Dirac's Representation Theory as a Framework for Signal Theory. I. Discrete Finite Signals

We demonstrate that the Dirac representation theory can be effectively adjusted and applied to signal theory. The main emphasis is on orthogonality as the principal physical requirement. The particular role of the identity and projection operators is stressed. A Dirac space is defined, which is spanned by an orthonormal basis labeled with the time points. An infinite number of orthonormal bases is found which are labeled with frequencies, they are distinguished by the continuous parameter a. In a way, similar to one used in quantum mechanics, self-adjoint operators (observables) and averages (expectation values) are defined. Non orthonormal bases are discussed and it is shown, in an example, that they are less stable compared to the orthonormal ones. A variant of the sampling theorem for finite signals is derived. The aliasing phenomenon is described in the paper in terms of aliasing symmetry. Relations between different bases are derived. The uncertainty principle for finite signals is discussed.