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.

翻译：我们研究了在多个用户通过多个未知信道传输消息，并由单个基站（BS）接收其叠加信号场景下的非网格盲反卷积与盲分离（GB2D）问题。该场景出现在多种通信领域，包括无线通信、物联网、空中计算以及集成感知与通信。在此设定中，每个用户的消息通过与由若干缩放和延迟的狄拉克尖峰形成的多径信道进行卷积。基站接收这些卷积信号的线性组合，其目标是从基站处压缩的测量向量中恢复未知的振幅、连续索引的延迟以及传输波形。然而，在缺乏传输消息和信道任何先验知识的情况下，GB2D通常高度复杂且难以处理。为解决此问题，我们假设每个用户的消息遵循一种已知低维子空间中的特定调制方案。通过利用这些子空间假设以及不同用户多径信道的稀疏性，我们将非线性GB2D问题转化为从少量线性测量中恢复矩阵元组的问题。为此，我们提出了一种半定规划优化方法，该方法利用矩阵元组的特定低维结构，从基站接收的单个信号中恢复不同通信路径的消息和连续延迟。最后，我们的数值实验表明，在样本数量充足的情况下，所提方法能有效恢复所有传输消息及信道的连续延迟参数。