HOUND: High-Order Universal Numerical Differentiator for a Parameter-free Polynomial Online Approximation
This provides a practical solution for engineers and scientists needing real-time derivative estimation from noisy measurements without manual parameter tuning.
The paper tackles the problem of numerical differentiation for noisy polynomial signals by introducing HOUND, a parameter-free high-order differentiator that operates online without requiring signal-specific tuning. The method achieves zero error convergence for polynomial signals with additive white noise and bounded error for more general cases when the highest derivative is bounded.
This paper introduces a scalar numerical differentiator, represented as a system of nonlinear differential equations of any high order. We derive the explicit solution for this system and demonstrate that, with a suitable choice of differentiator order, the error converges to zero for polynomial signals with additive white noise. In more general cases, the error remains bounded, provided that the highest estimated derivative is also bounded. A notable advantage of this numerical differentiation method is that it does not require tuning parameters based on the specific characteristics of the signal being differentiated. We propose a discretization method for the equations that implements a cumulative smoothing algorithm for time series. This algorithm operates online, without the need for data accumulation, and it solves both interpolation and extrapolation problems without fitting any coefficients to the data.