Bipolar cusp-like pulse shaping algorithm

The interaction between the scintillator material and incident particles results in the emission of light, which is subsequently collected by a Photomultiplier Tube (PMT) to produce an electrical signal. This signal is generally characterized by a bi-exponential model. Following digitization by a high-speed ADC with sampling period ΔT, the bi-exponential model in the discrete-time domain can be expressed as(1)where τ₁ and τ₂ represent the decay time constants for the slow and fast components, respectively, and are determined by factors such as the properties of the scintillator material and the design of the PMT and preamplifier. Here, A₁ and A₂ denote the pulse amplitudes, which also govern the signal width. The subsequent derivation is based on this standard bi-exponential model.

The unfolding synthesis technique enables the construction of virtually any pulse shape, either exactly or as a close approximation [32]. It finds applications in linear signal processing systems, whether they are time-invariant or time-varying. The bipolar cusp-like pulse-shaping algorithm consists of two main components: an unfolding system and a synthesis system.

The purpose of the unfolding system is to deconvolute the nuclear signal into unit impulses. The unfolding equation for a single-exponential signal is provided in [32]. In the discrete-time domain, a single-exponential signal is defined as(2)(3)where a is the exponential base [32], a constant gain determined by the ADC sampling rate and the decay time constant of the single-exponential signal. Equation 2 can be rewritten in the following recursive form:(4)Equation 4 shows that the single-exponential signal defined in Eq. 2 can be unfolded via appropriate scaling and shifting. The operational algorithm for unfolding a single-exponential signal is described by the following equation [32]:(5)For n>0, Eq. 5 becomes(6)For n=0, Eq. 5 becomes(7)Based on the definition of a single-exponential signal in Eq. 2, all sampling points before n=0 are zero. Thus, is defined as(8)which is the same as the definition of δ(n). Therefore, the unfolding equation for a single-exponential signal is defined as(9)However, the actual nuclear signal is better approximated by a bi-exponential model than by a single-exponential model. To address this, [40] demonstrated that using two sequential unfolding equations can significantly reduce the width of a nuclear signal, allowing it to approximate a unit-impulse signal. As shown in Fig. 1, Eq. 9 is applied twice in the unfolding system. In this process, a₁ and a₂ are determined using τ₁ and τ₂ from Eq. 1, respectively. Implementation of the unfolding system thus requires only the determination of the two decay time constants of the bi-exponential signal.

Fig. 1Full-screen View

Block diagram of the bipolar cusp-like shaping algorithm constructed in MATLAB Simulink

The function of the synthesis system is to reconstruct the unit impulse into a target pulse shape through addition, multiplication, and accumulation. This reconstruction requires determining the target pulse shape, followed by differentiating it to obtain a linear combination of weighted and shifted unit impulses. Based on this linear combination, the synthesis equations for the target pulse shape are derived through accumulation. A bipolar cusp-like shape was selected for this study because of its narrow width, which is well suited for high count-rate measurements, and its bipolar nature, which helps prevent baseline drift. The target bipolar cusp-like signal can be expressed as(10)where L determines the duration of the rising and falling edges. Equation 10 is a piecewise function comprising two quadratic curves, as shown in Fig. 2f. Applying a discrete difference to Eq. 10 results in(11)To remove the discontinuity at n=L-1, the differentiated result is adjusted as(12)and the piecewise function representation becomes(13)which is considered the first-order difference of Eq. 10 in this paper. As shown in Fig. 2e, exhibits the shape of an isosceles trapezoid. Differentiating Eq. 13 yields(14)The unit-step function u(n) is used to represent the interval of the piecewise function in Eq. 14 in a discrete form, which can be expressed as(15)whereas the third-order difference is expressed as(16)which is a linear combination of weighted and shifted unit impulses. The equations for the synthesis system can be derived by using the δ(n) signal obtained from the unfolding system to construct Eq. 16, followed by three accumulation operations:(17)(18)(19)

Fig. 2Full-screen View

(a) Input bi-exponential signal and (b) its unfolding result. (f) Target bipolar cusp-like signal and its derivatives: (e) first, (d) second, and (c) third derivatives

