Non-linear least squares estimation for harmonics in multiplicative and additive noise

We consider the problem of estimating the frequency of a complex harmonic in Doppler radar signals in the presence of additive and multiplicative noise, which may be colored and non-Gaussian. Two nonlinear least-squares (NLLS) estimators, NLLS₁ and NLLS₂, are proposed, which consist of matching the data and the squared data, respectively, with a constant amplitude harmonic. Expressions for the asymptotic covariances of the NLLS estimators are derived. It is shown that for high SNR, NLLS₂ should be used instead of NLLS₁, regardless of the value of the intrinsic SNR (ISNR) of the random amplitude. On the other hand, for low SNR, there is a trade-off between NLLS₁ and NLLS₂. The former is superior to the latter when the ISNR is below a threshold which is a function of the SNR and the kurtosis of the additive noise