Our numerical results also show that the BP threshold $R_{BP}(\Delta)$ converges to some value for different $J$ as $\Delta\rightarrow 0$. Analyzing these observations rigorously is left for future work.

![MMSE in dB as a function of measurement rate $R$ and number of jointly sparse signal vectors $J$ (noise variance $\Delta=-35$ dB).[]{data-label="fig:MMSE_R_J"}](MMSE_vs_R_J.pdf){width="8cm"}

BP simulation {#sec:AMPsim}
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After obtaining the theoretic MMSE for MMV, as well as the predicted MSE for BP, we run some simulations to estimate the $\underline{\s}^j$ of channel  in a Bayesian setting. The algorithm we use is approximate message passing (AMP) [@DMM2009; @Montanari2012; @Bayati2011; @Krzakala2012probabilistic; @krzakala2012statistical; @Barbier2015], which is an approximation to the BP algorithm; related algorithms have been proposed by Ziniel and Schniter [@ZinielSchniter2013MMV] and Kim et al. [@KimChangJungBaronYe2011]. In the SMV case, when the measurement matrix and the signal have i.i.d. entries, AMP has the state-evolution (SE) property [@DMM2011; @Bayati2011; @JavanmardMontanari2012; @Donoho2013; @Bayati2015] that tracks the evolution of the MSE at each iteration. Recently, Javanmard and Montanari proved that SE tracks AMP rigorously in an SMV setting with a spatially coupled measurement matrix [@JavanmardMontanari2012]. According to our transform in Fig. \[fig:channel\], we can see that the proof [@JavanmardMontanari2012] could be extended to the MMV setting. Note 