Xilinx System Generator (XSG) is a powerful tool for developing digital signal processing (DSP) systems using Xilinx FPGAs. It is an essential component of the Xilinx design suite and offers a high-level graphical environment for algorithm development and hardware implementation. The block diagram of the bipolar cusp-like shaping algorithm, defined by Eqs. 9, 16, 17, 18, and 19, is illustrated in Fig. 1. The proposed algorithm comprises only a few adders and multipliers, rendering its structure simple and conducive to real-time FPGA implementation. The output bit-width limits (labeled in Fig. 1) for each block indicate that the algorithm operates in fixed-point arithmetic. Consequently, each time the signal passes through a block, quantization noise is introduced due to rounding. Under high count-rate conditions, L was fixed at five in this study, resulting in a short rising and falling edge, as shown in Fig. 2f. For different types of signals from various detectors, only a₁ and a₂ need to be adjusted.

Setting A₁ and A₂ to 1000, ΔT/τ₂ to 1/10, and ΔT/τ₂ to 1/6 in Eq. 2 artificially produces a double exponential signal as:(20)The parameters of the unfolding system, shown in Fig. 1, correspond to the decay time constants in Eq. 20. Processing Eq. 20 with the algorithm illustrated in Fig. 1 produced the results shown in Fig. 2, which display the unfolding and synthesis process for a bi-exponential signal. As shown in Fig. 2b, the standard bi-exponential signal is deconvoluted into a unit impulse through the unfolding system containing two decay time constants. A noticeable deviation from the ideal shape appears during the shaping process, as the amplitude fluctuates around 0 for n>1, rather than reaching 0 precisely. This occurs because all mathematical operations in the algorithm are implemented using fixed-point arithmetic to achieve fast computation speeds. The parameter a, which acts as a gain constant in the unfolding system, has restricted precision due to its fixed-point representation. Consequently, despite the real-time DSP advantages offered by FPGAs, nonfloating-point operations may incur some precision loss, leading to computational results that deviate from those obtained via computer calculations. Using higher-end FPGAs with greater resources and improved performance can allow for an increased bit width in each operation, thereby rendering the precision loss more tolerable.

Particle classification criteria and amplitude recovery

The neutron signal is narrower than the gamma signal in the output of the NaIL detector used in this study. Figure 3a illustrates the neutron and gamma signals with distinct pulse widths, along with artificially constructed pile-up events formed by combining these two signals. In these experiments, one signal was delayed by eight clock cycles and added to the other to create a pile-up event. Because neutron and gamma signals have different decay time constants, a pulse-shaping algorithm optimized for one decay constant produces distinct shaping results for each signal type.

Fig. 3Full-screen View

(Color online) (a) Input neutron, gamma, and pile-up signals. (b) Signal processing flow on the FPGA. (c) Shaping result obtained by the PSAG. (d) Shaping result obtained by the PSAN

The blue curve in Fig. 3c shows the shaping result of the red curve in Fig. 3a when processed using the pulse shaping algorithm for gamma (PSAG). PSAG is designed to match the decay time constants of the gamma signal, rendering the gamma signal closer to a bipolar cusp-like shape after shaping. However, the neutron signal, which decays faster than the gamma signal, does not conform to PSAG’s parameters and exhibits an overshoot when returning to the baseline. In n+γ or n+n pile-up cases, the later pulse is superimposed on this overshoot, leading to inaccurate amplitude recovery.

Conversely, the green curve in Fig. 3d displays the shaping result of the red curve in Fig. 3a when processed with the pulse shaping algorithm for neutron (PSAN). Unlike PSAG, PSAN shapes the neutron signal into a bipolar cusp-like form; however, when PSAN is applied to the gamma signal, the slower decay prevents it from returning to the baseline within five clock cycles. In γ+n and γ+γ pile-up events, the amplitude of the latter signal is lower than its true value because the previous gamma signal does not return to the baseline quickly. Thus, PSAG produces an overshoot for neutron signals while PSAN yields a slow decay for gamma signals. Both overshoot and slow decay distort the preceding pulse, preventing the subsequent pulse’s amplitude from recovering to its original value.

