Super-Resolution With Noisy Measurements: Reconciling Upper And Lower Bounds

This paper considers the problem of lower bounding the mean-squared-error (MSE) of unbiased super-resolution estimates. In literature, only upper bounds on the MSE are available which scale with the so-called super-resolution factor (SRF). However, the upper bound does not indicate whether the MSE indeed exhibits noise amplification that increases with the target resolution. The main contribution of this paper is to derive the Cram\'er-Rao Bound for noisy super-resolution problem and understand its scaling as a function of the super-resolution factor. We compare our lower bound with the upper bound established in prior work and show that the dependency of MSE on SRF is fundamental. Our analysis can be applied to other unbiased estimates in the problem of super-resolution. Numerical experiments are conducted to demonstrate our theoretical claims.