Strong consistency of the over- and underdetermined LSE of 2-D exponentials in white noise

We consider the problem of least squares estimation of the parameters of two-dimensional (2-D) exponential signals observed in the presence of an additive noise field, when the assumed number of exponentials is incorrect. We consider both the case where the number of exponential signals is underestimated, and the case where the number of exponential signals is overestimated. In the case where the number of exponential signals is underestimated, we prove the almost sure convergence of the least squares estimates (LSE) to the parameters of the dominant exponentials. In the case where the number of exponential signals is overestimated, the estimated parameter vector obtained by the least squares estimator contains a subvector that converges almost surely to the correct parameters of the exponentials.