The reconstruction of discontinuous piecewise polynomial signals

• Published in: IEEE transactions on signal processing Vol. 53; no. 7; pp. 2603 - 2607
• Main Author: MacInnes, C.S.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.07.2005; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Accuracy; Chebyshev approximation; Denoising; Discontinuity; discrete Fourier transform; Discrete Fourier transforms; Fourier series; Fourier transforms; Gegenbauer polynomials; Gibbs oscillations; Gibbs phenomenon; Mathematical analysis; Mathematical models; Noise reduction; Oscillations; Polynomials; Reconstruction; Signal analysis; Signal processing; Signal processing algorithms; Signal reconstruction
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2005.849217

The Gibbs phenomenon was recognized as early as 1898 by Michelson and Stratton. Gibbs oscillations occur during the reconstruction of discontinuous functions from a truncated periodic series expansion, such as a truncated Fourier series expansion or a truncated discrete Fourier transform expansion. Recent theoretical results have shown that Gibbs oscillations can be removed from the truncated Fourier series representation of a function that has discontinuities. This is accomplished by a change of basis to the set of orthogonal polynomials called the Gegenbauer polynomials. In this correspondence, a straightforward numerical procedure for the denoising of piecewise polynomial signals is developed. Examples using truncated Fourier series and discrete Fourier transform (DFT) series demonstrate the effectiveness of the numerical procedure.