Digital data acquisition systems are popular due to their advantages over analog electronics(Al-Adili et al., 2010; Mitra et al., 2004). These systems apply numerical algorithms to digitized signals. They are flexible systems due to an unlimited choice of pulse shape analysis methods.

What’s the catch?

Extracting time information when the sampling period doesn’t match the signal. Algorithms exist but they need tailored to the digitized signal. Nyquist filters(Nyquist, 1928) and ADC sampling frequencies affect the signal response. The timing problem becomes a deconvolution problem. The major challenge is determining the response function.

I tested three numerical algorithms : a digital constant fraction discriminator (DCFD), a fitting algorithm (FA), and a weighted average algorithm (WAA). I present a new method to verify the algorithms. Finally, I’ll show you that “slow” digital systems can instrument fast scintillators.

## Testing Setup and Software Analysis

This work used an 100 MHz Tektronix AFG3102C Arbitrary Function Generator (AFG). The jitter in the arrival time of signals produced by the AFG is on the order of 50 ps as measured by a 2.5 GS/s oscilloscope. I connected AFG outputs to adjacent channels in XIA Pixie-16 digitizers.

The Pixie-16 systems process digitized signals using trapezoidal filters. The filters provide energy and time information about the digitized waveform (trace). The signal arrival time consists of two components. The first component is a 48-bit timestamp. It is the real-time counter value of the FPGA latched at the time of the valid trigger. The FPGA’s sampling

frequency limits this time resolution. We’ll call this the trigger time.

We calculate the second part using the digitized waveform. I call this the phase ($\phi$). It has a linear relationship with the trigger time. The phase provides sub-sampling time information about the waveform. We can now calculate the signal arrival time by summing the trigger time and phase.

As Madurga(Madurga et al., 2011) demonstrated, we’re interested in the waveform’s derivative. Linear algorithms only work if the derivative is constant. Assume local linearity if the digitization frequency is ~10x smaller signal length. For example, GS/s digitization of a 50 ns long signal. Linear timing algorithms breakdown when the time scales are similar. Only non-linear algorithms will produce accurate results.

Bardelli(Bardelli et al., 2004) and Fallu-Labruyere(Fallu-Labruyere et al., 2007) demonstrated the DCFD with digitizers. The DCFD applies the following equation to the digitized waveform

$$DCFD[k] = Fy[k]-y[k-D].$$

D is the delay. F is the fraction of the original trace. y[k] is the baseline subtracted amplitude of the trace at bin *k*. The figure to the right shows an example of the output of the DCFD algorithm. In the figure, D = 1 and F = 0.75. The trace comes from the 250 MS/s system.

Figure 1 shows a sample trace from the 250 MS/s system using the signal produced by the AFG (solid line). The dashed line represents the DCFD for this trace. The position of $\phi$ is given by the vertical dashed line and is given by zero crossing of the DCFD calculated using Equation 1 with D = 1 and F = 0.75. The horizontal dashed arrow denotes the region used to determine the baseline.

The second method, is a fit to the digitized waveform. The analytic model approximates the response of the digital system. For this work the function took the following form

$$f(t)=\alpha e^{-(t-\phi)/\beta} (1-e^{-(t-\phi)^4 / \gamma}).$$

$\beta$ is the decay constant for the exponential. An inverted-squared Gaussian describes the leading edge. γ fixes the width of the Gaussian. α normalizes the fit to the signal. Finally, $\phi$ provides the phase. We optimize $\beta$ and $\gamma$ by fitting waveforms and averaging the results. They now define the shape of the response function. We hold them constant for the time analysis. I implemented the fit using the Gnu Scientific Library.

I performed an error analysis on the shape parameters using 16k traces. $\beta$ and $\gamma$ had a standard deviation of 0.01 ns and 0.001 ns^{4} for the pulser. For scintillator signals, the standard deviation is 0.3 ns and 0.1 ns^{4}. The error analysis assumes a normal distribution.

The final approach uses the weighted average of the trace. We’ll call this the Weighted Average Algorithm (WAA) described by

$$ \phi = \frac{ \sum_{i=\alpha}^{\beta} (y_i – \bar{b})i }

{\sum_{j=\alpha}^{\beta} (y_j – \bar{b}) }.$$

y_{i} is the value of the trace at bin i or j. $\bar{b}$ is the average value of the baseline of the trace. $\alpha$ is the starting bin for the weighted average, and $\beta$ is the final bin. We choose $\alpha$ and $\beta$ to optimize the resolution of the system.

