  We consider the problem of gridless blind deconvolution and demixing (GB2D)
in scenarios where multiple users communicate messages through multiple unknown
channels, and a single base station (BS) collects their contributions. This
scenario arises in various communication fields, including wireless
communications, the Internet of Things, over-the-air computation, and
integrated sensing and communications. In this setup, each user's message is
convolved with a multi-path channel formed by several scaled and delayed copies
of Dirac spikes. The BS receives a linear combination of the convolved signals,
and the goal is to recover the unknown amplitudes, continuous-indexed delays,
and transmitted waveforms from a compressed vector of measurements at the BS.
However, in the absence of any prior knowledge of the transmitted messages and
channels, GB2D is highly challenging and intractable in general. To address
this issue, we assume that each user's message follows a distinct modulation
scheme living in a known low-dimensional subspace. By exploiting these subspace
assumptions and the sparsity of the multipath channels for different users, we
transform the nonlinear GB2D problem into a matrix tuple recovery problem from
a few linear measurements. To achieve this, we propose a semidefinite
programming optimization that exploits the specific low-dimensional structure
of the matrix tuple to recover the messages and continuous delays of different
communication paths from a single received signal at the BS. Finally, our
numerical experiments show that our proposed method effectively recovers all
transmitted messages and the continuous delay parameters of the channels with a
sufficient number of samples.
