  Compressed sensing is a signal processing technique in which data is acquired
directly in a compressed form. There are two modeling approaches that can be
considered: the worst-case (Hamming) approach and a statistical mechanism, in
which the signals are modeled as random processes rather than as individual
sequences. In this paper, the second approach is studied. In particular, we
consider a model of the form $\boldsymbol{Y} =
\boldsymbol{H}\boldsymbol{X}+\boldsymbol{W}$, where each comportment of
$\boldsymbol{X}$ is given by $X_i = S_iU_i$, where $\left\{U_i\right\}$ are
i.i.d. Gaussian random variables, and $\left\{S_i\right\}$ are binary random
variables independent of $\left\{U_i\right\}$, and not necessarily independent
and identically distributed (i.i.d.), $\boldsymbol{H}\in\mathbb{R}^{k\times n}$
is a random matrix with i.i.d. entries, and $\boldsymbol{W}$ is white Gaussian
noise. Using a direct relationship between optimum estimation and certain
partition functions, and by invoking methods from statistical mechanics and
from random matrix theory (RMT), we derive an asymptotic formula for the
minimum mean-square error (MMSE) of estimating the input vector
$\boldsymbol{X}$ given $\boldsymbol{Y}$ and $\boldsymbol{H}$, as
$k,n\to\infty$, keeping the measurement rate, $R = k/n$, fixed. In contrast to
previous derivations, which are based on the replica method, the analysis
carried out in this paper is rigorous.
