Nonlinear estimation of frequency and phase of a sinusoid in noise

We present an approach to the problem of estimating the frequency and phase of a sinusoid in the presence of noise. Analytical expressions for the discrete Fourier Transform (DFT) of a finite sequence containing a sinusoid are first obtained for each frequency component. With noise, these DFT components are functions of frequency, amplitude and phase of the sinusoid and of the noise power as well as the number of samples. Nonlinear operations (multiplication and division) on the DFT components to eliminate certain variables then yield an estimator for the frequency and phase. Means and variances of the estimator are derived and simulation results are presented.