A heuristic derivation of the uncertainty for frequency determination in time series data

Context. Several approaches to estimating frequency, phase, and amplitude errors in time-series analyse have been reported in the literature, but they are either time-consuming to compute, grossly overestimating the error, or are based on empirically determined criteria. Aims. A simple, but realistic estimate of the frequency uncertainty in time-series analyses is our goal here. Methods. Synthetic data sets with mono- and multi-periodic harmonic signals and with randomly distributed amplitude, frequency, and phase were generated and white noise added. We tried to recover the input parameters with classical Fourier techniques and investigated the error as a function of the relative level of noise, signal, and frequency difference. Results. We present simple formulas for the upper limit of the amplitude, frequency, and phase uncertainties in time-serie analyses. We also demonstrate the possibility of detecting frequencies that are separated by less than the classical frequency resolution and of finding that the realistic frequency error is at least 4 times smaller than the classical frequency resolution.