Nevertheless, the γ+n event shown in Fig. 3c and the n+γ event in Fig. 3d exhibit satisfactory amplitude recovery. In Eq. 21, the variable ε quantifies the amplitude recovery of the pile-up signal, with smaller values indicating better recovery. The amplitude of the preceding signal is typically unaffected (), and the later signal in a pile-up event can only be approximately restored if the preceding signal returns to the baseline without overshoot or slow decay (). In the PSAG results, the single-shaped neutron signal has an amplitude of 1095. Because PSAG matches the gamma decay constants, the gamma signal returns to the baseline without overshoot in the γ+n event, and the recovered neutron amplitude is approximately 1093, yielding an ε value of 2. Similarly, in the PSAN results the single-shaped gamma signal has an amplitude of 904, and in the n+γ event the recovered gamma amplitude is about 902, also yielding an ε value of 2.(21)In both PSAG and PSAN, the troughs of the shaped neutron signals are markedly larger than their peaks due to the faster decay of neutron signals. The ratio of the peak to the sum of the peak and trough serves as the criterion for particle-type discrimination, as defined in Eqs. 22 and 23. The values of λ₁ and λ₂ were computed using the Xilinx Vivado Divider IP core operating in Radix-2 mode, which divides two 12-bit unsigned numbers to yield a 24-bit result that includes a 12-digit fractional part. This process results in a latency of 26 clock cycles. Because both λ₁ and λ₂ must be less than 1, the 12-digit fractional part is used as the horizontal coordinate of the PSD plot. The PSD measurements allowed us to obtain the λ values (λ_standard) used by PSAN and PSAG to distinguish between neutron and gamma signals. In Eq. 24, the variable σ represents the discrimination power between neutron and gamma signals, with higher values being preferable.(22)(23)(24)Both ε and σ are used as evaluation metrics in the multi-parameter optimization, and only the λ value is computed on the FPGA for classifying neutron/gamma signals.

In summary, although both PSAG and PSAN can discriminate between the two signals, accurate amplitude recovery requires dual-channel shaping utilizing both PSAG and PSAN. Thanks to the FPGA’s parallel-processing capability, dual-channel shaping does not introduce delays. Figure 3b shows the signal processing flow on the FPGA, where the 12-bit signal from the ADC is processed in parallel through PSAG and PSAN to produce two 16-bit outputs. The number of pile-up events in the signals processed by PSAG and PSAN is minimal, primarily due to the reduction in signal widths. A feature extraction module simultaneously extracts signal characteristics (signal type, peaks, and troughs) from the outputs of the PSAN and PSAG algorithms, while a pile-up detection module identifies pile-up events by calculating the interval between the current and previous signal peaks.

First, both algorithms produce a classification result for the same signal using λ_standard determined from the PSD plots; the discrimination result is considered valid only if both algorithms yield the same classification outcome. Following classification, the valid amplitude output is selected. In Figs. 3c and d, the valid amplitude output is marked with a triangle symbol. For isolated signals (where the current signal peak is more than 50 sampling points away from the previous peak), the gamma signal amplitude from PSAG is valid, while the neutron signal amplitude from PSAN is valid. When the current signal peak is less than 50 sampling points away from the previous peak and the preceding signal is a neutron signal, the current amplitude from PSAN is considered valid. Conversely, if the previous signal is a gamma signal, the current amplitude from PSAG is taken as valid. Therefore, γ+n and γ+γ outputs from PSAG are valid, and n+γ and n+n outputs from PSAN are valid.

For the same neutron signal, PSAG yields an amplitude of 1095 while PSAN produces an amplitude of 1403, indicating that PSAN has a higher output amplitude. This selection approach enhances the overall neutron signal amplitudes, thereby mitigating misclassification between neutron and gamma signals in the low-energy region. In addition, due to the generally higher amplitudes of the PSAN outputs, the valid neutron amplitude from PSAG is multiplied by a gain factor of 1.282 for the neutron response spectrum, and the valid gamma amplitude from PSAN is multiplied by a gain factor of 0.865 for the gamma spectrum.

