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 the presence of additive and multiplicative noise. Two non-linear 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 at high SNR, NLLS2 should be used instead of NLLS1, regardless of the value of the coherent to non-coherent power ratio of the multiplicative noise. On the other hand, at low SNR, there is a trade-off between NLLS1 and NLLS2. The latter should be preferred when the coherent to non-coherent power ratio is below a threshold which is a function of the SNR and the kurtosis of the additive noise.

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