HOUND: High-Order Universal Numerical Differentiator for a Parameter-free Polynomial Online Approximation

• URL: http://arxiv.org/abs/2411.00794v1
• Date: Fri, 18 Oct 2024 13:42:01 GMT
• Title: HOUND: High-Order Universal Numerical Differentiator for a Parameter-free Polynomial Online Approximation
• Authors: Igor Katrichek,
• Abstract summary: This paper introduces a numerical differentiator, represented as a system of nonlinear differential equations of any high order. We demonstrate that, with a suitable choice of differentiator order, the error converges to zero for signals with additive white noise. A notable advantage of this numerical differentiation is that it does not require tuning parameters based on the specific characteristics of the signal being differentiated.
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• Abstract: 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.

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