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Adaptive Filtering via Canonical Systems with Time-Varying Hamiltonians

Keshav Raj Acharya, Pitambar Acharya

Abstract

In many practical applications, signals and environments are time- varying, which makes fixed filters unreliable. Adaptive filtering, on the other hand, updates in real time to suppress noise, track nonstationary signals, and identify unknown systems. This paper investigates an adaptive filtering frame- work based on canonical systems with time-varying symmetric positive semi- definite Hamiltonian matrices. The proposed method adapts the Hamiltonian matrix using a gradient-based scheme designed to minimize the squared er- ror between the system output and a desired reference signal. We establish theoretical stability guarantees via Lyapunov analysis, ensuring boundedness of system trajectories and convergence of the error signal under suitable as- sumptions. Furthermore, we present numerical integration schemes preserving the underlying Hamiltonian structure and projective techniques to maintain positive semidefiniteness of the Hamiltonian matrix. Extensive simulations on synthetic nonstationary signals illustrate the effectiveness and robustness of the proposed adaptive filter.

Adaptive Filtering via Canonical Systems with Time-Varying Hamiltonians

Abstract

In many practical applications, signals and environments are time- varying, which makes fixed filters unreliable. Adaptive filtering, on the other hand, updates in real time to suppress noise, track nonstationary signals, and identify unknown systems. This paper investigates an adaptive filtering frame- work based on canonical systems with time-varying symmetric positive semi- definite Hamiltonian matrices. The proposed method adapts the Hamiltonian matrix using a gradient-based scheme designed to minimize the squared er- ror between the system output and a desired reference signal. We establish theoretical stability guarantees via Lyapunov analysis, ensuring boundedness of system trajectories and convergence of the error signal under suitable as- sumptions. Furthermore, we present numerical integration schemes preserving the underlying Hamiltonian structure and projective techniques to maintain positive semidefiniteness of the Hamiltonian matrix. Extensive simulations on synthetic nonstationary signals illustrate the effectiveness and robustness of the proposed adaptive filter.
Paper Structure (14 sections, 7 theorems, 90 equations, 3 figures, 1 table)

This paper contains 14 sections, 7 theorems, 90 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Structured Adaptive Filtering via Canonical Systems
  3. Model Formulation: Canonical-System Adaptive Filter Model
  4. Discretization and Maintaining Positive Semidefiniteness
  5. Case 1: $\lambda_{\min}(t) > 0$.
  6. Case 2: $\lambda_{\min}(t) = 0$.
  7. Stability of the Adaptive Gradient Flow
  8. Sensitivity Bounds
  9. Numerical Simulation and Analysis
  10. Algorithm Implementation
  11. Conclusion
  12. Author Contributions:
  13. Funding Declaration:
  14. Conflict of Interest:

Key Result

Lemma 2.1

Let $H(t)\in \mathbb{R}^{2\times 2}$ be a differentiable real symmetric matrix valued function. Suppose that for some $t_{0}$ the smallest eigenvalue $\lambda_{\min}(t_{0})$ is simple. Let $v_{\min}(t)$ be the corresponding unit eigenvector chosen smoothly near $t_{0}$. Then the function $\lambda_{\

Figures (3)

  • Figure 1: Flow chart of the Hamiltonian adaptation algorithm. At each time step, the Hamiltonian matrix is updated via a gradient-based rule and projected onto the positive semidefinite cone, followed by canonical state evolution under symplectic dynamics.
  • Figure 2: Adaptive canonical system filter output tracking the desired nonstationary signal. The filter output $u(t) = C y(t)$ successfully tracks the reference signal $r(t)$, demonstrating the adaptation capability of the Hamiltonian matrix. The initial transient period shows rapid convergence from the initial conditions to the desired trajectory.
  • Figure 3: Tracking error $e(t)$ over time. The error magnitude decreases significantly after the initial adaptation phase and remains bounded throughout the simulation, indicating stable learning behavior. The occasional spikes correspond to abrupt changes in the nonstationary signal frequency.

Theorems & Definitions (15)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 2.4: Finite-Difference Approximation
  • proof
  • Lemma 2.5
  • proof
  • ...and 5 more