Blind Deconvolution Meets Blind Demixing: Algorithms and Performance Bounds

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.