CIF: Small:Analog-to-Analog Compression: Fundamental Limits and Constructive Schemes

对于高维度的模拟信号，依赖于整个信号的采集和量化的压缩方法可能并不总是可行的。因此，感兴趣的是开发用于直接从模拟到模拟（A2 A）域压缩信息的方法。两种哲学已经证明成功地解决了通过模拟介质传送模拟信息的一般问题：香农理论方法和压缩传感方法。本研究探讨了第三种方法，它融合了香农理论方法的通用性与压缩感知方法的实用性。A2 A压缩的基本限制在单信号和多信号设置进行了研究，并提出了建设性的计划，以实现这些限制。 A2 A压缩的挑战是实现最大的降维，即，- 通过利用关于信号的先验知识，每个测量的信号维度的带宽减小因子，所述先验知识可以包括但不限于稀疏性。 Wu-Verdú证明了Renyi信息维（RID）是i.i.d.的基本极限。已知分布的信号。RID是一种非常粗略的信息度量，因为许多不同的信号导致相同的RID。本研究调查的其他信号功能，影响A2 A的压缩性能超过RID，特别是在非渐近制度。它研究了没有i.i.d.的信号的A2 A压缩。已知的分布和多信号设置，其中可以利用信号之间的相关性。最后，它提出了建设性的方案，以实现使用极化码的基本限制。

For analog signals of high dimension, compression methods relying on the acquisition and quantization of the entire signal may not always be possible. It is hence of interest to develop methods for compressing information directly from the analog to the analog (A2A) domain. Two philosophies have proven successful in addressing the general problem of conveying analog information through an analog medium: the Shannon theoretic approach and the compressed sensing approach. This research investigates a third approach, which blends the generality of the Shannon theoretic approach with the practicality of the compressed sensing approach. The fundamental limits of A2A compression are investigated in single and multi-signal settings, and constructive schemes are proposed to achieve these limits. The challenge of A2A compression is to achieve the maximal dimensionality reduction, i.e., a bandwidth reduction factor of the signal dimension per measurement, by exploiting prior knowledge about the signal, which may include, but it is not limited to sparsity. Wu-Verdú have shown that the Renyi Information dimension (RID) is the fundamental limit for i.i.d. signals with known distributions. The RID is a very coarse measure of information, as many different signals lead to the same RID. This research investigates other signal features that influence the A2A compression performance beyond RID, particularly in the non-asymptotic regime. It investigates the A2A compression of signals which do not have an i.i.d. known distribution, and of multi-signal settings, where correlations among signals can be exploited. Finally, it proposes constructive schemes to achieve the fundamental limits using polar codes.