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:[1] where
δ
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 (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. Since
, where: Then per the convolution theorem, the starred transform is equivalent to the complex convolution of
, hence:[1] 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
{\displaystyle {\frac {1}{1-e^{-T(s-p)}}}}
in the right half-plane of p. The result of such an integration would be: Given a Z-transform, X(z), the corresponding starred transform is a simple substitution: This substitution restores the dependence on T. It's interchangeable,[citation needed] Property 1:
is periodic in
with period
2 π
2 π