Below is a document describing a Pulse Shape Discrimination (PSD) technique based on two integrated areas (QSHORT and QLONG) rather than the peak‐to‐area approach. It also includes the relevant formulas for QSHORT, QLONG, and the resulting PSD ratio as functions of the decay constant $\tau$.

1. Introduction

In many radiation detectors (e.g., organic scintillators or photomultiplier systems), the output pulse can be discriminated based on its temporal shape. A common technique, known as Pulse Shape Discrimination (PSD), involves measuring two integration windows of the same signal and forming their ratio. Specifically, we define:

QLONG: A “long” integration window that captures (nearly) the entire pulse.
QSHORT: A “short” integration window that covers only the initial portion of the pulse.

By taking the ratio $\text{PSD} = \frac{QSHORT}{QLONG}$, one obtains a shape indicator. Pulses with a faster decay (or larger prompt component) tend to have a larger fraction of their total area in the short window, hence a higher ratio. Conversely, slower pulses have more of their energy in the tail, producing a lower ratio.

1.1 Exponential Pulse Model

Assume the pulse can be approximated by: $$ V(t) = A e^{-t/\tau}, \quad t \ge 0, $$ where:

$A$ is the peak amplitude at $t=0$.
$\tau$ is the decay time constant.

1.1.1 Integrating the Exponential

The integrated area from $t = 0$ to $t = T$ is: $$ \int_{0}^{T} A e^{-t/\tau} dt = A\tau \bigl[1 - e^{-T/\tau}\bigr]. $$ Hence:

QLONG $\approx$ $\displaystyle A\tau[1 - e^{-T_{\text{Long}}/\tau}]$
QSHORT $\approx$ $\displaystyle A\tau[1 - e^{-T_{\text{Short}}/\tau}]$

1.1.2 PSD Ratio

$$ \text{PSD} = \frac{QSHORT}{QLONG}=\frac{1 - e^{-T_{\text{Short}}/\tau}}{1 - e^{-T_{\text{Long}}/\tau}}. $$

Note that $\text{PSD}$ increases as $\tau$ decreases (faster decay ⇒ larger short fraction) and decreases as $\tau$ increases (slower decay ⇒ relatively smaller short fraction).

To avoid to provide fractional number Sci-Compiler IP provide in output the 1/PSD ratio.

2. Block Diagram Overview

Below is the main block diagram illustrating how the PSD module and the associated logic are connected:

A Trigger detects when the signal crosses a specified threshold.
The PSD Block initiates two integrations:
- QSHORT in a short time window,
- QLONG in a longer window covering the entire pulse.
Pile‐up Rejection logic inhibits measurements if a second trigger arrives too soon, ensuring valid pulse capture.
At the end of the integration windows, the PSD block provides:
- QLONG and QSHORT (both 32‐bit and 16‐bit outputs),
- The 1/PSD ratio $\tfrac{QLONG}{QSHORT}$ (16 and/or 32 bits, depending on configuration).
Additional digital gain (scaling) blocks may be used downstream to map these values into suitable ranges for real‐time plotting (e.g., 1D spectrum and 2D histograms).

2.1 The PSD Block

The PSD block performs:

Integration Control
- Starts QLONG and QSHORT when the trigger is asserted.
- Each window (short and long) accumulates its respective charge until the programmed number of samples is reached.
- At the end of each window, the accumulated charge is latched to the output.

Pile‐up Rejection
- If a second trigger arrives within the QLONG interval of the first event, both events are discarded (no valid data).
- If a second trigger arrives after QLONG completes but within the PostInib interval (an additional inhibition time), only the second event is discarded.
- This ensures non‐overlapping pulses, preserving integrity of the QSHORT/QLONG measurements.
Outputs
- QLONG[31:0] and QSHORT[31:0]: Full‐precision 32‐bit outputs (no internal digital gain).
- QLONG[15:0] and QSHORT[15:0]: Scaled or reduced 16‐bit versions.
- PSD Ratio = $\text{QLONG}/\text{QSHORT}$ (32 bits).