However, the dual-channel shaping algorithm fails to respond adequately to multiple consecutive short-interval pulses. In the γ+n+γ+n event shown in Fig. 3c, the first two pulses exhibit amplitude recovery similar to that observed in the n+γ event, but the two subsequent pulses cannot be accurately recovered due to the overshoot produced by the neutron signal. In this scenario, the algorithm erroneously considered the amplitude of the last neutron signal (1138) as valid, although it should have been 1095. Similarly, in the n+γ+n+γ event shown in Fig. 3d, because the preceding signal to the last gamma signal is a neutron signal, the last gamma signal amplitude of 917 is taken as valid, although the expected value is 904. Thus, even though dual-channel shaping rectifies the inaccurate amplitude recovery associated with two pile-up signals, the failure of the pulse to rapidly return to baseline in these events leads to poor amplitude recovery for any subsequent signals.

Multi-objective optimization for parameter determination

As with other pulse-shaping algorithms, the decay time constants are the key parameters for this algorithm. Typically, a preliminary fitting process based on an analytical exponential model is conducted on a computer to obtain these parameters. However, discrepancies between the actual nuclear signal and the ideal bi-exponential model often render complete fitting impossible. Consequently, the fitting process and subsequent algorithm tests are frequently performed by trial and error, and the decay time constants derived from such fitting may not yield smooth pulse shaping (as illustrated in Fig. 2), thereby necessitating manual adjustments to achieve optimal discrimination.

To circumvent the need for manual adjustments, Gaussian mixture models and deep-learning transformer models have been employed to identify pulse shape parameters [41, 42]. In this study, multi-objective optimization was employed to determine the optimal decay time constants, ensuring both effective neutron/γ discrimination (quantified by σ) and accurate amplitude recovery (quantified by ε).

The multi-objective optimization of the proposed algorithm was performed using the MATLAB function gamultiobj from the Global Optimization Toolbox. This function generates a set of optimal solutions along the Pareto front in the space of decision variables and employs a controlled elitist genetic algorithm—a variant of the Non-dominated Sorting Genetic Algorithm II (NSGA-II) [43]. In genetic algorithms, the controlled elitist approach is notable for its emphasis on preserving population diversity, even at the expense of selecting individuals with lower fitness. This strategy is pivotal for steering convergence toward an optimal Pareto front, where the solutions balance the conflicting objectives (σ and ε). Maintaining diversity within the population plays a crucial role as the algorithm unfolds.

Two key mechanisms, namely ParetoFraction and DistanceMeasureFcn, are instrumental in controlling elitism. ParetoFraction regulates the number of individuals designated as elite members on the Pareto front, ensuring a balanced representation of optimal solutions. Meanwhile, the DistanceMeasureFcn actively promotes diversity along the Pareto front by favoring individuals that are spatially separated, thereby mitigating the risk of premature convergence toward suboptimal solutions.

A key stopping criterion for the algorithm is the spread assessment—a metric that monitors the extent of movement along the Pareto front. If the spread diminishes to a predefined threshold, indicative of stagnation or limited exploration of the solution space, the algorithm terminates. This approach not only safeguards against premature convergence but also encourages a thorough exploration of the solution landscape, ultimately fostering the discovery of robust and diverse Pareto-optimal solutions. After performing multi-objective optimization based on the amplitude recovery and discrimination performance for the four pile-up events (γ+γ, γ+n, n+γ, n+n), the optimal decay time constants were selected from the Pareto front.

Validation of the pulse shaping algorithm

Pile-up events occur occasionally during nuclear radiation detection. At low count rates, these events are rare and may be regarded as part of statistical fluctuations. However, at high count rates, high-frequency pile-up events become significant and cannot be ignored. The closer the two pulse signals are spaced, the greater the impact; for example, the amplitude of the second pulse in a pile-up event may be falsely identified as being higher. Moreover, when two overlapping signals are extremely close, pile-up correction becomes even more challenging.

Figures 5a–c depict neutron and gamma signals with different amplitudes, along with artificially constructed pile-up events generated by adding one signal to another with a delay of eight clocks (160 ns). Such pile-up events severely distort the pulse waveform, causing two overlapping pulses to appear as a single pulse. The second pulse, which overlaps with the first, is erroneously detected by the peak detection algorithm with an exaggerated amplitude. Consequently, the standard peak detection algorithm not only detects the pile-up event as a pulse but also incorrectly extracts it with a large amplitude.

