large system limit. Hence, the three settings have the same free energy expression and their MMSE’s are the same under the same noise variance $\Delta$ and measurement rate $R$ in the large system limit.

Extension to complex SMV {#sec:complex}
------------------------

MMV with jointly sparse signals is a versatile model that can be adapted to other problems. As an example, we show how the MMV model can be used to analyze the MMSE of a complex SMV.[^7] Consider the complex CS channel, $\y^{\mathcal{C}}=\F^{\mathcal{C}}\s^{\mathcal{C}}+\z^{\mathcal{C}}$, where $\s^{\mathcal{C}}=\s^{\mathcal{R}}+i\s^{\mathcal{I}}\in\mathbb{C}^N$, $\F^{\mathcal{C}}=\F^{\mathcal{R}}+i\F^{\mathcal{I}}\in\mathbb{C}^{M\times N}$, $\z^{\mathcal{C}}=\z^{\mathcal{R}}+i\z^{\mathcal{I}}\in\mathbb{C}^M$, $\y^{\mathcal{C}}=\y^{\mathcal{R}}+i\y^{\mathcal{I}}\in\mathbb{C}^M$, $i=\sqrt{-1}$, and $\mathcal{R}$ and $\mathcal{I}$ refer to the real and imaginary parts, respectively. The real and imaginary parts of the entries of $\z^{\mathcal{C}}$ both follow a Gaussian distribution, $z_l^{\mathcal{R}}, z_l^{\mathcal{I}}\sim\mathcal{N}(0,\Delta), l\in\{1,...,M\}$. Assume that the complex signal $\s^{\mathcal{C}}$ is comprised of two jointly sparse signals, $\s^{\mathcal{R}}$ and $\s^{\mathcal{I}}$, that satisfy the $J=2$ dimensional Bernoulli-Gaussian distribution . We can extend the analysis of Section \[sec:set1\] to two settings of complex CS: ([*i*]{}) the measurement matrix $\F^{\mathcal{C}}$ is real, and ([*ii*]{}) $\F^{\mathcal{C}}$ is complex.[^8]

[**Real measurement matrix:**]{} Suppose that $\F^{\mathcal{C}}$ is real, $\F^{\mathcal{C}}=\F^{\mathcal{R}}\in \mathbb{R}^{M\times N}$, and the entries of $\F^{\mathcal{R}}$ follow a Gaussian distribution, $F^{\mathcal{R}}_{\mu l}\sim \mathcal{N}(0,\frac{1}{N})$. Complex CS with a real measurement matrix can 