Almost sure identifiability of multidimensional harmonic retrieval

Two-dimensional (2-D) and more generally multi-dimensional harmonic retrieval is of interest in a variety of applications. The associated identifiability problem is key in understanding the fundamental limitations of parametric high-resolution methods. In the 2-D case, existing identifiability results indicate that, assuming sampling at Nyquist or above, the number of resolvable exponentials is proportional to I + J, where I is the number of (equispaced) samples along one dimension, and J likewise for the other dimension. We prove in this paper that the number of resolvable exponentials is roughly I J/4, almost surely. This is not far from the equations-versus-unknowns bound of I J/3. We then generalize the result to the N-D case for any N > 2, showing that, under quite general conditions, the number of resolvable exponentials is proportional to total sample size, hence grows exponentially with the number of dimensions.