Compressed sensing deals with recovery of sparse signals from low dimensional projections, but under the assumption that the measurement setup has infinite dynamic range. In this letter, we consider a system with finite dynamic range, and to counter the clipping effect, the measurements crossing the range are folded back into the dynamic range of the system through modulo arithmetic. For this setup, we derive theoretical results on the minimum number of measurements required for unique recovery of sparse vectors. We also show that recovery using the minimum number of measurements is achievable by using a measurement matrix whose entries are independently drawn from a continuous distribution. Finally, we present an algorithm based on convex relaxation and develop a mixed integer linear program (MILP) for recovering sparse signals from the modulo measurements. Our empirical results demonstrate that the minimum number of measurements required for recovery using the MILP algorithm is close to the theoretical result for signals with low variance.

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关于来自模压缩传感测量的稀疏向量的可识别性

压缩感知处理从低维投影中恢复稀疏信号，但假设测量设置具有无限动态范围。在这封信中，我们考虑了一个具有有限动态范围的系统，为了对抗削波效应，跨越该范围的测量值通过模运算折回到系统的动态范围内。对于这种设置，我们推导出了稀疏向量的唯一恢复所需的最少测量次数的理论结果。我们还表明，通过使用其条目独立于连续分布的测量矩阵，可以实现使用最少测量次数的恢复。最后，我们提出了一种基于凸松弛的算法，并开发了一个混合整数线性程序 (MILP)，用于从模测量中恢复稀疏信号。我们的经验结果表明，使用 MILP 算法进行恢复所需的最小测量次数接近于低方差信号的理论结果。