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This work investigates the parameter estimation performance ofsuper-resolution line spectral estimation using atomic norm minimization. Thefocus is on analyzing the algorithm's accuracy of inferring the frequencies andcomplex magnitudes from noisy observations. When the Signal-to-Noise Ratio isreasonably high and the true frequencies are separated by $O(\frac{1}{n})$, theatomic 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 ofthe true frequencies. Here $n$ is half the number of temporal samples and$\sigma^2$ is the Gaussian noise variance. The analysis is based on aprimal-dual witness construction procedure. The obtained error bound matchesthe Cram\'er-Rao lower bound up to a logarithmic factor. The relationshipbetween resolution (separation of frequencies) and precision or accuracy of theestimator is highlighted. Our analysis also reveals that the atomic normminimization can be viewed as a convex way to solve a $\ell_1$-normregularized, nonlinear and nonconvex least-squares problem to globaloptimality.

Approximate Support Recovery of Atomic Line Spectral Estimation: A Tale of Resolution and Precision