  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.
