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

We consider the problem of least squares estimation of the parameters of 2D 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 under-estimated, and the case where the number of exponential signals is over-estimated. In the case where the number of exponential signals is under-estimated we prove the almost sure convergence of the least squares estimates to the parameters of the dominant exponentials. In the case where the number of exponential 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 exponentials.