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

TL;DR

This paper introduces simple formulas to estimate the uncertainty in frequency determination from time series data, showing that realistic errors are smaller than classical resolution estimates and enabling detection of closely spaced frequencies.

Contribution

It provides a heuristic, computationally efficient method for estimating frequency errors in time series analysis, improving upon existing approaches.

Findings

Realistic frequency errors are at least 4 times smaller than classical resolution.

The method allows detection of frequencies separated by less than the classical resolution.

Simple formulas for upper limits of amplitude, frequency, and phase uncertainties are presented.

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.