Signal pre-processing

To accommodate the discrepancies across channels, the channel-wise baseline and noise must be calculated within each packet. The baseline was determined by averaging the first 10 sample points of each channel, and the noise was calculated using the root-mean-square value of these points. In the selection process, if any sample point exceeds the minimum signal level required for electronics processing, the operation is stopped, and the region is deemed an effective signal segment. The baseline and noise were recalculated by moving back 20 points (adjusted for the pulse width). Considering both the baseline and noise, the threshold was measured as the sum of the baseline and four times the noise. If the sample point exceeded the threshold, the threshold was subtracted, whereas a sample point that was less than or equal to the threshold was replaced with zero.

Only signals with more than three consecutive sample points above the threshold were considered as valid pulse signals. The packet payload records the total quantity of these overthreshold sample points (Clust_L), the serial number of the first overthreshold points in the packet (Clust_T), and the corresponding data (D). Multiplying Clust_T by the ADC sampling interval (50 ns) gives the offset time of the pulse relative to the coarse timestamp in the packet header. Following zero compression, the total quantity of data (ChLen) in each channel should include all pulse data points and the corresponding descriptive information (Clust_L and Clust_T). This approach significantly reduced the amount of data.

Feature extraction

When operating in trigger-based mode, event screening is based on trigger signals; while in trigger-less mode, it relies on high-precision time information. Therefore, implementing a timing module is essential for enhancing the accuracy of the inferred Clust_T. The constant fraction discriminator (CFD) [37] divides the original signal into two paths: one delays the signal for a certain time, whereas the other inverts and weakens the signal. By merging these two paths, constant fraction timing is converted into zero-crossing timing, thereby effectively mitigating the effects of time wandering. Because the ADC operates based on discrete sampling, the overthreshold time derived from CFD is often imprecise. Therefore, two data points were located before and after the overthreshold time (Va and Vb), and a linear equation or interpolation technique was applied to achieve a more precise time measurement. During the initial validation of the software simulation, we derived a linear equation for the amplitude and time of the superimposed signals based on Va and Vb: The exact moment of overthresholding corresponds to when the amplitude is zero. In the firmware, we use an interpolation method to approximate the threshold, which follows the principle of dichotomy [38]. We first calculated the mid-amplitude point (V_mid) based on Va and Vb by dividing the original time interval into two equal sub-regions (a and b). The original time interval is equivalent to the ADC sampling interval. If V_mid multiplied by Vb is negative, we can determine that the overthreshold moment is in sub-region b; otherwise, it is in the other sub-region. Therefore, we reduced the time interval corresponding to the overthreshold moment to half that of the original. By repeating this process n times, we can increase the timing accuracy by a factor of 2ⁿ compared to the original. Finally, the timing outcome was superimposed on Clust_T to provide a more precise pulse time. Clust_T is a 16-bit variable, and its high 10 bits are used to record the serial number of the data points within the packet (ranging from 0 to 1000) as the offset time relative to the coarse timestamp. The remaining six bits are utilized to record high-precision timing outcomes, which subdivides 50ns into 64 equal parts, thereby improving the time resolution.

Events can accumulate at high counting rates. The peak method can better distinguish overlapping signals compared to the area integration method. When several adjacent sample points first increase and then decrease, the peak (Clust_E) is at the vertex. After the energy extraction, only Clust_T and Clust_E were recorded for each pulse. We used the ChLen field to denote the pulse count for each channel to facilitate subsequent sorting.

