Joint Autoregressive Framework

• Joint Autoregressive Framework is a modeling scheme that uses autoregressive dependencies to decompose and track interleaved, time-dependent processes.
• It employs algorithmic structures like stacks and FIFO queues to achieve real-time instance identification with constant time and space complexity.
• Applications include embedded interrupt analysis, radar pulse scheduling, particle tracking, and TI-ADC calibration, demonstrating robust and efficient performance.

A joint autoregressive framework refers to algorithmic schemes in which multiple interacting components or processes are modeled or analyzed by tracking their interleaved, temporally dependent executions in real time, typically for the purpose of decomposition, tracking, or resource scheduling. Such frameworks provide precise instance identification, enable correct disentanglement of interleaved activities, and support efficient, low-overhead analyses in applications ranging from interrupt-driven embedded systems to radar signal scheduling and advanced particle tracking.

1. Formal Definition and Instantiation

The essential structure of a joint autoregressive framework is characterized by explicit modeling of entities (procedural instances, tasks, or signal channels) whose executions or outputs depend autoregressively on their own past states but are also interleaved with those of other entities. A canonical example is the formalization of Interrupt Procedure Instances (IPIs) in interrupt-driven embedded programs:

Given an interrupt handler $IH$, define:

• $Cal(IH)$: The closure of all functions and tasks reachable from $IH$,
• $Tasks(IH)$: All tasks possibly posted from $Cal(IH)$.

An IPI is a tuple: $IPI = \left( \mathit{id},\, i,\, E,\, X \right)$ with unique identifier $\mathit{id}$, originating interrupt $i$, execution trace $E$, and subset of posted tasks $X \subseteq Tasks(IH)$. All observable execution events are modeled as a sequence: $$\tau = e_1\, e_2\, \dots\, e_n,\quad e_k \in \Sigma$$ with $\Sigma$ denoting the event alphabet (e.g., handler entry/exit, task posting, task entry/exit). This formalization allows precise association of every instruction or event with its unique instance, and tracking of context switching and interleavings without ambiguity (Sun et al., 2018).

2. Real-Time Interleaved Tracking Algorithms

Efficient identification of interleaved procedural or signal instances is central in joint autoregressive frameworks. A typical algorithm maintains:

• A stack for preempted instances,
• A FIFO queue for pending tasks,
• Scalars for the current instance and its state (START, END, INTERM).

At each program point or signal sample, the algorithm associates the current instruction with an instance and emits whether it marks a START, END, or intermediate point of the instance. The event-driven update logic ensures that handler entries push and switch context, handler exits pop context, task postings enqueue, and task entries dequeue—and all bookkeeping is performed in $O(1)$ time and space due to hardware-bounded limits on interrupt depth and queue size (Sun et al., 2018).

3. Formal Analysis: Correctness and Complexity

Correctness follows by mapping the hardware-level call stack and OS-level task scheduler onto algorithmic data structures, with key invariants:

• Handler resumption (stack pop) restores exactly the instance preempted at the corresponding entry,
• Task scheduling (queue dequeue) preserves FIFO ordering relative to posting,
• Detecting instance END points is determined by the emptiness of the pending task queue.

No lookahead or rollback is required, yielding single-pass, real-time identification. For $n$ total events, time complexity is $O(n)$ and the auxiliary space is strictly bounded, i.e., $\Theta(1)$ for all practical WSNs or embedded deployments (Sun et al., 2018).

4. Applications Across Domains

Joint autoregressive interleaving and tracking appear in several engineering domains:

• Interrupt-driven program analysis: Disentangling overlapping IPIs to support dynamic analysis, profiling, and verification of WSN and IoT programs (Sun et al., 2018).
• Pulse interleaving in radar scheduling: Packing multiple target tracking tasks across pulse repetition intervals while satisfying receive/transmit constraints, utilizing integer or heuristic scheduling that leverages interleaved, autoregressive instance management (Jang et al., 2014).
• Particle tracking in experimental mechanics: Iterative scale-invariant particle tracking combined with global compatibility enforced via augmented Lagrangian steps, with local/global steps interleaved and optimized for rotation and scale invariance (Yang et al., 2022).
• TI-ADC calibration: EKF-based correction of interleaved channel mismatches interpreted as an online tracking problem, where each channel is an autoregressive process estimated in the presence of time-varying, multicomponent interleaving (Sung et al., 13 Mar 2025).

5. Empirical Evaluation and Comparative Advantages

Empirical results in the context of IPI tracking show that memory usage remains constant (≈1 MB), and runtime overhead is negligible (≈0.03 s), in contrast with legacy approaches whose resource requirements grow with execution duration (up to 0.6 s and unbounded memory at 150 s) (Sun et al., 2018). Similar constant-time and space behavior has been demonstrated in radar pulse scheduling, where polynomial-time heuristics with $O(N_t\log N_t)$ or $O(N_t^2\log N_t)$ complexity—depending on the digital beamforming architecture—enable real-time, large-scale scheduling infeasible with previous brute-force algorithms (Jang et al., 2014). In particle tracking, interleaved-invariant approaches robustly achieve >95% tracking ratio and sub-pixel accuracy even for large rotations and deformations (Yang et al., 2022).

6. Implications, Assumptions, and Extensions

These frameworks rely on:

• Predictable hardware or scheduler context switching semantics (e.g., enforced by TinyOS or radar processing logic),
• Hardware-bounded state for efficient stack/queue management,
• Explicit modeling of all possible interleavings at the event or sample granularity,
• For some applications, the ability to distinguish and correlate “known” and “desired” signals, as in tracking-based TI-ADC calibration (Sung et al., 13 Mar 2025).

A plausible implication is that, where such visibility and hardware bounds exist, joint autoregressive frameworks provide a provably minimal-overhead and formally correct foundation for high-precision, real-time analysis and scheduling, supporting advanced program analysis, signal calibration, and large-scale experimental tracking under real-world resource constraints.

7. Summary Table of Key Features

Application Domain	Interleaved Entity	Instance Identification Method
IoT/WSN Programs	IPIs (handler/tasks)	Stack and FIFO queue; $O(1)$ RT
Pulse-Doppler Radar	Tracking Looks	Integer-program/heuristic scheduling
Particle Tracking (SerialTrack)	Displacements	Interleaved local/global ADMM
TI-ADC Calibration	Channel Parameters	EKF state tracking + FIR correction

These results delineate the foundational role of joint autoregressive, interleaved-tracking frameworks across diverse technical areas, with consistent principles of decomposing, labeling, and tracking temporally interdependent executions with real-time, resource-optimal algorithms.