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:^{[1]}
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 (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.
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Relation to Laplace transform[edit]
Since , where:
Then per the convolution theorem, the starred transform is equivalent to the complex convolution of and , 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 in the right half-plane of p. The result of such an integration would be:
Relation to Z transform[edit]
Given a Z-transform, X(z), the corresponding starred transform is a simple substitution:
- ^{[2]}
This substitution restores the dependence on T.
It's interchangeable,^{[citation needed]}
Properties of the starred transform[edit]
Property 1: is periodic in with period
Property 2: If has a pole at , then must have poles at , where
Citations[edit]
References[edit]
- Bech, Michael M. "Digital Control Theory" (PDF). AALBORG University. Retrieved 5 February 2014.
- Gopal, M. (March 1989). Digital Control Engineering. John Wiley & Sons. ISBN 0852263082.
- Phillips and Nagle, "Digital Control System Analysis and Design", 3rd Edition, Prentice Hall, 1995. ISBN 0-13-309832-X