通信信道的自由度：使用广义奇异值
Degrees of Freedom of a Communication Channel: Using Generalised Singular Values
摘要：
任何通信系统中的一个基本问题是：给定发送器和接收器之间的通信信道，它们之间可以交换多少“独立”信号？本文研究了可由赋范空间之间的线性紧信道算子映射描述的任意通信信道。（众所周知的）概念的自由度在水平$ε$和必要的尺寸，这样的渠道是在这个一般的设置。我们认为，在$ε$级的自由度和基本尺寸从根本上限制了发射机和接收机之间可以交换的独立信号的数量。我们还概括了紧算子奇异值的概念，适用于定义在任意赋范空间上的紧算子，这些空间不一定具有Hilbert空间结构。我们展示了如何这些广义奇异值可以用来计算在$ε$级的自由度和描述通信渠道的紧凑型运营商的基本尺寸。我们描述的物理现实的渠道，需要这样的一般渠道模型。
英文摘要：
A fundamental problem in any communication system is: given a communication channel between a transmitter and a receiver, how many "independent" signals can be exchanged between them? Arbitrary communication channels that can be described by linear compact channel operators mapping between normed spaces are examined in this paper. The (well-known) notions of degrees of freedom at level $ε$ and essential dimension of such channels are developed in this general setting. We argue that the degrees of freedom at level $ε$ and the essential dimension fundamentally limit the number of independent signals that can be exchanged between the transmitter and the receiver. We also generalise the concept of singular values of compact operators to be applicable to compact operators defined on arbitrary normed spaces which do not necessarily carry a Hilbert space structure. We show how these generalised singular values can be used to calculate the degrees of freedom at level $ε$ and the essential dimension of compact operators that describe communication channels. We describe physically realistic channels that require such general channel models.
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