Digital Pulse Timing with Semiconductor Gamma-ray Detectors Using a Wavelet Transform Technique

M. Nakhostina)

Department of Physics, University of Surrey, Guildford, GU2 7XH, UK

ABSTRACT

Obtaining precise timing information from semiconductor gamma-ray detectors is of great interest for a variety of applications such as high-resolution positron emission tomography (PET). However, pulse timing with these detectors through the common constant-fraction discrimination (CFD) method is strongly affected by the time-walk error that results from the inherent variations in the shape of the detectors’ pulses. This paper reports on the use of the wavelet transform for minimizing the time-walk error in digital CFD pulse timing with semiconductor gamma-ray detectors. The details of the method are described and the experimental results with a 1 mm thick CdTe detector are shown. It is demonstrated that, by using the Haar wavelet transform of the digitized preamplifier pulses, the original tailed time spectrum of the detector with a time resolution of 8.22±0.12 ns at full-width at half-maximum (FWHM) in the energy range of 300-550 keV improves to a symmetric time spectrum with a time resolution of 3.39±0.02 ns (FWHM).

Keywords: Pulse timing; Semiconductor detectors; Wavelet transform; Time resolution

a)Electronic mail: m.nakhostin@surrey.ac.uk

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I. INTRODUCTION

Precise pulse timing with semiconductor gamma-ray detectors is of great importance for various applications. In high-resolution positron emission tomography (PET) systems using compound semiconductor detectors such as CdTe, improving the time resolution of the detectors is highly desirable for reducing the number of chance coincidences that degrade the image quality1-3. In nuclear structure experiments using germanium detectors, a good time resolution is fundamental for purposes such as the discrimination of background neutrons and other unwanted events by using time-of-flight measurements4,5. The common method of determining the time of occurrence of a gamma-ray event from a semiconductor detector is to use the CFD method6. In recent years, the CFD method has been also implemented in the digital domain7, 8. However, a major limitation by the CFD method is the time-walk error that results from the variations in the shape of the detector’s pulses. The variations in the shape of output pulses is inherent to the semiconductor gamma-ray detectors and originates from a difference in the mobility of charge carriers and the variations in the distribution of the weighting potential in the detectors. This paper reports on the use of the wavelet transform for minimizing the time-walk error in digital CFD pulse timing with semiconductor gamma-ray detectors. The principles of the method are described and the performance of the method is demonstrated for a planar CdTe detector.

II. THE WAVEFORM TRANSFORM

The wavelet transform is a method for the analysis of a signal simultaneously over both time and frequency9. This method can be considered as a generalization of the Fourier transform where a signal is evaluated in frequency space only. The wavelet transform is obtained by using wavelets that are waveforms of effectively limited duration and average value of zero. A wavelet is capable of both translation and scaling (i.e. dilating and compressing). Scaling allows the characterization of the frequency contents of the signal, while translation enables the localization of different frequency contents. A wavelet for every scale and translation can be calculated from a mother wavelet φ, as

φu ,s=1√ sφ( t−u

s) , (1)

where t is time, s is the scaling factor that either dilates or compresses a signal and u is the translation factor. Larger scales correspond to dilated, and small scales correspond to compressed wavelets. The continuous wavelet transform (CWT) of a signal f(t) at time u and scale s is defined as

Wf (u , s)=∫−∞

+∞

f (t) 1√ sφ¿( t−u

s)dt , (2)

where W represents the wavelet transform of the signal f(t), φ is the mother wavelet, and the star indicates the complex conjugate. There are numerous choices for the mother wavelet, and the proper choice depends on the application. In general, the choice of the wavelet should relate closely to the feature being sought from within the signal. The result of the wavelet transform at

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different times and scales can be expressed as a two-dimensional array of coefficients of both time and scale that are typically presented as an image referred to as scalogram.