3. Detailed Operation

Trigger
- The trigger block monitors the input waveform (converted to digital samples) and asserts a valid trigger when the signal crosses the set threshold (e.g., THRESHOLD = 2500).
QSHORT & QLONG Integration
- QSHORT is integrated over a short window (e.g., 30 samples).
- QLONG is integrated over a longer window (e.g., 200 samples) to capture the full pulse.
- Each sum is internally stored until the window ends, then latched out.
Pile‐up Rejection
- If a new event arrives before the first event’s QLONG window is finished, we declare a pile‐up and discard both events.
- If the new event arrives after QLONG but within PostInib, only the new event is discarded.
- The PSD block thus acts as a non‐parallelizable machine, ensuring one clean measurement at a time.
Outputs
- The 32‐bit versions of QSHORT/QLONG are unscaled (gain = 1).
- The 16‐bit versions are typically scaled or clipped for real‐time plots by GAIN input (GAIN = 0xFFFF = * 1).
- The ratio $\frac{QLONG}{QSHORT}$ is computed and can be output in 16 bits (or 32 bits, if configured).

4. 2D Histogram: QLONG vs. PSD Ratio

A common way to visualize PSD data is a 2D histogram:

X‐axis = QLONG (pulse charge / energy).
Y‐axis = $\text{PSD Ratio} = \frac{QLONG}{QSHORT}$.

Varying Amplitude:
- If only the amplitude changes but the shape is constant, $\tau$ remains the same.
- QLONG scales with amplitude, but the ratio $\tfrac{QLONG}{QSHORT}$ stays nearly constant, forming a horizontal band in the histogram.
Varying $\tau$:
- If $\tau$ changes, QSHORT and QLONG each respond differently.
- Shorter $\tau$ ⇒ more charge in the short window ⇒ higher ratio ⇒ data points move up.
- Longer $\tau$ ⇒ relatively less charge in the short window ⇒ lower ratio ⇒ data points move down.
- Thus, pulses with different decay times separate along the Y‐axis.

5. Configuration Guidelines

Trigger Threshold: Choose a value that reliably detects pulses above noise.
Short/Long Windows (QSHORT, QLONG): Must be set according to the pulse shape (e.g., short window is a fraction of the total, long covers the full pulse).
Gain/Scaling:
- The PSD block does not apply digital gain to the 32‐bit outputs, preserving full resolution. An external gain is applied by FX block in order to fit the histogram 2D y bins (64).
- 16‐bit outputs typically require user‐defined scaling (GAIN) to avoid saturation in real‐time histograms. A 5 bit shift is applied to fit the histogram 2D x bins (128).

By plotting QLONG vs. the ratio in a 2D histogram, one can effectively separate pulses by energy (X‐axis) and decay characteristics (Y‐axis), enabling robust real‐time discrimination.

6. Example Results

This is the procedure to test the PSD module using the DT1260 and DT4800 modules. The DT4800 module is a detector emulator that generates pulses with variable amplitude and decay time.

6.1 Setting Up the Detector Emulator

Configure the DT4800 module as follows:

Rate = 1kcps
Shape = variable between 0.8us e 2us. Set 1.8us
Amplitude = 6000

6.2 Setting Up the DT1260 Module

Connect the output of the detector emulator (ie DT4800) to the input of the DT1260 module. Open Resource Explorer and connect to the DT1260 module.

You will see the following resource tree:

Configure the DT1260 registers as follows:

GAIN = 2000 means a scale of factor $\frac{2000}{65535}= 0.03$ on the QLONG integration for the measurement of the charge.
PSD_GAIN = 1000 means a scale of factor $\frac{1000}{16384}= 0.06$ on the PSD ratio.

Check the signal on the Oscilloscope. Set signal as Signed Data Type because the PSD data_out port is SIGNAL-BASELINE. You should see a pulse, with baseline oscillating around 0. The signal should be above the threshold and generating a trigger pulse. You should see the following signal on the oscilloscope:

D0 is the trigger signal.
D1 is the QLONG signal.
D2 is the QSHORT signal.
D3 is the data ready signal used to trigger the histogram.

6.3 Understanding the behavior of the PSD module

Sweep the tau between 0.8us and 2us.

$ Q_{\mathrm{LONG}} $ depends on $\tau$, consider $ T_{\mathrm{LONG}} = 4.5\mu\mathrm{s} $ and $\tau$ ranging from $0.8$ to $2.0\mu\mathrm{s}$.

