Symbolic calculus on the time-frequency half-plane

Details Symbolic calculus on the time-frequency half-plane Symbolic calculus on the time-frequency half-plane

1997-08-24

arxiv.org/abs/physics/9708027v2

J.Math.Phys. 39 (1998) 4071-4090

Physics

Mathematical Physics

25 pages, Latex, minor changes and more references; to be published in the "Journal of Mathematical Physics" (special issue on

Scientific paper

10.1063/1.532484

The study concerns a special symbolic calculus of interest for signal analysis. This calculus associates functions on the time-frequency half-plane f>0 with linear operators defined on the positive-frequency signals. Full attention is given to its construction which is entirely based on the study of the affine group in a simple and direct way. The correspondence rule is detailed and the associated Wigner function is given. Formulas expressing the basic operation (star-bracket) of the Lie algebra of symbols, which is isomorphic to that of the operators, are obtained. In addition, it is shown that the resulting calculus is covariant under a three-parameter group which contains the affine group as subgroup. This observation is the starting point of an investigation leading to a whole class of symbolic calculi which can be considered as modifications of the original one.

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