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

Abstract

Context: Several approaches to estimate frequency, phase and amplitude errors in time series analyses were 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. 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-series analyses. We also demonstrate the possibility to detect frequencies which are separated by less than the classical frequency resolution and that the realistic frequency error is at least 4 times smaller than the classical frequency resolution.

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