1. Core Idea of Position Sensing
For a neutron-sensitive tube read out at both ends (Side A and Side B), the principle is:
\[ \text{Position} = \frac{P_{sideA}}{P_{sideA} + P_{sideB}} \quad\text{and}\quad P_{\text{sum}} = P_{sideA} + P_{sideB}. \]
- \(P_{sideA}\) and \(P_{sideB}\) are the peak amplitudes (or integrated charges) measured at each end of the tube when a particle interacts somewhere along its length.
- If the two amplitudes are equal (i.e., \(P_{sideA} = P_{sideB}\)), the position is 0.5, meaning the interaction occurred in the center of the tube.
- If \(P_{sideA} \gg P_{sideB}\), the event is closer to side A; if \(P_{sideB} \gg P_{sideA}\), the event is closer to side B.
The PSA block:
- Performs baseline corrections and filters on each of the two input channels (A and B).
- Detects the peak amplitude for each channel and for the sum \(P_{\text{sum}}\).
- Provides triggers, flags (pile-up, overflow) and final amplitude outputs for each side, plus the sum.
2. Digital Processing Stages
Each channel (A or B) is processed by a chain of up to five sub-blocks (all implemented in fixed-point arithmetic):
- Baseline Correction (Block 1)
- Pole-Zero (PZ) Compensation
- First second-order Gaussian Filter Section
- Second second-order Gaussian Filter Section
- Final Baseline Correction (Block 2)
At the end of this chain, the signal is shaped approximately like a “quasi-Gaussian” pulse, with zero baseline and minimal low-frequency drift.
2.1 Baseline Correction
The baseline correction aims to remove DC offset and ensure the signal baseline is around zero. One typical approach is:
- Break the data stream into chunks of length \(N\).
- In each chunk, find the minimum sample value, denoted \(i[n]\).
- Filter this minimum value with a first-order low-pass:
\[ w[n] = \frac{1}{k} \, i[n] \;+\; \Bigl(1 - \frac{1}{k}\Bigr) \, w[n-1] \]
where \(k\) is typically a power of two (e.g., 32).
4. Subtract this estimate \(w[n]\) from the samples in the next chunk to shift the baseline:
\[ u[N \cdot n + m] \;=\; i[N \cdot n + m] \;-\; w[n], \quad 0 \le m < N. \]
This “running minimum” approach can slightly under-correct in the presence of noise, but ensures that pulses remain well above zero.
2.2 Pole-Zero Compensation
A pole-zero compensation corrects for a large decay-time constant \(\tau_l\) in the incoming signal—often due to a charge amplifier or a preamplifier with a slow exponential decay. We want effectively a much shorter decay time \(\tau_s\).
The continuous-time PZ transfer function is often written as:
\[ H(s) \;=\; \frac{s + \tfrac{1}{\tau_l}}{s + \tfrac{1}{\tau_s}}, \]
where \(\tau_s\) is chosen to be significantly shorter than \(\tau_l\). We can rewrite:
\[H(s) = \frac{s + \tfrac{1}{\tau_l}}{s + \tfrac{1}{\tau_s}}= 1 \;+\;\frac{\tfrac{1}{\tau_l} \; -\; \tfrac{1}{\tau_s}}{s + \tfrac{1}{\tau_s}}. \]
When applying a discrete-time approximation (for instance via an impulse-invariance or matched-\(z\) transform), one arrives at a two-stage approach:
- A first-order IIR filter to approximate \([1/(1 + s\tau_s)]\).
- A direct feed-through plus a weighted version of that IIR to implement the difference \(\frac{1/\tau_l - 1/\tau_s}{s + 1/\tau_s}\).
In discrete form, one can approximate \(e^{-T_s/\tau_s} \approx 1 - T_s/\tau_s\) (assuming \(\tau_s \gg T_s\)). Then the equations become something like:
\[ w[n] \;=\; i[n] \;+\; \Bigl(1 - \frac{T_s}{\tau_s}\Bigr)\, w[n-1], \] \[ u[n] \;=\; i[n] \;+\; \Bigl(\frac{T_s}{\tau_l} \; -\; \frac{T_s}{\tau_s}\Bigr)\, w[n]. \]
These filters typically work in fixed point by scaling the coefficients (\(T_s / \tau_s\), etc.) by a power of two (e.g., \(2^{15}\)).