Multiplicity screen

When particles enter the detector, there is usually an angle between the incoming direction and detector plane. Transverse diffusion occurs after the particle incidence, enabling multiple readout strips to produce induction signals. The signals have consistent timestamps, but different levels of magnitude across multiple readout channels, which is called particle multiplicity [22] as shown in Fig. 5a. Thus, a multiplicity screen was required to recover the original hit position of the particle. When merging the particle multiplicity, the first step is to arrange the pulses of all channels within the packet in chronological and channel order. Because all the pulses in a packet share the same coarse timestamp (TCode), they only need to be sorted according to Clust_T, which is known as fine timestamp sorting. Figure 5b shows the workflow of fine timestamp sorting. Since the pulse data of each channel are stored in separate containers (FIFOs or arrays) in chronological order, only the first pulse data in each container were considered for comparison. Subsequently, the pulse data corresponding to the smallest moment obtained from this comparison were moved from the old container to the new container. In the second round, the second pulse data from the old container were compared with the first pulse data from the other containers. This process continues until all data in each container are compared and stored in the new container. If the pulse times in the two containers are equal, they are arranged in the channel order. If the time interval between the adjacent channel’s pulse signals is shorter than the designated time window, they are categorized as the same event and can be merged according to the channel order, as illustrated in Fig. 5c. The Clust_T value was determined based on the fine timestamp of the central channel, and a count field was added to record the total number of pulses in each event. This information is then used in the CG to identify the hit’s position of the incident particle.

Fig. 5Full-screen View

(Color online)Schematic diagram of (a) Particle multiplicity,(b) Fine timestamp sorting, and (c) Fine timestamp merging

Using CG (in Eq. (1)), the hit’s position (X_hit) of each particle on the readout strip was determined based on the previously obtained channel-fired ID and energy value. The channel-fired ID was used as the xi coordinate, and the energy value acted as the weight Ei. The drift time of the electrons is measured as the sum of the coarse timestamp and fine timestamp. Both the hit’s position and drift time from the twin TPCs were relayed to the hit-matching module, and the two-dimensional trace of the incoming particles was ultimately reconstructed. This process also helps discard invalid or irrelevant data. $X_{\text{hit}}={\displaystyle \sum x_i}\cdot E_i/{\displaystyle \sum E_i}$ (1)

Hit matching

In Eq. (2), the drift distance (L) and drift speed (v_drift) of electrons for twin TPCs are predetermined. As a result, the sum of the drift time (t_cs) remains fixed and can be used as a constraint to achieve hit-matching [22] of twin TPCs. Although the plastic scintillator detector (T₀) can provide a highly accurate reference time (t₀), a large fluctuation in the particle flight time can negatively impact its accuracy. Moreover, t₀ information is transferred directly to the master DAQ, and system-wise event building is achieved by combining the data from the subsystem servers, which may reduce timeliness. Considering the aforementioned factors, we propose a novel hit-matching algorithm for reconstructing particle tracks without t₀ information. This allows event building to be performed on the subsystem server, thus enabling additional data compression and easing the computational burden on the master DAQ. $t_{\text{cs}}=t_u+t_d-2t_0=\frac{L}{v_{\text{drift}}}$ (2) where t_u, t_d is the absolute drift time of electrons from the twin TPCs and t₀ is the reference time from detector T₀.

The matching algorithm can be classified into two methods based on whether the factor t₀ is incorporated: the relative time method and the absolute time method.

1. the relative time method

(a) Search for all hits from twin TPCs in the range [$t_0-5{\sigma }_t\mathrm{,}{t}_0+t_{\text{dmax}}+5{\sigma }_t$] according to t₀.

(b) Time matching: For the searched hits, use their drift times and t₀ to calculate t_cs, then select the hit combinations within 5σ_tcs.

(c) Position matching: For hits that meet the time matching conditions, choose the combination with the smallest hit distance.

(d) Positioning: According to the drift time of the chosen hit, combined with the drift distance and drift speed, calculate the hit position in the Y direction.

2. the absolute time method

(a) Search for all all hits from the second TPC in the range [$t_u-t_{\text{dmax}}-6{\sigma }_t\mathrm{,}{t}_u+t_{\text{dmax}}+6{\sigma }_t$] according to tu.

(b) Time matching: For the searched hits, calculate $|t_u-t_d|$ and select the hit combinations within 12σt.

(c) Position matching: For hits that meet the time matching conditions, choose the combination with the smallest hit distance and minimize the energy difference.

(d) Positioning: According to the drift time of the chosen hit, combined with the drift distance and drift speed, calculate the hit position in the Y direction. ${\sigma }_{t_{cs}}=\sqrt{2{\sigma }_t{}^2+4{\sigma }_{t_0}{}^2}$ (3) where ${\sigma }_t\mathrm{,}{\sigma }_{t_0}\mathrm{,}{\sigma }_{\text{tcs}}$ are the time resolutions of the TPC detector, T₀ detector, and t_cs variable, respectively.

