  This work investigates the parameter estimation performance of
super-resolution line spectral estimation using atomic norm minimization. The
focus is on analyzing the algorithm's accuracy of inferring the frequencies and
complex magnitudes from noisy observations. When the Signal-to-Noise Ratio is
reasonably high and the true frequencies are separated by $O(\frac{1}{n})$, the
atomic norm estimator is shown to localize the correct number of frequencies,
each within a neighborhood of size $O(\sqrt{{\log n}/{n^3}} \sigma)$ of one of
the true frequencies. Here $n$ is half the number of temporal samples and
$\sigma^2$ is the Gaussian noise variance. The analysis is based on a
primal-dual witness construction procedure. The obtained error bound matches
the Cram\'er-Rao lower bound up to a logarithmic factor. The relationship
between resolution (separation of frequencies) and precision or accuracy of the
estimator is highlighted. Our analysis also reveals that the atomic norm
minimization can be viewed as a convex way to solve a $\ell_1$-norm
regularized, nonlinear and nonconvex least-squares problem to global
optimality.