On the energy spectrum, you should observe a nearly log‐like behavior as $\tau$ increases linearly, because the energy spectrum here is effectively the histogram of $Q_{\mathrm{LONG}}$, where

$$ Q_{\mathrm{LONG}}(\tau) = A \tau \Bigl[1 - e^{-T_{\mathrm{LONG}}/\tau}\Bigr] $$

depends on $\tau$ such that for very short $\tau$, the exponential term rapidly decays within the integration window, while for larger $\tau$, the pulse is more spread out, increasing $Q_{\mathrm{LONG}}$ but at a diminishing rate.

In other words, $Q_{\mathrm{LONG}}$ grows monotonically with $\tau$, but it tends to plateau as $\tau$ becomes very large compared to the fixed integration time $T_{\mathrm{LONG}}$.

Plot Example ($T_{\mathrm{LONG}} = 4.5\mu\mathrm{s}$, $\tau \in [0.8, 2.0]$)

Suppose the pulse amplitude $A$ is normalized to 1. Then: $$ Q_{\mathrm{LONG}}(\tau)=\tau \Bigl[1 - e^{-4.5/\tau}\Bigr]. $$ Let’s evaluate $Q_{\mathrm{LONG}}$ at a few points:

$\tau$ ($\mu\mathrm{s}$)	Expression	$Q_{\mathrm{LONG}}$ (a.u.)
0.8	$0.8[1 - e^{-4.5/0.8}]$	$\approx 0.80$
1.0	$1.0[1 - e^{-4.5/1.0}]$	$\approx 0.90$
1.5	$1.5[1 - e^{-4.5/1.5}]$	$\approx 1.35$
2.0	$2.0[1 - e^{-4.5/2.0}]$	$\approx 1.79$

At $\tau=0.8\mu\mathrm{s}$, the pulse decays quickly within the 4.5 µs window, so $Q_{\mathrm{LONG}}$ approaches $\approx 0.8$.

At $\tau=2.0\mu\mathrm{s}$, the pulse is more stretched, so the integrated area is larger ($\approx 1.79$), but we see it’s starting to plateau—further increases in $\tau$ yield diminishing growth in $Q_{\mathrm{LONG}}$.

6.4 Energy spectrum example

Load now an energy spectrum in the detector emulator

You will observe the Co60 spectrum.

Because we have a fixed decay time, the energy spectrum appear as a bend of points in the 2D-histogram. You can clearly see the two peaks of the Co60 spectrum.

Now change the decay time to 0.8us and you will see:

the energy spectrum is shifted to the left, because the QLONG integrated charge is smaller than before.
the PSD y is both compressed and shifted to lower values, because the 1/PSD ratio is smaller than before.

Spectrum changing the decay time to 0.8us

Below is a conceptual explanation of why, in a certain regime, plotting $\text{PSD}$ vs. $Q_{\mathrm{LONG}}$ yields an approximately linear dependence on the decay constant $\tau$. This relies on the standard exponential‐pulse assumptions and on choosing integration windows so that one window is “much longer” and the other is “somewhat shorter” than $\tau$.

7. Linear approximation of the PSD ratio vs $\tau$.

For an (ideal) exponential pulse of amplitude $A$ and decay constant $\tau$: $$ V(t) = A e^{-t/\tau}, \quad t \ge 0, $$ we define two integrations over fixed time windows:

Long integration ($T_{\mathrm{LONG}}$) $$ Q_{\mathrm{LONG}}(\tau) =\int_{0}^{T_{\mathrm{LONG}}} Ae^{-t/\tau}\mathrm{d}t =A\tau \bigl[1 - e^{-T_{\mathrm{LONG}} / \tau}\bigr]. $$
Short integration ($T_{\mathrm{SHORT}}$) $$ Q_{\mathrm{SHORT}}(\tau)=A\tau \bigl[1 - e^{-T_{\mathrm{SHORT}} / \tau}\bigr]; $$

The PSD ratio is: $$ \text{PSD}(\tau)=\frac{Q_{\mathrm{SHORT}}(\tau)}{Q_{\mathrm{LONG}}(\tau)}=\frac{1 - e^{-T_{\mathrm{SHORT}}/\tau}}{1 - e^{-T_{\mathrm{LONG}}/\tau}}. $$

The “Linear” Regime

Although $Q_{\mathrm{LONG}}$ and $\text{PSD}$ in general each follow a nonlinear function of $\tau$, there is a common practical regime:

