Discrete-Valued Sparse Signals: Theory, Algorithms, and Applications

在过去的十年中，压缩感知（CS）已经获得了巨大的关注，无论是从理论的角度来看，并从其各种应用。压缩感知的关键是在未知向量稀疏的假设下求解欠定线性方程组，即，在数字通信中使用压缩感知中开发的思想和工具是非常有吸引力的。示例性场景是发射机侧信号优化（例如，峰均功率比降低）、具有小占空比的多址方案、信源编码方案和雷达应用。然而，在这些场景中，要恢复的矢量/信号（从噪声测量）可能不仅是稀疏的，而且其元素取自离散集合是有益的。因此，离散稀疏信号在数字通信系统和信号处理中是极其相关的。不幸的是，这样的信号和相应的恢复算法尚未充分研究-如果在所有-在literation.Consequently，这个建议解决了应用压缩感知方法的离散值稀疏信号的分析。要从数学的角度从根本上理解这个问题，必须付出努力。为此，我们的目标是开发一个全面的理论，从几何的角度来看，并通过采用分析结果和工具的离散稀疏信号的恢复。此外，我们设计了量身定制的恢复算法，从而解释离散压缩传感之间的联系，经典的压缩传感和多输入/多输出解码任务。最后，将讨论离散稀疏信号在通信、传感器网络和信道运营商识别中的应用。

Over the last decade, compressed sensing (CS) has gained enormous attention, both from a theoretical point of view and from its various applications. The key point in compressed sensing is to solve underdetermined systems of linear equations under the assumption that the unknown vector is sparse, i.e., a signal where only a few non-zero components are present.It is very attractive to use ideas and tools developed in compressed sensing in digital communications. Exemplary scenarios are transmitter-side signal optimization (e.g., peak-to-average power ratio reduction), multiple-access schemes with small duty cycles, source coding schemes, and radar applications. However, in these scenarios the vector/the signal to be recovered (from noisy measurements) may not only be sparse, but it is beneficial that its elements are taken from a discrete set. Hence, discrete sparse signals are extremely relevant in digital communication systems and signal processing. Unfortunately, such signals and the respective recovery algorithms are not yet studied adequately---if at all---in the literature.Consequently, this proposal addresses the application of compressed sensing methodology to the analysis of discrete-valued sparse signals. Effort has to be spent to fundamentally understand the problem from the mathematical side. To this end, we aim to develop a comprehensive theory for the recovery of discrete sparse signals, both from a geometric viewpoint and by adopting analytical results and tools. Moreover, we devise tailored recovery algorithms, thereby interpreting discrete compressed sensing as a link between classical compressed sensing and a multiple-input/multiple-output decoding task. Finally, the application of discrete sparse signals in communications, sensor networks, and for the identification of channel operators will be addressed.