Fig. 5Full-screen View

(Color online) Manually constructed pile-up events include: (a) similar amplitude pulses, (b) small-amplitude pulses added to large-amplitude pulses, and (c) large-amplitude pulses added to small-amplitude pulses. The decay time constants for PSAG are 103.79 ns and 195.886 ns. (d), (e), and (f) show the bipolar cusp-like shaping results of (a), (b), and (c) using PSAG. The decay time constants for PSAN are 43.864 ns and 209.4 ns. (g), (h), and (i) display the bipolar cusp-like shaping results of (a), (b), and (c) using PSAN

Figures 5d–f and Figs. 5d–g show the bipolar cusp-like shaping results using PSAG (blue curve) and PSAN (green curve), respectively. The thick red line highlights the successfully separated pile-up signals, indicating that signals with different amplitudes can be restored to their original shapes without compromising particle discrimination and amplitude recovery. Accurate amplitude recovery is possible only if the first-shaped signal in a pile-up event quickly returns to the baseline. Regardless of the signal amplitude, the shaped neutron signals consistently exhibit a significant trough amplitude. The separated signals maintain the same characteristics as the individually shaped signals, making particle discrimination more reliable than amplitude recovery. PSAG and PSAN each utilize their own criteria for classifying neutron and gamma signals due to the differences in their shaping results. Figures 5d and g show that PSAG has a greater advantage in distinguishing the two types. In particular, the large trough in the neutron signal produced by PSAG yields a peak-to-valley ratio of 1094/1933 in Fig. 5d, which is smaller than that in Fig. 5g (1402/1686).

Figure 5d–i shows that the amplitudes of the shaped pileup signals exhibit slight deviations compared to those of the individually shaped signals. In the case of n+ pile-up events shown in Fig. 5i, for instance, the amplitude of the gamma signal (901) is reduced by 3 channels compared to its amplitude in isolation (904), resulting in a three-channel shift in the spectrum. The primary cause of these deviations is fixedpoint arithmetic errors. As illustrated in Fig. 1, each operator in the algorithm produces a maximum of 16 fractional bits, while the final output is a 16-bit integer; fractional bits are dropped to produce integer channels. In addition, the use of a bi-exponential model for algorithm derivation introduces discrepancies, as the actual nuclear signal does not perfectly adhere to this model. Even after adjusting the decay time constant parameters, the shaped signal fluctuates around the baseline, so that the subsequent signal in a pile-up event is superimposed on this fluctuation rather than on a flat baseline. These slight deviations are considered tolerable and can be treated as statistical fluctuations in the energy spectrum measurements when accurate neutron counting is ensured.

Multi-objective optimization of the decay time constants

Before implementing the pulse-shaping algorithm on the FPGA, the decay time constant parameters were obtained using MATLAB. Two decay time constants, ranging from 0 to 400 ns, were used as decision variables. The multi-objective optimization results for PSAG, using these two decay time constants, are shown in Fig. 6. As Fig. 5 suggests, optimization for PSAG primarily considers the γ+n and γ+γ pile-up cases. The discrimination ability σ, calculated using Eq. 24, is directly proportional to τ₁ and τ₂ within a certain range (Fig. 6a). In particular, σ approaches 0.3 as τ₁ increases to 240 ns and τ₂ to 120 ns. Although a larger σ generally indicates better discrimination, excessively large decay time constants may distort the bipolar cusp-like target shape and hinder amplitude recovery.

Fig. 6Full-screen View

(Color online) (a) Variation of the discrimination ability σ in γ+n pile-up event as a function of two decay time constants. Amplitude recovery (b) in the γ+γ pile-up event (λ₁) and (c) in the γ+n pile-up event (λ₂) vary with two decay time constants. (d) Pareto optimal front of two decay time constants

Figure 6b and c illustrate the second signal’s amplitude recovery ability in γ+n and γ+γ pile-up cases. A smaller λ value indicates better amplitude recovery, and only specific combinations of τ₁ and τ₂ yield λ values close to zero. In these cases, τ₁ and τ₂ best fit the bi-exponential model of the gamma signal, likely resulting in the target bipolar cusp-like shapes seen in Figs. 5d–f. Moreover, amplitude recovery primarily depends on τ₂, which characterizes the fast component of the signal. τ₂ must be approximately 100 ns to ensure good amplitude recovery (small λ) while maintaining a sufficiently high σ (above 0.2). By considering λ₁ and λ₂ as optimization objectives, decay time constants that best fit the bi-exponential model of the gamma signal can be identified, giving amplitude recovery a greater weight than discrimination ability.

