An algorithm is presented for one-dimensional NMR systems that employs nonlinear, non Fourier methods to convert noisy time dependent free induction decay (FID) data to a denoised frequency spectrum that gives reliable chemical shifts and coupling constants when the spectrum is Lorentzian. It is formulated in a way that increases frequency sensitivity and resolution and, for nuclei of low natural abundance, potentially avoids enrichment totally or in part. The algorithm should also be of use in analytical chemistry where enrichment is not possible. In effect, the useful limit of detection is significantly lowered. The algorithm uses new "phasing and "feature stability upon accumulation" methods to reliably separate signal from noise at low signal-to-noise ratios where the Fourier spectrum requires many more transients to be definitive as to what is signal and what is noise. The long-standing problem of "false features" that plagued many prior attempts to employ nonlinear methods is thereby resolved for Lorentzian spectra. Examples are reported, and the limitations of the algorithm are discussed.