Model order selection rule for estimating the parameters of 2-D sinusoids in colored noise

We consider the problem of jointly estimating the number as well as the parameters of two-dimensional sinusoidal signals, observed in the presence of an additive colored noise field. In this framework we consider the problem of least squares estimation of the parameters of 2-D sinusoidal signals observed in the presence of an additive noise field, when the assumed number of sinusoids is incorrect. In the case where the number of sinusoidal signals is under-estimated we show the almost sure convergence of the least squares estimates to the parameters of the dominant sinusoids. In the case where the number of sinusoidal signals is over-estimated, the estimated parameter vector obtained by the least squares estimator contains a sub-vector that converges almost surely to the correct parameters of the sinusoids. Based on these results, we prove the strong consistency of a large family of model order selection rules.