Dirac's Representation Theory as a Framework for Signal Theory: II. Infinite Duration and Continuous Signals

✍ Scribed by Alexander Gersten

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 422 KB
• Volume: 262
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1006/aphy.1997.5741

✦ Synopsis

In the preceding paper we dealt with discrete signals of finite duration. Here we generalize previous results and demonstrate that the Dirac representation theory can be effectively adjusted and applied to continuous or discrete signals of infinite time duration. The role of the identity and projection operators is emphasized. The sampling theorem is viewed from the point of view of orthogonal physical states. An orthogonal basis which spanned the time space ceases to be orthogonal and becomes overcomplete if the domain of frequencies is restricted in a bandwidth. In this case there exists an infinite number of sub-bases of discrete times which are orthogonal and complete. The relation between the overcomplete bases and a complete one is the essence of the sampling theorem. The signal theory is reformulated in the framework of the Dirac bra-kets. The case of signals existing for positive time is treated in detail.

1998 Academic Press 3. The transition to the continuous case is more complicated, as the limits can be distributions (generalized functions) [6 8].

Most of the developments in this paper are based on the assumption of Dirac [9], made in 1930, that a given self-adjoint operator A can be presented as