime distortions are present and accurate estimation of the true input
can be performed. Accurate estimations were performed by using the time widths and the timeinterval distributions of the pulses from the fast channel.Moreover, the DPP output results, provided in timed packed listing mode, together with the housekeeping data, allow
corrections even for variable or transient radiation sources, with time resolutions depending on the ICR and the chosen number of radiation events.We stress that the digital system allows, after a simple parameter setting, different and sophisticated procedures for deadtime correction, traditionally implemented in complex/dedicated systems and timeconsuming setups.
Acknowledgements
This work was supported by the Italian Ministry for Education, University and Research (MIUR) under PRIN Project No. 2012WM9MEP.
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