2.3 Gaussian Filtering
Each side is then put through a 4th-order Gaussian filter, implemented in two sequential 2nd-order IIR stages. The analog Gaussian filter’s transfer function can be approximated by:
\[ H(s) \;=\; \frac{4.899}{ 4.899 \; +\;11.42\,s\; +\;10.87\,s^2\; +\;5.073\,s^3\; +\;s^4 }, \] with poles at \(-1.35536 \pm j\,0.32795\) and \(-1.18108 \pm j\,1.0604\).
For a specified shaping time \(\tau\), one must divide these poles by \(2 \pi \tau\). To bring that into the discrete domain, a common approach is the matched-\(z\) transform (or impulse invariance for single exponentials). Each pair of complex-conjugate poles yields a 2nd-order IIR section:
\[ (1 - a\,z^{-1} + b\,z^{-2}), \]
where:
\[ a_1 = 2\, e^{-\alpha_1\,T} \,\cos(\beta_1\,T), \quad b_1 = e^{-2\alpha_1\,T}, \] and similarly for \(a_2, b_2\). Here:
- \(\alpha_1,\beta_1\) come from the real and imaginary parts of the first pair of poles, \(-\alpha_1 \pm j\beta_1\).
- \(\alpha_2,\beta_2\) come from the second pair.
If we let \(T = \tfrac{T_s}{2\pi\tau}\) with \(T_s\) the sampling period (e.g., \(T_s = 1/F_{\text{ADC}}\)):
\[ a_1 = 2\, e^{-1.355\,T}\,\cos(0.3279\,T), \quad b_1 = e^{-2.711\,T}, \] \[ a_2 = 2\, e^{-1.181\,T}\,\cos(1.0604\,T), \quad b_2 = e^{-2.362\,T}. \]
One then implements the recursive difference equation (per 2nd-order section):
\[ u[n] \;=\; i[n] \;+\; a_k\,u[n-1] \;-\; b_k\,u[n-2], \] where \((a_k, b_k)\) correspond to \((a_1, b_1)\) or \((a_2, b_2)\). After the first 2nd-order stage, the output is fed into the second 2nd-order stage to achieve the 4th-order response.
Note: Because we skip the explicit finite zeros that might appear in the full partial fraction expansion, the implementation is simpler (fewer multiplications), but still yields a near-Gaussian impulse response for typical shaping times.
3. Trigger and Peak Detection
Once the shaped signals from Side A and Side B are generated, the PSA logic can:
- Sum the two channels:
\[ P_{\text{sum}}(n) = \text{Filter}_A(n) + \text{Filter}_B(n). \] - Compare \(P_{\text{sum}}\) (or another selected trigger source) with a threshold (\(\text{TRG\_THRES}\)).
- When the threshold is crossed, it starts a window for peak finding:
- The system searches for the global maximum in \(P_{\text{sum}}(n)\) within the gate window.
- At that same time index (where \(P_{\text{sum}}(n)\) is max), it records the side A and side B shaped amplitudes:
\[ P_{sideA} = \text{Filter}_A(n{\text{max}}) \]
\[ P_{sideB} = \text{Filter}_B(n{\text{max}}) \]
- The ratio \(\frac{P_{sideA}}{P_{sideA} + P_{sideB}}\) then gives the position.
Further logic checks for pile-up (another pulse arriving within the same gate), overflow (signal saturations), etc.
4. Practical Steps in Simulation
-
Signal Generator Settings
- Two channels, A0 and A1, each with a certain amplitude, decay time, and offset.
- For example, if A0 = A1 in amplitude, you expect the measured position to be ~0.5.
-
Filtering and PZ
- In the simulation, the baseline correction removes the DC offset.
- The pole-zero stage compensates for preamp decay times if needed.
- The 4th-order Gaussian filter shapes the pulse to a clean “bell-like” shape, easier to measure its peak.
-
Changing Amplitudes
- By adjusting the amplitude of A0 or A1, one can see how \(P_{sideA}\) vs. \(P_{sideB}\) changes and, correspondingly, how \(\tfrac{P_{sideA}}{P_{\text{sum}}}\) shifts the detected interaction point.
-
Outputs
- The final shaped peaks for side A and side B are latched at the moment the sum is at maximum.
- The simulation or hardware readout then delivers \(P_{sideA}\), \(P_{sideB}\), \(P_{\text{sum}}\), plus flags and time stamps.
