
Starred transform

In applied mathematics, the starred transform, or star transform, is a discrete-time variation of the Laplace transform, so-named because of the asterisk or "star" in the customary notation of the sampled signals. 
The transform is an operator of a continuous-time function formula_1, which is transformed to a function in the following manner:
where is a Dirac comb function, with period of time T.
The starred transform is a convenient mathematical abstraction that represents the Laplace transform of an "impulse sampled" function , which is the output of an "ideal sampler", whose input is a continuous function, formula_1.
The starred transform is similar to the Z transform, with a simple change of variables, where the starred transform is explicitly declared in terms of the sampling period (T), while the Z transform is performed on a discrete signal and is independent of the sampling period. This makes the starred transform a de-normalized version of the one-sided Z-transform, as it restores the dependence on sampling parameter "T".
Relation to Laplace transform.
Since , where:
Then per the convolution theorem, the starred transform is equivalent to the complex convolution of and </math>}}, hence:
This line integration is equivalent to integration in the positive sense along a closed contour formed by such a line and an infinite semicircle that encloses the poles of X(s) in the left half-plane of "p". The result of such an integration (per the residue theorem) would be:
Alternatively, the aforementioned line integration is equivalent to integration in the negative sense along a closed contour formed by such a line and an infinite semicircle that encloses the infinite poles of </math>}} in the right half-plane of "p". The result of such an integration would be:
Relation to Z transform.
Given a Z-transform, "X"("z"), the corresponding starred transform is a simple substitution:
This substitution restores the dependence on "T".
It's interchangeable,
Properties of the starred transform.
Property 1:  formula_11 is periodic in formula_12 with period formula_13
Property 2:  If has a pole at formula_15, then must have poles at formula_16, where formula_17