Simultaneous blind deconvolution and blind demixing via convex programming

Abstract

Suppose that one receives the superposition of r signals and each of them passes through an unknown channel, can we correctly recover the signals and their corresponding channels simultaneously from only one received signal? This problem, which intertwines blind deconvolution with blind demixing, appears in many fields of science and engineering such as wireless communication, imaging sciences, audio signal processing, etc. We will present a robust, reliable and efficient approach to this challenging problem by solving a semi-definite program. The exact recoveries of both signals and channels are guaranteed under mild assumptions. We give an upper bound of how many measurements are required to accomplish this task. While our theory is suboptimal, numerical experiments demonstrate remarkable performance, even if the number of constraints is close to the degree of freedom.