Figure 6d presents the Pareto front obtained with a population size of 500 using MATLAB’s gamultiobj function, which finds the optimal decay time constants with high precision within the range of 0 to 400 ns. Although the Pareto frontier provides a series of optimal high-precision solutions, the hardware implementation of the pulse-shaping algorithm runs in fixed-point arithmetic. Therefore, arbitrarily choosing one of these high-precision solutions does not affect the final shaping results because of the rounding that occurs after each operation (as shown in Fig. 1).

Spectra restoration

During the high-count-rate online experiment, pile-up events involving multiple pulses were specifically extracted using an Integrated Logic Analyzer (ILA), as shown in Fig. 7. The NaIL detector in this study produced output signals with widths ranging from 200 to 500 ns. Neutron signals typically have widths of less than 300 ns, while gamma signals are usually wider (exceeding 300 ns). Given the higher prevalence of gamma signals, pile-up events are more likely to consist primarily of gamma pulses. The real-time shaping results observed on the FPGA were consistent with those in Fig. 5; both algorithms effectively shortened the pulse width and separated the accumulated signals. After pulse shaping, as shown in Figs. 7b and c, a signal with a notably smaller peak-to-valley ratio was observed and classified as a neutron signal by the algorithm.

Fig. 7Full-screen View

(Color online) (a) Inverted piled-up signals from the NaIL detector output. Shaped signal outputs from (b) PSAG and (c) PSAN

The online implementation of the algorithm exhibits the same inaccuracies in magnitude recovery as seen in Fig. 5, leading to slight deviations in channel addresses. These inaccuracies result from several factors: (1) the shaping algorithm is an approximation derived from a bi-exponential signal; (2) the FPGA implementation relies on fixed-point computation, reducing accuracy; and (3) the input signal contains noise and other non-deterministic elements.

In our previous study, the NaIL detector was positioned outside a boron-containing polyethylene shield with a 6-mm-thick lead layer, resulting in a lower count rate. At low count rates, the Figs. standard energy spectrum of the NaIL detector response to an ²⁴¹Am-Be source (shown by the green lines in Fig. 8c) was not processed by the filter-shaping algorithm and was affected by only a small number of pile-up events, resulting in three distinct peaks that served as a benchmark for energy spectrum restoration.

Fig. 8Full-screen View

(Color online) PSD and energy spectrum analyses. (a) PSD plot obtained using PSAG and measured over one minute. (b) PSD plot obtained using PSAN and measured over one minute. (c) Energy spectrum distortion due to increased count rate: the green spectrum was measured over five minutes at 10584 cps, whereas the purple spectrum was measured over one minute at 59257 cps. (d) Restoration and separation of the spectrum applying both PSAG and PSAN in a one-minute measurement: the gamma spectrum was measured at 58374 cps and the neutron response spectrum at 2804 cps. Only counts above 244 keV were recorded in these spectra

As shown in Fig. 4, in the present study the NaIL detector was positioned inside a boron-containing polyethylene shield in close proximity to the source, leading to a high count rate and a substantial number of low-energy gamma rays from ²⁴¹Am. In practice, this abundance of low-energy gamma rays causes baseline fluctuations. Although a lead shield was wrapped around the NaIL scintillator to block gamma rays (as shown in Fig. 4), low-amplitude gamma signals were still observed (see Fig. 7), causing baseline instability.

The purple line in Fig. 8c shows the distorted energy spectrum at a high count rate. Owing to extensive pile-up events, the spectral peaks become less pronounced and the counts shift toward the high-energy region. Although the total counts in the two spectra of Fig. 8c are similar, the number of false counts increases significantly in the energy region above 5000 keV. Because low-energy gamma counts are very high and neutrons are nearly absent in the low-energy region, an energy threshold of 244 keV was set so that signals below this threshold were excluded.