III. THE METHOD

The wavelet transform has been commonly used for the detection of singularities and irregular structures in one and two-dimensional signals10. A common approach for this purpose is to use the scalogram and is based on the fact that at the location of an abrupt change such as singularity, wavelet coefficients create a cone-like appearance in the scalogram, whose boundaries coverage at small scales on the location of the change. In principle, the scalogram method may be used for the determination of the start-time of pulses from radiation detectors because at the start-time a sudden transition of the signal from the baseline happens. However, an event-by-event use of the scalogram is computationally intensive, and also the accuracy of the method can be strongly affected by small fluctuations such as noise in the signal that could create a local maximum in the wavelet transform. Here, our aim is to use the wavelet transform to convert preamplifier pulses of different shapes into pulses of similar leading-edges, thus minimizing the time-walk error in the time pickoff with the conventional digital CFD method. In the digital framework, a pulse captured from a charge-sensitive preamplifier is represented as a digital signal with data points at regular time intervals. For a digitized pulse, the discrete wavelet transform (DWT) is used as the digitized version of CWT. It is worth mentioning that the DWT can be easily implemented on embedded platforms such as field programable gate arrays (FPGAs). As mentioned earlier, the wavelet should be chosen according to the information sought in the signal. Since the CFD method favours pulses of fast rise-time, the Haar wavelet represents a good choice due to its capability of detecting high-frequency features11. The Haar wavelet is shown in Fig. 1. The wavelet has a symmetric shape that ensures no phase distortion in the filtered signal11. This wavelet is also the simplest wavelet and easy to implement. Fig. 2 shows the Haar wavelet transforms of two noiseless pulses of different rise-times. It is apparent that, in spite of the different rise-times of the pulses, the wavelet transforms of both pulses show the same rise-time which is determined by the scale value of the wavelet. It is important to note that a delay between the start-time of the wavelet transform of the pulse and that of the original pulse appears, but the delay is constant for both pulses (due to the absence of phase distortion in Haar wavelet transform), and thus, the original timing information is preserved. In the bottom of Fig. 2 the wavelet transforms of the pulses are normalized to their amplitudes to mimic the behaviour of a CFD. The same time pickoff is produced for both pulses, in spite of the difference in the rise-times of the original pulses.

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FIG. 1. The Haar wavelet. The sharp edge of this wavelet makes it suitable for the detection of sudden transitions in signals.

FIG. 2. (A) Two noiseless pulses of different rise-times. The different rise-times of the pulses can cause a significant time-walk in a CFD output. (B) The Haar wavelet transform of the pulses. The transformed pulses show the same leading-edge, regardless of the different rise-times of the pulses. (C) The amplitude-normalized wavelet transforms of the pulses to mimic the time pickoff in the CFD method. The same time pickoff is produced for both the pulses.

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IV. EXPERIMENTAL RESULTSA. Experimental setup

We have examined the performance of the wavelet transform timing method by using a CdTe detector due to its importance for PET applications. The experimental arrangement is composed of a planar Ohmic CdTe detector (5×5×1 mm3) from Acrorad Co. Ltd, Japan. The detector is placed against a cylindrical LaBr3(Ce) scintillator detector (1″ diameter by 1″ height) for coincidence timing measurements. The CdTe detector was biased with a voltage of 100 volt and was used with a fast charge-sensitive preamplifier (model A250 from Amptek Inc). The measurements were taken at room temperature and with a 22Na point source, placed between the two detectors. The outputs of the preamplifier and the anode signals of the LaBr3(Ce) detector were simultaneously digitized by means of a fast waveform digitizer with 10 bit resolution at 1 GS/s sampling rate (model DC252HF from Agilent Technologies Inc). The waveforms were transferred to a personal computer for offline analysis. All of the analysis was carried out using MATLAB and its Wavelet Toolbox.

B. Wavelet transform of the detector’s pulses

In a CdTe semiconductor detector, the mobility of holes is about ten times smaller than that for electrons12, and this difference causes pulses of large difference in their rise-times, depending on the interaction location of gamma-rays in the detector. Fig. 3 shows three pulses due to gamma-ray interactions at different locations in the CdTe detector, together with the Haar wavelet transforms of the pulses with a typical scale value of 20 ns. The pulse due to the interaction close to the cathode shows the fastest rise-time while the pulse due to the interaction close to the anode has the longest rise-time. The pulse due to the interaction between the electrodes has a fast electron component followed by a slow component due to the holes. The wavelet transforms of all the pulses exhibit a fast-leading edge, determined by the scale of the waveform transform. In fact, the wavelet transform performs a differentiation of the smoothed input pulses13, and thus, the wavelet transform of a preamplifier charge pulse gives a representation of the induced current in the detector. For a given scale value, the leading-edges of the wavelet transform of all pulses have the same shape but their signal-to-noise ratio depends on (i) signal-to-noise ratio of the original pulse and (ii) the time scale of the original pulse, i.e. faster pulses produce larger Haar wavelet transform coefficients. The effect of the scale value on the wavelet transform of a typical preamplifier pulses is shown in Fig. 4. For small scales, the transformed pulse is very close to the instantaneous induced current, but for larger scales the current pulse is further smoothed, reducing the noise at the cost of increasing the rise-time of the pulse.

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FIG. 3. The wavelet transforms of the CdTe detector’s pulses of different rise-times. (A) A pulse due to an interaction close to the cathode. (B) Interaction between the electrodes. (C) Interaction close to the anode. The wavelet transforms of all the pulses exhibit a fast leading-edge but with different degrees of signal-to-noise ratio.

FIG. 4. The effect of the scale value on the wavelet transform of a typical preamplifier pulse. By increasing the scale of the wavelet transform, noise on the pulses decreases at the cost of increasing the rise-time. Therefore, the choice of the optimum scale is a trade-off between the rise-time and noise.

