Error analysis through residual frequency spectra in the fast Padé transform (FPT)

Most critical to an overall validity of any estimator of spectra in both parametric and non-parametric signal processings is an adequate error analysis. Working within the fast Padé transform (FPT), we presently design a stringent error analysis based upon three successive verifications of the obtained results. The theoretical predictions for all the spectral parameters (positions, widths, heights and phases of each resonance) as well as the shape of the computed spectrum are considered to be acceptable only if they stabilise while varying the fractions of the total signal length in the two variants of the FPT that converge inside and outside the unit circle. Stabilisation is considered as achieved if the successive differences of parameter estimations as a function of the signal length are within the prescribed accuracy threshold, such that the shape spectrum is indistinguishable from random background noise. These differences can be graphically displayed as the so-called residual spectra or the error spectra. It is found that all of these strict criteria are successfully satisfied by the FPT which, therefore, emerges as a powerful parametric and non-parametric processor with a robust error analysis. The present numerical illustrations prove that the computed error spectra are negligible with only a small fraction between 14 and 12 of the full signal length. This virtue of the FPT could be of a paramount importance in medical diagnostics where unequivocal quantification of time signals (possibly short) encoded from patients now become feasible with the reassuring benefit of the accompanying error analysis of proven validity.