This Article introduces a new computationally efficient noise-tolerant signal processing method, referred to as phased spectrum deconvolution method (ΦSDM), designed for Fourier transform mass spectrometry (FT MS). ΦSDM produces interference-free mass spectra with resolution beyond the Fourier transform (FT) uncertainty limit. With a presumption that the oscillation phases are preserved, the method deconvolves an observed FT spectrum into a distribution of harmonic components bound to a fixed frequency grid, which is several times finer than that of FT. The approach shows stability under noisy conditions, and the noise levels in the resulting spectra are lower than those of the original FT spectra. Although requiring more computational power than standard FT algorithms, ΦSDM runs in a quasilinear time. The method was tested on both synthetic and experimental data, and consistently demonstrated performance superior to the FT-based methodologies, be it across the entire mass range or on a selected mass window of interest. ΦSDM promises substantial improvements in the spectral quality and the speed of FT MS instruments. It might also be beneficial for other spectroscopy approaches which require harmonic analysis for data processing.