Nuclear magnetic resonance spectroscopy is one of the key methods for studying the structure of matter on different levels (sub-nuclear, nuclear, atomic, molecular, cellular, etc). Its overall success critically depends upon reliable mathematical analysis and interpretation of the studied data. This is especially aided by parametric signal processing with the ensuing data quantification, which can yield the abundance or concentrations of the constituents in the examined matter. The sought reliability of signal processing rests upon the possibility of an accurate solution of the quantification problem alongside the unambiguous separation of true from false information in the spectrally analysed data. We presently demonstrate that the fast Padé transform (FPT), as the unique ratio of two polynomials for a given Maclaurin series, can yield exact signal-noise separation for a synthesized free induction decay curve built from 25 molecules. This is achieved by using the concept of Froissart doublets or pole–zero cancellations. Unphysical/spurious (noise or noise-like) resonances have coincident or near-coincident poles and zeros. They possess either zero- or near-zero-valued amplitudes. Such spectral structures never converge due to their instability against even the smallest perturbations. By contrast, upon convergence of the FPT, physical/genuine resonances are identified by their persistent stability against external perturbations, such as signal truncation or addition of random noise, etc. In practice, the computation is carried out by gradually and systematically increasing the common degree of the Padé numerator and denominator polynomials in the diagonal FPT. As this degree changes, the reconstructed parameters and spectra fluctuate until stabilization occurs. The polynomial degree at which this full stabilization is achieved represents the sought exact number of resonances. An illustrative set of results is reported in this work to show the exact separation of genuine from spurious information by reliance upon Froissart doublets and stabilization of reconstructions. The FPT for optimal quantification of the physical constituents of the studied matter and the denoising Froissart filter for unequivocal signal-noise separation is expected to significantly aid nuclear magnetic resonance spectroscopy in achieving the most reliable data analysis and interpretation.