Figures 8a and b illustrate the PSD performance of PSAG and PSAN, respectively. The Figure of Merit (FoM), first introduced by Winyard et al. [46], is given by:(25)where S represents the distance between the centroids of the gamma and neutron peaks, and FWHM_γ and FWHM_n are the full widths at half maximum of the gamma and neutron peaks, respectively.

Figures 8a and b show that while both PSAG and PSAN yield similar distributions, the neutron and gamma peaks are more widely separated in the PSD plot obtained by PSAG, indicating superior PSD capability. As previously mentioned, the pulse shaped by PSAG has a greater peak-to-valley ratio, resulting in a higher FoM. After determining λ_standard from the PSD plots, signal types can be classified based on λ_standard to obtain the separated energy spectra shown in Fig. 8d. For example, PSAG output signals with λ less than 0.492 (represented as 0x07DF on the FPGA) are classified as neutron signals.

It is noteworthy that Fig. 8b shows more total counts than Fig. 8a, even though both PSAG and PSAN process the same input signals in parallel. This is because the same channel-address threshold was imposed in the hardware, and PSAN tends to increase the overall amplitude of the signals, resulting in more signals exceeding the threshold. However, due to the presence of a coincidence system, excess counts from PSAN were not included in the final energy spectrum.

As shown in Fig. 8d, the coincidence system combining PSAG and PSAN is capable of restoring the normal neutron and gamma energy spectra from the pile-up-affected spectrum shown in Fig. 8c. However, the restored spectra in Fig. 8d exhibit noticeable noise and lack smoothness, resulting in fluctuations and irregularities in the spectral peaks. This noise may stem from statistical fluctuations in detection, electronic noise in the measurement system, or inaccuracies introduced by the FPGA implementation of the shaping algorithm.

Despite these inaccuracies, the application of the shaping algorithm resulted in more pronounced spectral peaks. The ²⁴¹Am-Be source used in this study primarily emitted a large number of low-energy gamma rays. Although this alone did not cause severe distortion at a count rate of 10 kcps, a noticeable number of pile-up events still occurred in the low-energy region. Consequently, the spectral peaks in the low-energy region (as seen in Fig. 8c) are not distinct when no filtering is applied, but become more pronounced after applying the proposed narrow shaping algorithm, demonstrating the algorithm’s capability to improve the signal-to-noise ratio.

The highest-energy gamma-ray peak corresponds to the full-energy peak at 4438 keV from the de-excitation of ^12*C produced in the reaction α(⁹Be, n)^12*C [47]. Adjacent to this full-energy peak, a single escape peak occurs at 3927 keV and a double escape peak at 3416 keV. In addition, hydrogen nuclei in the polyethylene shield captured thermal neutrons, producing gamma rays at 2230 keV. In the low-energy region, ²¹⁴Pb and its beta-decay product ²¹⁴Bi emit gamma rays at 351 keV and 609 keV, respectively, and a 511 keV peak due to positron annihilation is also observed. These low-energy peaks become visible after pulse shaping.

In a NaIL scintillator, neutrons are indirectly detected through their reaction with ⁶Li [44]:(26)This reaction produces tritium and alpha particles with a total energy release of 4.78 MeV. The high neutron response energy captured by ⁶Li precludes particle discrimination in the low-energy region. Due to the finite fluorescence efficiency of the charged particles in NaIL scintillators, a single neutron response peak appears at 4319 keV in the gamma energy spectrum. The position of the neutron response peak in the gamma spectrum depends on the experimental setup. Lai et al. reported that the luminous efficiency of alpha and tritium particles in NaIL crystals was approximately 70%, resulting in a neutron response peak close to the double escape peak [48]. Figure 5 demonstrates that a pulse-shaping algorithm produces different gains for neutron and gamma signals. In [40], signals processed using a trapezoidal shaper with different gains for neutrons and gammas resulted in a neutron response peak close to a single escape peak. In Fig. 8c, the amplitudes at the NaIL detector output are recorded without filtering or shaping. As shown in Fig. 8d, the thermal neutron response energy is centered at 4319 keV, which is close to the energy released by ^12*C. The separated neutron response spectrum also indicates that the NaIL detector, when placed adjacent to the ²⁴¹Am-Be source, recorded 2804 neutron events per second.