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We consider the problem of recovering the superposition of $R$ distinctcomplex exponential functions from compressed non-uniform time-domain samples.Total Variation (TV) minimization or atomic norm minimization was proposed inthe literature to recover the $R$ frequencies or the missing data. However, itis known that in order for TV minimization and atomic norm minimization torecover the missing data or the frequencies, the underlying $R$ frequencies arerequired to be well-separated, even when the measurements are noiseless. Thispaper shows that the Hankel matrix recovery approach can super-resolve the $R$complex exponentials and their frequencies from compressed non-uniformmeasurements, regardless of how close their frequencies are to each other. Wepropose a new concept of orthonormal atomic norm minimization (OANM), anddemonstrate that the success of Hankel matrix recovery in separation-freesuper-resolution comes from the fact that the nuclear norm of a Hankel matrixis an orthonormal atomic norm. More specifically, we show that, in traditionalatomic norm minimization, the underlying parameter values $\textbf{must}$ bewell separated to achieve successful signal recovery, if the atoms are changingcontinuously with respect to the continuously-valued parameter. In contrast,for the OANM, it is possible the OANM is successful even though the originalatoms can be arbitrarily close.  As a byproduct of this research, we provide one matrix-theoretic inequalityof nuclear norm, and give its proof from the theory of compressed sensing.

Separation-Free Super-Resolution from Compressed Measurements is Possible: an Orthonormal Atomic Norm Minimization Approach