Suppose the random matrix $\mathbf{X}\in {{C}^{{{N}_{b}}\times {{N}_{a}}}}$, $\mathbf{X}=diag\left[ {{\mathbf{X}}_{1}},{{\mathbf{X}}_{2}},\cdots ,{{\mathbf{X}}_{R}} \right]$. The submatrix ${{\mathbf{X}}_{i}}\in {{C}^{{{N}_{b,i}}\times {{N}_{a,i}}}},i=1,2,\cdots ,R$ is assumed to have i.i.d. $CN \left( 0,\sigma _{i}^{2} \right)$ entries. ${{\mathbf{X}}_{i}}$ is independent of ${{\mathbf{X}}_{j}}$ for $i\ne j$. Thus, we have $E\left[ \mathbf{X}{{\mathbf{X}}^{H}} \right]={{\mathbf{\Omega }}_{X}}=diag\left( \sigma _{1}^{2}\mathbf{I},\sigma _{2}^{2}\mathbf{I},\cdots ,\sigma _{R}^{2}\mathbf{I} \right)$.
The matrix $\mathbf{F}\in {{C}^{{{N}_{a}}\times {{N}_{c}}}}$ is deterministic, and the entries in $\mathbf{F}$ could be real or complex. In general, ${{N}_{c}}<{{N}_{a}}<{{N}_{b}}$.
The matrices $\mathbf{X}$ and $\mathbf{F}$ are assumed to be independent of each other.
Assume $\mathbf{F}=\left[ \begin{matrix} \mathbf{F}_{1}^{r} \\ \mathbf{F}_{2}^{r} \\ \vdots \\ \mathbf{F}_{R}^{r} \\ \end{matrix} \right]$. Then \begin{align} & E\left[ \mathbf{XF}{{\mathbf{F}}^{H}}{{\mathbf{X}}^{H}} \right] \\ & =E\left[ \begin{matrix} {{\mathbf{X}}_{1}}\mathbf{F}_{1}^{r}\mathbf{F}{{_{1}^{r}}^{H}}\mathbf{X}_{1}^{H} & \cdots & \mathbf{0} \\ \vdots & \ddots & \vdots \\ \mathbf{0} & \cdots & {{\mathbf{X}}_{R}}\mathbf{F}_{R}^{r}\mathbf{F}{{_{R}^{r}}^{H}}\mathbf{X}_{R}^{H} \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} E\left[ {{\mathbf{X}}_{1}}\mathbf{F}_{1}^{r}\mathbf{F}{{_{1}^{r}}^{H}}\mathbf{X}_{1}^{H} \right] & \cdots & \mathbf{0} \\ \vdots & \ddots & \vdots \\ \mathbf{0} & \cdots & E\left[ {{\mathbf{X}}_{R}}\mathbf{F}_{R}^{r}\mathbf{F}{{_{R}^{r}}^{H}}\mathbf{X}_{R}^{H} \right] \\ \end{matrix} \right] \end{align}
My question is: Considering the feature of $\mathbf{X}_{i}$, how to further simplify $E\left[ {{\mathbf{X}}_{i}}\mathbf{F}_{i}^{r}\mathbf{F}{{_{i}^{r}}^{H}}\mathbf{X}_{i}^{H} \right]$?