Degrees of Freedom of a Communication Channel: Using Generalised Singular Values

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 $\epsilon$ and essential dimension of such channels are developed in this general setting. We argue that the degrees of freedom at level $\epsilon$ 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 $\epsilon$ and the essential dimension of compact operators that describe communication channels. We describe physically realistic channels that require such general channel models.