Adaptive Filtering via Canonical Systems Withtime-varying Hamiltonians

This paper investigates an adaptive filtering framework based oncanonical systems governed by first order differential equations with time-varying symmetric positive semidefinite Hamiltonian matrices. The proposedmethod adapts the Hamiltonian matrix online using a gradient based schemedesigned to minimize the squared error between the system output and adesired reference signal. We establish theoretical stability guarantees via Lyapunov analysis, ensuring boundedness of system trajectories and convergenceof the error signal under suitable assumptions. Furthermore, we present numerical integration schemes preserving the underlying Hamiltonian structure andprojective techniques to maintain positive semidefiniteness of the Hamiltonianmatrix. Extensive simulations on synthetic nonstationary signals illustrate theeffectiveness and robustness of the proposed adaptive filter.