An Exact Direct Method of Sinusoidal Parameter Estimation Derived From Finite Fourier Integral of Differential Equation

In this paper, we propose a novel method for estimating the parameters (frequency, amplitude, and phase) of real sinusoids. To derive the estimator, we start from the characteristic differential equation of a sinusoid. To remove differentials and obtain an algebraic relation for frequency, we introduce finite-period weighted integrals of the differential equation, which become equivalent to the differential equation when a sufficient number of weight functions are applied. As weight functions, we show that Fourier kernels have excellent properties. Terms related to integral boundaries are readily eliminated, observations are provided by Fourier coefficients, and the relation becomes independently accurate for multiple sinusoids if they are sufficiently spaced. We solve the obtained equations in two ways: one is for approaching to the Cramer-Rao lower bound (CRLB), and the other is for enhancing the interference rejection capability. Also, methods are proposed to calculate the weighted integrals from sampled signals with an improved accuracy. Proposed algorithms are examined under noise and sinusoidal interference. Error variances are compared with the CRLB and other fast Fourier transform (FFT)-based methods.