C. Time resolution measurement results

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The time resolution of the CdTe detector was determined from the histogram of the time difference between the pulses from the CdTe and LaBr3(Ce) detectors. Since the LaBr3(Ce) detector is a very fast detector (time resolution < 500 ps), the time resolution of the coincidence measurements is practically determined by the time resolution of the CdTe detector. The time resolution was first determined by performing the time pickoff from the CdTe detector through the common procedure of pulse timing with semiconductor gamma-ray detectors14. This include the application of a digital integrator-differentiator timing filter followed by a digital CFD. The time pickoff from the LaBr3(Ce) detector was also carried out by using the standard digital CFD method whose details can be found elsewhere15. For both detectors, a linear interpolation was used when the time pickoff lies between two successive samples. To obtain the time spectrum by using the wavelet transform of the pulses, the scale of the Haar wavelet transform was optimized for the best timing performance through a trial-and-error procedure. The time spectra obtained by using the common method of pulse timing and by using the wavelet transform of the pulses are shown in Fig. 5. The common procedure of pulse timing with semiconductor detectors leads to a tailed time spectrum that is a clear sign of the time-walk error due to the slow pulses. This asymmetric time spectrum is well fitted with an exponentially modified Gaussian function that gives a time resolution of 8.22±0.12 ns (FWHM) in the energy range of 300-550 keV. The time spectrum obtained by using the wavelet transform of the pulses has a resolution of 3.39±0.02 ns (FWHM) that indicates an improvement by a factor of 2.4. This time resolution was achieved by using a scale of 28 ns and with a CFD fraction of 20 %. The time spectrum also shows a symmetric shape as evaluated by the Chi-square value of the Gaussian fit (χ2=0.98) which indicates the time-walk error is negligible. The time resolution is also quite consistent with the results obtained for the same detector by using a time-walk correction method (3.29±0.02 ns)14.

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FIG. 5. (Top) The time spectrum of the CdTe detector against a fast LaBr 3(Ce) detector, obtained by using the common method of pulse timing with semiconductor gamma-ray detectors that includes a digital integrator-differentiator filter followed by a digital CFD14. The time spectrum has an asymmetric shape and its FWHM as determined through a Gaussian fit is 8.22±0.12 ns. (Bottom) The time spectrum obtained by using the Haar wavelet transform of the pulses. The time spectrum shows a symmetric shape with 3.39±0.02 ns FWHM. The energy range of the measurements was 300-550 keV.

V. Discussion

In this section we discuss how the variations in the shape of pulses affects the accuracy of pulse timing with the wavelet transform of the pulses. In the absence of the time-walk error, the time resolution due to the electronic noise, expressed in standard deviation σt, is given by:

σ t=σ e

( dvdt

) , (3)

where σe is the noise level and dv(t)/dt is the signal slope at the time of discrimination. By assuming a linear leading-edge, the slope of the pulses can be estimated as:

dvdt

= At°

, (4)

where A is the amplitude of the pulse and t˳ is the rise-time of the pulse. The wavelet transform produces timing pulses of the same rise-time (t˳) but the amplitude of the

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wavelet transform of the pulses (A) will depend on the shape and amplitude of the original pulses. Larger and faster pulses produce larger wavelet coefficients, thus larger amplitudes. Therefore, while the wavelet transform removes the time-walk error, different levels of time jitter will be resulted for pulses of different shapes and amplitudes. Nevertheless, it is important to note that the ultimate time resolution is determined only by the level of electronic noise. Fig. 6 shows the time spectra for two different energy ranges. By increasing the energy threshold, the time resolution clearly improves due to the improvement in the amplitude of the wavelet transform of the pulses. Further improvement of the timing resolution would be possible by reducing the noise level through cooling the device and capacitance matching of the detector and the preamplifier.

FIG. 6. The effect of energy threshold on the timing performance. The time resolution improves with the energy of pulses.

VI. SUMMARY AND CONCLUSION

We have shown that by using the Haar wavelet transform of the preamplifier pulses, the time-walk error due to the variations in the shape of pulses from semiconductor gamma-ray detectors can be significantly minimized. By using the Haar wavelet transform, the preamplifier pulses of different shapes are transformed into pulses of the same leading-edge, and thus, the time resolution is only limited by the effect of electronic noise. The method was tested with a CdTe compound semiconductor detector. A time resolution of 2.6±0.02 ns for the energy range of 500-550 keV was achieved with a 1 mm thick CdTe detector which is the best ever resolution reported for these detectors. Further work will continue to study the performance of the wavelet timing method by applying the method to other semiconductor detectors such as germanium detectors.

ACKNOWLEDGMENT

We acknowledge support from the UK STFC (ST/L005743/1)

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