  This paper develops a new mathematical framework for denoising in blind
two-dimensional (2D) super-resolution upon using the atomic norm. The framework
denoises a signal that consists of a weighted sum of an unknown number of
time-delayed and frequency-shifted unknown waveforms from its noisy
measurements. Moreover, the framework also provides an approach for estimating
the unknown parameters in the signal. We prove that when the number of the
observed samples satisfies certain lower bound that is a function of the system
parameters, we can estimate the noise-free signal, with very high accuracy,
upon solving a regularized least-squares atomic norm minimization problem. We
derive the theoretical mean-squared error of the estimator, and we show that it
depends on the noise variance, the number of unknown waveforms, the number of
samples, and the dimension of the low-dimensional space where the unknown
waveforms lie. Finally, we verify the theoretical findings of the paper by
using extensive simulation experiments.
