3.1 Traditional algorithm

Outputted by a typical radiation detector, a nuclear pulse is a bi-exponential signal with a sharp rising-edge and a relatively slow falling-edge. It is fitted frequently by functions of the uni-exponential model and the bi-exponential model [14]. In this paper, the pulse signal is characterized by a uni-exponential,.

where A is the pulse amplitude, τ is the time constant of the pulse falling-edge. As shown in Fig. 2, the traditional pulse shaping obtains the cusp-like shape through the convolution of the original pulse and the impulse response function of the cusp-like shaper. In fact, this shaper works as a digital finite impulse response (FIR) filter [6,11,15].

Fig. 2Full-screen View

Transfer function model of cusp-like pulse shaping.

The discrete equation of convolution is:

where, a_k are the coefficients of the filter, and N = τ_s/T_s is the number of coefficients with τ_s being the finite time of y[n] and T_s the sampling period. This convolution needs too many multiplication and memory units to be practical.

3.2 Recursive algorithm

We use the algorithm of cusp-like pulse shaping based on the recursive difference equations in time domain. This recursive algorithm is suitable for real-time implementation in hardware system [16]. As shown in Fig.3, the decayed tails of original pulse signal s[n] is removed to obtain the impulse signal δ[n] through deconvolution. Then, δ[n] is synthesized to the pulse sequence p[n] through the operations of both delay and reverse. A symmetrical T-shaped pulse q[n] is obtained by the integration of p[n]. The continuous integration of q[n] can obtain a bipolar triangular pulse r[n]. Lastly, the cusp-like shape y[n] can be obtained by the integration of r[n].

Fig. 3Full-screen View

Recursive method of cusp-like pulse shaping using difference equations.

Current pulses of a nuclear detector are not ideal as they have a certain lasting time. The partial electric charges are dissipated in the charge collection. Ballistic deficit and charge trapping effects can noticeably reduce energy resolution of spectrometers when shapers producing short pulses with sharp peaks are used. It has been demonstrated that the effects can be reduced by using flat-topped shapes [3,5]. Therefore, it is necessary to design a cusp-like shaper with the finite flat-top. As long as the width of flat-top is greater than the lasting time of the current pulse, the flat-topped cusp-like shaper can effectively overcome the ballistic deficit. Equations (3–7) are recursive difference equations of cusp-like and flat-topped cusp-like pulse shaping.

where, d = exp(T_s/τ), with T_s being the ADC sampling period; A is the number of points on the rising edge, and B is the number of points on the flat-top. When B is not equal to 0, it is a flat-topped cusp-like shaper.

5.1 Implementation in hardware system

The low power dissipation AD9224 (12-Bit and 40 MSPS, ADI) and low-cost EP4CE6 series FPGA (ALTERA) are used in this paper, which are the good trade-off between the cost and performances. For convenience in high-speed ADC sampling, the original pulse signals from nuclear detectors are adjusted by the signal conditioning circuits. Then, the FPGA is used to implement the digital pulse processing algorithm by its parallel operation. The modules of pulse height analyzer (PHA) in FPGA consist of data buffers, cusp-like shaper, pile-up rejection, baseline restorer, amplitude extraction, external interface and so on [12,18]. In addition, a parameter module, which is a register array, stores the parameters of other modules. The parameters of pulse shaping can be modified by changing the registers. The logic diagram of cusp-like pulse shaping algorithm in FPGA is shown in Fig. 6.

Fig. 6Full-screen View

The logic diagram of cusp-like pulse shaping algorithm in FPGA

The parameter d in Eq. (4) is of the constant float-point type, but the float-point operations in FPGA should be avoided, for saving the operation speed and data memory volume. This parameter can be converted into a fraction, so that the float-point operations can turn into integer multiplication and integer division.

In the experiment, a NaI(Tl) crystal (ϕ76 mm ×76 mm) coupled with a PMT (76 mm CR160-01, Hamamatsu) is used as the front-end detector, and a ¹³⁷Cs source is placed at 10 cm away from the detector.

