Exact Constant-Time Recursive Realization of Polynomial FIR Filters | Zenodo
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Published July 13, 2026
| Version 1.0
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Exact Constant-Time Recursive Realization of Polynomial FIR Filters
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Di Michele, Pierpaolo<br>(Researcher)1
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Independent researcher
Description
An exact recursive realization with constant computational cost O(1) per sample is presented for causal FIR filters having a truncated polynomial impulse response of arbitrary degree m and finite length k. The filter output admits a representation as a linear combination of m+1 moving weighted sums, referred to as moments, defined on the monomial basis of the polynomial impulse response. The finite difference operator, applied iteratively to these moments, progressively cancels their contribution, leaving a linear constant-coefficient difference relation of order m+1. The recursion coefficients depend exclusively on the polynomial coefficients, on the degree m, and on the window length k, and are independent of the input signal. Closed-form expressions for the recursive coefficients are provided for m=0, 1, 2, 3.<br>The same formulation extends naturally to the estimation of the polynomial derivatives and to evaluation at an arbitrary point, in each case retaining the O(1) cost per sample. Moving polynomial regression turns out to be a particular case of the class considered.<br>The correctness of the algebraic derivation is verified by means of a minimal analytical case and extensive numerical simulations. The numerical analysis further highlights the effects of the recursive realization in floating-point arithmetic, attributable to the imperfect pole-zero cancellation at the multiple root z=1 associated with this recursive structure.
Notes
(En)
Attribution Note
The included Python code implements the algorithms described in this working paper. Any use of the code or of the analytical formulas derived from it requires proper attribution to the original source (DOI) and must comply with the CC BY‑NC‑ND 4.0 license. This ensures correct acknowledgment of the intellectual authorship of the research.
Notes
(En)
Editorial Note
The table of contents contains two section titles (“Sommario” and “Introduzione”) inherited from the original Italian drafting template. These correspond respectively to “Table of Contents” and “Introduction” in the English version. A fully uniform English table of contents will be provided in a future revision.
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Additional details
Additional titles
Alternative title
The Recursive Truncated Polynomial Filter (RTPF), with Applications to Moving Polynomial Regression
Related works
Is derived from
Working paper:
10.5281/zenodo.17053349
(DOI)
Working paper:
10.5281/zenodo.18381278
(DOI)
Working paper:
10.5281/zenodo.18518657
(DOI)
Working paper:
10.5281/zenodo.20574595do.20574595
(DOI)
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2026-07-13
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Keywords and subjects
Keywords
Recursive Polynomial Regression
Truncated Polynomial FIR Filters
Constant-Time Algorithms
Recursive Filter Realization
Weighted Moving Sums
Finite Difference Operators
Polynomial Impulse Response
IIR–FIR Hybrid Architecture
Digital Signal Processing
Time Series Processing
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DOI
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DOI
10.5281/zenodo.21336669
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Resource type<br>Working paper
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En
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Creative Commons Attribution Non Commercial No Derivatives 4.0 International
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Copyright
Copyright (C) 2026 Pierpaolo Di Michele
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Created
July 13, 2026
Modified
July 13, 2026
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