called successful if both the relative MSE of signal recovery and the MSE of frequency recovery are less than $10^{-12}$. For each combination $\sbra{K,\Delta_f}$, the success rate is measured over 20 Monte Carlo runs.

We plot the success rates of ANM and RAM with $L=1$ in Fig. \[Fig:phasetrans\_1\], where it is shown that successful recovery can be obtained with more ease in the case of a smaller $K$ and a larger frequency separation $\Delta_f$, leading to a phase transition in the sparsity-separation domain. By comparing the two images, we see that RAM significantly enlarges the success phase and enhances sparsity and resolution. It is worth noting that the phase transitions of both ANM and RAM are not sharp. One reason is that, a set of *well separated* frequencies can be possibly generated at a small value of $\Delta_f$ while we only control that the frequencies are separated by *at least* $\Delta_f$. It is also observed that RAM tends to converge in less iterations with a smaller $K$ and a larger $\Delta_f$.

![Sparsity-separation phase transition of ANM (left) and RAM (right) with $L=1$, $N=64$ and $M=30$. The grayscale images present the success rates, where white and black indicate complete success and 