5.2 Ballistic deficit correction

To verify whether the cusp-like shaper in FPGA hardware is consistent with the theoretical design, SignalTap logic analyzer is used to analyze the real-time data in FPGA memory through a JTAG hardware debugger. Fig. 7 shows an original pulse signal, the corresponding cusp-like pulse shaping (A=120, B=0) and the flat-topped cusp-like pulse shaping (A=100, B=40). Due to influence of the lasting time of current pulse, the rising edge of original pulse is not fast enough, hence the ballistic deficit in the corresponding cusp-like pulse shaping, and the gradual height increases till the maximum at completion of the charges collection. The flat-topped cusp-like shaper can eliminate the influence of amplitude deficit. The average of flat-top can be set as the peak value, which can reduce the influence of noise. However, the overlarge flat-top width would increase the probability of pulse pile-up, so the trade-off of between ballistic deficit and pulse pile-up should be carefully considered.

Fig. 7Full-screen View

Original pulse and cusp-like pulse shaping form FPGA

5.3 Baseline drift restoration

Figures 8(a) and 8(b) show the original pulse signals and the baseline drifts over the time. Since it performs differential operations, the recursive process of pulse shaping amplifies the noises from the original pulse signals, so that the baseline drifts [18]. Thus, all the differential operations must be placed at the final stage by changing the order of transfer functions in the linear system. Fig.8 (c) shows the improved baseline fluctuates in a small range of a long-term test. Besides, the large numeric value can be obtained through the integration operations, which effectively reduce the round-off error of division operations in the back-stage.

Fig. 8Full-screen View

The original pulse signals(a), baseline drift of pulse shaping(b) and baseline restoration after adjusting the order of transfer functions(c).

5.4 Pile-up rejection

Under high count rates, pulse pile-up is serious. The pile-up rejection method plays an important role in distinguishing the dead-time signals whose peak intervals are less than half of the shaping time constant. The fast pulse shaping can generate an ultra-narrow pulse signal by using a very small shaping time constant, which can be used for pile-up rejection. As the ultra-narrow pulse signal position is the same as the rise edge of its corresponding original pulse signal, a threshold can be set to trigger this ultra-narrow pulse signal. Fig.9 shows the pulse pile-up of three signals, the cusp-like pulse shaping (A=120, B=0, τ = 6 μs) and the ultra-narrow pulse signals of the fast pulse shaping (A=6, B=0). It can be seen that the time intervals of the firstly two ultra-narrow pulses are greater than 3 μs, so the first cusp-like shape can be used for peak extraction. However, the last two ultra-narrow pulses are gapped by less than 3 μs, so they could be discarded.

Fig. 9Full-screen View

The pile-up correction method of the cusp-like shaper

5.5 Energy spectrum measurement

Energy spectra of the ¹³⁷Cs source measured by traditional analog quasi-Gaussian shaper and the digital cusp-like shaper are shown in Fig.10. The FWHM of photoelectric peak at 662 keV is 7.2% and 6.6% for the analog quasi-Gaussian shaper and the digital cusp-like shaper, respectively. The analog pulse shaper has no immunity to ballistic deficit so that its energy spectrum of photoelectric peak at 662 keV has serious asymmetry tails. Besides, as it is susceptible to noises, a higher pulse triggering threshold is set to reduce the error counts in low-energy region, and thus the X-ray peak of ¹³⁷Ba, which is the decay product of ¹³⁷Cs, cannot be detected. However, the digital flat-topped cusp-like shaper overcomes the ballistic deficit, presents the clear Compton scattering spectrum, detects the X-ray peak of ¹³⁷Ba, and has good linearity.

Fig. 10Full-screen View

Measured energy spectrum of ¹³⁷Cs by using analog quasi-Gaussian shaper and digital cusp-like shaper.