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

在过去的十年中，压缩感知（CS）无论是从理论角度还是从其各种应用中都得到了极大的关注。压缩感知的关键是在假设未知向量是稀疏的情况下求解欠定线性方程组，即只有少数非零分量存在的信号。在数字通信中使用压缩感知的思想和工具是非常有吸引力的。示例性场景包括发射机侧信号优化（例如，峰值-平均功率比降低）、小占空比的多址方案、源编码方案和雷达应用。然而，在这些情况下，要恢复的矢量/信号（从噪声测量中）可能不仅是稀疏的，而且它的元素是从一个离散集合中获取是有益的。因此，离散稀疏信号在数字通信系统和信号处理中具有极其重要的意义。不幸的是，这些信号和相应的恢复算法尚未在文献中得到充分的研究——如果有的话。因此，本提案解决了压缩感知方法在离散值稀疏信号分析中的应用。必须努力从数学的角度从根本上理解这个问题。为此，我们的目标是从几何角度和采用分析结果和工具，为离散稀疏信号的恢复开发一个全面的理论。此外，我们设计了定制的恢复算法，从而将离散压缩感知解释为经典压缩感知与多输入/多输出解码任务之间的联系。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.