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Some Results of Designing an IIR Smoothing Filter with P- Splines

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Requirements for modern digital filters are high and depend on some selective properties of such filters. IIR filters can ensure the narrowest transition width of signal omission and suppression with the same DF order. This study proposes to design and analyze a software-based IIR filter, a digital filter (DF) that is based on simple mathematical basis facilitates subsequent interpretation of data (estimation of derivatives, trends, forecast of time series, etc.). A modified penalty spline has been used as a model for such digital software-based IIR filter. A recurrent form of the P-spline computation has been created for the the IIR spline filter. The new modified recurrent P-spline can be used for stand-alone or group data. The recurrent spline filter has been investigated by the methods of linear dynamic systems. The obtained difference equation of the spline transform corresponds to a linear recursive DF with constant parameters. Frequency and time characteristics of the DF, being the main characteristics system and instrument functions, have been obtained analytically. The system function has been used to investigate the frequency characteristics of the spline filter and it has been found out that the spline filter is low frequency throughout the entire variation range of the spline parameters. Such filter characteristics correspond to a second-order aperiodic segment or a system with delay, which is typical for grouped data processing. The selective properties of the spline filter are clearly illustrated with the width of the instrument function, which depends on the spline parameters. Studies have shown that the passband bandwidth is mostly influenced by the group data size (the number of non-recursive terms of the difference equation). The stability of the recursive spline filter has been also estimated analytically based on the characteristic equation and the Hurwitz criterion. DF is always stable only when the conjugation of recurrent coefficients occurs at the beginning of a spline segment. Frequency characteristics of the spline filter have been confirmed in the time domain by histograms of the remainders of two recovered model functions.
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P-Spline; Recurrence Algorithm; IIR Filter Stability; Frequency Response; Instrumental Function

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