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 afunction $\boldsymbol{f}_i$. The receiver records the function$\boldsymbol{y}$, given by the sum of all these convolved signals. When andunder which conditions is it possible to recover the individual signals$\boldsymbol{g}_i$ and the blurring functions $\boldsymbol{f}_i$ from just onereceived signal $\boldsymbol{y}$? This challenging problem, which intertwinesblind deconvolution with blind demixing, appears in a variety of applications,such as audio processing, image processing, neuroscience, spectroscopy, andastronomy. It is also expected to play a central role in connection with thefuture Internet-of-Things. We will prove that under reasonable and practicalassumptions, it is possible to solve this otherwise highly ill-posed problemand recover the $r$ transmitted functions $\boldsymbol{g}_i$ and the impulseresponses $\boldsymbol{f}_i$ in a robust, reliable, and efficient manner fromjust one single received function $\boldsymbol{y}$ by solving a semidefiniteprogram. We derive explicit bounds on the number of measurements needed forsuccessful recovery and prove that our method is robust in the presence ofnoise. Our theory is actually sub-optimal, since numerical experimentsdemonstrate that, quite remarkably, recovery is still possible if the number ofmeasurements is close to the number of degrees of freedom.
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