  Line spectral estimation is a classical signal processing problem that aims
to estimate the line spectra from their signal which is contaminated by
deterministic or random noise. Despite a large body of research on this
subject, the theoretical understanding of this problem is still elusive. In
this paper, we introduce and quantitatively characterize the two resolution
limits for the line spectral estimation problem under deterministic noise: one
is the minimum separation distance between the line spectra that is required
for exact detection of their number, and the other is the minimum separation
distance between the line spectra that is required for a stable recovery of
their supports. The quantitative results imply a phase transition phenomenon in
each of the two recovery problems, and also the subtle difference between the
two. We further propose a sweeping singular-value-thresholding algorithm for
the number detection problem and conduct numerical experiments. The numerical
results confirm the phase transition phenomenon in the number detection
problem.
