Symbolic calculus on the time-frequency half-plane

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 operations of the Lie algebra of symbols, which is isomorphic to that of the operators, are obtained. The calculus is shown to be a very special member of a large family which is briefly described.