The reconstruction of discontinuous piecewise polynomial signals

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.