Accurate and fast estimation of the Fourier coefficients of periodic signals disturbed by trends

In this paper, a linear least-squares estimator is presented for the estimation of the Fourier coefficients and trend components of a periodic time series disturbed by a trend. In opposition to existing methods, the removal of the trend is not separated from the estimation of the Fourier coefficients. Based on the mathematical description of the proposed algorithm, a computationally efficient implementation is given. It is demonstrated under some weak assumptions (mixing condition on the measurement noise) that the estimates of the Fourier coefficients are consistent (converge to the true Fourier coefficients of the periodic signal), asymptotically efficient (lowest possible uncertainty), and asymptotically normally distributed. Moreover, using simulations it is shown that the asymptotic properties of the estimator are reached for practical measurement problems. Finally, the estimator is applied to the measured load current and voltage of a traction battery.