In this chapter, several systems of digital filters are presented. The first system consists of regressive smoothing filters, which are a direct consequence of the least squares polynomial approximation to equally spaced observations. Descriptions of some particular univariate cases of these filters have been published and applied (see, for example, Anderson, 1971; Berezin and Zhidkov, 1965; Kendall and Stuart, 1963; Lanczos, 1956), but the study presented in this chapter is more general, more elaborate in detail, and more fully illustrated. It gives exhaustive information about classical smoothing, differentiating, one- and two-dimensional filtering schemes with their representation in the spaces of time, lags, and frequencies. The results are presented in the form of algorithms, which can be directly used for software development as well as for theoretical analysis of their accuracy in the design of an experiment. The second system consists of harmonic filters, which are a direct consequence of a Fourier approximation of the observations. These filters are widely used in the spectral and correlation analysis of time series. The foundation for developing regressive filters is the least squares polynomial approximation (of equally spaced observations), a principal notion that will be considered briefly.
Keywords: Bernoulli polynomials, Fourier coefficients, Gauss-Markov Theory, Student statistic, Tukey filter, derivative estimation, frequency characteristics, ground truth problem, harmonic filter, inverse normal matrix
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