5. Summary of Key Formulas
-
Position: \[ \text{pos} = \frac{P_{sideA}}{P_{sideA} + P_{sideB}}. \]
-
Pole-Zero (continuous): \[ H(s) = \frac{s + \tfrac{1}{\tau_l}}{s + \tfrac{1}{\tau_s}} \;=\; 1 \;+\; \frac{\tfrac{1}{\tau_l} - \tfrac{1}{\tau_s}}{s + \tfrac{1}{\tau_s}}. \] (Discrete approximation done via first-order IIR.)
-
Gaussian Filter
Two 2nd-order IIR sections, each of the form: \[ u[n] = i[n] \;+\; a\,u[n-1] \;-\; b\,u[n-2], \] with \[ a = 2\,e^{-\alpha\,T}\,\cos(\beta\,T), \quad b = e^{-2\alpha\,T}, \] for each conjugate pole pair \(-\alpha \pm j\beta\). -
Baseline Correction:
- Min-based approach: \[ w[n] = \frac{1}{k}\, i[n] + \Bigl(1-\frac{1}{k}\Bigr)\, w[n-1], \quad \text{then} \quad u[n] = i[n] - w[\lfloor n/N\rfloor]. \]
6. Block Diagram Commentary and Simulation Guide:
In the block diagram, the two analog inputs (A0 and A1) feed into the Position Sense Detector (U0), which applies sequential stages—baseline correction, pole-zero compensation, multi-stage Gaussian filtering, and peak detection—to produce the outputs \(P_{sideA}\), \(P_{sideB}\), and their sum \(P_{\text{sum}}\).
To run the simulation, first configure the “A0” and “A1” signal generators by specifying amplitude, delay, and decay time (\(\tau\)) and click “Generate” to create the pulse events.
Then launch the simulation, where the shaped signals will appear as exponential decays.
When each pulse reaches its peak, U0 captures \(P_{sideA}\), \(P_{sideB}\), and \(P_{\text{sum}}\) numerically in the waveform viewer. By changing the amplitude on A1 (for example, raising the “Amplitude (ADC)” from 600 to a higher value), you will see \(P_{sideB}\) increase and the resulting position ratio \(\tfrac{P_{sideA}}{P_{sideA} + P_{sideB}}\) shift accordingly—moving the detected event closer to side B. In this way, you can verify how altering the pulses on A0 and A1 affects the simulated position along the detector, confirming the expected change in the peak values and output ratio.
Coefficient Calculation
Below is an example of how to compute the filter and pole-zero coefficients for a 1 µs decay time (\(\tau = 1\,\mu\mathrm{s}\)), 1 µs preamplifier decay, and a 125 MSPS (125 MHz) ADC clock. We define:
\[ \mathrm{CLOCK\_FREQUENCY} = 125000000.0 \quad,\quad \tau = 1\times 10^{-6} \quad,\quad pz = 0.1\times 10^{-6}. \]
First, compute the parameter \(TS\): \[ TS = \frac{2\,\pi}{\tau \,\times\, \mathrm{CLOCK\_FREQUENCY}}. \]
Then the four Gaussian filter coefficients are calculated by:
\[ \text{coeff11} = \Bigl\lfloor \exp\bigl(-2.71072\,TS\bigr)\;\times\;16384 \; +\; 0.5 \Bigr\rfloor, \] \[ \text{coeff12} = \Bigl\lfloor 2 \;\times\; \exp\bigl(-1.35536\,TS\bigr)\;\times\;\cos\bigl(0.327948\,TS\bigr)\;\times\;16384 \; +\; 0.5 \Bigr\rfloor, \] \[ \text{coeff21} = \Bigl\lfloor \exp\bigl(-2.36216\,TS\bigr)\;\times\;16384 \; +\; 0.5 \Bigr\rfloor, \] \[ \text{coeff22} = \Bigl\lfloor 2 \;\times\; \exp\bigl(-1.18108\,TS\bigr)\;\times\;\cos\bigl(1.06037\,TS\bigr)\;\times\;16384 \; +\; 0.5 \Bigr\rfloor. \]
Lastly, the pole-zero compensation coefficient is: \[ \text{pzcoeff2} = \Bigl\lfloor \frac{32768}{\mathrm{CLOCK\_FREQUENCY}\,\times\,pz} \Bigr\rfloor. \]
For the parameters specified above (i.e., 1 µs decay, 125 MHz sampling), these integer values typically evaluate to:
- coeff11 = 0x37d9
- coeff12 = 0x778e
- coeff21 = 0x38d6
- coeff22 = 0x7873
- pzcoeff2 = 0x106
7 PSA I/O and settings user guide
Below is a summary of the main inputs, outputs, and configuration parameters for the PSA (Position Sense Amplitude) core. The signals are grouped by function (Inputs, Outputs, and Configuration). Where applicable, enumerated parameter values are shown in tables.
