the main diagonal (other than entries along the main diagonal itself) are $w_3=\mathbb{E}_{\F,\z}[v_{\mu j}^a v_{\mu j}^b]$. The main diagonals of other blocks have entries $w_2=\mathbb{E}_{\F,\z}[v_{\mu j}^a v_{\mu\eta}^a]$, and other entries in these blocks are $w_4=\mathbb{E}_{\F,\z}[v_{\mu j}^a v_{\mu \eta}^b]$. We now calculate each of these values as follows for MMV-1, MMV-2, and complex SMV with a complex measurement matrix.

![Covariance matrix $\G_{\mu}\in\mathbb{R}^{nJ\times nJ}$. Each block in $\G_{\mu}$ has a size of $n\times n$. The entries in the heavily marked blocks take the value $w_3$, except that entries along the dashed diagonal are $w_1$. The entries in the lightly marked blocks take the value $w_4$, except that entries along the dotted diagonal are $w_2$.[]{data-label="fig.cov"}](MMV_cov_mat.png){width="3.5cm"}

[**MMV-1:**]{} We begin by calculating the diagonal entries of the covariance matrix $\G_{\mu}=\mathbb{E}[\v_{\mu}\v_{\mu}^T]$, $$\label{eq:vVar1}
\begin{split}
    &w_1=\mathbb{E}_{\F,\z}\!\l[(v_{\mu j}^a)^2\r]\!=\!{\sum_{l,k=1}^{N,N}}\!\Bigg[(\s_l-\x_l^a)^T\times\\
    & \mathbb{E}_{\F}\l\{\F_{\mu+M(j-1), l}^T\F_{\mu+M(j-1), k}\r\}(\s_k-\x_k^a)\Bigg]+\Delta.
\end{split}$$ In , $\mathbb{E}_{\F}\l\{\F_{\mu+M(j-1), l}^T\F_{\mu+M(j-1), k}\r\}=\frac{\delta_{k,l}}{N}\widetilde{\mathbf{I}}_J$ (cf. Fig. \[fig:channel\]), where $\widetilde{\mathbf{I}}_J$ is a $J\times J$ matrix with only one 1 located at row $j$ and column $j$, and $\delta_{k,l}=1$ when $k=l$, else zero. Hence,  becomes $$\begin{aligned}
 w_1&=& \mathbb{E}_{\F,\z}\l[(v_{\mu j}^a)^2\r]=\frac{1}{N}{\sum_{l=1}^{N}} (s_{l,j}-x_{l,j}^a)^2+\Delta\label{eq:vVar2}\\
  &=&\!\frac{1}{NJ}\!{\sum_{l=1}^{N}} \!(\s_l-\x_l^a)^T (\s_l-\x_l^a)\!+\!\Delta,\label{eq:vVar2_1}\end{aligned}$$ where $s_{l,j}$ and $x_{l,j}^a$  denote the $j$-th entries in super symbols $\s_l$ and $\x_l^a$, respectively, and  holds because all $J$ entries within the same 