Simulation study

We use the Monte Carlo method [39] to generate the twin TPCs’ simulation packet. When particles enter the twin TPCs, electrons generated by ionization drift up and down, with the drift time based on their hits’ position in the Y direction, the total drift distance remains unchanged. The fired readout channels were determined by the hits’ position on the readout strips, incidence angle, and transverse diffusion, with a maximum of five channels. Therefore, the drift time and fired channels can be set according to the length and width of the TPCs’ incidence cross section. The FEAM chip exhibited a peak time of 160 ns and a falling edge at approximately 320 ns. As a result, the CR-(RC)³ characteristic equation is employed to derive accurate pulse data, and the signal amplitude range is defined by the selected ADC bit width. When the system’s counting rate reaches 1 MHz, a particle is expected to hit the twin TPCs every 1000 ns on average. Therefore, two sets of simulation waveform data were simultaneously produced every 1000 ns (t₀). The amplitude, fired channels, and drift time (with t₀ as the reference) for each waveform were randomly generated to mimic a real scenario accurately. Baseline noise obtained from existing FEE measurements was added to the simulation data to enhance authenticity. In addition, simulation data were produced for each TPC using only one FEE consisting of 32 channels.

Figure 6 shows the overall distribution of the simulation data for a pair of complete packets for the twin TPCs. Multiple simulation waveforms were created within a 50us period, traversing 32 different channels with varying peak amplitudes and covering most of the ADC range (bit width of 10). Figure 7(a) shows the simulated waveforms of the twin TPCs’ central channel fired at a specific time, which exhibit high similarity with only minor amplitude variations. The phase relationship between the two drift times covers the three cases of overrun, approach, and lag that satisfy the simulation requirements. Furthermore, Figure 7(b) shows the simulated waveforms after removing the baseline, which dropped to zero. The two TPC packets contained 219 and 210 pulses, respectively. In other words, an average of 6~7 pulses were generated on each readout channel in 50us time for both TPCs, so the average counting rate greater than 100 kHz/channel as expected, indicating that the simulation data closely resembles the actual situation.

Fig. 6Full-screen View

(Color online)Distribution of (a) The 1st TPC’ simulation data and (b) The 2nd TPC’ simulation data

Fig. 7Full-screen View

(Color online) Simulated waveforms of the twin TPCs’ central channel fired. (a) Raw simulated waveforms; (b) Waveforms after subtracting the baseline

After merging the multiplicities, the first TPC collected 50 sets of event data and the second TPC recorded 49 sets of event data. Subsequently, the hits’ positions and drift times of all event data are transmitted to the hit-matching module, where the absolute time method is employed for matching. 49 data sets were successfully matched, and only one event dataset was discarded due to matching failure. Finally, based on the matched drift time pairs, the drift distances were calculated, and the particle hit positions in the Y-direction were recovered, as depicted in Fig. 8. When the incident particle strikes the twin TPCs, three drift distances are possible: the first TPC’s drift distance may be greater or less than that of the second TPC, or the two drift distances may be equal; however, the sum of the drift distances in the two TPCs is always the maximum drift distance. Figure 8 shows three typical cases: (a) when the incident particle collides with the center of the twin TPCs and the two drift distances are equal; (b) when the particle impacts a position close to the first TPC’s readout strip, the second TPC experiences the maximum drift distance; and (c) if the hit is offset to the second TPC’s readout strip, the first TPC undergoes a larger drift distance than the second TPC. Theoretically, the particle should hit the same position on the readout strips of both TPCs. However, in practice, a slight deviation between the two hits’ positions is possible due to oblique incidence. Using the relative-time approach, the same datasets produced consistent results, as shown in Fig. 8.

Fig. 8Full-screen View

(Color online)Matched results obtained using the absolute time method from the twin TPCs’ simulation packets. (Two different symbols of the same color represent the hits’ positions of the same incident particle in two TPCs, respectively.)

