complete failure, respectively.[]{data-label="Fig:phasetrans_1"}](Fig_phasetrans_ANM_1.pdf "fig:"){width="1.65in"} ![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 complete failure, respectively.[]{data-label="Fig:phasetrans_1"}](Fig_phasetrans_RWANM_1.pdf "fig:"){width="1.63in"}

We also consider the MMV case with $L=5$. The success rates of ANM and RAM are presented in Fig. \[Fig:phasetrans\_5\]. Again, remarkable improvement is obtained by the proposed RAM compared to ANM. In fact, we did not find a single failure in our simulation whenever $K\leq20$ and $\Delta_f\geq\frac{0.3}{N}$. By comparing the results in Figs. \[Fig:phasetrans\_1\] and \[Fig:phasetrans\_5\], it can be observed that improved signal and frequency recovery performance can be obtained by increasing $L$, as reported in [@yang2014continuous; @yang2014exact].

![Sparsity-separation phase transition of ANM (left) and RAM (right) with $L=5$, $N=64$ and $M=30$. The grayscale images present the success rates, where white and black indicate complete success and complete failure, respectively.[]{data-label="Fig:phasetrans_5"}](Fig_phasetrans_ANM_5.pdf "fig:"){width="1.65in"} ![Sparsity-separation phase transition of ANM (left) and RAM (right) with $L=5$, $N=64$ and $M=30$. The grayscale images present the success rates, where white and black indicate complete success and complete failure, respectively.[]{data-label="Fig:phasetrans_5"}](Fig_phasetrans_RWANM_5.pdf "fig:"){width="1.63in"}

Application to DOA Estimation {#sec:simulation_DOA}
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We apply the proposed RAM method to DOA estimation. In particular, we 