Adaptive Regularization Parameter Selection for Discrete Numerical Differentiation of Noisy Signals

Estimating derivatives from noisy discrete-time measurements is a classical ill-posed inverse problem. Regularization-based approaches, particularly those inspired by total variation principles, have been widely employed to stabilize the solution. Many existing formulations originate at the functional level, where the optimization problem is posed in the continuous domain. In these formulations, minimization of the functional via the Euler-Lagrange formalism leads to partial differential equations (PDEs), which must then be solved numerically. In contrast, in this paper we propose a fully discrete regularization framework which bypasses the functional-PDE route entirely by formulating the problem directly in a finite-dimensional discrete form. For a fixed value of the regularization parameter, the resulting optimization problem is convex and possesses a closed-form solution that can be obtained via standard linear algebraic operations. The primary contribution of the work is an adaptive strategy for selecting the regularization parameter automatically from the data. We demonstrate that the regularization path contains distinct plateau regions, which reflect the trade-off between noise suppression and signal fidelity. Taking advantage of this behavior, we develop a plateau-guided parameter selection method capable of estimating an optimal or near-optimal regularization level without exhaustive search. Furthermore, the method is extended to a sliding-window setting in which the parameter is updated locally, enabling adaptation to nonstationary noise and time-varying signal features. Numerical experiments demonstrate that the proposed adaptive scheme improves derivative reconstruction accuracy while maintaining computational efficiency, making it suitable for practical large-scale and real-time applications.