To verify the performance of event-building algorithm at a high counting rate, the hit probability of the readout channel was adjusted from a uniform to Gaussian distribution. Furthermore, the probability of the readout channels in the central region being fired increases. As illustrated in Fig. 9a, four to five channels within the two TPC packets were fired more than 25 times in 50us, indicating a counting rate of 500 kHz/channel. Three sets of packets were analyzed and the matching results obtained using both methods were consistent, as shown in Fig. 9b. It was observed that both matching algorithms could effectively recover the two-dimensional traces of incident particles under normal conditions. However, the relative time method is more suitable for severe cases because it employs a more rigorous equation as a constraint. Therefore, the absolute time method can be utilized in the subsystem server to initially match hits, remove inconsistent data, and further reduce data volume. In addition, the relative-time technique can be employed in the master DAQ to accomplish system-wise hit identification.

Fig. 9Full-screen View

(Color online) (a) Number of fired for each channel in the twin TPCs; (b) Particles hit positions in the Y-direction obtained by hit matching in the 1st TPC: The red symbol represents the results acquired by the relative time method, while the blue symbol shows the outcomes obtained using the absolute time method

As an illustration, the current simulation parameters, that is, a single data packet containing 32 channels with 1000 sample points each and an additional 16-bit descriptive information as the raw data volume were used to evaluate the data compression capability for the event-building algorithm. The evaluation was based on the following assumptions.

1) A total of 50 particles are incident in 50 microseconds and each particle hits 5 readout channels, generating a total of 250 valid pulses.

2) After removing the baseline, an average of 15 sampling points were retained for each valid pulse.

3) The fine timestamps and energy information were each 16 bits.

4) The position information for the hits was 10 bits in both the X (readout strip) and Y directions.

The compression ratios for several stages are listed in Table 2, which indicates that the event-building algorithm can shrink 99% of the raw data and 97% of the zero-compressed data.

Test Verification

As shown in Fig. 10, a primary test system was set up in the laboratory using existing electronic devices. Note that the chassis of the remote server and slave DAQ contained within it are not visible in this figure. A signal generator (keysight33522B) [40] was employed to create a trigger signal and two-pulse signals. The two pulse signals were fed into the two FEAM chip’s analog inputs on the FEE to emulate the signals from two TPCs, respectively. The input signals were first converted into digital form by the ADC, and then transmitted using an optical fiber to the slave DAQ for aggregation and packaging. Finally, the data were sent to a remote server via a PCIe interface. The drift time of the electrons generated by the primary ionizing particles was simulated by adjusting the delay time of the input signal relative to the trigger signal. Multiple pulse signals collected by the FEE were then spliced together to mimic a trigger-less mechanism. With a sampling frequency of 50 MHz, the ADC collects 100 samples per pulse signal, implying a pulse period of 2 μs and a counting rate of 500 kHz/channel. Figure 11(a) illustrates the initial waveforms of the two FEE channels’ output at specific times. Next, the adaptive baseline subtraction module analyzes and removes any baselines from the data. The outcomes of this procedure are shown in Fig. 11(b). The results indicate the effectiveness of the module in cleaning the baseline noise.

Fig. 10Full-screen View

(Color online) Electronics test system

Fig. 11Full-screen View

(Color online) (a) Original waveforms; (b) Waveforms after subtracting the baseline

To verify the performance of the event-building algorithm using experimental data, we selected eight typical delay time pairs and five different amplitude combinations of input signals to conduct 40 tests, each consisting of eight pulses. The additional delay caused by the readout system must be subtracted before data processing. After processing, 320 test pulses were sent to the hit matching module, all of which were accurately matched using either absolute or relative time methods. The Y-direction positions where the particles hit were determined using 320 pairs of drift times. From this dataset, 20% of the data were extracted and plotted in Fig. 12, which shows that the results of the two matching algorithms were highly consistent. The key test parameters and compression ratios are shown in Table 3. The preliminary evidence shows that the event-building algorithm can effectively compress data.

Fig. 12Full-screen View

(Color online)Particles hit positions in the Y-direction obtained by hit matching in the 1st TPC: The red symbol represents the results acquired by the relative time method, while the blue symbol shows the outcomes obtained using the absolute time method