  Suppose that we have $r$ sensors and each one intends to send a function
$\boldsymbol{g}_i$ (e.g.\ a signal or an image) to a receiver common to all $r$
sensors. During transmission, each $\boldsymbol{g}_i$ gets convolved with a
function $\boldsymbol{f}_i$. The receiver records the function
$\boldsymbol{y}$, given by the sum of all these convolved signals. When and
under which conditions is it possible to recover the individual signals
$\boldsymbol{g}_i$ and the blurring functions $\boldsymbol{f}_i$ from just one
received signal $\boldsymbol{y}$? This challenging problem, which intertwines
blind deconvolution with blind demixing, appears in a variety of applications,
such as audio processing, image processing, neuroscience, spectroscopy, and
astronomy. It is also expected to play a central role in connection with the
future Internet-of-Things. We will prove that under reasonable and practical
assumptions, it is possible to solve this otherwise highly ill-posed problem
and recover the $r$ transmitted functions $\boldsymbol{g}_i$ and the impulse
responses $\boldsymbol{f}_i$ in a robust, reliable, and efficient manner from
just one single received function $\boldsymbol{y}$ by solving a semidefinite
program. We derive explicit bounds on the number of measurements needed for
successful recovery and prove that our method is robust in the presence of
noise. Our theory is actually sub-optimal, since numerical experiments
demonstrate that, quite remarkably, recovery is still possible if the number of
measurements is close to the number of degrees of freedom.
