Estimation of the complex frequency of a harmonic signal based on a linear least squares method

In this study, we propose a simple linear least squares estimation method (LLS) based on a Fourier transform to estimate the complex frequency of a harmonic signal. We first use a synthetically-generated noisy time series to validate the accuracy and effectiveness of LLS by comparing it with the commonly used linear autoregressive method (AR). For an input frequency of 0.5 mHz, the calculated deviations from the theoretical value were 0.004‰ and 0.008‰ for the LLS and AR methods respectively; and for an input 5 × 10−6 attenuation, the calculated deviations for the LLS and AR methods were 2.4% and 1.6%. Though the theory of the AR method is more complex than that of LLS, the results show LLS is a useful alternative method. Finally, we use LLS to estimate the complex frequencies of the five singlets of the 0S2 mode of the Earth's free oscillation. Not only are the results consistent with previous studies, the method has high estimation precisions, which may prove helpful in determining constraints on the Earth's interior structures.