A method of producing a mass spectrum from a time-varying transient signal detected in a mass spectrometer, the method comprising: performing a Fourier transform of the transient signal to produce a first set of complex amplitudes wherein each of the complex amplitudes corresponds to a respective frequency of a first set of frequencies; generating a second set of complex amplitudes, wherein each of the complex amplitudes corresponds to a respective frequency of a second set of frequencies with a minimum spacing less than the inverse of the duration of the transient signal; optimizing the second set of complex amplitudes to produce an improved second set; generating a mass spectrum from at least some of the improved second set of complex amplitudes; wherein optimizing the second set of complex amplitudes to produce an improved second set of complex amplitudes is based on an objective function subject to some phase constraints.