1. Inputs
| Signal | Size | Description |
|---|---|---|
| AN_IN0 | 16 bits | Analog data from ADC – Side A. |
| AN_IN1 | 16 bits | Analog data from ADC – Side B. |
| CNTRL_RUN | 1 bit | When set to 1, enables the core. |
| TRIG_EXT | 1 bit | External trigger input (always enabled). |
| TRIG_VETO | 1 bit | Global trigger veto input. |
| COINC_IN | 1 bit | Coincidence-validation input. If coincidence is required (via register), this must go high for at least 1 clock cycle during the gate window. |
| TIMESTAMP | 64 bits | Free-running external timestamp input. |
| CLK | 1 bit | System clock (e.g., 80 MHz or 125 MHz, depending on hardware). |
| RESET | 1 bit | Resets the filtering process (global reset). |
2. Outputs
| Signal | Size | Description |
|---|---|---|
| AN_PROBE0 | 16 bits | Analog probe for side A (selected by PRB_SEL). |
| AN_PROBE1 | 16 bits | Analog probe for side B (selected by PRB_SEL). |
| D_PROBEx_0 | 16 bits | Digital probe for side A/B (Trigger signal). |
| D_PROBEx_1 | 16 bits | Digital probe for side A/B (Over threshold). |
| D_PROBEx_2 | 16 bits | Digital probe for side A/B (Gate). |
| D_PROBEx_3 | 16 bits | Digital probe for side A/B (Energy Ready). |
| OR_TRIGGER | 1 bit | Single-clock pulse triggered from side A OR side B. |
| TOTs | 2 bits | Over-threshold signals from trigger circuit. |
| TRIGGER | 1 bit | Trigger output for coincidence purposes (width set by TRG_OUT_W). |
| SIDE_A | 16 bits | Energy measured on side A (valid only when DV=1). |
| SIDE_B | 16 bits | Energy measured on side B (valid only when DV=1). |
| SIDE_B (sum) | 16 bits | Sum of side A + side B energies (valid only when DV=1). (Sometimes labeled SIDE_SUM.) |
| PILEUP_A | 1 bit | Pile-up flag on side A (a second trigger occurred during the gate window). Valid only when DV=1. |
| PILEUP_B | 1 bit | Pile-up flag on side B (a second trigger occurred during the gate window). Valid only when DV=1. |
| OVR_A | 1 bit | Overflow occurred on side A (invalid energy). Valid only when DV=1. |
| OVR_B | 1 bit | Overflow occurred on side B (invalid energy). Valid only when DV=1. |
| TIMESTAMP | 64 bits | Timestamp captured on trigger edge. |
| DATA | 128 bits | Output packet in “list format.” |
| DV | 1 bit | Data Valid indicator. |
3. Configuration Parameters
These parameters control the internal filtering, trigger modes, coincidences, and readout. Most are 32-bit registers.