How do we know if they’re performing as expected?

## Algorithm Validation

We analyze the time difference between two signals. The first signal has a phase $\phi_1$ and the second signal has a phase $\phi_2$. We vary the delay ($\Delta$) between the signals in steps smaller than the sampling frequency. This varies the phase space that we’re analyzing. With a $\Delta$ of n times the sampling frequency, a bias of the timing algorithm would be impossible to detect. The test setup ensures $\phi_1$ is random with respect to the sampling clock. A plot of $\phi_1$ vs. $\phi_2$ immediately indicates the accuracy of the algorithm. A nonlinear relationship reflects deficiencies in the timing algorithm.

Figure 2 displays the results for the three algorithms. For a 0 ns delay between the signals (top row), all three of the algorithms produce similar results. The delay is equal to an integer multiple of the sampling frequency. In the second row, $\Delta$ moves to 2 ns (0.5 * sampling frequency). The DCFD fails and produces peaks. This demonstrates a bias determining $\phi$. The FA and the WAA both produce a single peak.

Figure 3 displays the phase diagrams for the three timing algorithms (columns). The first row, $\Delta$=0 ns, shows all algorithms having linear linear results. The DCFD bias shows in Panel (f). The linear behavior of the system is not reproduced. The FA and the WAA are linear regardless of the delay. Both algorithms use non-linear methods to calculate phase. We can see roll over to the next filter time stamp in the short lines that are 8 ns from the main distribution.

A projection of the phase space should be uniform. Uniformity arises from the random nature of the start signal. The algorithm demonstrates bias if the phase space isn’t uniform. Figure 4-(a) shows the projection of the phase-phase diagram for the FA with 2 ns delay. Figure 4-(b) is the projection for the DCFD with 2 ns delay. The FA displays a uniform distribution, and the DCFD shows bias. Results for the WAA are like the FA.

The DCFD’s nonlinear results are due to linear interpolation of the zero crossing. For these cases the zero crossing is non-linear(Madurga et al., 2011), the DCFD would produce accurate results with a faster digitizer. The FA approximates the response function using a non-linear function. Not fixing the response function shape results in bias like the DCFD. Let’s take a closer look at the FA and WAA.

## Performance With An Arbitrary Function Generator

I tested two systems: 100 MS/s and 250Ms/s. $\Delta$ varied between 0 and 4 ns. The resolution, seen in Figure 2, comes from the FWHM of the time distribution. It’s denoted by $\xi$ in the plots. I studied input voltages from 20 mV to 1 V. I measure the signal-to-noise ratio (SNR) to be 64 dB for 1 V signals, and 30 dB for 20 mV.

In all cases, the FA calculated the proper $\Delta$ and the phase space remained linear. Timing performance between the 100 MS/s and 250 MS/s systems are quite different. In Figure 5, the 100 MS/s resolution degrades below 400 mV reaching a value of 1.7 ns for 20 mV signals. The 250 MS/s system yields a factor of 2 improvement for signals smaller than 400 mV. This factor increases to 3.5 for 20 mV signals.

Performing the same analysis with the WAA maintained the expected phase space. For an optimized choice of averaging window, the WAA performs like the FA. Figure 6 shows that the WAA maintains excellent resolution over the full range. Again, the 250 MHz system outperformed the 100 MHz system, leading to a factor of 2.7 improvement at 20 mV.

The WAA has worse resolution than the FA. It’s advantage is its simple implementation. It would work on-board with the DSP. An on-board analysis removes the need to store digitized signals. This means less data, and increased throughput.

## Summary

I hope you’re convinced that “slow” digitizers can provide excellent timing. We tested this with both a pulser and PMT signals. You can apply these algorithms to any signal. Keep in mind that the limitations are on the hardware and detector. You’ll never get great timing out of a slow detector.

If you’d like to read more check out the full article!

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- Bardelli, L., Poggi, G., Bini, M., Pasquali, G., Taccetti, N., 2004. Time measurements by means of digital sampling techniques: a study case of FWHM time resolution with a , digitizer. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 521, 480–492. https://doi.org/10.1016/j.nima.2003.10.106
- Fallu-Labruyere, A., Tan, H., Hennig, W., Warburton, W.K., 2007. Time resolution studies using digital constant fraction discrimination. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 579, 247–251. https://doi.org/10.1016/j.nima.2007.04.048
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