Least squares estimation of 2-D sinusoids in colored noise: Asymptotic analysis

This paper considers the problem of estimating the parameters of real-valued two-dimensional (2-D) sinusoidal signals observed in colored noise. This problem is a special case of the general problem of estimating the parameters of a real-valued homogeneous random field with mixed spectral distribution from a single observed realization of it. The large sample properties of the least squares (LS) estimator of the parameters of the sinusoidal components are derived, making no assumptions on the type of the probability distribution of the observed field. It is shown that if the disturbance field satisfies a combination of conditions comprised of a strong mixing condition and a condition on the order of its uniformly bounded moments, the normalized estimation error of the LS estimator is consistent asymptotically normal with zero mean and a normalized asymptotic covariance matrix for which a simple expression is derived. It is further shown that the LS estimator is asymptotically unbiased. The normalized asymptotic covariance matrix is block diagonal where each block corresponds to the parameters of a different sinusoidal component. Assuming further that the colored noise field is Gaussian, the LS estimator of the sinusoidal components is shown to be asymptotically efficient.