Abstract

This paper investigates the dynamical modeling of Haar wavelet coefficient estimation in signal processing. The problem is formulated as the minimization of a strictly convex quadratic energy functional composed of data-fidelity and regularization terms. By applying continuous-time Euclidean gradient flow, the estimation problem is transformed into a linear first-order dynamical system. Under the assumption that the system matrix M+W is symmetric positive definite, spectral analysis and Lyapunov theory are employed to establish the existence of a unique equilibrium point and to prove global exponential convergence toward the minimizer. The exponential decay rate is explicitly characterized by the smallest eigenvalue of the system matrix. Numerical experiments on both synthetic and ECG signals validate the theoretical analysis and demonstrate stable and robust convergence behavior. The proposed framework provides a rigorous connection between wavelet-based coefficient estimation and dynamical systems stability theory.