A Full Rank Pileup Deconvolution Scheme Suitable for Calorimeter Online Trigger Primitive Generation

This paper presents a full-rank pileup deconvolution scheme for calorimeter online trigger primitive generation that eliminates the need for mathematical assumptions like sparse representation by leveraging a determined system where the number of ADC samples matches the number of unknowns.

The Gist

The Big Problem: The "Echo" in the Room

Imagine you are in a large, echoey hall. Every time someone claps, the sound doesn't just stop; it lingers and fades out slowly. If someone else claps just a split second later, their new clap mixes with the fading echo of the first one.

In high-energy physics experiments (like those at the Large Hadron Collider), detectors act like that hall. When particles hit the detector, they create a signal (a "clap"). But because the detector is so sensitive, the signal from one hit takes a little while to die down. If another particle hits while the first signal is still fading, the two signals pile up and blend together into a messy wave.

Scientists need to know exactly when each particle hit and how strong it was. But right now, they are looking at a messy, blended wave and trying to guess where the individual claps happened.

The Old Way: Guessing with Math

Usually, when scientists try to untangle this mess (a process called deconvolution), they don't have enough information. Imagine trying to figure out who clapped and when, but you only have a recording of the last 5 seconds, while the echoes from the 6th and 7th seconds ago are still hanging in the air.

Because they are missing the "past" data, they have to make mathematical guesses. They assume, for example, that "claps are rare" (a concept called Sparse Representation). They force the math to find the solution that uses the fewest possible claps to explain the noise. It's like solving a puzzle where you are missing half the pieces, so you have to guess what the missing picture looks like based on the idea that the picture is usually simple.

The New Idea: The "Full Recording"

This paper proposes a new way to solve the puzzle, specifically for online triggers (the fast computers that decide which data to save in real-time).

The author, Jinyuan Wu, points out a key difference between "offline" analysis (looking at data later) and "online" processing (looking at data right now):

• Offline: You only get a small window of data. You are missing the past.
• Online: The computer (FPGA) is connected directly to the detector. It has access to every single sample of data as it happens, not just a small window.

The Analogy:
Imagine you are trying to figure out who spoke in a conversation.

• The Old Way: You only hear the last 10 seconds of the conversation. You have to guess who spoke before that based on the assumption that people don't talk too much.
• The New Way: You have a recording of the entire conversation from the very beginning. You know exactly when the silence started. Because you have the full history, you don't need to guess. You can mathematically calculate exactly who spoke and when, without any assumptions.

How It Works: The "Sliding Window"

The paper describes a method to untangle the signals step-by-step:

1. Find a Quiet Spot: The system waits for a moment when the accelerator is silent (a "beam gap"). At this moment, we know for a fact that no particles hit the detector. The "echo" is zero.
2. Solve the First Puzzle: Using this quiet starting point, the computer solves the math for the next few seconds. It figures out exactly what the signals were.
3. Pass the Baton: Once it solves the first chunk, it takes the "tail end" of that solution and uses it as the "past history" for the next chunk of time.
4. Repeat: It keeps sliding forward, using the known past to solve the present.

Because the system has a "full rank" matrix (a fancy math way of saying the puzzle has a unique, perfect solution) and doesn't need to guess, it can separate signals that are very close together, even if their peaks look merged.

The Results: Clean and Stable

The author tested this with a computer simulation:

• The Test: They created a fake detector signal with random "claps" (particle hits) and added some static noise (like radio interference).
• The Result: The new method successfully separated the claps, even when they were right next to each other.
• The Noise: While the static noise created tiny "ghost" signals (fake claps), they were so small they didn't matter.
• Long-term Stability: The biggest fear with this "passing the baton" method is that small errors might pile up over time, making the result worse and worse. However, the simulation showed that because the detector signals die out quickly, the errors do not pile up. The system stays stable even over long periods.

The Bottom Line

This paper presents a way to clean up messy detector signals in real-time without needing to make mathematical guesses. By using the fact that online computers have access to the full history of data, they can solve the puzzle perfectly, separating overlapping particle hits just by doing the math, rather than guessing. The next step for the author is to build this into the actual hardware (FPGA) to see if it works in the real world.

