Локальная интерполяция и аппроксимация в задачах эвристического синтеза цифровых фильтров

Abstract

The article has an overview character with elements of generalization, but at the same time it includes information brought to the level of immediate practical implementation. Approaches to heuristic and functional synthesis of digital filters based on local nterpolation and approximation are considered. Digital differentiators based on local polynomial interpolation, digital integrators based on local polynomial and spline interpolation and digital filters with a U-shaped amplitude-frequency characteristic based on polynomial OLS approximation are obtained. The obtained results can be used in practice according to their functional purpose, taking into account the recommendations mentioned. The idea of numerical differentiation based on interpolation is not new, but the deep connection between this approach and digital filtering is not often reflected in literature due to the fact that this concept is limited to numerical differentiation based on finite differences. The gap is filled in this work. Numerical integration is usually considered on the basis of stepwise or piecewise linear or piecewise parabolic interpolation (method of rectangles, trapezoids, Simpson). However, regardless of the choice of the interpolation method, the digital integrator has a certain generalized structure, and approaches to numerical integration can be based on more effective methods of interpolation, which is shown in the article. The OLS filters (Savitzky-Golay) are described in the literature. However, the steps for their practical implementation are clearly insufficient. Traditionally the general description of filters does not develop further 4th degree of the approximating polynomial and is often limited to the moving average filters. The frequency properties of the filters are also not fully described. In this work the description of the OLS filters obtained at the degree of the approximating polynomial 0-8 was given, their frequency properties were studied, and sufficient information was given to obtain digital filters based on them with a U-shaped amplitude-frequency characteristic without pulsations in the passband. Thus, the field of filters application is extended beyond the OLS smoothing.