  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.