Technical Summary

Technical Summary: A Full Rank Pileup Deconvolution Scheme for Calorimeter Online Trigger Primitive Generation

Problem Statement
In modern high-energy physics experiments, increasing luminosity leads to significant "pile-up" issues in detectors such as calorimeters. When the ADC sampling clock is synchronized with the accelerator RF (e.g., 40 MHz or 160 MHz in the LHC), the readout data represents a convolution of the detector's impulse response (IR) and the train of event hits. When multiple hits occur within a few beam crossings (BX), their signals superimpose, merging peaks and obscuring individual contributions.

Traditional deconvolution approaches used for offline analysis face a fundamental mathematical limitation: the Data Acquisition (DAQ) system typically records only a limited timing window around a triggered event. Consequently, the number of unknowns (actual hits, including those just outside the window that contribute to the signal) exceeds the number of equations (ADC samples). This results in an underdetermined, rank-deficient system. To solve this, offline methods rely on mathematical pre-assumptions, such as Sparse Representation, to constrain the solution space. However, these assumptions may not be suitable or necessary for online trigger primitive generation, where signal availability and computational constraints differ significantly.

Methodology
The paper proposes a deconvolution scheme specifically designed for online processing in Field-Programmable Gate Arrays (FPGAs) directly connected to ADCs. The core premise is that in an online environment, all ADC readout values are available, not just those within a specific DAQ window. This availability allows the system of equations to be arranged such that the number of equations equals the number of unknowns, creating a determined, full-rank system.

The methodology models the calorimeter as a Linear Time-Invariant (LTI) system with a Finite Impulse Response (FIR), $h[i]$. The ADC output $y[j]$ is the convolution of the impulse response and the input impulse train $x[i]$.

1. Matrix Formulation: The process defines a time window of $N+1$ samples. The relationship between the output vector $\mathbf{y_0}$ and the input vector $\mathbf{x_0}$ (impulses within the window) is expressed as $\mathbf{y_0} = \mathbf{H_0}\mathbf{x_0}$, where $\mathbf{H_0}$ is a Toeplitz matrix constructed from the impulse response.
2. Handling History: To account for impulses occurring before the current window (history), a perturbation term $\mathbf{y_1} = \mathbf{H_1}\mathbf{x_1}$ is introduced, where $\mathbf{x_1}$ represents the history of $n$ previous impulses.
3. Recursive Solution: The unknown current impulses are calculated using the formula $\mathbf{x_0} = (\mathbf{H_0})^{-1}(\mathbf{y} - \mathbf{H_1}\mathbf{x_1})$.
   • The process initiates at a sufficiently long beam gap (e.g., accelerator revolution gap) where history is known to be zero ($\mathbf{x_1} = 0$).
   • The calculated impulses for the current window become the history ($\mathbf{x_1}$) for the next window, creating a recursive algorithm.
   • To mitigate error accumulation, the algorithm includes a logic to force $\mathbf{x_0} = 0$ if calculated hits are indistinguishable from random noise.

Key Contributions

• Elimination of Mathematical Pre-assumptions: Unlike offline methods, this scheme does not require Sparse Representation or other mathematical guessing because the system is fully determined by the availability of all ADC data in the online stream.
• Full-Rank Determined System: By utilizing the full stream of ADC data, the convolution matrix $\mathbf{H_0}$ becomes a full-rank squared matrix, ensuring its inverse exists and allowing for a direct solution without regularization assumptions.
• Recursive Online Algorithm: The paper outlines a practical recursive implementation suitable for FPGA hardware, starting from a known zero-history state and propagating the solution forward.

Simulation Results
The authors validated the scheme using a simulation with a finite-length impulse response (featuring fast decay and tail ringing) and random noise levels (~9% peak-to-peak).

• Recovery Accuracy: The simulation demonstrated that actual impulses were correctly recovered, even when hits were closely spaced with smeared peaks.
• Noise Handling: While noise introduced small "ghost" impulses (positive and negative), these were sufficiently small compared to actual signals.
• Noise Accumulation: A long-window simulation was conducted to test for error accumulation in the recursive process. The results showed no visible noise accumulation, suggesting that with a fast-decaying impulse response, the noise gain across iterations remains less than 1, effectively suppressing accumulation.

Significance and Claims
The paper claims that this approach offers a robust, assumption-free method for pileup deconvolution specifically tailored for online trigger primitive generation. By leveraging the full availability of ADC data in FPGA-based systems, the method transforms an underdetermined problem into a determined one, removing the need for complex mathematical priors used in offline analysis. The authors state that the next step toward practical application is the actual FPGA implementation of this scheme. The work is presented as a principle discussion and simulation validation, establishing the feasibility of a full-rank deconvolution scheme for high-luminosity environments.