$T_{\mathrm{LONG}} \gg \tau$ (the long gate is sufficiently long compared to $\tau$):
- Then the exponential term $e^{-T_{\mathrm{LONG}}/\tau}$ is very small, so
  $$ Q_{\mathrm{LONG}}(\tau) \approx A\tau, $$ i.e., $Q_{\mathrm{LONG}}$ grows $\propto \tau$ (a linear dependence on $\tau$ in this range).
$T_{\mathrm{SHORT}}$ is comparable to or somewhat larger than $\tau$, but still $T_{\mathrm{SHORT}} < T_{\mathrm{LONG}}$.
- Then $Q_{\mathrm{SHORT}}(\tau)$ is neither fully saturated nor negligible, and the fraction
  $\displaystyle \frac{Q_{\mathrm{SHORT}}}{Q_{\mathrm{LONG}}} $
  varies fairly smoothly as $\tau$ changes.

Parametric Plot: PSD vs. $Q_{\mathrm{LONG}}$

If we treat $\tau$ as the parameter and plot:

X‐axis: $Q_{\mathrm{LONG}}(\tau) \approx A\tau$,
Y‐axis: $\text{PSD}(\tau) = \tfrac{Q_{\mathrm{SHORT}}(\tau)}{Q_{\mathrm{LONG}}(\tau)}$,

then in the region where $Q_{\mathrm{LONG}}\approx A\tau$:

Changing $\tau$ linearly shifts $Q_{\mathrm{LONG}}$.
$\text{PSD}(\tau)$ also changes smoothly (almost linearly or near‐linearly) as $\tau$ goes from smaller to larger values, because the short gate captures a fraction of the total exponential that is neither saturating to 0 nor to 1.

The result can look roughly linear on a 2D scatter/line plot (i.e., $\text{PSD}$ vs. $Q_{\mathrm{LONG}}$), especially over a moderate range of $\tau$. In other words, as $Q_{\mathrm{LONG}}$ increases “in step” with $\tau$, the PSD ratio also increases (or decreases) in a manner that often appears straight over the limited interval you measure.

Illustrative Example

Long gate: $T_{\mathrm{LONG}} = 5\mu\mathrm{s}$.
Short gate: $T_{\mathrm{SHORT}} = 1\mu\mathrm{s}$.
Amplitude: $A = 1$ (normalized).

For $\tau$ from 0.5 µs to 2.0 µs, one finds numerically:

$\displaystyle Q_{\mathrm{LONG}}(\tau) \approx \tau[1 - e^{-5/\tau}]$.
$\displaystyle \text{PSD}(\tau) = \dfrac{1 - e^{-1/\tau}}{1 - e^{-5/\tau}}$.

For $\tau$ in this range, $Q_{\mathrm{LONG}}(\tau)$ grows nearly linearly with $\tau$, and $\text{PSD}(\tau)$ changes smoothly. Plotting these against each other (with $\tau$ only as an internal parameter) often yields a curve that is close to linear for part of the domain.

Conclusion

Although $Q_{\mathrm{LONG}}$ and the PSD ratio each follow a nonlinear exponential dependence on $\tau$ in general, in the common practical regime (where $T_{\mathrm{LONG}}\gg\tau$ but the short gate is not too small), the quantity
$$ Q_{\mathrm{LONG}}(\tau) \approx A\tau \quad\text{and}\quad \text{PSD}(\tau) \approx \frac{Q_{\mathrm{SHORT}}(\tau)}{Q_{\mathrm{LONG}}(\tau)} $$ can produce an approximately linear relationship when plotted (PSD vs. $Q_{\mathrm{LONG}}$) over a moderate range of $\tau$. This is because $Q_{\mathrm{LONG}}$ is scaling almost linearly with $\tau$, while the PSD fraction also changes in a somewhat linear fashion in that specific range—thus giving the appearance of a straight line in the 2D plot.

Repository

You will find the full project in the Gitlab repository.

\(\tau\) (\(\mu\mathrm{s}\))	Expression	\(Q_{\mathrm{LONG}}\) (a.u.)
0.8	\(0.8[1 - e^{-4.5/0.8}]\)	\(\approx 0.80\)
1.0	\(1.0[1 - e^{-4.5/1.0}]\)	\(\approx 0.90\)
1.5	\(1.5[1 - e^{-4.5/1.5}]\)	\(\approx 1.35\)
2.0	\(2.0[1 - e^{-4.5/2.0}]\)	\(\approx 1.79\)

Lab 18 - Implementing a Pulse Shape Discriminator (PSD) Based on Ratio between areas