3.1 Polarity and Offset
| Parameter | Size | Meaning |
|---|---|---|
| POL | 32 bits | Polarity of input signal. See table below. |
| OFFSET | 32 bits | Offset subtracted from the 12-bit input before polarity inversion. |
POL (Polarity) Values:
| Value | Meaning |
|---|---|
| 0 | Positive |
| 1 | Negative |
3.2 Trigger-Related
| Parameter | Size | Meaning |
|---|---|---|
| TRG_SOURCE | 32 bits | Selects trigger data path (shaper output or raw input). See table below. |
| EN_COINC | 32 bits | Require external validation on COINC_IN. |
| TRG_MASK | 32 bits | Mask each side’s internal trigger. |
| TRG_MODE | 32 bits | Internal trigger mode (sum, no trigger, OR, AND). See table below. |
| TRG_THRES | 32 bits | Trigger threshold (0–65535). |
| TRG_HIST | 32 bits | Trigger hysteresis (subtracted on falling edge to avoid re-trigger on slow decay). |
| TRG_HOLD | 32 bits | Trigger holdoff time (in clock cycles), max 4095. |
| TRG_OUT_W | 32 bits | Width (in samples) of the output TRIGGER signal. |
TRG_SOURCE:
| Value | Meaning |
|---|---|
| 0 | Use shaper output for trigger |
| 1 | Use input signal for trigger |
EN_COINC:
| Value | Meaning |
|---|---|
| 0 | No external validation (DV always active) |
| 1 | External validation required (COINC_IN) |
TRG_MASK:
| Value | Meaning |
|---|---|
| 0 | Both sides enabled |
| 1 | A side masked |
| 2 | B side masked |
| 3 | Both masked |
TRG_MODE:
| Value | Meaning |
|---|---|
| 0 | Use sum (A + B) for trigger |
| 1 | No internal trigger |
| 2 | Use OR of side A & side B |
| 3 | Use AND of side A & side B |
3.3 Filter Settings
| Parameter | Size | Meaning |
|---|---|---|
| FLT_CFG | 32 bits | Bitfield configuration for filtering. |
| COEFF11 | 32 bits | 1st Gaussian filter coefficient (see formula). |
| COEFF12 | 32 bits | 1st Gaussian filter coefficient (see formula). |
| COEFF21 | 32 bits | 2nd Gaussian filter coefficient (see formula). |
| COEFF22 | 32 bits | 2nd Gaussian filter coefficient (see formula). |
| PZCOEFF | 32 bits | Pole-zero compensation coefficient (see formula). |
| PRB_SEL | 32 bits | Select analog monitor output (probe point). |
FLT_CFG is typically started at 0x1F33 and includes bits for:
- bits [3..0]: Attenuation (Section 1)
- bits [7..4]: Attenuation (Section 2)
- bit 8: Enable baseline correction (input)
- bit 9: Enable pole-zero compensation
- bit 10: Enable first second-order Gaussian filter section
- bit 11: Enable second second-order Gaussian filter section
- bit 12: Enable baseline correction (output)
PRB_SEL (Analog Monitor Selector):
| Value | Probed Signal |
|---|---|
| 0 | Signal with corrected offset and polarity |
| 1 | Signal with corrected offset only |
| 2 | Main filter output |
| 3 | Sum of both side main filter outputs |
| 4 | Peak detector output |
| 5 | Peak detector output on sum signal |
| 6 | Input baseline-correction algorithm output |
| 7 | Pole-zero compensation output |
| 8 | First filter algorithm output (divided by 2) |
| 9 | Second filter algorithm output (divided by 2) |
| A | Output baseline algorithm (divided by 2) |
3.4 Clock and Pole-Zero Formulas
-
COEFFxxare computed as: \[ \text{int}\bigl(\exp(\dots)\times 16384 + 0.5\bigr), \quad \text{int}\bigl(2\times \exp(\dots)\times\cos(\dots)\times 16384 + 0.5\bigr), \dots \] depending on the Gaussian poles and sample rate. -
PZCOEFFis typically: \[ \text{int}\Bigl(\frac{32768}{(\mathrm{CLOCK\_FREQUENCY}\times pz)}\Bigr). \]
3.5 Notes on Use
- Configure FLT_CFG to enable/disable the relevant processing blocks (baseline, pole-zero, Gaussian).
- Set TRG_THRES and TRG_HIST carefully to avoid multiple triggers on a single pulse.
- Adjust offset and polarity if the incoming signal is negative.
- Monitor AN_PROBEx signals for diagnostic checks (to ensure no saturation occurs in the filter chain).
Conclusion
Using this chain (Baseline → PZ → Gaussian), each channel’s raw pulse is converted into a clean, peaked shape whose maximum is straightforward to measure. The system then captures the peak amplitudes simultaneously on both ends of the tube, and the charge (or amplitude) ratio provides the position of the incident neutron. By adjusting the amplitudes in the simulator (e.g., via A0, A1), one directly observes how \(P_{sideA}\) and \(P_{sideB}\) vary, and thus how the measured position shifts.
This approach is widely used for position-sensing He-3 or other gas-filled tubes and can be extended to multi-wire proportional chambers or other detectors requiring charge-division readout.