In this lab, we demonstrate how to discriminate pulses by computing the ratio of a pulse’s peak amplitude to its integrated area. This approach, known as Pulse Shape Discrimination (PSD), is often used in nuclear or radiation detection to distinguish between pulses of different decay times. For exponential pulses, the ratio of amplitude to area is directly linked to the decay constant, allowing us to classify signals based on their shape.
1. Introduction: Theoretical Basis of PSD (Amplitude vs. Area)
Why Use Amplitude vs. Area?
- Pulse Shape: Many detector outputs (e.g., from photomultiplier tubes or charge‐sensitive preamplifiers) approximate an exponential decay, often with a very fast rise time.
- Exponential Model: Suppose the pulse has the form \[ V(t) = A e^{-t/\tau}, \quad t \ge 0, \] where \(A\) is the peak amplitude (at \(t=0\)) and \(\tau\) is the decay time constant.
- Integrated Area: The total area (integral) under this exponential from \(t=0\) to \(\infty\) is \[ \text{Area} = \int_{0}^{\infty} A , e^{-t/\tau} , dt= A\tau. \]
- Amplitude/Area Ratio:
\[ \frac{\text{Amplitude}}{\text{Area}} = \frac{A}{A \tau}= \frac{1}{\tau}. \] Hence, measuring the peak amplitude and area of an exponential pulse gives a ratio inversely proportional to the decay time \(\tau\). In other words, shorter decay pulses yield a larger amplitude‐to‐area ratio; longer decay pulses yield a smaller ratio.
Practical Usage
In Pulse Shape Discrimination, we might measure the amplitude \(A\) and the integrated charge \(\int V(t),dt\). By comparing their ratio, we can classify pulses originating from different physical processes (e.g., neutron vs. gamma in scintillators, different nuclear transitions, etc.). Here, we focus on implementing that ratio measurement in Sci‐Compiler.
2. Block Diagram
Below is the main block diagram for our PSD approach:
ANALOG IN PIN (A0) goes to a Trigger LE Hyst block. The triggered window resets the Peak Detector and Accumulator (for integration). After the time window elapses (Timer), we latch both the peak and the integral. A final division block computes the ratio.]
Component Descriptions
- Trigger LE Hyst: Detects when the input pulse crosses a threshold, issuing a trigger to reset and enable measurement.
- Peak Detector: Tracks the maximum amplitude of the pulse during the measurement window.
- DSP Accumulator: Integrates (sums) the pulse samples, effectively measuring area or “charge.”
- Timer (Start‐Stop): Defines how long to measure the pulse. Once it expires, the final peak and area are latched.
- Edge Det Rise: Ensures a clean rising‐edge signal to strobe the latched values.
- Divider (Fixed‐Point): Divides the 16‐bit amplitude by the 32‐bit area, producing a 32‐bit result where the high 16 bits are integer and the low 16 bits are fractional.
3. Timer Configuration
We configure the Timer in periodic mode but also Wait for Start, so it does not begin counting until the Trigger block fires:
4. Divider Configuration
Because the amplitude is 16 bits and the area is 32 bits, we use a fixed‐point divider to produce a 32‐bit output, where:
- Upper 16 bits: integer part,
- Lower 16 bits: fractional part.
5. Detector Emulator Settings
We generate an exponential signal using the Detector Emulator. For the first simulation run, we choose:
- Amplitude = 1000 (ADC counts)
- Decay Time = 1 µs
- Rise Time = 0 µs
- Event Rate = 100 kHz
The emulator continuously produces pulses at 100 kHz, each with the specified shape.
6. Simulation Results (First Run, \(\tau=1,\mu s\))
From the logs:
- Peak \(\approx 992\)
- Area \(\approx 68564\)
- Ratio \(\approx 0.0144\)
In fixed‐point format, the ratio is stored as 948 in the lower 16 bits. Since \(0.0144 \times 2^{16} \approx 948\), that matches perfectly.
7. Simulation with \(\tau=0.8,\mu s\)
Now we change the decay time in the Detector Emulator to 0.8 µs:
Results
Observations:
- Peak \(\approx 990\). (Slightly below 1000 due to sampling. The emulator randomly offsets the pulse time, so the maximum might not always be caught exactly at the peak sample.)
- Area \(\approx 62905\).
- Ratio \(\approx 0.0157\). In fixed‐point, \(\text{ratio} \times 2^{16} = 0.0157 \times 65536 \approx 1031\).
Comparing 1 µs vs. 0.8 µs decay times, the ratio is higher for the shorter decay, consistent with the formula \(\text{Amplitude}/\text{Area} = 1/\tau\).
8. Conclusion
By measuring both the peak and the integral of an exponential pulse, we obtain a ratio inversely proportional to the pulse’s decay constant \(\tau\). This is the core principle behind Pulse Shape Discrimination (PSD) in many detector applications.
Key Takeaways:
- Peak Detector: Captures amplitude.
- Accumulator: Integrates area.
- Ratio: \(\approx \tfrac{1}{\tau}\) for an exponential pulse.
- Adjusting Decay: Changing \(\tau\) modifies amplitude/area, letting us distinguish different pulse shapes.
This Lab 5 design can be extended to real PSD tasks (e.g., distinguishing neutron vs. gamma events in liquid scintillators) by measuring amplitude/area (or short‐gate vs. long‐gate integrals) and comparing the ratio to known thresholds.