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, 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,.
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, 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, 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 in the left half-plane of p. The result of such an integration 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 in the right half-plane of p. The result of such an integration would be:

Relation to Z transform

Given a Z-transform, X, 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: is periodic in with period
Property 2: If has a pole at, then must have poles at, where