the signal. Consider a [*super symbol*]{} $\s_l=[\underline{s}_l^1,...,\underline{s}_l^J]^T,\ l\in\{1,...,N\}$, where $[\cdot]^T$ denotes the transpose. The super symbol $\s_l$ follows a $J$-dimensional Bernoulli-Gaussian distribution, $$\label{eq:jsm}
P(\s_l)=\rho \phi(\s_l)+(1-\rho)\delta(\s_l),$$ where $\rho$ is the sparsity rate, $\phi(\s_l)$ is a $J$-dimensional Gaussian distribution with zero mean and identity covariance matrix, and $\delta(\s_l)$ is the delta function for $J$-dimensional vectors.

\[def:jointly\_sparse\] [*Ensembles of signals that obey  are called jointly sparse.*]{}

[**Measurement models**]{}: Each signal $\underline{\s}^j$ is measured by an i.i.d. Gaussian measurement matrix $\underline{\F}^j\in\mathbb{R}^{M\times N}$, $\underline{F}_{\mu l}^j \sim \mathcal{N}(0,1/N)$, where $\mu$ refers to the row index and $l$ is the column index. The measurements $\underline{\y}^j$ are corrupted by i.i.d. Gaussian noise $\underline{\z}^j$ consisting of entries $\underline{z}_{\mu}^j\sim \mathcal{N}(0,\Delta)$, $$\label{eq:MMVmodel}
\underline{\y}^j=\underline{\F}^j\underline{\s}^j+\underline{\z}^j,\quad j\in\{1,\cdots,J\}.$$ When the number of signal vectors $J=1$, this MMV model  becomes an SMV problem. Our analyses in this paper are readily extended to other i.i.d. matrices, jointly sparse signals , and other i.i.d. noise distributions.

\[def:MMV\_set1\] [*The setting MMV-1 refers to the measurement model in  with all matrices $\underline{F}^j$ being different.*]{}

\[def:MMV\_set2\] [*The setting MMV-2 refers to the measurement model in  with all matrices $\underline{F}^j$ being equal.*]{}

In the signal model  and measurement model , the sparsity rate $\rho$, channel noise variance $\Delta$, and number of channels 